A11100 TTSSbM NBS

PUBLICATIONS

NBSIR 82-2454

Polarized Tagged Photons

An Analysis of the Differential

Cross Section for Polarized

Bremsstrahlung in the Range of

Interest for a Tagged Photon

System

January 1 982

DNAL 3UREAU OF STANDARDS

DEPARTMENT OF COMMERCE

RATIONAL BUREAU

OB BTAWDABDS

LXBRART

0<t •> NBSIR 82-2454

MAR 4 1982

flo£ (y. CC- - Cl

POLARIZED TAGGED PHOTONS

AN ANALYSIS OF THE DIFFERENTIAL

CROSS SECTION FOR POLARIZED

BREMSSTRAHLUNG IN THE RANGE OF

Leonard C. Maximon

U.S. DEPARTMENT OF COMMERCE

National Bureau of Standards

Center for Radiation Research

Nuclear Radiation Division

Washington, DC 20234

Eric Ganz

Physics Department

Stanford University

Stanford, California 94305

Thierry Aniel and Arlette de Miniac

Centre d Etudes Nucleaires de Saclay

Department de Physique Nucleaire

Service de la Metrologie et de la Physique Neutroniques Fondamentales

91191 Gif —sur —Yvette Cedex, France

January 1 982

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

NATIONAL BUREAU OF STANDARDS. Ernest Ambler, Director

INTEREST FOR A TAGGED PHOTON

SYSTEM
§881 -. $Af$
POLARIZED TAGGED PHOTONS

An analysis of the differential cross section for polarized

bremsstrahlung 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

Eric Ganz

Physics Department

Stanford University

Stanford, California 94305

Thierry Aniel and Arlette de Miniac

Centre d' Etudes Nucleaires de Sac! ay

Department de Physique Nucleaire

Service de la Metro! ogie et

de la Physique Neutroniques Fondamentales

91191 Gif-sur-Yvette Cedex, France

ABSTRACT

We consider in detail the differential cross section for polarized

bremsstrahlung for angles and energies in the range of interest for a tagging system and derive a high energy, small angle approximation for

this cross section. For photons polarized perpendicular and parallel

to the production plane these are given by eqs (1.2) and (1.3). We use these approximations to determine the maxima and minima of the cross sections for these two polarization states, da, and da,, and to

evaluate these cross sections at the extrema. It is shown that both

da

x and da, have a very sharp dip in the region of small momentum

transfers. However, their behavior in the region of the dip, as a function of the azimuthal angle <f>, is quite different over most of

the photon spectrum (condition (VI. 33)). The cross section da

x behaves similarly to the cross section for unpolarized photons in that

as 0 increases, the sharp dip vanishes, the minimum fuses with the

second maximum, and the cross section then has only a single maximum.

In contrast, the sharp dip in the cross section da, remains as <J> increases, provided condition (VI. 33) is satisfied. This results in

rather large polarizations in the region of the dip as shown in

figs. 3 (a ) -3 (h ) . Coulomb corrections to the Born approximation are considered, and do not fill in these dips.

Key Words: Bethe-Heitler cross section; bremsstrahlung monochromator

;

photonuclear research; polarized bremsstrahl ung differential

cross section; polarized photon beams; 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 Bergere and Larry Cardman for their helpful discussions,

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

Roland Bergere for the gracious hospitality accorded him during his

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

done.
TABLE OF CONTENTS

ABSTRACT i

ACKNOWLEDGEMENTS i i

NOTATION iv

I. INTRODUCTION 1

II. POLARIZATION 6

III. KINEMATICS 9

IV. BORN APPROXIMATION DIFFERENTIAL CROSS SECTION FOR

POLARIZED PHOTONS EMITTED BY UNPOLARIZED ELECTRONS .... 12

V. HIGH ENERGY SMALL ANGLE APPROXIMATION TO THE

DIFFERENTIAL CROSS SECTION FOR POLARIZED PHOTONS ..... 19

VI. DETERMINATION OF THE MAXIMA AND MINIMUM OF THE

POLARIZED BREMSSTRAHLUNG CROSS SECTIONS dc

L AND da

y

. . 47

VII. COULOMB CORRECTIONS TO THE DIFFERENTIAL CROSS SECTION

FOR POLARIZED BREMSSTRAHLUNG 66

REFERENCES 68

FIGURE CAPTIONS 70

FIGURES 75
NOTATION

Physical Constants

e charge of the electron

m mass of the electron

c velocity of light

2 me rest energy of the electron

ft Planck's constant

Z atomic number of the nucleus

Kinematics and Cross Section

2 Note : Unless otherwise specified, all energies are in units of me and all momenta are in units of me.

k_ momentum of the photon

k photon energy

£ momentum of the electron

p-j for the initial electron

P

2 for the final electron

pL component of £ perpendicul ar to k_ Pn for the initial electron

P 2i

for the final electron

p-j = | p | magnitude of £

p-j for the initial electron

P

2 for the final electron

iv
e electron energy

e-j for the initial electron (E^, in MeV, in the figures)

e

2

for the final electron

0 polar angle in system with z-axis in the direction of k

0-| between the photon and the initial electron direction

0

2

between the photon and the final electron direction

<p azimuthal angle between the plane defined by the directions

of the photon and the initial electron and the plane defined

by the directions of the photon and the final electron

£

q

F(q

)

e_

Q

dft ?2

dk

dtt. dS2 dk

k p

2

momentum transferred to the nucleus

^ component perpendicular to the photon direction

q

z component in the photon direction

magnitude of £

atomic form factor for screening ( = 0 in this report)

polarization vector of the photon

angle between e and the production plane defined by the

directions k_ and £,

solid angle in which the photon is emitted

solid angle in which the final electron is emitted

2 energy bin of the photon in me

differential cross section for bremsstrahl ung in units of

b/(_sr)

2 /(mc

2

)

v
Special Notations of this Report

da, = d

3a =

d

3a

dft dft.dk

p

2

k

differential cross section for bremsstrahlung

for an angle of polarization ip

da

x = d

3a

x = A(* = !)

dft dft, dk

value of da for ip = J

(perpendicular

p 2

k polarization)

2 d (j[ip = 0)

da.. = d a,, =

11 11 dtt dft. dk

p 2

k

value of da^ for ip = 0 (parallel polarization)

ft 32 e

2 / e2 \ [l-F(q)] 2

1 p

2

A \

2

. , < \ quantity defined by da = r — I —~ J 2 — < >

* ' he \mc / q k p

1 \2tt /

1 ’

A, B, C

{

L - c

{ III A

quantities defined by

| = Acostp + 2Bcosx/;s i nip + Csin‘

(exact expressions are given in eq (IV. 11) page 17).

yalue of

^ ^ for ip = j

value of

<j J> for ip = 0

degree of polarization, P^ = da , - da ,

,

,

Ip Ip+^TT

da , + da . ,

,

Ip Ip+%1T

value of P for ^ J

, P

=

daj^ - dO||

da

x + da,. [Mi

(

), *

!

).

C-A

C+A

angle ip for which P^ reaches its maximum for fixed values

of £-j , k 5 6

-j j 0

2

, 4>

vi
^- =

fi.u

v =

u = | u_| = p-|Stn0i

v = |v| « p 2

sin©

2

magnitude of u

magnitude of v

z = v-u

u

d^ - e-j - p-|COS0^

d

2

— £" 2—

p

2

cos0

2

K * 2e^d^

n

= 2e

2 d2

=

a

1 0 + in

1 0 + v

2

)

Hi = n. n

6 =

k 2£-j£ 2
f tv)

g(v)

G

P

quantities: defined by

{ } = fCv] (y - u + <r) 2 + g(v)sin

2

|

+ p

f

i (vl value of f(v) for ip

TT "

2

, given in eq (V. 38} p. 45

f

ll (vl yalue of f(y) for ip = o, given in eq (V.41) p. 45

9i Cvl yalue of g(y) for ip

TT ~

2

, given in eq (V.39) p. 45;; (approxi

mat ion for' q < O(u0)„ and <f> « 1 given in eq (VI .4) p. 48)

9

|

(vl yalue of g ( v ). for ip = o. given in eq (V.42) p. 45; (approxi

mat ion for• q

< Q(u6) and 4 « 1 given in eq (VI .5) p. 48)

g

l is zero

CTjj value of a for \p = Q, given in eq (V.43) p. 46; (approximation

for q < 0(u9l and <p « 1 given in eq (VI. 6) p. 49)

Pj_ is zero

Pjl value of p for ip = 0, given in eq (V.44) p. 46

f Oil = (v)

dv v-u value of the derivative of f(v) with respect to

the variable v at the point v = u f[(ul

fl Cul

value of

value of

f ' (u)

f(u)

for

for

, TT

Ip = J

ip = 0

yalue of the derivative function of g(v) with

respect to the variable v at the point v = u g|(u] yalue of g'(u) for ip =

IT

g'tul-2|M

V=U
2

critical yalue of k. k 51 4£i£q1 r

j Q 1 A. .4 u +1

d> critical value of 4 for which the minimum and the second

c maximum of the unpolarized cross section fuse in an inflection

point

<

p

Q

when <p approaches the value <f> Q

, the approximation for the

unpolarized cross section becomes invalid

<j>^ same as <t> but for perpendicularly polarized cross section

same as <|> but for perpendicularly polarized cross section,

expression given in eq (VI. 16] p. 53

*cll same as <f> y

c but for parallel polarized cross section

*01! same as <P 0

but for parallel polarized cross section, used

for k > ko

; expression is given in eq (VI. 26) p. 58

*01, same as *0 but for parallel polarized cross section, used

for k <

z . approximation for the position of the minimum of the unpolarized

jm 1 1

1 2 cross section in the variable z in the region z = 0(6^)

z mini same as z . but for perpendicularly polarized cross section,

expression given in eq (VI. 18) p. 53

z mi n|| same as z

nl i n but f° r P^ral lei polarized cross section;

expression available for k> k , given in eq (VI. 28) p. 59.

For k < k replace <f> 2

by - ¥

2 o

P

^o||

J v

o||

ix
max approximation for the positions of the maxima of the

2 unpolarized cross section in the region z = 0(0 )

'maxi same as zmgx

but for perpendicularly polarized cross

section, expression given in eq (VI. 20) p. 55

'maxll same as zmax

but for parallel polarized cross section,

expression given in eq (VI. 30) p. 61

( 2 ) quantity appearing in the expression of » expression

given in eq (V 1.22) p. 56

( 2 ) quantity appearing in the expression of zmax( j , expression

given in eq (VI. 32) p. 61

z maxi zeroth order approximation for zmaxi , expression given

in eq (VI. 19) p. 55

‘maxll zeroth order approximation for zmax(| , expression given

in eq (VI. 29) p. 60

x
I. INTRODUCTION

In a recent report [1]^ we presented a detailed analysis of the

differential cross section for tagged photons in the range of energies

and angles of interest for experiments currently being undertaken at the

ALS (Accelerateur Lineaire de Saclay) and other linear electron accelerator

laboratories [2-4]. In that report we derived, starting from the BetheHeitler differential cross section for bremsstrahlung , an approximate

expression for the cross section, valid for high energies, c, of the

initial and final electron, and small angles, 0, between intial electron,

final electron, and photon. (See fig. 1 and section III for kinematics.)

2

In that approximate expression we neglected terms of relative order 1/e

2

and 0 . Terms of relative order 1/e and 0 were retained throughout.

We then used this approximate expression to examine in detail the cross

section in the region of particular interest for experiments utilizing

tagged photons, viz., the region of very small momentum transfers, of the

order of the minimum allowed momentum transfer. We saw that although in

general the differential cross section was very large in this region,

there was a sharp dip in the cross section corresponding to momentum

transfers very nearly in the direction of the emitted photon (qx « q ).

(See fig. 2(a)).

In the present report we extend our investigation of the

differential cross section by examining the polarization of tagged

photons emitted by a beam of unpolarized electrons. Paralleling our

recent report, we now start from the differential Born approximation

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

paper.
cross section for polarized bremsstrahl ung. We write this cross

section in section IV, eqs (IV. la) and (IV. lb), in terms of the polarization vector, e_, of the photon, in a form that is quite similar to

that of the Bethe-Heitler cross section. Again paralleling [1], we

next write this cross section in a form which explicitely takes account

of the large cancellations which occur for small momentum transfers in

the expression for the cross section as it is generally written. This

is done without making any high energy or small angle approximations,

and is presented in eq (IV. 5). For the consideration of the polarization we then write the cross section in the form

A

dfi„ dfi.dk

p

2

k

k p

1 (2tt )'

x |acos 2

iJ> + 2Bcos^sim|j + Csin^| . ( 1 . 1 )

(See eqs (IV. la), (IV. 5), and (IV. 6).) Here ip is the angle between

the photon polarization vector e and the production plane (defined

by the momentum of the initial electron, £-| , and the photon momentum,

k_). The quantities A, B, and C, which depend on the energies and

angles of the electrons and photon, are given by eq (IV. 11). The cross

sections dc^ and da^ , appearing in the conventional definition of

polarization, follow directly from eq (1.1): The cross section do^

,

for photons whose electric vector is perpendicular to the production

plane, is given by eq (1.1) with ip = J

, ({ } = { }± = C). The cross

2
section dc^ , for photons whose electric vector is in the production

plane, is given by eq (1.1) with ip - it (or 0), ({ }=={ }^j = A). The

polarization is discussed in section II, and in figs. 3(a)-3(h) we plot

the polarization as a function of the angle 0

2

between the final

electron and photon momenta for various values of the azimuthal angle <|>,

defined in fig. 1, and for two different values of the photon energy, in

the region of small momentum transfers.

