Distorted wave Born approximation calculation of pair production cross section for 12.5 MeV photon

ARTICLE IN PRESS

Radiation Physics and Chemistry 75 (2006) 631–643 www.elsevier.com/locate/radphyschem

Distorted wave Born approximation calculation of pair production cross section for 12.5 MeV photon K.K. Suda,, D.K. Sharmab a

Department of Physics, College of Science Campus, M. L. S. University, Udaipur-313002, India Department of Physics, Faculty of Science, Jai Narayan Vyas University, Jodhpur-342001, India

b

Abstract We present in this communication the theoretical results of our calculation in distorted wave Born approximation (DWBA) in point Coulomb potential of the differential and total electron-positron pair production cross section by 12.5 MeV photons on different targets (Z ¼ 1, 30, 50, 68, 82 and 92). We also present differential pair production crosssection in the tip region ðE
¼ 1:008 mc2 Þ for photon energies 5.0–40.0 MeV for a large number of targets. We compare the interpolated DWBA pair production cross section with the experimental electron-positron pair creation cross section in the energy range 5.0–12.5 MeV on Ta and Bi. r 2005 Elsevier Ltd. All rights reserved. Keywords: Pair production cross section; PWBA; DWBA; Positron spectrum

1. Introduction Accurate knowledge of the total gamma rays (photon) absorption cross section is needed in atomic and molecular physics as well as in a number of applications involving the absorption of photon by material (e.g. radiological safety analysis, radiation shielding, industrial irradiation and gauging). There has been considerable interest in obtaining accurate atomic absorption cross section data as it provides a global check on all elementary fundamental processes (photoelectric effect, pair production, triplet production, Compton effect, etc.) constituting it. Furthermore, the knowledge of accurate atomic absorption cross section is needed to obtain nuclear photo-absorption cross section which is the difference of the total photon absorption cross section and atomic absorption cross section. We refer Corresponding author. Tel.: +91 294 418123; fax: +91 294 413150. E-mail address: [email protected] (K.K. Sud).

the reader for more details to the widely used tabulations of the total atomic cross sections for Z ¼ 1–100 for photon energies from 1 MeV to 100 GeV by Hubbell et al. (1980). In the intermediate energy range (5–50 MeV) the atomic absorption cross section is dominated by the contribution due to the pair production process. For lead, e.g., 74% at 10.0 MeV and 87% at 20.0 MeV of the total atomic cross section is due to pair production in the field of the atomic nucleus. However the tabulations have been compiled by using the empirical Coulomb correction to the pair production cross section in the intermediate energy range 5–50 MeV, as the exact distorted-wave Born approximation (DWBA) results in this range were not available. We refer the reader for up to date status of the pair production cross section data by photons to the work of Hubbell and Seltzer (2004) and for a historical overview of the pair production process to the work of Hubbell (2004). The DWBA calculations of the electron-pair-production cross section in the point Coulomb field of the atomic nucleus have been performed by Jaeger and Hulme (1936),

0969-806X/$ - see front matter r 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.radphyschem.2005.09.003

ARTICLE IN PRESS 632

K.K. Sud, D.K. Sharma / Radiation Physics and Chemistry 75 (2006) 631–643

Øverbø et al. (1968, 1973), Tseng and Pratt (1971), Dugne and Proriol (1976), Sud et al. (1979), Sud and Sharma (1984), Sud and Soto Vargas (1991, 1994), Wright et al. (1987) and Selvaraju et al. (2001). The existing DWBA calculation of the differential and total pair production cross section are up to 10.0 MeV photon energy for Z ¼ 1; 30; 50; 68; 82 and 92 and at 20.0 MeV for Z ¼ 92. The present calculation of the differential and total pair production cross section at 12.5 MeV is in distorted wave Born approximation. For photon energy $ ¼ 12:5 MeV and higher the contribution due to the lepton partial waves higher than 150 is significant. We have developed and used a different technique to incorporate the contribution to the differential pair production cross section from the higher partial waves. We discuss the technique in detail in Section 3. We present the status of the Coulomb corrections to the pair production cross section in Section 2 and the DWBA calculations of the pair production cross section in Section 3. We also provide an outline of the technique developed by us to evaluate the radial integrals needed in the DWBA calculation of the pair production cross sections. We present in Section 4 the results of our DWBA calculation of pair-production cross section for 12.5 MeV photons for atomic numbers (Z ¼ 1; 30; 50; 68; 82 and 92). We also present the results of the DWBA calculation in the tip region ðE
¼ 1:008 mc2 Þ of the positron spectrum for photon energies 5.0–40.0 MeV for a large number of atomic numbers. We have compared the results of our calculation with the Bethe and Heitler (1934), Øverbø (1977), Maximon and Gimm (1978) and Berger and Hubbell (1987). We also compare the interpolated DWBA pair production cross sections with the experimental data on Ta and Bi of Sherman et al. (1980).

2. Status of the Coulomb corrections to the electron pair production The first theoretical treatment of the pair production process was carried out by Bethe and Heitler (1934) in plane wave Born approximation (PWBA). In this approximation the interaction between the leptons and the Coulomb field of the nucleus, is treated as perturbation. This results in an expansion of the transition probability in power of aZ (where a ¼ e2 =_c and Z is the atomic number of the target nucleus), the lowest order nonzero terms in the expansion constitute the plane wave Born approximation. We refer the reader to the review article by Motz et al. (1969) for explicit expressions in PWBA for computing pair production cross sections. The plane wave Born approximation is valid for 2paZE  =p c51, E7 and p7 are the energy and momentum of the leptons, respectively. The cross section in the PWBA can be computed for the entire

