IL NUOVO CIMENTO

VOL. 99 A, N. 1

Gennaio 1988

Study of 147pm Beta-Induced Bremsstrahlung Spectra (*). K. GOPALA, B. RUDRASWAMY, P. VENKATARAMAIAHand H. SANJEEVIAH Department of Physics, University of Mysore - Mysore 570006 India (ricevuto il 25 Novembre 1986)

Summary. - - External bremsstrahlung produced in thick targets of A1, Cu, Mo, Cd and Pb by the beta-particles of ltTpm has been investigated using a (4.55 x 5.08) cm2 NaI(T1) scintillator. The measured spectra after unfolding have been compared with the Bethe-Heitler, Elwert-corrected BetheHeitler, Morgan-corrected Bethe-Heitler and Tseng and Pratt theories. The experimental spectra agree fairly well with the EBH theory up to about 120 keV and deviate positively thereafter from all the theories. PACS. 23.40. - ~-decay; electron and muon capture.

1. -

Introduction.

External bremsstrahlung (EB) is the emission of electromagnetic radiation produced when charged particles are deflected in the Coulomb field of other charged particles. Classically, the intensity of bremsstrahlung radiation is proportional to the square of the target charge and inversely proportional to the square of the mass of the incident particle. Thus the EB produced by light particles like electrons is much more than that produced by heavier ones like mesons or protons. The electron-electron bremsstrahlung is very small. According to the field theoretical description, the interaction of an electron with the electromagnetic field of a nucleus results in the emission of a photon with a finite probability ~, the fine-structure constant. However, every collision does not result in the emission of a photon. The energy of the photon can vary from zero to a maximum equal to the kinetic energy of the incident particle. (*) To speed up publication, the authors of this paper have agreed to not receive the proofs for correction. 55

K. GOPALA, B. RUDRASWAMY, P. VENKATARAMAIAH and H. SANJEEVIAH

56

2. - Theory. Sommerfeld (1) gave a nonrelativistic theory of bremsstrahlung to explain the continuous X-ray spectra. He neglected the screening of the nuclear charge by the atomic electron cloud. Bethe and Heitler (2) formulated the relativistic theory of bremsstrahlung using the Dirac's relativistic equation for the electron. They used Born approximation in their theory which renders it less accurate for high Z-material and high-momentum transfer collisions. Their expression for the bremsstrahlung cross-section differential in photon-energy is

dz Z2~ 1 P [4_2E E [p] +p2~ ~oE ,Eo

(1)

d k - 137 k po [~

o t ~

) +7-o .4 p3

8EoE -~ k2(E~E 2 +po2p2) + L / 3pop po~pS

~o~ tPoP k

[[EoE+p~

7 `0-

EoE + p2~

j'+

2kEoE\7]

where

(

L = 2 In Eo E + PoP - 1

k

)

~o=

'

~Eo-Po]

and Z = atomic number of the target material, ro = classical electron radius, Po, P = initial and final momentum of the electron in m0 c units, E0, E = initial and final total energy of the electron in m0 c2 units. Born approximation considers the wave function of the electron to be plane, thus disregarding the nuclear Coulomb effect on the scattered electron. Elwert(3) compared the nonrelativistic Bethe-Heitler theory with the exact nonrelativistic theory of Sommerfeld and arrived at a multiplicative Coulomb correction factor fe to be applied to the BH cross-sections. Obviously this holds well for low electron energies. Brysk et al. (4) used exact Coulomb wave functions instead of plane waves for (1) (2) (8) (')

A. H. G. N.

SOMMERFELD:Ann. Phys. (N.Y.), 11, 257 (1931). BETHE and W. HEITLER: Proc. R. Soc. London, Ser. A, 146, 83 (1934). ELWERT: Ann. Phys. (N.Y.), 34, 178 (1930). BRYSK, C. D. ZERBY and S. K. PENNY: Phys. Rev., 180, 104 (1969).

