A modified electron optical system and the gamma-electron angular correlation of gold-198.

University of Windsor University of Windsor

Scholarship at UWindsor

Scholarship at UWindsor

Electronic Theses and Dissertations Theses, Dissertations, and Major Papers

7-17-1965

A modified electron optical system and the gamma-electron

A modified electron optical system and the gamma-electron

angular correlation of gold-198.

angular correlation of gold-198.

Murray Lawrence Trudel University of Windsor

Follow this and additional works at: https://scholar.uwindsor.ca/etd

Recommended Citation Recommended Citation

Trudel, Murray Lawrence, "A modified electron optical system and the gamma-electron angular correlation of gold-198." (1965). Electronic Theses and Dissertations. 6406.

https://scholar.uwindsor.ca/etd/6406

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ProQuest Information and Learning

A MODIFIED ELECTRON OPTICAL SYSTEM

AND THE - Sl ANGULAR CORRELATION OF Au1 9 8

3Y

MURRAY LAWRENCE TRUDEL

A T h e s i s

S u b m i t t e d t o t h e F a c u l t y o f G r a d u a t e S t u d i e s t h r o u g h t h e D e p a r tm e n t o f P h y s i c s i n P a r t i a l F u l f i l l m e n t

o f t h e R e q u i r e m e n t s f o r t h e D e g re e o f M a s te r o f S c i e n c e a t

The U n i v e r s i t y o f W in d so r

W in d s o r, O n t a r i o

UMI Number:EC52587

UMI*

UMI Microform EC52587

Copyright 2007 by ProQuest Information and Learning Company. All rights reserved. This microform edition is protected against

unauthorized copying under Title 17, United States Code.

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P.O. Box 1346 Ann Arbor, Ml 48106-1346

APPROVED:

E. E. H a b ib

F . H o l u j

S. K a l r a

ABSTRACT

The e l e c t r o n o p t i c s u s e d i n a n g u l a r c o r r e l a t i o n m e a s u r e m e n t s w ere m o d i f i e d i n o r d e r t o m e a s u r e b o t h t h e A2 a n d t h e A^ c o r r e l a t i o n

c o e f f i c i e n t s o f t h e c o r r e l a t i o n f u n c t i o n :

W(d) = £ _{^2K ^2K ( c o s ®-)}

K

A new e x p e r i m e n t a l m eth o d f o r t h e s t u d y o f t h e e l e c t r o n c o n v e r s i o n p r o c e s s

i n v o l v i n g t h e b2 p a r t i c l e p a r a m e t e r i s o u t l i n e d .

The m o d i f i e d o p t i c s s y s t e m was em ployed i n a ^ - JL

d i r e c t i o n a l c o r r e l a t i o n e x p e r i m e n t f o r t h e 2+--- ^ 2+---^ 0+ c a s c a d e i n ^ g A u ^ ® . The r e s u l t s o f t h e e x p e r i m e n t w e re a n a l y s e d a c c o r d i n g t o t h e

new m eth o d a b o v e .

From t h i s e x p e r i m e n t , t h e v a l u e o f t h e b2 p a r t i c l e p a r a m e t e r o b t a i n e d was:

b2 = 1 . 1 9 £ 0 . 1 0

T h i s r e s u l t a g r e e s w i t h t h e t h e o r e t i c a l c a l c u l a t i o n o f b2

o b t a i n e d by B i e d e n h a r n an d Rose ( 1 9 5 3 ) w i t h i n t h e l i m i t s o f e x p e r i m e n t a l e r r o r .

1 1

ACKNOWLEDGEMENTS

I would l i k e t o e x p r e s s my a p p r e c i a t i o n t o D r . E. E. H a b ib

f o r h i s g u i d a n c e a n d i n s t r u c t i o n t h r o u g h o u t t h i s w ork.

My t h a n k s go t o Mr. W. Grewe f o r t h e p a r t h e p l a y e d i n t h e

c o n s t r u c t i o n o f t h e m o d i f i e d e l e c t r o n o p t i c s s y s t e m . I would l i k e t o t h a n k M iss L i n d a Cummings f o r t y p i n g t h e m a n u s c r i p t . I am i n d e b t e d t o t h e

TABLE OF CONTENTS

ABSTRACT i i

ACKNOWLEDGEMENTS i i i

CHAPTER I ANGULAR CORRELATIONS IN NUCLEAR SPECTROSCOPY

i ) I n t r o d u c t i o n 1

i i ) A n g u l a r C o r r e l a t i o n o f S u c c e s s i v e R a d i a t i o n s 1 i i i ) The P u r p o s e o f A n g u l a r C o r r e l a t i o n s 2

i v ) T h e o r y o f A n g u l a r C o r r e l a t i o n s 3 . a ) I n t r o d u c t i o n 3 b ) The T h e o r e t i c a l D i r e c t i o n a l C o r r e l a t i o n F u n c i i o n 4 v ) P a r t i c l e P a r a m e t e r s 9 v i ) The New T e c h n i q u e 11 a ) I n t r o d u c t i o n 11 b ) The New Method F o r A c c u r a t e b2 D e t e r m i n a t i o n s 11 CHAPTER I I CORRECTIONS FOR FINITE SOLID ANGLE

i ) C o r r e c t i o n s f o r t h e C h a n n e l 1 4 i i ) C o r r e c t i o n s f o r t h e j l~ C h a n n e l 20

i i i ) The P<j, P2 , P4 B a f f l e s 24

i v ) A F i n a l N o te 27

CHAPTER I I I THE BAFFLE SYSTEMS

i ) The Gerholm I n s t r u m e n t 28 i i ) R in g F o c u s a n d B a f f l e s 28 i i i ) R e s o l u t i o n a n d T r a n s m i s s i o n 31 i v ) The A2 a n d A4 B a f f l e S y s te m s 32 a ) S p e c i a l Remarks 32 b ) C h o i c e o f E m i s s i o n A n g le 3 3 v) A4 B a f f l e C o n s t r u c t i o n 3 4 a ) The Cam era a n d S p e c i a l B a f f l e s 3 5 b ) The P h o t o g r a p h y 3 3 c ) The P4 B a f f l e 4 3 CHAPTER IV PRELIMINARY TESTING OF THE BAFFLES

i ) The A4 B a f f l e S y s te m 4 4 a ) The Shadow E f f e c t 4 4 b ) R e s o l u t i o n v s T r a n s m i s s i o n 4 4 i i ) T e s t i n g o f t h e P4 B a f f l e 47

i v

CHAPTER V THE Au1 9 8 EXPERIMENT

i ) I n t r o d u c t i o n 50 i i ) P r e p a r a t i o n 50 a ) S o u r c e P r e p a r a t i o n 50 b ) f r - P r o b e C e n t e r i n g 50' c ) The S p e c t r o m e t e r S e t t i n g s 53 d) The E x p e r i m e n t a l A r r a n g e m e n t 53 i i i ) E x p e r i m e n t a l P r o c e d u r e 56 i v ) T r e a t m e n t o f D a t a and C a l c u l a t i o n s 56

v) E r r o r s 59

v i ) C o n c l u s i o n s 59

BIBLIOGRAPHY ' 62

L IST OF TABLES

TABLE NO. PAGE NO.

I

,

3

5

II

47

III

57

IV

Experi ment al R esu lts

60

V

Gerholm's R e su lts

60

VI

L IST OF ILLUSTRATIONS

1 . 1 S c h e m a t i c o f a D i r e c t i o n a l C o r r e l a t i o n M e a s u re m e n t 5

-1 . 2 A f - y C a s c a d e 8

1 . 3 A P a r a m e t r i c P l o t 10

2 . 1 A G eom etry o f Two O D e t e c t o r s 15 2 . 2 G ra p h s o f J2 / J q -1-8 2 . 3 G ra p h s o f / Jq 19 2 . 4 A Geom etry o f a n SL~ C h a n n e l 21 2 . 5 G ra p h s o f P2 ( c o s * ) a n d Pi* ( c o s « 0 23 2 . 6 The P o , P2« p 4 B a f f l e s 2 5 ,2 6 3 .1 b A S c h e m a t i c o f t h e S p e c t r o m e t e r 29 3 . 1 a A S c h e m a t i c o f t h e j2_‘- g- S p e c t r o m e t e r 29 3 . 2 T r i a n g u l a r F i e l d D i s t r i b u t i o n . . . . 30

