'W^ INSr. OF STAND & TECH

NBSIR 78-145(0

AlllOfc 17107']

A 11 I Q 1 ~7 E b 5 fl b

Comments on the Analysis of

Total Photoabsorption

Measurements in the Energy

Range 10 - 1 50 MeV

L. C. Maximon

Center for Radiation Research

National Measurement Laboratory

National Bureau of Standards

Washington, D C. 20234

and

H. A. Gimm

Max-Planck-lnstitute for Chemistry

Nuclear Physics Division

D-6500 Mainz, W. Germany

June 1978

U S. DEPARTMENT OF COMMERCE

“QC —

100

. U56

73-1456

NATIONAL BUREAU OF STANDARDS

ICaffonal Bursas ef Stasik mr i -i 1979

NBSift 78-1456

COMMENTS ON THE ANALYSIS OF

TOTAL PHOTOABSORPTION

MEASUREMENTS IN THE ENERGY

RANGE 10-150 MeV

L. C. iVSaximon

Center for Radiation Research

National Measurement Laboratory

National Bureau of Standards

Washington, D.C. 20234

and

H. A. Gimm

Max-Planck-lnstitute for Chemistry

Nuclear Physics Division

D-6500 Mainz, W. Germany

June 1 978

U.S. DEPARTMENT OF COMMERCE, Juarsita M. Kreps.

Dr. Sidney hatrmam, Under Secretary

Jordan J. Baruch, Assistant Secretary for Science and Technology

NATSONAl BUREAU OF STANDARDS, Ernest Amio>a©r„ Director

4aIJLAQM/Y

ir I

Abstract

This note deals with details of the procedure used to extract photonuclear

cross sections from total photon absorption measurements. We examine closely

some of the approximations implicit in the expressions available for the

purely electromagnetic cross sections, most especially pair production, triplet

production and the Compton effect, which must be subtracted from the measured

photoabsorption in order to obtain photonuclear cross sections. We single out

those aspects of the expressions for these electromagnetic processes which

most warrant further theoretical research in that they constitute the principal

source of uncertainty in the extraction of nuclear data.

Key words: Atomic screening corrections; Coulomb corrections; pair production;

photonuclear data; total photon absorption cross section; triplet production.

1

The recent report of Gimm and Hubbell [1] presents details of the

procedure used to extract photonuclear cross sections from the total

photon absorption measurements made at the MPI in Mainz by Ahrens et al

[2]. This note is intended as a supplement to that report. It is our

primary intention here to examine more closely some of the approximations

implicit in the expressions available for the purely electromagnetic cross

sections, most especially pair production, triplet production and the

Compton effect, which must be subtracted from the measured photoabsorption

in order to obtain photonuclear cross sections. Furthermore, we wish

thereby to single out those aspects of the expressions for these electromagnetic processes which most warrant further theoretical research in that

they constitute the principal source of uncertainty in the extraction of

nuclear data.

We first present a brief resume of the salient aspects of the processes

entering these measurements. The total cross section for photon absorption

has been measured in a series of elements ranging from Li to Pb , in the

energy range 10 MeV to 150 MeV . The results for light elements (Li to Ca)

are presented in [2]. More recent, as yet unpublished, measurements have

been made for heavier elements, viz., Cu, Sn, Ta and Pb. (In the case of

Li and Be, the measurements go to 220 MeV, and for Pb to 160 MeV, but the

bulk of the measurements are in the range 10 150 MeV.) Although the

primary purpose of these measurements is to obtain photonuclear cross

sections, they constitute at the most only 5% of the total cross section

(this in the region of the giant dipole resonance, where they are largest:

-- for other energies they are even smaller). Thus, depending on the

particular energy, between 95 and 100% of the total photoabsorption cross

1
section is due to atomic processes: Compton effect, pair production (in

the field of the nucleus), triplet production (pair production in the

field of the atomic electrons) and photoelectric effect. The cross

sections for these electromagnetic processes, which are subtracted from

the total measured cross section to obtain the photonuclear cross section,

must, therefore, be known with great accuracy if useful nuclear data are

to be obtained. In those regions where the photonuclear cross section is

zero or negligible, these measurements constitute above all a very accurate

determination of the atomic cross sections. We will return to this point

in our final comment at the end of this note. This is the case, for

example, for heavy elements at high energies (Z ^ 50 and 100 - 150 MeV)

,

and for all elements below particle threshold. This then suggests a

number of questions, which are the starting point for these comments.

1) For those regions in which the photonuclear cross section is not

negligible, what are the errors in the available expressions for the

atomic cross sections, and what limits do these place on the accuracy to

which the nuclear cross sections may be known.

