Hand Book of Thermoluminescence

Handbook ofThermo uminescence

Handbook of

ThermoluminescenceClaudio Furetta

Physics DepartmentRome University "La Sapienza"

Italy

V f e World Scientificw B New Jersey • London • Singapore • Hong Kong

Published by

World Scientific Publishing Co. Pte. Ltd.

5 Toh Tuck Link, Singapore 596224

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Library of Congress Cataloging-in-Publication DataFuretta, C, 1937-

Handbook of thermoluminescence / Claudio Furetta.p. cm.

Includes bibliographical references and index.ISBN 9812382402 (alk. paper)1. Thermoluminescence-Handbooks, manuals, etc. I. Title.

QC478 .F87 2003535'.356-dc21 2002038068

British Library Cataloguing-in-Publication DataA catalogue record for this book is available from the British Library.

Copyright © 2003 by World Scientific Publishing Co. Pte. Ltd.

All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means,electronic or mechanical, including photocopying, recording or any information storage and retrievalsystem now known or to be invented, without written permission from the Publisher.

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This book is printed on acid-free paper.

Printed in Singapore by Uto-Print

I am deeply grateful to my wife Maria Clotilde for herconstant and loving support to my work. This book isdedicated to her.

PREFACE

This book on thermoluminescence (TL) is born from the idea to provide toexperts, teachers, students and technicians practical support for research, study,routine work and terminology.

The term "handbook" of the title does not mean that this book is a "summa"of thermoluminescence. Actually, the goal is to be dynamic, fluid and of easyconsultation on several subjects.

This book collects a certain number of subjects, mainly referring to thethermoluminescence models, to the methods for determining the kinetic parameters,to the procedures to follow for characterizing a thermoluminescent dosimetric systemand to the definition of terms commonly used in TL literature. Furthermore, theanalytical treatments of the various TL models are fully developed.

Subjects concerning solid state physics as well as TL dating are notconsidered because they are widely treated in many fundamental books which can beeasily found in the market.

In general, the subjects considered here are dispersed in specialized journalswhich are not always available to everyone.

The arguments are given in alphabetic order to make the research easy.

ACKNOWLEDGMENTS

The author is grateful to Prof. Juan Azorin, of the Physics Department ofUniversidad Autonoma Metropolitana (UAM), Iztapalapa, Mexico D.F., for hissincere help.

A special thank is due to Dr. Teodoro Rivera Montalvo, of the sameInstitution, for his full assistance in computing the text.

CONTENTS

CHAPTER A

Accuracy (definition) 1Activation energy (definition and properties) 1Activator 3Adirovitch model 3Afterglow 8Aluminium oxide (A12O3) 8Annealing (definition) 9Annealing (general considerations) 9Annealing procedures 11Anomalous fading 19Anomalous thermal fading 20Area measurement methods (generality) 20Area measurement method (Maxia et al.) 21Area measurement method (May and Partridge: general order) 24Area measurement method (Muntoni et al.: general order) 24Area measurement method (Moharil: general order) 2 5Area measurement method (Moharil: general order, s=s(T)) 26Area measurement method (Rasheedy: general order) 31Arrhenius equation 3 5Assessment of random uncertainties in precision of TL measurements (general) 3 6Atomic number (calculation) 3 9

CHAPTER B

Basic equation of radiation dosimetry by thermoluminescence 43Batch of TLDs 45Braunlich-Scharmann model 45

CHAPTER C

Calcium fluoride (CaF2) 5 5Calibration factor Fc (definition) 5 5Calibration factor^ (procedures) 56Competition 58Competitors 60Computerized glow curve deconvolution (CGCD): Kitis' expressions 60

XII CONTENTS

Condition at the maximum (first order) 69Condition at the maximum (first order): remarks 70Condition at the maximum (general order) 71Condition at the maximum (second order) 72Condition at the maximum when s'=s'(T) (second-order kinetics) 74Condition at the maximum when s"=s"(T) (general-order kinetics) 76Condition at the maximum when s=s(T) (first-order kinetics) 77Considerations on the heating rate 78Considerations on the methods for determining E 85Considerations on the symmetry factor, fi, and the order of kinetics, b 91Correction factor for beam quality, Fm (general) 95Curve fitting method (Kirsh: general order) 97CVD diamond 99

CHAPTER D

Defects 101Delocalized bands 105Determination of the dose by thermoluminescence 105Dihalides phosphors 106Dosimeter's background or zero dose reading (definition) 107Dosimeter's background or zero dose reading (procedure) 107Dosimetric peak 108Dosimetric trap 108

CHAPTER E

Effect of temperature lag on trapping parameters 109Energy dependence (procedure) 110Environmental dose rate (calculation) 112Environmental dose rate (correction factors) 116Erasing treatment 117Error sources in TLD measurements 117

CHAPTER F

Fading (theoretical aspects) 123Fading factor 137Fading: useful expressions 138First-order kinetics when s=s(T) 147Fluorescence 148

CONTENTS XIII

Fluoropatite (Ca5F(PO4)3) 149Frequency factor, s 149Frequency factor, s (errors in its determination) 150Frequency factor and pre-exponential factor expressions 151

CHAPTER G

Garlick-Gibson model (second-order kinetics) 157General characteristics of first and second order glow-peaks 159General-order kinetics when s"=s"(T) 163Glow curve 163

CHAPTER I

In-vivo dosimetry (dose calibration factors) 165Inflection points method (Land: first order) 166Inflection points method (Singh et al.: general order) 168Initial rise method when s=s(T) (Aramu et al.) 171Initialization procedure 172Integral approximation 175Integral approximation when s=s(T) 176Interactive traps 176Isothermal decay method (Garlick-Gibson: first order) 176Isothermal decay method (general) 177Isothermal decay method (May-Partridge: (a) general order) 178Isothermal decay method (May-Partridge: (b) general order) 179Isothermal decay method (Moharil: general order) 180Isothermal decay method (Takeuchi et al.: general order) 182

CHAPTER K

Keating method (first order, s=s(T)) 185Killer centers 188Kinetic parameters determination: observations 188Kinetics order: effects on the glow-curve shape 194

CHAPTER L

Linearization factor, Flin (general requirements for linearity) 197

XIV CONTENTS

Linearity (procedure) 200Linearity test (procedure) 202Lithium borate (Li2B4O7) 204Lithium fluoride family (LiF) 206Localized energy levels 209Lower detection limit (Dyi) 209Luminescence (general) 209Luminescence (thermal stimulation) 210Luminescence centers 212Luminescence dosimetric techniques 212Luminescence dosimetry 213Luminescence efficiency 213Luminescence phenomena 214

CHAPTER M

Magnesium borate (MgO x nB2C<3) 215Magnesium fluoride (MgF2) 216Magnesium orthosilicate (Mg2Si04) 216May-Partridge model (general order kinetics) 217Mean and half-life of a trap 219Metastable state 223Method based on the temperature at the maximum (Randall-Wilkins) 223Method based on the temperature at the maximum (Urbach) 224Methods for checking the linearity 224Model of non-ideal heat transfer in TL measurements 228Multi-hit or multi-stage reaction models 231

CHAPTER N

Nonlinearity 233Non-ideal heat transfer in TL measurements (generality) 240Numerical curve fitting method (Mohan-Chen: first order) 241Numerical curve fitting method (Mohan-Chen: second order) 243Numerical curve fitting method (Shenker-Chen: general order) 244

CHAPTER O

Optical bleaching 247Optical fading 247Oven (quality control) 247

CONTENTS XV

CHAPTER P-l

Partridge-May model (zero-order kinetics) 255Peak shape method (Balarin: first- and second-order kinetics) 256Peak shape method (Chen: first- and second-order) 260Peak shape method (Chen: general-order kinetics) 272Peak shape method (Christodoulides: first- and general-order) 276Peak shape method (Gartia, Singh and Mazumdar: (b) general order) 279Peak shape method (Grossweiner: first order) 280Peak shape method (Halperin-Braner) 282Peak shape method (Lushchik: first and second order) 292Peak shape method (Mazumdar, Singh and Gartia: (a) general order) 295Peak shape method (parameters) 299Peak shape method when s=s(T) (Chen: first- second- and general-order) 300Peak shape method: reliability expressions 312

CHAPTER P-2

Peak shift 323Perovskite's family (ABX3) 325Phosphorescence 326Phosphors (definition) 329Photon energy response (calculation) 329Photon energy response (definition) 332Phototransferred thermoluminescence (PTTL) (general) 333Phototransferred thermoluminescence (PTTL): model 334Post-irradiation annealing 340Post-readout annealing 340Precision and accuracy (general considerations) 340Precision concerning a group of TLDs

of the same type submitted to one irradiation 344Precision concerning only one TLD undergoing

repeated cycles of measurements (same dose) 345Precision concerning several identical dosimeters

submitted to different doses 346Precision concerning several identical dosimeters

undergoing repeated and equal irradiations (procedures) 349Precision in TL measurements (definition) 357Pre-irradiation annealing 357Pre-readout annealing 357Properties of the maximum conditions 357

XVI CONTENTS

CHAPTER Q

Quasiequilibrium condition 359

CHAPTER R

Radiation-induced defects 361Randall-Wilkins model (first-order kinetics) 361Recombination center 364Recombination processes 364Reference and field dosimeters (definitions) 365Relative intrinsic sensitivity factor or

individual correction factor Si (definition) 365Relative intrinsic sensitivity factor or

individual correction factor Sj (procedures) 368Residual TL signal 374Rubidium halide 375

CHAPTER S

Second-order kinetics when s'=s'(T) 377Self-dose in competition to fading (procedure) 378Sensitization (definition) 379Sensitivity (definition) 379Set up of a thermoluminescent dosimetric system (general requirements) 380Simultaneous determination of dose and time elapsed since irradiation 381Sodium pyrophosphate (Na4P2O7) 390Solid state dosimeters 391Solid state dosimetry 391Spurious thermoluminescence: chemiluminescence 391Spurious thermoluminescence: surface-related phenomena 392Spurious thermoluminescence: triboluminescence 392Stability factor Fst (definition) 392Stability factor Fst (procedure) 393Stability of the reading system background 395Stability of the reading system background (procedure) 396Stability of TL response 396Standard annealing 397Stokes' law 397Sulphate phosphors 397

CONTENTS XVII

CHAPTER T

Temperature gradient in a TL sample 401Temperature lag: Kitis' expressions for correction (procedure) 403Temperature lag: Kitis' expressions for correction (theory) 406Test for batch homogeneity 411Test for the reproducibility of a TL system (procedure) 415Thermal cleaning (peak separation) 417Thermal fading (procedure) 418Thermal quenching 420Thermally connected traps 421Thermally disconnected traps 421Thermoluminescence (thermodynamic definition) 422Thermoluminescence (TL) 424Thermoluminescent dosimetric system (definition) 424Thermoluminescent materials: requirements 425Tissue equivalent phosphors 426Trap characteristics obtained by fading experiments 427Trap creation model 429Trapping state 430Tunnelling 430Two-trap model (Sweet and Urquhart) 431

CHAPTER V

Various heating rates method (Bohum, Porfianovitch, Booth: first order) 435Various heating rates method (Chen-Winer: first order) 435Various heating rates method (Chen-Winer: second and general orders) 437Various heating rates method (Gartia et al.: general order) 439Various heating rates method (Hoogenstraaten: first order) 440Various heating rates method (Sweet-Urquhart: two-trap model) 440Various hetaing rates method when s=s(T)

(Chen and Winer: first- and general-order) 441

CHAPTER Z

Zirconium oxide (ZrQ) 445

AUTHOR INDEX 447

XVIII CONTENTS

SUBJECT INDEX 457

AAccuracy (definition)

Errors of measurement are of two types, random and systematic. For agiven set of measurement conditions a source of random error is variable in bothmagnitude and sign, whereas a source of systematic error has a constant relativemagnitude and is always of the same sign.

The accuracy is affected by both systematic and random uncertainties.Accuracy is related to the closeness of a measurement, within certain limits,

with the true value of the quantity under measurement. For instance, the accuracy ofdose determination by TLD is given by the difference between the measured valueof the dose (TL reading) and the true dose given to the dosimeter.

A method of combining systematic and random uncertainties has beensuggested in a BCS document: both systematic and random errors are combined byquadratic addition but the result for systematic errors is multiplied by 1.13. Thisfactor is necessary to ensure a minimum confidence level of 95%.

ReferenceBritish Calibration Society, BCS Draft Document 3004

Activation energy (definition and properties)

It is the energy, E, expressed in eV, assigned to a metastable state or levelwithin the forbidden band gap between the conduction band (CB) and the valenceband (VB) of a crystal. This energy is also called trap depth. The metastable levelcan be an electron trap, near to the CB, or a hole trap, near the VB, or aluminescence centre, more or less in the middle of the band gap. The metastablelevels are originated from defects of the crystal structure. A crystal can containseveral kinds of traps and luminescence centers. If E is such that E > several kT,where k is the Boltzmann's constant, then the trapped charge can remain in the trapfor a long period. For an electron trap, E is measured, in eV, from the trap level tothe bottom of the CB. For a hole trap, it is measured from the trap to the top of theVB.

Figure 1 shows the simplest band structure of an isolant containing defectsacting as traps or luminescence centers.

Bombarding the solid with an ionizing radiation, this produces free chargeswhich can be trapped at the metastable states. Supposing the solid previously excitedis heated, a quantity of energy is supplied in the form of thermal energy and the

2 HANDBOOK OF THERMOLUMINESCENCE

trapped charges can be released from the traps. The rate of such thermally stimulatedprocess is usually expressed by the Arrhenius equation which leads to the conceptof the activation energy, E, which can be seen as an energy barrier which must beovercome to reach equilibrium.

Considering the maximum condition using the first order kinetics:

P-E ( E \——- = s exp

CB

DEFECTS

VB

Fig. 1. A simple band structure of an isolant withdefect levels in the band gap.

it is easily observed that TM increases as E increases. In fact, for E » VTM , TMincrease almost linearly with E. This behavior agrees with the Randall-Wilkinsmodel where, for deeper traps, more energy and, in turn, a higher temperature, isrequired to detrap the electrons [1-4].

References1. Braunlich P. in Thermally Stimulated Relaxion in Solids, P. Braunlich

editor, Spring-Verlag, Berlin (1979)2. Chen R. and McKeever S.W.S., Theory of Thermoluminescence and

Related Phenomena, World Scientific (1997)3. Chen R. and Kirsh Y., Analysis of Thermally Stimulated Processes,

Pergamon Press (1981)4. McKeever S.W.S., Thermoluminescence of Solids, Cambridge University

Press (1985)

CHAPTER A 3

Activator

There are several luminescent materials but not all are efficient enough forpractical purposes. To enhance the luminescence efficiency of the material it isnecessary to add an element, called activator (i.e., Dy in CaSO4), to the host crystal.The activator then acts as luminescent center.

Adirovitch model

Adirovitch, in 1956, used a set of three differential equations to explain thedecay of phosphorescence in the general case. The same model has been used byHaering-Adams (1960) and Halperin-Braner (1960) to describe the flow chargebetween localized energy levels and delocalized bands during trap emptying. Theenergy level diagram is shown in Fig.2.

^ ~| 1 1" C B

An > < S

Y I

n NAm

1 r

m

VB

Fig.2. Energy level diagram concerning the phosphorescence decayaccording to Adirovitch. The meaning of the symbols is given in the text.


With the assumption that retrapping of electrons occurs in the trappingstates of the same kind as those from which they had been released, the intensity ofemission, /, is given by

wherem is the concentration of recombination centers (holes in centers), (cm3);nc is the concentration of free electrons in the conduction band, (cm3);Am is the recombination probability (cm3 sec"1).

This equation states that the recombination rate is proportional to thenumber of free electrons, nc, and the number of active recombination centres, m.

A second equation deals with the population variation of electrons in traps,n (cm"3), and it takes into account the excitation of electrons into the conductionband as well as the possible retrapping. Then we have:

-jt = -sn expj^- —J + nc(N- n)An (2)

where An (cm3 s"1) is the retrapping probability and N (cm"3) is the total concentrationof traps. Am and An are assumed to be independent of temperature.

The third equation relates to the charge neutrality. It can be expressed as

dnc dm dn

~d7 = ~dt~~dt

or better, using Eqs. (1) and (2), as

dn ( E\—^• = sn exp|^- — J - ncmAm - nc(N - n)An (4)

Equation (4) states that the rate of change of nc is given by the rate ofrelease of electrons from N, minus the rate of recombination in m and retrapping inN.

While Adirovitch used the previous equations to explain the decay ofphosphorescence, Halperin and Braner were the former to apply the same equationsto the case of thermoluminescence, that is to say when the light emission is

(1)

T(3)

CHAPTER A 5

measured during the heating of the sample, when one trapping state and one kind ofrecombination center are involved.

Two basic assumptions have been made for solving the previous set ofequations:

nc « n (5)

and

dn. dn

The condition (6) means that the concentration of carriers in the conductionband does not change; that is to say

dnc = 0 (7)

In this case Eq.(4) gets

5"e x pr^Jn<-mAm+(N-n)An (8)

and then the intensity is given by

dm S"°X*(-1f) A

'—t-mA.+W-nW'*- <9>

Introducing the retrapping-recombination cross-section ratio

a = - = - (10)

Eq. (9) becomes

(6)

(8)

(9)

(10)

Introducing the retrapping-recombination cross-section ratio


dm ( E^\ a(N-n)/ = - — = n s e x d - — 1-——; : (11)

dt *\ kTJ\_ a(N-n) + m\

Equation (11) gets the general one-trap equation (GOT) for the TLintensity. The term preceding the square brackets is the number of electronsthermally released to the conduction band per unit time. The term in square bracketsis the fraction of conduction band electrons undergoing recombination. From thisequation it is possible to obtain the first and second order kinetics equations.

Indeed, the first order kinetics is the case when recombination dominatesand this means that

mAm»(N-n)An (12)

or

a = 0 (13)

The equation of intensity then becomes

dm ( E)

The assumption (7) gives

dm dn-— = —— or m = n + constdt dt

and so Eq.(14) becomes

dn I E\

that is the same as the equation of the first order kinetics.The second order expression can be derived from Eq.(ll) using two

assumptions which both include the restrictive assertion m = n.Remembering the Garlick and Gibson's retrapping assumption, the first

condition can be written as

(14)

(15)

CHAPTER A 7

™Am«(N-n)An

and then the intensity is given by

dt (N-n)An

Secondly, assuming that the trap is far from saturation, which means N»n,we obtain

mAns expdm m \ kT)

/ = - - — = ^ '- (17)dt NAn

Using the condition m = » the last equation becomes

/ = "^ = m, (I8>

which, with s' =• sAn/NAn, is the Garlick and Gibson equation.Assuming now equal recombination and retrapping probabilities Am = An, as

suggested by Wrzesinska, one obtains the same equation of Garlick and Gibson withs' = s/N:

ReferenceAdirovitch E.I.A., J. Phys. Rad. 17 (1956) 705

(16)

(17)

(19)


Afterglow

Afterglow is the term used to indicate the luminescence emitted from a TLphosphor immediately after irradiation. If this effect is thermally dependent,according to the equation

x -s exp —

it is more properly termed phosphorescence.The emission spectrum of the afterglow is the same as that of

thermoluminescence: this fact indicates that the same luminescence centres areinvolved. Zimmerman found a correlation between the anomalous (athermal) fadingand the afterglow [1-3].

References1. Zimmermann D.N., Abstract Symp. Archaeometry and Archaeological

Prospection, Philadelphia (1977)2. Zimmermann D.N., PACT 3 (1979) 2573. Visocekas R., Leva T., Marti C , Lefaucheux F.and Robert M.C., Phys.

Stat. Sol. (a) 35 (1976) 315

Aluminium oxide (A12O3)

Chromium substituting for some of the aluminum atoms in A12O3 changessapphire into ruby, which exhibits TL properties studied since the 60s [1-5].

Investigations on the TL of ruby, whose effective atomic number is 10.2,are performed by using synthetic crystals of A12O3 containing various knownconcentrations of Cr2O3 (typically 0.01 to 0.2 wt%).

TL glow curve of ruby consists of a main glow peak at 347°C (shiftingtoward lower temperatures for high exposures) and a less intense peak at 132°C (inthe same region as the peak reported for sapphire). High chromium concentrationscause a relative increase in the lower temperature portion of the glow curve.

References1. Gabrysh A.F., Eyring H., Le Febre V. and Evans M.D., J. Appl. Phys. 33

(1962) 33892. Maruyama T., Matsuda Y. and Kon H., J. Phys. Soc. Japan 18-11 (1963)

3153. Buckman W.G., Philbrick C.R. and Underwood N., U.S. Atomic Energy

Commission Rep. CONF-680920 (1968)

CHAPTER A 9

4. Hashizume T., Kato Y., Nakajima T., Yamaguchi H. and Fujimoto K.,Health Phys. 23 (1972) 855

5. Watson J.E., Health Phys. 31 (1976) 47

Annealing (definition)

Annealing is the thermal treatment needs to erase any irradiation memoryfrom the dosimetric material.

Some thermoluminescent material required a complex annealing procedure.LiF:Mg,Ti is one of them. It requires a high temperature anneal, followed by a lowtemperature anneal. Generally speaking the high temperature anneal is required toclear the dosimetric traps of residual signal which may cause unwantedbackground during subsequent use of the dosimeters. The low temperature anneal isrequired to stabilize and aggregate low temperature traps in order to enhance thesensitivity of the main dosimetry traps and to reduce losses of radiation-inducedsignal due to thermal or optical fading during use. The combination of these twoanneals is termed standard anneal.

For lithium fluoride the standard annealing consists of a high temperatureanneal at 400°C during 1 hour followed by a low temperature thermal treatment for20 hours at 80°C. In some laboratories, annealing at 100°C for 2 hours has beenused instead of the longer anneal at 80°C.

The TL properties exhibited by a phosphor strongly depend upon the kindof thermal annealing experienced by it prior to the irradiation. It is also true, ingeneral, that more defects are produced ay higher temperatures of annealing.

The number of defects also depends on the cooling rate employed to coolthe phosphor to the ambient temperature.

Once the best annealing procedure has been determined, i.e. the highest TLresponse with the lowest standard deviation, the same procedure must always befollowed for reproducible results in TL applications [1,2].

References1. Driscoll C.M.H., National Radiological Protection Board, Tech. Mem.

5(82)2. Busuoli G. in Applied Thermoluminescence Dosimetry, ISPRA Courses,

Edited by M.Oberhofer and A. Scharmann, Adam Hilger publisher (1981)

Annealing (general considerations)

Before using a thermoluminescent material for dosimetric purposes, it hasto be prepared. To prepare a TL material means to erase from it all the information


due to any previous irradiation, i.e., to restore in it the initial conditions of thecrystal as they were before irradiation. The preparation also has the purpose ofstabilizing the trap structure.

In order to prepare a thermoluminescent material for use, it is needed toperform a thermal treatment, usually called annealing [1,2], carried out in ovenor/and furnace, which consists of heating up the TL samples to a predeterminedtemperature, keeping them at that temperature for a predetermined period of timeand then cooling down the samples to room temperature. It has to be stressed thatthe thermal history of the thermoluminescent dosimeters is crucial for theperformance of any TLD system.

There is a large number of thermoluminescent materials, however theannealing procedures are quite similar. Just a few materials, like LiF:Mg,Ti, need acomplex annealing procedure.

The thermal treatments normally adopted for the TLDs can be divided intothree classes:

~ initialisation treatment: this treatment is used for new (fresh or virgin)TL samples or for dosimeters which have not been used for a long time.The aim of this thermal treatment is to stabilise the trap levels, so thatduring subsequent uses the intrinsic background and the sensitivity areboth reproducible. The time and temperature of the initialisationannealing are, in general, the same as those of the standard annealing.

~ erasing treatment or standard annealing (also called pre-irradiationannealing or post-readout annealing): this treatment is used to eraseany previous residual irradiation effect which is supposed to remainstored in the crystal after the readout. It is carried out before using theTLDs in new measurements. The general aim of this thermal treatment isto bring back the traps - recombination centres structure to the formerone obtained after the initialisation procedure. It may consist of one ortwo thermal treatments (in latter case, at two different temperatures).

~ post-irradiation or pre-readout annealing: this kind of thermaltreatment is used to erase the low-temperature peaks, if they are found inthe glow-curve structure. Such low-temperature peaks are normallysubjected to a quick thermal decay (fading) and possibly must not beincluded in the readout to avoid any errors in the dose determination.

In all cases, value and reproducibility of the cooling rate after the annealingare of great importance for the performance of a TLD system. In general, the TLsensitivity is increased using a rapid cool down. It seems that the sensitivity reachesthe maximum value when a cooling rate of 50-100°C/s is used. To obtain this, theTLDs must be taken out of the oven after the pre-set time of annealing is over and

CHAPTER A 11

placed directly on a cold metal block. The procedure must be reproducible andunchanged during the whole use of the dosimeters.

It must be noted that the thermal procedures listed above can be carried outin the reader itself. This is important for TL elements embedded in plastic cards asthe dosimeters used for large personnel dosimetry services. In fact, the plastic cardsare not able to tolerate high temperatures and the in-reader annealing is shortened toa few seconds. However, its efficiency is very low when high dose values areinvolved. The in-reader annealing procedure should be used only if the dosereceived by the dosimeter is lower than 10 to 20 mGy. Driscoll suggests in this casea further annealing in oven during 20 hours at 80°C for cards holding LiF:Mg,Ti; atthis temperature the plastic holder does not suffer any deformation. Any way,excluding cards, for bare TL solid chips or TL materials in powder form, theannealing must be performed in an oven.

References1. Busuoli G. in Applied Thermoluminescence Dosimetry, ISPRA Courses,

Edited by M.Oberhofer and A.Scharmann, Adam Hilger publisher (1981)2. Drisoll C.M.H., Barthe J.R., Oberhofer M., Busuoli G. and Hickman C ,

Rad. Prot. Dos. 14(1) (1986) 17

Annealing procedures

When a new TL material is going to be used for the first time, it isnecessary to perform at first an annealing study which has three main goals:

~ to find the good combination of annealing temperature and time toerase any effect of previous irradiation,

~ to produce the lowest intrinsic background and the highestsensitivity,

~ to obtain the highest reproducibility for both TL and backgroundsignals.

The suggested procedures are the following:

Is' procedure

~ irradiate 10 TLDs samples to a test dose in the range of the fieldapplications,


"• anneal the irradiated samples at a given temperature (e.g., 300°C)for a given period of time (e.g., 30 minutes),

~ read the samples,

~ repeat the first three steps above increasing the annealingtemperature of 50°C each time up to the maximum value at whichthe residual TL (background) will remain constant as thetemperature increases,

~ plot the data as shown in Fig.3. As it can be observed, after athreshold temperature value, i.e., Tc, the residual TL signal remainsconstant,

~ repeat now the procedure, keeping constant the temperature at thevalue Tc and varying the annealing time by steps of 30 minutes andplot the results. The plot should be similar to the previous one,

"• choose now the best combination of temperature and time,

~ carry out a reproducibility test to verify the goodness of theannealing, in the sense that background must be unchanged duringthe test.

RTL 1

\ . BACKGROUND LEVEL

ANNEALING TEMPERATURE

Fig.3. Decrease of TL response, after irradiation,as a function of the annealing procedure.

CHAPTER A 13

Td procedure

This procedure has been suggested by G.Scarpa [1] who used it for sinteredBeryllium Oxide. With this procedure both informations concerning annealing andreproducibility are obtained at once. The procedure consists of changing thetemperature, step by step, at a constant annealing time. After annealing at a giventemperature, the samples are irradiated and then readout. For each temperature 10samples are used, cycled 10 times. So that each experimental point in Fig.4 is basedon 100 measurements. From the figure it can be seen that the best reproducibility,i.e., the lowest standard deviation in %, is achieved at around 600 °C, whereas theabsolute value of the TL output is practically constant between 500 and 700 CC. Thesame procedure can now be carried out for a constant temperature and changing theannealing time. Finally, as before, the best combination of time and temperatureshould give the optimum annealing procedure.

To be sure that the annealing procedure is useful at any level of dose, it issuggested to repeat the procedure at different doses, according to the specific use ofthe material.

Figures 5 and 6 gave other examples of this procedure [2]. Eachexperimental point corresponds to the average over ten samples. The annealing timeat each temperature was 1 hr.

The following Tables la, lb and lc list the annealing and the post-annealing procedures used for most of the thermoluminescent materials.

70 6 0 • • paak area 7100 R Co 8——a % standard deviation

60 6 z

250 5 5

9 S~40 4 *

5ao i N " S — — i $ 3 i5 20 ["• -* Y 2 ™

10 T 1

400 500 600 700ANNEALING TEMPERATURE i t )

Fig.4. TL emission and corresponding S.D.% vs annealing temperatures.


700 j jO.45

f 600 - ?T~~-~-^ _Zi^-----^^* "—~* " °"4_o • / \ — 0.35

| 5 0 ° - X - \ -0.3.E 400 -- " ^ \ -* 0.25 £

g 300 - v • ^ -- 0.2 ^Q.,n r , -HB^TL-output ^ - 0 . 1 5(o 200 --£ — •- -%STD - 0.1pi 1 0 ° - - 0.05

0 -I 1 1 1 1 1- 0100 200 300 400 500

Annealing temperature in °C

Fig.5. Behavior of the TL response and the corresponding standard deviationas a function of the annealing temperature (Ge-doped optical fiber).

70 -| j- 0.45

I so -• ^ r . - -•

f 4 0 - ^ T L - o u t p u t • - . .-' 0 , 5 0o 30 - -•- - %STD - - 0.2 jo

% - 0 . 1 5

1 0 0.05

0 -I 1 1 1 1 1- o

100 200 300 400 500

Annealing temperature (°C)

Fig.6. Behavior of the TL response and the corresponding standard deviationas a function of the annealing temperature (Eu-doped optical fiber).

CHAPTER A 15

I material annealing procedure

| 1 in oven | in reader |

LiF:Mg,Ti I 1 h at 400°C + 2 h at 100°C [4] or I 30 sec at 300-(TLD100,600, 1 h at 400°C + 400°C

700) 20 h at 80°C [4] (+ 20 h at 80 °Cfast anneal: in oven)

15 min at 400°C +10 min at [3]100°C [5]

LiF:Mg,Ti in I 1 h at 300°C + 20 h at 80°C [6] I 30 sec at 300°CPTFE (+ 20 h at 80°C in

(polytetrafluoroethylene) oven)

LiF:Mg,Ti,Na I 30 min at 500°C + fast cooling I(LiF-PTL) [7]

LiF:Mg,Cu,P I 10 min at 240°C [8-11] or 15 min I 30 sec at 240°C(GR-200A) at240°C[12]

CaF2:Dy I 1 h at 600°C or I 30 sec at 400°C(TLD-200) 30minat450°Cor

1.5 hat400°Cor1 h at 400°C or

lha t400 o C + 3hat l00°C[13,14]

CaF2:Tm (TLD-300) I 1V2 - 2 h at 400°C or I3Ominat3OO°C [15]

| CaF2:Mn (TLD-400) | 30-60 min at 450-500°C [16] | |

j CaSO4:Dy (TLD-900) | V2 -1 h at 400°C | |

CaSO4:Tm 30 min-1 h at 400°C

(PTFE: 2 h at 300°C)

| BeO (Thermal ox 995) | 15 min at 400 or 600 °C [17,18] | 30secat400°C |

| Li2B4O7:Mn (TLD-800) | 15 min - 1 h at 300°C | |

| Li2B4O7:Mn,Si | 30 min at400°C [31] | j

| Li2B4O7:Cu | 3Ominat3OO°C [31] [ j

| Li2B4O7:Cu,Ag | 15 min-1 h at300°C | |

| Li2B4O7:Cu,In | 30 min at 300°C [311 | |

Table l.a. Annealing treatments [3]


material annealing procedure

in oven in reader

q-Al2O,:C ' 1 hat400°C + 16 hat 80°C~Al2O3:Cr 15 minat 350°C

Mg2Si04:Tb 2 - 3 h at 500°C1 h at 300°C

MgB4O7:Dy/Tm 1 h at 500-600°Cn9,20]

MgB4O7:Dy,Na 30 min at 700°C + 30 min at800°C or2hat550°C

[21,22]lhrat400°C[32]

CVD Diamond ' '/2hat300°C [23]KMgF3 lhra t400°C

(various dopants) [24-28]semiconductor- several seconds at 400 ° C

doped Vycorglass

RbChOH" 30 min at 600 ° C [33]RbCl:OH- | I

Table l.b. Annealing treatments [3]

CHAPTER A 17

material pre-readout treatment(post-irradiation anneal)

in oven in readerLiF:Mg,Ti 10 min at 100°C 20 sec at 160°C

(TLD-100,600,700)LiF:Mg,Ti in 10 min at 100°C 10-20 sec at 160°C

PTFELiF:Mg,Na 10 sec at 130°C(LiF-PTL)

LiF:Mg,Cu,P 10 min at 130°C 20-30 sec at 160°C(GR-200A) [29] [29]

CaF2:Dy 10 min at 110°C or 16 sec at 160°C(TLD-200) 10minatll5°C

CaF2:Tm 30minat90°C or 16secatl60°C(TLD-300) 10 min at 115°CCaSO4:Dy 20 - 30 min at 100°C 16 - 32 sec at 120°C(TLD-900) or 5 min at 140°CCaSO4:Tm 20- 30 min at 100°C 16 - 32 sec at 120°C

BeO 1 min at 140°C(Thermalox 995)

Li2B4O7:Mn 10 min at 100°C(TLD-800)

Li2B4O7:Mn,Si 20secatl60°CLi2B4O7.Cu,Ag 20 sec at 160°C

Al2O3:Cr 15minatl50°CMgB4O7:Dy/Tm few sec at 160°C [301

KMgF3 30 - 60 min at 50°C(various dopants) [24-28]

Table 1 .c. Post-irradiation treatments

References1. Benincasa G., Ceravolo L. and Scarpa G., CNEN RT/PROT(74) 12. Youssef Abdulla, private communication


3. Driscoll C.M.H., Barthe J.R., Oberhofer M., Busuoli G. and Hickman C ,Rad. Prot. Dos. 14(1) (1986) 17

4. Scarpa G. in "Corso sulla termoluminescenza applicata alia dosimetria"University of Rome "La Sapienza", Italy, 15-17 February 1994

5. Scarpa G. in "IV incontro di aggiornamento e di studio sulla dosimetria atermoluminescenza" ENEA, Centro Ricerche Energia Ambiente, S.Teresa(La Spezia), Italy, 18-19 June 1984

6. Horowitz Y.S. "Thermoluminescence and thermoluminescent dosimetry"Vol. I, CRC Press, 1984

7. Portal G., Francois H., Carpenter S., Dajlevic R., Proc. 2nd Int. Conf. Lum.Dos., Gatlinburg USAEC Rep. Conf. 680920, 1968

8. Wang S., Cheng G., Wu F., Li Y., Zha Z., Zhu J., Rad. Prot. Dos. 14, 223,1986

9. Driscoll C.M.H., McWhan A.F., O'Hogan J.B., Dodson J., Mundy S.J. andTodd C.D.T., Rad. Prot. Dos. 17, 367, 1986

10. Horowitz Y.S. and Horowitz A., Rad. Prot. Dos. 33, 279, 199011. Zha Z., Wang S., Wu F., Chen G., Li Y. and Zhu J., Rad. Prot. Dos. 17,

415, 198612. Scarpa G. private communication 199113. Binder W. and Cameron R.J., Health Phys. 17, 613, 196914. Portal G., in Applied Thermoluminescence Dosimetry, ed. M. Oberhofer

and A. Sharmann, Adam & Hilger, Bristol, 198115. Furetta C. and Lee Y.K., Rad. Prot. Dos., 5, 57, 198316. Ginther R.J. and Kirk R.D., J. Electrochem. Soc, 104, 365, 195717. Tochilin E., Goldstein, N.and Miller W.G., Health Phys. 16,1, 196918. Busuoli G., Lembo L., Nanni R. and Sermenghi I., Rad. Prot. Dos. 6, 317,

198419. Barbina V., Contento G., Furetta C , Molisan C. and Padovani R., Rad. Eff.

Lett. 67, 55, 198120. Barbina V., Contento G., Furetta C , Padovani R. and Prokic M., Proc Third

Int. Symp. Soc. Radiol. Prot. (Inverness) 198221. Driscoll C.M.H., Mundy S.J. and Elliot J.M., Rad. Prot. Dos. 1, 135, 198122. Furetta C , Weng P.S., Hsu P.C., Tsai L.J and Vismara L., Int. Conf. Rad.

Dos. & Safety, Taipei, Taiwan, 199723. Borchi E., Furetta C , Kitis G., Leroy C. and Sussmann R.S., Rad. Prot.

Dos. 65(1-4), 291, 199624. Furetta C, Bacci C , Rispoli B., Sanipoli C. and Scacco A., Rad. Prot. Dos.

33 107,199025. Bacci C , Fioravanti S., Furetta C , Missori M., Ramogida G, Rossetti R,

Sanipoli C. and Scacco A., Rad. Prot. Dos. 47, 1993, 27726. Furetta C , Ramogida G., Scacco A, Martini M. and Paravisi S., J. Phys.

Chem. Solids 55, 1994, 1337

CHAPTER A 19

27. Furetta C, Santopietro F., Sanipoli C. and Kitis G., Appl. Rad. Isot. 55,2001,533

28. Furetta C , Sanipoli C. and Kitis G., J. Phys D: Appl. Phys. 34,2001, 85729. Scarpa G., Moscati M., Soriani A. in "Proc. XXVII Cong. Naz. AIRP,

Ferrara, Italy, 16-18 Sept., 199130. Driscoll C.M.H., Mundy S.J. and Elliot J.M., Rad. Prot. Dos. 1 (1981) 13531. Kitis.G, Furetta C. Prokic M. and Prokic V., J. Phys. D: Appl. Phys.

(2000) 125232. Furetta C , Prokic M., Salamon R. and Kitis G., Appl. Rad. Isot. 52 (2000)

24333. Furetta C , Laudadio M.T., Sanipoli C , Scacco A., Gomez Ros J.M. and

Correcher V., J. Phys. Chem. Solids 60 (1999) 957

Anomalous fading

The expected mean lifetime, x, of a charge in a trap having a depth E isgiven by the following equation, according to a first order kinetics:

where 5 is the frequency factor and T is the storage temperature.For many materials it is often found that the drainage of traps is not

accounted for by the previous equation: i.e., the charges are released by the trap at arate which is much faster than those expected from the equation and thephenomenon is only weakly dependent on the temperature. This kind of fading isknown as anomalous fading and it is explained by tunnelling of carriers from thetrap to the recombination centre [1,2].

The anomalous fading is observed in natural minerals, as well as in TLmaterials as ZnS:Cu, ZnS:Co, CaF2:Mn, KC1:T1, etc.

The characteristic of the anomalous fading is an initial rapid decay followedby a decrease of the decay rate over long storage periods.

The experimental way for detecting a suspected anomalous fading is toperform a long-term fading experiment in order to accumulate a measurable signalloss and to compare the experimental amount of fading to the one calculated takinginto account the quantities E, s and the storage temperature.

References1. McKeever S.W.S., Thermoluminescence of Solids, Cambridge University

Press (1985)


2. Chen R. and Kirsh Y., Analysis of Thermally Stimulated Processes,Pergamon Press (1981)

Anomalous thermal fading

This effect has been encountered in dating of meteorites. When very longperiods of storage/irradiation are involved, a thermal fading of the TL correspondingto very deep traps becomes significant. This means that the thermal detrapping ofthese traps takes place at the same time of their filling due to the irradiation. In caseof very long periods it can be possible to reach an equilibrium condition between thefilling rate and the detrapping rate although a saturation level has not reached.

Area measurement methods (generality)

The area methods are based on the measurements of the integral of theglow-peak; it can be applied when a well isolated and clean peak is available.

Assuming a first order kinetics, the Randall-Wilkins relation in the timedomain gives

\Idt = n = Y~ 0)sexp(-—)

In the temperature domain the same relation leads to

/ s £-jjr = " iTe xP(-T^) (2)J, IdT P kT

l n h ^ — ] = ln^-~^ (3)J, IdT P kT

where the integral is calculated as shown in Fig.7.

CHAPTER A 21

Again, the In term is a linear function of 1/T and can be plotted on asemilog paper to obtain the slope -E/k and the intercept ln^/p").

I

T Teo T

Fig.7. How to calculate the integral comparing in Eq.(3).

Area measurement method (Maxia et al.)

Maxia et al. [1] have suggested a rather complicated area method for theevaluation of both activation energy and frequency factor. It essentially stems fromthe fact that the filled traps density, at any temperature T, is proportional to theremaining area of the glow-peak. The main assumption is that the various peaks in acomplex glow-curve result from the escape of electrons from a single trap and theirrecombination into various recombination centres. The mathematical treatment isbased on the equation proposed by Antonov-Romanovskii [2]:

dm. ( E\ Binmi

-*- = -SeX*Cla;)A(N-n) + Blml 0 )

where, in particular

nii is the concentration of luminescent centers,A is the probability factor for retrapping,Bt is the probability for recombination,N is the electron trap concentration,

(1)


n is the trapped electron concentration.

Using a constant heating rate one can write

£-«/(r> (2)where a being a constant depending on the light collection efficiency and the usedunits.

From Eq.(2) one has

m,(T) = a^ I{T')dT = aS(T) (3)

and

mi0=affI(T')dT'=aS0 (4)

in which

To = temperature at the beginning of the glow-peak,Tf= temperature at the end of the glow-peak,T = temperature between To and T/,So = total area from To to 7},S = area from T and Tf.

Introducing now the areas CJ and S defined as

n0 = ac (5)

N = dL (6)

from a single glow-peak one obtains

n0 = mi0 (7)

and then

CHAPTER A 23

a = So (8)

where n0 and mi0 represent initial values.By straight-forward calculations, using Eq.(l) to Eq.(6), one gets

, f , Xcose+S(7>in8i E

Tml5(mw'r"tf+' (9)where

B-AtanO = — — (10)

A(L-a+S0)

A=a-S0 (11)

» = -\\ntij-J[AiZ-a+S0)2+(B-A)4 (12)

The plot of Eq.(9) will be rectilinear if an appropriate value of 0 is chosen.The angle 9 depends on unknown parameters, as one can see from Eq.(lO). Todetermine 0 one can put

, f , cosG +S(T)sine"^ = T ( r ) SHT) + S(T)A J <B)

x=- (14)

The searched value for 0 is the one corresponding to a linear behavior of yversus x. Such a behavior can be carried out considering the covariance, cov(x,y),and the variances, V(JC) and v(y), for x and y respectively, hi this way the correlationcoefficient can be numerically computed as a function of 8. Then, using the angularcoefficient of the straight line described by Eq.(9), the activation energy can befound as


cov(x,7) , cov(x,>;)

£ = -£—n—= -«—rr~ (15)

The authors claim that this method is also applicable when overlappedpeaks are present.

References1. Maxia V., Onnis S. and Rucci A., J. Lum. 3 (1971) 3782. Antonov-Romanovskii V.V., Isvest. Akad. Nauk. SSSR Fiz. 10 (1946) 477

Area measurement methods (May and Partridge: general order)

May and Partridge proposed the area method in the case of a general orderb. In this case the equation is

/ Eln(^-) = ln(s)- — (1)

which is graphically processed in the usual way. By visual examination of suchplots, the value of b which gives the best straight line can be ascertained.

ReferenceMay C.E. and Partridge J.A., J. Chem. Phys. 40 (1964) 1401

Area measurement method (Muntoni et al.: general order)

Muntoni and co-workers suggested a method base on the glow peak areaand on the fact that the filled traps density, at any temperature T, is proportional tothe remaining area of the glow-peak S(T). They used the general order kineticsequation in the form

I = -a— = aAmbexp(-—) (1)

where m is the concentration of recombination centres, A is a proportionality factor,b represents the kinetic order and a is a constant. The integral area, S(T) in theinterval from 7 to 7\in the glow-curve, is given by

CHAPTER A 25

S(T) = £ ' IdT = -afi ^dm = apm (2)

from which, considering the order b,

m =V^\ (3)

Equation (1), using Eq.(3), yields to

ln[^f = C"^ (4)

A plot of the first term against 1/T gives a straight line when the best valuefor b is chosen.

ReferenceMuntoni C , Rucci A. and Serpi A., Ric. Sci. 38 (1968) 762

Area measurement method (Moharil: general order)

This method [1] uses the peak area and it is based on the Antonov-Romanovskii equation [2]:

dn Bnm E

I = - ^ = SBn + A(N-n)eXp(-l¥) (1)

Considering that:«0 is proportional to the area under the total peak (= Ao); n is proportional to theremaining area, AT , under the glow peak, from a given temperature T to the end ofthe peak. If n0 = JV, saturation case, Ao is proportional to JV: in this case the area isdenoted by As instead of AQ_

Equation (1) transforms in, with the conditions n = m, n = «o

(1)


BAl , E

The intensity at the maximum, /M, and the half maximum intensities, /, andI2, from Eq.(2) can be derived from the following two expressions:

l n 2 = ~T T~"r" + l n "P"' / ^TV r" V s M B-> (3)

For different values of the ratio A/B, E can be calculated. The correct valueof A/B is the one which gives the same value of E from both equations.

The same procedure can be carried out in the case of non-saturated doses[3].

References1. MoharilS.V., Phys.Stat.Sol.(a) 73(1982)5092. Atonov-Romanoski V. V., Bull. Acad. Sci.USSR Phys.Res. 15 (1951) 6733. Moharil S.V. and Kathurian S.P., J.Phys.D: Appl.Phys. 16 (1983) 2017

Area measurement method (Moharil: general order, s = s(T))

Moharil suggested a new method to obtain the trapping parameters whichdoes not require a priori knowledge of the order of kinetics. Furthermore, heassumes a temperature dependence of the frequency factor.

Starting from the general order equation, he modifies it as follows, takinginto account the temperature dependence of the frequency factor:

/ = -•§• = VVexp(-^) (1)

(2)

CHAPTER A 27

and assumes that the glow-curve consists of a single glow-peak corresponding totraps of only one kind.

If it is assumed that at the end of the glow-curve all the traps are emptied,the number of traps populated at the beginning of the glow-curve, ng, is proportionalto the area under the glow-curve. As a consequence, the number of traps emptied upto the temperature T is given by the area under the glow-curve up to T. Hence, thenumber of populated traps at temperature T is proportional to the area AT which isequal to the total area under the glow-curve less the area under the curve betweenthe initial point and T. So, Eq.(l) becomes

I = Bs0TaAhTexp(-~;) (2)

where B is a constant.Let TM be the temperature at which the TL intensity is maximum and 7

and T2 be the temperatures at which the intensity falls to half of its maximum bothon the low and high temperature sides of TM, respectively. Figure 8 shows thedifferent parameters. Equation (2) then gives

IM=BsJ«Alexp[-1^-j

Il=^BSJ1aAbnexp[--j^J (3)

/2^or2°4exP(--!)

From the previous expressions one obtains

(4)


The terms containing a can be neglected because ^(T^/T^ and ln^j^T,)

are of the order of 10 and a lies between -2 and +2. Using Eq.(4) E and b can thenbe obtained. The value of a is obtained by Eq.(2):

In/ = ln5 + a l n r + 6 1 n 4 - - - ^ (5)

Because at T=TM, d(lnI)/dT=0, one gets

where d(\nAT)/dT is known experimentally and will be negative. The sign of adepends upon whether E/kTu is larger or smaller than [i7>Md(hL4T)d7] calculated at

r=rM.After having determined the values of a, b and E, one can now obtain the

frequency factor starting from the general order equation including thetemperature dependence of the frequency factor:

- ^ = -«\rexpf-—} (7)dt ° \ kT)

which can be written as, using a linear heating rate 3 = ATI At

_drt=^r exp(_A)Jr (8)

nb 0 VK kT '

Integrating this equation between 0 and T and using the condition n = ng atT=0:

W pJ \ kT'J

(6)

(5)

CHAPTER A 29

and the expression of n is obtained:

so that the expression of the intensity/is now

1 = nlsj* exp(-AjT, + (*-»°"" J r expf--?-Vl^

where sono T is similar to the frequency factor 5 of the first-order equation.Using the substitution

RTa = s with R = s0nb0~l

the previous equation for the intensity becomes

/ = n0R r exp(-£-{l + fc* J r expf- A U ' l ^

(10)

Since d7/c/r = 0 at r=7'w,-Eq.(10) gives

a +

(H)

(9)


where the integral is evaluated by graphical method or using the Newton-Raphsona

method. The frequency factor at any temperature is then calculated by s=RT .

500 r

1 400 /

to 300 ^" /

100 200 300 400 500 600 700 800 900

CHANNELS

500

O /

H ^ ^ ^ ^ /

100 200 300 400 500 600 700 800 900CHANNELS

500 .

jrf. 400 /

to 300' . /tO A2 /

\ 200. r\ /

100 / \ J

100 200 300 400 500 600 700 800 900 1000CHANNELS

Fig.8. In this figure the channel numberis proportional to the temperature.

CHAPTER A 31

ReferenceMoharil S.V., Phys.Stat.Sol.(a) 66 (1981) 767

Area measurement method (Rasheedy: general order)

M.S.Rasheedy developed a method of obtaining the trap parameters of acomplex TL glow-curve including several peaks [1]. His method is based on adevelopment of Moharil's method. The trap parameters are obtained starting fromthe higher temperature glow peak. The procedure begins by determining the order ofkinetics, b, of the higher temperature peak of the glow curve.

It is assumed, at first, that the glow curve consists of a single glow-peak,corresponding to only one kind of traps. Furthermore, it is also assumed that all thetraps are emptied at the end of the glow-curve. As usual, the concentration of thetrapped charges, at the beginning of the glow-curve, ng, is proportional to the area Aunder the glow-curve; then, the concentration of trapped charges at any temperatureT., during the read out run, is proportional to the area At under the glow-curvebetween T. and the final temperature, T. at which the TL light falls to zero. Figure 9gives a sketch of an isolated peak with indication of the different parameters.

Taking into account the equation for a general order kinetics written asfollows [2]:

dn nb , £ N

1 = = — T T 5 e x P ( ) (!)

and considering the maximum intensity of the peak, IM , the previous equationbecomes

r < { E 1/ = —-^r-sexp (2)

m N"-1 { kTm)According to the quantities shown in Fig. 9, the following equations can

also be written

(3)

(2)


where A2 and A4 are the areas under the glow peak from T2 to Tf and from T4 to T,

respectively.Making now the logarithm of Eqs. (2) and (3) one gets

[\n2-bln(^)]kTMT2

E = ^ (4)TM~T2

[In2-bH^f-)WMT4

E=—r^r (5)

Eliminating E from the two previous equations, it is easy to find an expressionwhich gives the kinetics order b:

;ztiFig. 9. An isolated glow-peak with the parameters of interest.

CHAPTER A 33

b_ T2(TM-T4)\n2-T4(TM-T2)\n4 (6)

T2(TM-T4)ln(^f)-T4(TM-T2)\n(^f-)A2 A4

The previous equations can be arranged to determine E and b using anyportion of the descending part of the glow-peak. Let us indicate Ix the TL intensity ata temperature Tx of the descending part of the peak and T'2 and T'4 the temperaturesat which the TL intensities are equal to IJ2 and IJ4 respectively; the new equationsare then

[\n2-b\n(^))kTX

Tx-T2

[]n2-b\n(^)]kTxT2 (7)

E = 4TX~T4

b_ T2(TX-T4)\n2-T4(TX-T2)ln4 (8)

T2\Tx-T4)ln(^f-)-T4(Tx~T2)ln(^)A2 A4

The same method is then applied to glow-curves having peaks more thanone. In particular, the author applies his method to BeO which presents a glow-curvewith two well resolved peaks. The first step of the method consists of determiningthe trap parameters of the higher temperature peak. The value of b of the peak isevaluated at different intensities of the descending part of the peak starting from Tu

The pre-exponential factor

s-=s(^r (9)

and the relative value n0 are estimated by the equation

P£exp(-^)s,, = *!±M (io)

M7£-£(2>-l)<Dexp(-f-)K1 M

and where

Eq.(lO) is obtained by equating the derivative of the following equation to zero

1

Now, substituting 5" into Eq.(l 1) one obtains

V " e " P ( - 4 ) (.2)(1+[fc>K:rexp(_Avr]F[ L p ^0 FV fcr' JJ

Furthermore, the relative value of ng can be found using the maximumintensity 1M In this case the procedure is the usual one, which means making thelogarithm of Eq.(ll), then its derivative with respect to the temperature T andfinally to equate to zero the derivative at T = TM In this way the maximum of theintensity, 7^, is given by the following expression:

(11)

CHAPTER A 35

"</ 'exp(--—) (13)J K1M

1M ~ _b_

'kTlbs" / E >->

from which the value of ng is obtained

b

E r ~\b-\I}A eXP(jLT ) hfrfl „

„ klM °MMS (14)0 " e» E

P^exp(—r)Klu

References1. Rasheedy M.S., J. Phys. D: Appl. Phys. 29 (1996) 13402. Rasheedy M.S., J. Phys.: Condens. Matter. 5 (1993) 633

Arrhenius equation

The Arrhenius equation gives the mean time, T, that an electron spends in atrap at a given temperature T. It is

where 5 is the frequency factor (in the case of thermoluminescence the frequencyfactor is also called attempt-to-escape frequency), E is the energy differencebetween the bottom of the CB and the trap position in the band gap, also called trapdepth or activation energy, k is the Boltzmann's constant.

Equation (1) can be rewritten as

p=x~l (2)

which gives the probability^, per unit of time, of the release of an electron from thetrap.

(14)

(1)


According to Eq.(l), if the trap depth is such that at the temperature ofirradiation, let us say Th E is much larger than kTh electrons produced by irradiationand then trapped will remain in the trap for a long period of time, even after theremoval of the irradiation. The Arrhenius equation introduces the concept of anactivation energy, E, seen as an energy barrier which must be overcome in order toreach equilibrium.

ReferenceBube R.H., Photoconductivity of Solids, Wiley & Sons, N.Y. (1960)

Assessment of random uncertainties in precision of TL measurements (general)

The reproducibility of TL measurements depends on the dose level. Figure10 shows how the standard deviation, in percentage, behaves as a function of thedose.

From the figure it can be observed that the relative standard deviation inpercentage decreases very fast as the dose increases. As the dose increases, therelative standard deviation assumes a minimum constant value.

This behavior is justified by the competition of two components:

~ the intrinsic variability of the TL system, given by the standard deviation ofthe zero dose readings (background),

~ the variation of the TL system at high doses, expressed in terms of standarddeviation.

Burkhardt and Piesh [1] and Zarand and Polgar [2,3] used a mathematicalformalism to describe the effect of the two components so far introduced. Theyproposed the following expression

VD=^IKG+GIDD* (1)

where

CT D = standard deviation of the evaluated dose D,

cr BKG = standard deviation of the zero-dose readings, expressed in unit of dose,

OrD = relative standard deviation of the readings obtained at the dose D,

relatively high.

CHAPTER A 37


CD _ qBKG . 2 o .

From Eq.(2) it can be observed that:

™ the ratio a BA:G /£) becomes almost zero for doses quite large with respect

to G BKG and then G D j D ~ G r D , which takes into account the minimum

and constant value observed,

" for very little doses, the term G r D becomes negligible and Eq.(2) assumes

the form

y = - (3)x

™ expression (3), on a log-log scale, is a straight line having a unity negativeslope (in the region 1-10 |aGy of Fig. 11).

™ furthermore, modifying Eq.(2) as in the following

g BKG

°D _ CTBKG , _ 2

or better

(2)

(4)


From Eq.(4) a D /D is independent by the standard deviation, expressed in

terms of dose, of the zero-dose readings, but only depends by R and a r D, as is

shown in Fig. 12.

120-r

100 |

r so JP 60-H5 1UJ 1tC T

o f V ^ ? , ? , ? ^, ? , , , , ? , , , , ?0 50 100 150 200

DOSE(nGy)

Fig. 10. Behavior of the Rel. Stand. Dev. of the TL readings as a function of dose.

l.OE+03 -r 1 1 1

g'l.OE+02 • 5—

" l.OE+Ol • — ^ ^ —

1.0E+00 -I • •— I • •— I • •—• • • • • ! !

1.0E+0O 1.0E+01 1.0E+02 l.OE+03DOSE (tiGy)

Fig.l 1. Same plot of Fig. 10 but in log-log scale.

CHAPTER A 39

1 0 0 0 • I i i i i i i i - m— —

j?

i 100 <. B, * °-°5 J - - = —a ^ = = = = E : : : ^ = = - = = E : : ^ B = 0.04 -EEi:H \ . ——— r- /| —' M i l -(u ' 1 k j - — 7 - - / — B = 0 .0 3 "

I =i5^=:::::===::"/::=zr^B-o.o2° . I - ^ U '—.-/—A L y L B - 0 . 0 1

g i o . . — = ^ = = : = ; L /__-z.._7Z=Z=_^=:

S T ^ ~-^3j-l|'-OH>«mHHh-l

1 10 1 0 0 1 0 0 0

R

Fig. 12. Behavior of — (%) as a function of £ =D aBKG

fo r g i v e n v a l u e s o f B = a rD.

References1. Burkhardt B. and Piesh E., Nucl.Instr.Meth. 175 (1980) 1592. Zarand P. and Polgar I., Nucl.Instr.Meth. 205 (1983) 5253. Zarand P. and Polgar I., Nucl.Instr.Meth. 222(1984)567

Atomic number (calculation)

For some practical dosimetric applications, as the wide range ofradiological dosimetry, two properties of the TL dosimeters are advantageous forprecise measurements. These are high sensitivity and tissue equivalence. Highsensitivity thermoluminescent phosphors (i.e. CaF2 and CaSO4) have high effectiveatomic numbers, Zeff, so that at photon energies below about 100 keV, the responseto a given absorbed dose of radiation becomes significantly greater than that athigher energies. In this region the photoelectric effect is predominant and the cross


section per atom depends upon approximately Z4 for high atomic number materialsand on Z4'8 for low Z materials. Since each atom contains Z electrons, the coefficientper electron depends upon Z3 and Z3 8 for high and low Z materials respectively.

It is important to know a priori the effective atomic number of athermoluminescent material, Z, for getting an idea of the expected TL response atdifferent energies. The behaviour of different materials to X and gamma raysdepends on the atomic number of the constituents and not on the chemicalcomposition of these constituents.

Z = ^a 1 Z; c +a 2 Z 2 x +_ (1)

2>,(z,)i

n,=NA-Z, (3)

where a,, a2,... are the fractional contents of electrons belonging to elements Z;, Z2 ,... respectively, w, is the number of electrons, in one mole, belonging to eachelement Z, and NA is the Avogadro's number. The value of x is 2.94.

A numerical example concerning LiF is given below:1 mole of compounds contains 6.022 -1023 atoms so that 1 mole of LiF has6.022 • 1023 atoms of Li and 6.022 • 1023 atoms of F. Now, the number of electronsbelonging to each element in 1 mole of compound is given by the atomic number ofthe element multiplied by the number of atoms:

for Li: 3 • 6.022 • 1023 = 1.81 -1024 electrons,for F: 9 • 6.022 • 1023 = 5.41 • 1024 electrons.

The total amount of electrons in LiF is then 7.23 • 1024.The partial contents, ai, are respectively

flil=Mli^ = 0.257.23-1024

(2)

CHAPTER A 41

5.411024 nnc

aF = ,7 = 0.757.23-1024

Then

Z2.94= 32.94= 2 5 2 8

Z2.94= 92.94= 6 3 8 9 6

from which

Z 2.94 r -3O„ Li = 6 - 3 2

aF-ZFM = 479.22and finally

Z«8.2

Alternatively, the number of electrons per gram can be calculated asfollows

where N& is the Avogadro's number, Aw,i is the atomic number, W\ is the fractionalweight and Z\ is the atomic number of the i-th element in the compound.

The following table shows the atomic number of the main TL materials.

phosphors effective atomic numberLiF:Mg,Ti LiF:Mg,Ti,Ma LiF:Mg,Cu,P 8J4Li2B4O7:Mn Li2B4O7:Cu Li2B4O7.Cu,Ag 1_AMgB4O7:Tm MgB4O7:Tb 8;4CaSO4:Dy CaSO4:Tm CaSO4:Mn 153CaF2:Dy CaF2:Mn CaF2 (nat) CaF2:Tm 163BeO JA3A12O3 10;2ZrO2 3^6KMgF3 (various activators) 13.4CVD diamond 6Ca5F(PO4)3 14MgF2 10Mg2SiO4 HNa4P2O7 11

(4)


Reference.

Mayneord W.V., The significance of the Roentgen. Acta Int. Union AgainstCancer 2 (1937) 271

BBasic equation of radiation dosimetry by thermoluminescence

A certain amount of the ionizing radiation energy absorbed by an insulatingmedium, i.e., a thermoluminescent material, provokes the excitation of electronsfrom the valence band (VB) to the conduction band (CB) of the material. The freeelectrons in the CB may be trapped at a site of crystalline imperfection (i.e.,impurity atom, lattice vacancy, dislocation). The trapped electrons have a certainprobability per unit of time, p, to be released back into the CB which depends on thetemperature (7) and on the activation energy (£). This probability is given by theArrhenius equation rewritten as

p^expf-A] (•>

where s is a constant for each kind of insulator, called frequency factor, in s"1, E isthe activation energy, called trap depth, in eV, given as a difference between the traplevel and the bottom of the CB, k is the Boltzmann's constant (0.862KT4 eV/K), T isthe temperature in K.

By heating of the sample, the filled traps can be evacuated by thermalstimulation of the trapped electrons which rise to the CB. From here the freeelectrons have a certain probability to recombine with a hole at some sites, calledluminescent or recombination centres. The recombination event results in theemission of visible light. This emission of light is called TL glow curve which isformed, in general, by some peaks. Each peak reflects a trap type having a definedactivation energy. The wavelength spectra of the emitted light gives informationabout the recombination centres.

Let us define N as the concentration of empty traps in the material. Duringirradiation at a dose rate dD/dt the filled traps are

Nf=N-n (2)

where n is the concentration of the remaining empty traps. So the rate of decrease ofn can be written as

dn dD~—- = A-n~- (3)

dt dtwhere A is a constant of the material, called radiation susceptibility.

Making the assumption that no trapped electrons are thermally releasedduring the irradiation (i.e., the filled traps are deep enough to resist to a thermaldrainage), Eq.(2) can be integrated as follows, with the initial condition that at t=0,n=N

}dN I dtfrom which

n = Nexip(-A-D) (4)

where D is the total irradiation dose received by the material during the irradiationtime t.

