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U.S. DEPARTMENT OF COMMERCE/ National Bureau of Standards

Electromagnetic Interactions

from 5 to 500 MeV

and Nuclear Research—

A Position Paper as of March 1977
NATIONAL BUREAU OF STANDARDS

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4 1977

Electromagnetic Interactions from 5 to 500 MeV

and Nuclear Research—A Position Paper as of

53 March 1977

7

r i Tec \ <f

Leonard C. Maximon

Institute for Basic Standards

National Bureau of Standards

Washington, D.C. 20234

^T °F o

0t

^fAU 0?

U.S. DEPARTMENT OF COMMERCE, Juanita M. Kreps, Secretary

Dr. Sidney Harman, Under Secretary Jordan J. Baruch, Assistant Secretary for Science and Technology

'JS. NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Acting Director

Issued August 1977
National Bureau of Standards Technical Note 955

Nat. Bur. Stand. (U.S.), Tech. Note 955, 51 pages (Aug. 1977) CODEN: NBTNAE

U.S. GOVERNMENT PRINTING OFFICE

WASHINGTON: 1977

For sale by the Superintendent of Documents, U.S. Government Printing Office Washington, D.C. 20402

Stock No. 003-003-01816-3
CONTENTS

Page

Introduction 1

1. Total photoabsorption cross section measurements 4

2. Elastic photon scattering and Delbruck scattering 8

3. Electron scattering and its corrections - radiative, recoil, dispersion and relativistic 18

4. Bremsstrahlung spectrum tip 28

5. Virtual Photon Theory - The comparison of electron

and photon induced reactions 35

Concluding observations 37

Acknowledgements 38

References 39

in
Abstract

We present the current status of both experimental measurements and

theoretical expressions for cross sections for the majority of the

electromagnetic processes that are intimately connected with nuclear

research using electromagnetic probes, emphasizing in particular those

theoretical papers which are both reliable and useful for an analysis

of the experiments, and indicating those regions in which the theoretical

calculations are inadequate. Serving as a structure for this presentation,

four areas of experimental research have been chosen, and we discuss in

reasonable detail the status of the latest experiments in these areas:

Total photoabsorption cross section measurements, Elastic photon

scattering and Delbruck scattering, Electron scattering and its correctionsradiative, recoil, dispersive and relativistic-,and the Bremsstrahlung

spectrum tip. In addition, a guide to the more pertinent articles

(experiment and theory) on the subject of virtual photons is presented.

Key Words: Bremsstrahlung spectrum tip; Delbruck scattering;

electromagnetic interactions; electron scattering;

radiative corrections; total photon absorption cross sections; virtual photon theory.

IV
Electromagnetic Interactions from 5 to 500 MeV and Nuclear Research —

A Position Paper as of March 1977

Leonard C. Maximon

National Bureau of Standards

Washington, D.C. 20234

Introduction

The use of electromagnetic probes, electron and photon beams in particular,

as a tool in nuclear research has been widespread for the past few decades.

Their great usefulness is generally spoken of as being due both to the smallness

of the coupling constant, a ~ 1/137, and to the fact that quantum electrodynamics is considered to provide a well-understood theoretical basis for

calculating the electromagnetic processes that are associated with the use of

these probes. While this view is not without substance, it has perhaps also

been partially responsible for the relative paucity, during the last decade,

of useful and pertinent calculations of those electromagnetic processes which

enter unavoidably any experiment using electrons or photons to learn something

about the nucleus. In contrast, the accuracy of experimental measurements has

increased considerably during the past ten or fifteen years: Total photon

absorption cross sections have been measured to a few tenths of a percent,

absolute cross sections for electron scattering to better than one percent.

The available theoretical expressions for the accompanying electromagnetic

processes such as radiative corrections to electron scattering, or pair

production, most of which were calculated at a time when no such accuracy was

needed, are simply not adequate for an extraction of the nuclear cross sections
at this level of precision. The objective of this report is, therefore, to

present the current status of both experimental measurements and theoretical

expressions for cross sections for the majority of the electromagnetic processes

that are intimately connected with nuclear research using electromagnetic

probes, emphasizing in particular those theoretical papers which are both

reliable and useful for an analysis of the experiments, and indicating those

regions in which the theoretical calculations are inadequate. Serving as a

structure for this presentation, four areas of experimental research have been

chosen, and we discuss in reasonable detail the status of the latest experiments

in these areas: Total photoabsorption cross section measurements, Elastic

photon scattering and Delbriick scattering, Electron scattering and its

corrections-radiative, recoil, dispersive and relativistic- and the Bremsstrahlung spectrum tip. A fifth, closely allied, subject might have been

discussed: In the context of this paper the importance of the bremsstrahlung

tip is that it has been the main source of (real) photons for nuclear research

for over twenty years. On the other hand, there are a significant number of

experiments on electrodisintegration of nuclei and electroproduction of pions.

Here the electron beam serves as the source of photons (virtual). In addition,

a number of electrodisintegration experiments have been performed using

positron beams. It is then of interest to relate the cross sections for

electron induced reactions to those for photoreactions , this forming the subject

of virtual photon theory. It is our opinion that this subject is of sufficient

importance that it warrants a more complete review than is currently to be

found in the literature, and one that would require more original material than

is presented here. We have therefore restricted the scope of the last section;
it is merely intended to be a guide to what we consider to be the more

pertinent articles on this subject.

We have thus aspired to present a survey, both of the latest experiments

that will be of help to the theoretician and of the most recent or most useful

calculations that will serve the experimenter. In contrast to the style of

review that lists every existent publication on the subject, a conscious,

subjective, choice has been made in the references listed, with the aim of

presenting the current "state of the art" in theory and experiment. Thus, for

example, direct reference to calculations or measurements which have been

superseded by later research has been generally omitted, particularly if the

more recent papers include an historical review.

Although the primary objective in each of the experiments was to obtain

nuclear information, that was not always the sole or even the most important

information that resulted, given the intertwining of electromagnetic and

nuclear processes. For example, in the case of the total photoabsorption

measurements, while the electromagnetic processes are generally thought of as

an unwanted background to the sought after nuclear absorption, these experiments in fact constitute, over much of the energy region, (in which the

electromagnetic "background" is more than 95% of the total cross section) , a

very accurate measurement of the various electromagnetic cross sections: pair

production, Compton, triplet, and photoelectric. In the case of elastic

scattering of photons, particularly in the experiments in the energy range

8-11 MeV, while one wants to obtain the nuclear scattering amplitudes, the

contribution to the differential cross section at 90° may, for the appropriate

nucleus and energy, come more or less equally from nuclear scattering and
Delbruck scattering. In view of the fact that only in the last two or three

years have there appeared some clear experimental measurements of Delbruck

scattering and a correct theoretical calculation for comparison, one might

well consider the electromagnetic process to be the more interesting one in

this case.

The conclusion to which we are lead at the end of this presentation is

that for each of the electromagnetic processes involved, the energy region

10 - 100 MeV, of primary concern to nuclear physics, is lacking, certainly in

accurate and reliable theoretical calculations and often in experimental data.

1. Total photoabsorption cross section measurements.

One of the clearest examples of the difficulty of extracting nuclear

cross sections from the concomitant electromagnetic processes occurs in

connection with measurements of the total cross section for the absorption of

photons. Of particular interest for nuclear research is the photonuclear cross

section, but to obtain this from the total cross section one must subtract the

numerous cross sections for atomic (electromagnetic) processes, and these

contribute at least 95% of the total, even in the region of the giant dipole

resonance, where the photonuclear cross section has its greatest value. This

implies that the atomic cross sections must be known very accurately if one is

to derive any meaningful information on the photonuclear cross section. The

most precise measurements of the total absorption cross section for photons

are those of Ziegler, Ahrens and coworkers [1] at the Max-Planck Institute in

Mainz, in the energy range 10 - 160 MeV, in elements ranging from lithium to

lead, an effort that goes back over 15 years [2]. The situation is well
illustrated by the figure below (taken from fig. 2, p. 361 of the article of

Ahrens et al [3]) showing the energy dependence of the partial atomic cross

sections and the total cross section in H„0 as a function of photon energy.

