Chapter 1

Introduction and literature survey

1.1 General introduction

Bremsstrahlung is the fundamental process, in which the photon emission occurs due to

the scattering of an electron from an atom. Bremsstrahlung plays an important role in all

branches of physics: atomic and nuclear physics, solid state physics, plasma physics and

astrophysics. It has a wide range of application in many areas of experimental and theoretical

physics research. Until 1970s, bremsstrahlung was considered in the domain of an acceleration

of electron in the static screened coulomb field of the target nuclei. In early 1970s, several people

[Buimistrov and Trakhtenberg (1975, 1977) and Amusia et al. (1985)] consider the dynamic

response of the target atom, that can be polarized by the incident electron and the photon

emission occurs. This mechanism of photon emission is known as polarization bremsstrahlung.

The total bremsstrahlung (BS) amplitude is the sum of ordinary bremsstrahlung (OB) and

polarization bremsstrahlung (PB) amplitudes. Ordinary bremsstrahlung is the process by which

the photon is emitted by the electron decelerating in the static field of the target atom.

Polarization bremsstrahlung is the process by which the photon is emitted by the target as a result

of its polarization by incident electron. During the collision of the incident electron and the atom,

the internal structure of the atom is deformed or polarized and an electric dipole moment is

induced. Being time-dependent, it becomes a source of continuous electromagnetic radiation,

called polarization bremsstrahlung. It is more complicated than the ordinary bremsstrahlung. In

addition to the electron-photon interaction, one has to consider the dynamic response of the

target atom created by the action of the two fields created by the incident electron and the

emitted photon. The polarization bremsstrahlung plays an important role particularly at lower

and medium photon energy regions and its contribution in the total bremsstrahlung spectra must

be taken into account, while comparing the theoretical and experimental results.

A beam of mono-energetic electron passes through the material medium suffers elastic,

inelastic scattering and multiple scattering. The electron loses energy through the excitation and

ionization of the absorbing atoms of the material media. Total energy loss by an electron in the

target includes the energy lost by the electron due to its inelastic and radiative collisions with the

atom in a material medium. The scattering of electrons through the finite angle always

accompanied by the emission of electromagnetic radiation termed as bremsstrahlung. The

behavior of the continuous beta particles is same as that of the mono-energetic electrons except

continuous nature of the beta particles, whose energy spread over from zero to maximum end

point energy of the beta radioactive source. The continuous spectrum of beta particles or

electrons produces electromagnetic radiation on suffering deflection through acceleration or

retardation from the static coulomb field of the nuclei in material medium is termed as external

bremsstrahlung or ordinary bremsstrahlung (OB). However, there exist an important difference

between the mono-energetic electron and continuous beta particles of radioactive beta source. In

case of beta emitter, an electromagnetic radiation is emitted along with the originated electrons

or continuous beta particles interaction with the coulomb field of the daughter nucleus. This

electromagnetic radiation is termed as internal bremsstrahlung and it is the characteristic of beta

emitter. The mode of production of ordinary bremsstrahlung is relatively different from the

production of the internal or inner bremsstrahlung due to electron capture. Historically, the first

measurements of the ordinary bremsstrahlung produced by the continuous beta particles of the

beta emitter were reported by Gray (1911, 1912) and Chadwick (1912). Later, Gray (1922)

measured the ordinary bremsstrahlung spectra in targets of iron, lead and paper produce by the

absorption of continuous beta particles of beta emitter. The phenomenon of internal

bremsstrahlung was first discovered by Aston (1927) in beta decay of radioactive source. Later,

the detailed experimental studies have been carried out by Gray and Hinds (1936) and Droste

(1938).

1.2 Description of ordinary bremsstrahlung

In classical electrodynamics an accelerated charged particle passing through the static

field of the target nucleus emits a photon; this is termed as ordinary bremsstrahlung. The

ordinary bremsstrahlung amplitude is proportional to the acceleration produced by nucleus, on a

particle of charge ‘Ze’ and its mass ‘m’. The bremsstrahlung intensity is directly proportional to

the square of atomic number of the target element or absorbing material and varies inversely as

the square of the mass of the projectile. Therefore, the bremsstrahlung intensity is more for the

light particle like electron and it is relatively small for the heavy particle like proton, alpha

particles etc. According to the conservation laws of energy and momentum we have,

We = W + k + Wq (1.2.1)

Pe = P + Pk + q (1.2.2)

Here, We and W are the total energies of the incident and scattered electron, and Pe and P are the

momentum of the incident and scattered electron, Wq and q be the total energy and momentum

of the recoil atom. k and Pk be the energy and momentum of the emitted photon.

Fig. 1.21 shows the process of production of ordinary bremsstrahlung. This depicts the

production of photon by the interaction of the incident electron with the interaction of the

nucleus of the target atom.

Figure: 1.21 Kinematics of elementary process of bremsstrahlung

Therefore, in individual interaction of the incident electron with the target nucleus, the incident

electron losses energy from 0 to maximum amount of its total kinetic energy Te=We-1. The

maximum photon energy kmax at the short wavelength limit of continuous X-ray spectrum is

given by

kmax = Te = We -1 (1.2.3)

This relation was experimentally established by Duane and Hunt (1915) and is known as Duane

and Hunt’s law.

The Feynman diagram, Fig. 1.22 represents the process of ordinary bremsstrahlung.

(W, P)

(k, Pk)

k

Incident electron

e

Scattered electron

Target nucleus

(We,Pe)

Recoil (Wq, q)

Emitted photon

Figure: 1.22 Feynman representation of ordinary bremsstrahlung

Initially, two possible intermediate states can occur, the photon is emitted and then the electron is

scattered afterwards, or the sequence is reversed. In one of the cases the intermediate state is a

virtual state, since photon cannot be emitted at least violating one of the conservation law.

