The Scattering of Fast β-Particles by Electrons

28

THE SCATTERING OF FAST ,a-PARTICLES BY ELECTRONS

ByK. C. KAR

AND

MRS. C. BASU

AIJBTRACT. In Sec. A a general relativistic treatment is given of the problem of collision of two particles of unequal masses, one being initially at rest; and general expressions connec- ting energies, momenta, angles of deflection •... etc., ill C- and L- systems are derived. In Sec. n the relativistic wave statistical formula for the intensity of electron-electron scattering is derived. It is shown to be in decidedly better agreement than Moller's with the experiment of Champion. In Sec. C a general relativistic theory of scattering for unequal masses has been developed.

In the present paper we propose to develop the relativistic wave-statistical theory of scattering of fast ,a-particles by electrons ('Vide Sec. B) and also of a particle of mass m 1 by a second particle of mass 1n2, under coulomb field ('Vide Sec. C). Before going into the problem of scattering it appears useful to give ('VIde Sec. A) a systematic theory of relativistic collision of two particles of unequal masses and the relations between the centre of gravity (C·) and laboratory (1,-) systems. As far as we are aware, a systematic treatment of these problems has not been given before. Only a few results which were found useful have been derived so far (Champion 1932, Bethe 1932 and Wolfe 1931).

Sec. A: Theory of relativistic collision

Suppose that the C-system for two colliding particles moves with velocity

u and that the second particle of rest mass m02 is initially at rest. It is thus evident that the relative velocity between C- and L- systems is u. Therefore using the well-known relativistic addition theorem of velocities and denoting the C-syste1l1 by , star' we have the following rigorous relations between energies and momenta in C- and L-systems, the first particle being supposed to move along x"axis before collision.

Before Collision First particle

PIZ*='Y*(Plz-~El

⁾

Plr*==Ph=O Pl,*==Pl'==O

E1*='Y*(El-uPlz)

Ih =

1no 1 'Yc s

PI

.=mOl'Y'V=1n01C

V

'Yi-I

4-1S1SP-4

Second particle

PilZ*=Y*(Ph- ~Es)

Ps,*=Pu==o P2'*=PS'=0

E2*=y*(Es ^-uPu) ES==1nOlC^Il

224 • K. C. Kar and C. Basu

Or, the re~iprocal relations, viz.,

Ph='Y*(P1x*+:~lh*),

^etc.,

Or, the reciprocal relations

P2X='Y*(P2i*+ :2E2*).

etc., where

y*= JI -2i

^v the velocity of

th~

first particle before collision and

1 - .. u 2 e

I

y=

J -

^-v2'

1 - -2

If we suppose that the scattering takes place in xz"plane, we have·

c

for the corresponding relations after collision denoted by dash

!liter Collision First particle

P *,=y*(p

^I x ) x

'-:'!:.E')

2 ,)

C

PlY*'=PI /= ⁰

Pl. *'=PI

,'::/=0

El *'=y*(EI'-uP 1 x') Or, the reciprocal relations

P

1%

,=y*('p

).~

*'+~.EI*')

e2 ' • ^etc "

Second particle 1)9%*'

=Y*(P2 . .,/ - ~~iE2')

P2y*'=P2r/=O

1)2. *'

⁼ P2'

'=1=0

E2*'=')I*(E2'-UP2x') Or, the reciprocal relations P ^2>:

'=y*(p

^{2x :j:'} +

~E

^{(,2 2}

*,)

^{, etc "}

Let ')11'=

J-

^{r ,2 and 'Y2'=}

^J

^{I '2 where VI' and 112'} arc the· velo-

1 _:!'.L I -~ - .

e² e² .

cities of first and second particles after collision, in L-systellJ,

In our general treatment we shall always consider three special cases, 'Viz"

(I) tno ^{1= tn02} (relativistic) which has been discussed by Moller (1932), (II) m02» mOl and (III) mOI=Fmo2 (non-relativistic) which has been discussed before by Kar (1941).

Now, from the condition of conservation of momentum in C-system we have

(I)

On substituting the values of the momenta from the relativistic equations given before we get the general relation

and hence

u=--'---v 'Y

')I+mog mOl

(1'1)

.. ' '

Scattering of Fast f3-particles by Electrons

'Case I :-mo ^I = mall (relativistic) u=-X-^'V

Y+1

y*.= Jy;

¹

}

225

Neglecting the effect of relativity

i.e.,

taking y=l, we have

u=h

and y*=1.

Case 11 :-m02 » m O l ' In this case we have

u=o

I

Y*=1

Case III :-mo 1 =/=m02 (non-relativistic). In this case

y=

^{I and so}

tt

= -~Q.L...

