A program to calculate virtual photon spectrum in second order born approximation

Computer Physics Communications 32 (1984) 291—307 North-Holland, Amsterdam

291

A PROGRAM TO CALCULATE VIRTUAL PHOTON SPECTRUM IN SECOND ORDER BORN APPROXIMATION P. DURGAPAL

*

The George Washington University, Washington, DC 20052, USA

and D.S. ONLEY

**

Department of Physics and Astronomy, Clippinger Research Laboratories, Ohio University, Athens, OH 45701, USA

Received 1 December 1983

PROGRAM SUMMARY Title of program: SOVPS Catalogue number: ABPN

calculated in second order Born approximation for magnetic dipole (Ml), electric dipole (El) and electric quadrupole (E2) transitions.

Program obtainable form: CPC Program Library, Queens’ Uni-

Method of solution

versity of Belfast, N. Ireland (see application form in this issue)

An analytic expression for the differential inelastic electron scattering cross Section is evaluated using the techniques given in ref. [1]. The distortion effects due to the charge of the 2/hc) nucleus are taken into account to first order in aZ (a = e and the finite size effects are taken into account by including a nuclear inelastic form factor.

Computer: IBM 370/158 Operating systems: VS1 Programming language used: FORTRAN IV

Validity of the results High speed storage required: 48000 words No. of bits in a word: 32

No. of lines in the program: 1422

Except for very low electron energies (E~<5 MeV) and very heavy nuclei [approximately A > 100 + 1.6E (MeV) for El, A >60 + 2.OE (MeV) for Ml and E2J the results obtained using the program SOVPS compare very well with distorted wave calculations [2]. In the low energy domain the program VIRTSPEC [3] is preferred.

Keywords: virtual photon spectrum, second order Born ap-

proximation, inelastic electron scattering Nature of physical problem

Nuclear size and charge corrections to the virtual photon spectrum accompanying electrons or muons of either charge scattered from a nucleus of mass number A and charge Z are

* **

References

[1] P. Durgapal and D.S. Onley, Phys. Rev. C27 (1983) 523. [2] D.S. Onley, Bull. Am. Phys. Soc. 26 (1981) 1129. [3] L.E. Wright and C.W. Soto Vargas, Comput. Phys. Cornmun. 20 (1980) 337.

Present address: WELEX, P.O. Box 42800, Houston, TX 77242, USA. Supported in part by a grant from the US Department of Energy.

OO1O-4655/84/$03.OO © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

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Virtual photon spectrum in Born approximation

LONG WRITE-UP 1. Introduction If we consider that the effect of the electromagnetic field of a charged particle passing near a nucleus or other target is equivalent to the incidence of a burst of radiation, the analysis of electrodisintegration data may be related to the corresponding photodisintegration processes. If a’(w) is the photoabsorption cross section for photons of energy co and multipolarity TI (T being the label E or M for electric or magnetic), then the integrated cross section for electrons of kinetic energy Ee is assumed to be of the form ~7e(Ee)

=fEc~G;I(~)Nrt(E ~

(1)

~

TI

In order to use this expression we need to know the virtual photon spectrum NTI( Ee, The earliest quantum mechanical calculations [4] of the virtual photon spectrum were done using the plane wave Born approximation (PWBA) and the long wavelength limit. The authors of ref. [4] give explicit expressions for El, E2, E3 and Ml spectra which have been widely used. Expressions for general multipole order is given in ref. [1]. We refer to these as the conventional spectra. Coulomb distortion effects on the electron wavefunction were taken into account by Gargaro and Onley [5], who showed that these result in an enhancement of the virtual photon spectrum in comparison to the conventional spectrum, which increases with the atomic number Z and also with multipolarity 1. Ref. [5] also uses the long wavelength limit or its equivalent in that the nucleus is considered to be of negligible extension. A program incorporating these techniques called VIRTSPEC is reported by Wright and Soto [3]. In order to correct for both finite size and Coulomb distortion we have here included form factors for both elastic and inelastic scattering and carried out calculations in second order Born approximation [6] (SOBA). In ref. [1] we have derived an expression for the differential cross section applicable with any electron wavefunction. In first order calculations, a study of finite size effects of the nucleus was made to determine to what extent the shape of the spectrum depends on the specifics of the inelastic form factor, also expressions for the virtual photon spectrum for El and Ml radiation in SOBA were derived. In the current work we include an expression for E2 transitions derived in the same way. As a result of the survey to determine the relevant parameter of the inelastic form factors we are able to choose functions with the appropriate asymptotic behavior (both for q 0 and q cc) and for which the transition radius [1] can be matched to some known value, but which need not conform in detail to form factors determined in inelastic electron scattering. Accordingly, it is possible to choose functions with the merit that all integrals over intermediate momentum transfer can be carried out analytically. The detailed expressions for the virtual photon spectra have been presented in ref. [1]; here we present only a brief summary. (i)).

—~

—

2. Virtual photon spectra for Ml transitions The expression in second order Born approximation for magnetic dipole transitions can be written as a sum of three terms 1(Ei,ca)

NM

2 ~N,Ml(E

(2)

1,t~.,), i=O

where E1

=

incident electron energy and

w

=

photon energy.