(We, Pe)

(W, P)

(k, P k)

(0, q)

n´

(W, P)

(0, q)

(k, P k)

(We, Pe)

n´´

(a) (b)

1.3 Theoretical aspect of ordinary bremsstrahlung

Theoretical investigation of bremsstrahlung process was started on the basis of classical

electrodynamics. The initial attempts were made by Kramers (1923) to develop the

bremsstrahlung theory on the basis of semi-classical calculation, by using correspondence

principle. The first Quantum mechanical cross-section formulae for the elementary process of

bremsstrahlung was derived by Sommerfeld (1931), in the non relativistic dipole approximation

including retardation for non relativistic electrons without taking accounts the nuclear screening

effects. For the relativistic case, by using the Dirac theory, Sauter (1931) and Bethe and Heitler

(1934) obtained independently an analytical expression for the OB cross-section, by using the

first order Born approximation. They neglected the coulomb field effects on the wave function of

incident and scattering electrons on the nucleus. Elwert (1939) gave the multiplicative coulomb

correction factor ( ElwertF ) for Bethe-Heitler OB cross-section. Tseng and Pratt (1971) developed

a quantum theory for the bremsstrahlung for relativistic electrons by using screened self

consistent field wave function. They gave the exact screened calculations of atomic field

bremsstrahlung by considering the modified Hartree-Fock Slater potential to incorporate the

effect of screening in the production to the electron-nucleus bremsstrahlung. Pratt et al. (1977)

published extensive tables of OB cross-section for incident electron energy from 1 to 2000

keV and for Z values between 2 and 92. Later, Seltzer and Berger (1986) incorporated the

contribution of electron-electron bremsstrahlung to electron- nucleus bremsstrahlung, given

by Pratt et al. (1977). There are extensive reviews on the theory of OB by Koch and Motz

(1959), Pratt and Feng (1985), Seltzer and Berger (1985) and Pratt et al. (1995). They considered

the interaction of electron with the static field of nucleus.

The expression for ordinary bremsstrahlung cross-section in Born’s approximation is known as

Bethe-Heitler (1934) formula. The Bethe-Heitler OB cross-section ( )),,( ZkWeBH differential in

photon energy k is given as,

)(2

3

4

137),,(

3322

2222

BALPP

EE

P

WE

P

WE

PP

PPWW

P

P

k

dkrZZkW

e

eee

e

e

e

e

e

e

o

eBH (1.3.1)

Where

A=33

22222 )(

3

8

PP

PPWWk

PP

WW

e

ee

e

e

B=

223

2

3

2 2

2 PP

WkWE

P

PWWE

P

PWW

PP

k

e

ee

e

e

ee

e

ro= classical radius of electron= 2.818×10-13

cm

We= W+k

12 ee WP

12 WP

ee

eee

PW

PWE

ln

PW

PWE

ln

kWPPP

kWPPPL

eee

eee

2

2

ln =k

WWPP ee )1(ln2

We, W = initial and final total energy of electron

Pe, P = initial and final momentum of electron

In the non relativistic limit following approximations have been used

(i) Non screened : 137Z-1/3

>>(WeW/k)

(ii) Born’s approximation : ee pZW /2 ; pZW /2 << 1

(iii) Non relativistic : 1e

e

W

p

The total OB cross-section follows as limit of eq. (1.3.1) is given as,

PP

PP

Pk

dkrZZkW

e

e

e

o

e ln1

3

16

137),,(

2

22

(1.3.2)

The multiplicative coulomb correction factor ( ElwertF ) for Bethe-Heitler OB cross-

section ( ),,( ZkWeBH ) is given by

)]/2exp(1[/

)]/2exp(1[/

PZWPW

PZWPWF

ee

ee

Elwert

(1.3.3)

This correction factor was derived on the basis of a comparison between the non-relativistic

Born-approximation and non-relativistic calculations.

1.4 Description of polarization bremsstrahlung

The idea of polarization bremsstrahlung was first given by Buimistrov and Trakhtenberg

(1975, 1977). Later, this mechanism of polarization bremsstrahlung was also demonstrated by

Amusia et al. (1985). This mechanism was applied in discussing gas discharge phenomenon by

Zon (1977) and resonance photoemission from solids by Wendin and Nuroch (1977). The

theoretical and experimental studies of polarization bremsstrahlung have been reviewed

extensively by Tystovich and Ojringel (1993), Korol and Solov'yov (1997), Korol et al. (2004),

Korol and Solov’yov (2006) and Amusia (1988, 2006).

The process of polarization bremsstrahlung is illustrated in Fig. 1.41. Here, the photon emission

occurs due to the virtual excitation or polarization of the target atom by the incident electron.

Virtual excitation of the target atom is equivalent to the polarization of the target atom. The total

induced dipole moment of the system alters during the collision resulting into the emission of

photon.

Figure: 1.41 Kinematics of process of polarization bremsstrahlung

The first qualitative and quantitative estimates were made for the role of the target polarization

into the forming the bremsstrahlung spectrum in an electron-atom collision, in the photon energy

ranges close to the atomic ionization potentials by Buimistrov and Trakhtenberg (1975, 1977) .

The effect of a dynamic descreening (stripping) of a many electron subshell of the atom for the

photon energy larger than its ionization potential was formulated by Amusia et al. (1985). On the

basis, the additional asymmetry of the experimentally measured bremsstrahlung spectrum was

also explained.