3 . 3 R in g F o c u s 3 1

3 . 4 A P e a k P r o f i l e 3 1

3 . 5 The Camera 36

CHAPTER I

ANGULAR CORRELATIONS IN NUCLEAR SPECTROSCOPY ( i ) I n t r o d u c t i o n

I f a n u c l e u s u n d e r g o e s e m i s s i o n o f tw o r a d i a t i o n s i n s u c c e s s i o n ( n u c l e a r ^ o r - r a d i a t i o n , o r s e c o n d a r y r a d i a t i o n s f ro m t h e

s u r r o u n d i n g e l e c t r o n s h e l l s ) , t h e i r c o r r e l a t i o n s a r e an a t t r a c t i v e a n d u s e f u l f i e l d o f s t u d y . Such s t u d i e s c a n be d i v i d e d i n t o e s s e n t i a l l y two

k i n d s : s p a t i a l c o r r e l a t i o n s a n d t i m e c o r r e l a t i o n s . The f i r s t o n e s

( a n g u l a r c o r r e l a t i o n s ) c o n s t i t u t e t h e t o p i c o f t h i s c h a p t e r . F o r c o m p l e t e n e s s ,

h o w e v e r , i t may be s a i d t h a t t i m e c o r r e l a t i o n s s t u d i e s c o n c e r n t h e t i m e d e l a y b e tw e e n t h e tw o t r a n s i t i o n s ( d e l a y e d c o i n c i d e n c e s ) a n d t h a t i n many c a s e s ,

t h e p u r p o s e o f t h e s e s t u d i e s i s t h e d e t e r m i n a t i o n o f t h e c o i n c i d e n c e i n t e n s i t y o f two k i n d s o f t r a n s i t i o n s , i . e . t h e n u m b er o f t r a n s i t i o n s o f t h e f i r s t k i n d

f o l l o w i n g ( o r p r e c e d i n g ) t h e o t h e r . i n t h e c o n s t r u c t i o n o f d e c a y s c h e m e s . O f t e n , o n l y a v e r y r o u g h k n o w le d g e o f c o i n c i d e n c e i n t e n s i t i e s i s n e c e s s a r y

f o r t h i s l a s t p u r p o s e .

( i i ) A n g u l a r C o r r e l a t i o n o f S u c c e s s i v e R a d i a t i o n s

The p h o t o n s e m i t t e d by a s a m p le i n w h ic h a l a r g e num ber o f n u c l e i a r e u n d e r g o i n g i d e n t i c a l - r a y t r a n s i t i o n s w i l l be i s o t r o p i c i n t h e l a b o r a t o r y

c o o r d i n a t e s . T h e r e i s no p r e f e r r e d d i r e c t i o n o f e m i s s i o n f o r t h e - r a y p h o t o n f ro m t h e i n d i v i d u a l t r a n s i t i o n 1^--- £ ---- ^ 1^ b e c a u s e t h e a to m s a n d

n u c l e i a r e o r i e n t e d a t ra n d o m . The same i s t r u e f o r - r a y , - r a y and c o n v e r s i o n e l e c t r o n e m i s s i o n . I f t h e t r a n s i t i o n I a.--- * ? Ib i s f o l l o w e d by

a s e c o n d t r a n s i t i o n l b --- ^ l c > t 'ne i n d i v i d u a l r a d i a t i o n s from t h e s e c o n d t r a n s i t i o n a r e l i k e w i s e i s o t r o p i c i n t h e l a b o r a t o r y c o o r d i n a t e s .

1

2

How ever, i n a tw o s t e p c a s c a d e t r a n s i t i o n , s u c h a s

I a —--- > I j j ---^ — ^ I c t h e r e i s o f t e n a n a n g u l a r c o r r e l a t i o n b e tw e e n

t h e d i r e c t i o n s o f e m i s s i o n o f two s u c c e s s i v e - r a y p h o t o n s a n d^.a , w hich a r e e m i t t e d f r o m t h e same n u c l e u s . O f t e n t h e r e a r e s i m i l a r a n g u l a r

c o r r e l a t i o n s f o r o t h e r p a i r s o f s u c c e s s i v e r a d i a t i o n s , s u c h a s ( w h e re JL m eans a c o n v e r s i o n e l e c t r o n ) , ... The e x i s t e n c e o f a n a n g u l a r c o r r e l a t i o n a r i s e s b e c a u s e t h e

d i r e c t i o n o f t h e f i r s t r a d i a t i o n i s r e l a t e d t o t h e o r i e n t a t i o n o f t h e

a n g u l a r momentum 1^ o f t h e i n t e r m e d i a t e l e v e l . T h i s o r i e n t a t i o n c a n be e x p r e s s e d i n t e r m s o f t h e m a g n e t i c a n g u l a r momentum quantum num ber m-Q

w i t h r e s p e c t t o some l a b o r a t o r y d i r e c t i o n s u c h a s t h a t o f t h e f i r s t r a d i a t i o n .

I f Ifc i s n o r z e r o , a n d i f t h e l i f e t i m e o f t h e i n t e r m e d i a t e l e v e l i s s h o r t e n o u g h so t h a t t h e o r i e n t a t i o n o f p e r s i s t s , t h e n t h e d i r e c t i o n o f e m i s s i o n o f t h e s e c o n d r a d i a t i o n w i l l be r e l a t e d t o t h e d i r e c t i o n o f I-D an d h e n c e t o t h a t o f t h e f i r s t r a d i a t i o n .

( i i i ) The P u r p o s e o f A n g u l a r C o r r e l a t i o n s

The a n g u l a r c o r r e l a t i o n o f r a d i a t i o n s e m i t t e d o r a b s o r b e d i n

n u c l e a r p r o c e s s e s h a s b e e n o n e o f t h e p r i n c i p a l t o o l s a v a i l a b l e f o r t h e

n u c l e a r s p e c t r o s c o p y o f e x c i t e d s t a t e s .

The i n f o r m a t i o n t h a t c a n be o b t a i n e d f ro m a n g u l a r c o r r e l a t i o n

work d e p e n d s on t h e t y p e o f r a d i a t i o n o b s e r v e d ( B j ^ ; J2-~ on t a e p r o p e r t i e s t h a t a r e s i n g l e d o u t by t h e e x p e r i m e n t ( d i r e c t i o n , p o l a r i z a t i o n , e n e r g y ) , a n d on t h e e x t r a n u c l e a r f i e l d s a c t i n g oh t h e n u c l e u s .

I f we a s su m e t h a t t h e d e c a y i n g n u c l e i a r e f r e e , i . e . t h a t no

3

an d d i r e c t i o n a l c o r r e l a t i o n s y i e l d t h e s p i n s o f t h e n u c l e a r l e v e l s , b u t n o t t h e p a r i t i e s . The r e l a t i v e

p a r i t i e s c a n be d e t e r m i n e d , h o w e v e r , i f o n e o b s e r v e s i n a d d i t i o n t o t h e d i r e c t i o n , t h e p o l a r i z a t i o n o f t h e - r a y s , o r i f one m e a s u r e s t h e d i r e c t i o n a l c o r r e l a t i o n b e tw e en c o n v e r s i o n e l e c t r o n s (.£ ) . JL~ — ^

c o r r e l a t i o n s p r o v i d e new i n f o r m a t i o n c o n c e r n i n g t h e d e t a i l s o f n u c l e a r s t r u c t u r e , v i z , t h e n e w ly d i s c o v e r e d " n u c l e a r p e n e t r a t i o n m a t r i x e l e m e n t s "

[ C h u r c h a n d W e n e s e r, ( 1 9 5 6 ) ] . A n g u l a r c o r r e l a t i o n s t h a t i n v o l v e e i t h e r t h e

p o l a r i z a t i o n o f ^ - p a r t i c l e s o r t h e c i r c u l a r p o l a r i z a t i o n o f ^ - r a y s , y i e l d i n f o r m a t i o n on t h e i n t e r a c t i o n i n p - d e c a y .

The i n f o r m a t i o n t h a t c a n be o b t a i n e d f ro m t h e i n f l u e n c e o f

e x t r a n u c l e a r f i e l d s on t h e n u c l e a r a n g u l a r c o r r e l a t i o n i s a l s o m u l t i f o l d .

V e ry o f t e n , one c a n d e t e r m i n e t h e q u a d r u p o l e c o u p l i n g f ro m t h e c h a n g e o f t h e c o r r e l a t i o n du e t o e x t r a n u c l e a r f i e l d s . I n many c a s e s , one c a n m e a s u r e

t h e g - f a c t o r o f a n e x c i t e d n u c l e a r s t a t e by o b s e r v i n g t h e d i r e c t i o n a l

c o r r e l a t i o n a s a f u n c t i o n o f a n e x t e r n a l m a g n e t i c f i e l d . From t h e g - f a c t o r

o n e g e t s t h e m a g n e t i c moment i f t h e s p i n o f t h e n u c l e a r s t a t e i s known, e . g . f r o m t h e u n p e r t u r b e d d i r e c t i o n a l c o r r e l a t i o n .