8oo —

p 6/mb

i p -i r T T

Photon Energy /MeV

1o 12 15 2o 25 3o 4o 5o 6o 8o loo 12o 15o

Figure 2: Energy dependence of the partial atomic cross-sections

and a in H, 0. tot 2

The atomic cross section is a sum of the partial cross-sections:

0=0+0+0+0 a x c p t '

where the subscripts T, c, p and t denote the absorption due to photoelectric, Compton, pair-production (in the field of a nucleus) and triplet

(pair-production in the field of an electron) processes respectively. (The

photoelectric cross section in HO in this energy region is negligible and is

5
not shown in the figure.) In their most recently published work [1] the

authors estimate the error in the sum of all non-nuclear cross sections to be

smaller than 0.1% for elements with Z =20 in the energy range 10 - 160 MeV.

While this is probably the accuracy that is needed in the atomic cross sections

in order to obtain useful photonuclear cross sections, it is not evident that

such accuracy is in fact currently available - even using their combination of

empirical and theoretical corrections - for example in obtaining the Coulomb

corrections to the pair production cross section. To be sure, their calculations of the various atomic cross sections have been refined over the past

five years, as may be seen from their various interim reports (compare for

example, [1] and [3]), and significant refinements of the procedure used in [1]

have been made subsequent to the publication of that paper. Nonetheless, the

lack of sufficient accuracy in our knowledge of the electromagnetic cross

sections in the region of 10 - 100 MeV is clear.

For example, in the case of pair production in the field of the nucleus,

there are uncertainties in both the screening correction and the Coulomb

correction. With regard to the Coulomb correction, this is given for photon

energies below 5 MeV by the very fine work of 0verb?4 [4] (who does not consider

screening) and for photon energies above approximately 100 MeV by the work of

Davies, Bethe and Maximon [5]. However, there exists no theoretical treatment

of the Coulomb correction to the total pair production cross section in the

intermediate region between 5 and 100 MeV. Given this lack of theoretical

knowledge, the Mainz group has determined the Coulomb correction experimentally

by attenuation measurements in high-Z elements (Cu, Sn, Ta, Pb) , assuming that

the nuclear cross section for these elements is either known (from (Y» n )

experiments [6]) or negligible (above 50 MeV). But they recognize that "the

lack of reliable theoretical calculations for the Coulomb correction between

10 and 50 MeV certainly restricts the range in Z available for absolute

private communication
O measurements to Z = 20." With regard to the screening correction, it

Y > * too presents problems, given the accuracy required in these experiments.

Although the latest analysis of the Mainz group has made use of the more

accurate (and older) expression of Jost, Luttinger and Slotnick [7] for the

total Born approximation cross section for pair production with screening in

place of the high energy expression of S^renssen [8] used in their latest

published work [1], there remain at least two problems. The first is the

question of just what form factor should be inserted in these expressions

which have the form of a weighted integral over the square of the atomic form

2 * factor, or, more exactly of (l-F(q)) . The Mainz group has done the analysis

using several different form factors, each purporting to be the result of a

highly accurate, relativistic, self-consistent Hartee-Fock calculation.

Nonetheless, some of the form factors do not give results consistent with the

total absorption measurements. Indeed, as has been recognized, for high-Z

elements, for which the uncertainties in the theoretical expressions for the

screening and Coulomb corrections to the pair production cross section constitute the largest uncertainty in the determination of the nuclear cross

section from the measured total photon absorption cross section, the measurements of the Mainz group, particularly those above 50 MeV, constitute more

truly an accurate determination of the atomic form factor than a measurement

of the photonuclear cross section. In addition to the lack of sufficiently

accurate atomic form factors, there is an additional uncertainty not dealt

with in their analysis. There, as is generally done, the screening and Coulomb

corrections are added independently to the unscreened Born approximation

(Bethe-Heitler) cross section. The justification for this procedure at high

energies was given by Davies, Bethe and Maximon [5]. It is quite probably a

reasonable approximation in the intermediate (10 - 100 MeV) region, where the

screening correction itself becomes less important with decreasing energy.

However, it is far from clear that this separation is valid to a level of

0.1% — such accuracy was never claimed in any of the theoretical works.

*private communication 7
In the case of triplet production I do not know of any treatment of the

effects of screening and atomic binding with the accuracy required in these

total absorption experiments. Although refinements subsequent to [1] include

a screening correction using very recent tabulated incoherent scattering

functions [9], the basic unscreened cross section used [10] is that of

Borsellino and Ghizzetti [11], which has errors of approximately 1.2%, as has

been shown in the most recent and comprehensive theoretical treatment of

bremsstrahlung and pair production in the field of a free electron, by E. Haug

[12]. The most recent experimental determination of the triplet cross section

is that of J. Augerat e_t al_. [ 13 ] , who also gives reference to earlier experiments

In summary, at energies above, say, 100 MeV, there exist high energy

theoretical expressions for the electromagnetic cross sections. For energies

below 10 MeV, roughly, exact numerical calculations are feasible. In the

region 10 - 100 MeV only the Born approximation cross section is available

without further (high-energy) approximation, — and in the case of triplet

production, even this only as recently as last year.

2. Elastic photon scattering and Delbriick scattering.

The scattering of light in the field of a potential, known as Delbriick

scattering, is an essentially quantum electrodynamical process: An incident

photon creates a virtual electron-positron pair which scatter in the potential

field and then annihilate, creating a photon having (apart from the kinetic

energy of the recoiling nucleus, neglected in all theoretical calculations)

the same energy as the incident photon, but an arbitrary direction, the cross

section being peaked strongly in the forward direction. The explanation for

this process was first proposed by Delbriick in 1933, but it is only within

the last two or three years that there have appeared both a correct theoretical

8
evaluation of the experimentally measureable cross section and measurements

having an accuracy sufficient to make a significant comparison with the

theoretical calculations.

The lowest order ("Born approximation") Feynman diagrams for Delbriick

scattering are

Examples of the higher order ("Coulomb corrections") diagrams are

It is to be noted that they all have an even number of interactions with the

potential — a fact known as Furry 's Theorem. The cross section for this

process is thus of order

. l/^T.7) mc

J

= (Za)

4

r

2

where r = —r- is the classical electron radius. From these diagrams it is mc
clear that Delbriick scattering is closely related to other processes for which

the initial and final states consist only of photons:

Photon-photon scattering:

^

s X

Photon coalescence: X-.

n

Photon splitting: \__y and

/

/

A Y

(For a survey of theory and experiment for photon splitting, see [22].)

Attempts to measure Delbriick scattering prior to 1969 were largely

unsuccessful. (The reasons for this, as well as a review of earlier experiments, are given quite clearly in the Introduction (pp. 1153-1154) of the

article by Jackson, Thomas and Wetzel [14a] and in the survey of Papatzacos

and Mork [15] (see pp. 109-116) who also give a fine review of the various

theoretical calculations.) Recent experiments which provide an evaluation of

the Delbriick scattering cross section all consist of measurements of the

differential cross section for the elastic scattering of monochromatic photons

from high-Z targets (necessitated by the fact that the lowest order cross

4 section is proportional to (aZ) ) . They have been performed at a number of

energies and at various laboratories. The separation of the Delbriick cross

10
section from these measurements is always complicated by the presence of the

other coherent electromagnetic and nuclear scattering amplitudes: Thomson,

Rayleigh and nuclear resonance scattering. Use is made, however, of the

different energy and angular dependences of these processes, so that for a

judicious choice one or another of them may dominate.

