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NBSIR 81-2262

TAGGED PHOTONS

An Analysis of the Bremsstranlung

Differential Cross Section in the

Range of Interest for A Tagged

Photon System

, Leonard C. Maximon

Nuclear Radiation Division

Center for Radiation Research

National Measurement Laboratory

U.S. Department of Commerce

National Bureau of Standards

Washington, DC 20234

Arlette de Miniac and Thierry Aniel

Centre d'etudes Nucleaires de saclay

Division de la Physique

Department de Physique Nucleaire

Service de la Metrologie et de la Physique Neutroniques Fondamentales

91190 Gif-sur-Yvette, France

April 1981

40* °' %

/ w %

DEPARTMENT OF COMMERCE

)NAL BUREAU OF STANDARDS

RATIONAL BURKA l OF FTAKDARU8

LIBRARY

JUL 2 0 1981

NBSIR 81-2262

TAGGED PHOTONS

AN ANALYSIS OF THE BREMSSTRANLUNG

DIFFERENTIAL CROSS SECTION IN THE

RANGE OF INTEREST FOR A TAGGED

PHOTON SYSTEM

Leonard C. Maximon

Nuclear Radiation Division

Center for Radiation Research

National Measurement Laboratory

U.S. Department of Commerce

National Bureau of Standards

Washington, DC 20234

Arlette de Miniac and Thierry Aniel ,

Centre d'etudes Nucleaires de saclay

Division de la Physique

Department de Physique Nucleaire

Service de la Metrologie et de la Physique Neutroniques Fondamentales

91 190 Gif-sur-Yvette, France

April 1981

U.S. DEPARTMENT OF COMMERCE, Malcolm Baldrige, Secretary

NATIONAL BUREAU OF STANDARDS. Ernest Ambler, Director
1
TAGGED PHOTONS

An analysis of the bremsstrahl ung differential cross section

in the range of interest for a tagged photon system

Leonard C. Maximon

Nuclear Radiation Division

Center for Radiation Research

National Bureau of Standards

Washington, D.C. 20234

Arlette de Miniac and Thierry Aniel

Centre d' Etudes Nucleaires de Saclay

Department de Physique Nucleaire

Service de la Metrologie et

de la Physique Neutroniques Fondamentales

91 1 91 Gif-sur-Yvette Cedex, France

ABSTRACT

We consider in detail the differential cross section for

bremsstrahl ung for angles and energies in the range of interest for a

tagging system. We derive a high energy, small angle approximation for

the differential cross section for bremsstrahl ung, eq (1.1). We use

this approximation to determine the maxima and minimum of the cross

section and to evaluate it at these extrema. It is shown that the

differential cross section has a very sharp dip in the region of small

momentum transfers. Coulomb corrections to the Born approximation are

considered, and do not fill in this dip.

Key Words: Bethe-Hei tier cross section; bremsstrahl ung differential

cross section; bremsstrahl ung monochromator; photon beams;

photonuclear research; tagged photon method
ACKNOWLEDGEMENTS

We wish to express our gratitude to Gloria Wiersma, to whom we

are indebted for the skillful and expeditious typing of this report.

We thank Roland Bergfere and Larry Cardman for their helpful discussions,

encouragement, aid and counsel. One of us (L. C. M.) wishes to thank

Roland Bergfere for the gracious hospitality accorded him during his

stay at the Centre d' Etudes Nucldaires de Saclay, where this work was

done.

A
TABLE OF CONTENTS

ABSTRACT i

ACKNOWLEDGEMENTS ii

I. INTRODUCTION ..... 1

II. CONSIDERATIONS OF A GENERAL NATURE CONCERNING DIFFERENTIAL

AND INTEGRATED CROSS SECTIONS 7

III. KINEMATICS 16

IV. HIGH ENERGY, SMALL ANGLE APPROXIMATION OF THE

BREMSSTRAHLUNG CROSS SECTION; GENERAL CONSIDERATIONS ... 22

V. APPROXIMATION OF q

2

, THE MOMENTUM TRANSFER SQUARED,

FOR HIGH ENERGIES AND SMALL ANGLES 26

VI. HIGH ENERGY, SMALL ANGLE APPROXIMATION OF THE DIFFERENTIAL

CROSS SECTION FOR BREMSSTRAHLUNG . 39

VII. DETERMINATION OF THE MAXIMA AND MINIMUM OF THE

BREMSSTRAHLUNG CROSS SECTION . 72

VIII. SUMMARY OF ESSENTIAL FORMULAS DERIVED IN SECTIONS V, VI,

AND VII 110

IX. COULOMB CORRECTIONS TO THE DIFFERENTIAL CROSS SECTION

FOR BREMSSTRAHLUNG 122

REFERENCES 125

FIGURE CAPTIONS . 127

FIGURES 129
' . ' ' , . .
I. INTRODUCTION

During the last twenty years there has been a strong trend toward

the utilization of "monochromatic" photon beams rather than bremsstrahlung photon beams for the measurement of photonuclear reaction cross

sections. This trend clearly represents progress toward the "ideal"

photon beam which would have the characteristics outlined in the recent

survey of Beil and Bergkre [1]:^

1) An energy resolution as small as possible,

2) A flux within the desired energy resolution as high as

possible,

3) Low counting rates associated with unwanted photons, both

within and outside the desired energy range, and

4) A well defined, and easily and continuously adjustable photon

energy.

At present there are two main experimental techniques which have

had the greatest success in achieving good compromises between these

sometimes conflicting requirements; the positron annihilation-in-flight

technique, and the bremsstrahl ung monochromator or photon "tagging"

technique. The tagged photon technique detects the electron after

bremsstrahl ung to determine the emitted photon energy. This detected

electron is put in time coincidence with a nuclear decay product

(i.e., n, p, or y') following the photon induced reaction. The

laboratory at Saclay has specialized in the production and utilization

of photon beams obtained using the positron annihilation-in-flight

^Figures in brackets indicate literature references at the end of this

paper.
technique, first at the AL60 accelerator [2], and now using the

positron beams obtainable at the ALS (Accdl£rateur Lindaire de Saclay)

accelerator [3]. The photon tagging technique has become increasingly

popular as electron beams of duty cycle approaching 100% have become

available. At present there are photon monochromators installed both at

the University of Illinois [4] and at the University of Mainz [5]. Both

of the installations are at present restricted to modest photon energies

due to the maximum electron beam energies available (67 MeV at Illinois

and 14 MeV at Mainz). However, the first results obtained at these

installations provide a clear indication of the advantages of the photon

tagging technique with high duty cycle electron beams. A monochromator

capable of tagging photons of energies up to 390 MeV has been installed

[6] at the University of Bonn, utilizing the internal beam of their 5%

duty cycle, 500 MeV electron synchrotron. The excellent characteristics

of the tagged photon beam obtained at Bonn despite the restriction of a

5% duty cycle has led the Saclay group to consider the possibility of

developing a photon tagging apparatus using the 2% duty cycle beams

available at the ALS.

In order to investigate the experimental possibilities of tagged

photon beams at intermediate energies without disrupting the ongoing

program of photonuclear experiments using the positron annihilation-inflight photon beams, the Saclay group has adapted the magnet normally

used to dump the positron beam so that it can also function as a photon

monochromator. To further reduce conflicts with the ongoing program,

the group has chosen to test the monochromator using the "parasite"

electron beam which can be obtained in the low energy (BE) experimental

2
hall whenever the ALS is producing a high current beam by collecting

electrons scattered through a few degrees from a small tungsten wire

inserted into the main beam. The development of a photon tagging

system which operates within the constraints imposed by the use of this

parasite beam, which has high emittance and low current, and by the use

of an existing magnet, which was not designed for use as a tagging

spectrometer, necessitates a careful study of the theory of the tagging

process. Specifically, one must understand whether the available magnet

can provide a practical tagging system with good efficiency and energy

resolution when a large emittance electron beam is used. It is also of

interest to understand how the characteristics of the available system

would improve if the low emittance electron beam obtainable directly

from the accelerator were used in place of the "parasite" electron beam.

Finally, a more complete understanding of the tagging process can be

expected to provide important information for the design and optimization

of future photon monochromators.

These considerations lead naturally to a study of the bremsstrahl ung

process itself. In the remaining sections of this report we consider in

detail the differential cross section for bremsstrahl ung for angles and

energies in the range of interest for a tagging system. For a given

photon energy and direction the angular distribution of the tagging

electrons is precisely this differential cross section. In this report

we consider the cross section summed over both electron spins and photon

polarization. In a later report we will consider in detail the

bremsstrahlung cross section for polarized photons, still, however,

summing over electron spins. One of the more interesting aspects of the

problem is the pronounced structure, a sharp dip, which exists in the

3
differential cross section but is absent from both the total cross

section and the cross section integrated over the angles of either the

photon or the final electron. This dip is shown in figs. 4-10 where one

may see that, for the energies and angles considered there, the region

of the dip extends over an angular region of approximately 0.2° in the

polar angle, 0^, and 0.5° in the azimuthal angle (0 < <J> < 0.5°) of

the final electron. This marked difference between the differential and

integrated cross section may be observed in a number of elementary

electrodynamic processes, and is of importance both from a theoretical

point of view and for the analysis of experiments using electrons,

positrons, and photons as probes. A detailed knowledge of the differential cross section in the region of the dip is of particular importance

for the analysis of experiments using finite size detectors. An evaluation of the cross section for a particular set of angles, especially

for <p = 0°, could then give totally incorrect values for the cross

section integrated over the finite detection solid angles. We therefore

devote the next section to this point, with the aim of situating the

present specific problem in a larger context. In the following sections

we present the details leading to our high energy, small angle, approximation to the Bethe-Heitler differential cross section for bremsstrahlung.

This is given by eqs (IV. 1) and (VI. 49):

2

[1 - F(q)]

2

1 p

2

6Q, dft, dk

p

2

k q 4

k pp-, ( 2tt ) 2

( 1 . 1 )

4
where f(y), g(y), p, and a are given in eqs (VI. 44), (VI . 45),

(VI. 47), and (VI. 48). All other yariables are defined in section III.

This expression serves as the basis for our detailed description of the

cross section in the region of interest for the present tagged photon

facility, viz., the region of very small momentum transfers, of the

order of the kinematically allowed minimum momentum transfer. When we

speak of eq (1.1) as being a high energy, small angle approximation to

the differential cross section, we mean that the terms neglected in

arriving at eq (1.1) are of relative order 9 , i.e., relative to

those terms retained in eq (1.1), for all angles and energies, including

the region of the dip. The expression (1.1) has in addition the virtue

the very large cancellations which occur in the original expression,

(IV. 1). Specifically, in section III we present the kinematics to be

used throughout this report. We also present there the relations

between the angles in the system with z-axis along the incident beam

and the angles in the system with z-axis along the emitted photon

direction. It is this latter system that will be used in all the

expressions presented in this report. In section IV we make some

general remarks concerning the subject of high energy, small angle

approximations to the cross section. In section V we develop in detail

2 our high energy small angle approximation to the expression for q ,

the square of the momentum transfer. This section also serves as

introduction to the procedures used in section VI, where we develop in

detail the high energy, small angle approximation to the Born approximation (Bethe-Hei tier) differential cross section. In section VII we

that each of the three terms It is thus free of

5
use the approximation developed in section YI in order to obtain

analytic expressions for the maxima and minimum of this cross section

in the region of very small momentum transfers. In section VIII we

present a summary of the essential formulas derived in sections V, VI,

and VII for the reader who may be more concerned with applying these

formulas than with their derivation. In section IX we discuss the

Coulomb corrections to the differential cross section.

We wish to emphasize that this report is intended to be a working

paper. The inclusion of mathematical details in the text is intentional.

6
II. CONSIDERATIONS OF A GENERAL NATURE CONCERNING DIFFERENTIAL

AND INTEGRATED CROSS SECTIONS

In the analysis of photonuclear experiments, the fundamental electrodynamic processes of bremsstrahlung and pair production intervene inevitably.

Indeed, in some of these analyses it is the lack in our knowledge of the

theoretical cross sections for the electrodynamic processes that constitutes

the major uncertainty in the determination of the desired nuclear information.

These processes may, moreover, enter in the analysis in various guises. The

field in which the bremsstrahlung or pair production takes place may be

provided either by the nucleus or by the atomic electrons. In the case of

bremsstrahlung, the incident beam of interest may be either electrons or

positrons. And in each case, the differences in the nature of the target or

of the beam are reflected in significant differences in the details of the

cross section. Furthermore, in the analysis of a given experiment, one may

be interested either in the differential cross section for the particular

process or in an integrated cross section (integrated over either angles

alone or both angles and energy). Thus, for example, in the use of a

positron beam to provide "monoenergetic" photons by annihilation-in-flight

([1], [3]), the relative importance of the background bremsstrahlung photons

may be diminished by utilizing photons emitted at an appropriate (relatively

large) angle with respect to the incident beam of positrons. For the

determination of the relative intensity of the background bremsstrahlung,

the differential cross section for the process is required. Similarly, in

connection with the present experimental arrangement in which monoenergetic

photons are produced by tagging the associated scattered electrons, it is

the differential cross section for bremsstrahlung that is of concern. On

7
the other hand, in the use of the entire forward cone of the bremsstrahl ung

spectrum as a source of photons, we are concerned with cross sections

integrated over angles. And in the case of total photoabsorption measurements, the principal background to the photonuclear absorption is due to

pair production. In this case we require the cross section integrated over

both angles and energies.

It may appear on first consideration that this distinction between the

differential and integrated cross section is an essentially trivial one,

implying nothing more profound than an integration over the unobserved

quantities. We would like to stress that this is decidedly not the case.

The differential cross section on the one hand, and the integrated cross

section on the other have each very distinctive properties that are not

shared by the other, as we will illustrate by a number of examples. The

integrated cross section in general will reflect the completeness of the

functions over which one integrates. It may show invariance properties and

is associated with sum rules. On the other hand the integrated cross

section only reflects that part of the differential cross section which

contributes significantly to the integral. The characteristics of the

differential cross section in a region of angles and energies which does not

contribute significantly are not manifest in the integral cross section.

Likewise, any fine structure in the differential cross section, even if it

appears in an otherwise important region, is lost in the integration. It is

precisely such structure, or other particular properties occurring for

special angles and energies, that may be utilized in an experiment, which are

specific to the differential cross section. Let us illustrate these

statements with a few examples, the last of which will be the present case

of tagged bremsstrahl ung photons.

8
For our first example we consider jointly the processes of bremsstrahlung and pair production. As is well known £7], in first Born approximation

(the Bethe-Heitler cross section) the differential cross section for either

process can be obtained from that of the other by a simple reversal of sign

of the appropriate momenta and energy (and, of course, an appropriate modification of the density of final states). However, once the Coulomb corrections

(required for high Z target nuclei) are included, this is no longer true

[8]

, The expression for the differential cross section for bremsstrahlung

is then totally different from that for pair production. This results from

the asymptotic condition which must be imposed on the wave function for each

of the particles [8], In pair production, both the electron and the positron

go out from the reaction and hence both are characterized asymptotically by

a plane wave plus incoming spherical waves. In the case of bremsstrahlung

the final electron goes out, but the initial electron is incident upon the

reaction. The final electron is therefore characterized by a plane wave

plus incoming spherical waves, whereas the initial electron is characterized

by a plane wave plus outgoing spherical waves. This difference in the

asymptotic character of the two wave functions results in a matrix element

quite different from that for pair production, where both particles have the

same asymptotic character [8]. However, if we consider the cross section

for each of these processes integrated over the angles of the final electron,

then the two cross sections are indeed related as in the Born approximation

[9]

. This result is a reflection of the fact that the set of wave functions

with either ingoing or outgoing spherical waves separately form a complete

set. By integrating over the direction of the final electron we have summed

over the functions of the complete set, with the consequence that the

particular radiation condition — ingoing waves for the final particle —

9
is no longer reflected. In the integrated cross sections for the two

processes we thus have only one wave function with a specified radiation

conditions rather than two. The corresponding matrix elements are then

simply related by the changes of sign of the appropriate momenta and

energies. Thus to study or make use of characteristics which distinguish

bremsstrahlung from pair production one must measure the completely

differential cross section. For example, at high energies the Coulomb

corrections for these two processes arise in different regions: for

relatively large momentum transfers, q ~ me, in the case of bremsstrahlung, and for very small momentum transfers, q ~ mc/E, for pair

production. This distinction can be observed only in a measurement of

the completely differential cross section. If one of the final particles

is not observed, then one effectively integrates over the momentum

transfer. Since the Coulomb corrections to the integrated cross sections

are identical, there is then no significant difference between the two

processes

.

