Non-relativistic form factor program for compton scattering of gamma rays by bound electrons

Computer Physics Communications 11(1976) 363—368 © North-Holland Publishing Company

NON-RELATNISTIC FORM FACTOR PROGRAM FOR COMPTON SCATFERING OF GAMMA RAYS BY BOUND ELECTRONS F. SMEND and M. SCHUMACHER II. Physikalisches Inst itut der Universitlt, D-3400 Goettingen, Fed. Rep. Germany Received 22 June 1976

PROGRAM SUMMARY Title of program: COMPTON CROSS SECTIONS

tion of the cross-section profile for Compton scattering of

Catalogue number: ACWW

gamma rays by the electrons of a complete subshell of an atom on the basis of non-relativistic hydrogen-like one-elec-

Program obtainable from: CPC Program Library, Queen’s University of Belfast, N. Ireland

(see application form

issue). Computer: UNIVAC 1108;

tron wave functions [1].

in this

Method of solution Installation: GWDG Goettingen

Operating system: EXEC 8 Programming language used: FORTRAN IV High speed storage required: 12 000 words No. of bits ma word: 36 Overlay structure: none No. of magnetic tapes required: none Other peripherals used: card reader, printer No. of cards in combined program and test deck: 761 Card punching code: BCD Keywords: atomic, gamma ray scattering, Compton scattering, form factor, cross-section profile, spectroscopy. Nature of physical problem The program Compton cross sections performs the calcula-

Closed-form expressions for the cross-section profile derived

in ref. [1] are used.

Restrictions on the complexity of the problem For energies of the scattered gamma rays near the free-electron value convergence may be too slow.

Typical running time On the Univac 1108 computer less than a few minutes are required in most cases.

Unusual features of the program Different effective atomic numbers may be used for the scattering electron in the initial bound state and the final con-

tinuum state, respectively. Reference [1] M. Schumacher, F. Smend and I. Borchert, J. Phys. B: Atom. Molec. Phys. 8 (1975) 1428.

F. Smend, M. Schumacher/Compton scattering of gamma rays

364

LONG WRITE-UP 1. Introduction The program calculates the cross-section profile d2o/d&~dk2for the Compton scattering of gamma rays by the electrons of a complete subshell of an atom. The subshell is described by the principal quantum number n and the 2 orbital momentum In quantum numberthe 1. The incoming and the scattered and k angular 2, respectively. the following relativistic system of units~ = mphotons have the energies k1m0c 2 m0c 0 = c = 1 is used. During the scattering process the momentum km0c is transferred to the atomic system, and the electron, 2 > 0 in the initial state, leaves the atom with the asymptoticscattering momentum pm having the binding energy bm0c 0c. The momentap and k are given by 2 _l]h/2, k = [k~ ÷ — 2k p = [(k1— k2 + 1—b) 1k2 cos 011/2, (1, 2) with 0 denoting the scattering angle. Using the form factor approximation and non-relativistic hydrogen-like wave functions the following expression for the cross section profile was derived in ref. [1]: 2lA~i=A~

d~7dk2=2r~(l+cosO)~~~p(l+p2)1/2 ~

~ I

íS,~ m~

(3)

1~,

with s = 1—1’~,il—i’i

+

2,..., l+l’, 1” = il—si, il—si + 2,..., i+s,

mi ~

S.

The quantity r0 = 2.818 X 1O~13 cm is the classical electron radius. The quantities A51~are zero unless 2g = s + 1+ 1’ is an even number, and i—i’i are given by

~

s~ 1

+

1’ and ml ~ 1. They

((s irni)! (1 + imi)! 2 +1 21+1 \1/2 1’! (_1)g—l (2g—21’)! g! ~ (s + mi)! (i—mi)! ~ ~ ~ ~ (g-i)! (g-s)! (g-i’)! (2g+l)! —

~ (s+[iml+t])! (l+[1’—imi—t])! X

(—1) (s—[imi +tj~)! (1— [1’—imi-t])!

1

(4

(l’_t)! t!’

m lfl

where tmin=

max (0,!’—!— imi),

tmax

=

mm (s— ml, i’÷i—imi,i’).