In section V we derive an approximate expression for the cross

section given in eq (1.1), i.e., for the quantities A, B, and C, valid

for high energies and small angles, the neglected terms being of 0(l/e )

and 0(0 ). Once again, terms of 0(l/c) and 0(0) are retained

throughout. These are given in eqs (V.21), (V.22), and (V.25). We show

explicitely for the high energy small angle expressions, that A > 0,

C > 0, and AC - B >0, which insures that eq (1.1) is positive. From

eqs (V.22) and (V.25) we then write the numerators appearing in the cross

sections do^ and dCy in a form similar to that given for the numerator

of the cross section for unpolarized photons in eq (VI. 49) on p. 66 of

[1]. These are given in eqs (V. 37) and (V.40). We then have our high

energy small angle approximations for the cross sections dc^ and dc^ ,

which have a form quite similar to that of the cross section for

unpolarized photons, given by eq (1.1) on p. 4 of [1]:

2

[1 - F(q)]

2

1 p

2

1

q 4

k p, (2tt) 2

3
and

2

[1 - F(q)]

2

1 p 2

da,

k p

1 (2tt) 2

|f„ Cy)(v-u+a

(| ) 2 + g1( (v )si

n

2

^d> + P||

|

. (1.3)

The functions f

L

, gx

, fj| , , and are defined in eqs (V .39)-

(V.44). All other variables are defined in section III. In section VI

we use eqs (1.2) and (1.3) to obtain analytic expressions for the maxima

and minimum of each of these cross sections in the region of small momentum

transfers, following closely the detailed analysis given in section VII of

[1]. We find that although the cross sections da

x and da^ each exhibit

a sharp dip in the same region as that found for the cross section for

unpolarized photons in [1] Cq± « q ), the minima occur for slightly

different values of the angle Og. [See figs. 2(b) and 2(c).) The polarization therefore varies rapidly in the region of the dip, as shown in

figs. 3(a)-3(h). Furthermore, the behavior of do^ and dc^ in the

region of the dip, as a function of the azimuthal angle 4, is quite

different over most of the photon spectrum (that region given by condition

(VI. 33)). The cross section do

x behaves similarly to the cross section

da for unpolarized photons (discussed in [1]), in that as <j> increases,

the sharp dip vanishes, the minimum fuses with the second maximum, and

the cross section then has only a single maximum. In contrast, the

sharp dip in the cross section do^ remains as <p increases, provided

condition (VI. 33) is satisfied. This results in rather large polarization

in the region of the dip, as shown in figs. 3(a)-3(h).

4
Although the point was discussed in detail in [1], we stress again

that we are dealing here with the completely differential cross section;

once one integrates over the angles of the final electron, as required

for the analysis of earlier experiments [5-9] as well as several current

experiments using polarized bremsstrahlung [10,11], the sharp dip

considered here and in [1] no longer appears.

5
II. POLARIZATION

The required expressions for the polarization may all be obtained directly

from eq (1.1). Relating this to the conventional definition of polarization,

the cross section for photons whose electric vector is perpendicular to the

production plane, dc^ , is given by eq (1.1) with ip = j

. The cross section

for photons whose electric vector is in the production plane, da^ , is given

by eq (1 . 1 ) with ip = tt. The polarization, P, is then defined by

P = do^ - dC||

do

x + da,. (II. 1)

C - A

C + A

(II. 2)

In the expression (II. 1) for the polarization, and in figs. 3(a)-3(h) giving

the polarization as a function of 6

2

, it is assumed that the directions of

the incident beam and the emitted photon are defined with infinite precision,

and thus that the production plane is perfectly defined. In an actual

experiment, however, the incident beam has a finite angular spread and the

photon detector has a finite size. The observed polarization will then

depend on the details of the experimental set-up. Although we will not

enter into these considerations in this report, it should be noted that they

are undoubtedly of importance for those angles for which the polarization

varies very rapidly (see figs. 3(a)-3(h)). If, however, it were possible to

perform a coincidence measurement which was essentially differential in

the angles of the final electron as well as the emitted photon, then we

6
could define a polarization referred to two axes which are orthogonal,

but arbitrarily oriented with regard to the plane of scattering:

da , - da , ,

,

p = ^ ijj+SglT ^ da. + da. t Ip lp+%IT

where da^ is the cross section given by eq (I.

this same cross section with ip replaced by ip

P, defined in eq ( 1 1 . 1 ) is then

Then from eq ( 1 . 1

)

[Acos^ip + 2Bcos^simp + Csin^] - [Asin^if>

p = ^ [Acos^ij; + 2Bcosipsim|i + Csin^] + [Asin^

(A-C)cos2i|i + 2Bsin2i|i

A + C

• /(A- C)Z

a : f C0S2(^0

) .

where

A-C = /(A^C)^+4B^ cos 2^

(11. 3)

)» and dV^ is

%ir. The polarization,

(11. 4)

2

2Bsinipcosip + Ceos 41 ]

2 2Bsini(icosi(j + Ceos i|i]

(II. 5)

(II. 6 )

7
2B (II. 7) = j/(A- C)

2 + 4B2 sin2^

tan2*

0

- ^ (II. 8)

Thus if we choose ip = ipQ or ip = \pQ + %rr then we have

= /(A-C)

2 + 4B2

A + C

/(A+C)

2

- 4(AC-B

2

)

(A+C)

4(AC-B^)

(A+C)‘

(II. 9)

This is the maximum polarization that can be obtained for specified

energies and momenta of the final electron and photon. We note from

eqs (II. 2) and (II. 9) that

p i = i =

1 y

o

(II. 10)

In figs. 3(a)-3(h) we plot P as a function of the angle between

the final electron and photon momenta for various values of the

azimuthal angle <|>, defined in fig. 1, in the region of small momentum

transfers.

8
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

£p£-j : Energy and momentum of the incident electron

2 2 (or positron) (e^ -

£-| =1).

, 2_2 : Energy and momentum of the final electron

2 2 (or positron) (eg - £g

= !)•

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

£ = £-| " £2 " —: Momentum transferred to the target nucleus.

The energy transferred to the nucleus, which is in general

qg

= e-j - £g

“

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^,cj)^), the angles of £^ are (e^,^). These vectors are

9
shown in fig. 1. The components of £| and ^

are then

“

=

£-11

1

=

£-21

The angle between the vectors £ and £, which

perpendicular to k_, is then

(P $2

~ ^i

The component of £ perpendicular to k_ is

^

= u - v .

We have then

2 2 2 qi

= u - 2uv cost}) + v

The component of £ in the direction of k_ is

q

z = Pi cos 6

-j - P 2

<:os 02 - k .

perpendicular to k

(ni.i)

lie in the plane

(III. 2)

(III. 3)

(HI. 4)

(HI. 5)

The magnitude of the vectors and v_ is given by
u = p-j si n 0 -j

(III. 6 )

u

v = lyj = P2Sin02

Throughout the analysis given in this report we use the coordinate

system with z-axis in the direction of k.

11
IV. BORN APPROXIMATION DIFFERENTIAL CROSS SECTION

FOR POLARIZED PHOTONS EMITTED BY UNPOLARIZED ELECTRONS

The differential Born approximation cross section for polarized

photons emitted by unpolarized electrons was first given by May [12]

and by Gluckstern, Hull, and Breit [13]. However, we choose to write

this cross section in terms of the polarization vector e of the

photon. Its form is then similar to that of the Bethe-Heitler cross

section, and the fact that the sum over polarization states gives the

Bethe-Heitler cross section may then be seen by inspection (compare

eq (IV. 1) on p. 22 of [1]):

(IV. la)

where

(£.,. e)

2 (4e

2

2

- q

2

) + (£2

* e)

2

(4e., 2

2

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

2

- p

2 cos0

2

)

(e-j - p-|COS0^)(e

2

- p

2 cos©2

)

+ (IV. lb)
We note here that the first three terms depend on the photon polarization

but that the last term does not. This last term therefore differs from

the corresponding term in the Bethe-Heitler cross section (eq (IV. 1) on

p. 22 of [1]) by a factor 2, which appears only after summing over

polarization states. Further, and more significantly, we note that the

first Born approximation cross section for photons emitted by unpolarized

electrons does not have any terms correlating the momenta of the

2 particles with the circular polarization of the photon (of the form

(ie_ x e*) • (u x v), where £ and v_ are the components of £j and

£2 perpendicular to the photon momentum, JO . Thus in first Born

approximation we have only linearly polarized photons. This result

follows from the invariance of the cross section for electromagnetic

interactions under both space and time reversal, as has been discussed

in detail by Kolbenstvedt and Olsen [14] and Olsen and Maximon [15].

Although in principal one can have a correlation between the circular

polarization of the photon and the momenta of the particles in higher

order Born approximation, this effect vanishes for bremsstrahl ung at

high energies and small angles, [16]; i.e., if we neglect terms of

0(l/e) and 0(0). The circular polarization of photons emitted by

unpolarized high energy electrons at small angles is thus negligible.

Only in the case of pair production can one produce strongly polarized

particles from an initially unpolarized particle at high energies [17].

We return now to the expression appearing in the numerator of the

cross section,

| as given in eq (IV. lb), and, in analogy with

section VI of [1], write it in the form

13
where

2e 0 u*e 2e^ _v*.e

- q

2

/iL*ji v.*6n

d

l d

2

+ k

(u-v)'

2

^

d

l d2

(IV. 2)

d

l = £

1 ” P-|COS0^

d

^ ^2 "* P2 C0S ®

2

(IV. 3)

Again we define

€

= 2e

i di

n

= 202^2

(IV. 4)

The expression for

j

may now be written very simply in terms of

£ and n

:

| |

= 4k^e-j£2(M.“V.)^?n + 16 e-| ^£

2

^(u.*e£ - y_*eji)^

2 2

- 4q (e-| u_-e£ - £2 v_-en) (IV. 5)

14
It is worth noting from this expression that for u_ = v_ (u = v and

<t> = 0), i.e., for qx = 0, we can write

| | = |l6ej

2

e

2

2

U-n)

2

- 4q

2 (e^ - e

2 n)

2J

(u-e)

2

.

The linear polarization, P, defined by eq ( 1 1 . 1 ) is therefore -1 for

qi = 0, since then do, = 0 (e l u). Although the point at which

qx = 0 does not correspond exactly to the minimum of the cross section,

the dip discussed in [1], it occurs in the region of the dip. We may

thus expect strong linear polarization of photons emitted in the

region of the dip where q

x « q . (See figs. 3 (a) -3(h).)

Finally, we note that the expression for

j

j

, eq (IV. 5), can be

written in terms of the polarization angle ip in the form

| |

= Acos

2^

+ 2Bcostpsin^ + Csin 2^ . (IV. 6)

Here we have defined the polarization angle ip by

u/£ = u cosi|> (|ej = 1) , (IV. 7)

i.e., we measure the angle of the polarization vector e, which lies

in the plane perpendicular to Jc, from the plane defined by £j and

k (see fig . 1 )

.

Then

v*e = v cos(ip - $) . (IV. 8)

15
The expressions for

| j

appearing in eqs (1.2) and (1.3) in the

numerator of the cross sections dc^ (for which ip = j ) and dO|

( for which ip = tt) follow at once from eq (IV. 6):

For da

x

• = C . (IV. 9)

I

= A . (IV. 10)

II

Proceeding now to the explicit evaluation of the coefficients A,

B, and C in eq (IV. 6), we substitute eqs (IV. 7) and (IV. 8) in

eq (IV. 5), which gives

u/e£ - v/eri = u^cos^ - vncos(i|; -

(J>)

= (u^ - vncos(j))cosip - vnsin^simj;

and

e-j u_*e_£; - =

( £ -] u 5

" £2 vriC0S ^^ C0S^“ £; 2

vr|S1

' n(®)S1

' n^ *

2 2 Thus, multiplying the first term in eq (IV. 5) by (cos ip + sin ip)

,

{ H

For day

< H

16
4k^£20i " v.)^n(cos

2

ip + sin^)

2 2 2 + 16e-j £2 I(u£-vncos4>)cos^ - vrisin^sin^]

2 2

-4q [(e^u^-^uncos^cosij; - £2Vqsin<J)Sini|/3

2 2 = Acos ip + 2Bcos^sim|; + Csin ip

where

2 2 2 2 2 2 2

A = 4k ^ ^ + ^e

i e

2 ( u ?_Vfl COS(l

) ) " 4q (£-|U£-£2VncoS(|))

2 2 2

B = [- 1 6e-j £2 (u^-vncos<J))vri + 4q (e-j uC-egVncos^EgVTiJsi

C = 4k^£i£2Cu-v.)^?n + 1 6 e

1 ^£2^v^n^sin% - 4q

2 £22 v

2n

2 sin

2 $ . (IV. 11)

We note that if one integrates the cross section over the angle <j>

of the final electron, then the integral of the term with the factor B

is zero. This follows from the fact that this term has the factor sin<|>

and that its only other dependence on <p (e.g., through q ) is a

function of cos4>. Thus on integrating over the angles of the final

electron we have an expression of the form

Acos

2^ + Csin 2 ^

= A + (C - A)sin 2

4» , (IV. 12)

as should be expected [5],

17
In the following section we derive the high energy small angle

approximation to

j j

, i.e., we obtain approximations for A, B,

and C valid for high energies and small angles.