For our second example we consider the bremsstrahlung spectrum

produced by a beam of electrons (or positrons), of energy e, (in units

of me ) incident upon a target, of atomic number Z. If the entire

forward cone of photons is utilized, then at high energies, in first

2

Born approximation, the cross section consists of two parts: Z

from bremsstrahl ung produced in the nuclear field, Zo^ from bremsstrahlung produced in the field of the atomic electrons, the total being the

familiar expression Z(Z + 1 ) 0

bh

w*iere °BH 1S t*ie Bethe-Heitler

cross section for a point change, e. The spectrum has the familiar

shape shown in figure 1, and the tip of the spectrum is given by

10
k.max

= e - 1

Now, however, let us consider the experimental setup described in the

introduction, where one utilizes the quasi -monoenergetic photons produced

by annihilation-in-fl ight of a beam of positrons. There the background

radiation, the bremsstrahlung produced by the beam of positrons, is

diminished relative to the annihilation photons by using photons emitted

at an angle 6 (relative to the incident beam) which, although fairly

small, is nonetheless large compared to 1/e. In this case we require,

for a description of the background, the differential cross section for

the bremsstrahlung emitted at an angle 6, rather than the integrated

cross section. The tip of the spectrum for photons produced in the

nuclear field is

as before. However, the tip of the spectrum for photons produced in

the field of the atomic electrons is [1], [10],

kmax

e - 1 £ + 1

£ - I

For high energies and small angles (0 « 1, but not necessarily

0 ~ 1/e), this can be written

11
e - 1 k u = p0 . emax

1 +

1 + u Ze

5

This formula serves to illustrate the point made earlier. For those

angles which contribute significantly to the integrated cross sections

(viz, 0 ~ 1/e), k

„ is very close to k . differing only by max max

terms of relative order 1/e. Thus if the experimental setup is such

that one observes essentially the cross section integrated over angles,

e N then k „

k . the spectra are similar in shape and the cross max max r r

section is, approximately, Z(Z + 1 )°BH‘However » f° r an 9l es 6 » l/e»

which do not contribute significantly to the total cross section,

e N

k v may be significantly smaller than k . Thus, for example, for max max

an incident beam of 100 MeV (e = 200) and an angle 0=4°, we have

k^x - 67 MeV. The two spectra are then as shown in figure 2. The

cross section is then approximately Z(Z + 1) da^^/dfi for k < k

mQx

and Z

2 da

BH/dn for k*ax

< k < k^.

Thus, the characteristics of the bremsstrahlung spectra that are

distinctive to the mass of the target (nucleus or electron), are

manifest in the differential cross section, but not in the integrated

cross section. This observation can be inverted; by utilizing photons

emitted at angles 0 » 1/e (more precisely, for angles 0 > /(1/e) ),

we have at our disposal a reasonably wide range of energies,

e N

k v < k < k . in which the bremsstrahlung photons are emitted

solely in the nuclear field. This part of the spectrum can then serve

either to study the properties of nuclear bremsstrahlung by itself, or

as a source of photons whose cross section is accurately known: [A

12
fairly simple analysis shows that for this portion of the spectrum, the

associated momentum transfers, q, are such that q

> 1 (in units of

me). There are as well, therefore, essentially no effects of screening

in this region of the spectrum.]

Our final example is provided by the analysis of the differential

cross section for bremsstrahlung in the range of angles of interest for

the tagged photon system which constitutes the primary concern of this

report. This system is characterized by high energies (e » 1) of

the scattered electron and emitted photon, and small angles (6 « 1)

between the directions of the scattered electron, emitted photon, and

incident beam. With these restrictions, the momentum transfer, q,

to the nucleus, lies in the range

qmin

= + u2 ) < 9 ^ °( u ) »

where u = P-]©-]* and 6 = k/^e^), in which and k are the

energies (in units of me ) of the initial electron, final electron and

emitted photon, jd-j

, £g, and k_ are the corresponding momenta (in

units of me), and 9^ is the angle of emission of the photon with

respect to the incident beam (in radians).

Now in the region q

. < q < 0(u) all momentum transfers contribute

significantly to the cross section integrated over the angles of the

final electron, the momentum distribution being essentially of the

form dq/q, [11]. Thus we have equal contribution from the region of

small momentum transfers, q ~ qm

.. , and from the region of "large"

momentum transfers, q ~ u. However, the solid angle of the final

2 2 electron corresponding to q ~ qm

- is much smaller, of order Pm -j n

/ u

13
relative to the solid angle corresponding to q ~ u. Consequently the

cross section in the region q ~ q. is (for a given solid angle of

the final electron) larger by u /qmin than that from the region q ~ u.

Thus, from the experimental standpoint, it is the region of small

momentum transfers, q ~ q . , that is of interest, the cross section

being so much larger there. The primary concern of this report is a

detailed examination of the differential cross section for bremsstrahlung

in the region of momentum transfers q ~ q

. . When we examine the

differential cross section in detail in the region of small momentum

transfers, q ~ qm^ n

> what we find is that the cross section, considered

as a function of the angle ©2 between the photon and the final electron,

has a very sharp dip for sufficiently small values of the azimuthal

angle <J> between and ^ ( see fig- 3a). The ratio of the cross

section in the dip to that at the neighboring maxima, is (as will be

shown in the analysis that follows) given (for the case in which

= 0 and u = e -j 6

-j » 1) approximately by

If we consider this expression for the values used in most of the figures

presented here, viz., e-j = 140 MeV, k = 95 MeV, 6 = 1°, then we have

This sharp dip for <j> = 0 may be seen in figure 4.

14
As in the two previous examples, this structure is particular to

the totally differential cross section. The region over which it

occurs is sufficiently small that it gives a negligible contribution to

the cross section integrated over the angles of either the final

electron or the emitted photon. The familiar bremsstrahlung spectrum

(in which the final electron is not observed) thus shows none of the

structure particular to the spectrum of tagged photons.

15
III. KINEMATICS

In this section we present the details pertinent to the kinematics

and define the various quantities which are used throughout this report.

Unless specified otherwise, we take energies to be in units of the

2 electron rest energy, me , and momenta in units of me. We have

e-j ,£i

: Energy and momentum of the incident electron

2 2

(or positron) (e-j - £| =1).

e

2 ,£2

: Energy and momentum of the final electron

2 2

(or positron) (e^ - £2 = !)•

k, k : Energy and momentum of the emitted photon.

3. = £] " £2 ” —: Momentum transferred to the target nucleus.

The energy transferred to the nucleus, which is in general q Q

= e-j - - k,

is taken throughout this report to be zero, as in the Bethe-Heitler cross

section. This is equivalent to the assumption of an infinitely heavy

target nucleus. We thereby neglect the effects of the recoil of the target

nucleus, both kinematic and dynamic (this latter due to photon emission

by the nucleus). For high energies and small angles (in which case

q < 0(u)), these effects are completely negligible, since they give

contributions of relative order qZ(m/M) in the region of the dip, and

less elsewhere, M being the mass of the target nucleus.

In the system with z-axis in the direction of k_, the angles of

£j are (0-|><j>-|), the angles of £2 are These vectors are

shown in fig. 3a. The components of £j and £2 perpendicular to k

are then

16
The angle between

perpendicular to

The component of

We have then

The component of

The magnitude of

the vectors u_ and v , which lie in the plane

k^, is then

<}) - <J>2 -

4> 1

£ perpendicular to k is

^

= u - v .

2 2 2

qx = u - 2uv cos$ + v

£ in the direction of k is

q z

= p-jCOS0-j - P2cos0

2

- k .

the vectors u_ and is given by

u = |u| = p-|Sin0-j

v = |vj = P2sin©2 .

17
Throughout the analysis given in this report we use the coordinate

system with z-axis in the direction of k. However from the experimental standpoint, it may be convenient to use the coordinate system

with z-axis in the direction of the incident beam (jd.j). We therefore

present here some details of the relation between these two systems.

In the system with z-axis in the direction of jd-j we denote the

polar angle of by The P° lar angle of k_ is again 0-j . The

azimuthal angle between the vectors ^

and K in this system is

denoted by co. These vectors are shown in fig. 3b. It should be noted

that if <p, the azimuthal angle of relative to £-| in the

k-oriented system, is small (0 < <p « 1) then, co, the azimuthal

angle of relative to k_ in the p^ -oriented system, will be near

tt. Since we will be concerned with small 4>, it is useful to write

CO = TT + CO* ,

in which case 0 < f « 1 implies 0 < co' « 1 .

The angles just defined are related by

cosip = cose-|COS0

2 + sin0-jsin02cos<{> (III.!)

cos ©2 = cos0-|COSt|; + sin0-|Simpcosco

= cos0-jcosip - $in0-jSim|jcosoj' . (III. 2)

18
It is useful to express the angles of the final electron in the

j<-oriented system (©^

)

in terms of those in the £-| -oriented system

Oj>,w), and vice versa, and to do this in a form which lends itself to

the small angle approximation. From the above equations we have, first,

from eq (III.2),

cos6

2 = cos(<Jj + 6-j) + sin6^simp(l - cosoo') (III. 3)

and from eq (III . 1 )

,

costp

- cose,cose

2

COS(j) = —

sin0-|Sin6

2

cosi|> - cos0-|lcos0-]cos^ - sin0-|Sin^cos(ja']

sin0-|Sin0

2

sin0-|cosi^ + cos0iSin^cosa)'

sin0

2

si n (xp + 0-j ) - cos0-|Sim|)(l - cosw')]

sin©

2

19
We then have

2 2 2 2

sin e

2 sin <j> = sin 0

2

(1 - cos $)

2 2 = sin e

2

- (sin0

2

cos<t>)

2 2

= sin 0

2

- [si n (i^ + 6-j) - cos©.] si nip ( 1 - cosw')]

o

= 1 - cos 0

2

- [sin(ip + 6-j) - cos6^simjj(l - cosw')

2 = 1 - [cos(^ + 0-j) + sin0-|Simp(l - cosw')]

2

- [s i n

0-j) - cos0^simp(l - cosw

1 )]

= 2simJj(l-cosw' )[sin(^+0.| }cos0^ -cos(^+0-| )s i n©

-j ]-s

i

2 = sin ip(l - cosw')(l + cosw')

2 2 = sin ^sin w'

Thus, finally,

sin0

2

sin<}) = simpsinw'

^(1-cosw 1

(III. 4)

20
Thus if we have, initially, the angles 0

2

,4> fn the jc-oriented

system, then the angles in the ^-oriented system are obtained

from eq (III.l) (which gives ip) and eq (III. 4) (which then giyes co

1

in terms of ©

2

,<j> and ip. If, on the other hand we start with the

angles then the angles 0

2

,4> are obtained from eq (III. 2)

(which gives 0

2

) and eq (III.4) (which then gives <p in terms of

ip ,w

1 and 0

2

).

When all of the angles, 0-j , 0

2

, and 4> , are small, we have, from

eqs (III. 3) and (III. 4),

© 2

w ip + 0-j

and

0

2

<J) ^ ipa)

1

21
IV. HIGH ENERGY, SMALL ANGLE APPROXIMATION OF THE BREMSSTRAHLUNG

CROSS SECTION; GENERAL CONSIDERATIONS

We begin our consideration with the first Born approximation

differential cross section for bremsstrahlung as given originally by

Bethe and Heitler (see references [7], [12], [13]). This expression

accounts for the exchange of a single photon between the electron (or

positron) and the target in whose field the bremsstrahlung process takes

place. The cross section is thus proportional to Z , the square of

the nuclear charge, and hence does not include Coulomb corrections. A

brief discussion of the contribution of Coulomb corrections in the

region of energies and angles with which we are concerned here is given

at the end of this report, in section IX.

Our starting point is, as we have said, the first Born differential

cross section

(IV. 1)

where

(e-j - p-|COse-|)(e2 " P2 COS0

2

^

(c^ - p^cos0-|)(e

2

~ P2COS0

2

^

22
Here F(q) is the atomic form factor, discussed, for example, in

Sec. 1 1 E ( 3 ) of [13]. As it stands, this expression has no approximations in energy or angles. One could, therefore, deal with it in the

form just given. There are several reasons for not doing this, but

rather making high energy, small angle approximations. The first is

that the cross section without approximations, eq (IV. 1), is sufficiently

complicated that any of the structure we have mentioned is not at all

evident from the expression just given. Second, as we will see in the

course of this analysis, there are, for particular energies and angles,

extremely large cancellations of different terms in the expression.

Thus in a straightforward numerical evaluation of the expression (IV. 1)

4 there are cancellations leaving a remainder which is of order 1/e

relative to the individual terms. For the energies of concern here

(e-j « 140 MeV » 280 mc^) we note that 1/e-j 4 w 10"^°. One may thereby

very easily lose of the order of 10 or more significant digits unless

certain precautions are taken to write it in a form which takes account

of these cancellations explicitly, as in eq (1.1). While it is possible

to do this without making any approximation (in fact we do this, and

present the resulting expression in eq (VI. 4), as the first step toward

our final approximate form), it is nonetheless after making the approximations of high energy and small angles that this explicit cancellation

is performed most clearly. Finally, it is only after making these

approximations that we have an expression of sufficient simplicity that

it permits a detailed, analytical analysis of the structure of the cross

section, including a determination of the maxima and minimum of the

cross section.

23
A few preliminary remarks concerning these approximations are in

order, however. In most "high energy" analyses, it is assumed that the

energies, e, are large (e » 1) and that the angles are small, of

the order of 1/e. This is particularly appropriate if one wants

finally to obtain an integrated cross section, since at high energies

most of the contribution to the cross section integrated over angles

comes from angles 0 ~ 1/e. However, this is not the case of interest

here; for our present purpose we need the totally differential cross

section, at angles which are small (0 « 1), but not necessarily of

order 1/e. The first point is therefore that we will consider

throughout, for all energies and angles, e » 1 and 6 « 1 , but not

require 6 ~ 1/e. With these assumptions we then have several independent small parameters, 1/e-j, l/e

2

, 0-j , 0

2

> in the expression for

the cross section. The error in our approximate cross section will be

determined by the largest of these, which is 0^ for the experimental

conditions considered here, namely, 0^ > 1/e-j, and q * qm-j n

: as we

will see, q ~ qmin requires « £

2

^ 2

* ^rom w* 1

'' c * 1 e

2 > ^e

2 and 0

2

** (e-j/eg)

©i

>

0^ > 1/e-j. Our largest small parameter is thus

©2 w

^i/ e 2)

e l* ^ further point essential to our analysis is that we

retain, in the cross section, all terms of order 0 2> neglecting only

2

terms of order 0

2

relative to those which are retained. Thus our

high energy, small angle approximation is in error by less than 1% for

0

2 < 6°. (In fact the relevant parameter seems to be closer to %0

2

.)

Had we neglected terms of the order of 0 2> the errors in the resulting

expression for the cross section would have been of the order of 20% for

these same In particular, the expression for the cross section in

24
which terms of order $ 2

have been neglected is strictly zero when the

component of the momentum transfer perpendicular to the photon direction

is zero: qx

= 0, as may be seen from earlier calculations [14]. It

does not, therefore, give a good representation of the differential

cross section in the region of the minimum.

The expression (IV. 1) for the cross section has two important

in the denominator. In the next two sections we examine each of these

factors in detail and derive approximate expressions for them, valid

for high energies and small angles.

factors, (1) the expression in

/ x 4

numerator, and (2) q ,

25
V. APPROXIMATION OF q

2

, THE MOMENTUM TRANSFER

FOR HIGH ENERGIES AND SMALL ANGLES.

? In this section we examine the expansion for q

2

particular we derive an approximate expression for q ,

energies e-j and an<^ small angles 6^ and 6g.

2

We start with the exact expression for q ,

q

2 = (£, - £2

- k)

2

,

and write

q

2 2

where

= £li " £21

= £

- V

is the component of £ perpendicular to Ik

and

q z = Pi cose

1 - P2COS62 - k

is the component of £ in the direction of k.

SQUARED,

n detail . In

valid for high

(V.l)

(V. 2 )

(V. 3 )

(V. 4 )

26
Here

and

u = |uj = p-j si n6-j

v

s |v| « p 2

sin0

2

( V . 5

)

q z = PiCOse

1 - p

2 cose

2

- k

= (e

2

- p

2

cos0

2

) - (e^ - p^cose^)

= d

2

- d

1

, (V. 6)

where

e

l "

e

2

“ p

2

cos0

2

• (V. 7)

We next obtain expansions for d-j and d

2

which are useful for high

energies and small angles.

d

1 = e-j - p-j cos0-j

2 2 2

e-j -

p-j cos 0-j

e-| + p-j cos0-|

e

]

2

- P]

2 + P-| 2 0 - cos

2 ©^

£-| + p-jCOS0-|

27
Therefore

d

i

2 2

1 + p-j sin e

e-j + p-jCOsG-j

1 + u‘

e-j + p^cosG-j

1 + u

2e

1 - (e-j - p-jCOsG-j

)

1 + u‘

2e, - d

1

- + 1 + u

2 = 0 ( V . 8

)

and hence

d-j = e-j ± ^-j

2

- (1 + u 2

)

.

Since for small angles and large energies d-j = 0(l/e), it is clear

that we must take the minus sign. Thus

28
Since it will be of use later on, we note that

29
q

z

1+v

2

_ (1+u

2

) 2e

2

~ 2e^ ( V . 1 3

)

(1+v

2

) 2

(1+U

2

) 2

cL d2

We note at22this point that the expansion parameters for — and — £

] e

2 are and • In order to obtain a better appreciation of

e

l e2

these expressions, we examine them for high energies and small angles

We then have

e

l 61 £ 2

® 2

30
Thus for

®1 ~ V^-j » 1+u 1 2 ^ 2

£

i e

i

and for

e

i £ Ve-j

1+u ~ ft 2

2

6

1

Similarly, for

©2 ^ 1/^2 * 1+v ^ 1 2 ~ 2

£ 2 ^2

and for

0 2 ~ ^

/

e

2 *

1 +v

c ^ n 2

2

0

2

For the experimental situation of interest here, 0-j > l/e-|.

we are concerned with small momentum transfers, q

« 1, for

v «* u, i.e., e

2

e 2

w £

1 0

1

» or

£

1 0

1 > 0 - since < £]

(Also ©2 > 1/^2 > since e-j0-| > 1.)