The quantities ‘sl’ are, for the general case, radial integrals containing the radial functions R~1(r) and f1. (r) of the scattering electron in the initial bound state and the final continuum state, respectively: 2 isis(kr)Rni(r)f~(r)dr. ‘s,l’

=f

(5)

r

In the present work, R

01(r) and f1~(r) are wave functions for a pure Coulomb field. Then, ~

can be written in

the following closed form: ~

D n~-l A~~

H~k’~ (1+l’+1+~—j)![i1~’(Za/n

X 2Fi(i’+l_iZ’a/P~ 2+1’+i+v—f, 21’+2;

where

—

i(~+

ZcE/n—i(p + k))÷c.c.]~

(6)

365

F. Smend, M. Schumacher/Compton scattering of gamma rays —n+l+1+v 1 2Za 2l+2+v v+1 n’

H50= 1; —1

H~ = ~ (s+p)(s—p+1)

(7)

(8)

1L1

exp (irZ’a\ r’(I’+1+iZ’a/p) (2i’)~ (21+ (—2ip)1’ 1)! \(n_ ( (n+l)! l—1)!2nJ\1/2 (2Za\’~3I2 \ n /

)

~

Z and Z’ are the atomic numbers (or effective atomic numbers) for the initial-state and final-state wave function

of the scattering electron, respectively, and a = 1/137.037 is the fine structure constant.

2. Numerical methods The hypergeometric function 2F1 (a, b, C; x) in eq. (6) has to be evaluated for complex a with Re (a) being an integer, and integer values of b and c. The argument x is confined to the lower right quadrant of the complex plane with 0.1 lxi ~ 1.2. For ixi <0.9 Gauss’s hypergeometric series is used. For lxi ~a’0.9 the linear transformation (15.3.6) of ref. [2] is applied. With a=i’+l+iy,

y=—Z’a/p,

m= 1+i+v—f,

c=21’+2

b=i’+l+m,

the following transformation formulae result:

(a) For l’~m ~ x ((_l)Im I ~

(1+n+m+z)’)1._m(n+l)1~÷ m(1~Y~

+ (l_X)_m_IY(_l)m ~

~ (1 +n—m— ~3’)l+m(~l+1)i~_m (1_X)n);

(10)

(b)Fori’
2F1 (l’+1+~y,1’+l+m, 21’+2; x)

=

r(r÷l÷~v) (l~)_m~iy

~

(

)fl(Y)fl(!)n

with (r)~denoting Pochhammer’s symbol: (r)0=1,

(r)n=r(r+1)(r+2)X ...X(r+n—1).

The case Z’ = 0 is not covered by the linear transformation used here. However, very simple analytic expressions for the cross-section profile are obtained if the outgoing electron is represented by a plane wave (ref. [1]). Double-precision arithmetic is used in parts of the calculations for two reasons: (a) In eq. (10) the expression in curly brackets, which is the small difference of two large numbers, has to be evaluated with sufficient adcuracy. (b) In Univac FORTRAN the range of single-precision floating-point numbers is restricted to 1038_ 1038. Although the explicit calculation of factorials and gamma functions is avoided whenever possible, arithmetic over-

F. Smend, M. Schumacher/Compton scattering of gamma rays

366

flow may occur at some places for large values of 1’. In Univac double-precision arithmetic the range of floatingpoint numbers is lO_308_10308. Some machines offer a greater numerical range than lO_38_l038 in single precision arithmetic. For this case, one might change from double-precision to single-precision arithmetic in all parts of the program except the subroutine DSUM. In calculating the cross-section profile for given values of k 1 and 0, k2 is varied from (k1 — b) down to about 0.05. The convergence of the sum over i’m eq. (3) depends strongly on k2, being slow fork2 close to the energy k2 (free) = (1— cos 0 + 1/k1)’ of a photon scattered by a free electron at rest.2 = 279.2 keV through the angle take, as this an example, of gamma the energy c 0 =We 135°.For case k the scattering 2 = 144.5 keV. Ifrays the of scattering of k1m0 these gamma rays by the 3s electrons of 2(free)m0c copper is considered, good convergence of the sum over i’ is found for energies k 2 outside the interval 110— 2 m0c 170 keV. With i’max ~ 20 good accuracy is obtained. Within the interval 140—150 keY no convergence is found while for the intervals 110—140 and 150—170 keV the convergence is rather slow. Even using i’max = 35, which is the practi~a1limit of the program, the result has to be extrapolated. Fortunately, this lack of c ‘nvergence is pronounced only in the case of the outer shells of low-Z atoms where, for k 2 in the vicinity of k2 (free), the outgoing electron is well approximated by a plane wave. For this approximation the cross section profile is given by simple analytic expressions derived in ref. [1].