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Virtual photon spectrum in Born approximation

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The first term, NØMI, is obtained from plane wave Born approximation using long wavelength limit. It can be expressed in terms of the kinematic parameters as NoM1(E~,o)=~(12)ln(~), ~=E

(3)

1E2+PiP2me ~

p~

rr

where a is the fine structure constant, me is the electron mass and E1(E2) and P1(P2) refer to the incident (final) electron energy and momentum. In the second term of eq. (2), N1MI, a nuclear form factor, F.~’(q), has been included to account for finite nuclear size but the electrons are still assumed to be plane waves.

w)=-~-.~—1~dz~ ~ 2rr p2~

M1(E N

2L~ ~JIF~1(~)I2~\

~(~2w2)2

kIF.r~(o.)I2 w2J’

~2_~2J

where ~ is the physical momentum transfer with minimum and maximum values, zXmn and ~max~ The third term of eq. (2) provides the correction due to distortion effects of the finite nuclear charge to order aZ. In its simplest form the expression can be written as NMI(E 2 1’

2~rrp 2

~

1 ~2’rr (~2_~2) F.~’(L~)5M ~ IF.~(co)I2-’~,,.

(5)

~

1

where r I n ~‘Mk’~)

—

IA\ t.~*)

,12 i0M

2 3meJ

64

1 0

~ A

uq

q

2~’M1( \ T ~.l2 2 2 ~ t2M

~q, ( w—q

..~.

y21 t2M)’

—Ji.1~ F(I~ 2 qI) 2 ~ ‘~ TM)’ ( ç’12 \ (~—q) +X —

(7

U~q

—

2E —a.(~—q)

~

2

1

2

2

u~u1aY 17(~)ur.

m

j,~

q +2qP2—(P,

(8)

—P2)—1K

In eq. (6), I~ is obtained from I~ by the interchange of labels 1 and 2, i.e. q~-—q, P1~-’P2, E1~-*E2.

~-*—~,

(9)

(We shall refer to quantities with superscript ‘21’ as the interchanged term of quantities with superscript ‘12’.) In eq. (7), F(q) is the elastic scattering form factor and A is a convergence factor, ultimately set to zero, introduced to suppress the singularity arising from the infinite range of the Coulomb interactions. In eq. (8), }~7~(~) is a vector spherical harmonic, a is the Dirac spinor operator and u5(1~.)represents a four component column matrix corresponding to the two positive energy solutions of the Dirac equation. The label s is set equal to ‘i’ or ‘f’ to denote the initial and final electron spin states. The sums over final electron spin f and average over initial spin states i are straightforward and can be evaluated using the trace techniques discussed in Bjorken and Drell [7]. The sum over the magnetic quantum number m involves somewhat lengthy and tedious manipulations, the details of which are given in ref. [1]. The resulting integrals in eq. (7) can then be evaluated analytically provided the elastic electron scattering form factor F(q) contains only poles. For this reason we choose a Yukawa form factor which has the simple form 2/(q2 + y2), (10) F(q)

=

‘y

where y is the Yukawa parameter. Having found an expression for I~ we find that by making an appropriate choice for the inelastic form

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Virtual photon spectrum in Born approximation

factor, it is possible to evaluate the expression forflM by contour integration. In ref. [1] we have shown that the virtual photon spectrum is insensitive to the details of the inelastic form factor and depends only on the transition radius for a large range of electron energies. In view of this we choose a form factor which contains only poles: for Ml transitions we set F~1(q)=[N~q/(a2+q2)4~(1+C

(11)

2q2), 5M

(see eq. (6)) in

C2 being an adjustable parameter, and Nm gives the normalization. We can then express terms of the following integrals: ln

dqq2~ 0

(a2+q2)4(~2_q2)

d

~(k,y,g)=f 0

qq

(q_~)2+y2

2k—I

ln

— ~

(a2+q2)4(g2_q2)

(

(13)

,

q+y

1

dqq2k_l 4(~2_q2)~P~q4+uq2+v

~‘0

(12)

,

(q+~)2+y2

n

(q2_~2)v ~ 2+y2vi+2q~p~q4+uq2+v

(a2+q2)

(14) ~(k,y)

2 ‘~

=

2

where k is an integer (k

~) —~(k,y, i)],

[~(k,y,

(~5)

1, 2, 3, 4) and

=

2+P~—P~, V2=q u=—2~2P~+y2(P~2+Pfl—y2~2, J7

1=q2_(p12_pfl,

v= P~ +

+

~2[~2(p2

~fl—(~~— p2)2]

(16)

+p~’~.