^{'V }}

mOl + ^m02 y*= ^I

Relations between lhe angles 0/ Scattering of the Colliding 1)articles in C- and L-systems

(1'4)

Let 01

,°

¹

^*

^{and O2 , O2}

^*

be respectively the angles of scattering of the first (colliding) and second particles in L- and C-systems. Now, from the princi- ples of conservation of energy and momentum we have (in

L-

system)

and

P2,2 =P1 2 + Pl,2 -

2PIPI'COSOI ;

P

ls=Pl (momentum) PI,2=

PI2 + P2,2 - 2Pl

Pl.l'cosl12 (momentum) Eqs, (2'1) and (2'2) may be transformed to

•.. (2) (2'1) (2'2)

)'11,11_

^{I =} (mo

1)2

^(y2

+

1'1'2 _ 2) -

2 (_?no 1)2 v'

(1'2 - I){Yl" -,I).COSI11'" (2'3)

, m02 ' m02

and

1',,2 -

1

^=1'2_

I ⁺ ('!to 2)2

(1'2,2 - I) -

2( ^{!1l02) v' (y2 -}

^I){Y2,2 -I).eos02"· (2'4)

mOl mOl ,

On eliminating

y;'

^{from (2) and (2'3)} and then solving the quadratic equation in 'h' we get

-(I ⁺ m02y)(y ^{+ ~Q2) ±}

^{(1'2 - IlcosOJ}

^v'

^cos

2 8- I ^{+ {mg2/mSlf}

'YI'= ____ ^{rn Ol} m O l . _ _ _ _ " .' (2'5) (')'2 _ l)coSll61 _

(i'

⁺

^mOl).

^:;I

mOl

226

^~.

c. ^{Kar an'd C. Basil}

Similarly on eliminating -Yl' from (2) and (2'4) and then solving the quadratic equation in "1t' yve get

( "1 +

tliQ..!)

^{2 + (-y' - I)cos282}

mOl

"19'= - - - - ( "1 + m02) - (-y2 - I)cos²81l

mOl

Case 1 :-mOl =m02 (relativistic), We lJave from (2'5) and (2'6) ,_ I +-y+ (-Y-I)cos²81

"11 - - - ... " "

. I+-y-(-y-t)COS²01 ,_ I + -y+ (-y-I)coS201l -Ys - - - .

1+ -y- (-y-I)cos²89

1 t

J

Case 11 :-m02~::>' mOl, We have from (:1'5) and (2'6) -Yl'=-y

t

-Y2'= I

f

Case III :-mol=Fmo2 (non-relativistic), We have from (2'5) and (2'6)

", (2'9)

Now, on using the relativistic relations between momenta in C-and T^{4 -} systems and remembering that PIz*'=PI *'cus81

*,

PI z *'=PI*'sin81

*,

PI/= 1'1' cos8Jo pl.'=pl'sin81 and similarly for the second particle, we have

cot82*=-y*(cot82-~

^cosec8s . . ^{1 -y-:: )} c ^'V -Y9 - I

}

These

are general relations which may be transformed into simpler forms for the three special cases, by substituting the corresponding values of -Yl' and

"'12' obtained before, However, the second equation of (3) may be reduced to the following form without any simplifying assumption, On substituting the

,

general value of u from (1'1) and that of ,,--!-ft--I from (2'6) we have after easy

transformation' 'Y2

cot8.* .• ·"1*( cot281 -

~

^{: : cot8g ) ,}

After easy trigonometric transformations we have sin 8 *".-y* sin28J _

J

=- .

cosiB. +yt'sins6.

Scattering 01 Fast /3~particles by Electrons 227

sin fJ2

*

dfJ2*=sin 2fJ 2d0² 2,U _

(COS202

+

"dsin 2( 2 ) 2

These relations have been obtained by Moller (i.e.) for the special case

11101

=

^m02' But it is evident that Eqs. (3.1)-(3.4) are perfectly general for the second particle. ^!

Case 1 :-mOI =m02 (relativistic)~ On substituting the value of )'1' from (2·7) in (3) we get for the first particle

cot 01

*=,,*

(cot 201-~(u2/c2) cot (1 )

and so after trigonometric transformations

sin

81*=}*~~s26IsT"t2~n28;

^\

etc., etc.

as obtained by Moller. The corresponding relations for the second particle are perfectly general and are already given in (3.2)-(3.4).

For non-relativistic case ,,*= ¹ and from (3.6) 01

* =

281 and similarly 0.*=202 ,

Case II :-1n09»mOI. Using the conditions (1.4) for this case we have at once fron1 (3)

which also follows from the fact that in this case

C-

and

L-

systems are identical.

Now, because for the second particle ^)'2'=1 ('Vide (2.8», the second equation of (3) cannot be used in this case. We get from any of Eqs. (3.1)-(3.3)

It should be noted that in this case the second particle is at rest. So O2

*.

