UBC Theses and Dissertations

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UBC Theses and Dissertations

AC relaxation in the Fe8 molecular magnet Rose, Geordie 2000

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A C R E L A X A T I O N IN T H E Fe8 M O L E C U L A R M A G N E T By Geordie Rose B. Eng. (Engineering Physics), McMaster University, 1994 A T H E S I S S U B M I T T E D IN P A R T I A L F U L F I L L M E N T O F T H E R E Q U I R E M E N T S F O R T H E D E G R E E O F D O C T O R O F P H I L O S O P H Y in T H E F A C U L T Y O F G R A D U A T E S T U D I E S D E P A R T M E N T O F P H Y S I C S A N D A S T R O N O M Y We accept this thesis as conforming to the required standard T H E 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 January 2000 © Geordie Rose, 2000 I n 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 a n a d v a n c e d d e g r e e a t t h e 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 , I a g r e e t h a t t h e L i b r a r y s h a l l m a k e 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 a n d s t u d y . I f u r t h e r a g r e e 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 p u r p o s e s m a y b e g r a n t e d b y t h e h e a d o f m y d e p a r t m e n t o r b y h i s o r h e r r e p r e s e n t a t i v e l y I t isfc u n d e r s t o o d t h a t c o p y i n g o r 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 financial g a i n s h a l l mot b e a l l o w e d - w t t h Q u t m y w r i t t e n p e r m i s s i o n . D e p a r t m e n t o f P h y s i c s a n d A s t r o n o m y T h e 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 6 2 2 4 A g r i c u l t u r a l R o a d V a n c o u v e r , B.C., C a n a d a V 6 T 1 Z 1 D a t e : Abstract W e i n v e s t i g a t e t h e l o w e n e r g y m a g n e t i c r e l a x a t i o n c h a r a c t e r i s t i c s o f t h e " i r o n e i g h t " (Fe8) m o l e c u l a r m a g n e t . E a c h m o l e c u l e i n t h i s m a t e r i a l c o n t a i n s a c l u s t e r o f e i g h t Fe3+ i o n s s u r r o u n d e d b y o r g a n i c l i g a n d s . T h e m o l e c u l e s a r r a n g e t h e m s e l v e s i n t o a r e g u l a r l a t t i c e w i t h t r i c l i n i c s y m m e t r y . A t s u f f i c i e n t l y l o w e n e r g i e s , t h e e l e c t r o n i c s p i n s o f t h e Fe3+ i o n s l o c k t o g e t h e r i n t o a " q u a n t u m r o t a t o r " w i t h s p i n S = 10. W e d e r i v e a l o w e n e r g y e f f e c t i v e H a m i l t o n i a n f o r t h i s s y s t e m , v a l i d f o r t e m p e r a t u r e s l e s s t h a n Tc ~ 3 6 0 mK, w h e r e Tc i s t h e t e m p e r a t u r e a t w h i c h t h e F e 8 s y s t e m c r o s s e s o v e r i n t o a " q u a n t u m r e g i m e " w h e r e r e l a x a t i o n c h a r a c t e r i s t i c s b e c o m e t e m p e r a t u r e i n - d e p e n d e n t . W e s h o w t h a t i n t h i s r e g i m e t h e d o m i n a n t e n v i r o n m e n t a l c o u p l i n g i s t o t h e e n v i r o n m e n t a l s p i n b a t h i n t h e m o l e c u l e . W e s h o w h o w t o e x p l i c i t l y c a l c u l a t e t h e s e c o u - p l i n g s , g i v e n c r y s t a l l o g r a p h i c i n f o r m a t i o n a b o u t t h e m o l e c u l e , a n d d o t h i s f o r Fes- W e us e t h i s i n f o r m a t i o n t o c a l c u l a t e t h e l i n e w i d t h , t o p o l o g i c a l d e c o h e r e n c e a n d o r t h o g o n a l - i t y b l o c k i n g p a r a m e t e r s . A l l o f t h e s e q u a n t i t i e s a r e s h o w n t o e x h i b i t a n i s o t o p e e f f e c t . W e d e m o n s t r a t e t h a t o r t h o g o n a l i t y b l o c k i n g i n F e 8 i s s i g n i f i c a n t a n d s u p p r e s s e s c o h e r e n t t u n n e l i n g . W e t h e n u s e o u r l o w e n e r g y e f f e c t i v e H a m i l t o n i a n t o c a l c u l a t e t h e s i n g l e - m o l e c u l e r e l a x a t i o n r a t e i n t h e p r e s e n c e o f a n e x t e r n a l m a g n e t i c f i e l d w i t h b o t h A C a n d D C c o m p o n e n t s b y s o l v i n g t h e L a n d a u - Z e n e r p r o b l e m i n t h e p r e s e n c e o f a n u c l e a r s p i n b a t h . B o t h s a w t o o t h a n d s i n u s o i d a l A C fields a r e a n a l y z e d . T h i s s i n g l e - m o l e c u l e r e l a x a t i o n r a t e i s t h e n u s e d a s i n p u t i n t o a m a s t e r e q u a t i o n i n o r d e r t o t a k e i n t o a c c o u n t t h e m a n y - m o l e c u l e n a t u r e o f t h e f u l l s y s t e m . O u r r e s u l t s a r e t h e n c o m p a r e d t o q u a n t u m r e g i m e r e l a x a t i o n e x p e r i m e n t s p e r f o r m e d o n t h e F e g s y s t e m . i i Table of Contents Abstract ii Table of Contents iii List of Tables viii List of Figures ix Acknowledgements xix 1 Introduction and Overview 1 1.1 A n I n t r o d u c t i o n t o Fe& 4 1.1.1 G i a n t S p i n s a n d Q u a n t u m E n v i r o n m e n t s 8 1.1.2 A R e l a t e d S y s t e m i s C h a r a c t e r i z e d 9 .1.1.3 M a g n e t i c C h a r a c t e r i z a t i o n o f F e 8 13 1.1.4 R e s u l t s i n t h e Q u a n t u m R e g i m e 14 1.2 A n I n t r o d u c t i o n t o R e l a x a t i o n E x p e r i m e n t s 18 1.2.1 D C F i e l d R e l a x a t i o n i n P o l a r i z e d Fe8 1 9 1.2.2 D C R e l a x a t i o n o f A n n e a l e d C r y s t a l s 1 9 1.2.3 H o l e D i g g i n g a n d t h e T i m e - D e p e n d e n t I n t e r n a l L o n g i t u d i n a l B i a s D i s t r i b u t i o n 21 1.2.4 A C R e l a x a t i o n o f A n n e a l e d C r y s t a l s 2 3 1.2.5 E x t r a c t i o n o f T u n n e l i n g M a t r i x E l e m e n t s 2 3 1.3 T h e s i s O v e r v i e w 2 7 i i i 2 Effective Hamiltonians 29 2.1 T h e Fe3+ F r e e I o n H a m i l t o n i a n 3 0 2.2 T h e E f f e c t o f t h e C r y s t a l l i n e E n v i r o n m e n t 3 2 2.3 T h e S i n g l e I o n E f f e c t i v e H a m i l t o n i a n 3 4 2.3.1 F i r s t O r d e r P e r t u r b a t i o n T h e o r y 3 5 2.3.2 S e c o n d O r d e r P e r t u r b a t i o n T h e o r y 3 6 2.3.3 H i g h e r O r d e r s P e r t u r b a t i o n T h e o r y 3 7 2.4 T h e S i n g l e M o l e c u l e E f f e c t i v e H a m i l t o n i a n 3 9 2.4.1 I n c l u s i o n o f E x c h a n g e a n d S u p e r e x c h a n g e T e r m s 4 0 2.4.2 " O f f s i t e " D i p o l a r a n d Q u a d r u p o l a r C o n t r i b u t i o n s 4 0 2.4.3 I n t r a - N u c l e a r S p i n C o u p l i n g s 4 5 2.4.4 C o u p l i n g s o f t h e N u c l e a r B a t h t o E x t e r n a l M a g n e t i c F i e l d s .... 4 5 2.4.5 C o u p l i n g t o P h o n o n s 4 5 2.4.6 C o u p l i n g t o P h o t o n s 4 7 2.4.7 B r i n g i n g a l l t h e T e r m s T o g e t h e r - T h e B a r e Fe$ H a m i l t o n i a n ... 4 7 2.5 E x c h a n g e / S u p e r e x c h a n g e a n d t h e G i a n t S p i n P i c t u r e 4 9 2.6 I n v e s t i g a t i o n o f t h e G i a n t S p i n H a m i l t o n i a n i n t h e A b s e n c e o f E n v i r o n - m e n t a l C o u p l i n g s 5 1 2.6.1 E x a c t S o l u t i o n f o r T u n n e l i n g M a t r i x E l e m e n t s v i a D i a g o n a l i z a t i o n 5 2 2.6.2 T u n n e l i n g M a t r i x E l e m e n t s v i a P e r t u r b a t i o n T h e o r y 5 9 2.6.3 T u n n e l i n g M a t r i x E l e m e n t s v i a W K B M e t h o d s 6 0 2.6.4 T u n n e l i n g M a t r i x E l e m e n t s v i a I n s t a n t o n T e c h n i q u e s 6 0 2.6.5 C o m p a r i s o n o f A p p r o x i m a t e M e t h o d s t o E x a c t S o l u t i o n s 6 0 2.7 B a c k t o t h e F u l l H a m i l t o n i a n - S e p a r a t i o n o f T u n n e l i n g E n e r g y S c a l e U s i n g a n I n s t a n t o n T e c h n i q u e 6 2 2.8 O f f - D i a g o n a l T e r m s a n d t h e I n s t a n t o n M e t h o d 6 8 i v 2.8.1 R e v i e w o f t h e M e t h o d o f T u p i t s y n e t . a l 6 9 2.8.2 T h e T u n n e l i n g L a g r a n g i a n 7 0 2.8.3 A n A s s u m p t i o n i s M a d e 7 2 2.8.4 S o l u t i o n f o r t h e F r e e I n s t a n t o n T r a j e c t o r y 7 3 2.8.5 I n c l u s i o n o f t h e E x t e r n a l M a g n e t i c F i e l d a n d t h e N u c l e a r S p i n s . 