The expression for 5M can then be written as (~2 1E

2

+

+

E1

~

JMNM~

~1Em(P~

—

(~ -

E2 ‘~M(~min)

E1

~

‘~MC~max)

~L)_Ep~2I[2.~M(~)_9M]

~)~M(~min)

-

(~

+ ~).~aM(ZImax)

~

+5PM)}}~

(17)

where we have defined the quantities ~M_(l)+’~2D(2), ~M(Y) =

=

~M(Y)_(1,Y,~)+C2(2,Y,~),

~ [C2~(4,y) +

(1

+ C2K1)~(3,y)+

E1C2Sf(4) +(E1 + C2K3)~I’(3)+(K3

(K~+ C2K2)~(2,y)+ K2~(1,y)],

+ C2K4)~9(2)+ K4.2~(1),

(18)

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Virtual photon spectrum in Born approximation

295

with K1

2 + P~— E,~)], =

2[~2 + 2(P1

(19)

~

K

2, 3E1K1 +E~y

K 4=E1K2+y2[Ep~2+2Em(Pi~_Pfl+4E1(2P~_Ei~jJ+E2y4.

In these equations E~and Em refer to the sum and difference of the initial and final election energies, i.e. Ep=E1+E2andEm=Ei—E2.

3. Virtual photon spectra for El and E2 transitions The expressions for electric transitions are obtained by a method similar to that for magnetic transitions but in this case, besides the transverse component, we also have a longitudinal component and a component due to interference between the Coulomb and the transverse terms. Again we write the expression for the corrected virtual photon spectrum as 2

~P~’(E1,~~),

NEI(E1,w)=

(20)

i= I

with I

=

1 for El transitions and 1= 2 for E2 transitions. The first-order term is given by 4~d~ p2 ~—f~

NIE/(EI, w)= 2n -~-

+2 (E~_~2) ~

+

~

f(~max _~2)(~2 ~(~22)2

IF.~’(~)I2

(~\)~

(21

~l+ hf IF~(c~)I2’

where F.~’(i.~) is the transverse electric form factor for transitions of multipolarity E/. The correction due to nuclear charge is expressed as N~’(EI,

~

a2Z ~

=

/(2’rr)2 p2

1

{

J4”~’dz.~

F.~(w)I2

~

1

~2

(/

+

1~2~1~/+i[5IE/+

2

~Et

—

i~J)F~(~)

(22)

5EI gives the correction to the transverse term, the term In the above equation, term containing containing 5~gives the the correction to the longitudinal term and the terms containing ~11E/ and J~ are corrections due to the interference. The transverse electric form factors are taken to be

NEI

F.~(q)=

2)

2

(23)

(a +q2) 4(1+C2q

and F~2(q)=

N~ 2q (a +q

)

(24)

296

/

P. Durgapal, D. S. Onley

where NE/ are normalization constants, C2

the same as eq. (11). For El transitions

~

5E1

(z~

+2(P~2 _p2)[(~+~)~v

+

(~

where ‘~E1’ 8’El’ follows: ~E1

is

Virtual photon spectrum in Born approximation

=

—

~)~E1(~min)

—

(~

+

) —

(~

~fr~i(~max)

~‘)s~El(zimax)]

-

+4~(2~~ — ~El)

—(~E1 +~E1)}~

(25)

etc. can be written in terms of the integrals defined by the set of eqs. (12)—(l5) as

~El’

C 2E~.~(2, w)+[E~, — C2Em(P~ ~,

—

Pfl].~(i,

~,

w)—

Em(P~

—

Pfl~(o,~,~ (26)

~E1(Y)

=~(0,y, ~)

~El(Y)

=

=

+

C2~(l,y,

~),

/1C2~(3,y)+ (‘~+ C2l2)~(2,~) + (/~+ C2/3)~(l,y) + /3~(O,y),

16~(0)+(15 + C2I6)1E(l) +(14

and the quantities

+ C215)2’(2) + C2/4~2’(3),

12, etc. depend on kinematic parameters and are given by

~

2—E~,~ ~ =2L~ /3= _~2[~2E~+2(p~_pfl2],

/ 4=E1[/1—2(P~—Pfl],

/5=El[/2+4L~t2(P~_Pfl1+y212~2(EI+Ep)_Ep(p~_Pfl__2EiE,~,I, 2(P —2 ~2) 2E(p2 P2)1 16 ~ p2)1 + E~~ 1 —

(27)

—

=

The constituent integrals of 51E1 are similar in nature to those of ~E1’ 2+ y2)]D(1) + C 2(~ 2~(2), 2).~(1, w)+ C c.~)+(l+ C2~ 2~(2,

+ y2)~(O)

~1E

=

(~2

~1E

=

~2,~-(O

+[i

+

Thus, if we define

C

~,

~,

~,

w),

(28)

+IEm+ C2t2Ep(P~_Pfl_Em(L~2+y2)}1.~(l,y,W)+C2Em~(2,Y,to), i32’(O) + (i2 + C2i3).2’(l) + (i1 + C2i2)2’(2) + i1C2~~(3),

~‘1E

=

i1

EI(E~ + 2E1),

with =

~2=

~

~

(29)

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Virtual photon spectrum in Born approximation

297

then the integral 51E1 can be expressed as 1E 5tEI

2

~

Ei’)

(E2

~

E1~

(E 2

E1)~ max)~

2EmEp 1 ~ (2~IE_~IE)-~°IE+~IEj.