2) In those regions in which the photonuclear cross sections are

negligible, does the accuracy of the present measurements permit one to

clarify or resolve any of the uncertainties in the theoretical expressions

for the cross sections for the atomic processes.

3) Are there, currently available, theoretical expressions for the

atomic cross sections of greater accuracy than those employed in the

analysis described in [1].

4) In what direction do these absorption measurements suggest further

research that would improve upon the currently available theoretical

expressions for the atomic cross sections.

2
W< start by presenting some figures and tables that contain the

essentials for our comments. In figs. 1-4 we show curves of the three

major atomic processes occurring here: pair production, Compton effect,

and triplet production, as well as the total atomic cross section, as a

function of the photon energy, for four elements: Li, Ca, Sn, and Pb

(Z = 3, 20, 50, and 82). Table I gives, for these same elements, at each

of three different energies, the cross sections for each of these three

atomic processes as percentages of the total absorption cross section

(o ). Table II gives, for the same elements and energies as in Table I, tot

the screening and Coulomb corrections, again as percentages of a

The photoelectric cross section has been included in the actual analysis,

but is not shown in these figures and tables since it is always less than

1 % of C

tot

Next we note the essential formulae used in the analysis:

anuc 1 O -(0,4-0 + 0 4- 0 ) tot pn c t p (1)

o

p (a

BH

- AS - AC)

rad ( 2 )

Hero

nuc 1

0

tot

0

c

°m- as -

photonuciear cross section

total measured photoabsorption cross section

0 , O are the cross sections for the atomic processes:

t P photoelectric, Compton, triplet, pair production,

respectively

.

AC and f^ all refer to the pair production cross

section, o : P

3
o is the Born approximation (Bethe-Heitler) cross section for an BH

unscreened (point Coulomb) potential.

AS is the Born approximation screening correction,

(AS = a - a , , both in Born approximation,

screened unscreened

AC is the Coulomb correction in the absence of screening,

(AC = O . , _ - ,

. - 0DU , both for an with Coulomb corrections BH

unscreened, point Coulomb potential).

f , is the radiative correction to the pair production cross section

rad

As is clear from the figures and tables, the screening and Coulomb

corrections, AS and AC, are of particular significance for heavy

elements (Z ^ 50) , and for these the absorption is dominated by the

pair production cross section. Thus, if we wish to use these measurements

as a check on the theoretical expressions for the screening and Coulomb

corrections, we must consider heavy elements. Put otherwise: We don't

learn anything new about the pair production cross section from measurements on the light elements (Z ^ 20). For these, the Coulomb correction

is very small and the screening correction may be calculated to sufficient

accuracy using form factors derived from non-relativistic charge distributions since the difference between the screening correction obtained

from relativistic and non-relativistic charge distributions is within

the experimental error of these measurements (~ 0.1%) for light elements.

Next, we note from the expressions (1) and (2) used to obtain the

photonuclear cross sections that the screening and Coulomb corrections

appear together as AS + AC. Thus, for a given element and at a given

energy, we cannot distinguish, from the measurements, errors in the

4
screening correction from those in the Coulomb correction. Although in

principal these corrections might be separated experimentally by a series

of measurements since they differ both in their energy dependence and in

their Z dependence, a much greater experimental accuracy would be

required for such a separation. Moreover, such a separation, in which

these two corrections are simply added, is itself an approximation which

is not fully justified at the level of accuracy aspired to in these measurements. We will comment on this point later in greater detail.

We now examine in some detail a number of the approximations implicit

in the theoretical expressions used for the atomic processes of importance

here, and consider first the pair production cross section, o , as

given by (1)

.

1) The expression used for the radiative correction to the total

cross section for pair production, f is that given by Mork and Olsen

[3], which is the best available calculation. However, Mork and Olsen

only derive the radiative correction to the first Born approximation cross

section; the question of the radiative correction to the higher Born terms

(the Coulomb correction) is not examined. Since the radiative correction

to the first Born approximation cross section is of the order of 1% and

the Coulomb correction (e.g., for Pb) is of the order of 10%, the term

AC • f r ~ 0.1%. Thus if this term is wrong by a factor 2, this would

introduce an error of the order of 0.1%.

2) Again referring to the radiative correction to pair production,

we note that the calculation of Mork and Olsen uses the Weizsacker-

* Williams method, which is, in general, valid for very high energies, k.

* Throughout, unless otherwise indicated, energies are given in units of

o the electron rest mass, me , and momentum in units of me.