Nucleus

Electron

Cloud

Induced dipole moment

Projectile

Emitted

photon

1.5 Theoretical aspect of polarization bremsstrahlung

In polarization bremsstrahlung, the dynamic response of the target atom and the atomic

dynamic polarizability is needed for the calculation of polarization bremsstrahlung amplitude

and such calculations have been presented by several authors Korol et al. (1995, 1996, 1997,

2001), Amusia et al. (1985), Amusia and Korol (1993). Born approximation and distorted partial

wave approximation (DPWA) are used to calculate the polarization bremsstrahlung (PB). For

non-relativistic electron energies, in the Born approximation, Amusia et al. (1985) has described

that PB can be added with OB in a stripped atom approximation (SAA). The stripped

approximation is efficient for obtaining the BS spectra for photon energies greater than the

ionization potential of the outer shell electrons of the target atom. In SAA, the decrease of OB

due to screening of outer shell electrons is completely compensated by additional PB produced

by the same outer shell electrons. Therefore, the total bremsstrahlung (BS) is described simply

by an ion containing the outer shell electrons. As the emitted photon energy exceeds the

ionization potential of the inner most shell (1s), the bremsstrahlung occurs on the bare nucleus.

The difference between the OB from an ion and the bremsstrahlung on bare nucleus gives the

contribution of PB in the BS spectra. Korol et al. (2002) and Avdonina and Pratt (1999) have

given the equivalent method for the BS spectra in SAA. SAA approach neglects the specific

structure of the bremsstrahlung cross section near each sub shell threshold, where polarization

bremsstrahlung often becomes large in comparison with OB.

Avdonina and pratt (1999) modifies the Elwert corrected (non relativistic) Bethe and

Heitler (1934) theory for OB and described the BS spectra i.e. (OB+PB) over a wide range of

photon energy region, by applying the SAA. They further described that in the non relativistic

case, the PB decreases with increasing photon energy in the same way as the screening

contribution to OB, leading to the coulomb behavior of the spectrum. They also described that

for relativistic electron energies, the contribution of PB to the soft photon region of the

bremsstrahlung spectra is larger than in the non relativistic case.

Avdonina and pratt (1999) described the bremsstrahlung energy spectrum in terms of

Guant factor G (k) = ((dσ(We,k,Z)/dk)/σkr), where σkr = (16Z²π ⁄3√3βf²α³) is the Kramers (1923)

bremsstrahlung cross-section and ),,( ZkWe is the total bremsstrahlung cross section which

include the contribution of PB into OB and singly differential in radiated photon energy, which

depend on the energy of incident electron energy We , photon energy k and atomic number (Z) of

the target. They modified the Elwert corrected (non relativistic) Bethe and Heitler theory by

replacing relativistic velocity if by relativistic momentum 2if if ifP T T (where Tif is the

initial and final electron kinetic energy) in the Elwert factor (1939). The modified Elwert factor

is given by

)]/2exp(1[

)]/2exp(1[mod

ff

ii

PZP

PZPF

(1.5.1)

Further, a higher order born-approximation factor C (Ti, Z) was used to improve the accuracy of

the modified OB cross-section ( ),,(mod ZkWe ),

),,(),,( modmod ZkWFZkW eBHe (1.5.2)

4

)2()(1),( 2 i

i

TZZTC

(1.5.3)

The corrected modified OB cross-section ( ),,( ZkWecor ) is given by

),,(),(),,( mod ZkWFZTCZkW eBHiecor (1.5.4)

In the soft photon energy part of the spectrum, PB is important and can be incorporated by

applying SAA. A simple analytic expression which include a screening parameter 2 0.798o Z

for neutral atom for describing the PB is given by

)]()ln()ln(2[3

)(22

2

22

22

22

22

22

2

ooo

oB

q

q

q

qZ

q

qZ

q

qZ

Zk

(1.5.5)

Here fi PPq is the momentum transfer. Further, Avdonina and Pratt (1999) has proposed

to use a composite expression for bremsstrahlung cross-section ( ),,( ZkWe ), if the

parameter 1i

Z

, which include polarization bremsstrahlung in SAA with ordinary

bremsstrahlung,

),,()ln(3

)(),,( ZkWq

qkZkW ecorBe

(1.5.6)

This expression gives the total bremsstrahlung cross-section which includes the contribution of

polarization bremsstrahlung into ordinary bremsstrahlung.

1.6 Bremsstrahlung in material medium

The various theories for OB and BS spectra discussed in the previous sections (1.3 and

1.5) are applicable to thin target only, in which the monoenergtic electron has only a single

radiative interaction. In the case of thick target, processes such as electron scattering, excitation

and ionization and multiple scattering that compete with bremsstrahlung are required to be taken

into account. In this case an electron loses a significant part of its energy while coming to rest in

a target. Bethe and Heitler (1934) gave an expression for the bremsstrahlung spectral distribution

),,( ZkWn e in a sufficiently thick target to absorb an electron of energy eW with N atoms per unit

volume given by

e

e

ee

W

k

e dWdxdW

dkZkWdNZkWn

)/(

/),,('

),,(1

(1.6.1)

At lower photon energies in thick targets, the correction due to absorption of bremsstrahlung

photons in the target and electron backscattering from the target can not be neglected. Semaan

and Quarles (2001) have reported that the correction for the self absorption of bremsstrahlung

photons in the target and electron backscattering are required for ),,'( ZkWn e , in case of low

energy thick target bremsstrahlung. The bremsstrahlung spectral distribution )],,'([ ZkWn ecor

after absorption correction and electron backscattering correction in thick target is given by

)exp()/(

/),,('

),,'(1

xdWdxdW

dkZkWdRNZkWn e

e

ee

W

k

ecor

(1.6.2)

Here dkZkWd e /),,( is the singly differential cross section taken from the different theoretical

models, ‘ dxdWe / ’ is the total energy loss per unit path length of an electron in a target

material taken from the tabulations given by Berger and Seltzer (2000). Where )(exp x is

the absorption factor, is the mass attenuation coefficient for the given target element taken

from the tabulations given by Chantler et al. (2008) and '' x is the optimum thickness of the target

which is equal to the range of the beta particle in a target. ‘R’ is the electron backscattering

factor given by Semaan and Quarles (2001)

2

2

),(1

),(1

e

e

e

W

kZW

ZWR

(1.6.3)

Here, max4.0 WWe , maxW is the end point energy of beta particles and ),( ZWe is the total

backscattering factor given by August and Wernisch (1989). The bremsstrahlung spectral

distribution in a thick target obtained on complete absorption of beta particles of end point

energy maxW is expressed as number of photons of energy k per unit 2cmo per beta

disintegration for continuous beta particle is given by S(k,Z)

max

1

')'(),,'(),(

W

k

eeecor dWWPZkWnZkS (1.6.4)

Here '' )( ee dWWP is the beta spectrum of the beta source under study.

