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PB 161582

^ecUnlcai rlote 81

AN EVALUATION OF KACSER'S SECOND

ORDER BORN APPROXIMATION TO THE

BREMSSTRAHLUNG DIFFERENTIAL

CROSS SECTION

U. S. DEPARTMENT OF COMMERCE

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NATIONAL BUREAU OF STANDARDS

technical Ntete

JUNE 1961

AN EVALUATION OF KACSER'S SECOND

ORDER BORN APPROXIMATION TO THE

BREMSSTRAHLUNG DIFFERENTIAL

CROSS SECTION

George S. Ofelt

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Price $0.75

An Evaluation of Kacser's Second Order Born Approximation

to the Bremsstrahlung Differential Cross Section

George S. Ofelt

The second order term, as derived by C. Kacser

[Proc. Roy. Soc. A253 (1959)]> of the Born approximation

series for the bremsstrahlung cross section differential

with respect to photon energy, photon direction, and final

electron direction averaged over initial and summed over

final polarization states has been put in a form that admits

to numerical evaluation for the coplanar case. The results

are valid for relativistic as well as non-relativistic incident

electrons.

Completely general (non- coplanar) expressions are included for the first order (Bethe-Heitler) and second order

terms. However, the latter has not been given in a form

admissible to numerical evaluation.

Tabulated values are given for the case of the incident

electron kinetic energy = 500 kev, photon energy = 450 kev,

and photon direction 20° from the incident electron direction.

For this case the second order term increases the differential

cross section by about 50 percent for Al and about 200 percent

for Au for the various angles of electron emission. A rough

approximation to the integrated (over final electron direction)

cross section is included and a comparison is made with

experimental values. The inclusion of the second order

term brings the theoretical cross section closer to the

experimental values.

1. Introduction

Measurements of the bremsstrahlung cross section, differential

with respect to photon energy and direction in the region where the

incident electron's kinetic energy is of the order of the electron rest

energy, show the predictions of the first order Born approximation

theory to be too low by a considerable factor L1,2J . This discrepancy

increases with increasing atomic number of the scattering material and

also with increase of photon energy for a given incident electron energy.

The inclusion of screening effects does not remove the discrepancy.

At extreme relativistic incident electron energies the effect is

reversed- -Born approximation theory overestimates the experimental

1
values. Therefore, the second order term of the Born approximation

series should be evaluated.

The Born approximation is assumed to be valid for the following

simultaneous conditions on the initial, 3 » and final, $, electron

velocity ratio to the velocity of light: —*—- < < 1 and —n < <• 1

which are not satisfied for high values of Z. However, the first order

Born theory gives at least order of magnitude estimates of the cross

section for high values of Z. Therefore a second order calculation might

be expected to give reasonable results.

Kacser [3] has calculated the second order term of the Born

approximation series for the differential cross section with no screening

corrections which is averaged over initial and summed over final

polarization states.

The purpose of this paper is to give the second order term, as derived by Kacser, in a form that is readily calculable. In particular

only the coplanar case- -final electron momentum vector lying in the

plane defined by the initial electron momentum and the photon momentum

vectors-is considered.

The cross section obtained is differential with respect to photon

energy, photon direction, and final electron direction. Therefore an

integration over the final electron direction's solid angle would have to

be performed in order to compare the contribution of this second order

term with existing experimental results. However, this cannot be done

easily since only the coplanar case is reduced to a reasonable form.

A rough indication of the magnitude of this is given in section 8 (for Jthe

tabulated case of section 7).

2. Symbols, Constants, and Energy- Momentum Relations

The system of units and symbols used follow those of Koch and

Motz [2] with new symbols being introduced as needed.

2

T = incident electron kinetic energy in mc units.

E = incident electron total energy in mc units.

E = final electron total energy in mc units.
2 k = photon energy in mc units.

-_^ P = incident electron momentum in mc units.

_Sk o P = final electron momentum in mc units.

k = photon momentum in mc units.