It is now possible to define the constant A considering that if Dm is theradiation dose needed to fill half of the empty traps, from Eq.(3) we obatin

. 0.693A =

A/2

The filled traps at the end of the irradiation is given by

Nf=N[l-exp(-A-D)]

The heating phase of the irradiated sample, for obtained thermoluminescence, can beexpressed as follows

dN, ( E \f~ = p-Nf =Nf -s-expl

dt f f { kT)and the intensity of thermoluminescence, I(D,T), is then given by

l{D,T) = -Cd^ = C-s- N[\ - exp(- A • Z))]expf - ~) (5)at \ kTJ

lfAD< 1 for small values of D, l-exp(-^Z>) can be approximated to AD and thenEq.(5) becomes

CHAPTER B 45

l(D,T)=C-s-N-A-D-&J- —) (6)

from which it is easily observed that the TL intensity at a given temperature, i.e., theglow peak temperature, is proportional to the received dose D.

Batch of TLDs

A batch of TLDs is defined as the whole number of dosimeters of the samekind of material and activator(s), as obtained from the manufacturer, having thesame thermal and irradiation history and, possibly, produced at the same time (thislast requirement is not imperative). Before using a new batch of TLDs, it has to besubmitted to an initialization procedure.

Braunlich-Scharmann model

A more satisfactory physical interpretation of the TL kinetics can be basedon a more complex description of the TL centers in the forbidden gap. Braunlich andScharmann (1966), wrote a set of differential equations describing the traffic of thecharge carriers, during the thermal excitation, making reference to the energy levelscheme proposed by Schon. This scheme contains one electron trap, one hole trapand retrapping transitions of the freed carriers back into their respective traps. Thefollowing Fig.l shows the band model used to describe the traffic of the carriers.

Explanation of the symbols:

"" nc = concentration of electrons in CB,

~ nv = concentration of holes in VB,

~ n = concentration of trapped electrons,

~ JV = concentration of electron traps,

"* m = concentration of trapped holes,

~ M = concentration of recombination centers (hole traps),

~ An= retrapping probability for electrons in N,

" Amm = recombination probability for electrons in M,


"" Ap= retrapping probability for holes in M,

~ Anp - recombination probability for holes in N,

I A IPn A n

m n \ r

N A nM T mAp ^ PP A n p

J r

V B n

Fig. 1. The energy level scheme proposed by Schon.

( En\" Pn=S« eXP

is the thermal excitation probability for electrons from N to CB,

Fp " \ kT)is the thermal excitation probability for holes from Mto VB,

~ En = electron trap activation energy,

~ Ep = hole trap activation energy.

The set of the differential equations is:

~- = npn-ncAn(N-n)-ncmAmn (la)

CHAPTER B 47

-^ = mpp- nvAp (M-n)- nvnAnp (lb)

— = -npn + ncAn (N-n)- nvnAnp (lc)

dm . ,, r .— = -mpp + nvAp (M-m)- ncmAmn (Id)

Considering that, in the most general case, both recombination transitionsare radiative, the total TL intensity is given by

dnc dn

~dt~Yt= "c m" + "v "p (2)

Writing the previous equation for the intensity, it has been considered thatthe transitions of conduction electrons into traps and of holes from the valence bandinto recombination centers (hole traps) are non-radiative.

Two parameters have to be defined now:

K = ~ (3a)Amn

R m = ^ (3b)

which express the ratio of the retrapping probabilities compared to recombinationfor both electrons and holes.

The neutrality condition is given by

nc+n-nv+m (4)

and furthermore, with the assumptions that

nc «n, nv «m (5)

the following relation is also valid:


n » m (6)

Four cases can be analyzed now:

a) Rn*0, Rm*0

b) Rn »• 1, Rm »• 1

c) Rn « 0, J?M 1

d) Rn »• 1, Rm « 0

Case a) concerns a situation where recombination prevails over trapping, incase b) retrapping prevails over recombination and the two other cases areintermediate.

The quasi-equilibrium assumption is valid for both electrons and holes:

dn dnv

—c~ = —- « 0 (7)dt dt

Case (a)The retrapping rate for both electrons and holes is very small. Then the

retrapping terms can be neglected. Furthermore, taking into account the quasi-equilibrium condition the previous Eqs. (la,b,c,d) become

dnr

—r^nPn-ncmAmn (8a)

at

^ = mpp-nvnAnp (8b)

dn .-jt=-npn-nMnP (8c)dm .— = -mpp-ncmAmn (8d)

Because n « m, from Eqs. (8a) and (8b) we obtain, taking into account relation (7)

(8b)

(8c)

(8d)

CHAPTER B 49

nc*-j»- (9)

Ppnv~~- (10)

A»P

Eq.(8c) then reduces todn

-Jt=-<Pn+PP) (")

Considering a constant heating rate $=dT/dt, Eq.(l 1) becomes

±~^it «>2,n 3Megration of Eq.(12) yields

n = noexVl-jj\pn+pp}iA (13)

Going back to Eq.(2), it can be rewritten, using Eqs. (9) and (10), as:

I = pnm + ppn (14)

and using the relation (6) n » m

I = n(Pn+Pp) (15)

which can be rewritten, using Eq.(13)

I = "\pn +P , ] exp \ - j \Pn +Pp\iT'\ 06)

which is similar to the Randall-Wilkins first order equation. Neglecting thetransitions to the valence band, i.e. pp = 0, the Randall-Wilkins equation is obtained.

(12)

(14)

(15)

Case (b)The retrapping of charge carriers prevails over the recombination

transitions.Equations (la,b,c,d ) become now

-r± = npn-ncAn(N-ri) (17a)at

^ - = mpp-nvAp(M-n) (17b)

- ~ = -npn + ncAn (N-n)- nvnAnp (17c)

dm , / w x A

— = -mpp + nvAp (M-m)- ncmAmn (17d)

Using now the quasi-equilibrium condition, i.e.

dnc dn™ = « 0 (18)dt dt

and the neutrality condition in the form

dn dm

* " • * ( 1 9 )

from Eqs. (la,c) we get

dn dn dn^+d*^'-A™"'m-A">>"-n < 2 0 )

which becomes, using n « m,

-^ = -n(ncAmn+nvAnp) (21)

Eq.( 17a) becomes

(19)

CHAPTER B 51

npn-ncAn(N-n)*O (22)

Because n«N(far from saturation), Eq.(22) gives

nc « -*-*- (23)

ANSimilarly, considering m « M, we obtain for nv

™PPn**Tir (24)

ApMSubstituting expressions (23) and (24) into Eq.(21), we obtain, using n a m:

^VJIA^+PAA (25)

dt \ AnN ApM)

Using as before a linear heating rate, we get by integration

n = -. p Y (26)

[«„ P l i ^ A,N ) { ApM ) \ J

In conclusion, the TL intensity is given by

I=_dfL= \ [PnAmn , PpAnP )

dt f , , U A \ (A n M V\ AN AM

\n0 $*•.]{ A.N) {ApM)\ J(27)


This equation is similar to the second order equation given by Garlick andGibson. It becomes identical to it by neglecting the probability for transitions intothe valence band, i.e. supposing pp = 0.

Case (c)The new equations are now:

dn~^ = npn-ncmAmn (28a)

- ^ = mpp-nvAp(M-m) (28b)

-^ = -npn + nc An (N-n)- nvnAnp (28c)

~ = ~mpp + nvAp (M-m)- ncmAmn (28d)

From Eqs. (28a) and (28b) we get

n mpn

n c * ^ , nv= ^ (29)

From (29) and (28d) we obtain

dm— = -mpn (30)

dtand then

m«moexpl-^^pndT'} (31)

The thermoluminescence intensity is

I = nMmn+nvnAnp (32)

which transforms in, using (29), « « m and M» m

CHAPTER B 53

I = mpn+m2^r (33)ApM

and then, the explicit form for / is the following

/ = m,p, <J-1 f p,dT] + *l£- expf- \[ p.dr] (34)

which is again the Randall-Wilkins equation for pp = 0.

Case (d)Equations (la,b,c,d) reduce to

dr^ = npn-ncAn{N-n) (35a)at

d ^ = mpp-nvnAnp (35b)

- 7 = -«/>„ + ncAn (N-n)- nvnA (35c)

- ^ = - w p , + « v ^ ( M - m) - «cw^m n (35d)

Assuming the quasi-equilibrium condition, i.e.

^ « 0 , ^ « 0 (36)dt dt

and TV >> n, nv and «c very small, i.e. m««,we get from (35a)

n « ^ (37)

C ANand from (35b)

nv«?f- (38)

a

Then

dn-r-npn+ncA^N-n)-nvnAnp=-dr^-h^ = -npp (39)

from which, by integration

/i = /ioexp[--^-£/Vflr'J (40)

The TL emission is then given by

I = ncmAmn+nvnAnp (41)

which transforms, using approximations (37), (38) and n « m, in the followingexpression

j _ n Pn mn ( 4 2 )

AnN Fp

Using Eq.(40), we get the final expression for the intensity:

1 -%MT ip'dT)+n"''exp(T I"'*1") <43)This equation becomes again the Randall-Wilkins equation of the first

order, ignoring the thermal release of trapped electrons, i.e. pn = 0.

ReferenceBraunlich P. and Scharmann A., Phys. Stat. Sol. 18 (1966) 307

cCalcium fluoride (CaF2)

CaF2, activated by various dopants, is a TL phophor widely used in manydosimetric applications. It is used as natural CaF2 or with different activators as Mn,Dy and Tm [1-10].

Preparation of CaF2:Mn is carried out using the precipitation techniquefrom a solution of CaCl2 and MnCl2 in NH4F. The precipitate is dried and heated inoven with inert atmosphere at 1200°C, then it is powdered and graded. The finalmaterial can be pressed and sintered. Its atomic number is 16.57. Its sensitivity at the30 keV of photon energy is 15 times greater than the sensitivity at the 60Co energy.

The linearity of CaF2 natural is observed up to 50 Gy. CaF2:Mn, producedby Harshaw under the name TLD-400, gives a linear response up to 2 KGy.CaF2:Dy has been commercialized by Harshaw under the name TLD-200; it presentsa complicate glow curve consisting of six peaks. The TL response is linear up to 1KGy. CaF2:Tm, known as TLD-300, shows three resolved peaks, high stability andselective peak sensitivity to the radiation quality.

References1. Schayes R. and Brooke C , Rev. MBLE 6 (1963) 242. Ginther R.J., CONF 650637 (1965)3. Binder W., Disterhoft S. and Cameron J.R., Proc. 2nd Int. Conf. Lumin.

Dos., Gatlinburg (USA), 19684. Furetta C. and Lee Y.K., Rad. Prot. Dos. 5(1) (1983) 575. Furetta C , Lee Y.K. and Tuyn J.W.N., Int. J. Appl. Rad. Isot. 36(11)

(1985) 8966. Furetta C. and Tuyn J.W.N., Rad. Prot. Dos. 11(4) (1985) 8937. Furetta C. and Lee K.Y., Rad. Prot. Dos. 11(2) (1985) 1018. Furetta C. and Tuyn J.W.N., Int. J. Appl. Rad. Isot. 36(12) (1985) 10009. Furetta C. and Tuyn J.W.N., Int. J. Appl. Rad. Isot. 17(5) (1986) 45810. Azorin-Nieto J., Furetta C. and Gutierrez A., J. Phys. D: Appl. Phys. 22

(1989)458

Calibration factor Fc (definition)

The so-called calibration factor, Fc , allows to translate the TL emissionfrom a given phophor to the dose received by the phosphor itself. This factorincludes both reader and dosemeter properties.


Many experiments carried out in the field of thermoluminescent dosimetryhave well demonstrated that a reduction of uncertainties in the dose determinationcan be attained using a calibration factor of the dosimetric system.

At first we can introduce an individual calibration factor, Fci, defined for agiven quality of the calibration beam. Therefore, an unknown dose D is given by thefollowing relation

D = Fci • M.net (1)

whereD is the unknown absorbed dose and,Minet is the TL signal, corrected by background, of the ith dosimeter.

The experimental determination of the calibration factor can be carried outin principle in two different ways, according to the methodologies. The first methodconsists of the determination of a single value of the calibration factor, delivering toTLD a calibration dose Dc which is chosen in the linear region of the TL response ofthe material used. The second approach consists of determining a calibration curve,obtained with three or more points of dose, always in the linear region.

Calibration factor Fc (procedures)

Is' procedureLet us show now the first procedure consisting of the determination of only

one calibration factor. In this case it is necessary to introduce a group of referencedosimeters (m > 10), belonging to the same batch of the field dosimeters. As statedbefore, it is very important that the reference and the field dosimeters have the samethermal and irradiation history.

The reference dosimeters have to be prepared and then irradiated with acalibration dose (for every dosemeter the intrinsic background is known), Dc, chosenin the linear range of the TL response. From the calibration factor definition

D = Fci-Minet (1)

we obtain the following expression:

(2)

CHAPTER C 57

where Sr is the intrinsic sensitivity factor.Using Eq. (2), an unknown dose D will be given by

D=Minet-SrFcr (3)

Comparing Eqs. (1) and (3) one can observe that

Kj = Si-Fc, (4)

The previous relation means that the individual calibration factor Fci isdepending on two different quantities: the first one is the relative intrinsic sensitivitySi, which is quite stable during time and then it has to be checked no more than twotimes per year; the second quantity is the calibration factor Fcr, obtained using thereference dosimeters, whose response can vary tremendously from a reading cycle toanother because the delay between the moment of Fcr determination and the periodof field TLDs measurements, which means that any instability in the readerelectronic, for instance due to environmental variations and/or different periods ofswitch-off/switch-on of the reader, is not taken into account. It has been proved thatthe Fcr factor can vary significantly along a period of a few months and provokelarge errors in the dose determination. Then it is recommended to check thecalibration factor before any reading session.

In the case of radiotherapy measurements where the accuracy in the dosedetermination must be within 2 or 3%, it is imperative to determine the F c r factorjust before a cycle of TLDs readings. In this case, the reference dosimeters areirradiated to the proper value of calibration dose and read together with the fielddosimeters to avoid any effect of the TLD system instability.

2nd procedureThe second procedure consists of getting a calibration curve at each reading

session. The calibration curve is obtained using three or more points of dose.The procedure is the following.

~ choose three different values of dose in the linear range, possibly in alogarithmic scale, noted here as Dcj, Dcj and Dcj.

~ prepare a group of reference dosimeters, at least 5 for each level of dose, andirradiate them.

™ read all the dosimeters and correct the readings for background and relativeintrinsic correction factor.

~ the 5 corrected readings corresponding to the dose Dcl are then averaged.


~ call these averaged values as

Mc,i , Mc,2 , Mc,3 (5)

™ for each value and each dose one obtains

Mc,\ Mc,l Mc,3

with the condition

Fc,l=Fc,2=Fc,l (7)

The previous suggested procedures for the determination of the calibrationfactor must be, in principle, repeated at each reading session. In this way thepossible variations in the efficiency of the TLD reader are neglected. However, thestability of the system has to be checked periodically for detecting any possiblevariation due to environmental conditions and/or related to the reader itself.

Competition

Various traps (competitors) may be in competition among them fortrapping the free carriers produced during irradiation or heating. The process ofcompetition has been used to explain the enhancement of the TL sensitivity and thenthe phenomenon of supralinearity [1-4].

Figure 1 shows the competition during irradiation of the TL sample andFig.2 shows the mechanism of competition during heating.

During heating (readout), the electrons released from N\ could be retrappedin N2 or recombine in M. At higher dose levels, N2 could saturate and then thereleased electrons can be involved in the recombination process.

Both models have also been used, among other models, to explain thesupralinearity phenomenon.

(6)

CHAPTER C 59

CB

M y

VB

Fig. 1. Competition during irradiation. Ni = active trap (TL signal),N2 = competing trap having a trapping probability larger than

that of Ni, M = recombination center.

CBI v

—*—N,

M I

VB

Fig.2. Competition during heating.


References1. Suntharalingam N. and Cameron J.R., Report COO-1105-130, USAEC

(1967)2. Aitken M.J., Thompson J. and Fleming S.J. in 2 Conf. Lumin. Dosim.,

Gattlinburg, Tennessee )1968)3. Kristianpoller N., Chen R. and Israeli M., J. Phys. D: Appl. Phys. 7 (1974)

10634. Chen R., Yang X.H. and McKeever S.W.S., J. Phys. D: Appl. Phys. 21

(1988) 1452

Competitors

The term competitors indicate traps which are in competition over freecarriers during irradiation or heating the thermoluminescent samples

Computerized glow curve deconvolution (CGCD): Kitis' expressions

The computerized glow curve deconvolution (CGCD) analysis has beenwidely applied since 1980 to resolve a complex thermoluminescent glow curve intoindividual peak components. Once each component is determined, the trappingparameters, activation energy and frequency factor, can be evaluated.

The main problem is that the basic TL kinetics equations, i.e. the Randall-Wilkins equation for the first-order kinetics, and the Garlick-Gibson equation for thesecond-order, give the glow peak TL intensity, /, as a function of variousparameters:

I = l{no,E,s,T) (1)

wheren0 = initial concentration of trapped electrons (cm3)

E = activation energy (eV)s - frequency factor (s"1)T = absolute temperature (K)

The values of n0 and s are unknown.

Some approximated functions have been proposed for resolving acomposite glow curve into its components: i.e. Podgorsak-Moran-Cameronapproximation [1], Gaussian peak shape, asymmetric Gaussian functions and othersreviewed by Horowitz and Yossian [2].

CHAPTER C 61

From a historical point of view, the PMC approximation was the first.Although it was found that the approximation of PMC function is rather poor, it isthe only one which transforms Eq.(l) into the following

I = I{IM,E,TM,T) (2)

where IM and TM are the TL intensity and temperature at the glow peakmaximum.

The advantage of Eq.(2) is evident: in fact it has only two free parameters,namely 1u and TM , which are obtained directly from the experimental glowcurve.

Kitis [3,4] has proposed new functions for describing a glow peak which,keeping the advantage of the PMC equation, have the same accuracy of the basic TLkinetic equations.

First-order expressionThe TL intensity of a single glow peak following a first-order process is

given by the equation

I(T) = sn0 expj^- -^ exp - ~ jexp^- A jdT (3)

The integral comparing in Eq.(3) cannot be solved in an analytical form,but using successive integration by parts, in a second-order approximation (integralapproximation) it becomes

I { kT') E { E ) \ kT)

Hence, Eq.(3) becomes

( E^ f skT2(. 2kT) f E\]I(T) = sn0 exp exp 1 exp (5)

from which the condition at the maximum is given as

(4)


-§* J C J - * - l (6)kT2 \ kT IK1M \ K1MJ

or

s = -^-exp M M (7)IrT2 \ IrT

~ Inserting Eq.(6) into Eq.(5) one obtains

or better

/^^expKl-Aj] (8)

where A M = ^ .


^L/Mexp(l-Aj (9)

~ Inserting Eq.(7) into Eq.(5), after a little algebra, one obtains

7(70 = ^ 4 - ^ 1

(10)

CHAPTER C 63

with A = .E

~ Equation (9) can now be inserted into Eq.(lO) for getting the finalexpression of the form I(IM ,E,TM,T):

/(n = /M=xp[.^.^-|.(.-A,)exp(|.^)-A;

(11)

Second order expressionThe second-order kinetic equation is

I(T) = sna expf - — | 1 + — fexpf - — \ / T (12)° \ kT)[ prJ \ kT'J J

Inserting the integral approximation given by (4) in Eq.(12), one gets

/(D = OTo exp(- A J ^ l (1 - A)exP(- A ) + 1 ] " «,3)

from which the condition at the maximum is given by

s^.-^J^] (14)

or in another form

jai_^J|_.M.._L_ (15)\ kTu) «3 l + iM

Furthermore, Eq.(13) can be rewritten for the peak at the maximum:

/«=^-^)[*f(.-Ajexp(-^) + 1[2 ,,6,

~ The insertion of Eq.(14) into Eq.(13) gives the following expression forI(T):

im-nÊ l CJE iT-TAkT2M 1 + AM \kT { TM )_

\T2 I - A \E (T-TM\\ 1x<^— exp — + U (17)

[T* 1 + AM \kT { TM ) \ \

~ Inserting Eq.(15) into Eq.(16) we get a more simplified expression for IM:

I = « o M l ( 2 YM kT2M \ + ^M {l + A M )

which can be rewritten as

wop£ 1 f 2 Y

^'Â^HÂ^J (18)

~ Eq.(18) is now inserted into Eq.(17) for getting the final expression for theTL intensity:

T(T\-dT PYJ ^ M\VkT TM )

CHAPTER C 65

x l l f i - A j e J - ^ . ^ l + l + A J (19)

General orderThe equation of the TL intensity for a glow peak following a general-order

process is:

im = «. exp(- ± f 1 + { exp(- A]^. ]"^ (20)

It transforms in the following equation using the approximation (4):

The intensity at the peak maximum is then given by

The maximum condition, obtained from (21), is

withZM=l + {b-l)AM (24)

Eq.(23) can be rewritten in two different ways:

IC1M ^M \K1MJ

or

(21)

(22)

(23)

(25)


,eJ- -l = -f- (26)kT irT2 7

\ KiM ) klMLM

~ Inserting Eq.(25) into Eq.(21) we obtain the following expression for theintensity:

kT 7 kT TK1MZjM \K1 1M )

Jiz>4 ( 1 _ 4 ) e x p rA. iz^i + 1 i^ (27)\ 7 T2 \ IrT T I

"* Inserting Eq.(27) into Eq.(22) we get, after arrangement, the expression forthe intensity at the maximum:

b

7"=^HH (28)from which

b

k^=lMW) (29)

K1MZjM \AM J- Insertion of Eq.(29) into Eq.(27) gives the final equation for the TL

intensity:

/(r) = U^e*p(|.^)

b

(30)

CHAPTER C 67

Equations (11), (19) and (30) are equations in the form1(1 M ,E,TM, T) which has only two free parameters, IM and TM , directlyobtained from the experimental peak.

A further develop allows to transform Eqs.(l 1), (19) and (30) fromAE

the I(IM,E,TM,T) space into the I(IM,(O,TM,T) space, where co = .

With the assumption «0 = 1, Eq.(8), first-order, and Eq.(28),

general-order, become

A/=^-exp(Aj (31)

and

/ - ^ \ b T^ (32)

The two quantities

c ,=-exp(A M ) (33)e

andb

i r b i *-!C* = Z^ [1^(6-1^] (34)

vary extremely slowly in a large range of both E , from 0.5 to 2.5 eV, and s ,from 105 to 1025 s'\ so that they can be considered as constants. In turn, Eqs. (31)and (32) assume the following general form

IM-{cx,cb)^ (35)K1M

Equation (35) can be solved with respect to the activation energy, giving

E = 3 ^ , (36)


In the given range of £ and s values, Kitis found that the quantity IM / p

can be expressed as

^ L = ^ (37)p co

where cd is practically constant.

So, Eq.(36) can be transformed in

IrT2

E = c f ^ - (38)co

where

It must be noted that Eq.(38) is equivalent to the Chen's peak shapeformula based on the FWHM. Cy assumes in this case a mean value of 2.4.

Equations (11) and (30) can be transformed using Eq.(38) as follows, usingthe substitutions:

A_2kT _ 2Ta

A ^

C/TM

E T-TM =cfTM{T-Tu)

kT TM Tco M/

So, Eq.(l 1) transforms in

[ *M y CfIMj Cf1M)

and Eq.(30) transforms in:

(39)

CHAPTER C 69

I(T)

= /M(^-, exp(W/j i + (6-il i — z r b r e W / M * - 1 ) - ^ -V C/1MJIM Cf1M

(40)

Kitis investigated the variations of cfr and c^ as a function of ln(s)

and reported that, in case of first-order kinetics, pairs of E and s can be accepted

if cb , or Cj, are within the following limits

0.38 <cb< 0.4

2.3 < C / < 2.44

taking into account that the minimum value of Cy corresponds to the maximum for

References1. Podgorsak E.B., Moran P.R. and Cameron J.R., Proc. 3rd Int. Conf. on

Luminescence Dosimetry, Riso, 11-14 October, 19712. Horowitz Y.S. and Yossian D., Rad. Prot. Dos. 60 (1995) (special issue)3. Kitis G., Gomez-Ros J.M. and Tuyn J.W.N., J Phys. D: Appl. Phys. 31

(1998) 26364. Kitis G., J. Radionalyt. Nucl. Chem., 247(3) (2001) 697

Condition at the maximum (first order)

An important relationship is obtained by the first order equation

HT) = V=xp(- £ ) exp[- i- (exp(- r f r ] (!)by setting

dI a • T T

~dT = ° at T = TM


For practical purposes, the logarithm derivative is considered:

djlnl) 1 dldT ~ T dT

From Eq.(l) we obtain

then

fc/(ln/)l E s ( E )I dT ]T=Tm kTM P v kTM)

which yields to the expression

$E f E ]- ^ = ,expl- — I (2)

From Eq.(2), the frequency factor is easily determined

$E ( E \

Condition at the maximum (first order): remarks

From the equation at the maximum

k^=sexArw (1)

we can obtain some interesting remarks:- for a constant heating rate TM shifts toward higher temperatures as E increases or sdecreases;

(3)

CHAPTER C 71

- for a given trap (E and s are constant values) Tu shifts to higher temperatures asheating rate increases;- 7V is independent of no.

Condition at the maximum (general order)

The condition of maximum emission for i-order kinetics can be lookedfrom the general order equation:

where s = s"n0 expressed in sec"1.

The logarithm of I(T) is:

ln[/(D] = ln(,«0) - A _ JL J I + ffc1) f expf_ A V ' lL J ° kT b-\ [ P '• I kT'J J

then

^/(In7)T-TM

tr^ 6-i[ p y. \ kr) J p \ kTu)

from which we obtain

kTlbs ( E\ s(b-l) fu ( E\ m

From the last equation it is possible to obtain the expression for the pre-exponential factor. Rearranging Eq.(2), we obtain:

(1)

(2)


Using the integral approximation, we get

JtT^expf-—] , xx

»E { E )

expressed in sec"1.Considering s :

or

which is expressed in cm'^'^sec"1.

Condition at the maximum (second order)

The condition at the maximum is obtained by differentiating the secondorder equation for the intensity

(3)

(4)

(5)

CHAPTER C 73

, N nh'expl„ _ J« 2 , ( E \ ° \ kT)I(T) = = «Vexp = (l)

L P o \ kT'Jby setting

As usual, the logarithm derivative is considered:

ln(7) = ln(»02*') - — - 2 In 1 + f ^ - l f expf- — V '

^ o . ( _ J L )

Then

and rearranging

From this expression, the pre-exponential factor can be determined:

(2)


Using the integral approximation, the previous expression becomes

p£exp| — L _, .

* " o ^ L ^ Jwhich becomes, introducing s = s'n0

P£exp| — | r n .\tTu)\, 2kTMV

s = \ y 1 + ^ (4)

expressed in sec"'.

Condition at the maximum when s'=s'(T) (second-order kinetics)

To obtain the maximum condition we consider the I=I(T) equation:

4^raexp(--^)I(T) = T—; / F N i2 (1)

and its logarithm:

(3)

CHAPTER C 75

ln(/) = ln(^) + alnr--|-21n[^l + ^.{rexp(--|7)^j(2)

The derivative equal to zero yields

(3)

Using the integral approximation in the case of s' temperature dependent, we get

TM kTM [ p E L £ J I * ^ J J

x^r«expf--f] = 0

and rearranging, using h.M=2kT}JE:

— + — - • 1 + - 2 - 2 -— M — 1 - 1 + — AM exp

P " I kTM)

from which the pre-exponential factor can be derived:

n,kT^ \kTM(4)

Condition at the maximum when s"=s"(T) (general-order kinetics)

To obtain the maximum condition, the logarithm derivative of Eq.(2) givenin General-order kinetics when s"=s"(T) will be carried out as follows:

HI)

kT o - l [_ P ° V "T)

Considering

dT k r *

and using AM=2kT/yE, one obtains the maximum condition

Using now the integral approximation when s"=s"(T), we obtain

<6r«+2£exp(-^V|) t r 2 ^"^A 1^ IrTa+2V KIM J _i i S0[p-l) K1M

which can be rearranged for determining the pre-exponential factor:

(1)

CHAPTER C 77

-1 -1

JE^ l-A,Q + ,Xl-*) eJ^l (2)

It must be noted that when a —> 0 Eq.(2) becomes the non-temperaturedependent expression for the pre-exponential factor.

Condition at the maximum when s=s(T) (first-order kinetics)

The condition of maximum TL emission is obtained by the logarithm of theequation

7(7) = nosja exp(-A)expL^ j>« expC-™)^! (1)

i.e.

In I = ln(«050 ) + a In T - — + - ^ ( Ta exp(- — )dT'

and its derivative, d(hi/)/dr equal to zero. Then we have:

T irT2 ft JrT1M K1M P V K1Mj

and the final expression is then


_ P _ exp("^;)Tr=(aT^ + 1 ) (2)

V s0 ksj

Rearranging Eq.(2) and using b^lkT^E one obtains

r^ = exp(~ ~T~*—T (3)

From Eq.(3), the frequency factor is obtained

RE (t a . "\ ( E \sa =— T 1 + — AM exp (4)

K1M \ L J \K1Mj

Considerations on the heating rate

Because the great importance of the heating rate (H.R.) in any kind ofthermoluminescent measurements, it is better to report here the most relevantobservations on this experimental parameter.

Kelly and co-workers [1] discussed about the validity of the TL kinetictheories when high heating rates are involved: they found that heating rates up to 105

°C/s do not invalidate the Shockley-Read statistics on which kinetic theories arebased. Gorbics et al. [2] reported studies on thermal quenching of TL by varying theH.R. between 0.07 and about 11 °C/s. They found the following results:

~ the maximum glow-peak temperature, TM, is shifted to higher temperaturesas the H.R. increases.

~ the TL intensity, measured by both integrating and peak height methods,decreases as the H.R. increases.

Other papers, not specifically dedicated to the effect of H.R. on the TL

CHAPTER C 79

intensity, report experimentally results not always in agreement among them.The H.R. effect on TL glow-peaks has been largely discussed by G.Kitis [3]

who considers the H.R. as a dynamic parameter rather than a simple experimentalsetup variable. His study has been carried out on single, well separated glow peaks,considering the following experimental characteristics: i.e., TM, full width at halfmaximum (FWHM), peak intensity and peak integral. The first thing to beconsidered is a possible delay between the temperature monitored by thethermocouple, fixed on the heating planchet, and the sample. Furthermore, thepossibility of temperature gradients within the measured sample must be consideredtoo. To avoid, totally or partially, these effects, special care has to be taken: i.e., theuse of powder instead of solid samples diminishes greatly the gradient effectswithin the sample as well as between the heating planchet and the sample.

40 - ~7~

h • XK E 20 - X

r /o -j , , , u

0 40 80Heating Rite (°C/sec)

(reader)

Fig.3. The temperature gradient between heating tray and sample,(a) heating rate on the tray, (b) heating rate on the sample [3].

To ensure a good thermal contact between the heating strip and the powdersample, the following rules have to be taken into account:

~ dimensions of powder grains in the range of 80 - 140 mm

~ use no more than 4 mg in weight of powder

~ fix the powder on the heating element with silicon oil.


However, a certain gradient between sample and heating strip is emergingwhen high heating rates are used. Figure 3 shows that temperature gradients emergefor heating rates greater than 50°C/s.

3 F I

•

i - / JA«

LJm, .10 100 200

TEMPERATURE (°C)

Fig.4. Change of the peak shape and shift in the peakposition as a function of the heating rate. From (a)

to (h) = 2, 8, 20, 30,40, 50, 57, 71°C/s [3].

The TL reader used a TL analyzer type 711 of the Littlemore Companywith a planchet of nicochrom of thickness 0.8 mm. The experimental results ofFig.3 have been obtained by measuring directly the H.R. on the planchet and on thesample separately with Cr-Al thermocouples fixed on them. The main results of thisinvestigation concern the influence of the H.R. on the TL glow-peak and aresummarized by the following figures. From Fig.4 one can observe the behavior ofthe shape of the experimental glow-curves for the 110°C glow-peak of quartz,obtained using various heating rates between 2°C/s and 70°C/s. As the H.R.increases, the peak height decreases and the peak temperature shifts towards highvalues of temperature. The shift of TM is better seen in Fig.5, showing the dataconcerning the Victoreen a-Al2O3:C which has a well isolated main glow-peak [4].The dashed lines are the theoretical values calculated using the trapping parametersE and s determined with the lowest possible H.R. The solid lines are obtained as thebest fit (Minuit program) of the experimental results. The experimental results

CHAPTER C 81

follows exactly an equation of the form

Tu = a-Pr (l)

where /? is the heating rate and a and y are constants, a stands for the TM valueobtained with the lower heating rate. The same equation can fit the value of T\ andT2 which are the low and high half maximum temperatures respectively. Thetheoretical behavior is obtained using the general order equation for the heatingrate:

260 I |T

240 ^ - * ^ W * * * S * * r t r W "

160 O^

140 I I0 5 10 15 20 25 30 35 40 45 50

Heating rate (°Cs>)

Fig.5. Behavior of Tu T2 and Tu as a function of theheating rate. The dashed lines show the theoreticalbehavior and the solid lines the experimental one.

As above reported, the theoretical behavior has been obtained using thetrapping parameters as calculated using the lowest heating rate: i.e., E - 1.339 eV, s= 1.13-1014 s"1, b = 1.45. The experimental values have been fitted according toEq.(l) where a = 443.7 for H.R. = l°C/s and y= 0.025. The plots in Fig.5 give a

(2)


measure of the discrepancy between the experimental behavior and the oneexpected from the kinetic model according to Eq.(2). Figure 6 shows the behaviorof FWHM for the peak in a-Al2O3:C as a function of the heating rate.

:| Z^\65 ^

1" f45 /

40 i

35 f30 I '

0 5 10 15 20 25 30 35 40 45 50

Heating rate fC.s1)

Fig.6. Behavior of FWHM as a function of the heating rate [4].

Also in this case the experimental points can be fitted by an equation similar toEq.(l):

FWHM = a-p1 (3)

where a = 36.5 and y = 0.165.

More other important data are also reported in the same paper [4]. One ofthese is concerning the evolution of the integral and the peak height as a function ofthe H.R. Figure 7 shows the TL response of A12O3 normalized to the response at thelower H.R. (0.6°C/s) as a function of H.R. for both integral (•) and peak height(A).

CHAPTER C 83

1.0 !

u 0.8 L

I 0.6 k

Z 0.2 ^^v^*~~-

0 l i ' i i i i T I

0 10 20 30 40 50

Heating rate (°C.s')

Fig.7. TL response of A12O3 as a function of H.R.The response has been normalized to the one

obtained with the lowest H.R. [4].

The experimental points have been fitted by the equation

n = (4)\ + afir

where n is the TL emission (integral or peak height) normalized to that at the lowerH.R., /?is the heating rate, a and /are constants (a = 0.366 and / i s equal to 1 in thecase of integral and equal to 1.103 for the peak height). As it can be observed fromFig.7, there is a drastic reduction of TL as the heating rate increases. From a kineticpoint of view, the peak integral is expected to remain constant as the heating rateincreases. On the other hand, the peak height is expected to decrease as the heatingrate increases, because the FWHM increases, so that the integral is constant.

The experimental evidence of the reduction of the TL as a function ofheating rate is a general phenomenon and it has been observed in many differentmaterials [5-8]. This reduction has been attributed to thermal quenching effect,whose efficiency increases as the temperature increases [2]: since the glow peakshifts to higher temperatures it suffers from thermal quenching. The results indicatethat thermal quenching can be a very good explanation of the TL reduction with theheating rate. In fact, the luminescence efficiency of a phosphor, r\, is given by

"-7TT (5)1 r 1 nr


and where PT is the probability of luminescence transitions, temperature independent,and Pm is the probability of non-radiative transitions, which is temperaturedependent.

According to [4], Eq.(5) can be rewritten as

--—hur: (6)l + c e x p ( - — )

having replaced the efficiency rj with the obtained TL emission, n, where c is aconstant and the Boltzmann factor exp(-AE/kT) replaces Pm owing its dependencefrom temperature. Using then Eq.(6), the final expression for the luminescenceefficiency, related to the maximum temperature TM, is now

» = j r- (7)I M I1 + c exp

\ kafi")

Using the values for a and y above reported, Eq.(7) gives an excellent fit ofthe TL response vs heating rate. The very good fit of the exponential data obtainedusing Eq.(7) allows to attribute the TL response reduction with H.R. to thermalquenching effect.

References1. Kelly P., Braunlich P., Abtani A., Jones S.C. and deMurcia M., Rad. Prot.

Dos. 6 (1984) 252. Gorbics S.G., Nash A.E. and Attix F.H., Proc. 2nd Int. Conf on Lum. Dos.,

Gatlinburg, TN, USA, 587 (1968)3. Kitis G., Spiropulu M., Papadopoulos J. and Charalambous S., Nucl. Instr.

Meth. B73 (1993) 3674. Kitis G., Papadoupoulos J., Charalambous S. and Tuyn J.W.N., Rad. Prot.

Dos. 55(3) (1994) 1835. Kathuria S.P. and Sunta CM., J. Phys. D: Appl. Phys. 15 (1982) 4976. Kathuria S.P. and Moharil S.V., J. Phys. D: Appl. Phys. 16 (1983) 13317. Vana N. and Ritzinger G., Rad. Prot. Dos. 6 (1984) 298. Gartia R.K., Singh S.J. and Mazumdar P.S., Phys. Stat. Sol. (a) 106 (1988)

291

CHAPTER C 85

Considerations on the methods for determining E

A critical survey on the methods for determining E, points out at first howeach of them is applicable considering one or more of the physical consideredparameters. A graphical approach is often made possible by the analytical featuresthe glow-curve can show locally or on its whole. Because of its particularmathematical shape, the unitary order kinetics case is commonly apart from theothers; the general aim of the analytical techniques is to extend the domains ofapplication as long as possible.

The ways the glow-curve is taken into account vary: its analysis may belocal or general; it may regard the peak alone or the whole line; finally the curve orthe area it subtends may be, for each case, considered. The temperatures of mostinterest are however the peak ones and, eventually, those where the curve inflects.The tangents are then pointed out by Ilich [1] as auxiliary plots which might usefullybe applied to achieve, as described above, a knowledge of the involved energy.

More in detail, it is possible to group these methods in main sections:a) Methods based upon maximum temperatures,b) Methods based upon low temperatures side analysis,c) Variable heating rates methods,d) Area measurements methods,e) Isothermal decay method,f) Inflection points method,g) Peak shape geometrical methods.

It is evident, therefore, how any of the analytical features of the glow-curvecan give, if suitably manipulated, useful information on the quality of thephenomena which the thermoluminescent emission is an overall effect of.

The simplest procedure is that searching for a linear relationship betweenglow temperature and activation energy. This has led Randall-Wilkins [2] andUrbach [3] to their formulas; it is on the other hand to be noted how thecorresponding solutions are approximated; this is due to the fact that they have beencomputed starting from, as previously said, already known values of s, which theexpressions are independent of. For instance, the expression of Urbach (E = 2V5OO)is a very rough guide and then it is of limited accuracy. As reported in [4], the use ofthe Urbach's expression is equivalent to the assumption E/kTM = 23.2 and givesenergy values which may be wrong by up to a factor of two.

Consider section b), the initial rise method makes use of the existence, inthe glow-curve, of a temperature range where, while the integral exponential factorremains practically unitary, the Boltzmann probability factor increases with T andtherefore rules the curve shape. A semilog plot of / vs 1/T, acting as linearizingtransformation, gives an E evaluation which doesn't depend on s. It is worthwhile toremark that, when the method is extended to non-unitary kinetics orderconfigurations, and, therefore a knowledge of n is required, it is possible to associate


this last one with the glow-curve area, thus introducing an integrated parameter;finally, when the order b is unknown, the only way to proceed is to adjust it and, bya repeated procedure, to determine the value giving the best linear fit; thus a relatedstatistical analysis is, for the present situation, required, and, eventually, theapplication of convenient tests regarding the goodness of fit may constitute a usefulnumerical tool.

The tangent method is related to the initial rise technique, as far as it startsfrom the same equation; more emphasis is however attributed to the role of thetangent, the plot of which is important in computing the expression for E.

An eventual limitation of the initial rise method is given by the risk tounderestimate the actual E value. This might be caused by non-radiative eventswhich could lead to a computation of an apparent energy, differing from the real oneby an amount W connected to the characteristic non-radiative contribution depth.Wintle [5], analyzing the E values obtained by different methods, founddiscrepancies among them in the sense that the activation energies obtained withinitial rise method were always much less than the E values obtained with othermethods. Indeed, the initial rise method does not take into account the luminescenceefficiency expressed by

where Pr is the probability of radiative emission and it is independent oftemperature, and Pnr is the probability of non-radiative transition, which istemperature dependent and rises with increasing temperature. The resulting decreaseof efficiency with temperature rise is called thermal quenching. Wintle suggestedthat a better expression for the initial rise part is

( E-W\I = snex^-—j^rj (2)

Then the Eir value derived from initial rise measurements will be smallerthan E by an amount W.

The thermal quenching is experimentally demonstrated by theluminescence emission during irradiation at different temperatures. The W valuesobtained are the same as the discrepancy observed using different methods.

Other methods make use of the dependence of the glow-peak shape on theheating rate. When increasing it, a shift toward higher temperatures is observed,together with an increase in the peak height. The former effect is mathematicallyexpressible through the glow-peak numerical condition, giving, as a solution, the

(1)

CHAPTER C 87

value for E. This computation can be carried out apart from an s preliminaryknowledge, by writing down the equations for two different heating rates andreplacing in them the experimental data. By combining the two expressions, 5 can bedropped, and therefore an independent estimate of E is possible. The frequencyfactor may be found, after E, by substitution in either expression. It is to beremarked that E, as computed by means of the double heating rate technique doesn'tdepend on the existence of the non-radiative contribution W, described for the initialrise method. Therefore, by this latter procedure it is feasible to estimate the apparentE; by the double heating rate method, on the other hand, a "true" value for E may befound out; therefore a suitable combined use of both systems may give usefulinformation on the W amount.

By generalizing the present method, after Hoogenstraaten [6] and Chen-Winer [7], it is possible to make use of several heating rates; by manipulating, insuch cases, the general equations ruling the various kinetics, it is feasible to obtainquite simple shaped plots, respectively for unitary and non-unitary configurations. Itis to be observed how this technique marks out a graphical approach to thenumerical solutions. Its domain of application includes whatever order kineticscases, within the theoretical limitation seen above.

Moreover, the heating rate itself may be time dependent, although, ifconstant, the plotting procedure is made quite simpler. Even configuration with anunknown b may be analyzed in this way: in such cases, only an approach byattempts is feasible, and the best statistical value for b is consequently reckoned onthe basis of statistical tests.

Finally, it is to be noted how the double heating rate method itself can beextended to non-unitary order cases. The choice of the heating rate value is arbitrary,though tied to the practical limits.

The area measurement methods are independent of the glow-curve shape,and only the surface subtended by it, between two given temperatures, is required.An analytical survey on this procedure starts again from first order kinetics, andpasses then to include the possible variants and extensions. In the b = 1 case agraphical study appears simple and feasible. The analytical remarks regard the use ofa linearising logarithmic function, which leads to a parallel E and s evaluation. As inother methods, an expansion to more general configurations is of particular physicalinterest, and is attainable by referring back to the I(T) expression for the general-order kinetics, where the overall effect of the involved phenomena is considered andsynthesized in terms of a first order differential equation. From the May-Partridgearea method applied to the case of general-order kinetics [8], it is clear how thisextension bears the apparition of a power b in both members. This allows for aprocedure theoretically analogous to the unitary order situation. E and s are stillfound out by means of a suitable plot and their computations are independent ofeach other. Again, to an unknown b value, an optimization statistical problemcorresponds. The method allows for some kind of variants: Muntoni and others [9]


for this purpose start using a general order equation and a graphical estimation of Eis attainable. Finally Maxia [10] postulated a singularity in the electron trap leveland a multiplicity in the recombination centers.

The isothermal decay technique [11], apart from the details of the adoptedthermal cycle, analyses in particular the phenomenon of trapped electron decay, thatis to say of their rising to the conduction band. The magnitudes of physical interestare the temperature of the sample stored at and the time elapsed; after these data, agraphical estimate of E and s may be carried out. The isothermal decay method isalso appropriately extended to situations where the unitary order kinetics hypothesis,initially assumed, is no longer true; thus, the procedure can be applied toconfigurations where b is both determined and unknown. In this latter case atechnique "by attempts" must be followed.

The Land's method [12] of inflection points, makes primarily use of twoadditional experimental parameters, defined as the temperature values where theglow-curve inflects. To their experimental determination, an analytical expressioncorresponds, computed by deriving twice the glow-curve equation, as defined for afirst order kinetics, which this technique is applicable to. The accuracy availablewith this method is directly connected with the precision that may be reached in theexperimental evaluation of the graphical variables of interest.

Several analytical procedures make use of the peak geometrical features.These parameters are derived by studying the glow-curve data, mainly as regard thetotal width, the left and right half-width and the maximum itself. The ratio betweenthe two half-widths yields a measure of the degree of symmetry characterizing thepeak on its whole. Lushchik [13] and Grossweiner [14] outline two procedures eachfurnishing estimate of E and s, based upon the experimental knowledge of the glowand half width temperatures, as well as their associate errors. On the other hand, theHalperin and Braner technique [15] makes use of the maximum temperature, andboth the half width ones. The relative theory starts from a delineated investigationabout the two main phenomena which the electron-hole recombination is a result of.They assume that the recombination radiative event may occur both via theconduction band, or directly as a result of a tunnelling between the electron trap andthe recombination center under consideration. An analytical survey points out howthe activation energy is connected to the glow temperature and to the abovedescribed geometrical parameters. These relationships show also the tie between thekinetic order and the curve symmetry or asymmetry; furthermore, it is remarkablethat all the pertinent equations can be elaborated only in an iterative way, because ofthe presence of an ^-dependent term in the second members.

A more straightforward method, simplifying the E evaluation, has beenoutlined by Chen [16,17]. This method is not iterative and the evaluation of E iscarried out by means of an expression, the form of which can be unified for variousconfigurations differing from one another for the kinetic order and the kind ofgeometrical parameter involved.

CHAPTER C 89

A detailed critical review of the various expression based on the peak shapemethods, giving the E/kTM range of validity for each expression, is given in [18] andit is reported shortly here.

"* the Lushchik's formula gives an error by 3.3% for E/kTM= 10, reducing to1.7% for E/kTM = 100. However, in all cases the formula gives a highervalue of E than the actual one.

™ the Halperin and Braner's formula, based on x value, underestimates E by4.2% for EMM = 10, is exact for E/kTM « 11, over-estimates E by 12% forE/kTu = 20 and by 17% for E/kTM = 100.

" Grosswiener's expression overestimates E by 10.4% for E/kTM = 10, by7.1 % for E/kTM = 20 and by 4.1 % for E/kTM = 100.

~ the Keating's expression is valid in the range 10 < E/kTM < 18 and itoverestimates E by 3% at E/kTM = 10, by a maximum 10% at E/kTM =20; itis exact at E/kTM = 60 and underestimates E by 12.5% for E/kTM = 100.

~ the Chen's formula, based on co, valid for E/kTM between 14 and 42,underestimates E by 4% at E/kTM = 10, by 1.6% at E/kTM = 14; it is exact atE/kTM = 20 and overestimates E by 1.6% at E/kTM = 40 and by 2.4% atE/kTM = 100. Chen also corrected the Lushchik's equation so that the errorsbeing less than 0.5% for E/kTM between 14 and 40 and less than 0.8% whenE/KTM is low as 10 or as high as 100. The Chen's formula based onx underestimates E by 5.3% at E/kTM = 10, by 2.5% at E/kTM = 14; it is exactat E/kTM = 22 and overestimates E by 2% at E/kTM = 43 and by 3.2% atE/kTM= 100.

Some authors have also underlined the feasibility of computerized glow-peaks [19,20] analysis. A general program can be written: the input is given by theexperimental data and by rough estimates of the physical parameters. These latterones can be iteratively adjusted and each set of values gives a theoretical glow-curve. This plot can be statistically put in comparison to the experimental one, andso, the parameter optimization can kept on until a fair agreement is attained on thebasis of statistical tests.

At the beginning of the '80 studies on computerized glow-curvedeconvolution (CGCD) began to appear in the scientific literature. The CGCDprograms are normally developed by each research group according to the particularneeds and the material studied. A very useful review on this subject is appeared in1995 [21].

In all the previous methods the hypothesis of constant s has been tacitlyassumed. In some cases, however, there is evidence for a T-dependence of s and s'.


From a mathematical standpoint, this temperature dependence affects the numericalsolution of the integral comparing in all the equations.

Finally, it is to be noted how a convenient statistical treatment is of greatpractical interest, In the above discussed methods it has been often necessary tooperate linear best fittings as well as to check their applicability. The procedure mostcommonly adopted is the last square method, by which, after the experimental dataconsideration, the slope and the intercept of the resulting line are computed, togetherwith their errors. A first check on the actual linearity is given by the correlationcoefficient; a more accurate way is the application of a so-called "goodness of fit"statistical test, which the data are submitted to, and which can point out, within agiven probability level, the opportunity to accept or to reject the linearity hypothesis.

Concerning the Moharil's methods [22-25], finally, it must be pointed outthe quantity A/B, which varies from 0 to 1, which is physically more relevant thanthe general order of kinetics b.

References1. Ilich B.M., Sov. Phys. Solid State 21 (1979) 18802. Randall J.T. and Wilkins M.H.F., Proc. Roy. Soc. A184 (1945) 3663. Urbach F., Winer Ber. Ha 139 (1930) 3634. Christodoulides C , J. Phys. D: Appl. Phys. 18 (1985) 15015. Wintle A.G., J. Mater. Sci. 9 (1974) 20596. Hoogenstraaten W., Philips Res. Repts 13 (1958) 5157. Chen R. andWiner S.A.A., J. Appl. Phys. 41 (1970) 52278. May C.E. andPartridge J.A., J. Chem. Phys. 40 (1964) 14019. Muntoni C, Rucci A. and Serpi A., Ricerca Scient. 38 (1968) 76210. Maxia V., Onnis S. and Rucci A., J. Lumin. 3 (1971) 37811. Garlick G.F.J. and Gibson A.F., Proc. Phys. Soc. 60 (1948) 57412. Land P.L., J. Phys. Chem. Solids 30 (1969) 168113. Lushchik L.I., Soviet Phys. JEPT 3 (1956) 39014. Grossweiner L.I., J. Appl. Phys. 24 (1953) 130615. Halperin A. and Braner A.A., Phys. Rev. 117 (1960) 40816. Chen R., J. Appl. Phys. 40 (1969) 57017. Chen R., J. Electrochem. Soc. 116 (1969) 125418. Christodoulides C , J. Phys. D: Appl. Phys. 18 (1985) 150119. Mohan N.S. and Chen R., J. Phys. D: Appl. Phys. 3 (1970) 24320. Shenker D. and Chen R., J. Phys. D: Appl. Phys. 4 (1971) 28721. Horowitz Y.S. and Yossian D., Rad. Prot. Dos. 60 (1995) 122. Moharil S.V., Phys. Stat. Sol. (a) 66 (1981) 76723. Moharil S.V. and Kathurian S.P., J. Phys. D: Appl. Phys. 16 (1983) 42524. Moharil S.V., Phys. Stat. Sol. (a) 73 (1982) 50925. Moharil S.V. and Kathuria S.P., J. Phys. D: Appl. Phys. 16 (1983) 2017

CHAPTER C 91

Considerations on the symmetry factor, ft, and the order of kinetics, b

The order of kinetics, b, and the symmetry factor, ju=S/a>, are two importantparameters. After the Chen's work [1], the graphical picture of dependence of thesymmetry factor fionb has been utilised to determine easily the order of kinetics.It has to be stressed that the order of kinetics b still remains a topic of controversyand matter of debate, even in the case of the most widely studied material, i.e., LiF[2-4]. Indeed it must be noted the fact that for a given value of b, the symmetryfactor n is not unique. Chen, in his work [1], has pointed out that ju is dependent onthe thermal activation energy E and the frequency factor s, and for a given value ofb and for extreme values of £ and s, the maximum deviation in JJ. can be as high as±7%. Therefore, without an a priori knowledge of E and s the absolutedetermination of* from the value of p is not possible. The following mathematicaltreatment, as given in [5], allows to find a general expression for //, considering anyposition selected on the glow-peak, in terms of the variable u - E/kT and of itsvalue at the peak temperature, um = E/kTM.

The equation for a general order peak can be written as

( E\\ {b-l)s ff { E\ TC

where s = s"«o ' a s usual. It has to be reminded that the above equation is valid forKb^2.

Remembering the condition for the maximum intensity

(b-l)sT" ( E\ (sbkTJi) ( E ]

and replacing E/kT by u , E/kTM by uM and To by 0, we get

|_ bexp(-uM)-(b-\)JMuM

where

(1)

(2)

(3)

°?exp(-w') m. exp(-w')

u U uu U

The intensity at the maximum is given then by

b

L^expCMM)-^-!)^!/^

Expressing J and 7^ in terms of second exponential integrals [6]:

E2(u) = u]^^dz (6)a Z

one can writeb

— = exp(wA/-w) 1 - F ( « , M M ) (7)

where

^^^^^expC^/^^-^Ml (8)

V UM « J

Equation (7) gives the TL signal / as a function of temperature when 1M anduM are given. For a given value of uM, the ratio ///^ depends only on u.

Equation (7) can be transformed using any temperature value on the peak,i.e., Tx, for which l/IM=x:

toG9=<"'-"<)+Mi-Tlf(""'"")]

In order to solve the last expression, an iteration procedure is used, writing

a) TX=T; for TX<TM

(4)

(5)

(9)

CHAPTER C 93

b) TX=T: for TX>TM

For case a) the (ux - uM) term dominates:

ux =uM+\nI- - — - I n 1 — — ^ e x p ( ^ ) 2V M^ ^ _ ^W *-l [ i [ M M u~ J

(10)

For case b) the logarithm term in square brackets dominates:

ft-i

[-Jexp(ww-M^) -1 + —«wexp(«w)£2(MM)

— - W 2 M e x p ( M ; ) ^ ^

(11)

The above equations are valid for b > 1. For b = 1, the analogousexpressions are given in Christodoulides method.

Knowing u and uy it is possible to calculate the value of the symmetricfactor as:

x ~Tx "x ~Ux

Figure 8 shows the variation of p(x), for x = 0.5, as a function of uM forvarious order of kinetics. It is clear that to a given value of uM not only one value of(x corresponds. This means that it is not possible to find out the true value of theorder of kinetics. It is suggested to check the value of | j at various points on theglow-peak, x, to get an estimation of b. Table 1 gives the values of \x(x) for someparticular values of w^and various order of kinetics b, ranging from 0.7 to 2.5.

(12)


055 - \ ^ 5 _ _ ^

\^Z0

0.50 - \

^ ^ 1 . 5

0.45 -

0.40 -

0.35 I 1 ->0 50 u_10O

Fig.8. Variation of fi(x) for* = 0.5 as afunction of um for various order of kinetics.

order uM n(0.2) u(0.5) n(0.8)0.7 20 0.311 0.372 0.426

30 0.302 0.365 0.42240 0.297 0.362 0.420

1.0 20 0.389 0.426 0.45830 0.378 0.418 0.45340 0.372 0.415 0.451

1.5 20 0.481 0.485 0.49130 0.468 0.477 0.48740 0.461 0.473 0.485

2.0 20 0.544 OJ526 0.51430 0.531 0518 0.51040 0.524 0.514 0.508

2.5 20 0.592 0.557 0.53130 0.579 0.549 0.527

1 40 | 0.572 I 0.545 1 0.5215

Table 1. Values of /x as a function of the kinetics order and u m.

CHAPTER C 95

References1. Chen R., J.Electrochem.Soc. 116 (1969) 12542. Kathuria S.P. and Sunta CM., J.Phys.D: Appl.Phys. 15 (1982) 4973. Kathuria S.P. and Moharil S.V., J.Phys.D: Appl.Phys. 16 (1983) 13314. Vana N. and Ritzinger G., Rad.Prot.Dos. 6 (1984) 295. Gartia R.K., Singh S.J. and Mazumdar P.S., Phys. Stat. Sol. (a) 106 (1988)

2916. Abromowitz M. and Stegun I.A., Handbook of Mathematical Functions,

Dover, N.Y. (1965)

Correction factor for the beam quality, Fm (general)

This factor must be evaluated when high atomic number thermoluminescentmaterials are used. In this case, the TL response at photon energies below about 100keV becomes significantly greater than that, at the same dose, at higher energies.

The first step is then to calculate the effective atomic number of thedosimetric material, Ze/f, to check the possibility of an over estimation of the dose atlow energies (see Atomic number: calculation).

The second step consists of the theoretical calculation of the energyresponse (see Photon energy response: theory) and, finally, the third step is theexperimental determination of the energy response (see Energy dependence:procedure).

From the theoretical point of view, the absolute sensitivity, X, of a TLD,considered as the ratio between its net TL emission and the air absorbed dose D, atwhich the dosimeter has been exposed, is defined, in the linear range of the TLresponse of the given TL material and for a given energy E of the radiation, as

\ a J £

where the index "cT is referred to the TL material, "a" stands for air. The sameEq.(l) can be referred to the tissue; in this case D, substitutes Da.

Taking into account that the absorbed dose in a material is a function of itsmass energy absorption coefficient, the previous relation can be written as follows

(1)

which is derived from the Bragg-Gray cavity theory applied to a large cavity.Because (\ien/p)J is commonly referred to a compound of different

elements, it must be substituted by the expression

H -zf-V- <3)I P JTW < \ P ) ,

where PF; is the fraction by weight of the i-th element.60 137

Considering the values relative to a reference energy Eo (i.e. Co or Cs),one has the so-called Relative Energy Response (RER):

"sfrKM*£*=!*-4 ; " ; • \c (4)

HL V P J . J£.

The behaviour of Zf/Z^as a function of £ gives the energy dependence ofthe TL response. As a consequence of this energy dependence, the calibration factorFc also depends on the energy of the calibration source used for its determination.

The same calculation can be done for electrons, considering now the masscollision stopping power:

(2)

CHAPTER C 97

*« * TFT—

A p ),\ENormally the calibration factor is determined using a 60Co beam and in

several situations also the TLDs used for applications are irradiated with gammashaving the same energy. In this case the factor Fen is equal to unity. On the otherhand, in much more practical situations the batch of TLDs is used in a radiation fieldhaving an energy different from the one used for calibration. Generally speaking, ifwe indicate Fc as the calibration factor obtained with a reference source and Fq thesimilar factor obtained with another beam quality, the Fen factor is defined as

F--Fi

Curve fitting method (Kirsh: general order)

Y.Kirsh proposed an alternative approach to the curve fitting method,transforming the whole peak into a straight line. It may be regarded as an extensionof the initial rise method and it can be applied to the whole curve rather than to theinitial part of the curve [1,2].

Starting from the general order equation

and remembering that

*' = 4r P)Eq.(l) can be rewritten as

(5)

(6)

(1)


/=(^)O T oexp(-|) ,3,

where n0 is the initial value of n at t = 0.Taking now the logarithm on both sides of Eq.(3), we obtain

hiI = ~ + blJ—\ + h(sn0) (4)kT {nj

Taking now on the experimental glow curve any two points, i.e. (/], T\, n{)and (72, T2, n2), Eq.(4) can be written as

In/, = - — + Mn ^ + \n{sn0 ) (5)kT, {nj

ln/2 = + felrJ — + ln(.s«0) (6)

Subtracting Eq.(5) from Eq.(6) we obtain

••'•-'- {t)-bfe)Hflii)which can be written as

Aln/ , (E} \T)—r^=b~ IT r\ (7)

where A represents the difference between any two points on the glow curve.A plot of the left hand side of Eq.(7) against the part in the square brakets

should give a straight line with slope of -E/k and an intercept of b at the y-axis.

CHAPTER C 99

Using this method one can simultaneously determine both the order ofkinetics, b, and the activation energy, E. The frequency factor can then bedetermined by the maximum condition.

References1. Kirsh Y., Phys. Stat. Sol. (a), 129 (1992) 152. Dorendrajit Singh S., Mazumdar P.S., Gartia R. and Deb N.C., J. Phys. D:

Appl. Phys. 31 (1998) 231

CVD diamond

Chemical Vapor Deposition (CVD) diamond is a very interesting materialas a thermoluminescent detector of ionizing radiations. Its atomic number is Z = 6and then it can be considered a tissue equivalent material (effective atomic numberof soft human tissue is Zeff = 7.4). CVD diamond can be used in-vivo clinicaldosimetry because it is non-toxic and chemically stable against all body fluids.

The growth technique for obtaining CVD diamond have been recentlyreviewed in [1]. The role played by the impurities atoms has been studied andreported in [2]. According to this paper, a boron concentration of 1 ppm is theoptimum for obtaining a linear TL response Vs dose from 20 mGy to 10 Gy. Moredata about CVD TL properties are available in [3-9] .

References1. Sciortino S., Rivista del Nuovo Cimento 22 (1999) 32. Keddy R.J. and Nam T.L., Radiat. Phys. Chem. 41 (1993) 7673. Avila O. and Buentil A.E., Rad. Prot. Dos. 58 (1995) 614. Biggeri U., Borchi E., Bruzzi M., Leroy C , Sciortino S., Bacci T., Ulivi L.,

Zoppi M. and Furetta C , Nuovo Cimento A, 109 (1996) 12775. Borchi E., Furetta C , Kitis G., Leroy C. and Sussmann R.S., Rad. Prot.

Dos. 65(1996)2916. Borchi E., Bruzzi M., Leroy C. and Sciortino S., J. Phys. D, 31 (1998) 17. Furetta C , Kitis G., Brambilla A, Jany C , Bergonzo P. and Foulon F., Rad.

Prot. Dos. 84 (1999) 2018. Furetta C , Kitis G. and Kuo C.H., Nucl. Instr. Meth. 8160(2000) 659. Marczewska B., Furetta C, Bilski P. and Olko P., Phys. Stat. Sol. (a) 185

(2001) 183

DDefects

Materials of interest in thermoluminescent dosimetry are principallyinsulators in which conduction electrons are entirely due to absorbed radiationenergy. Examples of such insulators are the cubic structured alkali halides, such LiFandNaCl.

A crystal is an agglomerate of atoms or molecules characterized by a 3-foldperiodicity. To describe completely a crystal one has to define the positions of atoms(or molecules) inside a unit cell, built a three-vector 5, (i = 1, 2, 3) of arbitraryorigin. All the atoms of the crystal will be obtained from the atoms of the unit cellby all the translations t :

F = £a,a, (1)

where cij represents all the positive and negative integers.A crystal defined by Eq.(l) is termed ideal. Thermal vibrations disturb the

periodicity and make it impossible to obey Eq.(l), so the crystal is now calledimperfect. A further limitation to Eq.(l) is the finite crystal size. Crystals are limitedby free surfaces which are the first type of crystal defect. A crystal which has freesurfaces and probably other defects, is a real crystal.

