UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

Some magneto-optical studies of paramagnetic salts at low temperatures Rieckhoff, Klaus Ekkehard 1959

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Item Metadata

Download

Media
831-UBC_1959_A6_7 R4 S6.pdf [ 9.55MB ]
Metadata
JSON: 831-1.0085919.json
JSON-LD: 831-1.0085919-ld.json
RDF/XML (Pretty): 831-1.0085919-rdf.xml
RDF/JSON: 831-1.0085919-rdf.json
Turtle: 831-1.0085919-turtle.txt
N-Triples: 831-1.0085919-rdf-ntriples.txt
Original Record: 831-1.0085919-source.json
Full Text
831-1.0085919-fulltext.txt
Citation
831-1.0085919.ris

Full Text

-SOME MAGNETO- OPTICAL STUDIES -OF PARAMAGNETIC SALTS •AT LOW TEMPERATURES*! h y .KLAUS EKKEHARD RIECKHOFF ,-B.So., U n i v e r s i t y o f B r i t i s h C o l u m b i a , 1958 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF -MASTER OF SCIENCE i n t h e D e p a r t m e n t o f P h y s i c s -We a c c e p t t h i s t h e s i s as c o n f o r m i n g t,o t h e r e q u i r e d s t a n d a r d THE UNIVERSITY OF B R I T I S H COLUMBIA September, 1959 SOME MAGNETO-OPTICAL STUDIES OP PARAMAGNETIC SALTS AT LOA7 TEMPERATURES ABSTRACT S h o r t r e s u m e s o f t h e t h e o r i e s o f p r o p a g a t i o n o f e l e c t r o -m a g n e t i c waves i n a n a n i s o t r o p i c medium, o f t h e F a r a d a y e f f e c t , and o f t h e i n f l u e n c e o f p a r a m a g n e t i c r e s o n a n c e on t h e F a r a d a y e f f e c t a r e g i v e n . The P o i n c a r e s p h e r e i s i n t r o d u c e d t o d e s c r i b e p o l a r i z e d l i g h t . A p a r a m a g n e t i c r e s o n a n c e s p e c t r o m e t e r i s d e s c r i b e d , w h i c h was m o d i f i e d so as t o a l l o w t h e s t u d y o f m a g n e t o - o p t i c a l phenom-ena u n d e r t h e i n f l u e n c e o f p a r a m a g n e t i c r e s o n a n c e . The s p e c t r o -m e t e r o p e r a t e d i n t h e 2-band u s i n g a 2K39 K l y s t r o n . The s a m p l e s were l o c a t e d i n a t r a n s m i s s i o n t y p e c a v i t y o p e r a t i n g i n t h e TE 101 mode, a n d im m e r s e d i n l i q u i d h e l i u m . The c a v i t y was p r o v i d e d w i t h h o l e s a l l o w i n g t h e p a s s a g e o f l i g h t t h r o u g h t h e sample i n a d i r e c t i o n p a r a l l e l t o t h e e x t e r n a l m a g n e t i c f i e l d . An o p t i c a l s y s t e m p r o v i d e d p l a n e - p o l a r i z e d m o n o c h r o m a t i c l i g h t ( ^ = 5461 A) i n c i d e n t on t h e sa m p l e . The l i g h t e m e r g i n g f r o m t h e sample p a s s e d t h r o u g h a Glan-Thompson p r i s m a n a l y z e r . The r e l a t i v e i n t e n s i t y o f t h e l i g h t p a s s i n g t h e a n a l y z e r c o u l d b e m e a s u r e d by a p h o t o m u l t i -p l i e r c i r c u i t a nd c o u l d b e d i s p l a y e d a s a f u n c t i o n o f t i m e on a n o s c i l l o s c o p e . E x p e r i m e n t s a r e d e s c r i b e d i n d e t a i l i n w h i c h t h e s p i n -l a t t i c e r e l a x a t i o n t i m e was m e a s u r e d as a f u n c t i o n o f t e m p e r a t u r e and e x t e r n a l m a g n e t i c f i e l d . I n t h e s e e x p e r i m e n t s , f o r a g i v e n t e m p e r a t u r e and m a g n e t i c f i e l d , t h e F a r a d a y r o t a t i o n was r e d u c e d b y p u l s e s o f m i c r o w a v e power o f v a r y i n g l e n g t h a p p l i e d t o t h e c a v i t y . The r e t u r n o f t h e F a r a d a y r o t a t i o n a s a f u n c t i o n o f t i m e t o i t s e q u i l i b r i u m v a l u e a f t e r t h e m i c r o w a v e power was c u t o f f c o u l d be i n f e r r e d f r o m t h e i n t e n s i t y v e r s u s t i m e r e l a t i o n s h i p o f t h e l i g h t t r a n s m i t t e d by t h e a n a l y z e r . P h o t o g r a p h i c r e c o r d s o f t h i s i n -t e n s i t y v e r s u s t i m e r e l a t i o n s h i p were o b t a i n e d and t h e r e l a x a t i o n t i m e was d e d u c e d f r o m t h e s e r e c o r d s . R e s u l t s o f t h e measurement o f t h e s p i n - l a t t i c e r e l a x a t i o n t i m e o f neodymium e t h y l s u l f a t e f o r f i e l d s b e t w e e n 780 a n d 2540 O e r s t e d t and t e m p e r a t u r e s between 1 . 3 8 ° K and 4.22°K a r e g i v e n . The r e l a x a t i o n t i m e s m easured w e r e o f t h e o r d e r o f .001 t o .1 s e c o n d s . The r e l a x a t i o n t i m e a p p e a r e d t o b e i n v e r s e l y p r o p o r t i o n -a l t o t h e t h i r d power o f t h e t e m p e r a t u r e and showed o n l y s m a l l f i e l d d e p e ndence, e x c e p t f o r a l a r g e d i p a t a f i e l d c o r r e s p o n d i n g t o t h e r e s o n a n c e f i e l d f o r t h e microwave f r e q u e n c i e s u s e d . W i t h i n t h e a c c u r a c y o f t h e e x p e r i m e n t s no e f f e c t o f t h e l e n g t h o f t h e mic r o w a v e p u l s e s on t h e r e l a x a t i o n t i m e c o u l d be o b s e r v e d . An e x p e r i m e n t on c e r i u m e t h y l s u l f a t e i s d e s c r i b e d , w h i c h showed t h a t t h e s p i n - l a t t i c e r e l a x a t i o n t i m e must b e s m a l l e r t h a n 1 m i l l i s e c o n d f o r t h i s s a l t . No 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 r e -l a x a t i o n time c o u l d be made i n t h i s c a s e . M e n t i o n i s made o f a n " o v e r s h o o t e f f e c t " o b s e r v e d i n one p a r t i c u l a r c r y s t a l o f neodymium e t h y l s u l f a t e . A p o s s i b l e e x p l a n -a t i o n f o r t h i s e f f e c t i s g i v e n , b y a s s u m i n g t h a t t h e c r y s t a l i n q u e s t i o n was t w i n n e d . I n t h i s c a s e one may i n f e r t h a t t h e r e l a x a -t i o n t i m e i s s t r o n g l y d e p e n d e n t o n t h e o r i e n t a t i o n o f t h e o p t i c a l a x i s o f t h e c r y s t a l w i t h r e s p e c t t o t h e e x t e r n a l m a g n e t i c f i e l d . iv, Tne results were found to disagree with present-day theor-ies of paramagnetic relaxation. Assumptions of doubtful va l i d -ity in the theory are discussed as possible reasons for such dis-agreement . In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i 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 an advanced degree at the U n i v e r s i t y o f B r i t i s h Columbia, I agree t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . .I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y purposes may be g r a n t e d by t h e Head o f my Department o r by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g or p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Department of p h y £ i C £  The U n i v e r s i t y of B r i t i s h Columbia, Vancouver 8, Canada. Date Sep I , 19 s 9  V ACKN OlYLEDGSMENTS The research described in this thesis was supported by the National Research Council of Canada through research grants to Dr. J.M. Daniels and the award of a Bursary (1958-59( and Studentship (1959-60) to the author. I am greatly indebted to my research supervisor, Dr. J.M. Daniels, who suggested the research and helped me in a l l phases of the work. His help, advice, and contagious enthusiasm made this thesis possible. I owe.-.' him a special word of thanks for providing me with an assistant, Miss Id, Jay, in the f i n a l phases of the work. Grateful acknowledgement i s made to Miss M. Jay, who not only drafted a l l the diagrams for this thesis, but also did most of the work involved in reducing the hundreds of photographs con-taining the experimental results to numerical data. Her help was Invaluable, To the members of the faculty of the University of British Columbia, who - be i t in the classroom or in private discussions -provided me with knowledge and enthusiasm, I also owe a great debt of gratitude. I am very grateful to the technicians of the Department of Physics for their help, especially to Mr. J. Lees, Mr. W. Maier, and Mr, H. Zerbst. I also wish to express my appreciation to a l l my fellow students, who helped me i n many ways, and - last not least - to my wife Marianne, who contributed to this thesis by her encouragement and many personal sacrifices. v i TABLE OF CONTENTS page C h a p t e r I : I n t r o d u c t i o n 2. 1) P r o p a g a t i o n o f E l e c t r o m a g n e t i c Waves i n an A n i s o t r o p i c Medium 1 2) F a r a d a y R o t a t i o n and B i r e f r i n g e n c e ( 1 s t p a r t ) 5 3) P o i n e a r e ' s R e p r e s e n t a t i o n o f P o l a r i z e d L i g h t 10 4) F a r a d a y R o t a t i o n a n d B i r e f r i n g e n c e (2nd P a r t ) 13 5) F a r a d a y E f f e c t , P a r a m a g n e t i c R e s o n a n c e , and S p i n - L a t t i c e R e l a x a t i o n 16 6) S p i n - L a t t i c e C o u p l i n g 21 C h a p t e r I I : E x p e r i m e n t a l A r r a n g e m e n t 23 1) D e s c r i p t i o n o f the a p p a r a t u s 23 2) D e s c r i p t i o n o f a n E x p e r i m e n t 29 3) P r e p a r a t i o n o f Samples 39 C h a p t e r III: Measurements 43 1) L i n e a r i t y , R e p r o d u c i b i l i t y , e t c . , o f t h e A p p a r a t u s 43 2) Some T y p i c a l R e s u l t s .. 52 3) E f f e c t o f P u l s e - L e n g t h 54 4) Measurements o f R e l a x a t i o n Time a t V a r i o u s F i e l d s a n d T e m p e r a t u r e s 55 5) Experiment;* w i t h C e r i u m E t h y l s u l f a t e 63 6) " O v e r s h o o t " Phenomenon 64 C h a p t e r I V : D i s c u s s i o n 65 B i b l i o g r a p h y 73 T v i i . ' . L I S T OF ILLUSTRATIONS f o l l o w i n g page F i g u r e I : V i e w s o f the m o t i o n o f t h e t i p o f t h e e l e c t r i c v e c t o r f o r and 11 F i g u r e i l : D e c o m p o s i t i o n o f e l l i p t i c a l l y p o l a -r i z e d l i g h t i n t o c i r c u l a r l y p o l a r i z e d components 11 F i g u r e I I I : P o i n c a r e s p h e r e I S F i g u r e I V : P r o p e r t i e s o f t h e P o i n c a r e s p h e r e .... 12 F i g u r e V : E f f e c t o f c o m b i n e d F a r a d a y e f f e c t and b i r e f r i n g e n c e 15 F i g u r e V I : View o f a p p a r a t u s w i t h dewars removed 23 F i g u r e V I I : V i e w o f a p p a r a t u s i n o p e r a t i n g c o n d i t i o n 23 F i g u r e V I I I : View o f o p t i c a l s y s t e m , s o u r c e s i d e .. 23 F i g u r e IX : View o f o p t i c a l s y s t e m , d e t e c t o r s i d e 23 F i g u r e X : M i c r o w a v e s y s t e m 23 F i g u r e X I : P u l s e m o d u l a t o r c i r c u i t 24 F i g u r e X I I : O p t i c a l s y s t e m 27 F i g u r e X I I I : C i r c u i t s a s s o c i a t e d w i t h o p t i c a l s y s t e m 27 F i g u r e XIV : C r y s t a l g r o w i n g a p p a r a t u s 40 F i g u r e XV : V e r d e t c o n s t a n t v s . - p - ! 013* 4 6 F i g u r e s XVI t o : T y p i c a l v i e w s o f t h e r e l a x a t i o n X X V I I : p r o c e s s ' 53 v-ii-i following page Figure XXVIII •: Rotation vs. time on semi-log scale -53 Figures XXIX : Typical views of effect of pulse-to XXX : Length ;' 53 Figures XXXI : Typical views of "overshoot t o XXXII : phenomenon" 53 Figures XXXIII: Relaxation time vs. magnetic f i e l d ... 61 to XXXIV : • • Figures XXXV : Relaxation time vs. temperature 62 to XXXVI : Figure XXXVII : Poincare' representation of overshoot effect 69 C h a p t e r I : I n t r o d u c t i o n I n t h e e x p e r i m e n t s d e s c r i b e d i n t h i s t h e s i s t h e i n f l u e n c e o f p a r a m a g n e t i c r e s o n a n c e on t h e F a r a d a y e f f e c t i s u t i l i z e d t o measure t h e s p i n - l a t t i c e r e l a x a t i o n t i m e o f p a r a m a g n e t i c i o n s i n s i n g l e c r y s t a l s a t l i q u i d h e l i u m t e m p e r a t u r e s . The f o l l o w i n g s e c t i o n s o f t h i s c h a p t e r g i v e t h e t h e o r y o f t h e phenomena i n v o l v e d , i n s o f a r a s i t i s r e q u i r e d f o r a p r o p e r u n d e r s t a n d i n g o f t h e e x p e r i m e n t s . I n C h a p t e r I I t h e a p p a r a t u s u s e d f o r t h e e x p e r i m e n t s i s d e s c r i b e d a n d so a r e t h e e x p e r i m e n t a l t e c h n i q u e s u s e d i n c o n -n e c t i o n w i t h t h e e x p e r i m e n t s . C h a p t e r I I I g i v e s a d e t a i l e d a c c o u n t o f t h e r e s u l t s o b t a i n e d i n t h e e x p e r i m e n t s . C h a p t e r IV f i n a l l y g i v e s a d i s c u s s i o n o f t h e r e s u l t s i n t h e l i g h t o f p r e s e n t day k n o w ledge. 1) P r o p a g a t i o n o f E l e c t r o m a g n e t i c Waves i n a n A n i s o t r o p i c Medium The b a s i c e q u a t i o n s d e s c r i b i n g t h e b e h a v i o u r o f e l e c t r o -m a g n e t i c waves i n g e n e r a l a r e ( i n G a u s s i a n u n i t s ) ( l . l a ) (1.1b) c url E L <}B (1.1c) = 0 ( l . l d ) d i v D. The symbols have t h e i r u s u a l meaning, i . e . H_ i s t h e m a g n e t i c f i e l d , J B t h e m a g n e t i c i n d u c t i o n , JE1 t h e e l e c t r i c f i e l d , J^. t h e e l e c t r i c d i s p l a c e m e n t , t h e c u r r e n t d e n s i t y , g t h e c h a r g e d e n s i t y , and c t h e v e l o c i t y o f l i g h t i n v a c u o • 2 JTor a l i g h t wave p r o p a g a t e d i n a n a n i s o t r o p i c d i e l e c t r i c we have t h e s u b s i d i a r y e q u a t i o n s (1.2a) ± - 0 ( l . B b ) g = 0 (1.2c) B = H. (1.2d) Dj. = € Lj E j where (l„2d) i s w r i t t e n i n t e n s o r n o t a t i o n , u j a r e dummy i n -d i c e s , s t a n d i n g f o r t h e c o o r d i n a t e s x,y,2 . An i n d e x a p p e a r i n g t w i c e i n d i c a t e s summation o v e r a l l i t s v a l u e s , i s t h e p e r m i t t i v i t y t e n s o r o f t h e medium. U s i n g e q u a t i o n s ( 1 . 2 ) , t h e e q u a t i o n s (1.1) c a n be w r i t t e n ( i n t e n s o r n o t a t i o n ) (1.3a) ( c u r l H\ = ± eL] § | j ( l . 3 b ) ( c ^ L E ^ ^ (i.3c) d i v H = 0 C o n s i d e r i n g a p l a n e wave p r o p a g a t e d i n t h e 2 d i r e c t i o n o f a C a r t e s i a n c o o r d i n a t e s y s t e m , l e t us p u t tcot -^ e i.«t-/Z (1.4a) E x =» T e (1.4b) Ey * r[ e F o r s u c h a p l a n e wave, w h a t e v e r t h e f o r m o f E 2 , we h a v e ( 1 . 4 c ) = = 0 S u b s t i t u t i n g (1.4) i n t o (1.3b) and s o l v i n g f o r t h e Mc we o b t a i n HA = o E q u a t i o n (1.3c) i s s a t i s f i e d i d e n t i c a l l y b y ( 1 . 5 ) . S u b s t i t u t i n g (1.4) and (1.5) i n t o ( 1 . 3 a ) , we g e t (1.6a) -r-gp ^  e ° ~ ~ C " 6 U*xf + €xy^+€xz5,y ( i . 6 b ) ctr^ie =~ * \*Y*Z*eyY'VsY*5' (1.6c) 0 = — e 9 \ £ ? x ^ 6 2 ^ t 6 2 2 y where c.^ ~ ^ € E q u a t i o n ( 1 . 6 c ) i s t h e same a s i s o b t a i n e d when s u b s t i t u t i n g t h e v a l u e s f o r EL and »H i n t o e q u a t i o n (1.3d) Hence (1,7) S u b s t i t u t i n g (1.7) i n t o (1.6a) and (1.6b) we f i n a l l y g e t .a. , A (1.8a) ( SxK-^^i^)^ + (£xy-§JTTrX)'ri =• -E q u a t i o n s (1.8 ) r e p r e s e n t a n e i g e n v a l u e p r o b l e m and g i v e t h e p o l a r i z a t i o n s o f t h o s e waves w h i c h a r e p r o p a g a t e d w i t h o u t change o f p o l a r i z a t i o n , and a l s o t h e i r p r o p a g a t i o n c o n s t a n t s ^ 4 N o t e t h a t H l i e s p e r p e n d i c u l a r t o t h e d i r e c t i o n o f p r o p a g a t i o n , b u t EL does n o t . I f t h e components o f t h e £ - t e n s o r a r e a l l r e a l , t h e n so a r e t h e c o e f f i c i e n t s o f t h e e i g e n v a l u e e q u a t i o n s ( 1 . 8 ) , a n d t h e e i g e n v e c t o r s a r e l i n e a r c o m b i n a t i o n s o f ^ and w i t h r e a l c o e f f i c i e n t s . Thus t h e s t a b l e p r o p a g a t e d w a v e s a r e p l a n e p o l a r i z e d , b u t t r a v e l w i t h d i f f e r e n t v e l o c i t i e s . T h i s i s known as b i r e f r i n g e n c e . I f , however, t h e e- t e n s o r h a s i m a g i n a r y components, we have a F a r a d a y e f f e c t . F o r t h e waves t o b e p r o p a g a t e d w i t h o u t a t t e n u a t i o n , t h e e i g e n v a l u e s must be r e a l , and hence t h e t e n s o r must be H e r m i t i a n . W r i t i n g t h e e i g e n v a l u e e q u a t i o n (1.8) a s (1.9a) (1.9b) -loc^ + S - — ^~4s We have (1.10) - = 6 toe The e i g e n v e c t o r s a r e ^ £ iy^ , w h i c h r e p r e s e n t c i r c u l a r l y p o l a r i z e d waves. A p l a n e p o l a r i z e d wave e n t e r i n g s u c h a medium i s decomposed i n t o two c i r c u l a r l y p o l a r i z e d waves t r a v e l l i n g w i t h d i f f e r e n t v e l o c i t i e s , h ence a p h a s e d i f f e r e n c e i s i n t r o d u c e d b e -tween t h e two c i r c u l a r l y p o l a r i z e d components, and t h e s e re combine on e m e r g i n g t o g i v e a p l a n e p o l a r i z e d wave w i t h a new p l a n e o f p o l a r i z a t i o n . 5 2) F a r a d a y R o t a t i o n and B i r e f r i n g e n c e (1st p a r t ) As shown i n s e c t i o n 1) o f t h i s c h a p t e r , t h e p r o p a g a t i o n o f l i g h t i n a c r y s t a l i s i n g e n e r a l g o v e r n e d h y t h e £ - t e n s o r . The s ~ t e n s o r as i n t r o d u c e d i n e l e c t r o m a g n e t i c t h e o r y i s a p h e n o m e n o l o g i e a l p r o p e r t y o f t h e s u b s t a n c e . T h i s t e n s o r c a n be c a l c u l a t e d d i r e c t l y f r o m t h e p r o p e r t i e s o f t h e i o n s a n d an o u t l i n e o f how t h i s may be done i s now g i v e n : C o n s i d e r a n i o n i n a s t a t e !'%>> , whose e n e r g y i s z e r o , w i t h a s e t o f e x c i t e d s t a t e s I ' Y K^' o f e n e r g y E - k . L e t ^ = M < From t i m e d e p e n d e n t p e r t u r b a t i o n t h e o r y , t h e s t a t e - f u n c t i o n o f t h e i o n , when s u b j e c t e d t o a h a r m o n i c p e r t u r b a t i o n -T O ' tCO"t . . 3 c 0 e i s g i v e n b y (2.1) TOW Y o - i - i ; <k?u 4- c o — l e ~ e ) \ F o r i n t e r a c t i o n o f r a d i a t i o n w i t h m a t t e r , as i s w e l l known, we q A c a n r e p l a c e t h e momentum o p e r a t o r _|0 h y _ £ r^r- , a n d t h e p o t e n t i a l V b y V +• oj , where cj i s t h e a l g e b r a i c e l e c t r o n -i c c h a r g e , and A a n d y a r e t h e p o t e n t i a l s o f t h e e l e c t r o -m a g n e t i c f i e l d . We c a n c h o o s e a g a u g e i n w h i c h » o l i v A 3 0 . W r i t i n g e f o r t h e n u m e r i c a l e l e c t r o n i c c h a r g e , t h e H a m i l t o n i a n f o r an atom, w h i c h was o r i g i n a l l y o f t h e f o r m becomes (2.3) Be = ae 0-fa A* To t h e f i r s t o r d e r , we c a n d r o p t h e t e r m i n J\. . A p l a n e - p o l a r i z e d e l e c t r o m a g n e t i c wave p r o p a g a t e d i n t h e ^ - d i r e c t i o n may he e x p r e s s e d (2.4) E_ = & E o siv>{co(t--§•)} Where E 0 i s t h e s t r e n g t h o f t h e e l e c t r i c f i e l d , a nd i s a u n i t v e c t o r i n i t s d i r e c t i o n ( i f t h e wave i s n o t p l a n e p o l a r i z e d , we n e e d two s u c h t e r m s ) . F o r t h i s wave S i n c e t h e w a v e l e n g t h o f any l i g h t w h i c h i s l i k e l y t o h e u s e d i s much g r e a t e r t h a n a n a t o m i c d i m e n s i o n , we can a p p r o x i m a t e (2.6) e~ c =• I Hence (2.7) A = §: ( e +e ) U s i n g t h i s v a l u e o f A i n t h e e x p r e s s i o n (2.3) f o r t h e p e r t u r b a t i o n H a m i l t o n i a n a n d s u b s t i t u t i n g i n t h e e x p r e s s i o n (2.1) f o r t h e p e r t u r b e d s t a t e - f u n c t i o n we g e t The e l e c t r i c moment M i s g i v e n b y •^^lexl'M'^ I n t h i s we n e g l e c t : i ) A l l t e r m s o f d e g r e e > | i,n E l 0 £ ccot i i ) a l l t e r m s w i t h a t i m e dependence o t h e r t h a n e 7 and we w r i t e (2.9) < ^ O 1 ^ 1 Y K > = - I r*«>«<%\r tYk> as i s a l s o w e l l known. Then + comp l e x c o n j u g a t e M The r a t i o " ^ r - d e f i n e s a component o f t h e p o l a r i z a b i l i t y t e n s o r OC^j and t h e p e r m i t t i v i t y t e n s o r £ ^ i s r e l a t e d t o t h i s b y (2.11) 6cj 5 5 <&j •+* VtcoCtj To f i n d , e . g . t h e x y component o f odtj , we c o n s i d e r t h e component o f M p a r a l l e l t o y , and 6 p a r a l l e l t o x . Then - w t o c x y - X < X l y l Y k > | ^ ( z ^ + ^ ^ ) < % l x | Y o > ^ As an example o f how t h e F a r a d a y e f f e c t comes a b o u t , l e t u s c o n s i d e r an i o n whose g r o u n d s t a t e i s a K r a m e r s d o u b l e t , 1Vo> and , and whose e x c i t e d s t a t e i s a n o t h e r K r a m e r s d o u b l e t , 1 a > and 1 C > . Assume f o r s i m p l i c i t y , t h a t o n l y t r a n s -i t i o n s OL and V, c a r e p e r m i t t e d f o r d i p o l e r a d i a t i o n and a b s o r p t i o n . Take as a x i s o f q u a n t i z a t i o n t h e a - a x i s . B o t h IYo> and I cc> c a n be w r i t t e n a s a s e r i e s o f s p h e r i c a l h a r m o n i e s , and X + Ly i s a s p h e r i c a l h a rmonic p r o p o r t i o n a l t o 8 e 1 ^ and x - ty i s a s p h e r i c a l h a r m o n i c p r o p o r t i o n a l t o e . Then <Yo| X+tylcO* v a n i s h e s u n l e s s t h e r e a r e components o f 'Yo and ct f o r w h i c h t h e m a g n e t i c quantum numbers m d i f f e r by 1. I t i s e a s i l y s e e n t h a t <Y0lx+tylo<.