STUDY OF 'aPm BETA-INDUCED BREMSSTRAHLUNG SPECTRA

57

calculating the matrix elements. This involves a partial wave expansion for the incident and scattered electron and for the photon. Since the incident electron energy determines the number of matrix elements required for a solution of the problem, computer precision and storage became limiting factors for high energies. Morgan(5) compared the Bethe-Heitler theory with the theory of Brysk et al. and arrived at a multiplicative Coulomb correction factor fM to the BH theory which he showed would hold well from 0.1 to 2 MeV. Tseng and Pratt(e) have made calculations of electron bremsstrahlung describing the process as a single-electron transition in different central potentials, viz. point Coulomb, Thomas-Fermi, modified Thomas-Fermi and modified Hartree-Fock-Slater. Pratt et al. (7) have tabulated the EB crosssection differential in photon energy, using relativistic self-consistent screened potential up to 2 M~V. The above theories are applicable to thin target spectra wherein only one or very few elementary processes take place. When one uses thick targets all other processes like electron scattering, excitation and ionization competing with bremsstrahlung must be taken into account. This finally leads to an integral that includes the energy loss per unit path length. Bethe and Heitler(~) gave an expression for the thick target bremsstrahlung when all the incident electrons are absorbed in the target, as Eo

n(k, E o ) = N f

(2)

l+k

dz/dk -

"~-Tdx

dE ~

'

where n(k, Eo) = the number of EB photons produced per unit energy interval at the energy k, E0 = energy of the incident electron, - d E / d x = total energy loss of the electrons per unit path length in the target, N = number of atoms per cc in the target material. When beta-particles are used, the calculations are further complicated by their continuous energy distribution. The EB spectrum produced by beta-particles

(~) (6) (7) At.

S. H. MORGANJr.: Report NASA, TN-D-6038 (1970). H. K. TSENG and R. H. PRATT: Phys. Rev. A, 1, 528; 3, 100 (1971). R. H. PRATT, H. K. TSENG, C. M. LEE, L. KISSEL, C. MCCALLUMand M. RILEY: Data Nucl. Data Tables, 20, 175 (1977).

58

K. GOPALA, B. RUDRASWAMY, P. VENKATARAMAIAH

and

H. SANJEEVIAH

can be written as Em

f n(k, Eo) P(Eo) d Eo (3)

S(k) = ~÷~

~

,

f p(Eo)dEo

l+k

where S(k) = the number of EB photons produced per unit energy interval at the energy k per beta-particle, P(Eo) = beta spectrum, Em = maximum total energy of the beta-particles in m~ c2 units. The EB spectrum evaluated according to eq. (3) using BH cross-sections in eq. (2) has been called ,,BH theory,, in the present paper. The spectra evaluated using the Elwert-corrected BH cross-sections and Morgan-corrected BH crosssections have been called ,,EBH theory,, and ,,MBH theory,,, respectively. When the tabulated values of Pratt et al. C) cross-sections are used, it has been called ,,Pratt Theory,,.

3. - E x p e r i m e n t .

Even though EB can be generated either by mono-energetic electrons or by beta particles, historically it was the beta-induced EB that was first studied (8). Since then many investigations have been made regarding beta-induced EB (,19). All the EB sPectral investigations show that there is no complete agreement

(8) j. A. GRAY: Proc. R. Soc. London, Ser. A, 85, 131 (1911).

(9) j. CHADWICK:Philos. Mag., 24, 594 (1912). (lo) E. STAHEL and J. MASSA: Helv. Phys. Acta, 14, 325 (1941). (11) M. GOODRICH,J. S. LEVINGER and W. PAYNE: Phys. Rev., 91, 1225 (1952). (12) K. LIDEN and N. STARFELT: Phys. Rev., 97, 419 (1955). (13) C. BUSSOLATI:Nuovo Cimento, 13, 909 (1959). (14) T. S. BUSTARDand J. SILVERMAN:Nucl. Sci. Eng., 27, 586 (1967). (15) K. GOPALA, P. VENKATARAMAIAHand B. SANJEEVAIAH:Nucl. Phys., Solid State Phys., India, 12, 433 (1969). (le) T. S. MUDHOLEand N. UMAKANTHA:Am. J. Phys., 40, 591 (1972). (17) R. PRASADBABU, K. NARASIMHAMURTYand V. A. NARASIMHAMURTY:J. Phys. G, 3, 273 (1975). (18) M. S. POWAR, S. AHMADand M. SINGH: Phys. Rev. A, 21, 1884 (1980). (19) SHIVARAMU:Phys. Rev. A, 30, 3066 (1984).

