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COMPUTERISED CURVE DECONVOLUTION OF TL/OSLCURVES USING A POPULAR SPREADSHEET PROGRAMD. Afouxenidis1,2,*, G. S. Polymeris1, N. C. Tsirliganis1 and G. Kitis2

1Archaeometry Laboratory, Cultural and Educational Technology Institute (C.E.T.I.), R.C. ATHENA’,Tsimiski 58, 67100 Xanthi, Greece2Nuclear Physics Laboratory, Aristotle University of Thessaloniki, 54124 Thessaloniki, Greece

*Corresponding author: [email protected]

Received March 3 2011, revised May 12 2011, accepted June 17 2011

This paper exploits the possibility of using commercial software for thermoluminescence and optically stimulated luminescencecurve deconvolution analysis. The widely used software package Microsoft Excel, with the Solver utility has been used toperform deconvolution analysis to both experimental and reference glow curves resulted from the GLOw Curve ANalysisINtercomparison project. The simple interface of this programme combined with the powerful Solver utility, allows the analy-sis of complex stimulated luminescence curves into their components and the evaluation of the associated luminescence

parameters.

INTRODUCTION

The computerised curve deconvolution (CCD) analy-sis of thermoluminescence (TL) glow curves andoptically stimulated luminescence (OSL) decaycurves into their individual glow peaks andcomponents respectively have been recognised overthe last 30 y to be of major importance (1 – 5). Theinformation by the CCD concerning the trap energydepth (E ), frequency factor (s), kinetic order (b),photoionisation cross section (s ), etc, is very useful

in order to understand the luminescence mechanismof materials.The capabilities of several computer codes used by

the various research groups for CCD and theirassessment of the glow curve parameters were testedby the GLOw Curve ANalysis INtercomparison(GLOCANIN) project. Participants in the GLOCANIN project were asked to analyse the so-calledreference glow curves using their computer codes.The results on synthetic as well as on experimentallymeasured glow curves can be found in references(6, 7).

The application of the CCD consists essentially of two steps: (i) deciding on the mathematical model that

describes a single TL/OSL peak and (ii) assessing the

values of the curve parameters that minimise the sum

of squares of the differences between the fitted model

and the experimental curve(8). Concerning the first

step, there is a number of well-established analytical

expressions in the literature describing single TL/OSL

peaks(6, 9, 10). Although a great amount of work exists

on deriving analytical single peak expressions and

despite the significant information that the CCD

analysis yield, the latter is not widely adopted by the

TL/OSL community as a basic tool, while the

majority of TL/OSL publications does not involveany CCD analysis of relevant data. In authors’opinion, this could be partly attributed to the lack of available commercial software; so every researcher hasto write his/her own program.

The aim of the present work is to offer a solutionto the aforementioned problem that will enable anyresearcher to analyse easily and accurately even themost complex TL/OSL curves consisting of manyoverlapping individual peaks, employing the com-monly used spreadsheet software package Microsoft

Excel, along with its Solver, add-in utility.

SINGLE TL/OSL PEAK ANALYTICALEXPRESSIONS

The analytical expressions used in the present workare the well-known expressions for general andmixed order kinetics, named GOK and MOK,respectively hereafter(7, 11 – 14). The selection of theseexpressions was based on the following:

(1) The GOK expression, although empirical isalmost exclusively used in the literature(7).

(2) The MOK expression, although rarely used inthe literature, is physically meaningful(11).

(3) The GOK expression for b ¼ 2 coincides withsecond-order kinetics, whereas for b ! 1 coincideswith first-order kinetics.

(4) Similarly, the MOK expression for a ! 1coincides with second-order kinetics and for a ¼ 0coincides with first-order kinetics.