> <'Volx-eylot> L e t <YoU+d/U> = ^ , , < Yo I x-ty 1 o c ^ s . ^ a n d a r e b o t h r e a l b e c a u s e t h e c o -e f f i c i e n t s o f t h e e x p a n s i o n o f Y© a n d CL a r e r e a l . The c o -e f f i c i e n t s i n t u r n a r e r e a l b e c a u s e t h e H a m i l t o n i a n c o n t a i n s no i m a g i n a r y t e r m s . Then 2.13) < Y 0 U U > = < Y „ l y l o t > = I /*ir Hence t h e p r o d u c t < Y© l y I ot> K. &> IX I Yo > i s a p u r e i m a g i n a r y number. Thus when t h e e x p o n e n t i a l s i n e x p r e s s i o n (2.12) a r e added, t h e t i m e d e p e n d e n c e o f o6x^ i s a s cos cot , wh e r e a s t h e ti m e d e p e n d e n c e o f was a s w t ( s e e e q u a t i o n ( 2 . 4 ) . T h i s 9 0 ° phase s h i f t i s t o be r e p r e s e n t e d by an i m a g i n a r y e l e m e n t o f t h e oC ( o r € ) t e n s o r . The o t h e r component o f t h e K r a m e r s d o u b l e t i s o b t a i n e d by r e v e r s i n g t h e s i g n s o f t h e v*\ • s o f t h e o r i g i n a l component. Hence <Y, \x-Ly\c> - X , <V, Ix 1 c > yu, Thus f o r t h i s component we r e p l a c e t by - I i n t h e e x p r e s s i o n (2.13) and h e n c e t h i s component h a s a F a r a d a y e f f e c t o f t h e o p p o s i t e s i g n . I n g e n e r a l b o t h components a r e o c c u p i e d and t h e t o t a l F a r a d a y e f f e c t i s a s t a t i s t i c a l a v e r a g e o f t h e e f f e c t s o f t h e two components, a n d i s t h e r e f o r e p r o p o r t i o n a l t o t h e d i f f e r e n c e s i n 9 p o p u l a t i o n ( a n d t o o t h e r t h i n g s b e s i d e s ) . Now t h e r e i s i n g e n e r a l a s e t o f a x e s i n w h i c h t h e r e a l p a r t o f t h e 6 - t e n s o r i s d i a g o n a l . Then we have f o r t h e s t a b l e modes o f p r o p a g a t i o n : (2.14a) (e + COCY^= (2.14b) - i * ^ where i n g e n e r a l p} ^ « g Hence (2.15) ^ = and t h e r e f o r e o r The phase d i f f e r e n c e 6 i n t r o d u c e d p e r u n i t p a t h l e n g t h i s p r o p o r t i o n a l t o t h e d i f f e r e n c e b e t w e e n t h e two p r o p a g a t i o n c o n s t a n t s , i . e . (2.18) S~ . where /A. i s an a v e r a g e r e f r a c t i v e i n d e x . I n t h i s r e p r e s e n t a t i o n ft g i v e s t h e c o n t r i b u t i o n t o £ r e s -p o n s i b l e f o r b i r e f r i n g e n c e , w h i l e oc g i v e s t h e c o n t r i b u t i o n r e s p o n s i b l e f o r t h e F a r a d a y e f f e c t . 10* 3) P o i n o a r e ' s R e p r e s e n t a t i o n o f P o l a r i z e d L i g h t (a) R i g h t and l e f t c i r c u l a r l y p o l a r i z e d l i g h t . C o n s i d e r a r i g h t handed s y s t e m o f a x e s o r • Suppose l i g h t i s b e i n g p r o p a g a t e d i n t h e d i r e c t i o n o f +• 2 , and t h e e l e c t r i c v e c t o r l i e s a t some t i m e t s 0 i n t h e (y,2) p l a n e . L e t t h e l i g h t be o b s e r v e d b y a n o b s e r v e r l o o k i n g i n t h e d i r e c t i o n o f —"2 ; i f t h e t i p o f t h e e l e c t r i c v e c t o r moves on a c i r c l e i n c l o c k w i s e d i r e c t i o n , i t i s s a i d t o b e " r i g h t h a n d c i r c u l a r l y p o l a r i z e d " (RHCP), i f i n t h e o p p o s i t e d i r e c t i o n , i t i s s a i d t o b e " l e f t h a n d c i r c u l a r l y p o l a r i z e d " ( L H C P ) . I n t h e q u a n t u m - m e c h a n i c a l d e s c r i p t i o n o f a l i g h t wave t h e m a g n e t i c v e c t o r p o t e n t i a l ,A. p l a y s t h e r o l e a n a l o g o u s t o a s t a t e - f u n c t i o n . I t i s c o n v e n i e n t t o t a k e a s b a s i c s t a t e s t h e c i r c u l a r l y p o l a r i z e d components o f l i g h t , s i n c e t h e s e a r e e i g e n -f u n c t i o n s o f a n g u l a r momentum. They a r e (3.1) l+-> = + i_p l - > = ^ - t_p ( B o t h m u l t i p l i e d b y e c ^ i f t h e t i m e d e p e n d e n c e i s t o be s t r e s s e d ) . S i n c e (3.2) J S ^ c u r l A . c a r l £ = - ^ = ^ c u r l A_ we have I t i s s e e n t h a t r e p r e s e n t s RHCP l i g h t a n d I—> r e p r e s e n t s LHCP l i g h t . 11 L e t u s n o r m a l i z e t h e e x p r e s s i o n f o r EL. a n d d e f i n e s t a t e s I r> and I l > w i t h p h a s e s c o n s i s t e n t w i t h t h o s e o f I *> and I -> ( 3 . 4 ) I r> = ~^=i ( j_ -L j j . ) \l> = ^ t i) ( s e e d i a g r a m F i g u r e I) N o t e t h a t l r * l > i s p l a n e p o l a r i z e d (PP) p a r a l l e l t o x ( o r ± ) and l r - l > i s PP p a r a l l e l t o y ( o r _ j _ ) . (h) D e c o m p o s i t i o n o f e l l i p t i e a l l y p o l a r i z e d (EP) l i g h t i n t o c i r c u l a r l y p o l a r i z e d (CP) components. (See d i a g r a m F i g u r e I I ) L e t t h e h a l f - a x e s o f t h e e l l i p s e b e A+ B and A— B , t h e n t h e e l l i p s e i s made up o f A1l> +- B \\r& . The p h a s e s o f t h e two com-p o n e n t s d e t e r m i n e t h e d i r e c t i o n o f t h e a x e s o f t h e e l l i p s e . I n t h e d i a g r a m i t i s s e e n t h a t i f t h e r e i s a phase d i f f e r e n c e o f " Y , t h e m a j o r a x i s o f t h e e l l i p s e i s i n c l i n e d a t a n a n g l e -sj-t o t h e L - a x i s . The n o r m a l i z e d e x p r e s s i o n f o r t h i s e l l i p s e i s t h e n cos QlC> + s ln Se I r > where G i s a p a r a m e t e r s u c h t h a t ( 3 . 5 ) A - cos 0 B = s l n 0 e T h e r e f o r e (3.6) IA l 3 - + I B ^ = l e n s u r i n g n o r m a l i t y . The e l l i p t i c i t y o f t h e l i g h t c a n a l s o be d e s c r i b e d b y t h e a n g l e <c> i n s t e a d o f b y A and ft o r ' b y 0. Views of the motion of the t ip of the e lec t r i c vector for I I > and Ir >. FIGURE H Decomposition of ellipticolly polarized light into circularly polarized components. F o l l o w i n g page 11 Then . I A | - 161 c o s O - s t ^ 0 (3.7) t<x* vj> = | A U I B \ ~~ sen© and t h e r e f o r e (c) The P o i n c a r e s p h e r e * (See d i a g r a m F i g u r e I I I ) E a c h s t a t e o f p o l a r i z a t i o n ( s t a t e - f u n c t i o n ) i s d e s c r i b e d b y a p o i n t on a s p h e r e . The p o l e s o f t h e s p h e r e L and R r e p -r e s e n t LHCP and RHCP l i g h t r e s p e c t i v e l y . The e l l i p t i c a l l y p o l a r i z e d l i g h t j u s t d e s c r i b e d i s r e p r e s e n t e d b y t h e p o i n t E. . The p o i n t X r e p r e s e n t s l i g h t p o l a r i z e d i n t h e x ( o r ^ ) p l a n e . P P l i g h t i s r e p r e s e n t e d by a p o i n t on t h e e q u a t o r . LHEP l i g h t i s r e p r e s e n t e d by a p o i n t o n t h e u p p e r h e m i s p h e r e , RHEP on t h e l o w e r h e m i s p h e r e . N o t e t h a t t h e r e i s no way o f r e p r e s e n t -i n g t h e phase o f a p o l a r i z e d l i g h t beam - p h a s e has no p h y s i c a l s i g n i f i c a n c e h e r e , o n l y p h a s e d i f f e r e n c e s have p h y s i c a l s i g n i f i c -a n c e , a n d o n l y phase d i f f e r e n c e s a r e r e p r e s e n t e d on t h e P o i n c a r e s p h e r e . (d) P r o p e r t i e s o f t h e P o i n c a r e s p h e r e . i ) A s t a t e o f p o l a r i z a t i o n l*> = CosO\l> +stinQ« \r> i s r e p r e s e n t e d a s a p o i n t on t h e s p h e r e whose s p h e r i c a l p o l a r c o o r d i n a t e s a r e ^0,°Y . i i ) Two s t a t e s w h i c h a r e o r t h o g o n a l ( i n t h e q u a n t u m - m e c h a n i c a l s e n s e ) a r e r e p r e s e n t e d b y o p p o s i t e ends o f a d i a m e t e r : F I G U R E HI L X R P O I N C A R E S P H E R E F o l l o w i n g page 12 F I G U R E EZ" D Properties of the Poincare' Sphere F o l l o w i n g page 12 I f l u > = c o s @ l L > + s i r t Q e 1 , | r> t h e n t h e o r t h o g o n a l s t a t e lv> - sthOe l l > - c o s © l r > i s t h e same s t a t e as I v> = - s tnOl l> + cos 0 e t Y ! r > = cos • ® ) l l > +sin (*+0) e ' Y l r > Thus lu> and I have t h e same *Y , h u t t h e i r ' s d i f f e r b y Tf , i . e . t h e y a r e l o c a t e d on o p p o s i t e e n d s o f a d i a m e t e r o f t h e P o i n c a r e s p h e r e . i i i ) The r e p r e s e n t a t i o n o f a s t a t e a s a c o m b i n a t i o n o f two o r t h o g o n a l s t a t e s i s i n d e p e n d e n t o f t h e b a s i c s t a t e s c h o s e n : e . g . i f lu.> and l v > a r e two o r t h o g o n a l s t a t e s and l w > i s a n y o t h e r s t a t e , t h e n we c a n w r i t e I w> = cos 0 I u t > + slr\ © e i v> where 2. 0 ' i s t h e g r e a t c i r c l e a n g l e b e t w e e n t h e p o i n t s r e p -r e s e n t i n g lut> and lw> ; i v ) An a n a l y z e r w h i c h p a s s e s o n l y l i g h t o f a g i v e n p o l a r i z a t i o n s t a t e ( I ot> say) can be r e p r e s e n t e d b y a p o i n t ct on t h e s p h e r e . I f a beam o f l i g h t r e p r e s e n t e d b y a p o i n t w on t h e s p h e r e i s i n c i d e n t on t h e a n a l y z e r , t h e a m p l i t u d e t r a n s m i t t e d i s cos ^ , where 2-^ i s t h e g r e a t c i r c l e a n g l e between ot a n d w . 4 ) F a r a d a y R o t a t i o n and B i r e f r i n g e n c e (2nd p a r t ) We c a n d i s c u s s t h e e f f e c t o f a m i x t u r e o f F a r a d a y e f f e c t and b i r e f r i n g e n c e r a t h e r s i m p l y u s i n g t h e P o i n c a r e s p h e r e , C o n t i n u i n g f r o m t h e end o f s e c t i o n 2 ) : The e i g e n v e c t o r s a r e g i v e n b y (4.1) (/S + T/^ +o6* ) ^  + i o c y ^ = 0 o r (4.2) § = - - ~ ^ ( ^ V ^ ? ) ^ where D i s a n o r m a l i z i n g d e n o m i n a t o r (4.3) D3- = Xiot^^T /SVcFT^) and Ioc> a n d l v > t h e e i g e n v e c t o r s a r e ^ l x > •+• ^ l y ^ where l x > and l y > r e p r e s e n t l i g h t p l a n e p o l a r i z e d a l o n g t h e and y a x i s r e s p e c t i v e l y , t h e p r i n c i p a l a x e s o f t h e r e a l p a r t t h e € - t e n s o r . I f we w r i t e (4.4) |u> = cos 0 U > + s L n 0 e l y> t h e n (4.5) T a k i n g t h e u p p e r s i g n and l e t t i n g ( 4 . 6 ) "^T = stnh^ +1 = cosh^ t h e n - X (4.7) t a n Q = - e and h ence 15 14 .8 ) c o s a Q = T T T ^ = t o t « l l X = j ^ = r = ^ = ^ The r a t i o -TP- i s a p u r e i m a g i n a r y , i n d e p e n d e n t o f ot and T h e r e f o r e t h e p o i n t r e p r e s e n t i n g l u . ^ (and I V> ) l i e s on t h e g r e a t c i r c l e t h r o u g h Ix^ , I l > , l y > , and I , and i s d i s t a n t f r o m lx.> b y a n a n g l e 2 , $ . The e f f e c t o f c o m b i n e d F a r a d a y - e f f e c t and b i r e -f r i n g e n c e can now be e a s i l y v i s u a l i z e d b y r e f e r r i n g t o t h e d i a -gram F i g u r e V. The c o n s t r u c t i o n i s a s f o l l o w s : i ) F i n d t h e p o i n t s lx> a n d ly> w h i c h r e p r e s e n t t h e s t a b l e waves o f t h e b i r e f r i n g e n c e i i ) Draw t h e g r e a t c i r c l e t h r o u g h lx> , |y> a n d \ i i i ) lu.> a n d l i e on t h i s c i r c l e and t h e g r e a t c i r c l e a n g l e I u.> , l x ^ i s e q u a l 2, 0 , where i v ) I f p l a n e p o l a r i z e d l i g h t i s i n c i d e n t o n t h e c r y s t a l , t h i s w i l l be r e p r e s e n t e d b y a p o i n t on t h e e q u a t o r o f t h e s p h e r e . L e t t h i s p o i n t be v) The b i r e f r i n g e n c e and t h e F a r a d a y e f f e c t c o m b i n e d w i l l i n t r o d u c e a phase d i f f e r e n c e b e t w e e n t h e two waves p r o p o r t i o n a l t o + ^ . Hence i f t h e phase d i f f e r e n c e f r o m e a c h s e p a r a t e l y i s known, t h e t o t a l p h a s e d i f f e r e n c e cf i s e a s i l y c a l c u l a t e d . v i ) The e m e r g e n t l i g h t i s r e p r e s e n t e d b y lo> . To f i n d lc\> c o n s t r u c t a s m a l l c i r c l e w i t h lu.> a s t h e p o l e , p a s s i n g t h r o u g h I t> . Go r o u n d t h i s c i r c l e a n a n g l e cT . F I G U R E Y l> X - - - - - - - - _ \ l y > lu> / / / lx> li > / \ o> 2 0 > ' 0 ' N ) | v > i a > : / lr > E f fec t of combined Faraday Ef fect and Birefr ingence F o l l o w i n g page 15 16 i v i i ) I f t h e a n a l y z e r p a s s e s l i g h t r e p r e s e n t e d b y l a > , t o f i n d t h e i n t e n s i t y p a s s e d b y t h e a n a l y z e r , c o n s t r u c t t h e g r e a t c i r c l e t h r o u g h I o > a n d I <x> . L e t t h e g r e a t c i r c l e a n g l e b e t w e e n \ci> and I0> he O'. Then t h e i n t e n s i t y p a s s e d by t h e a n a l y z e r i s c o s 0') . ¥/hen oc a n d v a r y i n t i m e , n o t o n l y of b u t a l s o w i l l v a r y , e x c e p t f o r t h e c a s e /3 - 0 (no b i r e f r i n g e n c e ) , where lu.> w i l l c o i n c i d e w i t h r e g a r d l e s s o f <X, . I n t h e l a t t e r c a s e t h e n , 6 w i l l v a r y with t i m e i n a n i d e n t i c a l manner as oc and l o > w i l l t r a c e o u t a p a t h a l o n g t h e e q u a t o r . The i n t e n s i t y p a s s e d b y t h e a n a l y z e r w i l l t h e n b e a f u n c t i o n o f t i m e o f t h e f o r m £ { I + c o s (6> Wet))} where Q i s t h e g r e a t c i r c l e a n g l e b e t w e e n 1 i > and I a> . F o r t h e c a s e ^j=0 , however,, s i n c e b o t h I u.> a n d J v a r y i n a manner w h i c h i s no l o n g e r s i m p l y r e -l a t e d t o t h e t i m e d e p e n d e n c e o f oc a n d ^ , we w i l l h a v e w i t h i n c r e a s i n g / 2 p r o g r e s s i v e l y l a r g e r d e v i a t i o n s f r o m t h e i n t e n s i t y d ependence, s i n c e t h e p o s i t i o n o f t h e p o i n t l o > as a f u n c t i o n o f t i m e w i l l t r a c e o u t a p a t h b e a r i n g a somewhat complex r e l a t i o n s h i p t o t h e t i m e dependence o f oc (and/3 ) . I n c h a p t e r I I I i t i s shown t h a t f o r s u f f i c i e n t l y s m a l l i n i t i a l v a l u e s o f /3 ( s u c h a s a r e e n c o u n t e r e d i n t h e e x -p e r i m e n t s ) t h e d e v i a t i o n o f t h e i n t e n s i t y f r o m t h e ^ O + C o s © ' ) r e l a t i o n s h i p may be n e g l e c t e d . 5) F a r a d a y E f f e c t , P a r a m a g n e t i c R e s o n a n c e , and S p i n - L a t t i c e  R e l a x a t i o n The e f f e c t o f p a r a m a g n e t i c r e s o n a n c e on t h e F a r a d a y 17 > e f f e c t was f i r s t p r e d i c t e d b y K a s t l e r ( I , 1 9 5 1 ) . A quantum-m e c h a n i c a l t h e o r y was g i v e n b y G p e c h o w s k i ( I I , 1 9 5 5 ) , and t h e e f f e c t i t s e l f was f i r s t o b s e r v e d by D a n i e l s and Wesemeyer ( I I I , I V , 1 9 5 8 ) . A m a c r o s c o p i c t h e o r y f o r t h e c a s e i n v e s t i g a t e d by t h e l a t t e r two was g i v e n b y S k r o t s k i i , Z y r i a n o v a n d I z i u m ov ( V , 1 9 5 8 ) . The f o l l o w i n g i s a s h o r t resume o f t h e t h e o r y o f t h i s e f f e c t , s t r e s s i n g o n l y t h o s e p o i n t s n e c e s s a r y f o r t h e u n d e r s t a n d -i n g o f t h e e x p e r i m e n t s d i s c u s s e d i n a l a t e r c h a p t e r . I n s e c t i o n 2) i t was shown how t h e F a r a d a y e f f e c t a r i s e s f o r t h e case o f a p a r a m a g n e t i c i o n h a v i n g as g r o u n d s t a t e a K r a m e r s d o u b l e t and l i k e w i s e f o r t h e f i r s t e x c i t e d s t a t e . I t was s e e n t h a t t h e F a r a d a y r o t a t i o n i s p r o p o r t i o n a l t o t h e d i f f e r e n c e i n t h e p o p u l a t i o n s o f t h e two l e v e l s o f t h e g r o u n d s t a t e . Now l e t us c o n s i d e r a s y s t e m o f i o n s h a v i n g a. m a g n e t i c d i p o l e moment due t o s p i n S =• x s i t u a t e d i n an e x t e r n a l m a g n e t i c f i e l d H , Then t h e g r o u n d s t a t e w i l l be a s p l i t d o u b l e t w i t h e n e r g y l e v e l s s e p a r a t e d by 2 y U „ H . I f t h e s p i n s y s t e m i s i n t h e r m o d y n a m i c e q u i l i b r i u m w i t h i n i t s e l f , i t may be s a i d t o have a s p i n t e m p er-a t u r e T s . I f m o r e o v e r i t i s i n t h e r m o d y n a m i c e q u i l i b r i u m w i t h t h e c r y s t a l l a t t i c e , we have T s = T L t h e l a t t i c e t e m p e r-a t u r e . The p o p u l a t i o n s o f t h e l e v e l s w i l l t h e n be g i v e n b y t h e B o l t z m a n n d i s t r i b u t i o n ( w h i c h i n f a c t i s u s e d t o d e f i n e T s ). ( 5 , i ) NZ =» N, e k T * where and N, a r e t h e number o f i o n s i n t h e u p p e r and l o w e r 18 J s t a t e r e s p e c t i v e l y . The i n t e n s i t y o f p a r a m a g n e t i c r e s o n a n c e due t o a b s o r p -t i o n o f a quantum o f e l e c t r o m a g n e t i c r a d i a t i o n o f e n e r g y "fceuaJl^H depends o n t h e d i f f e r e n c e i n p o p u l a t i o n o f t h e two s t a t e s b e -tween w h i c h t h e t r a n s i t i o n s t a k e p l a c e . L e t t h i s d i f f e r e n c e i n p o p u l a t i o n be (5.2) y\ = N, -t h e e x c e s s number i n t h e l o w e r s t a t e . y\ i s r e d u c e d when mic r o w a v e t r a n s i t i o n s between t h e l e v e l s c o n c e r n e d o c c u r and c o n s e q u e n t l y T$ becomes l a r g e r t h a n Tt. . When t h e mic r o w a v e f i e l d i n d u c i n g t h e t r a n s i t i o n s i s s w i t c h e d o f f , Yi i n c r e a s e s due t o s p i n - l a t t i c e r e l a x a t i o n t o i t s e q u i l i b r i u m v a l u e Y^© ( i . e . t h e s p i n - s y s t e m l o o s e s e n e r g y t o t h e l a t t i c e a n d t h u s " c o o l s down" t o t h e t e m p e r a t u r e o f t h e l a t t e r , a t w h i c h p o i n t e q u i l i -b r i u m b e t w e e n s p i n - s y s t e m and l a t t i c e i s r e s t o r e d ) . S i n c e t h e F a r a d a y r o t a t i o n i s p r o p o r t i o n a l t o Y\ i t w i l l a l w a y s b e a measure o f t h e i n s t a n t a n e o u s s p i n t e m p e r a t u r e , p r o v i d e d t h e s p i n -s y s t e m i s i t s e l f i n e q u i l i b r i u m . The r a t e o f change o f VI i s s u p p o s e d t o b e p r o p o r t i o n a l t o Wo-V* and a l s o d e p e n d e n t on t h e c o u p l i n g b e t w e e n s p i n -s y s t e m and l a t t i c e . The c o u p l i n g i s e x p r e s s e d b y a p a r a m e t e r Tj , c a l l e d t h e s p i n - l a t t i c e r e l a x a t i o n t i m e . We have t h e n f o r t h e r a t e o f change o f Y I <{<r> Y>O - v\ ( 5 - 3 ) 2 t = " T T When mic r o w a v e r a d i a t i o n o f a p p r o p r i a t e f r e q u e n c y i s p r e s e n t , a t e r m Stoa P h a s t o be s u b t r a c t e d on t h e r i g h t h a n d s i d e , where 19 ' P i s t h e t r a n s i t i o n p r o b a b i l i t y p e r u n i t t i m e and t h e f a c t o r 2- r\ r e s u l t s f r o m t h e f a c t t h a t e a c h t r a n s i t i o n c h a n g es r\ by St, . Thus we have (5.4) - ~ A t e q u i l i b r i u m we have = 0 a n d hence (5,5) ^ a I +a,PT, P Now a s s u m i n g a r e s o n a n c e l i n e o f w i d t h & f we c a n w r i t e P a j | -where Po i s a c o n s t a n t . I t i s customars?