5
In particular, it generally neglects terms of order unity relative to

those of order £n k. One can therefore ask what errors are made by

applying it in the energy range of interest here: 10 - 150 MeV. If the

radiative correction given by this method is itself off by 10 - 20% at the

lower energies, which is not unreasonable, then this would again introduce

errors of 0.1 - 0.2% in the total pair cross section.

3) As we have already noted, the screening and Coulomb corrections,

AS and AC are, in the present analysis, [1], calculated independently

and added to the unscreened Born approximation cross section. The rationale

for this procedure was given sometime ago by Davies, Bethe and Maximon [4],

viz.

,

that for very high energies the Coulomb correction to the total pair

cross section comes from momentum transfers q ~ 1, whereas the screening

correction comes from relatively small momentum transfers, q ~ Y2T'

Moreover, an estimate of the error committed by this separation was also

given there — it was not then of significance, but for the present measurements it is no longer totally negligible. From p. 791, Eq . (41) of [4] we

have the estimate of the error committed by neglecting the Coulomb correci ZT

tions for momentum transfers q ^ 3 = ~12I’ v ^ z, »

3

2 (£n3 2

) « 0.2% of a

p for Pb, (a = Z/137). (3)

This should only be regarded as an estimate, however. A proper calculation

would contain the higher order terms in a screened Coulomb potential

directly. Such an approach was given by Olsen, Maximon and Wergeland [5].

There, however, the two significant regions of momentum transfer, q ~ 1

anc q -r,-, were separated, and wavef unctions and matrix elements appropriate to each of these regions individually were calculated. The problem

6
ol the overlap of Coulomb and screening corrections did not therefore

arise: in the region q the Coulomb corrections did not enter —

h

the matrix elements were shown to be identical to those calculated in Born

approximation in that region. And in the region q ~ 1, there were no

screening corrections — one could use the wavef unctions and matrix

elements calculated for a point Coulomb field. What is required, therefore,

if one is to examine the overlap of screening and Coulomb corrections, are

wavef unctions and matrix elements valid throughout the range of significant

momentum transfers, from q ~ ~ to q ~ 1, approximate only in that

terms giving contributions of relative order

^ to the final cross

section are neglected.

4) The next question concerns the form factors used in calculating

the screening correction, AS. It should be noted that in order to

calculate the total pair production cross section, the range of momentum

transfers for which the form factor must be known is

k + /k'2 -4

T q

" k + /k 2 -4 (4)

Thus for large photon energies, k.

2 „ k

= q 2k (4a

i.e., the form factor must be known both for very small and for very large

momentum transfers, although, as is well-known, the contribution to the

screening correction from q £ 0(1) is very small. The most accurate

form factors available in the literature for this range of momentum

7
transfers are those given by I. 0verb«$ [6], and these have been used for

calculating the screening correction in the present analysis. These form

factors have been calculated using a combination of the results of the

relativistic Hartree-Fock calculations of Liberman, Cromer and Waber [7a] and

of Doyle and Turner [7b]. Details of these calculations are given in [6],

[7a] and [7b], and need not be dwelt upon here. Two observations are of

particular significance for the present analysis, however. The first is that

for heavy elements (e.g., Pb) and at high energies (~ 150 MeV) , the difference

between the screening correction calculated from these relativistic charge

densities and those calculated from non-relativistic charge densities is

approximately 1.2% of the total pair cross section (see fig. 5). This is an

order of magnitude larger than the experimental errors claimed for these

measurements. The question of the errors implicit in the form factors used

here is thus a very important one. In this connection we note first the

comment in [6] (p. 332) that "for light elements, really accurate charge

densities can only be obtained by treating exchange according to the HartreeFock scheme," though for these elements non-relativistic wavefunctions should

be "fairly accurate." The question thus remains of estimating the errors in

the screening correction obtained from the form factors of ref. [6], Although

no "exact" calculation is possible, one can obtain an estimate of these

errors by using charge densities more accurate than those of ref. [7],

which formed the basis of the form factors given by 0verb$ in [6]. A

number of fully relativistic Dirac-Har tree-Fock calculations (in which the

exchange is also treated relativistically) have appeared within the last

few years. Papers relating to two of these should be noted:

8
a) That of J. B. Mann and J. T. Waber [8], on "Self-Consistent

Relativistic Dirac-Hartree-Fock Calculations on Lanthanide Atoms." The

codes for calculating the relativistic DHF wave functions used in that

* paper are available from J. B. Mann and could be used to calculate form

factors and screening corrections.