"T = P -"k o q = T- p momentum transfer to nucleus.

9 = angle between P and k in general case.

cp = angle between " and " in coplanar case.

9 = angle between P and " in general case.

cp = angle between " and " in coplanar case.

= angle between the planes (P ,k) and (P,k). x = <p -<p • dQi = element of solid angle in direction of k.

dQ = element of solid angle in direction of P.

-13

r = 2.82 x 10 cm classical electron radius.

°

1

ex = . y- fine structure constant.

Z = atomic number of target material.

2 m c = . 511 Mev electron rest energy,

oE = T +1.

o o E = E -k.

° 2 1/2

P =(E2-1) 1/2

.

3 = 2. initial electron velocity's ratio to velocity of light. op

3 = -=- final electron velocity's ratio to velocity of light.

Symbols Used Only in Section 5-

A

n = unit time like 4 vector (0, 0, 0, 1).

(1 ll=u=1, 2, 3.

[-1 |j,= u=4.

P =(P , E ) o

x o o P = ith component of P • o .. r o (Pv P)= (P »P-E E) scalar product.

o ' v_^o ^ o ' r (PP) = (P .P+E E).

(P n) = -E . v o

' o
Equation No.

3.2

5.1

5.2

5.3

5.4

5.5, 5.6

5.7, 5.8, 5.9

5.10

5.11

5.12

5.13

5. 15

5.17, 5.18

5.23

5.29

5.33-5.35

5.37-5.39

3. General Formulation for the Differential Cross Section Formula

The Born approximation development of the differential cross

section in a power series of aZ can be expressed by the following:

da = Ci|r 1 +Zt 2 + • • • H. T. ]Z 2 ^dQ^Qp (3.1)

where the \|/'s are defined to be independent of Z. Therefore the i|r's are

functions of E , k, ,9, and

o o

dCTi *

x = (3.2)

z2 fdf5k

d"p

where do is the Bethe-Heitler cross section (first order) term, then \|f_

is second order in relation to \|f 1 and is the quantity to be calculated in

this paper.

Symbols Defined in Text

l

*2

X

T

ij

J.. T

ij i

l • T

2 j

1

., ^ j

3

. R

—

S

v X

Y

COj, ' a> 2

, ( o 3

, A.., B..

a, b , c, d

w, x, y, z; w, x", y» z; c,, C2' C

3

c

1

, c

2

, c

3
4. The First Order Term (Bethe-Heitler)

A. General Case

This term is given in the same form as in reference (2).

(2tt) 2 P q

4 ME-Pcos6) Z

2 2 P sin 9 , o o //f T-2 2. + r (4E -q ) (Ex o-P

o cos9

o'p

(E-Pcos9)(E x ' x -P cos9 ) o o o'

[k

2 (P2 sin2 9+P 2 sin2

6 ) L o o'

- PPo sin9sine

o ((4EEo-q

2 )+2k

2

)cos j

(4.1)

where

q

2 ={p 2 +P2 +k2 -2P kcos9 +2Pkcos9 ^ L o o o

- 2P P(cos 6 cos9+sin6 sin9cos0)f (4.2)

o

x o o 'J

x

' k is taken as the polar axis and the angle is determined by which side

of k the projection of P on the P , k plane lies, -y > —0 if the proJjection of P lies on the same side of k as P and tt 20 > — if on the

o

**. 2

opposite side; (tt >: 6 >0, n > 9 ^0).