18
V. HIGH ENERGY SMALL ANGLE APPROXIMATION

TO THE DIFFERENTIAL CROSS SECTION FOR POLARIZED PHOTONS

We now consider the high energy small angle approximation to

| as given by eq (IV. 5). Noting the similarity of eq (IV. 5) to the

expression (IV. 4) in [1], we may follow step by step the analysis given

there with very little modification. For the last term in eq (IV. 5) we

then find, from pp. 50-54 of [1], terminating in expression (VI. 28) of

[1], the high energy small angle approximation

Again following [1], we have, from eq (VI. 29) on p. 54 there, the high

energy small angle approximation to the first term in eq (IV. 5), viz.,

(V.l)

(V. 2)

where

5 «0 5

n = n

0

- ni (V.3)

and

19
1 1 1

n

0

%

9

2e-j (e-j+p-jCOsS-i ) 4e-|‘‘

1 1 n

= as —

2£2(£2+P2 cos0 2^ ^e

2

(V.4)

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

second term in eq (IV. 5). Substituting eq (V.3) in (IV. 5) we have

2 2 2 2 2 2

1 6e-| (£•.££ * v-en) = 16c, e 0 (u*e£ n •1 c - v*en n ) 2

v - -s 0

--'

0

'

2 2

- 32e-| e

2 (_u-eC

Q

- u*en

0

) (.u^ - v/en-,

)

2 2 2 + 16e-j £2 )

2 2 2

16e-j £

2 (u/eCg “

+ 8k(£-

1 +£2 )(£-eC

0

- v.-en 0

)(_u*e)

k(£

] +£2

)

£

1 £2

(u-e) . (V.5)

2 2

Here we have neglected terms of order 1/e and 0 relative to those

retained, as on pp. 54-59 of [1]. The last three lines are,

respectively , of 0(e

4 5^q

1

2

), OCA^), and 0(u

2 ), as in eq (VI. 20)

on p. 48 of [1].

20
We now have, in eqs (V.2), (V.5), and (V.l), the high energy small

angle approximations to the first, second, and third terms in eq (IV. 5),

respectively. We thus have

positive. This aids later in the analytical determination of the maxima

and minima of the polarization as a function of the angles of the final

electron and photon. It also eliminates the large cancellations which

otherwise arise in the numerical evaluation of the cross section. To

this end we write

(V.6)

We next write the expression for < > in a form that is manifestly

(V . 7)

Substituting eqs (IV. 7) and (IV. 8) in (V.6) we have

21
U V

u/e^Q - v^-erin = j

C0Sl^ 2 cos ^ ” 1+u 1+v

and thus

j

costy j

cosi|;cos<j> o sim|>sin<f>

Itu 1+v 1+v

U V V V

s- costp j cosiJj + j

(l-cos<|))cos^ s i mps i n<$> 1+u 1+v 1+v 1+v

u-e?

0

- v-en

0 .

u v \ 2vcosi\> 2 2v

ojcosip +

p— s

i

-k sin4isin^<j)COS%4>

1+v / 1 +v

<” 1+v"1

= ((u-v)(l-uv)^

0n0 + 2vn0

sin

2^ cos ip

- vrigSin^simj; (V.8)

Substituting eqs (IV. 7), (V.7), and (V.8) in (V.6) we obtain

| | * 4k

2

£-j£ 2 ^ori 0^ u " v ^ 2 + 4uvs1n2^40

,, 2 2 + loe-j £2

p

^ ( °i

"*"^2 ^

(u-v)(l-uv)£nn n + 2vn n sin h<\> +

2—2“ I 00 "'"0 cos ^

- vrigSin^sin^j

4£-j £2

2 2 , U COS if) (V.9)

22
At this point we note that if this high energy small angle expression

for

| |

is summed over the polarization states of the emitted photon

(adding its value for a given ip to its value with \p replaced by

ip + Jgir) then the result is independent of ip and is equal to the

expression obtained earlier, in II]. For the first term in eq (V.9) we

then obtain

8k

2

£-j£ 2^QTiQ[(u-v)

2 + 4uvsin 2^] , (V.10)

which is the term given in [1], p. 54, eq (VI. 29). For the last term

in eq (V.9) we have

2 2

u cos (ip + %rr) = (V.ll)

which is the term given in [1], p. 54 eq (VI. 28). Finally, we note

that the second term in eq (V.9) is of the form

(acosij; + bsin^)'

Summing over polarization states then gives

2 2

(acosij; + bsimjj) + (acos(4i + ) + bsin(4> + %tt))

2 , 2 = (acosV; + bsimp) + (- asimj; + bcos^)

2 2

= a + b

23
Therefore from the second term in eq (V.9) we have

,, 2 2

loci

, 2 2

I o£-j

2 k(ei+C2)u \ p 2 2 (u-v)(l-uv)£

0

n

0 + 2vn0

sin ^ ^—f

+ v n

Q

sin <j> £2

,2. 2 2 2

(u-v) (1-uv) C 0 n

0 + 4(u-v)(l-uv)c

0

n

0 vn0

sin ^

2 2 4 2 2 2 + 4v n

0

si n %4> + v n

0

sin <t>

/ 2 \

2I<( £

i +£2

^ u + ((u-v)(1-uv)€

0

n

0 + 2vn0

sin %4>)

' ' ^

4e-j £2

, 2, A s2 2' k ( £ -j +£ 2

) U ,,4 4

I 0 £ 1 £2

( v . 1 2

)

In the first term in the square bracket above we may write

( 1 - uv

)

2

= 1 - 2uv + u

2 v2

= (l+u

2 )(l+v

2

) - (u+v)

2

and hence

(u-v)

2

(l-uv)

2 5 Q

2 n

0

2

= (u-v)

2

?

0n

0

- (u

2 -v2

) C 0

2n

Q

2

. ( V . 1 3

)

The next three terms in the above square bracket are

24
2 2 2 4 2 2 2 4(u-v)(l-uv)^

0

n

Q vn

0

sin h<$> + 4v n Q

sin h<$> + v n

Q

sin <p

The last two terms here combine to give

4v

2n

0

2 sin\(j) + v

2n

0

2 sin 2

(}> = 4v

2n

0

2

si n

2 %4>(si n2

^4> + cos\cp )

2 2 2

= 4v rig sin ^4

Adding this to the first term we have

2 2 2 2

4(u-v)(l-uv)£

0n0 vn0

sin h<\> + 4v n

Q

sin

= 4vn

0

2 sin 2%4>[(u-v)(l-uv)?

0 + v]

4v£

0n0

2 sin2%(j>[(u-v)(l-uv) + v(l+u

2 )]

* 4vS

o

n

0

Z sin2!^[u(l+v 2 )]

2 4uvC

0

n 0

sin %j>

Adding this to the expression (V.13) we have, for the first four terms

in eq (V.12)

£

5 0

n 0

[('J - v ) 2 + 4uvsin 2 ^] - (u

2 -v

2

) C

0

2 n

Q

2

= (u-v)

2 ? 0

n

0

- (u

2 -v2

) ?

0

2n

0

2

, (V.14)

25
which is the expression in square brackets in £1], p. 59, eq (VI. 38).

Next we observe that the last term in eq (V.12) is, together with

2 2 the factor 16e-| z^ outside the square bracket.

k(c, + c

2 )\

2

2 (V.l 5)

which is the last term in [1], p. 59, eq (VI. 38).

Finally we have the remaining term in eq (V.12), viz., (including

2 2 the factor 16e^ z^ ),

8k(£-|+£

2

) u ((u-v)(l-uv)£

0n0 + 2vn

0

sin

2^ . (V.16)

2 2 We note that the term here with factor sin is of 0(1 /z ) relative

to the term with this same factor in eq (V.14) (when we include the

overall factor 16e^ z^ in this latter term). It may therefore be

neglected. We are then left with

8k(c1 +£

2

) u(1-uv)CqT1q(u-v) . (V.17)

Here we note that we may, in the factors multiplying (u-v) in eq (V.17),

2

set v = u, again neglecting terms of 0(l/e ) relative to terms

retained in eq (V.14). Thus we may write

1 - uv » — (

u

2

— 1 )

26
and

(u+v)

giving, for eq (V. 17)

,

- 4k(e

1 +£

2 )(u

2

-l )£ 0n

0

(u

2 -v2

) (V. 18)

which is the second term in [1], p. 59, eq (VI. 38).

Our expression (V.9), when summed over any two orthogonal polarization states, thus indeed gives the expression for

|

j* derived in

2 2

[1], with neglect of terms of relative orders 1/e and 0 .

We now return to the task of writing the expression for

| |

in

a form that is manifestly positive. It is clear that the original

expression for

|

eqs (IV. la) and (IV. lb), before making the high

energy small angle approximations, can not be negative since the cross

section can not be negative. We wish to show, however, that our high

energy small angle approximation does not introduce errors such that

the expression for -j |

then becomes negative for certain values of

the variables. Since this involves rather tedious algebraic manipulation,

the reader may wish to skip to p. 44. To this end we note that eq (V.9)

may (after multiplying the first term there by (cos ip + sin i>)) be

written in the form

Acos

2

ip + 2Bcosips i nip + Csin 2ip (V. 1 9)

where

27
A = 4e

1 £

2 ^

0

n

0

Ik

2 + 4£-|£

2 Cl-uv)

2

^ 0

n

0J(u-v)

2

+ 32£

1

2

£

2

2

(l-uv)C

0

riQ

« k(£-|+£p)u

2vn

Qsin%j> +2 "2

4£

i ^£

2

(u-v)

, le; 2 2 + I D£

i £

2

2 k(£,+£~)u

2vr)gSin +

12

A 2 2

4£.j £

2

2 2 + 4k £-|£ 2

^qT1q • 4uvsin

2 2 B

= - 16e

1 £ 2

vrigSinc}) 2

k ( £ -I +£ 2

) u (u-v)(l-uv)C

0

n 0 + 2vn

Q

sin k<P +

2—

A 2 T2

4£-j £

2

C = 16£-|

2 £

2

2 v2

riQ 2 sin

2

(t) + 4k

2

£-| £

2

E>q1q[ ( u-v)

2 + 4uvsin 2

^cj)]

Clearly C > 0. Let us next show that A > 0. Completing the

in eq (V.20) we have

(V.20)

, (V.21)

(V.22)

square

28
A = 4e

1 e2S0

n

0

[k

2 + Ae^O-uv)] 2?^]

4e-|£

2

( 1 -uv

) |2vriQSin +

k±C

€

u-v +

/, 2 2

4e

^ £

2

k

2 + 4£

1 £

2

(l-uv)

2 ^

0

n

0

4£

1 £2 ^

Q

n

0 [4£

1 £

2

(l-uv)]'

2

k ( £ -j +£ o ) U 2vn

Q

sin^(j) +

2

- 2

12

4 e

^ £

2

k

2 + 4£

1 £2 (l-uv)

2 5 0

n

0

-,r 2 2 + ID£-| £

2

2 k(£,+£2

)ll 2vn

0 sin%j> +

2 2

4e ^£2

12

2 2 + 16k ^E^QfiQUVSin %j> -

; l e

2 )

Adding the second and third terms in eq (V.23) we have

(V.23)

29
,c 2 2

loe

;^ £2

2

k (e-. "*"£o )u

2vrigSin^ +

2—2“

4 e

-| £2

k 2 + 4£

1 £2

(l-uv)

2 5 0

n

0

k

2 + 4£

1 £

2

(l-uv)

2 ? 0

n

-4£

1 £

2 (1-UVJ^qTIq

,,,2 2 2 16k £-| £2

2

k ( £ -I +£ 2 ) U 2vn

0

sinS<f> + — 2 4e, £2

k

2 + 4£

1 £2(l-uv) 2

C 0

n

0

We now have

A = 4£

1 £

2 ^ 0

n

0

[k

2 + 4£

1 £

2

(1 -uv)] 2

C 0

tiq]

u-v +

4e^£

2 (1-uv)

2 k(£,+£2)u

^vrigS 1 n 'S’r ^ 2 2 4e^ £2

k

2 + 4£-j £ 2

(l-uv)

2

^onQ

,,.2 2 2

16k £-j £2

2

k ( £ -i +£ 2

) u 2vn

Q

sin h<p + ^—2 ~i 2

4e-j £2

k

2 + 4£-|£

2 (l-uv)

2

5

0n

0

2 . 2 + 16k £

-j £2^on O uvs i n ^ "

30
Here we add the second and fourth terms and obtain

2 2 2

16k' e

2

o k ( £ -I o)u

2vn

0

sin ^ +

2—2“

4ei £2

k

2 + 4£

1 e2

(l-uv)