energy range easily. But one has to incorporate a number of corrections to the cross sections obtained in PWBA. First, there is a screening of nuclear Coulomb field by the charge distribution of the atomic electrons. Secondly, there is deviation from the Coulomb field for a point nucleus which arises due to the finite extent of the nuclear charge densities. The finite nuclear size correction is important for the photon energies for which the de Broglie wavelength is equal to or smaller than the size of the nucleus. Two small corrections to the pair production are the nuclear recoil corrections and radiative corrections. For details of the screening and radiative corrections, we refer the reader to the work of Hubbell et al. (1980) and the references cited therein. The Coulomb correction is the result of distortion in the lepton wave functions due to the Coulomb field of the atomic nucleus. The Coulomb correction to the pair production is very significant for low and intermediate energy photons. However at very high energies (i.e. w4200 MeV) the Coulomb correction is not important since the distortion due to Coulomb field at such energies are negligible. Bethe and Maximon by using Furry–Sommerfeld–Maue wave functions obtained expression for the Coulomb correction in high energy and small angle (angle between the photons and both leptons) approximation. The error in the cross sections are estimated to be of the order ðaZÞ2 lnðE=mÞðE=mÞ
1=2 . This is because in each partial wave, the terms of the order of ðaZÞ2 =½k (for partial wave k) are neglected. At high energies and small angles the contribution of high partial waves dominates so approximation should become better. Fink and Pratt (1973) obtained the expression for pair production cross sections by using the full Furry–Sommerfeld–Maue wave functions for electrons and positrons. The total cross section was obtained by numerical integrations of the differential cross section without making the high energy small angle approximation. The calculation was performed in the low energy region (Fink and Pratt, 1973). Borie by using the code of Fink and Pratt (1973) calculated the cross section for 10, 25, and 50 MeV for a few atomic numbers (Z ¼ 13–82) and found that the Coulomb corrections to the cross sections obtained by using the Davies–Bethe–Maximon (1954) expression (DBM) to be inadequate. Roche et al. (1968) have developed a theoretical formalism to compute higher order Coulomb corrections to the cross sections obtained by using the Furry–Sommerfeld–Maue wave functions. They have calculated corrections to the cross sections by keeping next order terms in (aZ) and making approximations in 1/E and 1/r. This correction leads to improvement for intermediate Z but is still not sufficient for higher Z at low energies. Borie (1981) has also calculated the correction terms due to Roche et al. (1968) to the Fink–Pratt’s calculations. Roche and Jousset (1975) have obtained a correction to the total cross

ARTICLE IN PRESS K.K. Sud, D.K. Sharma / Radiation Physics and Chemistry 75 (2006) 631–643

section (Fink and Pratt) by using the expansion term in 1/r but keeping all orders in aZ. But no numerical results at higher energies are available. At higher energies (photon energy o450 MeV) the calculation of Davies– Bethe–Maximon using the Sommerfield–Maue approximation for lepton wave functions are expected to be accurate. Thus there exists an energy range 5–50 MeV or more where the Coulomb corrections to the Bethe– Heitler are reasonably large and uncertain. Øverbø (1977) and Maximon and Gimm (1978) have constructed semi-empirical formulae for Coulomb correction to the Bethe–Heitler cross section by choosing different reasonable analytic functions of Z and o along with a number of arbitrary parameters such that these expressions approach the DBM form in high energy limit. The parameters of the two formulae are fitted to the low energy DWBA pair production cross sections of Øverbø et al. (1968, 1973). Two formulae give different results in the intermediate energy range. Thus there exists an energy range from 5 to 50 MeV where the Coulomb distortion correction to the PWBA is large and uncertain. This has led to a number of calculations of the pair production cross section in distorted wave Born approximation in the intermediate photon energy range. We present the status of the DWBA calculation of the pair production cross section and also outline of the technique used in the present work to evaluate the needed DWBA radial integrals in Section 3.

3. DWBA calculations In distorted wave Born approximation calculation of the pair production cross sections the matrix elements are obtained by using the electron (positron) wave functions which are obtained by solving the Dirac equation in the static nuclear Coulomb field. A number of DWBA calculations of the pair production cross sections are available in the literature e.g. Jaeger and Hulme (1936), Øverbø et al. (1968, 1973), Dugne and Proriol (1976), Tseng and Pratt (1971), Sud et al. (1979), Sud and Sharma (1987), Sud and Soto Vargas (1991, 1994), Wright et al. (1987) and Selvaraju et al. (2001). The first DWBA calculation of the electron pair production in point-Coulomb potential was performed for two photon energies and a few elements by Jaeger and Hulme (1936). Øverbø et al. (1968, 1973) have computed the DWBA electron pair production cross section for photon energies ranging from threshold to 5.0 MeV for a large number of atomic numbers. Øverbø et al. (1968, 1973) obtained the analytic expression for the DWBA differential cross section by using the relativistic Coulomb wave functions for the leptons. In the DWBA formalism the radial integrals for the pair production process can be expressed in terms of the Appell’s hypergeometric function F2. We refer the readers for the explicit

633

expression for the DWBA differential pair production cross section to the reference Øverbø et al. (1968, 1973) and Sud et al. (1979). Appell’s hypergeometric function F2 which are required to compute the pair production cross section for 5.0 MeV photons, are very slowly convergent series and their convergence further deteriorates for higher energies. Infact it is not possible to evaluate accurate F2 functions by summing in its series representation. This has restricted Øverbø’s technique for computing the pair production cross section up to 5.0 MeV photons. The new impetus in the DWBA calculation was provided by the development of a technique by Sud et al. (1979), Sud and Sharma (1984) and Wright et al. (1987) to evaluate the Dirac–Coulomb integrals. In this technique the radial integrals are obtained from the elements of the matrix G function. The recurrence relation satisfied by the matrix G function is used to reduce the number of radial integrals required for computation. The matrix G function also satisfies a differential equation in photon-energy like parameters. The accurate radial integrals for the pair production cross section above 5.0 MeV are obtained by integrating the matrix differential equation and by using the initial G matrix at the photon-energy like parameters where its accurate evaluation is possible. The initial matrix G function is obtained by evaluating either F2 functions or G matrix at a larger photon-energy parameter at which its convergence is better. We refer the reader to the work of Wright et al. (1987) for the computational details and explicit expression for the pair production cross section in terms of the elements of the matrix G function. The technique has already been used by Selvaraju et al. (2001) for the computation of the differential and total pair production cross sections for photon energies 5.0–10.0 MeV in the photon energy step of 0.5 MeV for a large number of atomic numbers (Z ¼ 1, 30, 50, 68, 82 and 92). We present in the following paragraphs of this section an outline of the technique developed by Sud et al. (1979), Sud and Sharma (1984) and Wright et al. (1987) to evaluate the Dirac–Coulomb integrals needed to compute the DWBA pair production cross section in the intermediate energy range. The radial Dirac equation for a lepton in a point Coulomb potential can be expressed as
 duðxÞ A ¼
B uðxÞ, (1) dx x where, x ¼ rp (p7 is the lepton’s momentum) and A and B matrices are given as qffiffiffiffiffiffiffiffi 1 0 Eþm
 0
E
m
k aZ C B qffiffiffiffiffiffiffiffi A¼ and B ¼ @ A, E
m
aZ k 0 Eþm

(2) where, k is the eigen value of the Dirac operator;

ARTICLE IN PRESS K.K. Sud, D.K. Sharma / Radiation Physics and Chemistry 75 (2006) 631–643

634

~ þ 1Þ, E ¼
E þ or E
for a positron and k~ ¼ bð~ s:L electron, respectively. The B-diagonal solution of Eq. (1) is in terms of Whittacker function M as 1 ðBÞ u ¼ pffiffiffiffiffiffiffiffiffiffiffi 2ip r 0

B @

M

ð2ip rÞ 1
2
iy ;g

M1

2
iy ;g

ðg
iy Þ
ðg  þiy Þ

ð2ip rÞ

M

M

1

ð2ip rÞ 1
2
iy ;
g C

ð2ip rÞ 1
2
iy ;
g

A.