B. The Coplanar Case, P* • (P x k)=0

In this case is either zero or tt. We define the angles cp and cp in the photon emission plane as the angles between P and k, and P and

k respectively (2n^cp ^0, 2TT^cpiO). Therefore equation 4. 1 becomes:

,2 . 2

. a 2 P f P sin cp IA „2 2 X *1=

? r Q 4 I 2" (4Eo" q ) (2tt) 2 ° P q

4 L (E-Pcoscp) Z

P

2

sin

2°cp o

T +

o ,a-^2 2 X r (4E -q ) (E -P coscp ) 2

r, 2,-2 . 2 „2 2 , k (P sin cp+P sin cp )

ro'

2

(E-Pcoscp)(E -P coscp )

- Po Psincp

o sincp((4EoE-q2

)+2k

2

) ]} (4.3)
and 4. 2 becomes:

2 ( £ 2 2

"1 q^ = JP +P +k -2P kcoscp +2Pkcosm-2P PcosXf

l o o r o ^ o J (4.4)

5. The Second Order Term

A. General Case

In the units used here Kacser's second order term* is given by

(5.1)

2 P P ' l 2TT °P q

2 o o^

where i|f , is the first order quantity defined in section 4 and Xis defined

x =

2 ii 1 -*• —v 1 -^ -* £^T J J..+ —— R • X — S • Y

" n

3 1J (Pk) (P k)

(5.2)

.ij

In order to facilitate calculations in the coplanar case T

J and J., are

defined in several parts

T^T^+T^1 12 3

J..= J..+J .+J..

!J !J !J 1J

(5.3)

(5.4)

where,

T^g^{(P^ )[l^-^)]-(nk) [^ + _L]} (5.5)

(Pk) (P

Q

k)

T^ = PV ( Pn) +pV ( Pon) . pipJ ( nPo)

(Pk) (P

Q

k)

(Pk) (P k) (Pk)

+PipJ (Pn) P^ r (nPp) (Pn) "j (5.6)

TT

1J 4(P

oxP)'

«PoXP)xP

o

). (P T + TP ) L iC

,

' =s v,

+1AyJ ^ t(P t+P • T)

o o

(P XP)'(P x"T) x P (T Z 6..-T.T.)(P T .+TPn )(P T. + Tp )

x o '

x o ' /^ ov 1J 1 j O 1 °j O J °

\ \

P ^ T

3 (P T+P ."T) T^(P T+p .^) 2 ^

+ { with Po^ P} - { T% q] - {P

o

^, p ; t% q} (5.7)

*c f. the equations following equation 50 of reference (3).

6
J

2.= .

TT"

1J 4(P xP)i[{

(t*« P xP) 2 (t 2

6 ..- t.t.) .

v o ' v ij 1 j

' j

2T 3 (PT+p.^)(p T+P ."T) x /x o o '

+ {P

o ^ P} - {"^q} - [P ^ P;^-> q} (5.8)

J

3

. = -

rr _ , -(P PT + P P« T + PP • T+ TP »P)

f * o o o o ' 4(P

q

xP)2 L L 2(P

oP+Po *P) 2 (PT + P*^f)2 (Tp +f. p ) 2

[i- (P

o P + Po .P)("x((PxP)x T))i + (tx((P xP)x(P P+PPo ))) i

[i-» j] }+ {^V-rP}- {Ht}-{P VP)t^q (5.9)

R = P q

2(P

oq+P

Q -q) P

Q

s = - |J_tx(Px^)

2(Pxf)2

T

qx(Pxq)

j

(5.10)

(5.11)

_> ^r(Po

n) (kn) (P

o P)(P

o

n)

x=p Ltptt

+ fp-ry

+ 2(p

on) + "tptej—

_^ r (P n) (kn) (P

oP)(P

Q

n) (kP)(kn)

+poLtp^t + rp^y +(po

n) + -TP-^

v o'

+ 'k

(P

o

n) (Pn)

L WW ' 2 7P~kT + (p

o

n) "

•(P

QP)(Pn)

Y = X | P <-» P kM «-> -k;,

\i o (j, M- p,

(5.12)

(5.13)

B. The Coplanar Case, P »(Pxk) = The notation used here is the same as that for the coplanar

case of the first order term (sec. 4-B).