- t o d e f i n e a s p i n -s p i n r e l a x a t i o n t i m e 7^ by "J^ = T a and h e n c e we may a l s o w r i t e P 3 Po "Hi Then a t e q u i l i b r i u m (5.6) ft = i-t-ap0T;T; At low t e m p e r a t u r e s and f o r c o n c e n t r a t e d s a l t s cff t e n d s t o b e q u i t e l a r g e (due t o s p i n - s p i n i n t e r a c t i o n s ) and hence we h a v e T, » , e . g . i n t h e s a l t s d i s c u s s e d i n t h i s t h e s i s we have a t l i q u i d H e l i u m t e m p e r a t u r e s T a o f t h e o r d e r o f 10 sec w h e r e a s T 4 i s o f t h e o r d e r o f 10 ' s e c . A t h i g h t e m p e r a t u r e s TV,"** T , and i n t h i s c a s e e q u a t i o n (5.6) may be w r i t t e n as n 0 (5.7) ua.PoT,* F o r t h e s a l t s i n v e s t i g a t e d (5.7) i s a p p l i c a b l e a t l i q u i d n i t r o g e n t e m p e r a t u r e s . S i n c e P 0 i s p r o p o r t i o n a l t o t h e s q u a r e o f t h e a m p l i t u d e o f t h e m a g n e t i c f i e l d o f t h e mi c r o w a v e r a d i a t i o n (we a r e c o n -c e r n e d w i t h m a g n e t i c d i p o l e t r a n s i t i o n s o n l y ) , i t i s s e e n t h a t f o r s u f f i c i e n t l y s t r o n g microwave r a d i a t i o n t h e r e i s a n a p p r e c i -a b l e change o f v\ f r o m i t s e q u i l i b r i u m v a l u e 'Ylo , and hence a c o r r e s p o n d i n g change i n t h e F a r a d a y r o t a t i o n due t o t h e micr o w a v e r a d i a t i o n . I f t h e microxvave r a d i a t i o n i s s u d d e n l y c u t o f f a t a t i m e t a 0 , we w i l l t h e n o b s e r v e a change i n F a r a d a y r o t a t i o n p r o p o r t i o n a l t o as g i v e n i n e q u a t i o n ( 5 . 3 ) , w h i c h may a l s o be w r i t t e n ot (n»-y\Ck)) nop - TO(t) ( 5 . 8 ) = ~ T> I f we w r i t e f o r t h e i n s t a n t a n e o u s F a r a d a y r o t a t i o n (5.9) §Ct) a KvOO where K i s a c o n s t a n t o f p r o p o r t i o n a l i t y , we f i n d (5.10) = K { V 0 - = K{y,e^^C0)} e T ' Hence b y o b s e r v i n g ^ (.i) a f t e r t h e mic r o w a v e power has b e e n c u t o f f , one may f i n d T ( , t h e s p i n - l a t t i c e r e l a x a t i o n t i m e . N o t e t h a t we have i m p l i c i t l y assumed t h a t t h e l a t t i c e t e m p e r a t u r e T u r e m a i n s c o n s t a n t t h r o u g h o u t , l i e . t h a t t h e l a t t i c e r e p r e s e n t s an i n f i n i t e h e a t s i n k . T h i s a s s u m p t i o n i s v a l i d o n l y i f t h e c o u p l i n g between t h e l a t t i c e a n d t h e s u r r o u n d -i n g l i q u i d h e l i u m b a t h i s s u c h t h a t t h e r a t e o f c o o l i n g o f t h e 21 l a t t i c e by the bath exceeds the rate of heating by the relaxing spin-system. T h i s assumption was j u s t i f i e d w ithin the orders of magnitude observable i n the experiments described. 6) S p i n - L a t t i c e C o u p l i n g The mechanism by which energy i s transferred from the spin-system to the l a t t i c e i n an ioni c c r y s t a l and vice-versa, i s agreed to be the following: L a t t i c e vibrations (phonons) modulate the c r y s t a l l i n e e l e c t r i c f i e l d and the spin-system f e e l s the e f f e c t o f t h i s mod-u l a t i o n v i a the spin-orbit i n t e r a c t i o n , since the s p l i t t i n g of the o r b i t a l l e v e l s i s i n general determined by the c r y s t a l l i n e e l e c t r i c f i e l d (at least i n the cry s t a l s investigated i n the ex-periments described i n t h i s thesis}. Van V l e c k ( V I , 1940) made some detailed c a l c u l a t i o n s of the s p i n - l a t t i c e relaxation time f o r v ^ v and T t ions i n the complex OGH^O , and Ti • <o H^O as found i n K C r C S O ^ I ^ O and Cs Tt ( SOO^/ 12. H^ O on the basis of t h i s model. He distinguished between two processes: A "d i r e c t process" i n which spin-system absorbs or emits a quantum of l a t t i c e energy and a "Ramann process" i n which a quantum of l a t t i c e energy i s I n e l a s t -i c a l l y scattered, i . e . the spin-system absorbs a quantum of one energy and emits one with a diff e r e n t energy simultaneously. W h i l e the "direct process" i s the important one at very low tem-peratures(the relaxation time being proportional to T i n general), the "Ramann process" i s dominant at higher temperatures (where the relaxation time i s proportional to a much higher power of, ~Y » e«S» I f o r the case of C T and T t ) V a n V l e c k also obtained a dependence of the rel a x a t i o n time on 22 • t h e e x t e r n a l m a g n e t i c f i e l d H. However t h e d e t a i l e d c a l c u l a -t i o n s a r e o f g r e a t c o m p l e x i t y and depend t o a l a r g e e x t e n t o n t h e p a r t i c u l a r s a l t and i o n s u s e d , and Van V l e c k ' s r e s u l t a g r e e d „ . w _ y i o n . H i s r e s u l t s on t h e f i e l d and t e m p e r a t u r e d e p e n d e n c e o f t h e r e -l a x a t i o n t i m e were n o t e x p e r i m e n t a l l y c o n f i r m e d , t h o u g h i n gen-e r a l i t i s known t h a t t h e r e l a x a t i o n t i m e d e c r e a s e s v e r y r a p i d l y w i t h i n c r e a s e i n t e m p e r a t u r e . Van V l e c k a l s o d i s c u s s e s t h e d e p e n d e n c e o f r e l a x a t i o n t i m e o n t h e e x t e r n a l m a g n e t i c f i e l d . F o r t h e " d i r e c t p r o c e s s " a t l e a s t , t h e r e l a x a t i o n t i m e s h o u l d d e c r e a s e v e r y r a p i d l y as t h e f i e l d i s i n c r e a s e d . T h i s i s b e c a u s e : i ) W i t h a l a r g e r f i e l d , t h e e n e r g y o f t h e l a t t i c e v i b r a t i o n s w h i c h c a n i n t e r a c t d i r e c t l y i s g r e a t e r , a n d l a t t i c e modes o f h i g h e r e n e r g y a r e more numerous. i i ) The a p r i o r i t r a n s i t i o n p r o b a b i l i t y f o r h i g h e r e n e r g y l a t t i c e modes i s g r e a t e r , s i n c e t h e w a v e l e n g t h i s s h o r t e r . C o n t r a r y t o t h e s e two e f f e c t s , an i n c r e a s e o f t h e mag-n e t i c f i e l d i n c r e a s e s t h e s p e c i f i c h e a t o f t h e s p i n s y s t e m . How-e v e r , t h i s e f f e c t i s s m a l l compared i ) and i i ) , a n d t h e r e l a x a t i o n t i m e i s s t i l l e x p e c t e d t o d e c r e a s e r a p i d l y as t h e mag-n e t i c f i e l d i n c r e a s e s . T h i s p r e d i c t i o n i s n o t b o r n o u t b y e x p e r i m e n t . T h e r e a r e many a s s u m p t i o n s i n V a n V i e e k ' s t h e o r y , some o f w h i c h w i l l be d i s c u s s e d i n C h a p t e r IV, w h i c h g i v e doubt f o r i t s v a l i d i t y . Some o f t h e s e a r e d i s c u s s e d i n two p a p e r s b y V a n V l e c k ( V I I , 1 9 4 1 ) . 23 The f a c t t h a t s i n c e V an V l e c k 1 s p a p e r s i n 1940 and 1941 no d e t a i l e d c a l c u l a t i o n s o f t h e s p i n - l a t t i c e r e l a x a t i o n t i m e o f p a r a m a g n e t i c i o n s i n t h e s o l i d s t a t e have b e e n made, e m p h a s i s e s t h e t h e o r e t i c a l d i f f i c u l t i e s i n h e r e n t i n t h e p r o b l e m o f o b t a i n i n g a f i r m q u a n t i t i t i v e u n d e r s t a n d i n g o f t h e r e l a x a t i o n - p r o c e s s e s , i t a l s o e m p h a s i s e s t h a t p a r a m a g n e t i c r e l a x a t i o n h as n o t b e e n f a s h i o n a b l e , s i n c e e a s i e r t o p i c s f o r r e s e a r c h have b e e n d i s c o v -e r e d i n t h e meantime, e . g . p a r a m a g n e t i c r e s o n a n c e . C h a p t e r I I : E x p e r i m e n t a l A r r a n g e m e n t 1) D e s c r i p t i o n o f t h e A p p a r a t u s The a p p a r a t u s u s e d ( s e e p l a t e s F i g u r e V I t o IX) c o n -s i s t e d e s s e n t i a l l y o f a c o n v e n t i o n a l p a r a m a g n e t i c r e s o n a n c e s p e c t r o m e t e r , w h i c h had b e e n m o d i f i e d t o a l l o w m a g n e t o - o p t i c a l s t u d i e s . The m a j o r c o n s t i t u e n t s o f t h e s p e c t r o m e t e r were t h e f o l l o w i n g : a) A m i c r o w a v e s y s t e m c a p a b l e o f d e l i v e r i n g up t o 140mw o f mi c r o w a v e power i n t h e X -band (3 CM w a v e l e n g t h ) , e i t h e r p u l s e d o r c o n t i n u o u s l y , t o a r e s o n a n t c a v i t y c o n t a i n i n g t h e sample u n d e r i n v e s t i g a t i o n . (See b l o c k d i a g r a m F i g u r e X ) . A r e f l e x - k l y s t r o n p r o v i d e d t h e s o u r c e o f t h e microwave power. I t s f i l a m e n t was s u p p l i e d b y a 6v car. b a t t e r y , t h e o t h e r s u p p l y v o l t a g e s were o b t a i n e d f r o m an e l e c t r o n i c a l l y s t a b i l i z e d DC power p a c k f e d v i a a S o l a c o n s t a n t v o l t a g e t r a n s -f o r m e r f r o m t h e 115v AC m a i n s . The o u t p u t f r o m t h e k l y s t r o n p a s s e d v i a w a v e g u i d e t h r o u g h a v a r i a b l e s u s c e p t a n c e a n d an a t t e n u a t o r t o a d i r e c t i o n a l coupler-; where some o f t h e power was FIGURE V I APPARATUS WITH DEWARS REMOVED FIGURE V I I APPARATUS IN OPERATING CONDITION To f o l l o w page 23 FIGURE V I I I OPTICAL SYSTEM, SOURCE SIDE jrlGURE EC OPTICAL SYSTEbi, uETECTuR SIDE To follow page 23 F I G U R E X STABILIZED POWER SUPPLY KLYSTRON PULSE MODULATOR WAVE METER DUMMY LOAD MICROAMMETER W i l l i ATTENUATOR | I VARIABLE SUSCEPTANCE © C R Y S T A L DIODE TUNABLE END M SAMPLE OSCILLOSCOPE GALVANOMETER CAVITY M I C R O W A V E S Y S T E M 24' d e v i a t e d t o a b r a n c h c o n t a i n i n g a PRD 529 -B p r e c i s i o n wave-m e t e r and a c r y s t a l r e c t i f i e r w i t h m i c r o ammeter. F o l l o w i n g t h e d i r e c t i o n a l c o u p l e r was a n o t h e r a t t e n u a t o r , a n d f o l l o w i n g t h i s t h e m i c r o w a v e power was c o u p l e d f r o m w a v e g u i d e t o c o a x i a l c a b l e . The c o a x i a l c a b l e f e d t h e m i c r o w a v e s t o t h e t r a n s m i s s i o n t y p e r e s o n a n t c a v i t y o p e r a t i n g i n t h e TE 101 mode. C o u p l i n g i n t o c a v i t y was p r o v i d e d b y a s m a l l o p en l o o p , w h i c h c o u l d be b e n t t o a c h i e v e a s u i t a b l e d e g r e e o f c o u p l i n g . The l o a d e d Q* o f the c a v i t y a t l i q u i d h e l i u m t e m p e r a t u r e s was o f t h e o r d e r o f 5000 t o 9000 d e p e n d i n g on t h e sample u s e d . The c a v i t y was p r o v i d e d w i t h h o l e s o f 2mm d i a m e t e r i n i t s b r o a d f a c e s t o a l l o w p a s s a g e o f l i g h t t h r o u g h t h e s a m p l e w h i c h was mounted a t t h e t o p f a c e , between i n p u t and o u t p u t c o u p l i n g l o o p s , o n a c h o k e - p l u n g e r w h i c h c o u l d be r o t a t e d a r o u n d a v e r t i c a l a x i s d u r i n g an e x p e r i -ment. The o u t p u t c o u p l i n g was i d e n t i c a l t o t h e i n p u t c o u p l i n g , and c o a x i a l c a b l e f e d t h e o u t p u t t o a w a v e g u i d e s e c t i o n c o n t a i n -i n g a t u n a b l e l o w n o i s e s i l i c o n d i o d e d e t e c t o r . The o u t p u t f r o m t h e s i l i c o n d e t e c t o r c o u l d be f e d t o a g a l v a n o m e t e r o r v i a a v i d e o - a m p l i f i e r t o a n o s c i l l o s c o p e . I n t h e i n v e s t i g a t i o n s d e s -c r i b e d i n t h i s t h e s i s t h e o u t p u t was u s e d o n l y f o r t u n i n g and m o n i t o r i n g p u r p o s e s . P u l s e m o d u l a t i o n o f t h e k l y s t r o n was p r o v i d e d b y a m u l t i v i b r a t o r ( s e e d i a g r a m F i g u r e XI) c a p a b l e o f d e l i v e r i n g t o t h e r e f l e c t o r o f t h e k l y s t r o n e s s e n t i a l l y r e c t a n g u l a r p u l s e s v a r i a b l e i n l e n g t h i n s t e p s b e t w e e n 2 msec and 124 msec and i n a m p l i t u d e c o n t i n u o u s l y b e t w e e n 0 and 45 t o 8 5 v . The p u l s e s w e r e n o r m a l l y r e p e t i t i v e a t a r a t e o f a p p r o x i m a t e l y 1 p u l s e e v e r y 2 B 220 V P U L S E O U T P U T A ' M A N U A L l w S 4 7 K / C O N T R O L _ i I W 1 A. M — T R I G G E R O U T P U T S I N G L E P U L S E S F I G U R E X I P U L S E M O D U L A T O R C I R C U I T 25 s e c o n d s . However, t h e r e p e t i t i o n r a t e c o u l d be i n c r e a s e d b y a f a c t o r o f a b o u t 20 ( d e p e n d i n g on t h e p u l s e l e n g t h ) , a n d a manual s w i t c h was p r o v i d e d , a l l o w i n g t h e a p p l i c a t i o n o f s i n g l e p u l s e s . S y n c h r o n i z i n g p u l s e s , f o r t r i g g e r i n g t h e sweep o f t h e o s c i l l o s -cope .were a l s o o b t a i n e d f r o m t h e m u l t i v i b r a t o r b y d i f f e r e n t i a t i o n o f t h e l e a d i n g and t r a i l i n g edges o f t h e p u l s e s . The a m p l i t u d e o f t h e s e t r i g g e r p u l s e s was i n d e p e n d e n t o f t h e a m p l i t u d e c o n t r o l f o r t h e m o d u l a t i n g p u l s e s , b u t s i n c e i t was r a t h e r s m a l l f o r s t a b l e t r i g g e r i n g , a n a m p l i f i e r was u s e d between t h e t r i g g e r o u t -p u t and t h e o s c i l l o s c o p e . 6) A w a t e r - c o o l e d i r o n c o r e e l e c t r o m a g n e t , c a p a b l e o f p r o d u c i n g a m a g n e t i c f i e l d o f up t o 6000 o e r s t e d t w i t h t h e p o l e - p i e c e s l e a v i n g a gap o f 2.15 i n c h e s . The magnet was s u p p l i e d v i a a n e t w o r k o f w a t e r c o o l e d r h e o s t a t s f r o m a l l O v DC g e n e r a t o r . An a x i a l h o l e a t t h e c e n t r e o f t h e p o l e p i e c e s was p r o v i d e d f o r t h e p a s s a g e o f l i g h t i n t h e d i r e c t i o n o f t h e m a g n e t i c f i e l d . c) A c r y o s t a t a l l o w i n g a l l m easurements t o be made a t t e m p e r a -t u r e s down t o a p p r o x i m a t e l y 1.4° K. T h i s c r y o s t a t c o n s i s t e d o f a " s o f t " i n n e r dewar f l a s k t o p p e d b y a b r a s s cup w h i c h c o u l d be f i l l e d w i t h l i q u i d a i r . . T h i s dewar, w h i c h c o u l d b e f i l l e d w i t h l i q u i d h e l i u m v i a a s y p h o n i n a c o n v e n t i o n a l manner, c o n t a i n e d t h e r e s o n a n t c a v i t y o f t h e s p e c t r o m e t e r a s w e l l as a c o n s t a n t volume h e l i u m t h e r m o m e t e r , c o n n e c t i o n s t o p r e s s u r e g a u g e s , o u t -s i d e a i r , a n d a h i g h s p e e d r e t a r y vacuum pump. I t was s e a l e d a t t h e t o p b y a r u b b e r s l e e v e l u b r i c a t e d w i t h g l y c e r i n e , w h i c h s l i p p e d o v e r t h e r i m o f t h e b r a s s cup a n d t h e e n d o f t h e dewar. T o a l l o w f r e e p a s s a g e o f l i g h t , t h e t a i l e n d o f t h e dewar, h o l d i n g t h e c a v i t y , was l e f t u n s i l v e r e d . S u r r o u n d i n g t h e i n n e r dewar was a h a r d vacuum o u t e r dewar f l a s k t o h o l d l i q u i d a i r a s p r e c o o l a n t . I t s t a i l e n d was l i k e w i s e l e f t u n s i l v e r e d t o a l l o w p a s s a g e o f l i g h t . B u b b l i n g o f t h e c o o l a n t i n t h e o u t e r dewar c a n n o t be c o m p l e t e l y a v o i d e d , p a r t i c u l a r l y i n t h e u n s i l -v e r e d t a i l e n d . Y e t b u b b l e s p a s s i n g t h e a c t u a l l i g h t p a t h i n -t r o d u c e n o i s e i n t h e s i g n a l i n v e s t i g a t e d o f a t y p e a n d s t r e n g t h , w h i c h w o u l d make measurements o f t h e k i n d d e s c r i b e d i n t h i s t h e s i s n e a r l y I m p o s s i b l e . To e l i m i n a t e s u c h b u b b l e s i n t h e l i g h t p a t h and a t t h e same t i m e r e d u c e r a d i a t i o n o f h e a t i n t o t h e i n n e r dewar, a c y l i n d e r o f c o p p e r - f o i l was w r a p p e d a r o u n d t h e t a i l o f t h e i n n e r dewar. At t h e a p p r o p r i a t e p o s i t i o n s r e c -t a n g u l a r o p e n i n g s o f a b o u t 4 b y 4 mm were c u t i n t h e f o i l t o " a l l o w p a s s a g e o f l i g h t . S q u a r e windows o f t h e same i n n e r d i -m e n s i o n were c u t o u t o f S c o t c h b r a n d No. 33 e l e c t r i c a l t a p e and i n s e r t e d between t h e c o p p e r f o i l a n d t h e o u t e r w a l l s o f t h e i n n e r dewar, u n t i l a t t h e windows t h e t h i c k n e s s was s u c h t h a t a t i g h t f i t o f t h e c o p p e r f o i l t o t h e i n n e r w a l l s o f t h e o u t e r dewar was o b t a i n e d when t h e l a t t e r was p u t i n t o p o s i t i o n . I n t h i s manner a t u n n e l i n t h e d i r e c t i o n o f p r o p a g a t i o n o f l i g h t was c r e a t e d b e t w e e n t h e two dewar w a l l s w h i c h r e m a i n e d f r e e o f c o o l a n t a n d hence f r e e o f b u b b l e s . B u b b l i n g i n t h e h e l i u m b a t h does n o t o c c u r b e l o w t h e X - p o i n t (2.19°K ) , t h e r e g i o n i n w h i c h most measurements were made. By means o f t h e h i g h - s p e e d r o t a r y pump p r e s s u r e i n t h e i n n e r dewar c o u l d be r e d u c e d o v e r t h e l i q u i d h e l i u m b a t h t o r e a c h 27 t e m p e r a t u r e s a s low as 1.38° K . T e m p e r a t u r e s b e l o w 4 . 