2

C 0

rio

1CI 2 2 2

16k e-| £2

22 n k(£i+£2 )u2 4v rig sin $ + ^

2— vrigS i

n

£

1 £

2

4 / x v2 2

k ( £ 1 "^£ 2

) u

2 2 £

1 £2

k 2 + 4£

1 £

2

(l-uv)

2

5 0n

0

1

[k

2 + 4£

1 £2(l-uv)

2

C 0n

0

]

2 2 2 16k £-j £2

224 k(£,+£2

) 2

4v rig sin ^

y~ uvn

Q

sin h<$>

- 4^1

2

^ ^ ~uv ) £ 0n

0

I u

e

l £1

,.4 2 4k e^£2 u

2 2 £

1 £1

31
The last two terms in the last square bracket in the last expression are

4kV

£

1 £2

1 - (l-uv)‘

0+uZ )(l+v

2

)

-,4 2

4k u

£

1 £

2

? 0

n

0 [(l+u

2

)(l+v

2

) - (1-uv)

2

]

4k

4

u

2

C

Q

n

0 (u+v)

2

£

1 £

2

Thus we have

A = 4£

1 e2^0

n

0

[k

2 + 4e

1 £2 (l-uv)

2

C 0

rig]

4e^c

2

(1-uv)

u-v +

2 k(£-,+£2 )u

I 4- , — ... 2vn

Q

sin k<$> +

2 2 4e.j e

2

k

2 + 4£

1 £

2 (l-uv)

2

5

0n0

2 2 + 16k E-jE^QfigUVsin k<t>

,,.2 2 2

16k £

-| c

2

[k

2 + 4e^£

2 (l-uv)

2 ^ 0noJ

2 2 4

k(c-| +c

2

)

. 2

4v n Q

sin %<p +

2— uvn

Q

sin

£ 1 z

2

k

2

u

2 g

Q

n 0

(u+v)

2

4 e

1

3

e

2

3

32
This expression may be simplified somewhat. In the last square bracket

2

in eq (V.24) the term with factor sin h<\> gives a contribution to A

2 of order 1/e relative to the second term in eq (V.24), which also

2

has a factor sin h<p. It may therefore be neglected. In this same

2

square bracket the last term gives a contribution to A of 0(u ). It

2 4

is therefore of 0

(

4> /e ) relative to the second term in eq (V.24).

It need be retained, therefore, only if cj> < 0(l/e ). But in that case

it is negligible relative to the first term in eq (V.24), unless

u-v < 0(u/e ). We may therefore set v = u and n

0

= Cq in the last

term in eq (V.24). We then obtain

A = 4e

1 £

2 C

0

n 0

[k

2 + 4£

1 e

2

(l-uv)

2

?

0n0

]

4e-j£

2

(1-uv)

u-v +

2

k ( £ -I +£ 2

) u 2 vn

0 sin%j> + ^^2“

il 2

4£-j £

2

k

2 + 4£

1 £2

(l-uv)

2

C

0n

0

2 2 + 16k

£

i

£

2 ?

qTI g u v s i

n

t 2 2 2

16k £

-j £

2

[k

2 + 4£

1 £2 (l-uv)

2^

0n

0

]

,2 C 2

4' 224 K ’0 u 4v n

Q

sin%j> + ^

3 £

1 £

2

(V. 25)

33
Thus we also have A > 0. We may now write eq (V.19) in the form

= [ ( Acosi|^ + Bsimj;)

2 + (AC - B2

)sin

2

ijj] , (V. 26)

from which we see that the expression for

| ^ will be positive

provided that

AC - B > 0 . (V.27

)

We therefore return to the expressions ( V . 20 )- (V . 22 ) for A, B,

2

and C, and show directly that AC - B >0. From eq (V. 22) we may

wri te

AC - B

2

= 4k^£-j£2?ori 0^ u

" v ^ + 4uvsin 2

%4>] * A

, 2 2 2 2 . 2, . d 2 + 1 6e

-j £2 v Dq sin 4 • A - B ,

and from eqs (V. 20) and ( V . 21

)

A = 4k

2 £

1 £2 £ 0

n 0

[(u-v)

2 + 4uvsin 2^] - u
2 2 B = 16e-j e

2

vrigSincf)

« k(e-j+e

? )u

(u-v)(l-uv)^

0

nQ + 2vn

0

sin +

2—2 "

4e-|

and

16e

22222 2 22222

1 e2 y n0

sin * A - B = 16^ e

2 v nQ

sin <|>

4k

2 £

1 £

2 ^on O^ U " V ^ 2+4uvSln2^^

/ 2 \

2

/ r \ 2

u

V£

l £

2,

Therefore

AC - B

2 = 4k

2 2 2

£

l £ 2^0n0^ u

” v ) + 4uvsin

^ * A

+ 16£-

I

2

£

2

2 v2n

0

2 sin 2

(j) ^k2

£

1 £

2 C

0

n

0

[(u-v)

2 + 4uvsin 2

%cf)] -

= 4k

2

£

1 £2 C 0nQ[(u-v)

2 + 4uvsin 2

ii(j)]

X <

,, 2 2

1 0£-j £

2

(u-v)

2

(l-uv)

2 5 02n

0

2 + 2(u-v)(l-uv)C

0

n

0 (2vn0

sin

2

%4) +

] 22

2

22 4 9

^ £ l p

) U + 4v n

Q

sin %<{) + 4vn

Q

sin • ~i

±

T~

4e-j £ 2

4k

2

£

1 £

2C

0n

0

[(u-v)

2 + 4uvsin 2^]

+ 4k

2

£-|£ 2

5 0nQ[(u-v)

2 + 4uvsin 2^]

k 2 C£

] +£

2

) 2

u

2

2 2 £

1 £

2

4e -j £ 2

+ 16k

2

£-j 2 £

2

2 v2

riQ

2

sin

2

(l) • 4k

2

£-j

£

2 ^griQ[ ( u- v

)

2 + 4uysin 2

^4>]

2

2 22 2.2, l°£-j £

2

v rig sin 4>

• (44 (V. 28)

35
2 2 ^(e, +

&2 ^

In the expression just given for AC - B , the term 4vriQSin %<p * ^

2~

4e, ^2

in the large square bracket within the curly brackets is of order

2 2 1/e relative to other terms with a factor sin %f> and thus may be

neglected. Next we note that the terms in the second square brackets

in the second line may be simplified in that

k 2

(e

1 +e

2

) 2 u

2

(

k 2

2 2 1 £ 1 £ ^

\ £ 1 £

2

The sum of the second and fourth lines in eq (V. 28) is therefore

4k

2

u 2 4k

2

e^£2CQriQ(u-v)

2 + 16k

4

u

2 [CQriQ4uvsin

2

^(j) - v

2q

Q

2 si

n

2

(f>3 . £

1 £

2

The second line here is

4 2 2 2 16k u • 4sin hp • vtiq[u£q - vrigcos

4 2 2 2

= 16k u • 4sin h<p • vn

Q

[(u£

0

- vtiq) + vrigSin h<p]

Here u6

Q

- vn

0

= 0( (u-v )C q

)

•

Therefore we have terms here of

0(e%2(u-v)/u) and 0(e

4

cf> 4

). But we have other terms in AC - B

2

, of

0(e

8

?

0

2

(}> 4). Therefore the terms of 0(e%4

) may be dropped. We also

p

C

p have terms in AC - B of order e cp ^(u-yj/u. Therefore the terms

2

u =

n ,2 2 4k u

£

1 £2

36
4 2

of 0(e <j> (u-v)/u) may be dropped. Thus, neglecting terms of

2 2

relative orders 1/e and 0 , the only term to be retained from the

second and fourth lines in eq (V. 28) is

2 2

4k eI e

2 ?0ng(u-v)

4k

2

u

2

e

l e

2

We may therefore now write

AC - a

2 2

2—

2—2—2—2

= £( u - v ' + 4uvsin^$<J>J

16k e-j £

2 ^0 n

0

X ^£^2

(u-v)

2 (l-uv)2

C 0

n

0

p

k (e-. ) u + 2(u-v)(l-uv) { 2vn

0

si n h<\> +

2—T~

4e-| e

2

2 2 4v n—0 4

si n%4>

?

0n0

+ k

2

[(u-v)

2 + 4uvsin 2

^<j>]

+ (u-v)‘

r,2 2

k u ~ 9

2 2

O+u )(l+v

2

) £

1 e

2

.2 2 _,.. 2 k4 + I

1 • 4e^£2tCu-v)

2 + 4uvsin 2

^(})]

v n

0

Sln r

’CTO

(V.29)

37
In eq (V. 29) the terms without <j> are

o o o / (l-uv)k(£ 1 +e ? )u

[k + 4£

1 £

2 (l-uv)^

0

n

0]( u -v) (u-v + g

£.|£ 2

[k +4e

1 £

2

(1-uv) 50

n

0

]

9 o ( l - UV)

2

(£,+£

?

) 2

(l+u

2

)(l+v

2

) -

1 2

[k

2 +4£

1 £2

(1-uv)

2 ^ 0

n

0

]

Here in the curly brackets we have

k

2 (l+u

2

)(l+v

2

) - k 2 (l-uv)

2

[k + 4e

1 e2

(1-uv) ? 0

n

0

] I ^^(i-uv) 2 + 4e

1 e2

(1-uv)'

k 2 (u+v)

2

k

2 + 4£-|E

2

(l-uv)

2 ^ 0nQ

Thus the terms without <f> are

o O o I (l-uv)k(e 1 +£

9 )u

[k + 4£

1 £

2 (1-uvr5

0

n

0 ](u-vr (u-v + ^

£

1

£

2

[ k +4e

1 e

2 (1-uv) 5

0n0

]

(u-v)'

( u+v

)

1

[k

2 + 4£-j£

2 (l-uv)

2 ^

0

n 0

]

(V. 30)

38
2 Next the terms in eq (V. 29 ) with factors (u-v) and sin are

4 uvsin^<f) • (u-v)^[ 2 k^ + 4e^£20 -uv )^or| 0^

(

2 2 2 2 2 4

v nn sin <j> 4v n n sin 9 + + 2(u-v)(l-uv) • 2vrigSin

50

n

0

?

o

n

o

2 + 4£-j£

2

* 4uvsin ^<j> • 2(u-v) (1 -uv)

k(£,+£2)u

2

2—2~ + 2uvrigSin^^4)

fe

l £2

Here the second line may be written as

2

o /

vrln 2

Vr| 0 ? \

4e^£2(u-v) • 4vrigSin %<J>[ cos + sin ^ + (u-v)(l-uv)J

\'C 0

5

0

n

0 /

= 16£-|£2vriQ(u-v)^sin^^4)(v(l+u^) + (u-v) ( 1 -uv

))

= 1 6e-| £2VriQ(u-v)^si n^^(u(l +v'^)^

2 2 = 1 6£-j £^

(

u-v ) uvsin h<p

2 Thus the terms in eq (V. 29 ) with factors (u-v) and sin h<P are

39
2 4uvsin %<{>

2 + 4uvsin i

Here we write

O-uv)

2 ?

Thus we have.

2 •

+

+

+

2<P

(u-v)

2

[2k

2 + 4e

1 e

2 (l-uv}2

? Q

n Q + 4e^s

2

']

• 2 (u-v) (1-uv) • 4e^e

2

p kCe1 +e2 )u

2vn

Q

sin h<P + T^T

4e-j e

2

( n

0 + 1 = 2(l-uv)

2 ?

Qn0

+ 1 - (l-uv)

2

?

0n Q

= 2(l-uv)

2 ^ 0

n

0 + ?

0n0

[(l +u

2 )(l+v2

) - (1-uv)

2

]

= 2(1-uv)

2 CqT1q + C 0

n

0

(u+v)

2

.

2 for the terms with factors (u-v) and sin ,

4uvsin 2

%4> • (u-v)

2

[2k

2 + 4e.