It can be shown that T^J-Uo

1 ij

J

2.=

(T*j + T^J3

. =
Therefore equation 5.2 becomes

it 3 c 1] (Pk) (P k) J

Q

For convenience the following are defined

(i)

2

c

3 = -

(Pk)

1

3 2 ij — R • X

S* Y

(Pk) v o ' then

X = (JUj+CU^+ODo

i. Formulae for uj 1

(5.14)

(5.15)

(5.16)

Two new terms A., and B.. are defined from equation 5.7;

1 rr 1 _^ _^ -| (tP .+ t.p ) A

ij = ~

3^ Li [(P xP)xP

o j i °J J _,° + i^ j}

1J (P XP)2 Ll P

L °

° J1 T(P T +p -t)

+ 1 P

Q

*-r p} - { t!» q} - 1p

o^

p; t -» q] (5.17)

B..=

r. r r (T^6 ..- T.T.)

[{(P

oXP)'(P

oXT)[

(P t.+ tP .)(P t.+ tP .) I 7

x O 1 Ql /X O J OJ7

j J

1J (PnxP) 2 T (

x P T+ P • T ) o o '

P T (P T + P .^T) 2 o

x o o ' £?„*-* P}- ItW q}-lP ^y P;

o q) (5.18)

Then cu, can be expanded, noticing that A., and B.. are symmetric

where as TJ is not.

w

l = " 4[T

2

1

< A11 +B 11>+< T

22+T

2

1

>< A

12 +B 12>

+ Tf(A22+B

22 )

Equation 5. 6 becomes after some algebra

(5.19)
EP coscp E Pcoscp

11 f O To O

E- Pcoscp E -P cosm

(P coscp +Pcoscp) f EP coscp v o

T o \ ° ° K E-^coscpo

E Pcoscp 1 -2 _;

E-Pcoscp

(5.20)

T»+T*

, p EP sincp E Psincp

E-Pcoscp E -P coscp T o o o (P coscp +Pcoscp) /-EP sincp v o

T o

T/

J o

To

+

{

E Psincp

K E -P coscp E-Pcoscp

o o

T o

T

(P sincp +Psincp) rEP coscp

K

{JL.X- LUS>CD o ro E Pcoscp , o T 1

22. [

' Po

sinIPo+Psln[P ) r 2

= L K

l

E -P coscp E-Pcoscp

o o

T o

T

EP sincp o

T o E Psincp

o

T E -P coscp E-Pcoscp

o o

T o r

M (5.21)

(5.22)

In order to simplify the equations for the A..'s the following are

J defined

a = [P t(t+P -Kcoscp )]"

r> v n To' -1

And

b= [Pt(t+P cosx -Kcoscp)]

-1

c = CPoq(q+P

o-Kcoscp

o-PcosX )]

-1

d= [Pq(q-P+P cosx -Kcoscp)]

-1

t 2

= (P

2 +K2 -2P Kcoscp )

* o o

T o'

q

2 = (T2 +P2 +2PKcos9-2P PcosX)

(5.23)

(5.24)

(5.25)

where equation 5.25 is the same as 4.4.

Then equation 5. 17 becomes after some algebra
(coscpo

coscp-sin<p o

sincp) j j^

2 f „

Ksin*Po v Ksin<p o

+ sm cp jc(l+ )-a

Psinx Psinx }

-sin2

cp{(d-b)(l + ^x )}]

12= 2

[

(5.26)

(coscp sincp +coscp sincp) * t P Po

- sincp coscp -j c( 1 +

+ sincp coscp -j (d-b)(l +

Ksincp

Psinx

Ksincp

->-a

J- Psinx

P

Q

s inX -»}] (5.27)

The equation for A-- need not be given as it combines with B22 m a

rather simple manner

<a

2 2+B22>-[- <aii +b ii )

{« 1 +

Ksincp

Psinx

)-a

Ksincp

Psinx

}-{(d-b)( 1 + Ksincp

P

Q

sinX

>}] (5.28)