O2

represent the direction in which the momentum is imparted to the second particle by the first.

Case 111 :-mol=Fmo9 (non-relativistic). Because in this case from (2.9) we find

"1'= I

to a first approximation, we have to take the second approxima- tion value to evaluate

)'1 '

in (3). We have then

'Y

i ' +

I "" 2 to a first approx.

'! , _

1-. (1I11lcl){COSOJ.£...-{~oS2!J:::!_~~~'I/ma-J)1I

1

LI +

(mOIl/mo 1)]

to a second approx. from (2.5)

228 K. C. KGr dnd C. Basu

Hence, we have

;'1' 1 + (mos/mod

,,; ;,)'2-- 1 = (c/v)' cos 01 + ,,;

COS20~-~I.j.. (m62/m~l)

and so from (3)

cosec

(it

cot

°

^{*=cot (J -} -.--.--::....----cc=_,--,-~c-- .-.".."",.

===

1 I cos 8)

+ ,,;

cos²8} - I + (m~2/m~d or. after trigonometric transformations

sin 81

* =

^{sin OJ. (mO) / IH02){COS 81}

+,,;

^cos281 - 1+ (m~2 ^/ m~ ^IJ) cos 0) *= (mol / 1Il02){ - sin²61

+

cos 81

";(:05

²

8

1 - I

+

(mg2/m~I)}

Therefore. we have

which is the formula derived by Kar (1941). For the second particle we get immediately from (3.1-(3.3)

the same as (3.8) of case II. For IHOI=m02. (3.13) gives 81*=2°1 while for m02»mOl,8)*=81•

We shall next find the angle between the paths traced by the scattered and scattering particles after collision ill C-aud L-systems. This will be useful in the theory of scattering to be discussed in Sec. B.

From conservation of momentum in C-system (vide

Eq.

(1)) we have Pl:r*'+P2z*'=0~

p).*'+P2,*'=0\

From

W

we get Or •. we have

. which is evidently a general relation. So we need not consider the tl1ree special cases. 'l'he relation between 81 and 82 may be easily obtained by solving the triangle of momentum in L-system. We have

Because PI =molc";'Y2-I, etc., we have from (4.3) sin 8, cot 8g _. + cos 81 = _{. . )'1}

V7

_{- I}

in whicJ:i the general value of 'rt' should be taken fr0P1(2.s).

SCQtte'~ng

. of F Qstf3-particles by Electrons 229

Ca.se;l :~lnOl =mOI (relativIstic). On substituting the value of Yl' from (2.7) in (4.4) we get

• J

y2_ I

'V

'l'1,2_ I

Hence from (4'4) we have

() 'Y+l ' cot 2 = - - tan OJ

2

For tion-relativistic case 'Y~ I, so that

Case IT :-11102»11101' In this case 'Yl' is given in (2.8), so that the right hand side of (4.4) becomes unity. Hence after easy, trigonometric transformations we-have

This also follows from the geometry. It should be of conrse noted here that the second particle is at rest, so that O2 g-ivcs the direction of the momentum imparted to it.

Case HI :-mO] :j: 11102 (non-relutiyist,ie). JD this em,e 'Y=1, 'YI'=l to first approximation. So to evaluate the right hand side of (4.4) we have to use the second approximation values of ')II' ('I.'ide (3.9)) and -y. We have

I~

=

~-+J~'!Q2/mOI)., _

'\t

^'Yl'2

=i

^{cos (Jl +}

~ cos:l(J~~

^{J +}

('I1S.mS'J ...

^(4·8)

tIle ne~ative sign in (3.9) being rejected as it gives zero value for the momentulll in the case lIlo I = 1Ho 2. On combining (4.4) and (4.8) and after easy trigono- metric transformations, we have

c~H6.J+~~)=

(IIlP2/mO) cof 61 '-cosec OJ

~c()~:"6J. .. ~T+~~;i;;;~)

^(4.9)

. I+(m02/ m OI)

giving (z) -61 + 62

=

^{goO for 111,>1} ~ mo and (ii) 62 +

°

^{1 /2}

⁼

^{goO for} 1l102»mO] , as obtained before in (4.6) and (4 . .7)

The energies of Colliding particles in (.'- and L-syslems

From the relativistic equations !&ven before, we find for the energies of the first and second particles in C-systerrl before collis-i on,

El~.:=11l01'Y'Y*. (clJ-U'v)l ,E,*=molli'*c^{il. . \}

Using the general value of

u

given in (1.1), we have from' (5), for the total energy in C-system

'd'*='d' lie

E *.6 .'* ~.

1+ 2y(mOll/mol)+;(mOSl/mOl)2

.,1:1" ,1:1,1

+

I · " , . ' mOlr C • . i'

+ (

^mOil

I

mOl

)

(_ • '0

I i a \~ • • ;'

230

^~.

