Nuclear Instruments and Methods in Physics Research A 664 (2012) 332–335

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Nuclear Instruments and Methods inPhysics Research A

0168-90

doi:10.1

n Corr

E-m

journal homepage: www.elsevier.com/locate/nima

Dependence of spectral shape of bremsstrahlung spectra on atomic numberof target materials in the photon energy range of 5–30 keV

Tajinder Singh, K.S. Kahlon, A.S. Dhaliwal n

Department of Physics, Sant Longowal Institute of Engineering & Technology, Longowal (Sangrur), Punjab 148106, India

a r t i c l e i n f o

Article history:

Received 16 September 2011

Accepted 24 October 2011Available online 12 November 2011

Keywords:

Total bremsstrahlung

Ordinary bremsstrahlung

Polarization bremsstrahlung

Z-dependence

02/$ - see front matter & 2011 Elsevier B.V. A

016/j.nima.2011.10.030

esponding author. Tel./fax: þ91 1672 253186

ail address: asdhal@yahoo.com (A.S. Dhaliwa

a b s t r a c t

Dependence of spectral shape of total bremsstrahlung spectra i.e. the sum of ordinary bremsstrahlung

(OB) and polarization bremsstrahlung (PB), on the atomic number (Z) of target materials (Al, Ti, Sn and

Pb), produced by continuous beta particles of 90Sr and 204Tl, has been investigated in the photon energy

region of 5–30 keV. It has been found that the spectral shape of total bremsstrahlung spectra, in terms

of S (k, Z) i.e. the number of photons of energy k per moc2 per beta disintegration, is not linearly

dependent on the atomic number (Z) of the target material and rather it is proportional to Zn. 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. It is clear that the index ‘n’ values obtained from the modified

Elwert factor (relativistic) Bethe–Heitler theory, which include the contribution PB into OB, are in

agreement with the experimentally measured results using X-PIPS Si(Li) detector. Hence the contribu-

tion of PB into the formation of a spectral shape of total bremsstrahlung spectra plays a vital role.

& 2011 Elsevier B.V. All rights reserved.

1. Introduction

The total bremsstrahlung amplitude is the sum of ordinarybremsstrahlung (OB) and polarization bremsstrahlung (PB). Theprocesses by which the photon emission occurs due to theinteraction of electron with static field of the target is termed asOB, while in PB, the photon is emitted by the target as a result ofits polarization by incident electron. The PB phenomenon wasfirst demonstrated by Buimistrov and Trakhtenberg [1] and later,the calculation of PB amplitude has been reported by severalauthors [2–5]. Amusia et al. [2] has described that PB can beadded with OB in a stripped atom approximation (SAA). Thestripped approximation is efficient for obtaining the total brems-strahlung spectra for photon energies greater than the ionizationpotential of the outer shell electrons of the target atom. In SAA,the decrease of OB due to screening of outer shell electrons iscompletely compensated by additional PB produced by the sameouter shell electrons. Therefore, the total bremsstrahlung isdescribed simply by an ion containing the outer shell electrons.As the emitted photon energy exceeds the ionization potential ofthe inner most shell (1s), the bremsstrahlung occurs on the barenucleus. The difference between the OB from an ion and thebremsstrahlung on bare nucleus gives the contribution of PB in

ll rights reserved.

.

l).

the total bremsstrahlung spectra. Avdonina and Pratt [5] modifiesthe Elwert corrected (non relativistic) Bethe and Heitler [6] theoryfor OB and described the total bremsstrahlung spectra i.e.(OBþPB) over a wide range of photon energy region, by applyingthe SAA.

For mono-energetic electron, the dependence of OB cross-section on atomic number (Z) of the target has been reported byHippler et al. [7] and Semaan and Quarles [8]. Hippler et al. [7]reported that the OB cross-section at various electron energies fordifferent target elements shows discrepancy in order of factor of2 at low photon energies, particularly for high Z element andfurther suggested that the difference at the larger Z numbers isattributed to the screening of atomic electrons, which is impor-tant at low photon energies. Further, Hippler et al. [7], Avdoninaand Pratt [5] and Portillo and Quarles [9] pointed out that thedependence of bremsstrahlung on photon energy and atomicnumber of the target is of complex nature, particularly at lowerphoton energies. This is due to the screening of electron, inter-ferences between OB and PB, absorption of photons and electronbackscattering in a target. These factors may play a vital rolewhile studying the Z-dependence of total bremsstrahlung and OBin the photon energy region of 5–30 keV. Avdonina and Pratt [5]further pointed out that the contribution of PB at the lowerphoton energy region of the bremsstrahlung spectra in therelativistic regime is higher than in the non relativistic case.Therefore, the contribution of PB should decrease with increasingphoton energy.