3. Program structure The program consists of a main program and 15 subprograms. Of these, SIGMA performs the summations over I’, 1”, s, and m in eq. (3). ALEG supplies the A~1and CISL, together with 11 associated subprograms, calculates ‘sI - SGN is an auxiliary function.

In principle, the program offers, the possibility of using radial functions Rnl Q’) and f1~(r) different from those for a pure Coulomb field. Then, CISL with its associated subprograms has to be replaced by a subprogram performing the integration of eq. (5) explicitly. The radial functions Rni (r) and )“1’ (r) may be provided, for example, in tabular form.

4. Description of an input data deck Card 1

Z

FORMAT(2F10.2, 215, lOX, FlO.2, 215, FlO.6, 12)

Atomic number or effective atomic number for the initial bound state of the scattering electron. Z must be greater than zero. ZP Effective atomic number for the final continuum state of the scattering electron. ZP must be greater than zero. N Principal quantum number of the atomic subthell. L Orbital angular momentum quantum number of the atomic subshell. B Binding energy of an electron in the subthell (eV). LU Starting value of the summation index i plus 1. If left blank, LU = us used. LO Final value of I’ plus 1. If left blank, LO = 19 is used. EPS Convergence criterion for the sum over!’ in eq. (3). If left blank, EPS = 0.01 is used. Regardless of the value of EPS, the sum over 1’ is carried out to at least 1’ = 18. KTEST Parameter controlling the printout of intermediate results during the calculation. May be 0, 1, or 2. See section 5 for details.

F. Smend, M. Schumacher/Compton scattering of gamma rays Card 2

FORMAT(2F10.2)

W THET

Energy of the primary gamma ray (eV). Scattering angle 0 (degrees).

367

Card 3 FORMAT (2F10.2, IS)

WP1 WP2 NWP

Starting value in the sequence of energies of the scattered gamma ray for which the cross-section profile will be calculated (eV). Final value in the sequence. May be greater than, equal to, or less than WP1. Number of energy values in the sequence, including WP1 and WP2.

After having performed the calculation for the parameters given on Cards 2 and 3, the program expects a new set of Cards 2 and 3 unless a blank card is found. Then, a new Card 1 with subsequent sets of Cards 2 and 3 may follow or, alternatively, a second blank card terminating the run.

5. The test run The scattering of 279.2 keY gamma rays by the 3p electrons (n = 3,1 l)of copper through the angle 0 = 135° is chosen for the test run. For the bound and for the outgoing electron equal effective atomic numbers Z = Z’ = 17.75 are chosen. The empirical binding energy of a 3p electron in the copper atom is bm0c2 = 80.67 eY. Two energies of the scattered gamma rays are selected: (a) k 2 = 200 keV which corresponds to the absolute value ix I = 0.799 for the argument of the hyper2m0c geometric function. Gauss’ hypergeometric series is used. (b)k 2 = 75 keY. Now ixi = 1.196 and the hypergeometric function is evaluated using the linear transformation,2m0c eqs. (10) and (11). In the output the input parameters given on Cards 1—3 are listed first. After that the following data are printed for each value of the energy of the scattered gamma rays: (1) The asymptotic momentum p of the outgoing electron and the momentum transfer k; (2) for KTEST = 0 the final value of the sum over I’ of eq. (3), for KTEST = 1 the subtotal of that sum for each value of!’ and the final value, for KTEST = 2 in addition to the foregoing the ouput of the subroutine CISL and the subtotals of the sums over s and i” of eq. (3); (3) the energy k 2 of the scattered gamma rays and the double-differential scattering cross section in units of 2m0c cm2/(sr X m 2). 0c Acknowledgement The authors wish to thank Professor Dr. A. Flammersfeld for his support of this work.

References [1] M. Schumacher, F. Smend and I. Borchert, J. Phys. B: Atom. Molec. Phys. 8(1975)1428. [2] M. Abramowitz and I. Stegun (Eds), Handbook of Mathematical Functions, 5th edn (Dover, New York, 1968).

368

F. Smend, M. Schumacher/Compton scattering of gamma rays

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t.’

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Si. St • 15 SII$T~TAL Sji

LP FROM

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