7 5 2.9 T h e F i n a l S i n g l e M o l e c u l e E f f e c t i v e H a m i l t o n i a n 7 7 3 Nuclear Spin Couplings in Fe$ and the Isotope Effect 78 3.1 U n i t s a n d C o n s t a n t s 7 9 3.2 T h e P o i n t D i p o l e A p p r o x i m a t i o n 7 9 3.2.1 M a g n e t i c F i e l d a t f d u e t o a " P o i n t D i p o l e " a t 0 8 0 3.2.2 M a g n e t i c F i e l d a t fv d u e t o E i g h t " P o i n t D i p o l e s " a t fFei 8 .... 8 0 3.2.3 I s o t o p i c C o n c e n t r a t i o n s , N u c l e a r ^ - f a c t o r s a n d Q u a d r u p o l a r M o - m e n t s i n Fe8 8 0 3.2.4 D e f i n i t i o n a n d E v a l u a t i o n o f 7 ^ , 7 ^ , u^. a n d ufc 81 3.2.5 C o n t a c t H y p e r f i n e C o u p l i n g E n e r g i e s f o r 5 7 F e 3 + 8 2 3.2.6 C a l c u l a t i o n o f wjj a n d u^r f r o m K n o w l e d g e o f A t o m i c P o s i t i o n s . . 8 4 3.2.7 C a l c u l a t i o n o f t h e O r t h o g o n a l i t y B l o c k i n g P a r a m e t e r K 8 5 3.2.8 C a l c u l a t i o n o f E0 9 3 3.2.9 C a l c u l a t i o n o f T o p o l o g i c a l D e c o h e r e n c e P a r a m e t e r s AkND a n d A . 9 7 3.3 U s i n g F r e e Fe3+ H a r t r e e - F o c k W a v e f u n c t i o n s t o M o d e l A c t u a l S p i n D i s - t r i b u t i o n s 9 8 3.4 T a b l e s o f N u c l e a r P o s i t i o n s , F i e l d s a t N u c l e i a n d H y p e r f i n e C o u p l i n g E n - e r g i e s 1 1 2 4 A n Introduction to the Generalized Landau-Zener Problem 122 4.1 I n t r o d u c t i o n t o a n d E x a c t S o l u t i o n o f t h e L a n d a u - Z e n e r P r o b l e m .... 122 v 4.1.1 A l t e r n a t e M e t h o d o f S o l u t i o n f o r t h e T r a n s i t i o n P r o b a b i l i t y I. A l l O r d e r s P e r t u r b a t i o n E x p a n s i o n 1 2 6 4.1.2 A l t e r n a t e M e t h o d o f S o l u t i o n f o r t h e T r a n s i t i o n P r o b a b i l i t y I I . D y c h n e ' s F o r m u l a 1 2 8 4.1.3 A n a l y s i s o f t h e T r a n s i t i o n F o r m u l a 1 3 0 4.1.4 G e n e r a l i z a t i o n o f t h e T w o - L e v e l L a n d a u - Z e n e r P r o b l e m I. E x a c t S o l u t i o n f o r A(t) ~ Vj|(t) 1 31 4.1.5 G e n e r a l i z a t i o n o f t h e T w o - L e v e l L a n d a u - Z e n e r P r o b l e m I I . E x a c t S o l u t i o n b y M a p p i n g t o R i e m a n n ' s D i f f e r e n t i a l E q u a t i o n 1 3 7 5 The Landau-Zener Problem in the Presence of a Spin Bath 142 5.1 T h e A d d i t i o n o f a n E n v i r o n m e n t t o t h e L a n d a u - Z e n e r P r o b l e m : G e n e r a l C o n s i d e r a t i o n s 1 4 2 5.2 T h e Q u a n t u m R e g i m e E f f e c t i v e H a m i l t o n i a n : I n c l u s i o n o f a S p i n E n v i - r o n m e n t 1 4 3 5.3 G e n e r a l A C F i e l d S o l u t i o n i n F a s t P a s s a g e 1 4 7 5.3.1 A L i s t o f A p p r o x i m a t i o n s I n v o k e d i n t h e C a l c u l a t i o n s T h a t F o l l o w 1 4 9 5.3.2 G e n e r a l S t r a t e g y f o r C a l c u l a t i n g R e l a x a t i o n R a t e s 1 51 5.3.3 P r o c e s s i n g o f t h e T r a n s i t i o n A m p l i t u d e 1 5 5 5.3.4 P r o c e s s i n g o f t h e T r a n s i t i o n P r o b a b i l i t y ( i ) T h e F o r m a l E x p r e s s i o n 1 5 9 5.3.5 P r o c e s s i n g o f t h e T r a n s i t i o n P r o b a b i l i t y ( i i ) A v e r a g i n g o v e r t h e R a n d o m l y F l u c t u a t i n g Ti N o i s e 1 61 5.4 S o l u t i o n W i t h o u t S p i n B a t h 1 6 3 5.5 S o l u t i o n F o r a S p i n B a t h w i t h n o Q u a d r u p o l a r C o n t r i b u t i o n 1 6 9 5.5.1 P u r e O r t h o g o n a l i t y B l o c k i n g 1 6 9 v i 5.5.2 T h e G e n e r a l C a s e ; I n c l u s i o n o f T o p o l o g i c a l D e c o h e r e n c e 1 8 0 5.6 T h e G e n e r a l S i n g l e M o l e c u l e R e l a x a t i o n R a t e i n Fe$ 1 9 4 5.6.1 E f f e c t o f t h e N u c l e a r S p i n E n v i r o n m e n t o n t h e L a r g e A S i n g l e M o l e c u l e R e l a x a t i o n R a t e i n Fe8 2 0 2 5.7 S u m m a r y a n d D i s c u s s i o n o f R e s u l t s 2 0 3 6 A C Relaxation in a Crystal of Molecular Magnets 206 6.1 P r e a m b l e 2 0 6 6.2 T h e G e n e r a l i z e d M a s t e r E q u a t i o n 2 0 9 6.3 S h o r t T i m e D y n a m i c s 2 1 0 6.3.1 S t r o n g l y A n n e a l e d S a m p l e s a n d t h e L a r g e A L i m i t 2 1 2 6.3.2 G e n e r a l S o l u t i o n N e a r t h e N o d e s 2 1 4 7 Summary and Outlook 217 Appendices 221 A Bias Distribution in a Dilute Solution of Dipoles 222 B T i m e Evolution of Nuclear Spin States 228 Bibliography 231 v i i List o f Tables 2.1 P e r t u r b a t i o n t h e o r y r e s u l t s f o r s o m e s i m p l e H a m i l t o n i a n s , f r o m [70]. ... 59 3.1 N u c l e a r s p i n i n f o r m a t i o n f o r n u c l e i o c c u r i n g i n Fes- F r o m [48] 81 3.2 P o s i t i o n s o f t h e i r o n i o n s , u n i t s i n A n g s t o m s 86 3.3 D a t a f o r H y d r o g e n 113 3.4 D a t a f o r H y d r o g e n 114 3.5 D a t a f o r H y d r o g e n 115 3.6 D a t a f o r H y d r o g e n 116 3.7 D a t a f o r B r o m i n e 117 3.8 D a t a f o r N i t r o g e n 118 3.9 D a t a f o r C a r b o n 119 3.10 D a t a f o r I r o n 120 3.11 D a t a f o r O x y g e n 121 5.1 Q u a n t i t i e s c o m i n g f r o m o r t h o g o n a l i t y a n d d e g e n e r a c y b l o c k i n g 195 5.2 Z e r o e x t e r n a l field v a l u e s f o r r 0 a n d u0 f o r t h e t h r e e s p e c i e s s h o w n . U n i t s a r e i n MHz 196 5.3 Q u a n t i t i e s c o m i n g s o l e l y f r o m t o p o l o g i c a l d e c o h e r e n c e e f f e c t s 196 5.4 Z e r o field v a l u e s o f t h e t o p o l o g i c a l d e c o h e r e n c e t e r m s f o r s p e c i e s i n Fe$. . 197 5.5 T o p o l o g i c a l d e c o h e r e n c e t e r m s f o r t h r e e v a r i e t i e s o f Fes 201 5.6 Q u a n t i t i e s t h a t c o m e a b o u t d u e t o i n t e r p l a y b e t w e e n o r t h o g o n a l i t y b l o c k - i n g , d e g e n e r a c y b l o c k i n g a n d t o p o l o g i c a l d e c o h e r e n c e e f f e c t s 201 v i i i L i s t o f F i g u r e s 1.1 T h e n u m b e r o f a t o m s n e e d e d t o r e p r e s e n t o n e b i t o f i n f o r m a t i o n a s a f u n c t i o n o f c a l e n d a r y e a r . E x t r a p o l a t i o n o f t h e t r e n d s u g g e s t s t h a t t h e o n e a t o m p e r b i t l e v e l i s r e a c h e d i n a b o u t t h e y e a r 2 0 2 0 . A d a p t e d f r o m [1]. 1 1.2 C l o c k s p e e d ( H z ) v s . c a l e n d a r y e a r . A d a p t e d f r o m [1] 2 1.3 E n e r g y ( p i c o - J o u l e s ) d i s s i p a t e d p e r l o g i c a l o p e r a t i o n a s a f u n c t i o n o f c a l - e n d a r y e a r . T h e 1 kT l e v e l i s i n d i c a t e d b y a d a s h e d l i n e . A d a p t e d f r o m [1] 3 1.4 A 2-D p r o j e c t i o n v i e w o f t h e Fes u n i t c e l l o n t o t h e y — z p l a n e . D i s t a n c e s s h o w n a r e i n A n g s t r o m s . L e g e n d : R e d , i r o n ; P u r p l e , b r o m i n e ; L i g h t B l u e , o x y g e n ; G r e e n , n i t r o g e n ; Y e l l o w , c a r b o n ; a n d D a r k B l u e C r o s s e s , h y d r o - g e n . N o t e t h e c e n t r a l m a g n e t i c c o r e , s u r r o u n d e d b y a s h i e l d o f o r g a n i c s p e c i e s 5 1.5 A v i e w o f t h e F e 8 u n i t c e l l i n t h e x — z p l a n e 6 1.6 A v i e w o f t h e Fe& u n i t c e l l i n t h e x — y p l a n e . H e r e w e a r e l o o k i n g r i g h t d o w n t h e " e a s y a x i s " o f t h e m o l e c u l e (see c h a p t e r 2) 7 1.7 P r o j e c t i o n o f t h e M n i 2 u n i t c e l l o n t o t h e x — y p l a n e . H e r e w e a r e l o o k i n g d o w n t h e e a s y a x i s o f t h e c r y s t a l . T h e a x e s s c a l e s a r e i n A n g s t r o m s . L e g e n d : R e d , m a n g a n e s e ; P u r p l e , o x y g e n ; Y e l l o w , c a r b o n ; D a r k B l u e C r o s s e s , h y d r o g e n . N o t e t h e i n n e r a n d o u t e r " r i n g s " o f m a n g a n e s e i o n s . . 