Table 2The Z-dependence index ‘n’ values of the total bremsstrahlung production as a

function of photon energy k for 204Tl beta particles.

Photon energy(keV)

Index values (n)

EBHtheory

Fmod BHtheory

Fmod BHþPBtheory

Experiment

5.0 2.12 2.23 2.9 3.0170.13

7.5 2.04 2.13 2.7 2.7370.11

10.0 1.96 2.04 2.51 2.4570.10

12.5 1.84 1.92 2.36 2.2970.07

15.0 1.73 1.82 2.2 2.1470.07

17.5 1.65 1.73 2.05 2.0270.07

20.0 1.57 1.63 1.88 1.9170.06

22.5 1.43 1.5 1.66 1.7270.05

25.0 1.3 1.35 1.45 1.5270.04

27.5 1.27 1.30 1.38 1.4570.04

30.0 1.25 1.26 1.33 1.3870.04

Table 1The Z-dependence index ‘n’ values of the total bremsstrahlung production as a

function of photon energy k for 90Sr beta particles.

Photon energy(keV)

Index values (n)

EBHtheory

Fmod BHtheory

Fmod BHþPBtheory

Experiment

5.0 2.49 2.59 3.02 3.0370.14

7.5 2.28 2.40 2.76 2.8070.12

10.0 2.08 2.22 2.51 2.5670.10

12.5 1.85 1.98 2.26 2.2970.08

15.0 1.63 1.76 2.01 2.0370.08

17.5 1.55 1.68 1.95 1.9270.08

20.0 1.47 1.61 1.90 1.8270.07

22.5 1.46 1.56 1.85 1.7870.07

25.0 1.46 1.52 1.81 1.7570.05

27.5 1.39 1.43 1.74 1.6070.05

30.0 1.31 1.34 1.67 1.5670.05

T. Singh et al. / Nuclear Instruments and Methods in Physics Research A 664 (2012) 332–335 333

For the continuous beta particles, earlier Evans [10] reportedthat the shape of bremsstrahlung is independent of atomicnumber Z and each beta emitter has its own bremsstrahlungspectrum. However, later Dhaliwal [11–13] has reported thedependence of OB on atomic number (Z) of the target, particularlyat photon energy above 30 keV using the NaI(Tl) scintillationdetector. It has been reported that the shape of ordinary brems-strahlung spectrum produced by continuous beta particles is notlinearly dependent on the atomic number of the target material.However, the Z-dependence of the spectral shape of total brems-strahlung spectra, produced by continuous beta particles have notbeen reported so far in the photon energy region of 5–30 keV,where the contribution of PB plays a vital role in the formation ofa spectral shape. Therefore, systemic study of the dependence ofspectral shape of bremsstrahlung on atomic number (Z) of thetarget is required in the energy region of 5–30 keV. It is expectedthat this study may check various factors like screening of atomicelectron and interference of OB and PB in the formation ofspectral shapes of bremsstrahlung at lower energy regions. Forcontinuous beta particles the bremsstrahlung spectral photondistribution is expressed as the number of photons of energy k

per unit moc2 per beta disintegration is given by

Sðk,ZÞ ¼

Z Wmax

1þkncorðW

0e,k,ZÞPðW 0

eÞdW 0e ð1Þ

Here PðW 0eÞdW 0

e is the beta spectrum of the beta source understudy, ½ncorðW

0e,k,Z� is the bremsstrahlung spectral distribution in

a sufficiently thick target to absorb an electron of energy W ’e with

N atoms per unit volume [19]. It is not clear from this relationthat how the shape of bremsstrahlung spectral photon distribu-tion is dependent on the atomic number (Z) of the targetmaterials.