The advantage of those expressions is thatonly one expression accounts for both first- andsecond-order kinetics, and furthermore it can

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Radiation Protection Dosimetry (2012), Vol. 149, No. 4, pp. 363 –370 doi:10.1093/rpd/ncr315Advance Access publication 15 July 2011



additionally account for the intermediate orders.Concerning the decision between GOK and MOK,one has to know that their greater difference, as itwas shown by Kitis et al .(12), is observed at aboutb¼1.6 and this difference is ,4 %. Moreover theirdifference is mainly at the high temperature range of the TL glow-peak. Taking into account that thesedifferences were obtained using numerically gener-ated peaks, one can conclude that these differencesare rather difficult to be seen in an experimental TLpeak. Therefore, in most practical experimentalcases yielding an error higher than 1 %, there is noreal dilemma in choosing between GOK and MOK,since they will both produce similar results.

Another feature of the selected analyticalexpressions is that they are of the form of I (I m, T m,E , b or a, T )(7, 11) and I (I m, tm, b or a, t )(13, 14), forTL and linearly modulated (LM)-OSL, respectively,instead of the form I (n0, s, E , b or a, T ) and I (n0,l, b or a, t) of the original equations(9, 10), where

variable T is the temperature, T m is the temperatureat maximum TL intensity, I m is the maximum inten-sity, n0 is the initial concentration of trapped elec-trons, tm is the time at maximum LM-OSL intensityand l is the LM-OSL stimulation wavelength. Theadvantage of the proposed presentation is that thequantities I m, T m and tm can be evaluated directlyand accurately from the experimental TL/LM-OSLcurves, whereas it is not possible to have any knowl-edge about the probable values of n0, s and E appearing in the original form of the equations. The

advantage of the I (I m, T m, E , b or a, T ) represen-tation is not easily seen in the cases of experimentalcurves with one single peak. However, it becomesclear in cases of experimental curves consisting of many overlapped peaks.

It must be noted that the proposed expressionsinvolving I m, T m result as transformations of theoriginal expression involving n0 and s. The onlyapproximation in these expressions is the usualapproximation of the exponential integral appearingin TL theory. The analytical TL expressions arederived using two terms of the asymptotic seriesapproximation (ASA) of the exponential integral.The transformed equations can include more termsof the ASA through the term Dm, (see later text),which should be simply replaced by the ASAexpressions above the third term. Details about theASA approximation can be found in reference(15)

SPREADSHEET PREPARATIONThe steps needed to perform a CCD analysis withan Excel spreadsheet are shown in Table 1. Theexample used is that of a complex TL glow curveconsisting of two individual peaks using the GOKexpression.

In the first step, one ascribes the temperature andTL intensity of the experimental glow curve to thecolumns A and B. In the second step, the subsequentcolumns C and D are ascribed to each one of theindividual peaks 1 and 2, respectively.

Table 1. Calculations for TL data under GOK.

A B C D E F

1 Heat rate (K/s)2 134 Peak #1 Peak #25 I max 0.04 0.056 T max (K) 417 4567 E (eV) 1.3 1.58 b 1.0001 1.00019 M.F. 1000a 1000a

10 Integral11 11 098 16 904

12 Freq factor (s – 1)13 7Eþ15 2.9Eþ151415 FOM (%) 1.2161718 Temp I(T)exp Peak 1 Peak 2 S I(T)fit jI(T)exp –I(T)fitj19 301.01 0.0003 0.0003 4.8E206 0.0003 3.6E20620 302.02 0.0004 0.0004 5.9E206 0.0004 4.2E20621 303.02 0.0005 0.0005 7.1E206 0.0005 4.5E206

aThe order of magnitude of maximum intensity Imax.

D. AFOUXENIDIS ET AL.

364



For the fitting procedure, one has to insert initial,arbitrary but meaningful values for the parametersI m, T m, E and b for each single glow-peak. As canbe seen in Table 1, these values are inserted in therows 5–8 respectively in column C for the first peakand column D for the second peak.

When the curve fitting procedure is completed, itgives the net values of I m, T m, E and b. However,the evaluation of additional quantities such as theintegral of each glow-peak, the frequency factor andfigure-of merit (FOM) values (see later text) is alsofeasible.

In the next step, the raw experimental data areinserted in the same spreadsheet in a way that thefirst data point appears in the 19th row of columnsA and B, corresponding to the temperature andsignal respectively.