10 1.8 P r o j e c t i o n o f t h e M n i 2 u n i t c e l l o n t o t h e x — z p l a n e 11 1.9 P r o j e c t i o n o f t h e Mnyi u n i t c e l l o n t o t h e y — z p l a n e 12 i x 1.10 M a g n e t i z a t i o n s t e p s i n t h e h y s t e r e s i s c u r v e o f M n 1 2 . F r o m [17] 15 1.11 T h i s d a t a s h o w s t h e l o g o f t h e r e l a x a t i o n t i m e v s . 1 / T i n Fe8. A t h i g h t e m p e r a t u r e s t h e r m a l a c t i v a t i o n i s o b s e r v e d , w h i l e f o r T < ~ 3 6 0 mK r e l a x a t i o n b e c o m e s t e m p e r a t u r e i n d e p e n d e n t . F i g u r e o b t a i n e d f r o m [14]. 15 1.12 R e l a x a t i o n o f t h e m a g n e t i z a t i o n m e a s u r e d a t H — 0 a f t e r f i r s t s a t u r a t i n g i n —* a field o f % = 3.5 T z. A s i n d i c a t e d i n figure 1.11, t h e c u r v e s s u p e r i m p o s e f o r T < 3 6 0 mK. S h o w n i n t h e i n s e t a r e r e l a x a t i o n c h a r a c t e r i s t i c s i n t h e q u a n t u m r e g i m e f o r s o m e H ^ 0, a p p l i e d a l o n g t h e e a s y (z) a x i s . F i g u r e f r o m [50] 2 0 1.13 S h o r t t i m e r e l a x a t i o n o f a s i n g l e c r y s t a l o f F e 8 , m e a s u r e d a t 150mK. H e r e —* s e v e r a l d i f f e r e n t D C b i a s fields H w e r e a p p l i e d a l o n g t h e e a s y a x i s o f t h e c r y s t a l . N o t e t h a t t h e d a t a i s p l o t t e d a g a i n s t s q u a r e r o o t t. T h e i n s e t s h o w s t h e s l o p e o f e a c h o f t h e s e l i n e s a s f u n c t i o n s o f t h e D C b i a s field. F i g u r e f r o m [55] 21 1.14 H e r e w e i n c l u d e s o m e d a t a f r o m a d i f f e r e n t k i n d o f m o l e c u l a r m a g n e t , t h e Mnyi s y s t e m . H e r e w e a g a i n s e e t h e c l e a r s q u a r e - r o o t r e l a x a t i o n c h a r a c t e r - i s t i c . H o w e v e r i n t h i s c a s e t h e r e l a x a t i o n r a t e s a r e t e m p e r a t u r e d e p e n d e n t . F i g u r e f r o m [57] 22 1.15 H e r e i s d a t a f r o m a n e x p e r i m e n t o n a n Fe8 s a m p l e t h a t w a s a n n e a l e d i n z e r o field, g i v i n g i t z e r o i n i t i a l m a g n e t i z a t i o n . T h e s a m p l e w a s t h e n e x p o s e d t o l o n g i t u d i n a l D C fields o f v a r i o u s m a g n i t u d e s . W e s e e h e r e r e l a x a t i o n a w a y f r o m M = 0, i n t h e d i r e c t i o n o f t h e a p p l i e d f i e l d , w i t h t h e s a m e s q u a r e r o o t t e m p o r a l d e p e n d e n c e a s i n t h e i n i t i a l l y p o l a r i z e d c a s e . F r o m [49] 2 3 x 1.16 F i e l d d e p e n d e n c e o f s h o r t t i m e s q u a r e r o o t r e l a x a t i o n r a t e s Tsqrt(Hz). T h e i n i t i a l d i s t r i b u t i o n i s l a b e l l e d w i t h Min = — 0 . 9 9 8 Ms w h e r e a s t h e o t h e r s a r e d i s t r i b u t i o n s o b t a i n e d b y t h e r m a l a n n e a l i n g . T h e l a t t e r a r e d i s t o r t e d a t h i g h e r fields b y n e a r e s t n e i g h b o u r l a t t i c e e f f e c t s . F i g u r e f r o m [49]. . . 2 4 1.17 Q u a n t u m h o l e - d i g g i n g . F o r e a c h p o i n t , t h e s a m p l e w a s first s a t u r a t e d i n a field o f -1.4 T a t a t e m p e r a t u r e o f T ~ 2 K a n d t h e n c o o l e d t o 4 0 m K . T h e s a m p l e w a s t h e n a l l o w e d t o r e l a x f o r t i m e s t0. A f t e r t h i s t i m e h a d e l a p s e d , a D C field Hz w a s a p p l i e d , a n d rsgrt w a s m e a s u r e d . N o t e t h e r a p i d d e c r e a s e i n r e l a x a t i o n r a t e n e a r Hz — 0. F i g u r e f r o m [49] 2 4 1.18 Q u a n t u m h o l e d i g g i n g , a s i n figure 1.17, b u t n o w f o r a s a m p l e t h a t h a s b e e n a n n e a l e d t o Min = —0 . 2 Ms. T h e r e s u l t i n g e v o l u t i o n s h o w s a v e r y n a r r o w h o l e (see i n s e t ) . N e a r z e r o b i a s t h e h o l e d e v e l o p s v e r y r a p i d l y a l t h o u g h t h e r e s t o f t h e d i s t r i b u t i o n h a r d l y c h a n g e s a t a l l . F i g u r e f r o m [49]. 2 5 1.19 H e r e i s p l o t t e d t h e d i f f e r e n c e b e t w e e n t h e r e l a x a t i o n r a t e s a t t = 0 (Tinit) a n d a t t0 — 1 6 s ( T ^ ) , f o r s e v e r a l d i f f e r e n t a m o u n t s o f a n n e a l i n g . N o t e t h a t f o r \Min\ < 0.5 t h e h o l e w i d t h b e c o m e s i n d e p e n d e n t o f | M ; „ | , w i t h a n i n t r i n s i c w i d t h o f ~ 0.8 mT. F i g u r e f r o m [49] 2 5 1.20 T h e q u a n t i t y A h e r e i s r e l a t e d t o t h e r e l a x a t i o n r a t e o f t h e c r y s t a l ' s m a g - n e t i z a t i o n v i a (1.3). H e r e i t i s s h o w n a s a f u n c t i o n o f t h e m a g n i t u d e o f t h e t r a n s v e r s e D C field \H\ = + H% f o r s e v e r a l o r i e n t a t i o n s o f t h i s field f = t a n - 1 (Hy/Hx). I n t h i s c a s e t h e l o n g i t u d i n a l D C field w a s t a k e n t o b e z e r o (Hz = 0 ) . F i g u r e f r o m [51] 2 6 x i 1.21 T h e q u a n t i t y A s h o w n f o r ip = 0, a s a f u n c t i o n o f \H\. S h o w n h e r e a r e r e s u l t s f o r t h r e e d i f f e r e n t v a l u e s o f Hz. T h e l o w e s t c u r v e w a s o b t a i n e d f o r Hz = 0; t h e m i d d l e c u r v e f o r Hz = 0.22T, a n d t h e u p p e r c u r v e f o r Hz = 0 . 44T. I n t e r m s o f t h e e n e r g y l e v e l s t r u c t u r e o f t h e Fe8 m o l e c u l e ' s s p i n H a m i l t o n i a n p r e s e n t e d i n c h a p t e r 1, t h e s e a p p l i e d fields c o r r e s p o n d t o r e s o n a n c e s i t u a t i o n s b e t w e e n | — S | + S >, | — £ > < - > • 1 + 5 — 1 > a n d | — S >«-»• | + 5 — 2 > r e s p e c t i v e l y . N o t i c e t h a t a p a r i t y e f f e c t i s o b s e r v e d . F i g u r e f r o m [51] 2 7 2.1 E x c h a n g e p a t h w a y s i n Fe8 i n t h e i s o t r o p i c m o d e l o f D e l f s e t . a l . [18]. F i t s t o s u s c e p t i b i l i t y d a t a g i v e J i 2 ~ 35K, J i 3 ~ 180K, J 1 5 ~ 22K a n d J 3 5 ~ 52K, w i t h a l l c o u p l i n g s a n t i f e r r o m a g n e t i c 41 2.2 A l l o f t h e u n i t c e l l s ( a f t e r A s h c r o f t a n d M e r m i n [138]). ( i ) C u b i c , ( i i ) T e t r a g o n a l , ( i i i ) O r t h o r h o m b i c , ( i v ) M o n o c l i n i c , ( v ) T r i c l i n i c , ( v i ) H e x a g - o n a l a n d ( v i i ) T r i g o n a l 5 4 2.3 V a r i a t i o n o f As-s w i t h a^/D f o r f o u r d i f f e r e n t |5| v a l u e s ( c l o c k w i s e f r o m t o p l e f t , \S\ = 2 , 6, 10, a n d 14); t e t r a g o n a l s y m m e t r y . O n t h e x a x i s i s p l o t t e d a±S2/D a n d o n t h e y a x i s l o g 1 0 As-s- H e r e w e h a v e t a k e n t h e e x t e r n a l field t o b e z e r o 5 5 2.4 V a r i a t i o n o f A s _ s w i t h Hx/D f o r aAS2/D = 0.25 f o r f o u r d i f f e r e n t |5| v a l - u e s ( c l o c k w i s e f r o m t o p l e f t , |5| = 2 , 5, 10, a n d 15); t e t r a g o n a l s y m m e t r y . O n t h e x a x i s i s p l o t t e d HX/DS2 a n d o n t h e y a x i s l o g 1 0 A ^ - s 5 5 2.5 V a r i a t i o n o f As-s w i t h a2/D f o r f o u r d i f f e r e n t \S\ v a l u e s ( c l o c k w i s e f r o m t o p l e f t , \S\ = 2 , 5, 10, a n d 15); o r t h o r h o m b i c s y m m e t r y . O n t h e x a x i s i s p l o t t e d a2/D a n d o n t h e y a x i s l o g 1 0 A s _ s . H e r e w e h a v e t a k e n t h e e x t e r n a l field t o b e z e r o 5 7 x i i 2.6 V a r i a t i o n o f As-s w i t h Hx/D f o r a2/D = 0.25 f o r f o u r d i f f e r e n t |5| v a l u e s ( c l o c k w i s e f r o m t o p l e f t , |5| = 2 , 5, 10, a n d 15); o