2 ° K c o u l d be m e a s u r e d b y a m e r c u r y - a n d an o i l - m a n o m e t e r ( A p i e z o n o i l B), w h i c h i n d i c a t e d t h e v a p o u r p r e s s u r e o f t h e h e l i u m i n t h e i n n e r dewar. The c o n s t a n t volume t h e r m o m e t e r was u s e d o n l y t o o b t a i n an i n d i c a t i o n o f t h e t e m p e r a t u r e i n t h e i n n e r dewar d u r i n g p r e c o o l i n g p r i o r t o t h e t r a n s f e r o f l i q u i d h e l i u m , d) An o p t i c a l s y s t e m c a p a b l e o f p r o d u c i n g m o n o c h r o m a t i c p l a n e -p o l a r i z e d l i g h t w h i c h c o u l d be p a s s e d a l o n g t h e a x i s o f t h e mag-n e t i c f i e l d t h r o u g h t h e sample and s u b s e q u e n t l y a n a l y z e d . (See d i a g r a m F i g u r e X I I ) , A G e n e r a l - E l e c t r i c m e r c u r y a r c t y p e H 100-4A s e r v e d as l i g h t s o u r c e . T h i s was s u p p l i e d f r o m a DC g e n e r a t o r g i v i n g a p -p r o x i m a t e l y 190v v i a a f i l t e r n e t w o r k d e s i g n e d t o r e d u c e g e n e r a -t o r r i p p l e a nd t o s t a b i l i z e t h e c u r r e n t f l o w i n t h e a r c ( s e e d i a -gram F i g u r e X I I I ) . The l i g h t s o u r c e was f o l l o w e d b y a c o n v e r g -i n g l e n s s e r v i n g a s a c o l l i m a t o r . No p a r t i c u l a r p a i n s were t a k e n t o make t h e l i g h t e m e r g i n g f r o m t h e c o l l i m a t o r t r u l y p a r a l l e l , s i n c e t h e p o l a r i z e r u s e d i n t h e e x p e r i m e n t c o n t a i n e d a G l a n -Thompson p r i s m g i v i n g c o m p l e t e p o l a r i z a t i o n f o r l i g h t h a v i n g an a n g l e o f i n c i d e n c e up t o 3 0 ° . F o l l o w i n g t h e c o l l i m a t o r t h e l i g h t p a s s e d t h r o u g h a c o l o u r e d g l a s s f i l t e r s e l e c t i n g t h e g r e e n m e r c u r y l i n e ( X=5461 A ) and t h e n t h r o u g h t h e p o l a r i z e r a G l a n -Thompson p r i s m o f 1 b y 1 cm c r o s s - s e c t i o n and 2 cm l e n g t h . A f t e r p a s s a g e t h r o u g h t h e p o l a r i z e r a s y s t e m c o n s i s t i n g o f 1 d i v e r g i n g and 1 c o n v e r g i n g l e n s s e r v e d t o f o c u s t h e p l a n e o f t h e p o l a r i z e r t h r o u g h t h e a x i a l h o l e i n t h e magnet i n t o t h e sample i n t h e r e s o -H g - A R C P O L A R I Z E R F I L T E R F I G U R E 2 H C R Y S T A L A N A L Y Z E R C A V I T Y M A G N E T C O R E P H O T O -M U L T I P L I E R O P T I C A L S Y S T E M F I G U R E H E 5 Hy 1500 A h V W H GENERATOR 4 0 UF 8 0 UF AMMETER CURRENT REGULATOR Hg -ARC 5819 BATTERIES • 90 V E A C H 'SHUNT GALVANI 10 K r W V H O M E T E R .01 UF — O S C I L L O S C O P E UNIT P U L S E R C I R C U I T S A S S O C I A T E D W I T H O P T I C A L S Y S T E M F o l l o w i n g page 27 28 n a n t c a v i t y . From t h e sample t h e e m e r g i n g l i g h t p a s s e d t h r o u g h t h e a x i a l h o l e i n t h e s e c o n d h a l f o f t h e magnet, t h e n t h r o u g h a t e l e s c o p i c s y s t e m o f two c o n v e r g i n g l e n s e s f o l l o w e d by t h e a n a l y z e r - a g a i n a Glan-Thompson p r i s m i d e n t i c a l t o t h e p o l a r i z e r mounted i n a d i v i d e d r i n g w i t h v e r n i e r s c a l e -, a n d f i n a l l y o n t o t h e p h o t o c a t h o d e o f an RCA (5819) p h o t o m u l t i p l i e r t u b e . To a s s u r e l i n e a r i t y o f t h e r e s p o n s e o f t h e p h o t o m u l t i p -l i e r i t was s u p p l i e d b y r a d i o B - B a t t e r i e s d i r e c t l y w i t h o u t t h e u s e o f a d i v i d e r n e t w o r k . 180v were a p p l i e d b etween p h o t o -c a t h o d e and dynode no. 1, 90v b e t w e e n e a c h o f t h e f o l l o w i n g dynodes and t h e f i n a l dynode and t h e p l a t e . The p l a t e was s u p -p l i e d t h r o u g h a 10 KSX l o a d r e s i s t o r s h u n t e d by a . O l y u F m i c a -c a p a c i t o r a n d a g a l v a n o m e t e r o f 381 r e s i s t a n c e w h i c h c o u l d be s h o r t e d , o r s h u n t e d b y a r e s i s t a n c e box o f 0 t o 99 JCL . The p l a t e - s i d e o f t h e l o a d r e s i s t o r was g r o u n d e d , and t h e s i g n a l dev-e l o p e d a c r o s s t h e l o a d was f e d v i a c o a x i a l c a b l e t o t h e V 4 DC-a m p l i f i e r i n p u t o f a C o s s o r m o d e l 1049 d o u b l e beam o s c i l l o s c o p e e q u i p p e d w i t h a C o s s o r o s c i l l o s c o p e camera m o d e l 1428. The camera was l o a d e d w i t h I l f o r d BP-3 f i l m . The o u t p u t f r o m a G e n e r a l - R a d i o t y p e 1217-A u n i t - p u l s e r was c o n n e c t e d t o t h e i n -p u t o f t h e o s c i l l o s c o p e t o p r o v i d e t i m e m a r k e r s on t h e s e c o n d t r a c e . ( F o r a f u l l d i a g r a m o f t h e l i g h t - d e t e c t i n g c i r c u i t see d i a g r a m F i g u r e X I I I ) . The .01 yuF c a p a c i t o r a c r o s s t h e p l a t e - l o a d o f t h e p h o t o -m u l t i p l i e r p r o v i d e d t h e o u t p u t w i t h a t i m e - c o n s t a n t o f .1 - , i : m i l l i s e c o n d s , t h u s s e r v i n g t o r e d u c e t h e n o i s e b a n d w i d t h o f t h e 29 ' - s y s t e m , e f f e c t i v e l y e l i m i n a t i n g h i g h - f r e q u e n c y s h o t n o i s e f r o m t h e p h o t o m u l t i p l i e r and t h e l i g h t s o u r c e , w h i l e f a i t h f u l l y r e p r o -d u c i n g a l l s i g n a l components f r o m 0 t o 1000 c y c l e s p e r s e c o n d and t h u s a l l o w i n g c o r r e c t r e p r o d u c t i o n o f d e c a y - p r o c e s s e s w i t h t i m e -c o n s t a n t s ^ 1 m i l l i s e c o n d . The i n t e r n a l sweep o f t h e o s c i l l o -s c o p e was t r i g g e r e d f r o m t h e l e a d i n g e d g e s o f t h e m i c r o w a v e p u l s e s a p p l i e d t o t h e c a v i t y as m e n t i o n e d u n d e r cn, )', -A d e t a i l e d a c c o u n t o f some o f t h e m e c h a n i c a l f e a t u r e s o f t h e s p e c t r o m e t e r , e . g . c a v i t y , vacuum s y s t e m , e t c . h a s b e e n g i v e n b y H. Wesemeyer ( V I I I , 1 9 5 8 ) , 2) D e s c r i p t i o n o f en E x p e r i m e n t P r i o r t o e a c h e x p e r i m e n t a s u i t a b l e s a m ple h a d b e e n p r e -p a r e d , mounted o n a c h o k e p l u n g e r , a n d i t s p r o p e r o r i e n t a t i o n i n t h e c a v i t y a p p r o x i m a t e l y e s t a b l i s h e d . Some 8 t o 10 h o u r s b e f o r e t h e a c t u a l measurements were p l a n n e d t o b e g i n , t h e p h o t o m u l t i p l i e r was s w i t c h e d on and a n amount o f l i g h t e q u i v a l e n t t o t h e m a x i -m u m-light i n t e n s i t y e x p e c t e d d u r i n g t h e a c t u a l e x p e r i m e n t was a l l o w e d t o f a l l o n t o t h e p h o t o c a t h o d e ( u s u a l l y p r o d u c i n g a p h o t o -c u r r e n t o f t h e o r d e r o f 50 ^ amp.). T h i s was done t o a l l o w p o s -s i b l e f a t i g u e e f f e c t s a t t h e f i n a l d ynodes t o come t o e q u i l i b r i u m b e f o r e any measurements were made. S i n c e t h e s e e f f e c t s a r e s l o w , t h e y do n o t t e n d t o p r o d u c e sudden c h a n g e s i n t h e s e n s i t i v i t y o f t h e p h o t o m u l t i p l i e r , once e q u i l i b r i u m has been e s t a b l i s h e d a t t h e a p p r o x i m a t e l e v e l o f i n t e n s i t y u s e d d u r i n g an e x p e r i m e n t . The s y p h o n u s e d f o r t h e h e l i u m t r a n s f e r was a t t a c h e d t o t h e a p p a r a t u s t h e c h o k e - p l u n g e r w i t h t h e sample removed f r o m t h e c a v i t y , a n d t h e o p e n i n g t h r o u g h w h i c h i t c o u l d be i n s e r t e d f r o m t h e t o p o f t h e a p p a r a t u s was c l o s e d b y a r u b b e r s t o p . Then t h e i n n e r dewar and 30 t h e i n n e r dewar w a l l s w e r e e v a c u a t e d f o r s e v e r a l h o u r s . A b o u t 1 t o 3 c m m e r c u r y - p r e s s u r e o f a i r were a d m i t t e d t o t h e i n n e r dewar w a l l s f o r t h e p u r p o s e o f c o n d u c t i n g h e a t away f r o m t h e i n s i d e d u r i n g p r e c o o l i n g . T h i s a i r was f r o z e n o u t when l i q u i d h e l i u m was t r a n s f e r r e d , t h u s c r e a t i n g t h e n e c e s s a r y h a r d vacuum when nee d e d , From a c y l i n d e r h e l i u m g a s was t h e n p a s s e d i n t o t h e i n n e r dewar t h r o u g h a c h a r c o a l t r a p immersed i n l i q u i d a i r , u n t i l t h e p r e s s u r e r e a c h e d a p p r o x i m a t e l y 2 t o 3 cm m e r c u r y above atmos-p h e r i c p r e s s u r e . A t t h i s p o i n t t h e r u b b e r s t o p c l o s i n g t h e c h a n n e l t h r o u g h w h i c h t h e c h o k e - p l u n g e r w i t h sample c o u l d be i n -s e r t e d was removed and t h e c h o k e - p l u n g e r w i t h t h e sample a t t a c h e d was q u i c k l y i n s e r t e d . S i n c e t h e h e l i u m i n s i d e h a d b e e n u n d e r o v e r p r e s s u r e , d i f f u s i o n o f a i r i n t o t h e s y s t e m d u r i n g t h i s p r o -c e s s was k e p t to a minimum, a n d i f i t a c t u a l l y t o o k p l a c e , i t n e v e r made i t s e l f f e l t d u r i n g a n y o f t h e numerous e x p e r i m e n t s . The choke p l u n g e r was i m m e d i a t e l y o r i e n t e d i n t o t h e p r e d e t e r m i n e d p o s i t i o n , a n d the o u t e r dewar as w e l l as the. b r a s s cup on t o p o f t h e i n n e r dewar were f i l l e d up w i t h l i q u i d a i r and k e p t f u l l u n t i l t h e end o f t h e e x p e r i m e n t . W i t h i n 1 t o 1 § h o u r s t h e i n n e r dewar c o n t e n t s were c o o l e d t o l i q u i d a i r t e m p e r a t u r e . D u r i n g t h i s t i m e h e l i u m g a s was c o n t i n u a l l y a d m i t t e d i n t o t h e s y s t e m t o m a i n t a i n an i n s i d e p r e s s u r e s l i g h t l y a bove t h a t o f t h e atmos-p h e r e . T h i s e l a b o r a t e p r e c o o l i n g p r o c e d u r e was n e c e s s a r y b e c a u s e t h e c r y s t a l s u s e d d i d n o t s t a n d up u n d e r vacuum a t e l e v a t e d t e m p e r a t u r e s and a l s o t e n d e d t o c r a c k when c o o l e d t o o s u d d e n l y . 1, ft 31 Once t h e i n n e r dewar c o n t e n t s had b e e n c o o l e d t o l i q t i i d a i r t e m p e r a t u r e , t h e s y s t e m was opened t o t h e h e l i u m r e t u r n l i n e , l e a d i n g t o a r e c o v e r y s y s t e m , and l i q u i d h e l i u m was t r a n s -f e r r e d f r o m a s t o r a g e c o n t a i n e r i n t o t h e i n n e r dewar b y p r e s s i n g i t t h r o u g h t h e s y p h o n . U s u a l l y 2.5 t o 3 l i t r e s o f l i q u i d h e l i u m were t a k e n f r o m t h e c o n t a i n e r d u r i n g a t r a n s f e r . Of t h e s e .5 t o .8 l i t r e e v a p o r a t e d w h i l e t h e i n n e r dewar c o o l e d down t o 4,2° K . As s o o n as t h e i n n e r dewar was f i l l e d w i t h l i q u i d h e l i u m t h e r e t u r n l i n e was c l o s e d and s i m u l t a n e o u s l y t h e s y s t e m opened t o a i r . The s t o r a g e c o n t a i n e r was removed f r o m t h e s y p h o n , and t h e s y p h o n was e i t h e r removed f r o m t h e dewar o r i t s f r e e e n d c l o s e d ( i n c a s e a s e c o n d t r a n s f e r was c o n t e m p l a t e d f o r a l a t e r t i m e ) . Then t h e s y s t e m was c l o s e d t o t h e a i r and t h e r e t u r n l i n e o p ened s i m u l t a n e o u s l y . F o r t h e e x p e r i m e n t s done a t a t e m p e r a -t u r e b e l o w 4 . 2 ° K t h e r e t u r n l i n e was c l o s e d a g a i n a n d t h e s y s -tem was opened t o t h e h i g h s p e e d r o t a r y pump. When pumping away t h e h e l i u m v a p o u r , a b o u t .4 t o .5 l i t r e s o f h e l i u m e v a p o r a t e d b e f o r e t h e s y s t e m came t o e q u i l i b r i u m a t t h e d e s i r e d t e m p e r a t u r e b e l o w t h e A - p o i n t . The t e m p e r a t u r e c o u l d be a d j u s t e d by v a r y -i n g t h e pumping s p e e d . T h i s was a c h i e v e d b y a d j u s t i n g two p a r a l l e l v a l v e s i n t h e pumping l i n e . U s u a l l y i t was n o t f o u n d t o o d i f f i c u l t t o m a i n t a i n t h e t e m p e r a t u r e c o n s t a n t w i t h i n .02° K t h r o u g h o u t t h e measurements a l t h o u g h some m a n i p u l a t i o n o f t h e v a l v e s was r e q u i r e d when c h a n g e s i n t h e amount o f mi c r o w a v e pow-e r d i s s i p a t e d i n t h e c a v i t y t o o k p l a c e . 32, W h i l e t h e c r y o s t a t was pumped down, t h e m e r c u r y a r c was s w i t c h e d on as f o l l o w s : A 10 Amp m e t e r was p l a c e d i n s e r i e s w i t h t h e a r c and ' 190V DC were a p p l i e d t o t h e c i r c u i t . T h i s was a p p r o x i m a t e l y t h e p o t e n t i a l r e q u i r e d t o i g n i t e t h e a r c . Upon i g n i t i o n t h e a r c drew 4 t o 5 Amp o f c u r r e n t . As the a r c h e a t e d up, t h e c u r r e n t g r a d u a l l y d e c r e a s e d . When i t r e a c h e d a p p r o x i m -a t e l y 1.1 Amp. t h e s w i t c h s h u n t i n g t h e c u r r e n t r e g u l a t o r t u b e was o pened and a 10V m e t e r c o n n e c t e d a c r o s s i t . The v a r i a b l e r e s i s t a n c e i n t h e c i r c u i t was t h e n a d j u s t e d u n t i l t h e v o l t a g e d r o p a c r o s s t h e r e g u l a t o r was a p p r o x i m a t e l y 6 t o 7V. T h i s i s t h e b e s t o p e r a t i n g r e g i o n f o r t h i s t y p e o f t u b e . S u b s e q u e n t l y t h e v o l t a g e was c h e c k e d f r o m t i m e t o t i m e and t h e r e s i s t a n c e r e a d -j u s t e d i f n e c e s s a r y t o m a i n t a i n t h e p r o p e r o p e r a t i n g p o i n t o f t h e t u b e . As s o o n as t h e t e m p e r a t u r e h a d s t a b i l i z e d , a f i n a l v i s u a l a l i g n m e n t o f t h e o p t i c a l s y s t e m was made t o a s s u r e p a s s a g e o f a maximum i n t e n s i t y o f l i g h t t h r o u g h t h e c a v i t y a n d s a m p l e . T h i s a l i g n m e n t had t o be done a t t h i s s t a g e s i n c e t h e p o s i t i o n o f t h e c a v i t y was n o t e x a c t l y t h e same a t l i q u i d h e l i u m t e m p e r a t u r e a s a t room t e m p e r a t u r e . A l s o pumping on t h e i n n e r dewar s h i f t e d t h e l a t t e r and h e n c e t h e c a v i t y s l i g h t l y i n a d i r e c t i o n p e r p e n d i c u l a r t o t h e d i r e c t i o n o f p r o p a g a t i o n o f l i g h t . A f t e r a l i g n m e n t o f t h e o p t i c a l s y s t e m t h e f i n a l a l i g n m e n t o f t h e c r y s t a l , so as t o h a v e i t s o p t i c a x i s a s n e a r l y as p o s -s i b l e p a r a l l e l t o t h e d i r e c t i o n o f p r o p a g a t i o n o f l i g h t , t o o k p l a c e . T h i s was done i n t h e f o l l o w i n g manner: The p h o t o m u l t i p l i e r was p l a c e d i n p o s i t i o n b e h i n d t h e 33 a n a l y z e r and i t s o u t p u t d i s p l a y e d o n t h e o s c i l l o s c o p e a s a l i n e by u s i n g a f a s t , u n t r i g g e r e d , r e p e t i t i v e sweep. The s e c o n d beam was u s e d as a r e f e r e n c e l i n e . A b l a c k c l o t h was d r a p e d a r o u n d t h a t p a r t o f t h e o p t i c s y s t e m e x t e n d i n g f r o m t h e magnet t o t h e p h o t o m u l t i p l i e r a n d a l l l i g h t s i n t h e room were s w i t c h e d o f f , e x c e p t a s i n g l e i n c a n d e s c e n t b u l b i n a f a r c o r n e r o f t h e l a b o r -a t o r y , w h i c h p r o v i d e d v e r y s u bdued i n d i r e c t ' l i g h t t o t h e r e g i o n o c c u p i e d b y t h e a p p a r a t u s . By t h e s e p r e c a u t i o n s i t was made s u r e t h a t no m e a s u r a b l e amount o f l i g h t f r o m unwanted s o u r c e s r e a c h e d t h e p h o t o m u l t i p l i e r . Now t h e a n a l y z e r was r o t a t e d un-t i l t h e p h o t o c u r r e n t was a t a minimum a s e v i d e n c e d b y t h e p o s i -t i o n o f t h e l i n e on t h e o s c i l l o s c o p e , r e p r e s e n t i n g t h e p h o t o -m u l t i p l i e r o u t p u t . A d j u s t m e n t s w e r e made o n t h e o s c i l l o s c o p e t o p o s i t i o n t h e b o t t o m edge o f t h e l i n e c o r r e s p o n d i n g t o t h e p h o t o c u r r e n t a b o u t . 5 mm above t h e t o p edge o f t h e r e f e r e n c e l i n e . I n t h i s way v e r y s m a l l c h a n g e s i n t h e i n t e n s i t y o f l i g h t f a l l i n g o n t o t h e p h o t o m u l t i p l i e r c o u l d be d e t e c t e d r a p i d l y and a t l e a s t as a c c u r a t e l y as by o b s e r v a t i o n o f t h e g a l v a n o m e t e r m e a s u r i n g t h e p h o t o c u r r e n t . T h i s p r o c e d u r e was f o l l o w e d when-e v e r a d j u s t m e n t s f o r minimum i n t e n s i t y o r f o r c o m p a r i s o n o f i n -t e n s i t y l e v e l s were made. P a r t i c u l a r l y n e a r i n t e n s i t y m i n ima t h e i n t e n s i t y v e r s u s a n a l j r z e r s e t t i n g c u r v e i s e x t r e m e l y b r o a d and f l a t due t o t h e c o s & r e l a t i o n s h i p , m a k i n g d i r e c t v i s u a l d i s c r i m i n a t i o n o f m i n i m a w h i c h a r e n o t i d e n t i c a l l y z e r o e x t r e m -e l y d i f f i c u l t . H a v i n g s e t t h e a n a l y z e r f o r minimum i n t e n s i t y t h e c r y s t a l was c a r e f u l l y r o t a t e d i n a h o r i z o n t a l p l a n e , c o n t a i n -i n g t h e d i r e c t i o n o f p r o p a g a t i o n o f l i g h t , u n t i l no f u r t h e r improvement o f t h e minimum b y a d j u s t m e n t o f e i t h e r t h e a n a l y z e r o r t h e c r y s t a l c o u l d be made. The f i n a l a d j u s t m e n t o f t h e c r y -s t a l had t o be s u c h t h a t t h e s l i g h t e s t r o t a t i o n c l o c k w i s e as w e l l as c o u n t e r c l o c k w i s e r e s u l t e d i n an i n c r e a s e i n i n t e n s i t y ( o t h e r -w i s e , b y t u r n i n g t h e c r y s t a l 9 0 ° o f f a x i s and t h u s p r e s e n t i n g i t s s i d e s c o v e r e d w i t h p l a s t i c i n e t o t h e l i g h t , an e x c e l l e n t minimum was a l s o o b t a i n a b l e ' . } . W i t h t h e t e m p e r a t u r e o f t h e c a v i t y s t a b i l i z e d , t h e op-t i c a l s y s t e m and t h e c r y s t a l p r o p e r l y a l i g n e d , a c t u a l m e a s u r e -ments were begun, a f t e r f i r s t m a k i n g s u r e t h a t c o o l i n g - w a t e r was s u p p l i e d t o magnet, r h e o s t a t s , and k l y s t r o n . A t f i r s t t h e F a r a d a y r o t a t i o n a s a f u n c t i o n o f f i e l d was m e a s u r e d w i t h no m i c r o w a v e power p r e s e n t i n t h e c a v i t y , i . e . w i t h t h e k l y s t r o n e i t h e r n o t on a t a l l y o r o p e r a t i n g a t a mode w i t h s m a l l o u t p u t way o f f t h e r e s o n a n c e f r e q u e n c y o f t h e c a v i t y and w i t h b o t h a t t e n u a t o r s s e t f o r maximum a t t e n u a t i o n . The p o s i t i o n o f t h e a n a l y z e r f o r minimum i n t e n s i t y w i t h no c u r r e n t f l o w i n g t h r o u g h t h e magnet was r e c o r d e d , t h e n w i t h v a r i o u s c u r r e n t s , a r r a y s s t a r t i n g a t 35 amp. and w o r k i n g down t h e h y s t e r e s i s c u r v e b a c k t o 0 amp. A t y p i c a l c u r v e may be t a k e n f r o m 0 Amp, 35 Amp, 25 Amp, 15 Amp., 10 Amp.,5 Amp., 0 Amp. A t e a c h f i e l d 5 r e a d i n g s o f t h e a n a l y z e r s e t t i n g f o r minimum i n -t e n s i t y were o b t a i n e d a n d a v e r a g e d . A t t h i s s t a g e a c a l i b r a t i o n o f t h e r e s p o n s e o f t h e s y s -tem t o v a r i o u s a n g l e s b e t w e e n i n c o m i n g p o l a r i z a t i o n and s e t t i n g o f t h e a n a l y z e r was u s u a l l y made,.to c h e c k on t h e co ^ r e l a t i o n -s h i p , t h e l i n e a r i t y o f t h e p h o t o m u l t i p l i e r and o s c i l l o s c o p e 35 r e s p o n s e , and a l s o t o e s t a b l i s h an u p p e r l i m i t f o r t h e e l l i p t -i c i t y o f t h e l i g h t e m e r g i n g f r o m t h e c r y s t a l . The c a l i b r a t i o n was done as f o l l o w s : The r e f e r e n c e t r a c e on t h e o s c i l l o s c o p e was p o s i t i o n e d n e a r t h e b o t t o m o f t h e c a t h o d e r a y t u b e , t h e