] £

2

(l -uv

)

2

? 0

nQ

]

2 2 2 4uvsin h(p • (u-v) • 4e^e

2

(u+v) ^qOq

« k(e,+e«)u

2 • 4uvsin ^(f> • (u-v) • (1-uv)

£

1 £

2

2 2 4uvsin %<p • 2(u-v)(l-uv) • 4e^e

2

• 2vriQSin h<p

40
k(e,+£

? )u

= 2(u-v)[u-v + (1-uv) « e

1 e2

[ k +4£i£ 2

(l-u v )^^

0

r>o^

2

, 4uvsin 2

^c|)[k

2

+4£

1 £

2 (l-uv)

2

?QnQ]

+ 4uvsin 2 ^[(u-v) 2

• 4£^£ 2

(u+v)

2

5 0nQ + 2(u-v)(l-uv) • 4£^£ 2

• 2vn

0

sin

2

^4>] . (V.31

)

Finally, we have the terms in eq (V.29) without a factor (u-v)

2 4uvsin k<p

2 2 4v n

Q 4 2 4

£

i

£

2

• sin ^ + k

£

0

e

0

2 2 v n

Q 2 + 4£-j£

2

• sin (j) 5

0

n 0

2 4uvsin %<p

2 4uvsin k<P

2 2

4v rin 2 2 2 4£

l e2

* sin %4> + k • 4uvsin Jg<j> C 0n

0

4

16uvsin h(p

v Hq

2

4e -I £p + k uv

. «o

(V.32)

We must now combine the terms given in (V.30), (V.31), and (V.32). For this

purpose it is convenient to designate by (V.30a) and (V.30b) the first and

second terms in (V.30), and, in similar fashion, to designate by ( V . 31 a ) and

( V . 31 b ) the first and second terms in (V.31). We then have

41
(V. 30a) + (V. 31a)

[k

2+4ei £

2 (l-uv)

2

C0n

0

]

(u-v) u-v + (l-uv)k(e.j+E

2 )u

2 2 e

1 e2

[k +4e

1 e

2

(l-uv) ^

0

n

0

J

2 + 4uvsin

- [k

2 +4£

1 e

2

(l-uv)

2 5 0

n 0

] (4uvsin 2^) 2 (V.33)

Adding (V.32) to the second term in (V. 33) gives

4 16uvsin k4>

v n

Q 2

4e i £o 4£-j£ 2

(1-uv) CqHqUV 12

=0

4

64£-|£

2 uvsin h<P

v n

0

- (l-uv)

2

? 0

n 0 uv L

and adding this to (V.31b) gives

4

64£-|£

2 uvsin

+ 4uvsin 2 %(j)(u-v)

2

• 4£^£

2

(u+v)

2

?qT1q . (V.34)

The expression in the square brackets in (V.34) may be simplified:

We have there

2 v n n

o (1-uv) ?

0nQ uv + (u-v) ( 1 -uv)vn

0

- 5 0

42
vn

o (1-uv)

2

u£ 0 + (u-v)(l-uv)

5

°

- v^

0n

0IvO +u 2

) 2

- 0-uv) 2

u + (u-v)(l-uv)(l+u

2 )]

= v^Qf|Q[v+2vu 2+vu^-u+2u

2 v-u2 v2

+u-u

2

v+u

3 -u4

v-v+uv

2 -vu2+v2

u

2

]

= uv^qTIqEu

2 + 2uv + v2

]

o = uv£

0n0

(u+v)

Thus the expression (V.34) may be written in the form

4e

1 e2

(4uvsin 2

%((>) 2 £QnQ(u+v)

2 + 4£

1 £2

(4uvsin 2^)£0nQ(u-v)

2 (u+v)

2

.

Adding this to ( V - 30b ) then gives

4£

1 £

2£

0

n

0

(u+v)‘

(4uvsin 2^c|))[(u-v)

2 + 4uvsin 2

^cf>]

(u-v

)'

(V. 35)

2 2 4£

1 e

2 ?0n

0

[k + 4e.

] £

2

(l-uv) ?

Q n

0

]

Finally, combining (V.35) with the first term in (V. 33) gives our desired

2 form for AC - B , given previously by (V.29), viz..

43
16 k

2

£l

2 2 r 2 z

2 h n

2

0

[k

2

+4e

1 £

2

(l-uv)

2

CQn0

]

(u-v)lu-v +

(l-uv)k( £ .j+£ 2

)u

2 2 e^fk +4e-,£ 2

(l-uv) £q^q3

2 + 4uvsin %<p

X 4

4

£l

£

2

(u+v)

2

£

0n0

(4uvsin

2

%<j>)[(u-v)

2 + 4uvsin 2

%<j>]

k u(u+v) (u-v)‘

£-|£ 2 y

Ck "** 4£ -j£ 2

(1-UV) CqHqI

(V.36)

2

This shows that AC - B >0, which, along with A > 0 and C > 0,

completes the task of writing the high energy small angle expression

for in a form that is manifestly positive.

We now write our high energy small angle approximation for the

numerators

| |

and

|

j> appearing in the cross sections dc^ and

da,| (eqs (1.2), (1.3), (IV. 9), and (IV. 10)), in a form similar to that

given for the numerator of the cross section for unpolarized photons in

eq (VI. 49) on p. 66 of [1]. There we wrote

| |= f (v)(v-u+cr)

2 + g(v)sin 2 ^ + p

From eqs (IV. 9) and (V. 22) we have

{ |

= f

i

(v)(v-u)

2 + g

x (v)sirv

2

%<i> , (V . 37)

44
where

and

fj_ (v) = 4k^e-j£2

(l+u

2

)(l+v

2

)

9l {v) = 16k £-j£2 uv

( l+u

2

)(l+v

2

)

2 2 2 2 64£

] £ 2 V COS %(})

(l +v

2

) 2

and from eqs (IV. 10) and (V.25) we have

{ }„

= f

|| (vHv-u+cr,, ) 2

o + 9i, (v)sin h<p

where

(v) -

A 2 2

4£-j £2

(l+u

2 )(l+v2

)

k

2

( uv- 1

)

2 + 4 2 o~ £^2 (l+iT)(l+v )

%

(v) =

2 2 2 2 16k £1 £ 2

^ v 64k sin k<t>

(l+u

2 )(l+v2

) (1+v

2

}

2

k 2

. , (uv-1)

2 H

? ? £^2 0

+U ) (1+v )

(V.38)

(V.39)

(V.40)

( V . 41

)

(V.42)

45
= a„ (v) =

4(uv-l)

2 k(e1 +£2)u

2vTi

0

sin + 22

4e,

k

2

(uv-1)

2 + 4

£^2 (l+u

2

)(l+v

2

)

(V.43)

and

16

Pn =

ke

l £

2,

? 2 [ k

2

/u

2 -l\

2

]

(1+u

2

) £

1 £2 \

u2+1

/

(V.44)

In comparing eq (V.44) with eq (V.25) it will be noticed that in

eq (V.44) we have set v = u throughout. Since

pjj is of 0(1), it

is negligible unless both u-v = 0(u6 ) and <j> = 0(0 ). We may

therefore set v = u in

pjj as it appears in eq (V.25), with neglect

2 of terms of relative order 0 .

The expressions (V.37) and (V.40) for the numerators

| |

and

< > will be of particular use in the following section, where we

( 'II determine the maxima and minimum of the cross sections do^ and da^

in the region of small momentum transfers, q < O(u0) (i.e., q

= O(u0)

and q = O(u0^))

46
VI. DETERMINATION OF THE MAXIMA AND MINIMUM OF THE POLARIZED

BREMSSTRAHLUNG CROSS SECTIONS da

L AND da

(|

In this section we consider the cross sections dc^ and da^ as

functions of v (i.e., © 2

) for fixed values of the other variables

(e-j , £

2

’ 9

i and $) and determine the value of v for which these

cross sections achieve their maximum and minimum values. In order to

locate the extrema of the cross sections and evaluate the cross sections

at these extrema we use the high energy small angle approximations for

|

1 and |l given in eqs (V.37) and (V.40), and for q^, given

where 5 = k/(2e^e^). We will follow very closely the analysis presented

in detail in section VII of [1], and will therefore make frequent reference

to equations and discussion given there. As discussed there on pp. 72

and 73, we also neglect screening in this analysis in determining the

location of the extrema of the cross sections, setting F(q) = 0 in

eqs (1.2) and (1.3). The determining equations for the extrema are

then (see eq (VII. 2a), p. 73 of [1])

in eq (V.15) of [1], viz.

,

(VI. 2)

and

}. (VI. 3)

47
As in [1], we are looking for maxima and minima in the region of small

2 2

q, i.e., q < O(u0), and hence also qx < 0(.u6). From q£ = (u-y_) =

(u-v) + 4uvsin k<P we have | u- v | < qx and u$ < 0 (

q

±

)

. Thus for

qx < O(u0) we have < 0(6) and $ < 0(0). We therefore assume

in this section, where q < 0(u6), that <j> < 0(6), neglecting terms of

o relative order <f> < 0(0) in our high energy small angle approximations

for q^

, | | |||

, and their derivatives with respect to v.

This results in considerable simplification in the expressions for

| |x and

| |,|

(eqs (V.37) and (V.40)), i.e., in the eqs (V.38) and

(V.39) for f

± (v) and gi (v), and in eqs (V.41 ) - ( V . 44 ) for f

||

( v )

,

o

gjl (v), ^(v), and p]( . Thus, neglecting terms of relative order <jr

in eqs (V.39) and (V.42), we now have, for q < O(u0),

9j_(v) =

2 2 2 2 16k e-|£2 uv 64ei v

(l+u

2

)(l+v

2

)

+ (1+v

2

) 2

and

(VI. 4)

g„(v) =

1 6k c-j £2 uv

(l+u

2

)(l+v

2

)

(VI. 5)

2 Referring to eq (V.43), we see that the function Oy (v) is of 0(u6 ) 2

for (p < 0(0). Moreover, 0

| ( (v) appears in the term (v-u + a^) in

eq (V.40), where now v-u < O(u0). Now if, for v-u = O(u0), we set

v = u in the expression (V.43) for a^v), then we introduce errors

of 0( (v-u )/u ) = 0(0) relative to ^(v). These errors are thus of

3 2

O(u0 ); i.e., they are of 0(0 ) relative to v-u, and hence may be

2

neglected. On the other hand, for v-u = O(u0 ), setting v = u in

48
2

eq (V.43) introduces errors of order 6 relative to Cy 1 - Once again

these may be neglected. Therefore, for q < O(u0) we may, in eq (Y.43),

2

set v = u and neglect terms of relative order <p . We then obtain

u(u-l)

a„ =

2(p k(£n+£p)

jr +

C

l+u‘

2 2

s

1 e

2

+ 41 ^u 2

-l

. 2

£

-j £2 \U +1

(VI. 6)

The expression (VI. 6), valid in the region q < O(u0), now closely

resembles the expression for a given in eq (VI. 48) on p. 66 of [1].

The two expressions are independent of v, and are of the same order

of magnitude, viz., 0(u8 ).

Thus for q < O(u0), f

x (v) and f,j (v) as given in eqs (V.38-)

and (V.41) are each of the same form as f(v), given in eq (VI. 44) on

4 2 p. 62 of [1], all three being of 0 (c £ ), and gx (v) and ( v ) as

given in eqs (VI. 4) and (VI. 5) are each of the same form as g(v),

given in eq (VI. 45) on p. 62 of [1], all three being of 0(£

4

£

2

u

2

).

Further, as just noted, again provided that q < 0(u6), as given

in eq (VI. 6) is of the same order of magnitude as a, given in

eq (V.48) on p. 66 of [1], and p

!( as given in eq (V.44) is of the

same form and order of magnitude as p, given in eq (VI. 47) on p. 65

of [1]. (The fact that for large u » 1, p is larger than

p^ by

2

a factor u is of no significance for the analysis. In both cases,

2 2 they need be retained only when both v-u = C(u0 ) and p

= 0(0 ).)

It thus follows that if we define

49
CT, = 0 (VI. 7)

and

P

x = 0 (VI. 8)

then we can, for q < 0(u6), write

j

and

j

in the form

given previously for

| |

in eq (VI. 49) on p. 66 of [1]:

{ }

i ,|| = fl ,||( v )( v

- u+0

i ,||)

2 +

9^,1 (vising + pi(,|

- (VI. 9)

And, since the form and the order of magnitude of the quantities

appearing here are the same as those in eq (VI. 49) of [1], we can

utilize without modification most of the analysis presented in sec. VII

of [1]. There we defined ((VII. 3) in [1])

z = ^ (VI. 10)

and looked for solutions of the equation defining the extrema ((VII. 2a)

in [1], eqs (VI. 2) and (VI. 3) here), first in the region z < 0(0^),

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

0, the equation defining the extrema, considered as a function of z

o (or v), had one solution in the region z < 0(0 ), corresponding to

the minimum of the cross section, and two solutions in the region

z = 0(0), corresponding to the maxima of the cross section, one for

z < 0 (v < u) and one for z > 0 (v > u). Further, we found that as

(p increases, the position of the minimum moves to larger values of v.

50
and the position of the second maximum (the one for z > 0) moves to smaller

values of v. When <J> reaches a critical value, <J> , the minimum and the

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

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

behavior of the extrema as a function of <p was shown in figures 11 and 12

in [1], For polarized photons, identical statements may be made for do^

.

However, the behavior of dc^ depends on the photon energy, k. For k

satisfying (VI. 25) it is the same as that of do

1

, whereas for k satisfying

(VI. 33) dO|j has a sharp dip for all values of (p. (See figs. 4(a)-4(d) and

5 (a )-5 (d ) .

