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NBSIR 84-2354

Angular Distribution of High

Energy Electrons Following

Radiation

Leonard C. Maximon

U.S. DEPARTMENT OF COMMERCE

National Bureau of Standards

Center for Radiation Research

Gaithersburg, MD 20S99

and

Alfred Lepretre

Centre d'Etudes Nucleates de Saclay

Division de la Physique

Departement de Physique Nucieaire

Service de la Metrologie et de la Physique Neutroniques Fondamentales

91131 Gif-sur-Yvette Cedex, France

October 1 985

U.S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS

LIBRARY Drtyo

NBSIR 84-2854

ANGULAR DISTRIBUTION OF HIGH

ENERGY ELECTRONS FOLLOWING

RADIATION

Leonard C. Maximon

U S. DEPARTMENT OF COMMERCE

National Bureau of Standards

Center for Radiation Research

Gaithersburg, MD 20899

and

Alfred Lepretre

Centre d'Etudes Nucleates de Saclay

Division de la Physique

Departement de Physique Nucleaire

Service de la Metrologie et de la Physique Neutroniques Fondamentales

91191 Gif-sur-Yvette Cedex, France

October 1 985

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

NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Director

ENERGY-ANGLE DISTRIBUTION OF HIGH ENERGY ELECTRONS

FOLLOWING BREMSSTRAHLUNG

Leonard C. Maximon

Center for Radiation Research

National Bureau of Standards

Gaithersburg, MD 20899

Alfred Lepretre

Centre d' Etudes Nucl Zaires de Saclay

Division de la Physique

D<?partement de Physique Nucl ^ai re Service de la Mdtrologie et de la Physique Neutroniques Fondamental es 91191 Gif-sur-Yvette Cedex, France

ABSTRACT

We derive an expression for the angular distribution of high energy

electrons which have undergone scattering and radiated a photon,

integrated over the directions of the emitted photon, in the region of

small scattering angles, for which the atomic form factor must be taken

into account but the nuclear structure may be neglected. This

distribution is analogous to Schiff's high-energy small-angle distribution

for photons, integrated over the final electron angles. We show that the

correction to the energy-angle distribution of electrons due to atomic

screeing is identical in form to the correction to the energy-angle

distribution of photons. This correction involves an integral over the

atomic form factor, and is evaluated in closed form for the Thomas-Fermi

Moli£re model. A very simple expression is obtained for the case of

complete screening.

Key words: angular distribution of scattered electrons; atomic screening

effects in electron scattering; bremsstrahl ung ; high energy

electron scattering; small angle electron scattering cross

section; Schiff energy-angle di stri buci on.
ACKNOWLEDGMENTS

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

indebted for the skillful and expeditious typing of this report.

i i
CONTENTS

Abstract . . i

Acknowl edgments i i

I. Introduction. 1

II. The Cross Section Integrated Over Photon Directions. 3

III. Technical Preliminaries 8

IV. Derivation of the Screening Correction 16

V. Screening Correction in the Thomas-Fermi -Mol i$re Model 34

Appendix 37

References 47

INTRODUCTION

In this note we derive an expression for the angular distribution of

high energy electrons which have undergone scattering and radiated a

photon, integrated over the directions of the emitted photon. This cross

section has been obtained previously under conditions pertinent to large

angle, large momentum transfer electron scattering experiments [1-3].

Here we are concerned with this same cross section, but under conditions

appropriate to a photon tagging system, for which the important contribution comes from small angle scatterings, corresponding to momentum transfers q < 0(mc). Although momentum transfers q > 0(mc) give a negligible

contribution if one subsequently also integrates over the angles of the

scattered electron, our angular distribution for the scattered electron is

in fact valid for the wider range of momenttm transfers, qm

< q « 4,

where q m

is the minimum possible momentum transfer and R is the nuclear

radius, i.e., it is valid provided that the nuclear form factor does not

enter significantly. We therefore neglect target recoil, and in place of

the nuclear form factor which appears as g

2

(q) in the completely differential cross section involving large momentum transfers, we now have the

atomic form factor, appearing as [l-F(q)] 2

. (See [1], p. B1345, comments

following (5).) We start therefore with the Bethe-Heitler (Born approximation) differential cross section for bremsstrahlung. We note that the

integration of this cross section over the angles of the final electron,

assuming small momentum transfers, is well-known [4], and in the case of

complete screening results in the Schiff energy-angle distribution [4,5].

1
The integration of the bremsstrahlung cross section over final electron

directions has also been performed [6], including Coulomb corrections, in

the region of small momentum transfers q < 0(mc), and some of the analysis

given in [6] will be of use to us in the present work.

However, we have not found, in the available literature, an expression for the bremsstrahlung cross section integrated over photon

directions that is useful for the region of small momentum transfers (the

equivalent of Schiff's expression, but integrated over photon directions

rather than final electron directions). This lack is quite understandable. The major interest in the cross section for small angles (low

momentum transfers) has been to obtain the angular distribution of

bremsstrahl ung (integrated over electron directions) since the photons are

emitted preferenti al ly in the forward direction. On the other hand, when

the integration over photon directions has been performed, it has been in

connection with high energy electron scattering experiments, which are

performed at large angles and large momentum transfers in order to obtain

nuclear structure information.

2
II. THE CROSS SECTION INTEGRATED OVER PHOTON DIRECTIONS

We turn now to the Bethe-Heit ler (Born approximation) cross section

integrated over photon angles, but with no approximations concerning the

energies of the incident and final electron or the angles of the scattered

electron. This has been given by Maximon and Isabelle [1] in connection

with the radiative tail in elastic electron scattering (eqs (4) and (5)

on p. B1345 of [1]). We need only replace the nuclear structure function

£F( q) in eq (4) of [1] by l-F(q), where F(q) is the atomic form factor. We

then have

where

The most direct way of arriving at the result given in (2) is the

procedure used in [3] (rather than that followed in [1]). In [3] the

integration over photon angles is performed in a coordinate system with

z-axis in the direction of £1 -P 2

» in which the photon angles are e

k

, 4^.