Further,

which

31
The largest of the parameters 1/e-j , 0-j , l/e

2

^

, ©

2 2

U

1 °i

in the present study, 0„ . We note further that in both — and -p-

/ d 0 * e„\ e

l d

l

is, therefore,

d, eand similarly for

factor Thus the parameter in this expansion is

and the first correction has an additional

1 n

2

4

0

2

^1 0

]

N

and, where this is dropped, an exact calculation indeed shows the error

to be given very accurately by this parameter. Note that for 6^ = 1.2°

The neglect of these terms in the expression for q z thus introduces

(for these values of 0-j, and eg) an e^ror of 10” . In

q

222 = qx + q

z this introduces an error of 2 x 10

-34 , and in q

(which appears in the denominator of the cross section), and thus also

-3

in the cross section, an error of 4 x 10“ (** h°l) •

2

We will neglect these terms of order 0

2

in this analysis. We

then have

2 2

1+v 1+u

q z ' 2e

2

” 2e

]

We note here that

32
9i

2

= (u - v)

2

= (u - v)

2 + 2uv(l - cos<f

)

2 2 2 2 Thus unless both (v - u) and <J> are small, of order 0

2

, q

z 2 2 will be much smaller, of order 0

2

, relative to

qj^

. Therefore, in

the expression for q

z we may consider v - u « u, and write

v

2 = (v - u + u)2 = u 2 + 2u(v - u) + (v - u)2

,

so that

where

=

1 + u

2 + 2u(v - u) + (v - u)

2

1 + u

2

2e, 2en

> „ , » a- (, . ul , liU-al

2 e -|£2

= «(1 + u 2

) + (v - u) +

(v-2

~ V). 2 ,

6 = 2e^e

2

(V. 14)

Therefore

33
I 2 = (u-v)

2 + 2uv ( 1 -cos0 ) +

|fi(l+u

2

) t (v-u)

+ j

Here, in the last term, q

z

, we have

6

2

(l+u

2

) + 26(Hu2) u ^"u )

+ /iL\

2

(v . u )2 +

{v- u]

4 + u(v-u)

3 + 6(l+u

2

)(v-u)

2

4e,

2

The terms in the last line here are of order 02 , or smaller, relative

2 2

to the term (u - v) 1 n q and may therefore be neglected:

<S(l+u

2

) . k(Hu2) .. u

2

. 1 n 2

^

"

* 2 . 2 4 ®2

2^2 ^

e

i e

2 ^e2

2

We note that the second term in the expression for q , viz.,

2

z

2(5 ^ + u ^ y(v-~- -u) is, for v - u » 6(1 + u

2

) , of order — relative

e

2 9 e 2-3 2

to 6 (1 + u ) , i.e., of order 0

2

- Since ©

2

% 4 x 10 , we have

-2

©2 % 6 x 10 and therefore we keep this term. As we will see, the

cross section achieves its maximum for |v - u| % 6(1 + u ). Thus in

this region neglect of this term would introduce errors of the order

of 10% in the cross section.

34
We thus have, finally, our high energy, small angle approximation for

the square of the momentum transfer:

q

2 = (v-u)

2 + 2 uv(l-cos<j>) +

2

-6ilt. ^.. ) .u,(,v-u) + 6

2

(1+u

2

) (V. 1

5

) e

2

2

In figs. 4 and 5 we show q considered as a function of v ( i . e . , 62)

for fixed values of e-| , 0 -j , and <}>. Note that the neglected

2

terms are of relative order 0

2

(and never larger) for al

1

values of

2 v-u (i.e., for v-ueO, v-u=O(u0), v - u = O(u0), and

2

v - u = 0 (u)). Since we will require derivatives of q with respect

to v for our further calculations, we note that

(q

2 )' = 2 ( v - u) + 2u(l - cos<f>) + 26(1 + u

2

) —e

2

(q

2

)" = 2 ( V . 1 6

)

2 Again, since we have kept the principal terms multiplying both (v - u)

2 2 and (v - u), the errors in (q )' and (q )" are also of relative

2

order •

2 For the minimum value of q considered as a function of v

(i.e., 0

2

), for fixed values of e-j , £2, 0-| , and <j>, we have

(q

2

)' =

2

jv - u + u ( 1 - coscf>) +

u- —

j

= 0

or

V - U = -U f( 1 - COScj) ) + ^0 +-U_JL

e

2 .

35
or

v = u jl

- £(1

- cos<j>) +

U ^ jj

, (V.

2

and since (q )" = 2, this is indeed a minimum.

With e

] = 1°, e-j = 140 MeV, e

2

= 45 MeV (1 me

2 = 0.511 MeV)

2 we get 0^ = 3.1094° for the position of the minimum of q .

Note that q“ as given by this approximation is of the form

q

2 = (v - u)

2 + 2B( v - u) + C

with

B = u ( 1 - cos4>) + 6(1 + u

2

) —e

2

2

C = 6

2

(1 + u

2

) + 2u

2

(1 - cosd>) .

Thus we can write

q

2 = (v - u + B)

2 + C - B

2

.

The minimum value is thus clearly achieved when v - u + B = 0 (as we

have just seen), since C and B are independent of v. The minimum

2

value of q is then

36
q

. = C - B‘ Hmin

6

2

(1+u

2

) + 2u

2

(l-cos<J>) - u

2

j(

1 -coscj) ) + --O. +u. jj

6 2 (1+u

2

)

2

jl - + 2u

2

(l -cose))) ^

- -6

- Otu.2 )^

u

2

( 1 - COS(())

2

Here — = • —U- < A • —At . Thus the corrections in the two large

e

2 2e-j c

2

2 2 e

2

2

2

6(l"f"

2

l 22 parentheses, and —

£

- u

, are of relative order = 0(e

2

). c

2

2 c

2

Since we have neglected terms of this order multiplying C /specifically.

2 2

2 \

V multiplying 6 (1 + u ) ) these terms should also be neglected.

Thus we have

qmin

= ^ + + u2 ^ " C0S(*

) H 2 • 0 - cos4»)

)

Here

(1 - COSCp ) ( 2 - (1 - COS({>)) = (1 - COScf))(l + COScj)

)

. 2 = sin <()

37
Therefore

qmin

= ^ + ^ + • wi n (V. 18)

For <J> = 0, 0-j = 1°, e-| = 140 MeV, and k = 95 MeV, this gives

qf. = 8.45222 x 10" 3

wi n

( V . 1 9

)

Note that if we keep the factor [1 - we obtain, for the value

2 V

£

2 / 2 of q . given directly by our approximation for q , we have

ql. = 8.4273 x 10" 3

, win

_3

a difference of approximately 3 x 10 . However, as we have noted,

2

there are other terms of this order in the exact expression for q . . r win

38
VI. HIGH ENERGY, SMALL ANGLE APPROXIMATION OF THE DIFFERENTIAL

CROSS SECTION FOR BREMSSTRAHLUNG

In this section we examine in detail the expression in the

numerator of the cross section, in

j

eq (IV. 1). In particular we

derive an approximate expression for

| |, valid for high energies

e-j and £ 2> and small angles 0-j and 0

2

: e-j » 1

» ^

>:> ^

6-j « 1 , e 2

« 1

.

We start with the exact expression for

| |

, given in ( I V . 1

)

.

With the notation as defined in section III describing the kinematics

we then have

2.2 2.2

/ 1 p n sin 9, 0 0 p 0 sin 0 O 0 0

{ } = \ ( 4e

2

) + , (4e

l ^

}

(e

1 -p-,cos01

)‘ (c

2 -P2 cose 2^

y 2p

1 p

2 sin0^sin0

2 coscJ)(4e^e2 _(:

i )

(s:-|-p^cos0i ) (£2-P2COS0

2

)

2 2 2 2 2

2k (p-j sin 0-|+p

2

sin 0

2 -2p^p

2 sin0^sin0

2

cos(J))

(e-|-p-]COS0.| )(£

2 "P2 cose 2^

where

(VI. 1)

d-j =

El - P-j COS0-|

d

2 = e2

- P

2 cos02 . (VI. 2)

39
Next we define

£

= 2e

l d

l

11 =

2£

2

d

2 (VI. 3)

The expression for

| |

given in eq (VI. 1) may be written very simply

in terms of £ and n:

{ } 8k

2 e-|£2(iL'V.)^£n + T 6e

-j

2

^2^ ( u.£-vn

) ^ - 4q2

(e^u£-e2Vri)

2

. (VI. 4)

We will examine each of the three terms appearing here, obtaining an

estimate of their relative magnitudes for high energies and small angles.

First, however, let us consider £ and n in detail. We have

1 £

=

2^ 1

(

£

1 ~

£-j + P

1 COS0

1

2^1 ( £ -| _

P-j cos 0

1

)

2e^ - (e^ - p

] cos01

)

2e

l ~ P'1 ^

_J _

1

1 + u

2

2 £-j (e^ + p-jCOs6-|

)

(VI. 5)

40
since

p p p p p p p

e-j - p

1 cos 6-! = e-j -

p-j + p

] (1 - cos 6

]

)

2 2 = 1 + p-| sin e.j

= 1 + u .

In similar fashion

n

1 1 p

"

1 + v 2e2^2 + P2COS0 2^

(VI. 6)

The second term in the expressions (VI. 5) and (VI. 6) for £ and n is

2 2

small, of order 9 (or of order 1/e if 0 < 1/e) relative to the

first term:

1 + u

2

2e-j (e-j + p-|COS0-|

)

-i—L

1 + u

2

\ 2e^(e^ + p.|COS0-|)

2 2

1 + u 1 + u 1 « p

as p + -jr ©I

2e-|(e-| + p-|COS0-|) 4e^

41
Thus we may write

5 = «o

- 5,

n = n

0

- n-| (VI. 7)

where

Sn

" O » s 1

1 + u 2e-j (e-j + P

1 COS01 )

n

= —-—p

» hi = • (VI. 8)

1 + v 2^2 (^2 + P2cos6 2^

2

Then 5 Qand n

Q

(and hence also 5 and n) are of order 1/u

for u > 1 (6 > 1/e) and of order 1 if u < 1 (0 < 1/e). (As may

be seen from eq (V.15), q

« 1 implies u *» v and hence also £ * n

0

from eq (VI. 8)). Moreover, as we have just shown, C-j and are of

2 2 order 0 relative to 5

Q

and o

0 (or of order 1/e if 0 < 1/e).

Returning now to eq (VI. 4) we note that the first term there is

Sk^e-^fu. - v)^5n = Sk^e-^q^

Writing

e-^jk = 0(e)

u,v = 0(u)

5,n = 0(5)

e r

0

2 = 0(0)

42
4 2 2 we may then say that this first term is 0(e % qx

). Now, as we will

show, not only this first term, but < > as well is 0(e £ qx

) over

a very wide range of values of qx

, viz., for "large" qJL

: qx = 0(u),

for "small" qx

: qx = 0(6u) (or qx = 0(u/e) if 0 < 1/e), and for

qx

= 0(0 u) (or qx = 0(u/e ) if 0 < 1/e), this last region,

2

qx = 0(0 u), being the region of the dip in the cross section, indicated

2

in figures 4 through 10. For qx < 0(0 u), the second and third terms in

eq (VI. 4) do not decrease further, so that

j |

remains of the order

/ 2 x 4 2 4 2

of magnitude it has for qx = 0(0 u), viz., of order e £ 0 u .

Thus a numerical evaluation of eq ( I V . 1 ) , the initial expression

for { }, shows significant cancellations for values of © 2

and (p

corresponding to very small qx

, so that in this dip the value of

| | is of order 0

4

(or 1/e

4

if 0 < 1/e) relative to its value

for qx = 0(u). However, in contrast to eq (VI. 4), where the nature

of this cancellation is manifest, in eq ( IV . 1 ) it is quite obscure.

To demonstrate these order of magnitude statements we return to

eq (VI. 4) and proceed with our examination of the terms there. As we

4 2 2

noted, the first term there is 0(e £ qx ). In the second term we

write

(u? - vn)

2 = (u - v)hn + (u

2 £

- v

2n)(? - n) (VI. 9)

Let us consider the last term here in detail. From eqs (VI. 5) and

(VI. 6) we have

43
1 1

or

Hence

i =

1 + u 2c^ (2e-| - d-j

)

1 + u

2

‘ 2

4 £

1 “ 2e

i d

i

1 1 2 ", 2 1

1 + u 4e

1 i

Thus

2

- 2

1 + u 4e, £

- 1

5 1 +

4e.j

2

£

- l) 1 + u

2

5 =

1 / 4e l^' 1

1 + u

2

\ 4e

2

£

1

1 -

1

1 + u 2

46^5

44
and

2 2 2 1 C + u € = K

4eso that, finally.

uV = 5(1 - 5) - j V

(VI. 10)

and, in similar fashion.

v

2n2 = n(l - n)

^

4e

2

^ (VI. 11)

Thus for one of the factors in eq (VI. 9) we have

u

2 S

- v

2 n

= u 2

(£ - n) + n(u

2

- v

2

)

Hence, from eq (VI .9)

,

(u£-vri)

2 = (u-v)

2 ?n + u2

(^-n)

2 + n(u2 -v2

)(£-n) (VI. 12)

Now from eqs (VI. 5) and (VI. 6)

K - n

=

1 + u 1 + v —

j

+ 0(l/e

2

)

i2-2-~- v-2^ -2 + 0(l/e

2

) (1 + u

2

)(l + v

2

)

45
Thus from eqs (VI. 5) and (VI. 6)

Further, from

we have

Thus

Note here that q

the terms denoted

% - n| = 0(|u 2

- v

2

|£n) + 0(l/e

2

)

= 0 ( u

| u - v|£ 2

) + 0(1 /e

2

) . (VI. 13)

qx

2 = (u - v)

2

= (u - v)

2 + 2uv(l - cos<J>)

2 2 1 = (u - v) + 4uv sin j 4

|u - v| < 0(qA

) . (VI. 14)

1C - n| < 0( qi u£

2

) + 0(l/e

2

) . (VI. 15)

can go to zero; therefore we can not simply neglect

here as 0(l/e

2

).

46
From eq (VI. 15) we have

U - n)

2 = 0( qi

2

u

2 ?

4

) + 0( qi

u£

2

/e

2

) + 0(l/e

4

) . (VI. 16)

Referring to the terms in eq (VI. 12), we then have

(u - V)

2

5n = 0(qi

2

C

2

)

u

2

(S - n)

2 = 0(qx2 uV) + O^uV/e2

) + 0(u

2

/e

4

) . (VI. 17)

Here we note that

0(q1

2 uV) = 0(q12

C

2

(u

2

C)

2

) < Ofq^V)

and

0( qi u 3

?

2

/e

2

) = 0( qi u?(u

2?)/e2

) <0( qi u£/c

2

)

since u

2 £ < 0(1) from eq (VI. 5).

Thus

u

2 (S - n)

2 < O^V) + 0( qi u£/e

2

) + 0(u

2 /e

4

) . (VI. 18)

Finally, from eq (VI. 14) and (VI. 15),

47
n(u

2

- V

2

)U - n) = Otq^uV) + Ofq^uC/e

2

)

= 0(qx

2

C

2

(u

2

5)) + 0(qx uC/e

2

)

< 0(q1 V) + 0(qx u5/e

2

) (VI. 19)

Substituting eqs (VI. 17), (VI. 18), and (VI. 19) in eq (VI. 12), we have,

for the second term in eq (VI.4),

16e

1

2 e

2

2

(u^-vq)

2 = OfeVq^) + 0(e2

uCqi

) + G(u

2

) . (VI. 20)

p Now, as we will show shortly, the third term in eq (VI.4) is Q(u ),

2

and does not cancel the terms of 0(u ) in eq (VI. 20). Thus the terms

in eq (VI.4) give contributions of 0(e

4

C

2

qi

2

)» 0(e

2 u?q

i

), and 0(u

2

)

to

{ }

. We may write these as

o(u

2 (ch J-)

2

)

, o(u

2

(e

2

Z 5l))

, and 0(u

2 ). (VI. 21)

Note here that

6 > 1/e

6 < 1/e

48
We may thus distinguish three regions of q:

1) "large" q, for which qj - 0(u)

2) "small" q, with qx = 0(u6) = 0(q 7 ) for 0 > 1/e

(or q^ = 0(u/e) for 0 < 1/e).

3) small q with qx = 0 ( u0

2

) « q

z for 0 > 1/e

(or qx = 0(u/e

2

) for 0 < 1/e).

For "large" q, the first term in eq (VI. 21) is the largest; the

? other two can be neglected since they are of relative order 0 and

4 /

' N 2 4 / 0 (for 0 > 1/e) or of relative order 1/e and 1/e (for

9 < 1/e). For "small" q as given in 2) the first term is still

the largest, but the second term in eq (VI. 21) must also be kept since

it is of relative order 0 (for 0 > 1/e) or of relative order 1/e

(for 0 < 1/e). The third term may still be neglected, since it is of

2 2 order 0 (or 1/e ) relative to the first term in eq (VI. 21).

Finally, for small q as specified in 3) all three terms in eq (VI. 21)

are of the same order of magnitude and must all be retained.

We have thus demonstrated the statement made earlier, that over a

2 very wide range of values of qJ[ , specifically for 0 u < qx < u,

the order of magnitude of

| |

is that of the first term in eq (VI. 4),

4 2 2 namely 0(e E, q^ ). Since we want an approximate expression for < >

2 2 in which neglected terms are of relative order 0 or 1/e (relative

to the terms retained) for al

1

qi

, we must retain the small terms of

2 order u , which are the only ones that remain when qi

-> 0. Only

2 2 terms of relative order 0 (or 1/e ) for al

1

qi

will be neglected.