The DWBA radial matrix element is obtained from the elements of the matrix G function and which are Appell’s hypergeometric functions F2. The matrix G function satisfies a recurrence relation AGðA; BÞ ¼ BGðA þ 1; BÞ.

This relation has been used to reduce the number of the radial integrals to be calculated. The Hankel function has finite series expansion;

ð3Þ The A7 and B7 matrices for this particular solutions are ! g
iy iy A ¼ and g þ iy
iy !
i 0 0 ð4Þ B ¼ p B ¼ p 0 i where y ¼ 

aZE  p

and

hLð1Þ ðorÞ ¼ eior

Lþ1 X

an ðLÞr
n .

ðE þmÞ

(11)

n¼1

By using the Hankel function expansion we can express the required integral in the following form: 9 8   ðI 2  D
1  D
1 > > þ Þ > > = < ! GB ða
L; b
ioÞ , GðA þ 1; BÞ ¼ Re > > X > > B ; : G ða
L þ 1; b
ioÞ (12)

g ¼ ½k2
ðaZÞ2 1=2 .

The transforming matrix D is ðuB ¼ D u Þ, 0 iðg
iy Þ 1 ðg
iy Þ  pffiffiffiffiffiffiffiffiffiffiffi p ffiffiffiffiffiffiffiffiffiffiffi 0 þmÞ 0
mÞ ðE ðE 1 B C D ¼
@ ðk
ib Þ ðk
ib Þ A, p ffiffiffiffiffiffiffiffiffiffiffi ðg þ iy Þ2
pffiffiffiffiffiffiffiffiffiffiffi 0 0

(10)

where Z

1

GðA
n; B
ioÞ ¼ 0

(5)

ðE
mÞ

where b ¼ azm=p and E 0 ¼
E þ or þ E
. We need integral over the integrand which is direct product of the solutions ! jL ðorÞ W ðA; B : rÞ ¼  u
 uþ jL
1 ðorÞ ! ( ) h0L ðorÞ    u
 uþ , ð6Þ ¼ Re h0L
1 ðorÞ

eior  ðu  uþ Þ dr, rnþ1


A ¼ A
 I 2 þ I 2  Aþ ; B ¼ B
 I 2 þ I 2  Bþ ;

X¼

X ðLÞ 0

0 X ðL
1Þ

(13)

(14)

! (15)

and Lþ1 X

ALþ2
n ðLÞ

where jL(or) and h L(or) are spherical Bessel function and Hankel function, respectively. The product function W ðA; B : rÞ satisfies a matrix differential equation as in Eq. (1) with the following redefined A and B matrices

X ðLÞ ¼ ALþ1 ðLÞI 4 þ

A ¼ Ag  I 4 þ I 2  A
 I 2 þ I 4  Aþ , B ¼ Bg  I 4 þ I 2  B
 I 2 þ I 4  Bþ ,

As we require only first column of matrix (Eq. (12)), we can evaluate only first column of Eq. (13) as gamma vector. The matrix gamma vector as given in Eq. (13) can be evaluated, by using the following power series expansion in A-diagonal representation:

0

ð7Þ

where In is the unit matrix of the order of its suffix, Ag and Bg are matrices corresponding to Bessel function solution of Eq. (1). The product function W ðA; B : rÞ also satisfies a useful recurrence relation a
br

r e

W ðA; B : rÞ ¼ W ðA þ aI; B þ bI : rÞ.



n
1 Y

fðBðBÞ
ioÞ
1 ðAðBÞ
L þ m þ 1Þg.

ð16Þ

m¼1

ðAÞ ðAÞ u
 uþ ¼

X

ðAÞ

V n rnþa ,

(17)

(8)

The integral over the integrand W ðA; B : rÞ is defined as the matrix G function (Sud et al., 1976) Z 1 W ðA; B : rÞ dr. (9) GðA þ 1; BÞ ¼ 0

n¼2

where the coefficients are obtained from the recurrence relation; fV n gi ¼

fbðAÞ V n
1 gi ; ai
a1 þ n

fV 0 gi ¼ fIgi;1 .

(18)

ARTICLE IN PRESS K.K. Sud, D.K. Sharma / Radiation Physics and Chemistry 75 (2006) 631–643

We can express the matrix gamma vector Eq. (13) in the following form; GðaðAÞ
n; b
ioÞ ¼

GðaðAÞ
nÞ ð
ioÞa1
n
k 1 X i  Vk ðaðAÞ
nÞk , o k¼0

4.1. Differential pair production cross section

qG ¼ T i G ði ¼ 1; 3 and k1 ¼ pþ ; k2 ¼ p
or k3 ¼ oÞ. qki (20) The matrices Ti are: I 2  Aþ
I 2  B0 :B
1 ðA þ I 4 Þ, pþ A
 I 2 T2 ¼
B0  I 2 :B
1 ðA þ I 4 Þ, p

ðn þ 1Þ
iI 4 :B
1 ðA þ I 4 Þ. T3 ¼ o T1 ¼

We have used the following relations; (21)

and B0 ¼ p B to derive Eq. (20). We define G-vector equation (12) by replacing o by o0 (a free parameter) such that o0 ¼ o þ D. The G-vector for fixed electron and positron energy satisfy partial differential equation in energy like parameters; qGðo0 Þ ¼ TG, qo0

(22)

where T¼

tion cross sections with the available experimental data (Sherman et al., 1980) in the energy range 5–12.5 MeV for Ta and Bi in Section 4.4.

ð19Þ

where G is the mathematical gamma function, ai’s are the diagonal elements of the matrix A, and (X)k is the pochammer symbol. The matrix G function satisfies a number of partial differential equations;

qu ðp rÞ qu ðp rÞ ¼r p qp qr

635

1 1 AgðBÞ  I 4
0 BgðBÞ  I 4 :BðBÞ
1 ðAðBÞ þ 1Þ, o0 o (23)

Eq. (22) can be integrated to obtain the matrix function G(o) for photon energy o by using G(o0 ) at some arbitrary energy o0 as the initial value.