Following that, the GOK analytical expression(7)

I ðT Þ ¼I m bb=b1

exp

E

kT

T T m

T m

ðb 1Þ ð1 DÞ T 2

T 2m

exp E

kT

T T m

T m

þ Z m

b=b1

where

D ¼2kT

E ; Dm ¼

2kT m

E ; Z m ¼ 1 þ ðb 1Þ Dm

has to be written in Excel format in cell C19. It willbe used to reproduce the first (lower T max) peak incolumn C by dragging it to the entire columnC. Therefore a peak for the entire temperatureregion is calculated, according to the temperatureT (K ) and its four trapping parameters that were pre-selected. This procedure is repeated for the nextglow peak in column D, by copying and pasting thesame expressions to cell D19 and dragging it to theentire corresponding column.

In that way, two arbitrary glow peaks arecreated without any other limitation. However, thesum of these peaks should be similar to the total

experimental curve. This similarity is usuallychecked by the linear regression coefficient (x 2),which however does not provide an immediateclue regarding the goodness of fit. In order to cir-cumvent this problem, another mathematical indexwas selected, which is termed the FOM(16) and isdefined as

FOM %ð Þ ¼ 100

P p TLexp TLfit

P p TLfit

The FOM index value provides a measure for thegoodness of fit; the lowest its value, the best fit.Therefore, every fitting attempt should result inminimising the FOM index value, which isachieved by changing the set of the parametervalues of each glow peak. This is achieved byusing certain optimisation software packages, theSolver, the power full add in of Excel. A fulldescription for the latter is presented in the follow-ing section.

Returning to spreadsheet preparation procedure,two more columns are required, for instance E andF, in order for the sum of the fitted (calculated) glowpeaks and the absolute value of the differencebetween each experimental and calculated datapoint to be respectively presented. The main objec-tive is to minimise the values in column F. Finally,showing the FOM index value among the initialcells of the spreadsheet is suggested.

The spreadsheet of Table 1 described earlier

stands as an example for the deconvolution of theglow curve into its two overlapping glow peaks, inorder for the parameters associated with the indi-vidual peaks to be estimated. Only the first 21rows are shown for the sake of brevity. Columns(A19:A21) and (B19:B21) contain the experimentaldata points for the TL glow curve, while columns(C19:C21) and (D19:D21) contain the fitted datapoints using the GOK model. Row nine holds thevalue of a multiplier factor that is necessary forfurther analysis and represents the order of peakheight intensity.

The heating rate used throughout the measure-

ment is given in Cell B2 since it is a common par-ameter for all the glow peaks. Furthermore, the TLintegral of each glow peak (sum of the data incolumn below row 19) is given in row 11 of the cor-responding column, while in row 13, the correspond-ing frequency factors are shown. The frequencyfactor values are calculated from the respectiveequation(7) using the values of E , T m and b obtainedfrom the curve fitting and using the heating rategiven in Cell B2.

ABOUT SOLVER

The Solver(17, 18) is an Excel Add-in, a softwareprogram that could be found in the Tools menu; itcan be installed by checking the Solver Add-in fol-lowing the path: Tools !Add-Ins. Solver is ageneral-purpose optimisation package that is used inorder to find a maximum, minimum or specifiedvalue of the target Cell. The Solver code is aproduct of Frontline Systems Inc. The Solver can beused to minimise the sum of squares of residuals(differences between yobsd and ycalc) and thusperform a least- square fitting. It can be used to

COMPUTERISED CURVE DECONVOLUTION

365



perform either linear or non-linear least-squarefitting.

The Solver uses the Generalised ReducedGradient non-linear optimisation code. For each of the changing cells, the Solver evaluates the partialderivative of the objective function (the target cell)with respect to the changing cell a i, by means of thefinite-difference method.

The Solver uses a matrix including the partialderivatives to determine the gradient of the responsesurface and thus decides how to change the valuesof the changing cells in order to approach thedesired solution.

The Solver Parameter dialog box is presented inFigure 1. In the Set Target Cell box the cellcontaining the quantity that is going to be mini-mised, here the FOM (%), must be typed. Becauseof the minimisation process, the Min button mustbe checked. In the By Changing Cells box, theCells that contain the fitting parameter values

must be typed—here the values of I max, T max, E and b.