49
From eq (VI. 21) the terms to be neglected are of order

and

2 2 uV 5

or of order

and

We therefore begin our approximation of

^ |

by neglecting, in

2 2

eq (VI. 4), these terms of relative order 6 or 1/e . We consider

first the last term in eq (VI. 4):

4q

2

(e

1 ^-e

2

XJl) 2 = - 4(q,

2

+q

z

2

) (e^ u£-e

2yn)

2 (VI. 22)

and note that

(e

1 uC-e2 vn)

2

= 0(e

2

u

2

£

2

) = O(e

4

0

2 ^2

)

.

2 Thus for the term in eq (VI. 22) with factor we have

- 4qi

2 (e

1 u?-e

2 vn)

2

= OfeVq^e2

) ,

2 which is of order 6 relative to the first term in eq (VI. 4), and

hence may be neglected. We are then left with

50
4q

z

2

(e-j u^-e

2yn)^ = - 4e

1 e

2 q

z

2

(u£-vn)

2

’ 4q

z

2 [(e

1

2 -e

1 £

2

)u

2

?

2

t (e^-e^JvV] (VI. 23)

The first term on the right-hand side of eq (VI. 23) may be compared with

the second term in eq (VI. 4); it is of relative order

42

= “(v)= °\ e

4 ) =O(04) + 0(1/e4)

and hence may be neglected, since

1 ^ - o'1

^2e

2n 2e-|£

2 r 2

e K

(VI. 24)

We are thus left with the remaining term on the right-hand side of

eq (VI. 23) which may be written as

4q

z

2 k(e.|U2

?

2

- e

2 v

2n2

)

= - 4kq

z

2 [(£.,-£ 2 )uV + £

2 (uV-v2n

2

)]

= - 4kq

z

2 [kuV + £

25

2 (u2 -v

2

) + £

2 v

2 (?2 -n2

)]

. (VI. 25)

Consider next the second term in eq (VI. 25):

- 4kq

z

2 e

2C

2 (u2 -v

2

) = 0(e

2 Cuqi

. (q

z

2?))

51
from eq (VI. 14). But from eq (VI. 24)

^z + o(e

2

)

o 9

Thus the second term in eq (VI. 25) has terms of 0(0 ) and 0(l/e )

times the second term in eq (VI. 21), and hence may be neglected. In

the third term in eq (VI. 25) we have, from eq (VI. 13),

e 2

v

2

(£

2 -n

2

) = e

z ^{K+r])U-r])

e

2 v (S+rl) [-

(U

2 -v2

) (l+v

2

)(l+v

2

)

+ OO/e2

)']• (VI. 26)

The first term here is thus of 0(e:u

2

£ • |u

2 -v2

|£

2

) < 0(e£

2

|u

2 -v2

| );

it is therefore of the same order as the second term in eq (VI. 25) and

may also be neglected. We are thus left with the remaining term in

eq (VI. 26), of order u £/e. Compared with the first term in eq (VI. 25)

this is of relative 0(l/(e 2

£)) = = 0(6

2

) + 0(l/e 2) and thus

may also be neglected. We are thus left with the first term in (VI. 25),

viz.

,

4k

2 uVqz2 [

kU5

(

e

2 n

" e

l ?

)]

*

• h5-e2 n)]

52
Here we write

giving

e-jC - e

2n

=

( £ -|“ e 2^ n +

,.2 2 r

2 2

4k u c q

z

fl

k

(V 1.27)

The second term here gives, on expanding the square, contributions of

0(u

2 (tH)) and 0 (u2 (fcH)

2

)

< 0(u

2(ta)) since ^-o(^)<0(l).

From eq (VI. 15) we have

2

The first term on the right-hand side here is 0(0 ) relative to the

second term in eq (VI. 21) and may thus be neglected. The last term here

may be written

0((1+u

2

)6

2

) = 0(u

2

e

2

) + 0(u

2 /e2

)

2 2

and thus makes contributions of 0(6 ) and 0(l/e ) relative to the

last term in eq ( VI . 21 ) ; it may therefore also be neglected.

53
2

Finally, therefore, neglecting terms of relative order 6 and

1/e

2 in

| |, we see that the last term in eq (VI. 4) may be replaced

by

(VI. 28)

We continue our high energy, small angle approximation, considering

next the first term in eq (VI. 4),

2 2 2 2

8k e

^ g

^

(M. - v.) Cn = 8k e-jC^

From eq (VI. 7) we substitute £

s

£ 0

- €j and n = n

Q

- n-j in this

term, and recall from the discussion surrounding eq (VI. 7) that ^

and n-j are of order 6 relative to and n

Q

(or 0(l/e ) for

6 < 1/e). We may therefore neglect ^

and in this term, and

replace the first term in eq (VI. 4) by

8k

2

El e

2

(u - v)

2

S0

n

0

. (VI. 29)

Finally, we obtain the high energy, small angle approximation for

the second term in eq (VI.4),

16e-|

2

e

2

2

(uC-vn)

2 = 16e

1

2e

2

2 [(u-v)

2^n + u 2

(£-n)

2 + n(u2 -v2

) (£-n)] (VI. 30)

54
from eq (VI. 12). In the first term in eq (V 1.30) we may replace

by $ 0n

0

» as we just did in arriving at (VI. 29), again with neglect

2 2 of terms of relative order 6 and 1/e . In the remaining two terms

in eq (VI. 30) we substitute £ = ^

and n = n

0

- n-j from

eq (VI. 7). The terms with subscript zero are then

“

2 W2 + Vu2- v2 )(s0-V = [“2 (W + Vlj2 - v2);K Vno

)

= (u

2

?0

-v

2n

0 )(50

-n0

) •

Now from eqs (VI. 8), (VI. 10), and (VI. 11),

u\ - v\ *

-

(1 ’V

= - <w •

Thus the terms with subscript zero are simply.

( W = 1+u

2

l+v

2

y

, 2 2x

(u -v ) ,(l+u

2 )(l+v2

)

- ( u2 - v2 ) «0 V (VI. 31)

The terms with subscripts zero and one are, since from eq (VI. 7)

5

= 5

0

- C, , n = n

0

- n,

55
- 2u

2

(^0 -n0) (C

1 -n-j ) - n

0

(u

2 -v

2

)(C

1 -n

1

) - n-| (u

2 -v

2

) CC

Q -n0

)

2 2 Writing - n

Q

= - (u - v )£0nQ> these terms can be written as

(e-j-n-, )(u

2 -vz

) n

0

(2u

2 c0-1) + (u2 -v2

) . (VI. 32)

From eq (VI. 8) and n-i are 0(l/e

2

); from eq (VI. 14) u

2 -v2 = 0(uq

x

)

Therefore, for the second term here we have

(u

2 -v2

) C

0

n

0

n, = 0(u2C

2

qx 2

/e2

) - 0(e

2

C

2

qx

2

)

J It is thus of 0(0 ) relative to the first term in eq (VI. 30) and may

thus be neglected.

In the first term in eq (VI. 32) we write

2u^

0

- 1 _ 2u‘ W

- 1

uLi

u

2

+l

= OZ-lfc,

We thus retain, of the terms with subscripts zero and one,

(5

1 -n

1 )(u

2 -v2

)5 0n

0

(u

2

-l) (V 1.33)

56
Finally, the terms with subscript one are

u

2

(C

1 -n1

) 2 + n-, Cu

2 -v2

)(5

1 -it, ) . (VI. 34)

2 4

The first term in eq (VI. 34) is 0 ( u /e ). It therefore gives a

contribution of 0 (u ) to eq (VI. 30) and must be retained. Together

with the first term in eq (VI. 30), of 0(e

4

g

2

qj. 2

), we thus have

contributions from these two positive terms of

0(e

4

C

2

qi

2

) + 0(u

2

) > Ote^u) . (VI. 35)

(Note that a

2x2 + b2

y

2 > c2

(x

2 + y2

) > 2c

2

xy where c

2 = smaller of

a and b .) The second term in eq (VI. 34) is, from eqs (VI. 14) and

(VI. 37) of 0(uq

i /c

4

). It therefore gives a contribution to eq (VI. 30)

2 2 2

of 0(uq

i

) = 0(u £uq

x

) = 0(e £qx u) • 0(6 ), i.e., a contribution of

2

0(6 ) relative to that from the first term in eq (VI. 34) together with

the first term in eq (VI. 30), shown above in eq (VI. 35). We may

therefore neglect the second term in eq (VI. 34). Thus of the terms

with subscript one we retain only

u 2

(?

1 -n1

) 2

. (VI. 36)

In eq (VI. 33) and (VI. 36) the factor l-j-n-j may be simplified further,

2 2 again neglecting terms of relative order 1/e and 6 . From eq (VI. 8)

57
Cl =

1 "

2e-| (e^tp^cos0^

)

^7L2e

1 ”TI1 "^sT1')I

1 4

£l

2

(l - d^/2e-|

)

1

4ei

2

(l - l/(4e^

2

5))

Z

Neglecting the term

we have

= 0(l/e

2

) + O(0

2

)

as

1

and, similarly.

from which
5i - n, ~ 1/_L

*w

1

'

*7,

k(e

] + e

2

)

, 2 2 4ei e

2

(VI. 37)

Substituting this in (VI. 33) and (VI. 36) gives, together with

eqs (VI. 30) and (VI.31), the high energy, small angle approximation for

the second term in eq (VI.4):

16ei

2

e

2

2

(u£-vn)

2 ^ 16e-j2 e

2

2

(u-v)

2

C0% - (u

2 -v2

) K

0\

Z

- 4k(e

1 +e

2 )(u

2 -v2

)6 0n

0

(u

2

-l

)

+

k ( c -|+£ 2 )\ 2

> £

1 £

2 /

(VI. 38)

We now have, in eqs (VI. 29), (VI. 38), and (VI. 28), the high energy,

small angle approximations to the first, second, and third terms in

eq (VI.4), respectively. The last term in eq (VI. 38) may be added to

(VI. 28) to give

k(ei+e 2

) \

2

u

2

/k2u V

v £

1 £

2 /

U \

G

1 e2

/

4k

2

u 2

e

l e

2

(VI. 39)

59
The first term in eq (VI. 38) may be added to (VI. 29) to give

80^2 ( ei

2 +e

2

2

)(u-v)

2

C

0n0

•2 "a. u (VI. 40)

We thus have

| } « 8e

1 e

2

^

e: l 2+e

2

2 ^-‘-^2

^o

T1 o

' 16e /

£

2

2 ^2

* v2 ^ ^o\

4k(e 1+e

2

)(u

2 -v2 K0

n

0

(u

2

-l) (VI. 41)

These four terms are indeed of the order of magnitude given in eq (VI. 21):

The first is

0(8% V) = 0 9

the second is

0(e

4

u

2

qi V) = 0(e\ 2

£: 2

(uC)

2

) < 0(e\ 25

2

)

since uE, < 0(1) for all u, the third is

0(c

2

uq

i ?

2

(u

2

-l)) < 0(6^5) = 0

60
since £(u

2

-l) < 0(1) for all u, and the fourth term is Q(u

2

).

We next write eq (VI. 41) in a form that is manifestly positive.

To this end we substitute

(u-v)

2

= (u-v)

2 + 2uv(l-cos(j>)

= (u-v)

2 + 4uvsin 2

|-

2

in eq (VI. 41) and obtain, separating terms with factors (u-v) and

(u-v),

{ ) “ 8e

l e

25

ono[el 2 + e

2

2

- 2e

l e2 (u+v)2?

o

T1 o]

(u- v)2

- 4k(

£l +e2 )C 0

ri 0 (u+v)(u

2

-l)(u-v) +

+ 32e

1 e

2 (£

1

2

+£

2

2

)C 0n

0

uvsin

2

|

. (VI. 42)

In the first term here we may write the expression in square brackets

so that it is clearly positive:

£

2 + £2

- 2£.£

? (u+v)

2 £ n = k

2 + 2£,£

2

(l y~

1 2 1 2 0 0 1 2

\ (l+u

2

)(l+v

2

)

= k + 2enC (uv-1

V

1 2 (l+u

2

)(l+v

2

)

61
Keeping in mind that we will wish to consider the cross section as a

function of the angles of the final electron, ©2 and

alternatively, as a function of the variables v and <J>,

write eq (VI. 42) in the form

{ } ~ f(v)(v-u)

2 + 4k(e

1 +e

2 )C 0n

0 (u+v)(u

2

-l)(v-u)

+ g(v)sin2

|

where

and

f(v) =

q 2 2 °e

l e

2 + 2 [j_ - .(uydi;

(l+u

2

)(l+v

2

) [e^ (l+u

2

)(l+v

2

)

g(v) = 32e-j£2( £ ]|^+e 2^^ uv

(l+u

2

)(l+v

2

)

Completing the square in eq (VI. 43) now gives

{ } - f (v) (v-

[2k(e

1 +e2

)C0n

0

(u+v)(u

2

-l)]

2 4k2

u

2

f(v) 0^2

+ g(v)sin2

|

.

u + 2k(e

1 +e

2 )£o

n

0

(u+v)(u

2

-] ) \

2

f(v) /

» or,

we may now

2

2

(VI. 43)

(VI. 44)

(VI. 45)

(VI. 46)

62
4 2

Now from eq (VI. 44), f(v) = 0(e £ )• The second term above is thus

0(u ), of the same order as the third term. Furthermore, if we

consider the second term above as a function of v, then setting

v = u everywhere in this term implies neglect of terms of relative

order (v - u)/u:

v = u + (v - u)

2

Thus, setting v = u in this term, which is of order u , introduces

errors of order u

2

• (v - u)/u = 0(uq

1

) in

^ These terms may be

neglected, however, since they are

0(l/(e

2

c)) = 0 (^rj = °(

1

?)

+ o(0: )

relative to the third term, of 0(e Cuq^, in the expression (VI. 41)

for { }• We therefore set v = u in the second term in eq (VI. 46),

which then becomes

16k

2 (e

1 +e

2

) 2 ?

0

4

u

2

(u

2

-l)

2

f(u)

2k

2

(e

] +e

2

) 2 u

2

/u

2

-l

2 2

£

1 £

2 oAl,

k

2 + 2

/u

2 -l\2

£

1 £

2 ai 2+l

from eq (VI. 44).

63
Adding this to the third term in eq (VI. 46) then gives

Here in the numerator we have, multiplying

( e

l +e ?)

2

4 !—£_

e

l e2

(e-|-e 2

) 2

e

l e

2

e

l e2

The expression in the numerator above thus becomes

64
For the sum of the second and third terms in eq (VI. 46) we now have

(VI. 47)

Finally, we consider in eq (VI. 46) the last term in ( ) , which is of

Juju-m < 0 ( u(u +1)

) „ „/ U

e

2

S

4 2 since f(v) = Q(e Z ). After taking the square, this term gives

contributions to

| |

of 0(e^uCqi

) from the cross term and of O(u^)

from its square; they are therefore both retained, as we have discussed.

Now setting v = u in the last term in ( ) introduces errors of

relative order (v - u)/u < 0(qi /u), as we noted following eq (VI. 46);

/ u ql\ /

ql\

these errors are thus 01 -g— * u~)

= ^(“

2“) anc* thus 9 1

* ve > ^rom the cross

term with v - u, contributions to

j |

of 0 (e £ *q

±

• ~

2~) = 0(e gqj.

and, from the cross term with the last term itself, evaluated at v = u,

contributions of o(c -4J— • = 0(uqi

). These errors thus give

\ z Z/

contributions of 0(l/(e^)) = Q ^

+^—

j

= 0(l/e

2

) + O(0

2

) relative to

4 2 2 2 the retained terms of 0(e Z ) and 0(e u^), and may therefore be

2

neglected. Thus in the last term in ( ) in eq (VI. 46) we may set

v = u, giving, for this term,

65
2 k(e

1 +e

2

) 50

2

2 u(u

2

-l)

f(u)

k(e

1 +e2 )(u^-l )u

9

2 2

2e

i £i

k

2

| 2/

u

2

-l^

2t = a

4 2 ,, e ^2 \u +1

(VI. 48)

We thus have, finally, our high energy, small angle approximation to

| |

, defined initially by eq (IV. 1):

|

j- = f(v)(v-u+a)

2 + g(v)sin2

%j> + p (VI. 49)

where f(v) > 0 and g(v) > 0 are given by eqs (VI. 44) and (VI. 45)

and p = p(u) > 0 and a = a(u) are given by eqs (VI. 47) and (VI. 48).

At this point a few observations concerning this approximate

expression for

j | are in order. We note first that all the terms

here are positive. Thus our approximate expression for

| j

is

positive, as it should be since the cross section is, apart from a few

simple factors, equal to

| }/c^’

Next, we note that

| j as given without approximation in

eq (VI. 4), simplifies directly to one given some time ago, [15], if we

consider either the region of "large" momentum transfers, q

** qx = 0(u)

or that region of small momentum transfers in which %

~ 9 Z

= O(u0),

and if in addition we drop terms of relative order 0 as well as those

2

of relative order 0 . In that case, as we have seen, we can set

4 = £ , n = n

Q

in the first two terms and drop the last term in

eq (IV. 4) entirely, giving

| |

= 8e.j£

2 |V(u-v)5

0 n

0 + 2e

1 e

2

(u£

0 -yn0

) 2

]

• (VI. 50)

66
This expression also provides a satisfactory starting point if one

wishes to integrate over the angles of either the final electron or the

emitted photon, since the significant contributions to the cross section

integrated over angles come from the regions of momentum transfers in

which qx = 0(u) and qx ~ q

£ = O(u0). The expression given in

eq (VI. 50) does not, however, provide a useful approximation to the

cross section in the region of the sharp dip, i.e., for small momentum

o transfers with qx < O(u0 ) « q z

. In this region the accurate

2 approximation, eq (VI. 49) is, as we have shown, of 0(u ). Since

4 2 ? eq (VI. 50) is of 0(e £ qx

), the relative errors in eq (VI. 50) in

this region are of 0(u^/(e^2

qi ^)) = O(l/(qi /u0‘

:') ) for 0 > 1/e.