The momentum transfer to the atom,

q

2 = (Pj-j^-jO 2 = (£r£2 ) 2 + k2 * 2kj£1 -p 2 |cos0k

is independent of the azimuthal angle, <t> k

, of the photon. The integration over *

k is then straightforward and leads to the expression given

below in (2), while the integration over cos@

k may be written as an integration over q

2

, as in (1).

Note that we have replaced dk in [1] by de 2

. In the absence of

recoil this makes no difference since then k = e,-e 2 and hence dk = -dc

2

. However, since in the present investigation the final electron is

observed, we find it more reasonable to use de 2

.
2k

(2k + k

2 )^

1 1 \/

ql+ + ^ “ 4cJ 2

( £

l 2+e 2

2 "^ ” l 6e l e

2

D

1 o| / \ 2k - q

2

(4 ci - q

2

) + 2k —-- — [2\( \-k e 9 ) - ( \+k e , ) q

2

1 D, 2 / 2

(4 ci - q

2

) - 2k [2\(\+ke 1

) - (\-ke 2 )q

2

] d

2 3/2

with

\ = e

1 €2

- p x p 2 COSd - 1 = ^(|£: - p^i 2 - k

2

)

^0 = €

l e2

“ P 1^

2

~ ^

Dj_ = {p !

(

q

2 - q 22

) + 2p 2 \

0 cosd} 2 + 4k 2 p 2 2 sin 2 £

D

2 = {p 2 (q

2 - q x 2

) + 2p 1 \

0 cos^} 2 + 4k 2 p 1 2 sin 2

ft

Pm

= l£l * P.2 1 - k = + k2 )^ - k

Pm = ( £i - £2

1 + k =

( 2x + k2 )

i2 + k

q 2 = 2p 2 sin%d

q L = 2p 1 sin^d

6 = p 1 - p 2 - k

k = £

l “ £

2

.

( 2 )

( 3 )

4
Here dQ = sind da d<j> refers to the final electron, with polar and azimuthal

angles a,* in a coordinate system with z-axis in the direction of the

initial electron.

€

X

, p L are the energy and momentum of the incident electron.

e

2 , £2 are the energy and momentum of the final electron,

k = ei - e

2 is the energy of the emitted photon.

All of the energies and momenta are measured in the laboratory system, in

units of me 2 and me, respect i vely.

Having performed the integration over photon directions, giving (1)

and (2), we are now confronted by a rather complicated looking expression

which cannot be integrated analytically because of the presence of the

screening function, which is in general only given numerically. However,

if we consider the screening correction rather than the cross section

itself, then a considerable simplification results: For the cross section

with screening we write

da = [da -da 1 + d a

sc r

v scr unscr j unscr

= Aa + d a

scr unscr

(4)

In the integral in (1) we now write

( l-F) 2 = [( l-F) 2 - 1] + 1 . (5)

The term [(l-F) 2 - 1] gives the screening correction, Ao ; the last

term, 1, gives the unscreened cross section, da . This separation,

unscr

which was used in [6] in integrating over the angles of the final

5
electron, has two important consequences. First, the integration required

to obtain the unscreened cross section can then be performed in closed

form. In the present case, in which we integrate over photon angles, the

unscreened cross section is given in [1] (p. B1346, eq (6)). Second, in

the screening correction only very small values of q are significant,

viz.

,

q < 0( 8) , (6)

where 3 is the inverse atomic screening radius. (In the Thomas-Fermi

jl/3

model, 3 » -^y « 1.) This may be seen by noting that we can write,

approximately ([6], p. 897, eq (6.30))

q

2 q

2 + 8

2

and hence

fl - FI 2 - 1 = _ 2 q

2 B2 + B 4

. (8)

q

4 (q

2 + B

2

) q

4

Thus in the inteqral for the screening correction, the contribution from

q > 0(1) is of order B

2 and may be neglected. The only non-negl igible

contribution to the integral comes from q < 0( S) and, as we will show, a

remarkable simplification of the integrand in (1) results in this region

of very small momentim transfers. In fact, we find that the screening

correction to the angular distribution of the scattered electron

(integrated over photon angles) is the same as the previously obtained [6]

6
screening correction to the angular distribution of emitted photons

(integrated over final electron angles), once one has made the appropriate

change of variable from photon scattering anqle to electron scattering

angle.

The explicit expressions for the screening correction and the

unscreened cross section (the angular distribution of scattered electrons,

integrated over photon angles) derived in this report are given in the

text by eqs (61), (66), and (67) (for the correction for intermediate

screening), by eqs (61), (66), and (71) (for the correction for complete

screening, and by eqs (63a) and (63b) for the unscreened cross section.

7
III. TECHNICAL PRELIMINARIES

We first make a few preliminary observations concerning the variables

defined in (3) which are useful for the simplification of the integrand,

(2), for the screening correction. In order to simplify the order of

magnitude observations that follow, we will not only assume high energies of

both the incident and final electron,

>> 1 , €

2

>> 1 (9)

but will assume as well that the photon energy is of the same order of

magnitude as these energies:

k = 0(e) (10)

A more detailed analysis is required to show that our final results are

valid for somewhat smaller photon eneroies, but we will not present that

here.

In the following analysis we show that, as a consequence of (9) and

(10), the variables in (3) must satisfy certain restrictive conditions in

order that we have a non-negl i gi bl e screening correction. A very

significant simplification of the integrand then follows as a result of

these conditions, which are

i) e> 0(1/0)

ii) a

2 0(0/ e) < 0( 02

)

i ii ) 0( 1) < \ < 0( e(3)

8
From iii) it follows, then, since the significant values of q in the

integrand for the screening correction are q < 0(8), that we can set

< 0( P

2

) « 1 (11)

in the integrand. Most importantly, this permits us to expand the

denominators D^ 72

, D

2 1/2

, D^ 72, and 0

2 372 in powers of as a result

of which we find a very simple form for the integrand.