We present in this section the results of the present calculation of the differential pair production cross section for photon energy of 12.5 MeV for atomic numbers Z ¼ 1, 30, 50, 68, 82 and 92 in Table 1. We are not giving here the explicit expression for the DWBA differential pair production cross section used for evaluating it. The outline of the technique used for evaluating the needed Dirac–Coulomb radial integrals has been discussed in Section 3 and the reader is referred for further details to Wright et al. (1987) and the references cited therein. However, we would like to mention that in the Wright et al. (1987) formalism the expression for the differential pair production cross section has been expressed as a partial wave expansion and each term in it corresponds to a combination of lepton partial wave quantum numbers k+ (positron) and k
(electron). The differential pair production cross section expression is given as 1 X ds ¼ T q, dE þ q¼1

(24)

where q ¼ jkþ j þ jk
j
1 and Tq is the partial DWBA differential cross section (see Wright et al. (1987) for details). We have computed Tq for q values from q ¼ 1–150 for positron energy E þ ¼ 0:75 to 11.75 MeV in the step of 0.25 MeV for six Z values. We discuss in brief the procedure followed by us to include the contribution of the terms having large q values (i.e. for q values greater than qmax ¼ 150) to the differential pair production cross section. The variation of log(Tq) as a function of q has been found to be linear for a large values of q. We can thus express the differential pair production cross section as qX max ds ¼ T q þ R, dE þ q¼1

(25)

where the remainder R is given as ea 1
ea

4. Calculation and results

R ¼ T qmax

We present the results of our calculation of the differential pair production cross section in Section 4.1 and total pair production cross sections in Section 4.2 by 12.5 MeV photon for a number of targets. We also present differential pair production cross section in the tip region for photon energies 5.0–40.0 MeV in Section 4.3. We compare our interpolated DWBA pair produc-

and ‘a’ is the slope of log(Tq) versus q curve at q ¼ qmax . In the earlier calculation performed by Selvaraju et al. (2001), the contribution from the terms beyond qmax have been obtained by using Eq. (26). However, we have evaluated the differential pair production cross section for Z ¼ 1 by including the contribution of q values beyond q ¼ qmax by first extrapolating log(Tq) to

(26)

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636

Table 1 Differential pair production cross section [ð1=Z2 Þ ds=dE þ in mb/MeV] for different Z (Z ¼ 1, 30, 50, 68, 82 and 92) for 12.5 MeV photons for different positron energies (E+) E+ (MeV)

0.75 1.00 1.25 1.50 1.75 2.00 2.25 2.50 2.75 3.00 3.25 3.50 3.75 4.00 4.25 4.50 4.75 5.00 5.25 5.50 5.75 6.00 6.25 6.50 6.75 7.00 7.25 7.50 7.75 8.00 8.25 8.50 8.75 9.00 9.25 9.50 9.75 10.00 10.25 10.50 10.75 11.00 11.25 11.50 11.75 11.98491

ð1=Z2 Þ ds=dE þ (mb/MeV)

dsBH =dE þ (mb/MeV)

Z¼1

Z ¼ 30

Z ¼ 50

Z ¼ 68

Z ¼ 82

Z ¼ 92

0.08726 0.12813 0.15691 0.17844 0.19469 0.20713 0.21670 0.22408 0.22973 0.23404 0.23729 0.23969 0.24143 0.24264 0.24347 0.24402 0.24442 0.24473 0.24501 0.24522 0.24532 0.24532 0.24531 0.24533 0.24534 0.24525 0.24505 0.24479 0.24448 0.24410 0.24356 0.24275 0.24155 0.23983 0.23744 0.23421 0.22992 0.22429 0.21695 0.20740 0.19499 0.17855 0.15731 0.12865 0.08813 0.01227

0.07351 0.11978 0.15032 0.17240 0.18900 0.20171 0.21154 0.21917 0.22510 0.22969 0.23322 0.23592 0.23796 0.23948 0.24061 0.24142 0.24201 0.24243 0.24273 0.24295 0.24313 0.24328 0.24342 0.24355 0.24367 0.24377 0.24383 0.24383 0.24373 0.24348 0.24302 0.24229 0.24119 0.23962 0.23744 0.23447 0.23052 0.22530 0.21847 0.20956 0.19796 0.18279 0.16280 0.13611 0.09972 0.05592

0.06196 0.11156 0.14318 0.16549 0.18204 0.19464 0.20437 0.21192 0.21780 0.22237 0.22590 0.22862 0.23069 0.23226 0.23343 0.23430 0.23495 0.23543 0.23580 0.23609 0.23634 0.23657 0.23679 0.23700 0.23722 0.23743 0.23761 0.23774 0.23778 0.23771 0.23745 0.23696 0.23614 0.23490 0.23312 0.23064 0.22726 0.22275 0.21678 0.20892 0.19860 0.18500 0.16692 0.14254 0.10896 0.06843

0.05120 0.10249 0.13485 0.15724 0.17366 0.18609 0.19566 0.20310 0.20889 0.21340 0.21691 0.21962 0.22170 0.22329 0.22449 0.22540 0.22608 0.22661 0.22701 0.22735 0.22764 0.22791 0.22818 0.22845 0.22873 0.22901 0.22927 0.22949 0.22965 0.22971 0.22962 0.22933 0.22876 0.22784 0.22644 0.22445 0.22170 0.21796 0.21298 0.20637 0.19765 0.18607 0.17055 0.14930 0.11899 0.07942

0.04330 0.09480 0.12749 0.14984 0.16610 0.17834 0.18776 0.19506 0.20077 0.20522 0.20869 0.21138 0.21346 0.21505 0.21627 0.21719 0.21789 0.21842 0.21884 0.21919 0.21949 0.21978 0.22006 0.22034 0.22063 0.22093 0.22121 0.22147 0.22167 0.22179 0.22177 0.22158 0.22116 0.22042 0.21927 0.21762 0.21531 0.21217 0.20798 0.20244 0.19513 0.18545 0.17246 0.15453 0.12783 0.09099

0.03816 0.08931 0.12209 0.14436 0.16045 0.17253 0.18180 0.18900 0.19462 0.19901 0.20244 0.20511 0.20717 0.20875 0.20996 0.21088 0.21158 0.21212 0.21253 0.21288 0.21317 0.21344 0.21371 0.21398 0.21426 0.21454 0.21481 0.21506 0.21525 0.21537 0.21536 0.21519 0.21481 0.21413 0.21310 0.21161 0.20954 0.20675 0.20305 0.19821 0.19189 0.18363 0.17266 0.15757 0.13421 0.10178

0.08771 0.12846 0.15730 0.17876 0.19509 0.20764 0.21735 0.22487 0.23067 0.23513 0.23853 0.24110 0.24300 0.24439 0.24538 0.24606 0.24652 0.24681 0.24699 0.24709 0.24714 0.24716 0.24717 0.24716 0.24714 0.24709 0.24699 0.24681 0.24652 0.24606 0.24538 0.24439 0.24300 0.24110 0.23853 0.23513 0.23067 0.22487 0.21735 0.20764 0.19509 0.17876 0.15730 0.12846 0.08771 0.01060

ds=dE þ is the result of the present DWBA calculations and dsBH =dE þ is the plane wave Bethe–Heitler cross section.

additional 40 q-values by using cubic fitting to the calculated log(Tq) and then making linear extrapolation. The maximum contribution through remainder is about

6% of the sum of the term up to q190 for calculation involving almost evenly shared energy by positron and electron. The Bethe and Heitler results (see Table 1) for