a n a l y z e r s e t f o r minimum i n t e n s i t y , and t h e s i g n a l t r a c e p o s i t i o n e d s l i g h t l y above t h e r e f e r e n c e t r a c e . W i t h t h e g a i n o f t h e o s c i l l o s c o p e s e t t o t h e r a n g e b e s t s u i t e d t o t h e i n t e n s i t i e s o b t a i n e d i n a. g i v e n e x p e r i m e n t a n d medium b r i g h t n e s s o f t h e t r a c e s , a one s e c o n d e x p o s u r e was made on t h e camera. The a n a l y z e r was t h e n r o t a t e d c l o c k w i s e by 10 o r 15 d e g r e e s ( d e p e n d i n g on t h e o c c a -s i o n ) and a n o t h e r e x p o s u r e was made on t h e same f r a m e ; t h i s was r e p e a t e d u n t i l a f u l l 9 0 ° had b e e n c o v e r e d , w h e r e u p o n t h e f i l m was moved t o t h e n e x t f r a m e and a n o t h e r s e t o f e x p o s u r e s was t a k e n , c o v e r i n g t h e r a n g e f r o m 0 ° t o 9 0 ° c o u n t e r c l o c k w i s e . At e a c h e x p o s u r e a l s o t h e p h o t o c u r r e n t , as m e a s u r e d on t h e s u i t a b l y s h u n t e d g a l v a n o m e t e r was r e c o r d e d . N e x t t h e k l y s t r o n was s w i t c h e d on and a l l o w e d t o warm up f o r a t l e a s t h a l f an h o u r b e f o r e m a k i n g f u r t h e r m e a s u r e -m e n t s . The k l y s t r o n and t h e w h o l e microwave s y s t e m was t h e n c a r e f u l l y t u n e d t o t h e r e s o n a n c e f r e q u e n c y o f t h e c a v i t y , f o l -l o w e d by d e t u n i n g t h e k l y s t r o n i t s e l f by l o w e r i n g t h e p o t e n t i a l o f t h e r e f l e c t o r by a p p r o x i m a t e l y 40V. I t was a l w a y s a t t e m p t e d t o t u n e t h e k l y s t r o n as f a r as p o s s i b l e away f r o m r e s o n a n c e , w h i l e s t i l l b e i n g a b l e t o p u l s e i t b a c k i n t o r e s o n a n c e w i t h t h e p u l s e s s u p p l i e d b y t h e m u l t i v i b r a t o r . U s u a l l y t h e k l y s t r o n was 36 t h e n a b o u t 2 0 , m e g a c y c l e s o f f resonance;, and e i t h e r v e r y l i t t l e o r no i n d i c a t i o n o f a s i g n a l c oming t h r o u g h t h e c a v i t y c o u l d be ob-t a i n e d a t t h e o u t p u t on t h e g a l v a n o m e t e r . Next t h e m u l t i v i b r a t o r was a d j u s t e d t o t h e d e s i r e d p u l s e - l e n g t h ( u s u a l l y 8, 17, 35, 63 or 124 m i l l i s e c o n d s ) , a n d i t s o u t p u t i n c r e a s e d u n t i l a maximum o f power was d e l i v e r e d t o t h e c a v i t y d u r i n g t h e d u r a t i o n o f t h e p u l s e . An i n d i c a t i o n o f t h i s c o n d i t i o n was o b t a i n e d b y o b s e r v -i n g t h e s w i n g on t h e g a l v a n o m e t e r due t o t h e b u r s t s o f power f r o m t h e o u t p u t o f t h e c a v i t y . The magnet c u r r e n t was now i n c r e a s e d t o 35Amp. and t h e n r e d u c e d t o t h e v a l u e c o r r e s p o n d i n g t o t h e f i e l d s t r e n g t h a t w h i c h measurements w e r e t o be made. The m u l t i v i b r a t o r p u l s e s w e r e t e m p o r a r i l y c u t o f f and t h e a n a l y z e r a d j u s t e d f o r minimum a s b e -f o r e ( a g a i n 5 r e a d i n g s were t a k e n and a v e r a g e d ) . T h i s minimum d i d n o t a l w a y s c o r r e s p o n d t o t h e n o r m a l F a r a d a y r o t a t i o n , s i n c e e v e n t h e s l i g h t e s t amount o f m i c r o w a v e power e n t e r i n g t h e c a v i t y u n d e r "no p u l s e " c o n d i t i o n r e d u c e d t h e n o r m a l r o t a t i o n , p a r t i c u -l a r l y n e a r t h e r e s o n a n c e f i e l d . The p u l s e s were t h e n a g a i n a p p l i e d , t h e o s c i l l o s c o p e sweep s e t t o i t s e x t e r n a l l y t r i g g e r e d p o s i t i o n a t a s u i t a b l e s p e e d ( u s u a l l y c o v e r i n g 1500 o r 500 msec i n a s i n g l e sweep, but a t t i m e s g o i n g down t o 150, 50, o r e v e n 15 ( f o r f a s t e r r e l a x a t i o n t i m e s ) . The t i m i n g p u l s e s were a p p l i e d t o t h e s e c o n d beam ( u s u a l l y 30 p u l s e s p e r s e c o n d , lOmsecwide) . I f a r e a s o n a b l e s i g n a l , i . e . change i n i n t e n s i t y due t o change i n t h e F a r a d a y r o t a t i o n u pon a p p l i c a t i o n o f p u l s e d power t o t h e c a v i t y , was o b s e r v e d , t h e a p p r o x i m a t e change i n r o t a t i o n 37, was d e t e r m i n e d b y p o s i t i o n i n g t h e b a s e o f t h e t i m i n g t r a c e so a s t o c o i n c i d e w i t h t h e p e a k o f t h e s i g n a l , and t h e n r o t a t i n g t h e a n a l y z e r u n t i l t h e b a s e o f t h e s i g n a l c o i n c i d e d w i t h t h e p o s i t i o n p r e v i o u s l y o c c u p i e d by t h e p e a k o f t h e s i g n a l . The a n g l e t h r o u g h w h i c h t h e a n a l y z e r had t o be r o t a t e d t o a c h i e v e t h i s , was t h e n a measure o f t h e change i n r o t a t i o n p r o d u c e d b y t h e p u l s e s . T h i s was f r e q u e n t l y r e p e a t e d f o r a n a l y z e r s e t t i n g s d i f f e r e n t f r o m t h o s e g i v i n g a minimum f o r t h e b a s e o f t h e s i g n a l , s i n c e t h e n t h e s i g n a l c o u l d o f t e n be made t o sweep o v e r t h e more l i n e a r r e g i o n o f t h e cos? Q c h a r a c t e r i s t i c , r e s u l t i n g i n g r e a t e r a c c u r a c y o f t h e measurements. E x p o s u r e s o f t h e s i g n a l s due t o s i n g l e p u l s e s where t h e n made on t h e camera, w h i l e t h e s c r e e n o f t h e o s c i l l o -s c o p e was o b s e r v e d v i s u a l l y . T h i s c o u l d e a s i l y be done m a n u a l l y , due t o t h e low r e p e t i t i o n r a t e u s e d . O b s e r v a t i o n o f t h e o s c i l -l o s c o p e d u r i n g e x p o s u r e s d i d n o t cause a n y d i s t u r b i n g amount o f o u t s i d e l i g h t t o f a l l o n t h e f i l m , s i n c e t h e w h o l e r e g i o n o f t h e l a b o r a t o r y o c c u p i e d b y t h e a p p a r a t u s was i n n e a r d a r k n e s s ( r e a d i n g s on t h e a n a l y z e r s e t t i n g and o t h e r a d j u s t m e n t s were made w i t h t h e h e l p o f a f l a s h l i g h t ) . E x p o s u r e s were made a t v a r y i n g a n a l y z e r s e t t i n g s ( b a s e o f s i g n a l a t minimum, 4 5 ° o f f minimum o r o t h e r w i s e ) and w i t h d i f f e r e n t sweep s p e e d s . Whenever p o s s i b l e t h e s i g n a l was a l s o made t o sweep e n t i r e l y o r m a i n l y o v e r t h e l i n e a r r e g i o n o f t h e c o s ^ G c h a r a c t e r i s t i c ( a p p r o x i m a t e l y 3 0 ° t o 6 0 ° o f f t h e minimum). 4 t o 6 e x p o s u r e s were u s u a l l y made a t e a c h a n a l y z e r s e t t i n g . T h r o u g h o u t a l l measurements t h e h e l i u m v a p o u r p r e s s u r e was c h e c k e d a n d a d j u s t m e n t s o f t h e pumping s p e e d made when n e c -e s s a r y . The v a r i o u s p a r a m e t e r s (magnet c u r r e n t , h e l i u m v a p o u r p r e s s u r e , p u l s e l e n g t h , a n a l y z e r s e t t i n g s , sweep s p e e d , o s c i l l o -s cope g a i n ) a s s o c i a t e d w i t h e a c h e x p o s u r e w e r e r e c o r d e d . A t t i m e s e x p o s u r e s were a l s o made o f s i n g l e p u l s e s s p a c e d a t i n t e r v a l s o f s e v e r a l m i n u t e s and o f t h e f i r s t 5 t o ' 1 2 p u l s e s o f a seq u e n c e o f p u l s e s a t t h e n o r m a l r e p e t i t i o n r a t e ( ^ 1 e v e r y 2 s e c o n d s ) t o see w h e t h e r l o n g t i m e e f f e c t s w e r e n o t i c e a b l e i n t h e r e l a x a t i o n p r o c e s s . Once s a t i s f i e d t h a t enough e x p o s u r e s were o b t a i n e d f o r a p a r t i c u l a r s e t o f p a r a -m e t e r s , measurements u n d e r d i f f e r e n t c o n d i t i o n s ( d i f f e r e n t mag-n e t i c f i e l d , a n d / o r p u l s e l e n g t h , a n d / o r t e m p e r a t u r e ) were made i n an i d e n t i c a l f a s h i o n . A t y p i c a l s u c c e s s f u l e x p e r i m e n t y i e l d e d 200 t o 300 o r more e x p o s u r e s , r e p r e s e n t i n g r e l a x a t i o n p r o c e s s e s u n d e r 10 t o 25 e s s e n t i a l l y d i f f e r e n t s e t s o f c o n d i t i o n s ( f i e l d , t e m p e r a t u r e , p u l s e - l e n g t h ) . S u b s e q u e n t t o an e x p e r i m e n t t h e e x p o s e d f i l m s w e r e d e v e l -oped f o r 10 m i n u t e s i n Kodak D -76 d e v e l o p e r a n d f i x e d i n Kodak A c i d F i x e r w i t h H a r d e n e r f o r 10 t o 20 m i n u t e s , e x t e n s i v e l y washed, d r i e d a t room t e m p e r a t u r e and f i n a l l y e v a l u a t e d a s f o l -l o w s : The f r a m e s were p l a c e d f l a t b e t w e e n a g l a s s - p l a t e and m i l l i m e t e r g r a p h - p a p e r under a b i n o c u l a r low-power m i c r o s c o p e , and f o r e a c h d e c a y t h e i n t e n s i t y v e r s u s t i m e r e l a t i o n s h i p was m e a s u r e d and t a b u l a t e d . By means o f t h e e x p e r i m e n t a l d a t a and the c a l i b r a t i o n c u r v e s t h e i n t e n s i t y - t i m e r e l a t i o n s h i p was con -v e r t e d t o r o t a t i o n v e r s u s t i m e d a t a , e x c e p t when t h e s i g n a l was e n t i r e l y i n t h e l i n e a r i n t e n s i t y - r o t a t i o n r a n g e . The r o t a t i o n 3 9 , v e r s u s t i m e d a t a was p l o t t e d o n t o s e m i l o g g r a p h - p a p e r . The r e -s u l t i n g p o i n t s were f o u n d t o l i e to a f i r s t a p p r o x i m a t i o n o n a s t r a i g h t l i n e and t h e i n v e r s e o f t h e s l o p e o f t h i s l i n e was de-f i n e d a s t h e r e l a x a t i o n t i m e , w h i c h was t h e n t a b u l a t e d as a f u n c t i o n o f t h e v a r i o u s p a r a m e t e r s . Some t y p i c a l r e s u l t s a r e d i s c u s s e d i n C h a p t e r I I I . 3) P r e p a r a t i o n o f Samples The s a m p l e s u s e d i n t h e e x p e r i m e n t s were s i n g l e c r y s t a l s o f c e r i u m - and neodymium e t h y l s u l f a t e r e s p e c t i v e l y , i . e . C e ( C x H s S O > ) i - ? H a , 0 and Nd (C^ Hs 0 They were p r e p a r e d f r o m a q u e o u s s o l u t i o n s o f m o l e c u l a r e q u i v a l -e n t amounts o f t h e p a r t i c u l a r s u l f a t e and b a r i u m e t h y l s u l f a t e , BctfCjfcHsSO^a/ 2,Ha,0 a c c o r d i n g t o t h e r e a c t i o n ; (virtiere X s t a n d s f o r C e o r Mol r e s p e c t i v e l y ) . B a r i u m s u l f a t e f e l l o u t o f t h e m i x t u r e and was a l l o w e d t o s e t t l e a t t h e b o t t o m , t h e s o l u t i o n o f t h e r a r e - e a r t h e t h y l s u l f a t e was d e c a n t e d , c a r e f u l l y f i l t e r e d t o e l i m i n a t e a l l t r a c e s o f b a r i u m s u l f a t e a n d t h e s a l t s were o b t a i n e d b y l e t t i n g t h e s o l u t i o n e v a p -o r a t e a t room t e m p e r a t u r e i n a d e s i c c a t o r u n d e r r e d u c e d p r e s s u r e . The s a l t s were washed w i t h s m a l l amounts o f d i s t i l l e d w a t e r and t h e n k e p t i n a r e f r i g e r a t o r u n t i l f u r t h e r u s e ( s i n c e t h e y t e n d t o decompose s l o w l y a t r o o m - t e m p e r a t u r e ) . The p r e -p a r a t i o n has b e e n d e s c r i b e d i n d e t a i l b y l" !aeger ( IX, 1 9 1 4 ) . The b a r i u m e t h y l s u l f a t e u s e d i n t h e r e a c t i o n was a l s o p r e p a r e d i n t h e l a b o r a t o r y a c c o r d i n g t o B e i l s t e i n (X, 1 9 1 8 ) . 40 C o m m e r c i a l l y a v a i l a b l e p u r i f i e d c e r o u s s u l f a t e C e ^ ( SO*)* S H ^ O f r o m t h e F i s h e r S c i e n t i f i c Co. was u s e d i n t h e p r e p a r a t i o n o f t h e c e r i u m s a l t . The neodymium s u l f a t e , NOIJL ( ^ O i j . ^ * S H t 0 was p r e p a r e d i n t h e l a b o r a t o r y f r o m 99 % p u r e neodymium o x i d e s u p p l i e d by t h e L i n d s a y C h e m i c a l Co. The i m p u r i t i e s i n t h e neodymium o x i d e a r e c l a i m e d t o be t h e f o l l o w i n g : .1 t o .4% praseodymium, .1 t o .4% samarium, .5% o t h e r r a r e e a r t h o x i d e s . S i n g l e c r y s t a l s o f t h e d e s i r e d e t h y l s u l f a t e s were o b t a i n e d i n a v a r i e t y o f ways f r o m c l e a r d u s t f r e e s a t u r a t e d s o l u t i o n s o f t h e s a l t s : a) S i m p l y l e t t i n g a f a i r l y l a r g e amount ( ^ 30 cm ) o f s o l u t i o n e v a p o r a t e i n a d i s h , p l a c e d a t room t e m p e r a t u r e and a t m o s p h e r i c p r e s s u r e i n a d e s i c c a t o r , y i e l d e d , b e s i d e s u n u s a b l e s m a l l and t w i n n e d c r y s t a l s , some n i c e l y d e v e l o p e d l a r g e s p e c i m e n s . I n t h i s c a s e i t was e s s e n t i a l t o l e a v e t h e s o l u t i o n c o m p l e t e l y u n -d i s t u r b e d . b) F i l l i n g s m a l l b e a k e r s (1 t o 5 c m ) w i t h s a t u r a t e d s o l u t i o n , l e t t i n g a s e e d c r y s t a l e i t h e r r e s t a t t h e b o t t o m or f l o a t o n t h e s u r f a c e o f t h e s o l u t i o n , c o v e r i n g t h e b e a k e r s so a s t o admit a c c e s s o f a i r y e t p r e v e n t i n g d u s t p a r t i c l e s f r o m f a l l i n g i n t o t h e s o l u t i o n s , and l e t t i n g e v a p o r a t i o n t a k e p l a c e a t room t e m p e r -a t u r e . 30% t o 50% o f t h e s e e d s t e n d e d t o d e v e l o p i n t o s u f f i c i e n t -l y l a r g e s i n g l e c r y s t a l s . T h i s method had p r e v i o u s l y b e e n u s e d i n l a r g e r b e a k e r s b y H. Wesemeyer ( V I I , 1 9 5 8 ) . c) U s i n g a " c r y s t a l g r o w i n g m a c h i n e " o f own d e s i g n (See d i a g r a m F i g u r e X I V ) . T h i s m a c h i n e c o n s i s t e d o f two s m a l l g l a s s b e a k e r s j o i n e d a t t o p a n d b o t t o m by . .5 c m d i a m e t e r g l a s s - t u b i n g . 10 n o f r e s i s t a n c e w i r e were wound below t h e b e a k e r h a v i n g a l a r g e r F I G U R E 2L¥ Heater wi re ^ — 7 i ^ o o o o Saturated solution \\\\\\| M e r c u r y p o o l Ground glass cover Crystal seed C ylinder of f ilte r paper containing c rys ta ls C R Y S T A L GROWING A P P A R A T U S F o l l o w i n g page 40 d i a m e t e r . A t t h e b o t t o m o f t h e l o n g e r b e a k e r w i t h s m a l l e r d i a -m e t e r a p o o l o f mercury- ( *^ l c t v \ 3 ) was l o c a t e d . I n o p e r a t i o n t h e s y s t e m was f i l l e d w i t h s a t u r a t e d s o l u t i o n , a s e e d was i n t r o -d u c e d so t h a t i t f l o a t e d on t h e m e r c u r y p o o l a n d t h e b e a k e r s were c o v e r e d b y a g r o u n d g l a s s p l a t e t o a v o i d e v a p o r a t i o n . Below t h e g l a s s p l a t e a s m a l l c y l i n d r i c a l cup 1.7e»v> d i a m e t e r , 1.5c*y» l o n g ) made o f p e r f o r a t e d f i l t e r p a p e r a n d f i l l e d w i t h raw c r y s t a l s was g l u e d , so t h a t i t was a l m o s t c o m p l e t e l y immersed i n t h e s o l u t i o n i n t h e l a r g e d i a m e t e r b e a k e r . When a s m a l l c u r r e n t o f t h e o r d e r o f .1 Amp. was p a s s e d t h r o u g h t h e h e a t e r w i r e , a c o n v e c t i o n c u r r e n t s t a r t e d t o f l o w i n t h e s y s t e m . A t t h e same t i m e a temp-e r a t u r e g r a d i e n t d e v e l o p e d , t h e l a r g e d i a m e t e r b e a k e r c o n t e n t s b e i n g a b o u t 1 t o 2 d e g r e e s c e n t i g r a d e warmer t h a n t h e s o l u t i o n i n t h e l o n g s m a l l d i a m e t e r b e a k e r . The warmed s o l u t i o n d i s s o l v e d some o f t h e raw c r y s t a l s i n t h e f i l t e r p a p e r cup a n d became s u p e r s a t u r a t e d when p a s s i n g i n t o t h e c o o l e r b e a k e r , a l l o w i n g c r y -s t a l g r o w t h t o t a k e p l a c e . The m a c h i n e worked i n p r i n c i p l e and p r o d u c e d one good c e r i u m e t h y l s u l f a t e c r y s t a l . However, d i f f i c -u l t i e s were e x p e r i e n c e d i n k e e p i n g t h e s o l u t i o n c l e a n and r e -s t r i c t i n g c r y s t a l g r o w t h t o t h e s e e d . The c r y s t a l o b t a i n e d i n t h i s manner was o p t i c a l l y s u p e r i o r t o t h o s e grown by t h e o t h e r m e thods, p r o b a b l y b e c a u s e no d e p o s i t i o n o f a i r i n t h e c r y s t a l s was p o s s i b l e . No f u r t h e r r e f i n e m e n t s o n t h i s method were u n d e r -t a k e n , s i n c e enough c r y s t a l s h a d b e e n o b t a i n e d a l r e a d y b y t h e o t h e r methods. F o r u s e i n t h e e x p e r i m e n t s t h e o p t i c a x i s o f t h e c r y s t a l s , w h i c h a r e h e x a g o n a l i n s t r u c t u r e , had t o be i d e n t i f i e d , and p l a n e s h a d t o be c u t and p o l i s h e d p e r p e n d i c u l a r t o i t . The 42 a p p e a r a n c e o f t h e s e c r y s t a l s has b e e n d e s c r i b e d i n d e t a i l by J a e g e r ( I X , 1 9 1 4 ) . I d e n t i f i c a t i o n o f t h e p l a n e s o f t h e c r y s t a l s was d i f f i c u l t , p a r t i c u l a r l y i n s m a l l s a m p l e s . The s u r e s t way o f e s t a b l i s h i n g t h e i d e n t i t y o f t h e p l a n e s f o r m i n g t h e h e x a g o n a l p r i s m s was f o u n d t o be b y t h e i r c l e a v a g e p r o p e r t i e s . I t was f o u n d t h a t p e r f e c t c l e a v a g e o f t h e s e p l a n e s , and t h e s e p l a n e s o n l y , t o o k p l a c e when a r a z o r b l a d e was p r o p e r l y o r i e n t e d and p r e s s e d g e n t l y i n t h e d i r e c t i o n o f t h e o p t i c a x i s . The a p p r o p r i a t e p l a n e s w e r e g r o u n d f l a t w i t h f i n e corundum p a p e r o f p r o g r e s s i v e l y s m a l l e r g r a i n , and p o l i s h e d w i t h l e a t h e r a n d r o u g e . A b r a s s c y l i n d e r , h a v i n g a 1 2 0 ° s e c t o r c u t o u t a l o n g p a r t o f i t s a x i s and s l i d i n g i n a h e a v y s t a i n l e s s s t e e l c y l i n d e r , h e l d t h e c r y s t a l w h i c h was t i e d a nd p o s i t i o n e d b y p l a s t i c i n e , so t h a t one o f i t s 1 2 0 ° o u t s i d e a n g l e e d g e s r e s t e d i n t h e s e c t o r , w h i l e t h e p a r t t o be g r o u n d o f f p r o t r u d e d above t h e f l a t t o p o f t h e c y l i n d e r . Corundum p a p e r o r l e a t h e r was p l a c e d o n a f l a t g l a s s p l a t e , t h e s t e e l c y l i n d e r s e t o n t o p , a n d t h e b r a s s c y l i n d e r i n s e r t e d i n t o t h e s t e e l c y l i n d e r , so t h a t t h e p r o t r u d i n g edge o f t h e c r y s t a l r e s t e d on t h e g r i n d i n g s u r f a c e and was p r e s s e d down s l i g h t l y b y t h e w e i g h t o f t h e b r a s s c y l i n d -e r . By g e n t l e r o t a t i o n a l and t r a n s l a t i o n a l m o t i o n o f t h e s t e e l b l o c k , t h e c r y s t a l was g r o u n d down u n t i l i t s f a c e was f l u s h w i t h t h e t o p o f t h e b r a s s a n d s t e e l c y l i n d e r s , a n d t h u s p e r p e n d i c u l a r t o t h e o p t i c a x i s . The s a m p l e s used, i n t h e e x p e r i m e n t s v a r i e d i n l e n g t h ( a l o n g t h e o p t i c a x i s ) b e t w e e n 3 and 5 mm, and i n c r o s s - s e c t i o n a l st a r e a b etween 4 and 10 mm . The o p t i c a l q u a l i t y o f t h e c r y s t a l s l e f t much t o be d e s i r e d . E v e n t h o u g h t h e b e s t , i . e . t h e c l e a r e s t , 43 p a r t s were c u t o u t o f t h e s i n g l e c r y s t a l s a n d u s e d f o r s a m p l e s , t h e y s t i l l h a d some i n c l u s i o n s ( m a i n l y a i r b u b b l e s and s m a l l r e g i o n s o f i m p e r f e c t g r o w t h a s s o c i a t e d w i t h t h e l a t t e r ) . These i m p e r f e c t i o n s t e n d e d t o o b s t r u c t p a r t i a l l y t h e l i g h t p a t h , and .produced some d i f f u s e s c a t t e r i n g o f t h e i n c i d e n t l i g h t , r e -s u l t i n g i n a component o f r a n d o m l y p o l a r i z e d l i g h t i n t h e emerg-i n g beam. The c r y s t a l s had t o be h a n d l e d w i t h extreme c a r e , s i n c e t h e y a r e n o t o n l y v e r y s o f t , b r i t t l e and e a s i l y s o l u b l e , b u t a l s o decompose r a p i d l y a t t e m p e r a t u r e s above 3 5 ° C . C h a p t e r I I I : Measurements The e x p e r i m e n t a l r e s u l t s q u o t e d i n t h i s c h a p t e r r e f e r t o e x p e r i m e n t s p e r f o r m e d • w i t h neodymium e t h y l s u l f a t e e x c e p t where c e r i u m e t h y l s u l f a t e i s e x p l i c i t l y m e n t i o n e d . 