Since alkali halides and their imperfections are particularly suitable forunderstanding luminescence phenomena, they will be used to discuss the behavior ofa real crystal, all defects of which can potentially act as traps for the charge carrierscreated by secondary charged particles during irradiation.

The alkali halides structure consists of an orderly arrangement of alkali andhalide ions, one after another, alternating in all three directions. Figure 1 shows thestructure of two ideal crystals.

At the contrary, a real crystal possesses defects which are basically of threegeneral types:The intrinsic or native defects.

They can be:a) vacancies or missing atoms (called Schottky defects). A vacancy is a

defect obtained when one atom is extracted from its site and not replaced.b) interstitial or Frenkel defect. It consists of an atom X inserted in a crystal

X in a non-proper lattice site.c) substitutional defects: for example, halide ions in alkali sites.d) aggregate forms of previous defects.Figure 2 depicts the previous mentioned defects.


Fig.l. The three-dimensional structure of an ideal crystal:

(a) structure of LiF ( .Li, ° F) ; (b) structure of CaF2 ( . Ca, ° F).

1 + - + - + 1 L - • - J I • - + -• I- + - + - —, - + - + -

+ - + - + - + - + - + _ + _ +- + - + _ + - + Q + _ + _ + _

Frenkel and — + — •+• — Frenkel andSchottky Schottkydefects Schottky defect defects

Fig.2. Structures of a real crystal with intrinsic defects : i.e. LiF.

+ alkali ion (Li+), - halide ion (F"), \±A alkali ion vacancy, Q halide ion vacancy,

© interstitial alkali ion, © interstitial halide ion

Extrinsic or impurity defects, like chemical impurities Yin a crystalX.They can be:a) substitutional impurity: an atom Y takes the place of an atom X.

CHAPTER D 103

b) interstitial impurity: an atom Y is inserted in an additional site notbelonging to the perfect crystal.

These impurities either add into the crystal structure from the melt, ordiffuse or implant at a later stage. As an example, Fig.3 shows the behavior of thedivalent cation Mg2+ in LiF: it substitutes a Li+ ion.

To understand the mechanism of chemical impurities, one can see theinfluence of a divalent ion on vacancy concentration, as shown in Fig.4 (a): in orderto compensate for the excess positive charge of impurity, an alkali ion must beomitted; furthermore, since the divalent cation impurity is a local positive chargeand the cation vacancy is a local negative charge, the two attract each other givingrise to a complex as shown in Fig.4 (b).

- + - + -

+ - + - +

- + - + -

+ - v t ' • +

- + - + -

+ - + - +

Fig.3. Substitutional divalentcation impurity Mg2+.

- + - + - + - + - ++ - + - + - + ._ + -

- + . : + - , x + - ( + \ +

s •'£•- + (a) - + \ g V (b)• + - + - + . + ."* +

+ - + - + . + . + .

Fig.4. (a) an alkali ion missing ; (b) attraction of ions to form a complex.


Ionizing radiation produces further defects in alkali halides.These defects are called color centers which are absorption centers,

coloring ionic crystals. For example, negative ion vacancies are regions of localizedpositive charge, because the negative ion which normally occupies the site ismissing and the negative charges of the surrounding ions are not neutralized. As aresult of ionizing radiation, an electron is free to wonder in the crystal and it can beattracted by a Coulomb force to the localized positive charge and can be trapped inthe vacancy. This system or centre is called F center. Similarly, a positive ionvacancy represents a hole trap and the system is called V center, but no experimentaldata are known about it. Other types of hole centres are possible:

the Vk centre is obtained when a hole is trapped by a pair of negative ions,the V3 centre which consists of a neutral halogen molecule which occupies

the site of a halogen ion: in effect two halide ions with two holes trapped.All the previous defects are shown in Fig.5.

+ - + - + _ + _ + _ + _ + _ + .- + - + - + _ + _ f - + - * -+ - + - + _ */--* + - + - + ( = ^- + - + - +Li4 _ + _ + _ ; ' _+ - + - + _ + _ + _ + _ + _ +

V center Vfc center V3 center

Fig.5. V, Vk and V3 centers in a real crystal.

We have to outline the importance of the defect production duringirradiation because high dose levels can induce unwanted effects in TL materials,generally called radiation damage, which are important in the set up andmaintenance of a thermoluminescent dosimetric system, i.e., lowering in sensitivity,saturation effects and so on. Furthermore, to study the color centers using variousluminescence techniques, i.e., photoluminescence, can improve the knowledge ofthe thermoluminescent phenomena itself.

For this reason a phenomenological short feature of radiation damage incrystals is given below.

Photons, electrons, neutrons, charged and uncharged particles can createdefects by displacement in the sense that the bombarding radiation displaces thecrystal atoms from their normal position in the lattice, producing vacancies and

CHAPTER D 105

interstitials. The number of defects produced is proportional to the flux of irradiationand to the irradiation time. However, during a long irradiation the number of defectsproduced will gradually decrease because the possibility of vacancy-interstitialrecombination increases.

The irradiation can also produce negative ion vacancies by a process calledionization damage. This mechanism is related to the recombination of ionizationelectron and holes. During recombination a bound electron-hole pair (exciton) canbe trapped on a negative lattice ion. The energy released during recombination istransferred to the negative ion which produces collisions leaving vacancies andinterstitial atoms. The final result is the production of F centers and interstitialatoms.

Delocalized bands

Conduction band (CB) and valence band (VB).

Determination of the dose by thermoluminescence

The main algorithm which can be used to convert the light emissionobtained during the readout of a thermoluminescent detector to the absorbed dosecan be expressed by the following relationship

D = M-FC (1)

whereMis the TL signal (integral light or peak height), andFc is the individual calibration factor of the detector.

The previous Eq.(l) can be generalized by inserting in it all the parameterswhich can influence the dose determination during the preparation of the detector,its irradiation, the possible period of time elapsed between the end of irradiation andreadout and the readout itself.

A more general relation can then be written as follows

Dm = Mnet • St • Fc • Fst • Fen • Flin • Ffad (2)

where

~ Dm is the absorbed dose in the mass m of the phosphor,


"" Mnet is the net TL signal (i.e., the TL signal corrected for the intrinsicbackground signal Mo'. Mnet = M- Mo),

" 5, is the relative intrinsic sensitivity factor or also called individualcorrection factor concerning the ith dosimeter,

~ Fc is the individual calibration factor of the detector, relative to the beamquality, c, used for calibration purposes,

~ Fsl is the factor which takes into account the possible variations of Fc due tovariations of the whole dosimetric system and of the experimental conditions(electronic instabilities of the reader, changes in the planchet reflectivity,changes in the light transmission efficiency of the filters interposed betweenthe planchet and the PM tube, temperature instabilities of the annealingovens, variation of the environmental conditions in the laboratory, changesin the dose rate of the calibration source, etc.),

~ Fen is the factor which allows for a correction for the beam quality, q, if theradiation beam used is different from the one used for the detectorcalibration,

~ Fiin is the factor which takes into account for the non-linearity of the TLsignal as a function of the dose,

" Ffad is the correction factor for fading which is a function of the temperatureand the period of time between the end of irradiation and readout.

Dihalides phosphors

Dihalides have the general formula AXY, where A is an alkaline earthmetal and X and Y are two halogens. Single crystals of dihalides are obtained bygrowth using different known techniques. From a melt containing a mixture of ametal halide and a dopant (i.e. rare earth of heavy metal ions).

The systems studied are BaFCl, BaFBr, SrFCl and SrFBr doped by Tl orGd ions [1-3].

References1. Somaiah K., Vuresham P., Prisad K.L.N. and Hari Babu V., Phys. Stat.

Sol.(a)56(1979)7372. Somaiah K. and Hari Babu V., Phys. Stat. Sol.(a) 79 (1984) 2373. Somaiah K. and Hari Babu V., Phys. Stat. Sol.(a) 82 (1984) 201

CHAPTER D 107

Dosimeter's background or zero dose reading (definition)

The dosimeter's background, also called zero dose reading, is obtainedfrom repeated measurements carried out on unirradiated dosimeters. This quantity isparticularly important when the dosimeters are used for low dose measurements. Asthe dose increases, the background and its variation become less important and canthen be neglected at high doses.

The TL signal related to the background is due to variouscomponents:

™ spurious signals from tribo- and chemi-luminescence,

" stimulation of the TL phosphor by UV and visible light,

" infrared emission of the heating element and its surroundings,

~ dark current fluctuations of the PM tube,

™ residual signals from the TL phosphor due to previous irradiations.

All the above given effects can be reduced or eliminated using appropriateprocedures during handling and use of the TL dosimeters.

Dosimeter's background or zero dose reading (procedure)

The dosimeter's background, also called zero dose reading, is obtainedfrom repeated measurements carried out on annealed and unirradiated dosimeters.This quantity is particularly important when the dosimeters are used for low dosemeasurements. As the dose increases, the background and its variation become lessimportant and can then be neglected at high doses.

Quantitatively speaking, the zero dose reading is the result of two maincomponents:

"" reading without dosimeter: Lo (dark current)

~ reading of unexposed dosimeter: Lu

Several readings Lo and Lu have to be performed for getting the average valuesLo and Lu . Then, the mean value of the zero dose reading is given by

LBKG = LO + LU (1)


with the corresponding standard deviation, CT BKG .

In the lower dose range, the mean value LBKG given by Eq.(l) has to besubtracted from the irradiated dosimeter readings.

Dosimetric peak

It indicates a very well resolved peak in the glow curve structure, having a highintensity and a good stability, i.e. it is not or almost not affected by fading. Thesecharacteristics allow an accurate determination of the given dose.

Dosimetric trap

It is usual to indicate as dosimetric trap the trapping center related to aparticular peak in the glow curve and called dosimetric peak. This peak is used fordosimetric purposes.

EEffect of temperature lag on trapping parameters

The effect of temperature lag on the determination of the trappingparameters can be determined using the Randall-Wilkins model for the first-orderkinetics [1,2]. The equation of the TL intensity is given by

/(0 = «5exp--Jr (l)

with the usual meaning of the symbols and where Tx, the temperature of the heatingelement, is given by

7 ; = r o + p . /

Considering the temperature lag, Eq.(l) becomes

' ^ " K - K ^ y <2>The exponential in Eq.(2) can be developed around the temperature of the

maximum TL intensity, TM, into powers of (7J — AT — TM ) . Thus, Eq.(2) can be

approximated, as a first approximation, by

_/, 2AT\

7(0 = ns exp ^ exp - -A TJL± (3)

Comparing Eq.(l) with Eq.(3), it can be observed that, if the temperaturelag is ignored, the activation energy and the logarithm of the frequency factor areoverestimated by the quantities


_ 2EAT AEAE = Alns =

TM 2kTM

According to the experimental results on the peak 4th of TLD-100, errors inE and ln(s) can be of the order of 6% and 3% respectively if the temperature lag isneglected.

References1. Piters T.M. and Bos A.J.J., J. Phys. D: Appl. Phys. 27 (1994) 17472. Piters T.M., A study into the mechanism of thermoluminescence in a

LiF:Mg,Ti dosimetry material (Thesis, 1998),D.U.T.

Energy dependence (procedure)

~ prepare n groups (as many points of energy as possible for one intends touse) of at least 6 TLDs each;

" each group of TLDs is inserted in plastic bags and the bags are irradiated inair, using the appropriate built-up thickness for each point of energy;

"" irradiate each group with a reference dose at one energy;

~ read each group;

" correct each reading by individual background and by individual sensitivityfactor.

An example of data concerning the energy dependence of LiF:Mg,Ti (TLD-700) is reported in the following Table 1. The irradiations have been carried out in

137 60

air with the appropriate built-up for Cs and Co. Six TLDs were inserted in ablack plastic bag for each point of energy. The average TL readings were alreadycorrected for background and sensitivity factor. Figure 1 shows the relative TLresponse as a function of the energy.

We have to mention that the procedures of irradiation for the energydependence can be different according to the aim of the application, such as clinical,environmental and personal dosimetry.

CHAPTERE 111

10 T .

I3

10 100 1000 10000

Erwgy (I»V)

Fig. 1. Energy dependence for LiF :Mg,Ti (TLD-100).

Coming back to the correction factor Fen, if the calibration of the system60

was carried out using a Co source and then the batch has been used at a differentenergy, i.e. 58 keV, the correction factor will be different from unity because at thatenergy the dose is overestimated. In this case we get the following correction factor

F ^ 5 L = J _ = o.78e" F5S 1.28

Energy Dose Avcor/mGy a Ei/ECo

(keV) (mGy) (TL/mGy)23 7JS4 146.4 14J5 1.1931 636 164,8 14J 1.3458 6T7 158.0 14^9 1.28104 9.15 133.0 17^ 1.08662 10.36 120.5 24A 0.981250 I 10.96 I 123.1 | 10.9 | 1.00

Table 1. Energy dependence of LiF:Mg,Ti.


It is better in principle and when it is possible, to determine the calibrationfactor with the same quality beam used for applications. This is easily done inclinical dosimetry, and in radiological or therapeutic monitoring. In situations likepersonal dosimetry the monitored radiation field is normally unknown and adifferent approach must be considered.

Environmental dose rate (calculation)

Considering the escape probability rate per second for electrons trappedin a trap

P—^-jj) (1)

whereE = the trap depth (eV),k = Boltzmann's constant (8.61013 eV/K),T = the absolute temperature (K),s = the frequency factor (s"1) depending on the frequency of the number of hits in thetrap which can be considered as a potential well.

The reciprocated p is the mean life of the trapped charges in their sites:

therefore/? itself is the fading factor related to the rate of fading rate, dnjdt, where

n is the number of trapped charges, when the temperature is kept constant.The fading factor can be determined using a fading experiment, under

controlled environmental conditions.In the present calculation only an isolated thermoluminescent peak is

considered, without retrapping phenomenon (first order kinetics). In this case, therate of release of electrons from the trap is given by

dn- = -p-n. (2)

In the assumption of constant temperature throughout the experimentalperiod, the integration of Eq.(2) gives

n = n0 e x p ( - p • t) (3)

where «0 is the initial number of the trapped charges.

CHAPTER E 113

Since the TL intensity is proportional to the release rate of the trappedcharges

/««-£ (4)at

we obtain

I(t) = IQexp{-p-t) (5)

where 70 is the TL intensity at time t = 0.From a read-out system the integral TL light is normally obtained; whence,

introducing the function <b(t), expressing the total TL light or the area below the TLpeak, the function <J>(7) is related to /(/) by the following relation:

O(t)=)l(t)dt (6)o

Then, <!>(?) coincides numerically with n and Eq.(3) can be rewritten as

O(0=O 0 exp( -p - r ) (7)

from which the fading factor p is obtained as

The previous Eq.(8) gives then the fading factor in the case a singleirradiation is performed at the beginning of the experimental period.

The experimental situation during the measurement of the environmentaldose rate is described by a continuous irradiation of the dosimeter so, while thefading is equivalent to a progressive extinction of the stored information, theenvironmental contribution leads to a signal increasing. The two competing effectscan be described modifying Eq.(2) as follows

dn dB— = a - — -p-n (9)

dt dt

(8)

A

where a is the dosimeter sensitivity (i.e. reader units/dose) and dB/dt is the rate ofincrease of the background dose due to the environmental radiation.

The integration of Eq.(9) gives

( \ a dBn = cexp(- p-t) + (10)

p dtSetting up the initial condition «0 = 0 at the initial time of the environmentalmonitoring, the constant c is given by

p dt

which can be substituted into Eq.(10):

a dBr, , VIn = —ll-exp(-/>•/)] (12)

p dtAs already done before, the substitution of <Z> instead of n can be operated, obtaining

o(/) = - — [l-exp(-p-/)] (13)p at

If tw indicates the whole monitoring time during the environmental dose

determination, the previous equation yields

. / \ a dB F- / YI

<Mv)=---rli-exp(-/>-'*r)J (14)

p atfrom which the environmental dose rate per day, corrected by fading, is obtained

dB <£)w r, / w i—- = p—^[l-exp^p-^jj (15)

dt aThe accumulated environmental dose, B, is then

(11)

CHAPTER E 115

„ dBB = — -tw (16)

at

The dose rate per hour is then given by

A(dB\ (dt}

=-gL- (17)

The following Table 1 shows the numerical evaluation of p at differenttemperatures for LiF:Mg,Ti (TLD-100), using the data E = 1.36 eV and s =2.20-10 s"1, corresponding to the dosimetric peak in LiF.

Temperature p (day)'1 p% x 1 year x (year)(K)273 1.3-10"7 0.005 20693275 2.0-10"7 0.007 13583278 3.8-10"7 0.01 7304280 5.6-10"7 0.02 4865283 1.0-10"* 0.04 2673285 1.5-10"6 0.1 1807288 2.7-10"6 0.1 1014290 3.9-10"6 0.2 694293 6.9-106 0.3 397295 9.9-10"6 0.4 275298 1.7-10"5 0.6 161300 2.4-10"5 0.9 113303 4.310"5 1.6 64308 9.910'5 3.6 28313 2.3-10-4 8.4 12318 4.9-10"4 17.9 6323 1.1-10"3 402 3

Table 1. Calculated values of the fading factor at various temperatures.

Environmental dose rate (correction factors)

The environmental dose rate per day is given by

f=p^[i-*v(-p-<w)r CDat a

according to the environmental dose rate calculation [see Eq.(15)].The previous equation has to be corrected as follows.The correction factor to be considered is the zero dose reading or

background of the TL detector. We can denote this value as b. Then, the actualreading <t>, as well as the initial value <X>0 have to be corrected by the backgroundvalue b, subtracting it from both the previous values. This correction has to be donein both fading and environmental experiments.

Correction in the fading experimentTo take into account the zero dose reading in the fading experiment, a set of

annealed and undosed dosimeters, called control dosimeters, have to be used. Onegroup of these dosimeters has to be read out immediately after annealing to checkthe background. The second group will be read at the end of the fading experimentalperiod to measure both the background and the possible environmental signals. Letus indicate with P this environmental signal. Then, the correct readings in the fadingexperiment will be:

V0=®0-b (2)

and

vF = (D-(6+p) (3)

The fading factor is given by

1 Yp = —In (4)

Correction in the environmental measurementThe equation giving the TL reading after the environmental experiment is

given by Eq.(l). In that expression, W represents the sum of the environmental aswell as the dosimeter background signals:

CHAPTER E 117

<£>w=b + <S>IVnet

from which

®Wne,=®W-b (5)

then Eq.(l) becomes

^ = ^ [ l - e x p ( - p . V ) r (6)

Correction of the sensitivity factor, a.The sensitivity factor is obtained using a calibration dose, Do. After

irradiation of the calibration dosimeters with the calibration dose, the averagereading will be So- The sensitivity is then given by

Owing to the dosimeter background b, the previous equation has to be corrected asfollows

S0-ba.—

Erasing treatment

The erasing treatment is the thermal procedure used to empty the traps of aphosphor. In some way it is different from the thermal annealing. More precisely,the annealing also has the function to stabilize the traps; the erasing procedure is justused to empty the traps and then it could be carried out in the reader.

Error sources in TLD measurements

There are many sources of error in a thermoluminescence dosimetry system anda considerable effort can be done to reduce the effects of uncertainty on the accuracyand precision of the system [1-3].

(7)

(8)


First of all we have to list the commonly encountered sources of error that affectthe precision and accuracy of the system.

Both systematic and random sources of error can be originated from thecharacteristics of the thermoluminescent detector, or by the TL reader, or they comeout by the incorrect heat treatment during readout or during the anneal process. In allcases it is essential to carry out the whole procedure in a very high reproduciblemanner.

Sources of error due to the dosimeterThey can be enumerated as follows:

™ variation of transparency and other optical properties of the dosimeter;

~ variation of the optical properties of the covering material of the TLDelement if this material and the phosphor make a single body during readout(it is the case of some type of TLD cards);

~ effects due to the artificial light and/or natural light (optical fading);

~ effects due to the energy dependence of the thermoluminescent response;

~ effects due to the directional dependence of the incident radiation on thethermoluminescent response;

~ abnormal high values of the irradiation temperature;

~ non-radioactive contaminations of the phosphor and/or the detector;

"" non-efficient and non-reproducible procedure for cleaning the dosimeter;

~ variations in the mass and size of the TL material;

~ non-uniform distribution of the TL material on the reader tray when powderis used;

~ variations in sensitivity owing to radiation damage of the TL material;

~ loss of TL signal owing to thermal fading;

~ increase of the TL background due to environmental radiations.

Several of the previous sources of error can be avoided by taking a considerablecare during handling of the detectors. For instance, avoid any accidental contactbetween the TLD material and the fingers of the operator and/or the body of apatient during radiological inspections or therapy treatments.

CHAPTER E l 19

The use of metal tweezers can provoke crashes on the TL element surfaceand/or detachment of fragment material; use vacuum tweezers.

Pay attention to the radiation history of each detector and reject the dosimeterswhich have received an abnormal high dose.

Take a considerable care in the annealing procedure and be sure that the settemperature is the correct and appropriate value for a given annealing treatment.Check also if the actual temperature matches the set temperature value. A particularattention must be paid to the temperature distribution inside the anneal oven.Inappropriate lower annealing temperatures can leave high residual TL signals dueto previous irradiations. On the other hand, higher annealing temperatures candamage the crystal lattice and destroy traps and recombination centers.

Thermoluminescent materials are, in general, sensitive to light, especially to theultraviolet component. The rate of fading can be increased substantially in the caseof intense UV irradiations; in some cases the background can also be increased. It isalways a good procedure to keep the TLDs away from any light sources, shieldingthem either during use or storage. A black plastic box is enough to avoid lighteffects.

Pay attention to any radiation sources which can occasionally be in the TLDlaboratory. During storage after annealing, the TLDs must be located in appropriatelead box to avoid any radiation effects due to radioactive elements in some buildingmaterials (e.g. concrete) and/or from natural environmental radioactivity.

As it can be easily observed, many types of error can be avoided by making useof appropriate and accurate handling procedures.

One of them concerns the variations in sensitivity of TLDs within a batch.Variations of sensitivity within a batch of TLDs are quite inevitable even with afresh batch of phosphors. These variations can increase with time due to loss of thephosphor material, changes in the optical properties and other damages, and theintroduction of systematic errors in the measurements. Several methods are in usefor limiting the effects of sensitivity variation in accuracy and precision for themeasurements. The best approach is to divide the TLDs into batches each havingsimilar sensitivity and then to use appropriate sensitivity factors, as it will bedescribed later on. Further improvements can be obtained, if an individualcalibration is carried out for each detector. For specific uses where an extremeaccuracy is required, i.e. in clinical applications, the best procedure would be tocalibrate the detectors before and after each measurement. The choice of a specificprocedure depends strongly on the accuracy required. However, a check must bedone frequently during the time of use of a given batch.

Large errors can be introduced in the dose determination when the dosimetersare exposed to photons of unknown energy, mainly in a range around 100 keV andbelow because in this region the photoelectric effect is predominant and then thedosimeters could overestimate the dose. This kind of error can be minimized byusing the tissue equivalent phosphors which present a small variation in response to


energy. Another method is to calibrate the detectors with a well-known beam qualityand then use them with the same kind of beam. In some cases this procedure is notpossible because the field dosimeters are used in personnel or environmentaldosimetry where the energy field is not known. In these cases the errors can beminimized by using a combination of tissue equivalent materials and non-tissueequivalent materials so that information on the radiation energy can be obtained andcorrections can be made.

The thickness of the dosimeter is another factor to be taken into consideration.For low energy photons and for beta irradiation a thick dosimeter can give an under-response owing to the self-absorption effect. On the contrary if the detector is toothin, it can give an under-response at high photon energies because of a lack ofelectron equilibrium.

Before using any TL material, it is necessary to perform an accurate thermalfading experiment simulating the real conditions of the field measurements. Fadingdepends on the depth of the trap corresponding to the dosimetric peak; the stabilityof the trap is a function of the annealing procedure which, in turn, depends on thecharacteristic of the anneal oven.

Errors generated in the readerErrors associated with the reader can be generated by an unsuitable or instable

readout cycle, as well as by non-reproducibility of the detector position in the readertray.

In readers using planchet as heating element, an error is generated by a poorthermal contact between detector and heater.

If a built-in reference light source is used to check the stability and thebackground of the reader, attention should be paid to its performance which canchange as a function of time and temperature.

The use of TL powder can provoke contamination of the PM tube or of the filterinterposed between the PM and the tray and then their opacity. Irradiated powderlost in the reading chamber produces abnormal high background signals duringsuccessive use of the reader.

Concerning the background signal of the undosed TLDs or their zero dosereading, its effect on the dose evaluation is large when low doses have to bemeasured. It is very important to determine the standard deviation associated withthe average background of the undosed detectors. It is easily observed that as thedose increases, the effect of the background and its variation becomes less and lessimportant.

The light collection efficiency of the reader can change if the reflectance ofthe heater element changes; it is imperative to keep all parts of the reading chamberclean.

Another error can arise during the readout; the reading cycle must notinclude all the glow curve but only the dosimetric peak. Including low temperature

CHAPTER E 121

peaks provoke errors due to their high fading rate. Use the pre-heat technique isnecessary, both in oven or in the reader, to erase these low temperature peaks.

Errors due to the annealins proceduresIt has well been demonstrated that the non-reproducibility of the annealing

procedure can provoke large variations in the sensitivity of the TL materials.It is recommended to carry out thermal erasing procedure in oven. An in-

reader anneal can be done just in the case where very low irradiation doses havebeen detected and also in that case to be sure about the efficacy of the procedure interms of reproducibility in the measurements.

The in-reader anneal procedure is normally done for some type of TLDcards where the phosphors are covered by plastic transparent materials and thecovering material and the phosphors cannot be separated. In cases where the cardshave received a high dose, the in-reader anneal is not efficient and the cards must berejected.

For each TL material the proper annealing procedure must be determinedand checked, both in temperature and time. The best combination of temperature andtime will produce an effective depletion of the traps. Repeated cycles of irradiationand annealing-readout will show the precision of the thermal procedure as a functionof the residual TL emission. Repeated cycles of annealing-irradiation-readout willshow the precision of the TL response.

Another important factor which can introduce error in the dosedetermination concerns the cooling rate after annealing. As the cooling rate changes,the sensitivity changes dramatically. This effect is observed in any kind of TLDs.The best way is always to use the same procedure for cooling the TLDs. It must alsobe checked if a fast or low cooling rate is better for a given TLD material.

References1. Busuoli G. in Applied Thermoluminescent Dosimetry, ISPRA Courses,

Edited by M.Oberhofer and A. Scharmann. Adam Hilger Publisher (1981)2. Marshall T.O. in Proc. of the Hospital Physicists' Association. Meeting on

Practical Aspects of TLD. Edited by A.P.Hufton, University of Manchester,29th March, 1984

3. Nambi K.S.V., Thermoluminescence: Its Understanding and Applications.Instituto de Energia Atomica, Sao Paolo, Brasil, INF.IEA 54 (1977)

FFading (theoretical aspects)

To study, theoretically, the fading effects in various situations simulatingpractical cases, it is possible to consider the simple TL system, shown in Fig.l, inwhich only one kind of electron trap and one recombination center are present. Insuch a case, the system of equations describing the traffic of charges between thetrapping levels and the conduction and valence bands is the following [1-5]:

~ = -nX + Ann*(N-n) (la)dt

dn*= nX-Ann*(N-n)-Amn*m + nj (lb)

dt— = -Amn*m + Ahm*(M-m)-mXh (lc)dt

dm * . *.(,. \ -= -Ahm*\M-m)+trikh + «,. (Id)

dt

wheren is the trapped electron concentration (cm3) at time tn* is the electron concentration (cm3) in the CB at time /m is the trapped hole concentration (cm3) at time /m * is the hole concentration (cm"3) in the VB at time tN is the total density of electron traps (cm3)M is the total density of recombination centers (cm'3)An is the probability factor for electron trapping (cm" sec"1)Am is the recombination probability of electrons from the CB with holes in

centers (cm"3 sec"1)Ah is the probability factor for hole capture (cm"3 sec"1)

X =sexp

I kT)


A. = sh exp -h " \ kT)

s is the frequency factor for electron traps (sec1)E is the thermal activation energy for electron traps (eV)Sh is the frequency factor for recombination centers (sec"1)Eh is the thermal activation energy for recombination centers (eV)k is the Boltzmann's constant (8.6-10~5 eV K"1)T is the absolute temperature (K)

fl( is the rate of production of electron-hole pairs due to an applied external

radiation field

conduction s Sexp(-EZkT) m

band j • i n (t)

— L » i — N,n(t)

—rO-, M,m(t)

^J_Jvalence • . . I iband O m W Shexp(-Eh/kT)

Fig. 1 .Processes considered for fading simulation.

The processes allowed in the system described by Eq.(l) are:

~ electron trapping and releasing from traps to the CB

™ capture and releasing of holes from centers to the VB

CHAPTER F 125

™ creation of free electron-holes pairs by the external radiation field

~ recombination of free electrons with holes in recombination centers

The probability of direct band-to-band transitions and direct recombinations oftrapped electrons and holes are both assumed negligible.

The evolution of the TL signal during storage is considered at a constanttemperature and the temperature dependence of the various parameters is notconsidered.

Equation (lab) can be rearranged with respect to n* :

- dn*/bl + fl;„*= : < L

An(N-n)+Amm

This expression can be inserted now in Eq.(la) to obtain the variation ofthe trapped electron density n(t) as

dt [A,(N-n)*A,m\ \_A,(N-n)+Amm^1 dt )

Equation (2) can be transformed in an explicit form if the usual conditionsfor free carrier densities are considered:

dn* dn—— « — , n*«n (3)

dt dt

dm * dm 4

« — , m*«m (3f)dt dt

(2)

The conditions expressed by (3) and (3') mean that electrons and holesremain most of the time in localized states rather than in their respective bands.

Moreover, n, n*, m and m* are not independent functions but they arerelated by the charge neutrality equation:

dn dn* dm dm*— + = + (4)dt dt dt dt

which becomes, using conditions (3) (3'),

dn dm-T = —r (5)

dt dt

and then, by integration,n + q = m (6)

where q is a constant which could be different from zero. The value of q representsthe net charge due to the presence of trapping centers not active at the consideredtemperature, i.e. disconnected traps.

Equation (2) can then be rearranged as follows:

dn [" c(N-n) "I dn*^dt \_<j(N-n) + m dt

A m 1 f o(N-n) 1= -nk\—t r + — • ^ ^— •«.

lo(N-n)+m] [a(N-n)+mj '

where

a = T L (8)

(7)

CHAPTER F 127

is the so-called retrapping-recombination cross-section ratio.The term

crJN-n)<r{N -n)+m

comparing in Eq.(7), is always positive and lower than 1; so that one obtains for theleft side of Eq.(7), assuming the conditions (3) (3'):

dn 1" <r(N-n) ~\dn*dt l<j(N-n)+m\ dt

dn*

dn . |" <j(N-n) 1 At

dt [cr(N-n)+m] dndt .

dn~ dt

Then, substituting m=n+q, one finally obtains from Eq.(7) and expression (9)

dn ,|" n + q 1 [ cr(N-n) 1— = -nM—, X^T x + —7 v s / x •ni (10)dt lo(N-n)+(n + q)\ [ a{N -n)+(n + q) J '

Equation (10) represents the form for a general order kinetics. The first andthe second order kinetics are both particular cases of Eq.(10) in case where noradiation-induced electron-hole pairs is present, i.e. rt, = 0. The first order kineticsis obtained for a small retrapping-recombination ratio and the second order when theratio is high.

The two limit situations can also be obtained when fii > 0.

"" Assuming a strong recombination (first order kinetics), namely

(9)

a (N - n) « m

and#i(0 « N

one obtains from Eq.(lO)

dn . cNni— = -rik + '- (11)

dt n + q"" On the other hand, if retrapping dominates (second order kinetics) this

means

a (JV - n) » m

Equation (10) can be rearranged as follows: considering both terms on its rightside, they can be written as

( i ) » + g = n + q l | n + q

(j(N-n)+(n + q) a(N-n)\_ a(N-n)

r "i2= " + g " + <7 + (U)

a{N-n) L°(W-«)J

which is similar to the power expression of the type X — X2 +....

<j(N-n) = r n + q 1 '

(ii) a{N-n)+{n + q) [ a{N-n)\ ( n )

= 1 n±^ +cs(N-n)

CHAPTER F 129

which is similar to the power expansion of the type 1 - X +....Now

n + q m

a(N-n)~a(N-~n)

but with the condition G{N — n)» m

m-FT: ^ « 1 (14)

a [N-n)so that the second order term can be rejected and finally one gets

n + q n+q

— r - ^ v « — / ^ (15)

G{N-n) + (n + q) a{N-n)

Eq.(19) becomes then, using (12) and (15):

*=-Mx.-£±^+[i--ft«_'L (16)dt o{N-n) L o(JNT-/i)J

If n(t)« N, far from saturation, Eq.(16) becomesdt I a{N- n)\

where

r = \ (is)

(15)

(16)

(17)


If retrapping and recombination rates are equal, a = 1, Eq.(lO) simplybecomes

dn . n + q N-n rtn^— = -nX — + ft, (19)dt N + q N + q

or better

dn=-nX.n^- + \\-n^-\ni (20)dt N + q [ N + qj

and finally

^ = -nV(n + q) + \l-"^-\ni (21)dt V H) [ N + q] '

where

r = -A_ (22)N + q

If fij = 0 and 9 = 0, Eqs. (11), (17) and (21) reduce to the well-known equations

of first and second order decay processes.

Expressions related to different situations

"" Instantaneous irradiation

A strong initial irradiation, very short compared with the storage duration, isnow considered. The possible background radiation is neglected, which is equivalentto saying /}, = 0 . Therefore, only the fading of an initial density of filled traps, i.e.

n(t = 0) = n0 is taken into consideration. Assuming also that the net charge q issmall enough compared with the density of the filled traps and then it can beconsidered equal to zero, Eq.(10) becomes

CHAPTER F 131

dn . f n 1— = - « X - 7 v (23)

dt [a(N-n)+njwhich can be rewritten as

dn n2X— = 7 r (24)dt n(l-a)+oN

which can be integrated as follows

(1-a) — + GN\ —r = -X\dt"o n •"» w *

giving the solution

wC/)1"0 exp - ^ ] = nl0^ exp - — exp(- Xt) (25)

If a = 0, Eq.(25) becomes

n(t) = n0 exp(- X • t) (26)

which is the equation describing the first order kinetics isothermal decay.If a = 1, Eq.(25) transforms in

( N) ( N) ( A

exp = exp exp(- X-t)V n) \ n0)

and then in

"="°rirj (27)

which is the second order kinetics equation.

~ Continuous irradiation

Considering the more general Eq.(lO), the density of trapped electrons «(/)is

the result of two competitive effects, acting simultaneously: the progressive storageof radiation-induced free electrons and the fading that leads to a progressive releaseof the trapped charges. As a consequence, the trapped electron concentration tendsto a steady value when thermal raiseng exactly compensate the trapping of freeelectrons. The limit of Eq.(lO), dnjdt -> 0, when t -» +00 , gives the equilibriumvalue

or better

nl = V . ~ (28)kq

Considering that the system is far from saturation, i.e. nx « N, and the net charge

q is small compared with the equilibrium value at a given dose, the equilibrium

density of the trapped electrons can be derived by Eq.(28) as:

< = - ^ (29)

In this case, it is possible to find an explict solution of Eq.(lO). Indeed, taking intoaccount expression (28) and the condition q « n, we get

— = J "2 1+J fo-*K 1.1dt |_aW + / i ( l - a ) J LCTAr + "(1~CT)J N

CHAPTER F 133

dn = [" k T 2 _ O^M"

and then, considering we are far from saturation

dn f A, lr 2 2 1

at°U + -a)f ""] <M)After that we have

Mt)aN + (\-a)n , l r i«W dn /. \fC) «<in ,I ^-^-dn = aN\ r + (l-a)| ^ r = -kt

(31)

The integrals in (31) have a singularity at n-nx and the «(?) must be

continuous; hence there are two possible solutions depending on whether n is

greater or lower than nm.

The two possible solutions are:

~ if «0 > nx, then n(t) is always greater than nx and n(t) decreases

with time according to the following expression

[n()+ni <->= (».2-«irf"^--]""exp(-2^)

™ if nQ < nx, then n{t) is always lower than nx and n(t) increases

with time according to

It has to be stressed that in both cases, n{t) tends asymptotically to the steady value

The previous equations can now be used for simulating various possiblepractical situations.

"* Instantaneous initial irradiation

This case is depicted in Fig.2The sample receives a short and strong initial irradiation, followed by a storage

period at a constant temperature. The background irradiation is neglected in this caseand so Eq.(25) can be applied.

I \

t

Fig.2. Instantaneous irradiation at thebeginning of the storage period.

(33)

CHAPTER F 135

~ Continuous irradiation

Figure 3 shows the situation. The sample is irradiated during a long period,with a constant rate of trapping electrons na . During irradiation the fading acts as

a competitive effect respect to the trapping. The conditions are n(t = 0) = 0 ,

fit = tla and to be far from saturation. The equation to be used in this case is

Eq.(32).

^

t

Fig.3. Continuous irradiation during storage.

~ Instantaneous initial irradiation followed by a continuous backgrounddurins storage

Figure 4 shows this case.The conditions are «, = na, n(t = 0)=n0 and fading during the storage

(aNn y2period. Equation (31) can be applied if «0 > — , and Eq.(33) if

n°

t

FigAInitial irradiation plus background.

" Long irradiation during storage under continuous backgroundirradiation

This case is depicted in Fig. 5.The TL sample is continuously exposed to background, na , and for a given

period of time between tx and tj, it undergoes a long and constant irradiation na .

For t <tx the trapped electron density follows Eq.(33) and it will be nx = n{t = tx)

at the end of this period.

K"

h (2 t

Fig.5. Strong irradiation superposedon background irradiation.

Between f, and t2 , the irradiation field is ni =na +n» . So, at time t = tx the

condition

CHAPTER F 137

A,

is fulfilled and Eq.(33) has to be again applied for calculation.For t >t2 two cases are possible:

if n(t2 ) > , Eq.(32) is applied;

\ X J

if n(t2 ) < , Eq.(33) is applied.

\ X JReferences

1. Levy P.W., Nucl. Tracks Rad. Meas. 10, 1985, 212. Furetta C , Nucl. Tracks Rad. Meas. 14, 1988, 4133. Delgado A. and Gomez Ros J.M., J.Phys. D: Appl. Phys. 23, 1990, 5714. Delgado A., Gomez Ros J.M. and Mufliz J.L., Rad. Prot. Dos. 45, 1992,

1015. Gomez Ros J.M., Delgado A., Furetta C. and Scacco A., Rad. Meas. 26,

1996, 243

Fading factor

Starting from the first order kinetics equation

by integration one obtains

( E\

or more simply

n =noexp(-pt)

where n and n0 are the trapped charges at time / and / = 0 respectively.Considering that n is proportional to the TL emission, let us say the glow

curve or peak area <Z> one gets

O = O0exp(-^)and then

'-Hi)Example: after irradiation of some TLDs, a part of them is immediately readout,giving an average TL reading of 1425 (reader units). The rest of the irradiated TLDsare stored in a lead box and readout after a period of 30 days, giving an averagereading of 1285 (reader units). Using the previous equation one obtains

p = 3A5-l0~3d-1

which means a lost per day of 0.345%.

Fading: useful expressions

In the following, some expressions for fading correction in practicalsituations will be given. They are based considering the first order process and thegeneral case in which, during the experimental period of time, two effects are incompetition between them: one is the trapping rate due to a continuous irradiationover all the experimental period, i.e. environmental background irradiation; thesecond one is the detrapping rate which takes place at the same time, i.e. thermalfading. Such a situation can be described by the following first order differentialequation:

d<$> ,_ D— = -Ad> + — (i)dt Fc

(1)

CHAPTER F 139

where• O is the total TL light of a given peak in the glow curve;• X is the fading factor and it is constant for a constant temperature. In case

of the kinetics parameters of the considered peak are known, i.e. E and s ,

f E)it can be expressed by S exp ;V kTJ

• t> is the dose rate of the irradiation field;• Fc is the calibration factor of the thermoluminescence system, expressed

in dose/TL.

Equation (1) represents a dynamic situation where two competing effectsare taken into account. This equation tends to an asymptotical limit as the fadingproduces a progressive extinction of the accumulated charges, whereas thecontinuous irradiation leads to an increase of them. Equation (1) only holds in thecase we are far from saturation.

The solution of Eq.(l) is then obtained as follows:

FcUsing the substitutions

X = -X® + — , dc = -Xd®Fc

we get

and then

-Im \ . ,

x -»••£from which

<D = O 0 e x p ( - ^ ) + - | - [ l - e x p ( - ^ ) ] (2)

Equation (2) depicts a situation where a non-zero charge population isalready trapped at the beginning of the experimental time, i.e. <X>0 * 0.

Considering the practical situation where the TLDs are annealed before use,all the traps are empty at the beginning of the experimental period. In such a caseEq.(2) becomes

O = -^[l-exp(-^)] (3)

When a very long time has elapsed, i.e. / —> oo, 0 gets more and moresimilar to the asymptotical value

« - - * - (4)" XFC

Such a value grows larger as the dose rate and /or the sensitivity (\IFC)increases, or as the fading effect decreases. The asymptotical value given by Eq.(4)may be explained assuming that, at infinity, a dynamical equilibrium is attained,providing the trapped charges to compensate at each instant those escaping owingthe fading phenomenon.

Discussion of some practical situations

1. Initial and instantaneous irradiation followed by fading at room temperatureFigure 4 depicts the situation. In this case the irradiation is delivered to the

dosimeters at the beginning of the experimental period and the duration ofirradiation, /, , is very short so that any fading effect during irradiation can beneglected. After irradiation the irradiated samples are stored, at room temperature orat any other controlled temperature, for a time ts »/,.

The situation depicted in Fig.4 is the usual case for fading studies. Equation(2) reduces to the simply expression

O(fs) = 00exp(-tos) (5)from which

CHAPTER F 141

_

I O(ts)

k • *,

storage time ts

Fig.4.Case 1. Initial irradiation followed by storage at R.T.

<D0=Ofo)exp(Ais) (6)

Through the calibration factor Fc, the initial deliverd dose is then obtained:

Do = FcO{ts)exp{Xts) (7)

1.1 -i 1

1 • v ^

-J ° '9 ^ \ X =6x10'3 day1

Q ^ v . F IO^GyATLg 0 8 ' ^ \ . *o=1O3

< ^ - - ^ D0=1Gy

i 0 7 ^ ^ ^ ^O ^ \ ^ ^

0.6 ^ ^ ^ ^ ^ ^

0.5 -

0.4 -I 1 1 1 1 10 20 40 60 80 100

DAYSFig.5. Case 1. Plot of Eq.(7). The imput data are given in the same figure.


2. Initial but not instantaneous irradiation, followed by fading at room temperatureAn initial irradiation is delivered at the beginning of the experimental

period, but the irradiation time, U , is so long that a fading effect can no longer beignored during the period of irradiation.

After the irradiation the samples are stored for the time ts. Figure 6 depictsthe situation.

D

N— t- -H^ 1 H1 1-1 • «-S

Fig.6. Case 2. Long irradiation followed by storage at RM.

During the irradiation time, Eq.(2) reduces to the following expression

^ , ) = rrr[l-exp(-H)] (8)

which gives the TL emission at the end of the irradaition time. As the irradiationstops, the samples are only subject to fading at room temperature, so that

®(ts) = q>(ti)exp{-Xts) (9)

Combining Eq.(8) and Eq.(9), we get

H{s) = ^ r I1 ~ exp(- kt, )]exp(- Xts) (10)kr c

CHAPTER F 143

from which the true deliverd dose is obtained, taking into account that D = D • ti,,

D = lFc<t>(ts >,. exp(?as \l - exp(- ty )]"' (11)

3. The irradiation is carried out over all the experimental periodThis is the case of environmental background measurements or self-dose

irradiation. See Fig. 8.The irradiation time, ti, is now equal to the storage time fc. The TLD

samples are prepared and exposed to the irradiation field, then the initial condition is<P0 = 0 and Eq.(2) reduces to the following expression:

O = —r-[l-exp(-X/)] (12)kr c

where ts =tt=t.

If D is the environmental background dose rate or the internal dose ratedue to the self irradiation of the samples, the total dose is obtained as

D = <DAJ^[l - exp(- Xt)]~l (13)

5000 F ^ ~ 1 P^MTdayH^ ~ ^ - - ^ ^ -m- TL (3 days)

4000 ^ 4 ^ ^ _a ^ ^ — ^ ^ _ _ ^ —*— TL(5days)

§ 3 0 0 0 • • - - - "~~"~-"~^^_

| ^^-^_^____^^ ' D = \Gyld

5 200° ~~~ —"—11 '/ = Wdays

K 1000 » — ^ — ^ _ » _ _ 9 L = 6 - 1 O - 3 C ? ^

-*' Fc =10"3 GylTL

0 20 40 60 80 100

DAYS

Fig.7.Case2.PlotofEq.(10).

D

k t = t = t Hi i s ij — i i

Fig.8. Case 3. The irradiation is carried outover all the storage time.

8 0 T •,

70 ^ ^

•5 -60 - ^ " ^

z 5 0 ^ ^

I40 / ^i=f 20 « ^

10 af

ok— , , , ,0 20 40 60 80 100

DAYS

Fig.9. Case 3. Plot of Eq.(13). Input data are:

D = l0'iGy/d,X =6-l(T3<r1 )Fc=l(r3 GylTL.

CHAPTER F 145

0.35 -r .: — • • • • <>

„ 0.3 •; r

3. 0.25 • fz •§ 0.2 • 2

w 0.15 i l_ i 7

0.1 • I

o.o5 r i i i i i i i i i i i i i i i i i i i i i i i i |

0 20 40 60 80 100

DAYS

Fig. 10. Case 3. This figure depicts the case of saturation after few days.The numerical values are the following:

D = \0~4Gy/d,k=3l0~ld'l,Fc =10~3Gy/TL

4. An initial and short irradiation is superposed to a background irradiationLet us indicate with DB the background irradiation, which acts over all the

period of storage, ts. Figure 11 shows the situation. The fading during the shortirradiation is neglected. The equation simulating this case is always Eq.(2), writtenin the following way:

<D = < D o e x p ( - ^ ) + - ^ [ l - e x p ( - ^ s ) ] (14)XFC

which gives, in explicit form

o = | ® - ^ - [ l - e x p ( - ^ ) ] l e x p ( ^ ) (15)

from which the initial delivered dose is obtained:

Do = FcO0 (16)


Do

n Da

N ts H

Fig.l 1. Case 4. Initial and short irradiation superposedto background irradiation.

S1-5 ^ * ^ (a)

0.5 J 1 , , ] ' T0 20 40 60 80 100

DAYS

Fig. 12. Case 4. Eq.(16) has been computed for two different sets of input data.

(a): DB =\Q-iGy/d,D0 =l0Gy,Fc =\0~3Gy/TL,X =10^d^

(b): DB=l0'3Gy/d,D0 =l00Gy,Fc =\0~3Gy/TL,X = 6-IQ'3d~l.

CHAPTER F 147

First-order kinetics when s=s(T)

The frequency factor s may be considered in some cases to be dependenton temperature, and proportional to 7", where a has various values in the range -2 <a<2[l-3] .

Let us suppose that s depends on the temperature according to thefollowing relation

s = sja (1)

The detrapping rate is now

£ = -*!-«*-£) (2,Using a linear heating rate /3 =dT/dt, the solution of Eq.(2) is

n = »0 exp|^- ^ £ T" exp(- ~)dT j (3)

and the TL intensity 7(7) will be expressed by

I(T) = nosoTa exp(-^)exp - ^ [j" exp(-^)dT (4)

Equations (3) and (4) can be further developed using the integralapproximation when s=s(T). In this case Eq.(3) becomes:

"=H~M^ha + 2 >fH-i)l) <3>)Equation (4) becomes

m -n^r exp(-|)expj_L. « ^ [ , _(. + 2 ) | ] e x p ( _ | |(4')


References1. Chen R. and Kirsh Y., Analysis of Thermally Stimulated Processes

(Pergamon Press, 1981)2. McKeever S.W.S., Thermoluminescence of Solids (Cambridge University

Press, 1985)3. Furetta C. and Weng P.S., Operational Thermoluminescence Dosimetry

(World Scientific, 1998)

Fluorescence

Fluorescence is a luminescence effect occurring during excitation. The lightis emitted at a time less than 10~8 s after the absorption of the radiation. This meansthat fluorescence is a luminescent process that persists only as long as the excitationis continued. The decay time of fluorescence is independent of the temperature: it isdetermined by the transition probability of the transition from an excited level Ee tothe ground state Eo. The process is shown in the following Fig. 13.

Ee

hv^.

Eo

Fig. 13. Fluorescence process.

CHAPTER F 149

Fluorapatite (Ca5F(PO4)3)

This material belongs to a class of compounds, mineralogically known asapatites. The effective atomic number is about 14.

Fluorapatite is prepared by synthesis from CaF2 and CaHPO4 throughelimination of hydrofluoric acid. About 160 mg of the resulting powder, coveredwith thin LiF crystals and contained im silver boats, are typical samples forthermoluminescence investigation.

The TL glow curve of synthetic fluorapatite powder exhibits peaks at 145,185,260and395°C.

ReferenceRatnam V.V., Jayaprakash R. and Daw N.P., J. Lum. 21 (1980) 417

Frequency factor, s

The frequency factor, s, is known as the attempt-to-escape frequency and isinterpreted as the number of times per second, v, that an electron interacts with thecrystal lattice of a solid, multiplied by a transition probability K, multiplied by a

term which accounts for the change in entropy AS associated with the transitionfrom a trap to the delocalized band, s may be written as

(AS)

where k is the Boltzmann constant [1,2].The expected maximum value of s should be similar to the lattice

vibrational frequency (Debye frequency), i.e. 1012 - 10u s"1. According to Chen, thepossible range for s is from 105 to 1013 s"1 [3].

Randall and Wilkins gave the following meaning to the frequency factor:they described the trap as a potential well and s should be the product between thefrequency with which the trapped electrons strike the wells of the potential barrierand the reflection coefficient. According to this definition, s should be expected tobe about of the order the vibrational frequency of the crystal, i.e. 1012 s"1.

Alternatively, s1 should be considered as connected with the capture cross-section, CT, of the trap by the following relation

s = ve7Vca


where ve is the thermal velocity of the electrons in the conduction band, Nc is the

density of states (available electron levels) near the bottom of the conduction bandand a the capture cross-section of the trap. In this case the values of 5 are rangingf r o m l 0 8 t o l 0 1 V .

In some cases, the frequency factor, as well as the pre-exponential factors,may be considered temperature dependent and proportional to 7* with a rangingfrom -2 to +2.

References1. Glasstone S., Laidler K.J. and Eyring H., The Theory of Rate Processes.

McGraw-Hill, New York, 19412. Curie D., Luminescence in Crystals, Methuen, London, 19603. Chen R. and Kirsh Y., Analysis of Thermally Stimulated Processes,

Pergamon Press, 1981

Frequency factor, s (errors in its determination)

It must be stressed that any error in the evaluation of the activation energyand of temperature introduces very high error in the determination of s. Makingreference to the expression of the condition at the maximum for the first orderkinetics

BE ( E )5 = ^—r-exp (1)

kT2 kTKiM \K1M)

its logarithm is

lns = lny9 + l n £ - l n & - 2 1 n r M + (2)

Assuming that the heating rate ft has no error, we can differentiate Eq.(2)to give

As = AE i2ATM 1 TMAE + EATM

* E TM k Tl (3)

CHAPTER F 151

Supposing an error of 2% on the evaluation of E and the same error on thetemperature determination, it is easy to see that the error on s is very large. Makingthe assumption that no error is done on the temperature measurement, Eq.(l) gives

s E kTu l

which still remains a large error.

Frequency factor and pre-exponential factor expressions

• Frequency factor: 1st order of kinetics

• Pre-exponential factor: 2nd order of kinetics

(3£exp| — L _, ,,_!_!!M+?«kr (2)

kn^M L E \

which becomes, introducing s = s'n0 ,

f E \

K1M L ^ J

(4)

(1)

(3)

• Pre-exponential factor: general kinetics order (l The frequency factor given by Eq.(5) acts as an effective frequency factor

expressed by the following equation

v = s"<x <6>

10l»l 1

t10"; Ji

10«» JT

\ io«» r jr

io«* jr

10" or

t

^ 1 0 . " 101 ' • 1 W 1 1 10* " 10* f

n>

Fig.l4.PlotofEq.(6), [1]

(4)

CHAPTER F 153

• Frequency factor vs temperature: 1st order kinetics

s = ^ I + __A e x p (7)kTa+2 [ 2 I \ kT )KlM V L J \K1Mj

where A^HcTn/E and -2<a<2.

• Pre-exponential factor vs temperature: 2nd order kinetics

s'o = — ^ ^ ^ e x p - ^ (8)

• Pre-exponential factor vs temperature: general kinetics order (l < b < 2)

-i-i

BE l-AM(l + a)(l-b) ( E \si = -JL-^r ~ ^ '- exp (9)

kTa+1 a \ IrT

2 M

• Remarks

It is easy to note that, except in the first order case, s' and s" are constantsfor a given TL sample and dose but would vary in the same sample as the dose isvaried.

In order to overcome this difficulty, it has been suggested in [2] to rewritethe general-order equation in the form

dn nb ( E\— = -s—1-,-exp (10)dt Nb~' I kT)

taking into account that the equations for the first- and second-order can be written,respectively, as


dn s ( E )— = -n—prexp (11)

dt N° { kT)

dn 2 s ( E \— = - / r — e x p (12)dt N \ kT)

In all cases s has units of sec"1, having eliminated any difficulty related tothe dimension problem of s' and s".

According to [1], the empirical expression (10) should also eliminate thevariation of s with respect to the variation of the absorbed dose. Indeed, this pointdoes not seem correct. In fact, the TL intensity obtained from the new expression(10), is

^ ^ L f ^ U - A V f (,3,Nb~l [ pNb~x I \ kT') J

Using the condition at the maximum and the integral approximation, thenew expression is

(N_r\kTici-S] I^JM^V (14)

from which the expression for the second-order process (b=2) is easily obtained.Although in Eq.(14) s is now expressed in sec"1, as in the first-order

kinetics, the dependence of s on the initial trapped charges, n0, still remains. Only inthe saturation case, i.e. «o = N, the s values are independent of the value of the initialtrapped charges. Furthermore, Eq.(14) includes the parameter N which cannot beeasily determined.

CHAPTER F 155

It is then evident that the suggested way to rewrite the rate equations doesnot eliminate the dependence on the dose. The new formulation only allows toexpress s in units of time in all cases.

References1. Kitis G., private communication.2. Rasheedy M.S., J. Phys.: Condens. Matter. 5, 1993, 633

GGarlick-Gibson model (second-order kinetics)

In 1948 Garlick and Gibson, in their studies on phosphorescence,considered the case when a free charge carrier has probability of either being trappedor recombining within a recombination center. The term second order kinetics isused to describe a situation in which retrapping is present.

They assumed that the escaping electron from the trap has equal probabilityof either being retrapped or of recombining with hole in a recombination center.Let us indicate:N= concentration of traps,n = electrons in N,m = concentration of recombination centers,n = m for charge neutrality condition.

The probability that an electron escapes from the trap and recombine in arecombination center is

m n7 x = — (!){N-n)+m N

So, the intensity of phosphorescence, /, is given by the rate of decrease ofthe occupied trap density, resulting in the recombination of the released electronswith hole in the recombination centers:

r M dn (n\(n\ n2 ( E\l(t) = = c — • - \ = c — sexp (2)

W dt \N) UJ N { kT)where T is the mean trap lifetime.


dn 2 ( E\

^ = -"Vexp[--J (3)The quantity s' = s/N is called pre-exponential factor and it is a constant

having dimensions of cm3sec"'. Equation (3) is different from that one obtained inthe case of first order kinetics, where the recombination probability is equal to 1,


since no retrapping is possible.From Eq.(3), by integration with constant temperature T, we obtain:

f> dn , ( E \ f ,™r = s exp \\dt

1 1 , f E )— — = -s'tex.p [- —nQ n V kTJ

r ( EW~1n = n0 l + s'notexp\j—J (4)

and then, the intensity /(/) is:

/(O = - - = «Vexp(--J=f 7-^f (5)

which describes the hyperbolic decay of phosphorescence.Otherwise, the luminescence intensity of an irradiated phosphor under

increasing temperature, i.e. thermoluminescence, taking into account that dt=dT/$,is obtained as follows:

dn s' ( E^ „

therefore

p dn s' f ( E \

and then

CHAPTER G 159

1 1 s' f ( E > \ J

The intensity 7(7) is then

dt l kTJ L^M-ALrf

Eq.(7) can be rewritten as

where s = s'n0. In this case s has units of s"1 like the frequency factor in the first-

order kinetics, but it depends on n0.

ReferenceGarlick G.F.J. and Gibson A.F., Proc. Phys. Soc. 60 (1948) 574

General characteristics of first and second order glow-peaks

Some general characteristics can be listed to distinguish between first andsecond order glow-peaks when a linear heating rate function is used.

First order peaks

— The first order peaks are asymmetrical and x = TM -T^ , the half-width at the

(6)

(7)

(8)


low temperature side of the peak, is almost 50% larger than 5 = T2 -TM , thehalf-width towards the fall-off of the glow-peak T ~ 1.58. The shape and thepeak temperature depend on the heating rate.

~ For a fixed heating rate, both peak temperature and shape are independent ofthe initial trapped electron concentration n0, as it can be observed from thecondition at the maximum

P£ ( E ]—z- = s expkT kTKIM \ K1M J

" The value of n0 depends on the pre-measurement dose.

~ The TL glow-curve obtained for any «o value can be superimposed onto thecurve obtained for a different n0 by multiplying by an appropriate factor(Fig.l).

"" A first order peak is characterized by a geometrical factor (a. = oVco = (7*2 -TM)/(TI -TI) equal to about 0.423.

~ For fixed values of dose and heating rate, the co value increases as Edecreases (Fig.2).

~ The decay at constant temperature of a first order peak is exponential.

Second order peaks

"" A second order peak is practically symmetrical (5 ~ x).

" To keep all other parameters constant, the shape and the peak temperaturedepend on the heating rate.

™ For a fixed heating rate, the peak temperature and shape are stronglydependent on the initial trapped charge concentration «0- Peaks obtained fordifferent initial trapped charge concentrations cannot be superimposed bymultiplying a factor.

~ The glow-peaks obtained for different n0 values tend to superimpose at thehigh temperature extremity of the glow-peak.

" An increase of «0 produces a decrease in the temperature of the peak,

CHAPTER G 161

according to the maximum condition.

R£ I" s \ ft, ( E V , 1 , ( E ^- e - T 1 + — - I exp W7" =5f«oexp2&T2 R *i I kT' \ kT \UijnL P ° V K1 J J \ KIM)

2*)0(! —VULUES Of I, f\ E-0.75 (tV)

S 10 15 20 25(10'*.l).) / \ TM.<5O(K)

^ 150(l ~ \ // /~\ \1

1 100°" \ ////Avi

300 350 400 450 500TEMPERATURE (K)

Fig. 1. A computed first order glow-peak showingthe linear increase of /M as a function of dose.

~ The isothermal decay of a second order peak is hyperbolic.

~ A second order peak is characterized by a geometrical factor n = 0.524.

~ Furthermore, a decrease in the temperature of the peak, TM, is observed as afunction of the kinetics order changing from 1 to 2. This effect is illustratedinFig.3.


500 —

Values of E(eV) /f|\; 0.5,0.75,1.0,1.25,1-5 I | \

0 300 410 SO0

TEMPERATURE (K)

Fig.2. A computed first order glow-peak showingthe increase of oo as E decreases.

m ; b = 2,1.75,1.5,1.25,1

0 300 350 400 450

TEMPERATURE (K)

Fig.3. Computerized glow-peaks showing the effect of thekinetics order on the position of the peak temperature.

CHAPTER G 163

ReferenceBacci C , Bernardini P., Di Domenico A., Furetta C. and Rispoli B., Nucl.Instr. Meth. A 286 (1990) 295

General-order kinetics when s"-s"(T)

In the case of general order kinetics, b, the TL intensity equation I=I(T) hasto be modified by substituting s" with the following expression [1-3]:

s' = s'Ja (1)

In this case the TL intensity, I(T) , becomes

I(T)

= <«or exp(-A)J1 + r i ) f > e x p f - ^ 1 "0 0 FV kT [ P *. *\ kT)

(2)




(World Scientific, 1998)

Glow curve

It is the plot of the thermoluminescence intensity, /, as a function of thesample temperature during read out. Each trapping level in the material gives rise toan associated glow peak; so, a glow curve may be formed by several peaks, eachrelated to different trapping levels. These peaks may or may not be resolved in theglow curve.

Considering the basic equation

1 = - 4 <•>dt

where the TL intensity is proportional to the detrapping rate, by its integration weobtain

n(t)-nx=™\I-dt' (2)

Dealing with only one peak, nx = 0 and therefore

n(t) = ]l-dt' (3)t

Furthermore, using a linear heating rate, Eq.(3) transforms in

n(T)=]-]l(T')dr (4)P T

From a practical point of view, n(T) can be evaluated from the area underthe peak, from a value T=Th initial rise region of the peak, to a temperature 7}, endof the peak (i.e., when the glow intensity is at its minimum. So, Eqs. (3) and (4) canbe rewritten as

'/ 1 Tf

n = \Idt' = - \IdT (5)', P T,

Because the trapped charge concentration, n, is proportional to the dosedelivered to the TL sample, the concept expressed by Eq.(5) is of great importancein radiation dosimetry.

IIn-vivo dosimetry (dose calibration factors)

It is strongly recommended to perform a separate calibration for eachradiation beam quality. If the TLDs can be identified, a calibration factor could begiven to each dosimeter and it is necessary to monitor the individual factors fromtime to time.

In practice, having a large number of dosimeters is possible to save a part ofthem for the purpose of calibration. The readings of the patient dosimeters can thenbe converted in dose by comparing their response to the ones of the calibrateddosimeters.

Entrance dose calibration factorThe entrance dose calibration factor, FIN, is defined as the factor with

which the TL signal, TL1N , of the TLD, positioned on the skin of the patient at theentrance surface, with the right built-up cap, must be multiplied to obtain theentrance dose, DIN:

F -D'»IN ~ TL

1LlIN

F1N is determined by positioning the TLD on the surface of a flat phantom, at

the entrance side of the beam. The TLD response ( TLIN ) is then compared with the

response of a calibrated ionization chamber (DIN ), positioned at depth dINnax.