C. KaT and C. Basu

Case 1:- m~l =mOl =mo (relativistic). From the value of u' given in (1.3) it may be easily shown that

y( 1 - U'II

I

CIl)-1

to second approximation. Thus from (5)

also from (5.1)

Hence for the non-relativistic case E*=2El*=2E2*=!11l0'll1.

Case 1I:- 1It02»mO I. From (1.4) and (5) we have El*=m^OI'^YC2

f

E2*=mOllc 2

..• (5.4)

The total energy may be obtained from (5.1) by taking values up to second approximation or from (5.4) by addition. For the non-relativistic case, we have

from (5.4)

El :1: = ~11l0 ^{1 '112}

f

E2*=O

Case I I I : - 11l01=Fmo2 (non-relativistic). From (Y.5) and (5) we have E,*=I.mo2/(mol + 11101).1'1'2

t

ESI*=t. 11101/(11101 +11102)' I^l'V2

~

... (5.6) where ^1'=11101 1JI02/(mOl +11102)' The total energy may be obtained from (5.6) or by subtracting the total rest energy, i.e., (mOl +mOIl)c2 from (5.1) and we have

Next we proceed to find the energies in C- and L·systems after collision, We have for the general values of energy in L-system

(6)

where "II' and 'Y~/ are given in (2.5) and (2'6). The total energy may be obtained by adding the energies given in (6) and substituting the values of "II', 'Yg', This is a laborious method. We may. however, get it directly front the principle of

conservation of energy _,

Ih'+E,'=E1 +E,=m01CII("I+ mOll).

m01

... (6,1)

Scattering of Fast (3-particles by Electrons 231

(6.2)

where

Case 1.-llIot

=

^{III 0 2} (relativistic). The values of E_{t ',} 1<:2' may be obtained from (6) by substituting the values of

y/,

Y2' from (2.7). It may be easily shown that in this case the colliding particle loses energy. Thus the scattering is inelastic. It shou!d be noted that the values of El', E2' may also be obtained directly fr0111 the reciprocal relativistic equations, when E I *1, E2

*',

PI

*',

^P'/~'

are known. We have

El ' ^{= 1IIoc2} [y*2 + (y*2 - I)cos(ll *]

I.

E2'=1/l2C2ly*~-(y*2_I)COS(l2*] ^\ Case JI.-11102»1Il01' We have from (6) and (2.8)

Thus the. energies of the particles are same as before collision and so the scaltering is eia.ltic.

Case lll.-mo 1 =/=m02 (non-relativistic). The energy of the first particle is in this case

On substituting the value of "11' from (;~.7) and after simplification, we get

El

'=

^4-lI1ot1,2.

lC _0s8

]l'

,vc;"'o:~,:;j: \

l~:::: r

where the minus sign should be rejected as it gives zero energy for For the special case mo ^t = ml) ~ we have from (6.6) _

and consequently the loss of energy is

~El'=i1/l_o'l!2sin28) .

(6.5)

(6.6)

(6.7)

(6.8)

The pl~ysical significance of the results in (6.7) and (6.8) may be ea~ily under- stood. For the second particle we have

(6.9)

232 K. C.

Kar and

C. Basu

This may be taken as the non-relativistic energy or the relativIstic

gain

of energy of the second particle. Eq. (6.9) may be easily reduced to r'Vide (6.2)]

... 6.10)

(p2 + ¹¹¹² C2)~

Or, because y= 01 I

mOl c 2 2

2 P cos ^&2

=2111()2c . I

[tl/02( + ^Ip2

+

m~ I c²)1

J

2 - p 2 COS 2 (J2

(6. II) It is evident from (6.ll) that the energy transferred is maximuIIl for (J2

=

⁰

i.e., for head 011 collision. Eq. (6.ld has been given by Rossi and Greisen (1941). Now, because p«mOlc, (6.10) may be further simplified and we have

E21=2m 02 • ( __ p2COS2B'l )>1 =

21/1021,2~

^{{ ('OS2(12} )"}") ...

mOl +mO:.l T + "'11102/11101 ~ (6. 12)

For the case mOl =m02=mO, (6.12) gives

which is easily found to be in agreement with the conclusions arrived at in (6·7) and (6.8) when it is remembered that in the nun-relativistic case 8₁ + 82 =90°.

It is thus evident that the maximum angle of scattering for 1!101=/II02 is 90°.

For the case m02»mOI we get frol11 (6.12) E/=o. as it should be since the second particle is at rest i1,idc (6.4)].

From the relativistic equations after collision, we fiml for the energies in C-system

,

E _I

*,-

_-Y *(E' _:',1 -ucos"'l ^II

P

1 -mOlY ( ^{1 ) - :1:.2\}

'IY!