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We have estimated the uncertainty in our calculation of differential pair production cross section for o ¼ 12:5 MeV to be less then 0.25%. This has been taken into account the difference between T qmax at Z ¼ 1 and 92, errors in the remainder and radial integrals. We depict the positron spectra i.e. (1/Z2)ds/dE+ in Fig. 1 and 1/Z2 Coulomb distortion ratio in Fig. 2 as a function of the positron energy for o ¼ 12:5 MeV for six Z-values (Z ¼ 1; 30; 50; 68; 82 and 92). It can be concluded from the Fig. 2 that the effect of the distortion is pronounced in the tip region as well as on the slow moving positron in the positron spectrum. Further the variation is very smooth and is amenable for accurate interpolation over positron energies as well as over photon energies in the range 5.0 to 12.5 MeV following the procedure used by Selvaraju et al. (2001). 4.2. Total pair production cross section We have obtained the total pair production cross section by using the positron spectra given in Table 1. The total cross section has been computed by numerical

(1/z2)dσDW /dE+ (mb/MeV)

0.25 0.20 ω = 12.5 Mev

0.15

(Z=1) (Z=30) (Z=50) (Z=68) (Z=80) (Z=92) (PWBA)

0.10 0.05 0.00

0

2

4 6 8 Positron Energy (MeV)

10

12

Fig. 1. DWBA differential pair production cross section DW ð1=Z 2 Þds dE þ in mb/MeV as a function of positron energy for 12.5 MeV photons for Z ¼ 1, 30, 50, 68, 82 and 92. The results obtained in PWBA is also depicted.

1.8 1.6 Coulomb Distortion Factor

the differential pair-production cross section are symmetric about the evenly shared energy points (i.e. E þ ¼ E
¼ 6:25 MeV), as expected. It can be seen from the Table 1 that the DWBA results for Z ¼ 1 are also almost symmetric about the evenly shared energy points and are very close to the results obtained by the Bethe–Heitler calculation. Our study shows that for larger q values Tq for Z ¼ 1 approaches the plane wave values. To confirm this we have compared the extrapolated DWBA T190 for E þ ¼ 8:5 and 4.0 MeV and found that the difference to be only 0.06%. We have estimated the total error in the DWBA differential pair production cross section for Z ¼ 1. The error is determined by estimating the error in the remainder by using the technique of Wright et al. (1987). The variation in the slope (a) of the log(Tq) versus q curve for larger q values the extrapolation to larger q-values may lead to error in our estimate of the remainder. So to estimate error we compare exp(a) at q ¼ 100, 150 and 190 and take these q values to generate the remainder and use it to estimate the error. We have estimated total uncertainty in our calculation for photon energy 12.5 MeV and Z ¼ 1 to be less than 0.1%. The total error has been estimated by taking into account the contribution of the remainder and in the calculation of the radial integrals. We calculate the differential cross section for other Zvalues by adding and subtracting the plane wave cross section sBH which is assumed to be represented by Z ¼ 1 in the partial wave representation. The expression for the differential cross section for any Z value is given as   qX max ds ¼ ½T q ðZÞ
T q ð1Þ þ sBH . (27) dE þ Z q¼1

637

1.4 1.2

ω = 12.5 MeV Z=1 Z=30 Z=50 Z=68 Z=82 Z=92 PWBA

1.0 0.8 0.6 0.4

0

2

4

6

8

10

12

Positron Energy E+ (MeV)

Fig. 2. Coulomb distortion ratio of the differential pair production cross sections ð1=Z2 Þ½ðdsDWBA =dE þ Þ=ðdsPWBA = dE þ Þ as a function of positron energy for 12.5 MeV photons for Z ¼ 1, 30, 50, 68, 82 and 92.

integration of the positron spectrum data by using Gauss quadrature. We use spline interpolation of the positron spectra to obtain the differential cross section at positron energy points required by the Gauss quadrature numerical routine. The results of the total pair production cross section calculation at photon energy 12.5 MeV are given in Table 2 for Z ¼ 1; 30; 50; 68; 82 and 92. We have also compared our DWBA total pair production cross sections with the results obtained from the semi-empirical formulas of Øverbø (1977), Maximon and Gimm (1978), plane wave Bethe–Heitler results (1934) and also results obtained by using the computer program XCOM of Berger and Hubbell (1987). The data obtained by using the XCOM

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638

Table 2 Total pair production cross sections for photon energy 12.5 MeV for different Z values Z

1 30 50 68 82 92

Total pair production cross section (b) sDWBA

sOv

sMG

sHU

sPWBA

2.4215E
03 2.1687 5.8954 10.5796 14.9184 18.3008

2.4707E
03 2.1932 5.9509 10.6739 15.0426 18.4237

2.4707E
03 2.1887 5.9224 10.5991 14.9203 18.2652

2.4710E
03 2.1947 5.9524 10.6678 14.9487 18.3413

2.4708E
03 2.2237 6.1769 11.4248 16.6134 20.9126

sDWBA represents the present DWBA results, sFv and sMG are the results obtained by using the XCOM program of Berger and Hubbell (1987) (divided by the factor frad[l
R] and sPWBA are the plane wave Bethe–Heitler results.

1.04

Distortion ratio

1.00

0.96

0.92 Z=92 Z=82 Z=68 Z =50 Z =30

0.88

0.84

4

6

8 10 Energy ω (MeV)

12

Fig. 3. Coulomb distortion ratio of the total pair production cross sections ð1=Z 2 Þ½dsDWBA =dsPWBA  as a function of photon energy for Z ¼ 1, 30, 50, 68, 82 and 92.

Z=92 Φverbφ Maximon DBM Our calculation

1.0 Distortion Ratio

computer program gives pair production cross sections corrected for radiative and screening corrections. So to compare with our results we have divided the XCOM data by the factor frad (1
R) where frad is the radiative correction factor given by Mork and Olsen (1965) and the nuclear field relativistic screening correction [1
R] has been obtained from the tabulation of Hubbell et al. (1980). We represent our DWBA results by sDWBA, results obtained from the semi-empirical formulas of Øverbø (1977) and Maximon and Gimm (1978) by sFv and sMG, respectively and the results obtained from the XCOM program (divided by the factor frad (1
R) of Berger and Hubbell (1987) by sHU. We refer the reader to the work of Selvaraju et al. (2001) wherein they present the results of their calculation of the total DWBA pair production cross sections, for photon energies 5.0 to 10.0 MeV in the step of 0.5 MeV for Z ¼ 1, 30, 50, 68, 82 and 92 and is presented there in Tables 13–18 (see Selvaraju et al. (2001)). Selvaraju et al. (2001) concluded from their analysis that the discrepancy between the DWBA results and the results obtained from the two interpolating formulas of Øverbø (1977), Maximon and Gimm (1978) and Berger and Hubbell (1987) varies with Z and has no specific pattern. We discuss in brief how the sDWBA, for the photon energy o ¼ 12:5 MeV, compare with the sFv, sMG and sHU. We observe that sFv and sMG values are almost identical for Z ¼ 1 but the difference increases to 0.76% of the Bethe–Heitler results at 12.5 MeV. The present DWBA results for Z ¼ 1, 30, 50, 68 and 82 are lower than sFv, sMG and sHU but for Z ¼ 92 it lies between the two values obtained from two interpolating formulas. In Fig. 3 we show the ratio of the total DWBA pair production cross section to the Bethe–Heitler result (Distortion ratio) as a function of photon energy (5.0–12.5 MeV) for different Z-values. The distortion ratio for photon energy range 5.0–10.0 MeV have been obtained from the work of Selvaraju et al. (2001). The variation of the distortion ratio as a function of the photon energy is very smooth and amenable for accurate