The a b o v e s p e c i f i e d i n f o r m a t i o n by no m eans e x h a u s t s t h e v a s t

am ount o f know ledge o b t a i n a b l e f r o m a n g u l a r c o r r e l a t i o n s . F u r t h e r

e l a b o r a t i o n , h o w e v e r , i s beyond t h e s c o p e o f t h i s w ork.

( i v ) T h e o r y o f A n g u l a r C o r r e l a t i o n s

( a ) I n t r o d u c t i o n

The t h e o r y o f a n g u l a r c o r r e l a t i o n i s h i g h l y d e v e l o p e d and g e n e r a l

e x p r e s s i o n s c a n be w r i t t e n dov.-n f o r e ve n v e r y com p le x s i t u a t i o n s . H ow ever, t h e g e n e r a l f o r m u l a t i o n o f t h e t h e o r y i s v e r y c o m p l i c a t e d and i s som ew hat

l e n g t h y . V ery good f o r m u l a t i o n s o f t h e t h e o r y a r e p r e s e n t i n t h e l i t e r a t u r e r a d i a t i o n s . To be more p r e c i s e , d - <.

4

[ B i e d e n h a r n and R o s e , ( 1 9 5 3 ) ] . V7e w i l l t h e r e f o r e p r e s e n t i n t h i s s e c t i o n a s i m p l e t r e a t m e n t o f d i r e c t i o n a l c o r r e l a t i o n t h a t , w h i l e r e s t r i c t e d i n

i t s u s e f u l n e s s , p r o v i d e s some i n s i g h t i n t o t h e c o r r e l a t i o n m e c h a n is m . The common f e a t u r e o f a l l c o r r e l a t i o n p r o b l e m s i s t o be f o u n d i n t h e f a c t t h a t t h e y i n v o l v e a n i n i t i a l n u c l e a r s t a t e o f s h a r p

a n g u l a r momentum and p a r i t y t h a t u n d e r g o e s s u c c e s s i v e t r a n s f o r m a t i o n s

e i t h e r e m i t t i n g o r a b s o r b i n g r a d i a t i o n s t h r o u g h i n t e r m e d i a t e n u c l e a r s t a t e s o f s h a r p a n g u l a r momenta I a , I b . . . . w i t h s h a r p p a r i t y a n d t e r m i n a t i n g a s a n u c l e u s w i t h s h a r p a n g u l a r momentum I2 an d p a r i t y . T h i s a s s u m p t i o n o f

s h a r p a n g u l a r momenta a n d p a r i t y i s r e s t r i c t i v e a n d d i s t i n g u i s h e s t h e c o r r e l a t i o n p r o b le m fro m s a y t h e c l o s e l y r e l a t e d p r o b l e m o f t h e a n g u l a r

d i s t r i b u t i o n o f n u c l e a r r e a c t i o n s w h ic h show c o h e r e n t m i x i n g o f t h e i n t e r m e d i a t e n u c l e a r s t a t e s .

( b ) The T h e o r e t i c a l D i r e c t i o n a l C o r r e l a t i o n F u n c t i o n

The s c h e m a t i c p i c t u r e o f a d i r e c t i o n a l c o r r e l a t i o n m e a s u r e m e n t i s shown i n F i g u r e 1 . 1 . A n u c l e a r c a s c a d e i n v o l v i n g s t a t e s a , b , c w i t h

s p i n s I a , 1^ , I c o c c u r s t h r o u g h t h e s u c c e s s i v e e m i s s i o n o f p a r t i c l e s ( o r p h o t o n s ) Rj_ an d R2 ( I t i s a s su m e d h e r e t h a t t h e s p i n s a n d p a r i t i e s o f

t h e s t a t e s a , b , c a r e w e l l d e f i n e d ) .

The m o s t c o n v e n i e n t f o r m i n w h ic h t o e x p r e s s t h e c o r r e l a t i o n f u n c t i o n W (& ) i s :

W( ©•) = £ AK PK ( c o s © ) ( 1 . 1 ) M o

The f u n c t i o n s a r e L e g e n d r e p o l y n o m i a l s . The c o e f f i c i e n t s A^ c o n t a i n

a l l t h e p h y s i c a l i n f o r m a t i o n . U s u a l l y one c h o o s e s Ag = 1 , s o t h a t t h e

c o r r e l a t i o n f u n c t i o n i n t e g r a t e d o v e r a l l a n g l e s i s u n i t y :

(

,

(1*2)

I a

b

c

5-n-C o u n t e r I

C o i n c i d e n c e s

S o u r c e

C o u n t e r ^

( a ) ( b ) ( c )

F i g u r e 1 , 1 S c h e m a t i c o f a D i r e c t i o n a l C o r r e l a t i o n M e a s u re m e n t ( a ) N u c l e a r c a s c a d e

( b ) W ( & ) A n. i s t h e r e l a t i v e p r o b a b i l i t y t h a t t h e r a d i a t i o n R]_ i s e m i t t e d i n t o t h e s o l i d a n g l e d-n. a t an a n g l e ©• w i t h r e s p e c t t o R2 .

( c ) The tw o c o u n t e r s 1 a n d 2 s u b t e n d an a n g l e &■ a t t h e s o u r c e . From t h e c o i n c i d e n c e c o u n t r a t e a s a f u n c t i o n o f a n g l e , C (d> ) , one o b t a i n s a f t e r s u i t a b l e c o r r e c t i o n t h e c o r r e l a t i o n f u n c t i o n W ( £ j .

6

At t h i s p o i n t we i n t r o d u c e t h e f o l l o w i n g n o m e n c l a t u r e f o r a d o u b l e c a s c a d e , v i z , I a ( ) 1 5 (L2) I c w here I a , Ijj a n d I c d e n o t e t h e a n g u l a r momenta o f t h e f i r s t , i n t e r m e d i a t e and f i n a l n u c l e a r l e v e l s ,

and Lj_, a n d L2 a r e t h e a n g u l a r momenta o f t h e two s u c c e s s i v e r a d i a t i o n s . T h r e e g e n e r a l r u l e s f o r t h e c o e f f i c i e n t s h o l d f o r a l l t y p e s o f p a r t i c l e s :

0

R ule 1 e x p r e s s i n g t h e f a c t t h a t o n l y e ven L e g e n d re p o l y n o m i a l s a p p e a r , h o l d s a s l o n g a s one m e a s u r e s o n l y d i r e c t i o n s and l i n e a r p o l a r i z a t i o n s

o f t h e r a d i a t i o n s . The o b s e r v a t i o n o f t h e c i r c u l a r p o l a r i z a t i o n , h o w e v e r , i n t r o d u c e s a l s o odd i n t e g e r s .

R u le 2 s t a t e s t h a t t h e h i g h e s t t e r m i-n The e x p a n s i o n i s s m a l l e r t h a n , o r e q u a l t o , t h e s m a l l e s t o f t h e t h r e e n u m b ers 2 1 ^ , 2Lj_,

2L2 w here L^_ a n d L2 a r e t h e a n g u l a r momenta c a r r i e d away by a n d R2 . T h i s r u l e i m p l i e s i s o t r o p i c c o r r e l a t i o n i f t h e a n g u l a r momentum o f e i t h e r o f t h e r a d i a t i o n s ' o r o f t h e i n t e r m e d i a t e s t a t e e q u a ls' 0 o r 1/2.

R u le 3 e x p r e s s e s t h e f a c t t h a t t h e c o e f f i c i e n t s A^ f o r a c a s c a d e c a n be b r o k e n up i n t o tw o f a c t o r s , e a c h f a c t o r d e p e n d i n g on o n l y

o n e t r a n s i t i o n o f t h e c a s c a d e . The s e c o n d f a c t o r Ak ( b , c ) f o r i n s t a n c e i s

e n t i r e l y d e t e r m i n e d by t h e p r o p e r t i e s o f t h e l e v e l s b and c o f t h e

A

r a d i a t i o n R2 .

E q u a t i o n 1 . 1 i s v e r y g e n e r a l i n d e e d , and w i t h t h e p r o p e r

e v a l u a t i o n o f A-^ i t a p p l i e s t o a l l two s t e ? c a s c a d e s ^ ~ J y e t c . a s w e l l a s t o n u c l e a r s c a t t e r i n g e x p e r i m e n t s and n u c l e a r d i s i n t e g r a t i o n s .