Exit dose calibration factorThe exit dose calibration factor, FOUT, is determined in a very similar way.

The TLD is positioned on the exit surface of the beam and its signal is compared tothe response of the calibrated ionization chamber positioned in the phantom atdOUTmsK from the exit surface:

F — OUT

OUT ~ TL


The phantom thickness should be variable to match the various thickness ofthe patients.

It is also suggested to determine the calibration factors for each particularkind of treatment.

The following Fig.l shows the experimental set up for the calibrationfactors determination.

buid N. \ \ / I on iza t ion / \up ^ j L \ / chamber / \

dmax + O-\ / I

I \ _ I \ dmax

• ' \

Fig.l. Experimental set up for determiningthe entrance and exit calibration factors.

Inflection points method (Land: first order)

This method, proposed and applied by Land, uses, in addition to thetemperature at the maximum, T^ the two inflection points in the curve of the TLemission. Using the Randall and Wilkins expression, the second derivative of the TLintensity can be written as

dr2 aryar) dry idi) dry dr J

CHAPTER I 167

The temperature values Tp corresponding to the inflection points, areobtained from Eq.(l) by quoting it to zero, for T=T,:

d T d2

—(In/) +^(ln/) = 0 (2)

Using now the logarithm of the intensity 1(T) one obtains

" d A y \ E (s) t E I2

^nI)Lr[w\irf(-^\ (3)d1 n n (2E) ( E)(s) , E^

Inserting Eq. (4) into Eq.(2) and using the condition at the maximum, oneobtains

E \lE( 1 _ \X\_ 3E r^fj__J_ll E _2

(5)

Using now the substitutions

(1)

(4)

(6)

the following final form is obtained

E.J&L.JA) (7)

\T,-TU\ ye)

with A = 0.77 if T{ < TM, or, A = 2.66 if T, > TM.

The frequency factor is then obtained from the condition at the maximum.This method is useful even in case of closed peaks and E and s can be

obtained for all peaks from a single glow-curve.

ReferenceLand P.L., J. Phys. Chem. Solids 30 (1969) 1681

Inflection points method (Singh et al.: general order)

Singh et al. presented the method of Land in a more simple form.Considering the equation

I(t) = snoexp(-~)\l + s(b-l)texpf-J;] " 0)

which gives the TL intensity function 7(7) for a general order peak, the first and thesecond derivatives of 7(7) with respect to Tare expressed as

dl

~df = I'f{T) (2)

d2l dl df(T)HF-df^^-dT (3)

where

CHAPTER I 169

EE ^exp(-—)

fiT) = W~ T E W

fll + 0-l)-^Jexp(-— )dT]if b * 1; and

f(T)=w-rM-h (5)if 6 = 1 .

dl/dT = 0 gives the peak temperature at the maximum, TM, and d 1/dT = 0

gives the inflection pints Tt of the glow-curve. Furthermore, T. = T. corresponds to+

the inflection point on the raising side of the glow-curve and T=T. is the inflectionpoint on the falling side.

According to Land, one can write

*-W • x--wu (6)

Because a good linear correlation exists between the following pairs ofvariables:

{x^~^r^)] ; [ X M ' ( ^ 7 ^ ) ] ; [XM'XM(X;'-X:)]

(7)

one can write

Xi XM

xM = A2^^-7 + B2 (8)XM ~ Xi


xfx-xM = A3 —^r—IT + B3

xM(Xi-xl)where the coefficients Aj and Bj depend on the order of kinetics b.

The previous equations can be rewritten in the following explicit form

A kT2

E=W^t)+B>kT"A JrT2

By using the method of non-linear least-square regression, the coefficientsAj and Bj can be expressed as a quadratic function of the kinetics order, for branging from 0.7 to 2.5:

Aj=a0J+aljb + a2jb2

(10)BJ=c9J+clJb + c2Jb2

The following Table 1 shows the numerical values of the coefficients using Eq. (10):

J aoi an a2i Co; C!j c2j

1 0.8730 -1.5619 0.1334 0.4489 0.5853 -0.07512 0.6676 -1.8493 0.1499 0.4479 0.5866 -0.07563 I 0.9394 I -1.7055 | 0.1422 | 0.8967 | 1.1721 | -0.1507 "

Table 1 .Values of the coefficients ak. and ck. in Eq.(10).

Figure 2 shows the behavior of /, dl/dT and d I/dT as a function oftemperature for an isolated peak at 320°C in KAlSi3O8 following a second orderkinetics.

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CHAPTER I 171

dl/dT / ' \ b ! \d!l/dT2 / ' \ ' \

-o.«l i i !J u_ i ITi V

IBO 2 6 0 MO 4 2 0T(°C)

Fig.2. Behavior of I, (a), dl/dT, (b)and d2I/dT2 (c) as a function of the temperature T.

ReferenceSingh T.C.S., Mazumdar P.S. and Gartia R.K., J. Phys. D: Appl. Phys. 23(1990) 562

Initial rise method when s — s(T) (Aramu et al.)

Aramu and his colleagues applied the initial rise method in the case of thefrequency factor s which is temperature dependent. In this case, the intensity / isproportional to the first exponential only:


/ocr-exp^j (l)

from which

ln/ = alnr- —kT

Comparing Eq.(2) with the following equation

or better with

~(ln/)=^ (3)dTy ' kT

one obtains

kT2 T kT2

from which

E = Eir-akT (4)

This means the need to correct E for a few percent.

ReferenceAramu F., Brovetto P. and Rucci A., Phys. Lett. 23 (1966) 308

Initialization procedure

The initialization procedure on a new batch of TLDs is recommended toreduce the possibility of variations in dosimeter performance characteristics duringusage [1,2].

The first stage of the procedure involves heating dosimeters inside afurnace using the optimum annealing parameters (temperature and time) indicatedfor the TL material under test. In another section of this book all the annealingprocedures used for different materials are listed. The dosimeters are placed inlidded crucible or in suitable annealing stacks (such as those made from quality

(2)

CHAPTER I 173

stainless steel or electroplated copper). Annealing stacks allow separation andidentification of dosimeter elements and are particularly useful if these elements areto be calibrated individually rather than in batches. The annealing stack containingthe dosimeters is placed in the furnace, preheated to the required temperature. Theactual duration of annealing will be longer than the required annealing time in orderto attain thermal equilibrium at the required temperature. This additional time shouldbe determined before all the setting up procedures as it will be indicated in thesection concerning the quality control of the furnaces.

After annealing, the dosimeters are cooled in their containers in areproducible manner. It is imperative to always use the same cooling procedure andthat this is reproducible because the glow-curve of the material is strongly affectedby the cooling.

The cooling may be accomplished by keeping the furnace door open afterthe heating has been stopped. In this manner the cooling will be more or less long,depending on the starting temperature.

Alternatively, the crucible or annealing stack may be removed from thefurnace immediately after the thermal treatment in order to allow the dosimeters tobe cooled much faster to room temperature. This can be obtained by laying theannealing container on a metal plate.

Tests should be made before initialization to find the most suitable meansof cooling for the user's particular requirements.

It is not recommended to switch to other methods once a cooling procedurehas been adopted.

In some cases the annealing procedure consists of two subsequentannealing (see the annealing section): the first is carried out at high temperature andthe second at low temperature. An example is given by LiF:Mg,Ti in the form ofTLD-100, 600 or 700, which needs a first annealing at 400°C during 1 hourfollowed by 2 hours at 100°C (or 24 hours at 80°C). In all the cases where theannealing procedure is formed by two thermal treatments, the first at hightemperature followed by one at low temperature, the dosimeters have to be cooled toroom temperature at the end of the first annealing and then placed in the preheatedoven for the second annealing. There are now several commercial programmedovens in which the thermal cycles can be programmed at the beginning of thetreatment; in this case the low temperature annealing is switched on when the hightemperature of the first annealing decreases until the lower temperature of thesecond one. However, the procedures of heating and cooling have to be always inthe same manner.

At the end of the annealing procedure the dosimeters are read to check thebackground signal. The background depends on the H.V. applied to the P.M. tube,on its age and on the room temperature: the stability of the TL reader must bechecked before and after any reading session.


The initialization procedure is repeated over three cycles. If thebackgrounds on the dosimeters have remained low over these cycles, theinitialization is terminated and the dosimeters are ready for the subsequent tests. Ifbackgrounds on the dosimeters are variable, the initialization can be continued forfurther two cycles of treatment. If backgrounds continue to remain high or variablethe efficiency of the readout system should be checked and/or the dosimetersrejected.

An example of the above initialization procedure is given for 10 TLD-100.The TLD reader was an Harshaw Mod. 2000 A+B with a heating rate of 5 °C/s. Nonitrogen flux was used. The following table shows the results obtained. Consideringthe negligible changes in the average values obtained through the three subsequentcycles (annealing + readout) one can consider the background to be stable and theinitialization ended.

The background values determined for each dosimeter have to be collected(i.e. memorized in a file concerned the batch under test) so that they can be used forthe successive tests. In many cases an average background value is considered forthe whole batch and then subtracted from each individual reading of the irradiatedTLDs. This procedure is valid when the background is very low and constant for thewhole batch. In other specific situations, as in radiotherapy where a high accuracy isnecessary, an individual background is used and checked periodically to avoid anypossible mistakes in the dose determination owing to large variations of thebackground. The following Table 1 shows an example of initialization procedure.

TLD I I3 I P I y3 I I (a u) I % c vNo. BKG BKG BKG '_

1 0.091 0.087 0.0902 0.099 0.101 0.0983 0.101 0.098 0.0994 0.087 0.091 0.0905 0.095 0.087 0.0916 0.107 0.095 0.0977 0.085 0.090 0.0888 0.083 0.087 0.0859 0.085 0.088 0.09110 0.093 0.091 0.089B 0-093 0.092 0.092 0.092 0.60

~%CV | 8.60 I 5.40 | 5.10 | [

Table 1. Example of initialization procedure (BKG = background)

CHAPTER I 175

References1. Driscoll C.M.H., National Radiological Protection Board, Tech. Mem.

5(82)2. Scarpa G. in Corso sulla Termoluminescenza Applicata alia Dosimetria,

Universita' di Roma "La Sapienza", 15-17 Febbraio 1994

Integral approximation

The integral comparing the thermoluminescence theory

cannot be solved in an analytical form.A method which is usually followed for evaluating the value of the integral

is by integration in parts, when the lower limit of integration is 0 instead of To. So,a good approximation is provided by the asymptotic series

F(T,E)= (expf-A^^rexpf-Dsff)'(-!)-„. (2,

The value of (1) is then given by

[tK^~)fT' = F(T,E)-F(Tn,E)

Since F(T,E) is a very strong increasing function of T, F(TO,E) is

negligible compared to F[T,E), the right hand side of Eq.(2) can be considered to

represent the integral value from To as well.In the practical case, a good approximation of the integral is given by the

second order approximation of Eq.(2):

K ' E 1 UK E )

(1)

(3)


Integral approximation when S = s(T)

« l - ( a + 2 ) — exp (1)E L ' E\ \ kT)

if T = 7"M , expression (1) becomes

Interactive traps

Electrons released by a shallow trap may be captured by a deep trap(thermally disconnected traps): in this way the traps are called interactive. Thedeep traps are in competition with the recombination centers for capturing electronsreleased by the shallow traps.

Isothermal decay method (Garlick-Gibson: first order)

Formerly the isothermal decay method was illustrated for the first-orderkinetics by Garlick and Gibson.

Let the initial integral light be So, while St_ will be the integral light at

time tf.

So =n0

Sti =«! =/ioexp(-pf,)

at T = constMaking the ratios

(2)

(1)

CHAPTER I 177

S, S,ln(—) = -;*,. • • -W-f-) = -ptn (2)

the graphs of ln(S,JS0) versus t-t is then plotted for data obtained at a given

storage temperature T. Using different storage temperatures (7 ) one can obtain a setof straight line of slopes

Emi = - s e x p ( - — ) (3)

from which

ln(m,) = l n ( - 5 ) - J 7 (4)

Therefore a plot of ln(w) versus 1/T yields a straight line of slope -E/k andintercept ln(-s) on the ordinate axis.

If the experiment is carried out with two different constant storagetemperatures, 7 and T2, two different slopes, mi and m2, are obtained and then fromthem

VW;J £v r 2 T2J

The last equation allows to calculate E. The frequency factor s is derived bysubstitution of the E value in Eq.(3).

ReferenceGarlick G.F J. and Gibson A.F., Proc. Phys. Soc. 60 (1948) 574

Isothermal decay method (general)

Isothermal decay of the thermoluminescence emission does not employ anyparticular heating. Strictly speaking, the isothermal decay technique is not a TLbased method but nevertheless is a general method to determine E and s. Theexperimental steps consist of quickly heating the sample, after irradiation, to a

(5)


specific temperature just below the maximum temperature of the peak under study,and keeping it at this constant temperature during a given time. The light output(phosphorescence decay) is measured and so it is possible to evaluate the decay rateof trapped electrons.

Isothermal decay method (May-Partridge: (a) general order)

May and Partridge suggested to apply the isothermal decay method in thegeneral case of any order. In this case it is also possible to find the order b. TheTL intensity, at any temperature, is given by the equation

dn h ( E\

whence

I^=-£ -^-F** (2)

By integration, the following expression is obtained

n'-b-n'ob E^ 3 ^ = -"exp(--) ,3,

which, with the substitution

Ec = -(\-b)s"exp(-—) (4)

reduces to

l

n = (a + cty-b (5)

Executing the derivative of Eq.(5) one gets

- = c — ( « + c/)- (6 )

(1)