The bremsstrahlung photon yield T for the target, with mink and maxk as the lower and upper limit

of photon energy of the bremsstrahlung spectrum respectively is given by

max

min

),(

k

k

dkZkST (1.6.5)

Computer programs are written to calculate the bremsstrahlung spectral photon distribution in

terms of the number of photons of energy k per unit 2cmo per beta disintegration, i.e. S(k,Z) by

using eq. (1.6.2-1.6.4) from various theories. The total photon yields T were obtained for

different targets from graphical integration of the bremsstrahlung spectra from the plots of S(k,Z)

versus photon energy k between kmin and kmax. The experimental and theoretical results were

compared in terms of the number of photons of energy k per 2

om c per unit total photon yield.

This method makes the results independent of source strength and removes the uncertainties

associated with its measurements.

1.7 Z-dependence of spectral shape of bremsstrahlung spectra

The bremsstrahlung spectral photon energy distribution depends upon the fundamental

cross-section for interaction of an electron with an atom of the target material. Bremsstrahlung

cross-section is proportional to the square of the atomic number of the target atom. The Z-

dependence of spectral shape of bremsstrahlung spectral photon energy distributions can be

studied as a function of photon energy and atomic number of the target element. For

monoenergetic electron, the Z-dependence of ordinary bremsstrahlung (OB) cross-section as a

function of photon energy, has been studied by Hippler et al. (1981) and Semaan and Quarles

(1982). Further, Hippler et al. (1981), Avdonina and Pratt (1999) and Portillo and Quarles

(2003) pointed out that the dependence of bremsstrahlung on photon energy and atomic number

of the target is of complex nature, particularly at lower photon energies. This is due to the

screening of electron, interferences between OB and PB, absorption of photons and electron

backscattering in a target. These factors may play a vital role while studying the Z-dependence of

spectral shape of BS and OB, in the photon energy region of 5-30 keV. Avdonina and Pratt

(1999) further pointed out that the contribution of PB at the lower photon energy region of the

bremsstrahlung spectra in the relativistic regime is higher than in the non relativistic case.

Therefore, the contribution of PB should decreases with increasing photon energy.

For the continuous beta particles, Evans (1955) reported that the spectral shape of

bremsstrahlung spectra is independent of atomic number Z and each beta emitter has its own

bremsstrahlung spectrum. The available theoretical model for OB and BS are adequate to explain

the bremsstrahlung spectral photon distributions as a function of photon energy. However, these

theoretical models are not adequate to describe the dependence of spectral shape of

bremsstrahlung spectra as a function of atomic number of the target material. Dhaliwal (2003a)

reported the inadequacy of the theoretical models to describe the shape of ordinary

bremsstrahlung in detail. The factors comprising S (k, Z) (eq. 1.6.4) do not clearly describes the

dependence of shape of bremsstrahlung spectra on atomic number of the target material.

Therefore, in order to investigate the Z-dependence of the spectral shape of bremsstrahlung, the

S(k,Z) (eq. 1.6.4) number of photons of energy k per unit 2cmo per beta disintegration at

the photon energy k can be expressed as a function of Z and has been reported by Dhaliwal

(2003a) i.e.

nZkKZkS )(),( (1.7.1)

Where ‘n’ is the index of the Z-dependence of a photon energy k per unit 2cmo per beta

disintegration and K(k) is the proportionality factor, which is independent of Z at particular

photon energy k. Knowledge of the index ‘n’ is essential for evaluating the Z-dependence of the

spectral shape of OB and BS spectra. Theoretical and experimental bremsstrahlung spectral

distributions were required for the determination of the Z-dependence index ‘n’ and the

proportionality factor K(k) defined in eq. 1.7.1 at different photon energies. These spectral

photon distributions were calculated on the basis of the continuous slowing down approximation

given by Seltzer and Berger (1986). In these approximations the rate at which the electron loses

energy has two components:

(i) the average energy loss per unit path length, due to inelastic collision with the bound

electron of the medium resulting in ionization and excitation.

(ii) the average energy loss per unit path length due to the emission of the bremsstrahlung

in the coulomb electric field of the atomic nucleus.

The dependence of spectral shape of OB and BS spectra as a function of atomic number (Z) of

the target materials can be studied through the Z-dependence index (n) of the bremsstrahlung

production, by using different beta emitters in the different thick target elements, in the photon

energy region of 5-30 keV.

1.8 Literature Survey

1.8.1 Experimental studies on ordinary bremsstrahlung

For mono-energetic electrons, the large number of measurements of ordinary

bremsstrahlung for thin and thick targets covering the energy range from few keV to few tens of

MeV is available in the literature. However, for continuous beta particles relatively fewer studies

for thin and thick targets were carried out to check the experimentally measured ordinary

bremsstrahlung spectra with the different theoretical models. It has been worthwhile to mention

here, that there is an important difference between the thin-target and thick target bremsstrahlung

studies. In thin target bremsstrahlung, the photon emission occurs due to the single interaction of

the electron with the single atom of the target. However, in thick target bremsstrahlung the

radiation photon is emitted due to the multiple interaction of an electron through which it loses

energy and comes to the rest. The studies of thin and thick target bremsstrahlung were

extensively reviewed by Evans (1955), Roy and Reed (1968) and Koch and Motz (1959), Nakel

(1994) and recently by Quarles (2000), Quarles and Portillo (2006) and Shanker (2006) in detail.