b) That of J. P. Desclaux [9], on "A Multiconfiguration Relativistic

Dirac-Fock Program." This work is similar to that of Mann and Waber in

that it is also a self-consistent Dirac-Hartree-Fock calculation. In

fact, both calculations are based on the work of I.P. Grant on the

"Relativistic Calculation of Atomic Structures," [10]. The calculation of

Desclaux is probably superior to that of Mann and Waber, however, in that

the latter uses pure jj coupling, which is not true for many atoms, whereas

in the work of Desclaux, an average is taken over all jj configurations

arising from a single LS configuration. The details of this procedure are

given by Desclaux, Mayers and O'Brien [11], and by Desclaux, Moser and

Verhaegen [12]. Theirs seems to be the best available procedure for

calculating the charge densities. It would clearly be of interest to

compare the form factors and screening corrections obtained with the codes

used by these authors with those obtained from the calculation of Mann and

Waber and with that of 0verb^.

5) The next question concerns the Coulomb correction, AC. For the

heavier elements, it is this correction which introduces the greatest

uncertainty in the photonuclear cross sections obtained from these measurements in the manner described here. Indeed, it was already noted in [2]

k

Joseph B. Mann, LASL, P.0. Box 1663, Los Alamos, N.M. 87545 (private

communicat ion)

9
that the "lack of reliable theoretical calculations for the Coulomb correction between 10 and 50 MeV restricts the range in Z available for absolute

o measurements to Z = 20." Specifically, there exist accurate

Y »T calculations of the Coulomb correction (the correction to the unscreened.

Born approximation total cross section for pair production in a point

Coulomb field) in two very separated energy regions. For low photon

energies — from threshold (2 me

2

) to 10 me

2 (5 MeV), extensive, accurate

(having errors of the order of 0.1%) numerical calculations of the Coulomb

correction have been given by 0verb^, Mork and Olsen [13]. (For the most

detailed account of this calculation one should consult the thesis of I.

0verb^ [14].) At the other extreme, for very high energies, the Coulomb

correction has been given by Davies, Bethe and Maximon [4]. Unfortunately,

there is at present no accurate method of determining the errors in their

high-energy-limit expression for the Coulomb correction when applied at

the energies of interest for these experiments — say 10 - 300 MeV.

A conservative limit on the error for an incident photon energy of 75 MeV

would be, for example for Pb, 10% of the Coulomb correction itself, or 1%

of a . This is, however, far from sufficient for the analysis of the

present measurements on heavy elements. In order to bridge the gap

between the region below 10 me

2 and the high energy limit, an empirical

analytic expression given by 0verb^ [15], has been used in the present

analysis. This expression was derived by representing the Coulomb correction by a simple, reasonable, analytic function of Z and k whose form

is chosen so that it goes manifestly into the high energy limit for very

large k, and containing a number of arbitrary parameters (which are

polynomials in Z

2 ), chosen to fit the known [13], [14] calculated values

10
in the low energy region, 3.5 rac^ r k ' 10 me

2

. With six parameters he

is able to fit the values of [13], [14] for all Z in this low energy

region within the errors (~ 0.1% of 0 ) of the values calculated in

P [13], [14]. This analytic expression is then assumed to give the Coulomb

correction for all higher energies (k > 5 MeV) by extrapolation. While

this is not an unreasonable procedure, given the lack of any calculation

of sufficient accuracy for heavy elements in the intermediate energy

region, 10-150 MeV, it does not enable one to make any statement about the

errors that should be associated with these extrapolated values in this

intermediate energy region. To elucidate this point we have repeated

0verb^'s procedure, but have chosen an alternate, equally reasonable

analytic function of Z and k to represent the Coulomb correction, also

of a form such that it goes manifestly into the high energy limit for

very large k, containing eight arbitrary parameters. Specifically, our

alternate analytic form has been chosen to have the same energy dependence

as that of the first few terms of the high energy expansion of the Born

approximation cross section for pair production [16], viz..

11
0 * °BH

+ AC

+ <l l^f - ^f)

^6 ln2k - i + k In 3 2k

2 3 - In 2 2k - ~7T

2 ln2k + 2£(3) +

- (!)' (it ln2k + s)

- (I/ ln2k - 27^12)

(5)

+ a

+ (|) (c^ ln 3 2k + C

2 In2 2k + C

3 ln2k + C

4 )

+ (!) (S ln2k + c

e)

+ \k) (

C

7 ln2k + C(

; t(3) - 1. 2020569,

12
The .itiH'-na* of a term of order ln2k in the Coulomb correction is perhaps

worth noting. It may be understood from the observation that this term

2 comes from the region of very small momentum transfers, q ~ q . = —

.

y hrn.n k

But for these momentum transfers the Born approximation is valid, there is

no Coulomb correction. Thus the term of order ln2k appears only in the

Born approximation. The constant C^ is given by the high energy limit

[4] , viz.