STUDY OF 147pm BETA-INDUCED BREMSSTRAHLUNG SPECTRA

59

between experiment and theory throughout the energy spectrum. Bethe-Heitler and Elwert-corrected Bethe-Heitler theories have been used for comparison with the experimental spectra by many investigators. Almost all t h e investigators have found that the theory underestimates the experimental results at high-energy region of the spectrum. The discrepancy increases with increasing energy and atomic number of the target. In the present investigation the authors have compared their experimental results with four theories including the most recent one, v/z. the theory of Tseng and Pratt (6). EB spectra produced by the beta-particles of 147pmin targets of A1, Cu, Mo, Cd and Pb, thick enough to stop all beta-particles, have been measured. 147pm is a low end point energy (225 keV) beta source with almost 100% beta emission and a half-life of 2.623 y. Its beta decay is classified as nonunique fncst forbidden (5J= O, 5~= yes and logfl=7.4). The experimental arrangement is similar to the one described earlier (20). 147pmsource was obtained from Bhabha Atomic Research Centre, Bombay, India in the standard form in which the backscattering from the source support is minimum. The source was placed in a perspex source holder that was slid into a perspex stand fixed to a perspex sheet. This stand was placed on a lead shield within which there was a (4.55 × 5.08) cm2 NaI(T1) crystal mounted on an RCA 8053 photomultiplier assembly with a preamplifier. The target, cut into circular shape, was placed between the source and the NaI(T1) crystal. The distance was so adjusted as to provide a good geometry arrangement. The pulses from the preamplifier were amplified in a PA521 pulse amplifier (Electronics Corporation of India Ltd., Hyderabad, India) and fed to an ORTEC model 7150 MCA set to 256 channels. First, the spectrum was recorded with the source and target in position and with a perspex sheet thick enough to absorb all the beta-particles of 14Vpm, between them. This gives the background (BG) plus source-dependent internal bremsstrahlung (IB) spectrum. Next the spectrum was recorded with the perspex absorber below the target. This gives the BG plus IB plus EB spectrum. The difference between the two spectra gives the raw EB spectrum. This is unfolded using the Liden and Starfelt (21)procedure that corrects for the various processes which take place in the crystal-photomultiplier assembly responsible for the dispersion of the actual incident spectrum on the crystal. The resulting distribution is further corrected for the absorption in the target, perspex beta absorber and the aluminium can covering the NaI(T1) crystal. The unfolded spectrum is finally divided by the source strength and represented as number of EB photons/moc2/~-particle. The source strength was determined following the procedure of Zumwalt (~). Figure 1 represents the unfolded thick target EB spectra of A1, Cu, Mo, Cd and (2o) B. RUDRASWAMY, K. GOPALA, P. VENKATARAMAIAHand H. SANJEEVIAH:J. Phys. G, 10, 1579 (1984). (~1) K. LIDEN and N. STARFELT:Ark. Fys., 7, 109 (1953). (22) L. R. ZUMWALT:Report AECU-567 Clinton Labs., 1950.

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STUDY OF 147pm BETA-INDUCED BREMSSTRAHLUNG SPECTRA

61

Pb compared with the theoretical spectra of BH, EBH, MBH and Pratt as detailed above.

4. - E r r o r s .

The main error contribution to the measured spectra is from counting statistics. The error is less than 1% at 25 keV and less than 22% at 200 keV. The other contribution to the error comes from the correction for finite-energy resolution which is less than 5% throughout the energy region. The errors due to Compton electron distribution at 25, 100 and 200 keV are less than 1%, 2% and 3%, respectively. The error involved in the estimation of the geometric efficiency of the NAI(T1) crystal which is due to both the inaccuracy in the experimental determination of peak-to-total ratio and the uncertainty in the values of the absorption coefficients of NaI(T1), is found to vary from 1% at 25 keV to 7% at 200 keV. Errors due to the absorption in the target material, beta stopper and aluminium can of the detector are estimated to be less than 3% throughout the energy region considered. The error due to source strength determination is found to be 8%. The overall error is estimated to be less than 10% at 25 keV and less than 25% at 200 keV.