0.9

0.8

10

ω (MeV)

100

Fig. 4. Distortion ratio; the ratio of the total DWBA cross section to the Bethe–Heitler result as a function of photon energy for Z ¼ 92, is shown for the interpolating formulae of Overbo (1977) and Maximon and Gimm (1978), Davies et al. (1954) and the DWBA calculations.

ARTICLE IN PRESS

Z ¼ 92

2.0372E
01 1.8012E
01 1.6128E
01 1.4591E
01 1.3315E
01 1.2241E
01 1.0178E
01 8.7063E
02 6.7489E
02 4.7112E
02 3.7573E
02

Z ¼ 82

1.7904E
01 1.5924E
01 1.4310E
01 1.2977E
01 1.1862E
01 1.0920E
01 9.0994E
02 7.7945E
02 6.0528E
02 4.2339E
02 3.4350E
02

Z ¼ 74

1.6334E
01 1.4585E
01 1.3137E
01 1.1932E
01 1.0918E
01 1.0058E
01 8.3892E
02 7.1900E
02 5.5851E
02 3.9035E
02 3.1661E
02

Z ¼ 68

1.5360E
01 1.3747E
01 1.2399E
01 1.1271E
01 1.0320E
01 9.5100E
02 7.9423E
02 6.8241E
02 5.2877E
02 3.7045E
02 3.0137E
02

Z ¼ 58

1.4040E
01 1.2598E
01 1.1377E
01 1.0350E
01 9.4798E
02 8.7380E
02 7.2949E
02 6.2546E
02 4.8601E
02 3.3992E
02 2.7474E
02 1.3171E
01 1.1826E
01 1.0682E
01 9.7164E
02 8.8984E
02 8.2008E
02 6.8432E
02 5.8650E
02 4.5558E
02 3.1800E
02 2.5552E
02 1.1940E
01 1.0706E
01 9.6562E
02 8.7724E
02 8.0256E
02 7.3896E
02 6.1562E
02 5.2697E
02 4.0859E
02 2.8452E
02 2.2683E
02 1.0954E
01 9.7977E
02 8.8202E
02 8.0008E
02 7.3108E
02 6.7247E
02 5.5920E
02 4.7804E
02 3.7023E
02 2.5838E
02 2.0233E
02 8.8977E
02 7.9180E
02 7.1016E
02 6.4235E
02 5.8562E
02 5.3769E
02 4.4564E
02 3.8010E
02 2.9339E
02 2.0288E
02 1.5845E
02 6.4658E
02 5.7270E
02 5.1192E
02 4.6188E
02 4.2025E
02 3.8528E
02 3.1836E
02 2.7102E
02 2.0867E
02 1.4745E
02 1.1154E
02 2.5528E
02 2.2462E
02 1.9982E
02 1.7965E
02 1.6301E
02 1.4910E
02 1.2273E
02 1.0420E
02 8.0155E
03 5.4800E
03 4.2217E
03

We compare the results of our calculation with the one obtained using Deck et al. (1969) expression in Fig 5. The DMA expression has been derived by using the modified Sommerfeld–Maue lepton’s wave functions. In this approximation a correction term to the Bethe– Heitler formula of the order of aZ is obtained for the limit electron momentum equal to zero. The present

5.0 6.0 7.0 8.0 9.0 10.0 12.5 15.0 20.0 30.0 40.0

þ 0:0083ðaZÞ4
0:002ðaZÞ6 g.

Z ¼ 50

f ðZÞ ¼ ðaZÞ2 f½1 þ ðaZÞ2 
1 þ 0:202
0:0369ðaZÞ2

Z ¼ 38

where Z is atomic number, r0 is classical electron radius, a is fine structure constant and m is electron mass. The Z-dependent function f(Z) is as given by Davies et al. (1954) as

Z ¼ 30

(29)

Z ¼ 19

2  2 2paZ
ds0 4 2 a 1
¼ 4pZr0 paZ f ðZÞ, dE þ m e2paZ
1 15

Z ¼ 11

where

Z¼1

We present in this section the results of our calculation of the DWBA pair production cross sections in the tip region (E
¼ 1:008 mc2 ) of the positron spectrum for photon energy range 5.0–40.0 MeV. The calculation has been performed for a few photon energy points (o ¼ 5:0 to 10.0 MeV in the step of 1.0 MeV and at o values 12.5, 15.0, 20.0, 30.0 and 40.0 MeV) for a number of targets Z ¼ 1, 11, 19, 30, 38, 50, 68, 70, 82 and 92). We refer the reader to the work of Sud and Sharma (1984) for details of the technique which has been used by us to evaluate the needed radial integrals for computing the differential pair production cross section. We have evaluated Tq (Eq. (24)) for each value of q for q ¼ 1–130. The contribution to the cross section from the terms greater than q ¼ 130 is included by linear extrapolation (Eq. (26)). We present the results of the present DWBA calculation in Table 3. We fit the calculated points by fitting it to the following semi empirical curve which has an energy dependence like that suggested by Deck et al. (1969) (see Eq. (37) in the reference and we refer it as DMA)   dsDWBA ds0 a b c (28) ¼ 1þ þ 2þ 3 , dE þ dE þ o o o

ð1=Z2 Þ ds=dE þ (mb/Mev)

4.3. Differential cross section in the tip region of the positron spectrum

o (MeV)

interpolation. We depict in Fig. 4 the distortion ratio obtained from the two interpolating formulas (Øverbø, 1977; Maximon and Gimm, 1978), DWBA calculation and Davies et al. (1954) as a function of the photon energy for Z ¼ 92. The ratios from our calculated values lie between the values obtained from the two interpolating formulas, but are slightly closer to the Maximon– Gimm curve.