The relative errors in eq (VI. 50) are thus of 0(1) for qx = O(u0 ),

and get larger for smaller qx

.

We now consider

| j as a function of v (i.e., ©

2

) for fixed

values of the other variables (e-j, Q-j and <|>) and determine

the value of v for which | | achieves its minimum as well as the

value of

| |

at the minimum. In figs. 4 and 5 we show

| j as a

function of for = 140 MeV, k = 95 MeV, 0.| = 1° and <|) = 0°.

We use our approximate expression, (VI. 49), which differs from the exact

2 2 expression only by neglect of terms of relative order 1/e and 0 ,

for all values of the variables. From eq (VI. 49) we have

tv { }

= { }

= (vMv-u+a)

2 + 2f(v)(v-u+a) + g' (v)sin 2^ = 0 ,

which we write in the form

67
Uf '(v)(^) (VI. 51)

2 + 2f(v)(^) + 31M sin

2^ . 0 .

First, we observe from eqs (VI. 44) and (VI. 45) that uf'(v), f(v),

and g'(v)/u are all of the same order, viz., 0(e £ ). Now we are

looking for a solution to eq (VI. 51) for small q, i.e., qx « u,

%# • i which implies « 1 and <p « 1 since | v-u | < 0(qi

) and

u<j> < 0(qi

). But from eq (VI. 48), a is small, of O(u0

2

) for

0 > 1/e (or 0(u/e ) for 6 < 1/e). Thus we are looking for a

solution to eq (VI. 51) such that v

~jj

+q « 1. Assuming qx < O(u0),

we have <j> < 0 ( 0 ). Then from eq (VI. 51) the solution is such that

v-~^+0

= 0(4> 2

) = O(0

2

). And from this it follows that the first term

in eq (VI. 51) is of order uf'(v)*<|> , i.e., that it is of order 0

relative to the second term in eq (VI. 51) and hence may be neglected.

2

We then have, neglecting terms of relative order <j> ,

v-u+g „ -g'(v)f

2

u 8uf (v)

Here too we may expand the right hand side about the point v = u.

Neglecting terms of relative order = O(0

2

) is thus equivalent to

setting v = u there. We then obtain

or

v-u+a = -a' Mfr 8f(u

68
( v (VI. 52) - u) min

= - 0

From eq (VI. 48) we have a, from eq (VI. 45) we have

g'(u) = 326^2 ( £ -|

( 1 +u

2

) 3

and from eq (VI. 44)

f(u) =

Q 2 2

8e, e-.

( 1 +u

2

) 2

Thus

(v-u) m

. mi n

k(e^+£2)u(u2

-l ) (e-| 2+e

2

2

) /u

2 -A

2 + — u| ——)<J) 9 2 2

2e, e, 2£

i £

2

+ 2 e

1 e

2 \u2+1

or

(VI. 53)

69
The value of

j (at this minimum is given by eqs (VI. 52) and

(VI. 49). From eq (VI. 52), recalling that a = O(u0

2

) and that

g'(u) = 0(uf (u) ) , we note that ( v-y ) rn

-j n = 0 ( u©

2

) f 0(u<j) 2

) = 0(u6

2

)

since we have assumed that <j> < 0(6). Thus, at the minimum, the first

term in eq (VI. 49) is 0(f(u) • (u0

2

) 2) = 0(u

2

f(u) • 6

4

). This is of

0(0 ) relative to the second term in eq (VI. 49), which is

0(g(v)4>

2

) = O(u

2 f(u)0

2

), since from eqs (VI. 44) and (VI. 45)

g ( v ) = 0(u f(u)). We may therefore neglect the first term in

2

eq (VI. 49), and in the second term set v = u (since v - u = O(u0 ) 2

at the minimum) and neglect terms of relative order <j) , thus again

2 neglecting terms of relative order 0 . We thus obtain, for the value

of

| | at its minimum,

{ } * 1 g(uH2 + P

' ; min

n , 2v 2.2 Se^lCi +&2 )u <p

(1+u

2

) 2

1 +

2u

+ 2

,u 2

+l

1 +u T~ . (VI. 54)

2

It may appear that the second term here, being of 0(u ), should be

2

(for 0 > 1/e). Thus for <f> = 0(0) the second term in eq (VI. 54) is

2 indeed of order 6 relative to the first term. However, since we

want this expression to be valid for all small <j>, proyided only that

<j> < 0(0), we must retain the second term, that being all that remains

when <p -* 0.

neglected since the first term is 0(e

4

u

2

£

2

4> 2

) =

70
It is worth noting that the minimum for

|

does not occur for

2

precisely the same value of v as the minimum for q , as is seen by

comparing eq (VI. 53) with eq (V.17). In both cases the value of v .j n

2 2 2 differs from u by terms of order <p , 0 , and 1/e , but the terms

are not the same for { } and for q

2

.

71
VII. DETERMINATION OF THE MAXIMA AND MINIMUM

OF THE BREMSSTRAHLUNG CROSS SECTION

In this section we consider the cross section as a function of v

(i.e., 0

2

) for fixed values of the other variables (c^, e

2

,

and 4> ) and determine the value of v for which the cross section

achieves its maximum and minimum values. In order to locate the

extrema of the cross section and evaluate the cross section at these

2

extrema we use the high energy small angle approximations for q and

•|

J> given in eqs (V.15) and (VI. 49). However, we begin our

considerations with the cross section as given in eq ( IV. 1

)

,3

d a da dft dk

k p 2

= Z 2 If e

2

hc w,

2

|[l-F(q)] 2 1

k

P2

1 { }

P

1 (2ir)

2

q

4 (VII. 1)

Clearly the location of the extrema of the cross section depends to

some extent on the atomic form factor, F(q), of the particular atom

in question. This dependence is very small, however, for the energies

and angles of experimental interest here, for which the minimum momentum

transfer, q. in eq (V.18), is relatively large compared with the

1 /3 inverse of the screening radius, 8 « Z

' /1 21 in the Thomas-Fermi

model for screening, [16]. Specifically, screening is important only if

2 2

qmi n < 8 . Even if we take the case which gives the largest screening

(qmi n as small as possible, 8 as large as possible), we have, from

eq (V.19), for <j> = 0, 6^ = 1°, e-j = 140 MeV and k = 95 MeV,

72
win

= 8.5 * 10

-3

whereas, for Z = 92,

3

2

= 1.4 x 10' 3

.

2 2

Thus in this case we have qm^ n > 8 . For smaller values of Z, and

2 2

for <j> > 0, q

. will be even larger, and 8 will be even smaller,

thus reducing further the effect of screening. We therefore neglect

screening in determining the location of the extrema of the cross

section, setting F(q) = 0 in eq ( VI I . 1

) . The extrema are then

determined from

fet-VM >] - [<iV‘

{ (]'

-IqV'j }' • 2(,

2 rV>'{ )

V

= «

2

>-T{ r-2<«

2

>'{ a

= 0 . (VII. 2)

Our determining equation is thus

q

2

{ }'=2(q2 )'{

}

. (VII. 2a)

73
However, we are looking for maxima and minima in the region of small

q, i.e., q < 0(u6) and hence also qi < 0(u6). Now from

2 2 2 2

qx = (u-v) = (u-v) + 4uvsin h<P we have |u-v| < qx and u<j> < 0(q

x

).

Thus for qL < O(u0) we have < 0(0) and $ < 0(0). We therefore

assume (p < 0(0), neglecting terms of relative order $ < 0(0 ) in

our high energy small angle approximations for q^, ( q

^

)

' , ^ |

and

| |

(eqs (V.15), ( V „

1

6 ) , (VI. 49) and (VI. 51)) and substitute

them in eq (VI I. 2a). Defining

z

= v-u

we then obtain

,2 + 2z(^ + + +

2 +

6

2

( Hu2

) 2

iZ Co

(VII. 3)

Ilf' (V) (z + 2

.

J

+ 2f ( v

) (z + 2) + |i

= 4 + ^ + m±h£i f(v)(z + 2)

z

v u/

u u 4

. (VII. 4)

o We then look for solutions of eq (VII. 4), first in the region z < 0(0 )

and then in the region z = 0(0). We find that for sufficiently small

<J>, the equation (VII. 4), considered as a function of z (or v) has

one solution in the region z < 0(0 ), corresponding to the minimum

74
of the cross section, and two solutions in the region z « 0(6),

corresponding to maxima of the cross section, one for z < 0 (v < u)

and one for z > 0 (v > u). However, as <J> increases the position

of the minimum moves to larger values of v, and the position of the

second maximum (the one for z > 0) moves to smaller values of v.

When <|> reaches a critical value, 6, the minimum and the second

maximum occur at the same point, which is then an inflection point of

the cross section. This point occurs for z = 0(0^). p or ^ ^

there is only one extremum, the maximum which occurs for v < u. This

behavior of the extrema as a function of may be seen clearly in

figures 11 and 12.

To guide us in the solution of eq (VII. 4), we note, from

g'(y)

eqs (VI. 44) and (VI. 45), that uf'(v), f(v), and are

4 2

^ all of the same order of magnitude, of 0(e £ ). For the other terms

in eq (VII. 4) we note that ^

,

2

e

2 u all of 0(0^ ) . (In these estimates we assume, for ease of presentation.

and ^are

is

that 0 > 1/e, so that u > 0(1).) Thus on the left hand side of

eq (VII. 4), in the first square bracket the term 2z^|- + -

2 5^(1 +m

** ) ^ smaller by 0(z) than the remaining terms, <J> + ^ ^

• And in

the second bracket, the term uf ' (v)(z+a; is smaller by Cmz +

than the next term, 2f(v)^z +

We first look for solutions of eq (VII. 4) for which z = 0(0 ),

but make no assumption as to whether <p = 0(6) or <i> < 0(0 ). Thus

if <j> = 0(0) the solution corresponds to qx

= O(u0), whereas if

<p < O(0

2

) the solution corresponds to qi = O(u0

2

). With z = O(0

2

),
the equation (VII. 4) simplifies considerably. Dropping terms of

? relative order 6 , in the first bracket on the left hand side only

2 6

2

n+u 2

) 2 the term <J> + —— remains. And in the second bracket on the

u .2

left hand side the term uf'(v)fz + may be neglected. The left

hand side of eq (VI I. 4) thus simplifies to

2 +<S 2

( 1 +u

2

) 2 »«’(* S) **¥ •’ = 0(f(v)9

4

)

Now on the right hand side of eq (VII. 4) the first bracket is of order

2

6

2

. In the second bracket, the first term, f(v)(z + is 0(f(v)e

4

),

and thus, with the first bracket as factor, gives a contribution of

O(f(v)0

6

), and thus may be neglected. In similar fashion, the second

term in the second bracket, , is of 0(1). This therefore gives

u

2 / e 2 \

a contribution to the right hand side of 0(0 ) = OSf(v) ~j

~j ) - /

„2 n2x2\ \ e 0(f(v) —»

4 “J

= ancS may therefore also be neglected.

We are therefore left with

z +

fr 2 + SQ+u2) al*l£

2 4

u

from the right hand side of eq (VII. 4). We note that in fact only if

<j> = 0(0) need we keep this term. For <p = 0(0 ) it too may be

neglected.

2 Thus, for z = 0(0 ), eq (VII. 4) simplifies, after neglect of

2 terms of relative order 0 , to

76
2f(v)(z

9 ' ( V

)

<fr 2

u 4

a

2

()tu

2

) 2 = 4 Z + L

£

2

£Q±u_l g(v)4>

4u

This equation may be simplified further: Since we have assumed

z = 0(6 ), the functions of v appearing here, f(v), g'(v), and

g(v), may each be expanded about the point v = u. Corrections to

their value at v = u, being of relative order - z = 0(6^),

can thus be neglected. We then obtain

[

2f(u)

(

z + £) + <m!f

u

- 4 z +

2

ad+u2

)1

g(u)4>‘

4u

(VII. 5)

2

We have here a linear equation for z. Thus for z = 0(6 ) there is

only one solution. We will show later, in the footnote following

eq (VII. 33), that this corresponds to a minimum of the cross section.

For the case <j> < 0(6 ), in which case qx = 0(u6 ), we have

2 4 2

<j> < 0(6 ) so that all terms with factor <j> may be neglected. The

equation for the minimum of the cross section then becomes extremely

simple, viz.

,

z + 2=0 .

77
From eq (VI. 48) we then have, for <J> < O(0

Z

),

zmin

k(e

1 +e

2

)(u^-l

)

0

2 2

2e.|

2 / 2

JL_ + 2

/u -i

21

e

l e

2

,u 2+l

(VII. 6}

We note that zmi

- as given here is indeed 0(0 ;, as initially

assumed.

For c}) arbitrary (but still assuming <j> < 0(0)), we have, from

eq (VII. 5),

r of(ul

+ ^2 +

6^Uu2J_

2

u u 8

aM + «o+u

2 )'

u‘ 2 \2

zmin jMc

u

2

2

(VII. 7)

The functions appearing here may all be obtained directly from

eqs (VI. 44), (VI. 45), and (VI. 48):

af(u) 4k (e^+e^) ( u^-1

)

(1+u

2

) 2

8e^e

2

(1+u ) 2~

2

( e

i +e 2^ <5 ^ 1+u '

78
g ' (u

)

u

1 3g(v)

u 3v v=u

32e

l e

2 2 l _ 2 \ o o Ui / (1+u

2

) 2 1 2

f(u) = 8e

1

2 e

2

2

(1+u

2

) 2 + 2 e

l e2

8e

l e

2 L 2,_ 2 0 _ 0 O I ^ 1 "“"Bq “ u£*j £n

(1+U

2

) 2 2 12

8£n £« U r\ a

^ ^ F

2 +F2

- ?F F

^ 1+u 2j2

e

l e 2

2C

1 £

2L...2,

(VII. 8)

g(u)

_ 328^2 t 2 , 2,

u

2 '

(1 +U

2

) 2 (£l

E2 1

Substituting these in eq (VII. 7) we have, in the denominator there.

f(u) (*2 + «Wi

u

. aM 4_

u

2

2

8e-j£2

iiW

. 2 2

? /

2u

£i "•"£ a "l£i £a! a

1 2 1 2 \l+u

?

6

2

(l+u

2

)

14> + o— - 2(e

1

2+e

2

2

)4>

8e-|e

2

e-j 2

+£2 2_2e

i e

2\

2u

,l+u‘

6

2

(1+u

2 )'

79
We thus have

0

2 <S 2

(l+u

2

) 2

u

2\ r

(e

1 +£

2

)fi(l+u

2

)-(E

1

2 +e

2

2 )^ [^)-2(ei2 +e

2

2

)^

2 (^ + iU±yli

2 J\u +1

z mi n L 2 +e 2 nu_

p

£

2 zei e 2\ 1+u

2

6

2

(1+u

2

) e

2+e2

+2e e /-^-u .. £

1 £

2

Z£

1 £

2^1+l| 2

(VII. 9)

2 4

We note that for 4) < 0(0) the numerator here is 0(e 0 ) and the denominator is,

2 2 2

in general, 0(e 0 ). Thus indeed z . is 0(0 ) as originally assumed. However,

the denominator in eq (VI I. 9) is zero for

6(l+y

2

)

so that eq (VII. 9) is clearly invalid for <j> > 4 . (Note, however, that <b = 0(0).)

In fact eq (VI 1. 9) becomes invalid for <p somewhat less than <j> . Once <f> is

2 2 2 sufficiently close to 4> 0

that - 4> is less than 0(0 ) then zmin is larger

than 0(0 ) and the assumptions made in deriving eq (VII. 9) no longer hold. As we

show shortly, as <j> approaches <J

> 0

(from below), the position of the minimum, z . ,

increases from O(0

2

) to 0(0^). when <p reaches the critical value <|> (<|> < <(> )

the minimum fuses with the second maximum of the cross section, becoming an inflection

point. For <j> > <|> the cross section no longer has a minimum. For 0^

= 1°,

e-j = 140 MeV, k = 95 MeV, we find, from a careful computer analysis of the cross

section as a function of v and <J> (see figs. 11 and 12),

e,

2+e 92 -2

1 2 1 2 \l+u 2

2 , 2

,

0 / 2u

e, +e 0 +2e,e

(VII. 10)

1 2 1 2l 1+u2

80
4> c

= 0.836492...°

From eq (VII. 10) we have, for these same values of 0-j, and k.

(J) = 1 .002958

We next look for solutions of eq (VI I. 4) under the assumption that

2

z = 0(6). In contrast to the previous assumption that z = 0(6 ), in

which case the small terms in eq (VII. 4) were of relative order 6 and

hence could be neglected, now the various terms in eq (VII. 4) are of

relative order 9 and hence must be kept. However, the nature of the

solution for z = 0(6) may be seen most easily if we first examine

eq (VII. 4) and neglect the terms of 0(0) as well. Then in the first

square bracket on the left-hand side of eq (VII. 4) we neglect the term

since it is OCO*

5

) while the other terms in this

bracket are 0(0 j. In the second square bracket on the left-hand side

we then keep only the single term 2f(v)z, all other terms in this

bracket being of relative order 0. In similar fashion, on the righthand side of eq (VII. 4), in the first square bracket we keep only the

first term, z. And in the second square bracket on the right-hand side

2 we keep only the terms f(v)z

2 + )_ • We then have, as a zerothu order approximation to eq (VII. 4) for z = 0(6),

2\2

z

2 + (b 2 + Ml±iO • 2f(y)z = 4z f(v)z2 +

u 2 4 .