Various theoretical models are available to describe the OB[6,14,15] and total bremsstrahlung [2–5] spectra in thin and thicktargets. Available theoretical models for OB and total bremsstrah-lung spectra are not adequate to describe the spectral shape ofbremsstrahlung spectra. This is due to the fact that the impor-tance of interference between OB and PB is neglected in SAA,which describes the total bremsstrahlung spectrum. Dhaliwal[12] reported the inadequacy of the theoretical models todescribe the shape of ordinary bremsstrahlung in detail. There-fore, in order to investigate the Z-dependence of the spectralshape of bremsstrahlung, the S (k, Z) number of photons of energyk per unit moc2 per beta disintegration at the photon energy k canbe expressed as a function of Z and is reported by Dhaliwal [12] i.e.

Sðk,ZÞ ¼ KðkÞZnð2Þ

where ‘n’ is the index of the Z-dependence of a photon energy k perunit moc2 per beta disintegration and K (k) is the proportionalityfactor, 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 bremsstrahlung. In the presentmeasurements efforts are made to check the Z-dependence of thespectral shape of total bremsstrahlung spectra, in the photon energyregion 5–30 keV, where the polarization bremsstrahlung plays avital role in the formation of the shape of bremsstrahlung spectra.

The spectral photon distributions in various targets arerequired for a given beta emitter for the determination of the Z

dependence index ‘n’ values. The theoretical bremsstrahlungspectral photon distributions were calculated for thick targets ofAl, Ti, Sn and Pb that were obtained from Elwert corrected (nonrelativistic) Bethe–Heitler theory (EBH), modified Elwert factor(relativistic) Bethe–Heitler theory (Fmod BH), which describes OBand modified Elwert factor (relativistic) Bethe–Heitler theory(Fmod BHþPB), which describes the total bremsstrahlung spectra

in SAA. In the present measurement, the experimental andtheoretical bremsstrahlung spectral photon distribution producedby continuous beta particles of 90Sr and 204Tl beta emitters intargets of Al, Ti, Sn and Pb reported elsewhere in Refs. [16–19]have been used to study the Z-dependence of spectral shape oftotal bremsstrahlung spectra and determination of Z-dependenceindex ‘n’ values. Now, using the relation (2) and a least-squarepower-function-fitting computer program, the Z-dependenceindex values were obtained at 5–30 keV photon energies, for OBand total bremsstrahlung spectra. Further, the values of propor-tionality constant factor K (k) were also calculated as a function ofphoton energy using relation (2) for bremsstrahlung spectralphoton distribution produced by continuous beta particles of90Sr and 204Tl beta emitters in targets of Al, Ti, Sn and Pb.

2. Experimental details

The details of experimental arrangement and the method ofmeasurement are given elsewhere by the authors [16]. Thebremsstrahlung spectral photon distributions in targets of Al, Ti,Sn and Pb produced by 90Sr and 204Tl beta particles in the photonenergy region of 5–30 keV were measured using a high resolutionX-PIPS (Canberra make) Si (Li) detector. A Perspex beta stoppertechnique was employed to determine the bremsstrahlung pro-duced in the target material. This technique eliminates thecontribution of internal bremsstrahlung (IB), bremsstrahlunggenerated in the source material and room background. The

T. Singh et al. / Nuclear Instruments and Methods in Physics Research A 664 (2012) 332–335334

experimental measured bremsstrahlung spectra for Al, Ti, Sn andPb targets were converted into a true spectrum by applying thecorrections due to self absorption of bremsstrahlung photons intothe target, electron backscattering and detector efficiency, thedetail is given elsewhere by the authors [16]. The measuredexperimental bremsstrahlung spectral photon distributions weredivided by the geometrical full-energy peak detector efficiency forthe detector and were reduced to the number of photons ofenergy k per unit moc2.