(30)

The corresponding quantities for the E2 transition are given in the appendix. In order to write an explicit expression for the integral .fl~and 5~ we need to define additional integrals 2k—i

~(k)=f~dq

q 0

4

(a2+q2) q2k_i

.~(k,y)=J

/ Ini\(q+~)2+y2)’ (q_~)2±y2) ln~—’~

dq 0

(31) (32)

Iq+yI’

(a2+q2)4

.00

dqq2~

(q2_~2)v

_____________ lnl 4~P~q4+uq2+v

0

2+y2v1+2q~p~q4+uq2+v _____________ .

(q2_~2)/

(a2+q2)

(33)

_2q~/P~q4+uq2+v

2+y2r~

In terms of these integrals, if we introduce the notation 9~:(y)=~(0,y) + C ‘ 2.~(l,y), +

(34) ~c(Y)

~

=

Y, z.~)

—(/7

+

C2/8)..~(l,y,zl) +(1 — C2/7)..~(2,y,z~)+ C2~(3,y,~),

with l7=2(E~_L~2), /82(2E~._~2), 2, lO=E l9=Ei/7~EmY

(35)

1l8+y2(2E1Ep2_E~I~2)+E2y4.

Then the expression for 5~is given by 1E 2 jcl

=NEi[Y2{(~_~i)3

1E2 -

~

E1 _~)~C(~min)

E1~

2E ~

m~n)_~+~)3c(L1max)J

+

(~+

~)1c(z~max)

-(~~

+5~)].

(36)

The expression for~ can also be written in terms of the integrals defined by relations (31)—(33). Thus, if we introduce the notation =

~ic(Y)

9(o) + c29(l),

2) + Em(L~2+ y2)] .~j0, y) [2E~(P~ P2 +1_Em + C 2 y2)}J~(1,y)_ 2(2E~(P~ pfl +Em(~ =

—

—

—

=

i5Z~(0)+(i4 + C2i5)s9~(1)+(E1Em

+

C2i4)Zj2)

+

C 2Em~(2,Y),

(37)

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Virtual photon spectrum in Born approximation

with i

38

4=_[2Ei{2Ei~2+Ep(Pi2_pn}+y2(21.~2_E,~jI,

(

~

then 5ici

=

~

—

(~ —

+4EmEp~(2~(~) ~

~)~(~min)}

~

—~ic].

(39)

The expressions for 5C2 and 51C2 are given in the appendix. The integrals over the intermediate momentum transfer q, defined by the eqs. (12)—(15), and (31)—(33) are evaluated analytically yielding exact values for the integrals ~ and 5~ where T = M, E or C. The integrals over the physical momentum transfer in eqs. (4), (5), (21) and (22) are carried out numerically using Simpson’s rule. There is a simple pole at ~ = w in the integrand of each of the above integrals contributing to the transverse part. The point ~ = ~ lies outside the integration range, but extremely close to the lower limit ~ = ~mjfl’ [~min w(1 + m~/E 1E2)], and in order to avoid problems in the numerical integration we subtract the pole contribution as follows. Suppose the term with singularity at ~ = w is fa~d~

h(~)

(40)

,

where h(z~)is an analytic function of

~.

Then we numerically integrate the non-singular function

~

(41)

between the limits ~ mm and ~ max’ and add in the explicit expression for the integral

J

~.

h(~) 0)

2

22

h(w) 1 nA

~max~0)

~max

—

~min~ mm

0) —

‘

()

4. Program structure The program SOVPS evaluates the virtual photon spectrum for magnetic dipole, electric dipole and electric quadrupole transitions in second-order Born approximation over a much wider range of energies than has been formerly possible. This program can be used for either sign of the charges and for both electron and muon mass. The system of units used has h = c = 1, and all energy and mass parameters are to be specified in units of MeV. A flow diagram of the program is given in fig. 1. The program is written in double precision. Quantities in common

Information is transmitted among the various parts of the program by the following COMMON variables.

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Virtual photon spectrum in Born approximation

299

MAIN

IM1~~

IELSPECI

______

(DIFE0M~-

_______

DINTM1

LDINTEI

I ______________________

I

I

I

__________

TELEC1~MDGAMJIELECT2! IPRVALT!

I

I

IDINTE2H ~ ~F L~ADDINT~

1

I

1

[ExAcr3I LTDINTILCOUL1J [~~AMJ IRESIDI rRTSUMI

I

I

I

I~BRANCHI (CHKPI LAFOUR1I ~AFOURCI Fig. 1. Flowchart for the programs SOVPS.

COMMON names

Variables

Meaning

PE

Ph, P2, El, E2, EME

VAR

QMN, QMX, EM, EP

CONST

ALFA, ALFAZ, W, WMN, P1, L. MORE

OMEGA

FW, SOC, FWSQ, DELIW, DIW

YUKAWA

GAM, K

PARAM

P1SQ, P2SQ, PM, PMSQ, GAMSQ, GRT, ASQ, WSQ

EQTN

U(3), DU(3)