From (3), the minimim momentum transfer, q , is given by

\

= l£i ' P.2 1 “ k l PrP 2 - k = 5

and from (9) and (10) (see appendix)

5

k Zc l c

2 0(l/e) .

Now if we are to have a non-negl igible screening correction, we must have

qm 1 °<

( 12 )

and hence also 5<£0(|3), from which we have restriction (i).

e > 0(1/3) (13)

Next, from (3) we also have

9
( 14 )

= (2X + k

2

) h - k

2X

(2x + k

2 )^ + k

Thus from (10) we have (2X + k

2

)

12 + k = 0( e) and q^

= 0(\/e).

order that qm

< 0(3) we now must require

X < 0( ep) .

But at high energies we have (see appendix)

u2

\ <* -z + 2e 1 e,sin 2 ^d

^ e

l e2

1 1

= 0 ( 1 ) + 0 (€ 2 *

2

) > 0 ( 1 )

.

Now from (15) and (16) we have

e

2

fl 2 < 0( eB) ,

which, together with (11), gives ( i i

)

:

d

2 < 0( 3/e) £ 0( 82

)

.

Again from (15) and (16),

0(1) < X < 0( eS)

which is (iii).

Again in

(15)

(16)

(17)

(18)

10
As a prelude to writing the explicit expression for the expansion of

the integrand fj

|

in (2)) in powers of q

2 /\, we examine the order of

magnitude of each of the terms there. We show that, individually, the

k

2 largest terms there are each of order -rj > O(l/0 2

'l » 1. However, taken

A.

'N' together they cancel each other almost completely, and as a consequence we

must keep the terms of order 1 in

|

j>, these being of order 8

2 relative

to the individually large terms. To see this clearly we first write the

expressions for D

1 and D2 in a form more suitable for an expansion in

powers of q

2 /\. (The expressions given in (3) are taken directly from

[1].) From the appendix of the present report -we have

D

x = 4p 2

2 \2 - 4[( €

X € 2 - 1)\ - k

2]q 2 + p x 2 q

4 (19a)

and

D

2 = 4p 1 2 \

2 - 4[(e 1 e

2

- 1)\ - k

2]q 2 + p 2

2 q

4

. (19b)

Now from (11) we note that in (19a) and (19b) the second term is of order

£ < 0(g2)

relative to the first term, and the third term is of order

(q

2/\) 2 < 0(34

)

relative to the first term. Thus we can -write

11
Dj* = 2p (20a) 2 x(l + 0(g2 ))

d

2 % = 2 Pl x(i + oo2

) )

. (20b)

Again referring to (2), we note from (11) and (18) that

+ 4

X

2 - 4q 2

( ~ 16ej_e 2\ -16e^€ 2 2X - q

: 2X

(1 + 0( 8

2

) ) (21)

[2X( X - ke 2

) - ( X + k

e

i ) q

=

2x( X - k £

2 ) (1 + 0( (3 2

) ^ (22)

[2x( X + ke^) - ( X - ke 2 )q

2 ]

= 2X( X + ke^fl + 0( f$ 2

) 1 • (23)

Furthermore, in (22) and (23) we can write, from (3),

X ** k e

2 — e i

e

2

“ p ^p 2 cosd 1 •

£

i

£

2 £ 2

“

= p 2 2 - p 1 p 2 COS^

= -p

2 (p 1 -p2

) + 2p 1 p 2 sin 2 ^

(24)

and

X + ke l = £

L £

2

- p 1 p 2 cos-5 - 1 + £

X 2 - £x £2

= Pi

2 - P X P 2 cos^

= D

1 (p 1 -p2

) + 2p 1 p 2 sin 2 ^d . (25)

12
Now for high energies, P!-P 2

25 e i" e

2 = k. Thus* assuming k = 0( e) and

noting that O

2 <^O(0/e) from (17), we have

X - ke 2 = -p 2 (p 1 -p 2 )(l + O(0/e)) (26)

X + ke L = p 1 (p 1 -p 2 ) (1 + 0(0/ e) ). (27)

We now collect the terms in

j ^ with a power of Dj_ or D

2 in the

denominator. From (20a), (20b), and (21) we have, for the terms with D

1 -h

and D.

-k

1 1 \ /q

4+4X2 -4q 2

( £^

2+e 22 -l)-16e^e 2N 4ke 1 e

2 / 1 1 V D 2

% ' 2\ - q

: , ,

. |fl+0( 3

2

) 1 2X \p 2 Pi,

(28)

4k(p 1 -p 2

) e^e 2

X

2 p x p 2

(1 + 0(32 ))

Next, substituting (26) and (27) in (22) and (23) and using (20a) and

(20b), we have, for the terms with D

1 * 3/2 and D

2

’ 3/2 :

2k(4e 2

2 -q 2

)

D

x 3 /2

[2X( X-k e

2

)

-2k(

p i-p 2) e2

^

( X+ke^q 2 ] i.f 1 + o( 3

2 )) (29)

X

2 P 2 2

and

-2kf4e, 2 -q 2

)

D

2

3 / 2 [2X( X+ke^

-2k(p

1 -p2

) £l

2

( X-k

e

2 )q

2 J

(1 + O( 0

2 )l . (30)

X

2 Pl

2

13
Thus the large terms in < >, those exhibited specifically in (28), (29),

* k

2 and (30), are indeed each of order -rj > 01/

p

2

) » 1, as mentioned.

However, when added they give

2k(p rp 2

) /e

l 6

2

\ 2 /k 2

k

2 \

(31)

since, as shown in the appendix.

Thus the "large" terms cancel, leaving a remainder of order

k 4- / k

2 \2 ~ 1 1

6 ’ VxT2 / T2 1

c

2

\£ 2 since from (16) —^2 > 0(1). This remainder can therefore be neglected;

the contribution to be retained thus comes from the terms of 0(1), i.e.,

from the terms indicated in (20a) - (23) as being of 0(

0

2

) relative to the

individually large terms. It should be noted that the very significant

cancellation, giving the factor

< 0 ( 1 /

£

4

)

14
results from our particular choice for the "large" terms in ( 28) -

(

30)

.