The development of high resolution solid state detectors and the use of Ge and Si (Li) detector

revolutionized the nuclear spectroscopy and opened up new window of studies of

bremsstrahlung. The experimentalist’s were looking forward, to check the evidence of the

existence of the contribution of polarization bremsstrahlung in the total bremsstrahlung spectra.

The experimental evidence for polarization bremsstrahlung by electrons has come exclusively

from different experiments. Their experiment that shows resonance enhancement in the X-ray

spectra near the 3d, 4d or 5d shells of some rare earth element and Xe targets reported by

Wendin and Nuroch (1977). The polarization bremsstrahlung is more pronounced near the

resonance condition of 3d, 4d or 5d shells of the target atom. However, it has been slowly

recognized that polarization bremsstrahlung also play its role in the total bremsstrahlung spectra

over a wide range of photon and incident electron energies. Recently, efforts were made to check

the contribution of polarization bremsstrahlung into the total bremsstrahlung spectra produced by

monoenergetic electron in gaseous targets by Portillo and Quarles (2003). They reported the

absolute differential cross-section of the total bremsstrahlung spectra produced by 25 keV and 50

keV incident electron energy and first time prove the definite existence of contribution of

polarization bremsstrahlung in the total bremsstrahlung spectra over a wide range of photon

energies. However, Williams and Quarles (2008) reported that the absolute bremsstrahlung

yields at backward angle of 135o from 53 keV electrons on gold targets of thickness in the range

of 66 to 28976 µg/cm2 agrees with the predications of ordinary bremsstrahlung, for radiated

photon energies from 5 to 53 keV and do not agrees with the predication of SAA, which include

the contribution of polarization bremsstrahlung. This conclusion is in contrast with the recent

free gas atom results reported by Portillo and Quarles (2003).

For continuous beta particles, the earliest studies of ordinary bremsstrahlung produced

by different beta emitter were carried out by the Gray (1911, 1912) and Chadwick (1912). Later,

Wu (1941) reported the ordinary bremsstrahlung spectra produced by continuous beta particles in

number of targets and reported that the ordinary bremsstrahlung spectra were independent of the

atomic number of the target atom. Similar results were reported by Edwards and Pool (1946).

The studies of ordinary bremsstrahlung spectra produced by continuous beta particles in different

targets were divided into two categories:

(i) Thin target bremsstrahlung studies

(ii) Thick target bremsstrahlung studies

1.8.2 Thin target bremsstrahlung studies

The study of thin target bremsstrahlung mainly deals with the evaluation of yield

constants for bremsstrahlung production, the data were fitted to establish the semi-empirical

relation to obtain the absorption parameter. The experimental data were obtained by varying

thickness of the targets to obtain the maximum ordinary bremsstrahlung yield.

Subharmanym et al. (1977), obtained yield constants values for different alloys by using

the varying end point energies of the beta emitters. The values of yield constants both for OB

photon and energy yield were compared with the energy yield values obtained by Evans (1955).

Mudhole and Umakhanta (1972), studied the OB intensity measurements by varying the

thickness of the different target material by using 90

Sr-90

Y beta emitter. They fitted their

experimental data in a semi-empirical relation of the form I=K Zn (t/A) (-ΣB t). Where, ‘I’

represents the OB intensity, Z, t and A represents the atomic number, atomic weight, thickness of

the target, ΣB represents the parameter giving the self-absorption parameter (cm2/mg), K is the

constant for a given experimental geometry and beta emitter and n is the index of the Z-

dependence of the target atom. They found that their relation fitted the experimental data up to

the target thickness ≤ 0.4 Ro (Ro is the range of beta particles).

Sarma and Murthy (1976), studied the integral growth of OB in targets of Al, Cu, Ag, Sn and Pb

of varying thickness for different beta emitters of 32

P, 91

Y, 204

Tl, 185

W and 169

Er. They found that

the values of the target thickness of maximum OB intensity were not constant and varied

between the ranges 0.21 Ro to 0.716 Ro. The values of the self absorption parameter were

considerably varied with the thickness of the target and beta end point energy.

Gopala et al. (1987), measured the bremsstrahlung yields in different target materials,

produced by the beta particles of 147

Pm and 170

Tm beta emitters. They found that the

bremsstrahlung yields show non-linear dependence on the atomic number of the target material.

Dhaliwal et al. (1990a, 1990

b) studied the growth and attenuation of ordinary bremsstrahlung in

targets of Al, Cu, Sn and Pb excited by beta particles of some forbidden beta emitters (204

Tl, 32

P,

147Pm and

170Tm) along with their spectral photon distributions. They found that the incident and

transmitted integral intensities are expressed in terms of attenuation parameters (ap) for an

absorber of thickness ta, by the relation )exp( ap

no

nt taT

T . They found further that the value of ap

decreased with the absorber thickness and confirm that the attenuation of ordinary

bremsstrahlung in an absorber did not conform to a single exponential law. Rather, it is a

combination of large exponential terms.

So, it may be concluded from the above studies that the values of yield constants and the

integral measurements of the OB intensity above 30 keV photon energies were considerably

varied with the nature of the experimental set up. The values of various semi-empirical relation

parameter obtained by different workers were different from one another. Therefore, no

worthwhile conclusions can be drawn from these measurements. Also, these measurements did

not provide the sensitive check of the different theories of bremsstrahlung spectra.