TEST

TH1, TH2, THE1, THE2, TH3, TH4, THE3, THE4, THW3, THW4, THA3, THA4,

Momenta and energy for the initial and final lepton states plus the value of electron mass Minimum and maximum value of the momentum transferred to the target nucleus. Difference and sum of initial and final energies Fine structure constant a, aZ, photon energy, parameters appearing in the nuclear form factor, minimum photon energy for which the virtual photon spectrum is calculated. Values of ‘rr, multipolarity and the nature of the transition Value of form factor evaluated at ~ = w, constant factor for the correction term, square of the form factor, values of the integrand at ~ = w (the values of various quantities at ~ = w are needed for subtracting the simple pole at ~ = w) Yukawa parameter, integer giving the value of k in eqs. (12)—(15) and (31)—(35) Quantities based on kinematic parameters used frequently in different subroutines (defined in MAIN) Quantities used in the numerical integration, DU is the integrand and U the integral In the evaluation of integrals (14) and (33) several terms involving acrtangent appear in the result (see ref. [1]). The arctan routine returns a value in

EPS, AA,

PP,

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P. Durgapal, D.S. Onley

COMMON names

/

Virtual photon spectrum in Born approximation

Variables

Meaning

THWE3, THWE4, THAE3, THAE4, THW1, THW2, THAi, THA2, THWE1, THWE2, THAE1, THAE2, THO1, THO2, THO3, THO4, THOE1, THOE2, THOE3, THOE4

the range, —‘c to ‘r, but not necessarily the appropriate value. In subroutine CHKPI the numerical values of these terms are checked for each value of the photon energy and momentum transfer so that continuity in its value is maintamed. This is achieved with the help of the variables defined here

Input to SOVPS

There are two READ statements in the program and both of them appear in MAIN. They are READ(5, 1000) READ(5, 2000)

A, Z, L, MORE Tl, WW

The corresponding FORMAT statements are 1000 2000

FORMAT (2F10.l, 215) FORMAT (2F10.O)

The meaning of these quantities is as follows: A Mass number of the nucleus; Z Charge number of the nucleus; L Multipole order; MORE A parameter specifying the character of the multipole, magnetic (MORE = 0) or electric (MORE =1); Tl Initial kinetic energy of the electron; WW Virtual photon energy. The value of electronic mass, EME = 0.511 MeV, appears in MAIN and can be changed to 105.7 MeV for the muon mass. After reading the first card, values of nuclear parameters such as the root-mean-square radius Rrms, the transition radius Rtr, etc. are calculated and printed out (units are changed to fm for this purpose). The program returns repeatedly to the second READ statement and a sequence of values of electron energies and virtual photon energies may thus be read. A negative value of electron energy returns the program to the first READ statement: a zero value is used for normal termination.

List of subroutines Name

Comments

M1SPEC (TM1, PW, VPS)

Sets up program for calculating magnetic dipole virtual photon spectra Sets up program for calculating electric dipole or quadrupole spectra Numerical integration performed using Simpson’s rule

ELSPEC (PTVPS, FIRST, SUM) SIMP (X, H, N, IN)

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Name

/

Virtual photon spectrum in Born approximation

301

Comments

DIFEQM (D)

Sets up non-singular function of the type defined by eq. (41) for evaluating second-order spectrum for Ml transitions DIFEQE (D) Sets up functions for evaluating transverse parts of El and E2 spectra as in DIFEQM and also the Coulomb part (which has no singularity) to be used in subroutine DIFEQM DINTM1 (D, DIQ) Evaluates integral DINTE1 (D, DIQ1, DIQ2) Evaluates integral ~ ~ 5!EI~ ~ to be used in subroutine DINTE2 (D, DIQ1, DIQ2) DIFEQE (I = 1 for El and / = 2 for E2 transitions) TELEC1 (QM, DINTO, DINT1, DINT2) Evaluates integrals.~(k,y, w) for k = 0, 1, 2 QMDGAM (D, GAMO, GAM1, GAM2) Evaluates integrals 9(k) for k = 0, 1, 2 ELEC2 (D, QM) Evaluates ~(k, y) for k = 0, 1, 2, 3, 4 PRVALT (D, QM, DIQM) Evaluates ~M(Y) and -~n,(y)’ / = 1, 2 EXACT3 (D, P1, P2, El, Calculates integrals 2’(k), ~2~(k) for k = 0, 1, 2, 3, 4 E2, DDI, CDI) BRANCH (D, AL, BR1, BR2, The contribution due to the branch cuts in the integrals ~9(k) BR3, BR4, P1, P2, El, E2) and Z~(k)are calculated CHKP1 (D, AL, TANG, TH1, This subprogram evaluates the value of the arctan terms apTH2, TH3, TH4) pearing in 2’(k) and ~(k), in the correct quadrant by adding or subtracting ‘rr as necessary to maintain continuity AFOURT (A0, Al, A2, A3) Calculates 1M

‘

d” ~

AFOURC (A0, Al, A2, A3)

(w2z2)(z+ai)4

for n = 0, 1, 2, 3 and k = evaluation of ~9(k) Calculates d” dz’7

FORM (Q, FF) TDINT (D, P1, P2, El, E2, QIA, EMDIA, DI) COUL1 (QM, DINTO, DINT1, DINT2) EXGAM (D, GAMO, GAM1, GAM2) RESID (D, QM, DIQM) RTSUM (D, P1, P2, El, E2, DINT, CDIA, CMDIA) CONVPS (TN, CN)

z=ai

0, 1, 2, 3,

4. These are needed in the

z2k~

(z+ai)4

z=ai

for n = 0, 1, 2, 3 and k = 0, 1, 2, 3. These are needed in the evaluation of ~ ( k) Calculates form factor for Ml, El and E2 transitions Evaluates ~“E/’ ~°IE/ and their interchanged terms 5~’M’

Calculates 3~(k,y) for k = 0, 1, 2 [eq. (32)] Calculates 9(k) for k = 0, 1, 2 [eq. (31)] Calculates ~ / = 1, 2 Calculates ~ and the corresponding interchanged terms, / = 1, 2 Calculates the conventional virtual photon spectrum for Ml, El and E2 transitions ~

302 5.