,

a C =

o 28

f (Z)

f(Z) = a

2 £

n=l

. 1 ~ J

n(n 2 +a 2

) (1 + a

2

) -1 + 0.20206 - 0.0369a ;

+ 0.0083a 4 - 0.002a 6 j

( 6 )

The parameters C^, . . . Cg are chosen to be second order polynomials in

a

2 = (Ze 2 /ftc)

2

, as in [15]:

C. = a. + 6.

a

2 + y.a

4 1111 . (i = 1,2, ’ * ...8) (7)

The constants a., 8- and Y. have been chosen to give the best

l i l °

possible least squares fit to the analytic expression [15] in the low

energy region*, within which it never differs from that result by more

In fact, we have made a least squares fit to [15] in the somewhat restricted

region 4 - k i 10 me

2

. Since these are, or represent, asymptotic

expressions for large k, they should not be expected to converge well

at all for very small k. Forcing them to do so would, in our opinion,

tend to determine constants giving a poorer representation of the function

in the intermediate energy region. The maximum difference with the

results of [15] just quoted, viz., 6 x 10 a , is for all Z 82

and for the energy region 4.15 - k

" 10 rac

2 ,^ but for most of this

region the errors are considerably less.

13
than 6 x 10 0 . Thus both in the low energy region and at the high energy

limit, the two expressions give essentially identical values. The numerical

values of the constants a_^, Y_. are given in Table III.

The significant difference between the analytic form used here and that

of [15] is that in this note the factors of the constants and log terms in

2 * AC are inverse powers of k , as in the Born approximation cross section

[16], whereas in [15] they are inverse powers of k. Thus, although the two

expressions agree within the low energy region and go to the same high energy

limit, they may be expected to give somewhat different results in the in

termediate energy region. To show this, is, in fact, the purpose of this

exercise. In particular, the expression for AC presented here will approach

the high energy limit more rapidly than that of [15] . It is not that either

expression has a particularly greater claim to validity in the intermediate

region, but rather that neither can claim to have an uncertainty which is less

than the difference of the values given by these two expressions. In fig. 6

we plot the ratio of the difference between the Coulomb correction calculated

using our analytical expression and that calculated using the expression in

[15] to the Born approximation cross section, G , as a function of photon

energy, for Sn and for Pb. Since this difference is essentially zero in the

low energy region, on the one hand, and at extremely high energies, on the

other, it must either be essentially zero for all intermediate energies or

else have a maximum somewhere in the intermediate region. From fig. 6, it is

seen that there is a maximum in the neighborhood of 20 MeV, and that the

difference there is slightly more than 0.9% of o , or slightly more than 1%

BH

of the unscreened pair cross section, o 4- AC. Since neither expression

Bn

* It is our belief, though this is at present only a conjecture, that the

asymptotic form of the Coulomb correction is also of this form, involving

only inverse powers of k

2

.

14
has, in tact, a basis in theory that would make it more accurate than the

other, this implies an uncertainty of at least 1 / in the pair cross section at

these energies, where, moreover, the photonuclear cross section is largest.

The fact that at higher energies, where the nuclear cross section is

essentially zero, there is agreement between the measurements and the pair

cross section calculated from [15], is not a test of the validity of that

expression in the region about 20 MeV : At higher energies (~ 150 MeV) , the

asymptotic value has almost been attained, and almost all reasonable

expressions will be within a few tenths of a percent of the correct value

there. In any event, one observation is eminently clear from fig. 6, namely,

that the low energy region, 1.75 k 5 MeV, provides far too small a basis

for an extrapolation to the region 10-150 MeV. This is all the more true in

that the Coulomb correction varies quite rapidly in the low energy region, as

may be seen from fig. 7, where the ratio of the Coulomb correction to the Born

approximation cross section is plotted as a function of energy. (Values for

AC are from [14] for k < 10 me

2 and from [15] for k > 10 me". Values for

O are from [16].) On the other hand, above 10 MeV this ratio varies

relatively little. This suggests that an extension of the numerical

calculation of 0verb$ [15] to 20 me

2 might well provide a sufficient basis

for an extrapolation to higher energies. Such an extension should not be of

unsurraountable difficulty, given the improvements in the capabilities of large

scale computers that have occurred within the almost ten years since that last

calculation.