Had we made a less auspicious choice, for example by writing

a. - ke 2 = -(<£ 2(1 + 0(p/e))

\ + keT = ke^(l + 0(f3/e))

instead of (26) and (27), then the large terms would only have cancelled

to relative order 1/e 2 (rather than to relative order 1/e 4

). The

remainder would then have been of order 1, and have had to be retained.

15
IV. DERIVATION OF THE SCREENING CORRECTION

We now proceed with the expansion of the terms in

| ^ in (2),

retaining all terms of 0(1), i.e., of order p 2 relative to the large terms

which have just been shown to cancel. However, we neglect terms which are

individually of 0(

p

2

) or of 0( 1/

e

2

) ; these terms are of order p

4 or of

order p

2 /e 2 relative to the large terms.

Referring to (2), we have first

-2k -2

{ 2 \ + k2) % '

(l + $)*

= -2 + 0(\/k 2

) . (33)

Here, from (18), (10), and (13), — < 0(p/c) < 0(p 2

) . (34)

k

2 ~

Thus, as just explained, we retain the term -2, but neglect the term of

0( x/k 2

) < 0((3 2 ), writing

-2k

r = -2(1 + 0(p 2 )) . (35)

(2\ + k

2 )*

Next, in the term with factors and D -h

2 we expand the factors

16
q

4 +4\2 -4q 2

( e

1 2+£ 2

2 ’ 1 )” 16e l e

2

2 \ - q

2

8 £

: c2 2q

2 (£ L 2 +£

2

2

) 4£ 1 £

2 q

2

_ _ + 2 \ - (l+0(,8 4 )) (36)

and, from (19a) and (19b),

-k

1 1 ,o7'^ 2p 7 \

1 ( €

1 £ 2 X * k2 )^

2

1 + -k — 2t,2

p 2

z x

(1 + 0 ( 8 4

))

2p x X

1 ( £

1 “ k2 )^

2 1+ ?

Pl

2 X2 (1 + 0(f3

4 ))

k (Pi-P 2

)

2 \ p^p 2

1 + k(p

1 2+p 1 p 2+p2 2 )q

2

2p 1 2 p 2 2 \

2 (1 + 0(f3

4

)) . (37)

Multiplying the factors in (36) and (37) then gives the expansion of the

term in (2) with factors D

x

2 and D

2

:

17
q

4+4x2 -4q 2

( e

1 2+e2 2 -l)~16e 1 e

2 2X - q

2

k ( P !*P 2

) 1 +

(e

1 £2 \-k 2 )(p 1 2+p 1 p 2+p 2

2 )q2

2Xp ^ 2

x X

2q 2

( «i

2+ £

2

2

)

4k(p 1 -p 2 ) £^£2

2X P 1 P 2

k(Pi-P 2 )

PiP 2

<<(Pl

_ P 2 )Q

2 2£ 1 €

2 (e

1 €2 X-k 2 )(p 1 2+p 1 p 2+p 2

2

) (£i

2+ £ 2

2

) ^ €

1 €

2 +

1 —— - — ... — — — — + — - ------- +

P L P 2 X L Pi

2 P 2

2 X3 X X2

.

+ terms of relative order p 4

. (38)

18
Finally, we have the terms in (2) with factors D

x ~ 3/2 and D

2

“ 3/2 . Again

we expand, substituting (24) and (25) in the factors in these terms:

2x( X-ke 2

) - ( \+k

e

x )q

2 = 2x[-p 2 (p rp 2

) + 2p 1 p 2 sin 2 ^>]

- [Pi(Pi-P 2

) + 2p 1 p 2 sin 2 ^]q 2

= [-2Xp 2 (p x -p 2

) + 4xp 1 p 2 sin 2 ^

- p 1 (p 1 -p

2 )q

2

] (l+0( p

2 /

e

2

) )

2\( X+keJ - ( X-k

e

2 )q

2 = 2x[p 1 (p 1 -p 2

) + 2p : p 2 sin 2 ^]

- [-p 2 (p rp 2

) + 2p 1 p 2 sin 2 ^]q 2

= [2Xp 1 (p 1 -p 2 ) + 4Xp 1 p 2 sin 2 ^d

+ p 2 (p rp 2 )q

2

] (l+0( 3 2/

e

2

}

)

We then expand Dj“ 3/2 and D

2

“ 3/2

, using (19a) and (19b), from which

D

3 /2 =

nU -3/2 =

2

1

( 2p 2 X) 3

1

J.

(2p X X) 3

1

1

(« 1 e

2 X-k 2 )q

2

P 2

2 X2

(

e

2 (q^

Pi

2 x2

(l+otp

1*))

(l+0( 3“)

)

(39a)

(39b)

(40)

(41)

19
Then, referring to the terms in (2), we have, from (39a) and (40),

(4, 2

2 -q 2 2k

D^2 [2\(\-ke 2 ) - (X+keJq 2

]

k e

2

2

P 2

3 X3 + 4

(e

1 e2 \-k 2

) 2%2 P/X

x [-2\p 2 (p 1 -p

2

) + 4\p 1 p 2 sin 2 ^d - p 1 (p 1 -p 2 )q

2

] (l+0(p 4 ))

2k(p rp 2

) £

2 2 ’ ~

p7

4kp

x e2 2 + si

n

2 ^d

X

2 P 2

2

k ( P i~P 2

) e 2*

X

4 P 2 4 q

2

[ 3( e

1 e 2 X-k‘1 ) + P i

P

2 X

]

+ terms of relative order p 4

. (42)

20
Finally, from (39b) and (41) we have

f4 £l

2 -q 2

) •2k [2\(\+k£ X

) - ( \-ke 2 )q

2J 0

2

3 ^ 2

^i 2

Pl

3 X3 1+ i

3 (£ 1 €

2 X-k 2

]