In the present measurement of bremsstrahlung spectralphoton distributions, the uncertainties are mainly due to thecounting statistics, full energy detection efficiency of the detector,electron backscattering and the attenuation of bremsstrahlungphotons in the target materials. The overall errors in the presentmeasurement were estimated to be less than 10% in the entirestudied photon energy region of 5–30 keV. The uncertainties in

5 10 15 20 25 301.2

1.4

1.6

1.8

2.0

2.2

2.4

2.6

2.8

3.0

3.2

Z-de

pend

ence

inde

x 'n

'

Photon Energy (keV)

1 EBH2 FmodBH

3 FmodBH+PB

Experiment

123

90Sr

5 10 15 20 25 30

1.2

1.4

1.6

1.8

2.0

2.2

2.4

2.6

2.8

3.0

3.2

Z-de

pend

ence

inde

x 'n

'

Photon Energy (keV)

1 EBH2 FmodBH

3 FmodBH+PB

Experiment

1

23

204Tl

Fig. 1. (a, b) Plots of Z-dependence index ‘n’ values versus photon energy (k) for90Sr and 204Tl beta particles.

proportionality factor values are also estimated to be within 10%.The errors in determination of index ‘n’ values obtained fromexperimental data mentioned in Tables 1 and 2 are mainly due tothe statistics of data and the uncertainty in the least squares fit.

3. Results and discussions

The experimental and theoretical values of the Z-dependenceindex ‘n’ values obtained from the spectral shape of bremsstrah-lung produced by 90Sr and 204Tl beta particles in thick metallictargets of Al, Ti, Sn and Pb as a function of photon energy in thestudied photon energy region 5–30 keV are given in Tables 1 and2 and have been shown in Fig. 1(a, b). It has been found that theZ-dependence index ‘n’ values obtained from experimentalbremsstrahlung photon distributions shows better agreement

5 10 15 20 25 30

104

1014

1024

1034

1044

1054

Pro

porti

onal

ity c

onst

ant K

(k)

Photon Energy (keV)

FmodBH+PB

FmodBH

EBH

90SrTheory

K(k) = 2.43E64 exp (-1.73xk)

5 10 15 20 25 30

1010

1020

1030

1040

1050

1060

Pro

porti

onal

ity C

onst

ant K

(k)

Photon Energy (keV)

Experiment

K(k) = 2.43E64 exp (-1.73xk)

90Sr

Fig. 2. (a, b) Plots of proportionality constant K (k) versus photon energy k for 90Sr

beta particles (symbols are the actual points and line is just to guide the eye,

errors are lying within the points).

5 10 15 20 25 30

104

1014

1024

1034

1044

1054

1064

1074

1084

Pro

porti

onal

ity c

onst

ant K

(k)

Photon Energy (keV)

FmodBH+PB

FmodBH

EBH

204Tl

K(k) = 2.43E64 exp (-1.83xk)

5 10 15 20 25 30

1010

1020

1030

1040

1050

1060

1070

1080

Pro

porti

onal

ity c

onst

ant K

(k)

Photon Energy (keV)

Experiment

K(k) = 2.43E64 exp (-1.83xk)

204Tl

Fig. 3. (a, b) Plots of proportionality constant K (k) versus photon energy k for204Tl beta particles (symbols are the actual points and line is just to guide the eye,

errors are lying within the points).

T. Singh et al. / Nuclear Instruments and Methods in Physics Research A 664 (2012) 332–335 335

with the index values ‘n’ obtained from theoretical total brems-strahlung photon distribution from the modified Elwert factor(relativistic) Bethe–Heitler theory (Fmod BHþPB), which includethe contribution of polarization bremsstrahlung (PB) into OB inSAA. It was observed that the Z-dependence index ‘n’ values arevarying with the photon energy. It has been further found thatthese Z-dependence index ‘n’ values are not constant anddecreases with increasing photon energy. The plots of proportion-ality factor K (k) as a function of photon energy have been shownin Fig. 2(a, b) for 90Sr beta particles and in Fig. 3(a, b) for 204Tl betaparticles. It has been found that the proportionality factor showsexponential decaying dependence on photon energy ‘k’. This

exponential decay dependency was found to be same for thetheoretical and experimental bremsstrahlung measurement for aparticular beta emitter. In case of 90Sr, the exponential functionexp (�1.73� k) fits the exponential curves showing the varia-tions of K (k) with k for theory and experiment. For 204Tl, theexponential function exp (�1.83� k) fits the exponential curvesshowing the variations of K (k) with k for theory and experiment.This indicates that the proportionality factor K (k) is directlyproportional to the end point energy of the beta particles.