It is worth noting that this same problem arises in connection with

the extraction of nuclear cross sections from the background radiative

tail in high energy electron scattering measurements. There it is possible

15
to determine experimentally the radiative tail in the region near the

elastic peak (below particle threshold) and again well beyond the giant

resonances, where the scattered electron has radiated most of its energy.

Again, a common procedure is to fit the form of the tail in the intermediate

region by an empirical analytic expression that fits the experimental

radiative tail at the two extremes. In those experiments, the nuclear

cross sections are sometimes no more than 5% of the total, so that

uncertainties of the order of a percent in the assumed radiative tail in

the region of the giant resonances (uncertainties which themselves vary

with energy) can contribute large errors in the extracted nuclear cross

sections.

6) Our final comments concern the triplet cross section. This is of

particular importance for light elements. For example, it is 15% of the

total photoabsorption cross section for Li at 100 MeV. Our comments here

concern in particular the screening correction as calculated in the present

analysis (p. 10, Eq. (37) in [1]). In [1], the recoil momentum distribution

for pair production in the field of the nucleus has been used (i.e., the

target is assumed to be infinitely heavy). Since the target is, for

triplet production, an electron, this procedure is somewhat in error,

though only slightly so, since, as was shown by Suh and Bethe [17], the

recoil momentum distribution is essentially independent of the target mass

for small momentum transfers, q << 1 , i.e., for momentum transfers of

importance for the screening correction. This error is then corrected

(Eq. (39) in [1]) by multiplying the screening correction by a factor,

f , taken to be the ratio of the unscreened triplet cross section to

the unscreened Bethe-Heitler pair cross section.

lb
A more correct procedure would be simply to use the recoil distribution proper to triplet production. This has been given by Borsellino [18].

His expression neglects only the y-e and exchange diagrams, whose

contribution is only of order 1% of the triplet cross section for the

energies of interest here (> 10 MeV) . (See Mork [19], and Haug [20],

for a discussion of this point.) We would therefore suggest that the

screening correction for triplet cross section be calculated using the

recoil momentum distribution given by Borsellino, and that the results

thus obtained be compared with those obtained in [1].

As mentioned in [1], the aim of the total absorption cross section

measurements for high Z elements is to check the nonnuclear cross

sections rather than to perform a precise measurement of the nuclear cross

section. For Z 50, the photonuclear cross section is almost entirely

due to (y,n), (y,2n), ... reactions, and the most accurate measurements

of these photoneutron cross sections are those performed at Illinois,

Livermore, and, most expecially, by Bergere et al at Saclay [21]. The

present total photoabsorption measurements may thus be used in conjunction

with the photoneutron measurements to obtain the best experimental

determination of the atomic cross section:

as was in fact suggested by the group at Mainz several years ago [22]. We

noLe further that for heavy elements the pair production cross section

dominates the nonnuclear cross section (it is over 65% of o for

tot

Z

' 50 and k 10 MeV). Thus, upon subtracting the relatively small

o

a a tot ( 8 )

17
cross sections for Compton, triplet and photoelectric effect (calculated

from theory), we may use the total photoabsorption measurements to obtain

the best experimental determination of the pair production cross section

as a function of photon energy:

0=0

p tot 0(y, n) + O (y , 2n) + O +0 + 0

ph (9)

This may then be compared directly with the best currently available

theoretical determination, given recently by 0verb^ [23]
References

[1] H. A. Gimm and J. H. Hubbell, Total Photon Cross Section Measurements,

Theoretical Analysis and Evaluations for Energies above 10 MeV, NBS

Tech. Note 968 (June 1978)

[2] J. Ahrens, H. Borchert, K. H. Czock, H. B. Eppler, H. Gimm, H. Gundrum,

M. Kroning, P. Riehm, G. Sita Ram, A. Zieger and B. Ziegler, Nucl. Phys

A251 , 479 (1975).

[3] Kjell Mork and Haakon Olsen, Phys. Rev. 140. B1661 (1965); Phys. Rev.

166, 1862 (1968).

[4] H. Davies, H. A. Bethe and L. C. Maximon, Phys. Rev. 9_3, 788 (1954).

[5] Haakon Olsen, L. C. Maximon and Harald Wergeland, Phys. Rev. 106 , 27

(1957).

[6] I. 0verbd, Nuovo Cimento B4Q , 330 (1977); Arkiv for Det Fysiske

Seminar i Trondheim (No. 4, 1978).

[7a] D. A. Liberman, D. T. Cromer and J. T. Waber, Computer Phys. Commun. 2_, 107, (1971).