Pl

2 X

2

x [2Xp

1 (p 1 -p 2

) + 4xp 1 p 2 sin 2 ^ + p 2 (p 1 -p

2 )q

2] (l+0(64 ))

2k( Pl -p 2

) £

2 ~

77

4 k P 2 e

i 2

. . sin 2 ^

X

2 Pl2

k ( P i”P 2) €

i‘

x

4 P L

‘ q

2 [3( e

1 e2 X-k 2

) + P : P 2 X]

+ terms of relative order 8

4

. (43)

Now the first term on the right hand side of each of the three expressions — (38), (42), and (43) — are the large terms we have already considered

(note eqs. (28), (29), and (30)). As mentioned, they are individually of

order k

2 /\2 = 0(e 2 ), but when added give a contribution of 0(l/e 2 ), which

we neglect (see eqs (31) and (32)1. We are thus left with the remaining

terms, of 0(1), in (38), (42), and (43), as 'well as the term -2 from

eqs (33) -(35). We first gather the terms without a factor q

2

. These come

from (35) and the second term on the right hand side of eas (38), (42),

and (43). They are

21
-2 - k(p rp 2 ) 4kp : e

2

2 4kp 2

e

x 2 + sin 2 ^d

p x p 2 X

2 p 2 2 X2 p x 2 si

n

2 ^d (44)

Since these terms are each of 0(1), we may now set e

1 « p 1# e2 a

p 2 throughout, neglecting terms of 0(l/e 2 ). We then have

k

2 4k 2 f£ 1 2+e 22

) 4k 2 -2 - + sin 2 ^ = - + —— sin 2 ^d . (45)

€ 1^2 X 2 €

2 e2 X

2

We next gather the terms with a factor q

2

. These come from the last term

on the right hand side of eqs (38), (42), and (43). They are

k(p2_—

P

2 ) 2 e

2 ( e

1 e2 ^-”k 2

1 (p ^ 2+p ^P 2+ P 2 2) (e^ 2+e 2

2

) 2e^c 2 q2 + + \PlP 2 P X 2 P 2

2 X3 X X2

k ( p i“P 2 ) , e

2

2 qX

4 P 2 4

( €

i € 2 X - ^

2

1 + P 1 P 2 XJ

^ ( P ]_~P 2

) e

1 2 q

2 [3( £l e

2 \ - k

2

) + p 1 p 2 \] . (46)

X

4 P X 4

As with the terms without a factor q

2

, listed in (44), we may now set

e

i " Pi* e2 ” P 2 throughout, again neglecting terms of 0(l/e 2 ). The terms

in (46) then become

22
2( e

1 e2 ^- k2 )( €

i 2+e i €2+€

2 2

) U^+e2

) 2 + + —

k

2 q2

. 2- 2 X

2

e

l e

2 K e

l e

2

3(€ 1 e

2 ^**<2 ) 3(e 1 €

2 \-k 2

) 1 / € 2

£

2

2 X2 2 \ 2 *1 * \ \ £ 2 € ^

k

2 q

2

( e

i e 2 X— k

2

) 2 2 2 — + +

2 .2

L

c

2

€

1

3 3

2 .2

i c2 n €

2 j

(ei 2+€

2

2

) + — +

\ e

l e

2 M)

k

2 q2

(

e

1 £2 \-k 2

) / 1 1 1 £j

2 +£~ 2

£ 1 2

£

2

2

£

i £ 2 / £ 1 £ 2^

( £l

2+£ 22 -2£

1 £2 )+

£ 1 £ 2

k

2 q2

( €i£ 2 X-k 2 )k 2 k

2 (£^ 2+£

2

2

) _ +

£

1 2

£

2

2 X2

£ 1

£

2 ^ Cl e2

k

2 q2 2k 2 k4 (e 1 2+e 2 2

) + +

€ 1 £ 2

^" €

i 2 €

2

2

^- 2 £ i12€

k

2 q2 (£ 2+£

2 2

) k

2 / k

2 2 \ -

£ l £ 2 X

2

£ i £

l c2 £ l £ 2

23
Now for high energies (see (A4) )

k

2 . o 2 \ 53 1 + 4e 1 e :,sin 2 ^d

e

l e 2

1 2

and thus we have above, in (46),

k V

\

2

e

l e2

4k 2 sin 2 ^d

\

2 (47)

We note that the terms in the factor [ ] here are identical to the terms

on the right hand side of (45), apart from a sign. Thus, adding (45) and

(47), we have, for the terms in { } in (2), the remarkably simple

expression

12

e 2+e 2

l +e 2

4k 2 6

1

e 12

si n

2 %d (48)

We now write this in a form which is similar to that obtained for the

screening correction considered in [6], where we integrated over final

electron directions. For high energies we have, from (A4),

_ k 2 Ae^eoSin 2 ^ » 2 \ -

1 2 £

1 £2

and thus, in (48),

24
4k 2 e

1 e2 sin 2 ^

k

2 /

» —* 2X

X

2 X

2 V

2k 2 / k

2 \

S3 4c

1 £2 6k

X

on substituting

thus be written

2c x £

2 from

in the form

(Al). The expression (48) for

{ }

Ca "

k

2

X 2 £

1 £

2

4e l £ 2— (49)

Now from (A7) and (A19) we have, for high energies and small scattering

angles.

j“ * 6(1 + w

2

) = ~ (50)

where w is defined by (A6):

w = — . (51)

k

Substituting (50) in (49) then gives

31^

' 4£l£2X(1 • x)

] f

2 • ~) • (52)

25
Finally, from the definition for x* (A19)

,

1 x

=

1 + w

2

we have

x( 1 - x) = W

2 x2

,

which, when substituted in (52), gives

IV-22 " 4e i e

2 w2 ^

2l n2 " "T| • (53)