)

Thus for z < 0(9 ) and <p arbitrary (but still assuming <j> < 0(0)),

we have, directly from eq (VII. 7) on p. 78 of [1], the position of the minima

for the cross sections dc^ and dc^ , z^^ and zm-j n|| > respectively

:

~ 4/3 of the cross section. This point occurs for z = 0(0 ' ). For <j> > <j>

o

L f

L (u) gx

' (u) <f> 2

u u 8

(VI. 11)

and

(VI. 12)

51
Noting from eq (VI. 7) that we defined = 0, the expression (VI. 11)

for z then 'simp! if ies somewhat and may be written in the form

zminl

= - 4>

9*' (u)

1*2

5

2

(l+u

2

) gx

(u) ^ 6(1+U

2

)'

811 ^(u) 2u

z f

1 (u) \2 £2 )

i'oTM1 )' *‘ lul

- ll»

!

v 2 u fj_ (u)

. (VI. 13)

The expression (VI. 12) for may be written in the closely similar

form

g,l

' (u )4> 2 Q||^/ 2

6

2 (1+u

2

) ^

gjj

( u )<J> 2 ^

4> 2 <5(l+u

2

)

, 8uf,, (u) u 2u f„

( u ) \2

zmi nil 6

2

(1+u

2 )' 9

» (U> -iL2

2u f, (u)

(VI. 14)

We consider first the expression (VI. 13) for z .. . . Noting from eqs (VI. 4)

and (V.38) that

9l (u) 2k

2 + 80^2

2 u

2 f

i (u) k

2 (VI. 15)
we define, in analogy with eq (VII. 10) on p. 80 of [1],

ol

5(l+u ) / gi (u)

u \2u 2

f

1 (u)

- 1

6(l+u

2

) / k

2

u \k + 8e-j£2

(VI. 16)

Further, from eq (VI. 4) we have

16k

2 e 1 e ;) u / 1-v2

> 9 ~

g,

' ( v ) = - '-2

jry] + 128e, e

2

1

(1+u

2

) \(l+v 2

) 2.

and thus, with f, (u) as given by eq (V. 38),

gj/tu)

uf

x (u)

= 4

k + 8£-j£2'

(VI. 17)

Substituting eqs (VI. 15), (VI. 16), and (VI. 17) in (VI. 13), we may write

the expression for z . . in the form

'mini

- <j)

(VI. 18)

53
As noted in [1] p. 80 in connection with z . , we observe here in min’

2

identical fashion that for <p < 0(0) we have, in general, z < 0(0 )

as originally assumed. (Since a

x

= 0, we have z = 0 (4> ) from

eq (VI. 18).) Again as observed for eq (VII. 9) in [1], so also in

eq (VI. 18) given here, the denominator is zero for 4

= d>^ , so that

this equation is clearly invalid for 4 > 4^

. In fact, as noted in

[1], it becomes invalid for (j) somewhat less than 4^. Once 4 is

sufficiently close to 4Ql

that 4^ -

<f> is less than 0(0 ),

then z ^ is larger than 0(0 ), and the assumptions made in

deriving eq (VI. 18) no longer hold. As shown in [1], as 4 approaches

^oi

(from below), the position of the minimum, 2m-j ^

> increases from

O(0

2

) to O(0

4//3

). When reaches the critical value 4 ^ ( 4 ^

< 4^),

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

3 an inflection point. For 4 > 4^ the cross section d no longer

has a minimum. For 0^=1°, e-j = 140 MeV, and k = 95 MeV we find,

from a careful computer analysis of the cross section as a function of

v and 4 (see fig 6(a)),

4^ = 0.3645° .

From eq (VI. 15) we have, for these same values of 0-j , c-j , and k.

oi

0.4293°
Having thus far examined in detail the extremum (the minimum) of

the cross section d ^in the region z < 0(0 ), we now look for

extrema (also solutions of eq (VI. 2)) of this cross section in the

region z = 0(0). Again we follow the analysis given in [1], pp. 81-93.

2

In contrast to the case z = 0(0 ), we now find two extrema. These

3 correspond to the two maxima of the cross section d In zeroth

order, i.e., neglecting terms of relative 0(0) as well as those of

O(_0

2

), we have, directly from eq (VII. 14) on p. 83 of [1],

(VI. 19)

(o) „ 6(l+u

2

) ‘maxi

= + 1 - oi/ J

3

The full solution for the maxima of the cross section d (i.e., with

2 neglect only of terms of relative 0(0 )) is, from eq (VII. 32) on p. 93

of [1],

zmaxi

+

6 ( 1 +u

2

) uf

x '(u) 6

2

( 1 +u

2

)

4f

x (u) u

2

(VI. 20)

Here f^(u) and ' ( u

)

3fj.(v)

3v v=u

are given by eq (V.38), from which

55
2u

(VI. 21)

1 (u)

f

x (u) 1+u^

and, from eqs (VII. 20) and (VII. 21) on p. 88 of [1], (and noting from

eq (VI. 7) that o

L =0),

3g,'(u)

?

<j> £

Su^ (u)

gi (u)f

i '(u)

2

2uf/(u)

uf/tu)

2fj_ (u)

+

«

2

(l+u

2

) (VI. 22)

As was the case for z . . , derived under the assumption that

2

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

v

2

- = °( 02

)

or

<f> oi - <P = 0(0) .

Again we see that the solutions given in eqs (VI. 19) and (VI. 20) break

down as cp approaches <J>^

.

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

3 cross section d , given by eq (VI. 18), and the two maxima, given by

eq (VI. 20), or, to zeroth order, by eq (VI. 19). We see that the form

3 of the cross section d in the region of small momentum transfers

56
(q < O(u0 ) ) and its behavior in this region as a function of v and <p

are very similar to that of the cross section for unpolarized photons

discussed in £1]. This similarity holds both in the region of the

minimum (z = 0(6 )) and in the region of the maxima (where z = 0(6))

(see figs. 7(a)-7(d)).

We now return to our consideration of the extrema of the cross

section d ajj . Given the similarity of the expressions (VI. 11) and

3

(VI. 12) for the positions of the minima for the cross sections d

3 3

and d , it might be expected that the analysis just given for d ,

paralleling that presented in [1] for the cross section for unpolarized

3

photons, could be repeated without modification for d . The

essential factor in that analysis, as may be seen from eqs (VI. 13),

(VI. 16), and (VI. 18), is the factor

gx (u)

~T 1 (u)

which appears in the denominator of z . . and in the numerator of mini

zmaxi ^ ec,s (VI. 19) and (VI. 20)). However, implicit in the definition

(VI. 16) for <|>^ is the assumption that

9, (u) —-f 1 > 0 . (VI. 23)

2u Tx (u)

This is clearly true from eq (VI. 15), and was also the case for —rjj-^ 1 as given in [1] p. 82, eq. (VII. 11a). However, the

2u

Z f(u)

3 corresponding term for d is, from eqs (VI. 5) and (V.41),

57
%

(u)

2u

2

f„ (u)

- 1 = 2k‘

^ + ^

(u

2

+I.

- 1

0 / 2

' k - 4 e-, £, u

'1 2 \ 2

\ii +!

o /

2 ,

N

k + 4e, e, u

1 2

Vu

2 + l

(VI. 24)

Thus if

k > 4e,e

u 2-l' E k ° (VI. 25)

we can again define, in analogy with eq (VI. 16),

5(1 +u ) / g„ (u)

oil u \2u f,. (u)

- 1

6 (

1 +u )

,2 . , /u

2 -l

k + 4e,e

1 2

\u

2 +1

k 2

- 4e,zJ

uZ- ]

11 2 \ 2

+1

(VI. 26)

58
Then paralleling the steps followed in going from eq (VI. 13) to eq (VI. 18)

we have, from eq (VI. 5),

3i,

' Cv) = 16k e^u

O+U2

)

n L 1 -V

W)

and thus, with fy (v) as given by eq (V.41),

9||

' (u)

ufy (u)

(VI. 27)

Substituting eqs (VI. 6), (VI. 24), (VI. 26), and (VI. 27) in (VI. 14) we may

then write the expression for z . „ in the form

(VI. 28)

The comments made in connection with z .^, following eq (VI. 18), are

equally applicable to zmi

- n|| for The present case in which the photon

energy is sufficiently large that the inequality (VI. 25) is satisfied.

59
p Thus we have, in general, z . n < 0(6 ) as originally assumed.

Further, the denominator in eq (VI. 28} is zero for <f> = ^ , so that

the equation is clearly invalid for <j> > <j> ^

. Again, as noted in [1],

it becomes invalid for $ somewhat less than <j>^ . Once 4> is

sufficiently close to <t>^ that <j>^ - <p is less than 0(0 ), then

zmi n|| l ar9er than 0(0 ), and the assumptions made in deriving

eq (VI. 28) no longer hold. As <p approaches (from below), the

position of the minimum, z^^ , increases from 0(0 ) to 0(0 ' ).

When (p reaches the critical value 4> c( | (t}^ < <J> 0( |), the minimum

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

inflection point (see fig. 5(f).) For (p > <|> ^ the cross section d

no longer has a minimum. Again following the analysis given in [1],

pp. 81-93 for the extrema of the cross section in the region z = 0(0),

we now find two extrema, corresponding to the two maxima of the cross

section d^a^ , as we just did for d^ (eqs (VI. 10) and (VI. 20)). In

zeroth order, i.e., neglecting terms of relative 0(0) as well as those

of O(0

2

), we have, directly from eq (VII. 14) on p. 83 of [1],

(o) _ ± 5(l+u

2

) , maxll u (VI. 29)

3

The full solution for the maxima of the cross section d (i.e.,

2 with neglect only of terms of relative 0(0 )) is, from eq (VII. 32)

on p. 93 of [1],

60
z

maxll

5(l+u

2

)

u

1 - ufy ' (u) 6

2

(1+u

2

)

4fuCu) u

2

2

Here f„ (u) and f„

1 (u) = (v)

9v are given by eq (V.41 )

,

v=u

'(u) 2u

f„(u) l+u‘

k

' +

and, from eqs (VII. 20) and (VII. 21) on p. 88 of [1],

a

( 2) = + 3g

ll

* ^

^2 _ g

|| ^

f

ll

' ^

^2 _ uf

ll

' ^

Lz +

5 ( 1+u

11 u 8uf

H (u) 2ufj| 2 (u ) 2f(1 (u) \ u

2

As was the case for zm^ nj| , derived under the assumption that

we note that here too the solution is valid only for (J> < <j> ^

.

we assume now that z = 0(0) we must again have

oil

' = 0(e2)

or

4

> 0|| " *

= °( 0 )

•

1 a

( 2 ) “ 2 A

ll

(VI. 30)

from which

(VI. 31)

(VI. 32)

z = o(e

2

),

And since

61
We have now found, for sufficiently small <j>, and assuming the

condition (VI. 25), the minimum of the cross section d a

( | , given by

eq (VI. 28), and the two maxima, given by eq (VI. 30), or, to zeroth order,

by eq (VI. 29). We see that, assuming the condition (VI. 25), the form

3

of the cross section d in the region of small momentum transfers

(q < O(u0)) and its behavior in this region as a function of v and 4>

3 are very similar to that of the cross section d as well as to that

of the cross section for unpolarized photons discussed in [1]. This

similarity holds both in the region of the minimum (z = 0(9 )) and in

the region of the maxima (where z = 0(0)). (See figs. 8(a)-8(d).)

3

We have considered the cross section d assuming the condition

3

(VI. 25) and noted its similarity to the cross section d a

L as well as

3

to the cross section summed over polarizations, d a, under this condition. We now consider the case in which (VI. 25) does not hold, viz.,

when

k

2 < 4e,e, ) i k 2

. (VI. 33)

1 2 \u 2

+l )

0

It should be noted that for most situations of experimental interest,

the condition (VI. 33) is the more likely to occur: In general we have

u = p^sin9-j 33 p-|0.j

» 1

Thus

62
From (VI. 33) we then have, approximately.

from which

or

k < 4c-| ^2

= 4e-j (e-j - k) ,

(k + 2e-| ) 2

< 8c-| 2

— < 2(/Z - 1) * 0.8

£

1 (VI. 33a)

Thus, if p-|9-| » 1, then for most of the spectrum (0 < — < 0.8) the

condition (VI. 33) holds. In this case, we see from eq (VI. 24) that

9

,, (u)

2u

2 fM (u)

k 2

- 4e,e

^

2

" ]

\ 2

- 1 =

21 2 x1 \u +1

.2 . „ /

u

2 -l

k + 4e,e,

< 0

'1 21 2 , \u +1

We can not then define as in eq (VI. 26). We may, however, retain

all of the expressions for and z max|| if we now define

63
6(l+u

2

)

*011 u

\'-

g

ll (u) \

2u

2 f

y (u) J

6(l+u

2

)

2 1

u p

t2UJ

2

- k

2

(VI. 34)

The expression (VI. 28) for zm^ ^

is then modified only in that the

2 —2

term must be replaced by -<j>

0|j

. The numerator of (VI. 28) is thus

2 2 unchanged, and the term 4> o( | - <p in the denominator is replaced by

In the region of small momentum transfers (q < O(u0), and thus also

o < 0(0)) we now find zm

-,- njj °f 0(0 ) for all <j>; the denominator

does not become smaller as cj) increases and the modified expression

for z . ,, does not break down at some critical angle. Similarly, the

mi nil 3

expressions (VI. 29) and (VI. 30) for zmax|| need only be modified by

2 —2

replacing (J)^ by . The modified expressions for z max|| are

then of 0(0) for all <j>; they do not break down at some critical angle.