K i n t e g e r , e v e n (1)

^ Kmax. ^ ( 2 I b> 2L1> 2L2> ( 2 )

7

N u m e r i c a l v a l u e s f o r t h e f a c t o r s Ax ( a , b ) f o r g i v e n

p r o p e r t i e s o f t h e l e v e l s ( a ) a n d ( b ) a n d o f t h e r a d i a t i o n R h a v e b e e n c a l c u l a t e d a n d t a b u l a t e d f o r m o st c a s e s o f i n t e r e s t .

As a n e x a m p le o f t h e a b o v e , c o n s i d e r t h e d i r e c t i o n a l c o r r e l a t i o n b e tw e e n s u c c e s s i v e g a r a a r a y s i n t h e c a s c a d e shown i n

F i g u r e 1 . 2 ( a ) ( I a = 2 + , l b = 2 + , I c <» 0+ ) . The s e c o n d gamma r a y i n

t h i s c a s c a d e m u st be p u r e q u a d r u p o l e , a n d f r o m t h e r e f e r e n c e s i n t h e

l i t e r a t u r e , we f i n d t h e c o r r e s p o n d i n g d i r e c t i o n a l c o r r e l a t i o n f a c t o r s A2 ( b , c ) = - 0 . 5 9 7 6 , A4 ( b , c ) = - 1 . 0 6 9 . I f we assum e t h e f i r s t t r a n s i t i o n

t o be p u r e a l s o , f o r i n s t a n c e d i p o l e r a d i a t i o n ( L i = 1) , we f i n d i m m e d i a t e l y A2 ( a , b ) = -0.4-183 a n d A4 ( a , b ) = 0 a n d h e n c e

A2 = 0 .2 5 0 0 an d Ai; = 0 . T h e s e v a l u e s h a v e t o be c o m p a red w i t h e x p e r i m e n t .

.j. «i_ j. G e n e r a l l y , h o w e v e r , t h e f i r s t t r a n s i t i o n i n a 2 —5 * 2 ---5 * 0 c a s c a d e w i l l be m ix e d , i . e . t h e r a d i a t i o n c a r r i e d away i n v o l v e s m ore t h a n o n e s i n g l e v a l u e o f a n g u l a r momentum. A n g u l a r momentum s e l e c t i o n r u l e s a l l o w Lj_ = 1 , 2 , 3 o r 4 . As a r u l e o n l y t h e tw o l o w e s t m u l t i p l e s ,

Lj_ = 1 , a n d = 2 , o c c u r w i t h m e a s u r a b l e i n t e n s i t y [ c f . F i g u r e 1 . 2 ( b ) ] , The c o e f f i c i e n t s A2 ( a , b ) a n d A4. ( a , b ) and h e n c e a l s o Ag an d At; now

become c o n t i n u o u s f u n c t i o n s o f t h e m i x i n g r a t i o £ , w hich i n t h i s c a s e i n d i c a t e s t h e r a t i o o f a m p l i t u d e o f t h e q u a d r u p o l e t o t h a t o f t h e d i p o l e

c o n t r i b u t i o n ( & i s d e f i n e d a s t h e r a t i o o f t h e t o t a l i n t e n s i t y o f t h e L* p o l e t o t h a t o f t h e L p o l e ) .

The m o st c o n v e n i e n t r e p r e s e n t a t i o n o f t h e d e p e n d e n c e o f A2 and A^ on i s a p a r a m e t r i c p l o t [ C o le m a n , ( 1 9 5 8 ) ] : o n e d r a w s t h e c u r v e

A2

=

)

=

(<r

F r a u e n f e l d e r ( 1 9 5 5 ) , 3 i e d e n h a r n an d Rose ( 1 9 5 3 ) , F e r e n t z an d R o s e n z w e ig

a

b

c

I a =2+

V 2+'

I c = ° 1

L, = 1

L i = 2

( a ) ( b )

F i g u r e 1 . 2 A c a s c a d e .

( a ) 2+--- ^ 2+--- s * Q + c a s c a d e ( w h e r e t h e

s u p e r s c r i p t + d e n o t e s e v e n p a r i t y ) , o c c u r r i n g f r e q u e n t l y i n e v e n - e v e n n u c l e i .

9

( 1 9 5 5 ) , a n d Arns a n d W iedenbeck ( 1 9 5 8 ) ] . A p a r a m e t r i c p l o t f o r t h e c a s c a d e

2 +--- ^ - 2 +--- s » 0 + i s shown i n F i g u r e 1 . 3 . The e x p e r i m e n t a l p o i n t s w i t h c o o r d i n a t e s A2j6Xp > and Ai|j6Xp^ m ust l i e on t h i s c u r v e a n d t h u s y i e l d

v a l u e s f o r s i g n a n d m a g n i t u d e o f t h e m i x i n g r a t i o

S ' .

At t h i s p o i n t we r e c a l l t h e f o r m o f t h e A^ c o e f f i c i e n t s , v i z ,

Ak = Ak ( a , b ) Ak ( b , c ) . From t h e t h e o r y o f a n g u l a r c o r r e l a t i o n s i t may be

shown [ B i e d e n h a r n a n d R o s e , ( 1 9 5 3 ) ] t h a t t h e i n d i v i d u a l f a c t o r s Aj< ( a , b ) and Aj^ ( b , c ) a l s o f a c t o r i n t o tw o s e p a r a t e f a c t o r s n a m e ly i n t o a

" n u c l e a r f a c t o r " w h ic h d e p e n d s on s p e c i f i c n u c l e a r p r o p e r t i e s , a n d a

" g e o m e t r i c a l f a c t o r " w h ic h d e s c r i b e s t h e g e o m e t r y o f t h e p r o b l e m .

I t i s c o n v e n i e n t t o a d o p t t h e a n g u l a r c o r r e l a t i o n f u n c t i o n

a s t h e s t a n d a r d r e p r e s e n t a t i o n o f t h e " g e o m e t r y " o f t h e c o r r e l a t i o n p r o c e s s

a n d t o e x p r e s s t h e a n g u l a r c o r r e l a t i o n f o r d i f f e r e n t r a d i a t i o n s by d e f i n i n g

a m u l t i p l i c a t i v e p a r t i c l e p a r a m e t e r bj< t o e f f e c t t h e n e c e s s a r y c h a n g e f ro m

t h e s t a n d a r d c o r r e l a t i o n . Of c o u r s e , t h e c h o i c e o f t h e c o r r e l a t i o n a s t h e s t a n d a r d o n e i s a r b i t r a r y . We c o u l d h a v e c h o s e n t h e

c o r r e l a t i o n w h i c h , f ro m t h e p o i n t o f v ie w o f a n a l y t i c a l s i m p l i c i t y an d

a v a i l a b i l i t y o f n u m e r i c a l r e s u l t s , would h a v e be e n a l m o s t e q u a l l y s u i t a b l e .

However, t h e f a c t t h a t t h e p a r i t y r e s t r i c t i o n p r e c l u d e s t h e m i x i n g o f e v e n a n d odd L m i l i t a t e s a g a i n s t t h i s c h o i c e . F o r ^ - r a y s t h e p a r i t y f o r g i v e n L d e p e n d s on t h e c h a r a c t e r o f t h e r a d i a t i o n ( e l e c t r i c o r m a g n e t i c ) and

m i x t u r e s o f a l l L, L* p a i r s ( c o n s i s t e n t w i t h a n g u l a r momentum c o n s e - r v a t i o n ) a r e p e r m i s s i b l e i n p r i n c i p l e .

P a r t i c l e p a r a m e t e r s b^ may be i n t r o d u c e d t o e x p r e s s t h e a n g u l a r

c o r r e l a t i o n b e tw e e n a n y tw o s u c c e s s i v e r a d i a t i o n s . The d e t a i l e d i n f o r m a t i o n o b t a i n a b l e f ro m t h e u s e o f t h e s e p a r a m e t e r s i s beyond t h e s c o p e o f t h i s w ork.

0.50 _0.8

-0.7 0.7,

-0.6

0.6

0.2 0,—

-0.5

as

IS8

; 0 -4 0.!0 0.4

-0.5

0.5

-0.2 _0.2

-0.1

■> A -0.30 -0 . 2 0 -0.10 0.10 *0.20 -030 -0.40

F i g u r e 1 . 3 A p a r a m e t r i c p l o t .