P. Durgapal, D.S. Onley

/

Virtual photon spectrum in Born approximation

Program output

The output of the program consists of a heading for identification of the multipole, the mass number and charge of the nucleus for which the run is made. The root-mean-square and transition radii are printed (in fm). Finally, the incident electron energy E1, the photon energy w, the conventional virtual photon spectrum (VPS in LWL), the first-order spectrum including finite size (VPS in PWBA) and the spectrum in second order Born approximation (VPS in SOBA) are printed out in five different columns.

6. Description of test run As a test run we have calculated the El and E2 spectra for ~Ca for incident electron energy E1 = 100 Mev and virtual photon energies w = 1—10 in steps of I MeV, w = 10—90 in steps of 10 MeV and = 90—99 in steps of 1 MeV.

Appendix The expression the E2 photon be obtained eq. (22) by putting I = of 2. The 5E2’for 51E2’ 5C2 virtual and 51C2 can spectrum be writtencan in terms of the from integrals defined by the set eqs. expressions for (12)—(l5) and (32)—(34).

5E2

+

‘~E2(~max) + 2~E2(~) ~9E2

~E2(~min)

+~E

(~ —

)~E2(~min)

—

(~ +

9~E 2+~

(Al)

2+K7JE,

where E2

E1

—

—±—

~E2(Y)4(’~11’2)

(A.2) (A.3) 9E2

= 8~[E~9(2)

+ E~y29(l) — Em(L~2+

y2)(P 1~

—

~E2

=

Pfl9(o)],

(A.4) (A.5)

2[K1P1(3,y) + K2~(2,y) + K3.~(l,y)+ K4.9(0,y)],

(A.6) 2+0)2) l6a6 8a4(a2+w2) 2a2(a2+o,2)2 (a2+0)2)3 _____ 5 3 1 1 .1E = 2a(a + + (A.7) In eq. (A.2), we use the upper signs for y = mm and lower signs for y = max The same will apply in eqs. (A.9) and (A.15). The constants appearing in eqs. (A.1)—(A.7) are —+

~

.

P. Durgapal, D. S. Onley

/

Virtual photon spectrum in Born approximation

303

2(E C2=~f_~2((Pj~_Pfl(E2E~+ 2E~P~)+4E1E~P~} +(P~—P~) 2E~+4E1P~) +2E2E~P~y2I,

K1

= ~2

K2 K 3

= ~

—

2 — 2(P~— ~fl2 + 4L12(P~+ —

=z14{zX2

K4 =

Pfl,

3E,~ —

3E,~+ 4(P~+ +

_~4[E~2

~J3=E1[—K1

2(P~

—

~fl1,

p2)2]

+2(P~—Pfl],

2{E~ 2(P~ —

+

(‘4 = E1[_2~

—

3~

nfl) — 8~2P~+ 2(P~

+ 5Ei)+6EmE~

— p2)2]

+

E

2

—

2(P1

~flj,

~ +

4E

Pfl+ 3EIE,~j, 2] +y2z14[E Pfl} +2(P~ Pfl 2(P~

1)+2E2(P~

—

C6 =

E1~4[~2(E~ +

2(P~

—

—

—

Pfl+6EmEfl

6(EiEm~P~+~fl,

—Pfl+3E1E~} —2E1(P~_p2)2] +Emy

+y4[~2{2E2(P~

E 2 2~ 1(7= —4E,~y The contribution of the pure Coulomb term is

—2E~[2~~

~

2—9~2] —

~

+

C2(~min)_(~

~)~C2(~max)]

+~(~2

+.~2)+K10J~,

(A.8)

where we have defined = (K8 ~

~2(y)

K8)~(l,y) + (K9 2.~(l,~)+ ~

= 3F~(2, z~)+ = ~C2(Y)

=

+

K9)~(0,y), ~),

2~

3D(2) + 2(~2 + 3y2)D(l) + 3(%~2+ y2)2D(o) ~

The constants appearing in eqs. (A.8)—(A.l3) are: 2+6E~, C 7= —5~ C9=3~4(_42+2E~),

(A.ll) (A.12) (A.13)

S’rr/32a7.

C

(A.9) (A.lO)

8=z12(_M2+4E~), 2(Ei+Em), Cio=~EiC7+3y

304

P. Durgapal, D.S. Onley

/

Virtual photon spectrum in Born approximation

C11= —E1C8+2~2y2(6E1—E2)—l2E1E~y2+3y4(E3—2E2), C12=_E1C9+3y2[~4(2E1_E2)_4E1E~2+~2y2(E1_2E2)_2E1E~y2_E2y4},

_2EmP221,

K8=~?~3[E2{Et2_(Pi2_P22)}

2+4E ~

+E2(P~_Pfl

2(p2p2)} K10=6~~2[_

1EP~2+

2E2y2P2~],

~~2~(p~_p

~

+4Ep].