In summary, then, for photon energies such that the condition (VI. 33)

3 holds, the cross section d has, for all (p , one minimum in the

region z = 0(0), provided, of course, that we remain in the region of

3

small momentum transfers. This difference in the behavior of d

and da. for photon energies satisfying the condition (VI. 33), shown

64
in figs. 9(a)-9(d), is clearly significant for the polarization: For

<f> > <j>^, has no sharp dip, as may be seen in figs. 9(c) and 9(d),

whereas d has a sharp dip regardless of the values of <j>. (We are

of course still assuming that <j> is small, of 0(9), so that we are

still in the region of small momentum transfers.) Then in this region

of the dip in d the polarization, defined in eq (1 1 . 1 ) , is very

large, i.e., the tagged photons are highly polarized perpendicular to

the production plane (see figs. 3(a )-3(h )

)

.

65
VII. COULOMB CORRECTIONS TO THE DIFFERENTIAL CROSS SECTIONS

FOR POLARIZED BREMSSTRAHLUNG

The analysis presented in this report thus far has dealt exclusively

with the first Born approximation differential cross sections for

polarized bremsstrahlung, da

x and da

() . In particular we have

considered the behavior of these cross sections in the region of small

momentum transfers.

As noted in £1], Sec. IX, the Coulomb correction to the differential

cross section for bremsstrahlung at high energies has been calculated

specifically in this region of small momentum transfers. In connection

with the cross section for unpolarized photons 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 [19]. In both

cases it was found that the entire effect of the Coulomb correction is

simply to mu! ti ply the Born approximation cross section by a factor

which, although it varies throughout the region of small momentum

transfers, is always of order unity. This work was extended to the

case of polarized photons by Olsen and Maximon [16], who showed that

the same multiplicative factor applies to each of the completely

3 3 differential cross sections for polarized photons, d and d .

This factor, denoted by J A | in [19] and by R in [18], is given

for the case of no screening on p. 123 of [1]. Thus also for the cross

3 3 sections d and d , the Coulomb corrections do not fill in the

sharp dips, as they might if they were additive. Moreover, since the

same mul tipi icati ve factor applies to both of the completely differential

66
3 3 cross sections, d and d , it does not enter in the polarization,

P, eq (II. 1). However, once we integrate over some region of angles,

as is required in the analysis of most experiments. Coulomb corrections

do indeed enter the expression for the polarization.

It must be recalled that in the analysis presented in references

£18], £19], and £16], there have 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], £19], and [16] must be extended to include the

terms of relative order 1/e.

67
REFERENCES

[1] Leonard C. Maximon, Arlette de Miniac, and Thierry Ariel, Tagged

Photons, An analysis of the bremsstrahlung differential cross section in the range of interest for a tagged photon system. National

Bureau of Standards Report NBSIR 81-2262, April 1981.

[2] Nuclear Physics Research with Microtrons Using Superconducting

Linacs. July 1980 Report of the Nuclear Physics Lab., University

of Illinois, Urbana, 111. (unpublished).

£3] 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 spectrom&tre Compton du MPI

de Mayence, Comptes Rendus des journees d'etudes de Saclay, 14-18 mars 1977.

[4] 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 journees d'etudes de Saclay,

14-18 mars 1977. This material is also available as a report of

the Physikal isches Institut of Bonn University, preprint no. BONN-HE-78-1 9 (April 1979).

[5] Christophe Tzara, Comptes Rendus Acad. Sci . 239 , 44 (1954).

[6] Jacques Miller, Polarization of bremsstrahlung , CEN Saclay Note

CEA-655, 1957.

[7] D. Jamnik and P. Axel, Phys. Rev. 117 , 194 (1960). This article

also gives other references to earlier experiments.

[8] Katsufusa Shoda, A new arrangement for experiments using polarized

bremsstrahl ung , L.A.L. Report 1135, April 1965 (Laboratoire de

1

' Accdlerateur Lin^aire, University de Paris, B.P. 5, Orsay, France).

£9] F. F. Liu, Phys. Rev. 138, B1443 (1965).

[10] A. M. Nathan, R. Starr, R. M. Laszewski , and P. Axel, Phys. Rev.

Lett. 42, 221 (1979).

68
[11] K. Wienhard, R. Schneider, K. Ackermann, K. Bangert, U. E. P. Berg,

G. Junghans, W. Naatz, and R. Stock, Photonuclear reactions with

linearly polarized photons, in Abstracts of contributed papers to

the International Conference on Nuclear Physics with Electromagnetic

Interactions, Mainz, Germany, June 5-9, 1979.

K. Ackermann, K. Bangert, U. E. P. berg, K. Kobras, W. Naatz,

R. Schneider, R. Stock, and K. Wienhard, pp. 3-9 of Tatigkeitsbericht

1979, Kernphysik, Strahlenzentrum, Justus-Liebig-Universitat Giessen,

D-63 Giessen, West Germany.

K. Ackermann, K. Bangert, U. E. P. Berg, C. Biasing, K. Kobras,

W. Naatz, D. Ruck, R. K. M. Schneider, R. Stock, and K. Wienhard,

Large bound El -strength in T=1 sd-shell nuclei, p. 274 of Vol . 1 Abstracts of the Proceedings of the International Conference on Nuclear Physics, August 24-30, 1980, Berkeley, California.

K. Wienhard, R. K. M. Schneider, K. Ackermann, K. Bangert,

U. E. P. Berg, and R. Stock, A study of ls

0(y,p)

15 N with linearly

polarized photons, p. 47 of Workshop on nuclear physics with real

and yirtual photons, from collective states to quarks in nuclei,

November 25-28, 1980.

U. E. P. Berg, K. Ackermann, K. Bangert, C. Biasing, K. Kobras,

K. Naatz, D. Ruck, R. K. M. Schneider, R. Stock, and K. Wienhard,

Nuclear resonance fluorescence measurements with linearly

polarized bremsstrahlung photons, p. 49 of Workshop on nuclear

physics with real and virtual photons, from collective states to

quarks in nuclei, November 25-28, 1980.

[12] Michael M. May, Phys. Rev. 84, 265 (1951). Note that in eq (1) on p. 266 of this article the sign of the term with factor q

2 is

incorrect -- it should be negative.

[13] R. L. Gluckstern, M. H. Hull, Jr., and G. Breit, Science 114, 480

(1951); Phys. Rev. 90, 1026 (1953).

[14] H. Kolbenstvedt and Haakon Olsen, Nuovo Cimento 22, 610 (1961);

note Sec. 2 and Appendix A.

[15] Haakon Olsen and L. C. Maximon, Nuovo Cimento 24, 186 (1962).

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

Eq (4.9), p. 892, which does not contain any correlations between

two momenta and only one polarization.

[17] Haakon Olsen and L. C. Maximon, Phys. Rev. Lett. ]3_, 112 (1964).

[18] H. A. Bethe and L. C. Maximon, Phys. Rev. 93, 768 (1954).

T 1 9] Haakon Olsen, L. C. Maximon, and Harald Wergeland, Phys. Rev. 106,

27 (1957).

69
FIGURE CAPTIONS

Fig. 1 Momenta of the initial and final electrons, ^

and and

photon momentum and polarization, J< and in the system

with z-axis in the direction of J<, and in the x-z plane.

Fig. 2(a) The cross section for unpolarized photons, ^ do

^

= Kf d^o/dfip dfi^dk, eq ( I V

.

1 ) , p. 22 of [l]j, and the

perpendicular and parallel components of the momentum transfer,

qx and q

z

, as functions of 02 for ^

= 140 MeV,

k = 95 MeV, 0

] = 1°, and $

= 0°.

2(b) The same as in 2(a), except for the cross section, which is

for photons polarized perpendicular to the production plane.

Fig. 3(a) The polarization, P = (dc^ - dc^ )/(do

i + da

)( ) as a function

of 02> for c i = 140 MeV, k = 95 MeV, 0-j = 1°, and

several values of <j>, viz., <j> = 0°, 0.01°, 0.02°, 0.03°,

and 0.1°.

3(b) The same as in 3(a), except that (p = 0°, 0.5°, and 1°.

2(c) The same as in 2(a), except for the cross section, which is

for photons polarized parallel to the production plane,

70
Fig. 3(c) The same as in 3(a), except that <J> * 2

9

, 6°, and 10°.

3(d) The same as i

3(e) The same as i

3(f) The same as i

3(g) The same as i

n 3(a), except that

n 3(a), except that

n 3(b), except that

n 3(c), except that

(p = 1° and 30°.

k = 30 MeV.

k = 30 MeV.

k = 30 MeV.

3(h) The same as in 3(d), except that k = 30 MeV.

Fig. 4(a) The cross section, da^ ^ d^/dS^ d^dk^j , for

photons polarized perpendicularly to the production plane, as

a function of Qg, for k < k . (From eq (VI. 25),

k

Q

3 = 4e-j£2 ( (u^-1 )/{u

3

+l )

)

3

. ) We choose e-j = 140 MeV,

k = 95 MeV, 6-, = 1°, and 4

= 0°, 0.25°, 0.5°, and 0.75°.

4(b) The same as 4(a), except that k = 130 MeV, so that k > k ,

and 4

= Q

9

, 2°, 4

9

, and 6°.

4(c) The same as 4(a), except that we plot the cross section,

V da,, (= I*- d

3

a,. /dti

to the production plane.

4(d) The same as 4(b), except that we plot the cross section,

j

ia

w (

= 7

d Vd“

P to the production plane.

Note that following the individual figures 4(a) - 4(d) we present

these four figures on a single page for easy comparison.

d^dkj, for photons polarized parallel

dft^dk for photons polarized parallel

71
Fig. 5(a) The positions, z = (v-u)/u, of the maxima, the minimum, and

the points of inflection of the cross section

-U- da, f = -U- d

3

a, /dft„ dft.dk) as a function of the azimuthal

Z

2 1

\ Z

2 1 p

2

k /

angle <J>, for k < k

Q

. We choose c-j = 140 MeV, k = 95 MeV,

and 0^ = 1°.

5(b) The same as 5(a), except that k > k Q

. (We choose k = 130 MeV.)

Note that in this figure the ranges of <J> and z are

0° < <p < 40°, and - 1.0 < z < 0.2

5(c) The same as 5(b), except that the ranges of <p and z are

0° < <j) < 10° and - 0.2 < z < 0.2.

5(d) The same as 5(a), except that we consider the cross section —1 - 1 ,3

do,, ( = ~2 d cT||/dftp dft^dk).

5(e) The same as 5(b), except that we consider the cross section

1 2

da

ll ("

z

2

d a

il / %2

dfi k dk

)

'

5(f) The same as 5(c), except that we consider the cross section

1 da„ —

d

3

a„ /dft dft. dk

7 2 II p 9

k

NOTE: Only in Fig. 5(d), pertaining to the cross section da

]t with k < k , do the minimum and second maximum remain

o as <f) increases.

72
Fig. 6(a) The positions, z = (v-u)/u, of the minimum and the second

maximum of the cross section da^3 ^2 d^/dfip dfi^dk^

as a function of the azimuthal angle, <p, for k < k .

Here e

] 3 140 MeV, k = 95 MeV, and 9] =1°.

6(b) The same as in 6(a), except that k > k . (We choose

k = 130 MeV).

6(c) The same as in 6(b), except that we consider the cross section

J2

da

ll (

= ^2

d a

|| /dfi p

2

d

^k

dk

)

*

Fig. 7(a) The cross sections do /V

dS^dkl

c-j = 140 MeV, k = 95 MeV, 6

] = 1°, and <p = 0°.

7(b) The same as in 7(a), except that cf> = 0.2°.

7(c) The same as in 7(a), except that <$> = 0.4°.

7(d) The same as in 7(a), except that (p = 0.6°.

Note that following the individual figures 7(a) - 7(d) we present

these four figures on a single page for easy comparison.

-U- do, ( = -U- d

2

a. /dft. Ti

\ V1

-U- d

2

cr/dfL dfi, dk) and

Z

2 P2 k / as functions of 0

2

, for

Fig. 8(a) The cross sections dCj(= d^/dfi dfi^dkj and

1 / 1 3

2 V2 2 /

~2 dajj d ajj/dfip dft^dkj as functions of ©

2

, for k > k

Q

.

Here e-j = 140 MeV, k = 130 MeV, 0-j = 1°, and cp = 0°.

8(b) The same as in 8(a), except that cj> = 0.2°.

8(c) The same as in 8(a), except that <j> = 0.4°.

73
Fig. 8(d) The same as in 8(a), except that <j> = Q.6 9

.

Note that following the individual figures 8(a) - 8(d) we present

these four figures on a single page for easy comparison.

/dfi or k < k .

Here e

] = 140 MeV, k = 95 MeV, 9

] = 1°, and 4

= 0°.

9(b) The same as in 9(a), except that <p = 0.2°.

9(c) The same as in 9(a), except that d> = 0.4°.

9(d) The same as in 9(a), except that 4

= 0.6°.

Note that following the individual figures 9(a) - 9(d) we present

these four figures on a single page for easy comparison.

NOTE: As illustrated also in fig. 5(d) for k < k , the minimum

Fig. 9(a) The cross sections

o

and second maximum for da,, remain as increases.