P a r a m e t r i c p l o t o f t h e d i r e c t i o n a l c o r r e l a t i o n c o e f f i c i e n t s

A2 an d A14 f o r t h e c a s c a d e 2+--- ^ 2+--- > 0+ . & i s t h e m ix i n g r a t i o ( q u a d r u p o l e t o d i p o l e r a d i a t i o n ) i n t h e f i r s t

t r a n s i t i o n . The c u r v e i s l a b e l e d w i t h t h e v a l u e s o f t h e

I :

1 1

Of p a r t i c u l a r i n t e r e s t t o u s h e r e i s t h e —j l a n g u l a r

c o r r e l a t i o n s i n c e t h e e x p e r i m e n t p e r f o r m e d i n t h i s t h e s i s was t h e

*• 198

679 Kev - 4-12 Kev jSl. a n g u l a r c o r r e l a t i o n i n Au . A new e x p e r i m e n t a l t e c h n i q u e i n v o l v i n g t h e b^ p a r t i c l e p a r a m e t e r s was e m p lo y e d .

( v i ) The New T e c h n iq u e

( a ) I n t r o d u c t i o n

A new e x p e r i m e n t a l t e c h n i q u e f o r t h e s t u d y o f t h e i n t e r n a l c o n v e r s i o n p r o c e s s h a s b e e n d e v e l o p e d . The new m ethod i m p l i e s a n i n d i r e c t

a n d a c c u r a t e d e t e r m i n a t i o n o f t h e p a r t i c l e p a r a m e t e r b2 f r o m an

e x p e r i m e n t a l m e a s u re m e n t o f t h e bi*. p a r t i c l e p a r a m e t e r . I t i s shown t h a t t h e a c c u r a c y i n t h e v a l u e f o r b2 t h u s o b t a i n e d i s c o n s i d e r a b l y h i g h e r t h a n w ould be p o s s i b l e o n t h e b a s i s o f a d i r e c t m e a s u r e m e n t o f b2 . On t h e b a s i s o f t h e t h e o r y o f G reen a nd R ose (1958)^ t h e p a r a m e t e r b2 when a c c u r a t e l y d e t e r m i n e d , may be u s e d t o d e r i v e e x p e r i m e n t a l v a l u e s f o r a n y

d y n a m ic c o n t r i b u t i o n s . t o t h e c o n v e r s i o n p r o c e s s i f t h e y a r e p r e s e n t .

H ow ever, t o be d e c i s i v e f o r t h i s a n a l y s i s b2 h a s t o be known t o w i t h i n a f e w p e r c e n t . An e x p e r i m e n t a l a c c u r a c y o f t h i s o r d e r h a s s o f a r n o t b een

p o s s i b l e t o a c h i e v e .

( b ) The New Method F o r A c c u r a t e 0 2 D e t e r m i n a t i o n s

The u s u a l e x p a n s i o n o f t h e c o r r e l a t i o n f u n c t i o n W (O ) i n t e r m s

o f L e g e n d r e p o l y n o m i a l s , r e a d s , i n t h e c a s e o f gamma - gamma d i r e c t i o n a l

c o r r e l a t i o n s :

W( f r y ) = £ A2i< ( f r y ) P2K ( c o s £ b ) ( 1 . 3 ) F o r p u r e gamma - e l e c t r o n ( £ ,- J L ) c o r r e l a t i o n s a s i m i l a r e x p a n s i o n a p p l i e s :

W ( ^ ^ j C ) = £ A2K P2K ( c o s S ') ^

= ^ b2K A2K ( f f ) P2K ( c o s &■)

1 2

Thus f ro m a s t u d y o f t h e gamma - e l e c t r o n a n d gamma - gamma c o r r e l a t i o n s m e a s u r e d w i t h t h e same s o u r c e a n d u n d e r i d e n t i c a l

e x p e r i m e n t a l c o n d i t i o n s we o b t a i n :

A2 ( £ a T ) ( 1 . 5 )

b 2 = _________

a 2 ( )

A4 ( ^ Jl' ) ( 1 . 6 ) b 4 = ____________

A4 ( >

T h i s c o m p a r i s o n m e th o d h a s t h e a d d i t i o n a l a d v a n t a g e t h a t t h e p a r t i c l e p a r a m e t e r s d e t e r m i n e d by ( 1 . 5 ) a n d ( 1 . 5 ) a r e u n a f f e c t e d by a l l

a t t e n u a t i o n s c a u s e d by e x t r a - n u c l e a r f i e l d s s i n c e t h e a t t e n u a t i o n c o e f f i c i e n t s

i f p r e s e n t d i s a p p e a r i n t h e r a t i o s .

The v a l u e f o r b2 t h u s o b t a i n e d c a n n o t b e s u f f i c i e n t l y

a c c u r a t e l y d e t e r m i n e d t o b e o f s i g n i f i c a n c e f o r a n a l y s i s o f d y n a m ic e f f e c t s . H o w e v e r, t h e . two p a r t i c l e p a r a m e t e r s a r e n o t m u t u a l l y i n d e p e n d e n t . As

shown by B i e d e n h a r n and Rose ( 1 9 5 3 ) b2 and b4 a r e i n t e r - r e l a t e d by means o f a r e c u r s i o n f o r m u l a . I n t h i s p a r t i c u l a r c a s e , w i t h t h e n u m e r i c a l

c o n s t a n t s i n t r o d u c e d , t h e r e c u r s i o n f o r m u l a r e a d s :

b2 = 1.4. - b4 ( 1 . 7 ) 2; 5

C o n s e q u e n t l y we may u s e o u r e x p e r i m e n t a l l y d e t e r m i n e d bi* v a l u e t o d e r i v e a c o r r e s p o n d i n g v a l u e f o r b2. M o r e o v e r , b e c a u s e o f t h e

a p p e a r a n c e o f t h e n u m e r i c a l f a c t o r s 1 . 4 an d t h e 2 . 5 i n t h e d e n o m i n a t o r t h e

r e l a t i v e e r r o r i n b2 i s c o n s i d e r a b l y s m a l l e r t h a n t h e c o r r e s p o n d i n g e r r o r i n b ^ . T h u s , p r o v i d e d bi+ c a n b e m e a s u r e d w i t h a r e a s o n a b l e d e g r e e o f

a c c u r a c y , t h e c o r r e s p o n d i n g v a l u e f o r b2 w i l l b e q u i t e p r e c i s e l y d e t e r m i n e d .

19 8

i s t h e c a s e f o r t h e 679 Kev - ^12 Kev gamma gamma c a s c a d e i n Au

T h i s i n d i r e c t way t o a r r i v e a t a b2 v a l u e from a

m e a s u r e m e n t o f b 4 , r e s u l t s i n an a c c u r a c y w h ic h i s c o n s i d e r a b l y h i g h e r t h a n i s p o s s i b l e t o a c h i e v e i n any d i r e c t d e t e r m i n a t i o n o f b2

< i * • ) .

from t h e m e a s u r e d A2 ( * V an d A2

CHAPTER I I

CORRECTIONS FOR FINITE SOLID ANGLE

( i ) C o r r e c t i o n s f o r t h e Gamma Chan n e l

The c o i n c i d e n c e c o u n t i n g r a t e p e r u n i t s o l i d a n g l e s i n &■ dio- cL<p

(w h e re t h e a n g l e s r e p r e s e n t t h e r e l a t i v e o r i e n t a t i o n o f t h e p r o p a g a t i o n

v e c t o r s o f t h e tw o r a d i a t i o n s ) f o r a n a n g u l a r c o r r e l a t i o n m e a s u r e m e n t i s p r o p o r t i o n a l t o :

W(S-) = ^2K ^2K ( c o s (2.1)

K

The m e a s u r e d c o r r e l a t i o n , h o w e v e r , i s l e s s t h a n t h e t r u e c o r r e l a t i o n b e c a u s e o f t h e f i n i t e s o l i d a n g l e t h a t t h e tw o d e t e c t o r s m u st s u b t e n d i n

a n y f e a s i b l e e x p e r i m e n t .

When t h e d e t e c t o r s f o r t h e two r a d i a t i o n s s u b t e n d f i n i t e s o l i d

a n g l e s a n d - r v ^ a t t h e s o u r c e , i t i s a d v i s a b l e t o m o d i f y t h e t h e o r e t i c a l

c o r r e l a t i o n a n d c o m p a re t h i s s m e a r e d c o r r e l a t i o n w i t h t h e m e a s u r e d o n e .