^I -~---~-"-" v(y*"z:::"I)(YJ':I-I) "~

Y* .

The total energy after collision lllay be obtained by adding the two equations in (7). However, it need not be determined, as from the conservation of energy in C-system

which is already gi yen in (5. I).

Case T.-mOl =m02=mO (relativistic). We have from (7), (1.3) and (2.7)

as obtained by Moller. Thus the ent:rgy of any particle in C.system remains unchanged by collision ('Vide (5.2)). Or, the collision is elastic in C-system.

Case 1l.-m02»mOl' We have from (7), (1.5) and (2'9)

E

1

*'=El*=E

1

'=E

1

Scattering 0/ F dsl (3-particles by Electrons i33

C~system: being identical with the L-system in this case.

Case III.-mol:f=mo2 (non-relativistic). On subtracting mOlc ll , m02c^\!

fr~m the two equations in (7) we get after easy transformation

Mome~ta of Colliding particles in C- and L-systems

From the relativistic equations given bef_e, we get the momenta of the first and' second l'artic1es in C-system be/ore collision (since Plz*=PI*, P2z*

-PB*)

Pl*= -P2*=mo2Y*u (8)

Case I.-mOl =m02=mO {relativistic}. We get at once from (8)

(8.1) Case tI.---'m02»m01' Decause, when m02-+ 00, u-+o, the momenta given by (8) is indeterminate. From physical consideration we get it to be ¹¹¹¹ 'V,

the initial momenta of the collidil1g particle.

Case IH.-m01 fm02 (non-relativistic). From (8) and (1.5) we have

Next, we proceed to find the momenta after collision ^{In C-} and L-systems.

We have for th~ general values of momenta in L-system p)'=mo ^I C ~'Y1'2 - I

P2'=m02 c ~y~'2

-

^{I (9)}

Case I.-ruo 1 =m02

=

rno (relativistic). On substituting the values of 'YI', 'Yll' from (2.7) we have from (9)

}

For the non-relativistic case 'Y-+1 and so PI'=PCOS()., P2'=P cos ()2' It should be noted that corresponding to (6·3) one may get the values of PI',

P2'

by

" . 1 1 ' " t' h n p

*,

P

*,

^E

*,

^E"

*,

l1sing the reClproca re atlVlstJc equa IOns, w e i , '}., " I , II are known:

"Case II.-m02»mOl' From (9) and (2.8) we have Pl'=P,

234 ^K. C. Kar and C. Basu

since the collision is elastic. Because from (2.8) y~.{+ I we have to take in this case second order values in (9) and we have

which is apparent if the geometry is considered.

Case 11 ^J .-mo ^I

:t

11102 (non-relativistic). In this case we have to take second order valw:s and we get from (9)

. , 1Il02 ()

P2

= -_._-- .

^{2j> cos 2}

/HOI + 11102

If we take 11102»11101 we have from (9.4) h'=P, P::/=2PCOS ()2 as in (9·2) and (9.3 L If, -again, we take mo 1 = 11102 we h~ve P " == j) cos ()1, p'/ == p cos O2 as in case I.

From the relativistic equations after collision, the momenta in C-sYbtem for the three cases may be easily obtained.

Case 1 :-11101 = 1fl02= ¹¹¹⁰ (relativisti(;). From the relativistic equations and Eqs. (g.I), (2.7), (3.3) and (3.6) we get

P *,

^{·x- () l<-}

f

-. 1'" = l/loY UCOS I

P_2x'K"= -1Il0Y'x,ucos f)2'N

(ID)

and hence

(10.1) as obtained by Moller.

Case II ^:-1n02» mOl. From the relativistic equations and Eq. (6.4) we have

Pi*'=h'=P~

P2*'=

-p \

the sign being determined from the condition of conservation of momentum, 'Viz.,

Pl*'+1J',l*'=0.

Case III :-mol=Fmo2 (non.relativistic). Because E_1'=1I101C²+E1 ' (non- relativistic) and similarly for E2', we get (10.2) in this case also, from the rela- tivistic equations already given neglecting the effect of relativity. Thus on substituting the value of u fr0111 (1.5) and those of PI:/,

P2 . ../

fro111 (9.4) we get from (10.2)

Scattering 0/ Fast fj.particles by Electrons 235

'Ie' 11/02 ( 2f) )

P23:

= -~-~.

-P

2COS 2-}

• nlOI +m02

The values of 1) I ,

*,

^and P2'

*,

may be easily obtained frolll (9.4) and the rela- tivistic equations. Therefore we get

and

-2c~Sf}I(cOSf))

^{-1 ••}

~~2O;-'='~+T;11~~/n;gd)+I r~

I

P2-*'= - _1I102 __

p=

-1"7)

1110 I

+

^{11'102 (10.6)}

Two special cases may llowbeconsidered. (a) If mOl =11102=1110 we have frol11 (10.5) and (10.6)

PI""=

-P2l(-'=~P (10.7)

which also follows from (10. I) if the correction for relativity IS neglected.