For the determination of the Delbruck scattering cross section, the

experiments of principal interest are the following: At 1.33 MeV there are

measurements by three different groups: Basavaraju and Kane [16], Hardie,

DeVries and Chiang [17], and Schumacher, Smend and Borchert [18].

Using 2.754 MeV photons scattered by Pb, the group at Gottingen

(Schumacher, Borchert, Smend and Rullhusen [19]) have measured the differential

cross section at angles between 15 and 150 .

The group at the Nuclear Research Center at Beer-Sheva (Moreh, Kahane and

Bar-Noy [20]) has performed elastic scattering experiments using monochromatic

photons obtained from thermal neutron capture in the energy range 7.9 - 11.4 MeV

•«-», k c - A5L 165u 175 T 181_, 232^, 237 M with a number of targets ( Tb, Ho, Lu, Ta, Th, Np,

238

U)m

.

At these and higher energies, the Rayleigh scattering amplitude is negligible.

Next, Jackson, Thomas and Wetzel [14a] at Argonne have measured differ238

ential cross sections for 10.83 MeV photons scattered from Pb and

i U at

angles between 20 and 150 , [14a], and for 11.4 MeV photons scattered from

14 L 159^, 165u

181 232^, , 238 TT on o . . „_o M/ ,, Pr, Tb, Ho, Ta, Th and U at 90 and 150 , [14b].

The measurements between 2.75 and 11.4 MeV have been of particular interest

in connection with a determination of the real part of the Delbruck amplitude,

the real and imaginary parts being of the same order of magnitude in this energy

range. At higher energies (> 100 MeV) the contribution of the real part to the

total cross section is negligible, while at lower energies (e.g., at 1.33 MeV),

where the real part is much larger than the imaginary, the entire contribution

of the Delbruck amplitude to the elastic scattering cross section is negligible,

being an order of magnitude less than the Rayleigh amplitude.

11
At energies above the 10 MeV region there are at present only two sets

of significant measurements:

1. That of Moffat and Stringfellow [21] (1960) in which photons of

87 MeV were scattered from a variety of targets (13 = Z S 92).

2. That of Jarlskog et al. at DESY [22] (1973) using photons of energies

from 1 to 7.3 GeV scattered from Cu, Ag, Au and U.

Now what is the current status of theoretical calculations and how do

theory and experiment compare? There are at present only two sources of

calculations of the cross section for Delbriick scattering that are both

reliable and present numerical or analytical results which are useful for the

analysis of experiments, — the Born approximation calculation of Papatzacos

and Mork (PM) [15,23,24], who give as well a fine review of earlier work, and

the high energy approximation calculation of Cheng and Wu (CW) [25,26]. This

is but a slight exaggeration. There is an earlier, correct, calculation of

the Delbriick cross section in Born approximation by Constantini, De Tollis and

Pistoni [27a]. Although they give no numerical or analytic results (other than

the low energy limit of the Delbriick amplitude) a recent revision of that work

by De Tollis, Lusignoli and Pistoni [27b] does give numerical values for one of

the amplitudes for circularly polarized photons at one energy (10.83 MeV) and

finds agreement in this case with the results of PM. Further, the most accurate

numerical calculations of the Delbriick amplitudes currently available are those

given by Bar-Noy and Kahane [28] in a computer program based on Constantini

er_ aJ . [27a] for the imaginary part of the Delbriick amplitude and on PM [15,23,24.

for the real part.

The cross section given by PM is calculated to lowest (non-vanishing)

order in the Coulomb interaction — "Born approximation" — and the final

result is in the form of four-fold integrals which are performed numerically.

12
The expression is valid at all energies. On the other hand, the cross section

given by CW is valid only for very high energies, but includes the Coulomb

interaction to all orders — "cross section with Coulomb correction." Their

final result is in the form of a five-fold integral which in general is to be

evaluated numerically. However, for photon energies, to, and momentum

transfers, A, satisfying A << m « oj, (i.e., scattering close to the forward

direction) or satisfying m « A « lo, they have given relatively simple analytic

expressions — for the former case in [25] and for the latter case (which is the

one required for comparison with the high energy experiments [22]) in [26].

Concerning the high energy limit of the amplitude (including the Coulomb

correction) note should be made of the paper by W. Czyz [29], who arrives at

the same expression for the amplitude as do CW, but uses an approach which

brings out more clearly the basic physics of the high-energy limit.

Before comparing the theoretical calculations with experiment, it would

be well to compare each of them with the other. As just stated, the cross

section given by PM is good at all energies but neglects Coulomb corrections.

The cross section of CW includes Coulomb corrections but is good only at

high energies. This is analogous to the situation for pair production in the

field of a nucleus, where the Born approximation, given by Bethe and Heitler

[30], is good at all energies but neglects Coulomb corrections, while the cross

section of Bethe and Maximon [31] includes Coulomb corrections but is good

only at high energies.

An error in sign in eq (7.7) of CW [25], p. 2453 is corrected in the

footnote on p. 103 of PM [15].

Actually the analysis of CW is closer to the derivation of the pair

production cross section given by Olsen, Maximon and Wergeland [32a]

See also, T. Jaroszewicz and J. Wosiek [32b].

13
There are thus two questions which come immediately to mind:

1) How large are the Coulomb corrections; i.e., what is the accuracy of

the Born approximation value given by PM.

2) How rapidly does the amplitude for Delbruck scattering approach the

asymptotic value given by CW with increasing energy; i.e., for a given

energy, what is the accuracy of the high energy limit given by CW.

The answer to each of these questions depends very strongly on the momentum

transfer, A (or, alternatively, on the scattering angle, 6). In the case

of identically forward scattering (A = G = 0) one can make a relatively

clear statement. This should not be surprising: From the optical theorem,

the imaginary part of the forward amplitude for Delbruck scattering is

(neglecting corrections of higher order in a) proportional to the total

cross section for pair production [33], and the real part of that forward

amplitude is related to the total pair production cross section by a dispersion

integral [34], and is (at least in Born approximation) smaller than the

imaginary part for oo > 10 MeV [35]. It should not, therefore, be surprising

that the forward scattering amplitude has, as in pair production [36], Coulomb

corrections of the order of 10% above 10 MeV. Indeed, the Coulomb correction

for high energy forward scattering given by CW [37] is precisely that given

earlier for pair production by Davies, Bethe and Maximon [38]. Similarly,

the approach of the forward scattering amplitude towards its asymptotic

expression is the same as for pair production [39], namely, errors in the high

energy expression [40] are (with Ui in units of the electron rest energy) of

relative order £n 2 co/w

2 — and this is true both for the real and imaginary

parts of the amplitude. The rapid approach to the asymptotic value for

0) > 40 MeV shown earlier for pair production [41] is noted again for the

forward Delbruck scattering amplitude [42]

14
Thus the qualitative, and, to a great extent, the quantitative features

of the Delbriick scattering amplitude in the forward direction, can be understood from the total pair production cross section. Unfortunately, from the

experimental point of view it is of very little interest — and once one gets

away from the forward direction (A ^ m) the situation changes drastically.