CHAPTER I 179

Since

T _ dn

~~~dtwe obtain

I = (a + ctn-b

that is

/* ={a + cAs''exd-^)

which becomes

1-6

/ * =A + B-t (7)

where1-6

A = a ^expC-—) * (8)

\-b

B = c 5ffexp(-—) * (9)

The I(t) function given by Eq.(7) is a linear function of the time; thus a plotof the left side versus time yields a straight line when by iterative procedure usingdifferent values of b the best b value is determined to fit Eq.(7).


Isothermal decay method (May-Partridge : (b) general order)

May and Partridge gave an alternative method to the one proposed by themfor the (a) general order case. Their method can be explained as follows.


By differentiation of Eq.(l) (see Isothermal decay method (May-Partridge: (a) general order) at constant temperature:

1-6

fb~ = A + B-t (1)

one obtains

\ - b h^dl— I » — = B (2)

b at

from whichIT 26-1

The logarithm of Eq.(3) yields

l n A = lnC + l n ( / ) (4)at b

thus a plot of ]n(dl/dt) versus ln(7) gives a straight line having a slope m=(2b-l)/bfrom which b can be evaluated.


Isothermal decay method (Moharil: general order)

Moharil suggests the isothermal decay technique for obtaining a parameterwhich is physically more relevant than the order of the kinetics [1]. The theory isbased on the Antonov-Romanovskii equation [2]:

dn Bnm E^ - ^ ^ B n + AiN-n)^-^ (1)

whereB = probability of recombination, A = probability of retrapping, m = number ofrecombination centers at time t,n = number of filled traps at time t,N= total numberof traps and the usual meaning for the other quantities. If n = m, Eq.(l) becomes

(3)

CHAPTER I 181

dn Bn2 E

which reduces to the first-order equation if A«B, and to the second-order equationfor A-B. When neither of the two conditions are satisfied, one has the general orderkinetics. In this case, the general order equation cannot be derived from Eq.(2) andthe kinetics order b cannot be related to the physical quantities A and B.

As it is suggested by Moharil, the ratio A/B can be obtained fromisothermal decay experiment and using Eq.(2). Rearranging this equation one has

Bn + A(N-n) , ( E^- 2 dn = -seW{--)dt (3)

Integrating between 0 and t, with the condition « = «0 for t = 0, we obtain:

vB-A r AN , f ( E\1dn + —jdn = - I s exp - -— \dt

\ Bn kBn2 * FV kTJ

(, A). n0 AN(no-n) f E^

The following hypothesis is now assumed:nQ is proportional to the area under the isothermal decay curve (= AQ); n isproportional to the remaining area under the decay curve after time t (A^). If nQ - N,saturation case, Ao is proportional to N: in this case the area is denoted A$ instead ofAQ. Equation (4) can be written as

O - D - t ^ ^ - - ' (-£) (5)A graph of the left-hand side of Eq.(5) against time should give a straight

line when the best value of A/B is chosen.

References1. Moharil S. V. and Kathuria P. S., J. Phys. D: Appl. Phys. 16 (1983) 4252. Atonov-Romanoskii V. V., Bull. Acad. Sci. USSR Phys. Res. 15 (1951)

673

(2)

(4)


Isothermal decay method (Takeuchi et al.: general order)

Takeuchi et al. reported a method slightly different from the one describedby May and Partridge. From the equations for general order:

I = s"nbexp{-^pj (1)

7(0 = s"nl expj^- A | i + (b - l)t expj^- j " (2)

keeping constant the temperature, one obtains:

/ 0=5"«0 Aexp(-_ |)

b

I, =s"4\l + S"nbo-i(b-l)texp(-^)}~b -exp(-J;)

where Io and no are respectively the initial intensity and the initial concentration oftrapped charges and /(is the intensity at time /. The ratio of the two equations gives

b

[ i V-1 r F

- ^ =\l + s(b-l)texp(-—)IJ I yK kT}\ (3)

with s = s"nbt~x.

The plot of the left side term versus time should then be a straight line whena suitable value of b is found.

Using different decay temperatures, a set of straight lines of slopes

Ew = j(*-l)exp(-—) (4)

is obtained and the activation energy E will be determined from the plot of ln(w)versus l/T:

CHAPTER I 183

\n(m)=his{b-l)-— (5)

ReferenceTakeuchi M., Inabe K. and Nanto H., J. Mater. Sci. 10 (1975) 159

KKeating method (first-order, s=s(T) )

Keating has proposed a method to determine E, for the first-order, when s issupposed to be temperature dependent [1].

The equation giving the TL intensity, when the frequency factor istemperature dependent, is the following:

I(T) = nosoTa exp(-—) -—[ra exV(-^)dT' (l)

Putting noso=Io and making the logarithm of Eq.(l), one gets

lnf-1 =a\nT- — - W f r a exp( -— )dT (2)

Differentiation of this equation with respect to T, and setting the derivativeat the maximum equal to zero, yields

^ « i UJM \ kTM)

from which

i f a E ") £7= pTr + 7^rJexp(—) 0)

Remembering that the integral in Eq.(l) can be evaluated by an asymptotic series, inthis case we have

ff E IcTa+2 E

jVexp(--)</rS—(l-A)exp(--) (4)

withkT

A = (a + 2 )—

Thus, Eq.(2) becomes

Inserting in Eq.(5) the expression (3), we get

-ln(f)

, _ E (TY+2(akTM X v ( E E]= -ahiT + — + — ^ + 1 (l-A)exp

kT \TM) I E / ; \hTM kT) (6)

Using now the temperature T, and T2 when I=IJ1, the followingparameters are defined:

T —TY — M i

I - j

T -Tr - 2 M

1 M

Hence, the following expressions, with T, and T2 respectively, can beobtained

(5)

(7)

CHAPTER K 187

= - ¥ 1 - a l n ( l - r 1 ) + (l-r1)flr+2(l + —^) ( I -A)exp(^ )h

= -Y2-aln(l-r2) + ( l - r 2 r 2 ( l + ) ( l -A)exp(T2 ) (8b)hi

with

Since A - « l for E/kT>lO the expressions (a+ 2^^ IE and

(or + 2)W2 / £ have been taken equal to A = (a + 2)kTM IE.

Equations (8a,b) can be resolved numerically for Tx and F2 for values of a= 0, ±2 and E/kTu between 10 and 35. Analysis of the data shows that E can befound by the following linear equation

fr - 0 7 5 VE = kTM y{L2T - 0.54) + 5.5-10"3 -1 - — J (10)

with

(8a)

(9a)

(9b)

y = r 1 + r 2 and r=S/x

Nicholas and Woods have found that Eq.(lO) holds true for 0.75 < T < 0.90 [2].

References1. Keating P.N., Proc. Phys. Soc. 78 (1964) 14082. Nicholas K.H. and Woods J., Br. J. Appl. Phys. 15 (1964) 783

Killer centers

The killer centers have been introduced by Schon and Klasens to explainthe thermal quenching of luminescence. At high enough temperatures, holes may bereleased from luminescence centers and migrated to other centers called "killers", inwhich the recombination between free electrons with the trapped holes is notaccompanied by emission of light due to phonon interaction. An increase in theconcentration of the killer centers provokes the decrease of the luminescenceefficiency.

Kinetic parameters determination: observations

The glow-curve computerized deconvolution analysis (GCD) is the mostrecent and widely used technique for determining the kinetics parameters. Anyway,it has to be emphasized that it is possible, in principle, to deconvolute a complex,and even a single peak, in a very large number of different configurations and tochoose that one or those which give the best figure of merit (FOM). Indeed, even inthis case many configurations may be obtained, each one with a different set of thetrapping parameters. Of course, this kind of result is not physically acceptable.

For this reason, trapping parameters obtained just using the GCD are notacceptable and some suggestions on how to proceed are given below:

~ Start the analysis using at least two classical methods which areindependent of the shape of the peak. The GCD depends, on the contrary,on the shape. The initial rise and the various heating rate methods maybe used for this purpose.

" Use now the GCD and compare the trapping data to the ones obtained inthe first point.

CHAPTER K 189

As an example, the following table reports the values of the activationenergy of two different kinds of lithium borate. The data are referred to the veryintense peak only [1].

Figures 1 and 2 show the glow curves of both materials. The experimentaldata are given by the open circles. In the same figures the deconvolution is alsoshown.

From Table 1 it is evident that there is the discrepancy between the dataobtained by IR and VHR methods and the results of the deconvolution. The valuesobtained in the last case are lower in comparison to the data resulting from IR andVHR.

Material I Initial Rise I Various Heating I GCD(IR) Rates (VHR)

Li2B4O7:Cu 1.56 ± 0.04 1.57 + 0.02 1.37 + 0.03Li2B4O7:Cu, In | 1.61+0.03 | 1.66 ±0.02 | 1.35 + 0.03

Table 1. Activation energy (eV)

~ Check which of the results should be the more realistic and physicallyacceptable. For this, one should apply a method which depends, as theGCD is, on the quantities characterizing the shape of the peak: i.e. one ofthe peak shape methods (PS), for instance the Chen's method. Thismethod should give results very similar to those obtained by GCD. Table2 shows the results obtained using the PS method.

Material I Er(eV) I Es(eV) 1 Em (eV) ~Li2B4O7:Cu 1.38 ±0.03 1.39 ±0.02 1.40 ±0.03

Li2B4O7:Cu, In | 1.38 ±0.04 | 1.40 ±0.04 | 1.40 ±0.04

Table 2. Activation energy values obtained by PS method.

From Table 2 results that the data obtained by the PS method are verysimilar to the data resulting from GCD.

"* Make the following assumption: it could be possible that the peaks underinvestigation are not single peaks but rather there is some satellite


peak/peaks that made their shape broader than a pure single peak. In turn,this should cause the activation energy to be lower than the real one inboth PS and GCD methods.

~ Look for a method which is again independent of the glow shape and,furthermore, which should allow to estimate the number and position ofindividual, not resolved peaks within the glow peak appearing as a singlepeak. This method is the modified IR method introduced by McKeever

[2].

~ Perform a second deconvolution according to the results obtained above.Figures 5 and 6 show the new deconvolution and Table 3 the new data.

The application of the McKeever method allows to obtain the followingplots showed in Figs. 3 and 4.

For Li2B4O7:Cu three distinct plateau can be observed, the firstcorresponding to the main peak, the second and third indicate the presence of twohigh-temperature peaks. Li2B4O7:Cu,In analysis shows the main peak, correspondingto the first plateau, and a possible second peak at higher temperature.

3000 | 1U2B4O7: Cu

2400 f\

=f1800 / \

"~ 1200 / \

600 / \

400 440 480 520 560 600Temperature (K)

Fig.l. Glow curve of Li2B4O7:Cu. The open circles indicate theexperimental data. The performed deconvolution is also shown.

CHAPTER K 191

Li2B4O7: Cu, In „

6000 / V

J. 4000 / \

2000 / \

400 425 450 475 500 525 550 575Temperature (K)

Fig.2. Glow curve of Li2B4O7:Cu,In. The open circles indicate theexperimental data. The performed deconvolution is also shown.


300 | 1Li2B4O7: Cu

275

250 j s

g 2 2 5 ^200 a B-a-» a B " " "

175

150 I 1150 170 190 210 230 250

Tstop (°C)

Fig.3.1.R. plot for Li2B4O7:Cu.

240 I 1Li2B4O7:Cu, In

230!

220

£210 1 1

200 5 a s °

190

180 I — 1150 170 190 210 230 250

Ts,op(°C)

Fig.4.1.R. plot for Li2B4O7:Cu,In.

CHAPTER K 193

3000 I 1Li2B4O7: Cu

2400 f\

-M800 J b

H1200 £ \

600 jbl \l

380 420 460 500 540 580Temperature (K)

Fig.5. The new deconvolution performed for Li2B4O7:Cu.

7500 I 1Li2B4O7 : Cu, In a

6000 jl V

~ 4500 J \

*" 300° I \1500 II \

380 420 460 500 540 580

Temperature (K)

Fig.6. The new deconvolution performed for Li2B4O7:Cu,In.


~ Material I E (eV)Li2B4O7:Cu 1.61+0.03

Li2B4O7:Cu, In | 1.62 + 0.02

Table 3. New GCD data.

Table 3 shows that the new data are now in a very good agreement with thedata obtained by IR and VHR methods.

The discrepancies observed before are now disappeared and it is possible totrust in the second deconvolution performed taking into account a more complexglow peak structure.

References1. Kitis G., Furetta C , Prokic M. and Prokic V., J. Phys. D: AppLPhys. 33

(2000) 12522. McKeever S.W.S., Phys. Stat. Sol. (a) 62 (1980) 331

Kinetics order: effects on the glow-curve shape

The practical effect of the order of kinetics on the glow-peak shape isillustrated in Fig.7, in which two glow-curves from a single type of trap arecompared.

In the case of second order kinetics TM increases by the order of 1% withrespect to the temperature at the maximum of a first order peak. The main differenceis that the light is produced at temperatures above TM because the trapping delays therelease of the electrons. Furthermore, for a fixed value of E, TM increases as /?increases or s' decrease; for a fixed value of fl, TM results to be directly proportionaltoE.

CHAPTER K 195

I 7 \V

TEMPERATURE

Fig.7. Glow-peak shapes for a first order (I)and a second order (II).The largest differenceis related to the descending part of the curve.

LLinearization factor, Flin (general requirements for linearity)

Let us define, at first, the yield or efficiency of the thermoluminescentemission, TJ, from a material having a mass m, as the ratio between the energy, s,released as light from the material itself, and the mass m multiplied by the absorbeddose D [1]:

em-D

In the range where the efficiency rj is constant, there is a linear relationshipbetween the TL signal, M, and the absorbed dose, D:

M = k-D (2)

where k is a constant.It is important in any thermoluminescent dosimetric application to have, if

it is possible, a linear relationship between the TL emission and the absorbed dose.The linearity zone, if exists, is more or less depending on the material as well as onthe reader.

A typical first-order relationship can be written as [2]

y = ax + b (3)

The linearity range, as already mentioned, depends on the particularthermoluminescent material. The plot of Eq.(3) is a straight line with slope "a" andintercept "b" on the Y-axis.

The physical meaning of the x and y variables, when using Eq.(3) todescribe the TL yield as a function of dose, are:- the independent variable x represents the absorbed dose D received by the TLdosimeter,- the depending variable y is the TL light emitted by the dosimeters irradiated at thedose D; it is expressed in reader units,- the slope "a" identifies itself with the absolute sensitivity of the dosimeter(expressed in terms of reader units per dose), or, with the inverse of the calibrationfactor Fc (expressed in terms of dose per reader units),

(1)


- the intercept on the Y-axis, "b", is the TL reading due to the intrinsic backgroundfor the same dosimeter just annealed and not irradiated.

Equation (3) can then be rewritten according to the symbols usedpreviously

M = — D + M o (4)Fc

where M is the TL signal at a given dose and Mo is the intrinsic background of thedosimeter.

Equation (4) is strictly valid only for a material having a relative intrinsicsensitivity factor (individual correction factor) equal to 1; if this is not true, the TLreading must be corrected consequently. In the following discussion the case of Sj#lis omitted to avoid a heavy formalism.

Considering the net TL response, Eq.(4) becomes

M-Mo=—D (5)

In this form Eq.(5) can be better defined as a proportionality relationshipbetween the TL emission and the dose.

Figure 1 shows the plots of both Eqs. (4) and (5), where

1tana = — (6)

tc

For practical reasons, the data concerning the TL emission vs. dose arenormally plotted on a log-log paper. In this way Eq.(5) becomes

log( M - Mo ) = log D + logl — J (7)

which is still the equation of a straight line having now a slope equal to one.Figure 2 shows, schematically, the behavior of the TL vs. dose for three

different materials. The dotted line represents the proportionality as indicated byEq.(6).

An unfortunate use of terminology has crept into the literature onthermoluminescence dosimetry which may easily mislead the uninitiated.

CHAPTER L 199

70 - sS

30 - yS S^

yS s^ tan a^l/F

10 - yS

U—l 1 1 1 1 1 ' 1 I I ' I2 4 6 8 10

Dase(D)

Fig.l. Plots of Eq.(4) and Eq.(5).

to* -Z^^—

I 04 ^ ^

% ,ff» I _ = v >^ J£

, , 4 ^ J_I X X T X 1 I I . I I I

icr' iff2 iff1 id* lo1 ic? lrfDosc(Gy)

Fig.2. TL response as a function of dose for threedifferent types of TLDs.

Calibration data for various dosimeter materials are usually presented, asalready stated, as a plot of the logarithm of thermoluminescence response vs. the


logarithm of the absorbed dose. It must be stressed that a straight line on full logpaper implies linearity only in the special case when it makes an angle of 45° withthe logarithm axes. Other straight lines imply some power relationship between thevariables. Then remember that a straight line on full log paper is not necessarylinear.


Edietd by M.Oberhofer and A.Scharmann, Adam Hilger Publ. (1981)2. Scarpa G. in Corso sulla Termoluminescenza Applicata alia Dosimetria,

15-17 February 1994, Rome University La Sapienza (I)

Linearity (procedure)

~ prepare a group of N dosimeters. For each detector one must know theindividual background and the intrinsic sensitivity factor.

"" the N dosimeters are divided into n subgroups (« = 1, 2, ..., i), eachsubgroup corresponding to a dose level. Each subgroup has a number m ofdetectors (m = 1, 2 , . . . , j> 5).

™ each subgroup is then irradiated using a calibration source possibly of thesame quality of the radiation used for the applications so that no correctionfactor for energy will be necessary.

~ the range of doses delivered to the dosimeters has to be chosen according tothe needs. In any case it is good to give increasing doses following alogarithm scale (i.e., ...0.1, 1, 10, 100 Gy,...).

" read all the dosimeters in only one session.

" correct the readings by background and sensitivity factors.

" for each subgroup, calculate the average value

M,=£— (1)7=1 m

where M\ stands for the average value of the ith subgroup and Mj standsfor the reading of the y'th dosimeter already corrected by background andsensitivity factor.

CHAPTER L 201

~ plot on a log-log paper the Mi values as a function of the doses.

~ test the linear behavior using a statistical methods.

The following Table 1 lists the data obtained after irradiation of CaF :Dy (TLD-200) samples to Co gamma rays at various doses in the range from 25 to 300 \xGy.Each experimental point is the average of the readings of five TLDs. The data arecorrected by subtraction of the individual background and by the intrinsic sensitivityfactor. For simplicity, Table 1 reports only the average values and the correspondingstandard deviations. The plot is shown in Fig.3.

Dose Average reading <j%foGy) (aAL) (a.u.)

25 0.340 2.450 0.644 2.175 0.980 2.2100 1327 1.8125 L605 1.6150 1.977 2.0200 2.695 1.5250 3.215 1.3

~ 300 I 3.972 1 1.1

Table 1. Example of TL responsevs. dose


10 j - — — — — 1

y * 0.0132X - 0.0088R» = 0.999

0 1 I ' ' • • • • — • • I . • • • • • • • • I

10 100 1000

OooOiOy)

Fig.3. Linearity plot for TLD-200.

ReferenceFuretta C. and Weng P.S., Operational Thermoluminescence Dosimetry,World Scientific, 1998

Linearity test (procedure)

The aim of this test is to verify if a TL system is linear as a function of thedoses used or, in other words if, within the reproducibility characteristics of thesystem, the net reading is proportional to the given dose.

~ select a random group of 10 TLDs from the batch,

~ anneal the TLD samples according to the appropriate standard annealing,

~ irradiated the samples at a test dose of 0.1 mGy,

~ read out the irradiated samples,

~ second read out for background determination,

~ repeat points 2 to 5 for different doses, i.e. 1 mGy, 10 mGy and 100 mGy,

CHAPTER L 203

" create the following table, use the net readings and correct them according tothe intrinsic sensitivity factors

T L D readings Si corrected readingsN.o Dose(Dc) m G y Dose(Dc) m G y

0.12 I 1 10 I 100 0.12 I 1 10 I 1001 242341 1943571 20227100 196833833 1.24 192211 1564170 16308952 1587337362 204128 1654501 16578730 164846367 1.02 196203 1618138 16253657 1616101643 208551 1584265 16833900 164535400 1.03 198593 1534238 16343592 1597392234 231101 1861617 18020780 183750000 1.11 204596 1673529 16234937 1655369375 226310 1888630 19638860 188766300 1.20 185258 1570525 16365717 1573019176 184280 1436054 15034580 147125067 0.92 195957 1556580 16341935 1599142037 155635 1278430 12628450 126211967 0.78 194404 1633885 16190321 1618050868 159326 1313236 13000080 129595100 0.80 194158 1636545 16250100 1619888759 160594 1234075 12814190 126674767 0.79 198220 1557057 16220494 16034274310 229475 | 1926944 [ 18099080 1 185755033 | 1.12 201317 1716914 16159893 165849137

Average 196092 1606158 16266960 161282202q 5265 59363 69915 2734799

m/Dc 1634097 1606158 1626696 1612822<E 1619943 1619943 1619943 1619943DCT 0.121 0.991 10.042 99.560oCT 0.003 0.037 0.043 1.688

„,?"* 1.028 1.018 1.007 1.008U. ICeJUc ^ ^ ^ ^ ^ ^ _ _ _ _ _ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^

. , ? " " 0.989 0.965 1.001 0.984u.ygey/Uc ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^

Where;

™ Dc is the given dose

~ Si is the relative intrinsic sensitivity factor, defined as

Si~ M

~ m is the average of the corrected readings over the ten dosimeters at eachdose

~ a is the corresponding standard deviation

1 4

- <t> = — ^ —— is the slope of the best fit straight line crossing the origin

of the axis in the plot of TL emission vs dose

mi

~ Dey is the evaluated dose = —0>

a~ <rev is the standard deviation of the evaluated dose —

o

The acceptability limit at each level of dose is given by

09<fo,±0-7O<n

where the coefficient 0.7 is given by the ratio

t

n being the number of measurements at each level of dose (10 in the example) and tis the t-Student value (equal to 2.26 for 10 measurements and a confidence level of95%).

ReferenceScarpa G. in Corso sulla Termoluminescenza Applicata alia Dosimetria.Rome University "La Sapienza", Rome (Italy)

Lithium borate (Li2B4O7)

The effective atomic number 7.3 makes Li2B4O7 a tissue equivalent materialand encourages studies on its TL properties for radiation dosimetry in general andpersonnel monitoring in particular. This is the reason why repeated investigationswere devoted in the last two decades to identify effective TL activators and tooptimize the method of preparation of lithium borate [1-6] .

Two different methods can be adopted to prepare L12B4O7 phosphor.

~ melting method: lithium carbonate Li2CO3 and boric acid H3BO3 are mixedin the stoichiometric ratio and sufficient aqueous solution of the desired

CHAPTER L 205

dopant (Mn, Fe, Co, Mo, Ag, Cu, in form of chlorides, nitrates, or oxides) isadded, to obtain an impurity content ranging from 0.03 to 0.5 wt%. Afterstirring and desiccation, the mixture is melted at 950°C in a silica orplatinum crucible, then rapidly cooled to room temperature. The resultantglassy mass is reheated at 650°C for 0.5 h, which assures a completecrystallization, and then ground and sieved to obtain a 100 to 200 meshcrystalline powder.

~ sintering method: an acetone or alcohol solution of the activator is added toraw IJ2B4O7 powder, and the mixture is stirred and dried. The resultingpowder is heated in air in a platinum container for 1 h at 910°C and thencooled to room temperature.

Good TL performances are reported for L12B4O7 doped with Cu, Ag, Mnand Cu+In impurities.

All these phosphors exhibit two glow peaks, the first one occurring at 110 to120 °C (very low temperature for dosimetric purposes), and the second one in therange between 185 and 230°C, depending on the activator.

Linearity of the TL response in Li2B4C>7:Cu and Li2B4O7:Cu,In is observedfrom 210"4 up to 103 Gy. The energy dependence of TL output in Li2B4O7 : Cu andLi2B4O7:Cu,In for photons is almost flat from 30 keV to Co60 energy.

Fading is very fast for the low temperature peak, but the dosimetric peakfades less than 10% after 3 months [7].

References1. Schulman J.H., Kirk R.D., and West E.J., Proc. 1st Int. Conf. Lumin. Dos.,

Stanford (USA), 19672. Moreno y Moreno A., Archundia C. and Salsberg L., Proc. 3rd Int. Conf.

Lumin. Dos., Riso (Denmark), 19713. Botter-Jensen L. and Christensen P., Acta Radiol., Suppl. 313 (1972) 2474. Takenaga M., Yamamoto O. and Yamashita T., Proc. 5th Int. Conf. Lumin.

Dos., San Paulo (Brazil), 19775. Takenaga M., Yamamoto O. and Yamashita T., Nucl. Instr. Meth. 175

(1980) 776. Takenaga M., Yamamoto O. and Yamashita T., Health Phys. 44 (1983) 3877. Furetta C , Prokic M., Salamon R., Prokic V. and Kitis G., Nucl. Instr.

Meth. A4S6 (2001) 411


Lithium fluoride family (LiF)

Lithium fluoride is among the most widely used TL phosphors indosimetric applications, because it provides a good compromise between the desireddosimetric properties. Its effective atomic number (8.14) is sufficiently close to thatof the biological tissue (7.4) so as to provide a response which varies only slightlywith photon energy. Thus it can be considered as tissue equivalent.LiF:Me.Ti

This phosphor is produced commercially by the Harshaw Chemical Co.,USA. LiF:Mg,Ti dosimeters are known as TLD-100, TLD-600, and TLD-700,depending on their preparation from natural lithium or lithium enriched with 6Li or7Li, respectively: 6Li 95.6% and 7Li 4.4% for TLD-600,6Li 0.01% and 7Li 99.99%for TLD-700. Harshaw patent [1] describes two preparation methods for LiF:Mg,TiTL phosphor powders: the solidification method and the single crystal method.

" in the solidification method, lithium fluoride (106 parts by weight),magnesium fluoride (400 parts by weight), lithium cryolite (200 parts byweight), and lithium titanium fluoride (55 parts by weight) are mixed in agraphite crucible. The mixture is homogeneously fused in vacuum and theproduct slowly cooled, then crushed and sieved between 60 and 200 \xn.

~ in the single crystal method, the above mixture is placed in a vacuum orinert-atmosphere oven to grow a single crystal by the Czochralski method ata temperature sufficiently high to obtain a homogeneous fusion mixture. Themixture is then slowly moved to a lower temperature zone to allowprogressive solidification (about 15 mm/h). Once the material is cooled, it iscrushed and sieved between 60 and 200 um.

In both cases the resulting TL phosphor powder is annealed at 400 °C duringsome hours and then at 80 °C during 48 h.

"" the same patent also describes the preparation of extruded LiF dosimeters.To obtain them, the LiF powder mixture is placed in a neutral atmosphereand pressed at 3.5 • 108 Pa at a temperature of 700 °C, pushing the mixturewith a piston through a hole which acts as a die. The bar obtained is cut intosections to prepare pellets of uniform thickness and finally the faces of thepellets are polished. The extruded dosimeters have identical TLcharacteristics as the TL phosphor powder.

™ another method [2] describes how to prepare sodium stabilized LiFdosimeters. In this method, 200 ppm of magnesium fluoride and 2 wt% ofsodium fluoride are added to the LiF powder. The powder mixture ishomogenized, put in an aluminum oxide crucible, and held at the

CHAPTER L 207

crystallization temperature for about 3 h in a nitrogen flow oven. Then, thetemperature is reduced to 60 °C in 45 min and the sample taken out of theoven to be cooled quickly. The product is finely pulverized and the treatmentrepeated. Finally the product is repulverized and sieved between 60 and 200lira. In order to favor the creation of traps, the product is annealed in anordinary oven at 500 °C over 72 h. The crystals are quenched by pouringthem on a cold metal plate. To make pellets, the TL powder is finely sieved,compressed at about 5 • 108 Pa in the desired form, and submitted to athermal treatment in a nitrogen oven at a temperature sightly lower than thatof fusion. Before using, the pellets must be annealed at 500 °C.

Other methods have been developed to prepare LiF:Mg,Ti phosphor powder,LiF:Mg,Ti + PTFE (polytetrafluoroethylene) and LiF sintered pellets [3].

~ the preparation of LiF : Mg, Ti phosphor powder is the following. A few mlof a solution 0.1 M of MgCl2 are added to 40 ml of a LiCl solution (0.9g/ml). Meanwhile, metallic titanium is dissolved in 50 ml of hydrofluoricacid (HF, 48 to 50%), then the first mixture is slowly added. Once LiF isprecipitated, the sample is centrifuged and washed repeatedly. Theprecipitate is dried in a Pt crucible at a temperature of 30 °C for 1 h. Thenthe material is cooled to room temperature adding a few ml of LiCl solution.This wet material is dried at 100 °C for 1 h, placed in a Pt crucible, and thenin an oven with nitrogen atmosphere at 300 °C for 15 min. After that thetemperature is raised up to 640 °C and kept constant for 1 h. The sample isslowly moved to a lower temperature zone (400 °C) to allow crystallization,and then taken out of the oven to be rapidly cooled to room temperature.Finally, the product is crushed and sieved to select powder with grain sizesbetween 80 and 200 | m.

~ To obtain LiF : Mg, Ti + PTFE pellets, a mixture 2:1 of the phosphorpowder and PTFE resin powder is placed in a stainless steel die to bepressed, at room temperature, at about 1 GPa. Pellets thus obtained (5 mmdiameter and 0.7 mm thickness), weighing approximately 30 mg, arethermally treated for a period longer than 5 h in a nitrogen oven at atemperature sightly lower than that of PTFE fusion.

~ Sintered LiF : Mg, Ti pellets are obtained by pressing the TL powder into astainless steel die at about 10 GPa. These compressed pellets undergo athermal treatment in a nitrogen oven at a temperature slightly lower than thatof LiF fusion to be sintered.


The TL glow curve of LiF:Mg,Ti, shows at least six peaks; it is quitecomplicated because of its complex trap dynamics. The main peak (indicated aspeak 5) normally used for dosimetric purposes, and then called the dosimetric peak,appears at a temperature of about 225 °C corresponding to a very stable trap level.The low temperature peaks 1, 2, and 3 are relatively unstable and must besuppressed by a thermal treatment.

The linearity is maintained from 100 mGy up to about 6 Gy, beyond whichsuperlinearity appears.

LiF containing 6Li is sensitive to thermal neutrons. Peak 5 shows a responsewhich deceases with increasing LET of ionizing particles (protons, a-particles, etc.).Peak 6 is particularly sensitive to a-particles. This difference in behavior is useful tomeasure thermal neutrons in a mixed radiation field.LiF.Me.Cu.P

LiF : Mg, Cu, P has been developed as a phosphor of low effective atomicnumber which exhibits a simple glow curve, low fading rate, and high sensitivity.

The preparations of this phosphor are the following:

~ LiF of special grade in the market, used as starting material, is mixed inwater with activators, CuF2 (0.05 mol%) and MgCl2 (0.2 mol%), and addedwith ammonium phosphate. The wet mixture is heated in a Pt crucible at1050 °C for 30 min in nitrogen gas after being dried at about 80 °C for 4 h.The melted LiF material is rapidly cooled to 400 °C during 30 min and thepolycrystalline mass is powdered and sieved. Powder of size between 80and 150 mesh is used as LiF: Mg, Cu, P TL phosphor [4].

~ another method [5] consists of obtaining first undoped LiF from thereaction LiCl + HF = LiF + HC1. Once LiF was precipitated, activatorsMgCI2, (NH4)2HPO4, and CuF2 in aqueous solutions are incorporated untilthe required concentrations are reached. The material obtained in this way isdried (70 to 80 °C for 4 h) and washed repeatedly. This dried material,placed in a Pt crucible, is oven heated in nitrogen atmosphere at 400 °Cduring 15 min. After that the temperature is raised to 1150 °C and keptconstant for 15 min, then lowered to 400 °C, and subsequently suddenly toroom temperature. The resulting polycrystalline material is crushed andsieved selecting powder with grain sizes between 100 and 300 nm. Thefinal product is the TL phosphor powder.

~ pellets of LiF: Mg, Cu, P + PTFE are obtained in the same way as those ofLiF: Mg,Ti + PTFE.

LiF: Mg, Cu, P obtained following the first reported preparation [4] showslinearity in the dose range between 5 • 10"5 and 10 Gy, beyond which the responsebecomes sublinear, a property quite different from superlinearity. The phosphor

CHAPTER L 209

prepared following the second suggested procedure [5] gives linear responsebetween 10"4 and 102 Gy.LiF:Cu2+

The growth of single crystals is carried out by Kyropoulos method fromMerck 99.6% powder. Doping with Cu2+ ions is obtained by adding to the meltvarious amounts of CuF2 according to the required dopant concentrations. The glowcurve of LiF:Cu2+ shows a very preminent and intense peak at 155°C (H.R.=3°C/s)and a minor peak at about 205°C overlapped, at high doses, by a third peak ataround 230°C [6,7].

References1. Patent Harshaw Chemical Co., USA2. Portal G., Rep. CEA-R-4943 (1978)3. Azorin J., Gutierrez A. and Gonzalez P., Tech. Rep. IA-89-07 ININ

(Mexico) (1989)4. Nakajima T., Morayama Y., Matsuzawa T. and Koyano A., Nucl. Instr.

Meth. 157(1978)1555. Azorin J., Tech. Rep. IA-89-08 ININ. Mexico (1989)6. Furetta C , Mendozzi V., Sanipoli C , Scacco A., Leroy C , Marullo F. and

Roy P., J. Phys. D: Appl. Phys. 28 (1995) 14887. Scacco A., Furetta C , Sanipoli C. and Vistoso G.F., Nucl. Instr. Meth.

B116(1996)545

Localized energy levels

Trapping levels within the material's forbidden energy gap.

Lower detection limit \Dldl)

The lower detection limit, DM, is defined as three times the standard

deviation of the zero dose reading:

Luminescence (general)

Luminescence [1-3] is the energy emitted by a material as light, afterabsorption of the energy from an exciting source which provokes the rise of an


electron from its ground energy level to another corresponding to a larger energy(excited level). The light emitted, when the electron comes back to its ground energylevel, can be classified according to a characteristic time, T , between the absorptionof the exciting energy and the emission of light.

If this time is less than 10~ sec, the luminescence is called fluorescence.The light is emitted with a wavelength larger than the wavelength of the absorbedlight owing to dispersion of energy by the molecule. If the time between absorptionand emission is larger than lO^sec, the luminescence is then calledphosphorescence. The process of phosphorescence is explained with the presence ofa metastable level, between the fundamental and the excited levels, which acts as atrap for the electron.

If the transition arrives at a temperature T and the energy difference E,between the excited and the metastable levels, is much larger than kT , the electronhas a high probability to remain trapped for a very long time.

Assuming a Maxwellian distribution of the energy, the probability ofescaping by the trap is given by

As a consequence, the period of time between the excitation and thetransition back to the ground state is delayed for the time the electron spends in themetastable state.

In the previous equation, the probability p is a function of the stimulation

method, which can be thermal or optical and will assume a different form accordingto the type of stimulation.

References1. McKeever S. W.S., Thermoluminescence of Solids, Cambridge University

Press (1985)2. Chen R. and McKeever S.W.S., Theory of Thermoluminescence and

Related Phenomena, World Scientific (1997)3. Chen R. and Kirsh Y., Analysis of Thermally Stimulated Processes,

Pergamon Press (1981)

Luminescence (thermal stimulation)

Let us define N to be the concentration of the metastable states occupied byelectrons. The intensity of luminescence, / , is proportional to the decrease, as a

CHAPTER L 211

function of time, of the concentration of the metastable states in the system (i.e. thecrystal structure) [1,2]:

dNthe quantity — can also be expressed as

dt

d N M— = -p-Ndt

In the case of thermal stimulation, the probability p is expressed as follows

( F]p = v K expF \ kT)

whereV is the vibrational frequency of phonons within the crystal structure,K is the transition probability,F is the free energy of Helmholtz andk is the Boltzmann's constant.Because the free energy can be expressed as

F = E-TAS

where AS is the entropy change during the transition and E is the thermal energyimparted to the electron, the probability can then be expressed as

(AS) ( E)p -v K • exp — • exp

\k) \ kT)

from which

(AS)S = V-K- exp —

\k )

where S is called frequency factor (sec1); it is also called "attempt-to-escapefrequency".


Because in this description N = n , where n is the concentration of thetrapped electrons, the detrapping rate is given by

dn— = —p-ndt

and then, the intensity of luminescence is

dnI = -c = c • p-n

dt F

with c a constant.

References1. Bube R.H., Photoconductivity of Solids, Wiley & Sons, N.Y. (1960)2. Mahesh K., Weng P.S. and Furetta C , Thermoluminescence in Solids and

Its Application, Nuclear Technology Publishing, England (U.K.) (1989)

Luminescence centers

The luminescent centres are atoms or group of atoms, called activators,positioned in the lattice of the host material and serve as discrete centers forlocalised absorption of excitation energy. In other words, a luminescent center is aquantum state in the band gap of an insulator which acts as a center ofrecombination of charge carriers when it captures a carrier and holds it for a periodof time until another carrier of opposite sign is also trapped and both combine. Therecombination causes the release of the energy in excess as photons or phonons.

Luminescence dosimetric techniques

The main luminescence dosimetric techniques are:(i) radio-thermoluminescence (RTL) or thermoluminescence (TL) which

consists in a transient emission of light from an irradiated solid whenheated;

(ii) radio-photoluminescence (RPL) which consists of the emission of lightfrom an irradiated solid by excitation with ultra-violet light;

(iii) radio-lyoluminescence (RLL) which consists of a transient emission oflight from an irradiated solid upon dissolving it in water or some othersolvent

CHAPTER L 213

Luminescence dosimetry

Luminescence dosimetry is an important part of solid state dosimetry andincorporates processes whereby energy absorbed from ionizing radiation is laterreleased as light.

Luminescence efficiency

The luminescence efficiency of a material, 77, is defined as the ratio of thetotal energy emitted (hv) in the form of light to the energy observed (hv0) by thematerial during the process of excitation:

hv Xn = -— = — (1)

hv0 Ao

The emission of luminescence following irradiation and the absorption ofenergy, depend on the relative probabilities of the radiative and non-radiativetransitions. Eq.(l) can be then expressed in another form:

radiative events Pr

77 = = — (2)total events Pr - Pm

where Pr is the probability of luminescent transitions and Pnr the probability of non-radiative transitions.

Experiments have shown that 7] is strongly temperature dependent: theefficiency remains quite constant up to a critical temperature beyond which itdecreases rapidly.

Equation (2) can also be written as follows:

because the radiative probability Pr is not affected by temperature, while the non-radiative probability Pm depends on temperature through the Boltzmann factor.

In the above Eq.(3), the quantity AE represents the thermal energyabsorbed by an electron, which is in an excited state at the minimum of energy, forrising to a higher excited state. From this higher state the electron can transfer to theground state without emission of radiation. The decrease of luminescence efficiency

(3)


as the temperature increases (thermal quencing) has been explained introducing theso called killer centers.

Luminescence phenomena

Luminescence is the emission of light from certain solids called phosphors.This emission, which does not include black body radiation, is the release

of energy stored within the solid through certain types of prior excitation of theelectronic system of the solid. This ability to store is important in luminescencedosimetry and is generally associated with the presence of activators.

The following table lists the luminescence phenomena and the methods ofexcitation.

LUMINESCENCE PHENOMENA I METHODS OF EXCITATIONBioluminescence Biochemical reactionsCathodoluminescence Electron beamChemiluminescence Chemical reactionsElectroluminescence Electric fieldPhotoluminescence U.V. and infrared lightPiezoluminescence Pressure (10 tons m ' )Triboluminescence FrictionRadioluminescence Ionising radiationSonoluminescence Sound wavesFluorescence Ionizing radiation, U.V. and visiblePhosphorescence lightThermoluminescenceLyoluminescence

In particular, when some of the radiation energy is absorbed by a material,it can be re-emitted as light having a longer wavelength, according to the Stoke'slaw. Furthermore, the wavelength of the emitted light is characteristic of thematerial.

MMagnesium borate (MgO x nB2O3)

This phosphor is a near tissue equivalent material with an effective atomicnumber for photoelectron absorption equal to 8.4.

The preparation of polycrystalline magnesium borate activated bydysprosium has been reported at first in 1974 [1]. A certain quantity of magnesiumcarbonate MgCO3, boric acid H3BO3, and dysprosium nitrate Dy(NO3)3' is placed ina quartz cup and dried at a temperature ranging between 80 and 100 °C. After thatthe material is annealed in a furnace, then cooled, ground, and screened.

The most sensitive material is obtained at the proportion of boric anhydrideand magnesium oxide 2.2 to 2.4 and at the dysprosium concentration of about 1 mg-atom per g-mol of the base.

The glow curve of such a material shows a single peak located in the regionfrom 190 to 200 °C.

The sensitivity is reported to be 10 to 20 times larger than that of LiF. Theenergy response at 40 keV is about 30% larger than that of LiF. The TL response Vsdose is linear from 10"5 to 10 Gy. Fading at room temperature is about 25% over aperiod of 40 days.

A development of the preparation method of magnesium borate activatedby Dy and Tm and other unknown impurities added as co-activators, was presentedin 1980 [2]. The sensitivity has been reported to be about seven times greater thanthat of LiF; other investigators reported a factor of four [3]. The glow curve ofMgB4O7:Dy is composed by a single peak ; the TL response is linear from 10'5 to102 Gy.

Further investigations [3,4] reported high variability of the TL featureswithin a batch as well as among different batches. This suggested the necessity ofimproving the material preparation in order to use such a phosphor widely inpersonnel and environmental dosimetry without problems of individual detectorcalibration.

A new production of MgB.407 , activated by Dy + Na shows very goodperformances: reproducibility within 2% from 1 mGy to 0.25 Gy and a linear rangefrom6-10"8Gyto40Gy[5].

References1. Kazanskaya V.A., Kuzmin V.V., Minaeva E.E. and Sokolov A.D., Proc. 4th

Int. Conf. Lumin. Dos., Krakow (Poland), 19742. Prokic M, Nucl. Instr. Meth. 175 (1980) 83


3. Barbina V., Contento G., Furetta C , Malisan M. And Padovani R., Rad.Eff. Letters 67 (1981) 55

4. Driscoll C.M.H., Mundy S.J. and Elliot J.M., Rad. Prot. Dos. 1 (1981) 1355. Furetta C , Prokic M., Salamon R. and Kitis G., Nucl. Instr. Meth. B160

(2000) 65

Magnesium fluoride (MgF2)

A mixture of MgF2 and individual dopant as Mn, Tb, Tm or Dy is heated at1200°C during 1 hr in a nitrogen atmosphere. The molten mass is then cooled toroom temperature. The atomic number of the obtained phosphor is about 10. Theglow curves of both pure or doped phosphors show 10 peaks from room temperatureand 400°C. The dopants enhance the thermoluminescence emission. The highestsensitive phosphor is obtained with Mn. The TL response is linear up to about 40 R[1-3].

References1. Paun J.( Jipa S. and Hie S., Radiochem. Radioanal. Lett. 40 (1979) 1692. Braunlich P., Hanle W. and Scharmann A.Z., Z. Naturf. 16a (1961) 8693. Nagpal J.S., Kathuria V.K. and Bapat V.N., Int. J. Appl. Rad. Isot. 32

(1981) 147

Magnesium orthosilicate (Mg2SiO4)

Doping of Mg2Si04 with terbium impurities produces a TL dosimetryphosphor, showing highest sensitivity and moderate photon energy dependence,particularly useful for dosimetry in high temperature areas. TL properties of thissystem, whose effective atomic number is about 11, are reported since 1970 [1-3]and are strongly dependent on the preparation procedure. Magnesium oxide MgO,freshly prepared by decomposition at 600 °C of Mg(NO3)2 and silica gel are mixedin the molar ratio 2: 1 and added of Tb4O7 dopant. After thorough stirring in distilledwater, the mixture is dried in an oven and then melted in a silica crucible by directlyblowing a petroleum gas-oxygen flame (temperature of about 2750 °C) over it.

The weight of dosimeter samples is typically 5 mg of powder. Solid discsare also available.

The TL glow curve of Mg2Si04:Tb contains distinct peaks at 50, 90, 170,300, 420 °C (with an extra peak at 485 °C for exposures greater than 12 KR), but95% of the total intensity belongs to the 300 °C peak. The sensitivity of this materialis 50 to 80 times higher than that of LiF TLD-100, depending on the sample quality.The exposure response is linear in the range from about 20 mR to 400 R.

CHAPTER M 217

Annealing at 500 °C for 2 to 3 h is necessary for re-using the TL detector.Mg2Si04: Tb exhibits intense TL under irradiation with 254 nm UV light.

This sensitivity to biologically active UV light (typical of germicidal lamps) can bevery useful for UV dosimetry.

References1. Hashizume T., Kato Y., Nakajima T., Toryu T., Sakamato H., Kotera N.

and Eguchi S., Adv. Phys. Biol. Rad. Detec. IAEA, Vienna (1971)2. Jun J.S. and Becker K., Health Phys. 28 (1975) 4593. Bhasin B.D., Sasidharan R. and Sunta CM., Health Phys. 30 (1976) 139

May-Partridge model (general order kinetics)

When the conditions of first or second order kinetics are notsatisfied, one obtains the so-called general order kinetics which deals withintermediate cases. May-Partridge (1964) wrote an empirical expression for takinginto account experimental situations which indicated intermediate kineticsprocesses. They started with the assumption that the energy level of traps is single,as already assumed for the first and second orders.

Let's assume that the number n of charge carriers present in a single energylevel is proportional to nb. Then, the probability rate of escape is:

where s" is the pre-exponential factor.Equation (1) is the so-called general order kinetics relation, and usually b is

ranging in the interval between 1 and 2. The pre-exponential factor s" is nowexpressed in cm3(b"1)sec"1. It has to be stressed that the dimensions of s" change withthe order b. Furthermore, s"reduces to s' when b=2.

From Eq.(l) we can deduce the relation describing the TL emission.Rearranging Eq.(l) we have:

dn ( E \ ,

(1)

(2)


n1-" = nl-"\l + s"nb0-l(b-l)texP[-^j

H = J 1 + s(b -1)/ exp(- —J ' * (3)

in which

s = s"nb0-1 (4)

where s has units of sec"1.With this definition the difficulty with respect to the variation of

dimensions has been bypassed. Anyway, the frequency factor s is constant for agiven dose and would vary when the dose is varied.

The intensity I(t) is then given by:

I{t) = -dn=s"nbQJ-E^\W dt \ kT)

b

= sn0 expj - ~j\ + s(b- l)t exp( - ~ j j '"* (5)

Assuming a linear heating rate dT=fklt, we obtain from Eq.(2):

Derivation of the root from both members and using expression (4) yields

CHAPTER M 219

The intensity 1(7) is now given by

It must be observed that two factors contribute to 1(7):

" the exponential factor which constantly increases with T;

"" the factor included in brackets, decreasing as T increases.

So we have again the explanation of the bell shape of the glow-curve asexperimentally observed.

To conclude, Eq.(7) includes the second order case (b=2). Equation (7),which is not valid for the case b=\, reduces to the first order equation when b->\.

It must be stressed that Eq.(l) is entirely empirical, in the sense that noapproximation can be found which is able to derive Eq.(l) from the set ofdifferential equations governing the traffic of charge carriers and so, as aconsequence, a physical model leading to general order kinetics does not exist.


Mean and half-life of a trap

The half-life (f1/2), at a constant temperature, of a trap and, as aconsequence, of the corresponding peak in the glow curve, is defined as the time forthe number of trapped electrons to fall to half of its original value.

Starting from the first order kinetics equation

dn ( E\— = -nsexpdt \ kT)

from which

(6)

(7)


— = sexp \dt

I n \ kT)\and then

0.693

The temperature effect on the half-life is showing in Figs. 1 and 2. Figure 1shows the variation of the half-life as a function of the activation energy for givenvalues of the frequency factor. Figure 2 shows the same plot for given values of theactivation energy.

The mean life of the decay process expressed by the equation:

« = woexJ-j-/-exp(-—j

can be easily calculated substituting in the equation the n value with n^e and usingT instead of t. So, the mean life for the first order kinetics is then obtained as

From (1) and (2) result

tv=T\n2 (3)71

(1)

(2)

CHAPTER M 221

10000 j 1

1000' * * v

U. ^ V ^ V

1 0 E*1.15eV V . X .•-1611 MQ-\ » v V

i4 , 1 1 . 1 .—~xj270 2K 290 300 3 0

T(K)

Fig.l. Variation of the half-life, Eq.(l), as afunction of E for given values of s [1].

The mean life concept cannot be applied to a second or general orderkinetics because the isothermal decay is not exponential any more. Furthermore, asit can be seen in the following calculations, in the hypothetical expression of the halflife for any order different from the first one, the value of n^xs always present [2].

I O O O D T — — 1

1000 ^ - s .

^ % . E-1.2 «V

•g IOO ^ » >

\ . ••1E11WC-1

'270 2M 290 300 1*0

T(K)

Fig.2. Variation of the half-life, Eq.(l) as afunction of s for given values of E [1].

The half-life for the second order process could be calculated as follows

f°/2dn , ( E)?y1 ~ = ~s QM~~^ I dt*o n V KTJ *

its integration gives

1 Nfvr ( E \ - { E) (4)

There is a substantial difference between the half-life for a first order andthe one for a second order. Indeed, the half-life in the case of the first order kineticsis independent of the initial concentration of the trapped charges, which means to beindependent of the dose. In the case of the second order kinetics, the situation istotally different because the half-life is dose dependent (i.e., no): so, for an initialvalue of «0, hn will have a given value; after a time from the initial one, «0 changesto a value n'o (w'o < «o) and the same does ty2 {f\n > t\a). So that, as the period oftime from the initial irradiation increases, the same does the half life. The samehappens for a general order case.

For the general order, starting from the general order equation, one has

from which

'ir^SJ'-^Hi) (5)References

1. Furetta C. and Weng P.S., Operational Thermoluminescence Dosimetry,World Scientific (1998)

2. McKeever S.W.S., Thermoluminescence of Solids, Cambridge UniversityPress (1985)

CHAPTER M 223

Metastable state

The metastable state is a level within the forbidden gap. This level isassociated to a trapping level.

Method based on the temperature at the maximum (Randall-Wilkins)

The intensity / of a first order thermoluminescence peak is given by

At the beginning the intensity rises exponentially with temperature; theconcentration of trapped electrons reduces and the intensity, after reaching amaximum at a temperature TM, begins to fall and reaches to zero when the traps haveemptied.

Randall and Wilkins did not solve Eq.(l) but they considered that at themaximum temperature the probability of electron escaping from a trap is equal tounity. So, they wrote

*exp|-^[l+ /(*,/?)] L l (2)

from which

E = TM[l + f(s, /3)]-k-\n(s) (3)

and where f(s, b) is a function of the frequency factor and the heating rate. Assumingthe average time t, during which the charge carrier remains in the trap, to be thereciprocal of the electron escape probability and plotting ln(7) against TM one obtainsthe linear relation

HO = TM L J K ' P - Ms) (4)

(1)


where T is the temperature at which the material is left to decay byphosphorescence. The value in double brackets corresponds to the slope of thestraight line and -ln(j) to the intercept. They showed from Eq.(3), using the values ofBunger and Flechng for s and E in KC1:T1 phosphor, that the function / is smallcompared to unity when the heating rate is in the range from 0.5 to 2.5°C/s.

9 -1

Equation (3) becomes, using s = 2.9-10 s :

E = 25kTM (5)

The E value determined in this way is very inaccurate because the value of

s which changes from peak to peak and from a material to another.

ReferenceRandall J.T. and Wilkins M.H.F., Proc. R. Soc. London, Ser. A184 (1945)

366

Method based on the temperature at the maximum (Urbach)

9 -1

Urbach gave the following relation using s = 10 s :

TE = -M-

500

The numerical factors in this equation depend upon the s value and hencethe value of E is only approximated because s may be different for each trap in thesame substance as well as for different materials.

ReferenceUrbach F., Winer Ber. Ha, 139 (1930) 363

Methods for checking the linearity

For checking the linearity of the experimental data, some methods aresuggested in the following [1]:

CHAPTER M 225

Graphical method.

The points of co-ordinates (D ; , /n , ) are reported on a log-log paper, eachwith the respective error bar. An interpolation with a straight line having a slopeequal to one. The best interpolation is obtained using the confidence interval,

2I( nti), associated to each average /w,, with

(S.D.)i

1 hwith tn_l is the value of the Student-t distribution for «,-l degree of freedom at the

confidence level required (95%-99%).UNI. IEC and IAEA methods.

Both UNI and IEC technical recommendations suggest to convert theaverage values mj in evaluated kerma (Kv0 with the relative errors and comparethese values with the conventional real kerma (K^i). The maximum error betweenthese two values for each group must not be larger than ±10%:

(S.Z).),.K\d - ' n , - l /

0.90 < — < 1.10 (2)Kci

The IAEA method suggests to use three groups, here numbered 1, 2 and 3,of ten dosimeters each. All groups are processed as already mentioned in point a)and irradiated at the specified doses, Ds, of 1, 10 and 100 mGy for groups 1, 2 and 3respectively. The readings are then converted in evaluated doses (De). Ds and De arethen substituted in the following expressions:

-r-(groupl)0.95 < —s- < 1.05

-^ (group!)

(3)

-^-(groupl)0.95 < - ^ <1.25

jj-(group!)

(1)


Regression analysis.This method allows to adapt the experimental values obtained with the

various TLDs groups to a regression straight line crossing the origin of the axis.The starting point is to consider the equation of the type

y = a-xk

and its logarithmic transformation

log y = log a + k log x

This equation, using the previous symbols, becomes

log m = log(—) + k log D (4)

The straight line described in Eq.(4) has the property to have a slope kequal to 1 in case of proportionality between dose and TL emission. The slope canbe calculated using the method of the least squares:

k

1=1

where

S, = (S.D.)X

*, =log£>,

yt =logw,.

1 *

x=—y x,ft , = 1

1 *ft ,=1 (6)

(5)

CHAPTER M 227

The standard deviation of k is then given by

fss -is )2Y/2

where

One cannot reject the hypothesis of linearity and proportionality if

\\-k\<th_2Sk (9)

in other words if k is not significantly different from 1.Analysis of variance.

Among the methods here outlined this is the more complex because itneeds a numerical analysis not only for the h avarages but also for all theexperimental N data.

Let us call y the net and corrected readings for each jth dosimeter

belonging to the ith group irradiated at the dose Xf. Let us indicate now «, the

number of dosimeters belonging to the ith group. We calculate now the followingquantities:

A =!>,(?,-J,') do)

with

\ "• 1*1"'

«/ j=\ M 1=1 *,- 7=1

and

1=1 7=1

(7)

(8)

(11)

(12)

Then calculate the Fischer's index, F:

F = ^ . ^ ( .3,h-2 D2

Let us say F t, the value relative to [(h-2)(N-h)] degree of freedom at thedesired confidence level (95 or 99%); if we get

F<Ftab

the hypothesis of linearity is accepted.

ReferenceScarpa G. in Corso sulla Termoluminescenza Applicata alia Dosimetria, 15-17 February 1994, Rome University La Sapienza (I)

Model of non-ideal heat transfer in TL measurements

An interesting model for heat transfer from the heating element to thesample and from the sample to the surrounding, assuming that all heat transmissionis due to conduction (neglecting the convection from the sample to the surroundings)has been treated in [1,2] and this model is reported below.

The following assumptions are made:

~ heat homogeneous distribution inside both heating element and TLD sample(temperature gradients are present if fast heating rate is used)

™ surrounding temperature constant

~ heat capacities at the interfaces (contact layers between heating planchet-sample and sample-surrounding gas) are zero

~ heat capacities and thermal conductance of all the elements are temperatureindependent.

Let us indicate T\ and T2 the temperatures of the heating element and of thesample respectively, the rate transfer through the contact layer between the planchetand the sample is

CHAPTER M 229

d^ = Hc{Tl-T2) (1)

where Qc (in J) is the energy transferred from the planchet to the sample and Hc isthe thermal conductance of the contact layer (in J K"1 s"1).

The change of the sample temperature is then given by

dt c, \ dt dt J

where Qd (in J) is the energy transferred from the sample to the surroundings and cs

is the heat capacity of the sample (JK1).The rate of heat transfer from sample to gas is

^- = H,(T2-Tg) (3)

where Qd (in J) is the energy transferred from the sample to the gas, Hd is thethermal conductance of the sample-gas interface and Tg the gas temperature.

The quantities expressed by Eqs. (1) and (3) can be substituted in Eq.(2):

^.^.(r.-r.J-^-r.) (4," ' Cs Cs

Considering now a linear heating rate /? , T\ = To + fl • t, Eq.(4) becomes

dt cs ( 5 )

where To is the sample and planchet temperature at time t=0.The solution of Eq.(5) is

(2)


H,+Hd H,*Hd [ \ c, I

,(H1+H,,y\

where a is a coefficient depending on the initial condition T2 at time t=0.A simulation of Eq.(6) shown that after a transit period (less than 10

seconds in the simulation) the factor in the first square brackets approaches to unity,so that Eq.(6) can be approximated by

or

T2(t) = Tt + j3'-t (7)

Eq.(7) means that after a transitory period, the temperature profile of the sample isthe same as that of the planchet but with the heating rate /? replaced by an effective

heating rate /?' and the initial temperature To replaced by an effective initial

temperature TQ.

The temperature lag, AT, between the sample temperature and theplanchet temperature is then

References1. Piters T.M. and Bos A.J.J., J. Phys. D: Appl. Phys. 27 (1994) 17472. Piters T.M., A study of the mechanism of thermoluminescence in a

LiF:Mg,Ti dosimetry material (Thesis, 1998), D.U.T.

(6)

(8)

CHAPTER M 231

Multi-hit or multi-stage reaction models

These models of thermoluminescence are based on the assumption that atrap may be subject to a two or more stage reaction before its activation in thethermoluminescent process.

The multi-hit models were introduced to explain the supralinear growth ofthermoluminescence, i.e., the thermoluminescence intensity, / , increases as a

function of D1, where D is the absorbed dose and / is not necessary one or aninteger.

Halperin and Chen [1] found a relation of the type I x D3 for thesupralinearity of semiconducting diamond, concluding that the growth of TLintensity as a function of the dose was ruled by a three-stage reaction. This modelneeds two intermediate energy levels where the electrons are increased by twosuccessive doses of irradiation. A third irradiation finally rises the electrons to theCB from where they are trapped.

In the works of Larson and Katz [2], Katz [3], Waligorski and Katz [4] atwo-hit model was presented. In this model a trap is only produced after trappingfirst one and then a second electron. With this model Katz and colleagues were ableto explain the supralinearity of certain peaks in LiF.

A similar model has been used in the works of Takeuchi et al. [5].

References1. Halperin A. and Chen R., Phys. Rev. 148 (1966) 8392. Larsson L. and Katz R., Nucl. Instr. Meth. 138 (1976) 6313. Katz R., Nucl. Track Detect. 2 (1978) 14. Waligorski M.P.R. and Katz R., Nucl. Instr. Meth. 172 (1980)5. Takeuchi N., Inabe K., Kido H. and Yamashita J., J. Phys. C: Sol. St. Phys.

11(1978)L147

NNonlinearity

The plot of the TL signal vs. dose may present different zones. Ahypothetical curve is shown in Fig.l. As it can be seen from the figure, the TLemission is not linear in the low dose region and is not linear anymore at high doses.To bypass some linguistic ambiguities concerning the terms superlinearity andsupralinearity, two universal indices have been proposed by Chen and MacKeever[1] to mathematically describe all forms of nonlinearity.

The first of these indices is called "superlinearity index", g(D); it gives theindication of change in the slope of the dose response in all cases.

2 / \£ /

" SY LINEARy f RANGE '

DOSE

Fig. 1. The various zones which could be observedin a plot of TL as a function of dose.

The second one is the well known "supralinearity index", or dose responsefunction, f(D), used to quantify the size of the correction required for extrapolationof the linear dose region.

As already discussed by Chen and Bowman [2], the term superlinearity isreserved to indicate an increase of the derivative of the M = M(D) function, whereM indicates, as usual, the measured TL signal, both the peak height at the maximumor the peak area. Let us indicate by M' the first derivative of M at a point D and M"the second derivative. Then, if

M"(D) > 0 -> M'(D) increases in D -> M(D) increases and then is superlinear;ifM"(D) < 0 -> M'(D) decreases in D -> M(D) decreases and then is sublinear;ifM"(D) = 0 -» M'(D) is constant in D -» M(D) is linear.

To quantify the amount of superlinearity (or sublinearity) the authors haveproposed the function

s(D)=hmrr (1)called the "superlinearity index".

The following cases are possible:

~ g(D) > 1 indicates superlinearity

~ g(D) = 1 signifies linearity

~ g(D) < 1 means sublinearity

The second quantity, the f(D) index, concerns the supralinearity effect. Theauthors have suggested a slightly modified definition of the old dose responsefunction. The old expression was

M(D)

/ ( Z ) ) = ^ | ) (2>

A

where D; is the normalization dose in the linear region. The authors have proposedthe following modified expression

M(D)-M0

f(D)= D (3)M(D,)-M0

D,

where MQ is the intercept on the TL response axis.

CHAPTER N 235

The advantage of the new Eq.(3) lies in the possibility of applying it tocases in which the supralinear region precedes the linear region. In this case Mo isnegative but is still valid since it has no physical meaning.

M(D) values above the extrapolated linear region produce f(D) to be largerthan 1, and the supralinearity appears in the TL response. M(D) values below theextrapolated linear region cause f(D) < 1 and underlinearity occurs [3]. When f(D)approaches to zero, saturation occurs. Of course f(D) = 1 means linearity. As alreadystated, f(D) monitors the amount of deviation from linearity; that is the quantityneeds for extrapolation to the linear region.

The main problem in the use of the previous indices concerns g(D) becauseit is not a trivial problem to fit the experimental values of a TL response vs. dosewith an analytical expression. Nevertheless, from a practical point of view the f(D)function is enough to characterize the TL vs. dose behavior.

In the following some examples are given for a better understanding on theuse of the new indices.

Figure 2 depicts a situation where the TL response at high doses is belowthe extrapolated linear range; on the contrary, at low doses the TL response is abovethe linearity. The experimental data are given in the following Table 1. The values inbold correspond to the linearity region. The third column corresponds to the TL netresponse. The dose dependence curve can be analytically expressed by the equation

M = 8.4539D4 - 70.873D3 + 170.74D2 - 27.930D + 0.4909 (4)

" Dose (Gy) " TL (a.u.) TLnet (a.u.)0.000 13.932 0.0000.001 13.932 0.0000.100 13.932 0.0000.120 13.990 0.0580.250 17.182 3.2500.500 34.553 20.6210.750 62.008 48.0761.000 95.691 80.7591.500 160.513 146.5812.000 209.951 196.0192.500 234.355 220.4233.000 238.495 224.5633.500 | 238.154 1 224.222

Table 1. TL vs. dose. TLnet correspondsto the reading minus background.

The linear region is given by the equation

M= 131.38Z)- 50327 (5)

In both equations M is the net TL response. Some points of the curve cannow be considered.

4S0 p - — I

«0 >^

J 1!» * T

MO ^ / ^

SO jf

^ ^ ^ 05 1 18 2 28 S 3S

-SO ^

DOM(C*]

Fig.2. Plot of TL vs. dose showing under-response athigh doses and over-response at low doses.

D = 2GvOne obtains:M'(2) = 75.0788 > 0 which indicates an increase of M in D = 2.M" (2) = - 103.2088 < 0 which means that the M(D) function has the concavityfacing the bottom in D = 2 and that M1 is decreasing at the same point.Then the values of the g(D) and f(D) functions areg(2) =-1.7493 <1f(2) = 0.9390 < 1

The value of g(D) indicates sublinearity of the M(D) function in D = 2 andthe value of f(D) depicts a situation of underlinearity or, in other words, it meansthat saturation starts to appear.

For the low dose region one can consider the valueD = 0.250 GvIn this case one obtains:

CHAPTER N 237

M'(0.250) = 44.6797 > 1 which means that M is an increasing function in D = 0 250Gy.M" (0.250) = 241.5109 > 1: M has the concavity facing the top in D = 0.250 Gy and,furthermore, M' is increasing. Then g(D) and f(D) areg(0.250) = 2.3513 > 1f(0.250)= 1.6385 > 1.

The above two values indicate superlinearity and supralinearity in theregion preceding the linear part of the curve.

For a value of D = 1 Gy, i.e., a dose value situated in the linear range of thecurve, both g(D) and f(D) give approximately 1.

A further example is the one given in Fig.3. The plot has been obtainedusing the following equation [4]:

M=Msat(l-e-aD)-\3De-aD (6)

where Msal is the TL response at saturation level (=4844 a.u.) and a = 2.8910"3Gy"1.The data (calculated using the previous equation) are given in the following Table 2.

001 • y ^ ^

O.0O1 y i I I I I I I I I0.001 0.O1 0.1 1 10 100 1000 10000 100000

Fig.3. Plot of TL vs. dose according to Eq. (6).

The linear zone, numbers in bold in Table 2, is given by the followingequation


M = -1.0472D + 9.4260 10"5 (7)

Dose (Gy) TL (a.u.)0.001 0.00110.005 0.00520.010 0.01100.050 0.05200.100 0.10500.500 0.52801.000 1.06502.000 2.16405.000 5.670010.00 12.20025.00 36.49050.00 91.57075.00 162.46100.0 246.67250.0 923.88500.0 2183.0750.0 3186.01000 3862.02000 4751.05000 4844.07500 4844.010000 4844.050000 1 4844.0

Table 2. Data calculated from Eq.(6).

Some points of the plot can then be analyzed:

D = 50 Gv;M' > 0 -> M is increasingM" > 0 -> M' is increasing and the concavity is facing the topg > 1 -> M is superlinearf > 1 —» M is supralinear

CHAPTER N 239

D = 500 Gv:M' > 0 -> M is increasingM" < 0 -> M' is decreasing and the concavity is facing the bottomg < 1 -> M is sublinearf > 1 -> M is supralinear

concavityS" S' S g behaviour supralinear underlinear

of S

> 0 incr

> 0 incr ] > 1 superlinear f > 1 t -~-"""

> 0 incr f .> 0 incr I > 1 superlinear f > 1 ^ ^ ^ — - ^ " ^

< 0 deer ^ _ _ -•;•

>0 incr I <1 sublinear f < 1 ^~^"^

< 0 deer ^ ^> 0 incr i I ^ - —

< 0 deer ^^^~> 0 incr ] 0 incr \ <1 sublinear f< 1 ~ ^

>0 incr ^ ^

lmeanty ^ ^

> 0 incr i- .-• '•

> 0 incr 1 1 t>l "^^'

saturation starts

< 0 d e e r . . . ••• '

>0 incr I 1 f<. \ ^saturation starts

Table 3. Summary of the various configurations

D = 104 Gv:M' > 0 -> M is increasingM" < 0 -> M' is decreasing and M has the concavity facing the bottomg < 1 -» M is sublinearf < 1 —> M is underlinear and approaches saturation.

Table 3 gives a summary of the various configurations which can be foundin case of nonlinearity TL response.

References1. Chen R. and McKeever S.W.S., Rad. Meas. 23 (1994) 6672. Chen R. and Bowman S.G.E., European PACT J. 2 (1978) 2163. Furetta C. and Kitis G. (unpublished data)4. Inabe K. and Takeuchi N., Jap. J. Appl. Phys. 19 (1980) 1165

Non-ideal heat transfer in TL measurements (generality)

There are various types of heating a thermoluminescent sample during readout. The most popular is the contact way realized using a planchet heating.

Because the temperature control is usually achieved by means athermocouple mounted on the back of the planchet, this method gives only a controlof the planchet's temperature and not of the sample. The temperature lag betweenplanchet and sample, as well as the temperature gradient across the TLD, canstrongly influence the analysis of the glow curve, specially in the calculation of thekinetic parameters, where an accurate temperature determination is absolutelynecessary.

The problem of non-ideal heat transfer has been studied by various authorsand corrections have also been proposed [1-7]

References1. Taylor G.C. and Lilley E., J. Phys. D: Appl. Phys. 15 (1982) 20532. Gotlib V.I., Kantorovitch L.N., Grebenshicov V.L., Bichev V.R. and

Nemiro E.A., J. Phys. D: Appl. Phys. 17 (1984) 20973. Betts D.S., Couturier L., Khayrat A.H., Luff B.J. and Townsend P.D., J.

Phys. D: Appl. Phys. 26 (1993) 8434. Betts D.S. and Townsend P.D., J. Phys. D: Appl. Phys. 26 (1993) 8495. Piters T.M. and Bos A.J.J., J. Phys. D: Appl. Phys. 27 (1994) 17476. Facey R.A., Health Phys. 12 (1996) 7207. Kitis G. and Tuyn J.W.N., J. Phys. D: Appl. Phys. 31 (1998) 2065

CHAPTER N 241

Numerical curve fitting method (Mohan-Chen: first order)

Mohan and Chen suggested the following method for first-order TL curves.Haake has given an asymptotic series for evaluating the integral comparing

in the expression of 7(7) for the first order:

Using only the first two terms of expression (1), one has

(extf-f^rfexpe^-r.fexp^ (2)

Since the first term on the right hand side is very strongly increasingfunction of T, it is conventional to neglect the second term in comparison to the firstone. In this assumption the equation of the first order kinetics

HT)=V«P(~) «p[-f £«P(- £ H (3)becomes

In Eq.(4) the term sE/fik can be approximated by the following way:using B = sE/f&. and x = E/kT, Eq.(4) can be written as

I(T) = Cexp[- x - Bx~2 exp(-x)]

Making the logarithm of the previous expression one gets

In I(T) = In C + [- x - Bx~2 exp(-x)]

and then its derivative at the maximum, for T=TM, is

(1)

(4)


[ — ] = -1 + 2Bx~* exp(-x) + Bx~2 exp(-x) = 0\dTJT T

which givesx3 exp(x)

so that

sE \kTu) E_ , : _ e x p ( _ ) ( 5 )

The intensity is then given by

l ^ ^ i 1 N \ N •TM

Fig.4. Comparison between experimental and theoretical glow-peaks.experiment, E is too high," " " " " E is too small

(6)

CHAPTER N 243

Expression (6) leads to a convenient method of fitting because only oneparameter, E, is free.

The procedure is now as follows: an experimental glow-curve is measuredand an E value is estimated by using one of the experimental methods reported.Then a theoretical glow-curve is plotted using Eq.(6) and the constant is adjusted sothat the intensity at maximum (IM) of the experimental and theoretical curves

coincide. The fitting of the remaining curve is then checked. If the chosen value of Eis too small or too high the theoretical curve will lie above or below theexperimental curve (except for the maximum) as shown in Fig.4. In these cases anew value of E is chosen and the procedure is repeated until the desired fit isobtained.

ReferenceMohan N.S. and Chen R., J. Phys. D: Appl. Phys. 3 (1970) 243

Numerical curve fitting methods (Mohan-Chen: second order)

In the case of a second-order kinetics, the Garlick and Gibson equation isused:

|_ p *. \ JO")

From Eq.(l) the maximum intensity I(TM) is found; after that the intensity I(T[)corresponding to a certain number N of temperatures T{ is chosen and the normalisedintensity is obtained by dividing each I(T.) by I(T^ as follows

exp(- AJ*! + (SS\ f eXp(- — )dr\XT.) P kT/ I p )k FV kTJ

— = =— (2)

!(TM) t E f (s'n^ f, , E x T

(1)


Using the condition for the maximum

J3E I" s'n0 ?M { E \ 1 ( E \

and the integral approximation

f exp( )dT s T—exp( ) -To —- exp( ) (4)*l FV kTJ E FV kTJ ° E JcT0J K)

The procedure for the curve fitting is similar to the numerical curve fittingfor the first-order case. However, a better fit may be expected if only points belowthe maximum temperature are taken, since the main difference between first- andsecond-order peaks is in the region above the maximum.

ReferenceMohan N.S. and Chen R., J. Phys. D: Appl. Phys. 3 (1970) 243

Numerical curve fitting method (Shenker-Chen: general order)