(b) If mo2

»

^{1110 I} we have from (10.5) and (10.6)

(10.8) being the sallie as (10' 3).

Section B. Theory of relativistic Scattering (equal masses)

We arc now in a position to discuss the theory of relativistic scattering of a particle by a second particle of the same mass being initially at rest. We shall take the interacting potential to be Coulombian.

Taking 1110 1= m02 = 1110, we have for the total energy of the particles in L-system

This total energy as it appears to the observer in C· system is E= 1noY* c²{y + 1).

(u)

(II.I)

It should be noted that it is different from the total energy in C-system, which is from (5' 3)

Now, if E t is the total e11erg:1,' as it appears to the observer in C-system. at the centre

tif

gravity, we have

;

Et =~(E-E)f):e.~mo'Y*cB('Y-I). (II.3)

i36 k. C. kar and C. Basu

On the other hand, because the total mass at c.g. for the observer in

C-

system is mo~*(Y+ 1), we have also

where u is the velocity of C-system being equal to c

JY-l

('Vide (1.3). On 'Y+¹

substituting this value of u in (11.4) we get the same result as in (n.3).

Now, to the observer in C-system the relativistic x-equations for the two.

particles are

~IXI

⁺

::;2{(lh -

V)2_E02}x1 =0 }

~2X2

⁺

·-4~~-{(1i2

- V)2 - Eo2}x2=O h²c²

E2=Eo=mo'Y*c2 •

In tbe relativistic theory of scattering we neglect V2-tennwitbin { to a first approximation. And so (12) may be written ill the form

a

1XI + _h:}c

~7T2

₂ ^El(l':'

-~Q~

_El ^{-2V)Xl =0 }}

~2X2

^{+ 4.}

7T~

^{E2(E2 -}

~!l2

^{- 2V)X2}

=

⁰

h²c² E2

On substituting for E1 , ESI we have

--~~

... - .alXl +

(Et - Il_~2 -

2V)Xl =0 } 47T'mo"Yi'* r<'1

h²

'~2X2+(E2-~E~9'~-2V)X2-o

47T2mo~ . \I

}-bracket,

... (12.1)

Because the total wave function

w=

^{X 1} X:h where X I, X2 are functions of xl, XII

respectively, (12.2) takes the form

.

hI! h² . ( Eo2 Eot .)

II ---.a1'lt+-\l-- all\}!+ E - - - 2 V W=o ... (12.3) 411" mo'Y'Y* 47T mOY* Et EIJ

where E:::;EI + E2. Or, after substituting the values of E ... etc.,

..

_~_~._al'lt+h2 ~2W+('11I0C!!Y*.

^y2-:} -2V)W=0 ...

47T!!moyy* 47T2mO'Y* Y

Let us take W, the joint wave statistical probability for. both the particles,

as the product ,

Scattering of Fast (3-particles by Electrons 237

where 1{1 is a function of (~. tj, ~) the coordinates of the centre of gravity and X is a function of x, ::V, z, in which

Thus, we have

X=XI-X2, ntoY*(Y+ I)~=mOy*yxl + moy*x2 Y= Yl - Y2, moY*(Y+ I)tj= moY*i'Yl + 11l0Y*Y2

a

²

a

^{2 I}

a

^{2 I 02}

---= - + --- ··---2~-

0:\:22

ox

^{2 (y} + ¹⁾² O~2 )' + ^I axa~

lIence in terms of the new variables x, ~ the wave equation (12.4) transforms into

h²(Y+I) hI) ( l! .¥;. y2_1 )

---:l-

*

flxX.1{I+ - 2 - - * ( · - - f l c1{l.X+ moc y'.---2V x1{I=o ... (I2.6) 411" mon 411" lItoY y+ I) ~ Y

On dividing the above equation by x1{I the variables are completely separated.

Since V the interaction potential, depends 011 the mutual distance, it should be taken with X. Thus we have for 1{1 and X equations

flxx+_4{f~;~_Y:)* (11I O

c

2

)'* )'2;1 -A-2V)X=O

where A is a constant. From elementary principles of wave-statistics, it follows that the fl equency and velocity of 1{I-waves are

and

moy*c²(y-r) ( .