1) L i n e a r i t y , R e p r o d u c i b i l i t y , e t c . , o f t h e A p p a r a t u s a) M icrowave s y s t e m : The f r e q u e n c y o f t h e microxvaves c o u l d be m e a s u r e d w i t h t h e h e l p o f t h e wavemeter t o an a c c u r a c y o f b e t t e r t h a n £ . 5 M c S i n c e t h e k l y s t r o n was immersed i n a w a t e r c o o l e d o i l b a t h and d r i v e n f r o m a w e l l r e g u l a t e d power s u p p l y i t d i d n o t show any d r i f t i n f r e q u e n c y a f t e r warm-up. The u p p e r l i m i t f o r d r i f t d u r i n g a p e r i o d o f 15 m i n u t e s c a n c o n s e r v a t i v e l y be g i v e n a s s m a l l e r * .215 M c . I n p r a c t i c e no f r e q u e n c y d r i f t was e v e r e v i d e n t a f t e r t h e i n i t i a l warm up, e v e n t h o u g h one s h o u l d h a v e n o t i c e d d r i f t s as s m a l l as .1 M c ( t h a t i s a p p r o x i m a t e l y 1 p a r t i n 1 0 5 ) a s f l u c t u a t i o n s i n t h e o u t p u t o f t h e t u n e d r e s o n -a n t c a v i t y . The f r e q u e n c y o f t h e k l y s t r o n d u r i n g t h e d u r a t i o n o f t h e p u l s e s , when p u l s e - m o d u l a t e d , c a n s a f e l y be s a i d t o have h a d t h e same a c c u r a c y , t h o u g h i n t h i s c a s e a d r i f t was o c c a s i o n a l l y n o t i c e d and had t o be c o r r e c t e d b y a d j u s t m e n t o f t h e h e i g h t o f t h e mod-u l a t i n g p u l s e s . The r e s o n a n c e f r e q u e n c y o f the c a v i t y c o n t a i n -i n g t h e c r y s t a l r e m a i n e d c o n s t a n t t h r o u g h o u t any e x p e r i m e n t , p r o v i d e d t h e t e m p e r a t u r e r e m a i n e d s t a b l e w i t h i n £ . 1° K and no b o i l i n g o f h e l i u m t o o k p l a c e i n t h e c a v i t y . The power d e l i v e r e d t o t h e c a v i t y may have v a r i e d g r e a t -l y f r o m e x p e r i m e n t t o e x p e r i m e n t , and a l s o t h r o u g h o u t any p a r -t i c u l a r e x p e r i m e n t . No a t t e m p t was made t o measure i t and no p a r t i c u l a r p a i n s w e r e t a k e n t o a s s u r e c o n s t a n c y . However, a t t e m p t s were made t o h a v e optimum m a t c h i n g a t t h e r e s o n a n c e f r e q u e n c y , so as t o o b t a i n s u f f i c i e n t power t o g e t a p p r e c i a b l e s a t u r a t i o n o f t h e e n e r g y l e v e l s a n d hence o b s e r v a b l e c h a n ges i n th e F a r a d a y r o t a t i o n , b) Magnet: I t maximum f i e l d t h e magnet drew a c u r r e n t o f 35 Amp. The w a t e r -c o o l i n g was f o u n d t o be s o e f f i c i e n t t h a t no m e a s u r a b l e change i n t h e c u r r e n t was o b s e r v e d - e v e n o v e r p r o l o n g e d p e r i o d s o f t i m e - i f t h e c u r r e n t d i d n o t e x c e e d 25 Amp. S i n c e a l l m e a s u r e -ments o f t h e r e l a x a t i o n t i m e were done w i t h f i e l d s r e q u i r i n g l e s s t h a n 11 Imp., t h e a c c u r a c y i n t h e d e t e r m i n a t i o n o f t h e f i e l d and i t s s t a b i l i t y were l i m i t e d o n l y b y the a c c u r a c y o f t h e c a l i b r a t i o n and t h e ammeter u s e d . The c a l i b r a t i o n o f m a g n e t - c u r r e n t v e r s u s f i e l d h a d been done b y a f l i p - c o i l and c h e c k e d b y p r o t o n r e s o n a n c e s a t v a r i o u s 4 5 , p o i n t s o f t h e upper p a r t o f t h e h y s t e r e s i s c u r v e . T h i s p a r t o f the c u r v e was u s e d e x c l u s i v e l y i n t h e e x p e r i m e n t s , i . e . t h e magnet c u r r e n t was i n c r e a s e d t o s a t u r a t i o n (35 Amp. a n d up) and t h e n r e d u c e d t o t h e v a l u e r e q u i r e d t o p r o d u c e t h e d e s i r e d f i e l d . The ammeter u s e d h a d a r e a d i n g a c c u r a c y o f — .1 Amp., so t h a t t h e maximum e r r o r i n t h e d e t e r m i n a t i o n o f t h e m a g n e t i c f i e l d was s m a l l e r t h a n ± 85 O e r s t e d t . c) C r y o s t a t : I n t h e c r y o s t a t t e m p e r a t u r e s between 1.38 and 4 . 2 ° K c o u l d be m a i n t a i n e d f o r p e r i o d s up t o 12 h o u r s w i t h i n ± .001°K i f t h e power i n p u t f r o m t h e m i c r o w a v e s y s t e m was k e p t c o n s t a n t . S i n c e t h e power i n p u t d u r i n g an e x p e r i m e n t a t a f i x e d t e m p e r a t u r e v a r i e d , o c c a s i o n a l a d j u s t m e n t s of t h e pumping s p e e d on t h e l i -q u i d h e l i u m w e r e r e q u i r e d t o m a i n t a i n t h e t e m p e r a t u r e c o n s t a n t . I n a l l m e a surements a t f i x e d temperatures., v a r i a t i o n s c o u l d be k e p t w i t h i n — .5%. The a b s o l u t e v a l u e o f t h e t e m p e r a t u r e was d e r i v e d f r o m t h e v a p o u r p r e s s u r e i n t h e c r y o s t a t on t h e b a s i s o f t h e "1958 s c a l e " ( 2 1 , 1 9 5 8 ) . M easurements were made a t 4.22°K a n d a t s e v e r a l t e m p e r a -t u r e s b e l o w t h e X - p o i n t . No r e l a x a t i o n t i m e m e a surements • c o u l d be made betw e e n t h e X - p o i n t and 4.2°K s i n c e v i o l e n t b o i l i n g o f h e l i u m c a u s e d b u b b l e s i n t h e c a v i t y t o s u c h a d e g r e e t h a t t h e i n t e n s i t y f l u c t u a t i o n s i n t h e l i g h t beam o b s c u r e d any s i g n a l . A c h e c k on t h e c o n s i s t e n c y o f t h e t e m p e r a t u r e m e a s u r e -ments was made b y p l o t t i n g t h e V e r d e t c o n s t a n t ( r o t a t i o n i n ° 46 ' p e r cm p a t h l e n g t h p e r O e r s t e d t m a g n e t i c f i e l d ) o b t a i n e d a t v a r i o u s t e m p e r a t u r e s a g a i n s t y- _ QJ^® , where T i s t h e temp-e r a t u r e i n °K (See d i a g r a m F i g u r e Z V ) , S i n c e t h e V e r d e t c o n -s t a n t i s p r o p o r t i o n a l t o t h e m a g n e t i c s u s c e p t i b i l i t y , w h i c h i n t u r n ( f o r neodymium e t h y l s u l f a t e ) i s p r o p o r t i o n a l t o l / T - .013°, we e x p e c t a s t r a i g h t l i n e p l o t t h r o u g h t h e o r i g i n . As t h e d i a -gram shows t h i s was i n d e e d t h e case f o r e a c h o f t h e two c r y s t a l p a r t s u s e d i n t h e e x p e r i m e n t s . The s l i g h t d i s c r e p a n c y i n s l o p e b etween t h e two c r y s t a l s i s r e a d i l y e x p l a i n e d b y t h e d i f f i c u l t y i n m e a s u r i n g i t s l e n g t h a l o n g t h e o p t i c a x i s e x a c t l y ( i t i s t o be remembered t h a t t h e c r y s t a l s a r e v e r y d e l i c a t e w i t h r e s -p e c t t o h a n d l i n g ) , d) O p t i c a l S y s t e m : S i n c e a l l measurements were b a s e d on i n t e n s i t y f l u c t u a t i o n s due t o r o t a t i o n o f t h e p l a n e o f p o l a r i z a t i o n i n t h e s a m p l e , c a r e h a d t o b e t a k e n t o i ) a v o i d i n t e n s i t y f l u c t u a t i o n s n o t a r i s i n g f r o m r o t a t i o n , i i ) o b t a i n r e l i a b l e m easurements o f t h e r e l a t i v e i n t e n s i t y o f l i g h t p a s s e d b y t h e a n a l y z e r a s a f u n c t i o n o f t i m e , i i i ) e s t a b l i s h f a i t h f u l l y t h e i n t e n s i t y - r o t a t i o n r e l a t i o n s h i p . The r e q u i r e m e n t i ) was s a t i s f i e d t o a s u f f i c i e n t d e g r e e . The s t a b i l i t y o f t h e DC - g e n e r a t o r i n c o n j u n c t i o n w i t h t h e f i l -t e r n e t w o r k and s t a b i l i z e r a s s u r e d t h a t t h e r e was no n o t i c e a b l e c o n t r i b u t i o n t o t h e s i g n a l n o i s e f r o m f l u c t u a t i o n s i n t h e l i g h t -o u t p u t o f t h e m e r c u r y - a r c . A l t h o u g h t h e m e c h a n i c a l s t a b i l i t y o f t h e o p t i c a l s y s t e m l e f t much t o b e d e s i r e d , w i t h p r o p e r c a r e FIGURE X Z VE&DET CONSTANT VS. I/(T - .0I3°) CRYSTAL I deg. cm" 0 o CRYSTAL H .02 .2 .3 , .4 7 (°Kj-F o l l o w i n g page 46 47 p ( t h a t i s a v o i d i n g m e c h a n i c a l v i b r a t i o n s o r d i s p l a c e m e n t s i n t h e s y s t e m w h i l e m a k i n g measurements) f l u c t u a t i o n s i n I n t e n s i t y a r i s -i n g f r o m s h i f t s I n t h e l i g h t beam c o u l d be a v o i d e d . The l i q u i d a i r was k e p t o u t o f t h e l i g h t beam, and t h u s c o u l d n o t cause f l u c t u a t i o n s o f i n t e n s i t y . The h e l i u m b a t h was o p t i c a l l y c l e a r b elow t h e % - p o i n t . Above t h e % - p o i n t , b u b b l e s i n t h e l i g h t p a t h c o u l d n o t be a v o i d e d e n t i r e l y and c o n t r i b u t e d c o n s i d e r a b l y t o s i g n a l n o i s e f o r t h e m easurements a t 4.22°K. R e q u i r e m e n t i i ) was s a t i s f i e d o n l y w i t h i n l i m i t a t i o n s . The p h o t o m u l t i p l i e r when p r e v i o u s l y f a t i g u e d gave a l i n e a r r e s -ponse t o t h e i n t e n s i t y . However, t h e r e s p o n s e o f t h e o s c i l l o -s cope d e p e n d e d o n t h e p o s i t i o n o f the t r a c e on t h e s c r e e n , d e v i a t i o n f r o m l i n e a r i t y b e i n g as l a r g e as 15%. A l t h o u g h p a r t o f t h i s n o n l i n e a r i t y was t a k e n c a r e o f b y t h e r o t a t i o n - i n t e n s i t y c a l i b r a t i o n , t h e r e n e v e r t h e l e s s r e m a i n e d an i n a c c u r a c y o f up t o * 10% i n t h e r e s u l t i n g measurements, s i n c e t h e p o s i t i o n o f t h e t r a c e s on t h e s c r e e n s e r v i n g f o r measurements c o u l d n o t b e made t o c o r r e s p o n d e x a c t l y t o t h e p o s i t i o n o f t h e t r a c e s s e r v -i n g f o r c a l i b r a t i o n p u r p o s e s . A t l o w i n t e n s i t y l e v e l s t h e random n o i s e o f t h e p h o t o -m u l t i p l i e r c o n t r i b u t e d t o a b r o a d e n i n g o f t h e t r a c e s , and a t h i g h i n t e n s i t y l e v e l s hum p i c k - u p o f t h e p h o t o m u l t i p l i e r c i r c u i t r e d u c e d t h e s i g n a l t o n o i s e - r a d i o somewhat. The f i n a l l i m i t a t i o n o f t h e r e s o l u t i o n was t h e s p o t - s i z e o f t h e o s c i l l o s c o p e , w h i c h u n d e r o t h e r w i s e f a v o u r a b l e c o n d i t i o n s o f the. s i g n a l t o n o i s e r a t i o was o f t h e same o r d e r o f m a g n i t u d e as hum a n d random n o i s e . On t h e t i m e s c a l e e r r o r s up t o £. 5% were i n t r o d u c e d due t o t h e i n h e r e n t v a r i a b i l i t y o f t h e t i m e b a s e o f t h e o s c i l l o s c o p e . 48 Any d i s t o r t i o n o f t h e i n t e n s i t y - t i m e r e l a t i o n s h i p due t o t h e p h o t o g r a p h i c p r o c e s s o f r e c o r d i n g t h e t r a c e s was f o u n d t o he c o m p l e t e l y n e g l i g i b l e . However, when r e d u c i n g t h e r e -c o r d e d t r a c e s t o n u m e r i c a l r e l a t i o n s h i p s b y measurement u n d e r a low-power m i c r o s c o p e , t h e n o i s e i n t h e s i g n a l s made i t p o s s i b l e t o g e t m i s r e a d i n g s c o r r e s p o n d i n g t o t 2° o r t 1 0 % , w h i c h e v e r i s g r e a t e r , o f t h e change i n r o t a t i o n i n a g i v e n t i m e i n t e r v a l , r e -s u l t i n g i n o c c a s i o n a l s e v e r e s c a t t e r o f t h e p o i n t s on t h e s e m i -l o g p l o t o f r o t a t i o n v e r s u s t i m e . R e q u i r e m e n t i i i ) was a l s o n o t c o m p l e t e l y s a t i s f i e d . E v e n t h o u g h i n e a c h e x p e r i m e n t a c a l i b r a t i o n o f t h e i n t e n s i t y v e r s u s p o s i t i o n o f t h e a n a l y z e r r e l a t i o n s h i p was p e r f o r m e d , i t has t o b e remembered t h a t t h i s was done w i t h l i g h t o f a f i x e d s t a t e o f p o l a r i z a t i o n e n t e r i n g t h e a n a l y z e r . I n t h e i d e a l c a s e o f a p e r f e c t c r y s t a l p e r f e c t l y a l i g n e d t h i s w i l l be 1 0 0 % p l a n e p o l a r i z e d , r e s u l t i n g i n a p u r e COS*"^ ~ 0 + C©s@) r e -s ' l a t i o n s h i p between r o t a t i o n b y an a n g l e and i n t e n s i t y , and t h e e f f e c t o f a n a l y z e r r o t a t i o n w i l l b e i d e n t i c a l t o t h e e f f e c t o f a c o r r e s p o n d i n g F a r a d a y r o t a t i o n . T h i s f o l l o w s f r o m what has be e n s a i d i n C h a p t e r I , S e c t i o n 3) (See a l s o d i a g r a m F i g u r e IV) on t h e r e s p o n s e o f a n a n a l y z e r t o p o l a r i z e d l i g h t ( i n t e n s i t y i s p r o -p o r t i o n a l t o s q u a r e o f a m p l i t u d e ) and t h e c o n s t r u c t i o n d e s c r i b e d i n C h a p t e r I , S e c t i o n 4) and i l l u s t r a t e d i n F i g u r e V. The a n a l y z e r i n o u r c a s e i s r e p r e s e n t e d b y a p o i n t l<x> on t h e e q u a t o r o f t h e P o i n c a r e ' s p h e r e ( s i n c e i t p a s s e s p l a n e p o l a r i z e d l i g h t o n l y ) , a n d i n t h e i d e a l c a s e u n d e r d i s c u s s i o n t h e p o i n t I o> r e p r e s e n t i n g t h e l i g h t i n c i d e n t on t h e a n a l y z e r l i e s a l s o on t h e 49 i e q u a t o r o f t h e P o i n c a r e s p h e r e . Now whether we r o t a t e a l o n g t h e e q u a t o r , k e e p i n g lo> f i x e d , o r whether I o> r o t a t e s , w h i l e l©t> r e m a i n s f i x e d , w e w i l l g e t t h e same i n t e n s i t y r o t a t -i o n r e l a t i o n s h i p ( i f t h e r e i s no b i r e f r i n g e n c e a n d he n c e l o > r e m a i n s a l w a y s on t h e e q u a t o r a s d i s c u s s e d , i n C h a p t e r I , S e c t i o n 4)). C l e a r l y an a n a l y z e r r o t a t i o n by c o r r e s p o n d s t o a movement o f lot> on t h e P o i n c a r e s p h e r e b y 0 ' . Npw i n p r a c t i c e t h e c r y s t a l i s n e i t h e r o p t i c a l l y p e r -f e c t n o r p e r f e c t l y a l i g n e d . As a c o n s e q u e n c e some b i r e f r i n g e n c e and d i f f u s e s c a t t e r i n g o f l i g h t t a k e s p l a c e i n t h e c r y s t a l , t h e f o r m e r m a k i n g t h e l i g h t e n t e r i n g t h e a n a l y z e r more o r l e s s e l l i p -t i c a l l y p o l a r i z e d , t h e l a t t e r i n t r o d u c i n g a n u n p o l a r i z e d compon-e n t . W h i l e t h e c a l i b r a t i o n t a k e s c a r e o f t h e u n p o l a r i z e d com-p o n e n t , i t d o e s n o t g i v e u s t h e e x a c t c o r r e c t i o n f o r t h e e l l i p -t i c i t y o f t h e l i g h t u n d e r t h e c o n d i t i o n s o f a c t u a l measurement. I n g e n e r a l r o t a t i o n o f l«.>will n o t g i v e t h e same i n t e n s i t y r o -t a t i o n r e l a t i o n s h i p as r o t a t i o n o f l o > s i n c e t h e two p o i n t s now move on two d i f f e r e n t c i r c l e s whose p l a n e s a r e i n c l i n e d t o e a c h o t h e r by a n a n g l e <p c o r r e s p o n d i n g t o t h e maximum e l l i p t i c i t y o b t a i n a b l e o f t h e l i g h t i n c i d e n t on t h e a n a l y z e r , a s t h e p o i n t I o> f o l l o w s i t s p a t h w i t h v a r y i n g 6 (See d i a g r a m F i g u r e V ) . I n t h i s c a s e ( t h e s t a b l e mode o f p r o p a g a t i o n ) l i e s a n a n g l e ^ away f r o m t h e p o l e o f t h e s p h e r e . M o r e o v e r t h e p o s i t i o n o f lu,> i t s e l f w i l l depend now on t h e F a r a d a y r o t a t i o n , and t h e b i r e f r i n g e n c e a l s o w i l l v a r y w i t h t h e a p p l i c a t i o n o f t h e m a g n e t i c f i e l d . To o b t a i n an e s t i m a t e o f t h e o r d e r o f m a g n i t u d e o f t h e e r r o r p o s s i b l y i n t r o d u c e d b y m i s a l i g n m e n t o f t h e c r y s t a l , t h e 50 maximum e l l i p t i c i t y a s w e l l a s t h e change i n e l l i p t i e i t y w i t h change i n r o t a t i o n were i n v e s t i g a t e d i n one p a r t i c u l a r e a s e . An u p p e r l i m i t on t h e e l l i p t i c i t y i s o b t a i n e d , b y c o m p a r i n g max-imum and minimum i n t e n s i t y o f l i g h t p a s s e d b y t h e a n a l y z e r and c o n s i d e r i n g t h e l i g h t t o b e c o m p l e t e l y e l l i p t i c a l l y p o l a r i z e d ( i . e . n e g l e c t i n g t h e u n p o l a r i z e d component) a s f o l l o w s : L e t <p be t h e e l l i p t i c i t y p a r a m e t e r a s g i v e n i n e q u a t i o n s (3.7) and (3.8) C h a p t e r I . L e t X and I be t h e minimum a n d maximum i n t e n s i t y p a s s e d b y t h e a n a l y z e r r e s p e c t i v e l y , a n d X t h e i n -c i d e n t i n t e n s i t y . T h en Hence i -5/1 ¥ = « r c cos { + z/T was f o u n d t o have a n u p p e r l i m i t a s l a r g e as 3 0 ° . S i n c e i n c l u s i o n s i n t h e c r y s t a l s c a u s e d c o n s i d e r a b l e d i f f u s e s c a t t e r -i n g i t i s more r e a l i s t i c a l t o e s t i m a t e t h e e l l i p t i c i t y <p t o be s m a l l e r 1 0 ° . T h i s e s t i m a t e was a l s o b o r n e o u t when c o n s i d e r -i n g t h e change i n e l l i p t i c i t y a s t h e f i e l d was < i n c r e a s e d a n d a * l a r g e F a r a d a y r o t a t i o n was i n t r o d u c e d . I t may s a f e l y be assumed ( u n l e s s g r o s s m i s a l i g n m e n t p r e v a i l s ) t h a t t h e oC o f t h e e q u a t -i o n s ( a . 1 4 ) t o ( 2 . 1 8 ) and t h o s e i n S e c t i o n 4 ) . o f C h a p t e r I , w h i c h i s r e s p o n s i b l e f o r t h e F a r a d a y r o t a t i o n , w i l l i n c r e a s e r a p i d l y w i t h i n c r e a s e I n f i e l d , w h i l e & ( c a u s i n g b i r e f r i n g e n c e ) 51 w i l l s t a y n e a r l y c o n s t a n t . T h i s means t h a t as t h e f i e l d i s i n -c r e a s e d oC becomes i n c r e a s i n g l y dominant and l t x > w i l l be s h i f t e d t o w a r d s t h e p o l e o f t h e s p h e r e . F o r a change o f r o t a t i o n o f 5 2 0 ° i t was f o u n d t h a t t h e e l l i p t i c i t y c hanged b y a p p r o x i m a t e l y 1 0 ° . Thus i t r e m a i n s t o be s e e n what e r r o r i s I n t r o d u c e d i f we a t t r i -b u t e t h e c h a n g e s i n r o t a t i o n o b s e r v e d e n t i r e l y t o oc* , w h i l e i n f a c t we have ^ 0 , and hence S = j/cj1 + oc The p o i n t lw.