In order to simplify the equations for the B.. 's the following are

defined:

w = (Px +t )coscp -K o ' Yo x = P coscp +t coscp -K

y = (P

Q

+q)coscp

o-Pcoscp-K

z = P coscp -(P-q)coscp-K

w= (P

o

+T)sincpo x = P s incp + t s incp

7"= (P

Q

+q)sincp o-Psincp

z"= P

q

s incp Q-(P-q) sincp

2 sin^f

O

P ^ sin^cpYo

10
2 (P sirup -Psinqj)

M= -2 °q^

P sincp (P coscp -K)

rr o ^ov o Y o

' L 2

T

(P sincp -Psincp)(P coscp -Pcoscp-K)

M= — - —- - (5.29)

q

2

By use of equation 5.28, only the B, , and B, ? terms need be given.

Ksin9o 2

11

=

[_ ~ l

Poaw " L

Psinx

Ksincp

- b (

1+ ——){pbx2- L

} P S "VX

^ Psinx ^

L °

'

Ksincp l

f 2

._, +d( 1+ )l pdz " M f | (5.30) V Po sinx ^ J

Ksincp Q

. .

. , -, = a i P aww+L [ 12 L _ . L o J Psmx

Ksincp.,

_b ( 1+ )] Pbxx+L r V P sin/ L

' Ksincp j- . -cfl + J|p cyy+M}

Psinx

^ Ksincp .

f

. dfv1++ )[Pdzz+M|j (5.31) J P sinx

ii. Formulae for co ?

11
U)2

= - 2 (E-Pcoscp) ^ K

{ — [(P +q)cosx-P-Kcoscp]c

i

+ —— (P +q)-Pcosx-Kcoscp C

+ [(Pb

+q)coscp o-Pcoscp-K]c

3 }

where

= {-«. [

3+-

(E E-P Pcosx )

+ K +

K(E -P coscp ) K(E-Pcoscp)

1 (Ex -P coscp ) o o yo'

}

C^Ct+E CIO+E+K

C

3 = {Eo[

(E-Pcoscp)

(Ert -Prt coscpo

) o o

K(E-Pcoscp)

..].«[

2-(E

oE-PoPcosx ) 1

K(E -P coscp ) }

The c in equation 5.32 is the c defined in equations 5.23.

iii. Formulae for uu

(5.32)

(5.33)

(5.34)

(5.35)

- 1 IP Ksincp P

(Do = - 2 K(E -P coscp )

* o o Yo c

sinx

t q

v

(Psinx + Ksincp )

2 u

t q

1 1 PoKsincp

o K

+ —- sincp + ( )C - T q

where

:».[-«{3 +

(P sinx + Ksincp)

1 (E E-P Pcosx )

} (5.36)

K(E -P coscp ) K(E-Pcoscp)

}- K

+ (E-Pcoscp)

(5.37)

12
-> t K(E -P cosm ) C^C^E +E- ° ° ^-

(5.38)

o

(5.39)

(E-Pcoscp)

3 r r

1

^

" f

2 -(e

oe-p Pcosx) r C = -E \l + +E 2 L L L K(E -P coscpQ

) J K(E-Pcoscp)

6. Discussion of the Second Order Coplanar Equation

It should be noted that there occur several points in the equations

for the second order term that are of the form -*r (for example, when

sin x= in the equation for aw (5. 19) and when P sinx+k sincp = in the

equation for ^o (5.36) ). However, the value of i|r_ is finite at each of

these points.

The calculation of the limiting values is rather involved and will

not be given here.

13
Tabulated Values for a Specific Case

We consider the following case where there is a considerable

difference between the first order theory and experimental values.

Thus the second order term would be expected to be significant.