For example, at photon scattering angles of interest in connection with the

experiments of Jarlskog e_t al_. at DESY [22] in the energy range 1-7 GeV, the

Coulomb "corrections" reduce the cross section by a factor of 10, as contrasted

with a correction of order 10% in the forward direction (CW, [26] p. 3084

Fig. 5 and Jarlskog et al. , [22] pp. 3820, 3821 Figs. 8 and 9). Although the

agreement of the Born approximation calculation of PM with experiment (Jackson

et al., [14]) at 10.83 MeV (PM, [24] p. 216, Fig. 6) would seem to indicate

that the Coulomb corrections are small at that energy, there exists no

theoretical calculation which provides an estimate of the Coulomb corrections,

at either lower energies (of, say, a few MeV) or in the region 10 MeV to

several hundred MeV. As we have mentioned, this latter region is almost

equally devoid of experimental data. Nevertheless, despite the striking

difference between the Coulomb correction to the forward scattering amplitude

and the Coulomb correction to the amplitude for non-zero scattering angle

(A ^ m) , the latter may also be understood qualitatively in terms of the

cross section for pair production. Consider, for example, the amplitudes for

Delbriick scattering in the high energy limit: For very small scattering

angles, those for which — ^ 0( — )

, the main contribution to the amplitude

comes from momentum transfers to the nucleus (from one or the other of the

intermediate state pair particles) of order — and of order m, as in the

case of the total cross section for pair production at high energy. The

15
Coulomb correction to the Delbruck amplitude is then identical with that of

the total pair production cross section, and this is the result found by

CW [25]. On the other hand, if m << A << to, then the main contribution to

the Delbruck amplitude comes from momentum transfers of order A to the

nucleus from one of the particles of the virtual pair in the intermediate

state. The Delbruck amplitude is, in this case, a convolution of two pair

production amplitudes, one of which is associated with a momentum transfer of

order A >> m, (the other being or order ^ m) , and consequently has a large

Coulomb correction, which is again the result found by CW [26].

With regard to the approach with increasing energy to the asymptotic

expression of CW, the region 10 - 200 MeV is poorly understood. The rather

rapid convergence to the asymptotic limit just mentioned in connection with

forward scattering no longer exists for non-forward scattering. In fact, the

difference between the values calculated by PM and those given by CW are

shown by PM ([24], p. 214, Table IX and p. 215 Table X) to diverge slowly,

1/2 albeit less rapidly than 0(oj/m), for a momentum transfer A = m(m/oj)

Moreover, this discrepancy between the two theoretical calculations is not

changed significantly by including the correction terms given by CW [43]

,

p. 1885, Eqs. (4.12), (4.13)). Their correction terms, being less than

0.1 • (A/m) 2 relative to the main term, are far too small to account for the

difference between the values of PM and those of CW. An examination of this

apparent disagreement will be given in a later note. In any event, the

theoretical calculations do not at present provide a clear answer to either of

the two questions posed earlier for the situation of experimental interest

(non-forward scattering). In particular, the theory is particularly lacking

in providing an estimate of the Coulomb corrections in the region 10 - 200 MeV.

16
Now let us look at the comparison between theory and experiment. For

those in the energy range 1.33 - 11.4 MeV, the only reliable calculations are

those of PM, and their analysis of these experiments should be consulted. The

only experiment that has appeared after that analysis is that of Schumacher

et al» [19] at 2.754 MeV. In the case of each of the three different experiments

at 1.33 MeV, the interference with Thomson and Rayleigh scattering is very

strong at this energy, and the theoretical knowledge of the Rayleigh scattering

available when the analysis was done was not such as to permit a significant

extraction of the Delbruck scattering cross section. In the analysis by

Schumacher et al of their experiment at 2.754 MeV, use was made of two very

recent theoretical studies of Rayleigh scattering:

a) W. R. Johnson and K.-t. Cheng [44]

b) V. Florescu and M. Gavrila [45]

The agreement between theory (using PM) and experiment is good at 15 , 120

and 150 , but shows a discrepancy of up to a factor 1.7 at intermediate

angles.

The experiments of Bar-Noy, Kahane and Moreh in the energy range

7.9 - 11.4 MeV were analyzed by PM. (It should be noted that the analysis of

PM differs, in some cases, from that given in the original experimental papers,

some of which used older, incorrect, theoretical calculations.) They find

181 good agreement between theory and experiment for Ta, but for Th and U the

experimental values are consistently below the theoretical ones for scattering

angles smaller than 90 .

In their analysis of the data of Jackson, Thomas and Wetzel at 10.83 MeV,

PM [24] find rather good agreement between their theory and the experiment.

The two exceptions to this are the bump in the experimental cross sections

for Pb at 50 - 60 , and that for U the theory seems to give values which

are too low for large angles.

17
In the energy region between 11.4 MeV and 986 MeV there is only one

significant measurement — that of Moffat and Stringfellow at 87 MeV [21].

Their experimental data were compared with both the PM calculation ([26]

p. 216, Table XI) and that of CW ([43], p. 1886, Fig. 4 and [22], p. 3821,

Fig. 9(a)). Unfortunately, neither the accuracy of the theoretical

calculations nor that of the experiment is such that a significant statement,

other than that of "general agreement", is possible.

Finally, the experimental measurements of Jarlskog e_t al . of the scattering

of photons of energies between 1 GeV and 7.3 GeV on U and Au nuclei show good

agreement with the calculations of CW for momentum transfers between 1 and

10 MeV, but are somewhat larger than the theoretical values for momentum

transfers between 10 and 20 MeV.

Again, as in the case of the closely related process of pair production,

it is the energy region 10 - 100 MeV in which there is the greatest uncertainty

in the theoretical calculations, and for which there are practically no

experimental data of sufficient accuracy.

3. Electron scattering and its corrections -

radiative, recoil, dispersion and relativistic.

High energy electron scattering and the various corrections that must be

applied (radiative, recoil, dispersion and relativistic) provide another case

of a quantum electrodynamic process for which the increase in the accuracy of

the experiments in the past decade has far exceeded that due to advances in

theoretical calculations. A striking example of this is provided by a number of

recent experiments [46,47,48,49] in which electrons in the energy range 50-250

208 197

MeV were scattered inelastically from various high-Z targets ( Pb, Au,

181

Ta,

150

Nd,

142

Nd,

141

Pr, Ce, and

139

La) in order to investigate giant multipole

18
resonances other than dipole. However, to obtain the nuclear cross sections

associated with these broad resonances, background radiation including the

radiative tail must be subtracted. The situation is quite analogous to that

described in connection with total photoabsorption measurements and depicted

there. Once again, a relatively large electromagnetic background must be

subtracted in order to get out nuclear data, as may be seen in the figures

below.

tO 15 20 25 30

EXCITATION ENERGY (MeV)

35 40 10 15 20 25 30

EJCITATlON ENERGY (MeV)

FIG. 1. Spectrum of 90-MeV electrons, scattered

inelastic ally from Pb and Au. The fitted background

which consists of the radiation tail and the machine

background is shown. The counting rate is corrected

for the constant momentum dispersion of the spectrometer. Thus the error increases with the excitation energy.

19
In the present inelastic electron scattering experiments, background subtraction is important in deducing the multipolarity and strength of the giant

resonances. However, the only existing theoretical expressions for the

radiative tail [50] have all been calculated in first Born approximation,

valid for low-Z targets, but off by as much as a factor of two for heavy

elements. The procedure that has been used generally [46,47,48] to calculate

the radiative tail of the elastic peak in the case of high-Z elements is

rather ad hoc: Starting with the Born approximation expression for bremsstrahlung (in some analyses [48,49] using in addition the peaking approximation),

the form factor (which is correct in first Born approximation) is replaced by

a pseudo form factor, defined as the cross section in the absence of radiation

(obtained, for example, from a DWBA calculation) divided by the Mott cross

section. While this procedure may be reasonable very near the elastic peak,

its validity in the region of the giant resonances is open to question and has

not been investigated. To this uncertainty in the radiative tail there is

compounded the complication of other sources of background, such as instrumental scattering and quasi-elastic processes. In some of the analyses

[46,47,49] this requires the addition of an empirical background term, which

may be a function of the electron energy, in order to match "theory" with experiment. Given that the nuclear cross sections obtained after subtracting the

radiation background often have errors of the order of 50%, an accurate

calculation of the radiative tail for some high-Z elements would certainly be

very useful as a guideline in the analysis of these experiments.