«

2 «i«2

L J

\ W

In order to obtain the screening correction, we must substitute (53) in

(1), and replace (1 - F) 2 in the integrand by (1 - F) 2 - 1, as indicated

in (4) and (5). Further, in the lower limit in (1) we can write, from

( A20)

,

and take the upper limit to be infinite since there is negligible

contribution to the screeinq correction from q > 0(1). We then have

26
In order to compare this expression with the screening correction given in

[6] we note from (50) and (51) that for small scattering angles, #,

dQ = sin# d# d<t>

* # d# dg

• *fe)

2 d(w2) d *

= 2

5

2d(w 2

) d<j>

5

2 = -2 — d x d(D . (56)

X

2

Then, since the expression (55) is independent of <j>, integration over this

variable merely gives a factor 2n. We then have, if we take q rather than

q

2 as the variable of integration,

e

2 / Ze 2

\ 2 de 2 da =4 — I I scr corr

ftc \mc 2 /

«i

2 k

dx 4 e l£ 2 W X^

r [(i-F ) 2 - 1]

X J q

3 5/

X

Comparison with the cross section integrated over final electron

directions, i.e., the angular distribution of the photons, given in [6],

p. 897, eq (7.2),

27
then shows that (57) is identical to the screening correction in [6],

once one has made the substitutions

dk *> de 2

u -* w

1 1 l = - = X . (58)

1 + u

2

1 + w

2

Having already noted that in the absence of recoil dk = -de 2

, the

significant replacement in (58) is

u w

where

u = p 1 9

1 with 0J = $

is the convenient variable when the photon is observed, and

P lP w = with 3 = * (P ].>

£

2

^ k

Note that the screening correction in [6] is contained in the terms in

(7,2) with factor r, defined in (6.29) and (6.28) in that reference. It comes, specifically, from the term £?(5/£) in r, as can be seen from

eq (6.28) there.

28
is the corresponding variable when the final electron is observed.

Indeed, a comparison of the two situations in the figure below

photon observed

ntegrate over electron angles

electron observed

integrate over photon angles

suggests that the analogous variables might be

ke l (photon observed)

and (59)

P 2 ^ (electron observed)

This would imply the substitution given in (58), since from

k9]_ * P 2 d

29
we have

Pi Pi «i €

u = p 1 e

1 = — (kex

) -> — (p 2a) * -

k k k w

and

1 1 5

= 7 ; = X • 1 + u

2

1 + w

2

Having shown the identity of the screening correction derived here,

eq (57), with that given earlier in [6] when integrating over electron

angles, under the substitution (58), we choose finally to express it in

terms of the variable q^. It then has the same form whether one

integrates over photon or electron angles, the only difference being the

definition of the minimum momentum transfer, q , in terms of the remaininq

variables, viz.,

q m

= |£ : - kj - p 2 photon observed (60a)

q m

= |£ x - £21 - k final electron observed (60b)

Substituting q m

as given in (54) into the expression (55) we have

30
d ascr corr

1 e

2 /Ze 2 '\ 2 de 2 dQ „c \mc 2

/

' c

2 k

£

l 2+e 2

2 - 4£ 1 £

2 — t 1 - q

’m m

,

x [(l-F(q) l 2 - i]

q

4

d(q 2

) ( 61 )

From (4) and (61) we now have our complete expression for the cross

section integrated over photon angles:

d cr da da —scr unscr , scr corr = — + — . (62)

dQ dQ dQ

As noted earlier, the unscreened cross section (for bremsstrahl ung in the

field of a point Coulomb potential) can be integrated in closed form over

the photon angles, and is given in [1], p. 81346, eq (6). The expression

given in that reference is for arbitrary energies and angles, and is

consequently rather lengthy. Since for our present considerations it is

quite appropriate to assume e

l >> 1 and e2 » 1, we give here the high

energy form of that expression without, however, making the small angle

approximation, 9 « 1. We then have

31
Ifunscr = Lffpf\2

l!i.

dQ 2 it he \mc 2 / k^ 2 Ze l e

2 (63a)

where

k

2 \ + 2 e i

£

2 ( \+l ) cos / , x r An < \ + 1 + [\( \+2) ] 2 > + 2 )V [ ) \

2 [\{ \ + 2 ) ]'

c

i( €i +€2

) €

i 2 (2 €2 ~k

)

S1 n

2 ^

- k + cos 2 %d +

£o \

2 An(2 £

2 )

£ 2 ( e, + £,) e 9

2 (2e 1 +k) si n

2 ^

k + cos 2 ^d + An(2e 1

)

k

2

2 £ ^ £ 2^- £^

2+£

2

2 ” 2( £

1 2+£2 2+ £^£9)COS 2 ^d

2k 2 si n

2

v> 1 e

2

( e

i 2+e 2

2 " £

i e2 ) ~ cos 2 ** . (63b)

The high energy cross section for electrons which have scattered

through an angle d and radiated a photon, integrated over photon angles,

is then given by eq (62), in which we substitute eqs (61) and (63). In

these equations the variables 6, q^, and \ are given by (Al), (A12), and

(A4), viz..

32
k

=r

\ = — + 2€i€oSin 2 ^d . (64)

It is to be noted that although the screening correction, eq (61),

has the same relatively simple form whether we integrate over photon or

electron angles, the unscreened cross section is very different in the two

cases. The rather simple form that results for high energies and small

angles when integratinq over electron angles f[5l, p. 924, Formula 28N(a)l

is unfortunately not obtained when integrating over photon angles, as is

clear from eq (63), even when we make the small angle approximation there,

viz., by setting sin 2 # » #

2

, cos 2 %# =* 1. The rather complicated arqument

of the logarithm in (63) does not simplify for small angles, where

\ = 0 ( 1 ).