1.8.3 Thick target bremsstrahlung studies

Thick target bremsstrahlung provides information about the spectral photon distribution

of bremsstrahlung, produced by continuous beta particles and their comparison with the different

theoretical models. In literature, numbers of measurements of thick target bremsstrahlung are

available for the different beta emitters having different end point energy. In thick target

bremsstrahlung, the contribution of internal bremsstrahlung generated from the source is

essential to be subtracting from the measurements of ordinary bremsstrahlung spectra in the

given experimental set up. So, the production of internal bremsstrahlung and its subtraction from

the ordinary bremsstrahlung creates different type of uncertainties in the given measurement,

particularly for the low Z targets. In the following paragraphs the brief review of the thick targets

bremsstrahlung studies for the different beta emitter has been given.

The earlier studies of thick target bremsstrahlung for the soft beta emitters like 35

S, 147

Pm,

and 45

Ca in different targets has been carried out by Liden and Starfelt (1955), Starfelt and

Svantesson (1955). Liden and Starfelt (1955) measured OB spectra produced by absorption of

continuous beta particles of 32

P in thick targets. They found a good agreement between the

experiment and theory for light elements while for high Z elements and higher photon energy

significant deviations were observed. Starfelt and Svantesson (1955) found good agreement

between the experimental and theoretical results obtained from Elwert (1939) corrected

Sommerfeld (1931) theory, particularly for the low Z target element. However, for high Z

element Lead, the deviation between experiment and theory has been found to be 60% at 60 keV

and 170% at 170 keV photon energies respectively.

Bussolati (1959) investigated the OB spectra produced by continuous beta particles of

90Y in thick targets of Al, Ag and Pb in energy range from 0.4 MeV to 2 MeV and

compared their experimental results with the Elwert corrected Bethe-Heitler (1934) theory. They

compared their experimental results with the theory by normalization method at 0.7 MeV for all

the targets. They reported a good agreement in low Z element, but appreciable difference for

high Z element.

Prasad Babu et al. (1975) has been reported a good agreement between the experimental

and theoretical results for low Z element C, in the photon energy region from 30 to 100 keV.

However, for high Z element they reported appreciable difference between the theoretical and

experimental results above 75 keV.

Sarma and Murty (1976) investigated the OB spectra, produced by continuous beta

particles of 204

Tl in targets of Al, Cu, Sn and Pb. They compared their experimental results with

the theoretical OB spectra obtained from the Elwert corrected Bethe-Heitler (1934) theory. These

workers reported a good agreement between the experimental and theoretical results for Al and

small deviations in case of Cu, Sn and Pb.

The study of ordinary bremsstrahlung spectra, produced by continuous beta particles

from 45

Ca in thick targets have been reported by Subrahmanyam et al. (1978). They reported the

effect of target thickness on the spectral photon distributions of OB and found a good agreement

for Al target for all thicknesses, while in case of other elements the difference between

experiment and theory was observed.

Ahmad et al. (1979) and Powar et al. (1980) studied the thick target bremsstrahlung

spectrum of 32

P and 204

Tl beta particles. They reported a good agreement between Elwert

corrected Bethe-Heitler (1934) theory and experiment for low Z target elements, while the

measurements with high Z element show discrepancy between theory and experiment.

Shivaramu (1984) investigated the spectral shapes for the ordinary bremsstrahlung

produced by absorption of continuous beta particles of 90

Y in thick targets of Cu, Cd, Ta and Pb

over the photon energy range of 600 keV to 2 MeV. The experimental results were compared

with the numerical calculation based on the formulation of Pratt el al. (1977), the Born

approximation theory of Bethe-Heitler (1934) and with the classical theory of Kramers (1923).

The experimental results showed a good agreement with theory of Pratt et al. (1977). However,

the results agreed with the Bethe-Heitler theory (1934) only for light element Cu and systematic

positive deviations were found for Cd, Ta and Pb targets. The results have poor agreement with

the classical theory of Kramers (1923) for the entire target elements.

Gopala et al. (1988) studied the external bremsstrahlung produced by the beta-particles of

147Pm in thick targets, measured by using a NaI (Tl) scintillator detector. The experimental

results were compared with the Elwert-corrected Bethe-Heitler (EBH) (1934) and Tseng and

Pratt (1971) theories. They reported that the experimental results agree fairly well with the EBH

theory up to about 120 keV and deviate positively thereafter from all the theories. Dhaliwal et al.

(1993) studied the external bremsstrahlung spectral photon distributions for OB, produced by

absorption of continuous beta particles of 147

Pm and 35

S soft beta emitter, in targets of Al, Cu, Sn

and Pb. The experimental and theoretical results were compared in the form of number of

photons of energy k per moc2 per unit total photon yield to overcome the uncertainty in the

source strength measurement and inherent inadequacy produced by the normalization procedures

used by the earlier workers. They reported that the experimental results were in agreement with

the Tseng and Pratt (1971) theory than with the Elwert corrected Bethe-Heitler (1934) theory,

particularly at the higher energy ends in medium and high Z element. However, for low Z

element both the theories were found to be accurate. Further, for 89

Sr beta emitter, Dhaliwal

(2003b) has reported the comparison of experimental OB spectra for thick target with the

theoretical ordinary bremsstrahlung distributions obtained from Elwert corrected (non-

relativistic) Bethe–Heitler (1934) theory, Tseng and Pratt (1971) theory and modified Elwert

factor (relativistic) Bethe–Heitler theory given by Avdonina and Pratt (1999). Again, it has been

found, that for low Z elements all theories are equally suitable throughout the energy region

studied. For medium Z elements, the Tseng and Pratt (1971) and modified Elwert factor

(relativistic) Bethe–Heitler (1934) theories are more accurate, particularly in medium and higher

energy regions. However, for high-Z elements, the modified Elwert factor (relativistic) Bethe–

Heitler theory shows better agreement with the experimental results.