22))

can be expressed in terms of the same integrals

5!E2

Of as the two terms contributing to the interference term, 4E E 51E2

~E2(~m~n)

~E2(~max)

~ [—2~~~ +

+

~IE2]

1E

2Ev ~~+p 2 E1

~1E2(~min)~p

1E2

+

‘E

-

+ KI

+~‘IE2 ~IE2

3JE + 2

~E2(Y)

=

p

~

E

)~LE2(~min)

—

1

c)~(2~~.

4(E2 ±

~IE2C~max)

E~ 2

with

E1

0))+y2(K1i

E p

~

2

~tE2(~max)

(A.l4)

‘

1

2(K ±~11)~(1,y,

0))

+y

12 ±k12)~(0,y,

0)),

(A.15) ~1E2

=~(2,y, w) + ~

91E2 =9(2) + 2(L~2+

y2)9(1)

w),

(A.16)

—2E~(P 2—Pfl.~(1,y, W)+~2[Em~2_ 1

~~E2(y)

(A.l7) (A.l8)

+ (~2 + y2)29(o)

2E~(P~—Pfl].~(0,y, 0)),

(A.19)

‘971E2 = C 13~’(3) +

C14~(2)+ c152(l)

+

C162’(O).

(A.20)

The constants appearing in eqs. (A.14)—(A.20): C13 = 2E1(3E1 + E2), C14= _2L~2(EIEm+2y2)+4EiEp(PI2_P~)y2(3+8EiEp+Pi~_P2E~), 4(3E 2y2(~2 +E C15 = — 2E1z~ 1+ E2) — 4~ 2Em) +y4[4E1E~ + 3E~— P~+ P~— 2(3E,~.+ P~— ~fl] C16 = 2EiEmL%~+z~4[—4E1E~(P~ — ~ +-y ~

K 11 ~

~21,

P. Durgapal, D.S. Onley

/

305

Virtual photon spectrum in Born approximation

~ K13

p2

2 =

(~2

— ~

+

—

p2

(~2

P2 1

+

4EmY

+ 4Ev].

—

The second contribution to the interference term comes from 51C2 which can be expressed in terms of the same integrals as 5C2 ~IC2(’~max)l

51C2 = ~y2I.4~IC2(~min)

+

8EmEpL~[25~c 2(Z~)~IC2l

—

)~IC2(~min)

—

+ ~~IC2(~max)] 92IC2~~92IC2+

_4Ep~21(~ +~)~C2(~min)_(~

—

~)~C2(~max)1

K 16J~, (A.21)

+

where ~‘IC2(Y)

=

(K14 ±k14).3l~(l,y)+ (K15

± K15).~(0,y),

(A.22)

(A.23) (A.24) ~

~1C2(Y)

(A.25) = C17~~(3) +

C18..2~(2)+ C19~(l) + C20z~(0).

(A.26)

The constants appearing in eqs. (A.21)—(A.26) are 2— E,~)I, K14 =_~[E2Em(~2_P~+P~)+2P~(2~ K 15= ~

‘~16= _4EmY2[~(L12

—

p~+ pfl+.~-~(z~~ + p~ —

2EiEm~

C17 = C 18 = C 19 =

2(3E,~ +P~—Pfl, _2z12[E1(3E1

+E2)+2y2J

—4E1E~(P~ Pfl+y —

2(P

2) ~442y2(E ~24’~(EiEm

+ 2y

2Em

+

~2)

—

8E1E~y

1~ F~)+ y~(3E~, P~+ —

—

Pfl,

~ 6. +y4[z12(4E1E~ + 3E~,— P~+ p22) —4E1E~(P~ — P1)1 ~2E2Emy

In all the above equations the quantities with the ‘bar’ on top correspond to the interchanged term. For example, SE 2 is obtained from SE2 by making the interchanges given by eq. (9) of the main text.

306

P. Durgapal, D.S. Onley

/

Virtual photon spectrum in Born approximation

References [1) P. Durgapal and D.S. Onley, Phys. Rev. C27 (1983) 523. [2] D.S. Onley, Bull. Am. Phys. Soc. 26 (1981) 1129. [3] L.E. Wright and C.W. Soto Vargas, Comput. Phys. Commun. 20 (1980) 337. [41J.A. Thie, C.J. Mullin and E. Guth, Phys. Rev. 87 (1952) 962. (5] W.W. Gargaro and D.S. Onley, Phys. Rev. C4 (1971) 1032. [61P. Durgapal, Ph.D. Dissertation, Ohio University, 1982 (unpublished). [7] iD. Bjorken and S.D. Drell, Relativistic Quantum Mechanics (McGraw-Hill, New York, 1965).