5. - R e s u l t s

and discussion.

Figure 1 shows that the experimental spectra agree fairly well with the EBH cross-sections up to a photon energy of about 120 keV. Thereafter the experimental values take a positive deviation from the EBH theory and later a complete disagreement with all the four theories can be observed. Even though the EBH and Pratt theories are closer, the agreement with EBH is better than that with the Pratt theory. Pratt cross-sections used in the present work are obtained by the interpolation of the ,~benchmark, data (~) for i and 2 MeV using a smooth spline fit. These are claimed to be accurate (7) to within at least 10%. With this the present measurements also show agreement with the Pratt theory within error limits. However, the increasing deviations with increasing atomic number of the target towards the end point energy of the spectrum persist. The increasing deviation of experimental values from theory at higher energies is probably due to the assumption, made in arriving at the experimental spectra, that the EB production in thick targets is isotropic. But for high-energy photons the electrons should have lost almost all their energy in the first collisions themselves and the EB will be forward peaked. Thus considering the

(23) L. KISSEL and C. MCCALLUM:PITT-179, University of Pittsburgh (1977).

62

K. GOPALA, B. RUDRASWAMY, P. VENKATARAMAIAHand H. SANJEEVIAH

E B emission to be isotropic one is overestimating the experimental value. H i g h e r the photon e n e r g y larger is the deviation.

The authors thank Prof. R. H. P r a t t for useful discussions and Prof. D. Krishnamurti, Head of the D e p a r t m e n t of Physics, University of Mysore, Mysore, for facilities. Thanks are also due to Dr. N. C. Shivaprakash, Indian Institute of Science, Bangalore, for assistance in the computations.

•

R I A S S U N T O (*)

I1 bremsstrahlung esterno prodotto in bersagli spessi di A1, Cu, Mo, Cd e Pb da particelle beta di 147pm~ stato studiato usando uno scintillatore NaI(T1) (4.55 x 5.08)cm 2. Gli spettri misurati dopo lo sviluppo sono stati paragonati con le teorie di Bethe-Heitler, di BetheHeitler corrette da Elwert, di Bethe-Heitler e di Tseng e Pratt corrette da Morgan. Gli spettri sperimentali sono abbastanza in accordo con la teoria EBH fino a 120 keV e deviano positivamente da questo valore in poi rispetto a tutte le altre teorie. (*) Traduzione a cura della Redazione.

Hcc~e~oBanxe cneKTpOB TOpMO3HOFO a3ayqenas, Hll~yi~llpOBallHOrO 6eTa-pacna~oM 147pm"

Pe3IoMe (*). - - Hcnom,ay~ NaI(Tl) cUm~TnnJinTop(4.55 x 5.08) CM2, HCCJIeayeTcnBneimlee

TOpuo3noe naayqerme, o6pa3oBannoe B TOaCTb~ mttuenax Al, Cu, Mo, Cd n Pb 6eTa-qacwn~aMn H3 147pm. I,I3MeperrHbie CneKTp~,I cpaBnI~a~oTc~t c TeoprmMn BeTeFaffraepa, BeTe-Fa~Taepa c nonpaBKaX~ 3aBepTa, BeTe-Fa~iTaepa c nonpaBKaX~ Moprana n qenra rt IIpaTa. 3KcneprrMeaTa~bnbte CneKTpbt y~oBaeTBOpnTeJIbnO onHCbIBamTCn TeopHe~ BeTe-Fa~Tsiepa c nonpaBKaUa ~gaBepTa BnaOTb aO 120KaB, nO o6napy~a~aioT OTKaOneHrmOT Bcex japyrnx Teopn~. (*) Hepe6ec)eno pec)ata~uet~.