It is evident from the literature survey that:-

(i) All the experimental studies for thin and thick targets, produced by continuous

beta particles are reported for the ordinary bremsstrahlung spectral photon

distributions only, particularly above photon energy 30 keV.

(ii) The experimental OB spectral photon distributions were measured by using NaI

(Tl) detector. Further, these measurements were compared with the OB spectral

photon distributions obtained from Kramers (1923) theory, Sommerfeld (1931)

theory, Elwert corrected (non-relativistic) Bethe–Heitler (1934) theory, Tseng and

Pratt (1971) theory and modified Elwert factor (relativistic) Bethe–Heitler theory

given by Avdonina and Pratt (1999).

(iii) In general, earlier measurements showed that the experimental and theoretical

results obtained from various theories for OB, agree only for low and medium Z

elements. However, for high Z element, there is a disagreement between theory

and experiment.

For mono-energetic electron, variety of calculations of polarization bremsstrahlung as

explained in section 1.5, are available in the literature. However, experimentally only few

measurements were reported by Portillo and Quarles (2003) and Williams and Quarles (2008) to

check the existence of PB in the BS spectra. However, for continuous beta particles, no

measurement has been reported in the literature to check these theories which describe OB and

BS spectra i.e. (OB+PB) in the photon energy region of 5-30 keV. So, there is need to study the

BS and OB spectra produced by continuous beta particles in thick targets by a high resolution X-

PIPS Si (Li) detector to check the contributions of polarization bremsstrahlung, particularly at

photon energy 5 keV to 30 keV.

1.8.4 Experimental studies of Z-dependence of spectral shape of

bremsstrahlung spectra

For mono-energetic electrons, Hippler et al. (1981) and Semaan and Quarles (1982) has

studied, the Z-dependence of spectral shape of ordinary bremsstrahlung spectra as a function of

photon energy. They predicted the Z2 dependence of bremsstrahlung cross-section in different

target materials and for the different incident electron energies. Hippler et al. (1981) investigated

the relative ordinary bremsstrahlung cross-sections for free atoms, with the atomic number

ranges from Z=2 to 92 at low incident electron energies of 2.5 keV and 10 keV. The

experimental results were compared with the theoretical values obtained from the Pratt et al.

(1977). They reported a good agreement between experiment and theory, except at lower photon

energies, where increases in bremsstrahlung intensity up to factor 2 have been observed for high

Z element. They suggested the process of two photon bremsstrahlung, which is not included in

the theory. Hippler et al. (1981) further, suggested that the difference at the larger Z numbers is

attributed to the screening of atomic electrons, which is important at low photon energies.

Semaan and Quarles (1982) studied the atomic field bremsstrahlung cross-sections for Z=80 and

the atomic number ranges from Z=10 to 54 at lower photon energies. They compared

experimental results with the theory of Pratt el al. (1977). The results show better agreement with

the theory for all the target elements and suggested that there is no evidence of two photon

bremsstrahlung, at lower photon energy. So, they reported that the bremsstrahlung cross-section

show Z2 dependence and there is no need to include the two photon process at lower photon

energies.

For continuous beta particles, Wu (1941) and Evans (1955) reported that the

bremsstrahlung intensity is linearly dependent on the atomic number Z of the target element.

Evans (1955) further, reported that the spectral shape of ordinary bremsstrahlung is independent

of the target element and each beta emitters has its own characteristic OB spectrum. However,

Subrahmanyam et al. (1977) reported that the bremsstrahlung intensity showed limited linearity

to atomic number Z. Mudhole (1973), investigated the Z-dependence of the ordinary

bremsstrahlung cross-section in thin foils of metallic targets, by using 90

Sr and 90

Y the beta

sources. It has been found, that the ordinary bremsstrahlung intensity is in close agreement with

the theoretical predications of Z2 dependence of the ordinary bremsstrahlung cross-sections

versus photon energy. Rudraswamy et al. (1983) has studied the Z-dependence of ordinary

bremsstrahlung intensity in thick targets, produced by continuous beta particles of 90

Sr-90

Y. They

suggested that, the relationship between the bremsstrahlung intensity and atomic number Z is not

strictly linear.

It is clear from the studies available in literature, that the Z-dependence of spectral shape

of ordinary bremsstrahlung spectral photon distribution has been studied only as a function of

photon energy. These studies did not explain the dependence of spectral shape of OB spectra on

the atomic number of the target materials. However, Dhaliwal (2002, 2003a, 2005) reported the

dependence of spectral shape of OB spectra, produced by beta particles of different beta emitters

(147

Pm, 32

P, 89

Sr, 35

S, 204

Tl and 170

Tm), as a function of atomic number (Z) of the target element,

above photon energy 30 keV. It has been reported, that S (k, Z) i.e. the number of photons of

energy k per unit 2cmo per beta disintegration at the photon energy k can be expressed as a

function of Z as S (k, Z) = K (k) Zn. Here ‘n’ is the index of the Z and K(k) is the proportionality

factor. The experimental Z-dependence index ‘n’ values were compared with the index ‘n’

values obtained from Tseng and Pratt (1971) theory and Elwert corrected Bethe-Heitler theory

(1934). It has been found that the values of index ‘n’ obtained from experiment and from the

theories of Tseng and Pratt (1971) and Elwert corrected Bethe-Heitler (1934) theory are not

constant. It has been observed, that the Z-dependence index ‘n’ values are varying with the

photon energy and increases with increasing photon energy. Further, it has been found that the

proportionality factor K (k) shows exponential decaying dependence on photon energy ‘k’ and

inversely proportional to the end point energy of the beta particles. These results clearly indicate

that the spectral shape of ordinary bremsstrahlung is not independent of the atomic number of the

target element, as reported by Evans (1955).