T = o 500 kev

k = 450 kev

^o

= 20

TABLE I

MS mb * mb kdaAl

Z

2 dkdQ

p

dQ k

kd a Au

9 ?1

ster/ster

^2

ster/ster Z

2

dkdfi

p

dQk

80° .050 .0016 .071 . 180

60° .125 .0043 .178 .449

40° .280 .0090 .397 .991

25° .357 .0115 .506 1.26

15° .355 .0114 .503 1.26

0° .295 .0095 .419 1.05

-20° .195 .0062 .276 .687

-60° .075 . 0025 .108 .273

Values for i|r. and i|L are plotted in Figure 1

14
8. Summary

The contribution to the differential cross section by the second

order term for the case calculated in section 7 leads to an increase of

about 50% for Al and about 200% for Au as would be expected as discussed in section 1. However, these values cannot be compared with

existing experimental data since these data are for the integrated (over

finial electron direction) cross sections.

Since the addition from the second order term (at least for the

above case) is so large, it appears that the third or even higher order

terms should be calculated or estimated as to magnitude and algebraic

sign. The case might be that the first order term of the series gives a

closer approximation to the limiting value of the series than the sum of

the first several terms. Or, conversely, the third and higher order

terms might make a negligible contribution.

Therefore, since at the present time a third order term has not

been calculated and no experimental data for the cross section

differential with respect to both final electron and photon directions

are available, it cannot be said exactly what the limitations on this

second order contribution are. Although, as pointed out above, the values obtained in section 7

cannot be compared to experimental data, a rough approximation to the

integrated cross- section for this case only can be made. It is noticed

that even in the non-coplanar case | - is the sum of two parts (see

equation 5. 1), one involving X (dependent on the final electron direction),

the other, a multiplicative factor of \jr , , independent of angles i.e.

E E

it a (-=- - —?

) which is always positive. In the case tabulated the former

ir ±0

is approximately 1/10 of the latter for all final electron angles. Therefore if it is assumed that the relationship between the two does not

change much as the plane containing P is rotated about P then the

integral of i|r _ over final electron direction

15
o r E E „ M'? dn r»=

l TTa < ) ^ 1 ^ t, +

«J £ p L p p Jip

O

a

2

2 Pk2 + r 2n ° P ,2 P

(8.1)

is the sum of two positive quantities that are approximately in the

ratio of 10 to 1. _ _, „

p p

Therefore Nr_dfi >na(

Then from equation 3. 1 Jdadfl = Z

2 •^dQ k J(t 1 +Zt 2 )dQ '

Substitution of 8. 2 into 8. 3 gives

'dado >{z2 dk

k"

dn k f 1 dQ }{l+iraZ(

E

(8.2)

(8.3)

(8.4)

Let

da (1st) = z

2 ^dn k

J t^

da (1st + 2d) = Z

2 ^IdQ.

k ( i + Z^JdQ.

da = the experimental value for the integrated cross section.

Then 8.4 becomes upon substitution of numerical values

da(lst + 2d) >1.029 Z da(lst) (8.5)

In reference (1) experimental values are given for T = 500 kev,

k = 450 kev; = 0°, 30°, 60°, and 90° for Z = 13 and 79, with and

o estimated error within 20%. Interpolation for =20° gives

approximately:

16
TABLE II

Z = 13 Z = 79

k da

exp

Z

2dkdQ k

2. ± 0. 4 mb

ster

3. 2 ± 0. 6 mb

ster

da 4. ± 0.8 6.4 ± 1.3

da (1st)

da(lst + 2d)

da(lst)

>1.38 > 3.29

da

exp

da(lst + 2d)

<2.9 ± 0. 6 <1.9 ± 0.4

therefore, even with this crude approximation of the integrated cross

section, the second order term brings the Born approximation into

better agreement with existing experimental data, (at least for the

particular case as tabulated in section 7).

17
References

(1). J. W. Motz, Phys. Rev. _K)°_» Pg» 156 ° (1955).

(2). H. W. Koch and J. W. Motz, Rev. M. Physics, _3_1> Pg. 920 (1959).

(3). C. Kacser, Proc. Roy. Soc. , A253, pg. 103 (1959).

18
T = 500 Kev

k = .9T

Fig. 1. Variation of Y and Y with angle, cp, from the data of TABLE I.

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