The g e o m e t r y e n v i s a g e d i s shown i n F i g u r e 2 . 1 . The d e t e c t o r s ( s c i n t i l l a t i o n c o u n t e r s ) a r e a s su m e d t o be c r y s t a l s c u t i n t h e fo rm o f r i g h t c i r c u l a r

c y l i n d e r s w i t h t h e b a s e o r i e n t e d t o w a r d s t h e s o u r c e . The s o u r c e , a t t h e

o r i g i n , i s on t h e i n t e r s e c t i o n o f t h e a x e s o f t h e c y l i n d e r s . I n t h i s c a s e ,

a s t h e f o l l o w i n g s h o w s , t h e f o rm o f t h e c o r r e l a t i o n f u n c t i o n i s u n c h a n g e d a n d e a c h c o e f f i c i e n t A2K becom es m u l t i p l i e d by a n a t t e n u a t i o n f a c t o r f o r

w hich o n e c a n o b t a i n an e x a c t a n d v e r y s i m p l e e x p r e s s i o n . T a k i n g i n t o a c c o u n t t h e a b s o r p t i o n o f t h e r a d i a t i o n i n e a c h c r y s t a l , we i n t r o d u c e t h e

f o l l o w i n g n o t a t i o n . The d i s t a n c e f ro m t h e s o u r c e t o t h e f r o n t f a c e o f e a c h c r y s t a l I s h , t h e t h i c k n e s s i s t , and r i s t h e r a d i u s o f t h e c r y s t a l . I f

x( B) i s t h e d i s t a n c e t r a v e r s e d by t h e r a d i a t i o n i n c i d e n t on t h e c r y s t a l a t

-rxtsj

t

I__

F i g u r e 2 . 1 A g e o m e t r y o f tw o ^ d e t e c t o r s .

G eom etry f o r f i n i t e r e s o l u t i o n i n a n g u l a r momentum. The n o t a t i o n a p p l i e s f o r s i n g l e s c o u n t s a s i n a n g u l a r d i s t r i b u t i o n m e a s u r e m e n t s . The a z i m u t h a n g l e s @t a nd a r e m e a s u r e d w i t h r e s p e c t t o ' . t h e c y l i n d e r a x e s a n d A2 r e s p e c t i v e l y .

16

f u n c t i o n would b e :

_ f j - s

W ( s . ' ( \ -

m

. r * ‘) ( l - 4 .

)

W f o J = 7 --- f f t t --- ? n --- ( 2 - 2) j J j L , d - n - s . ( ( - - “■ ' i ( I

-w h e re xj_ a n d X2 r e f e r t o t h e tw o c r y s t a l s , and ef-A-a a r e t h e s o l i d

a n g l e e l e m e n t s f o r e a c h r a d i a t i o n , a n d O- i s t h e a n g l e b e tw e e n t h e i r p r o p a g a t i o n v e c t o r s , w h i l e & i s t h e a n g l e b e tw e e n t h e c y l i n d e r a x e s .

The r e q u i r e d i n t e g r a l s a r e o f t h e f o r m :

X jI = ^ ^C 0 S ^ ^ I ' r X * ) ( 2. 3)

w h ic h by a p p l i c a t i o n o f t h e L e g e n d r e p o l y n o m i a l a d d i t i o n t h e o r e m r e d u c e s t o :

w here

I =

P ^ c o s &) J ^ t )

(

.

( 2 . 5 )

o

^ b e i n g t h e h a l f a n g l e s u b t e n d e d on t h e f r o n t f a c e , a n d (1) a n d (2)

r e f e r t o t h e r e s p e c t i v e c r y s t a l s . We h a v e t o e v a l u a t e Jjj. t o o b t a i n t h e

r e q u i r e d c o r r e c t i o n f a c t o r s . F i n a l l y , we h a v e :

V / { 6 )

( t. o i / ( To.u ) /

17

The a t t e n u a t i o n f a c t o r i s :

~ ' ( h

w

f o r s i m i l a r d e t e c t o r s , an d f o r ^ * r a y s w i t h s i m i l a r a b s o r p t i o n c o e f f i c i e n t s .

F o r a s i n g l e d e t e c t o r a s w ould be u s e d i n a n a n g u l a r d i s t r i b u t i o n m e a s u r e ,

w i t h t h e d a t a r e p r e s e n t e d by e q u a t i o n 1.1, t h e a t t e n u a t i o n f a c t o r would be s i m p l y :

U s - ) -

-

To

V a l u e s f o r

j

j

( F i g u r e s 2 . 2 a n d 2 . 3 ) by West ( 1 9 5 9 ) f o r c r y s t a l s i z e s o f 1 1 / 2 i n c h e s i n d i a m e t e r by 1 i n c h l o n g , 1 3 / 4 " x 2 " , 2" x 2 " , a n d 3" x 3" f o r s o u r c e t o

c r y s t a l f a c e d i s t a n c e s o f 3 , 5 , 7 and 10 cm.

In t h e c o r r e l a t i o n e x p e r i m e n t t o be d e s c r i b e d

( C h a p t e r V) a 2" x 2" c r y s t a l and a 1 3 / 4 " x 2 " c r y s t a l were u t i l i z e d a t a d i s t a n c e o f 10 cm. f ro m t h e s o u r c e . The g r a p h s ( F i g u r e s 2 . 2 a n d 2 . 3 )

t h e n y i e l d t h e f o l l o w i n g v a l u e s f o r t h e c o r r e c t i o n f a c t o r s :

i j t j - ) = 1 = ° - 96S

Jo f o r 2" x 2" c r y s t a l \

- 0 . 8 8 6 Jo

* 5 : = ° ‘ 9 7 3

To f o r 1 3 / 4 " x 2" c r y s t a l = J V = 0 . 9 1 2

To

i.00

0.S5

i n

j ;-■ i : 1: ; : | I 1;,1 • I

“ T - : ;

i7LFF::;lFIF'-i 7 iOc-ig'RTkv

[7 c rai-o::

:■: i ::;j;=:;• i :;: j j

'i; E E jiiiii:;;:: ip ;;i i Lix"

^ m m r m

0.90

mm.

W%

\

rr*jrr

~iE.hL

=~T~

m

O

tz=rr:*. 0 .S 5 TitL

■ ?:•: L iii_

yj m . | -; ! ;;: •:'1 ; ' .1

jii: 'Kc i:-;.: • :;..i;: ::;;) - y | ; : : j ; j L ' l - ^ p i i L L :' l i E E : F i i F i M 7 F F F ! “

t I-'--! :'i "Li-'u .iyfF'iy'Uy - y:-.

±Em

- .j;. •:, . - ; :::■ i :.r

■ -'-TV iv - ’ - lr j - r :— ' |-~ '■—r:: j

i ± :. x £ :

... ... ... LHHyNLLi/:

' ■ : ! t r.::-. .-... .. ■

T y ^ W -H i'- ; : - : :

= H | ^ : t = ; ; i ; ; : n !. ;• • •

' " “"7: 'F:;r-r;rrrr:r-rrr: rtrp^rrrrr;

0.75

^£VrT.;v--T -: •■• v : v / • : E ' ’ ■•' V " ' ' - ; : - - E E y E : ' : r ' Fi Fi Fi F' FyFv' :

7

;

'

..ELL.-.;;. 1 - : - JiLi- LiiiililliEEELF

E'i:*::*::!'

'I E — - i ■ l E -rrrF—— W i F —u ; ■--- r-.-— l.... E -v : ' ■■: E : v r.F :r r j H " ' : H ' : F: :;v'.',:r-f-r

■ ,•.’!'•••)•. ih :;.t;■ :*i: . •" ■ ...I ; / 1 .-.jr .j : • . ; • : ::: '!. :.*•.

ji f{ •: - ■-•:!- =• v:E1LL;L,.: V...

i ^ L : l i - i ' F:'-:-1 : ; ‘" :: I. : F:!:'i: :.--: :::v- :‘:r

I S B p l i l l i M E W s S S ! m &

■— jyri— ■y— ;— — • • ~ 1.!.1 T\ T1 ‘ T ’■ .' r*.'T ..!

' ' E - i y i E f f l K F

mm

liiiiiyLi'r-"!: v : 7 ;i:7 r-;. r i ii

:i i

0.70

Figure 2 .3

20

( i i ) C o r r e c tio n s f o r th e E le ctro n Channel

Apart from th e c o r r e c t io n f a c t o r s f o r th e

— ch an n el d is c u s s e d ab ove, we have in th e c a s e o f an e le c tr o n

c o r r e l a t i o n , a l s o t o a p p ly due c o r r e c t io n s f o r th e sm earing ou t o f th e

d i r e c t i o n a l c o r r e la t io n owing t o th e f i n i t e a p e rtu r e a n g le subtended by

th e a c c e p te d cone o f e l e c t r o n s in th e b eta sp ectr o m e ter .