It is clear from the survey of literature that the experimental studies of Z-

dependence of the spectral shape of OB and BS spectra on the atomic number of the target

materials, produced by continuous beta particles, have not been reported so far in the photon

energy region of 5-30 keV, where PB plays a vital role in the formation of BS spectra. Therefore,

a systemic study of the Z-dependence of the spectral shape of BS and OB spectra, i.e. with and

without the contribution of PB, produced by continuous beta particles is required in the photon

energy region of 5-30 keV, to check the role of PB in the formation of spectral shape of total

bremsstrahlung. This study will provide the information about the importance of various factors

like screening of atomic electrons, interferences between OB and PB, absorption of photons and

electron backscattering in a target.

1.9 Aims, scope and outcome of the study

In the present thesis:-

I. The total bremsstrahlung (BS) and ordinary bremsstrahlung (OB) spectral photon

distribution in thick metallic targets of Al (Z=13), Cu (Z=29), Sn (Z=50) and Pb (Z=82),

produced by soft beta particles of 147

Pm (Wmax=225 keV) and 45

Ca (Wmax=257

keV), and in thick targets of Al (Z=13), Ti (Z=22), Sn (Z=50) and Pb (Z=82), produced

by complete absorption of continuous beta particles of 90

Sr (Wmax=546 keV) and 204

Tl

(Wmax=765 keV) medium energy range beta emitters, have been investigated in the

photon energy region of 5-30 keV.

These beta emitters having different end point energies were chosen to check, how the

change in energy of beta particles affects the contribution of PB into OB and the role of

PB in the formation of BS spectra, while studying the BS and OB spectra, for different

target materials in the low, medium and high Z range in the photon energy region of 5-30

keV.

The experimental BS spectral photon distributions measured with X-PIPS Si(Li) detector

were compared with the theoretical bremsstrahlung spectral photon distributions obtained

from Elwert corrected (non relativistic) Bethe-Heitler (1934) and modified Elwert factor

(relativistic) Bethe-Heitler theories for ordinary bremsstrahlung (OB) and the modified

Elwert factor (relativistic) Bethe-Heitler theory, which include the polarization

bremsstrahlung (PB) into OB given by Avdonina and Pratt (1999).

The modified Elwert factor (relativistic) Bethe-Heitler theory, having the contribution of

polarization bremsstrahlung has been found accurate to describe the BS spectral photon

distribution in the photon energy region of 5-30 keV. This indicates that the contribution

of PB into the formation of BS spectra cannot be neglected.

Further, it has established that, the contribution of PB into OB decreases with increase in

the end point energy of beta emitter and the energy of the emitted photon. Further, the

contribution of PB into OB increases with increase in atomic number of the target atom.

This indicates the importance of PB in the formation of BS produced by continuous beta

particle.

It is expected, that this study shall provide vital information about the nature of spectral

photon distributions in thick metallic targets, produced by continuous beta particles of

beta emitters having different end point energies and the role of the polarization

bremsstrahlung in the formation of total bremsstrahlung spectra, in the photon energy

region of 5-30 keV. These studies shall check the accuracy of various theories available

to describe the phenomenon of OB and BS spectra in detail.

II. The factors comprising S (k, Z) i.e. the number of photons of energy k per moc2 per beta

disintegration do not clearly describes the dependence of shape of OB and BS spectra on

atomic number of the target material. Therefore, the dependence of spectral shape of OB

and BS spectra on the atomic number of the target material (Al, Cu, Sn and Pb), produced

by the continuous beta particles of beta emitters147

Pm and 45

Ca, and in the target

materials (Al, Ti, Sn and Pb), produced by beta particles of 90

Sr and 204

Tl have also been

studied in the photon energy region of 5-30 keV. These studies were carried out to

establish that, how the spectral shape of OB and BS spectra, produced by the beta

particles of beta emitters having different end point energies, depends on atomic number

(Z) of the target material.

It has been established, that the spectral shape of total bremsstrahlung spectra, in terms of

S (k, Z) is not linearly dependent on the atomic number (Z) of the target material. Rather,

this shows Zn dependency i.e. S(k,Z) = K(k)Z

n. Here, K(k) is the proportionality factor,

which is independent of Z at particular photon energy k.

The present results showed that the values of the Z-dependence index ‘n’, obtained from

the Elwert corrected (non relativistic) Bethe-Heitler (1934) theory and modified Elwert

factor (relativistic) Bethe-Heitler theories for OB and the modified Elwert factor

(relativistic) Bethe-Heitler theory, which include the PB into OB given by Avdonina and

Pratt (1999) and from experiment are not constant. These results clearly shows that the

spectral shape of OB and BS spectra are dependent on Z.

At lower photon energies, the index values ‘n’ of Z-dependence are much higher than

unity, which is due to the larger contribution of PB into OB. The decrease in ‘n’ values

with increase of photon energy is due to the decrease in contribution of PB into OB.

The index ‘n’ values obtained from modified Elwert factor (relativistic) Bethe-Heitler

theory, which include the contribution of PB into OB given by Avdonina and Pratt

(1999), are in agreement with the experimentally measured results by using X-PIPS Si

(Li) detector. This shows the importance of screening of atomic electron and the role of

PB in the energy region of 5-30 keV.

Proportionality constant K(k) factor shows exponential decaying dependency on photon

energy k and directly proportional to the end point energy of the beta particle.

It is expected that the values of the indices ‘n’ obtained from the theoretical and

experimental BS and OB spectral distribution will provide information about the

importance of screening of atomic electrons, interferences between OB and PB,

absorption of photons and electron backscattering in a target in the photon energy region

of 5-30 keV. These results shall be useful, while studying the dependence of spectral

shape of BS and OB spectra on atomic number of the target material in this energy

region. This study shall be useful to understand the phenomenon of bremsstrahlung more

rigorously.

The part of this work has already published by the authors Tajinder Singh et al. See (List

of Publications, Page 149).