The numerical curve fitting procedure for the case of general-order hasbeen carried out by Shenker and Chen.

The equation for the general-order case is the following

dn „ b ( E\- = -*n>exp^--J (1)

where s" is the pre-exponential factor, expressed in cm s~l and b is the orderof the kinetics, ranging from 1 to 2.

The solution of Eq.(l) is given by

(3)

(2)

CHAPTER N 245

where s = s"n0 , expressed in s .

Also in this case, since E/kT has values of 10 or more, the integral on theright-side of Eq.(2) can be resolved by using the asymptotic series. Equation (2) canbe normalized by dividing 1(T) by I(TjJ. The frequency factor s is found using thecondition at the maximum and then some points I(T!) have to be taken from theexperimental glow-curve and processed as for first and second cases (see numericalcurve fitting method for first- and second-order).

ReferenceShenker D. and Chen R., J. Phys. D: Appl. Phys. 4 (1971) 287

oOptical bleaching

Optical bleaching indicates the effect of light, of a specific wavelength, onirradiated TL samples, in the sense that charge carrier stimulation of a particulardefect center can be achieved via absorption of optical energy, resulting then in aphotodepopulation of the center. The charge carriers released may recombine withopposite sign carriers, emitting light during the illumination (bleaching light), ormay be retrapped in other trapping centers. Observing then the changes occurring inthe glow-curve resulting after the optical stimulation, relationships betweenthermoluminescence traps and optically activated centers can be obtained.

The term "beaching" is taken from the vocabulary of color centers: a crystalis colored by high dose of ionizing radiation and a subsequent illumination producesthe color fading, i.e., the sample is bleached.

Optical fading

The effect of light on an irradiated thermoluminescent sample consists of areduction of the TL signal, depending on the light intensity, its wavelength andduration of exposure.

For practical applications (personel, environmental and clinical dosimetry),the sensitivity to the light of different TL materials can be avoided by wrapping thedosimeters in light-tight envelopes. If this procedure is not applied, fading correctionfactors have to be determined carrying out experiments in dark and light conditions.

Oven (quality control)

The oven used for annealing should be able to keep predeterminedtemperature oscillations within well specified margins. However, it must be notedthat the reproducibility of the annealing procedure, concerning both heating up andcooling down processes, is much more important than the accuracy of thetemperature setting.

Temperature overshoots due to the high thermal capacity of the oven wallscan be minimized using ovens with circulating hot air. In this way the problemrelated to a non-ideal thermal conductivity of the annealing trays is also solved.


In some cases, when surface oxidation of chips is possible (i.e., in the caseof carbon loaded chips), it would be advantageous to operate the annealing underinert gas atmosphere. This facility could also reduce any possible contamination.

It would be better to use different annealing ovens depending on the variousneeds: one of them should be suitable for high temperature annealing, another onefor low temperature annealing and a third for any pre-readout thermal cycles.

As far as the trays where the TLDs are located for the annealing procedureare concerned, the following suggestions may be useful:

~ the tray should have between 50 to 100 recesses to accommodate thedosimeters,

~ each position in the tray should be identified,

~ the tray must be as thin as possible and with a flat bottom to get a very goodthermal contact,

~ the tray material can be ceramic (in particular porcelain), Pyrex and purealuminum. Ceramic is preferable for its chemical inertia and good thermalconductivity. Good results have also been obtained using Ni-Cu and anylight compound not oxidable,

~ it should be possible to insert in the tray a thin thermocouple to monitor theactual temperature of the tray as well as that of the dosimeters during theannealing cycle.

The quality control program of the annealing procedure should include thefollowing points:

~ determination of the heating rate of the oven from the switch-on time to thesteady condition,

"" determination of the temperature accuracy and setup of a correction factorwhich is needed,

~ check on the temperature stability,

~ check on the temperature distribution inside the oven chamber,

~ determination of the heating rate of the tray.

A quality control program concerning the ovens has been suggested byScarpa and takes into account the various quantities which have to be checked,displayed graphically in Fig.l. The accuracy is related to the difference between the

CHAPTER O 249

temperature set and the temperature monitored; the instability of the oven concernsthe oscillations of the temperature monitored.

Figure 2 shows an example concerning the heating up profile of a muffleoven. Because the heating time is a characteristic of each oven, it must be checkedaccurately. It is convenient to switch on the oven several hours before use.

T(°C) I

— — • ' — Tmax _

* - - •. i • Toven . J...

24» - - • • • • •— „ . j*f Tmin i

" • • - ACCURACY j INSTABILITY I244 - - I i :

I h PERIOD H240 -t Tset

J3S - -

I I I I I I I I 1 1 | I0 1 2 3 4 5 6 7 8 9

TIME (min)

Fig.l. Quantities to be checked for the quality control of the ovens.

Figure 3 depicts the temperature oscillations during the heating up phase(temperature set at 240°C) and successive Fig.4 shows a typical thermalconditioning for a ceramic tray, inserted in a preheated oven.

During the steady phase of the oven the temperature, normally, is notstable. The oscillations around the temperature set depend on the quality of the oven.This parameter has to be reported in the list of the characteristics of any new oven.As an example, Fig. 5 depicts the temperature oscillations during the steady phase(temperature set at 240°C).

Another effect to be taken into account is that one which arises when thedoor of a preheated oven is opened to put the tray inside; the temperature drops to alower value and then increases above the pre-set value. An example of this behavior,

measured for an oven without forced air circulation, set at a steady temperature of400°C and an opening time of the door of 60 seconds, is shown in Fig.6. Afterclosing the door, the temperature rises to about 410°C and then, slowly, goes back tothe pre-set value in about 30 minutes. Of course, it is not a good procedure to openthe oven during the annealing treatment.

According to the previous effects, it is convenient to use at least twodifferent ovens when the TL dosimeters need a complex annealing procedure, as inthe case of LiF :Mg,Ti which needs a high temperature annealing followed by a lowtemperature treatment.

Figure 7 shows the space distribution of temperatures inside an oven. Becausethe temperature gradients are always present inside an oven, the TLD tray must always bepositioned at the same place.

T(°C) I I

3H - -

2 4 0 • IIN w ~ ~ i ~ -

MO - ^ !

IN " " S " ^ \

IM - / \

n • - / j

7 I STEADY40 • „= HEATING UP PHASE > | < PHASE - ,

L_ 1 1 1 1—_H 1 1—!—1 1 11 2 J 4 S I 7 8 B

TIME (hours)

Fig.2. Heating up phase of a muffle oven.

CHAPTER O 251

"*'* I I j 1 HEATING UP PHASE ~ | I I I

|»J 131.5 1 1 •

, , . SWITCH ON TIME: 10.50.00

1 1 | — —3 2 7 . 5 T ' I ' I ' • • • I ' — I 1 . • i • | • i . i I i . i . |

11.91.4) ll.9t.lt ll.55.il 11.57.07 ll.St.lt 11.00.00 13.01.Jt la.01.S3 12.0t.lt

TIME

Fig.3. Temperature oscillations during heating up phase.

T(.c; ——

2» - •

MO - - ^ "*" j j

160 - - y^ ! j

tao - - y ^ i •

" J- I 9 S % —] j

* «: 1 w . / . i »tI 1 1 1 1 1—M i 1 1—I

TIME (min)

Fig.4. Heating rate of a ceramic tray inserted in a preheated oven.


"''' 1 1 I STEADY PHASE I 1 I I342 . .

m.i —

U HI

" " * ' * TEMPERATURE SET: 240°C

lit 1 1 1

117.«

317 t - I 4 - - • • I — • I n I — • I I

TIME

Fig. 5. Temperature oscillations in an ovenduring the steady phase.

* i o • • ' * * .

401 •

40ft • •

« 404 .

- 4(K

5 400 •>-

I s " ' '••. . • - . .

394 •

T I . I . . .1 , ,1 , .2 4 6 8 10 12 t4 10 IS 20 22 24 20 29 30 32 34 36

TIHC AFTCR CLOSING THC OVEN DOOft ( m i n i

Fig.6. Effect of "open door" on a preheated oven.

CHAPTER O 253

C = = 190 mm =>

-3.3 °C -4.5 °C

< 55—•>

96 290 mm

- 4 . 1 ° C - 3 . 9 ° C

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ > >

Fig.7. Space distribution of temperatures inside an oven.

ReferenceScarpa G. in Corso sulla Termoluminescenza Applicata alia Dosimetria,Universita' di Roma "La Sapienza", 15-17 Febbraio 1994

p-1(from Patridge-May model to Peak shape method: reliability expressions)

Partridge-May model (zero-order kinetics)

Partridge and May have reported some observations concerning an apparentkinetic order less than the first one (b<l).

If a single process is involved, one has the following equation

dn .I = -— = cnb (1)

atwhere c is the rate constant.

On the contrary, if b is less than 1, they explained such behavior by meansof two competing processes: a first radiative order and a zero order without radiativetransitions.

For such a model one can write:

dn- - ^ - = C,M + C2 (2)

where c2 is the zero-order rate constant.The integration of Eq.(2) gets:

r dn f- J = \dtJcln + c2 J

_ J_ ed{cxn + c2) _ r

c, •* cxn + c2 •*

from which

- — ln(c,H + c 2 ) = / + &' (3)c\

where k' is a constant of integration.


The intensity is given by

/ = c,fi (4)

Combining Eq.(3) and Eq.(4) one obtains

- l n ( / + c2) = c^ + A:" (5)

Partridge and May have reported that in isothermal decay experimentssome data fitted Eq.(5) better than an equation expressing order higher than the firstone.

ReferencePartridge J.A. and May C.E., J. Chem. Phys. 42 (1965) 797

Peak-shape method (Balarin: first- and second-order kinetics)

B alar in [1,2] deduced some expressions for determining the activationenergy, based on the quantity co = T2 - Tx, where T\ and T2 correspond to the

temperatures on either side of TM, corresponding to half intensity. In the followingthe Balarin's original symbolism is used.

He started from the following general rate equation

dc cr . ( E\

with Co = C\T0 ) and where

c = concentration of some kind of reactant,y= kinetic order,k0 = frequency factor,E = activation energy,r=time,T= temperature,k = Boltzmann constant.

(1)

CHAPTER P/l 257

The processes governed by Eq.(l) are enhanced when the temperature of

the system is raised continuously with a constant heating rate /? = — . Equation (1)dx

can then be transformed as

dc 1 cr . ( E\= r-^o-exP (2)

dT P cj"1 ° \ kT)

The maximum condition is then obtained

d2c k \ y_x E ( E \ .dT2 R rr~l kT2 kT

T=TM " o K1M \ K1M J

Considering the following quantities:

Eq.(2) becomes

Integration of Eq.(4), starting at t - 0,T = Q,y = 0 and Co = 1 up to C(T), gives,

for various values of y:

(3)

(4)

r-\ {c-' ) \ /- _n1 \ 1

Y = \ I n - = / 2 e x p f- -7]{y) (5)c / yM

y = 0 1 - C V )

where

1-2

»70')=i+^(-i)" •(»+!);'" (6)n=l

is a correction function which is always close to unity, rj(y) < 1.Using expression (6) in (5), we obtain

/ j_

/ v r )CM=C{TM) = l exp(-^M) r = l (7)

\ I - 7 M r = o

CHAPTER P/l 259

/ j _/ / - '

/ I

/ f1"1]/ x-O'-lW+O'-lV'exp—*-

/ L \ y

/ r fi-M"C(TM) / i t- ~ A = { exp //M -rjt2 exp — L

\

\ " f1-1)"\ l-^/2exp -\ ^ M

\ - -\ 1 - 7 *

(8)

The temperature positions 7} = Tj and 72 are obtained when the intensity

J(T) = is half of the maximum intensity. By means of Eq.(l) or (4), we getdT

'-XT*,) Ug>J K l T,)_

fl J-l

v )

where the subscript (y) indicates the individual solution for every distinct kineticorder.

Inserting (8) in (9), expressions for t] and T^ can be obtained. These

expressions contain the quantities TjM{y) and Tji which are polynomials in

k TyM = —— . Furthermore, the half-width O) = T2-Tl again depends on yM . It is

Ethen deduced that the quantity Ex a, divided by T^ is an invariant, different for

every kinetic order, and the following relations can then be obtained

T2

E = - — 1st order kinetic (10)4998-o>

T2E = 2nd order kinetic (11)

3542-<y

According to Balarin, the previous expressions allow to determine theactivation energy values accurately to within 0.5%. The experimental inaccuracy inthe determination of co then becomes dominant and limiting.

References1. Balarin M., Phys. Stat. Sol. (a) 31 (1975) Kl 112. Balarin M. J., Thermal Anal. 17 (1979) 319

Peak shape method (Chen: first- and second-order)

Chen [1] derived expressions for evaluating E using numericalapproximations. The Chen method is useful for a broad range of energies ranging

between 0.1 eV and 2.0 eV and pre-exponential factors between 10 sec and 10

sec and it does not make any use of iterative procedures. Furthermore, the method

(9)

CHAPTER P/l 261

does not need any knowledge of the kinetics order which is directly found from thepeak shape.

Its method is based on the shape of a TL peak, similarly to the Lushchik[2] and Halberin-Braner methods [3]. For smplicity, the parameters involved in awell resolved peak are here reported in Fig. 1,

I M - ^ K

IM _Z ffl | A

/ I x 8 | \

/ r*—*t^-H\

Ti TM T2

Fig. 1. Parameters characterising a single peak.

whereTM,Ti,T2: are respectively the peak temperature at the maximum and thetemperatures on either side of the temperature at the maximum, corresponding tohalf intensity,

r = TM - T{: is the half-width at the low temperature side of the peak,

S = T2 - TM: is the half-width towards the fall-off of the glow peak,

co = T - 7,: is the total half-width,

u = —: is the symmetrical geometrical factor.s eo

Total half-width peak method• First order kinetics

Starting from the following first order equation, giving the TL intensity as afunction of the temperature,

7(7-) = vexp(-- | )exp[-^£exp(-J 7 )rfr ' ] (1)

the equation for the maximum intensity, IM , is given by

'" = "'s e !"f H " 1 f exp(" F7)7"] <2>The integral on the right-hand side can be approximated by means of an

asymptotic expansion and a reasonable approximation is given by

ii-HyifHS1-^ <3)w i t h A M = ^ .u E

Then, Eq.(2) becomes

IM = nos exp exp - — —— exp (l - AM) (4)V K1MJ [\ PJ\ £ / V K1MJ

Using now the maximum condition for the first order kinetics, Eq.(4) becomes

Using the assumption of Lushchik [2] that the area of the second half of the peak isequal to that of a triangle having the same height and half width, one can write

— ^ - = 1 (6)P'nM

where nM is the number of the trapped charges at TM .

A similar assumption about the relation between the total glow peak and atriangle can be written as

CHAPTER P/l 263

Chen considers relation (7) as a constant different from 1 for obtaining aresult with higher precision:

Inserting now the quantity /? • n0 from Eq.(8) into Eq.(5), one obtains

£ e x p ( A M ) = ^ & - (9)CO

Since AM is quite small, we get exp(AM)« 1 +A M and then Eq.(9)

becomes

E"=UTui^dr)~l (10)

Inserting Eq.(lO) in the condition at the maximum, one obtains thefrequency factor as follows

Chen found Cm = 0.92, and so Eq.(10) becomes

i^=2£rJl.25^-lj (12)

Inserting Eq.(12) into the condition at the maximum

(7)

(8)

(11)


PE _ ( _E_\IrT2 \ IrTK1M V K1M )

one has

2.29^ (1.29TU\s — exp —

co y a> Jwhich can be rewritten as

CO

having taken into account that 2.29 is close to ln\0. Furthermore, 2.29 has beenchanged in 2.67 in order to compensate for additional inaccuracies.

• Second order kineticsAccording to the second order equation

KT) = ^ ^ ry (14)

the intensity at the maximum is given by

^"M-iH^lH-^H (i5)Inserting the maximum condition for the second order into Eq.(15), this becomes

2 , ( E )\2kT*nos' ( E YP

or better

(13)

(16)

CHAPTER P/l 265

7* 4(2*1) e xfe)Using the integral approximation (3) in Eq.(14), one obtains

{ fiE ) \ kTuf »' { PE } \ kTj

and rearranging

Inserting now Eqs. (8) and (18) into Eq.(17), one gets

CO

or better

*-y--l\ (19)

CO )

In this case Chen calculated the coefficient Ca equal to 0.878 and then, Eq.(19)

becomes:

£'fl,=2^fl.756-^-lJ (20)

High-temperature-side half peak• First-order kinetics

The method of Lushchik [2] is slightly modified here in order to obtainmore accurate values.

The assumption of Lushchik [2], given by the relation

(17)

(18)


^-M^^^^-'-w-'-f (2i)has been changed by Chen as follows

^4 = Q (22)

From equation

^ - = .sexp(-—1 (23)

using the maximum condition for the first-order kinetics and Eq.(22), Eq.(23)yields

CSP = Efi8 kT2

and hence

C kT2E=L^hL (24)

Chen calculated Cs to be 0.976, then Eq.(24) becomes

kT2E = 0 . 9 7 6 ^ (25)

• Second-order kineticsUsing the solution for n and replacing nM for n, one has

»,=«.[i+y^--jff'j (26)

CHAPTER P/l 267

Using now the expression for the intensity at the maximum, IM , and making

the ratio between IM and nM, one has

' ( E)i '"°expr^r- = f—'. M^V-n (27)

P y { kT'J

The insertion of the maximum condition for the second-order kinetics inEq.(27) yields

' ( E )sna exp

nM 2kT2Mnas' ( E \

EP \ kTM)and, rearranging

^ = ^ (2S,

Inserting in Eq.(28) the condition given by Eq.(22), the expression for E isfound

E = C{2k?) (29)Chen found Cs = 0.853, so that Eq.(29) can be rewritten as

E = 0.S53(2k^) (30)I S )


Low-temperature side half peakConcerning the low-temperature side of the peak, Chen gave a more

accurate expression for the activation energy with respect to the expression ofHalperin and Braner [3].

At first, Chen wrote that the Halperin-Braner expressions include twoinaccuracies:(i) the Lushichik's assumption Cs = 1

(ii) the approximation of fxM = —— by nM =

To give more accurate expressions, Chen introduced the quantity

hd. -r on

which means that the ratio between the first half of the peak and a triangle havingthe same height and half-width is a constant.

Equation (31) can be rewritten again as

IM • T = C

\»« )

and rearranging, as

^-1 = (—l-W (32)

• First-order kineticsThe number of trapped charges at the maximum is given by

«A/=«oexp-^^exp^-^rj (33)

Using the approximation for the integral compared in (33), one gets

CHAPTER P/l 269

Inserting now the condition at the maximum in (34), one obtains

- ^ = exp(l-Aj«(l-Aj-e (35)

Using now the following equation

-JL = sexp\

and inserting in it the condition at the maximum, one has

Iu EBt = l^ (36)

Inserting now Eqs. (35) and (36) in Eq.(32), one obtains

(1-Aj,).e-1 = - ^ — ^ 4which gives

E = CT-^L[e-(l-AM)-l\ (37)

Chen determined a value equal to 0.885 for the constant CT, so that Eq.(37)

becomes

kT2E = \.52^M~(l-\.5SAM) (38)

T

which is the Halperin and Braner's corrected expression.

(36)

(34)

Because this equation needs iterative calculations, Chen gave a new

expression, using another approximation for —— . In fact

Mo _, g

so that Eq.(33) becomes

_e 1 = _ r _ EJL1 + A^ CTP k-Tl

and then

E{j±^]=iJtli<\ < 3 9 )tl-O.58Aj { T )

because 0.58AM is very small, one can write

1 + A M «(l + A.,X1 + 0-58)«l + 1.58A./l-0.58AM V MA ' M

by neglecting the second power of A w . So, Eq.(39) becomes

£[1 + 1.58p^)] = 1.72(^ii)

and in final form

fkT2 ^ET = 1 . 5 2 - ^ -1 .58(2*rJ (40)

V T )where Cr =0.919.

Equation (40) is more useful than Eq.(38) because no iterative processes arenecessary.

• Second-order kineticsRemembering Eq.(26) and inserting in it the maximum condition, one has

CHAPTER P/l 271

Using the approximation given by Eq.(18), the previous equation gives

— = 7—T~ (41)

Inserting Eq.(41) into Eq.(32), one obtains

-2—J-LJLL) (42)

Inserting now Eq.(28) in Eq.(42), one has

2 T-E

1 + A 1C JcT2

which, using the following approximation

gives the Halperin and Braner's corrected expression

£ = 1.81/ 1(1-2* J (44)I T )

where Cr = 0.906.

As before, Chen found a new expression without any iterative calculation.Indeed, Eq.(43) can be rewritten as

£(1 + A j = 2CTkT2M

1-AW x

(43)

(45)


which can be changed using the approximation

so that Eq.(45) becomes

E J^ny^ ){ T )

from which, using CT - 0.906, the more convenient expression is obtained

kT2ET=l.$\3^L-4kTM (46)

T

References1. Chen R., J. Appl. Phys. 40 (1969) 5702. Lushchik C.B., Sov. Phys. JETP 3 (1956) 3903. Halperin A. and Braner A.A., Phys. Rev. 117 (1960) 408

Peak shape method (Chen: general-order kinetics)

Chen gave a method of calculating the activation energy for cases whoseorder is not necessarily first or second but rather may be a non-integer value [1].This method is again based on measuring the maximum and the half intensitytemperatures. From the equation proposed by May and Partridge for the generalorder

where b is the order of the kinetics and s" is the pre-exponential factor expressed incm^'W1.

The solution of Eq.(l) is given by

(1)

CHAPTER P/l 273

where s = s'ng'1 is in sec'1.

This equation can be rearranged using the condition at the maximum andthen solved numerically by approximating the integral by a certain number ofterms of the asymptotic series and using the iterative Newton-Raphson method.

The Chen's method consists of finding the temperature at the maximum,TM, by computer calculations for given values of b,s,E,P. The used values were

0.7 < b < 2.5

l O V 1 ^ . s ^ l O ' V 1

O.leV<E<l.6eVP = 0.5Ks'1

Once the value of TM is found, the intensity IM can be found by inserting the

temperature at the maximum in Eq.(2) and using again the asymptotic series. The

values TX,T2, the low and high temperatures of half intensity, can be calculated by

solving numerically the equation

I(T) = *-*-

Using the asymptotic series for the integral approximation, Chen found

Tx = 0.95TM

T2=l.05TM

After that, the geometrical parameters of the peak, i.Q.d,r,co,fig, are

found.Interpolating and extrapolating the constants appearing in the equations for

the first- and second-order, Chen gave a general expression which summarizes allthe previously given expressions.

The equations can be summed up as:

(2)


Ea=ca[^yba(2kTM) (3)

where a is T, SOT G>. The values of ca and ba are summarized as:

cr=l.5\ + 3.0(jug -0.42) bT = 1.58 + 4.2(//g -0.42)

cs = 0.976 + 7.3(//g - 0.42) bs = 0

^ = 2 . 5 2 + 10.2^-0.42) 6ffl=l

with/^=0.42 for 1st order

//g = 0.52 for 2nd order

Chen [1] calculated a graph of/^, ranging from 0.36 to 0.55 for values of*between 0.7 and 2.5 which can be used for the evaluation of b from a measured n g

(see Fig. 1).Another graph has been proposed by Balarin [2] which gives the kinetics

order as a function of y=8/v (Fig.2).Once the activation energy is obtained, one can find the frequency factor

using the following equation

(P\E\ 1 (E)

E

(4)

CHAPTER P/l 275

3T -7 1

• /m

oj . , , , , 10.3 0.4 0.5 0.6

GEOMETRICAL FACTOR(H)

Fig.l. Plot of the kinetics order b as a function of thegeometrical factor fig = S/w [3].

3-1 > • -7 1

• / a

Z 1" ^ ^ ^

1 ^

oJ . , . 1 . , . , . 10.4 0.6 0.8 1.0 1.2 1.4

GEOMETRICAL FACTOR ( y)

Fig.2. Plot of the kinetics order b as a functionof the geometrical factor y = S/r[3].

References1. Chen R., J. Electrochem. Soc: Solid State Science, 116 (1969) 12542. Balarin M., Phys. Stat. Sol. (a), 54 (1979) K1373. Furetta C. and Weng P.S. , Operational Thermoluminescence Dosimetry,

World Scientific Pub. (1998)

Peak shape method (Christodoulides: first- and general-order)

Christodoulides [1] developed some expressions for the determination ofthe activation energy, E, of a first order peak, using the widths or half-widths of thepeaks. These widths correspond to temperatures at which the signal level is 1/4, 1/2or 3/4 of the peak height, on both sides of the peak temperature at the maximum, TM.Fig.3 shows the various temperatures previously defined.

The expressions are valid in the region of E/JCTM values between 10 and100. Using the first-order kinetics equation giving the variation of light intensitywith temperature, and inserting in it the equation of the maximum for a constantheating rate, one gets the following expression in terms of the variable e = E/kT andits value at the peak maximum, SM = E/JCTM-

[ , _ f» exp(-f) , 1/ = nos expi -s-eM exp(f M ) | ——2— de > (1)

/L:iS, T, V, TM U.TiS, -T

1—f—I 1—hHSi r., <;, RM C J B J S I c = - E -

kT

Fig.3. A glow-peak and the various temperaturesused in the method.

CHAPTER P/l 277

The integral in Eq.(l) can be expressed in terms of the second exponentialintegral

^ , % r exp(-fiT) , r° exp(-z)

£ 2 ( s )=] ^2 '<ft = g f ^2 dz (2)So that Eq.(l) becomes

/ = nos expj - e - s2M exp(*M) - ^ - 1 (3)

which has a maximum value, for S = £ w equal to

IM = nos exp{- sM - £M exp(£M)£:2(fM)} (4)

and finally

/ = IM expi- (s-sM)-sl a&ej^ - ^ 1 } (5)I L £ % JJ

which gives the signal output / as a function of e for a TL peak of given IM and eu.For a given value of £ M, the ratio 1/IM depends only on E . An iteration procedure isneeded to solve Eq.(5) because one must be sure about the convergence of theprocedure itself. This is done using £ M as a starting value off . Equation (5) may berewritten as

(*-**)+[ — ] exP( A/ -eJie exp(»£2 O)]

= [*Mexp(*Ji<2(*J]+ln(^

Using tables of exponential integrals [2] , one finds that for 5 < £ < <x> thequantity £ • exp( £) • £2( £•) is ranging between 0.7 and 1: therefore it is of orderunity.-for T < TM, i-e. £ > £ M, we have exp( £ M - £ ) < 1 and the term (E -£ M) dominatesin Eq.(6):

£ = l\f) + £M^ + 6XP M 2 (£M - P M exp(*Jexp(-*fcexpME2(ff)] O)

(6)

because E • exp ( s ) • E2(e)=l and then exp(-s ) = e • E2(s):

e = \n(^ + sM[l + exv(SM)E2(£M)]-(£f] exp(^)^2(f) (8)

- for T > TM, i.e., e < e M, exp( s - s M) dominates over ( e - £ M) in Eq.(6):

£=£M- \n\ j / \v t \ I I 9

£2Mexp(£)E2(£)/e

In expressions (8) and (9) a rational approximation may be used for thetranscendental function E2( £). Such an expression is [3]

0.99997^ + 3.03962 . ,e x p ( ^ 2 ( g ) % 2 + 5 0 3 6 3 7 3 g + 4 l 9 l 6 ( ) + A ( g ) (10)

where \bie\ < 10"7 for e > 10.

The values off, corresponding to I/IM = V*, V* and V*, are defined in Tab. 1along with the corresponding temperatures.

I/IM £ T(K) peak zoneV* 8] Si low temperatureVz £ i T] side of the peak

V* h Ui1 £ M TM maximum

3A ^2 U2 h i g h tempera tu re

V2 £2 T2 side of the peak% I 8 2 [ S2 I

Table 1 - Definitions of the £ and temperatures.

Simple linear relations can then be searched for connecting pairs of the

quantities (£ u £M),(£2, £ M), (£ U £2), (<$>, £ id, (<%, £ »d, (^1. ^2), (Su &)

CHAPTER P/l 279

and (<%, £)• Similar expressions are also given which allow to know the width of apeak whose E and TM are known.

References1. Christodoulides C , J. Phys. D: Appl. Phys. 18 (1985) 15012. Abromowitz M. and Stegun I.A., Handbook of Mathematical Functions.

Dover, N.Y. (1955)3. Hastings C. Jr., Approximation for digital computers. Univ. Press

Princeton (1955)

Peak shape method (Gartia, Singh & Mazumdar: (b) general order)

These authors presented a new set of expressions for general order [1]. Theprior knowledge of the kinetics order is required. The method uses any points of apeak. The mathematical procedure is similar to the one already given in Mazumdar,Singh & Gartia peak shape method (a). Using Eq.(l) for b = 1, given in Peakshape method (Christodoulides: first- and general-order) [2], and Eqs.(12) and(13) for b =f= 1, given in Mazumdar, Singh & Gartia peak shape method (a) [3],and solving them by an iterative method, it is possible to write the followingexpression for the activation energy

CkT2ME= — + DkTM (1)

Tx-T,where

Tx-Ty\ = r,S,or a

The coefficients C and D are found using the method of least squares fordifferent order of kinetics b in the range from 0.7 to 2.5 and for x = 1/2, 2/3 and 4/3.For a particular value of x the coefficients result to be dependent on b and then canbe expressed as a quadratic function of b itself. So that, the previous equation canbe rewritten as

E = LJ ! f + (DQ + D{b + D2b2)kTM (2)Tx-Ty

Table 1 gives the coefficients for different values of x.


The authors claim the validity and the superiority of their method incomparison to those of Chen. Indeed, the E values obtained by using expression forx = 1/2 are more accurate than those of Chen. Furthermore, it is pointed out that En

Eg and Ea are in excellent agreement among themselves, whereas the Chen's valuesfor Eg and Em yield poor results.

ratio parameter Co Ci C2 Dp Dt D2

1/2 T 1.019 0.504 -0.066 -1.059 -1.217 0.1095 0.105 0.926 -0.048 0.154 -0.205 -0.128ea 1.124 1.427 -0.113 -0.902 -0.346 -0.061

2/3 T 0.684 0.426 -0.055 -0.720 -1.21 0.098S 0.146 0.683 -0.048 0.184 -.0.432 -0.094co 0.830 1.108 -0.103 -0.529 -0.607 -0.029

4/5 t Q.449 0.342 -0.043 -0.480 -1.184 0.085S 0.153 0.487 -0.041 0.180 -0.606 -0.062

| oa I 0.602 0.829 | -0.084" -0.293 -0.777 -0.006

Table 1 - Numerical values of the coefficients comparing in Eq.(2).

References1. Gartia R.K., Singh S.J. and Mazumdar P.S., Phys. Stat. Sol. (a) 114 (1989)

4072. Christodoulides C , J. Phys. D: Appl. Phys. 18 (1985) 15013. Mazumdar P.S., Singh S.J. and Gartia R.K., J. Phys. D: Appl. Phys. 21

(1988)815

Peak shape method (Grossweiner: first order)

Grossweiner [1] was the first to use the shape of the glow-peak to calculatethe trap depth and the frequency factor. His method is based on the temperature atmaximum and on the low temperature at half intensity, Ty Using the first order-kinetics one can write

IM = I(TU) = nos exp(- — ) exp - - [" exp(- —)dT (1)

CHAPTER P/l 281

/ M = ^ ) = "o^xp(-Jr)exp-^ exp(-J;>/r (2)

their ratio is

i r E(I i ]\ \s fM f £ v j- = exp exp — I exp \dT\ (3)2 \ k\Tx TM)\ *\fik \ kT) \

The integral in brackets can be resolved by asymptotic expansion asindicated before. By dropping terms after the first in the series, expression (3)changes in the following

1 F E f 1 1 )] \s kT2M . E , s M]2 , E ~— = exp exp ^ e x p ( ) ^-exp( )2 [ k\Tx TM)\ \ p E PV kTM} p E VK kT,

(4)

Doing the logarithm and rearranging using the maximum condition oneobtains

E (i i^i r r V E E

lir^h^-W exp(- + ;> (5)For E/kT larger than 20, the exponential of the last expression becomes

equal to 0.184. Furthermore, the term (T^/T^ may be neglected because it affects E

by less than 2% if s/fiis larger than 10 . These approximations get the final form

£ = 1 . 5 1 £ - ^ ^ (6)

T -T

This expression was empirically modified by Chen [2] with 1.41 replacing1.51 to get a better accuracy in the calculation of E, i.e.

(5)


T TE = \A\k M ' (7)T -T

lM A\

The frequency factor can be directly obtained from the followingexpression

1.41/xr. (IAITA

s=^t™{-^) (8)References

1. Grossweiner L.I., J. Appl. Phys. 24 (1969) 13062. Chen R., J. Appl. Phys. 24 (1969) 570

Peak shape method (Halperin-Braner)

Halperin and Braner [1] proposed a method, for determining the activationenergy, based on the temperatures on either side of the temperature at the maximum,corresponding to the half maximum intensity of the peak. They considered theluminescence emission as mainly due to two different kinds of recombinationprocesses. In the first one the electrons raise to an excited state within the forbiddengap below the conduction band and recombine with holes by tunnelling process(model A). In the second process, the recombination takes place via conduction band(model B). Figures 4 and 5 show the two recombination processes treated in the text.

The kinetics equations are formuled as shown below:

Model A

dm .--j7 = mneAm (la)

atdn

~^- = yn-sne (lb)dt

dn , t \--r = rn-ne\mAm+s) (lc)

dt

CHAPTER P/l 283

Model B

dm

~~dt=mn° m (2a )

-~tt=yn~nc{N-n)An (2b)

—%T = yn-ncmAm+{N-n)AH (2c)at

whereflg = concentration of electrons in the excited states Ne,

nc = concentration of electrons in the conduction band (CB),

n = number oftrapped electrons in the electron traps N,m = number of trapped holes in hole traps M,Am = probability of recombination,

CB

^^ m

VBMODEL A

Fig.4. Electrons raise from N to an excited state within the forbiddengap below the conduction band (Ne) and recombine with holes, in M,

by tunnelling process.


CBnc

A i.

1 2

3

• m

VBMODEL B

Fig. 5. Recombination process takes place via conduction band.

An = probability of retrapping,

( E}Y = s exp = probability of thermal excitation,V kTJ

s = frequency factor or probability per second of retrapping.

In Model B one assumes that s is temperature dependent, i.e. s = SOT .

Assuming that transition 2 or 3 is fast enough, one can put

dn n

— - = 0 for model A, (3)dt

and

dn—^ = 0 for model B (4)dt

The neutrality condition is expressed by

no-n = mo-m (5)

CHAPTER P/l 285

where n0 and m0 being the concentration at time /„ and n, m at time / .

Model AIn this case, from Eq.(lc), using the assumption of Eq.(3), one has

yn" * = J ( 6 )

mAm+sEq.( la) becomes

dm YnmAm

at mAm + sTo find an expression for the activation energy, E, one defines the ratio of

the initial concentrations of trapped electrons to trapped holes:

p = ^ > \ (8)m0

Let us introduce now the following paprameters:

m0

NX = ~ (10)

"oA = Am (11)

£ = — (12)m0

from the neutrality condition gives by Eq.(5) one obtains

n = «0 + m - m0

n = mo(p + ju-l) (13)

from Eq.(9) one obtains

(7)

(9)

dm du

dt ° dtand so Eq.(7) becomes

-m-di=Jr^- <14)dt mAm +s

Rearranging and inserting Eqs. (9), (11), (12) and (13), one gets

dt m0 (mA + s)or better

dfi = f4A(p + fi-i)

dt (pA + B)

From Eq.(la) one has nowdm du

/ = -—= -/«0-f (16)dt dt

Using a linear heating rate J5 = dTjdt, Eq.(16) becomes

/ = ~/KS 07)and again, using Eq.(15), one obtains

/ du [Ayu\p+u-\= —— = —^- — — - — (18)

fim0 dT { J3 ) juA + BNow, Halperin and Braner introduced the following parameters concerning

an isolated TL glow peak.According to the Fig.6, the defined parameters are:

TM,TX,T2: are respectively the peak temperature at the maximum and thetemperatures on either side of the temperature at the maximum, corresponding tohalf intensity,

T = TM - Tx: is the half-width at the low temperature side of the peak,

S = T2 - TM: is the half-width towards the fall-off of the glow peak,

(15)

CHAPTER P/l 287

co = T - 7\: is the total half-width,s

H = —: is the symmetrical geometrical factor.* co

I M

2M _/ a i A2 JT H

/ | x 8 | \X ' I \

I ^ i ! i \

Tj TM T2

Fig.6. The geometrical parameterscharacterizing an isolated peak.

CIM ~~ J.

T l TM T2 B

K-2S-H

Fig.7. A glow peak assimilated to a triangle.


Considering Fig.7, where a glow peak may be regarded as a triangle, theconcentration of the carriers at the maximum, nM, can be calculated, with a goodapproximation, as

i ^ T Ji"M = ~ r !dT * AREAUBC) = —IM 28 = - * - (19)

where IM is the maximum intensity.Then, from Eq.(18), calculated at the maximum of the glow peak, it follows

mM IM8m0 Pm0

Hence, Eq.(18), calculated at the maximum, becomes

LhL = lM^J_clH\ =ArMMMJP + MM-1) (21)

5 PmQ { dT)M P(MMA + B)

Taking now its logarithm:

lnf^-Uln^ + ln^ + -O-lnU/i + +lnf^l-J;

and equating its temperature derivative at maximum to zero, one gets

\ — \ • + + ^ = 0 (22)

ydTjTu [juM p + t*u-l AfiM+B) kTM

Inserting now Eq.(21) into Eq.(22) and rearranging, one gets

E =l( »H | AVM+B-AMU}

kT2M S{P + MM-1 AMM+B )

from which

(20)

CHAPTER P/l 289

kT2E = H/C^L ( 23 )

where

H = —!± + (24)P + VM-1 AftM+B

with the approximation

mM $MM=-JL= (25)

where co is the half-intensity width of the peak.

Model BFor this model, using the condition expressed in Eq.(4), one obtains

ynn = { r— (26)

mAn+{N-n)An

and therefore

dm . ymnAm- — = mncAm = -^-rrr^YT (27)

dt mAm+{N-n)An

The following expression are now introduced

S=Am-An (28)

S'=AH(pz-p + l) (29)

Also in this case it is considered p >- 1.

The expression N — n can be transformed using Eqs. (9) and (13):

M (N n) ( N 'wo0o + //-l)>jN-n-m\ = mow V-^-— =\m m) ymo{/ m0/u ) (30)

= mo(pX-M-P + l)

By introducing Eq.(30) into Eq.(27) one gets


\dt) mMm+An{p%-M-p + l)and rearranging

dju _yAmii{p + Li-\)dt ~ A'M + B' ( 3 1 )

Using a linear heating rate f3, Eq.(31) becomes

dr'lfi ) A'fi+B- ( )

Using Eq.(17) for the intensity, Eq.(32) gives

I = dju JyAm}n(p + v-\)J3mo dT { J3 ) A'M + B*

Using the approximation expressed by Eq.(20), Eq.(33), calculated at the maximum,becomes

8 /3m0 { dT)T=Tu HMA*+B'

The logarithm of Eq.(33) yields now

l n l — =ln// + In0o + -l)-ln(u4*+5*)+21nr- —+ cos/

and its derivative at maximum equating to zero is

f - u — i ^ - Y ^ l +{MM P + MM-1 MM^+B'XdT) (35)

+ — T 1 + *- =0kT2M\ E

(33)

(34)

CHAPTER P/l 291

Inserting Eq.(34) into Eq.(35), one has

2kTM

where AM = . Rearranging the last expression we getE

{kTMj U A P + ^ - I HMA +B J

by using the parameter

H= "» + B*P + MM-1 A'VM+B*

Eq.(36) becomes

^---f- '—V2 (31), T 2 " <• , , . \K1M W)

kTM S\\ + AM)Since AM •« 1, we can write

(l-Aj-'-l-A*and Eq.(37) becomes

^{^f}-^ (38)The values of H are different for the first and second order kinetics.

Exactly:

H = \.ll\ MM ( l - 1 .58 -A M ) first order (39)

H = l \ MM ( l - 2 - A w ) second order (40){}-MM)

(36)

Halperin and Braner also gave a very easy way to decide the type ofkinetics is involved in the process.

M**1-^ ( « )e

the process is of the first order, while if

»„>—'<- (42)e

the process is of the second order.Equations (39) and (40) can be changed by introducing the half-width at the

low temperature side of the peakX = CO-5 (43)

This is very useful because if it is easy to eliminate any interferring glowappearing at low temperature side, it is impossible to eliminate shouldering peaksat the high temperature side of the observed peak. Using Eq.(43) and fiM - S/co inEqs.(39) and (40), Eq.(38) becomes

1 72 kT2E = M (l - 2.58 • AM ) for the 1st order (44)

Xand

2 • kT2

E = — (l - 3 • A M ) for the 2nd order (45)T

The equations of Halperin and Braner require iterative process to find Eowing the presence of AM To overcome this difficulty a new approximated methodwas proposed by Chen [2] without any iterative process (see Peak shape method.Chen: first- and second-order).

References1. Halperin A. and Braner A.A., Phys. Rev. 117(1960) 4082. Chen R., J. Appl. Phys. 40 (1969) 570

Peak shape method (Lushchik: first and second order)

Lushchik [1] also proposed a method based on the glow-peak shape forboth first- and second-order kinetics. Introducing the parameter 8 = T2 - TM, a glow-peak can be approximated to a triangle as shown in Fig. 8.

CHAPTER P/l 293

In this case, with a good approximation, one has

where nM is the carrier concentration at the maximum.

C

W2—r~^lk

K-28-H

Fig.8. Approximation of a glow-peak in a triangle.

For the first-order kinetics, the equation

I = cpnat the maximum point becomes

iM f E )— = .sexp-— (2)"M V kTM)

Using the condition at the maximum

J3E ( E \—— = s expKIM V KlM )

Eq.(2) gets

(1)


LL = M. (3)"M kT2M

The substitution of expression (1) in (3) allows to obtain the Lushchik'sexpression for the activation energy for the first-order process:

E = MjL (4)

The Lushchik formula for a second-order kinetics is obtained using thesolution for n, valid for a second order kinetics, replacing n with n^

Using now the expression for the intensity at the maximum, IM, and doingthe ratio between 1M and HM, one has

The insertion of the maximum condition for the second order in Eq.(6)yields

' ( E 1s nQexpIM _ I kTM) = PEnM 2kT2Mn,s' ( E ^ 2kT*

PE \ kTM)

Using again Eq.(l) and rearranging, the expression of Lushchik for thesecond order is obtained:

(5)

(6)

(7)

CHAPTER P/l 295

E _ 2kTM (8)

8

Chen [2] modified the two previous equations for a better accuracy in theE value by multiplying by 0.978 Eq.(4) and by 0.853 Eq.(8), i.e

E = 0 . 9 7 8 ^ £ = 1 . 7 0 6 ^ -8 S

The frequency factor for the first-order process is obtained by the followingexpression:

s = 0.976[ ^ J expj 0.976 ^ - ] (9)

References1. Lushihik L.I., Sov. Phys. JETP 3 (1956) 3902. Chen R., J. Appl. Phys. 40 (1969) 570

Peak shape method (Mazumdar, Singh & Gartia: (a) general order)

A new set of expressions, to evaluate the thermal activation energy, E, of athermoluminescent peak following a general order of kinetics, are given byMazumadar et al. [1]. The work is an extension of the peak shape methodsuggested by Christodoulides [2]. The involved temperatures are now the ones atwhich the intensity of the peak is, respectively, 1/2, 2/3 and 4/5 of the maximum.The Authors claim that the selection of these points is based on the fact that theupper half of the peak, in general, is expected to be free from interference fromsatellite peaks.

Taking into consideration the intensity at any temperature, T, for a peakobeying a general-order kinetics, given by

and the condition for the maximum intensity given by

(1)


kT2Mbs { E \ . s(b-\) f* f E V_,,—-—exp =1 + — L exp -\dT (2)

fiE \ kTM) P k \ kT'JPutting To = 0 in it, as well as s=E/kT and eM = E/kTu , we have

^=£{ £i 1 „,13 £[iexp(-«)-(*-l)«i^J

where

p° exp(-z)

• / - = C -^ A (4)

Equation (1) then becomes

__b_

, Jbexp(-£/u) + (b-l)(J-JM)s2MYb-iI = sn0 exp(-£) , / , , , 1 W 2 (5)

[ 6exp(-fM)-(6-l)yM4 J

with

r°exp(-z)J = I FV ^ z (6)

is Z

The intensity at the maximum is then given by

_b_

f bexp(-sM) }'"-'Iu = sn0 e x p ( - ^ ) ^ " ' T 2 (7)

[bexp(-eM)-(b-\)JM£2M\

In both expressions the integrals are expressed in terms of the second-exponential integral, i.e., E2{s) = eJ.

Finally, one can write

CHAPTER P/l 297

b

i r \b-\\ l"*=i— = exp(£M-s)U-\~j-^F(e,£M)y (8)

where

F(e,£M) = £2M exp(£M) - ± ^ - - - ^ (9)

Putting now l/lM = x and e = ex, one gets

_b_

x = exp(£M -£x)\- j—^F(£X,£M) (10)

and then

( b \ \ b 1\nx = £M-£x-\^—)ln l- — F(ex,eu) (11)

The procedure is now very similar to the one already used inChristodoulides' method.

Indicating ex =s~ for T< TM:

£~x =£M-lnx- — ln l - - y - F(e; ,eM) (12)

Having now sx = sx for T> TM, one gets:

rro i* b-\I - I exp(f M -ex) -1 + - — £M exp(^w )£2 (fM )

^ = £M - In ^—; +

(13)


It must be noted that the above equations are not valid for b = 1. For thiscase the previous equations given by Christodoulides have to be used.

For a given value of the ratio MM the corresponding values of s~ and e*are then determined. The iteration procedure is the same already used byChristodoulides, using eM as a starting value of s.

Now, if / and j denote the intensity ratios, the expression for the activationenergy can be written as

TT T

for Tj > Tj where Tt and 7} are the temperature at a given ratio at the falling andrising side respectively of the peak. The values of coefficients C and D are listed inthe following Table 1.

Temperature b=1.0 b = 1.5 b = 2.0relation C ID C 1} C DT T 7941 14978 7124 10430 6584 8126

T+ T 11779 10001 8372 6351 6585 4 5 7 7

r+ T- 4742 11967 3846 7659 3289 5543•* 1/2»•* 1/2

T T~ 10965 14025 9659 9717 8816 75391M'12I3

r+ T 14816 10351 10926 6687 8817 489012/3'1 M

r+ T- 6299 11819 5124 7726 4405 569812/3 '•'2/3

T T 15444 13362 13375 9211 12065 71161M'14J5

r+ T 19304 10690 14653 6997 12067 51761A/5'1 M

r+ r- 8578 11765 6990 7796 6030 5822

•t4/S>J4/5 I I I I I ITable 1. - Numerical values of coefficients C and D comparing in Eq.(14).

References

1. Mazumdar P.S., Singh SJ. and Gartia R.K., J. Phys. D: Appl. Phys. 21(1988)815

2. Christodoulides C, J. Phys. D: Appl. Phys. 18 (1985) 1501

(14)

21

CHAPTER P/l 299

Peak shape method (parameters)

An isolated TL glow peak, obtained using a linear heating rate, can becharacterized by some parameters as can be seen in the figure below.

I M ^ ^ ^

1M / to ! A

/I*—*K-H\

I S i ! i \Ti TM T2

Fig.9. The geometrical parameterscharacterizing an isolated peak.

As a first approach, it is possible to check the symmetry properties of thepeak:

"" a first-order peak has an asymmetrical shape.

*" a second-order peak is characterized by a symmetrical shape.

According to the figure, the following parameters can be defined:

TM,TX,T2\ are respectively the peak temperature at the maximum and thetemperatures on either side of the temperature at the maximum, corresponding tohalf intensity,

r = TM - Tx: is the half-width at the low temperature side of the peak,

S = T2-TM: is the half-width towards the fall-off of the glow peak,

a = T - Tx: is the total half-width,

H = —: is the symmetrical geometrical factor.s co

It has to be noted that:

~ According to the asymmetrical property of a first-order peak, r is almost50% bigger than 8 ,

~ The geometrical factor ng is equal to 0.42 for a first-order kinetics, and

0.52 in the case of a second-order, hence, the following relation can bededuced

0.52- / i g | - |0 .42- / / g | (1)

and two possibilities can be obtained. If relation (1) is less than zero, asecond-order kinetics or a tendency has to be considered; if relation (1)results to be positive, a first-order or a tendency is possible.

•" ng is practically independent of E, in the range from 0.1 to 1.6 eV, and

of a", from 105 to 1 0 ' V ,

~ jug is strongly dependent on the kinetic order, b, in the range 0.7 < b <

2.5,

~ Another factor, namely y = — , is ranging from 0.7 to 0.8 for a first-x

order peak, and from 1.05 to 1.20 for a second-order.

References1. Grosswiener L.I., J. Appl. Phys. 24 (1953) 13062. Lushchik C.B., Sov. Phys. JETP 3 (1953) 3903. Halperin A. and Braner A.A., J. Appl. Phys. 46 (1960) 4084. Chen R., J. Appl. Phys. 46 (1969) 5705. Chen R., J. Electrochem. Soc. 106 (1969) 12546. Balarin M., Phys. Stat. Sol. (a) 31 (1975) Kl 11

Peak shape method when s=s(T). (Chen: first-, second- and general-order)

Total half-width of a peak• First-order kinetics

Starting from the TL intensity expression, I=I(T), given in case of *=s(T), havingused the integral approximation [1,2]:

CHAPTER P/l 301

I(T) = nosoT" exp(-- | )[-^ f Ta exp(-A)</r] (l)

the expression for the intensity at the maximum is given by

(2)

Inserting in Eq.(2) the condition at the maximum:

P / E fop

Eq.(2) becomes:

IM =- ° , 1 + - A M exp - 1 + Aw + — A w + — Aw

Neglecting in the above equation the second-order A M terms, one has

_nQ.fi-Ef a Yexp(Ajw~ it-Ti I 2 M) e (4)

Because

exp(AM)«l + AM

Eq.(4) becomes

(3)


p-k-T2 \ 2 1

which can be rearranged as

Remembering the Chen's assumption between the total glow area and a triangle

°^- = C. (6,

Eq.(5) becomes

I \ 2 ) \ co

from which the expression for the activation energy is obtained:

Ea=2kTM 1 . 26 -^ -^ l + | j j (7)

• Second-order kineticsThe condition at the maximum (see dependence of...) is given by

(8)

which becomes, using the integral approximation:

(5)

CHAPTER P/l 303

?<•' n Ta+2lr ( F \

Rearranging the above expression and neglecting the second-order AM

terms, we obtain

v ^ /

Inserting Eq.(9) in the expression of the intensity

l-T^-k [s'J \kTM)

and using expression (6) and rearranging, we obtain

is 4tr^

(9)

(10)


Neglecting again the second-order A M terms and substituting AM with

2kTM/E , we have

2?.=2*T^^--(l + |J| (11)

High-temperature side half peak• First-order kinetics

The maximum intensity can be expressed as follows

J*=»tf^expf--J-J (12)

and using the condition at the maximum we can write

2kTwhich, substituting AM = — , yields

E

(I kT2 }

Using the Lushchik assumption, modified by Chen,

¥*-C. (.4)Eq.( 13) becomes

ES=~-kT2M-a-k-TM (15)

(13)

CHAPTER P/l 305

• Second-order kineticsThe expression of nM is given by

and using the expression of the intensity as a function of the temperature

n]S'Ja exp(--|)I(T) = kT

l + -lVexpf-^V[ J3 k y\ kTJ

we obtain

KoOTexp- —AL = I kTu)

>ff k \ kV)

Inserting in this expression the condition at the maximum, one has

»M 2kTM

Using the assumption (13), we finaly obtain

2?,=^P--o*7'JI, (16)o


Low-temperature side half peak• First-order kinetics

The number of trapped charges, at the temperature at the maximum, is given by

nM = n0 exp - j £ T' expj^- j ^ d T ' (17)

Using the integral approximation, we obtain

Inserting in the above expression the condition at the maximum andrearranging, we have

nM _ ex J (kaTM+EXl-AM)'— C A D —• ™~

«o L Eor better

from which, neglecting the second-order AM terms, one has

^ e x p K l - A j ] * 1 ^ (18)«o e

Remembering the relation given by Chen (low-temperature side half peakmethod):

CHAPTER P/l 307

the insertion in the last expression of Eqs. (12) and (17) gives

e r E (t a . \1 = - • + - A W (19)

1 + A C kT2 I 2l^^M ^T K1M V z J

To obtain the Halperin and Braner's corrected formula, Chen used a~2,so that the previous equation becomes

_ CzkT2M\ e 1 1E = ~L—=- T — (20)

Eq. (18) can be simplified using the following approximations:

(I + AJJ

— l — « 1 - A w

hence obtained

(kT2 ^ET=\.5\5\—*- (1-2.58AW) (21)

I T )To obtain an expression without iterative calculations, one can start from theprevious Eq.(19):


1 + A M CTkT2MV 2 M)

1 . 7 1 8 ( l - 0 . 5 8 A M ) ^ x - E L | a^ \1 + A M CxkTl\ 2 M)

1.718 _ x-E ( | o:A "I(l + AA/)(l + 0 . 5 8 A w ) ~ C ^ l + 2 MJ

from which, neglecting the second-order AM terms. One has

£r =1.515(^Vff+ 1.58}(2Wj (22)l r J v2 ;

• Second-order kineticsInserting Eq.(8) into the equation which gives the intensity in the case of a

second order kinetics with the frequency factor depending on the temperature:

H0Vaexp(--^)I(T) = - kT 2 (23)

we obtain

Iu=syjZaql~ 2 I " ^ (24)

CHAPTER P/l 309

from which

-1-2

/ \

C D "\ c'lrTa+2 On

~YT~\—W^^^ (25)

I kTM)y 2 )_Coming back to Eq.(8) and using the approximation for the integral

comparing in that equation, we get

sX^"+2exp ——I f

The previous expression can be modified considering the followingapproximations:

^ *l + AM (26)

A M ^ ^ 1 (27)

It becomes then

i= L^ir2fi-£Aj#Vi+fi+£V-/5E L I 2 M J I 2 j M_

which, rearranged, yields

(25)


Inserting Eq.(28) into Eq.(25), allows one to obtain, using the approximations (26)and (27):

^ = «^[i+(i+«V,~4£T2 I 2 J™-/ ML V ^ /

To find the expression for the activation energy, we need the expression of n :

and insert it in Eq.(8) to obtain

J3E\\ + -AM} , .

n 2?'w lrTa+2 Jt-71

This last expression can be now inserted in Eq.(28), obtaining

or better

«n (l - A M )« o - « M = - J ^ y - ^ i (29)

(28)

CHAPTER P/l 311

Remembering the Chen's expression for the low-temperature side halfpeak, i.e.

du = c

P{na-nM)

and inserting it in Eq.(29):

from which the activation energy is obtained

Since this expression needs an iterative procedure, it can be expressed inanother way. Rearranging Eq.(30) as follows

2 kT I 2 IT

from which we get

^ ^ - f l ^ W (32)T \ 2)

Inserting in this equation the value 2CT =1.81, an expression without

resorting to iterative process is obtained.Chen gave a general expression for the activation energy, i.e.

(30)

(31)


Ey=cy[^ + by(2kTM) (33)

where yis T, SOT CO. The values of cr and br are summarized as:

cT = 1.51 + 3.0(//g - 0.42) 6r =1.58 + 4.2(//g - 0.42) + 1

^=0.976 + 7.3(^-0.42) ft,=|

c. =2.52 + 10.2(^1,-0.42) ^ = 1 + f

with fUg = 0.42 for 1st order

and Ar = 0.52 for 2nd order

References1. Chen R., J. Appl. Phys. 40 (1969) 5702. Chen R., J. Electrochem. Soc. 116 (1969) 1254

Peak shape method: reliability expressions

An important and widely used method for investigating the trapping levelsin crystals is based, among the various TL methods introduced during the years, onthe geometrical characterization of a TL glow peak, the well-known peak shape (PS)methods. In fact, for calculating the activation energy of the trapping levelcorresponding to a peak in the glow curve, one needs to measure three temperaturevalues on the peak itself: the temperature at the maximum, T^ and the first andsecond half temperatures, T, and Tr

The formulas proposed [1,2] for finding the activation energy usuallyinclude the following factors:

r=TM-Tx the half width at the low temperature side of the peak,

S=T2-TM the half width towards the falloff of the peak,

CHAPTER P/l 313

a = T2 - Tx the total half width (FWHM).

In the following a list of the various expressions is given, for both first andsecond order of kinetics, allowing for the activation energy determination. All theexpressions have been modified by Chen for getting a better accuracy in the Evalues.

Grosswiener (G)TT

1* ORDER (EG)T=lAlk^^- (1)T

TT2nd ORDER (EG)r =l.6$k-1^- (2)

rLushchik(L)

JcT2Ist ORDER {EL)S =0.976—*- (3)

S

kT22nd ORDER {EL)S = 1 . 7 0 6 - - ^ (4)

sHalperin & Braner (HB1

kT2V ORDER (EHB)X =1 .72-^( l -2 .58A M ) (5)

T

1kT22nd ORDER (EHB\= H\-2AM) (6)

T

where AM = 2*rj(/E:

Chen also gave two more expressions based on the at factor:

Chen's additional expressions (Caex)

1" ORDER E = 2.29k ^^ (7)

2nd ORDER £ „ = 2A;rM 1 1 . 7 5 6 ^ - - 1 1 (8)

Chen's expressions (general) (Cgex)

The previous methods were summed up by Chen, who considered generalorder kinetics, 1, ranging from 1 to 2, then giving the possibility of non-integer valueof the kinetics order. The general expression is

E « = 4 ^ ) A ( 2 W M )

where a is t, 8 or co. The values of ca and ba are summarized as below

cr = 1.51+3.0(^-0.42) bT = l58 + 4.2(ju-0.42)

cs = 0.976+ 7.3(ju-0.42) bs = 0

cw = 2.52 +10.2^ - 0.42) bw = 1

with

S _T2-TM

F co T2-T2

where n = 0.42 for a first order kinetics and n = 0.52 for a second order.

The previous general expression, developed just for a 1s t and a 2n<* order,gives:

V ORDER

kT2{Ec)t=\.5\-^-?>MkTM (9)

TkT2

(Ec)s = 0 . 9 7 6 - 5 (10)o

kT2{Ec)a=2.52~^-2kTM (11)

ft)

2nd ORDER

(£c)r = 1 . 8 1 ^ ^ - 4 ^ (12)r

CHAPTER P/l 315

kT2

(Ec)s=0.706^f- (13)o

kT2(Ec)a=3.54-^--2kTM (14)

CO

Furthermore, the following parameter, introduced by Balarin, is also used:

S = T2-TM

Using the previous parameters, some relations among them can be obtainedas follows:

1st order-kinetics:

^ = 0.42 y = 0.12 £ = 0.72r S = 0.42co

2nd order-kinetics:

/ / = 0.52 7 = 1.09 S = 1.09T 5 = 0.52CD

As a first approximation, the following relations among the peak'stemperatures can also be used:

r,=0.95rM and T2=l.05TM

The expressions so far given have been handled for getting a criteria ofreliability of the E values obtained using the PS methods. In most of the cases theChen's expressions have been used as reference because they have a more generalmeaning with respect to the others and also give more accurate values of E.

1st ORDER

( v \ 0.978£ -V~\ ^ \ = £- = 1.002

^Ec)s 0 . 9 7 6 ^

( x 1 . 4 1 * ^

{Ec) Tl 1.07l(rM-2.09r)V c 7 r 1.51^^-3.16/0^ V M '

xT

— ! 0 99151.07(2.097; - 1 . 0 9 T M ) ~

1 i?kT2

(E \ ~ -0-2.58A*) 1 _ 2 5 8 A

hHB T = 1 139 Z 3 S A M{EC)T l^Tl_3A6kTM • 1 _ 2 . 0 9 3 7 ^ 7 I

r TM

Limits:

1.139 1.042A w = 0 - > _ =

1-2.039-^ l- 1.915—*—1

0.742 0.679AM=O.I-> _ = _ ^ —

1-2.093 ^ ' 1 . 9 1 5 ^ - 1J M 1M

0-679 < f ^ l < !-042

1.915-^-1 l ^ c j r i.9i5_5__iT T

V A/ /AM=0.1 V XM /AM=0

fil lAlk^VL = 0.81987;

l ^ r L 7 2 ^ ( l _ 2 5 8 A j " ^ ( l - 2 . 5 8 A j

limits:

CHAPTER P/l 317

A M =0^0 .8198- 7 L

AM =0.1->l. 1048^-

fo.8198^-1 < [ ^ 1 ^f 1 - 1 0 4 8 ^]

2n d ORDER

/ F N 1 . 7 0 6 ^ ^\f^\ = 4 - = 0.998UcJ, inkTl

8

r

r ^ 2^a-3Aw)p « =__?^_ = 0 . 9 1 7 ^ ^ ^V^cA 1.81*^-4*^ 1-83-L-l

Limits:

^ i.83r,-rM

1.8371-7^

so that


{ 0-77J, ) JEJ^) ( 0-9177^ ^

ti.8371 -r* J^=01 [ £c Jr -[i.83r, -r* JAM=0

k ) . 1 ^ 7 V I o-84r,

Limits:

AM = 0 - ^ 0 . 8 4 ^ A v =0.1->1 .2^-1M 1M

so that

(0,4f) S(f)S(^f)

Some more expressions derived by the original ones, using the geometricalfactors n and y.

Grosswiener expressions given as a function of 8 and w:

TT(EG)g =1.0152k J ^ 1st order

8

(£G), =1 .8313*^- 2nd order8

(EG) =2A\l\k^- 1st orderCO

TT(EG) = 3.52 Ilk -L^- 2nd order

CO

CHAPTER P/l 319

Lushchik expressions given as a function of rand <o:

{EL\ = 1 . 3 5 5 6 ^ 1st order

kT2(EL\ =1 .5651^ 2nd order

JcT2(£^=2.3238—^ 1st order

eo

kT2(EL) = 3.2808 2nc/ orc/er

Halperin-Braner expressions given as a function of ff and a:

kT2(£/fl})<5=1.2384^(l-2.58Aw) 15/ order

SIcT2

{EHB )^ =2.1801 —^- (l - 3 A M ) 2«c/ order5

kT2(Em) =2.9487—^(l-2.58AjJ 15/ orJer

ft)

(£,„)„ =4.1929-^-(l-3A^) 2nd order

Comparison of the previous derived expressions to the corresponding Chen'sexpressions

Grosswiener's modified expressions related to Chen's expressions

TT

=£-] = f - - = 1.0402^- 1st order

Ec>s 0.916^ TM

S


TTf N 1.8313*-*-^ T

\=^\ = | - = 2 . 5 9 3 9 ^ - 2nd order

^Ec'* 0 . 7 0 6 * ^ TM

8

TT

[ \ 7 417H5- ' M

5= . J ^ ! ^ L _ . _ 1.20867; order

EA . ^ ' » — _ L76097; order

CO

Error analysis

According to the error propagation rules, having a function of variousindependent variables, i.e.

® = f(xi,x2,....,xn)the error is given by

[{ax, ') [dx, 2J [a«, ")

The previous expression can be applied, for instance, to the Chen'sequation

2 29-k -T2

0)

According to the error propagation one gets

CHAPTER P/l 321

The errors associated to the various expressions can be calculated in thesame way.

References1. Kitis G., private communication2. Furetta C , Sanipoli C. and Kitis G., J. Phys. D: Appl. Phys. 34 (2001) 857

P-2(from Peak shift to Properties of the maximum conditions)

Peak shift

The TL intensity for first (Randall and Wilkins model) and second(Garlick and Gibson model) order of kinetics are respectively

I(T) = n-s-exp\-~) (1* order) (1)

and

I(T) = n2 -s'- expf j (2nd order) (2)

where « is the trapped carrier concentration, 5 is the frequency factor, s' = S/N is

the so-called pre-exponential factor, with N the available trap concentration.Equation (2) can be rewritten as

I(T) = s"-n-^-^j (3)

with s" = s'n, which is equivalent to s in the first order case.Considering Eqs. (1) to (3), it can be seen that the peak temperature at the

maximum, TM, depends on E and s, s'or s"; then, if n changes, increase ordecreases, a first-order peak remains in the same position, but a second-order peakshifts, i.e., to higher temperatures as n decreases because the variation in s'.

Figures (1) and (2) show the different behaviors expected for glow peaksfollowing a first-order kinetics or a second-order.


5E11

0E11 {"*" ^ ^i I i

150 250TEMPERATURE (K)

Fig. 1.Glow curves of first-order kinetics as afunction of the given dose; no increases from (1) to (5).

I ' ' ' ' I5

150 400TEMPERATURE (K)

Fig.2. Behavior of the second-order glow curves asa function of the given dose; no increases from (1) to (5).

CHAPTER P/2 325

Perovskite's family (ABX3)

Perovskite compounds corresponding to the general formula ABX3 (whereA is an alkali metal, B is an alkaline earth metal, and X is a halogen, usuallyfluorine) constitute a class of TL phosphors with good performances, especiallywhen doped with proper activators.

Considerable experimental work has been carried out on these TLmaterials, pure or doped with rare earth or transition metal impurities [1-4].

Preparation of these materials in crystalline form is achieved by growingpolycrystals or single crystals from a melt, obtained by mixing fluorides of thedesired alkali and alkaline earth metals in the stoichiometric ratio. The dopant isusually added to the starting powder before the growth, which can be performedwith various techniques (Czochralski, Bridgman, slow cooling).

TL signals of undoped compounds are in general less intense than thoseobtained from doped samples. Rare earth impurities show high efficiency asactivators in perovskites.

For dosimetry purposes, KMgF3:Eu and KMgF3:Ce can be considered avery interesting phosphor. Its sensitivity is higher (about two to four times) than thatof LiF, the response to the radiation dose is linear up to 1 Gy, the most prominentpeak at 340 °C shows no fading effect in a time of 15 h. Since its effective atomicnumber (about 13) is higher than that of the biological tissue, a good applicationwould be in the environmental dosimetry.

References1. Altshuler N.S., Kazakov B.N., Korableva S.L., Livanova L.D. and Stolov

A.L., Soviet Phys.- Optics and Spectroscopy 33 (1972) 2072. Alcala R., Koumvakalis N. and Sibley W.A., Phys. Stat. Sol. (a) 30 (1975)

4493. Kantha Reddy B., Somaiah K and Hari Babu V., Cryst. Res. Technol. 18

(1983) 14434. Furetta C, Bacci C , Rispoli B., Sanipoli C. and Scacco A., Rad. Prot. Dos.

33 (1990) 1075. Scacco A., Furetta C , Bacci C , Ramogida G. and Sanipoli C , Nucl. Instr.

Meth. Phys. Res., B91 (1994) 2236. J. Phys. Chem. Solids, 55(11) (1994) 13377. Kitis G., Furetta C , Sanipoli C. and Scacco A., Rad. Prot. Dos. 65(1-4)

(1996) 5458. Kitis G., Furetta C , Sanipoli C. and Scacco A., Rad. Prot. Dos. 82(2)

(1999) 1519. Furetta C , Sanipoli C. and Kitis G., J. Phys. D: Appl. Phys. 34 (2001) 85710. Furetta C , Santopietro F, Sanipoli C. and Kitis G., Appl. Rad. Isot. 55

(2001) 533


11. Le Masson N.J.M., Bos A.J.J., Van Eijk C.W.E., Furetta C. and ChaminadeJ.P., to be published in Rad. Prot. Dos.

Phosphorescence

Phosphorescence takes place for a time longer than 10"8 s and it is alsoobservable after removal of exciting source. The decay time of phosphorescence isdependent on the temperature. Referring to Fig.3, one can observe that this situationarises when an electron is excited (e.g. by ionizing radiation) from a ground state Eo

to a metastable state Em (electron trap), from which it does not return to the groundlevel with emission of a photon (e.g. the transition from Em to Eo), because it iscompletely or partially forbidden by the selection rules.

I 1 Ee

T EmM ^ \ / \ ^ hv

1 * E0

Fig.3. Phosphorescence phenomenon.

If one supposes that a higher excited level, Ee, exists to which the systemcan be raised by absorption of the energy Ee - Em and that the radiative transition Ee -Em is allowed, then one can provide the energy Ee - Em by thermal means at roomtemperature. After that a continuing luminescence emission (phosphorescence) canbe observed even after the excitation source is removed. This emission willcontinue with diminishing intensity until there are no longer any charges in themetastable state. For a short delay time, let us say less than 10"4 s, it is difficult todistinguish between fluorescence and phosphorescence. The only way is then tocheck if the phenomenon is temperature dependent or not. If the system is raised to ahigher temperature, the transition from Em to Ee will occur at an increased rate;consequently the phosphorescence will be brighter and the decay time will beshorter due to the faster depopulation of the metastable state. The phosphorescence

CHAPTER P/2 327

is then called thermoluminescence. The delay between excitation and light emissionis now ranging from minutes to about 1010 years.

The delay observed in phosphorescence corresponds, then, to the time thetrapped charge (i.e. an electron) spends in the electron trap. The mean time spent bythe electron in the trap, at a given temperature T, is expressed by

*=s-exp(-|) (.,

where s is called frequency factor (sec"1), E is the energy difference between Ee and

Em, called trap depth (eV) and k is the Boltzmann's constant (8.62-lO^eV K"1).Once the electron is in the electron trap, it needs an energy E, provided by thermalstimulation, for rising to Ee from Em and then to fall back to Eo emitting a photon.

Randall and Wilkins in 1945 [1], presented the first mathematical treatmentof phosphorescence, which is also the foundation of the thermoluminescence theory,making the assumption that once the electron has done the transition Em —> Ee , the

probability of retrapping in Em is much less than the probability to reach Eo.

According to their formalism, the emission intensity of phosphorescence at any

instant, I(t), is proportional to the rate of recombination (i.e. rate of the transitions

Ee -> Eo); because these transitions depend on the Em -» Ee transitions, the

intensity of phosphorescence is proportional to the rate of release of electrons,

d"/drfrom Em:

*, N dn7(0 = - c — (2)

at

where c is a constant (which can be assumed equal to 1).Equation (2) can be rewritten as

W = c- (3)T

where X ~ is the probability per second, p, concerning the thermally stimulatedprocess and n is the concentration of the trapped electrons.

Using Eqs. (2) and (3) one gets, by integration


n = « o e x p — (4)

where n0 is the initial concentration of the trapped electrons.Equation (4), together Eq.(2), gives

/(O = /oexp^-£) (5)

where IQ is the intensity at time t = 0.Equation (5) is the equation of the phosphorescence decay at a given

constant temperature, which is an exponential decay, also termed first-order decay.Randall and Wilkins, in their theory, also postulated the probability that the

decay of phosphorescence is non-exponential. In fact, the electron released from thetrap may return to the trap (retrapping) or recombine at Eo. In this case, therecombination rate is proportional to both the concentration of the trapped electronsin Em and to the concentration of recombination sites in Eo. Assuming that theconcentrations are equal («in Eo =ninEm), the intensity is now given by

I(t) = -c--=a-n2 (6)at

where a (cm3 sec"1) is a constant.The solution of Eq.(6) is

/ ( 0 = T: ^ — ^ (7)(1+a •«„•/)

which is related to a hyperbolic decay of phosphorescence, termed second-orderdecay. The physical process is called bimolecular.

E.I.Adirovitch, in 1956 [2], used a set of three differential equations toexplain the decay of phosphorescence in a more general case.

References1. Randall J.T. and Wilkins M.H.F., Proc. Roy. Soc. A184 (1945) 3662. Adirovitch E.I., J. Phys. Rad. 17 (1956) 705

CHAPTER P/2 329

Phosphors (definition)

The term phosphors is used to design all solid or liquid luminescentmaterials. This term is also used, in particular, for thermoluminescent materials (i.e.,TL phosphors).

Photon energy response (calculation)

In any dosimetric applications in the field of photon radiation, the energyresponse is one of the main characteristics that must be known.

The energy response, or energy dependence, is the measure of the energyabsorbed in the thermoluminescent material in comparison to the energy absorbed ina reference material (i.e. air or human tissue), when irradiated at the same dose[1,2,3].

Let us indicate with S(E) the energy response; then, according to thedefinition so far given, S(E) is given by

f—1b(t)--y r (1)

HI P Jair

where 1 - ^ 1 and l - ^ - lyPJTW I P Jair

are the mass energy absorption coefficients for the TLD and for air respectively.Because the 60Co (1.25 MeV) is normally considered as the reference

photon source, it is convenient to introduce the relative energy response, RER, of theTLD material, at the photon energy E, normalized to the 60Co energy:

RER = ^ M L (2)

Since TLDs are complex media, the law of mixture can be applied:


M =XN -W, (3)

where —— is the mass energy absorption coefficient of the i-th element and W;I P Ji

is its fraction by weight.As an example, the RER has been calculated for Ge-doped optical fiber [4].

Table 1 shows the fiber composition detected by Scan Electron Microscope (SEM).

element ~Wj (%)Si 46.12O 53.64Ge 1 0.233

Table 1. Ge-doped opticalfiber composition.

Table 2 shows the mass energy absorption coefficients for each element andfor each energy [5].

Energy I jT 7 "(MeV) ^e/p (cm2/g)

Si O Ge Air0.015 9.794 1.545 62.56 1.3340.03 1.164 77729x10-' 11.26 1.537x10'0.05 ~2.43xlO'' ~4AUxWl 2.759 4.098x10'^0.1 4.513x10^ 2.355x10'^ 3.803x10' 2.325x10'^1.25 ^652x10'^ 2.669x10-' 2.353x10^ 2.666x10^

6 1827x10-' 1.668x10^ 2.027x10^ 1.647x10*10 1.753x10^ i483xlO'z "2!208xl0^ 1.45X10'220 I 1.757x10-' | 1.36x10'' | 2.452X10'21 1.311xlO'f

Table 2. Mass energy absorption coefficients for the elementsof Ge-doped optical fiber and for air.

CHAPTER P/2 331

Table 3 shows, at each energy, both the energy dependence, Eq.(l), as wellas the experimental and theoretical RER, Eq.(2), for Ge-doped optical fiber.

Energy Energy Dependence Relative Energy Response(MeV) Theoretical Experimental0.015 4.116 4.126 —0.03 426 4.269 —0.05 3.468 3.475 3.920.1 L44 L443 1.4971.25 0.997 1 1

6 L05 L052 1.0210 U 0 9 LU 0.9320 1 1.17 1 1.172 [ 1.11

Table 3. Energy dependence, Eq.(l), and relative energy response (RER),Eq.(2), for Ge-doped optical fiber.

Figure 4 shows the energy dependence according Eq.(l) and Table 3.Figure 5 shows the relative energy response (RER), both theoretical andexperimental results, according to Table 3.

f 4.5<g 4 * ^ \£ 3.5 \

8 3- \

i 2 -5 \

«jj 0.5g 0 j , , , 1

0.01 0.1 1 10 100

ENERGY (MeV)

Fig. 4. Energy dependence according to thedata given in Table 3.


4 5 1—3; 1'PS3.5 b

3 \K 2 5 I —o—SerielUJ Ia. 2 \ --•--Serie2

1.5 V1 - ^ • • * * *

0.50 -I 1 1 1 10.01 0.1 1 10 100

ENERGY(MeV)

Fig. 5. Theoretical and experimental relative energy response(RER) according to the data in Table 3.


Edited by M.Oberhofer and A.Scharmann (Adam Hilger Ltd, Bristol, 1981)2. Furetta C. and Weng P.S., Operational Thermoluminescence Dosimetry

(World Scientific, 1998)3. S.W.S.McKeever, Thermoluminescence of Solids (Cambridge University

Press, 1985)4. Youssef Abdulla (private communication)5. Hubbell J.H. and Selzer S.M., Int. J. Appl. Radiat. Isot. 33 (1982) 1269

Photon energy response (definition)

The energy response is a measure of the energy absorbed in the TL materialused in comparison to the energy absorbed in a material taken as reference (i.e., airor tissue), when irradiated at the same exposure dose.

The energy response is a characteristic of each thermoluminescent materialand its direct measurement is only possible when the TL sample is irradiated underelectronic equilibrium conditions.

The following Table 1 lists the energy response at 30 keV, normalized tothe response to 60Co and to 137Cs, for many different kind of phosphors.

CHAPTER P/2 333

phosphors 1 30 130 keV/J37CskeV/^Co

LiF:Mg,Ti 1.3 1.3LiF:Mg,Ti,NaLiF:Mg,Cu,PLi2B4O7:Mn 0.9 - 0.98 0.9Li2B4O7:Cu 0.8 - 0.98 0.8 - 0.9Li2B4O7:Cu,Ag 0.98MgB4O7:Dy 1.5 1.3-2.4MgB4O7:TmMg2Si04:Tb 4^5 -4 .5CaSO4:Dy 11.5 - 1 3CaSO4:Tm 11.5CaSO4:Mn 112CaF2 (natural) 14.5 - 1 5CaF2:Dy 15.6CaF2:Mn 154BeO 0.87-1.4 0.9-1A12O3 I 3.5 | - 4 . 5

Table 1. Normalized energy response

Phototransferred thermoluminescence (PTTL) (general)

Phototransferred TL technique is based on the phenomenon of re-excitedTL by UV illumination after annealing or read-out of a thermoluminescent sample.UV irradiation induces transfer of electrons from deeper traps (not involved in theannealing or read-out procedures) into shallower traps. The efficiency of thephenomenon is temperature dependent. The phototransfer effect was first observedbyStoddard(1960).

A practical use of phototransfer is in TL dosimetry and TL dating: i.e.,measurement of carriers accumulated in very deep traps as a measure of theabsorbed dose.

The phototransfer technique consists of giving to the sample a certainamount of UV light which allows the transfer of carriers from a deep trap to ashallower one. The TL intensity of the transferred peak is proportional to theoriginal concentration of the carriers in the deep trap.

ReferenceStoddard A.E., Phys. Rev. 120 (1960) 114


Phototransferred thermoluminescence (PTTL): model

The most simple model for phototransferred thermoluminescence (PTTL) isthe one which considers one shallow trap, one deep trap and one recombinationcenter [1-3].

Let us indicate:na = concentration of electrons in the shallow traps (acceptor),nj = concentration of electrons in the deep traps (donor),m = concentration of holes in the recombination centers,nc = electrons in the conduction band (CB),Nd= total number of deep traps (donor traps),Na = total number of shallow traps (acceptor traps),M= recombination centers,T = (n^m)'1 is a lifetime,Ad, Aa = retrapping probability for free electrons into empty traps,Am = recombination probability,/ = rate of loss of electrons from deep traps (donor) owing to light excitation,ya = thermal excitation from the shallow (acceptor) traps.

The initial conditions, at the end of the ionizing radiation and before the lightillumination (/ = 0), are".0 = 0ndo=mo

Considering now that the illumination excites electrons from the deep traps(donor traps) to the shallow traps (acceptor traps), one can write the following rateequations, valid during the illumination period (0 -1*):

dt dt dt dt

d^ = "cAa(Na-na) (2)

dt^ = ncAd{Nd-nd)-ndf (3)

dm J m— = -Ammnc = - (4)dt x

Considering the equilibrium condition

(1)

CHAPTER P/2 335

*L-«*=-,£ (5)dt dt dt

and the condition of no retrapping into the donor traps:

ndf^ncAdiNd-nd) (6)

Integration of Eq.(3), taking into account the condition (6)

•L dt *

gives

nd=nd0Qxp(-ft*) (7)

From Eq.(2), with the initial condition «ao = 0, we get

l°~n°rdn^=("cAadt* {Na-na) *

we get

na=Na[l-exp(-ncA/)] (8)

Finally, from integration of Eq.(4)

pn dm 1 / .

= I dt*» dt x *

we obtain

m = m0 exp - - (9)

where ticAa and T are approximately constant if dnjdt» 0.At the end of illumination, according to Eqs. (7), (8) and (9), a certain

concentration of charges will then be in traps and centers.


The heating phase of the sample follows the illumination phase. Theheating phase is similar to the situation of competition during heating, so that themathematical treatment is very similar.

The set of new equations is now:

dt dt dt dtdn-^r = Aanc(Na-na)-yana (11)

^ - = Adnc(Nd-nd) (12)at

I = -~t=Ammnc (13)