In t h i s c a s e , th e form o f th e jl~— c o r r e c tio n f a c t o r , denoted

5g _ I s i m i l a r t0 th e p r e v io u s ly d is c u s s e d c o r r e c tio n f a c t o r

s in c e th e g e o m e tr ic a l arrangem ents o f d e t e c t o r and so u rce in both

c a s e s a r e s im ila r (compare F ig u r es 2 .4 and 2 . 1 ) .

Then, f o r th e ch a n n el th e a p p r o p r ia te e x p r e s s io n f o r

( ^"J i s g iv e n by:

T

x J f ^ i c o s d ) Q ( d ) S l H s/<K

£ ( * “)= - = * --- ( 2 .9 )

j $ ( . d ) s i h d d oC

£K

where d . a n d r e p r e s e n t th e minimum and maximum e le c t r o n t a k e - o f f a n g le s I i

(F ig u r e 2 . 4 ) and (^(sO i s t h e d e t e c t o r e f f i c i e n c y f o r th e e m issio n a n g le c{.

Equation ( 2 .9 ) i s t o be compared w ith th e e x p r e s s io n f o r

^ » namely:

f

f n

f

- T * ( p ) s

j

) Po ( C o S $ ) i

} S l h & & 5

o

I t i s e v id e n t from a com parison o f e q u a tio n s ( 2 .9 ) and ( 2 .1 0 )

and o f F ig u r e s 2 .1 and. 2 .4 th a t th e e x p r e s s io n <p(«0s'*et in th e je.: c a s e i s

Lens

B a f f le - '

D e t e c t o r C r y s t a l

F igure 2 . 4 A geom etry o f an 4 . - ch a n n el.

22

e q u iv a le n t t o th e in t e g r a t io n o v er B from O— in th e £ - c a s e .

In g e n e r a l, t h e n , t h e e x p r e s s io n f o r th e e x p e r im e n ta lly

measured d i r e c t i o n a l c o r r e la t io n becom es:

Ni

W (© •) = 1 + A2 P2 (c o s © -) + A4 Pj| (c o s© -) ( 2 .1 1 )

where $■ and ^ are alw a y s £. J and r e p r e se n t th e t o t a l sm earing out o f th e

a n g u la r c o r r e la t io n p a tte r n due t o th e f i n i t e a p e r tu r e s in t h e two c h a n n e ls .

In p a r t ic u l a r , f o r an c o r r e la t io n :

Z ^ ^ ^ I (2.1 2)

, $v 2. /

R etu rn in g t o e q u a tio n ( 2 . 9 ) , i t may be n o ted t h a t f o r sm a ll

tr a n s m is s io n s e t t i n g s where i s very n e a r ly eq u a l t o o{^ > we o b ta in :

^

( £ )

t.JI. fa ( “ > * * ) ( 2 .1 3 )

^ ( 2 .1 4 )

T hese f u n c t io n s , ( i . e . P2 ( c o s cO and P4 ( c o s o() ) a re reproduced in F ig u re 2 .5 .

At h ig h e r tr a n s m is s io n s e t t i n g s th e approxim ate r e l a t i o n s ( 2 .1 3 ) and ( 2 .1 4 )

no lo n g e r a p p ly . Both f a c t o r s d e c r e a s e , more r a p id ly th an f As in c e th e

t r a n s m is s io n in c r e a s e s w ith ot,, and la r g e r o (-v a lu e s t h e r e f o r e dom inate th e

c o n t r ib u t io n s t o th e and i n t e g r a l s , an e f f e c t which i s more pronounced

f o r t h e f a c t o r than i t i s f o r th e f a c t o r . For t h i s r e a so n th e

e x p e r im e n ta lly measured and f a c t o r s a t h ig h e r tr a n s m is s io n s e t t i n g s no

lo n g e r corresp on d t o th e g e o m e tr ic a l mean v a lu e f o r o(. We u s e , o f c o u r s e ,

.0 .7 5 _

0 ,5 0

-20

15 10

F i g u r e 2 .5 Graphs o f P2 ( c o s c O and P4 ( c o s c O ,

2 4

( i i i ) The P0 , ?2 , P4 B a f f le s

Three s p e c ia l b a f f l e s , th e s o - c a l l e d Pq, P2 and P4 b a f f l e s

(F ig u r e 2 . 6 ) , a r e used t o e v a lu a te e q u a tio n s ( 2 . 8 ) and ( 2 .9 ) e x p e r im e n ta lly .

The ? 2 and P^ b a f f l e s have been made such t h a t th e y s im u la te

a and- a p4 d is t r i b u t i o n r e s p e c t i v e l y . The reduced c o u n tin g r a t e

t h e r e f o r e becomes a d ir e c t measure o f t h e sm earin g o u t e f f e c t s . The Pq

b a f f l e i s u se d , o f c o u r s e , fo r - n o r m a liz a t io n . The P^ b a f f l e and th e P2

b a f f l e have o p en in g s c u t in them each o f which has an a rea g iv e n by:

811

r e s p e c t i v e l y , where K = c o n s t a n t , w h ile th e P0 b a f f l e has o p e n in g s o f

sp ectro m eter e n tr a n c e , th e count r a t e s o b ta in e d w i l l be p r o p o r tio n a l t o ( 2 .1 5 )

* 1

( 2 .1 6 )

a rea :

( 2 .1 7 )

c£i

I f th e P2 or P4 , and th e Pq b a f f l e s a re now p la c e d in tu rn a t th e

th e a r e a s o f th e b a f f l e o p e n in g s. Hence:

«

H nJ m CM cu

•Q

<o CM 2 3 OS •H

u,

>4h

Mh

(0 n

o CL,

«o

B

a

f

f

27

and

o ] f Q U ) Pt+ ( C ° S << ) S i h f C J p(

r

h \

I <J____________________________________ f

(2.19)

1/ ( f t ) ^ *

| (5 («0

Sl h ck J <L

* (

where N(P2)» N (P i|), N(Pq) a r e t h e r e s p e c t i v e cou n t r a t e s w ith th e

?2» **4 an<* p0 b a f f le s . i n p la c e .

( i v ) A F in a l Note

I t i s n o ted h ere t h a t t h e a n g u la r c o r r e la t io n f u n c t io n i s

o f t e n w r itte n a s f o ll o w s :

W(d>)s l + ^ G2 A2 P2 ( c o s Pi| (cos<S«) ( 2 .2 0 )

where G2 and G4 a r e a tt e n u a t io n f a c t o r s . In th e c a s e o f £ - ja.”

c o r r e la t io n s th e r e a r e t h r e e p o s s i b l e k in d s o f a tte n u a tio n -m e c h a n ism s,

n a m e ly ^ a f t e r - e f f e c t s o f K - h o le fo r m a tio n , e le c t r o n s t r a g g li n g and s t a t i c

quadrupole in t e r a c t io n [Gerholm , Holmberg, P e t t e r s o n , ( u n p u b lis h e d ) ] . For

t h e exp erim en t in Au-1,98 t o be d e s c r ib e d , i t was found by t h e above a u th o rs

. t h a t G2 and Gj| a r e e s s e n t i a l l y u n it y .

CHAPTER I I I

THE BAFFLE SYSTEMS

( i ) The Gerholm Instrum ent

The sp ec tro m e te r used in t h i s ex p erim en t i s due to

T. R. Gerholm ( 1 9 6 1 ) . M agnetic momentum s e l e c t i o n i s p rovid ed in th e

e le c t r o n c h a n n el which c o n s i s t s o f an ir o n in c a p s u le d lo n g l e n s b eta ray

sp ec tr o m e te r w ith " tr ia n g u la r f i e l d " f o c u s s i n g . The in stru m en t i s a

m o d ific a tio n o f th e e l e c t r o n - e le c t r o n c o in c id e n c e sp e c tr o m e te r e a r l i e r

d ev elo p e d by Gerholm.

F ig u r e 3 .1 ( a ) i s a sch em a tic o f th e a ”--a." s p e c tr o m e te r ,

w hereas F igu re 3 .1 (b ) i s a sch em a tic o f t h e s p e c tr o m e te r . )

The " tr ia n g u la r f i e l d " d i s t r i b u t i o n i s d e p ic te d in F igure 3 .2 .

M agnetic f i e l d m easurem ents [Gerholm , ( 1 9 6 1 ) ] show t h a t th e so u r c e i s in

a f i e l d f r e e r e g io n and t h a t th e s tr e n g th o f t h e m a g n etic f i e l d in c r e a s e s

r o u g h ly p r o p o r tio n a l t o t h e d is t a n c e from th e s o u r c e , when measured a lo n g

t h e a x i s o f symmetry.