r t h o r h o m b i c s y m m e t r y . O n t h e x a x i s i s p l o t t e d HX/DS2 a n d o n t h e y a x i s l o g 1 0 As,-s 5 7 2.7 V a r i a t i o n o f AS-s w i t h ae/D f o r f o u r d i f f e r e n t \S\ v a l u e s ( c l o c k w i s e f r o m t o p l e f t , |5| = 2 , 6, 10, a n d 14); h e x a g o n a l s y m m e t r y . O n t h e x a x i s i s p l o t t e d aeS4/D a n d o n t h e y a x i s l o g 1 0 As-s- H e r e w e h a v e t a k e n t h e e x t e r n a l field t o b e z e r o 5 8 2.8 V a r i a t i o n o f As,-s w i t h Hx/D f o r a6S4/D = 0.25 f o r f o u r d i f f e r e n t \S\ v a l u e s ( c l o c k w i s e f r o m t o p l e f t , \S\ —2, 5, 10, a n d 15); h e x a g o n a l s y m m e t r y . O n t h e x a x i s i s p l o t t e d HX/DS2 a n d o n t h e y a x i s l o g 1 0 As-s 5 8 2.9 C o m p a r i s o n o f p e r t u r b a t i o n t h e o r y , W K B r e s u l t s a n d i n s t a n t o n r e s u l t s t o t h e e x a c t s o l u t i o n f o r t h e t u n n e l i n g s p l i t t i n g b e t w e e n t h e t w o l o w e s t l e v e l s o f t h e H a m i l t o n i a n o f o r t h o r h o m b i c s y m m e t r y w i t h S = 1 0 . P l o t t e d o n t h e h o r i z o n t a l a x i s i s a2/D, a n d o n t h e v e r t i c a l a x i s l o g 1 0 As,-s- L e g e n d : B l a c k , e x a c t s o l u t i o n ; G r e e n , i n s t a n t o n s o l u t i o n ; R e d , p e r t u r b a t i o n t h e o r y a n d B l u e , W K B 6 1 2.10 C o m p a r i s o n o f p e r t u r b a t i o n t h e o r y a n d W K B r e s u l t s t o t h e e x a c t s o l u t i o n f o r t h e t u n n e l i n g s p l i t t i n g b e t w e e n t h e t w o l o w e s t l e v e l s o f t h e H a m i l t o - n i a n o f t e t r a g o n a l s y m m e t r y w i t h S = 1 0 . P l o t t e d o n t h e h o r i z o n t a l a x i s i s otiSP/D, a n d o n t h e v e r t i c a l a x i s l o g 1 0 A s _ s . L e g e n d : Y e l l o w , e x a c t s o l u t i o n ; R e d , p e r t u r b a t i o n t h e o r y a n d G r e e n , W K B 6 1 2.11 Z - p r o j e c t i o n o f s p i n v e r s u s e n e r g y f r o m HGS f o r t h e Fe8 s y s t e m . T h e r e g i o n o f v a l i d i t y o f t h e m a p p i n g t o a t w o - s t a t e s y s t e m i s t h e r e g i o n w h e r e e x c i t e d s t a t e s a r e f o r b i d d e n ( t h i s r e g i o n i n s h a d e d g r e y i n t h e a b o v e ) . . . 6 4 x i i i 2.12 T y p i c a l e v o l u t i o n o f t h e p r o j e c t i o n o f t h e e x c e s s s p i n S(t) a l o n g t h e e a s y - a x i s . W e s e e t w o r e g i m e s ; o n e w h e r e S e v o l v e s w i t h o u t t u n n e l i n g ( d i a g o n a l i n f ) , a n d o n e w h e r e S t u n n e l s f r o m | + S | — S > ( o f f - d i a g o n a l i n f ) . N o t e t h e s e p a r a t i o n o f s c a l e s ; t h e t i m e b e t w e e n t u n n e l i n g e v e n t s i s m u c h g r e a t e r t h a n t h e t u n n e l i n g t i m e 6 8 3.1 (Jl f o r a l l n u c l e i i n Fes- L a b e l i n g i s a s i n d i c a t e d i n t h e t e x t . T h e d o t s r e p r e s e n t v a l u e s f o r 2H ( l a b e l s 1..120), slBr ( l a b e l s 121..128), a n d l5N ( l a b e l s 129..146) 8 6 3.2 1H, e m p h a s i z i n g l o w e n d o f t h e s p e c t r u m 8 7 3.3 1H, h i g h e n d o f t h e s p e c t r u m 8 7 3.4 2H, l o w e n d o f s p e c t r u m 8 8 3.5 2H, e n t i r e s p e c t r u m 8 8 3.6 7 9 . B r , e n t i r e s p e c t r u m 8 9 3.7 81Br, e n t i r e s p e c t r u m 8 9 3.8 1 4 i V , e n t i r e s p e c t r u m 9 0 3.9 15N, e n t i r e s p e c t r u m 9 0 3.10 5 7Fe, e n t i r e s p e c t r u m 91 3.11 1 3 C , e n t i r e s p e c t r u m 91 3.12 17O, l o w e n d o f s p e c t r u m 9 2 3.13 1 7 0 , e n t i r e s p e c t r u m 9 2 3.14 T h e o r t h o g o n a l i t y b l o c k i n g p a r a m e t e r K o v e r a l a r g e r a n g e o f e x t e r n a l f i e l d s a p p l i e d i n t h e x d i r e c t i o n f o r t h e Fe™ax m a t e r i a l 9 3 3.15 T h e p a r a m e t e r K f o r s m a l l v a l u e s o f e x t e r n a l f i e l d a p p l i e d i n t h e x d i r e c t i o n f o r t h e FeTx m a t e r i a l . 9 4 x i v 3.16 I n t r i n s i c l i n e w i d t h W d u e t o p a r t i c u l a r i s o t o p e s a s a f u n c t i o n o f Hx f o r 1 0 0 % c o n c e n t r a t i o n s o f t h e s e i s o t o p e s . N o t e t h a t u\ a n d t h e r e f o r e W d r o p s s l o w l y w i t h f i e l d . T h i s e f f e c t c o m e s a b o u t b e c a u s e a s t h e e x t e r n a l f i e l d i s r a i s e d , t h e t w o m i n i m a o f t h e c e n t r a l s p i n c o m p l e x a r e f o r c e d c l o s e r t o g e t h e r ( n o l o n g e r a r e t h e y a n t i p a r a l l e l ) . T h e c u r v e s h o w n a s " t o t a l " i s t h e t o t a l r e s u l t f o r a m a t e r i a l c o n t a i n i n g 1 0 0 % o f t h e i s o t o p e s s h o w n . . . 9 7 3.17 I n t r i n s i c l i n e w i d t h W a s a f u n c t i o n o f Hx f o r Fe8*, Fe8D a n d 5 7 F e 8 . ... 9 8 3.18 B i n n e d t o p o l o g i c a l d e c o h e r e n c e p a r a m e t e r s \AkND\ f o r a l l n u c l e i , a s s u m i n g 1 0 0 % c o n c e n t r a t i o n s o f ( c l o c k w i s e f r o m b o t t o m l e f t ) 1H, 79Br, UN, 1 7 0 , 1 3 C a n d 57Fe, u s i n g t h e p o i n t d i p o l e a p p r o x i m a t i o n . T h e b i n w i d t h h e r e i s 0.0001; p l o t t e d o n t h e x a x i s i s a n d o n t h e y a x i s " n u m b e r o f n u c l e i " . N o t e t h a t t h e c o n t r i b u t i o n t o f r o m 57Fe i s a l m o s t e n t i r e l y f r o m t h e c o n t a c t i n t e r a c t i o n 9 9 3.19 H a r t r e e - F o c k r e s u l t s f o r t h e f r e e Fe3+ w a v e f u n c t i o n 1 01 3.20 C o m p a r i s o n o f p o i n t d i p o l e a n d H a r t r e e - F o c k m e t h o d s ; z e r o field a>J| v a l u e s i n Fe™ax. T h e H a r t r e e - F o c k r e s u l t s a r e s h o w n a s d o t s 1 02 3.21 1H, H a r t r e e - F o c k , e m p h a s i z i n g l o w e n d o f t h e s p e c t r u m 1 0 3 3.22 lH, H a r t r e e - F o c k , h i g h e n d o f t h e s p e c t r u m 1 0 3 3.23 2H, H a r t r e e - F o c k , l o w e n d o f s p e c t r u m 1 0 4 3.24 2H, H a r t r e e - F o c k , e n t i r e s p e c t r u m 1 0 4 3.25 7 9 . B r , H a r t r e e - F o c k , e n t i r e s p e c t r u m 1 0 5 3.26 81Br, H a r t r e e - F o c k , e n t i r e s p e c t r u m 1 0 5 3.27 UN, H a r t r e e - F o c k , e n t i r e s p e c t r u m 1 0 6 3.28 15N, H a r t r e e - F o c k , e n t i r e s p e c t r u m 1 0 6 3.29 57Fe, H a r t r e e - F o c k , e n t i r e s p e c t r u m 1 0 7 3.30 1 3 C , H a r t r e e - F o c k , e n t i r e s p e c t r u m 1 0 7 x v 3.31 1 7 0 , H a r t r e e - F o c k , l o w e n d o f s p e c t r u m 1 0 8 3.32 1 7 0 , H a r t r e e - F o c k , e n t i r e s p e c t r u m 1 0 8 3.33 T h e o r t h o g o n a l i t y b l o c k i n g p a r a m e t e r K a s a f u n c t i o n o f Hx i n t h e H a r t r e e - F o c k w a v e f u n c t i o n p i c t u r e 1 0 9 3.34 T h e o r t h o g o n a l i t y b l o c k i n g p a r a m e t e r K a s a f u n c t i o n o f Hx i n t h e H a r t r e e - F o c k w a v e f u n c t i o n p i c t u r e , f o c u s i n g o n s m a l l fields 1 0 9 3.35 I n t r i n s i c l i n e w i d t h W a s a f u n c t i o n o f Hx f o r F e 8 » , Fe$D a n d 5 7 F e 8 i n t h e H a r t r e e - F o c k p i c t u r e I l l 3.36 B i n n e d t o p o l o g i c a l d e c o h e r e n c e p a r a m e t e r s | A ^ ) £ ) | f o r a l l n u c l e i , a s s u m i n g 1 0 0 % c o n c e n t r a t i o n s o f ( c l o c k w i s e f r o m b o t t o m l e f t ) 1H, 79Br, 14N, 1 7 0 , 1 3 C a n d 5 7 F e , u s i n g t h e H a r t r e e F o c k a p p r o x i m a t i o n . T h e b i n w i d t h h e r e i s 0.0001; p l o t t e d o n t h e x a x i s i s l A ^ I a n d o n t h e y a x i s " n u m b e r o f n u c l e i " . N o t e t h a t t h e c o n t r i b u t i o n t o | A ^ ) £ ) | f r o m 5 7 F e i s a l m o s t e n t i r e l y f r o m t h e c o n t