P. Durgapal, D.S. Onley

/

Virtual photon spectrum in Born approximation

TEST RUN OUTPUT SOVPS:— VIRrUAL PHI)TON SPECTRUM EVALUATED USING SECOND ORDER BORN APPROXIMATION. ELECTRIC TRANSITION OF MULTIPOLARITY L = 1 ELECTRON MASS =0.5110 MEV MASS ImUMEIER A = 40.0 RMS RADIUS = 3.42 FERMI ~.E. OF ELECTRON

CHARGE 7 r 20.0 TRANS. RADIUS

=

CAM = 141.1 MEV 4.35 FERMI

AA

PHOTON ENERGY

VPS IN LWL

VPS IN PWBA

VPS IN SOBA

100.0 100.0

1.00 2.00

4.4042040—02 4.0404120—02

4.3811340—02 3.8377940—02

4.4175680—02 3.8736660—02

100.0

3.00

3.6131150—02

3.5776430—02

3.6l31160—02

100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0

4.00 5.00 o.00 7.00 3.00 9.00 10.00 20.00 30.00 40.00 50.00 60.00 70.00 80.30 90.00 91.00 92.00 93.00 94.00 95.00 90.00 97.00 98.00 99.00

3.6425170—02 3.5334650—02 3.3946940—02 3.2301590—02 3.1962340—02 3.1005640—02 3.0215333—02 2.4372330—02 2.0411270—02 1.7421,650—02 1.5073240—02 l.319987D—02 1.1640270—02 1.0214870—02 8.5006540—03 8.2697850—03 8.0137650—03 7.7289920—03 7.4050170—03 7.0276220—03 6.5739790—03 6.0030500—03 5.2309330—03 4.0271990—03

3.3993110—02 3.2596030—02 3.1427010—02 3.0410610—02 2.9504940—02 2.B~83770—02 2.7929300—02 2.2435680—02 1.8773300—02 1.6035720—02 1.3907380—02 1.2220130—02 1.0844880—02 9.6197250—03 8.15141,40—03 7.9430650—03 7.7224550—03 7.4680300—03 7.1753980—03 6.8304030—03 6.4101350—03 5.3741190—03 5.1381950—03 3.9726960—03

3.434462D—02 3.2944680—02 3.1772950—02 3.0753950—02 2.9845540—02 2.9021650—02 2.8264880—02 2.2738220—02 1.0035950—02 1.6253470—02 1.4077220—02 1.2341060—02 1.0°1856D—02 9.6522230—03 8.1572300—03 7.952715D—03 7.7264120—03 7.4719050—03 1.1800390—03 6.8371000—03 6.4211780—03 5.8939520—03 5.1784570—03 4.0763670—03

=

263.9 ‘1EV

SOVPS:— VIREUAL °HOTIIN SPECTRUM EVALUATED USING SECOND ORDER 8DRN APPROXIMATION. ELECTRIC TRANSITION OF MULTIPOLARITY L = 2 ELECTRON MASS =0.5110 HEy MASS NUMBER A = 40.0 RMS RADIUS = 3.42 FERMI K.E. OF ELECTRON

CHARGE 7 = 20.0 TRANS. RADIUS

=

CAM 3.97

141.1 FERMI

MEV

AA

PHOTON ENERGY

VPS IN LWL

VPS IN PWBA

VPS IN SOBA

100.0

1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00 9.00 10.00 20.00 30.00 40.00 50.00 60.00 70.00 80.00 90.00 91.00 92.00 93.00 94.00 95.00 96.00 91.00

6.1385990 01 1.5072650 01 6.5867010 00 3.6469080 00 2.2996620 00 1.5751120 00 1.142504000 8.6444070—01 6.7562393—01 5.418386D—01 1.2346930—01 5.7887220—02 3.4392760—02 2.3733330—02 1.7895120—02 1.4226370—02 1.1569120—02 9.0698390—03 8.7748803—03 8.4539290—03 8.1152490—03 7.7342650—03 7.3016290—03 6.7942790—03 6.1713150—03

2.2275853 01 5.5237770 00 2.4417600 00 1.3695220 00 8.7601110—01 6.0933250—01 4.4931140—01 3.4591130—01 2.7530380—01 2.2497730—01 0.5239490—02 3.5188810—02 2.3945170—02 1.8204430—02 1.4717160—02 1.2335250—02 1.0480930—02 8.5603290—03 8.3166360—03 8.0511800—03 7.7573170—03 7.4255630—03 7.0416670—03 6.5826730—03 6.0078190—03

100.0

98.00

5.3482710—03

5.2328813—03

10u.O

99.00

4.0940700—03

4.0275120—03

2.5227340 01 6.241376D 00 2.7560900 00 1.5420730 00 9.8375990—01 6.8239650—01 5.0180920—01 3.8530530—01 3.0569129—01 2.4939710—01 1.1499390—02 3.8550720—02 2.6157530—02 1.969794D—02 1.5680680—02 1.2898000—02 1.0755020—02 8.6526190—33 8.3966720—03 8.1201440—03 7.9164990—03 7.4764630—03 1.O96196D—03 6.b236560—03 6.0503650—03 5.2891440—03 4.1411630—03

100.0 100.0 100.0 100.0 100.0 100.3 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0

=

263.0 MEV

307