With Solver, some physically imposed constrainscan be applied to the solution. For example, in thecase of GOK, the value of the kinetic parameter b islimited by the inequality 1,b 2 and in the case of MOK, the value of the kinetic parameter a islimited by the inequality 0 a , 1. So this con-strains must be used for the above cases.

Since the Solver operates by a search routine, asolution would be found most rapidly and efficientlyif the initial estimates that have been provided areclose to the final values. A chart of the data that dis-

plays both yobsd and ycalc is often useful to becreated, and then a good set of initial parameterscan be estimated by varying the parametersmanually.

It must be noticed that the initial values of thefitting parameters and the optimised parameter

values are placed in the same cell because the Solver

requires that the user should define the cells that

have to be modified (by changing cells, on the Solver

parameter dialog box) so that the quantity selected

in the set target cell (on the Solver parameter dialogbox) be minimised.

CURVE FITTING EXAMPLES ANDDISCUSSION

TL case

The synthetic glow curve REFGLOW.002 as well asthe measured glow curve REFGLOW.009 of theGLOCANIN project(19) are used next as examplesin order to estimate their parameters, using theearlier-mentioned procedure in Excel.

The synthetic glow curve REFGLOW.002 is thesum of four glow peaks obtained by solving directlythe differential equation with trapping parameterssuch as the glow curve structure of TLD-100, whichdescribes the charge transport in the TL materialaccording to the well-known Randall-Wilkins modelwith the Bulirsch-Stoer method. This glow curve is

included in order to evaluate the accuracy in thedetermination of parameters in the case of overlap-ping peaks.

Figure 2 shows the reference glow curveREFGLOW.002 in open cycles together with the cal-culated curve as a solid line passing through them,using the GOK model.

Table 2 gives the calculated parameters for thepeaks 2, 3, 4 and 5 for the GLOCANIN curveREFGLOW.002 as found by applying the solver Add-in on the functions for both GOK and MOK(7, 11),using the Microsoft Excel spreadsheet.

As can be seen from the low FOM values (0.0090

and 0.0093 % for the GOK model and the MOKmodel, respectively), the glow curve deconvolution

Figure 2. Curve fitting of REFGLOW.002 using excelspreadsheet and the Solver Add-in under GOK model. Inthe lower graph residual TL intensity versus temperature,

after curve fitting is presented.Figure 1. Solver parameter dialog box.


366



analysis process was very accurate. The similarvalues of peak temperature (T max), activation energy(E ) and Integral, for each individual glow curve pre-sented, as well as the same order of magnitude of

frequency factor (s), indicate the agreement betweenthe used models. These values are in great agreementwith those reported in the GLOCANIN project(19),indicating the computational power of Solver utility.

The value of the calculated parameter (b) thatstands for kinetic order according to the GOK hasbeen calculated as b¼1.00065 for peak 2, very closeto unity (b¼1) that stands for the first-order kineticsmodel that has been used in order to obtain theREFGLOW.002, according to the Randall-Wilkinsmodel. On the other hand, the value of the calcu-lated parameter (a) that stands for kinetic orderaccording to the MOK has been calculated asa¼7.261025. For this value (a¼0), the MOK

model is equivalent with the first-order kineticmodel as has been described by Kitis et al .(12).

Reference glow curve REFGLOW.009 stems for ahighly irradiated (D¼600 Gy) TLD 700 sample.Apart from the well-known glow peaks 2–5, it fea-tures a complex group of high temperature peaks.These overlapping peaks make the fitting proceduremuch more complicated.

Figure 3. Curve fitting of REFGLOW.009 usingexcel spreadsheet and the Solver Add-in underGeneral-Order Kinetics model. In the lower graph

residual TL intensity versus temperature, after curvefitting is presented.