a c t i n t e r a c t i o n I l l 4.1 E n e r g y l e v e l s o f t h e L a n d a u - Z e n e r H a m i l t o n i a n . S h o w n a r e b o t h t h e e i g e n s t a t e s o f az, w h i c h a r e l i n e a r i n t i m e , a n d t h e e i g e n s t a t e s o f H(t), E±(t) = ±(A2 + vH2)1'2 1 2 3 4.2 T r a n s i t i o n p r o b a b i l i t y (4.10) a s a f u n c t i o n o f t ( i n u n i t s o f A ) . H e r e w e h a v e t a k e n A2/v — 0.63, 1 a n d 5 f o r t h e s o l i d , d o t t e d a n d d a s h e d l i n e s r e s p e c t i v e l y 1 2 6 4.3 T r a n s i t i o n p r o b a b i l i t i e s f o r t h e p u l s e p o t e n t i a l 1 3 5 4.4 T r a n s i t i o n p r o b a b i l i t i e s f o r t h e s i n u s o i d a l p o t e n t i a l 1 3 6 4.5 T r a n s i t i o n p r o b a b i l i t y P ^ f o r t h e p u l s e / r a m p s c e n a r i o . P l o t t e d i s ATT/U o n t h e x a x i s a n d An/u o n t h e y a x i s 1 4 0 x v i 4.6 T r a n s i t i o n p r o b a b i l i t y f o r t h e r a m p s c e n a r i o . P l o t t e d o n t h e x a x i s i s A/u> a n d o n t h e y a x i s ^ 141 5.1 T r a n s i t i o n a m p l i t u d e a s a t r a i n o f b l i p s 1 5 6 5.2 T r a n s i t i o n p r o b a b i l i t y n o r m a l i z e d t o t h e s t a n d a r d L a n d a u - Z e n e r t r a n s i - t i o n p r o b a b i l i t y P^/Pff p l o t t e d a g a i n s t 2 £ / a ; f o r 2A/u = 0.1. T h e t o p ( b o t t o m ) g r a p h i s f o r t h e s i n u s o i d a l ( s a w t o o t h ) p e r t u r b a t i o n 1 6 6 5.3 T r a n s i t i o n p r o b a b i l i t y n o r m a l i z e d t o t h e s t a n d a r d L a n d a u - Z e n e r t r a n s i - t i o n p r o b a b i l i t y P^/Pff p l o t t e d a g a i n s t 2£/ui f o r 2A/UJ = 10. T h e t o p ( b o t t o m ) g r a p h i s f o r t h e s i n u s o i d a l ( s a w t o o t h ) p e r t u r b a t i o n 1 6 7 5.4 T r a n s i t i o n p r o b a b i l i t y n o r m a l i z e d t o t h e s t a n d a r d L a n d a u - Z e n e r t r a n s i - t i o n p r o b a b i l i t y P^/Pff p l o t t e d a g a i n s t 2£/o> f o r 2A/u = 500. T h e t o p ( b o t t o m ) g r a p h i s f o r t h e s i n u s o i d a l ( s a w t o o t h ) p e r t u r b a t i o n 1 6 8 5.5 D e p i c t i o n o f t h e p r e c e s s i o n o f t h e kth n u c l e a r s p i n d u r i n g a " b l i p " . T h e c e n t r a l s p i n i s s h o w n i n b l a c k , w i t h a s c h e m a t i c n u c l e a r s p i n u n d e r n e a t h . T h i s n u c l e a r s p i n f e e l s a field 7 ^ f o r t i m e s t <ti, w h i c h w e c h o o s e t o b e t h e a x i s o f q u a n t i z a t i o n . A f t e r t h e c e n t r a l s p i n flips a t tx t h e n u c l e a r s p i n f e e l s a d i f f e r e n t field 7 ^ w h i c h c o n t a i n s i n g e n e r a l t r a n s v e r s e c o m p o n e n t s . T h i s c a u s e s a p r e c e s s i o n o f t h e n u c l e a r s p i n . A f t e r t h e c e n t r a l s p i n f l i p s b a c k , t h e n u c l e a r s p i n w i l l b e i n a s t a t e t h a t h a s l e s s t h a n f u l l o v e r l a p w i t h i t s o r i g i n a l s t a t e 1 7 8 5.6 pXil ( d a s h e d ) a n d W ( 0 / i ) ( s o l i d ) f o r p =x H 1 9 7 5.7 plfl ( d a s h e d ) a n d W ( ( L J ( s o l i d ) f o r p =79 Br 1 9 8 5.8 plfl ( d a s h e d ) a n d W ( 0 M ) ( s o l i d ) f o r p = 1 4 N 1 9 8 5.9 plfl ( d a s h e d ) a n d W ( 0 M ) ( s o l i d ) f o r p = 5 7 Fe 1 9 9 5.10 pllt ( d a s h e d ) a n d W ( 0 M ) ( s o l i d ) f o r p = 1 7 0 1 9 9 x v i i 5.11 pXil ( d a s h e d ) a n d W ( 0 M ) ( s o l i d ) f o r (i =n C 2 0 0 5.12 F u l l w i d t h W f o r Fes* ( d o t t e d ) , Fe8D ( d a s h e d ) a n d 57Fe8 ( s o l i d ) 2 0 0 5.13 A 3 ( l e f t ) a n d A 4 ( r i g h t ) f o r t h e t h r e e v a r i e t i e s F e 8 * ( d o t t e d ) , Fe8D ( d a s h e d ) a n d 5 7 F e 8 ( s o l i d ) 2 0 2 5.14 P r e s e n t e d h e r e a r e (1) t h e b a r e r e s u l t A 0 | cos<I>| ( t h e l o w e r c u r v e ) a n d (2) t h e l a r g e A r e s u l t w i t h n u c l e a r s p i n s A 0 | c o s $ | ( t h e m i d d l e c u r v e ) p l o t t e d i n u n i t s o f K e l v i n . N o t e t h e l o g a r i t h m i c v e r t i c a l s c a l e . T h e h o r i z o n t a l a x i s i s Hx i n T e s l a - h e r e w e h a v e ip = 0 (Hy = 0) 2 0 4 6.1 T o p figure: Logw A 2 v s . H°, f o r 6 = s i n " 1 (H%/H$) = 0. C u r v e s a r e , f r o m b o t t o m t o t o p , WD — 0,10, 2 0 , 3 0 a n d 5 0 mT. B o t t o m figure: S a m e , b u t w i t h 6 = 1°. N o t e t h a t t h e s e r e s u l t s a r e o b t a i n e d f r o m o u r r e l a x a t i o n r a t e w h i c h w a s d e r i v e d a s s u m i n g t h e i n s t a n t o n a p p r o x i m a t i o n f o r t h e t u n n e l i n g a m p l i t u d e 2 1 5 6.2 F r o m [199]. H e r e t h e t u n n e l i n g m a t r i x e l e m e n t a n d i t s d e p e n d e n c e o n ex- t e r n a l field w e r e e x t r a c t e d b y e x a c t d i a g o n a l i z a t i o n , w i t h t h e n o d a l p h y s i c s d e t e r m i n e d a s d i s c u s s e d i n t h e t e x t . N o t e t h a t w h i l e t h e g e n e r a l f e a t u r e s o f t h e e x a c t r e s u l t m a t c h t h o s e o b t a i n e d b y t h e i n s t a n t o n c a l c u l a t i o n t h e y d i f f e r i n d e t a i l 2 1 6 B . l H e r e w e s e e a s y s t e m w i t h s e v e n e n v i r o n m e n t a l s p i n s t a t e s i n i t i a l l y p r e - p a r e d i n o n e o f t h e m e v o l v i n g v i a ( B . 3 ) 2 3 0 x v i i i Acknowledgements T h e r e a r e m a n y p e o p l e w h o h a v e p r o v i d e d s u p p o r t d u r i n g t h e c o m p l e t i o n o f t h i s w o r k . F i r s t l y I w o u l d l i k e t o t h a n k P h i l i p S t a m p , b o t h f o r i n t r o d u c i n g m e t o t h e c o m p l e x a n d i m p o r t a n t w o r l d o f t h e n u c l e a r s p i n b a t h a n d f o r a l l o w i n g m e s i g n i f i c a n t l e e w a y i n m y a p p r o a c h t o t h e d i f f i c u l t p r o b l e m s t r e a t e d i n t h i s d o c u m e n t . T h e e d u c a t i o n I r e c e i v e d f r o m h i m i n t h e a r e a o f q u a n t i t a t i v e t r e a t m e n t s o f d e c o h e r e n c e w i l l b e a b s o l u t e l y i n v a l u a b l e t o t h e s u c c e s s o f m y p o s t - d o c t o r a l p u r s u i t s . I h o p e t h a t w e w i l l b e w o r k i n g t o g e t h e r f o r m a n y y e a r s t o c o m e . I w o u l d a l s o l i k e t o t h a n k t h e o t h e r g r e a t t e a c h e r s I h a v e h a d t h e o p p o r t u n i t y t o l e a r n f r o m i n m y t i m e a t U B C , a n d i n p a r t i c u l a r I a n A f f l e c k , w h o h a s i n f l u e n c e d m y u n d e r s t a n d i n g o f s e v e r a l a s p e c t s o f c o n d e n s e d m a t t e r p h y s i c s , a n d H a i g F a r r i s , w h o h a s b e c o m e a g o o d f r i e n d , m e n t o r a n d b u s i n e s s p a r t n e r o v e r t h e p a s t y e a r . A s w e l l I w o u l d l i k e t o t h a n k c o l l e a g u e s a n d c o l l a b o r a t o r s M e h r d a d S h a r i f z a d e h - A m i n , T i m D u t y , S e b a s t i a n J a i m u n g a l , A l e x a n d r e Z a g o s k i n , J o n a t h a n O p p e n h e i m , S u r e s h P i l - l a i , M i c h e l O l i v i e r , M a r t i n D u b e , S t e p h a n i e C u r n o e , I g o r T u p i t s y n , N i k o l a i P r o k o f i e v a n d J e f f S o n i e r . A l s o I w o u l d l i k e t o t h a n k W o l f g a n g W e r n s d o r f e r f o r h i s a m a z i n g F e 8 d a t a . T h e c o m p l e t i o n o f t h i s d o c u m e n t w o u l d n o t h a v e b e e n p o s s i b l e w i t h o u t t h e s u p p o r t o f m y p a r e n t s D r . G e o r g e a n d ( s o o n t o b e D r . ) S a r a h R o s e , w h o I w o u l d l i k e t o d e d i c a t e t h i s t h e s i s t o - I finally d i d i t ! I w o u l d l i k e t o t h a n k C r a i g T h o m a s f o r t r y i n g t o k e e p u p w i t h m e o n t h e b e a c h v o l l e y b a l l c i r c u i t , R