Other recent examples of the increase in accuracy of experiments exceeding

advances in theoretical calculations are provided by the subject of elastic

electron scattering. In an experiment carried out recently at the Saclay

208

linac [51], electrons of 502 MeV were scattered elastically from Pb, and

the differential cross section was measured as a function of momentum transfer,

20
q, between 1.7 fm and 3.7 fm , the results showing excellent agreement in

the region of overlap with the data from Stanford (which covered 0.5 fm

< q < 2.7 fm ). In the analysis of this experiment, the expressions which

were used for the radiative correction (essentially the Schwinger correction,

2 exponentiated as in [52], p. 201), neglected terms of order a(aZ) In q

* relative to the cross section itself. These terms, which come from radiative

corrections (due to hard virtual photons) to second Born approximation elastic

scattering, may be of the order of a few percent for a lead target, energies

of a few hundred MeV, and large scattering angles, and this may easily be as

large or larger than the present experimental errors. Specific mention of

these terms was made almost a decade ago ([52], p. 202) but to my knowledge

they have still not been calculated. It would clearly be desirable to be able

to calculate terms of this form, and indeed, all terms of the form

a(aZ) In q (n = 1, 2, ...) for the case of high-Z targets. Unlike the

term in the radiative correction associated with soft real photons, of the

2 form a ln(E/AE) (In q -1), one cannot simply assume that terms of the form

a(aZ) In q should also be exponentiated. Their calculation would thus seem

to be extremely difficult. However, it is not inconceivable that one could

develop a procedure by which the high energy limit of the matrix element is

considered before performing any of the integrations, and the terms with a

2 factor In q are extracted early in the calculation, thus eliminating the

necessity of calculating the cross section for arbitrary energies (a truly

formidable task, once one goes beyond the Schwinger correction) and then

taking the high energy limit. With such a procedure, the neglected terms

would be of order ot(aZ), which is less than ^%, even for Z = 92.

Here, and in similar expressions, q is in units of mc

21
A detailed analysis of the errors involved in the determination of the

nuclear charge distribution from a finite number of elastic electron scattering

-, • • . , 12„ 208^ - cross section measurements, applied in particular to C and to Pb, has

been reviewed by Friar and Negele [53]. The aim of their paper is broader

than the corrections we are discussing here. It is concerned rather with the

larger question of what one actually learns about the nucleus from electron

scattering and muonic x-ray experiments.

A second illustration of the advance in experimental accuracy over that

of theoretical calculations is provided by the determination of the absolute

12 cross section for the elastic scattering of electrons from C in the energy

region 25 to 115 MeV, in progress at the National Bureau of Standards during

the past few years. The measurements cover the range of momentum transfers

from 0.12 fm to 1.0 fm , have errors somewhat less than 1%, and are

expected to resolve some of the inconsistencies between the earlier absolute

12 cross section measurements on C performed at Stanford, Mainz and Amsterdam.

In particular, the NBS data seem to indicate that the terms in the radiative

correction associated with soft real photons, viz.,

6 real

=

"IF ln (E/AE) [ln * ~ 1] '

should indeed be exponentiated. The proper radiative correction is then

-6

e

r (1 - 6 . .) rather than 1-6 - 6 ,, where 6 . , -, virtual real virtual virtual

includes both the contribution from virtual photons (the vertex correction)

and the vacuum polarization. (See [52], pp. 200-202.)

2 Although the terms of the form a(aZ) In q just discussed in connection

with the radiative corrections to electron scattering from high-Z targets

12 are unimportant for the measurements on C, the modification of the radiative

correction due to recoil must be included, at least to the extent that the

kinematics is treated correctly, since this is needed to establish the energy

22
from which the cut-off energy, AE, is measured. As has been noted elsewhere

([52], pp 202-203), the effects of recoil on the radiative correction may be

categorized as kinematic and dynamic. The kinematic corrections involve precisely

the same diagrams that enter when recoil is neglected, but take proper account of

the kinematics due to recoil. The dynamic corrections involve additional diagrams

in which the target nucleus may emit and reabsorb virtual photons or emit real

photons. Although no existing calculation of either the radiative correction or

the radiative tail treats the recoil corrections in a clear, unambiguous, manner,

the kinematic recoil corrections to both the radiative correction and the

radiative tail can certainly be done properly, and this alone would be worthwhile. A proper treatment of the dynamic recoil corrections is clearly much more

difficult, particularly if one wishes to treat the emission and reabsorption of

photons by a recoiling nucleus rather than, as is the case in most work on this

subject, a proton. (The distinction between nucleus and proton implied here is

that the nucleus is a composite system with structure that may be of relevance in

the electron scattering experiment, whereas the proton may be treated as a

elementary particle. Clearly for sufficiently high momentum transfers, the

proton structure is also of significance, so that it too presents all the

complications of a nucleus. And, at the other end of the scale, for sufficiently

small momentum transfers, the inner structure of the nucleus may be ignored, so

that it may be treated as a static charge distribution.) The difficulty inherent

in the problem of dynamic recoil corrections is intimately related to the fact that

they involve the exchange of at least two photons with the nucleus, introducing

the possibility of nuclear excitation in the intermediate state. In contrast, in

treating the kinematic corrections the main contribution comes from the exchange

of one photon (first Born approximation), and even if the exchange of more than

one photon is considered (as in an eikonal or DWBA approximation) , the nucleus is

treated as a static charge distribution: — its internal structure is neglected.

23
The difficulties just mentioned in connection with the dynamic recoil

corrections to the radiative correction form the subject of dispersion

corrections. They constitute the largest source of uncertainty in the analysis

of elastic electron scattering and are the least well understood of all the

corrections, involving not merely the problem of the proper wave function for

the electron, but the full complexity of the structure of the nucleus —

transition amplitudes and energy levels of the excited states as well as the

correct treatment of the center of mass motion of the nucleus. In treating

the intermediate excited states of the nucleus, all of the theoretical work

done prior to 1970 has made either of two significant approximations. In the

first, one considers the infinite, complete set of intermediate states of the

nucleus, but neglects the variation of the excitation energy in the energy

denominator (taking some "average" excitation energy, usually zero, — the

same one for all levels) and then uses closure to sum over intermediate states.