81
or

z

2 + ^2 + M1^ =2z

2 + _Mv]_^ . u 2 uf( v ) 2

Again neglecting terms of relative order G for this zeroth-order

equation, we may set v = u in the last term, thus giving

u \ 2u t(u)

/

(VII. 11)

where from eq (VII. 8) we have

i - _aM_= , . 2u

2

f(u)

2( ei

2 +e

2

2

)

E

e,

2

+£„

2

+2£,e ^ 2u

1 2 1 2 \

1 +l|

2 r

2 2 « / 2u ' 2

£-, +£~ ~2£-i £, t

1 2 1 2l 1+u2

(VII. 11a)

From eqs (VII. 10) and (VII. 11a) we may write the solution to the zerothorder equation, (VII. 11), as

z

2

e

l

e

l

2 2

,

+ 9 e

2 +2 e ^£ 2

2 . 2

+£ 9

2

-2

£l

£

2

(VII. 12)

82
or

?

6

2

(1+u

2

) z

= o 1 -I VJ

(VII. 13)

As for the case z = 0(6^), we note that here too the solution is

valid only for <J> < <j> . And since we assume now that z = 0(0), we

must again have

4

, o

2

- = o(e

2

)

or

<f> 0

- = 0(9) .

Again we see that the solution breaks down as <J> approaches <j> .

p In contrast to the case z = 0(6 ), however, we see from eqs (VII. 12)

or (VII. 13) that there are now two solutions; in zeroth order they are

z (o) = <S(1+u

2

) max u (VII. 14)

or

(v-u)

(o) max

= ± 6(l+u

2

) (VII. 15)

These correspond to the two maxima of the cross section.

83
We have now found, for sufficiently small <j>, the minimum of the

cross section (given by eq (VII. 7)) and the two maxima (given to zeroth

order by eq (VII. 14)). We could, at this point, continue with the

determination of the maxima of the cross section, obtaining the first

order corrections, of order 6 relative to the values given in

eq (VI I. 14). However, having noted that the solutions of eq (VI I. 4)

p determined thus far under the assumption that either z = 0(6 ) or

z = 0(6), both break down for <j> sufficiently close to <j) Q

, we now

return to eq (VII. 4) and solve it with no assumptions concerning either

z or <j> other than z < 0(0) and <p < 0(0), i.e., assuming only that

neither z nor $ exceed 0(0). To that end, let us first see what

terms in eq (VII. 4) are of relative order 0 for all z and hence may

be neglected. In the second bracket on the left-hand side of this

2

equation, the first term, uf‘(v)(z + , is of 0^z + < 0(9)

relative to the second term, 2f(v)^z + ^-) . In the first term we may

therefore neglect the terms uf'(v) • 2z^- and u-f'( v

)(^-) since they

are of 0^) = 0(6

2

) relative to the terms 2f(v)z and 2f(v)^j ,

respectively, in the second term. For the second square bracket on the

left-hand side of eq (VII. 4) we now have

uf'(v)z 2 + 2f(v)(z + ^)

+ 9'(v) 4> 2

u 4

(VII. 16)

Consider next the second square bracket on the right-hand side of

2

eq (VII. 4). In that bracket we have the term f(v)f^-j > which is

then multiplied with the terms in the first square bracket on the

84
right-hand side of eq (VII. 4), giving terms of O^zf(v)^

o(Vf(v)^j) and O^0

2

f(v)^j) )• These can all be neglected, since

they are of Q(0

2

) relative to terms of O(zf(v)0

2

), 0^(j)

2

f (v)^j)),

and o(0

2

f(v)(^)). We may thus neglect the term f( v

)(jj)

• Similarly,

in this same bracket, the term ^

= 0(1) may be neglected, since it

is of the same order as T(v)^^2 = 0(e

4

£

2

6

4

) = 0^

Thus for the second square bracket on the right-hand side of eq (VII. 4)

we now have

f(v)z

Z + 2f(v)z 2. + aM|_ (VII. 17)

Substituting eqs (VII. 16) and (VII. 17) in eq (VII. 4) and dividing both

sides of the resulting equation by 2f(v) we now have

[> * ^ Z

2 + Z + - + T u 8uf ( (v) ,2] fM * J

= 2 z + +

g. 0-+

jj 2

.) z 2 + 2 z 2 + JSLt 4u

2 f((v) J

(VII. 18)

Referring now to the second bracket on the left-hand side of eq (VII. 18)

we note again that the first term is 0 ( z ), i.e., that it is smaller

than the second term by 0 ( z ) < 0(0). Therefore in the first term we

may expand f'(v)/f(v) about the point v = u, and neglect terms in

this expansion of relative 0((v-u)/u) = 0 ( z ) < 0(0), since these give

85
a contribution of 0(6^) relative to the second term in the bracket.

For the first term we therefore now write. ^ z 2

. In similar

fashion, in this same bracket, we may expand the last term, 4> 2

,

about the point v = u. In this expansion, the terms of relative

0((v-u)/u) = 0(z) are of 0 ( z<f>^ ) < 0(z6

2

). They are thus of order

2

6 relative to the second term, z, in the bracket, and may thus be

neglected. For the last term we therefore now write 'gjj f<j> 2

. Next

consider the last term in the second bracket on the right-hand side of

eq (VII. 18), 9- 1—— = 0(cf> 2

) < O(0

2

). If we expand the factor

q(v)

4u t(v)

i n this term about the point v = u, then the terms in this

expansion, having successively higher powers of (v - u), are of

2 2 2 2

order <j> , z<j> , z 4> : The term of order z

2

<}> 2

< O(z

2

0

2

)

may thus be neglected, being of 0(0^) (or less) relative to the

2 first term in this bracket, z . However, the term of order

2 2

z4> < O(z0 ) must be kept, being of the same order as the term 2z

also in this bracket, if <J> = 0(0). We thus expand

aM = aM + (v-u) C f ( u )g‘ (») - 9(u)f'(u)] + fR nsy

(v u> f

2

(u)

ffuT

+ ZU

g' (u> . f(u)

ZU

(u) . f (u)

and keep only the terms shown here explicitely. Thus for the last

term we write

86

C|Q
_2ImL / + islIIm! / .

z 9( u )f ' ( u ) / . 4 u t(u) 4 uf(u) 4ur (u)

o We may thus write, neglecting terms of relative order 0 in eq (VI I. 18),

uf'(u)

2f(u)

+ z + ° g

1 (u

8uf (u

= 2 [z + + 6(l+u

2

) 2

z

2 + 22 ^

+ g(u)

4u^f (u)

2

<J> + z 4uf (u)

Mi/ z g(u)f ' (u) /]

4uf

2

(u) j

(VII. 19)

We have here a fourth-order algebraic equation for z. We multiply out the

terms appearing here, noting that the cross term coming from the second term in

the first bracket and the first term in the second bracket, viz.,

6(l+u ) uf'(u) ,2

TfTuJ 2 5

2 3

may be neglected since it is of 0(6 ) relative to the cross term z .

4 2 2 Similarly, cross terms of 0(zc}> ) or 0 ( z<j> 6 ) may be neglected since they

2 2 are of 0(0 ) relative to the cross term z<J> . We then have
2

uf'(u)

z4 + z 3

2f(u)

‘

+ 3a

+ 3^(uj ^2 _ g(u)f 1 (u) ^2

u 8uf(u) 2uf

2

(u)

uf 1 (u)

2f(u)

6

2

(l+u

2 )'

u

9(n) \

2u

2

f(u )/

z

+ + Mi+u2

)\ g(u) ^ .

,2 e

2 /

2u

2

f (u)

+

6

2

(1+u

2

)

u 8uf(u) /

= 0 (VII. 20)

4 3

We note here that the terms with z and z have coefficients of 0(1),

2 2

the terms with z and z have coefficients of 0(6 ), and the term

/ 4* without z is of 0(6 ). We note further that the coefficient of the

term with z is given by eq (VI 1. 10) and (VI 1. 11a). We may therefore

write eq (VI I. 20) in the form

4 3

a z + z +

o

A (2)z

2

- b

0

(<J> 0

2

- cf> 2 )z + y

(4)

= 0 (VII. 21)

( 2 ) 2 (4) where a Q

and b

Q are of order unity, \

K ' is 0(0 ), and y

v '

is O(0

4

). Specifically, from eq (VII. 20)

a o uf'(u)

“Winy (VII. 22)

and from eq (VII. 11a)

88
b (VII. 23)

o

2 , e-| +e

2

e-j +e

2

2

2

2

+2e^e2

*" 2 c

*|

> 1

In eq (VII. 21), A^ is the coefficient of z

2 in eq (VII. 20) and

is the term without z tn eq (VII. 20).

Let us first recapitulate our previous findings, and assume therefore,

2 2 2

for the moment, that (p Q

- cp = 0(0 ). Then if we look for a solution

2 with z = 0(6 ) we note that the first three terms in eq (VII. 21) are,

R f) fi respectively, of 0(0°), 0(0 ) and 0(0 ), and hence may be neglected

/ relative to the last two terms, which are each of 0(0 ). The solution

is then

z y

b (4

o

vs o

5

which we gave earlier in eqs (VII. 7) and (VII. 9). On the other hand,

if we look for a solution with z = 0(0) then we note that the

successive terms in eq (VII. 21) are O(0

4

), 0(0^), 0(0^), O(0

3

),

and 0(0 ). Thus if we desire the zeroth-order solution (i.e., neglect

terms of relative order 6), we need keep only the second and fourth

terms in eq (VI I. 21), and find

or

Z

3

- b

0

(cf. 0

2

-

<t> 2

)z = 0

(VII. 24)

89
which we gave earlier in eq (VI I. 12).

We note first that both of these solutions are included in the

4 third-order equation obtained from eq (VI 1. 21) by neglect of the z

2

and z terms there:

z J

' b0<*0

2

- <t' 2

) z + lj(4> = 0 • (VII. 25)

4 2

Next we observe that in fact the z and z terms are smaller than

2 2 2

the remaining terms for all z < 0(6), whether or not <j> -

<f> = 0(6 ).

4

That this is true for the z term is clear, since it is smaller than

z by a factor z < 0(6). With regard to the z term we note that

it is of 0(z

2

8

2

). Thus for z < 0(6^), it is of

0(e 8/3 • e

2

) = Q(e

4

•e

2 ^). It is therefore smaller by a factor of

0(6 2/3

) than the term y^, which is of 0(e

4

)„ On the other hand,

for z > 0(6 4/3

), it is, relative to the z

3

term, of

0(z

2

e 2/z

3

) = 0(e

2

/z) < 0(e 2/s

). Thus for all z < 0(6), the term in

eq (VII. 21) with factor z

2

is smaller by 0(6

2//3

) than either the (4)3 2

term y

x ' or the term z . Therefore this term with factor z , as

4

well as the term with z may be considered a perturbation, mdepen2 2 dently of the order of magnitude of <p Q

- <J> . We therefore assume a

solution of the form

z = z Q + z-j (VII. 26)

and choose z Q so that it satisfies the third-order equation (VII. 25):

90
(VII. 27)

3 -i / , 2

- ,2v , (4) _ n

<f> )z A + y

v - 0 . 0 0' T 0

Substituting eq (VII. 26) in the full equation, (VI 1 . 21 ) , we have

a o

( Vz

l )4 + ‘Vzl

> 3 + x(

2

)(zo+z

l )2 - b

o(*o

2

- $

2

)(zo+z

l

) + y(4) = 0 ’

which, in view of eq (VI 1.27), we may write in the form

[ 3z

o

2

- b

o

( *o

2 -*2) ^ z

l + 3 z

o

z

l

2

X (2)

z 2

0

(VII. 28)

Since we assume z-j small relative to z Q

, we will assume that in this

equation for z-j the terms which are manifestly of relative 0 ( z-j /z

Q

)

can be dropped. By solving the equation with these terms neglected, we

obtain the first correction, z-j , to the main term, z Q given by

eq (VII. 27):

[

3z

o

2

- b0<*0

2

- *

2

>] z

l + 3z

o

z

l

2

-

- Vo

4

- x(2)z

o

2

• < VII - Z9 >

o but we will have neglected corrections to z-j of 0(z^/z

Q

) relative

to the main term.

We begin our consideration of eq (VII. 29) by examining again the

2 2 2 cases already considered, assuming for the moment that <f> - <f> = 0(6 ).

Thus if z Q

= 0(6 ) then the largest term on the left-hand side of

eq (VII. 29) is - b

Q

(<}> 0

2

- <^)z^ = 0(e

2

z-|), the other terms there being

smaller: 3 z

q

2 z

1 = 0(6

4

z

1

) and 3 z q

z

1

2 = 0(e2

z.| 2

). On the right-hand

side of eq (VII. 29) the largest term is

91
since the other term is

- A<V - 0(9

6

) ,

- a Q

z o

4

“ O(0

8

). Thus for z-j we have

l /. 2 ,2\ - -» ( 2 ) 2

V*o - 4> ) Zl - - X

v 'z

Q

so that

*< 2

>z 2

z

1 g

-- ~2 - 0(e

n

) b

o<*o

- 0 )

Thus for z Q

= 0(9

2

), the first correction, z-j = 0(6

4

), is of

2 2 relative order 9 : z^/z

Q

= 0(6 ). In this case the correction may

therefore be neglected, as we noted earlier in deriving eg (VII. 7).

Next, if z Q

= 0(9) then on the left-hand side of eq (VII. 29) we

must retain

k

2

- v*

0

2

- *

2

>] z

i °< e S> •

o the other term there being 3 zq

z^ which is of Q(z^/z

o

) relative

2

to the term 3z

q

Zy On the right-hand side of eq (VII. 29) we now keep

all the terms.

“Vo - a^zq

2

= o(e

4

)

92
We now have the equation for Zy

K

2 - b,(*o

2 -*2 i]wo, - ,(2>z

»2

• (VII - 3°)

From eqs (VII. 24) or (VII. 14) we may substitute the zeroth-order

(o) solution, zQ

, given there as z

v

. From eq (VII. 24) we have

which, substituted in eq (VII. 30), gives

Z

1 = \

(a0Z

0

2 + X<2)) • (VII. 31)

Thus for z q

= 0(6) the first correction, Zy is of 0(9 ), i.e.,

it is of relative order 6 and is therefore retained: z^/z

Q

= 0(0).

( 2 ) 2

Here a

Q

is given by eq (VI I. 22) and A

v ‘ is the coefficient of z

in eq (VII. 20). With z Q as given in eq (VII. 14), we then have the

full solution, z = z Q + z-j , for the maxima of the cross section (i.e.,

with neglect only of terms of relative order 6 ) for 4 < 4

C < 40

and 4

q

2

- 4> 2 = 0(e

2

):

z max

+

2 _ uf 1 (u) 6

2 (1+u2

) i 4f(u) u 2

L

(VII. 32)

93
Having considered the extrema of the cross section for <p < 4> c < <j>

2 2 2

(and <|> -

(J) = 0(6 )), we now consider the case in which <f> is

sufficiently close to <f> Q

that <p Q

- <p < 0(6 ). This is the condition

for which the minimum of the cross section fuses with the second

maximum and becomes an inflection point. To determine the inflection

point we turn to eqs (VII. 2) and (VII. 2a), and require not only that

eq (VI I. 2a) be satisfied, but that the second derivative of the cross

section be zero as well. From eq (VII. 2) we have

• I ,

1 )'

3

p{

(' (

)]'

= 0

From eq (VI I. 2a) the first term on the right-hand side above is zero.

Thus we have as our added condition

[q

2

{

- 2(q

2

)'

{ }]

= 0 . (VII. 33)

If we now follow the analysis from eqs (VII. 2a) through (VII. 4), (VII. 18),

(VII. 19), (VII. 20), and arriving finally at eq (VII. 21), we see that,

apart from a factor independent of v, the polynomial on the left-hand

side of eq (VII. 21) is the negative of the expression in square brackets

94
2

in eq (VII. 33). Thus the inflection point occurs when both eq (VII. 21)

and its first derivative with respect to z are zero (recall, from

eq (VII. 3), that z = (v - u)/u):

a

Q

z 4 + A

( 2 ) z 3

- b

0

(<|> 0

2

- *

2

)z + y

(4) = 0

and

4a z

3 + 3z

2 + 2A^z - b (<J> 2

-

(f> 2) = 0

0 0 0 (VII. 34)

4 2 Earlier we showed that the z and z terms in the first equation

here were smaller than the z or y

v ' terms. Similarly, in the

3 2

second equation here, the z and z are smaller than the z term

for z > 0(0 ). We then have, in first approximation, at the inflection

point,

z 3

- b

0

(<t> Q

2

-

<t> 2

)z + y

<4) = 0

and

from which

3z

2

- b

Q

(<J) o

2

- <j> 2

) = 0 ,

(VII. 35a)

(VII. 35b)

2 It follows from this observation that, for the extremum which occurs for

2

z = 0(6 ) when <f> < co< <j> , the second derivative of the cross section

2 2 at this extremum is given simply from eq (VI 1. 21) by b (<J> -<|> ) > 0

(apart from a factor which is positive), since the terms in eq (VII. 21)

4 3 2 2 with z , z and z are all negligible, of relative order 0 . Thus

the point given earlier in eqs (VI I. 6) and (VI I. 7) is indeed a minimum.