Table 3 gives the calculated parameters for thepeaks 1, 2, 3, 4, 5, 5a, 6, 7, 8, 9, 10 and 11 of theGLOCANIN curve REFGLOW.009 as they wereestimated by applying the Solver add-in utility onthe function of the GOK(7), in the Microsoft Excelspreadsheet. These values are in great agreementwith those reported in the GLOCANIN project(19),indicating the computational power of Solver utility.

OSL case

As opposed to the case of TL, there is no projectsuch as the GLOCANIN for OSL. So, there are notany reference LM-OSL curves with their correspond-ing parameters in order to evaluate the capabilitiesof computer codes for fitting in LM-OSL cases.

Due to the lack of any reference LM-OSL curves,simulation was necessary in order to obtain data forreliable curve fitting. The latter was performed usingthe equation(10)

I ðtÞOSL ¼ n0sg t 1 þ b 1ð Þsg

2 t2

b=1b

where n0 is the initial concentration of electrons intraps, s the photoionisation cross section, g thestimulation increase rate and b the kinetic order par-ameter. To obtain numerical values for I(t) OSL, thevalues for n0¼100 000, s.g¼0.0001 andb¼1.00000001 were used. The resulting data wereused to perform a curve fitting using the equationsfor GOK and MOK of LM-OSL. Figure 4 showsthe simulated LM-OSL curve as open cycles and thecalculated curve as a solid line passing through themunder the GOK model.

Figure 3. Shows the reference glow curve REFGLOW.009in open cycles together with the calculated curve as a solid

line passing through them, using the GOK model.

Table 2. Estimated TL parameters for both GOK andMOK using Microsoft Excel Add-in, Solver forREFGLOW.002 as well as the references values as reported

in the CLOCANIN project.

TL parameters

Peak 2 Peak 3 Peak 4 Peak 5

General-order kinetic (FOM (%)¼0.008998)T max (K) 417.1 456.5 484 511.6E (eV) 1.383 1.483 1.584 2.0038b 1.0006 1.0009 1.0009 1.0006TL 11 100 16 914 27 382 47 302s (s21) 3.9Eþ16 1.6Eþ16 2.1Eþ16 4Eþ19Mixed-order kinetic (FOM (%)¼0.009339)tmax (K) 413.6 451.6 478.0 503.7E (eV) 1.383 1.483 1.583 2.0038a 7.2E205 0 0 0TL 11 098 16 896 27 397 47309s (s – 1) 5.6Eþ16 2.5Eþ16 3.3Eþ16 8.5Eþ19CLOCANIN(19)

tmax (K) 417.3 456.8 484 511.9E (eV) 1.383 1.483 1.583 2.0038s (s – 1) 3.9 Eþ16 1.6Eþ16 2Eþ16 4Eþ19TL 111 001 16 898 27 401 47 302

Note that T max for the CLOCANIN project is calculatedaccording to the Randall-Wilkins model from the givenheating rate, E and s values.


367



Table 4 gives the peak parameters for the simu-lated curve as found by applying the Solver add-inutility on the function of the GOK(13) and on thefunction of the MOK(14), using Microsoft Excelspreadsheet.

As can be seen from Table 4, the accuracy of thefitting procedure using Microsoft Excel Add-in,Solver, is extremely good, as described by the FOM

Table 3. Estimated TL parameters for General-Order Kinetics using Microsoft Excel Add-in, Solver for REFGLOW.009 aswell as the reference values as reported in the CLOCANIN project.

Peak 1 Peak 2 Peak 3 Peak 4 Peak 5 Peak 5a Peak 6 Peak 7 Peak 8 Peak 9 Peak 10 Peak 11

General-order kinetic (FOM¼0.78 %)E (eV) 0.95 1.23 1.30 1.56 2.08 1.77 1.46 1.37 1.86 1.97 1.75 1.41

s (s – 1) 11013 11015 21014 11016 51020 91016 11013 91011 71015 11016 31013 11010

GLOCANIN(19)

E (eV) — 1.25 1.3 1.6 2.02 — — — — — — — s (s – 1) — 11015 21014 81016 81019 — — — — — — —

Figure 4. Curve fitting of simulated LM-OSL curve usingMicrosoft Excel spreadsheet and the Solver Add-inaccording GOK model. In the lower graph values forresidual LM-OSL intensity versus stimulation time, after

curve fitting are presented.