33
V. SCREENING CORRECTION IN THE THOMAS- FERMI -MOL I ERE MODEL

Although the screening correction is given formally by eq (61) as an

integral over the atomic form factor, it is clear that a simple closed

form expression without integrals is useful for practical applications,

particularly if one must later perform an integration over this cross

section. We therefore perform the integration in (61) in closed form for

the Thomas-Fermi model as used by MoliSre [7], viz., with

i=l 3- 2 + q

2 (65)

where

CEj_ = 0,10 a

2 = 0.55 a3 - 0.35

(65a)

8

i = (Z 1/3 /121 )b • ; b

x = 6.0 , b2 = 1.20 , b

3 = 0.30 .

The result of this integration is then rather simple [6]:

34
Defining the function ^(q^) by

where

f fl-F(q) ) 2 - Vi (— - l\ d (q2) = - —C(qJ (66) f

~qM

2 \% m

2 m

’m

we f i nd

c{\] = S “i

2 tofl+B

i^

i =1

3 3

' 2EZvo i =1 j=l

1+B

.

B

iB

j

1 An[l+Bj ) + {

(67)

b . J!i

1 \ Q

\ Mm;

( 68 )

R • In the absence of screening, i.e., for « 1, eq (67) gives

Tn

C(q

m

) = 0 « 1 , (69)

as expected* For » 1. i.e., for complete screening, eq (67) gives

35
= An (70)

2l/3

111.8 qm

+ 0

Thus, if we write, in the limit of complete screening.

^compl ,scr. ^m

~ I

1 + G“-8 0 . (71)

then both the complete screening and the no screening limits, eqs (70) and

(69), are given by the single expression, (71). A comparison of C = C { q )

as given by (67) with C

cq ^ $cr

as given by (71) shows that the relative

error incurred by using (71), viz., (^com p] scr

- C)/C, is less than 1%

71/3 71/3

for tv' i - > 6.4 and less than 2% for ~o > 4.7.

111.8 q 111.8 qr "m ^m

For most practical applications requiring the screening correction,

the expression given in (67) may be used for the integral in (61).

1 1/3

Further, in the case of complete screeing, i.e., when

Yn

~8

~"

q

~ " ~ 5, one

may use the even simpler expression aiven in (71).

’m

36
APPENDIX

In this appendix we derive the high energy approximations used in the

body of the report, in particular for the variables 6 , x0

, and x defined

(32). In addition we derive approximations for x and q^, valid for both

high energies and small scattering angle, a.

By high energy approximation we mean e

1 » i and e

2 >> 1* an^ imp 1

y

neglect of terms of relative order l/e L 2 and l/e 2

2

, which we refer to as

neglect of terms of relative order 1/e 2

. By small angle approximation we

mean d « 1 and imply neglect of terms of relative order d

2

.

We start by deriving the high energy approximation for 5, given in

in (3), and for the quantity enters in eqs (31) and

(3):

6 = p rp 2-k

= P 1“P 2“ e

l + e2

=

( 2)

~ ^ e l"P 1

)

1 1 (since e

2 -p 2 = 1) €

2+ P2 £i +

Pi

e

l+Pl" e 2'P2

( e

1 +p 1 )( e

2+p 2 )

Here

37
(p 1 2 -p 2

2

) P 1~ P 2 '

P L +P 2

(n 2 ^2 2 ^

Pl+P 2

f €i + €

2

1

i e

l +e 2^

" k (Pi+P 2 )

Thus,

6 =

k 1 +

€

l +€2 \

P1+P2/

(c

1 +p 1 Ke2+p 2 )

k ( £

i +

P

i

+ €

2 + p 2

{ P i + P 2 ^( e

]_ + P]_lf ^2 + P 2

')

Thus far we have made no approximations, and have written 6 in a form in

which there are no cancellations. We can now set p : = e

1 (l+0( l/e^) 1 and

p 2 = e

2 fl+0( 1/

e

2

2

) ) and find, neglecting terms of relative order l/e^ 2 and

1/

e

2

2

.

5

k 2e l e 2 (Al)

38
Next we consider (see (3)

)

*0 = e

l e 2

" p l p 2

" ^

= ^[(p r p 2i 2

- Ur c

2 ) 2]

= ^[( Pl -P 2 ) 2 - k

2 ]

= %(p 1 -P 2 -k ) (p rp 2 +k)

= fc6(p 1 -p 2 -k+2k)

= 6 f k + ^6} .

Aqain we have an expression with no cancellations. We can now use our

high energy approximation for 6, eq (Al), and obtain

The second term is of relative order 1/e 2 and is thus neglected, giving

the high energy approximation

(A2)

39
Next, in eqs (31) and (32) we have the factor

/

€

1 g 2 \

2

( e

l p 2 -p

l e

2

) 2

\Pi P 2 / Pl

2 P 22

Here

(€,p 2 -p

1 62 ) 2 = e

1 2 e2

2 " £

l 2 2p 1 p 2 e1 e 2 + p 1 2 p 2

2+p 1 2

=

( c

l € 2“PlP2^ 2 “ ^

=

l €

i e 2”P lP 2”^ €

1 e 2~^ lP 2 + ^

= \

0 f\ 0 + 2) .

Using the high energy approximation for \

0 just given in (A2) we have

( ^iP 2 “P i e 2^

2 k

2 (k 2+4£

1 £

2

)

( 2e l e 2 ) 2

k

2 fe 1 + £ 2

) 2

C 2e i

2 l 2

We thus have the high energy approximation

li Ii\

2 „ k2 ( e

l + ~2 ]2

Pi P 2 /

(A3)

40
which is given in eqs (31) and (32) as

( A3a)

Next we have the important variables X and q^. From (3)

\ = e

l e2 - Pl p 2cos» - 1

= €

1 €2 -p1 p 2 -l + p 1 p 2 (l-cosd)

= \ 0 + 2p 1 p 2 sin 2 ^d .