Table 3 ð1=Z 2 Þ ds=dE þ DWBA differential pair production cross sections for electron energy E
¼ 1:008 mc2 and for photon energy o in the range 5.0–40.0 MeV for different Z-values

K.K. Sud, D.K. Sharma / Radiation Physics and Chemistry 75 (2006) 631–643

639

ARTICLE IN PRESS K.K. Sud, D.K. Sharma / Radiation Physics and Chemistry 75 (2006) 631–643

640

2

(1/Z )(dσ/dE+) (barns / MeV)

1.2x10-4 Z =92

1.0x10-4

DWBA DMA

8.0x10-5 6.0x10-5 4.0x10-5 2.0x10-5 0.0 0

20

40 60 80 Photon Energy (Mev)

100

Fig. 5. (1/Z2)(ds/dE+) the differential cross section for pair production for Z ¼ 92 as a function of photon energy in the tip region of the positron spectrum for electron energy E
¼ 1:008 mc2 . DWBA results of our calculation upto 40.0 MeV and extrapolated data obtained using the semiempirical fit of Eq. (28) for higher energy photons. The results obtained by using Deck et al. (1969) is shown by dashed line.

DWBA calculation is for E
¼ 1:008 mc2 , which corresponds to the parameters y ¼ E þ
1=o
2 ¼ 0:9998 for o ¼ 50:0 MeV, whereas the DMA calculation is for y ¼ 1. The extrapolated DWBA result approach the DMA results at a energy in the range of 100 MeV. The present investigation shows that even near the end point of the positron spectrum the influence of the Coulomb distortion to the differential pair production cross section is significant in the intermediate photon energy range. Further, the DWBA results are expected to approach the approximate DMA results at very high photon energy.

DWBA cross section evaluated for Z ¼ 1, 30, 50, 68, 82 and 92. Experimental data of Sherman et al. (1980) are at Z ¼ 73 and 83, so we have to further interpolate to obtain the DWBA cross section for Z ¼ 73 and 83 by using the energy interpolated DWBA cross sections at six different Z values for each experimental photon energy. The interpolated DWBA cross sections (sDWBA) are given in Tables 4 and 5. We have estimated the error in the interpolation over photon energy as well as Z value. To estimate the error in the energy-interpolation we use the calculated DWBA cross section for photon energies 5.0–12.5 MeV at six Z-values. We have estimated the error due to the energy-interpolation and Zinterpolation by carrying out the interpolations by omitting one or more calculated and energy-interpolated points and found the uncertainty to be less than 0.005%. The results (sDWBA) presented in Tables 4 and 5 are the DWBA pair production cross section at Z ¼ 73 and 83 in the Coulomb field of nucleus. However, the experimental results of Sherman et al. (1980) are obtained from the experimentally measured total absorption cross sections. Sherman et al. (1980) have used following procedure to obtain the pair production cross section from the total absorption cross section. The experimental atomic absorption cross section szexpt is obtained by subtracting the experimental photo nuclear absorption cross section sn from the total absorption Table 4 Total pair production cross sections for Z ¼ 73 for different photon energies Photon Energy (MeV)

4.4. Comparison with the experimental data In this sub-section we present a brief discussion on the comparison of our theoretical results with the available experimental data of Sherman et al. (1980). The experimental results of Sherman et al. (1980) are in the photon energy range 3 to 30.0 MeV and for atomic numbers Z ¼ 73 and 83. To compare the DWBA results with the experimental data we have used the DWBA total pair production cross sections evaluated by Selvaraju et al. (2001) at photon energies 5.0 to 10.0 MeV in the step of 0.5 MeV and the present DWBA total pair production cross sections at photon energy 12.5 MeV and for Z values 1, 30, 50, 68, 82 and 92. The available experimental results in photon energy range 5.0 to 12.5 MeV are at the following energy points 5.333, 5.837, 6.348, 6.870, 7.404, 7.936, 8.382, 8.936, 9.476, 9.992, 10.514, 11.039, 11.557 and 12.088 MeV. As the computed theoretical results are not at the available experimental photon energies, so first we perform interpolation over photon energy on the theoretical

5.333 5.837 6.348 6.87 7.404 7.936 8.382 8.936 9.476 9.992 10.514 11.039 11.557 12.088

Total pair production cross section (barns) sDWBA

sexp

sTheor

6.3853 6.9377 7.4644 7.9715 8.4615 8.9233 9.2921 9.7282 10.1311 10.4971 10.8478 11.1664 11.4202 11.6477

6.26 6.8 7.35 7.84 8.26 8.69 9.11 9.55 9.92 10.3 10.61 10.98 11.3 11.63

6.2701 6.8033 7.3105 7.7977 8.2683 8.7118 9.0659 9.4847 9.8716 10.2229 10.5595 10.8648 11.1069 11.3232

dsw

+0.16 +0.05
0.54
0.54 +0.10 +0.25
0.49
0.69
0.49
0.75
0.48
1.06
1.74
2.71

The subscript ‘DWBA’, ‘exp’ and ‘Theor’ correspond to the present calculation in point-Coulomb potential, the experimental results of Sherman et al. (1980) and the theoretical results obtained by multiplying sDWBA by frad and (1
R), i.e. including the radiative and screening correction in percentage difference (dsw) between the theoretical and experimental total pair productioncross sections.

ARTICLE IN PRESS K.K. Sud, D.K. Sharma / Radiation Physics and Chemistry 75 (2006) 631–643 Table 5 Total pair production cross sections for Z ¼ 83 for different photon energies Photon Energy (MeV)

5.333 5.837 6.348 6.87 7.404 7.936 8.382 8.936 9.476 9.992 10.514 11.039 11.557 12.088

Total pair production cross section (barns) sDWBA

sexp

sTheor

dsw

8.1892 8.8688 9.5174 10.1431 10.7484 11.3200 11.7771 12.3184 12.8190 13.2743 13.7207 14.1839 14.6954 15.2158

7.961 8.649 9.311 9.919 10.422 10.987 11.613 12.028 12.574 13.004 13.407 13.88 14.229 14.803

8.0211 8.6742 9.2951 9.8919 10.4690 11.0135 11.4487 11.9640 12.4405 12.8737 13.2985 13.7397 14.2278 14.7243

0.75 0.29
0.17
0.27 0.45 0.24
1.43
0.53
1.07
1.01
0.81
1.02
0.01
0.53

641

vary with photon energy. The experimental pair production cross sections of (Sherman et al. (1980)) have been extracted from the measured total atomic absorption cross sections by subtracting the theoretical absorption cross section (excluding the pair production cross sections) and photonuclear absorption cross sections. We depict the interpolated DWBA as well as experimental (Sherman et al., 1980) total pair production cross sections for Z ¼ 73 in Fig. 6 and Z ¼ 83 in Fig. 7 as a function of photon energy in the energy range 5.0–12.5 MeV. The accuracy of the atomic cross sections of the Hubbell et al. (1980) tabulations used to extract the sexp is better than 0.2%. The photonuclear cross section used is in terms of experimental photonuclear cross section. As discussed by Selvaraju et al. (2001) and in Section 4.1 the maximum error in the DWBA differential pair production cross section to be 0.25%. The discrepancy 13

cross section stot. sexpt z

¼ stot
sn .