A N D y f o r h o u r s o f r e v o l t i n g a m u s e m e n t , a n d S h a w n K e n n e y , D o n a n d K a m a r a L u c a s a n d C a m i l l e P a r e n t f o r p u t t i n g u p w i t h me. x i x F i n a l l y I w o u l d l i k e t o t h a n k m y w i f e V a l e r i e , f o r b e l i e v i n g t h a t s o m e d a y t h e r e - w r i t e I w a s d o i n g w o u l d b e t h e l a s t o n e . x x Chapter 1 Introduction and Overview Component sizes in commercially available semiconductor structures have been halving rather steadily every eighteen months or so since the early 1950s. This decrease in size, know as Moore's Law (named for Gordon Moore, one of the founders of Intel), is tracked by concommitant halvings in price and energy consumption and doublings of processor speed (see figures 1.1, 1.2 and 1.3) [1, 2, 3]. This continued shrinkage is producing much excitement and consternation in the high-tech world, for a very simple reason-it is clearly not sustainable. Naively one could say that this is because device sizes are limited to be larger than atomic length scales, Figure 1.1: The number of atoms needed to represent one bit of information as a function of calendar year. Extrapolation of the trend suggests that the one atom per bit level is reached in about the year 2020. Adapted from [1]. 1 Chapter 1. Introduction and Overview 2 which are on the order of Angstroms-at current shrinkage rates this barrier will be met in approximately 20 years. However it is not yet clear that the limitation on computing speed cannot be overcome in the near-term (10-30 years) with more efficient computer architecture-for example, stacking transistors horizontally [4], or even more exotic solu- tions such as the recent I B M SMASH proposal [5]. Figure 1.2: Clock speed (Hz) vs. calendar year. Adapted from [1]. Physicists have long been thinking about what will happen when component sizes become mesoscopic; that is, much larger than atomic length scales, but small enough so that at least in part they must be treated quantum mechanically [6, 7, 8, 9]. This line of thought has produced many extremely startling predictions. It is now well known that standard models of computation, based on the universal computing model or Turing model [10], contain an implicit assumption. This assumption is that the physical system which encodes and manipulates information evolves according to the classical laws of physics. This assumption can break down when components become "small enough". For example, quantum effects in mesoscopic normal metal rings [11], superconducting structures [12, 13] and in molecular magnets [14, 15, 16, 17] have been observed. Chapter 1. Introduction and Overview 3 If one rewrites computer science with quantum mechanics implicitly included from the outset it turns out that the range of tasks that computing machines can perform is increased. The most famous example of this is the solution of the factoring problem [1, 19] using a "quantum computer" (which is a theoretical machine which has the capa- bility of storing and manipulating coherent two level systems (quantum bits, or qubits)) in polynomial time, as opposed to superpolynomial time with classical computers. In addition to this, it is quite obvious that a quantum computer could in an analog fash- ion solve many important quantum mechanical problems, some of which are exceedingly important (such as pharmaceutical design), which are completely unsolvable using even the fastest imaginable classical supercomputers. Year Figure 1.3: Energy (pico-Joules) dissipated per logical operation as a function of calendar year. The 1 kT level is indicated by a dashed line. Adapted from [l]. These theoretical musings are now coming face to face with some very real physical problems, as engineers undertake to build quantum devices. If one wants to construct a quantum computer, there are several aspects of mesoscale condensed matter physics that must be understood wholly and completely. The most important of these is the Chapter 1. Introduction and Overview 4 p r o c e s s k n o w n a s decoherence, w h i c h i n v o l v e s t h e t r a n s f e r r a l o f p h a s e i n f o r m a t i o n f r o m a q u a n t u m b i t i n t o a n environment ( s u c h a s n u c l e a r s p i n s [20] o r p h o n o n s [21, 22]). D e c o h e r e n c e i s a n a t h e m a t o q u a n t u m c o m p u t a t i o n ( a n d i n t e r e s t i n g a l s o f r o m a p u r e l y t h e o r e t i c a l p e r s p e c t i v e ) , a n d y e t h o w i t w o r k s i n p r a c t i c e , q u a n t i t a t i v e l y , i s s t i l l n o t s a t i s f a c t o r i l y u n d e r s t o o d . T h e r e e x i s t s a c l a s s o f m e s o s c o p i c s y s t e m s w h e r e a n a t t e m p t c a n b e m a d e a t a q u a n - t i t a t i v e t h e o r y o f d e c o h e r e n c e . T h e s e a r e t h e s o - c a l l e d " m o l e c u l a r m a g n e t s " . O n e o f t h e s e m a t e r i a l s , w h i c h w e w i l l r e f e r t o t h r o u g h o u t a s " i r o n - e i g h t " (Fes), i s p a r t i c u l a r l y w e l l s u i t e d t o a q u a n t i t a t i v e s t u d y o f d e c o h e r e n c e d u e t o l o c a l i z e d e n v i r o n m e n t a l m o d e s , a n d i n p a r t i c u l a r n u c l e a r s p i n s . T h e d e v e l o p m e n t o f t h i s q u a n t i t a t i v e t h e o r y a s a t o o l t o b e u s e d i n f u t u r e i n v e s t i g a t i o n s o f m e s o s c a l e s y s t e m s i n t h e c o n t e x t o f d e v e l o p i n g s o l i d s t a t e q u a n t u m b i t s p r o v i d e s t h e m a i n m o t i v a t i o n f o r t h e w o r k p r e s e n t e d i n t h i s t h e s i s . 1.1 A n I n t r o d u c t i o n t o F e 8 Fes w a s first s y n t h e s i z e d i n 1 9 8 4 b y W i e g h a r d t e t . a l . [23]. T h i s m a t e r i a l , w i t h t h e r a t h e r i m p o s i n g c h e m i c a l f o r m u l a { [ ( t o c n ) 6 F e 8 ( / i 3 - 0 ) 2 ( /*2 - OH)12]Br7 • H20}®[Br • 8H20f (1.1) w h e r e tacn = 1,4,7 t r i a z a c y c l o n a n e , w a s t h e first o l i g o m e r w i t h g r e a t e r t h a n t h r e e F e 3 + i o n s p e r u n i t c e l l e v e r c h a r a c t e r i z e d . X - r a y c r y s t a l l o g r a p h y s t u d i e s p e r f o r m e d o n t h i s m a t e r i a l i n d i c a t e d t h a t s i x o f t h e Fe3+ i o n s w e r e b o n d e d t o a m i n e l i g a n d s FeNsOs a n d t h e r e m a i n i n g t w o w e r e s u r r o u n d e d b y a d i s t o r t e d o c t a h e d r a l a r r a y o f 6 O a t o m s . T h e i r o n i o n s a r e c o u p l e d v i a 12 \x2— h y d r o x o b r i d g e s a n d t w o / x 3 — o x o b r i d g e s . T h r e e v i e w s o f t h e u n i t c e l l a r e p r e s e n t e d h e r e , i n figures 1.4, 1.5 a n d 1.6. T h e p o s i t i o n s o f t h e i o n s s h o w n h e r e w e r e o b t a i n e d f r o m t h e C a m b r i d g e C r y s t a l l o g r a p h i c D a t a b a s e [24], w h i c h c o n t a i n s t h e o r i g i n a l X - r a y d a t a o b t a i n e d b y W e i g h a r d t e t . a l . [23]. Chapter 1. Introduction and Overview 5 1 0 8 - 6 4 - 2 0 - 2 - - 6 - + + + + + + + + + + + + + + w + + + + + + + 1 1 1 i 1 1 1 1 i 1 1 1 ' i 1 • 1 1 i ' • • 1 i 1 ' 1 1 i > 1 • 1 i ' > 0 2 4 6 8 1 0 1 2 Figure 1.4: A 2-D projection view of the Fe$ unit cell onto the y — z plane. Distances shown are in Angstroms. Legend: Red, iron; Purple, bromine; Light Blue, oxygen; Green, nitrogen; Yellow, carbon; and Dark Blue Crosses, hydrogen. Note the central magnetic core, surrounded by a shield of organic species. Chapter 1. Introduction and Overview 6 F i g u r e 1.5: A v i e w o f t h e Fe8 u n i t c e l l i n t h e x — z p l a n e . Chapter 1. Introduction and Overview 7 10- + s- G- •1- 4- 2- + • # + •» • + t + + w + * m * 0- • + + -2- • + # ** * + + + + , + <* * „ * + • • -6- • + + + + + -G -4 -2 0 2 4 6 S F i g u r e 1.6: A v i e w o f t h e F e 8 u n i t c e l l i n t h e x — y p l a n e . H e r e w e a r e l o o k i n g r i g h t d o w n t h e " e a s y a x i s " o f t h e m o l e c u l e (see c h a p t e r 2 ) . Chapter 1. Introduction and Overview 8 T h e a m i n e g r o u p s a r e c y c l i c a n d h y d r o p h o b i c . T h e Br a t o m s a r e b o n d e d e l e c t r o - s t a t i c a l l y t o NHe a n d OHe. T h e s p a c e g r o u p o f t h e m a t e r i a l i s P I ( t r i c l i n i c ) , w i t h l a t t i c e p a r a m e t e r s a = 10.522, b = 14.05 a n d c = 15.00 A n g s t r o m s w i t h u n i t c e l l a n g l e s a - 89.90, (3 = 109.65 a n d 7 = 109.27 [23]. T h e m o l e c u l a r w e i g h t o f t h e s u b s t a n c e p e r u n i t c e l l i s 2 2 5 0 . T h e l a t t i c e s t r u c t u r e i s o f t h e AB t y p e , w i t h t h e c a t i o n a n d a n i o n i n (1.1) o c c u p y i n g t h e A a n d B s i t e s r e s p e c t i v e l y ( t h i s i s s i m p l y a d i s t o r t e d N a C l s t r u c t u r e ) . 