[7b] P. A. Doyle and P. S. Turner, Acta Cryst. A24 , 390 (1968).

[8] J. B. Mann and J. T. Waber, Atomic Data _5 , 201 (1973).

[9] J. P. Desclaux, Computer Phys. Commun. 9_, 31 (1975).

[10] I. P. Grant, Adv. Physics 19^, 747 (1970).

[11] J. P. Desclaux, D. F. Mayers and F. O'Brien, J. of Physics _B4, 631

(1971).

[12] J. P. Desclaux, C. M. Moser and G. Verhaegen, J. of Physics B4, 296

(1971).

[13] I. 0verb$, K. J. Mork and H. A. Olsen, Phys. Rev. 175 , 1978 (1968);

Phys. Rev. A8, 668 (1973).

[14] Ingjald 0verbd, Ph.D. Thesis, Arkiv for Det Fysiske Seminar i Trondheim

No. 9, 1970.

[15] I. 0verb0, Phys. Lett. 71B , 412 (1977).

[16] L. C. Maximon, J. Res. Natl. Bur. Stand. (U.S.) B72 , 79 (1968).

[17] K. S. Suh and H. A. Bethe, Phys. Rev. 115, 672 (1959).

[18]

A. Borsellino, Nuovo Cimento 4_, 112 (1947); Rev. Univ. Nacl. Tucuman

A6, 7 (1947).

19
[19] K. J. Mork, Phys. Rev. 160 , 1065 (1967); S. Jarp and K. J. Mork,

Phys. Rev. D8, 159 (1973).

[20J E. Haug, Z. Natur forsch. A 30a , 1099 (1975).

[21] B. L. Berman, Atlas of Photoneutron Cross Sections Obtained with

Monoenergetic Photons, Lawrence Livermore Labs, Livermore, Cal.

UCRL-78482 (1976).

[22] J. Ahrens, H. Borchert, H. B. Eppler, H. Gimm, H. Gundrum, P. Riehn,

G. Sita Ram, A. Zieger and B. Ziegler, Tagungsberic’nt ElektronenBeschleuniger-Arbeitsgruppen, Universitat Giessen, AED-Conf .-71-400

(Sept. 23-25, 1971), pp 359-370.

[23] I. 0verb^, Arkiv for Det Fysiske Seminar i Trondheim (No. 10, 1977).

20
TABLE I

Total Cross Sections for Pair Production, Compton Effect,

and Triplet Production as Percentages of the Total Absorption

Cross Section, O , for Li, Ca, Sn and Pb

tot

20 MeV 80 MeV 150 MeV

pp 23 53 61

Li Compton 71 31 19

triplet 5.5 16 20

PP 66 88 91

Ca Compton 31 8 4.5

triplet 2.4 4 4.5

PP 82 94 96

Sn Compton 16.5 3.9 2.1

triplet 1.3 1.8 2.0

pp 87 96 97

Pb Compton 12 3 1.4

triplet .9 1 1.3
TABLE II

Screening and Coulomb Corrections (AS and AC) to the Total Pair Production

Cross Section, as Percentages of the Total Absorption Cross Section,

a . for Li, Ca, Sn, and Pb

tot

Li

AS

AC

Ca

AS

AC

Sn

AS

AC

Pb

AS

AC

20 MeV 80 MeV 150 MeV

1 4

1.5 8 14

.5 .7 .7

3.6 13 21

3.8 4.4 4.3

5.7 18 27

11 11.5 11.1
TABLE III

Numerical Values of Constants 3^, "Y^ (Eq (7)),

Used in Evaluating the Coulomb Correction Given in Eq (5).

a

i Bi Yi

20.692201 - 12.316591 .9068565

77.493212 30.661404 3.7151656

60.915963 - .74569210 - 7.7968410

59.942296 - 17.101498 - 8.4300644

30.472058 - 37.670672 - 1.8958007

26.309659 - 6.7556663 7.4799697

4.4037133 37.244493 26.205729

19.433380 30.092603 18.048061

Figure

1

Calculated

atomic

cross

sections,

a

a

(k),

(circles)

for

Li.

The

partial

cross

sections

for

Compton

effect,

a

c

(k),

for

pair

production

in

the

nuclear

Coulomb

field,

a

p

(k)

and

for

pair

production

in

the

electron

field,

Q^-(k),

are

also

given.

The

atomic

photoelectric

cross

section,

Qp

]1

(k),

is

too

small

to

be

displayed

in

this

scale.
i— 00- 03E

o

CO' 082 oo - OVZ oo- oo; 00 ' 001 oi* ( aw00-021

duo I S

00-06 00- Ofr 00-0

-

Figure

2

Calculated

atomic

cross

sections,

O

a

(k),

(circles)

for

Ca.