> i s s e p a r a t e d f r o m t h e p o l e o f t h e s p h e r e b y a g r e a t c i r c l e a n g l e o f 1 0 ° . We h a d p r e v i o u s l y I cos 3,0 =• A3 where £ L @ was t h e g r e a t c i r c l e a n g l e b e t w e e n Ix > and luc> ( s e e d i a g r a m F i g u r e V ) . Then, i n t e r m s o f t h e p a r a m e t e r vj> , we have s C" <f ~ TT = yr&f ^ i n T h e r e f o r e ^ \ = — and <S - Y ^ F ^ p 1 * * V \ = ^-VWi ~ I - O l c c w h i c h c o r r e s p o n d s t o a n e r r o r o f ~ 2%. E v e n i f y i s as l a r g e as 3 0 ° , by t h e same c a l c u l a t i o n an e r r o r o f o n l y 1 5 % i s o b t a i n e d . Thus i t i s s e e n t h a t e v e n t h o u g h t h e p o i n t lo> moves on a s m a l l c i r c l e t h a t " w o b b l e s " w i t h r e s p e c t t o t h e e q u a t o r i a l p l a n e , no s e r i o u s e f f e c t s on t h e a c c u r a c y o f our r e l a x a t i o n t i m e measurements a r e e x p e c t e d f r o m t h i s s o u r c e , a l t h o u g h i t i s n o t s u r p r i s i n g t h a t t h e r e i s some d i s a g r e e m e n t on r e l a x a t i o n t i m e s o b t a i n e d f o r d i f f e r e n t a n a l y z e r s e t t i n g s . A p o s s i b l e s m a l l m i s a l i g n m e n t o f t h e m a g n e t i c f i e l d w i t h r e s p e c t t o t h e c r y s t a l and t h e l i g h t p a t h w o u l d have no s e r i o u s 52 e f f e c t s on t h e r e l a x a t i o n t i m e measurements. The a b s o l u t e v a l u e s o f r o t a t i o n w o u l d t h e n c o r r e s p o n d t o t h e component o f t h e f i e l d p a r a l l e l t o t h e o p t i c a x i s , w h i l e t h e r e l a t i v e c h a n g e s w o u l d be u n a f f e c t e d . F rom t h e f o r e g o i n g i t a p p e a r s t h a t t h e r e s u l t s a r e r e -l i a b l e a t l e a s t w i t h i n t h e r a n g e o f t h e r o o t mean s q u a r e d e v i a -t i o n s f r o m t h e a v e r a g e s o b t a i n e d f o r a f i x e d s e t o f p a r a m e t e r s . T h i s g i v e s u s an u p p e r l i m i t f o r t h e e r r o r o f - 20%. C o m p a r i -son o f t h e r.mn.s.deviations o b t a i n e d make i t r e a s o n a b l e t o assume an o v e r a l l a c c u r a c y o f - 10% f o r t h e a v e r a g e d r e l a x a t i o n t i m e s . 2) Some T y p i c a l R e s u l t s . F i g u r e s X V I t o X X X I I on t h e f o l l o w i n g p a g e s r e p r e s e n t t y p i c a l d e c a y c u r v e s o b t a i n e d i n t h e e x p e r i m e n t s . F i g u r e s XVT and X V I I r e p r e s e n t d e c a y s c o r r e s p o n d i n g t o a change i n r o t a t i o n o f 8 ° due t o an e-vtuecpulse o f microwave power a t a f i e l d o f 2540 0 a n d t e m p e r a t u r e , o f 1 . 6 8 5 ° K a t two d i f f e r -e n t sweep s p e e d s . S i n c e t h e a n a l y z e r was s e t a t 5 3 . 5 ° frommin-imum f o r no p u l s e c o n d i t i o n , t h e i n t e n s i t y - r o t a t i o n r e l a t i o n f a l l s i n t o t h e l i n e a r r e g i o n . The r e l a x a t i o n t i m e s o b t a i n e d f r o m t h e two t r a c e s were 109 and 113 m i l l i s e c o n d s r e s p e c t i v e l y . F i g u r e s X V I I I and IXX show d e c a y s c o r r e s p o n d i n g t o a change i n r o t a t i o n o f a p p r o x i m a t e l y 4 4 ° due t o a 124msecpulse a t a f i e l d o f 2040 0 and a t e m p e r a t u r e o f 2 . 0 5 6 ° K a t two d i f f e r e n t sweeps w i t h t h e a n a l y z e r s e t a t minimum f o r n o - p u l s e c o n d i t i o n . F i g u r e XX and XXI show t h e same d e c a y s b u t w i t h t h e a n a l y z e r s e t 4 5 0 o f f minimum. The r e l a x a t i o n t i m e s o b t a i n e d f r o m t h e s e t r a c e s were 68.5, 78.1, 59.4, and 62.0 m i l l i s e c o n d s r e s p e c t i v e l y . F i g u r e s X X I I t o X X V I I show d e c a y s c o r r e s p o n d i n g t o a change 53 in rotation of approximately 8 9 ° , due to a 1 2 4 m s e c pulse at a f i e ld of 2040 # and a temperature of 1.381°K. Figures XXII and XXIII were pulsed from minimum intensity, Figures XXIV and XXV from 4 5 ° off minimum and Figures XXVI and XXVII from 8 0 ° off minimum. The relaxation times obtained from Figures XXII to XXV were 124, 128, 121, 126 milliseconds respectively. The graph on Figure XXVII shows a semilog plot of the intensity versus time relationship obtained by measuring the traces of Figures XVI, XX, and XXII. Figures XXIX and XXX are pictures of decays due to a 63 millisecond and 8 millisecond pulse respectively, with a l l other parameters ident ica l , i . e . a change of rotation approximately 3 1 ° , magnetic f i e ld 1540 , 0 ' , temperature 1,685° JC, pulsed from 4 5 ° off minimum. The relaxation times obtained were 63.0 and 69.2 milliseconds, showing no significant difference. The second trace on a l l these pictures(usually at the bottom) is a time-marker of 30 pulses per second 10 and 23w»secwide respectively. Figures XXXI and XXXII are evidence of the "overshoot" phenomenon observed i n one particular crystal - see section 6 of this Chapter - . Figure XXXI shows the decay due to a 124wsec pulse causing a change i n rotation of approximately 2 3 ° from 4 5 ° off minimum at a f i e ld of 1790 0 and a temperature of 1.38°K. The pulse i s seen to shift the trace up from near the centre of the picture. The decay then lets the trace swing be-yond the equilibrium position downward, before i t return again to the starting level of intensity. Figure XXXII displays this effect even more s tr ikingly . FIGURE X V I See T e x t To f o l l o w page 53 jj ' I G U H 3 Z Y I I I See I'ezt To f o l l o w p a g e 53 FIGURE XX £ T G U R E X X I I See Text To tollow page 53 S'lGUJOC X X I V See T e x t To f o l l o w page 5 3 FIGURE XXVI See T e x t irIGURS X X V I I To f o l l o w page 5 3 FIGURE X.XVIII ROTATION VS. TIME ON S E M I L O G S C A L E AS OBTAINED FROM • FIGURE • O FIGURE X X 3 FIGURE ~ W T 1 0 0 ° F 50 100 150 msec. F o l l o w i n g 53 T.Q f o l l o w page 5 3 F I G U R E T T T T 54 , "A 124 msec p u l s e a t a f i e l d o f 1540 # and a t e m p e r a t u r e o f a p p r o x -i m a t e l y 1.38°K i s e x p e c t e d t o cause a change o f r o t a t i o n o f r o u g h l y .5 t o 1 ° f r o m t h e minimum. T h i s c o r r e s p o n d s t o t h e r e c -t a n g u l a r p u l s e a t t h e l e f t o f t h e t r a c e . The d e c a y f o l l o w i n g t h e p u l s e s w i n g s t h e t r a c e as much as 15° b e y o n d t h e o r i g i n a l s t a r t i n g p o i n t b e f o r e r e t u r n i n g t o i t . S i n c e t h e o r i g i n a l i n t e n s i t y p r i o r t o t h e p u l s e was a minimum, t h e o v e r s h o o t i s d i s p l a y e d i n t h e same d i r e c t i o n as t h e p u l s e ( i n F i g u r e s XXXI and X X X I I i n c r e a s e o f i n t e n s i t y c o r r e s p o n d s t o a downward m o t i o n o f t h e t r a c e ) . No r e l a x a t i o n t i m e s w e r e " d e t e r m i n e d o n t h i s c r y s t a l . 3) E f f e c t o f p u l s e - l e n g t h I f t h e m i c r o w a v e power i s s u f f i c i e n t t o s a t u r a t e t h e e n e r g y l e v e l s , one may e x p e c t an e f f e c t o f p u l s e l e n g t h o n t h e • r e l a x a t i o n t i m e , s i n c e a l o n g p u l s e may c a u s e e x c i t a t i o n o f p h o n -on-bands, i . e . c r e a t e " h o t p h o n o n s " I n t h e l a t t i c e , w h i c h i n t u r n may e f f e c t t h e r e l a x a t i o n t i m e . An a t t e m p t was made t o d e -t e c t s u c h a n e f f e c t b y u s i n g p u l s e - l e n g t h s f r o m 8 msec t o 12* Msec u n d e r o t h e r w i s e i d e n t i c a l c o n d i t i o n s , and c o m p a r i n g t h e r e l a x a t i o n t i m e s o b t a i n e d f r o m t h e s e p u l s e s . No e f f e c t was o b s e r v e d , a l -t h o u g h any s y s t e m a t i c change o f more t h a n 10% s h o u l d have b e e n d e t e c t a b l e . E f f e c t s s m a l l e r t h a n 10% w o u l d n o t be n o t i c e a b l e . F i g u r e s XXIX and XXX, d e s c r i b e d i n t h e f o r e g o i n g s e c t i o n , a r e t y p i c a l o f t h e r e s u l t s o b t a i n e d . The f o l l o w i n g t a b l e shows t h e r e l a x a t i o n t i m e s o b t a i n e d a t a f i e l d o f 1540 flf and a t e m p e r a t u r e o f 1.685°K f o r v a r i o u s p u i s e - l e n g t h s , a l l o f them s u f f i c i e n t t o cause c o m p l e t e s a t u r a -t i o n o f t h e e n e r g y l e v e l s : P u l s e l e n g t h 124 63 35 17 8 R e l a x a t i o n Time T 74.2 68,9 63.0 84.0 78.2 67.5 68.7 82,0 73.4 69,2 The r e s u l t s a t d i f f e r e n t f i e l d s were s i m i l a r l y i n c o n c l u s i v e . 4) M e a s u r e m e n t s o f R e l a x a t i o n Time a t V a r i o u s F i e l d s and  T e m p e r a t u r e s The f o l l o w i n g t a b l e i s a summary o f t h e measurements o f t h e r e l a x a t i o n - t i m e o f neodymium e t h y l s u l f a t e at v a r i o u s t e m p e r a t u r e s a n d magnet f i e l d s . I t a l s o g i v e s t h e a v e r a g e v a l u e and r.m.s. - d e v i a t i o n f o r e a c h s e t o f p a r a m e t e r s . Temp. ( °K) 1.380 .322 M a g . F i e I d ( ) 780 1030 R e l a x . T i m e (msec) 146 150 165 176 165 173 135 157 171 185 186 207 186 198 T a v g . r.m.s.dev. (msec) (insefc) 162.5 11.1 178.1 21.8 Temp. ( °K) 1.380 1.542 .434 M a g . i ' i e l d ( ) 1290 1540 1790 2040 2290 2540 780 1030 1290 1540 R e l a x . T i m e (msec) 206 203 186 195 208 195 101 100 90 103 129 134 122 118 125 146 124 128 108 105 122 101 101 135 109 127 104 133 125 115 110 119 112 121 125 128 136 142 80 91 92 T a v g . r .rn.s.de-v (msec) (msec) 198.8 98.5 129.0 116.2 115.8 119.2 114.5 119.3 135.3 87.7 8.4 5.1 9.1 10.5 13.3 11.4 4.5 6.1 6.5 4.9 Temp. ( °K) 1.542 M a g . F i e l d ( ) 17.90 R e l a x . T i m e (msec) 95 99 102 95 T a v g . (msec) 97.8 r.m.s.dev, (msec) 2.9 2040 108 125 111 100 113 114 111.8 7.9 2290 95 98 100 137 121 110.2 16.2 2540 110 89 138 98 108.8 18.2 1.685 .522 780 109 127 99 130 114 129 107 130 111 130 112 118 100 104 115.7 11.3 1030 113 138 138 144 141 114 129 149 120 143 138 149 133.8 12.0 Temp. M a g . F i e l d R e l a x . T i m e T a v g . r.m.s.dev. ( o K j ( ) (msec) (msec) (msec) 1.685 1030 119 126 137 140 117 139 136 147 1290 121 146.0 16.2 139 •147 155 124 120 147 168 115 153 144 160 131 156 156 164 135 166 156 163 1540 69 72.9 6.3 74 63 84 68 78 69 82 69 73 1790 112 122.9 8.2 120 128 114 120 126 140 Temp. M a g . F i e l d R e l a x . T i m e T a v g . r.m.s.dev. ( OK) ( ) (msec) (msec) (msec) 1.685 2040 135 147.8 10.9 .5 22 162 140 159 136 142 139 163 143 159 2290 102 117.0 7.0 116 123 114 128 117 118 112 120 127 119 120 112 116 112 116 113 135 110 111 115 125 110 2540 110 108.2 5.5 109 105 112 104 120 94 100 98 104 112 110 118 116 109 109 107 Temp. M a g . F i e l d ( °) ( ) 1.685 .522 1.875 .629 2.060 .723 2540 780 1030 1290 1540 1790 2040 2290 2540 1030 1290 1540 1790 R e l a x . T i m e (msec) 107 103 106 113 109 113 55 48 63 66 69 70 51 52 59 61 61 64 74 57 59 52 66 63 70 68 64 65 61 62 69 42 40 37 38 60 51 55 51 51 50 Tm a v g . r.m.s.dev. (msec) (msec) 108.2 51.5 64.5 69.5 51.5 60.0 66.3 58.0 59.0 66.2 64.2 39.2 53.0 5.5 3.5 1.5 .5 .5 1.0 5.9 1.0 7.0 3.8 4.1 2.7 3.5 61 Temp. M a g . F i e l d R e l a x . T i m e T a v g . r.m.s.dev. ( °K) ( ) (msec) (msec) (msec) 2.060 2040 68 63.3 5.1 .723 78 59 62 59 64 62 59 59 65 61 2290 60 61.2 4.2 67 59 59 2540 58 60.8 5.6 65 53 • 67 4.22 1290 7.3 7.3 : . • . 1540 4.7 4.7 4.7 1790 6.0 5.8 .35 4.8 6.5 2040 2.8 , 2.7 .46 2.1 3.2 "tThe s e c o n d number i n t h e column Temp, i s I n T) • F i g u r e s X X X I I I and XXXIV show t h e f i e l d d e p endence o f th e r e l a x a t i o n t i m e a t d i f f e r e n t t e m p e r a t u r e s . A common f e a t u r e o f t h e r e s u l t s a t t h e v a r i o u s t e m p e r a t u r e s i s a n a p p r o x i m a t e l y -l i n e a r i n c r e a s e between 7800 and 1290 0 f o l l o w e d b y a p r o -n o u n c e d d i p a t 1540 0 , i . e . n e a r t h e " r e s o n a n c e f i e l d " where the s e p a r a t i o n o f t h e e n e r g y l e v e l s c o r r e s p o n d s t o t h e r e s o n a n c e f r e q u e n c y o f t h e m i c r o w a v e c a v i t y . The w i d t h o f t h i s d i p i s o f t h e same o r d e r o f m a g n i t u d e as t h e w i d t h o f t h e r e s o n a n c e l i n e FIGURE x x x in msec. 150 -100 50 RELAXATION TIME VS. MAGNETIC FIELD • T = I , 6 8 5 " K 3 T = 2 0 6 0 , , K O T = 4 . 2 2 ° K 9 3 O 3 9 O 9 a .9 K 0 F o l l o w i n g page 61 FIGURE XXXIV RELAXATION TIME VS. MAGNETIC FIELD msec. 200 100 • T = 1.380 °K O T= 1.542 °K 3 T=I.875°K G • o 0 • « 8 • o o o 3 9 « <• 3 3 3 ^ 3 3 K 0 F o l l o w i n g page 61 ( rvy 250j2f). At high e r f i e l d s , i . e . between 2040 # and 2540gr the f i e l d dependence i s somewhat obscured. The r e s u l t s from v a r i o u s temperatures do not gi v e i d e n t i c a l p a t t e r n s . I t appears, however, t h a t f i e l d dependence i n t h i s r e g i o n i s s m a l l , l e v e l -l i n g o f f at a value about midway between the h i g h e s t and lowest p o i n t o f the preceding d i p . The v a l u e s at a temperature o f 1.685°K appear somewhat high i n ab s o l u t e v a l u e s , except at the d i p , when compared w i t h those at the other temperatures. T h i s i s p a r t i c u l a r l y n o t i c e -able when stu d y i n g jJigures TKA7 and 2XCvT, i n vvhich the r e l a x a -t i o n t i - i e i s p l o t t e d versus temperature on a l o g - l o g s c a l e f o r v a r i o u s f i e l d s t r e n g t h s . ./hen making a l e a s t - s q u a r e s t r a i g h t l i n e f i t through the measured p o i n t s on the l o g - l o g p l o t , the f o l l o w i n g s l o p e s are obtained f o r the v a r i o u s f i e l d s . i-iagnetic f i e l d (0) Slope o f l o g T , Versus l o g T 780 - 3.48 1030 - 2.62 1290 - 2.99 1540 - 2.83 1790 - 2.88 2040 - 3.64 2290 - 1.94 2540 - 1.95 From these r e s u l t s i t can be concluded t h a t at low f i e l d s and i n the r e g i o n o f the d i p the r e l a x a t i o n tiiae i s p r o -p o r t i o n a l t o approximately 1 , whereas at high e r f i e l d s i t may be p r o p o r t i o n a l t o T . However, s i n c e p a r t i c u l a r l y at high f i e l d s t here i s a l a r g e amount o f s c a t t e r i n the measured d a t a , t h i s I dependence i s not very r e l i a b l e ( u n f o r t u n a t e l y no p o i n t s are a v a i l a b l e f o r 4.2°K at these f i e l d s ) . The da t a at 1 . 6 8 5 0 K , 2.06°K, 4.22°K and 1.380OK, 1.542°K, 1.875°K were 63 o b t a i n e d w i t h two d i f f e r e n t p a r t s o f t h e same c r y s t a l r e s p e c t i v e l y ( c r y s t a l I and c r y s t a l I I o f f i g u r e X V ) . When c a l c u l a t i n g t h e s e s e p a r a t e l y , a T dependence o f t h e r e l a x a t i o n t i m e a p p e a r s t o be t h e b e s t a p p r o x i m a t i o n a t a l l f i e l d s . 5) E x p e r i m e n t w i t h C e r i u m E t h y l s u l f a t e An e x p e r i m e n t was made w i t h a c r y s t a l o f c e r i u m e t h y l -s u l f a t e i n t h e hope o f e s t a b l i s h i n g t h e i n f l u e n c e o f paramag-n e t i c r e s o n a n c e on t h e F a r a d a y e f f e c t and m e a s u r i n g r e l a x a t i o n t i m e s as i n t h e c a s e o f neodymium e t h y l s u l f a t e . A n e x t r e m e l y l a r g e F a r a d a y r o t a t i o n (,277°per cno p e r ^  ) was o b t a i n e d a t a t e m p e r a t u r e o f 1.41 i ,01°K. A t f i e l d s up t o 6000 0 w i t h m icrowave power a p p l i e d t o t h e c a v i t y t h e t e m p e r a t u r e i n c r e a s e d t o 1.45°K and a s l i g h t d e c r e a s e o f a p p r o x i m a t e l y 2.4% o f t h e r o t a t i o n was o b s e r v e d . Upon sudden r e m o v a l o f t h e m i c r o -w a v e - i n p u t t o t h e c a v i t y t h e r o t a t i o n i n c r e a s e d a l m o s t i n s t a n t -a n e o u s l y ( w i t h i n l e s s t h a n .2 s e c o n d s ) by .6%; t h e n w i t h i n a p p r o x i m a t e l y 2 m i n u t e s , w h i l e t h e t e m p e r a t u r e d e c r e a s e d t o 1.42° K, i t r e t u r n e d t o t h e v a l u e i t had p r i o r t o t h e a p p l i c a t i o n o f microwave power. I t was c o n c l u d e d t h a t a change o f F a r a d a y r o -t a t i o n o f o n l y .6% c o u l d be a t t r i b u t e d t o t h e i n f l u e n c e o f t h e microwave r a d i a t i o n a t a f i e l d o f 5600 0 . T h i s change amounted t o o n l y 3 ° i n t h e sample u s e d and was i n s u f f i c i e n t t o o b t a i n a l a r g e enough s i g n a l f o r p u l s e - t y p e m easurements o f t h e r e l a x a t i o n t i m e . The who l e r e g i o n up t o 60000 was c a r e f u l l y i n v e s t i g a t e d w h i l e p u l s i n g w i t h m i c r o w a v e s . No s u f f i c i e n t l y l a r g e s i g n a l c o u l d be o b t a i n e d a t any f i e l d . The f a i l u r e t o f i n d a s t r o n g i n f l u e n c e o f t h e microwave r a d i a t i o n on t h e F a r a d a y r o t a t i o n may be a t t r i b u t e d t o a v e r y 64 s h o r t r e l a x a t i o n t i m e , and i n s u f f i c i e n t m i c r o w a v e power t o c a u s e s a t u r a t i o n o f t h e e n e r g y l e v e l s a t t h e e x t e r n a l m a g n e t i c f i e l d s u s e d . A t b e s t t h e n , one may u s e t h e r e l a t i o n (5.6) o f C h a p t e r I , S e c t i o n 5) t o o b t a i n an o r d e r o f m a g n i t u d e e s t i m a t e on t h e r e l a x a t i o n t i m e , m a k i n g s u i t a b l e a s s u m p t i o n s f o r t h e v a l u e . F o r c o n c e n t r a t e d c e r i u m e t h y l s u l f a t e B o g l e , Cooke and W h i t l e y , ( X I I , 1951) q u o t e a v e r y b r o a d r e s o n a n c e l i n e , ( f*j 700 0) ) a p p e a r i n g b e l o w 2.5°K w i t h flu^^O o f 3.70 ±- .02. Our measurements show t h a t t h e r e l a x a t i o n o f t h i s s a l t i s v e r y s h o r t ( l e s s t h a n 1 m s e c ) . T h i s i s c o n s i s t e n t w i t h t h e o b s e r v a -t i o n b y B o g l e , Cooke and W h i t l e y . 