1.1.1 Giant Spins and Quantum Environments E x p e r i m e n t a l i n t e r e s t i n t h e l o w t e m p e r a t u r e magnetic c h a r a c t e r i z a t i o n o f F e 8 s u r p r i s - i n g l y d i d n o t a r i s e u n t i l m u c h l a t e r [18]. H o w e v e r i n t h e e x p e r i m e n t a l l u l l b e t w e e n 1 9 8 4 a n d 1 9 9 3 m u c h t h e o r e t i c a l w o r k w a s b e i n g d o n e t h a t w o u l d l a y t h e g r o u n d w o r k f o r u n - d e r s t a n d i n g t h e l o w - e n e r g y m a g n e t i c n a t u r e o f t h i s s u b s t a n c e . T h e r e a r e t w o b a s i c t h e m e s t h a t n e e d e d t o b e d e v e l o p e d t o u n d e r s t a n d t h e e x p e r i - m e n t a l r e s u l t s t h a t w o u l d l a t e r a p p e a r . T h e f i r s t o f t h e s e i s t h e u n d e r s t a n d i n g o f t h e d y n a m i c s o f " g i a n t s p i n s " , t h a t i s s y s t e m s t h a t h a v e l a r g e s p i n q u a n t u m n u m b e r . O n e o f t h e m o s t o b v i o u s s y s t e m s t h a t c a n b e t h o u g h t o f a s a g i a n t s p i n i s a s i n g l e i s o l a t e d f e r r o - m a g n e t i c g r a i n , w e l l b e l o w i t s C u r i e t e m p e r a t u r e . A l l t h e e l e c t r o n i c s p i n s l o c k t o g e t h e r , a n d o n e c o u l d t h i n k o f t h i s o b j e c t a s a s i n g l e d e g r e e o f f r e e d o m , a l b e i t w i t h s p i n q u a n t u m n u m b e r a s l a r g e a s S = 1 0 8 . T h e i n t e r e s t i n g t h i n g a b o u t t h e s e s y s t e m s , a n d o n e o f t h e m a i n m o t i v a t i o n s f o r t h e i r e a r l y s t u d y , i s t h a t h e r e w e h a v e a n a d j u s t a b l e p a r a m e t e r ( 5 ) w h i c h a s i t i s i n c r e a s e d s h o u l d c a u s e t h e s y s t e m t o g o f r o m b e i n g q u a n t u m m e c h a n i c a l ( s a y f o r S — 1/2) t o b e i n g c l a s s i c a l ( s a y S = 1 0 8 ) i n a w a y t h a t w e c a n s t u d y w i t h s o m e i n t i m a c y . T w o p a p e r s t h a t w o u l d p r o v e t o b e i m p o r t a n t i n t h i s r e s p e c t w e r e t h o s e o f V a n H e m m e n a n d S u t o [25] a n d E n z a n d S h i l l i n g [26], b o t h i n 1 9 8 6 . T h e s e d i s c u s s e d t h e d y n a m i c s o f g i a n t s p i n s i n t h e W K B a p p r o x i m a t i o n . I n p a r t i c u l a r , t h e p r o b l e m o f h o w Chapter 1. Introduction and Overview 9 l a r g e s p i n o b j e c t s t u n n e l b e t w e e n e n e r g e t i c m i n i m a w a s t r e a t e d . T h e s e c o n d b a s i c t h e m e t h a t n e e d e d t o b e d e v e l o p e d w a s a n u n d e r s t a n d i n g o f t h e e f f e c t s o f " e n v i r o n m e n t s " o n t h e d y n a m i c s o f t h e d e g r e e s o f f r e e d o m t o w h i c h t h e y c o u p l e . I t w a s s h o w n b y F e y n m a n a n d V e r n o n [21] t h a t i f t h e r e e x i s t e n v i r o n m e n t a l c o u p l i n g s t h a t a r e w e a k , i n t h e s e n s e t h a t t h e i r e f f e c t c a n b e t r e a t e d i n s e c o n d - o r d e r p e r t u r b a t i o n t h e o r y , o n e c a n m o d e l t h e i r e f f e c t s b y c o u p l i n g t h e i n t e r e s t i n g d e g r e e o f f r e e d o m t o a n o s c i l l a t o r b a t h . I n a s o l i d a t l o w e n e r g i e s , i t i s u s u a l l y t h e c a s e t h a t d e l o c a l i z e d m o d e s , s u c h a s p h o n o n s a n d p h o t o n s , c a n b e m a p p e d t o o s c i l l a t o r b a t h s . T h i s i s b e c a u s e l o w e n e r g y m o d e s h a v e l o n g w a v e l e n g t h s a n d t h e r e f o r e t h e i r o v e r l a p w i t h t h e l o c a l i z e d i n t e r e s t i n g d e g r e e o f f r e e d o m i s s m a l l . V a r i o u s a s p e c t s o f t h e d y n a m i c s o f a c e n t r a l d e g r e e o f f r e e d o m c o u p l e d t o o s c i l l a t o r b a t h s h a v e b e e n i n v e s t i g a t e d [22]. P e r h a p s t h e m o s t f a m o u s o f t h e s e i s t h e t r e a t m e n t b y L e g g e t t e t . a l . o f t h e s o - c a l l e d s p i n - b o s o n p r o b l e m [27]. I n t h e s p i n - b o s o n p r o b l e m , t h e c e n t r a l d e g r e e o f f r e e d o m i s a t w o - s t a t e s y s t e m ( a s p i n ) . T h i s i s t h e n c o u p l e d t o a n o s c i l l a t o r b a t h a n d t h e d y n a m i c s o f t h e c e n t r a l s p i n a r e t h e n e x t r a c t e d . I t i s p o s s i b l e t o o b t a i n a n a l y t i c r e s u l t s f o r c e r t a i n c h o i c e s o f p a r a m e t e r s i n t h e s p i n - b o s o n H a m i l t o n i a n , b u t i n g e n e r a l d y n a m i c a l s o l u t i o n s a r e d i f f i c u l t t o o b t a i n . 1.1.2 A Related System is Characterized I n 1 9 9 1 a s u b s t a n c e t h a t w o u l d t u r n o u t t o b e r e l a t e d t o F e 8 u n d e r w e n t l o w t e m p e r a t u r e m a g n e t i c c h a r a c t e r i z a t i o n [28]. T h i s s u b s t a n c e i s u s u a l l y c a l l e d Mni2-acetate, o r s i m p l y Mni2, a n d h a s c h e m i c a l f o r m u l a [Mn12{CH3COO)16(H20)4012] • 2CH3COOH • 4H20 (1.2) C r y s t a l s o f t h i s s u b s t a n c e w e r e k n o w n [29, 30] t o h a v e t e t r a g o n a l s y m m e t r y o f s p a t i a l g r o u p 14 w i t h u n i t c e l l p a r a m e t e r s a = 17.3 A n g s t r o m s a n d b = 12.39 A n g s t r o m s . T h e Chapter 1. Introduction and Overview 10 t o t a l m o l e c u l a r w e i g h t p e r u n i t c e l l i s 2 0 6 0 . X - r a y c r y s t a l l o g r a p h y w a s p e r f o r m e d [30] o n t h i s s u b s t a n c e , a n d t h e a t o m i c p o s i t i o n s s t o r e d i n t h e C a m b r i d g e C r y s t a l l o g r a p h i c D a t a b a s e . T h i s d a t a i s p r e s e n t e d i n figures 1.7, 1.8 a n d 1.9. W e s e e t h a t t h e r e a r e t w o " r i n g s " o f m a n g a n e s e a t o m s . T h e i n n e r r i n g c o n s i s t s o f f o u r Mn4+ i o n s w i t h s p i n S = 3/2, w h i l e t h e o u t e r r i n g c o n s i s t s o f e i g h t Mn3+ i o n s w i t h s p i n 5 = 2. I t w a s p r o p o s e d t h a t a l l t h e s e i o n s c o u p l e t o e a c h o t h e r p r i m a r i l y v i a s u p e r e x c h a n g e t h r o u g h m e d i a t i n g o x o - b r i d g e s [30]. F i g u r e 1.7: P r o j e c t i o n o f t h e Mn-u u n i t c e l l o n t o t h e x — y p l a n e . H e r e w e a r e l o o k i n g d o w n t h e e a s y a x i s o f t h e c r y s t a l . T h e a x e s s c a l e s a r e i n A n g s t r o m s . L e g e n d : R e d , m a n g a n e s e ; P u r p l e , o x y g e n ; Y e l l o w , c a r b o n ; D a r k B l u e C r o s s e s , h y d r o g e n . N o t e t h e i n n e r a n d o u t e r " r i n g s " o f m a n g a n e s e i o n s . Chapter 1. Introduction and Overview F i g u r e 1.8: P r o j e c t i o n o f t h e M n 1 2 u n i t c e l l o n t o t h e x — z p l a n e . Chapter 1. Introduction and Overview F i g u r e 1.9: P r o j e c t i o n o f t h e M n 1 2 u n i t c e l l