The justification generally given for this approximation is that the energies

of the important excited states are much less than the incident electron

energy. In the second, almost antithetical approximation, it is assumed that

one, or perhaps a very few, low-lying excited states are of importance. The

energy of these particular states may be introduced explicitly, but transitions

to all other states are neglected. The justification for this approximation is

that these low- lying states have relatively large transition strengths. While

one or the other of these approximations may be valid for particular regions

of energy and momentum transfer, q, in fact neither is correct in general:

Many excited states contribute to the dispersion correction, and the energy of

each excited level must be taken into account. The reasons for this are, in

part, the very significant cancellations which occur in the total dispersion

correction. Thus, although for small q the contributions from successive

24
excited levels are of the same sign (and are largest from the lowest level)

,

in the case of moderate or large q the contributions from successive levels

alternate in sign, and the largest contributions come from more highly excited

states. A further source of cancellation comes from the correction for the

center of mass motion of the nucleus. This contribution may be of almost

equal magnitude, but of opposite sign to the other terms comprising the

dispersion correction. It is therefore extremely important that the various

contributions be calculated consistently, using a single model, and that center

of mass effects be considered. These points have been shown most clearly in

three short papers by de Forest, Friar and Rosen [54], who have calculated the

dispersion corrections to elastic electron scattering from C and 0. All

of these calculations employ a number of approximations: Only the longitudinal

(Coulomb) part of the electron-nucleus interaction (using the Coulomb gauge)

is included — the transverse parts are neglected. Only one- and two-photon

exchanges with the nucleus are considered (thus limiting the validity of

the cross section to light, low-Z, nuclei). And, most importantly, the

harmonic oscillator shell model is used for the nucleus, which permits both an

exact numerical calculation of the dispersion amplitudes and the inclusion of

corrections due to the center of mass motion. Particularly to be noted is

the fact that the dispersion corrections have a diffraction minimum at approximately the same position as the elastic form factor, with the consequence

that, relative to the first Born approximation cross section, the dispersion

corrections are not much different near the diffraction minimum than elsewhere — the diffraction minima are not strongly filled by the dispersion corrections.

The most careful, detailed, examination of the problem of dispersion corrections

is that given by Friar and Rosen [55] , where one may find a discussion of a

25
number of approximations that are made in all work on this subject, but not

referred to elsewhere. It is nonetheless clear that this subject is far from

being well-understood and that present calculations are not adequate for the

analysis of the most accurate electron scattering experiments. In particular,

all of the theoretical work just mentioned is limited to second Born approximation. For heavier nuclei, where this is insufficient, two sets of recent

calculations should be noted. The first is the work of Rosenf elder and Knoll

[56], who use a partial wave calculation for the non-dispersive part of the

electron scattering amplitude and an eikonal approximation for the electron

wave function to calculate the dispersive corrections. However, a somewhat

modified approximation of the usual closure approximation is utilized, and its

validity has not been investigated. In contrast to the second Born approximation results mentioned previously, the present work finds the dispersion

corrections to be relatively much larger near the diffraction minima. Clearly,

a more thorough investigation of this discrepancy in the conclusions of these

different calculations is required. The second and most recent work is that

of Ravenhall and Mercer [57], who use the coupled-channel method to calculate

the electron scattering cross section from deformed nuclei: samarium and

oriented holmium. While these appear to be the most thorough coupled channel

calculations to date, it must be noted that only a few intermediate excited

levels have been included. In view of the aforementioned conclusions of the

work of de Forest, Friar and Rosen [54], a direct comparison of the results

12

predicted by these two groups for a light, spin zero nucleus such as C or

would be instructive.

The comments just presented on dispersion corrections have been in

connection with elastic electron scattering. It should be noted that, under

the appellation "nuclear polarizability", this same effect, — the contribution

26
of virtual excitations of the nucleus when at least two photons are exchanged, — is also of great interest currently in connection with a precise determination of the energy difference between the 2P ,„ and 2S.. ,„ levels in the

4 +

muonic ion (u He) . The most recent and precise measurement is that of

Carboni, et al. [58], who refer as well to earlier measurements. The experimental errors are of the order of magnitude of the second-order vacuum

polarization corrections; this measurement constitutes, therefore, one of the

most precise tests of QED. Of the several effects which contribute to this

energy difference and must be calculated to analyze the experiment, the one

with largest uncertainty is (as in the case of electron scattering) the

nuclear polarizability. The lack of a sufficiently accurate theoretical

evaluation of this contribution is the major theoretical limitation in the

interpretation of this experiment. A discussion of the various theoretical

calculations of the polarizability correction and the discrepancy between them

is given by Bernabeu and Jarlskog [59] . The latest calculation of the nuclear

polarization, by Rinker [60], appears to have resolved these discrepancies; he

estimates errors in his theoretical evaluation of the energy difference to be

of the order of 10 to 20%. Also to be noted is the relation of this problem

to the discrepancy between theory and experiment in u-mesic x-rays, for

which one should see R. Barbieri [61] and the references given therein.

The complications which enter once one goes beyond one-photon exchange

with the nucleus are present also in the next set of correction terms that

should be considered: — the recoil correction. We are aware of only two

articles that attempt to treat recoil terms beyond the first Born approximation.

(For a detailed discussion of the recoil corrections in the first Born approximation, see the review articles on electron scattering and nuclear structure by

Drell and Walecka, deForest and Walecka, and Donnelly and Walecka [62], and

27
more recently, de Forest [63], and Friar [64].) In the most recent work in

which contributions to the second Born approximation are investigated, that of

Friar [64], terms of order E /M and q/M (E being the incident electron

energy, q the momentum transfer, M the mass of the target) are kept.

But not even all of these terms can be calculated at present; the analysis

remains very incomplete at this time.

Finally, mention should be made of the relativistic corrections to

2 electron scattering. These corrections, of relative order (q/M) , modify

the first Born approximation cross section, and have been considered (in this

approximation, and including only the longitudinal form factor) by Friar [65]

for elastic and inelastic electron scattering. Although these corrections

are, for many experiments, only of the order of 1%, it should be noted that

2 2 2 2 they also modify the argument of the form factor: F(q ) -> F(q /(l + q /4M )),

2 2 2 in which q = q - q . The relativistic corrections thus have the effect

vw o of shifting the non-relativistic diffraction minima out to higher values

, 2 of q . 4. Bremsstrahlung spectrum tip

For over twenty years, the main sources of photons for nuclear research

in the energy range 5 to 500 MeV has been the bremsstrahlung produced by

electron beams. In order to obtain a reasonably monochromatic source of

photons it is the tip of the bremsstrahlung spectrum (for which the emitted

photon has almost all of the incident energy, and the final electron has

relatively little energy) that is of particular interest; a comparison of the

various experimental techniques that are employed to this end may be found in

an article by Matthews and Owens [66]. The tip of the spectrum also distinguishes itself from the rest of the spectrum with regard to the existence of

reliable theoretical expressions for the cross section: Away from the tip,

when the final electron has more than a few MeV, the Born approximation

suffices for light elements, and for heavy elements, for which Coulomb

28
corrections are important, calculations have been performed in the high energy

limit [6,31,32], and are applicable when the final electron energy is greater

than about 50 MeV. Thus, outside the tip region theoretical expressions for

the cross section are well known and are satisfactory except for heavy elements

and moderate electron energies (5 - 50 MeV). In contrast, for the tip region

the Born approximation is not applicable even for light elements, and no

truly satisfactory theoretical calculations exist at present.

In considering the existing measurements of the bremsstrahlung cross

section it is important to categorize them according to the energy resolution

of the detector that is used. Different detection systems have energy

resolutions which differ by several orders of magnitude, and by this fact they

reveal very different aspects of the physics of the process.

Most of the existing experiments, at least those of the past few decades,

have used either a Nal scintillation crystal or a Ge(Li) solid state detector.

Measurements of the entire bremsstrahlung spectrum [67] (rather than just the

tip) made prior to 1970 have, for the most part, used Nal spectrometers, which

have the advantage of high efficiency, but their resolution (AE/E) is at best

of the order of a few percent, and large corrections must be made for the

spectrometer response near the bremsstrahlung tip, where the spectrum varies

rapidly. The tip cross sections obtained from these experiments are thus

extrapolations from lower points on the spectrum (where the photon energy is

less than 0.9 of the maximum) and are not very accurate. More recently [68,69],

Ge(Li) detectors have been used. ' These have a rather low efficiency, but

* Ref [68] uses a Ge(Li) detector for much of the experiment, but in measure- ments near the bremsstrahlung tip a Nal crystal is employed.