95
or

z (VII. 36)

Then from eq (VI I. 35) it follows that

4> 0

2

-

<t> 2

= 0(98/3

> (VII. 37)

at the inflection point, since b

Q

= 0(1) (eq (VII. 23)). Although in

principle we have in eqs (VII. 35a) and (VII. 35b) two equations for the

two unknowns, z and <|>, the critical value of <p for which the

inflection point occurs is most simply obtained directly from the first

of these two equations. It has [17] three real and unequal roots (the

minimum and two maxima already determined) when

It has three real roots, of which two (viz., the inflection point and

the second maximum) are equal when

(VII. 38a)

(VII. 38b)

And it has only one real root (a maximum) when

3

(VII. 38c)

96
We note from eq (VII. 37) that the right-hand side of eq (VII. 3$b) is

o m

0(6 ) . This indeed checks, since y

v

, on the left-hand side of

this equation, is Q(0

4

). In solving eq (VII. 38b) we recall that

y^ contains all the terms in eq (VII. 20) without a z-dependent

factor:

y

(4) = 6(1+u

2

)

| g(u)<j) 2

e 2 /2u2

f(u)

6

2

(l+u

2

) iW

8uf (u)

(VII. 39)

Now in substituting this in eq (VII. 38b) we note from eq (VII. 37) that

<j> Q

2

- (j) 2= 0(0^3) < o(e

2

). We will therefore set (f) 2 - <p Q

c in

eq (VII. 39), thus neglecting terms of relative order

(<j) a

2

-

<J> 2

)/<J> 0

2 = O(0 2/3

). With y

Q

(4) e y

( 4 )

(cf) 2 = cj^

2

), the critical

value of <p at which the minimum disappears is then given from

eq (VII. 38b) by

b ((j) 2

-

<f> 2 )"1 o

VY

o Tc

or

(VII. 40)
Now from eqs (VII. 10) and (VII. 23) we have so that

2 2 u1

c rQ

x 2/ti , 2x2

6 (Hu )

6

2 (Hu2

) 2 (VII. 41)

We thus indeed find <|> < <j> Q

, as noted earlier from our computer

determination of <J> for the specific values 6^

= 1°, = 140 MeV,

and k = 95 MeV.

The value of z at the inflection point is given at once by

eqs (VII. 35b) and (VII. 40), from which

z

2

0 b (<f> 2

- <b 2

) _ 2 _ ovy

o Y

c

' z c

" 3

or

(VII. 42)

It follows at once, from eqs (VII. 42) and (VII. 40), that this value

of z

c also satisfies eq (VII. 35a), if we replace there by

(4) (4)

y

Q

, as it should. The expression y

'

, which appears in

eqs (VII. 41) and (VII. 42) may be simplified somewhat. From eqs (VII. 10)

and (VII. 11a) we have

98
t

a

2

(i+u

2

)

2

g(u)<j> 0

2

2u

2

f(u)

and from eqs (VI I. 10) and (VI I. 23),

b d> o

y

o

6

2

(l+u

2

)

from which

$

2 + y

o

6

2

(1+u

2

)

1 g(u)4>

0

2

2u

2

f (u)

/^0 + A

62

( 1+u2 )

(VII. 43)

(VII. 44)

Substituting this in eq (VII. 39) with (p = <j> we have

(4) . /V1 ^

62

( 1+u

2

) ° ‘ -o

> »

2

<t> 0

2

d(l+u

2

) o g'(u)

2 + (J) e

2 u 8uf(u)

0

. (VII. 45)

Here, from eq (VI I. 8),

9

1 (u)

u

= _ I M_zi ] g(u)

vU 2+l / u

2

99
from which, with eq (VI I. 44),

g' (u)<f> o _ 8uf (u)

g(u)4>

0

2y

2

f(u)

^

6

2

(1+u

2

) (VII. 46)

And from eq (VI. 48)

a

u

k(e^+e

2 )(u

2

-1

)

2£

l e

2

kW '“

2- 1

11 2a\ 2

iT+l,

6 (e^ )

C

yZ

~l

)

e ^+e ^-2e e /_£u_

1 2

^ £

] £

2^1+u

2

(VII. 47)

Substituting eqs (VII. 43), (VII. 46), and (VII. 47) in eq (VII. 45) we have

(4) _ 6(l+u ) u — + —

2b u e 0 o 2

(e

1 +e

2

)u

e

l 2+e

2

2

- 2e

1 e2(^?

1 /

bo

+1\

«<“-!)

+ i

(VII. 48)

100
Defining

2 , 2, 0 ( 2u

£, = £ n +£ 0 ± 2 £,£

± " 1 2 —rzl 1+u

2

we have, from eq (VII. 23),

bo

= F

(VII. 49)

(VII. 50)

and

b

Q

+l

_ 2(

£l

2

+£

2

2

) b_ £

,

From eqs (VII. 48) and (VII. 50) we then have

(4 ) _ 2(£

7

2+£

2

2

) 6 3

(1+u

2

) 6(l+u

2

)e_ 1 (e-j +£

2

)

2u £+ £ 2 £

(£

1

2 +£

2

2

)<5(u

2

-1)

2u

2

£,

Adding the first and last terms in the square brackets here gives

101
—j— ^e_(l+u

2

) + (e

1

2

te

2

2

)(u

2

-l)^= ^(e^+e^Ju2

- 2e

] +e

2

<5 / ? ? 4e-,£

?

' ^ r i

^ f

* ** (’ &.

from which

fk2 + 2e,£. 12

^

Vu

2+1

6r k —_ + —

e+

G+

( 4 ) 2(e

1

2+e

2

2

) 6

3

(1 +u

2

)

3 r- 2

Sk 1 (VII. 51)

Substituting eq (VII. 51) in eqs (VII. 41) and (VII .-42 ) gives, finally.

c

B qO ’ 3

^/3

) (VII. 52)

102
and

z = 5U+u£l cl/3

c u

b (VII. 53)

It should be emphasized that eq (VII. 52) provides only a zeroth-order

approximation for 4> - <|> = 3<{> o

c

2^ 3

, and eq (VII. 53) provides only a

zeroth-order approximation for z Q

. As mentioned in the discussion

2/3 following eq (VI I. 39), we have neglected terms of relative order 6 :

— — = 3£ 2/3

(l + O(0 2/3

)) (VII. 52a)

*o

z

c

= C

1/3 0 + O(0 2/3

)) . (VII. 53a)

These approximations are thus much less accurate than expressions (VII. 9)

and (VII. 32) for z . and zmax

» respectively, in which the neglected

terms are of relative order 9 . The higher order corrections, indicated

above as O(0

2^ 3

) could be obtained in a straightforward manner by

continuing our perturbative approach for the region in which

<t> Q

2

- 4> 2 = O(0

8/ 3

), z - O(0^3

), but we shall not pursue this further

in this report.

103
Finally, we determine the value of the cross section at its

minimum and maxima.

The cross section is given by eq (VII. 1) in which{ } is given by

o eq (VI. 49) and q by eq (V.15). At the minimum of the cross section

2

z = z . = 0(0 ), given in eq (VI 1. 9), provided that <p is not too

2 2 2

close to <J> ; specifically provided that 4> Q

- 4> = 0(6 ). With

2

this assumption we may set v = u in the expression (V.15) for q , ? thus neglecting terms of relative order 0 , and obtain

q

2 = u 2

cf> 2 + 6

2

(1+u

2

)

2

. (VII. 55a)

Next, in eq (VI. 49) for

| J,

consider the first term, which is

2

f ( v )

( v~ u + a)

2 = u 2

f(v)(z + £) .

Here, z = z . is given by eq (VI I. 5), from which it is seen that

A

z + ^

is 0(<j) ) at the minimum. Thus, at the minimum, the first

term in eq (VII. 49) is 0(u

2

f (v)<J>^) , which is 0(<J) 2

) relative to

the second term in eq (VII. 49) and may therefore be neglected.

Similarly, in the second term itself we may neglect terms of higher

2 order in <j> , and, in addition, set v = u, again neglecting terms of

2 relative order 0 . We thus have, at the minimum of the cross section,

{ }

= \

g(u)*

2 + P . (VII. 55b)

104
It may be noted, on referring to eq (VI. 54), that the value just given

above in eq (VII. 55b) for

j | at the minimum of the cross section is

the same (to within terms of relative order 0

2

) as the value of { }

at the minimum of

|

j-. This follows from the fact that both minima

occur for z + ^

= 0(cj>

2

). (The minimum of

| }

ls 9 lven by ec! (VI. 52),

and that of the cross section by eq (VI I. 5).) However, they do not

occur at precisely the same value of z, and indeed cannot. The minimum

of the cross section is determined from eq (VI I. 2a), and there the

minimum of q

2 does not occur at the same point as the minimum of

| J\

From eqs (VII. 55a), (VI. 54), (VII. 55b), and (VII. 1) we then have,

for the cross section at the minimum (with F(q) = 0),

1

dft^dfL dk he \mc

2 / k p

1 (2tt) 2 [u

2

<J> 2 + 6

2

(l+u

2

) 2

]

2

o . ?

k

2

u

2

|l + /—^ x |(e

1

2+e

2

2

)u

2

(j) 2 + S

2

(l+u

2

) • ' 1+u

(l +u

)'

1 -2

2 . 2 9 e, +e 0 -2e,e

]

(VII. 56)

It should be noted here that unless $ is very small, i.e., unless

2

4> < 0(0 ), the first term in eq (VI 1. 56) is much larger than the second

2

term there. Thus for 0(0 )<<}>< 0(0) we can write eq (VII. 56) in the

2 2 2 much simpler form (recalling that here

qj^ = u 4> )

,

105
2

d

3a

dfi, dft dk

k p 2

he

1 P 2

1

k p

1 (2tt) 2

8e-j£2

a+u

2

) 2

I 2. 2x

(e^ +e

2 ' (VII. 57)

(See comment following reference [14] in the list of references.)

With regard to the cross section at the maxima, these values may

be obtained by substituting eq (VII. 32) in eq (VI. 49) (for

| jj and

in eq (V.15) (for q ), and then substituting these in eq (VII . 1 )

.

However, the expression (VI I. 32) for is rather complicated. On

the other hand, the average value of the cross section at the two

maxima is rather simple, and from the experimental viewpoint this

value, as well as the ratio of this average value to the value of the

cross section at the minimum is of almost equal interest. We therefore

determine this average value rather than the cross section at each of

the maxima. For this we need retain only the first term, of 0(0),

in eq (VII. 32). Since it has opposite sign for the two maxima, the

contributions from the remaining terms in eq (VI I. 32) (of order 6

relative to the first term) cancel in the average. In the same manner,

all terms of relative order 6 cancel in the expression for the

average of the cross section at the maxima. We may thus use only the

first term in eq (VI I. 32) and set v = u throughout, neglecting terms

of relative order 0 since they give contributions of relative order

2

0 to the average. From eqs (VI I. 49) and (V.15) we then have

106
{ }

= u

2

f(u)z

2 + {

g(u)<f>

2

and

* f(u) u

2

z 2 g(u)*

2 + —

2f(u)

(VII. 58a)

q

2 = u

2 z2 + u2

cf> 2 + 6

2

(1 +u

2

) (VII. 58b)

Here, from eq (VII .32) (retaining only the first term there).

u

2 z2

6

2

(1+u

2

)

and from eq (VII. 44)

g(uH0

2

2f(u)

u

2

(j>o

2 + 6

2

( 1 +u

2

)

(VII. 59)

Thus in eq (VI I. 58a) we have

p p g(u)4>

u z +

2u

2 f (u)

2

1 /<(> V _ . 2 /, ...2 f-1

= 6 < 1+u > y

o u\2+62 (1+u

2

)
The last line here follows from eq (VI I. 58b). We thus have, at the

maxima, neglecting terms of relative order 6,

so that

{ } - 1M

4

‘

9 2

q 2q

(VII. 60a)

(VII. 60b)

Substituting eq (VII. 59) in eq (VII. 58b) we have

q

2 = 2<5 2

(1+u

2

) - u 2

<J) 2 6

2

(1+u

2

) u

2*2

Here, from eqs (VII. 43), (VII. 49), and (VII. 50),

x 2, 1a 2\

6 (Hu ) i 2 2

" u iy

o

bo1

e+

- e

4e

i e^ e

l 2+c2

2+2c

1 e 2(^

108
Thus

q

2 = 2 <S 2

(l+u

2

) - u

2

<J) 2

2u W

2 , 2, 0

e-| +e

2 +2e-j£2

2u

1+u / J

(VII. 61)

Finally, from eqs (VII. 8), (VII. 60b), (VII. 61), and (VII. 1), we have

the average of the cross sections at the two maxima, neglecting terms

2

of relative order 0 ,

d

3a

dft. dft dk

k p

2

(VII. 62)

109
VIII. SUMMARY OF ESSENTIAL FORMULAS DERIVED IN SECTIONS V, VI, AND VII

In this section we present a summary of the formulas of sections M,

VI, and VII, which constitute the essence of this report. The details

of their derivation are given in those sections, but for the reader

more concerned with applying them than with their derivation, we have

abstracted them from those sections, and present them here with their

original equation numbers. One may thereby locate them easily in their

original context if one wishes to consider them in greater detail.

However, for the definition of the essential variables of the problem

one should see section III.

? In section V we derive an approximation for q , the momentum

transfer squared, for high energies and small angles. In that section

2 we start with the exact expression for q ,

(V.l)

and write

(V.2)

where

%

58 £-u £-21

u - v (V.3)

110
is the component of £ perpendicular to k_

and

q

z = p-jCOS0-| - P2COS02 - k

is the component of £ in the direction of k.

|uj = p-|Sin0-|

v = 1 v_| = p

2

sin 02

We define the energy denominators

(V.4)

(V.5)

d-j = e.j ~ p^cos0-j

d2 = £2 " P2COS0

2

* (V.7)

We obtain expansions for d-j and d

2

which are useful for high energies

and small angles:

(V.9)

111
In identical fashion we have

(V.11)

(V.12)

We then have

q

z = d2

” d

]

. T+v

2

(1+u

2

) ' 2e

2

' 2e-j

! (1+v

2

) 2

1 (1+u

2

) 2

(V.13)

For high energies and small angles we obtain the following approximation

, 2

for q :

q

2 = (v-u)2 + 2uv(l-cos<}>) +

2--^-+- L u ^ v

~ u) + 6

2

(1+u

2

)

2

. (V.15)

e

2

112
2

In this approximation neglected terms are of relative order 0

2

(and

never larger) for all_ values of v - u (i.e., for v - u = 0,

v - u = 0(u6

2

), v - u = O(u0), and v - u = 0(u)).

? We show that the minimum value of q considered as a function

of v (i.e., © 2

) » for fixed values of e-j , e

2

» 6 and 4>» occurs

for

(1 - cos<}>) + (V.17)

2

At this value of v, the minimum value of q is, neglecting terms

2

of relative order ,

o 9 9

c

9 9

q

inin = 6 (1 + u > + u sin 't’ • (V .18)

In section VI we derive the high energy, small angle approximation

of the differential cross section given in eq ( I V . 1

) . We define the

variables £ and n in terms of the energy denominators d-j and d

2

given in eq (V. 7):

i = 2e

i d

i n

1 2e

2

d

2

(VI. 3)

We note that the expression for -j J- given in eq (VI . 1 ) may be written

very simply in terms of £ and n:

{ } 8k

2

e-j£2(iL~v.)^^T1 + 16^1

^e 2^—

4q

2 (E^£-£2yn)^ • (VI. 4)

113
We note that in terms of the variables defined earlier, we may write

where

« 5

o

-

n = n

0

- n,

] + u‘ «1

2e^ (e-j + p^cosQ^ } 4e-j'

n

o

=

1 + V

] ^ 1 n

= «» 2

ZegCeg + P2COS0 2^ ^£

2

We derive the high energy, small angle approximation to

initial ly by eq (IV.l )

| |

- f(v)(v-u+o)

2 + g(v)sin2

%<J> + p

where

f(v) =

o 2 2

8e-j e

2

(l+u

2

)(l+v

2

)

+ 2

^1 (l+u

2

)(l+v

2

)

g(v) =

2 2

32e^£2( e

i +e

2 ) uv

(l+u

2

)(l+v

2

)

(VI. 7)

(VI. 8)

defined

(VI. 49)

(VI. 44)

(VI. 45)

114
(VI. 47)

and

o = k(e^ +^

2

) ( )u

9

2 2

2e

i Ei + 21 e

l e

2

'uLi

w

2

+l

(VI. 48)

We consider

| j as a function of v (i.e., e

2

) for fixed

values of the other variables (e-| , e

2

> e

i ancl 4>) and determine

the value of v for which

j j

achieves its minimum as well as the

value of | | at the minimum. From eq (VI. 49) we find

v . = u min

k(e1 +e

2 )(u

2 +l ) (e.|2+e

2

2

) ^ + cj) 9 2 2

2e

l e

l 2e-j£

2

+ 2

^u 2-l

£-|£ 2 \

U

2

+1

. (VI. 53)

115
We obtain, for the value of

| j

at its minimum,

II =1 9 (u )4> 2 + p

' 'min

o , 2X

2x 2,2

8e-j ^2

( +£2 )u <j> \fl e

2

1 +

2u

l+u‘

(1+u

2

) 2

e

l e 2

+ 2 | m

2

\U+]J

. (V 1.54)

We note that the minimum for

j j

does not occur for precisely the

2 same value of v as the minimum for q , as is seen by comparing

eq (VI. 53) with eq (V.17).