Eqs. (12) and (13) can be rewritten as

»C=-~4M^-«J] (14)Ad dt

nc=-——(lnm) (15)

Am dtwhich can be integrated taking into account that the initial values of m and nd (at

the end of illumination) are, respectively, m* and n*d .Then, the integration yields to

»H (16)Considering now the quasi-equilibrium condition (5), Eq.(lO) can be written as

dm dna dnd« — - + — - (17)

dt dt dt K '

By substitution of Eqs. (11), (12) and (13) in (17) we obtain:

(10)

(16)

CHAPTER P/2 337

-Ammne»Aanc(Na -na)-yana+ Adnc(Nd -nd)

from which we get an explicit expression for nc:

n — lag ,jg^

C Ad(N,-nd)+Aa(Na-nt,)+Amm

Then, the intensity is given by

r dm . Amrm n1 = = Ammn=—-, vm '" " r (19)

dt Ad(Nd-nd)+Aa{Na-na)+Ammwhere the first term in the denominator is the probability of retrapping in donorlevel, the second concerns the retrapping in the acceptor level and the third is therecombination probability.

The integration of Eq.(17) yields

m-m* =(na-n*)+(nd-n*) (20)

Substituting (16) and (20) in Eq.(19), we obtain

I(f) = ~=yaAmmF(m) (21)at

where

Aa{Na+Nd -i,; -nd+m')+(Am-AaiNd - < ( 4 J ""Assuming now that trapping in donor level is larger than both retrapping in acceptorlevel and in recombination center, this means

M*< -«<)>•>• *M, -»a)+Amm (22)

so Eq.(19) becomes

Furthermore, assuming that retrapping in the shallow traps is very little compared tothe rate of release of trapped electrons, i.e.

yanayyAanc{Na-na)

Eq.(ll) becomes

dnn- ^ = ^(ana (24)

which gives the following solution

na=natx^[-[yadxj

Substituting now Eqs. (16) and (24) into Eq.(23), we get

_ dm _^'">***{-k •*)

which can be integrated as follows

I — dKd . d)dm = [y exp -[y-dx }dt

Since the integral on the right of the previous expression is equal to 1, we get

r -iV

m = rrC 1 2 - —L *<-<_

(23)

CHAPTER P/2 339

which, using the approximation n*a <-< Nd — n*d , becomes

[ Ad{Nd-nd}_

The area 5 under the glow curve is then given by

S=[l(t)dt= ja-d—dt = m*-m

so that

o ~ Tn r— —\

which can be transformed using Eqs. (7), (8) and (9):

m0 expj - - \NaAm [l - exp(- ncAf)]g v T /

with the condition nd0 = rn0 at the end of irradiation and immediately before

illumination, the glow curve area becomes

«p(--V^-exp(-M/)]S = C ^ xJ =

J^-exp(-/f)LndO

which describes the PTTL peak produced by the shallow trap as a function of theillumination period 0 —t*.

References1. McKeveer S.W.S. and Chen R., Rad. Meas. 27 (1997) 625


2. Chen R. and McKeever S.W.S., Theory of Thermoluminescence andRelated Phenomena (World Scientific, 1997)

3. Alexander C.S. and McKeever S.W.S., J. Phys. D: Appl. 31 (1998) 2908

Post-irradiation annealing

The post-irradiation annealing is the thermal procedure having the aim toerase all the low temperature peaks which could be errors in the dose estimationbecause their high fading rate (see Annealing: general considerations)

Post-readout annealing

The post-readout annealing is another way to indicate the annealingprocedure, i.e. the standard annealing, to be used before using again thethermoluminescent dosimeters.

Precision and accuracy (general considerations)

Before of the identification of the sources of error in thermoluminescentdosimetry, to classify them and finally to give suggestions on the procedures to beused to optimize the experimental results, some general considerations should begiven [1,2].

The results obtained by a dosimetric evaluation, based onthermoluminescence phenomenon, present a large dispersion and then highuncertainty. To identify all the sources of uncertainty it is necessary to write thegeneral relationship between the dose D and the correlated TL emission signal.Several factors may be present in the dose determination

D = Mnet.SrFc.Fst.Fen-Flin-Ffad (l)where

"" D is the absorbed dose in the phosphor,

™ Mmt is the net TL signal (i.e., the TL signal corrected for the intrinsicbackground signal Mo: Mne, = M- Mo),

" Sj is the relative intrinsic sensitivity factor or individual correction factorconcerning the ith dosimeter,

CHAPTER P/2 341

~ Fc is the individual calibration factor of the detector, relative to the beamquality, c, used for calibration purposes,

~ Fst is the factor which takes into account the possible variations of Fc due tovariations of the whole dosimetric system and of the experimental conditions(electronic instabilities of the reader, changes in the planchet reflectivity,changes in the light transmission efficiency of the filters interposed betweenthe planchet and the PM tube, temperature instabilities of the annealingovens, variation of the environmental conditions in the laboratory, changesin the dose rate of the calibration source, etc.),

~ Fen is the factor which allows for a correction for the beam quality, q, if theradiation beam used is different from the one used for the detectorcalibration,

~ Fa,, is the factor which takes into account the non-linearity of the TL signalas a function of the dose,

~ Ffad is the correction factor for fading which is a function of the temperatureand the period of time between the end of irradiation and readout.

All the conversion and correction factors, let us say to be in the number ofm, can be indicated by using the general symbol a, . In this sense, the relationbetween dose and TL reading, Eq.(l), can be rewritten as

m

D = (Mi-M0)]Jaj (2)y=i

Before going into a deep discussion, we have to say that a measurement,which is the "reading" in the present case of thermoluminescence, can be affected bytwo types of errors: the random and the systematic errors.

The random errors are variable in both magnitude and sign. For randomuncertainties a statistical procedure can be applied since their probability distributionis known.

On the contrary, a source of systematic errors has a constant relativemagnitude and is always of the same sign. A statistical procedure cannot then beapplied because the distribution is not known.

Furthermore, two terms are very important to discriminate between errors.These two terms are "precision" and "accuracy ".

Precision is a term related to the reproducibility of a system and concernsstatistical methods applied to a number of repeated measurements. Low precisionmeans that random uncertainties are very high.

Accuracy concerns the closeness with which a measurement is expected toapproach the true value and includes both types of uncertainties. The value of aquantity is considered "true" either by theoretical considerations or by comparisonwith fundamental measurements. The true value is also called "actual value". Themeasured value is called "indicated value" or " reading". High accuracy means thatthe measured value and the true value are nearly the same.

Random uncertaintiesRepeated measurements follow a normal distribution, which is

characterized by the standard deviation a of the group of results. From a statisticalpoint of view, for an infinite number of results 95% of them fall within 2<r of themean. This is the commonly applied criterion to specify the reproducibility orprecision of a system.

Let us denote by X the measured value of any quantity. For a normaldistribution, the probability of X to have a value between X and X+dX is given by

P(X)dX = —\=exp - ^ ~ ? dX (3)CTV27T [ 2CJ2 J

where\i is a constant equal to the value ofXat the maximum of the distribution curve;CJ, the standard deviation, is a measure of the dispersion or width of the curve(FWHM). The quantity a2 is the variance of the distribution.

Performing N measurements of the same quantity X, the best estimate of \xis given by the mean value of the N measurements:

— 1 N

X = — Yxt (4)

The best estimate of a2 is the variance given by

i » 2

s2=—Y*,-Jr) (5)

In the practical situations the X comes from a limited number ofmeasurements. In this case, one can perform repeated determinations of the average,let us say M. It must be noted that if M is large, the average value will have adistribution very close to the normal one whatever the distribution of Xis.

CHAPTER P/2 343

It is now possible to define the standard deviation of the distribution of theaverage, called standard error:

s\x) = —l-—Y(x, -x)2 = to (6)J V ( M - l ) ^ M

In many cases, as in the one of Eq.(2), measurements involve several quantities.This means that the value XQ$ a physical quantity, e.g. the dose D, is a function

of other physical quantities, e.g. the parameters a,-. Each of the separate quantitieshas a proper variance, i.e.:

Sh ,) , 522( 2 ) , S2( 3 ) , ... (7)

The variance of Xis then given by

S\X) = A2S?(al)H^)2S22+... (8)dctj da2

A similar expression holds for the average.Systematic uncertaintiesLet us again consider a physical quantity X which depends on the independent

measurements of separate physical quantities <Xj. Because the distribution functionsfor each of the quantities oij are not known in the case of systematic errors, themethods for combining the individual systematic uncertainties are less well definedthan for the random uncertainties. Several methods can be used in practice tocombine the different conmponents in order to give the overall systematicuncertainty A/if.

The first method considers a simple arithmetic addition

8X r)XAX = (AX)ai +(AX)a2 +...= ~ - A a , + ^ A a 2 +. . . (9)

1 2 5 a , oa2

The second method is to combine them in quadrature

A =(A ,+(A ,...= (f~)Vg)W...(10)


Because the first method overestimate the total systematic uncertainty while thesecond tends to underestimate it, it has been suggested to multiply by 1.13 the resultof systematic errors. The factor 1.13 is necessary to ensure a minimum confidencelevel of 95%.

References1. Busuoli G. in Applied Thermoluminescent Dosimetry, ISPRA Courses,

Edited by M.Oberhofer and A. Scharmann. Adam Hilger publisher (1981)2. Marshall T.O. in Proceeding of the Hospital Physicists' Association.

Meeting on Practical Aspects of TLD. Edited by A.P.Hufton, University ofManchester, 29th March, 1984

Precision concerning a group of TLDs of the same type submitted to oneirradiation

One group of the same type of TLDs is annealed, irradiated and thenreadout. The variations in the precision are mainly due to the following causes:

~ variation in the mass among the TLDs group

~ variation in the optical transmission from sample to sample

"" instability of the TL reader during the period of the measurement

The precision is expressed by the following equation:

TOT ^ 1 0 0 J BKG

where

asis the percentage standard deviation of the dosimeter group irradiated at the

doseD

O BKG is the standard deviation of the background readings of the unirradiated

dosimeters.

CHAPTER P/2 345

ReferenceBusuoli G. in Applied Thermoluminescence Dosimetry, ISPRA Courses,Edited by M.Oberhofer and A.Sharmann, Adam Hilger Ltd., Bristol (1981)

Precision concerning only one TLD undergoing repeated cycles ofmeasurements (same dose)

A single TLD is repeatedly annealed, irradiated at exactly the same doseand read-out. All the experimental parameters must be kept constant. Variations inTL readings are then observed. The sources of the reading variations are mainly dueto:

™ dosimeter's background signal, or zero dose reading, and its variations;

"" electronic instability of the reading system

The precision may be expressed by the following expression giving the totalstandard deviation in a series of repeated measurements, at any dose D, carried outon only one single dosimeter

° r O T = i i o o D ] +G1KG

where

s/nn is the percentage standard deviation of the repeated

measurements when the background is negligible,D is the absorbed dose,

\KG is the variance of the readings of the unirradiated dosimeter

expressed in equivalent absorbed dose.

ReferenceBusuoli G. in Applied Thermoluminescence Dosimetry, ISPRA Courses,Edited by M.Oberhofer and A.Sharmann, Adam Hilger Ltd., Bristol (1981)

Precision concerning several identical dosimeters submitted to different doses

The equation given for the precision a group of TLDs of the same typesubmitted to one irradiation

can now be used to test the reproducibility of a TL system using a batch of TLDs(let us say 10 TLDs of the same type) which are irradiated consecutively to dosesstarting with the lowest detectable dose of the system, LDD, up to 1000 LDD. Thisis the approach used by Burkhardt and Piesh [1] and further developed by Zarandand Polgar [2, 3]. The relative standard deviations, obtained at different dose levels,are then compared to the theoretical two parameters Eq.(l). The lowest detectabledose, LDD, according to [4], is defined as three times the standard deviation of thezero dose reading of the non-irradiated batch. The characteristic shape of (oj/D)% asa function of dose D is shown in Fig.6.

The behavior of the plot can be interpreted considering the effect of twocomponents:

" aB, which is the main parameter affecting the reproducibility in the low doseregion (<100 LDD), and takes into account the intrinsic variability of the TLsystem (batch quality and readout technique), which is evaluated by thereadings of the non-irradiated TLDs;

" as takes into account the variation of the TL system as a function of theirradiation (batch calibration, annealing and reader quality).

To test the validity of Eq.(l), a set of 4 LiF:Mg,Ti (TLD-700) has beenirradiated at several doses from 0.3 mGy to 1 Gy. Each reading was then correctedby the individual background and the individual correction factor. After that theaverage value and the relative standard deviation were calculated. Table 1 shows theexperimental results.

Figure 7 shows the plot of the relative standard deviation, in percentage, asa function of the dose.

The following data were used:

aB = 3.4-10"2 mGy and as = 1.78-10"2 reader unitsso that

(1)

CHAPTER P/2 347

^ % = 1 0 0 X J ( C T 5 ) 2 + ( ^ - ) 2

D YD

V D 2

1 .OE+03 i 1

^ 1.0E+02 -:

£ -avi - •

05 1.0E+01 -j '— .• • • • • •••»

1 . O E + 0 0 -I • ' i ^ I1.0E+00 1.0E+01 1.0E+02 1.OE+03

DOSE

Fig.6. Theoretical behavior of the two parameters function given by Eq.l.

l.OE+02 -r 1

Il.OE+01 ° »RSD(TH)%g i ; •RSD(EXP)%

• • •

1.OE+00 I • " < — ' - A - ' " ' " " " I1.0E-01 1.0E+01 1.0E+03

DOSE

Fig.7. Comparison between experimental and theoreticalvalues of Eq. 1 as a function of dose.


~D I ~M ^ o % " a% ~D ~M ^ CT% a%

(wGy) (exP) (th) (/wGy) (exP) (th)

0.3 ' 1.194 20 ~69.6371.092 67.8570.991 68.6370.887 1.041 12.7 11.4 67.647 68.440 1.5 1.8

0.55 2.005 30 101.437L742 104.4372.017 103.1371.787 1.888 7.6 6.4 106.617 103.82 2.1 1.8

1 3.493 50 170.7373J37 169.9373.547 165.7373.357 3.530 4.5 3.8 164.737 168.00 1.8 1.8

2 6.863 100 332.0786.812 339.5187.017 331.4206.547 6.800 2.8 2.5 341.010 333.01 1.5 1.8

5 16.737 300 1027.0117.207 994.03116.437 985.07117.137 16.880 2.2 1.9 999.075 1001.3 1.8 1.8

10 33.837 1000 3371.5133.450 3371.0233.900 3500.15

I 35.437 I 34.406 I 2.2 | 1.9 [ | 3401.73 | 3411.1 | 1.8 | 1.8

Table 1. Comparison of experimental and theoretical data for a?.

References1. Burkhardt B. and Piesh E., Nucl. Instr. Meth. 175 (1980) 1592. Zarand P. and Polgar I., Nucl. Instr. Meth. 205 (1983) 5253. Zarand P. and Polgar I., Nucl. Instr. Meth. 222 (1984)5674. Busuoli G. in Applied Thermoluminescent Dosimetry, ISPRA Courses,

Edited by M.Oberhofer and A.Scharmann, Adam Hilger publisher, Bristol(1981)

CHAPTER P/2 349

Precision concerning several identical dosimeters undergoing repeated andequal irradiations (procedures)

First procedureAn accurate way for studying the reproducibility of a TLD system (detector +

reader + annealing + irradiation) can be performed using several dosimeters (i.e., 10- 20) of the same type [1]. All the dosimeters are annealed, irradiated and read outand the same procedure is repeated several times (10-20 times) as it is shown inFig.8. The analysis of the coefficients of variation allows us to determine thedifferent sources of variation: i.e., the variation of the system, the reader and the TLelements.

The following Table 1 shows, as an example, the matrix of the results obtainedfrom a sample of 10 TLDs (LiF:Mg,Cu,P).

From the data it is possible to calculate the following quantities:

• %CV which is the mean value of the percent standard deviations of eachdetector; it would give an indication of the reproducibility of the whole system.It is called here "system variability index" , SVI.

• %CV which is the percent standard deviation of the mean values of each cycleof readings; it would give an indication on the long term reader reproducibility.It is indicated here as "reader variability index", RVI.

• from the previous quantities it is now possible to define an index of variabilityonly concerning the TL detectors. This index, called "detector variabilityindex", DVI, is calculated as the square root of the difference between thesystem reproducibility index and reader reproducibility index, both squared:

DVI = ^{SVlf -{RVlf 0)

Using the data in the example, we get

SVI = %CV = 1.16%

RVI = %CV = 0.62%

DVI = 0.98%Second procedure

Another type of test allows a much more sophisticated statistical analysisconcerning a TLD system [2]. This kind of analysis allows us to prove if:• there are differences in sensitivity between the TL dosimeters,• there are differences between consecutive irradiations, at the same dose,

produced by sensitivity variations of the TL elements, instability of the readersystem and differences between irradiations.


These differences can be considered as systematic uncertainties if they arealways of the same sign (i.e., a dosimeters which systematically presents a higherresponse) or as random if they are variable in both magnitude and sign. Let us callthese kind of variability as "adjunctive variabilities". The following analysis iscarried out in two steps: at first the analysis is based on the %2 analysis, which allowsto recognize if there is an adjunctive variability or not. In a second step, using the Fdistribution, it is possible to know the cause of the adjunctive variability.

The analysis has been done using 10 selected and individually calibrateddosimeters LiF:Mg,Cu,P (GR-200A). The test dose was 130 mGy obtained from acalibrated 90Sr-90Y beta source. The TLDs were processed 10 times according to theprocedure: annealing, cooling, irradiation, prompt readout. The matrix of the resultsis shown in Table 2.

The data can be affected by both systematic and random errors caused by theTL elements, the TL reader unit, the irradiation system, and the thermal treatment.

Let us define the following quantities:m = number of TL elements = number of columns =10n = number of readings (cycles) = numbers of rows =10Xy = jth reading of the ith dosimeteri = index of column = l,...,mj = index of row = 1,..., n

i m n

X = average of the nxm data (=100) = / L / C Xu (2)

nmttP J

The variance of the BX/H data is given by

A statistical estimate of variance can be performed when both systematicindividual sensitivity differences among samples and the systematic changes ofsensitivity in repeated experiments are eliminated. This means that the possibility todetermine an interval, including the true value of the standard deviation, is with aprobability of 90%.

The following quantities can then be defined:

S = II(^-^)2 (4)' j

which takes into account all the variations within the values;

(3)

CHAPTER P/2 351

nTLDs

ANNEALING « ,

IRRADIATION

PREREADOUT! n C Y C L E SANNEALING |

1 F 1 '

READERINITIALIZATION

i

SEQUENTIALREADOUT OF '

ALL TLDs

Fig.8. Operational flow-chart for reproducibility measurements

SA=nfc(Xi-Xf (5)7=1

which takes into account variations between repeated irradiations (cycles) and Xrj

is the average value of the readings of all the dosimeters in reference to the jth cycle.

SB = nfJ(Xci-Xf (6)

which takes into account the differences in sensitivity between dosimeters and Xcj

is the average of the repeated readings of the same dosimeter.

S0 = S-SA-SB (7)

which takes into account the residual variations excluding the previous effects due todifferences among successive irradiations and among dosimeters. It takes intoaccount the variations due to the random effects only.

For each of the above quantities it is possible to calculate the degree offreedom, n, as the number of the elements which contributes to the quantity minus areference value which is the average one. Furthermore, the quantities

' > > \°)v vA v * v0

are the estimates of the corresponding variances with the following means:

2 " 0< J 0 = — (9)

Vo

which concerns with the intrinsic variation that one should have if only onedosimeter is irradiated several times;

°2A=^ (10)vA

takes into account two different components: the first is the intrinsic variation andthe second component is due to possible variations from one cycle to another (roweffect).

CHAPTER P/2 353

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CHAPTER P/2 355

The differences between the dosimeters (column effect) are not consideredbecause the average values XrJ are used.

Vl=— (11)

considers also two components: one is the intrinsic variation and the other concernsthe differences between detectors (column effect). The row effect turns out to beunimportant because the average values Xci are used.

Because So deals with the random uncertainties only, the quantity So I I

is a statistical variable which is %2 distributed with rio degrees of freedom. It is then

possible to determine a X95°/o(v),[x52%(v) , value such that the probability of

finding a value smaller, [larger], than it, in a single evaluation of So I I is

95%,[5%]. This is expressed by the following relation:

*j xs% (v)< ^ r < X95% (v) = 0.90 (12)

where P stands for probability. Rearranging, one finds for the standard deviation

/ f j - r2—<a 0 <j4^1 = OSH) (13)

The experimental data are shown in Table 3.

Table 3. Values calculated using the experimental data in Table 2.

quantity degree of freedom varianceS=2883029.00 n=nm-l=99 s2=S/n=29121.51SA=782366.25 nA=n-l=9 sA^=SA/nA=86929.58SB=307833.06 nB=m-l=9 sB2=SB/nB=34203.67

SQ=1792829.60 | I

Considering now that the %2 values for n0 = 81 are respectively

xl% = 61.262 and %l5% =103.009

the previous expression for the probability P gives

P[l31.926<a0 < 171.071]

or in percentage

P [ O . 9 O % < C J O % < 1 . 1 7 % ]

Considering now that the average of the nm data is

X = 14592.85

with an experimental standard deviation (%) equal to 1.16%, and comparing thisvalue with the percent standard deviation interval, it is easily observed that theinfluence of the systematic uncertainty is negligible.

We can now invoke the F distribution which allows us to recognize if thevariation in the observations depends on the dosimeters and/or on the cycles. The Ffunction takes into account the ratio between two experimental variances. Using asusual a confidential limit of 5%, the following two quantities can be calculated

2 2

^ f = 3.93 FB=-%- = 155 (14)CTo CTo

FA and FB account for the differences among columns and rows respectively.Having considered a confidential limit of 5%, from statistical tables a so-

called critical value Fcr is obtained as a function of the degrees of freedom of thevariances. In the present case Fcr = 1.93. Since FA>Fcr, there are statisticallysignificant variations from cycle to cycle: annealing, irradiations and/or reader. Onthe other hand, because FB is very close to Fcr, there is a slight tendency towards adifferent sensitivity among the dosimeters yet without statistically significantevidence at the 95% confidence level.

The same procedure can be applied to the residual (second reading) in all caseswhen an oven annealing cannot be carried out. It is the case of plastic cards whichcannot be annealed at high temperature.

CHAPTER P/2 357

References1. Scarpa G., Corso sulla termoluminescenza applicata alia dosimetria.

University of Rome "La Sapienza", Italy, 15-17 February 19942. Furetta C , Leroy C. and Lamarche F., Med. Phys. 21 (1994) 1605

Precision in TL measurements (definition)

The precision concerning with TL measurements is related to the randomuncertainties associated with the measurement itself. The standard deviation of a setof measurements may be used to quantify the precision.

Pre-irradiation annealing

The purpose of the pre-irradiation annealing is to re-estabilish thethermodynamic defect equilibrium which existed in the material before irradiationand readout.

Pre-readout annealing

This is another way to indicate the post-irradiation annealing procedure.

Properties of the maximum conditions

An interesting feature results from the equations giving the maximumconditions for the first-, second- and general-orders respectively:

^ = SCXP(~'kJr (1)

p£ I" s'n0 (ru ( E \ 1 , ( E ")

or

(2)

lalY P *• V a1) J \ kTu)

with s = s'n0, and

or

kT2Mbs ( E \ , s ( 6 - l ) <*v f £ ^ ,

~ Equation (1) does not include the initial concentration n0, therefore thefirst order peak is not expected to shift as a function of the irradiationdoses;

~ on the contrary, owing to the dependence of s on n0 for b# 1, and throughit, on the excitation dose, one should expect TM- from Eqs. (2') and (3') -to be dose dependent.

(2)

(3)

(3)

QQuasiequilibrium condition

The quasiequilibrium assumption [1-4] is expressed by the followingrelation:

dnr dn dm— - «—, (1)

dt dt dt

where

nc = free electron concentration in the conduction band (CB),

n = trapped electron concentration,m = hole concentration in the recombination centers.

The assumption (1) means that the number of free electrons in theconduction band is quasistationary. Furthermore, if the initial concentration of thefree electrons is assumed to be very small, (1) means that the free charges do notaccumulate in the conduction band. The quasiequilibrium assumption allows ananalytical solution of the differential equations describing the charge carriertransitions between the energy levels during thermal excitation.

References1. McKeever S.W.S., Thermoluminescence of Solids, Cambridge University

Press, 19852. Chen R. and Kirsh Y., Analysis of Thermally Stimulated Processes,

Pergamon Press, 19813. Chen R. and McKeever S.W.S., Theory of Thermoluminescence and

Related Phenomena, World Scientific, 19974. McKeever S.W.S., Markey B.G. and Lewandowski A.C., Nucl. Tracks

Radiat. Meas. 21 (1993) 57

RRadiation-induced defects

The radiation-induced defects are localized electronic states occupied bynon-equilibrium concentration of electrons.

Randall-Wilkins model (first-order kinetics)

In 1945, Randall and Wilkins used extensively a mathematicalrepresentation for each peak in a glow curve, starting from studies onphosphorescence. Their mathematical treatment was based on the energy bandmodel and yelds the well-known first order expression.

The simplest model used for the theoretical treatment consists oftwo delocalized bands, i.e. conduction band (CB) and valence band (VB),and two localized levels (metastable states), one acting as a trap, T, and theother acting as a recombination center (R). The distance between the trap Tand the bottom of the CB is called activation energy or trap depth: E. This energyis the energy required to liberate a charge, i.e., an electron, which is trapped in T.The probability p, per unit of time, that a trapped electron will escape from the trap,or the probability rate of escape per second, is given by the Arrhenius equation,having considered that the electrons in the trap have a Maxwellian distribution ofthermal energies

p = s-exp\-~ J (1)whereE is the trap depth (eV), k the Boltzmann's constant, T the absolute temperature (K),s the frequency factor (sec1), depending on the frequency of the number of hits of anelectron in the trap, seen as a potential well.

The life time, x, of the charge carrier in the metastable state at temperatureT, is given by

T = p~l (2)

If n is the number of trapped electrons in T, and if the temperature is keptconstant, then n decreases with time t according to the following expression:


Integrating this equation

[^ = -\p-dt (4)

one obtains

« = « o e x p - s e x p f - — \-t\ (5)

where no is the number of trapped electrons at the initial time to = 0.Assuming now the following assumptions:

~ irradiation of the thermoluminescent material at a low enoughtemperature so that no electrons are released from the trap,

~ the life time of the electrons in the conduction band is short,

~ all the released charges from trap recombine in luminescent center,

~ the luminescence efficiency of the recombination centers is temperatureindependent,

~ the concentrations of traps and recombination centers are temperatureindependent,

~ no electrons released from the trap is retrapped.

According to the previous assumptions, the TL intensity, I, at a constanttemperature, is directly proportional to the detrapping rate, dn/dt.

where c is a constant which can be set to unity. Equation (6) represents anexponential decay of phosphorescence.

Remembering Eq.(5), we obtain:

/(/) = nos exp(^- —J exp - st exp(^- —J (7)

(3)

(6)

CHAPTER R 363

Heating now the material at a constant rate of temperature, fi = dT/dt, fromEq.(4) we have:

and again

"•"•"{-iM-^H (8)Then, using Eq.(6)

1(T) - v exp(- JL) exp[- ( exp(- £ ) dr] (9)This expression can be evaluated by mean of numerical integration, and it

yields a bell-shaped curve, as in Fig.l, with a maximum intensity at a characteristictemperature TM-

lM 7j-\

Fig.l. Solution of Eq.(9). TM is independent of theinitial concentration of trapped electrons, n0.


Some observation can be done in Eq.(9):

~ J(T) depends on three parameters E, s and b,

™ E has values around 20kT in the range of occurence of TL peaks,

- exp is of the order of 10~7,

A kT)~ when T is slightly greater than of To , the argument of the second

exponential function is about equal to unity and decreases withincreasing temperature. I(T) is then dominated by the first exponentialand increases very fast as the temperature increases. At a certaintemperature, TM, the behavior of the two exponential functions cancel: atthis temperature the maximum temperature occurs,

~ Above TM, the decrease of the second exponential is much more rapidthan the increase of the first exponential and I(T) decreases until the trapsare totally emptied.

ReferenceRandall J.T. and Wilkins M.H.F., Proc. Roy. Soc. A184 (1945) 366

Recombination center

A recombination center is defined as the one in which the probability ofrecombination with an opposite sign charge carrier is greater than that of thermalexcitation of the trapped carrier.

Recombination processes

The recombination processes between electrons and holes govern allluminescence phenomena. The following figure shows the possible electronictransitions in an insulator, as a thermoluminescent material is, involving bothdelocalized bands and localized levels.

The band-to-band recombinations are termed "direct" and therecombinations involving localized levels are termed "indirect".

For getting luminescence, recombinations must be accompanied byemission of light, which means "radiative" transitions. A "non-radiative" transitionis accompanied only by phonon emission.

CHAPTER R 365

Reference and field dosimeters (definitions)

The main difference between the so-called reference dosimeters and thefield dosimeters is caused by their uses.

The sole function of the reference dosimeters is to provide a mean responseto which the response of the field dosimeters is normalized in order to produce theindividual correction factors. The reference dosimeters can be defined as a sub-batch of dosimeters which has a relative standard deviation smaller than 2-3%: thismeans that their responses are very close to the average value as defined in thehomogeneity test.

The field dosimeters are used to monitor the radiation in all dosimetricapplications and to calibrate the TLD readers.

The group of reference dosimeters, in a number of Nr depending on the sizeof the batch, is chosen from the previous batch itself; i.e. 10 dosimeters over a batchof 100 seems to be a proper sample. Their net TL signal must be much closer to theaverage value, calculated after an irradiation test, than those of all the samples. Theyare representative of the whole batch and will never be used for field applications(personnel, environmental or clinical dosimetry).

Only in the case of a very limited batch of dosimeters all of them can beused as reference dosimeters, in the sense that reference and field dosimeters are thesame.

After annealing, irradiation and readout, the average value of the responseof the Nr reference dosimeters is calculated as follows

^_ %(!*,-MjM=M (1)

Nr

The average is associated with the %CV, calculated as

Relative intrinsic sensitivity factor or individual correction factor 51, (definition)

The general definition of Sh where the index i stands for the ith dosimeterbelonging to a given batch of N dosimeters, is the following

(2)

where

- Mt - MOi = Miriet;

~ M, is the reading of the ith dosimeter annealed and irradiated at a well

defined dose D;

~ MOi is the background reading of the same dosimeter after annealing and

not irradiated;

~ M is the average of the net readings of the N dosimeters, annealed andirradiated at the dose D;

Using the previous definition of St, it becomes a multiplying factor of the actualnet reading. However, in many scientific reports the £, factor is defined as theinverse of the one defined by Eq.(l), so that it becomes a dividing factor of thereading.

It must be noted that the M, - Mm values should be distributed around the

average value M of all the readings, so that we should have

Sk<l<Sh (2) -

where Sk and Sh denote the individual correction factors for the kth and hthdosimeters respectively.

The S, factor is associated with the proper dosimeter and used as amultiplying factor (according to its definition) of the net reading, to correct thedosimeter response at any delivered dose:

Mi<net(cor) = MUm,-Si (3)

The Si factor is a correction factor which is needed to avoid any readingvariations owing to the individual sensitivity of each dosimeter which, generally canvary from one to another dosimeter even belonging to the same batch.

During the use of the dosimeters, the 5, factors could vary owing to theirradiation levels (especially if high doses are used which can provoke some damagein the crystal lattice of the TL material), the thermal history and environmentalfactors, i.e., humidity and storage temperature. Because small variations in thesensitivity factors can produce large errors in the dose determination, it is imperativeto check the 5, values for a given batch from time to time.

(1)

CHAPTER R 367

Tables 1 and 2 list the $ factors determined for a batch of 28 TLD-100 and

tested over a period of more than seven years. The factors have been calculated

according to Eq.(l)

I 01/07 I I 28/11 I I 28/01 I I 16/11 I I 22/12 I1988 1991 1994 1995 1995233.2 (?) 523.3 380.1 535.1

_ _ mR ^ ^ ^ ^ ^ ^ ^ ^ mR ^ ^ ^ ^ mR ^ ^ ^ ^ mR ^ ^ ^ ^

_ _ 'net ' 'net ' 'net ' 'net ' 'net '1 1.011 1.035 2.114 1.016 2.244 1.046 1.659 1.027 2.314 1.0372 0.918 1.139 1.905 1.128 2.070 1.134 1.494 1.140 2.076 1.1563 0.792 1.321 1.676 1.282 1.807 1.299 1.316 1.294 1.847 1.2994 0.796 1.314 1.608 1.386 1.785 1.315 1.291 1.319 1.807 1.3285 0.967 1.082 2.004 1.072 2.174 1.080 1.598 1.066 2.206 1.0886 0.996 1.050 1.938 1.108 2.230 1.052 1.593 1.069 2.233 1.0757 0.941 1.112 1.860 1.155 2.077 1.130 1.503 1.133 2.096 1.1458 1.040 1.006 2.093 1.026 2.264 1.037 1.668 1.021 2.352 1.0209 1.086 0.963 2.168 0.991 2.378 0.987 1.739 0.979 2.408 0.99710 1.070 0.978 2.096 1.025 2.324 1.010 1.703 1.000 2.341 1.02511 1.000 1.046 1.976 1.087 2.233 1.051 1.590 1.071 2.237 1.07312 0.933 1.121 1.863 1.153 2.086 1.125 1.511 1.127 2.087 1.15013 0.859 1.218 1.708 1.258 1.955 1.201 1.383 1.231 2.020 1.18814 0.937 1.116 1.886 1.139 2.110 1.112 1.538 1.107 2.135 1.12415 1.158 0.903 2.383 0.901 2.579 0.910 1.854 0.919 2.646 0.90716 1.236 0.846 2.564 0.838 2.782 0.844 2.027 0.840 2.879 0.83417 1.207 0.867 2.469 0.870 2.659 0.883 1.937 0.879 2.749 0.87318 0.909 1.151 1.869 1.149 2.068 1.135 1.478 1.152 2.108 1.13919 0.836 1.251 1.716 1.252 1.889 1.242 1.381 1.233 1.947 1.23320 1.259 0.831 2.572 0.835 2.890 0.812 2.009 0.848 2.861 0.83921 1.118 0.936 2.291 0.938 2.537 0.925 1.810 0.941 2.589 0.92722 1.230 0.850 2.491 0.862 2.728 0.860 1.976 0.862 2.831 0.84823 1.184 0.883 2.414 0.890 2.730 0.860 1.936 0.880 2.752 0.87224 1.075 0.973 2.208 0.973 2.470 0.950 1.787 0.953 2.568 0.93525 1.043 1.003 2.196 0.978 2.485 0.944 1.834 0.929 2.579 0.93126 1.321 0.792 2.725 0.788 2.968 0.791 2.219 0.767 3.139 0.76527 1.268 0.825 2.671 0.804 2.851 0.823 2.132 0.799 2.987 0.80328 1.087 0.962 2.671 0.804 2.344 1.001 1.725 0.987 2.409 0.996R 1.046 2.148 2.347 1.703 2.400

Table 1. Individual correction factors.

N ° | St±<T | o %

1 1.032±0.011 " 1.12 1.139±0.010 0.93 1.299±0.014 1.14 1.322±0.031 2.35 1,078*0.009 0.86 1.071±0.023 2.17 1.135±0.016 1.48 1.022±0.011 1.19 0.983±0.013 1.310 l.O08±0.02O 2.011 1.066±0.017 1.612 1.13S±0.015 1.313 1.219±0.027 2.214 1.120±0.013 1.215 0.908±0.007 0.816 O.84O±O,OO5 0.617 0.874±0.007 0.818 1.145±0.008 0.719 1.242±0.009 0.720 O.833±O.O13 1.621 0.933±0.007 0.822 0.856±0.007 0.823 0.877±0.012 1.424 0.957±0.016 1.725 0.957±0.032 3.326 0.781±0.013 1.727 0.811±0.012 1.528 I 0.950+0.083 | 8.7

Table 2. Average values of the correctionfactors given in the previous Table 1.

ReferenceData provided by Dr.V.Klammert of the Nuclear Engineering DepartmentofCESNEF, Milan, Italy

Relative intrinsic sensitivity factor or individual correction factor St

(procedures)

General procedures for the determination of the individual correctionfactors are given in the following.

CHAPTER R 369

Is' procedure

~ annealing of all the samples according to the standard anneal proceduresuitable for the material in use.

~ readout of the samples, using the appropriate readout cycle, for determiningthe intrinsic background value of each dosimeter, MOi.

" irradiation of the samples to a known dose, chosen in the region of the linearresponse and at a level which is supposed to be used for the dosimeters inthe applications.

™ readout of the irradiated samples, in only one session, using the samereadout cycle used in the second point and determine the values A/,.

~ calculate for each sample the quantity

MKnet = M,. - MOi (1)

and calculate the mean response of the batch as

M = ^ Z ( M , - M O , ) (2)

~ perform a new annealing of the samples and re-irradiate at the same dosealready delivered in the third point. Read all the samples and calculate a newaverage value

~ repeat the procedure 5 times.

~ calculate the quantity

where y stands for the number of irradiations performed for the samples.

~ calculate the average response for each sample of the batch according to theexpression

(3)


1^ = S ^ (4)where ;' indicates the i-th dosimeter.

~ calculate the relative intrinsic sensitivity for each single dosimeter as

S, = =- (5)' Mi

This factor is quite stable during time so that it only needs to be checked nomore than two times per year.

Td procedureThe procedure just given above is the best but it is not easy to adopt with a

large number of dosimeters, as it can be the case of a personnel dosimetry service.When the batch of dosimeters is quite big, the correction factors can be

calculated making use of a sub-group of dosimeters, the reference dosimeters,chosen from the same batch in use, and then normalize all the dosimeters to theresponse of the reference dosimeters.

Tables 1 (a,b) list the data concerning a batch of 78 dosimeters. From thebatch, five dosimeters have been chosen as reference dosimeters, labelled with (*).

The total average on the 78 dosimeters is

~Mm, = 7.556 ± 0.434 (5.7%)

The average of the reference dosimeters is

Irftt = 7.595 ± 0.040 (05%)

The sensitivity factor for each dosimeter of the batch is then calculated as

-rrt

' " ' Mi-Mo,

The effect of the correction factors is well proved by observing that the newaverage value of the remaining 73 dosimeters is now

'Mnet,cor =7.596 ±0.002

which means a %CV of 0.03%.

CHAPTER R 371

Dos. Mi,net Si Mincer) Dos. Mi,net Si Mi]M,(cor)

No. No.

1 8.468 0.897 7.596 40 7.836 0.969 7.593

2 8.076 0.940 7.591 41 7.167 1.060 7.597

3 7.808 0.973 7.597 42 7.912 0.960 7.596

4 7.085 1.072 7.595 43 7.946 0.956 7.596

5 7.231 1.050 7.593 44 6.765 1.123 7.597

6 7.601 0.999 7.593 45 6.771 1.122 7.597

7 7.587 1.001 7.595 46 7.531 1.009 7.599

8 7.346 1.034 7.596 47 7.657 0.992 7.596

9* 7.634 - - 4 8 _ 7.045 1.078 7.596

10 6.916 1.098 7.594 49 7.434 1.023 7.605

11 7.394 1.027 7.594 50 7.476 1.016 7.596

12 7.491 1.014 7.596 51 7.239 1.049 7.594

13 8.094 0.938 7.592 52 7.480 1.015 7.592

14* 7.600 - - 5 3 _ 6.704 1.133 7.596

15 7.854 0.967 7.595 54 7.656 0.992 7.595

16 7.509 1.012 7.599 55 7.118 1.067 7.595

17 7.428 1.023 7.599 56 7.167 1.060 7.597

18 7.329 1.036 7.593 57 6.699 1.134 7.597

19 7.963 0.954 7.597 58 8.047 0.944 7.596

20 [ 7.290 I 1.042 7.596 59 | 7.395 | 1.027 | 7.595

Table l(a) Effect of the use of reference dosimeters.

DOS. Miinn Sj Mi,n^cor) DOS. Mi,™, Si Mi, , ,^,)

No. No.21 7.676 0.989 7.592 60 7.984 0.951 7.59322 7.294 1.041 7.593 61 8.057 0.943 7.59823 8.387 0.906 7.599 62 8.014 0.948 7.59724 7.677 0.989 7.593 63 7.555 1.005 7.59325 8.232 0.923 7.598 64 6.968 t.090 7.59526 8.143 0.933 7.597 65 6.720 1.130 7.59427 7.839 0.969 7.596 66 8.320 0.913 7.59628 8.111 0.936 7.592 67 7.778 0.977 7.59929 7.464 1.018 7.598 68 7.487 1.014 7.59230 7.374 1.030 7.595 69 6.786 1.119 7.594

31* 7.539 - - 7 0 _ 8.433 0.901 7.59832 6.739 1.127 7.595 71 7.424 1.023 7.59533 7.411 1.025 7.596 72 7.812 0.972 7.59334* 7.574 - - 73 7.402 1.026 7.59435* 7.633 - - 74 7.620 0.997 7.59736 7.880 0.964 7.596 75 7.025 1.081 7.59437 7.783 0.976 7.596 76 7.934 0.957 7.59338 7.872 0.965 7.596 77 7.334 1.036 7.59839 I 7.568 | 1.004 | 7.598 | 78 | 7.580 | 1.002 | 7.595

Table l(b). Effect of the use of reference dosimeters


3rd procedureOn the use of the reference dosimeters it is very useful to follow the

procedure suggested by P.Plato and J.Miklos of the School of Public Health of theMichigan University. This procedure is well indicated when a large number ofTLDs, larger than 10000, is used for dosimetric purposes.

As claimed by the authors, their procedure should ensure that the individualcorrection factors take into account only variations among the TL samples of a givenbatch and not variations caused by the instability of the TLD reader.

The authors suggest to divide a new batch into two batches:

~ the reference dosimeters,

"" the field dosimeters.

As stated before, the aim of the reference dosimeters is to provide a meanresponse to which the response of the field dosimeters is normalized to obtain the St

factors. In this way the response of each field dosimeter will be the same as themean response of the reference dosimeters. The number of the reference dosimetersshould be about 2-5% of the whole batch, according to its size.

A problem can arise if some reference dosimeters are lost or their presencechanges in response owing to the age. To by-pass this potential problem, theprocedure suggests the use of subsets of reference dosimeters rather than the wholereference group.

It must also be noted that the correction factors could be affected byirradiation, if this is not done uniformly, due to room scatter or if the beam is notisotropic. So, the irradiation geometry must be carefully checked for obtaining thatall the dosimeters are irradiated uniformly.

The procedure consists of several steps and it is shortly reported here.

~ annealed, irradiated and read the reference dosimeters,

~ the same procedure is repeated at least three times,

~ calculate the mean values for each irradiation and the coefficient of variation(CV) associated with

Mmt

It must be noted that the mean values obtained are calculated without thecorrection factors being applied because these factors do not exist at thislevel of the procedure.

CHAPTER R 373

~ the individual correction factors for each reference dosimeter are nowcalculated for each irradiation:

Ks =with

N M

1 tr Nwhere

i=l,2,...N is the number of the reference dosimeters,j=l,2,3,... is the number of the repeated irradiations,Sy is the individual correction factor for the ith dosimeter obtained afterthe jth irradiation,My is the response of the ith dosimeter after the jth irradiation,

Mj is the mean response for all the reference dosimeters after the j-th

irradiation.It must be stressed that during the whole procedure involving the threeirradiations, the calibration of the TLD reader could change significantlyfrom one readout session to another. However, the calibration factors areunaffected since they are based on the mean of a given irradiation.

~ the average values of the correction factors are then calculated for eachsample along the three successive irradiations

1 inand the CV% is obtained as well.

~ once the averages of 5; have been obtained, it is important to examine theirdistribution as well as the distribution of the associated CV%. If one or bothof these quantities are abnormally large, it is better to reject the defectivesamples. It should be advisable to identify and eliminate all the elementshaving an 5, that is not within 20% of unity (the acceptable range is thenfrom 0.80 to 1.20) and the elements which have a CV% greater than 5%.


The limits given for 5, and CV% can be dependent on several factors, i.e., thelevel of the delivered dose, the light emission from the phosphors, the light detectionefficiency of the TLD reader. The CV is strongly dependent on the dose; one canexpect to have a large value of the CV% at low doses and a little one for high doses.However, the suggested limits can be changed according to the specific use of thedosimeters.

When a large number of field dosimeters has to be used, it is better, as suggestedby the authors, to divide the field dosimeters in sub-batches and to do the same forthe reference dosimeters. As a consequence, each sub-group of field dosimeters willbe related to a proper sub-group of reference dosimeters. This procedure is necessaryto ensure that the TLD reader response will remain stable during the readout which,using a small quantity of TLDs, can be carried out in only one session.

When sub-groups of reference and field dosimeters are used, the St factors forfield dosimeters are calculated using the TL response of the reference dosimeterscorrected by the 5, factors already existing (see above); in this way, the meanresponse of the sub-set of reference dosimeters is the same as the mean response ofthe of the whole reference group.

The response of each sample of the sub-group of field dosimeters is corrected bythe appropriate St calculated according to the following expression

' Mref

whereM{ is the response of the ith field dosimeter and

Mref is the mean response of the sub-group of reference dosimeters. Remember

that this value comes from a set of values already corrected by the appropriate S;.

ReferencePlato P. and Miklos J., Health Phys. 49(5) (1985) 873

Residual TL signal

It is so called the TL signal obtained after the annealing procedure or after asecond readout cycle of the same sample. The observation of a residual TL signalmeans that the annealing procedure or the second readout cycle has not obtained theeffect to erase all the phosphor traps.

Any unerased TL signal may interfere with further TL measurements usingthe same sample.

CHAPTER R 375

The lower detection limit as well as the reproducibility are strongly affectedby the residual signal. The residual signal depends on the phosphor type as well asits irradiation history.

Rubidium halide

RbCl and RbBr can be growth as single crystals from the melt by theKiroupoulos method. Doping was achieved by adding suitable amounts of KOH tothe melt. The suggested annealing, for getting high sensitive material, is 600°C for30 minutes followed by quick quenching to room temperature. The TL sensitivity ofRbCl.OH' is decreasing as the dopant concentration increases. At the lowest dopantconcentration, i.e. 0.13 mol %, the glow curve exhibits a single peak at about 100°C.RbBnOH- reveals a glow curve consisting of two peaks: one at 70°C and another,less intense, at about 175°C. After irradiation at more than 20 Gy, a third peakappears at 230°C. Any way, both materials are affected by high fading.

ReferenceFuretta C , Laudadio M.T., Sanipoli C , Scacco A., Gomez-Ros J.M. andCorrecher V., J. Phys. Chem. Solids 60 (1999)957

sSecond-order kinetics when s'=s'(T)

The detrapping rate in this case is given by [1-3]

Using a linear heating rate, Eq.(l) becomes

dn n2s> a ( E\-aT=-T~T e X P ^ ^ J (2)

and the solution is then

while the intensity is given by

n2os'oTa exp( -y- )

/ ( r ) = r ; / PN r (4)




(World Scientific Publ., 1998)

(1)

(3)


Self-dose in competition to fading (procedure)

The self-dose arises from the radioactive content of the thermoluminescentmaterials. With fading, the self-dose is an important factor in environmentalradiation monitoring and it is strongly dependent on the packing materialsconstituting the dosimeter. Self-dose and fading are two effects in competitionbetween them.

A precise estimation of self-dose effect needs an accurate experiment. Amethod of estimating accurately both self-dose and fading under conditions similarto the ones encountered in environmental monitoring applications consists of leavinga batch of TLDs in a sufficiently thick lead shield of about 5 cm to stop most of theexternal irradiations (only hard components of the cosmic rays will contribute to theradiation field inside the lead shield).

The experimental procedure is the same of the previous one and, as before,we have again three sub-groups of TLDs, group A, B and C.

After having done all the initial procedures, group A is irradiated at the testdose Dt and stored inside the lead shield together with the annealed one groups Band C but not irradiated.

At the end of the storage period, let us say 1 month, group B is irradiated atthe same dose Dt and all three groups of TLDs are read.

Now we have three quantities, the averaged readings, which are linked bydifferent equations:

MA=^ + (MB-^)cxp(-lta) (1)

A7c=*[l-exp(-\ffl)] (2)

MA-Mc=MBexp(-Xta) (3)

where ta is the storage period of time and B = B^ + Bs , with the first component

being the field dose rate inside the shield and the second the self-dose rate.The decay constant is now given by the following expression

J^-J-ln^^S: (4)

K MB

which substitutes in Eq.(2) gives:

CHAPTER S 379

B- mc ( 5 )

l-exp(-Xr,)

The component &,, the field dose rate inside the shield, can be measured

by a high pressure ionization chamber; after that the self-dose rate can easily beevaluated:

B^V^TY^-Bf (6)

l -exp(-X-O

Sensitization (definition)

Sensitization is a term used to indicate an increase of sensitivity in a TLsample due to a high dose of irradiation, usually followed by a heating treatment.This effect has been firstly found by Cameron in LiF [1-3].

References1. Cameron J.R., Suntharalingam N. and Kenney G.N., Thermoluminescence

Dosimetry, University of Wisconsin Press, Madison (1968)2. McKeever S.W.S., Thermoluminescence in Solids, Cambridge University

Press (1985)3. Chen R. and Kirsh Y., Analysis of Thermally Stimulated Processes,

Pergamon Press (1981)

Sensitivity (definition)

The sensitivity of a TLD, S, may be expressed, in general, as the TLresponse, in reader units, per unit of dose and unit of mass of the sample:

s = TLDm

Variations in sensitivity, among dosimeters belonging to the same batch,can be encountered in practice. The variations in sensitivity are mainly due to thefollowing reasons:

~ variation in the mass of the detectors,


~ variation in the optical density from sample to sample,

~ variation due to dirt contamination of the sample surface.

Set up of a thermoluminescent dosimetric system (general requirements)

For the setting up of the system, one needs to fill up some requirements forinitializing, characterizing and calibrating the TL material according to the use for.These operations consist of several tests and measurements:

~ initialization procedure,

" determination of the batch homogeneity,

~ to choose the reference dosimeters,

~ to determine the relative intrinsic sensitivity of each dosimeter,

~ the measurement of the threshold dose,

~ to determine the linearity range of the system and its calibration factor,

~ to carry out the reproducibility tests: of the reader, of the background, ofthe calibration factor, of the dosimeters,

~ to study the appropriate annealing procedure in case using a new TLmaterial for which no information in the scientific literature is available.

~ to carry out a quality control of the instruments for the thermaltreatments of the dosimeters (rise temperature curve, temperaturestability, temperature distribution inside oven, etc.).

The following points are very important prerequisites before starting theexperimental procedures listed above.

•" select dosimeter elements having approximately equal mass.

~ reject elements which are imperfect, discolored or dashed.

"" do not handle elements directly; use tweezers (preferably vacuumtweezers) or spatulas for TL powder. Avoid scratching the surfaces ofthe dosimeters.

CHAPTERS 381

~ do not leave the dosimeters uncovered in the laboratory. It is better tostore the dosimeters in opaque bags or containers.

~ some dosimeters are sensitive to sunlight, UV light or developbackground effects when exposed to UV light. It is advisable to usetungsten, filtered fluorescent lighting or red lamps and to keep thedosimeters away from direct sunlight.

~ keep the dosimeters away from heat and radiation sources duringstorage. It could be better to store the dosimeters in lead boxes to avoidany background irradiations.

Simultaneous determination of dose and time elapsed since irradiation

Information about the time which has elapsed since an external radiationexposure is useful in determining the time of occurrence of an abnormal exposure inpersonal and accident dosimetry.

This information can be obtained from certain irradiated thermoluminescentdosimeters and the method consists essentially of a glow-curve behavior study. Theglow-curve is a finger print of the radiation effect in a particular TL phosphor. Theglow-curve may consist of several peaks each having its maximum intensity atdifferent temperatures. Those peaks which occur in the temperature region above150°C are generally thermally stable, and the integrated area or the eight of suchpeaks is used to assess the absorbed dose from radiation exposure. However, in theglow-curve region below 150°C there are also less thermal stable peaks.

From a practical point of view, if one peak has faded and another hasscarcely faded, the peak area or the peak height ratios will be function of the timeafter exposure and then the time can be estimated throughout the ratios. By using thedecay rates of suitable peaks in the glow-curve and, in turn, the corresponding meantrap life times, t, the time elapsed since a single exposure may be determined.

The feasibility of the method has been theoretically investigated simulatinga glow-curve as shown in Fig.l. Furthermore, the simulation considers the combinedeffects of a single exposure superposed to a continuous background exposure [1]. Afurther work [2] gave a theoretical model of the method as well as a comparisonwith experimental results. The same subject has been investigated in [3-4].

• Theoretical modelExpressions for a sinele accidental exposure

The mathematical treatment starts from the first- and second-orderequations:

„ l2 r - p\ \< I / t \

•« r / ' \ i

Temperature !°KS

Fig. 1. A glow curve showing two well defined peaks.

dn ( E^— = -sn exp 1st order (1)dt \ kTj W

dn , 2 ( E \ A

— = -s'n2 exp 2nd order (2)dt \ kT)

Integration of the previous equations gives, respectively, the followingsolutions

( E\« = »oexp -stexp\-~\ (3)

/J = #J0 l + s\texA~— I (4)

The TL intensity is given by

T, N dn/(0«--7" (5)

dt

and then Eqs. (3) and (4) can be rewritten in the following way, respectively for the1st and the 2nd order of kinetics:

CHAPTER S 383

/(/) = nos expf- — 1 exp - st expf - — 1 (6)

W o V e x p (" j i ; ]7 ( 0 = r rS?

Taking into account the total TL light

O, = [l(t)dt (8)

and using Eq.(6), it turns out that o l + ^ o ' e x p f - - ^ j (10)

Introducing the mean trap lifetime for both 1st and 2nd order kinetics respectively

X=S~leXV{]tf) (H)

x*=(j'/io)-1exp^J (12)

Eqs. (9) and (10) can be written in the following way

O = <D0 expf - - I 1st order (13)

(7)


O = O0[ 1 + — 2nd order (14)

Expressions including a continuous irradiationThe second contribution to the final equations is the signal due to a

continuous irradiation, i.e. environmental irradiation background. The equationshave to take into consideration a progressive extinction of the initially storedinformation, i.e. the accidental irradiation signal, whereas the environmentalcontribution leads to an increase of the TL signal. Under this condition, the previousEqs. (1) and (2) assume the following forms

dn ^ n ,— = C X - Z ) - - 1st order (15)dt X

— =a-D-— 2nd order (16)dt x*

where £) is the environmental exposure rate and a is a constant, typical for eachtherrnolurninescent material and giving its sensitivity (TL per unit of dose andmass).

Integration of Eq.(15) gives

n = Cexp — +a Dx

which, using the initial condition n(0)=0 , becomes

n=a-Dt \-exp - ~ (17)

As the elapsed time becomes very large, n gets more and more similar to theasymptotic value

nx=a-Dt (18)

Equation (14) is explained assuming that, at infinity, a dynamicalequilibrium is attained, providing the trapped charges to compensate at each instantthose escaping owing to the fading phenomenon.

Equation (17) can be changed using the total light

CHAPTER S 385

< t > = a Z ) x 1-exp [ - - ] (19)

Integration of Eq.(16) yields to the final 2nd order expression

, + exp-^f.,Eq.(20) can be rewritten in a simpler way as

O^fa-D-x'-O^tanhl-0^-)2-/ (21)\y -Go)

Final expressionsThe equations related to a single accidental exposure and to a continuous

irradiation have now combined. The accidental exposure can be thought of asoverwhelming; then a characteristic time tt has to be introduced as the time intervalelapsed from the zero instant to the time of the accidental exposure. Figure 2 showsthe superposition of the accidental exposure on the background irradiation for a 1st

order. The accidental exposure has been assumed to occur in the middle of theobservation period, i.e. U = 15 days over a period of 30 days. Until the 15th day onlybackground is present. At the 15th day, as a consequence of the external accidentalirradiation, a sharp discontinuity occurs which is assumed to be as large as 1 Gy.

The overall equations can be written by combining Eqs. (13) and (19) forthe first order kinetics, and Eqs. (14) and (21) for the second order:

(20)

r 1 ———i 1 1

103 - r - _^^_^

101 - ]3 !

ca _

a ID - /

* /

£ I10° -

1ft-11 1 I I I0 10 20 30

Days

Fig.2. Effect of an irradiation superposed to thebackground irradiation.

(D = <Doexp l - ^ - +a-Z>-T 1-exp [ - - ] (22)

tD = <Dofl + l +(a-D-xt-O0Y2tanh\^-\ -t (23)

Assuming now a glow-curve having two peaks, it is necessary to define thearea ratio, R, between them. For the 1st order kinetics we get:

CHAPTER S 387

^ e x p - — +a2-Z>-c2 1-exp - —

».%.. L J ^ — L y (24)O01exp L + a , - 0 - T , 1-exp - —

and

/v = = p =-

(25)

for the second order kinetics.The indexes " 1 " and "2" refer to the first peak, i.e. high fading, and to the

second peak, no fading, respectively.The Equations (24) and (25) have been computed for some values of the

mean trap lifetimes. Figure 3 shows the trend of the peak area ratios for theparameters given in Table 1 and for different mean trap lifetimes.

Ooi <f 02 a , a 2 D t t-tj T 2 = T *

90 100 90" 100 "3x103 mGy/d 30 days 40~days 400 days'

Table 1. Parameters used for computing Eqs. (24) and (25).

From Fig.3 it is evident that for practical application one needs a steep line:only in this case an accidental exposure can be accurately backdated. If the lines aretoo flat, the error in time determination will be very high even when two elapsedtimes are very different.


, , 1 , , , , - , ,

500 h J

f 1st order kinetics Tt=1day

100 - / ">

/ 1st order kinetics Tj=,5days

50 " / yr "

*5 / £^inA order kinetics t*=1day (

\-i //V I / /

] 1/ 2ni order kinetics T*=50days

y/s^^"~"— 1s* orl'er toefki tjrSOdsys

^ 5_ .^ . ^ _ ^ ^ ^_

Elapsed time It-fjl (days!

Fig.3. Theoretical peak area ratios as a function of elapsed time.

CHAPTER S 389

I f37Cs - 200 ro% ,,^*~~~~~~-~*<cI

L v ^ W a f a 2I

••*** I— J F * * ™

- if / J"5 i / iji? j I I

* ' • ^4_——^-" -+~—f 101 I

II / ^ ^ Wata 1I i

5 5 10 S 20 ~ "~"2S 30 35EUpm4 fime ff-tjf {days]

Fig.4. Experimental data. Peak ratios vs elapsed time.

Observing Fig.4 it appears evident that the peak-area ratio is more usefulthan the peak-height ratio because, in the former case, the elapsed time afterirradiation can be estimated with a smaller error.


Looking at the peak-area ratio in Fig.4, the maximum error in timedetermination can be about ± 2 days when the accidental irradiation occurs in therange of 0 - 15 days. If the elapsed time between irradiation and readout is largerthan 15 days, the uncertainty becomes larger. In any case, the peak-area ratio givesbetter figures than the peak-height ratio.

A mathematical approximation has been done for fitting the experimentaldata. For this purpose, a polynomial approximation, using the Tchebychev's norm,

has been carried out. For example, the plots of 2 / l , labelled Data 1 and Data 2,

have been fitted by the following 4th degree polynomials:

t9 = -20.2 + 38flo - ZSRl + 5JRl - 037R^ Data 1

f* =33.9-33.7** +11.3/?* -1.5/J* +0.07i?; Data 2

References1. Furetta C , Pani V, Pellegrini R. and Driscoll V, Rad. Eff. 88 (1986) 592. Furetta C, Tuyn J.W.N., Louis F., Azorin-Nieto J., Gutierrez A. and

Driscoll C.M.H., Appl. Radiat. Isot. 39 (1988) 593. Furetta C. and Azorin J., Nucl. Instr. Meth. A280 (1989) 3184. Budzanowski M., Saez-Vergara J.C., Gomez-Ros J.M., Romero-Gutierrez

A.M. and Ryba E., Rad. Meas. 29 (1998) 361

Sodium pyrophosphate (Na4P2O7)

This material, whose effective atomic number is about 11, is suitable whendoped with dysprosium for obtaining TL dosimeters useful in accident monitoring.

The phosphor preparation consists of a mixing of commercially availablesodium pyrophosphate and dysprosium oxide Dy2O3 in the ratio of 1000: 1 byweight. The mixture is heated at 100 °C for one day under vacuum to remove allmoisture, melted at 880°C at a pressure of 1.33 x 10"2 Pa, and then cooled downslowly. The poly-crystalline mass is then grounded into particles ranging from 60 to100 mesh in size. The recommended annealing procedure is at 400 °C for 1 h beforeexposure.

The glow curves of Na4P2O7: Dy show three glow peaks at 90,181, and 228°C. The 90 °C peak fades away within a few hours after exposure, and the 228 °Cpeak has a negligible intensity.

Linearity of the response to y-rays is observed in the range between 1.6 KRand 13 KR. The sensitivity is comparable with that of LiF TLD-100. Thermalneutrons can also be detected.

CHAPTERS 391

The photon energy response is found to be not as good as that of LiF TLD-100.

ReferenceKundu H.K., Massand O.P., Marathe P.K. and Venkataraman G., Nucl.Instr. Meth. 175 (1980) 363

Solid state dosimeters

Common solid state dosimeters include:(i) the photographic emulsion, which darkens upon exposure to radiation;(ii) the silicon diode, which suffers radiation-induced changes in electrical

resistance under fast neutron irradiation;(iii) certain crystals which change color upon irradiation;(iv) crystals which present luminescence phenomena (see luminescence

dosimetry);(v) irradiated crystals which present, upon heating, a transient increase in

electrical conductivity (thermally stimulated conductivity, TSC);(vi) irradiated crystals which present, upon heating, a transient emission of

electrons from their surface (thermally stimulated exoelectronemission, TSEE).

Solid state dosimetry

Solid state dosimetry deals with the measurement of ionizing radiation bymeans of radiation-induced changes in the properties of certain materials (see solidstate dosimeters).

Spurious thermoluminescence: chemiluminescence

Chemiluminescence is another spurious TL emission which can alter theradiation induced TL response, especially in the range of very low doses.

Chemiluminescence effect has origin from impurities which cancontaminate the surface layer of the dosimeter. During readout of the TLD sample,the excitation of the impurities provokes a non-radiative signal which is superposedto the radiation induced signal.

The chemiluminescence effect is mainly produced by the oxidation of thesurface of the TL phosphors.


Spurious thermoluminescence: surface-related phenomena

The TL light emitted during readout of a sample may be contaminated bynon-radiation-induced signals (spurious thermoluminescence) which restrict thelower limit of detection.

Spurious thermoluminescence: triboluminescence

Triboluminescence indicates an emission of luminescence stimulated bymechanical stress, during readout of TL samples, and it is a spurious signal to beavoided otherwise it increases the detection threshold as well as the errors in thedose determination.

This phenomenon is much more evident in TL phosphors used in powderform than in solid chips. Furthermore it depends on the dose given to the dosimeter;in the range of high doses the phenomenon is less important.

Schulman and colleagues [1,2] carried out specific experiments to study theeffect and showed that the triboluminescence signal can be eliminated by justheating the TL sample in an oxygen-free atmosphere. The best results are obtainedperforming the TL readout in an atmosphere of inert gas, i.e. argon or nitrogen.

The effect of oxygen as well as of the inert gases on triboluminescence isnot understood and no theoretical explanation has been given until now.

References1. Schulman J.H., Attix F.H., West E.J., Ginther R.J. - Rev. Sc. Instrum. 32

(1960) 12632. Nash A.E., Attix F.H., Schulman J.H. - Proc. Int. Conf. Lumin. Dos.

(Stanford), 244 (1965)

Stability factor Fs, (definition)

This parameter is useful to check any possible variation in the stability ofthe reader and/or in the irradiation facility. Also in this case the procedures can bedifferent from one another according to the various laboratories. In the following isreported the most usual procedure and some suggestions are given for itsimplementation.

CHAPTER S 393

Stability factor Fs, (procedure)

The stability check of the reader and/or of the irradiation facility is carriedout using a group of reference dosimeters and the procedure is based on the controlof Fcr at any new session of readings. Making reference to the equation (seeCalibration factor Fc - 1st procedure)

K = -r^ <»

which is supposed to be determined at the beginning of the first use of a new batchof TLDs, one can use the same equation in all the period of use of the batch, andcalculate the value of Fcr at the beginning of each session. It must be noted that inthis way the period of time between a control and the subsequent readout could be ofseveral weeks. Therefore, before starting a new session of readings, a newcalibration factor is determined. Let us indicate it as

K,=T^T"— (2)

Note that the dose in Eq.(2) is the same as that in Eq.(l); also the same arethe Sr values. Only the TL response can be changed if variations occurred in thereader or/and in the irradiation facility.

The stability factor is then determined as a ratio between the two factorsFcr and F'cr as follows

If all the experimental conditions remain constant between the firstdetermination of F and any other subsequent determinations, the Fs, value will bemore or less equal to 1; differences within 1-2% among the lvalues confirm a verygood stability for both the reader and the irradiation system.

(3)


In some procedures it is suggested to determine Fcr, the former calibrationfactor, as an average of several factors obtained by irradiating the TLDs of thecontrol group several times, at least 5. On the contrary, the subsequent factors, F\are obtained after one irradiation only.

The previous procedure cannot be considered the best one because, asalready mentioned, the time between a control and the subsequent readout can belong enough, and sometimes it does not allow this kind of procedure. On thecontrary, the dose can be estimated using the actual calibration factor, F\ withoutany references to the previously F\ determinations. On the other hand, a stabilitycheck is very important if it is done during the readout session itself. This check isusually done at the beginning of the readout session and at the end of it. Dividing thereference dosimeters into two sub-groups, the first group is used as the reference atthe beginning of the session and the second group is read at the end of the session.The two factors are then compared with the same procedure just mentioned above. Ifthe number of dosimeters used for the field application is larger than 100, three ormore sub-groups of reference dosimeters can be used: one at the beginning of thesession, one at the end and the others during the session.

It must be noted that in this case the possible variations in the F values canbe attributed to the reader only, because all the reference dosimeters have beenirradiated altogether at the beginning and so there are no uncertainties due to theirradiation facility introduced in the whole procedure.

An example concerning the stability of the calibration factor is given here.This test has been carried out over a period of 5 weeks. Five TLDs have beenselected, prepared and irradiated at a dose of 12.44 mGy. The 5 TLDs have beenread immediately after irradiation. After one week the procedure was repeated andso on over the whole test period of 5 weeks. Table 1 shows the results obtained withthis test. For each dosimeter the individual background was determined afterannealing. After irradiation, the readings were corrected by background subtractionand by the individual correction factor. Note that the 5, factors here are dividingfactors.

Considering the first calibration factor, FCfl, determined at the beginning, asa normalization factor, one gets the Fsl values given in the Table. The average valueof the F d factors, over the five weeks, is 0.211 (0.5%) and the average value of Fs, is1.002(0.5%).

CHAPTER S 395

N- | M p o I Mml I SS I M^A MM{C) I CV I Fc H,(%)

1 59.7 0.155 59.55 T.005 59.25 " " ^ ^ ^ ^ ^2 58.6 0.119 58.48 0.999 58.543 59.1 0.137 58.96 0.991 59.504 60.1 9.144 59.96 1.016 59.025 57.1 0.102 57.00 0.979 58.22 58.91 0.9 0.211 1.0001 59.4 0.108 59.29 1.005 59.002 58.2 0.111 58.09 0.999 58.153 58.9 0.103 58.80 0.991 59.334 59.7 0.133 59.57 1.016 58.635 58.9 0.102 58.80 0.979 60.06 59.03 1.2 0.211 1.0001 S8.8 0.108 58.69 1.005 58.402 58.9 0.114 58.79 0.999 58.853 57.9 0.101 57.80 0.991 58.324 58.6 0.092 58.51 1.016 57.595 58.4 0.123 58.28 0.979 59.53 58.54 1.2 0.213 1.0091 60.4 0.227 60.17 1.005 59.872 59.1 0.157 58.94 0.999 59.003 58.0 0.241 57.76 0.991 58.284 60.2 0.122 60.08 1.016 59.135 58.3 0.278 58.02 0.979 59.26 59.11 1.0 0.210 0.9951 58.0 0.100 57.90 1.005 57.612 60.1 0.171 59.93 0.999 59.993 57.6 0.112 57.49 0.991 58.014 59.3 0.204 59.10 1.016 58.175 I 58.8 I 0.193 I 58.61 | 0.979 | 59.87 | 58.73 | 1.9 | 0.212 | 1.005

Table 1. Behavior of the stability factor over 5 weeks

Stability of the reading system background

The stability of the reding system depends on:

™ environmental conditions (i.e. temperature, humidity)

™ variations of the calibration light source placed inside the instrument,

" how long the instrument has been switched on before use,

" variations of the electronic stability during the use.

Stability of the reading system background (procedure)

~ n (n > 5) consecutive readings without dosimeter

- calculate the average value Ms

- repeat any time before using the TL reader (MS!)

- verify 0.80<^<1.20Ms

Stability of the TL response

The term stability referred to the TL response of a phophor means stabilityof the physicochemical properties of the phosphor. In other words, the repeated useof a phosphor, i.e. annealing - irradiation - readout cycles, should not change thephosphor's sensitivity and its glow curve. The stability can be checked on a group ofTLDs, chosen randomly from a batch. The following Fig.5 shows the stability plotobtained with 27 successive re-use cycles on LiF:Mg,Cu,P (GR-200A). Thereadings were obtained using a linear heating rate of 9°C/s. Figure 6 shows thereadings of 10 successive re-use cycles. In this case the readouts were undertaken bya plateau heating time at 230°C for 20 s.