Ref. [69] is a measurement of the bremsstrahlung polarization for initial

electron energies between 50 and 100 keV. The energy resolution of the

Ge(Li) spectrometer was of the order of 1%.

29
their resolution is roughly an order of magnitude better than that of Nal

crystals. (For Ge(Li), AE/E «* 10 for E of the order of a few MeV, and the

minimum observable AE is ~ 200 eV.) It should be noted, however, that

although the resolution of the Nal crystal is significant when one looks at the

spectrum of bremsstrahlung from an incident electron with fixed energy, it is

not of particular importance when an isochromat is utilized, as in the experiments

of Fuller, Hayward and Koch [70] and of Hall, Hanson and Jamnik [71]. In these

12

measurements of the bremsstrahlung tip, the 15.1 MeV resonance line in C was

used as an isochromat selector: The incident electron energy is varied (in

the neighborhood of 15.1 MeV) and the yield of 15.1 MeV photons is measured.

Since the 15.1 MeV line is very narrow (37 eV wide), it is not the detector

resolution which determines the structure that can be observed in the tip, but

the accuracy with which the incident electron energy can be measured (of the

order of 10 keV)

.

Within the last decade, there appears to be only one accurate measurement

of the bremsstrahlung tip cross section - that of Starek, Aiginger and Unfried

[72] for an incident electron energy of 1.84 MeV on a number of targets (Li,

A£, Cu, Ag, Au and Pb) , using a Ge(Li) detector with a resolution (FWHM) of

3.34 keV (AE/E w 0.2%). The agreement with the theoretical calculations cited

in their paper is poor. A later calculation of Pratt and Tseng [73] gives

moderate agreement with the experiment for high-Z targets, but poor agreement

for low-Z elements.

The discussion of the bremsstrahlung spectrum presented up to this point

has, as in the previous sections, been from the perspective of electromagnetic

interactions and nuclear research in the energy region 5 to 500 MeV. The tip

30
region has thus been used loosely to refer to the upper 10 keV to 1 MeV of the

spectrum. Also included in this perspective is the experiment of Starek e_t al .

[72], with incident electron energy 1.84 MeV and a detector resolution of

3.34 keV. However, interest in the tip region of the bremsstrahlung spectrum

is by no means limited to nuclear physics. It has been a significant subject

of research in atomic physics for almost forty years. There the initial

electron energies are generally in the keV region, and the tip generally refers

to the region within 100 eV of the endpoint of the spectrum. Although those

investigations clearly fall outside the main subject of this paper, we shall

nonetheless give a brief review of the work in that region, not only because

of its intrinsic interest, but also in the belief that even a consideration of

the bremsstrahlung tip that does not focus on the last 100 eV of the spectrum,

should be cognizant of the physics in that region. Measurements performed

there use crystal diffraction instruments, for which the resolution is

AE/E ~ 10 (an order of magnitude better than for Ge(Li) detectors) and have

a minimum observable AE « 1 eV. The crystals used most extensively in these

high resolution experiments are calcite, topaz, 0A0 (dioctadecyladipate)

,

OHM (octadecyl hydrogen maleate) , and KAP (potassium acid phthalate) . The

method of isochromats, already mentioned in connection with some of the measurements in the MeV region [70,71], is also the one employed in all of the

experiments using crystal diffraction instruments. For a rather clear and

detailed description of the differences between the classical bremsstrahlung

spectroscopy and a bremsstrahlung isochromat, there are two articles by K. Ulmer

[74] that are worth reading.

Having observed that among the experiments using Nal or Ge(Li) detectors

there is only one accurate measurement of the tip, and that there is disagreement between this experiment and the theoretical calculations, it is important

31
to distinguish the aspects of the physics that are elucidated with these

detectors from those investigated using crystal diffraction instruments. In

the measurements of the spectrum tip that have been made using either Nal or

Ge(Li) detectors, the final electron generally has an energy of a few keV, and

certainly has an energy of at least a few hundred eV, the minimum resolution

of the Ge(Li) detector. The final electron is thus, like the initial electron,

in a continuum state. The theoretical calculations with which one should

compare these experiments are, therefore, of the cross section for the

transition of an electron between two continuum states, with the emission of a

photon, in the screened field of a nucleus. The situation that pertains when

-4 one uses crystal diffraction instruments, with a resolution AE/E » 10 ,

and a minimum observable AE ~ 1 eV, is quite different. In such experiments

one can and does observe structure in the bremsstrahlung tip, mostly within

the first 20 eV, but extending out to roughly 100 eV from the high frequency

limit, and generally including a strong peak in the first few eV. None of

this structure is observed in experiments using Nal or Ge(Li) detectors, nor

is it predicted by the simple potential model calculations with the final

electron in a continuum state, as just mentioned. It is generally ascribed to

transitions in which the initial electron emits a photon and falls into one of

the unoccupied (usually d or f) bound states above the Fermi level. This

structure was first observed in 1940 by Ohlin [75] and has been the subject of

numerous experiments ever since [76-83]. It was originally of great interest

[75,76,78,80] in connection with an accurate determination of h/e, as given

by the high frequency limit of the bremsstrahlung spectrum, which limit was in

turn obtained by extrapolating the spectrum near the end point. Any structure

near the end point clearly modifies the limiting value thus obtained. More

recently, this structure has been of interest in connection with a determination

32
of the density of states above the Fermi level [78,81,82,83]. Although

numerous theoretical explanations of this structure have appeared in the

literature of the past 25 years [84-87], here too there is at present no truly

satisfactory theoretical calculation.

To summarize: Experiments in which the detector has an energy resolution

AE of more than a few hundred eV should be understandable in terms of a model

in which an initial, continuum state, electron in the field of a screened

potential makes a transition to a final continuum state in the field of the

same potential, with the emission of a photon. Additional such experiments,

as well as a theory in agreement with them, would be very helpful for an

understanding of the bremsstrahlung process. On the other hand, for experiments in which the detector has a resolution AE of the order of a few eV,

one must consider the details of the atomic structure of the target, including

the density of states above the Fermi level and matrix elements for transitions

to these states. This situation is inherently a more complicated one — and

much richer in its physical content.

One last point should be made with regard to the bremsstrahlung tip as

calculated from a simple potential, with initial and final electrons in

continuum states. As noted by Wigner [88] in his work on the behavior of

cross sections near thresholds (and again, later, by Jabbur and Pratt [89]

in connection with their calculation of the tip of the bremsstrahlung spectrum)

,

2 the bremsstrahlung cross section for an unscreened Coulomb potential (V = -Ze /r)

approaches a finite (non-zero) value at the high frequency limit, i.e., as the

momentum, p, of the final electron goes to zero. This follows as a consequence of the form of the Coulomb wave function at the origin, which is

1/2 inversely proportional to p for small p, and thus cancels (in the cross

section) a factor p coming from the density of final states. On the other

33
hand, for a screened potential the wave function remains finite as p

-> 0,

and hence the bremsstrahlung cross section goes to zero at the high frequency

limit. Nonetheless, one might, from some of the figures and tables in the

published literature [90], get the contrary impression, viz., that the cross

section goes to a finite, non-zero, limit for a screened potential. This

result in these articles comes from the fact that the limiting value there is

obtained by extrapolating from values close to the limit, as is stated

explicitely on p. 1801 of [73]. A vanishing cross section at the high

frequency limit is not necessarily in contradiction with the measurements of

the bremsstrahlung tip by Starek, Aiginger and Unfried [72]. As mentioned,

the energy resolution of the detector in that experiment was 3.34 keV (FWHM)

,

so that a falling of the cross section to zero within an energy region smaller

than, say, 1 keV, would not be observed as a vanishing cross section at the

end point. The discontinuous drop of the cross section at the limit is solely

a feature of the Coulomb potential (just one more of its many pathologies),

and even this feature disappears if one consider an "average" cross section

in the discrete region. This latter point has been made by Breit [91], and

discussed in detail by Newton and Fonda [92] in their consideration of the

energy dependence of cross sections in the neighborhood of the threshold for a

process in which the two emerging particles are oppositely charged. More

recently it has been discussed quite clearly and succinctly in a paper by

C. Tzara [93] on the effect of the Coulomb potential on Tr photoproduction

(see, in particular, the appendix in that paper). Since the Coulomb potential

has an infinite number of bound states arbitrarily close to the high frequency

limit, the final particle (here the it ; in the bremsstrahlung tip, an electron)

can occupy one of the bound levels rather than a continuum state. He constructs

explicitely an average cross section in the discrete region and shows that the

cross section is then continuous across the boundary between continuum and

discrete states.