Table 4. Estimated LM-OSL parameters for both GOKand MOK using Microsoft Excel Add-in, Solver for

simulated LM-OSL curve.

LM-OSL parameters

GOK MOK

tmax (s) 100.000013 99.99991708b 1.000000001 a 0.000002696Integral 99998.75394 99998.80336FOM (%) 0.00004929 0.000054287

Table 5. Estimated parameters for both GOK and MOKusing Microsoft Excel Add-in, Solver for Al2O3:C LM-OSL

curve structure.

Al2O3:C

Mixed-order kinetics [FOM (%)¼1.389]

Peak 1 Peak 2 Peak 3tmax (s) 93.552 415.368 1316.279a 0.950 0.909 0.914Integral 1 932 742 4 516 100 3 318 872

General-order kinetics [FOM (%)¼1.410]tmax (s) 93.471 415.367 1316.277b 2.000 2.000 1.916Integral 1 967 657 4 554 843 3 245 255

Deviations (%)tmax (s) 0.086 0.000 0.000Integral 21.806 20.858 2.218

Figure 5. Shows an experimental LM-OSL curve of thewidely used Al2O3:C dosemeter (open cycles) together withthe fitted curve (solid line) using the MOK model. Theindividual components of the Al2O3:C LM-OSL curve are

also shown in the same figure.


368



index. The respective FOM values are 0.000049 %for the GOK model and 0.000054 % for the MOKmodel. It should be noted that the estimated valuesof the parameters in both cases are almost the samewith very small deviations.

In Table 5, the calculated peak parameters for theexperimental LM-OSL curve of Al2O3:C are given, asthey were estimated by applying the Solver add-inutility on the function of the GOKand on the functionof the MOK, in the Microsoft Excel spreadsheet.

Figure 5. LM-OSL Curve fitting of Al2O3:C usingexcel spreadsheet and the Solver Add-in underGeneral-order kinetics model. In the lower graphvalues for residual LM-OSL intensity versus stimu-lation time, after curve fitting is presented.

As can be seen from the low FOM values (1.410and 1.389% for the GOK model and the MOKmodel, respectively), the curve deconvolution analy-sis process was very accurate. Deviations betweenthe estimated values of each of the components pre-

sented, in stimulation time at peak height (tmax), inIntegral, is also presented. These values are in greatagreement with those reported in the literature(20).

CONCLUSIONS

The deconvolution process for TL as well as linearmodulated optically stimulated luminescence curvesusing a common Microsoft Excel spreadsheet wasoutlined in this paper. Both GOK and MOK func-tions can be used for deconvolution. It should benoted that the same process can be used for analys-ing the sum of a number of exponentials and is not

limited to TL and LM-OSL cases presented expli-citly in this paper. For instance, CW-OSL curves canbe analysed in a similar way by applying the trans-formations proposed by Bulur.

The process uses the Solver utility, a MicrosoftExcel add-in, that is also briefly presented here;Specific examples of both TL and OSL compu-terised curve deconvolution analysis using thespreadsheet along with the Solver add-in, are given.These examples show that the luminescence trappingparameters can be estimated with a high accuracywith a simple procedure without the need of compli-cated and specialised computer codes. The accuracyof the results was further confirmed in the case of

TL by comparison with the results of theGLOCANIN project.

This work shows the power of using a simpleinterface commercial software such as MicrosoftExcel, in order to carry out complex scientific pro-blems, such as computerised curve deconvolutionanalysis. It should be noted that the authors stronglyrecommend the usage of the Solver utility because itis used within the familiar Excel environment; sonew commands and procedures do not have to belearnt.

Microsoft Excel spreadsheets for multi-componentluminescence computerised curve deconvolutionanalysis can be obtained on-line from the web site(www.ipet.gr) of the Archaeometry Department, of the Cultural and Educational Technology Instituteby filling a registration form with the contact infor-mation. The spreadsheets are offered free of charge,as long as they are not used for commercial purposesor as a part of a system for dose assessment(21).

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