( i i ) Ring Focus and B a f f le s

In a l e n s sp e c tr o m e te r , th e e l e c t r o n s le a v in g th e so u rce a t

an en tr a n c e a n g le t o th e a x i s e v e n t u a lly r e tu r n t o th e a x i s a g a in t o

form an im age. The s p h e r ic a l a b e r r a tio n o f t h i s image i s la r g e f o r many

f i e l d sh a p e s. A p o s it io n o f minimum s p h e r ic a l a b e r r a tio n g e n e r a lly o c c u r s

b e fo r e th e e l e c t r o n s c r o s s th e a x i s a t th e s o - c a l l e d r i n g f o c u s (F ig u r e 3 . 3 ) .

A g r e a t improvement in perform ance i s o b ta in e d by p la c in g th e d e f in in g

b a f f l e s a t t h i s f o c u s r a th e r th an a t th e d e t e c t o r [ N ic h o ls , (1 9 5 4 .)],

/ / ■

' / / L’Vyfia

/ / c ■(»««.

' / /_{' / /} IC*IU>T€ UCHt COUC1W _{fM-JHO!«XUT-n.H*} V / o,*tMr»i*Ka Mfi(

f / / u*r\£

F ig u r e 3 .1 ( a ) E le c tr o n - E le c tr o n S p ectro m eter

F igure 3 .2

125"

s »»•

-r Z Uml

31

D e te c to r S p ectro m eter A x is

S ource

F igu re 3 .3 Ring Focus

( i i i ) R e s o lu tio n and T ran sm ission

The r e s o lu t io n o f a sp ectro m eter i s u s u a lly d eterm ined by

o b s e r v in g i t s re sp o n se t o a so u rce o f m o n o -en erg etic e l e c t r o n s . When

t h e c o u n tin g r a t e i s p lo t t e d a s a f u n c t io n o f th e m agnetic f i e l d B, or

t h e momentum p , one o b ta in s r e s p o n se c u r v e s o f th e ty p e shown below

(F ig u r e 3 . 4 ) t

PO

p _ --- ^ F ig u re 3 .4

The r e s o lu t i o n R i s d e f in e d a s ( 4 J3/ p0 ) where Ap i s th e

momentum spread a t h a lf maximum and Pq i s t h e momentum a t th e peak c o u n tin g

r a t e . R i s independent o f p^. I t sh ou ld be n o ted t h a t each in stru m en t

h a s I t s own c h a r a c t e r i s t i c l i n e p r o f i l e determ ined by i t s own p a r t ic u la r

f o c u s s i n g p r o p e r t ie s , but t h i s d e f i n i t i o n o f R can be used f o r a l l o f them.

The tr a n s m is sio n to o f a sp ectr o m e ter i s d e fin e d a s th e

32

f r a c t i o n o f th e e l e c t r o n s o f momentum p em itte d from th e so u rce which

a r r iv e a t th e d e t e c t o r when th e in stru m en t i s fo c u s s e d on p. The

e x p r e s s io n ( / H-TT ) , where djn~ i s eq u al t o th e s o l i d a n g le subtended

by th e b a f f l e sy ste m , i s a good approxim ation o f w .

( i v ) The A2 and A^ B a f f le System s

( a ) S p e c ia l Remarks

The r e s o lu t i o n o f a m agnetic sp ec tro m eter i s e s s e n t i a l l y

d eterm ined by t h r e e f a c t o r s ; th e w idth o f th e s o u r c e , t h e s p h e r ic a l

a b e r r a t io n , and t h e w id th o f th e d e f in in g e x i t s l i t ( g e n e r a lly th e

a n n u la r r in g f o c u s e x i t ) . I f we n e g le c t t h e w id th o f th e so u rce and th e

s p h e r ic a l a b e r r a tio n t h e .r e s o l v i n g power i s determ ined s o l e l y by th e w idth

o f th e e x i t s l i t . Under t h e s e ap p ro x im a tio n s th e tr a n s m is s io n in c r e a s e s

p r o p o r tio n a l t o £ l h A » rt b e in g th e mean e m issio n a n g le . Thus, th e

f i g u r e s o f m erit o f a m agnetic sp ec tro m e te r would seem t o in c r e a s e w ith

in c r e a s in g e m issio n a n g le . The .gain in tr a n s m is s io n a t a g iv en r e s o lu t io n

i s , how ever, co u n terb a la n ced by an in c r e a se d s p h e r ic a l a b e r r a t io n . I f th e

s p h e r ic a l a b e r r a tio n i s ta k en i n t o a c c o u n t, th e tr a n s m is s io n Uj ( otj

c o n sid e r e d a s a f u n c t io n o f t h e e m issio n a n g le e( has a maximum f o r a

c e r t a in v a lu e o f «( w hich, o f c o u r s e , sh ould be ta k en a s th e most fa v o u ra b le

c h o ic e f o r th e e m is s io n a n g le . I t may be shown [DuMond, (1 9 5 7 )] th a t f o r

a uniform f i e l d , o > u j has a broad maximum around 40° t o 4 5 ° . Both

t h e o r e t i c a l and e x p er im e n ta l s t u d i e s o f th e " t r ia n g u la r f i e l d " show th a t

i t s s p h e r ic a l a b e r r a tio n i s somewhat la r g e r than th a t o f th e uniform f i e l d

and th e r e f o r e W (A ) in t h i s c a s e w i l l have i t s maximum a t a lo w er c ^ -v a lu e ,

30° sh ould be c l o s e t o th e optimum v a lu e .

Thus, th e n e t e f f e c t o f th e s p h e r ic a l a b e r r a tio n i s t o l i m i t

33

fa v o u r a b le em issio n a n g le . The p o in t t o be n oted th e n , i s th a t th e

tr a n s m is sio n ( a t a g iv en r e s o lu t i o n ) i s n o t a proper f ig u r e o f m e r it f o r

a d i r e c t i o n a l c o r r e la t io n in stru m e n t. With in c r e a s in g em issio n a n g le

t h e r e w i l l be a sm earing o u t o f th e d i r e c t i o n a l c o r r e la t io n owing t o

t h e f i n i t e a p ertu r e a n g le subtended by th e a c c e p te d cone o f e l e c t r o n s in

th e s p e c tr o m e te r .

f a c t o r s , t h e r e f o r e , do n o t s e r io u s ly a f f e c t th e e x p e rim en ta l r e s u l t s .

The c o r r e c t io n f a c t o r s f o r th e e le c t r o n c h a n n e l, how ever,

d e c r e a se r a p id ly w ith in c r e a s in g em issio n a n g le . T h is im p lie s , in th e

f i r s t p la c e , t h a t th e e x p e r im e n ta lly measured e f f e c t becomes s m a lle r and

t h e r e f o r e more d i f f i c u l t t o d e term in e . The g a in in s t a t i s t i c a l accu ra cy

o b ta in e d a s a con seq u en ce o f h ig h er tr a n s m is s io n i s in t h i s way

c o u n te r b a la n c e d . O b v io u sly , one sh o u ld ch o o se a v a lu e f o r th e em issio n

a n g le which i s th e most fa v o u r a b le compromise between s t a t i s t i c a l

a ccu ra c y ( ] an<3 sm earing o u t ( A~i °<l j • S eco n d ly ,

th e c h o ic e o f a f a i r l y la r g e em issio n a n g le im p lie s t h a t ^ (ji‘ ] and

w i l l become r a th e r sm a ll and t h e r e s u l t s , t h e r e f o r e , more

s e n s i t i v e t o th e u n c e r ta in ty in (A") and -f .

(b ) C hoice o f E m ission Angle

The problem c o n s i s t s in f in d in g th e v a lu e f o r o( which

than th e tr a n s m is sio n should be c o n s id e r e d a s th e proper f ig u r e o f m erit

f o r an electron-gam m a d ir e c t io n a l c o r r e la t io n sp ec tro m ete r . I f we n e g le c t

th e s p h e r ic a l a b e r r a tio n and th e broad en in g cau sed by th e w idth o f th e Under normal c ircu m sta n ce s th e ^ .-c o r r e c tio n f a c t o r s ^

and a r e f a i r l y c l o s e t o u n ity and minor e r r o r s in t h e s e c o r r e c t io n

m axim izes th e e x p r e s s io n T h is e x p r e s s io n r a th e r

s o u r c e , th e tr a n s m is sio n may be w r itte n ( dJ r Co /»J r. x

f u r t h e r , we r e s t r i c t o u r s e lv e s t o a sm a ll a n gu lar in t e r v a l %o( around th e