= --- from I I 3)

Iz

'V=U=CJY--I

)'+1 and that the 1/J-eq uation should be

Or, on substituting for v and 'v

t::..d+

411"21n5Y~,c2_(1'2-:!L1/J=0

~ 11²

On comparing (12.7) and (12.9) we get

}.-mo),*c'l(y-l),

238 K.C. KaT and C. Basu

Hence from (Iz.7)

6..,X + 471'2/11 YTiX ( . -0.. moc 2y*-'----= '\< - I -2V X=o )

h²(y+ I) Y

which gives the wave fUllction within the potential field. Outside the potential it becomes

• 471'2 m 2yd c2(Y-I) 6 xXo+ k2Xo=o; ,<2=

Tz2

·--.ll---:Y+I---~-

.

On proceeding in the usual manner we have for the differential first order scattering function

equatioll of the

Srr2 III 'Y"*

6,,1AIX 1) + k² (A I Xl)

=

L, 2 .-.!L+....L VXo

I Y I

On solving it in the usual way (Kar and others, Iq37) we have for the scattering function

'Y I c^ikr

~lXl = - -- _ .. - ._-.- cosec²},O-* -, - .~-~ FI fO)

2mOC~(Y-l)iK' - V'l! f

( 13·4) where for Coulomb repulsive force F(ro)= c²cosk'fO, k'=21<.sin§B*, I.: being given in (13.2).

Because in the present case the two colloiding particles are indistinguishable we have to consider the effect of exchange. In so doing we shall not introduce llew symbols but shall only use' dash' and 'double dash' for notations which are functions of 8*. In these lIew notations we have for the scattering functions before and after exchange (-"ide (4.2))

where

C i k r

--cosk'ro'

r

2 • k r

A1XJ"= - .. -.. --~£'I.-. see22B*...!.-=-.1.: ... cosk"ro"

ZlIloc 2(y-r)y-!(' VI' r I~'

=

² I,siD~ B*

f

,,"" = 2kcos~B'*

1

J

..-0

and ro', ro" the critical approaches before and after exchange are given by ro'= 2·7

x-.c-~~2

^{J2(.y_ ..}

TI)(cosec~O*

+ I) }

2n1o'l! y2

rol/= z.i x

~'~-2 J~:(,!,-+r)(sec~8*

+ ^I)

21110'l.' y~

(r.) .6)

On considering the effects of spin amI exchange in the usual way (Kar, 1942) we get for the total relative intensity of scattering after' substituting for 'y*

from (1.3)

Scattering

0/

F cist f3-particles

by

Electrons

239

After simple trigonometric transformations it may be written in the form

( e2

)2

^{2(1' f-1 ) ' .}

1=811"---' .--,_._/(&* 1'0' TO")Slllb*dB'1l-

2mOV2 1'2 ' . (13.9)

where

j(B*. TO', TO"!

=

~cos2k'T.Q~_,,!"_COS2 k~Tt),:') _ C~s2 ~~:.!!L2'~OS2~~~Q" + cosk' Yo'cosk" TO" ,

sm 46* sin²6*

2COt()-)(' " "

+~('COS:/~'TO' -cos2k"ro"),·· (r3'lO)

SID u "

Neglecting the wave statistical correction forthe criticai approach, we have from (13.10)

(r3·I l )

which is MOller's formula without his spin corredtioll tertII. 1'lllts om forml1la (13.9) for the relative intensity of scattering tak~ into account the wavestatis- tical correction for the critical approach but is free, to this order of accuracy, from the effect of spin·orbit interaction. It becomes in the laboratory co- ordinates

(14) Now, in order to verify (14) with th<: hell> of the observations of Champion (1932) in Wilson chamber. it would be necessary to write it in the following form. If i be the total track length and No the number of scattering centres per unit volume, we have for N e the 11l1mber of scattered f3-particles observed at an angle between fI and ^(j + dfl in laboratory coordinates

I , :I

N e =6411'f._e

-=--

^'Y~_) Nolf{B*) sin2BdB ,

\2/11.0',:2'

^{y (I+Cos 2}8+ysin:!B)2

As I in Champion) experiments denoles the total track length for the whole range of f3-ray spectrum used and as the number of !3 particles incident at dUll-rent velocities is different we are to m 111tiply ('4. ^I) by yet another factor " ^II denoting the fraction of {3.particles incident with velocities lying within the range of velo- cities cf3 and c({3 + df3L Thus we have finally

Ne=6411'N olnJ _{, \2I1luV~}

~. 'l'.~)2 _'Y

(r+cos2 (j+-YS1112(j)2

.. _fJ!i*.L~~~1]3~~B. ^__

^{'_i' (14'2)}

gtVlOg the number scattered for given velucity of incidence of the f3·partic1es and for the angular range 6 and (I t d9 ill laboratory coordinates.