These results clearly show that the Z-dependence index values ‘n’are continuously deviated from unity in the studied photon energyregion of 5–30 keV. The experimental index ‘n’ values are varying inthe range of 3.03–1.56 and 3.01–1.38 in case of 90Sr and 204Tl betaemitters, respectively. The index ‘n’ values obtained from Fmod

BHþPB theory, which includes PB, are varying in the range of3.02–1.67 and 2.99–1.33 in case of 90Sr and 204Tl beta emitters,respectively. The index ‘n’ values obtained from Fmod BH theory,which describes OB, are varying in the range of 2.59–1.34 and2.23–1.26 in case of 90Sr and 204Tl beta emitters, respectively. Atlower energies, the index values ‘n’ of Z-dependence are much higherthan unity, this may be due to the larger contribution of PB into OB.The decrease in ‘n’ values with increase of photon energy may be dueto the decrease in contribution of PB into OB, screening of electronand interference of OB and PB. However, Dhaliwal [11] reported thatthe index ‘n’ values are only 15–50% higher than unity in the photonenergy region of 100–500 keV. The present results also show that theshape of bremsstrahlung spectra is not linearly dependent on theatomic number Z of the target atom. The index ‘n’ values obtainedfrom Fmod BHþPB theory are higher than the ‘n’ values obtained fromFmod BH theory in the range 8–22% and 4–22% in case of 90Sr and 204Tlbeta emitters, respectively. It is concluded that the spectral shape ofbremsstrahlung spectra as a function of photon energy is notindependent of Z in the photon energy region of 5–30 keV and thelarger ‘n’ values for experiment and theory are attributed to theimportance of screening of atomic electrons, contribution of PB intoOB and interference of OB and PB at these photon energies. Hence thecontribution of PB into the formation of a spectral shape of totalbremsstrahlung spectra plays a vital role.

References

[1] V. Buimistrov, L. Trakhtenberg, Soviet Physics JETP 42 (1975) 54.[2] M.Ya Amusia, M.Yu Kuchiev, A.V. Korol, A.V. Solov’yov, Soviet Physics JETP 61

(1985) 224.[3] M.Ya Amusia, A.V. Korol, Journal of Physics B: Atomic, Molecular and Optical

Physics 24 (1991) 3251.[4] A.V. Korol, O.I. Obolensky, A.V. Solov’yov, Journal of Physics B: Atomic,

Molecular and Optical Physics 31 (1998) 5347.[5] N.B. Avdonina, R.H. Pratt, Journal of Physics B: Atomic, Molecular and Optical

Physics 32 (1999) 4261.[6] H. Bethe, W. Hietler, Proceedings of the Royal Society of London Series A 14

(1934) 83.[7] R. Hippler, K. Saeed, I. McGregor, H. Kleinpoppen, Physical Review Letters 46

(1981) 1622.[8] M. Semaan, C.A. Quarles, Physical Review A 26 (1982) 3152.[9] S. Portillo, C.A. Quarles, Physical Review Letters 91 (2003) 17.

[10] R.D. Evans, The Atomic Nucleus New York:McGraw-Hill. 1955.[11] A.S. Dhaliwal, Nuclear Instruments and Methods B 198 (2002) 32.[12] A.S. Dhaliwal, Journal of Physics B: Atomic, Molecular and Optical Physics 36

(2003) 2229.[13] A.S. Dhaliwal, Nuclear Instruments and Methods B 234 (2005) 194.[14] H.W. Kotch, J.H. Motz, Reviews of Modern Physics 31 (1959) 920.[15] H.K. Tseng, R.H. Pratt, Physical Review A 3 (1971) 100.[16] Tajinder Singh, K.S. Kahlon, A.S. Dhaliwal, Journal of Physics B: Atomic,

Molecular and Optical Physics 41 (2008) 235001.[17] Tajinder Singh, K.S. Kahlon, A.S. Dhaliwal, Nuclear Instruments and Methods

B 267 (2009) 737.[18] Tajinder Singh, K.S. Kahlon, A.S. Dhaliwal, X-Ray Spectrometry 39 (2010) 291.[19] Tajinder Singh, K.S. Kahlon, A.S. Dhaliwal, X-Ray Spectrometry (2011)

(Communicated).