74
—

p1

X

FIG. i

75
UNPOLARIZED

CROSS

SECTION

O

o

C^«0 d3JSNVyx WH1N3W0W

T f ? T

o o o o

?

o

CO

<0

CM

CD CD

CO

o

CO

76

(DEGREES)
CROSS

SECTION

>

PERPENDICULAR

POLARIZATION

m ywai WH1N3W0W

7 T T T t

o o o o o o o

CO

CO

r\

<n

tu

(JL1 QC

CD

UJ

a

CO

CO

o

CO

G>

CM

77

FIG.

2(b)
CROSS

SECTION

*

PARALLEL

POLARIZATION

cow) aajsNvai wniNawow

T 1 T T t

O O O O Q o o

CM — O I 1 1 J O O O Q O O O

"-DP,Z/l = CC,=""5/,C-,a 5/^5 NOI103S ssoao

78

a

<DEGREES>
POLARIZATION

i

o o o o o o o

I 1 I

o

00

A

CD

UJ

UJ

QL

CD

UJ

aV-'

CD

<D

<M

O

<!bp ^TbpO/C'V -Ibpj .NOIlVZiaVIOd

79

C

D

)8

*9IJ
POLARIZATION

1 I I Chp^hp -) / c

8-op-!bp 5 NOI1VZIdVlOd

SO

FIG.

3CkO
POLARIZATION

^ <0

UJ

CO Q£

CD

LU

a

CD

CM

CO

o

CO

CO

CM

cip+Spj/ctp-'ip? NOXlVZiaVIOd

81

FIG.

3(c)
cV+^P^/c'V-^P) NOIlVZiaVIOd

3.095

3.100

3.105

3.110

3.115

3.120

3.125

a

(DEGREES)
LO

CO

IU

Ui

QcL

CD

UiO

CO

CM

O

r\

D

CO

CDH

U_
POLARIZATION

LD

r\

CDU

UJ

Q' CD

UJ

o

*4

CP

co

<SJ

ID

O

Q

o

LD

O

I

O

I

C‘!op+Ibp } / <‘!op-Tbp } N0I1VZI3VlOd

84

FIG.

3Cf

)
POLARIZATION

O l0 Q 10 O —

• • • • • — o o o —

I I C'bp+Ibp } / cV-^bp 5 N0I1VZIdVlOd

35

FIG.

3Cq>
POLARIZATION

10

CM

r\

LU

Ld

O'

CD

'f LlI Q

OJ w

_ c\J o

r>- CM

CM

CM

CM

CM

O J

O

I i

C«op+!bp5/c, top-Tbp5 • NOIlVZIdVlOd

r\

-C;

( CO

CDH

U.

36
CROSS

SECTION

FOR

PERPENDICULAR

POLARIZATION

/\o XAX

O

r\

-0

'TOM

Ll.

37
CROSS

SECTION

FOR

PERPENDICULAR

POLARIZATION

r\

X

V

X

v

O

C\J - -

T-OPs Z/l ; NOIIC3S ssoao

r\

0

*

0

H

L.

88

{DEGREES}
CROSS

SECTION

FOR

PARALLEL

POLARIZATION

r\

(J) Id

Id

QL0

Id

a

<L

!loPs Z/l J NOI133S ssoao

89

FIG.

4Cc)
CROSS

SECTION

FOR

PARALLEL

POLARIZATION

o

CM

CO

r\

<f> UJ

UJ

QC

CD

UJ

a

CD

CD

CM

T^PZ Z/\ -• CC^DAC-1 *!)/^)

o

NQI133S

o

ssoao

90
CROSS

SECTION

W

OR

PERPENDICULAR

POLARIZATION

<

U

<

R.

>

co

UJ

LUsg

aT

<0M

U.

A

9

T-apt Z/l« N0I1039 SStttJO

COUJ

UJ

<fc o

UJ e a

91
EXTREMA

FOR

PERPENDICULAR

POLARIZATION

C

k

<

k

0

>

I

(J)

(DEGREES)

0 0 (0 £ £ «*» - - C

X C - 0-0

£ £ a

M- U

0 0 0 0

<0 » » X

C C C 0

0 0 0 —

— - - ^ -fa> c

1 1 s

o u?

*<r c> o

(d0

a.

Post Poet of

i

B I

uT

1

af 0 / a /

9 r9 /9 /

9 L/ 9 1

9 19 i

1/ 9 1• IV

/

9 /

f /

»A*.

1 V. « ~

\

v \

°

\

^ 0 ^

S' /0 19

%

L

0 0**

%\%\%\

92

0.04

-

0.02

0.0

0.02

0.04
EXTREMA

FOR

PERPENDICULAR

POLARIZATION Ck>k0 I| I

<MOOo

CM01 01

<o01

CO01 oT

N

r\

jO

LO

CDH

L93

=(v-u)/ u
EXTREMA

FOR

PERPENDICULAR

POLARIZATION Ck>k0 I

94 -

0.2-

0.1

0.0

0.1

0.2 2=

(v~u)/ U

FIG.

5Cc)
EXTREMA

FOR

PARALLEL

POLARIZATION £ £ ->

_ 3 X x O• J O I>

V-/

IIN

CMOOo o

CMO01 o

95

FIG.

5CcD
EXTREMA

FOR

PARALLEL

POLARIZATION Ck>

K>^

CM s\ 6 ?>

vx

I!N

Oo

CMo01

CO01

CO01 oT

FIG.

5C«3
EXTREMA

FOR

PARALLEL

POLARIZATION

r\*AX

v-/ 0 0# s s -** - - c X C - 0-0 £ £ a

V> <t- ^ c 0 0 0 0 # « # X C C C 0 0 0 0 — — - - <*-

-*» -u

C > >

2: z: o o

'T CO n n hi X

II d

C\J o D\

/-\DI>

IIN

r\

CO

Ld

LUX

LjJ Qv-/ “ • 0 0 0

a. a.

®0 v

a. o »1i ooo

97

FIG.

5Cf)
FUSION

OF

MINIMUM

AND

2nd

MAXIMUM

OF

CROSS

SECTION

FOR

PERPENDICULAR

POLARIZATION k<

k„

LO

CO

CO cn

LU

LUa

CD

bJav

<o

COo

CO

CO

CO• o

CM

CO

CO• ood

98

FIG.

6(a)
FUSION

OF

MINIMUM

AND

2nd

MAXIMUM

OF

CROSS

SECTION

FOR

PERPENDICULAR

POLARIZATION

AND

k

>

k.

<o LO 'T CO Oi

o O o o o

o’ d d o* o*

QCi

FIG.

ecb)
FUSION

OF

MINIMUM

AND

2nd

MAXIMUM

OF

CROSS

SECTION

FOR

PARALLEL

POLARIZATION

AND

k

>

(0 u> T CO CM

o o O o o

d d o

* o o

FIG.

6Cc)
UNPOLARIZED

AND

PERPENDICULARLY

POLARIZED

CROSS

SECTIONS

-op,Z/t Nonces ssoao

101

FIG.

7Ca)
UNPOLARIZED

AND

PERPENDICULARLY

POLARIZED

CROSS

SECTIONS

O

-OP, Z/ \ CC,OUJ 5/a C^3D/c?^ N0I103S SS030

FIG

.

7CbD
UNPOLARIZED

AND

PERPENDICULARLY

POLARIZED

CROSS

SECTIONS

o

CM CM *— <—

Z/\ ! N0I103S SSOdO

103

FIG.

7<c>
UNPOLARIZED

AND

PERPENDICULARLY

POLARIZED

CROSS

SECTIONS

o

04 <\J

r\

o

'w'

CDH

Li.

-ap, Z/\ •• CC^D/aC-!*}/^ N0I103S ssoao

1 04

CDEGREES)
UNPOLARIZED

AND

PERPENDICULARLY

POLARIZED

CROSS

SECTIONS

O o

N0CLL33S SSffiC

0

M0X1238 SS0M3

o

105
CcO

-

CROSS

SECTIONS

FOR

PERPENDICULAR

AND

PARALLEL

POLARIZATION

O U) O U3 O

CM - - O O

o o o o o

o o o o o

-®P,Z/\ 1 N0IJL33S SSO^O

o

CVI

<0

UJ

LlJ oc

CD

Ui

CO v

<0

M*

CM

O

1 OG

i

FIG.

8Ca>
<b>

-

CROSS

SECTIONS

FOR

PERPENDICULAR

AND

PARALLEL

POLARIZATION

• • • • • o o o o o

-op,Z/l • CC^^/iC-^/cD N0I133S SSO^O

O

CM

CO

r\

CO

Ui

UJ

O'

CD

Ui

a

CD

CO

CM

O

1 07

FIG.

8CfcO
Co>

-

CROSS

SECTIONS

FOR

PERPENDICULAR

AND

PARALLEL

POLARIZATION

CO

r\

CO

Li

Li

O'

o

Li

a

CM

CD

(O

CM

N0I103S SSOSO

1 08

0

00

C3H

L
CcDCROSS

SECTIONS

FOR

PERPENDICULAR

AND

PARALLEL

POLARIZATION

ON

00

r\

CO

Li

Li

O'

CD

Liav-/

<0*

CVi a

109

FIG.

8CcD
0

O

N

O

0

2

(DEGREES)

0-

(DEGREES)
C

Q

5

-

CROSS

SECTIONS

FOR

PERPENDICULAR

AND

PARALLEL

POLARIZATION

O

^P,Z/l * CC^^O/.C^/q} N0I133S SS033

1 1

1

FIG.

QCcO
CkO

-

CROSS

SECTIONS

FOR

PERPENDICULAR

AND

PARALLEL

POLARIZATION

I I 1 I I 1 .I 1 1 1... 1—1 I I I i 1 l.l 1 1 1 I 1 1. 1 l lil 1 I 1 L 1 1 I 1 1 U U O U) O U) O <M

ca - —

-°P,Z/ \ N0I103S SSO^fO

FIG.

9<b!>
Co)

-

CROSS

SECTIONS

FOR

PERPENDICULAR

AND

PARALLEL

POLARIZATION

CO

UJ

-ap, Z/l * NOIi.C3S ssoao

. 113

FIG.

9Cc>
Cd>

-

CROSS

SECTIONS

FOR

PERPENDICULAR

AND

PARALLEL

POLARIZATION

o

CM - —

*pz z/\ s N0H33S ssoao

FIG.

9Cd)
<o>

-

CROSS

SECTIONS

FOR

PERPENDICULAR

Atto

PARALLEL

POLARIZATION

<b>

-

CROSS

SECTIONS

FOR

PERPENDICULAR

AND

PARALLEL

POLARIZATION

O

-ow

9

N0X133S SSCRO ^•ZS » • N0T133S sstao

9v

9

9M

-»O.Zy 1 ' CC.'»«3/',C-»T/<0 M0UL338 SSOM3 >*••5

1 1 5
NBS-114A (REV. 2-401 U.3. DEPT. OF COMM.

BIBLIOGRAPHIC DATA

SHEET (See instructions)

1. PUBLICATION OR

REPORT NO.

82-2454

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

4. TITLE AND SUBTITLE

POLARIZED TAGGED PHOTONS

An analysis of the differential cross section for polarized

bremsstrahlung in the range of interest for a tagged photon system

5. AUTHOR(S)

Leonard C. Maximon,. Eric Ganz, Thierry Aniel, and Arlette de Miniac

S. 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.

S. Type of Report & Period Covered

9.

SPONSORING ORGANIZATION NAME AND COMPLETE AODRE5S (Street. City. Stote, ZIP

)

10.

SUPPLEMENTARY NOTES

| | 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 si gnificant

bibliography or literature survey, mention it here)

We consider in detail the differential cross section for polarized bremsstrahlung

for angles and energies in the range of interest for a tagging system and derive a high energy, small angle approximation for this cross section. For photons polarized

perpendicular and parallel to the production plane these are given by eqs (1.2) and

Cl. 3). We use these approximations to determine the maxima and minima of the cross sections for these two polarization states, daj_ and daj| , and to evaluate these

cross sections at the extrema. It is shown that both daj_ and dan have a very

sharp dip in the region of small momentum transfers. However, their behavior in the

region of the dip, as a function of the azimuthal angle cp, is quite different over

most of the photon spectrum (condition (VI.33)). The cross section dai behaves

similarly to the cross section for unpolarized photons in that as <f> increases, the

sharp dip vanishes, the minimum fuses with the second maximum, and the cross section

then has only a single maximum. In contrast, the shart dip in the cross section da., remains as <p increases, provided condition (VI.33) is satisfied. This results in

rather large polarizations in the region of the dip as shown in figs. 3(a)-3(h).

Coulomb corrections to the Born approximation are considered, and do not fill in

these dips.

12. KEY WORDS (Six to twelve entries; alphabetical order; capitalize only proper names; and separate key words by semicolons)

Bethe-Heitler cross section; bremsstrahlung monochromator; photonuclear research;

polarized bremsstrahlung differential cross section; polarized photon beams; tagged

photon method.

13. AVAILABILITY 14. NO. OF

PRINTED PAGES , Y^l Unlimited

| | For Official Distribution. Do Not Release to NTIS 127

! 1 Order From Superintendent of Documents, U.S. Government Printing Office, Washington, D.C.

15. Price 20402.

j Order From National Technical Information Service (NTIS), Springfield, VA. 22161 13.50

USCOMM-OC 9043-P8C

j

\IIil\i