In section VII we consider the cross section as a function of v

(i.e., 62 ) for fixed values of the other variables (e^ , £

2

* »

and 4>) and determine the value of v for which the cross section

achieves its maximum and minimum values. In this determination we

neglect atomic screening, since its effect is rather small for the

energies and angles of experimental interest. We show that for <f>

less than a critical value, <(> , the cross section has two maxima,

which occur for z = (v - u)/u = 0(6), one for v < u and one for

v > u, and in between these a very sharp minimum which occurs for

z = 0(0 ). For <|> = <|> the minimum of the cross section occurs at

the same point (z = z ) as the second maximum (the one for v > u),

and we then have a point of inflection at z = z . For <f> > <J> the

cross section has only one extremum, the maximum which occurs for

v < u. We derive a zeroth-order approximation for <j> and z

q

:

116
V1 - (VII. 52) 3?2/3

>

and

, _ 6(Uu2

) 1/3

c u (VII. 53)

(VII. 10)

(VII. 54)

with

e± = e]

2

+£

2

2

±2 * (VII. 49)

The relative errors in these zeroth-order approximations for <(> and

z c are 0(e 2/^ 3

) , as noted in eqs (VII. 52a) and (VII. 53a). For

<J> C < *

0 ( s P ec ifical ly, for <j> 0

- <j> = 0(0)), we show that the

position of the minimum is given by

117
<i> 2 6

2 (1+u

2

) 2 u

2\ ^(e^+e^dO+u2 )-^ 2*^2 )^

_Vi+l/ 1 L \2 e

2 y 'min

E

2X 2 9 e, +£o - 2£,£

1 ’-2 ‘- & r2^ 1 ..2 (

2u \

21 6

2 (]+u

2

) w] J u

2 e

i 2+e

2

2+2e

i e 2r^r

>l+u / .

(VII. 9)

and the position of the maxima by

z = ± max

6(l+u ) . (3lJ . uf 1 (u) 6Wfr _ a

4f(u) u

2

L VO

- h\^ . (VII. 32)

i o) ? where A

v ‘ is the coefficient of z in eq (VII. 20), viz..

( 2 ) _ 2/i .2'

3a

+

3g

1 (u

)

^2 _ g(u)f'(u) ^2 _ uf*(u) L2 +

6 (1+u )

u 8uf(u) 2uf

2

(u

)

2f(u)

see

(VII. 20)

and f(u), f'(u), g(u), and g'(u) are given from eqs (VI. 44), (VI. 45) and

(VII. 8) by

8e,e 9

f(u) = Xfc

(1+u

2

) 2

2 , 2 0 | £-| +e

2 ~2e-|e

2

|

2u

d+u‘

see

(VI. 44)

and

(VII. 8)

f

1 (u) = 3f(v)

8v v=u

64e-j

2e

2

2 u(u

2

-l

)

(1+u

2

) 5

16£

1 e2

U l 2+e

2

- 2et /-iy-'

2

(1+u

2

) 3

|

e

l e2

2£

1 £

2(1 +u

2.

118
g(u) =

2

32e^£2u (Hu2

) 2

(e

]

2+e

2

2

)

see

(VI. 45)

and

(VII. 8 )

g'(u) = 9g(v)

av v=u

32e^£2u (iW (e, +e 9 ) uii

,u 2

+l

The errors in the expressions just given for z . and zmaxare of

2 relative order 0 .

For the value of the cross section at the minimum (with F(q) = 0)

we find

d

3a

dft. dft dk

k p

2

2 / 2 x 2 e ( e \

1 p 91 1

he \mc2

/ k p 1 (2tt) 2

[u

2

4> 2 + 6

2 (l+u2

) 2]

2

8e

1 £

2 J, 2, 2 n 2,2 , ,2,,. 2 x x \Un +£9 )u <p +6 (

1 +u ) 0+u2

r

1 1 2

,2 2

k u 1 +

2 u

1+u"

2 2 9 £i +£o -2£t£

1 .c 2 2

(JK,

Vl+u‘

(VII. 56)

2

We note here that unless <|> is very small, i.e., unless <J> < 0(0 ),

the first term in eq (VI I. 56) is much larger than the second term there.

r\ Thus for 0 ( 9 ^) < <(> < 0(0) we can write eq (VII. 56) in the much simpler

form (recalling that here q

^

2 = u

2

(j) 2

).

119
With regard to the cross section at the maxima, these values may

However, the expression (VII. 32) for z is rather complicated. On

the other hand, the average value of the cross section at the two

maxima is rather simple, and from the experimental viewpoint this

value, as well as the ratio of this average value to the value of the

cross section at the minimum is of almost equal interest. We therefore

determine this average value rather than the cross section at each of

the maxima. For the average of the cross sections at the two maxima we

The errors in the expressions (VII. 56) and (VII. 62) are of relative

be obtained by substituting eq (VII. 32) in eq (VI. 49) (for

j J-) and

O in eq ( V . 1 5 ) (for q ), and then substituting these in eq (VII. 1).

find

x (VII. 62)

2

order 0 .

120
A simple picture of the overall behayior of the differential cross

section for bremsstrahlung follows from the analysis presented in

sections V, VI, and VII. For high energies and small angles we may

distinguish three regions which are characterized by the momentum

transfer vector, q, and illustrated in fig. 13. In the first region

| q | = 0 (u ) . We call this the "large" momentum transfer region. Here

qx » q

z

, so that q is essentially perpendicular to the photon

momentum, t. In the second region qx ~ q

z

. Thus in this region the

vector q may have any direction. We call this the region of "small"

momentum transfers. In this region, in general the differential cross

section is larger by a factor of Q(u /qm^ n

) = 0(1/0 ) than the

differential cross section in the region of large momentum transfers.

However, in the middle of the region of Small momentum transfers the

differential cross section has a very sharp dip (for <j> < (f> c

). The

region of this dip (the third region) is characterized by qi « q z> Thus the magnitude of q is of the same order of magnitude as in the

region in which ~ q

z

, but now the vector q is essentially parallel

to the photon momentum £. In the region of the dip the differential

p cross section is smaller by a factor of 0(0 ) than it is in the small

q region where q± ~ q z

, i.e., it is of the same order of magnitude

as in the large q region.

121
IX. COULOMB CORRECTIONS TO THE DIFFERENTIAL CROSS SECTION

FOR BREMSSTRAHLUNG

Thus far, all of the analysis presented in this report has dealt

exclusively with the first Born approximation differential cross section

for bremsstrahlung, the Bethe-Heitler cross section. We consider now

the question of the Coulomb corrections to this cross section. Having

noted and considered in detail the sharp dip in the cross section in

the region of small momentum transfers, the question of whether the

Coulomb correction might not fill in this dip arises in particular. In

fact the Coulomb correction to the differential cross section for

bremsstrahlung at high energies has been calculated specifically in

this region of small momentum transfers. It was first given for a pure

Coulomb field (no screening) by Bethe and Maximon [18], and later for

arbitrary screening by Olsen, Maximon, and Wergeland [8]. In both

cases it was found that the entire effect of the Coulomb corrections

is simply to multiply the Born approximation cross section by a factor

which, although it varies throughout the region of small momentum

transfers, is always of order unity. Thus the Coulomb corrections do

not fill in the sharp dip, as they might if they were additive.

Expressions for this factor are given in reference [8], both in terms

of the potential representing the screened atom and in terms of the

p atomic form factor. It is denoted by |A| in [8] and by R in [18],

p. 781, Eq. (8.20). For the case of no screening, it is given in both

[18] and [8]:

122
where

R

~ 2 = V

2

(x) + a

2

y

2

W

2

(x)

V

2

(l)

a = z ei

he

x

=

y

=

V

q

1 - X

V(x) = F(ia,-ia;l ;x)

W = F(l+ia,l-ia;2;x)

_L dV

2 » a dx

F being the hypergeometric function. In particular,

vm = sinhua

' } ~

ira

In the absence of screening it is shown in reference [18] that R is

a monotonically increasing function of x, that R = 1 for x = 1

and R < 1 otherwise.

123
Thus the effect of the Coulomb corrections is to deepen the dip

observed in the Born approximation. However, it must be noted that in

the analysis in reference [8], as well as that in [18], there have

2

been neglected throughout, not only terms of relative order 1/e , as

in the present report, but terms of relative order 1/e as well. Thus

if one wishes to evaluate the Coulomb corrections to the differential

cross section to the same level of accuracy as that pursued here for

the Born approximation, the analyses given earlier in [18] and [8]

must be extended to include the terms of relative order 1/e.

124
REFERENCES

[1] Beil, Hans and Bergdre, Roland, Monochromatic and Identifiable

Photons Used in Photonuclear Research, CEN Saclay Note CEA-N-2144,

July 1980.

[2] Audit, 6., de Botton, N., lamas, G., Beil, H. s Bergdre, R., and

Veyssidre, A., Nucl. Instr. and Meth. 79, 203 (1970).

[3] Vey ssidre, A., Beil, H., Bergdre, R. , Carlos, P., Fagot, J., and

Leprdtre, A., Nucl. Instr. and Meth. 165 , 417 (1979).

[4] Nuclear Physics Research with Microtrons Using Superconducting

Linacs. July 1980 Report of the Nuclear Physics Lab., University

of Illinois, Urbana, 111. (unpublished).

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

Gundrum, H., Kroning, M. , Riehn, P., Sita Ram, G., Zieger, A., and

Ziegler, B., Nucl. Phys. A251 , 479 (1975).

Borchert, H., Ziegler, B., Gimm, H., Zieger, A., Hughes, R., and

Ahrens, J., Description du double spectromdtre Compton du MPI

de Mayence, Comptes Rendus des journdes d'dtudes de Saclay,

14-18 mars 1977.

[6] Arends, J., Eyink, J., Hartmann, H., Hegerath, A., Mecking, B.,

Noldeke, G., and Rost, H., The tagged photon beam at the 500 MeV

synchrotron at Bonn, Comptes Rendus des journdes d'dtudes de Saclay,

14-18 mars 1977. This material is also available as a report of the

Physikalisches Institut of Bonn University, preprint no. BONN-HE-78-19

(April 1979).

[7] Heitler, W., The Quantum Theory of Radiation (Oxford Univ. Press,

London, 1954) 3rd ed.

[8] Olsen, Haakon, Maximon, L. C., and Wergeland, Harald, Phys. Rev.

106 , 27 (1957).

[9]

Olsen, Haakon, Phys. Rev. 99^ 1335 (1955).

[10] Oualifi, Bouchaib, Etude des spectres de rayonnement de freinage

dans la rdgion d'dnergie comprise entre 30 MeV et 120 MeV, en rapport avec les expdriences photonucldaires effectudes auprds

de l'ALS de Saclay, doctoral thesis, Universitd Pierre et Marie

Curie, Paris 6, 1980 (unpublished).

[11] Bethe, H. A., Cambridge Phil. Soc. 30, 524 (1934).

[12] Bethe, H., and Heitler, W., Proc. Roy. Soc. (London) A146 , 83 (1934).

[13] Koch, H. W., and Motz, J. W., Revs. Mod. Phys. 31_, 920 (1959).

Note p. 924, Formula IBS.

125
[14]

Maximon, L. C., and Olsen, Haakon, Phys. Rev. 128 , 316 (1962).

Note, in this paper, that after summing eq (4) over photon

polarizations, we have, from eqs (3) and (5),

, 2X

2v 2

(e-j +££ J q^

Although this paper deals with pair production, the same result

follows in bremsstrahlung, as may be seen in our eq. (VII. 57).

Note, however, from the discussion following eq (VII. 56), that this

result presupposes that 0 (

0

2

) < <j> < 0(e). The more correct result,

valid also when <J> < O(0 2 ), is given in our eq. (VII. 56).

[15] Olsen, Haakon and Maximon, L. C., Phys. Rev. 114 , 887 (1959).

From eqs (3.8), (3.8a), (4.4), and (4.9), after summing over electron

spins and photon polarization, we have, for our expression

j |

, in

Born approximation,

8e-,e

2

k

2 [(u£

0 -yn0

) 2 + (£ 0 -n0

) 2

] + Zc^O^-vt^) 2

2 2

With a bit of algebra, we find (u£ -vnJ + (S -nn ) - (M"l)?nnA , and thus arrive at eq. (VI. 50).

[16] Molikre, G., Zeits. f. Naturforsch. 2a, 133 (1947). See also

ref. 15 above, p. 897, eq (6.30).

[17] Condon, E. U., and Odishaw, Hugh, Handbook of Physics (McGraw-Hill

Book Co., New York, 1958). See Chap. 2, p. 1-11.

See also, Handbook of Mathematical Tables (The Chemical Rubber Co.,

Cleveland, Ohio (T964)) 2nd ed. Note p. 464.

[18] Bethe, H. A., and Maximon, L. C., Phys. Rev. 93, 768 (1954).

126
FIGURE CAPTIONS

Fig. 1. Theoretical integrated-over-angles thin-target bremsstrahlung

cross section multiplied by the photon energy. (The formula

used is listed as 3BS(e) in reference [13], with Z = 78.)

Fig. 2. Schematic representation of different partial cross sections

for a 100 MeV positron beam incident on a hydrogen target.

(a) Represents the "monochromatic" annihilation line, (b) the

positron-electron bremsstrahlung contribution, and (b') the

nuclear bremsstrahlung part. The spectra are observed along

a line making an angle of 4° with the line of arrival of the

incoming positrons. (Figure taken from reference [3].)

Fig. 3. Momenta of the initial and final electrons (jd-j and £2

) anc! photon (k), (a) in the system with z-axis in the direction

of k_; and with jd-j in the x-z plane, (b) in the system

with z-axis in the direction of jd-j

; and with

x-z plane.

k in the

Fig. 4. q

2

, | j, and the cross section as a function of 0

2 for

e

1 = 140 MeV, k = 95 MeV, 0

] = 1°, <f> = 0°,

between 0° and 4°.

for e

2 ranging

Fig. 5. q 2

, |

j, and the cross section as a function of 0

2

for

e-j = 140 MeV, k = 95 MeV, 0

] =1°, <p = 0°,

between 3° and 3.2°.

for 0

2 ranging

Fig. 6. The cross section as a function of 0

2

for e-j = 140 MeV,

k = 95 MeV, <f> = 0° and for 0^0.5°, 1°, 1.5°, and 2°.

Fig. 7. The cross section as a function of ©

2 for = 140 MeV,

k = 95 MeV, 0-j = 1° and for <j> = 0°, 0.5°, and 1°.

Fig. 8. The cross section as a function of 0

2 for e-j = 140 MeV,

k = 95 MeV, 0-j = 2° and for <j> = 0°, 0.5°, 1°, and 1.5°.

127
Fig. 9. The cross section as a function of ©

2

for e-j = 140 MeV,

k = 95 MeV, 0^0.5°, 1°, 1.5°, and 2°, and <j> = 0°,

0.5°, and 1°.

Fig. 10. The cross section as a function of 0

2

for e-j = 140 MeV,

k = 95 MeV, 0^0.5°, 1°, 1.5°, and 2

9

, and <j> = 0°,

0.5°, 1°, and 1.5°; linear scale.

Fig. 11. The positions, z = (v-u)/u, of the maxima, minimum, and

points of inflection as a function of the azimuthal angle

<t>, for e-| = 140 MeV, k = 95 MeV, and 0-j = 1°.

Fig. 12. The positions, z = (v-u)/u, of the minimum and the second

maximum as a function of the azimuthal angle <j>, for

e

1 = 140 MeV, k = 95 MeV, and 0] = 1°.

Fig. 13. The momentum transfer q corresponding to the different

regions of the cross section. See p. 118 at the end of

section VIII for a discussion of this figure.

128
BREMSSTRAHLUNG

CROSS

SECTION

X

PHOTON

ENERGY,

Mev

barns

/

Mev
130
>*
/xw/<» ) VpVp’ip/^p &v

\

ft

-3

*

tp

•H

Em

132
I }

133

Fig.

5

0£

(D
00

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141
NBS-114A (REV. 2-8C) U.S. DEPT. OF COMM.

BIBLIOGRAPHIC DATA

SHEET (See instructions)

1. PUBLICATION OR

REPORT NO.

NBSIR 81-2262

2. Performing Organ. Report No. 3. Publication Date APRIL 1981

4. TITLE AND SUBTITLE

TAGGED PHOTONS

An analysis of the bremsstrahlung differential cross section

in the range of interest for a tagged photon system

5. AUTHOR(S)

Leonard C. Maximon, Arlette de Miniac, and Thierry Aniel

6. PERFORMING ORGANIZATION (If joint or other than NBS, see instructions)

national bureau of standards

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

7. Contract/Grant No.

8. Type of Report & Period Covered

9.

SPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS (Street. City, State, ZIP)

10.

SUPPLEMENTARY NOTES

__j Document describes a computer program; SF-185, FIPS Software Summary, is attached.

11.

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 consider in detail the differential cross section for

bremsstrahlung for angles and energies in the range of interest for a tagging system. We derive a high energy, small angle approximation for

the differential cross section for bremsstrahlung, eq (1.1). We use

this approximation to determine the maxima and minimum of the cross section and to evaluate it at these extrema. It is shown that the

differential cross section has a very sharp dip in the region of small

momentum transfers. Coulomb corrections to the Born approximation are considered, and do not fill in this dip.

12. KEY WORDS (Six to twelve entries; alphabetical order; capitalize only proper names; and separate keywords by semicolons)

Bethe-Hei tier cross section; bremsstrahlung differential cross

section; bremsstrahlung monochromator; photon beams; photonuclear research;

t.aqqpd W phnt.nn mpt.hnd

13. AVAILABIL

'xx| Unlimited

| | For Official Distribution. Do Not Release to NTIS

FH Order From Superintendent of Documents, U.S. Government Printing Office, Washington, D.C.

20402.

^ Order From National Technical Information Service (NTIS), Springfield, VA. 22161

14. NO. OF

PRINTED PAGES

144

15. Price

$12.50

USCOMM-DC 6043-P80