" linear heatingat maKimum temperature: 270 *coa. i.os — A

UJ A / \f\

«a

_ , —- M " »« • *

(44 I • " - S - 2 ' - < • * * 3*-? E-05Q . - S,P, » 1,52 X 4 0.21

0,90 —I \ \ I i | i t t i t i i t t f

"» 8 12 is 20 2 4 28

N U M B E R O f C V C L E S

Fig.5. Stability of GR-200A using a linear heating rate.

CHAPTER S 397

UJ "plateau* heatingz 230 *C for 20 seconds

a.

ut \

u

ec . «-OK-I»- * • • • • I . IE -H l l.St-M

^ U.90 - *•»• " °-84 * * •••«

2 4 ft 8 tO 12 11

N U M B E R O r C V C L E S

Fig.6. Stability of GR-200A obtained with plateau readout.

Standard annealing

The standard annealing is the normal thermal procedure used for re-use ofthermoluminescent phosphors (see annealing general conditions and procedures).

Stokes' law

G.G.Stokes formulated in 1852 the law of luminescence. The law states thatthe wavelength of the emitted light is greater than that of the exciting radiation.

Sulphate phosphors

The sulphate phosphors family is composed of many different compounds.A short review is given below [1-10].

Calcium sulphate (CaSO^Two different kind of preparations can be used.In the first, analytical-reagent-grade CaSO4x2H2O and reagent-grade impurities(oxides of rare earths) are mixed in a proper ratio and dissolved in concentratedsulphuric acid to form a saturated solution of CaSO4. The solution is then heated atabout 300°C to allow the evaporation of the acid. Single crystals of doped calcium


sulphate appear during evaporation. After cooling, the crystals are ground to powderand sieved to obtain grains ranging from 100 to 200 (i in size.

Another method consists of the dissolution of Ca(NO3)2 in 225 cm3 ofconcentrated H2SO4. The dopants, in the required concentration, are added and thereagents thoroughly stirred in a flask, which is connected to a sealed condensersystem with constant air flow as carrier for the acid vapor. A beaker containing aNaOH solution captures and neutralizes the condensed acid. A hot plate provides theheat required to drive the reaction. An evaporation period of about 12 hs allows toobtain single crystals of CaSO4. The crystals are repeatedly washed to remove anyremaining acid, they are then placed in a Pt crucible and thermally treated for 1 h.After that the crystals are ground and sieved. The particle size ranges between 80and 200 \i. Pellets of calcium sulphate with PTFE may also be obtained.

Calcium sulphate (Zeff = 15.6) doped with Mn shows high sensitivity but a veryhigh fading rate because it presents only one peak at about 90°C. CaSO4:Dy andCaSO4:Tm show similar glow curves with three peaks at about 80, 120, 220°C and ashoulder at 250°C. The third peak, to most prominent, is the dosimetric peak. Fadingrate varies according to different authors and preparation technique: from 7% to30% in 6 months. The lower detection limit is about 1 (j.Gy and the TL response islinear up to 3 Gy for Tm doped material and up to 100 Gy for Dy actvated calciumsulphate.

Strotium and barium sulphates Dy activated (SrSO4:Dy, BaSO4:Dy)Analytical-reagent-grade SrSO4 (Zeff = 23) and BaSO4 (Zeff = 35) are dissolved

in sulphuric acid together with dysprosium oxide Dy2O3. Crystals are formed afterevaporation of the acid at 300°C. The crystals are then dried at 400°C during severalhours, ground and sieved. The powder is annealed at 400°C for 5 hrs; a secondannealing at 400°C increases the sensitivity of about 40%. Both materials show avery intense peak in the temperature region 130-140°C. Their relative sensitivities, atthe 60Co energy, compared with that of LiF TLD-100, are 11 for SrSO4 and 3 forBaSO4.

Mixed sulphates (K2Ca2(SO4)3, K2Cd2(SO4)3)For preparing K2Ca2(SO4)3, having an effective atomic number equal to about

14, K2SO4 and CaSO4 powders in the molar ratio 1:2 are mixed and heated in aquartz tube at 1000°C for 24 hrs. The compound is formed by a process of solid statediffusion. The molten mass is slowly cooled and then crushed and sieved to obtainparticles having a size of about 210|a. The glow curve shows four peaks in the regionbetween 80°C and 500°C. The dosimetric peak, very intense, at 447°C does not showany fading.

K2Cd2(SO4)3 is prepared using the solid state diffusion technique. K2SO4 andCdSO4 powders are mixed in the appropriate proportions and kept for 6 days at600°C. The obtained mass is powdered and then melted at 770°C. Aftter cooling the

CHAPTER S 399

powder is obtained as usual. This material has also been doped with Sm with anincrease in sensitivity by a factor of 40 with respect to the undoped material. Theundoped material presents a glow curve with two resolved peaks at 77°C and 200°Crespectively. The doped Sm material presents only one prominent peak at 157°C.

References1. Watanabe S. and Okuno E., Riso Rep. 249 (2) Danish AEC (1971) 8642. Yamashita T., Nada N., Onish H. and Kitamura S., Proc. 2nd Intern. Conf.

Luminescence Dosimetry, Gatlinburg (USA) (1968)3. Yamashita T., Nada N., Onish H. and Kitamura S., Health Phys. 21 (1971)

2954. Yamashita T., Proc. 4th Intern. Conf. Luminescence Dosimetry, Krakow

(Poland) (1974)5. Azorin J., Salvi R. and Moreno A., Nucl. Instr. Meth. 175 (1980) 816. Azorin J., Gonzalez G., Gutierrez A. and Salvi R., Health Phys. 46 (1984)

2697. Azorin J. and Gutierrez A., Health Phys. 56(1989)5518. Dixon R.L. and Ekstrand K.E., Phys. Med. Biol. 19 (1974) 1969. Sahare P.D., Moharil S.V. and Bhasin B.D., J. Phys. D 22 (1989) 97110. Deshmukh B.T., Bodade S.V. and Moharil S.V., phys. stat. sol. (a) 98

(1986)239

TTemperature gradient in a TL sample

In case high heating rates are used during readout, a temperature differencebetween the bottom and the top of a sample can be observed and the glow peakbecomes broader [1,2].

In case a temperature gradient across a TL sample is ignored, the TLintensity is given by

I(T) = mexV\- — I (1)

On the other hand, assuming a constant temperature gradient across a TLDsample, the emission can be written as

I\ri) = t x — sexA-—. -Adr (2)

whereT2 = temperature of the sample

«(r2 + r) = density of trapped charge carriers at temperature T2 + T at a givenposition within the sample.

Assuming a linear time dependence of temperature at each position in thesample and neglecting the energy dissipation to the surroundings, we can write, foreach position in the sample:

T2+r = T; + J3'{t + t') (3)

where To' and /?' are, respectively, the effective starting temperature and the

effective heating rate in the sample, and P't' = r .

Indicating with dn the difference of n between two positions,corresponding to a difference in temperature of dT at a certain time, we can write

dT p' dt

so that, the quantity n(T2 + T) can be approximated by

Using last Eq.(5), Eq.(2) becomes

I(T2) * n^)-8- f 'H-^expf-^llexpf- , * Afr

which can be approximated as

T(T\ <T ^ 5 ( £ 1 f^H", 5T ( E \\ f Et VI(T2)=n(T7) — exp I 1 exp • 1+ -\dx\u v 2 ; A r vy kT2)Wr\_ p' \ kT2)\{ kT22)

(6)and again, solving the integral:

I(T2) = n(T2 )s expl - — exp - S^AT) exp[ - — I (7)

The last exponential on the right of Eq.(7) can be developed into powers of

around :VT2 TM) TM

(4)

(5)

CHAPTER T 403

1 ( E) 1 ( E )

+ [—ex f —1 — ex f —11 f - - 1

and then Eq.(7) becomes

I(T2) = n(T2)SexP(-^)

\ (EsiAT2)) \( E) ( E) l] ( E\[

From the comparison between Eq.(l) and Eq.(8) it is easily seen that if thetemperature gradient across the TL sample is ignored, both activation energy andfrequency factor are underestimated by the quantities

AZ7 \Es{AT)2Y E ^ ( E \AE* ^—'— 2 exp

[l2TMfi'\{kTM ) \ kTM)

W [l2TM/3'\{kTM ) \ kTM)

References1. Piters T.M. and Bos A.J.J., J. Phys. D: Appl. Phys. 27 (1994) 17472. Piters T.M., A study into the mechanism of thermoluminescence in a

LiF:Mg,Ti dosimetry material (Thesis, 1998), D.U.T.

Temperature lag: Kitis' expressions for correction (procedure)

Kitis suggested the following procedure for corrected the temperaturevalues when a temperature lag is suspected to be in TL measurements [1] (seeTemperature lag: Kitis' expressions for correction (theory))

(8)


" make a few measurements at very low heating rates, i.e. 1 and 2°C/s inorder to evaluate the constant c from the relation

T —Tc= M2 m (1)

In 2

~ using the following equation

T(M,x)corr=Tm-clnU^) (2)

evaluate the real temperature, T(M,x)con at the maximum for the usedheating rate

~ evaluate the temperature lag at the heating rate Px

where TM,X is the peak maximum temperature of the glow peak with atemperature lag

•* using the following equation

TMx-T0~AT

fseff=J^r-^r—P (3)*M,x -*0

calculate the effective heating rate. To is the order of the roomtemperature (about 293 K)

Example

A glow peak shows the temperature at the maximum at 481.3 K when aheating rate of l°C/s is used, and at 488 K with a heating rate of 2°C/s. Supposingthat at those heating rates no temperature lag exists, we can calculate the constant cfromEq.(l):

e_488-481J_In 2

CHAPTER T 405

The same glow peak shows, using a heating rate of 40°C/s, a temperature atthe maximum of 518 K. The correct value is then, using Eq.(2)

TM,corr{40°Cls) = Tm - 9 . 6 7 - 1 1 ^ = 517*

The temperature lag at the heating rate of 40°C/s, is

Ar = 518-517 = 1A"

and the effective heating rate of the sample is, using Eq.(3):

5 1 8 - 2 9 3 - 1 4 0 = 3 9 8 O C / 5

eff 5 1 8 - 2 9 3

It must be stressed that:

"" Eq.(13) is valid in the range from l°C/s to 50°C/s. It is a generalequation, holding for every point of the glow peak. Each point of theglow peak shifts as a function of the heating rate, with its own constantc.

" The reference measurements at low heating rates need special attentionto avoid any temperature lag. This can be achieved: (i) using silicon oilof high thermal conductivity when solid TL samples are used, (ii) usingloose powder.

~ The temperature lag is a linear function of the heating rate. This is inagreement with the theoretical prediction [2-4].

References1. Kitis G and Tuyn J.W.N., J. Phys. D: Appl. Phys. 31 (1998) 20652. Gotlib V.I., Kantorovitch L.N., Grebenshicov V.L., Bichev V.R. and

Nemiro E.A., J. Phys. D: Appl. Phys. 17 (1984) 20973. Betts D.S. and Townsend P.D., J. Phys. D: Appl. Phys. 26 (1993) 8494. Piters T.M. and Bos A.J.J., J. Phys. D: Appl. Phys. 27 (1994) 1747

Temperature lag: Kitis' expressions for correction (theory)

Kitis has provided a simple method to correct the effect of temperature lagin TL measurements. He derived expressions for temperature lag correction for bothfirst- and general-order kinetics.

First-order kineticsThe equation describing the first-order kinetics is the following:

I(T) = nos exp(- A j exp[- j ( exp(- jdT'j (1)

which can be rewritten in the following way

l n / m = ln(/ioj)- — -—(expf - — W ' (2)

with the usual meaning of the various symbols.Using two differen heating rates, fix < P2 , the glow curve obtained with

the faster heating rate is shifted towards the higher temperature keeping its integral,any way, constant. Considering now the intensities of the two peaks at the same

fraction of their maximum intensity, IM , the following condition is verified:

</ln(/,) d\n(l2)^ = ^ = a (3)

dT dTwhere a = 0 at the peak maximum temperature.

From Eq.(2), according to Eq.(3), one then obtains

</ln[ln(/,)] E s ( E ")L ^ l ^ = — exp \ = a (4)

dT kT? A \ kTj

dln[ln(l2)] E s ( E)L^-2Ai = — exp \ = a (5)dT kTl p2 \ kT2

CHAPTER T 407

from which

E s ( E\— ; r - a = —exp (6)

kT? ft \ kTjE s ( E)

—r- - a = — exp (7)

From these two equations, the respective heating rates can be obtained as

A=4exp(-A] (8)

where

<ilcT2

E~akTx2

2 E-akTl

Eqs. (8) and (9) can be arranged as follows

A, exp

h A ( E)

and then

{A) UJ *r. «•.from which

(9)


Tt r, E yj E [AJ

Equation (10) holds true for every temperature point of the peak at the same

fraction of its maximum intensity and, of course, at IM . Equation (10) can now be

transformed as follows:

n = r , - ( r , r 2 ) i t a ( A ) + t e ) | , n ( A ] (11)

The shift of the peak from T{ to T2 as the heating rate increases, is given

by the sum of the last two terms on the right of Eq.(l 1). Any way, the contributionof the second term is less than 5% of the total shift and so this term can be omittedand Eq.(l 1) simplify to

T -T — TT — In — n2">E KPi)

r o \Taking fix =l°C/s and /?2=50°C/s, the extreme value of In ^~ is 4 and the term

\Pi)T{T2 increases only a few per cent in the range (1 - 50) °C/s; therefore, the term

kT{T2 — can be considered as a constant and Eq.(12) assumes the final form of

E

T2=T,-c\n[^\ (13)

The next step is to calculate the effective hating rate, f5eff , i.e. the rate of

heating of the sample.Let us indicate with Tg the peak maximum temperature of a peak received

with temperature lag, with TM the real value if there is no temperature lag, and with

KT -Tg-TM the difference. Both Tg and TM are given by

Tg=T0+j3-T(14)

TM=Tg-AT = To+p-eff.t

(10)

CHAPTER T 409

where P is the heating rate of the heating element and To is of the order of the

room temperature.From Eq.(14) we obtain

T-T0-AT

General-order kineticsThe intensity for a glow peak following a general-order kinetics is given by

which becomes

ln / ( r ) = ln(#io.s)- — —In 1 + * ^ ^ fexpf-^ldT' (17)

As already done for the first-order case, two heating rates fix < P2 are

considered. Hence

^ W = W = a (18)dT dT

From Eq.(17) we get

^lnC/1)=_£ b A I MjJ ^ ^

s(b-\) ( E)

d\n(I2)= E b fi2 { kT2) ^^

dT kT22 6 - 1 [ S(b-Y) f> ( E\T^\1 + — I exp \dT

A *i \ kT')and then

(15)

(16)


bsexp

(19)

foexp_ ^ I kT2)_a

kT22 J32s2

where £, and £2 are the expressions with integral in the denominators.

Solving Eq.s (19) with respect to/?! and /?2>w e obtain

( E \B. = A. expHx x \ kTJ (20)

( E \B? = A, exp

kT Iwith

bskT?

i^r22

Making the ratio and then its logarithm of Eq.(20), we finally obtain

1 1 k. (pA k, (AA— = —+ —ld-^- I d — (21)T T F \ R \ F \ A \

which is similar to Eq.(10) obtained for the first-order kinetics. Therefore, Eq.(13) isalso valid for the general-order kinetics.

ReferenceKitis G and Tuyn J.W.N., J. Phys. D: Appl. Phys. 31 (1998) 2065

CHAPTER T 411

Test for batch homogeneity

The batch homegeneity is concerned with the methods for the qualitycontrol on a new batch of dosimeters just received by users. Some quality tests canbe carried out, each giving a different precision.

The simplest method is the following. The user screens all the samples byirradiating them with a known dose from a calibrated radiation source showing agood beam uniformity and making sure that all the samples have been inside theirradiation field. Any TL sample outside the specified tolerance limits should berejected. The TL dosimeters can also be screened at periodic time intervals.

It must be noted that screening can only be used to determine acceptance orrejection of the samples. Indeed, there are two negative aspects of this procedure.Firstly, accepting a large range of responses (i.e. all responses which are within 20 -30% of the mean response), large precision errors are introduced in the dosedetermination. This is very dangerous when the dosimeters are used in clinicalapplications. Secondly, the replacement of the rejected TLDs is difficult when thereplacement dosimeters come from a different batch: a bias error can be introducedinto the whole procedure for the dose assessment.

However, this test remains valid as a first step to know the characteristicsof a new TLDs batch.

A quality control concerning the batch homogeneity for TLDs used inpersonnel dosimetry is suggested in the technical recommendations of theInternational Electrotechnical Commission (IEC) document. The procedure is givenbelow with some examples.Procedure for batch homogeneity.

All the N dosimeters of the same batch have to be annealed according to theannealing procedure used for the type of TL material under test.

At the end of the annealing procedure, all the dosimeters have to beirradiated using a calibrated gamma source under the appropriate electronequilibrium conditions. The given dose depends on the future use of the dosimeters;i.e., a dose of 5 mGy is suitable for personnel dosimetry, while 1 mGy is enough forenvironmental dosimetry.

Immediately after irradiation the TLDs are read to measure the TL emission(the readout cycle will be chosen as the best for the particular type of phosphor - seethe section concerning the readout cycles) of each dosimeter. Let us indicate thevalues of the TL emission as

Mt withi= 1,2, 3,.. . , N

The TLDs are now re-annealed and read again to measure the zero-reading(or the zero dose reading). This value should be the same as that already determinedduring the initialization procedure. In case the background levels are higher, thecharacteristics of the annealing oven must be checked (temperature uniformityinside the oven, correspondence between the temperature set and the actualtemperature, etc.). Let us indicate these background values as

MOi withi= 1,2, 3,.. . , N

The net readout is then defined as

Miinet=Mj - MOi with i = 1, 2, 3,.. . , N

In such a series, the maximum and minimum values have to be identifiedand substituted into the equation

A - = — - n r i r ^ 1 — 100£3° (1>where Amax represents the uniformity index for the given batch. If such expression isnot verified, namely the Amax of the batch is larger than 30, then some TLDs have tobe rejected. Figure 1 shows, as an example, a histogram obtained from the readingsof a batch of 1000 TL dosimeters.

The initial calculation of Amax gave:

n=1000 4H<« = 4 8 . 5 > 3 0

Since the uniformity index was larger than 30, some TLDs wereprogressively rejected. The results were:

Rejecting only 2 samples

n = 1000 - 2 Amax = 38.7 > 30 not acceptable

Rejecting 4 samples

n = 1000 - 4 Amax = 33.7 > 30 not acceptable

Rejecting 6 samples

n = 1000 - 6 Amax = 29A < 30 acceptable.

CHAPTER T 413

aoo I - • - • - - - " • - i

3 too- >• ' -

1 •llllllllll. 1Readings

Fig.l. Histogram of 1000 TLDs readings.

Another procedure can be used for this test (not included in the officialrecommendations). The average value of all readings is evaluated as

M = yL_L w = y_k!_ ( 2)i-i i=i

and the following two quantities are evaluated

M-ap and M+aP

where o> is a predetermined value of the standard deviation. All dosimeters whichexhibit a net TL readings outside the previous range are rejected.


Dos I TL I Dos. I TL I Dos. I TL I Dos. I TL.N. N. N. N.1 ~8.468M 21 7.601 41 6.765 61 144.9*2 7.808 22 7.346 42 7.531 62 7.9463 7.231 23 6.916 43 7.045 63 6.7714 7.587 24 7.491 44 7.476 64 7.6575 7.630 25 7.600 45 7.480 65 7.4346 7.394 26 7.509 46 7.656 66 7.2397 8.094 27 7.329 47 7.167 67 6.7048 7.854 28 7.290 48 8.047 68 7.1189 7.428 29 7.294 49 7.984 69 6.6991"10 7.963 30 7.677 50 8.014 70 7.39511 7.676 31 8.143 51 6.968 71 8.05712 8.387 32 8.111 52 8.320 72 7.55513 8.232 33 7.374 53 7.487 73 6.72014 7.839 34 6.739 54 8.433 74 7.77815 7.464 35 7.574 55 7.812 75 6.78616 7.539 36 7.880 56 7.620 76 7.42417 7.411 37 7.783 57 7.934 77 7.40218 7.633 38 7.836 58 7.568 78 7.02519 8.076 39 172.5* 59 7.872 79 7.58020 | 7.085 | 40 I 7.912 | 60 | 7.167 | 80 | 7.334

Table 1. Example of data for the homogeneity test. The superscripts M and mindicate the maximum and minimum values, respectively. * indicates abnormal

readings.

It can be noted here that it is not always possible or convenient to rejectsome dosimeters, i.e. when the batch is limited. In these cases all the samples arekept and their responses are corrected using the relative intrinsic sensitivity factor(also called individual correction factor). Any way, it has to be stressed that eithersome or more samples are rejected or all of the batch samples are considered, thecorrection factor must be calculated and used to achieve the best uniformity of thebatch response.

Another example is reported here. The test has been carried out for asample of 80 TLDs and the results show its usefulness in some particular cases. Itmust be noted that the background signal was obtained as an average value andsubtracted from each reading. Table 1 lists the net values and the correspondinghistogram is given in Fig.2; among them, the responses of two TLDs are evidently

CHAPTER T 415

abnormal and completely out of the range indicated by the test, so that their rejectionis obvious.

SOO --

400 - - ^ |

300 - • ^ ^ B l

100 - - ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ J

o J ^ — >«.« «.«

TL readings

Fig.2. Histogram of 80 readings.

ReferenceFuretta C. and Weng P.S., Operational Thermoluminescence Dosimetry,World Scientific, 1998

Test for the reproduribility of a TL system (procedure)

~ Select, randomly, a test group of 10 TLDs from a batch,

~ Anneal the TLD samples according to the appropriate standard annealing,

~ Irradiate the samples to a test dose of about 1 mGy: this dose is acompromise between high doses, which could give a residual TL in thesuccessive cycles, and lower doses which could lower the reproducibility,

~ Read out all the samples,


~ Repeat point 4 for the background acquisition,

~ Repeat the procedures 2 to 5 at least 10 times for statistical reasons,

~ Complete the following Table:

readingsTLD Cycles No. average S.D. CVj +No. 1 I 2 I 3-9 I 10 (mQ (q8) m1 1938734 1943571 (omissis) 2022710 1968338 47149 2.4%2 1633017 1654501 (omissis) 1657873 1648464 13483 0.8%3 1668407 1584265 (omissis) 1683390 1645354 53432 3.2%4 1848805 1861617 (omissis) 1802078 1837500 31338 1.7%5 1810473 1888630 (omissis) 1963886 1887663 76711 4.1%6 1474240 1436054 (omissis) 1503458 1471251 33801 2.3%7 1245084 1278430 (omissis) 1262845 1262120 16685 1.3%8 1274609 1313236 (omissis) 1300008 1295951 19630 1.5%9 1284749 1234075 (omissis) 1281419 1266748 28344 2.2%10 1835799 1926944 (omissis) 1809908 18575501 61475 [ 3.3%~

average 1601392 1612132 (omissis) 1628758rel. val. 1.000 1.007 (omissis) 1.017

The coefficient of variation, for the i-th TLD, is given by

cvt = ^ (i)mi

where a; and m, are the standard deviation and the average values of the 10 repeatedreadings of the i-th dosimeter.

The half-width of the confidence interval, juh is given by

^ ^ T ^ T i (2)

where n is the number of repeated cycles and t is the value of the student test.In the present case n = 10 and t = 2.26 at a confidence level of 95%. Then

CHAPTER T 417

M = CVX — = 0.53CVi (3)' ' 4.24

The reproducibility test, for each of the 10 dosimeters, is then acceptable if

CVi +//,< 7.5% (4)

which transforms, considering Eq.(3), in the following acceptable level

CVt < 5%

So, to define a TL system as "reproducible" each dosimeter included in thetest group should have a coefficient of variation no larger than 5%.

ReferenceScarpa G. in Corso sulla Termoluminescenza Applicata alia Dosimetria.Rome University "La Sapienza", Rome (Italy)

Thermal cleaning (peak separation)

The glow peaks in a glow curve are generally more or less overlapped.When the peaks are not too much overlapped, it is possible to use a thermaltechnique, called thermal cleaning, for getting a well defined and clearly separatedpeaks. This technique has been introduced and described by Nicholas and Woods(1964).

Let us imagine a phosphor showing a glow-curve with two, or more,overlapped peaks, each one having the maximum temperatures at T]<T2<...<Tn. Thesample is now heated to an appropriate intermediate temperature T just beyond themaximum temperature of the first peak and then rapidly cooled. This procedureerases the first peak, substantially emptying the traps responsible for it. Theprocedure is repeated for the subsequent peaks.

This technique is used when the energy acti vati onE has to be determinedby the initial rise metho d where a clean initial rise of the peak is needed. It mayalso be used in any cases when a clean rise part of a peak (low temperature side ofthe peak) is necessary as in some methods based on peak shape.

The thermal procedure has to be applied very carefully. In fact, when afirst-order process is supposed to be, the thermal cleaning allows to determine thetemperature of the maximum, TM, but not the intensity at the maximum because thecleaning thermal treatment could affect the number of the trapped charges. On the


other hand, when a second order is assumed, even the value of TM is not correctbecause the shift of the peak phosphor due to the reduction, as before, of the trappedcharges.

ReferenceNicholas K.H. and Woods J., Br. J. Appl. Phys. 15 (1964) 783

Thermal fading (procedure)

Several equations can be obtained to take into account thermal fading effectas well as fading in competition with other effects, i.e., fading in competition withself-dose, fading and background irradiation, fading in accidental exposure and soon. The following first experiment is performed in the simplest way, just todetermine the isothermal decay constant in a thermoluminescent material followinga first order kinetics (Randall-Wilkins theory) and without any other effect incompetition with fading.

" Determine the calibration factor of the reader as already explained inprevious paragraph.

~ Choose 30 similar TLDs that are expected to be used in the futuredosimetric application.

~ Identify each of them.

~ Anneal the dosimeters according to the proper annealing procedure.

"" At the end of the annealing the detectors have to be cooled in a reproducibleway up to the room temperature.

~ Read out the dosimeters with suitable heating cycle for the material chosenand determine the intrinsic background (zero dose reading).

~ Note the values of the individual background, M,o and calculate the

average value Mo and its standard deviation CJ0. If aQ is less than 2%, use

the average background value instead of the individual values.

™ Irradiate at a test dose D, for all 30 annealed dosimeters.

~ Read immediately after irradiation in one session only.

~ Determine the individual sensitivity factors for each of the 30 dosimeters

CHAPTER T 419

where

(=1 3V

(remember that Mj0 can be substituted by the average background value if ao<2%)

~ Repeat five times the same procedure to determine Sj (anneal, irradiation,

read) and calculate St and oy.

™ Divide the group of 30 TLDs into three sub-groups of 10 dosimeters each,called group A, group B and group C.

- Anneal all the 30 TLDs.

~ Irradiate group A only at a test dose Df (about 0.1 Gy).

~ Store all the three groups in a lead container in which the inside dose ratehas to be very low.

~ Temperature and relative humidity inside the container have to be monitoredduring the whole period of storage.

"" The storage period, ta, has to be chosen according to the specific needs forfuture dosimetric applications of the dosimeters.

~ At the end of the storage period, group B is taken out of the container andirradiated at the test dose Df.

"" All the dosimeters are now read in one session only.

~ Let us define the following quantities

(1)

(2)

i 10 _

MA=-^Z{MAi-Mm)SAi

2 io _

MB=TzYkM»-M**)S» <3>1 U i=ii 10 _

Mc = T^L{Ma-MCi0)Sa1 U i=i

where

MA gives the measure of the TL at the end of the storage period

MB gives the reference prompt TL response at storage time t = 0

Mc gives the measure of a possible increase of the TL emission, during the

storage period, due to environmental and/or self irradiation.

~ The isothermal decay constant is now calculated using the followingexpression

^f'-f^f ^ <4)la IVIA IVIC

It should be a good exercise to determine the value of A, as a function ofdifferent temperatures of storage.

Thermal quenching

Thermal quenching is the process such that the luminescence efficiencydecreases with temperature, due to the increased probability of non-radiativetransitions due to killer centers.

It has been shown that the luminescence efficiency can be expressed by thefollowing equation

l+e«p(--J(1)

CHAPTER T 421

where W can be evaluated from a plot of the luminescence intensity as a function of1/T.

Wintle discussed this problem extensively and she introduced the idea thatthe recombination process depends exponentially on the temperature: i.e., therecombination probability, Am, is given by

Am oc exp — (2)\kT)

where Wis the energy depth of a non-radiative recombination level.Equation (2) is a good approximation, in the high temperature range, of

Eq.(l).

Thermally connected traps

Traps which are considered thermally connected belong to overlapping trapstates having a very close energy difference and producing overlapped glow peaks.The thermally connected traps have been introduced by Sweet and Urquhart (twotrap model) to explain their experimental results concerning ZnS single crystals.

ReferenceSweet M.A.S. and Urquhart D., Phys. Stat. Sol. (a) 59 (1980) 223

Thermally disconnected traps

A thermally disconnected trap is an extra energy level introduced byDussel and Bube in 1967 for taking into account differences emerging in the resultsobtaining by simultaneous measurements of both thermoluminescence and thermallystimulated conductivity (TSC):

~ the maxima of the two species of signals do not occur at the sametemperature,

~ important differences in the shape of the signals.A thermally disconnected trap is a trap which can be filled by freed

electrons produced by irradiation, but which has a trap depth which is much greaterthan the normal trapping levels. Thus, during heating the sample, only electrons inthe shallower traps are freed, while the electrons trapped in the deeper levels(thermally disconnected) are not affected by heating. In other words, these trapping


sites have a thermal stability of the trapped charges which is greater than that of theshallow traps related to the TL signal.

From an experimental point of view, it is quite difficult to prove theexistence of the thermally disconnected traps because the limitation imposed by theblack-body radiation background signal of the detection system, including TLsample, heating strip and surroundings.

Any way, for a theoretical interpretation of the experimental data obtainedby both thermoluminescence and thermally stimulated conductivity measurements,it is realistic to include the thermally disconnected traps into the energy bandscheme and then to modify the rate equations describing trap filling and trapemptying processes.

ReferenceDussel G.A. and Bube R.H., Phys. Rev. 155 (1967) 764

Thermoluminescence (thermodynamic definition)

Thermoluminescence requires the perturbation of a system from a state ofthermodynamic equilibrium, via the absorption of external energy, into a metastablestate. This is then followed by a thermally stimulated relaxation of the system backto its equilibrium condition.

Figure 3 shows an energy diagram for a crystal having a certain number ofdefects distributed between the conduction band (CB) and the valence band (VB). Inthermal equilibrium condition, i.e. T = OK, all the defect levels, up to the Fermi levelF, are occupied by electrons. The other levels are empty (see Fig.la): electrons andcrystal lattice are in thermal equilibrium.

The system can now be perturbed by an ionizing radiation. Underirradiation the electrons in the defect levels or in the valence band gain energy andrise into higher levels, beyond the Fermi level (Fig.lb). After the irradiation aredistribution process takes place and the excited system goes back to equilibrium.The time required for going back to equilibrium may vary from milliseconds toyears, depending on the material, its defects and the temperature.

CHAPTER T 423

CB

p

o o o o_D_ _Q_ _Q_ _Q_

VB

Fig.3a. Energy diagram for a crystal having a certain number of defects distributedbetween the CB and the valence band VB. The open circles are the electrons.

CB

F

O Oo

O O VB

Fig.3b. Redistribution of the trapped electrons due to the irradiation.


Thermoluminescence (TL)

From a microscopic point of view, thermoluminescence consists of aperturbation of the electronic system of insulating or semiconducting materials, froma state of thermodynamic equilibrium, via the absorption of external energy, i.e.produced by an ionizing radiation, into a metastable state. This is then followed bythe thermally stimulated relaxation of the system back to its equilibrium condition.

Macroscopically, thermoluminescence is a temperature-stimulated lightemission from a crystal, after removal of excitation (i.e. ionizing radiation);thermoluminescence is a case of phosphorescence observed under condition ofsteadily increasing temperature. A plot of the light intensity as a function oftemperature is called glow-curve. A glow-curve may have one or more maxima,called glow-peaks, each corresponding to an energy level trap [1,2].

References1. Mckeever S.W.S. and Chen R., "Luminescence Models", Rad. Measur. 27

(5/6) (1997) 6252. Furetta C. and Weng P.S., "Operational Thermoluminescence Dosimetry"

World Scientific, 1998

Thermoluminescent dosimetric system (definition)

A thermoluminescent dosimetric system consists of several parts asfollows:

~ the passive elements: the TL dosimeters (or detectors)

~ a TL reader schematically consisting of a heating element, a PM tube,one or more electronic networks.

~ an appropriate algorithm to convert the TL signal (response of thereader) to dose.

~ ovens and/or furnaces to be used for thermal treatments of thedosimeters (annealing procedures).

~ any other complementary instrumentation or facility which can be usedfor the right setting up and working for the system and/or for theimplementation of the system (i.e. calibration sources; programme ableto deconvolute the glow-curve, to make an automatic estimation of thebackground, to calculate the average TL values and so on).

CHAPTER T 425

Thermoluminescent materials: requirements

Several properties have to be examined for the choice of a TL material withrespect to a specific application. In general, the more desirable properties of a TLDphosphor are listed as follows:

" a high concentration of traps and a high efficiency of light emissionassociated with the recombination process;

™ a good storage stability of the trapped charges, as a function of storage timeand temperature, so that a negligible fading affects the TL response. Thisshould also be true for opposite extreme temperature values (i.e., tropical orartic climates);

~ a very simple glow curve (i.e., a simple trap distribution) which allows theinterpretation of the readings as simple as possible, without any thermaltreatment after irradiation (post irradiation annealing). In case of more orless complex glow curve, the main peak (i.e., the dosimetric peak) should bewell resolved among other possible peaks in the glow curve;

~ a spectrum of the emitted TL light to which the detector system(photomultiplier and associated filters) responds well. A spectrumwavelengths between 300 and 500 nm seems the most desirable since itcorresponds to the commercially available detector systems. Furthermore,the black body radiation does not interfer in this spectral range even atrelatively high temperatures;

" the main peak should have a peak temperature at the maximum in the range180°C -^250°C. At higher temperatures the infrared emission from both TLDsample and TLD holder may interfere giving up to a source of errors in thereading interpretation;

™ good resistance against disturbing environmental factors as light (opticalfading), humidity, organic solvents, gases, moisture;

" the TL material should not suffer by radiation damage in the dose range ofapplications;

" the TL material should have a low photon energy dependence of response.For personnel and medical applications, tissue equivalent phosphors(effective atomic number of the tissue Zeff =7.4), or approximated tissueequivalent, should be used to avoid energy corrections;


~ a linear TL response over a wide range of doses is a desirable feature formost applications;

"" the TL material should be non-toxic: this is very important for in-vivomedical applications;

~ the TL response should be independent of dose rate and of the angle ofradiation incidence;

~ the lower limit of detection should be as low as possible for environmentalmonitoring;

~ low self-irradiation due to natural radionuclides in the TLD materials for allkind of applications;

~ the TLD phosphor should have a high/low thermal neutron sensitivityaccording to the specific use (i.e., monitoring around power plants,accelerators and so on);

™ a good LET sensitivity may also be useful in some cases;

"" high precision and high accuracy are required characteristics for any kind ofapplications;

~ in case of need, the TL detectors should be suitable for postal service.

The above list cannot be fulfilled by only one type of TL phosphor. As aresult, there is a serious limitation in the choice and the materials which can be usedfor dosimetry have properties which are a compromise among the variousrequirements. Any way, a material having very good performances for one or morespecific applications can be easily found.

Tissue equivalent phosphors

TL materials having an effective atomic number, Z ^ , similar to the one ofthe soft tissue (Z=7.4), are known as tissue-equivalent materials.

The tissue equivalence is a desirable feature for greater accuracy inbiomedical, clinical and personal monitoring.

Tissue equivalence for photons requires that the mass energy absorption

coefficients, —— , for the dosimetry material match those for the tissue in which theP

CHAPTER T 427

dose is to be measured.The cross-sections for photon interactions are directly proportional to the

atomic number, Z , raised to some numerical power for each element in the

dosemeter material, i.e. elemental cross-section oc Zx, where x depends on the typeof interaction occurring and varies between 1 and 5. It has a value closed to 4 for thephotoelectric effect [1,2].

A compound, as a thermoluminescent material is, may be regarded as a

single element with an effective atomic number, Z ~ , given by

where a, is the fractional electron content of element j-th in the compound.

For the photoelectric effect in muscle Z = 7.4, therefore materials withsimilar Z will have good tissue equivalence for low energy photons: their response

will vary with photon energy in the same way as —— for tissue.P

The requirements for tissue equivalence when a dosemeter is irradiatedwith neutrons or high LET particle is quite different. Neutrons entering tissueinteract with H, C, N and O releasing secondary charged particles like protons, alphaparticles and heavy recoil nuclei. For fast neutrons, the (n,p) reaction with Hpredominates contributing over 70% to the kerma, but few thermoluminescentmaterials contain H.

References1. Mayneord W.V., The Significance of the Roentgen in Acta Int. Union

Against Cancer 2 (1937) 2712. Driscoll C.M.H. in Practical Aspects of Thermoluminescence Dosimetry,

Proceedings of the Hospital Physicists' Meeting, University of Manchester(1984) Edited by A.P.Hufton

Trap characteristics obtained by fading experiments

A quick way for obtaining qualitative informations about the trapcharacteristics of a thermoluminescent material, consists in the comparison betweenglow-curves recorded at different time intervals after irradiation.

The following Fig.4 shows the glow-curves of CVD diamond sample,irradiated with UV light. The first glow-curve, labelled 1 min UV, has been recorded


immediately after UV irradiation; the second is the glow-curve recorded 23 hrs afterthe end of the UV irradiation. The third plot is the difference between the twoprevious glow-curves: this difference gives the indication of the TL lost duringfading as a function of the glow-curve temperature.

Supposing the fading is an isothermal decay at room temperature, it is givenby

/ = / o e x p ( - A - f ) (l)

where 70 is the TL emission recorded immediately after irradiation, / is the TL afterthe fading time t, and X is the decay constant of the process.

Hence

A = - - l n | — I (2)

8 3 E 4 B2 - UV fading

6.5E4 - /~\

1 min UV_^/ S~\

— 4 7 E 4 ' yS ^ * ^ \ra" - y^ / 23 hrs fading \

P! 2.9E4 - /~^y \

1.1E4 - / / ^ — ^ _ _ ^ ^ \^^y Difference ^ ^ ^ ^

-7.5E3 I 1 I I I—I—I 1—I 1—I—I 1—J 1—1—I0 50 100 150 200 250 300 350 400

Temperature (°C)

Fig.4. Glow-curves of CVD diamond and their difference.

CHAPTER T 429

10° E 1i B2-UV

§• 2 \& K T 8 " •

[ %

2 '.

10"4 - •

p i i i i i i i i i i i I I I i *

0 50 100 150 200 250 300 350 400Temperature (°C)

Fig.5. Decay constant as a function of the glow-curve temperatures.

Figure 5 shows the plot of the resulting values of X as a function of theglow-curve temperatures. In this figure it is possible to identify three differentregions. In the first region X decreases as the glowe-curve temperature increases; inthe second region a clear plateau is observed and, finally, a third region, above300°C, where the results are highly scattered because / and 70 are more or less quitesimilar.

The behavior of X in the first region is a clear indication of a continuousdistribution of trapping levels, whereas the plateau region indicates a single trappinglevel.

ReferenceKitis G. private communication

Trap creation model

The supralinearity is explained via trap creation during irradiation.


Cameron used this model to explain the supralinearity in LiF. The modelrequires that the new traps created by irradiation are the same as those originallypresented in the crystal srtucture. Furthermore, they suggested that luminescencecenters are also created by irradiation and that the new centers are the same type ofthe original ones because the emission spectra are unchanged.

ReferenceCameron J.R., Suntharalingam N. and Kenney G.N., ThermolurninescenceDosimetry, University of Wisconsin Press, Madison (1968)

Trapping state

A trapping state is that for which the probability of thermal excitation froma localized state into the respective delocalized band is greater than the probabilityof recombination of the trapped charge with a free charge carrier of opposite sign.

Tunnelling

An electron trapped in a level A of an atom (Fig.6) may recombine directlywith a hole in a level B of another atom without involving the delocalized bands.Mikhailov gave in 1971 a model for tunnelling process [1-3].

The defects responsible for levels A and B must be closed to each other.This can occur when traps and recombination centers are in a very highconcentration and when the two centers belong to the same defect site. Thistransition occurs through the potential barrier (tunnelling) which separates theelectron in A from the hole in B. The recombination results in the emission ofluminescence. The effect is athermal.

* \ ~ A

\ ^ B

Fig.6. Tunnelling between an electron in A and a hole in B.

CHAPTER T 431

The tunnelling process can explain the low temperature afterglow and therelationship between this and the TL lost in anomalous fading.

Visocekas et al (1976) [4] also considered the possibility that the electroncan be first excited to a higher energy state and then, not having still enough energyto escape from the trap, recombines via tunnelling. This type of process is calledthermally assisted tunnelling.

References1. McKeever S.W.S., Thermoluminescence in Solids, Cambridge University

Press (1985)2. Chen R. and Kirsh Y., Analysis of Thermally Stimulated Processes,

Pergamon Press (1981)3. Chen R. and McKeever S.W.S., Theory of Thermoluminescence and

Related Phenomena, World Scientific (1997)4. Visocekas R., Leva T., Marti C , Lefaucheux F. and Robert M.C., Phys.

Stat. Sol. (a) 35 (1976) 315

Two-trap model (Sweet and Urquhart)

This model has been proposed by Sweet and Urquhart to explain a situationwhere two peaks are so close that they appear as only one peak.

Let us define the following symbols:

Ex, E2X?- E jdepth of two very closed traps (eV),

N{, N2 concentration of trapping centers (m~3),

Hj, «2 concentration of trapped electrons (m"3),

M concentration of recombination centres (m'3),

m concentration of trapped holes in recombination centres (m3),

nc, mv concentration of free electrons in CB and free holes in VB (m3); both are

assumed negligible,

Anl,An2 trapping rate constants (m3 s"1),

Am recombination rate constant (m3 s" ),

Yx,y\ = Sj exp probability for electrons from trap to CB.

The following set of equations can be written:


d^- = -ylni+ncAnl{Nl-nl) (1)

^ t = "Wl + "cAn2 (^2 - «2 ) (2)at

dnc dn, dn2—- + —L + — - = -Anm (3)

dt dt dtThe condition of charge neutrality is now:

m = nc+nl+n2 (4)having considered mv = 0 .

Assuming a linear heating rate T =7) + fit, where T-, is the initial

temperature and fi = ^y, .

In order to solve numerically the previous equations, the followingapproximations are assumed:

dn dn,"c^ni^^—r (5)

dt dtdnr dn7

Inserting (5) and (6) in Eq. (4) one gets

m = «, + n2 (7)

Then, using Eqs. (3) and (4), we obtain

dm dn. dn, dn7

= - = Amnm (8)dt dt dt dt

Eq.(7) can be rewritten as

dm dn, dn7

« — L + — - (9)dt dt dt

(6)

CHAPTER T 433

and then, inserting (9) into Eq.(8). We obtain

£-0 00,at

Inserting now (10) in Eq.(3) we get

dn, dn,

1F+1<-+A'"'m = 0 <n)

Eqs. (1) and (2) can now be written as

~ + "^ r = -ri"i ~Yini + ncAnX(N{ -nx)+ncAn2(N2 -n2)

which can be rearranged, using (7) and (11), as

Amnc(#i, + n2) = / , « , + y2n2 -ncAnl(Nl -nx)-ncAn2{N2 -n2) (12)

from which

n = LA ! LA ? (\-i\AM+n2)+AnANX-nMAn2\N2-n2)

Because the glow-curve intensity is given by the decrease of the trapped holesduring recombination, the TL intensity is

KT) = -c^ (14)

at

Using Eqs. (7) and (8) and taking c = 1, Eq.(14) becomes

J(X) = Amncm = Amnc («, + n2)

= Am{ni+niXyini+r2n2) (16>

Am{^+n2)+AnXNi-nx)+An2{N2-n2)


It has to be stressed that Eq.(16) becomes the equation of the first-ordermodel by setting equal to zero the parameters with subscript 2 and considering thatrecombination dominates, i.e. Amnx »• Anl{Nx -nx). Equation (16) gives also thesecond order model considering the retrapping assumption and assuming to be farfrom the trap saturation, i.e. N\ >•>- nx.

Equation (16) has been computer-calculated for fitting the experimentalresults obtained from the study of ZnS. The parameters used for the best fit are

sl =s2 = 3 - 1 0 1 V I , ^ = 64 , A ^ = 100,£1 =lSmeV,E2 =22.5meV,

4. 4.Reference

Sweet M.A.S.and Urquhart D., Phys. Stat. Sol. (a), 59 (1980) 223

VVarious heating rates method (Bohum, Porfianovitch, Booth: first order)

-Bohum [1], Porfianovitch [2] and Booth [3], working independently,proposed a method based on two different heating rates for a first-order peak.Taking into account the condition at the maximum and using two different heatingrates one obtains:

-^-=SQX^S~) (i)K1 Ml K1 M\

~k^~ = sexp(-W~} (2)K1 Ml K1 Ml

from which, by eliminating s, E is obtained according to the following expression

E = kJMlMl^JA\(TM2_) (3)

TMi~Tm [fi2) \Tm)

Therefore, in the assumption b = 1, £ is easily evaluated by measuring thetwo peak temperatures corresponding to the maximum TL intensity for the twoheating rates. If TM can be measured within an accuracy of 1°C, the method yields Ewithin 5%. The value of s can then be calculated by substituting the numerical valueof E in one of the two equations (1) or (2).

References1. Bohum A.,Czech. J. Phys. 4 (1954) 912. Porfianovitch I.A.,J. Exp. Theor. Phys. SSSR, 26 (1954) 6963. Booth A.H.,Canad. J. Chem. 32 (1954) 214

Various heating rates method (Chen-Winer: first order)

Chen and Winer reported a method using an approximation for the integralwhich appears in the first-order expression of I(T). In fact, the integral can beapproximated as follows


if F icT^ Ffexp(-^)d7 ' = ( ^ r ) e x p ( - — )(1-A) (1)*o kT E kT

where2kT

A - — . 0 1

The insertion of (1) into the first-order expression for I(T) yields

/ = sn0 exp(- — ) exp - S— exp(- —)(1 - A) (2)

Inserting now the condition at the maximum in Eq. (2), one obtains

IM = nos exp - — - exp - (l - A M )

or

IM=— exp - y — exp(Aje { kTM)

and then

Even with large variations of ft, TM changes only a few percent andtherefore so AM and 1+A^ do consequently. Then, one can assume that the intensityis directly proportional to the exponential, considering as a constant the otherquantities.

The plot of ln(/M) against \ITM for various heating rates should get astraight line with a slope -E/k from which E can be found. Owing to theapproximated integration, which is true only for a linear heating rate, this method isvalid only in this case.

ReferenceChen R. and Winer S.A.A., J. Appl. Phys. 41 (1976) 5227

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CHAPTER V 437

Various heating rates method (Chen-Winer: second and general orders)

Chen and Winer showed how to apply the various heating rates method ifthe kinetics of any order is present, including the second-order kinetics. Theexpression of 7(7), in the general case, is given by

and the condition of maximum emission

kT2Mbs ( E ) , s(b-l) f ( E \ _ ,—-—exp = 1 + — [ exp UT (2)

fiE \ kTM) p k \ krywhere S = s"tl0 .

The maximum value of 7(7) is obtained by inserting Eq.(2) in Eq.(l):

'-—-x-ifJpSr-*-^]"* wwhich can be rewritten as

From the hi of this expression one obtains

in[c(f)}4+cwith c constant.

(1)

(4)

(5)


By means of this equation it is possible to evaluate the quantity on the leftside for different values of b and to obtain a set of experimental points. These pointsare then plotted as a function of \ITM on a semilog paper and fitted by a straight linewhose slope is E/k. Of course, one must find the value of b for which the plot best

approximates the linearity. In another way fP may be included in the constant.For the second order case, Eq.(5) gets

ln/J^U =^- + c (6)

[ U J J kTMThis method is useful only when b is appreciably different from unity since

for b = 1, TM is independent of the initial concentration «flof trapped electrons. Also

in this case p2 may be included in the constant.As also suggested by Chen and Winer, one can consider the condition of

maximum emission and the integral approximation. Remembering the conditionof maximum emission for a general-order kinetics given by Eq.(2), one obtains:

(kT^bs) E . . s(b-l)kT^ E .,. v^ ^ exp(-—-)£l+ v ' Mexp(----)(1-Aj

\ PE ) kTM PE kTM

(7)which gives the final form

li)s^-M^)+{b-lM (8)The quantity [ l+^-^A^] is close to unity and it can be considered as a

constant, so that the plot of different values of left side expression versus \ITM

should be well fitted by a straight line of slope -E/k. The new value of E can becompared with the former given by Eq.(6) to have an assessment of the errorintroduced by the integral approximation. The same procedure can be used in case ofa second-order kinetics.

ReferenceChen R. and Winer S.A.A., J. Appl. Phys. 41 (1970) 5227

CHAPTER V 439

Various heating rates method (Gartia et al.: general order)

A new method using two heating rates has been proposed by Gartia et al.[1]. It is analogous to the Booth method, which is strictly valid for a first order peak,but it is applied to a non-first order TL peak and it is based on the variation of lM

with b, which variation is much more faster than the variation of TM with b. Usingthe general-order expression [2], one can write:

E b \ sEEJuJ

I n /«=-^:+ I n O T»-^TIT+ (*-1 )-i^-j (1)where um=E/kTM and E2(um) is the second exponential integral [3]. The factor insquare brackets is very close to unity; hence, using two linear heating rates yS/and fcone obtains

Eln/ml S - — + \nsn0

KIml

(2)

In /^s—— - + lnsn0

K1rn2

which give

kT T 1E= ml ml I n — (3)

T —T 11m\ xm2 1m2

The authors pointed out that the maximum systematic error involved is lessthan 1% for any order of kinetics (1.1 < b < 2.5).

References1. Gartia R.K., Ingotombi S., Singh T.S.G. and Mazumdar P.S., J. Phys. D:

Appl. Phys.24(1991)652. Singh T.C.S., Mazumdar P.S. and Gartia R.K., J.Phys. D: Appl. Phys. 23

(1990) 5623. Gaustchi W. and Cahill W.F., Handbook of Mathematical Functions

(Dover, N.Y., 1972)


Various heating rates method (Hoogenstraaten: first order)

Hoogenstraaten, starting from the condition at the maximum for a firstorder kinetics, suggested the use of several heating rates to obtain a linear relation asthe following

In(7^) = (^)-1L + l n rA] (1)

The resultant plot should yield a straight line with slope E/k and anintercept ln(£M).

ReferenceHoogenstraaten W., Philips Res. Repts. 13 (1958) 515

Various heating rate method (Sweet-Urquhart: two-trap model)

Sweet and Urquhart propose a variation of the heating rate method, veryuseful when two or more peaks are so closely overlapped that they appear as onlyone peak. The method has derived by the two-trap model proposed by the sameauthors.

The procedure is based on the measurements of various glow curvesrecorded with different heating rates.

According to the two trap model, the TL intensity is given by

7(7) = 4»(wi+/l2X7y»i+7V»2)4 » («! + «2 ) + 4,1 (Nl ~ "l ) + 4,2 (N2 ~ n2 )

where

Nt, N2 concentration of trapping centers (mf3),

« i , n2 concentration of trapped electrons (m3),

4 i ' An2 trapping rate constants (m3 s"1),

Am recombination rate constant (m3 s"1),

( ( EA)Yi>Y2\ = si e x P probability for electrons from trap to CB.

I I kTjj

CHAPTER V 441

A AConsidering the temperature dependence of AnX, Anl, ——, — ^ , s l , s 2 to be

small, the previous equation for different heating rates y5, can be rewritten as

-? =r- = constant for all indices /

where Txi is the temperature at which the area under the glow curve on the low

Atemperature side of Txi is x% of the total area, i.e. x = — — . For different glow

A T O T

curves i, the temperature Tx! is chosen in such a way that the x value is always the

same. The corresponding values of I(Txi) are also found. Changing the heating rate,

Tx is seen to vary and a plot of InI(Txj) against \/Txj is a straight line whose slope

is E/k from which the activation energy can be found.According to the results obtained by Sweet and Urqhuart, this method

allows to determine the activation energies of two very close trapping states: when xis taken enough small (« 20%) the plot gives the shallow trap energy; on thecontrary, when x is large (« 80%), the slope gives the energy of the deeper trap.Reference

Sweet M.A.S. and Urquhart D., J. Phys. C: Solid State Phys., 14 (1981) 773

Various heating rate method when s = s(T) (Chen and Winer: first- andgeneral-order)

Chen and Winer developed the method of the various heating rate in thecase a temperature dependence of the frequency factor is suspected. Thetemperature dependence of the frequency factor is assumed as follows

s = s0Ta (1)

• First order kineticsIn the case of first order kinetics and for a-3/2, the maximum condition

when the frequency factor is temperature dependent can be written as follows fortwo different heating rates


T& E 1 + Um V Km)V 4 )

f \

v=^—\ e x p \~~l¥~\ (3)

V 4 J

from which, by eliminating So, one obtains

(A\(^fJ^e JLi)(j_n] (4)UJlr«,J 1+2A LI tJUv, Wj

4 M1

In the previous equation, the coefficient of the exponential term can beassumed equal to unity, without introducing any consistent error; Eq.(4) is thenreduced to the following form

E = kJjnTMi_in (A).(TMI_) 2 (5)

Tm-TM2 | y 2 J U , j jIn the general case of any value of OC, the relation becomes

E = kTM^f-la[^-\[YLY (6)1M\ LM2 \\P2J V-'A/l J

From Eq.(6), when a is known by some independent measurements, itdirectly gives the value of E. It must be noted that, if E is already known by other

(2)

CHAPTER V 443

methods, Eq.(6) can be used to evaluate a and hence s0 from either Eq. (2) or (3).In conclusion, a complete knowledge of the trap level under investigation ispossible.

Using several different heating rates one has to plot

JJL) „ -LTa+2 rp

which should yield a straight line, from whose slope the activation energy E isfound.

In the case of a general order kinetics, supposing 5" depending on thetemperature according to the relation

s" = slTa

by substituting the condition at the maximum when the frequency factor istemperature dependent:

slbT^kexA- — ] _ ..

0 M r /T ft /i 1 \ n I

in

we obtain

/v=s0V-"expU— ^ - v " y (7)

^ kl"J ^ I + I A ^ J

and rearranging

The logarithm of Eq.(8) is then

rri2b+a j-t

^ lbM • - * b - =T1T + ^TM + E) +const (9)P J KIM

from which the activation energy can be experimentally evaluated with the usual

procedure. In fact, since a < 2 and kTM - X E, the left-hand side of expression

(9) is a linear function of yL , with slope equal to *y-, .

ReferenceChen R. and Winer S.A., J. Appl. Phys. 41 (1970) 5227

(8)

zZirconium Oxide (ZrO2)

Zirconia has very much attracted the attention of technologists andscientists owing to its combined electrical, chemical, optical and mechanicalcharacteristics. All these properties make this material suitable for a large variety ofapplications, On the other hand, a little research has been done on its luminescentproperties [1-14].

In order to obtain material suitable for thermoluminescence dosimetry,Zirconium TL phosphors have been synthesized by blending zirconium oxichloride(ZrOCl:8H2O by Merck) and ethhyl alcohol. This solution is stirred for 15 minutes,then heated at 250°C for 30 minutes until full evaporation of the solvent. Theamorphous powder is then submitted to different thermal treatments in an oxidingatmosphere (air) in order to stabilize the trap structure. Then the powder is crushedand sieved to select grains having a size between 100 and 300 | . SinterdZ1O2+PTFE pellets of 5 mm in diameter and 0.8 mm in thickeness can alsoobtained.

The most attractive feature of ZrC>2 phosphor is its very high intrinsicsensitivity to UV radiation. The typical glow curve of ZrO2 , after UV irradiation,exhibits one single peak at 180°C.

After beta irradiation from '"Sr/9^ source, the glow curve presents tworesolved peaks at 200°C and 250°C respectively. The TL response is linear from 2 to60 Gy; the reproducibility, over several repeated cycles of annealing, irradiation andreadout, is better than 1.8% and fading at room temperature is 3.8% in one month.

The TL emission after X-ray irradiation of low energy, typically from 15 to60 KV, shows two peaks at about 200°C and 280°C. At 60 KV X-rays, the TLresponse is linear from 0.04 Gy to 1.12 Gy.

References1. Peters T.E., Pappalardo R.G and Hunt R.B., in Solid State Luminescence,

edited by A.H. Kitai (Chapman & Hall, London, 1993).2. Shionoya, in Luminescence of Solids, edited by D.R. Vij (Plenum Press,

New York, 1998)3. Bettinali C, Ferraresso G. and Manconi J.W., J. Chem. Phys. 50 (1969)

39574. Dhar A., Dewerd L.A. and Stoebe T.G., Med. Phys. 3 (1976) 4155. Iacconi P., Keller P. and Caruba R., Phys. Status Solid (a) 50 (1978) 275


6. Shan-Chou Chang and Ching-Shen Su, Nucl. Tracks. Radiat. Meas. 20 (3)(1992)511

7. Azorin J., Rubio J., Gutierrez A., Gonzalez P. and Rivera T., J. Thermal.Anal. 39(1993) 1107

8. Rivera T., Azorin J., Martinez E. and Garcia Hipolito M. Desarrollo denuevos materiales Termoluminiscentes para Dosimetria Personal yAmbiental de la Luz Ultravioleta. IV Congreso Regional SeguridadRadiologica y Nuclear IRPA, CUBA, 1998

9. Azorin J., Rivera T., Martinez E. and Garcia M., Radiat. Meas. 29 (1998)315

10. Azorin J. Rivera T., Falcony C, Martinez E. and Garcia M., Rad. Prot.Dos. 85(1999)315

11. Azorin, J. Rivera T., Falcony C , Garcia M and Martinez E., 10th Inter.Cong. Inter. Rad. Prot. Ass.. Hiroshima Japan (2000)

12. Rivera T., Azorin J., Falcony C, Martinez E. and Garcia M., Radiat. Phys.Chem. 61(2001)421

13. Rivera T. Estudio de las propiedades termoluminiscentes yfotoluminiscentes del ZrO2:TR y su aplicacion a la dosimetria de laradiacion ionizante. Tesis de Doctorado Universidad AutonomaMetropolitana. Mexico D.F.(2002)

14. Rivera T., Azorin J., C. Falcony, M. and Martinez E., Rad. Prot. Dos. (Inpress).

AUTHOR INDEX

AbdullaY.: 17,332Abromowitz M.: 95,278Abtani A.: 84Adams E.N.: 3Adirovitch E.I.A.: 3,7, 328AitkenMJ.:60AlcalaR.:325Alexander C.S.: 340AltshullerN.S.:325Antonov-Romanovkii V.V.: 21,24-26,180,181AramuO.: 171,172ArchundiaC.:205ArrheniusS.: 1,35,43,361Attix F.H.: 84, 392Avila O.: 99Azorin J.: 55,209,390, 399,446BacciC: 18,163,325Bacci T.: 99Balarin M.: 256,260,274,275,300, 315BapatV.N.:216BarbinaV.: 18,215BartheJ.R.: 18Becker K.: 217BenincasaG.: 17BergonzoP.:99Bernardini P.: 163Bettinali C: 445BettsD.S.:240,405BhasinB.D.:217,399BichevV.R.:240,405BiggeriU.:99BilskiP.:99Binder W.: 18,55Bodade S.V.: 399BohumA.:435Booth A.H.: 435BorchiE.: 18,99Boss A.J.J.: 110,230,240, 326,403,405Botter-Jensen L.: 205Bowman S.G.E.: 233,240

448 AUTHOR INDEX

BrambillaA.:99Brauer A.A.; 3, 4, 88-90, 261, 267, 271, 272, 282, 292, 300, 307, 313, 319Braunlich P.: 2, 45, 54, 84, 216Brooke C : 55Brovetto P.: 172Bruzzi M.: 99BubeR.H.: 36, 212,421,422BuckmanW.G.: 8Budzanowski M.: 390Buentil A.E.: 99Burkhardt B.: 36, 39, 346, 348Busuoli G.: 9, 11, 18, 121, 200, 332, 344, 345, 348, 427Cahill W.F.: 439Cameron R.J.: 18, 55, 60, 69, 379,430Carpenters.: 18Caruba R.: 445Ceravolo L.: 17Chaminade J.P.: 326Chang S.C.: 445Charalambous S.: 84Chen R.: 2, 20, 60, 88-91, 95, 148, 149, 150, 163, 210, 231, 233, 240, 243-245, 260,272, 274, 275, 281, 292, 295, 300, 302, 304, 307, 310-313, 315, 319, 320, 339, 340,359, 377, 379,424, 431,435-438, 441,444Cheng G.: 18Christiensen P.: 205Christodoulides C : 90, 93, 275, 278-280, 295, 297, 298ContentoG.: 18,215Correcher V.: 19, 375Curie D.: 150Dajlevic R.: 18DawN.P.J.: 149deMurciaM.: 84Deb N.C.: 99Delgado A.: 137Deshmukh B.T.: 399Dewerd L.A.: 445DharA.:445Di Domenico A.: 163Disterhoft S.: 55Dixon R.L.: 399DodsonJ.: 18Dorendrajit S.: 99

AUTHOR INDEX 449

Driscoll C.M.H.: 9, 11, 18, 19,175, 216, 390,427DusselG.A.:421,422Eguchi S.: 217Ekstrand K.E.: 399Elliot J.M.: 18, 19,216EvansM.D.:8Eyring H.: 8, 150Facey R.A.: 240Falcony C: 446Ferraresso G.: 445Fioravanti S.: 18Fleming S.J.: 60FoulonF.: 99Francois H.: 18Fujimoto K.: 9Furetta C : 18, 19, 55, 99, 137, 148, 163, 194, 202, 205, 209, 212, 215, 216, 222,240, 275, 321, 325, 326 ,332, 357, 375, 377, 390, 424Gabrysh A.F.: 8GautchiW.:439Garcia Hipolito M.: 446GarlickG.FJ.: 7, 67, 52, 60, 90, 157, 159, 176,177, 243, 323Gartia R.K.: 84, 95, 99, 171, 279, 280, 295, 298,445Gibson A.F.: 7, 84, 95, 99, 171, 279, 280, 295, 298,439Ginther R.J.: 18, 55, 392Glasstone S.: 150Goldstein N.: 18Gomez Ros J.M.: 19, 69, 137, 375, 390Gonzalez G.: 399Gonzalez P.: 209, 452Gorbics S.G.: 78, 84GotlibV.I.:240,405Grebenshicov V.L.: 240, 405GrossweinerL.L: 88-90, 280, 281, 300, 313, 318, 319Gutierrez A.: 55, 209, 390, 399, 446Haering R.R.: 3Halperin A.: 3, 4, 88-90, 231, 261, 267, 271, 272, 282, 300, 307, 313, 319HanleW.:216HariBabuV.: 106,325Hashizume T.: 9, 217Hastigs C. Jr: 278HickmanC: 11, 18Hoogenstraaten W.: 87, 90,440

450 AUTHOR INDEX

Horowitz A.: 18Horowitz Y.S.: 18, 60,69,90HsuP.C: 18Hubbell J.H.: 332Hunt R.B.: 445Iacconi P.: 445Ilich B.M.: 85, 90Hie S.: 216InabeK.: 183,231,240Ingotombi S.: 439Israeli M.: 60Jany C: 99Jayaprakash R.: 149Jones S.C.: 84JunJ.S.:217Kantha Reddy B.: 325Kantorovic L.N.: 240, 405Kathuria S.P.: 26, 84, 90, 95, 181, 21KatoY.:9, 217KatzR.:231KazakovB.N.:325Kazanskaya V.A.: 215Keating P.N.: 89, 185, 186, 188Keddy R.J.: 99Keller P.: 445Kelly P.: 78, 84Kenney G.N.: 379,430KidoH.:231KirkR.D.: 18,205Kirsh Y.: 2, 20,97, 99, 148, 150, 163, 210, 359, 377, 379, 431Kitamura S.: 399Kitis G.: 18, 19, 60, 61, 68, 69, 79, 84, 99, 155, 194, 205, 216, 240, 321, 325, 405,406,411,429Klammert V.: 368Korobleva S.L.: 325KoteraN.:217Kou H.: 8Koumvakalis N.: 325Koyano A.: 209Kristianpoller N.: 60KunduH.K.:391Kuo C.H.: 99

AUTHOR INDEX 451

Kuzmin V.V.: 215Laidler K.J.: 152Lamarche F.: 357Land P.L.: 88, 90, 166, 168LarrsonL.: 231Laudadio M.T.: 19,375LeFebreV.:8Le Masson N.J.M.: 326LeeY.K.: 18,55LefaucheuxF.: 8,431LemboL.: 18LeroyC.18,99, 209, 357Leva T.: 8,431Levy P.W.: 137Lewandowski A.C.: 359LiY.:18LilleyE.:240LivanovaL.D.: 325Louis F.: 390Lushchik L.I.: 88-90, 261, 262, 265,272, 292-295, 300, 304, 313, 318Mahesh K.: 212Manconi J.W.: 445Marathe P.K.: 391MarayamaT.:8, 209MarczewskaB.: 99Markey B.G.: 359Marshall T.O.: 121, 344Marti C : 8, 431Martinez E.: 446Martini M.: 18Marullo F.: 209Massand O.P.: 391Matsuda Y.: 8Matsuzawa T.: 209MaxiaV.:21,24, 87,90May C.E.: 24, 90, 178-180, 182, 217 ,219, 255, 256MayneordW.V.:42,427Mazmudar P.S.: 84, 95, 99, 171, 279, 280, 295, 298, 439McKeever S.W.S.: 2, 19, 60, 148, 163, 190, 191, 210, 222, 233, 240, 332, 339, 340,359,377,379,424,431McWhanAJF.: 18Mendozzi V.: 209

452 AUTHOR INDEX

Miklos L: 374Miller W.G.: 18MinaevaE.E.:215Missori M.: 18Mohan N.S.: 90, 240, 243, 244Moharil S.V.: 25, 26, 31, 84, 90, 95, 180, 181, 399MolisanC.:215MoranP.R.:60, 69Moreno A.: 399Moreno y Moreno A.: 205Moscati M.: 19MundyS.J.: 18, 19,216MuiiizJ.L.: 137Muntoni C : 24, 25, 87; 90NadaN.:399NagpalJ.S.:216NakajimaT.:9,209, 217Nam T.L.: 99NambiK.S.V.:121NanniR.: 18NantoH.: 183Nash A.E.: 84, 392Nemiro E.A.: 240, 405Nicholas K.H.:187, 188, 418O'HoganJ.B.: 18OberhoferM.: 18Okuno E.: 399Olko P.: 99Onish H.: 399Onnis S.: 24, 90PadovaniR.: 18,215Pani R.: 390Papadoupoulos J.: 84Pappalardo R.G.: 445ParavisiS.: 18Partridge J.A.: 24, 90, 178-180, 182, 217, 219, 255, 256PaunJ.:216Pellegrini R.: 390Peters T.E.: 445PhilbrickC.R.:8PieshE.:36, 39, 346, 348Piters T.M.: 110, 230, 240,403, 405

AUTHOR INDEX 453

Plato P.: 374PodgorsakE.B.: 60,69Polgarl.:36, 39, 346, 348Porfianovitch I.A.: 441Portal G.: 18, 208Prisad K.L.N.: 106Prokic M.: 18,19,194, 205, 215, 216Prokic V.: 19,194,205RamogidaG.: 18,325Randall J.T.: 20,49, 53, 54, 60, 85,90, 109, 223, 224, 323, 327, 328, 361, 364,418RasheedyM.S.:31,35, 155Ratnam V.V.: 149RispoliB.:18, 163, 325RitzingerG.: 84,95Rivera T.: 446Robert M.C.: 8, 431Romero Gutierrez A.M.: 390Rossetti R.: 18Rubio J.: 446Rucci A.: 24, 25, 90, 172RybaE.:390SaezV. J.C.:390Sahare P.D.: 399Sakamato H.: 217Salamon R.: 19, 205, 216Salvi R.: 399Salzberg L.: 205Sanipoli C : 18, 19, 209, 321, 325, 375Santopietro F.: 18,325SasidharanR.:217Scacco A.: 18, 137, 209, 325, 375Scarpa G.: 13, 18, 19, 175, 200, 204, 228, 253, 357, 417, 427, 429Scharmann A.: 45, 54, 216Schayes R.: 55Schon M.: 45Schulman J.H.: 205, 392Sciortino S.: 99Selzer S.M.: 332Sermenghil.: 18Serpi A.: 25,90Shenker D.: 90, 244, 245Shinoya S.: 445

454 AUTHOR INDEX

SibleyW.A.:325Singh S.J.: 84, 95, 168, 171, 279, 280, 295, 298Singh T.S.G.: 445Sokolov A.D.: 215Somaiah K.: 106, 325Soriani A.: 19Spiropulu M.: 84Stegun LA.: 95, 278Stoddard A.E.: 333Stoebe T.G.: 445Stokes G.G.: 397Stolov A.L.: 325Su C.S.: 445SuntaC.M.: 84,95,217Suntharalingam N.: 60,430Sussmann R.S.: 18,99Sweet M.A.S.: 421,436,431, 434, 440,441TakenagaM.:205Takeuchi M.: 182, 183, 231, 240Taylor G.C.: 240Thompson J.: 60TochilinE.: 18ToddC.D.T.: 18TorynT.:217Townsend P.D.: 240,405TsaiLJ.: 18Tuyn J.W.N.: 55, 69, 240, 390, 405,411Ulivi L.: 99Underwood N.: 8UrbachF.:85, 90, 224Urquhart D.: 421, 436, 439, 440, 441VanEijkC.W.E.:326VanaN.: 84,95Venkataraman G.: 391VismaraL.: 18VisocekasR.:8,435,431Vistoso G.F.:209VureshamP.: 106WaligorskiM.P.R.:231Wang S.: 18Watanabe S.: 399Watson J.E.: 9

AUTHOR INDEX 455

WengP.S.: 148, 163, 202, 212, 222, 275, 332, 377,424West E.J.: 205, 392Wilkins M.H.F.: 20, 49, 53, 54, 60, 85, 90, 109, 223, 224, 323, 327, 328, 361, 364,418Winer S.A.A.: 90,441-444,Wintle A.G.: 86, 90, 421Woods J.: 187,188,418Wrzesinska A.: 7WuF.: 18Yamaguchi H.: 8Yamamoto O.: 205YamashitaJ.:231YamashitaT.:205, 399YangX.H.:60Yossian D.: 60, 69,90Zarand P.: 36, 39, 346, 348ZhaZ.: 18ZhuJ.: 18ZimmermannD.N.: 8Zoppi M.: 99

SUBJECT INDEX

Accidental: 381Accuracy: 1,117,247,248,249,340Activation energy: 1,21,23,35,67, 85-90,109,189Activator: 2,3,41,45Afterglow: 7Aluminium oxide: 8Annealing: 8-16,121,173, 380,390, 394,396-398Area: 13,20,21,22,24-27,31Arrhenius: 1, 35,43Asymptotic series: 175,185,240,281Atomic number: 8,39-41

Background: 8, 9, 11, 12, 36, 56, 106, 107, 110, 113, 116, 120, 130, 134-136, 139,144-147,173,174,396

Band: 1-5,35, 43,45,47,49, 52,105,212Batch: 45,172,411Beam quality: 95,97,106,111Bleaching: 247

Calcium fluoride: 55Calibration factor: 55-58,96,97,105,106,139,165,166Capture cross-section: 151Chemical vapour deposition (CVD): 99Charge neutrality: 4Cleaning: 417Competition: 36,58,59,60Competitor: 58,60Complex: 103Computerized Golw Curve Deconvolution (CGCD): 60,89,188,189,191,193Condition at the maximum (see maximum condition)Connected traps: 421Continuous irradiation: 132,135,384

Dark current: 107Debye frequency: 151Decay: 3Defects: 1,2,9,101,102,104,105,361Delocalized bands: 3Detrapping: 20Diamond (CVD): 99

458 SUBJECT INDEX

Dihalides: 106Disconnected traps: 176,421Dosimetric peak: 108Dosimetric trap: 176,421Dosimetry:8,10,18,39

Efficiency: 83,84,86,197,213Energy: 1,3,21,23Energy dependence: 110,111,329Entrance dose: 165Erasing: 10, 117Errors: 110,117,118, 120,121,152,153,320Escape probability: 112Exitation: 4,46,284Exit dose: 165Exponential decay: 160Exponential integral: 277Extrinsic defects: 102

F center: 104F distribution: 356Fading: 7, 8,10,19,20,106,114,118,123,138,378,384, 387,398,418,427Fading factor: 113,115,116,137,139First order: 1, 5, 7, 19, 20, 49, 54, 61, 66, 69, 70, 77, 109, 112, 131, 137, 138, 148,

153,154, 166, 176, 185, 219, 240, 256, 260, 261, 275, 280, 292, 293, 361, 435,440,441

Fitting: 90, 97,240,242-245, 390Fluorapatite: 150Fluorescence: 149,150Free energy: 211Frenckel defect: 101Frequency factor: 28, 29, 60, 70, 78, 87, 91, 99, 109, 124, 148, 151-154, 159, 168,

171,185,211,218,244,281,441FWHM: 68,79, 82, 83

General order: 24-26,28,31,64,66,70,71,76,81,97,154,163,168,178-180,182,217,244,272,275,279,295,437,439,441

Geometrical factor (see Symmetry factor)Geometrical parameters: 261,273,286Glow curve: 163Glow peak: 163,195

Half life: 219-222

SUBJECT INDEX 459

Half width: 286Heat transfer: 228-230,240Heating rate: 70, 78-87,251,435-444Heating up: 247,249,250Homogeneity: 411-415Hyperbolic decay: 158,161

Individual correction factor: 57,107,110,203,365-374,419Inflection points: 166-169Initial irradiation: 141,142,146,147Initial rise: 171,188-190,192Initialization: 9,45,172,174Instantaneous irradiation: 130,134,141Integral approximation: 61,63,72,74-76,175,264,268,273, 302, 306Interactive traps: 176Interstitial impurity: 101-105Intrinsic defects: 101Intrinsic sensitivity factor (see Individual correction factor)Isothermal decay: 131,160,161,176-182

Killer: 188Kinetc order: 85, 88, 94,181,194Kinetic parameters: 188,261

Linearity: 197,200,202,205,207,208,224-228Linearization: 197Lithium borate: 204,205Lithium fluoride: 205-208Long irradiation: 136,143Luminescence: 209,210,213,214Luminescence center: 212

Magnesium borate: 215Magnesium fluoride: 216Magnesium orthosilicate: 216,217Maximum condition: 63, 65, 69, 70, 72, 74, 76, 77, 152, 160, 167, 243, 262, 263,

264,266,269,281,293,305,306, 357, 358Maximum temperature: 223,224Mean life: 219-222Metastable state: 1,223Multi-hit: 231

Native defects: 101

460 SUBJECT INDEX

Neutrality condition: 47, 50,126,157Nonlinearity: 233-239

Optical fading: 247-253Oscillations: 249-251Oven: 247

Peak parameters: 299-300Peak separation: 417Peak shape: 256,260,272,275,279,280,282,292,295,299,300,312Period249Perovskite:325Phantom: 165,166Phosphor: 329,431Phosphorescence: 209,326Phototransfer (PTTL): 333,334Post irradiation annealing: 10,340Post readout annealing: 10, 17,340Pre exponential factor: 33, 71, 73,75-77,151,153-155,157Pre irradiation annealing: 10,357Pre readout annealing: 10,17,357Precision: 36,117,340,345,346,349,357Pyrophosphate: 390

Quasi-equilibrium: 48,50,53,359Quenching: 78,83,85,86,420

Random uncertainties: 1,36, 342Recombination: 4, 5,7,19,47,48,123-129,157,176,337, 364Relative Energy Response (RER): 96,329-332Reliability: 312-321Reproducibility: 349-354,416Residual:l 1,12,374Retrapping: 3-7, 21,47,48, 50, 112,127-129, 328, 337Rubidium: 375

Schottky defect: 101Second order: 6, 52, 63, 72, 74, 132, 153, 157, 159, 219, 243, 256, 260, 292, 294,

377,437Self dose: 378Sensitization: 379Sensitivity: 117,379, 380,384, 391,396,398,399Shift: 323

SUBJECT INDEX 461

Spurious TL: 391,392Stability: 106,248, 380,392-396Standard annealing: 10Standard deviation: 12-14, 36-38,201,203,204,225,226, 344,345Steady phase: 250,251Stokes: 397Sublinearity: 234, 236Substitutional impurity: 101,102Sulphate: 397-399Superlinearity: 233, 234,237Supralinearity: 231,233-235, 237Symmetry factor: 90-94,160,161,273-275,287Systematic errors: 1, 343

Temperature gradient: 401-403Temperature lag: 109,110,403-411Thermal velocity: 151Thermoluminescence: 422,424Total half width: 261,286, 300Trap: 1-6,8-10,19-21,24,26,31,33,35,101,104,108,209,427,429,431-434Trapping: 3, 4,26, 123, 125, 132,209,430t-Student: 204Tunnelling: 19,283,430

Underlinearity: 235,236

V center: 104V3 center: 104Vacancy: 101,102, 104,105Variance: 227, 352, 355Variation coefficient: 39,372,416Various heting rates: 188,189VK center: 104

Zero dose: 107,116Zero order: 209,255Zirconium: 445,446

Hand Book of Thermoluminescence

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