The

partial

cross

sections

for

Compton

effect,

a

c

(k),

for

pair

production

in

the

nuclear

Coulomb

field,

Op(k)

and

for

pair

production

in

the

electron

field,

at

(k),

are

also

given.

The

atomic

photoelectric

cross

section,

a

Du

(k),

is

too

small

to

be

displayed

in

this

scale.
r

i

i

i~

00TJ91

O

G0*0M 00*031

T

O

o

o

_o

CvJ

o

o

o

CC1

o

o

_ *x>

o

o

o

0

0 ’ 0 0 i OO'OS 00*09

,oi* ( gw ) duois

00- Ofr 00-02

Figure

3

Calculated

atomic

cross

sections,

a

a

(k),

(circles)

for

Sn.

The

partial

cross

sections

for

Compton

effect,

0

c

(k),

for

pair

production

in

the

nuclear

Coulomb

field,

Op(k)

and

for

pair

production

in

the

electron

field,

Q

t

(k),

are

also

given.

The

atomic

photoelectric

cross

section,

CJph(k),

is

too

small

to

be

displayed

in

this

scale.
Figure

4

Calculated

atomic

cross

sections,

Q

a

(k),

(circles)

for

Pb.

The

partial

cross

sections

for

Compton

effect,

a

c

(k),

for

pair

production

in

the

nuclear

Coulomb

field,

Op(k)

and

for

pair

production

in

the

electron

field,

at

(k),

are

also

given.

The

atomic

photoelectric

cross

section,

Op^Ck),

is

too

small

to

be

displayed

in

this

scale.
Figure 5 Difference between screening corrections calculated from relativistic and non-relativistic charge densities for the cross section

for pair production in the field of a nucleus, in Born approximation, given as percentage of the total cross section, Ot- o£-, as a function of the photon energy, k, in MeV, for several

elements
o

<\io

E

>*

o

crUJ

LJ

O

h-O

C<N. —* o

a

Figure

6

Ratio

of

the

difference

between

the

Coulomb

correction

calculated

using

our

analytical

expression

(AC

MG )

and

'that

calculated

by

0verb^

using

the

expression

in

(15]

the

Born

approximation

cross

section

(^BH^

as

a^

unc

^^

on

photon

energy,

h,

in
Figure 7 Ratio of the Coulomb correction to the unscreened Born approxi

mation cross section as a function of the photon energy, k, in

me

2

, for Z = 82.
KAJ i 14A »« < 'in

u i oecT or conn I 1. H IT P 6 l'lo\ i >k Kt Pi 'hi !

• • i 2. A . . .on bibliographic data

| NBSIR 78-1456

SM € f. T

3. ipn e.t ’> Ar . . x. .on No

4. n i it: and st. in 1

u

Comment® on the Analysis of Total Photoabaorptlon

Measurements in the Energy Range 10 - 150 MeV

S. Publie.ition Date

6. Performing Organ! /at ton ( ode

7. ALT HOK(S)

L. C. Haximon and H. A. Gimm

8. Performing Organ. Report No.

9. PI Kl OH MING ORGAN l/.AITON NAML AND ADDRESS

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

10. Proji-i t / I a •• L W., r k Unit No. 2401104

11. ( ontr.u t 'Grant No.

12. Sponsoring Organization Name and Complete Address (Streit, City, State. 2.IP) 13. I ype of Kep.jrc & l J cin*d Covered

j

14. Sponsoring Agency ( ode

15. SUPPLEMENTARY NOTES

16. AHSTRA< .1 (A 200-word or less factual summary of most stgpifn ant information. If document includes a significant

bibliography or literature survey, mention it here.)

This note deals with details of the procedure used to extract photonuclear cross sections from total photon absorption measurements. We examine closely some of

the approximations implicit in the expressions available for the purely electromagnetic cross sections, most especially pair production, triplet production and

the Compton effect, which must be subtracted from the measured photoabsorption

in order to obtain photonuclear cross sections. We single out those aspects

of the expressions for these electromagnetic processes which most warrant

further theoretical research in that they constitute the principal source of

uncertainty in the extraction of nuclear data.

I

|

i

i

| 17. Kl Y WORDS (six to twelve entries, alphabetical order; capital i ze only the first letter of the first key word unless a firoper name; separated by semicolons

)

Atomic screening corrections; coulomb corrections; pair production; photonuclear data;

total photon absorption cross section; triplet production

i

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