In the following table are compared the theoretical values fwm different formulae with the experimental values of Champion (J932). In the second column are given the observations of Champion for different allgular ranges. In the third, fourth and fifth columns respectively are given the theoretical values from our formula (I4. 2 ) and the formulae of Moller and Mott. The ~heoretical values in the last two columns are taken from Champion's paper (1932). The

6-^I5^JSP-4

.~.

240 K. C. Kar and C. Basu

TABU; I

Angle ₈ Observed From formula _(14·2) Moller Mott

"._--.---^{.. ~.·,_,} __ • _ _ ~T~. _ _ _

--~---~~.---

IOo~-"20· 214 21 5 230 650

23·-3°· 26

i

^{26·7 30 lOS}

30·-max 10 9·7 ¹³ 28

.----.

!-

^~.--~---

Total 250 25 1 ·4 273 783

theoretical values from (14.2) for different angular ranges have been calculated by integrating (14.2) by the graphical method. Unfortunately, however, in Champion's paper cited above the value of No is not given. Accordingly, we had to go through the laborious process of finding it by graphically integrat- ing Moller's formula over the diffe:'cnt angular ranges and then cquating the values thus obtained with the Mciller's values giv'lll by Champion. The values of nil for different velocities of incidence has been taken from Champion's previous paper (1932). It should be noted that a formula which takes account of exchange should be minimum at 45°. ()f course, due to the effect of relativity the angle is slightly deviated. Thus, for example, whcn ill the Table, the scattering is considered between 10°-20° it IS evidently the total including those between the ang ular range 80°-goo. And so the maximullI angle means 45°

approximately.

It is evident from the above table that the formula derived by us is in reo markably good agreement with the experimental vdlnes. Moller's value is throughout higher than the experimental. The difference between our value and that of MCiller is mainly due to his additional spin term, although it is slightly affected by our taking the critical approach. It should be noted that the correction for the critical approach is very small in the" present ca!>e because of the high velocity of incidence and also because the Coulomb field is small, z being unity. Lastly, it lllay be mentioned that Mott's value is extremely high because his formula is applicable only for low velocity of incidence where the relativistic effect is negligihle.

SectIOn C. Theory of relati~istic Scattering (unequal ma.sses)

As, in the present case, the particles are distinguishable one should not consider the effect of exchange. Again, as the matter for discussion ill this section may be looked upon as an easy generaiisatioll of our treatment in Sec. B, we.$hall try to be as brief as possible.

It is evident that for unequal !nasses (12.3) should take the· form.

(IS)

Scattering of Fast !3-particles by Electrons

On substituting the values of Ii, E ... etc., we get from (IS)

~~-. ^u

^I

^'11

⁺

--.~":---:-

^U2

^'11

⁺

(mo

^{I Y*} C 2 • .1:'_2-I - '2 V

')'11 =

⁰

411' mOIYY~' 4rr:tmo2Y~' y

On changing variables as. in Sec. B to (x, y,z) and (~, '}'~) where, however, x=x) -X2 • ... etc.

but y*(moly+mO\!)~=y*ymolxl +y'''m02x2' ... etc., we get corrosponding to (12.7)

A_t

1f

+

~rr~'»lQJ_Y*'\(y

⁺

l1L()_~)",

^{= 0}

<; h² 11101

A",x + 4^rr_,

:r

^y

^*.y

⁺

1I~1;2/11:o

^)'-

(mo,

^{,,*c2 • .-1'2;!}

Now, the velocity of ",,-wave is

v=u=c.

Vy2~I

(vide (1.1)) y + ^{m02jm "J}

241

Again, the kinetic energy at the centre of gravity, as it appears to the ob- server in C-system, is

Et =h*(molY+ IH(2)U2=

~1Il0)

(21'*(,,2 -

rlj(Y+

^{1H(j2 ).}

, mOl Hence, the frequency of ""-wave is

2Et 1n0)C2y*(y2- I)

VI = h -Il-[y+(;';~~/;~~~)]' Thus corresponding to (12.9), we have

4rr2m r.ly*2c 2(y2-r) _

~~1/;+ ---h'J- - --1/;-0, On comparing (15.2) and (15.6) we get

,\=mo!y~c2(yi_I) (r5.7)

y+ (m02/motl .

On substituting the above value of,\ in (15.2) we have

Ax+4rr2~,,*

^.

_~..!":(,-2.

__ . _____

(~02y*c2(y_

^{2-:::-. .!) -ZV\=o, (15.8)}

h² y+ -m02/ m ol yly+m02/mlO)

J

giving the wave function within the potenlial ljeld, Ontside the potentiai field (15.8) becomes

(Is.g)

Thus proceeding in the usual manner we have for the first order scattering function

"-IXl

= -

^1'[ 'Y.:I=.

(mo~mJU1]cosec2!8*--..!c.

e ^{j k ..} F('oJ,

2m.OIlC 2(y2 - Ih* V v r (15.10)