34
5. Virtual Photon Theory - The comparison of electron

and photon induced reactions

The theoretical calculations which form the basis for most of the work in

this field are contained in the articles of Thie, Mullin and Guth [94] and of

Dalitz and Yennie [95]. Particularly to be recommended for its clarity is the

later paper of Eisenberg [96]. All of these calculations were performed in

Born approximation. Going beyond this, Gargaro and Onley [97] use distorted

waves for the electron in a Coulomb field. Other theoretical papers that should

be noted are those of Berman [98], who compares photoproduction and electroproduction by separating, in the cross sections for these processes, the

kinematic and dynamic aspects; Gibson and Williams [99], who consider virtual

photon theory in the long wavelength limit for two-body electrodisintegration,

and, most recently, Dressier and Tomusiak [100] , who examine in detail the

validity of those assumptions of virtual photon theory which must be made in

order to extract photodisintegration cross sections from electrodisintegration

data.

Experiments using virtual photon theory in their analysis have been

performed by several groups. Particularly to be noted are the measurements of

the total cross section for the electrodisintegration of deuterium in the energy

range 17 to 28 MeV by Skopik et_ aL in Saskatchewan [101], and the measurements

of both the total electrodisintegration and photodisintegration cross sections

for a number of nuclei ( C, F, " CI, Cu and Cu) by Wolynec e_t^ al. in

Slo Paulo [102]. The analysis of the latter group makes use of the distorted

wave treatment of Gargaro and Onley [97].

Finally, with regard to the use of positron beams and a comparison of

positron and electron cross sections, one should consult the fairly recent

theoretical paper of Nascimento and Wolynec [103], who analyze the existing

35
experiments, all of which measure the total cross sections for electrodisintegration by electrons and by positrons, on a number of nuclei. These have

been performed by two groups: Herring et aL [104] at 27 MeV on C, Cu,

Ag and Ta, and, more recently, Kuhl and Kneissl [105] in the energy

TO £q 10 7 range 20 to 30 MeV on C, Cu and Ag and Kneissl, Kuhl and Weller [106]

in the energy range 15 to 40 MeV on uranium.

36
Concluding Observations

We have concentrated our considerations in this report on the energy

range 5 - 500 MeV. This range of values has been chosen principally because

it comprises most of the experiments using electromagnetic probes for nuclear

research. It also represents a significant range from the viewpoint of

theoretical calculations. Below about 5 MeV, although Born approximation

calculations may have questionable validity, it is often feasible to use a

partial wave decomposition; for low energies a relatively small number of

partial waves enter. On the other hand, above a few hundred MeV, high-energy

approximations (eikonal or WKB, for example) can generally be applied. It is

just the energy range 5 - 500 MeV that is the least amenable to the wellknown

methods of mathematical physics, and hence the most lacking in reliable and

useful theoretical expressions for the cross sections we have considered.

Examples of this have been cited in the preceding sections, where it was noted

that the question of the magnitude of the Coulomb corrections to both Delbruck

scattering and pair production in the energy range 5 to 100 MeV is an open

one; the expressions for Delbruck scattering including Coulomb corrections

assume very high energies (of the order of 1 GeV) ; those for the total cross

section for pair production are valid only above about 50 MeV. The most

significant challenge to improve the calculations comes, however, as a result

of the increased accuracy of many experimental measurements. When these

become better than 1%, corrections to the main process of relative order

a (and with them all the difficulties of an additional set of diagrams) must

be calculated if comparison is to be made between theory and experiment.

37
Acknowledgments

In the production of this report I have benefited greatly from many

discussions with colleagues, both prior to the formulation of each of the

sections, and in the course of the several rewritings which the manuscript

has undergone. I am first of all indebted to R. Bergere for having asked.

It was his question concerning the possible importance of an accurate measurement of the difference between the bremsstrahlung spectrum from electrons and

positrons that engendered in response a letter which grew and grew until it

became this report. Several of those whose articles are listed in the

references were very helpful in providing, in discussions held with each of

them, an understanding of essential aspects of their work. For this I am

grateful to J. Ahrens, T. Bar-Noy, B. Frois and C. Tzara. The section on

dispersion corrections owes a great deal to several lengthy discussions with

J. L. Friar, and the part of the section on the bremsstrahlung tip dealing

with atomic physics would not have been formulated without several conversations with A. J. McAlister and with R. D. Deslattes whom I also thank for a

very thoughtful reading of the manuscript resulting in important improvements.

The manuscript also received careful and detailed readings by L. S. Cardman

and E. G. Fuller, whose suggestions have been very useful. The section on

electron scattering and its corrections owes much of its form and content to

innumerable discussions with J. W. Lightbody, X. K. Maruyama and S. Penner.

Their contribution to this section was very significant. And finally, a note

of thanks is appreciatively given to E. V. Hayward , who has been a sounding

board for almost every idea contained herein, who has given, throughout the

entire writing, unstinting counsel on both technical and editorial matters,

with patience and with humor.

38
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44
NBS-114A (REV. 7-73)

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1. PUBLICATION OR REPORT NO. NBS TN-955

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4. TITLE AND SUBTITLE

Electromagnetic Interactions from 5 to 500 MeV and Nuclear

Research — A Position Paper as of March 1977

5. Publication Date August 1977

6. Performing Organization Code

7. AUTHOR(S)

Leonard C. Maximon

8. Performing Organ. Report No.

9. PERFORMING ORGANIZATION NAME AND ADDRESS

NATIONAL BUREAU OF STANDARDS

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WASHINGTON, D.C. 20234

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16. ABSTRACT (A 200-word or less factual summary of most significant information. If document includes a significant

bibliography or literature survey, mention it here.) We present the current status of both experimental measurements and theoretical

expressions for cross sections for the majority of the electromagnetic processes that

are intimately connected with nuclear research using electromagnetic probes,

emphasizing in particular those theoretical papers which are both reliable and useful

for an analysis of the experiments, and indicating those regions in which the

theoretical calculations are inadequate. Serving as a structure for this

presentation, four areas of experimental research have been chosen, and we discuss

in reasonable detail the status of the latest experiments in these areas: Total

photoabsorption cross section measurements, Elastic Photon scattering and Delbriick

scattering, Electron scattering and its corrections-radiative, recoil, dispersive

and relativistic- and the bremsstrahlung spectrum tip. In addition, a guide to the more pertinent articles (experiment and theory) on the subject of virtual photons is

presented.

17. KEY WORDS (six to twelve entries; alphabetical order; capitalize only the first letter of the first key word unless a proper name; separated by semicolons)

Bremsstrahlung spectrum tip; Delbriick scattering; electromagnetic interactions;

electron scattering; radiative corrections; total photon absorption cross sections;

virtual photon theory . 18. AVAILABILITY " [^ Unlimited

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