(30)

where spe is photo electric, sC is Compton scattering, sR is Rayleigh scattering and st is triplet production cross section. The experimental values of total pair production cross sections sexp are given in Tables 4 and 5. The experimental values thus obtained are the total pair production cross sections including the screening and radiative corrections, whereas the sDWBA (in Tables 4 and 5) are the theoretical total pair production cross sections in the Coulomb field of the point nucleus. To compare the theoretical results with the experimental values we have to multiply the theoretical results by the multiplicative correction factors (1
R) for screening correction and frad for the radiative correction (Hubbell et al., 1980). The corrected theoretical total pair production cross sections (sTheor) are given in Tables 4 and 5. We have shown in Tables 4 and 5 the percentage difference (dsw) between the theoretical and experimental results. It is found that it is large in some regions and

10 9 8 7 6 5

6

7

8

9

10

11

12

Photon Energy (MeV)

Fig. 6. Interpolated DWBA and experimental (Sherman et al., 1980) total pair production cross sections for Z ¼ 73 as a function of the photon energy.

16 Our DWBA calculation Sherman et. al. PRC 21 (1980) 2328

15

Z=83

14 Cross section (b)

(31)

Z=73

11

The sum of the theoretical atomic cross sections other than the atomic pair production cross section is subtracted from szexpt to obtain the atomic pair production cross section sexp sexp ¼ sexpt
ðspe þ sC þ sR þ st Þ, z

Our DWBA calculation Sherman et. al. PRC 21 (1980) 2328

12

Cross section (b)

The subscript ‘DWBA’, ‘exp’ and ‘Theor’ correspond to the present calculation in point-Coulomb potential, the experimental results of Sherman et al. (1980) and the theoretical results obtained by multiplying sDWBA by frad and (1
R), i.e. including the radiative and screening correction in percentage difference (dsw) between the theoretical and experimental total pair production cross sections.

13 12 11 10 9 8 5

6

7

8 9 10 Photon Energy (MeV)

11

Fig. 7. Same as in Fig. 6 except for Z ¼ 83.

12

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between the DWBA theoretical cross section and experimental data of Sherman et al. (1980) is less than 0.5% for Z ¼ 73 and 83 for photon energies less than 8.0 MeV (except at o ¼ 5:333 MeV for Z ¼ 83, it is 0.75%) and between 0.5–2.7% for Z ¼ 73 and 0.5–1.0% for Z ¼ 83 (except for photon energy 11.557 MeV for Z ¼ 83). Thus the discrepancy between DWBA calculation and experimental result is larger than the experimental uncertainty. This is consistent with the finding of the Selvaraju et al. (2001) that the discrepancy between the theoretical and experimental data of Sherman and Del Bianco (1988) for 10.0 MeV photons, Z ¼ 92 is 1%.

5. Conclusions In conclusion, we would like to state that with the availability of the differential pair production cross sections in DWBA at photon energy 12.5 MeV, we now have a comprehensive database of the differential pair production in the photon energy range 5.0–12.5 MeV in the steps of 0.25 MeV for Z ¼ 1; 30; 50; 68; 82 and 92. The database can be used to generate accurate differential pair production cross section at the chosen photon and lepton energies for any Z value. Further this enables one to compute the total pair production cross section and Coulomb distortion ratio for any photon energy point and atomic number in the energy range 5.0–12.5 MeV. The discrepancies observed between the present theoretical and experimental results strongly suggest that the screening effect have not been properly calculated in this energy region and need further investigation. The results obtained from semi-emperical formulae of Øverbø (1977) and Maximon and Gimm (1978) deviate considerably from the one obtained in DWBA. Further the errors incorporated in using semi-empirical formulae do not follow any specific pattern in terms of photon energy and atomic number. So, we suggest that the results of the present study be used to modify and improve the existing interpolating formulae for Coulomb distortion. The nature of the variation of the Coulomb distortion ratio with photon energy suggests the need to perform DWBA calculation of the pair production cross section in the energy range 12.5–30.0 MeV. Finally we would like to mention that recently Lee et al. (2004) have presented a formalism to compute Coulomb correction to the pair production cross section in the intermediate energy range by using quasi classical electron Green’s function in an external electric field. They have expressed the Coulomb correction in the following form sC ¼ s0C þ s1C þ s2C þ   

(32)

where the leading term s0C is the Davies et al. (1954) high energy Coulomb correction term, snC has the form mn  m SðnÞ ln o o where S(n)(x) is some polynomial in x. The total pair production cross section obtained by using the first order Coulomb correction term s1C of Lee et al. (2004) show a large deviation from the experimental data in the intermediate energy range. Lee et al. (2004) have shown that on incorporating a semi-empirical second order Coulomb correction s2C the agreement with the experimental data for Z ¼ 82 and 83 improves. A comparison between the pair production cross section computed in DWBA and by using Lee et al. (2004) formalism would be interesting and will help to understand better the Coulomb correction to the pair production process in the intermediate energy range. We will be reporting the comparison of our calculated DWBA pair production cross section with the results obtained by using Lee et al. (2004) formalism as well as with the available experimental data in photon energy range 5.0 to 12.5 MeV on a large number of targets in the forthcoming communication (Sud et al., 2004).

Acknowledgements The support from the Special Assistance Programme of the University Grants Commission (UGC), New Delhi to the Physics Department, Mohanlal Sukhadia University, Udaipur, India is gratefully acknowledgement. We are thankful to Prof. R.H. Pratt for bringing to our notice the recent publication of Lee et al. (2004).

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ARTICLE IN PRESS K.K. Sud, D.K. Sharma / Radiation Physics and Chemistry 75 (2006) 631–643 Fink, J.K., Pratt, R.H., 1973. Use of Furry–Sommerfeld–Maue wave functions in pair production and bremsstrahlung. Phys. Rev. A 7, 392–403. Hubbell, J.H., 2004. Electron-positron pair production by photons: an historical overview. Rad. Phys. Chem. Hubbell, J.H., Seltzer, S.M., 2004. Cross section data for electron-positron pair production by photon: a status report. Nucl. Instrum. Methods Phys. B 213, 1–9. Hubbell, J.H., Gimm, H.A., Øverbø, I., 1980. Pair, triplet, and total atomic cross sections (and mass attenuation coefficients) for 1 MeV–100 GeV photons in elements Z ¼ 1–100. J. Phys. Chem. Ref. Data 9, 1023–1147. Jaeger, J.C., Hulme, H.R., 1936. On the production of electron pairs. Proc. Roy. Soc. (London) A 153, 443–447. Lee, R.N., Milstein, A.I., Strakhovenko, V.M., 2004. Highenergy expansion of Coulomb corrections to the e+e
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