6) " O v e r s h o o t " Phenomenon. I n one p a r t i c u l a r c r y s t a l o f neodymium e t h y l s u l f a t e an " o v e r s h o o t " e f f e c t was o b s e r v e d a s d e s c r i b e d i n S e c t i o n 2) and i l l u s t r a t e d i n f i g u r e s X XXI a n d X X X I I . T h i s e f f e c t a p p e a r e d c o n s i s t e n t l y u n d e r t h e c o n d i t i o n t h a t t h e " n o - p u l s e " m i c r o w a v e power c a u s e d an a p p r e c i a b l e amount o f s a t u r a t i o n . I n t h i s c a s e t h e F a r a d a y r o t a t i o n was r e d u c e d b e l o w i t s n o r m a l v a l u e . Upon a p p l i c a t i o n o f t h e m i c r o w a v e p u l s e s a f u r t h e r r e d u c t i o n t o o k p l a c e ( u s u a l l y on a c c o u n t o f c o m p l e t e s a t u r a t i o n ) ; i n t h e f o l l o w i n g r e -l a x a t i o n t h e r o t a t i o n i n c r e a s e d f i r s t b e y o n d th e v a l u e i t had p r i o r t o a p p l i c a t i o n o f t h e p u l s e • • : ( t h o u g h a l w a y s r e m a i n i n g b e l o w t h e no-power l e v e l ) b e f o r e i t r e t u r n e d t o t h e v a l u e c o r r e s -p o n d i n g t o t h e p r e v i o u s " n o - p u l s e " c o n d i t i o n . The e f f e c t was more p r o n o u n c e d w i t h p u l s e s o f l o n g d u r a t i o n (124msec) t h a n w i t h s h o r t e r p u l s e s , and a l s o i n c r e a s e d w i t h i n c r e a s i n g " n o - p u l s e " s a t u r a t i o n . C l o s e e x a m i n a t i o n o f t h e l e a d i n g edge o f t h e p u l s e r e v e a l e d no c o -65 r r e s p o n d i n g i n c r e a s e i n r o t a t i o n upon a p p l i c a t i o n o f t h e p u l s e . When t h e c r y s t a l on w h i c h t h i s e f f e c t had b e e n o b s e r v e d d e t e r i o r -a t e d a f t e r s e v e r a l e x p e r i m e n t s , so t h a t i t became u n u s a b l e , a t t e m p t s were made t o f i n d a s i m i l a r b e h a v i o u r i n o t h e r c r y s t a l s . These a t t e m p t s have f a i l e d so f a r , a n d no e n t i r e l y s a t i s f a c t o r y e x p l a n a t i o n o f t h e " o v e r s h o o t " phenomenon c a n b e g i v e n a t t h e p r e s e n t t i m e . C h a p t e r IV: D i s c u s s i o n A s was s t a t e d i n C h a p t e r I , t h e o n l y t h e o r y w i t h w h i c h t h e r e s u l t s g i v e n i n C h a p t e r I I I c a n b e compared i s t h a t o f Van V l e c k ( V I , 1 9 4 0 ) . Van V l e c k ' s t h e o r y s a y s t h a t t h e t e m p e r a t u r e dependence o f T J s h o u l d be a s T a t l o w t e m p e r a t u r e s , a n d a s T 1 a t h i g h t e m p e r a t u r e s . F u r t h e r T , s h o u l d d e c r e a s e v e r y r a p -i d l y w i t h f i e l d , b e i n g p r o p o r t i o n a l t o H f o r t h e low temp-e r a t u r e ( d i r e c t ) p r o c e s s . The m easurements r e p o r t e d i n t h i s t h e s i s show t h a t T 4 i s p r o p o r t i o n a l t o l , and t h a t t h e r e i s v e r y l i t t l e d epend-ence on f i e l d , e x c e p t f o r t h e d i p a t 1540 0 , ( a b o u t w h i c h we w i l l comment l a t e r ) . Thus agreement w i t h V a n V l e c k ' s t h e o r y i s very, p o o r . P r e v i o u s measurements on o t h e r s a l t s , e . g . by B e n z i e and Cooke ( X I I I , 1 9 5 0 ) , show a s i m i l a r p o o r agreement w i t h V an V l e c k ' s t h e o r y . I n . g e n e r a l a t low t e m p e r a t u r e s a t e m p e r a t u r e dependence o f Tj p r o p o r t i o n a l t o T w i t h Y\ a p p r o x i m a t e l y 2.5 i s o b s e r v e d . A l s o T , u s u a l l y shows o n l y l i t t l e dependence on H . O n l y E s c h e n f e l d e r a n d W e i d n e r (XIV, 1953) f o u n d a t l o w t e m p e r a t u r e s a t e m p e r a t u r e dependence o f T , p r o p o r t i o n a l t o T and t h u s i n a g r e e -ment w i t h Van V l e c k ' s t h e o r y . Thus, a l t h o u g h t h e r e s u l t s r e p o r t e d 66 i h t h i s t h e s i s do n o t a g r e e w e l l w i t h V a n V l e c k ' s t h e o r y , n e i t h e r do t h e r e s u l t s o f o t h e r s . I t s h o u l d be n o t e d t h a t t h e r e have been, u n t i l r e c e n t l y , -v e r y few measurements o f s p i n - l a t t i c e r e l a x a t i o n t i m e i n p a r a -m a g n e t i c s a l t s . I n t e r e s t h a s b e e n a r o u s e d l a t e l y i n t h i s s u b j e c t , b e c a u s e t h e d e t a i l s o f t h e r e l a x a t i o n p r o c e s s a r e i m p o r t a n t i n t h e f u n c t i o n i n g o f t h e 3 l e v e l s o l i d s t a t e M a s e r . A l m o s t a l l t h e measurements made up t o a b o u t 3 y e a r s ago were made on powder specimens- , w h e r e a s t h e r e c e n t measurements, i n c l u d i n g t h e o n e s r e p o r t e d i n t h i s t h e s i s , have been made on s i n g l e c r y s t a l s . M e asurements on s i n g l e c r y s t a l s u s u a l l y r e v e a l d e t a i l s n o t se e n i n powders, and r e l a x a t i o n p r o c e s s e s a r e no e x c e p t i o n t o t h i s g e n e r a l s t a t e m e n t . L e t u s r e v i e w t h e b a s i c a s s u m p t i o n s o f Van V l e c k ' s t h e o r y : i ) T h a t t h e r e i s t h e r m a l e q u i l i b r i u m i n t h e s p i n - s y s t e m . i i ) T h a t t h e r e i s t h e r m a l e q u i l i b r i u m i n t h e l a t t i c e . i i i ) T h a t t h e r e i s good h e a t c o n t a c t o f t h e l a t t i c e w i t h t h e h e l i u m b a t h . i v ) T h a t t h e f r e q u e n c y s p e c t r u m o f t h e l a t t i c e i s a Debye s p e c -trum. v) T h a t s p i n - s p i n i n t e r a c t i o n c a n be n e g l e c t e d I t i s r e a s o n a b l e , i n t h i s c a s e , t o assume t h a t ( i ) i s v a l i d . The r e a s o n i s t h a t t h e l i n e may be s a t u r a t e d , a n d t h e mag^ n e t i c moment o f t h e c r y s t a l r e d u c e d t o z e r o , when t h e e x t e r n a l m a g n e t i c f i e l d has a l m o s t any v a l u e b e l o w 2500 0 . T h i s i n d i c a t e s t h a t t h e l i n e i s homogeneously b r o a d e n e d , r a t h e r t h a n inhomogene-o u s l y b r o a d e n e d , a n d t h a t e n e r g y i s f r e e l y e x c h a n g e d between t h e s p i n s . T a k i n g t h e v a l u e 2 5 0 0 f o r t h e l i n e - w i d t h , we a r r i v e a t a v a l u e o f t h e s p i n - s p i n r e l a x a t i o n t i m e o f t h e o r d e r o f 10 67 s e c o n d s . Hence we e x p e c t t h e r m a l e q u i l i b r i u m t o be e s t a b l i s h e d w i t h i n t h e s p i n s y s t e m I n t i m e s v e r y much s m a l l e r (by an o r d e r o f ) t h a n t h o s e m e a s u r e d e x p e r i m e n t a l l y f o r T\ . A s s u m p t i o n ( i i ) i s much more open t o q u e s t i o n , and i n f a c t Van V l e c k ( V I I , 1941) c a l c u l a t e d t h a t , i n a t y p i c a l s a l t , t h e b o t t l e n e c k i n t h e h e a t t r a n s m i s s i o n f r o m t h e s p i n s y s t e m t o t h e h e l i u m b a t h s h o u l d o c c u r w i t h i n t h e s y s t e m o f l a t t i c e v i b r a -t i o n s , a n d n o t i n t h e t r a n s f e r o f e n e r g y between t h e i o n s and t h e l a t t i c e . However, a t t e m p t s t o d e t e c t s u c h an e f f e c t e x p e r i m e n t -a l l y have a l l f a i l e d so f a r . I n t h e e x p e r i m e n t s d e s c r i b e d i n t h i s t h e s i s i t was t r i e d t o o b s e r v e s u c h an e f f e c t by v a r y i n g t h e p u l s e l e n g t h o f m i c r o -wave power. The i d e a was t h a t a l o n g p u l s e o f m icrowave power w o u l d u p s e t t h e l a t t i c e e q u i l i b r i u m by o v e r p o p u l a t i n g one s m a l l f r e q u e n c y r a n g e i n t h e l a t t i c e f r e q u e n c y s p e c t r u m . T h i s o v e r -p o p u l a t e d r e g i o n ( o r " h o t phonon" band) m i g h t be e x p e c t e d t o d e l a y r e l a x a t i o n , s i n c e t h e i o n i c s y s t e m w o u l d see t h e l a t t i c e a s t h o u g h i t were a t a much h i g h e r t e m p e r a t u r e t h a n i t a c t u a l l y was. Ano-t h e r e f f e c t w h i c h m i g h t have b e e n s e e n , i s t h a t i f t h e r e w e r e two r e l a x a t i o n p r o c e s s e s i n s e r i e s , t h e r e l a x a t i o n o f t h e s p i n system w o u l d be a compound o f t h e two e x p o n e n t i a l s . T o t h e a c c u r a c y o f t h e s e e x p e r i m e n t s , t h e d e c a y i s a p e r f e c t e x p o n e n t i a l . B o t h t h e s e o b s e r v a t i o n s , t h e a b s e n c e o f a p u l s e l e n g t h e f f e c t a n d t h e p u r© e x p o n e n t i a l decay*,, i n d i c a t e t h a t t h e r e i s o n l y one p r e d o m i n -and r e l a x a t i o n p r o c e s s . I t c a n n o t be s a i d f r o m t h e s e e x p e r i m e n t s w h i c h i t i s . 68 It appears reasonable to consider assumption ( i i i ) to be valid, since the crystal is small and immersed directly i n the helium bath. The existence of a pronounced temperature gradient in the sample would moreover cause considerable deviation from the cos? O intensity versus rotation relationship, which was not observed. The correctness of assumption (iv) may be questioned. However, even major deviations from a true Debye spectrum would not seriously affect the relaxation process, since only a small fre-quency band of the la t t i c e vibrations Is instrumental in the re-laxation. The most important criticism i s that assumption (v) i s certainly not valid, i t has been shown experimentally that, when two energy level separations are equal, or one i s an integral multiple of the other, the relaxation time i s markedly reduced. This i s because, for example, one spin can turn over turning back two other spins of half the energy. The energy of the f i r s t spin is then communicated to the l a t t i c e via these other two spins. This, being a process in parallel with the straightforward process, reduces the relaxation time. Such processes are used'technically to control maser operation, and such a state of affairs can be brought about by adjustment of the f i e l d direction relative to the crystal axes, or by the inclusion of impurities. Neodymium has two isotopes, Nd 143 12.2% abundant X - \ and Nd 145 8>3% abundant X — also. The ions containing these nuclei have an extensive hyperfine structure, and the conditions for two and three spin processes are well satisfied. However, the energy-levels are so "broadened by spin-spin interaction that no sharp dips in T, as a function of magnetic f i e l d could be expected. The dip in T, at 1540 0 l i e s at the resonant f i e l d of the cavity. Its occurence at this f i e l d is too much of a coincid-ence to expect i t to be due to anything but some influence of the cavity and the microwave system. However, we have not been able to think of a mechanism which would shorten the relaxation time to the extent observed. A l l effects which we have thought of (e.g. that may be the klystron was not properly shut off) ought to in-crease T, rather than decrease i t . Further experiments on this topic are continuing. The "overshoot effect" described in Chapter III, Section 6) could not be investigated further, since the crystal on which i t was observed had decomposed. Hence a l l that can be said about i t at the present time is that in principle such an overshoot effect i s possible i f the following conditions are satisfied: i) The sample crystal is twinned so that i t consists of two parts, the optical axes of which are not parallel, i i ) The relaxation time depends strongly on the orientation of the optic axis with respect to the external magnetic f i e l d , (lihat this can be so has been verified in other experiments). How an overshoot may arise under these conditions can be visual-ized by considerations on the Poincare' sphere (See Figure XXXVII) . Suppose the sample crystal i s twinned such that i t con-sists of a part I, whose optic axis is parallel to the external mag-F IGURE 0 XXXVII (r> POINCARE R E P R E S E N T A T I O N OF OVERSHOOT E F F E C T F o l l o w i n g page 69 70 J n e t i c f i e l d , f o l l o w e d b y a p a r t I I , whose o p t i c a x i s i s p e r p e n d i c -u l a r t o t h e e x t e r n a l f i e l d . L e t l t > be t h e p o i n t on t h e s p h e r e r e p r e s e n t i n g t h e l i g h t i n c i d e n t on I , i t V t h e l i g h t e m e r g i n g f r o m I , and i n c i d e n t on I I . The g r e a t c i r c l e a n g l e b e t w e e n l i > and l t / > w i l l be d e t e r m i n e d b y t h e amount o f r o t a t i o n i n t r o -d u c e d i n I ( i n p r i n c i p l e we may c h o o s e t h e l e n g t h o f I a n d / o r t h e m a g n i t u d e o f H so t h a t t h e a n g l e U > t o U'> may have any de-s i r e d v a l u e ) . I n p a r t I I , now, we w i l l have combined b i r e f r i n g e n c e a n d F a r a d a y r o t a t i o n and t h e r e f o r e t h e c o n s t r u c t i o n d e s c r i b e d i n C h a p t e r I may be u s e d t o see what happens t o t h e l i g h t . Suppose t h e s t a b l e modes o f p r o p a g a t i o n i n I I a r e r e p r e s e n t e d b y l u > and l v > L e t t h e phase d i f f e r e n c e i n t r o d u c e d by t h e c ombined b i r e f r i n g e n c e and F a r a d a y - r o t a t i o n be s u c h t h a t t h e l i g h t e m e r g i n g f r o m I I i s r e p r e s e n t e d by l o / X / . Now i f u pon a p p l i c a t i o n o f a p u l s e o f m i crowave power t h e F a r a d a y r o t a t i o n i n I a n d I I i s b r o u g h t t o z e r o , t h e l i g h t i n c i d e n t on I I w i l l be o f t h e same t y p e as t h a t i n c i d e n t on I , i . e . r e p r e s e n t e d b y l t > . A l s o t h e s t a b l e modes o f p r o p a g a t i o n i n I I w i l l now be p l a n e p o l a r i z e d l i g h t l * > and l y ^ s a y . T h e r e w i l l o n l y be b i r e f r i n g e n c e l e f t i n I I , i n t r o -d u c i n g a p h a s e d i f f e r e n c e s u c h t h a t t h e l i g h t e m e r g i n g f r o m I I w i l l now be r e p r e s e n t e d b y l o > , s a y . Thus upon a p p l i c a t i o n o f t h e m i c r o w a v e s , t h e p o i n t r e p r e s e n t i n g t h e l i g h t e m e r g i n g f r o m t h e c r y s t a l w i l l s h i f t f r o m l o ' > t o l © > , a n d i f t h e m i c r o w a v e power i s s u f f i c i e n t l y l a r g e , t h e r o t a t i o n i n I and I I w i l l be d e s t r o y e d a t a p p r o x i m a t e l y t h e same r a t e and t h e p a t h t r a c e d o u t w i l l be v e r y n e a r l y a g r e a t c i r c l e p a t h , d e n o t e d b y 1. A s u i t a b l y p l a c e d a n a l y -zer, represented by l o t > , would pass during this time light of continuously decreasing intensity. As soon as the microwave power goes to zero at the end of the pulse, the rotation i n I end II w i l l reappear at a rate governed by the relaxation times of I and H respectively. Now suppose the relaxation time in I i s very short com-pared with that in II. Then rotation w i l l return in I while II exhibits only birefringence. The rotation in II w i l l then re-appear slowly. In this case the point representing the light emerging from the crystal w i l l trace out path 2 when returning from (o> to \o'> . When i t reaches the point le> along this path the intensity of the light passed by the analyzer w i l l be i d -entical to the intensity due to 1©'^ (the equilibrium position). Then the intensity w i l l f i r s t increase above i t s equilibrium value u n t i l a maximum is reached and then decrease again to the equi-librium level. This i s exactly what i s observed i n the case of the "overshoot" effect. However, i f the microwave pulse is short, we may have only p a r t i a l saturation in I, hence only partial re-moval of rotation in this part. Then i t is seen that as a result path 1 and path 2 would be closer together and the amount of over-shoot would be small, whereas with increasing pulse-length the path-difference and hence the amount of overshoot would increase, as was actually observed in the experiments. The extent and appearance of such an "overshoot" effect, while possible i n principle, i s seen to depend on many factors 72 ( c o m p a r a t i v e amounts o f r o t a t i o n i n t h e two p a r t s o f t h e t w i n n e d c r y s t a l , d i f f e r e n c e i n r e l a x a t i o n t i m e s , amount o f b i r e f r i n g e n c e , p o l a r i z a t i o n o f i n c i d e n t l i g h t , p o s i t i o n o f a n a l y z e r e t c . ) . I f t w i n n i n g o f t h e c r y s t a l i s a c c e p t e d as t h e cause u n d e r l y i n g t h e " o v e r s h o o t " e f f e c t , t h e n we must c o n c l u d e t h a t t h e r e l a x a t i o n t i m e Ty depends s t r o n g l y on t h e o r i e n t a t i o n o f t h e o p t i c a x i s w i t h r e s p e c t t o t h e e x t e r n a l m a g n e t i c f i e l d . F u r t h e r i n v e s t i g a t i o n s i n t h i s d i r e c t i o n a r e p l a n n e d . 73 BIBLIOGRAPHY R e f e r e n c e s a r e numbered and l i s t e d i n t h e o r d e r i n w h i c h t h e y f i r s t a p p e a r i n t h e t e x t . I . ) A. K a s t l e r , Compt.rend., 232, 953 ( 1 9 5 1 ) . I I . ) W. Opechowski, Rev.Mod.- P h y s . , 25, 264 ( 1 9 5 3 ) . I I I . ) J.M. D a n i e l s and H, wesemeyer, C a n . J o u r n . P h y s . , 36, 405 ( 1 9 5 8 ) . IV. ) H. Wesemeyer and J.M. D a n i e l s , Z e i t s c h r . P h y s . , 152, 591 ( 1 9 5 8 ) . V. ) G.V. S k r o t s k i i , P.S. Z y r i a n o v , and T.G. I z i u m o v , J . I x p t l . T h e o r e t . P h y s . ( U S S R ) , 35, 1471 ( 1 9 5 8 ) . V I . ) J.H. V a n V l e c k , Phys.Rev., 57, 426, (1941)., V I I . ) J.H. V a n V l e c k , Phys.Rev., 59, 724, 730 ( 1 9 4 1 ) . V I I I . ) H. Wesemeyer, U n i v . o f B r i t . V o l . , P h . D . - T h e s i s ( 1 9 5 8 ) . IX. ) M.P.M. J a e g e r , K e c u e i l d e s t r a v a u x c h i m i q u e s , 33, 342 ( 1 9 1 4 ) . X. ) B e i l s t e i n , Handb. d e r O r g a n i s c h e n Chemie, V o l . 1 , p « 3 2 5 , ( S p r i n g e r , 1 9 1 8 ) . X I . ) H. Van D i j k , M. Durieux, J.R. C l e m e n t and J.K. L o g a n , P h y s i c a 24, S 1 2 9 ( 1 9 5 8 ) . X I I . ) G.S. B o g l e , A.H. Cooke and S. ¥ h i t l e y , P r o c . P h y s . S o c . A, 64, 931 ( 1 9 5 1 ) . X I I I . ) R . J . B e n z i e and A.H. Cooke, P r o c . P h y s . S o c . A, 63, 201 (1950)-. XIV. ) A.H. E s c h e n f e l d e r and R.T. W e i d n e r , P h y s . Rev. 92, 869 ( 1 9 5 3 ) . ~ Views of the motion of the t ip of the e lec t r i c vector for I L > and I r >. F I G U R E TI Decomposition of elliptically polarized light into circularly polarized components. F I G U R E T H / i R P O I N C A R E S P H E R E F I G U R E 32T Properties of the Poincare' Sphere F I G U R E E E f f e c t of combined Faraday Ef fect and Birefr ingence FIGURE X STABILIZED POWER SUPPLY WAVE METER KLYSTRON T P U L S E MODULATOR mi MICRO A M M E T E R A T T E N U A T O R | | V A R I A B L E S U S C E P T A N C E 0 C R Y S T A L DIODE ~ ^ \ - T U N A B L E E N D M S A M P L E D U M M Y LOAD / G A L V A N O M E T E R O S C I L L O S C O P E CAVITY MICROWAVE SYSTEM ( MEG 1/2 6SL7 4.7 K< 100 K W V 10 K <!>o MANUAL | W > 4 * * K / CONTROL S_ IW - 1 FOR 220 K J T ,47 K S IOOK< B 220 V V<-[1 HI PULSE I S K ^ f - J , HEIGHT <? PULSE 9 oWtOTH I PULSE WIDTH I E ^ 0 / 2 6 S L 7 01 UF , PULSE OUTPUT <> > 4 7 K TRIGGER OUTPUT SINGLE P U L S E S FIGURE XE PULSE MODULATOR CIRCUIT Hg-ARC POLARIZER I FILTER FIGURE X E CRYSTAL ANALYZER CAVITY MAGNET C O R E PHOTO-MULTIPLIER OPTICAL SYSTEM F I G U R E H E 5 Hy 1500 A hAAAH 4_ GENERATOR 4 0 UF 8 0 UF AMMETER CURRENT REGULATOR H g - A R C C I R C U I T S A S S O C I A T E D W I T H O P T I C A L S Y S T E M F I G U R E XI2 — / ) Saturated solution \\\\\\| M e r c u r y p o o l Ground glass cover Crystal seed Cyl inder of f i l ter paper containing c rys ta ls o o o o C R Y S T A L GROWING A P P A R A T U S F I G U R E X E VERDET CONSTANT VS. l/(T-.OI3°) CRYSTAL I deg. cm"' 0"' O CRYSTAL H .08 .06 -.04 .02 J L F I G U R E XXVII I 1 0 0 ROTATION VS. TIME ON S E M I L O G S C A L E AS OBTAINED FROM • FIGURE O FIGURE 3 FIGURE 50 100 150 msec. FIGURE xxx m RELAXATION TIME VS. MAGNETIC FIELD msec. • T = I . 6 8 5 ° K 3 T = 2 . 0 6 0 ° K 150 -100 50 O T = 4 . 2 2 ° K 9 • • 3 c ( I 3 O o KO FIGURE xxxiv msec. RELAXATION TIME VS. MAGNETIC FIELD • T = I . 3 8 0 ° K O T= 1.542 °K 3 T-1.875 °K 200 O O o u ©8 100 h • O ° o 3 3 « » 3 3 3 ~ 3 3 K 0 msec. 2 0 0 150 100 5 0 -25 -10 -FIGURE X X X V RELAXATION T IME VS. TEMPERATURE ON LOG L O G S C A L E o H = 1290 0 , Slope =-2.99 € H = 1540 0 , Slope = - 2 . 8 3 ® H =1790 0 , Slope = - 2 . 8 8 • H = 2 0 4 0 0 , Slope = - 3 . 6 4 2.5 4 K FIGURE x x x v i RELAXATION TIME VS. TEMPERATURE ON LOG LOG S C A L E <• H = 7 8 0 0 , SLOPE = -3.48 , o H = 1030 0, SLOPE = -2 .62 1.2 1.4 1.6 1.8 2.0 2.2 °K POINCARE R E P R E S E N T A T I O N OF OVERSHOOT E F F E C T 

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
http://iiif.library.ubc.ca/presentation/dsp.831.1-0085919/manifest

Comment

Related Items