Thus from (A2) we have, at high energies (but arbitrary angle £)

,

k

2 X ~ + 2ei e 7 si n

2 %$

2e l £

2

1 2 (A4)

If in addition d « 1, as it is in the screening correction, then we have

X ®

k 2 e

i e 2^

2

2e l €

2 2

k

2

2 €

1 e2

1 +

£

l £2 d\

2

From (Al) this can be written as

X ® k8 1 + (A5)

41
This suggests that we define the variable w by

w

e

l e

2 d

k (A6)

from which

\ » k 5( 1 + w

2

)

or

£

" 6(1 + w

2

)

. (A7)

Equation (A7) provides a useful expression for the minimum momentum

transfer, q^, in the region of importance for the screening correction,

q m

< 0(3), since this condition, together with the assumption of high

energies, implies small scattering angles. This may be seen from qm

as

defined in (3): Starting with

qm= ( 2 X + k2) % - k < 0(p)

we write this as

(2\ + k

2 )^ - k < c8

where c is a constant of order unity. Thus

(2\ + k

2 )^ < k + c3 .

42
Squaring both sides we then have

2X < 2ck|3 + c

2

8

2

which may be written as

X < 0(kp) + 0( 8

2

)

With X as given at high energies in (A4), we have

k

2 X *-5 + < 0( k 3 ) + 0( |3 2

) ^ e

l e2

and hence

2e^£2 s ^ n2^ ^

0(k8) + 0( 8

2

)

or

sin 2 ^ <

'V n /ke\ . n /s 2 \

°\T7 j + « 1 . (A8)

Thus the condition q^ < 0(8) (together with tha assumDtion e

l » 1,

» 1 ) implies small scattering angles

S

2

( A9)

43
for all photon energies, k. (in the body of this report we derived this

result for k = 0(e); see eq (17).) We may thus use (A7) to obtain our

expression for in the region q^ < 0(6). Again referring to (3), we

have

a = (2\ + k

2

) h

- k m

'

from which

or

+ 2kcl m

2 \

qm

X

k (A10)

Thus if we assume that the photons are not too soft, specifically,

provided that

2k » qm

then

(All)

(A12)

Note that since q < 0(6), the condition 2k >> q^^ will m

7 1 / 3 1 provided 2k » 3 ~ C

- rr ~? < Thus to have 2k » q , 111 .8 ~ 25 m

to have k >> ~ me 2 35 10 keV.

be satisfied

it is sufficient

44
and from (A7)

" 5(1 + w

2

) . (A13)

It is worth noting the obvious similarity between (A13) and the high

energy, small angle approximation for the minimum momentum transfer when

one integrates over the final electron directions [6]. In that case

%,

* l£i - *1 * P 2

» 5(1 + u

2

) (A14)

where

u = p 1 Q

l (A15)

with

9

l = * (Pi.k) • ( A16)

When integrating over final electron directions [6] it was convenient to

define

l =

1

1 + u

2 (A17)

in terms of which

45
( A18

)

In the present situation, in which we integrate over photon directions, we

def i ne

Although in [6] the screening correction was expressed in terms of the

variables u and 5 , we note in the body of this report (see eq (61)1 that

the two screening corrections have the same form, identically, when

expressed in terms of q^, defined by q^

= |£: -

_k| - p 2 when the photon

is observed, and by q^ = |£ x - £2

| - k when the final electron is

observed.

1 X ( A19) =

1 + w

2

in terms of which

(A20)

46
REFERENCES

[1] L. C. Maximon and 0. B. Isabelle, Phys. Rev. 133 , B1344 (1964).

[2] H. Nguyen-Ngoc and J. P. Perez-y-Jorba, Phys. Rev. 136 , B1036 (1964).

[3] Y. S. Tsai, in Proceedings of the International Conference on Nuclear

Structure, 1963, edited by Hofstadter and Schiff (Stanford University

Press, Stanford, California, 1964), p. 221.

L. W. Mo and Y. S. Tsai, Rev. Mod. Phys. 41, 205 (1969).

Y. S. Tsai, SLAC-PUB 848, January 1971.

[4] L. I. Schiff, Phys. Rev. 83, 252 (1951).

[5] H. W. Koch and J. W. Motz, Rev. Mod. Phys. 3L, 920 (1959).

[6] Haakon Olsen and L. C. Maximon, Phys. Rev. 114 , 887 (1959).

[7] G. Molidre, Zeits, f. Naturforsch. 2a, 133 (1947),

47
H8S-I UA IRE v. 2-aCI U.S. OEPT. OF COMM.

BIBLIOGRAPHIC DATA

SHEET (See instruction s)

1. PUBLICATION OR

REPORT NO.

2. Performing Organ. Report No. 3. Publication Date

4. TITLE AND SUBTITLE

- Angular distribution of high energy ..electrons following radiation

5. AUTHOR(S) , , - Leonard C. Maximon and Alfred LePretre

i

1

6. PERFORMING ORGANIZATION (If joint or other thon N BS. see Jn struction s)

national bureau of standards

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

7. Contract^Grant No.

S. Type of Report 8. Period Covered

3. SPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS (Street. City. State. ZIP)

10. SUPPLEMENTARY NOTES ~

i

| 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 o si gnificant

bibliography or literature survey, mention it here)

We derive an expression for the angular distribution of high energy electrons which

have undergone scattering and radiated a photon, integrated over the directions of

the emitted photon, in the region of small scattering angles, for which the atomic

form factor must be taken into account, but the nuclear structure may be neglected.

This distribution is analogous to Schiff's high energy small angle expression for

the angular distribution of photons, integrated over final electron angles. We

show that the correction to the angular distribution of electrons due to atomic

screening is identical in form to the correction for the angular distribution of

photons. This correction involves an integral over the atomic form factor, which is

evaluated .in. closed form for the Thomas-Fermi -Moil ere model. A very simple

expression 4s obtained for the case of complete screening.

12. KEY WORDS (Six to twelve entries; alphabetical order; capitalize only proper names; and separate key words by semicolon s) Angular distribution of scattered electrons; atomic- screening effects in electron

scattering; bremsstrahlung; high energy electron scattering; small angle electron

scattering cross spr.f.inn: Schiff sppr.tnim

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