o n t o t h e y — z p l a n e . Chapter 1. Introduction and Overview 13 T h e 1 9 9 1 p a p e r o f C a n e s c h i e t . a l . d e s c r i b e d r e s u l t s o f A C s u s c e p t i b i l i t y , h i g h f i e l d m a g n e t i z a t i o n a n d E P R m e a s u r e m e n t s o n c r y s t a l s o f M n 1 2 w h i c h i n d i c a t e d t h a t e a c h M n 1 2 m o l e c u l e h a d a n S = 10 g r o u n d s t a t e . A m e c h a n i s m w a s d e s c r i b e d w h e r e b y t h e t w e l v e m a n g a n e s e i o n s i n e a c h u n i t c e l l l o c k t o g e t h e r v i a s u p e r e x c h a n g e a t l o w t e m p e r - a t u r e s i n t o a g i a n t s p i n . I n t h i s m a t e r i a l a n e a s y a x i s w a s o b s e r v e d , a n d w a s e x p l a i n e d a s b e i n g d u e t o c r y s t a l a n i s o t r o p y a n d / o r s p i n - o r b i t c o u p l i n g e f f e c t s . T h e s e r e s u l t s p r o - d u c e d m u c h e x c i t e m e n t , a s h e r e we h a v e w h a t s e e m s t o b e ( i n z e r o e x t e r n a l m a g n e t i c field a n d i n a l o w - e n e r g y l i m i t ) a t w o s t a t e s y s t e m (15 = + 1 0 > a n d \S — — 1 0 >, c o r r e - s p o n d i n g i n g i n a s e m i - c l a s s i c a l p i c t u r e t o t h e g i a n t s p i n p o i n t i n g p a r a l l e l / a n t i p a r a l l e l t o t h e e a s y a x i s r e s p e c t i v e l y ) w h o s e d y n a m i c s s h o u l d d e m o n s t r a t e q u a n t u m e f f e c t s o f s o m e k i n d ( a s S i s i n t h e f u z z y m e s o s c a l e r e g i o n ) . A n a n a l y s i s o f w h a t w a s k n o w n o f Mnu a n d s o m e o t h e r m o l e c u l a r m a g n e t s w a s p e r f o r m e d i n 1 9 9 3 b y S e s s o l i e t . a l . [31] a n d p r o v i d e s a n e x c e l l e n t r e v i e w . 1.1.3 Magnetic Characterization of Fes I n 1 9 9 3 a c o m p r e h e n s i v e c h a r a c t e r i z a t i o n o f t h e m a g n e t i c p r o p e r t i e s o f Fe% a t l o w t e m - p e r a t u r e s w a s p e r f o r m e d b y D e l f s e t . a l . [18]. I n t h i s s t u d y , A C a n d D C s u s c e p t i b i l i t y , m a g n e t i z a t i o n a s a f u n c t i o n o f e x t e r n a l field a n d E P R s t u d i e s w e r e p e r f o r m e d w h i c h i n d i c a t e d t h a t Fe% h a d , l i k e M n 1 2 , a s p i n 10 g r o u n d s t a t e . T h i s p a p e r d e s c r i b e d i t s r e s u l t s i n t e r m s o f a u n i t c e l l c o n t a i n i n g e i g h t S = 5/2 Fe3+ i o n s w h i c h c o u p l e t o e a c h o t h e r v i a e x c h a n g e a n d s u p e r e x c h a n g e v i a o x y g e n a n d h y d r o x o b r i d g e s . A t t e m p e r a t u r e s l e s s t h a n T ~ 20 K t h e s e l o c k t o g e t h e r i n t o a s p i n c o m p l e x w i t h S = 10. B e c a u s e o f s p i n - o r b i t c o u p l i n g s a n d c r y s t a l field a n i s o t r o p i e s t h e s p e c t r u m o f t h e s p i n 10 r o t a t o r i s s p l i t i n t o 10 d o u b l e t s a n d o n e s i n g l e t , w i t h ms = ± 1 0 b e i n g n e a r l y d e g e n e r a t e l o w e s t e n e r g y s t a t e s c o r r e s p o n d i n g i n a s e m i - c l a s s i c a l p i c t u r e t o t h e c e n t r a l s p i n o b j e c t p o i n t i n g " u p " o r " d o w n " a l o n g t h e e a s y a x i s s e l e c t e d b y t h e Chapter 1. Introduction and Overview 14 a n i s o t r o p y i n t h e s p i n H a m i l t o n i a n ( a r o u g h first a p p r o x i m a t i o n t o t h i s s p i n H a m i l t o n i a n i s g i v e n b y H = —DS2, w h e r e D i s a m e a s u r e o f t h e s t r e n g t h o f t h e v a r i o u s a n i s o t r o p i e s i n t h e c r y s t a l [32, 33, 3 4 ] - s e e c h a p t e r 2 f o r a c o m p r e h e n s i v e t r e a t m e n t o f t h e l o w e n e r g y e f f e c t i v e H a m i l t o n i a n f o r t h i s s y s t e m ) . T h e p a r a m e t e r D w a s r e p o r t e d t o b e o n t h e o r d e r o f D~0.3K [18]. 1.1.4 Results in the Quantum Regime T h e l o w t e m p e r a t u r e m a g n e t i c c h a r a c t e r i z a t i o n s o f Fes a n d Mnyi l e d t o a f l o o d o f i m - p o r t a n t r e s u l t s , b o t h f r o m t h e o r i s t s a n d f r o m t h e e x p e r i m e n t a l c o m m u n i t y . T h e o r i s t s w e r e p r e s e n t e d w i t h a t r u l y m e s o s c o p i c p r o b l e m w i t h a g r o w i n g n u m b e r o f e x p e r i m e n t a l r e s u l t s , a n d e x p e r i m e n t a l i s t s h a d a c c e s s t o s y s t e m s w h e r e t h e y c o u l d d i r e c t l y m e a s u r e m a c r o s c o p i c q u a n t u m e f f e c t s . T h e s t u d y o f Mriu p r o d u c e d m u c h w o r k o n t h e m a c r o - s c o p i c q u a n t u m t u n n e l i n g o f t h e c e n t r a l s p i n o f e a c h m o l e c u l e [35, 36, 37, 38]. A r g u a b l y t h e m o s t i m p o r t a n t e a r l y e x p e r i m e n t a l r e s u l t w a s t h a t o f T h o m a s e t . a l . [17] i n 1996, w h i c h p r e s e n t e d c l e a r - c u t e v i d e n c e f o r i n c o h e r e n t m a c r o s c o p i c t u n n e l i n g o f t h e m a g n e - t i z a t i o n i n M n i 2 (see figure 1.10). E v e n m o r e a s t o n i s h i n g w e r e r e s u l t s o b t a i n e d i n 1 9 9 7 o n t h e Fes s y s t e m w h i c h c l e a r l y d e m o n s t r a t e s t h e e x i s t e n c e o f a r e g i m e w h e r e m a g n e t i c r e l a x a t i o n r a t e s b e c o m e t e m p e r a t u r e i n d e p e n d e n t - t h e s o - c a l l e d q u a n t u m r e g i m e (see fig- u r e 1.11) [14]. T h e a u t h o r s a t t r i b u t e r e l a x a t i o n b e l o w T ~ 3 6 0 mK t o b e d u e t o p u r e l y q u a n t u m m e c h a n i c a l t u n n e l i n g b e t w e e n g r o u n d s t a t e s o f t h e Fes m o l e c u l e s i n t h e c r y s t a l s m e a s u r e d . R e s u l t s o f m a g n e t i c r e l a x a t i o n m e a s u r e m e n t s o n t h e s e s y s t e m s f o r t h e m o s t p a r t w a s a m e n a b l e t o a n a l y s i s w i t h i n e x i s t i n g t h e o r e t i c a l f r a m e w o r k s [40-48]. I n p a r t i c u l a r a f r a m e w o r k h a d b e e n c o n s t r u c t e d b y P r o k o f i e v a n d S t a m p [41, 45, 46, 47] t o e x p l a i n t h e p h y s i c s o f r e l a x a t i o n i n m o l e c u l a r m a g n e t s . I n t h e m o d e l s o f P r o k o f i e v a n d S t a m p , e a c h m o l e c u l a r m a g n e t i s t r e a t e d a s a g i a n t s p i n w h i c h c o u p l e s t o v a r i o u s e n v i r o n m e n t s , Chapter 1. Introduction and Overview 15 Figure 1.11: This data shows the log of the relaxation time vs. 1/T in Fe 8 . At high temperatures thermal activation is observed, while for T <~ 360 rnK relaxation becomes temperature independent. Figure obtained from [14]. Chapter 1. Introduction and Overview 16 t h e m o s t i m p o r t a n t o f w h i c h i t i s a r g u e d a r e l o c a l i z e d m o d e s , s u c h a s n u c l e a r s p i n s a n d m a g n e t i c i m p u r i t i e s . N o t e t h a t l o c a l i z e d m o d e s s u c h a s t h e s e c a n n o t i n g e n e r a l b e m a p p e d t o o s c i l l a t o r b a t h s a s t h e c o u p l i n g s t r e n g t h s a r e n o t i n g e n e r a l s m a l l ( f o r e x a m p l e , c o n t a c t h y p e r f i n e c o u p l i n g s i n r a r e e a r t h s c a n b e a s l a r g e a s 1 K [48]). T h e y s h o w t h a t t h e r e l a x a t i o n c h a r a c t e r i s t i c s o f g i a n t s p i n s i n c o n d e n s e d m a t t e r s y s t e m s s h o u l d b e s t r o n g l y i n f l u e n c e d b y t h e s e l o c a l i z e d m o d e s , a n d h a r d l y i n f l u e n c e d a t a l l b y o s c i l l a t o r b a t h s ( s u c h a s p h o n o n s ) . I n t h e q u a n t u m r e g i m e d e m o n s t r a t e d t o e x i s t i n F e 8 , t h e t e m p e r a t u r e i n d e p e n d e n c e i s e v i d e n c e t h a t t h e p h o n o n b a t h i s n o t p l a y i n g a s i g n i f i c a n t r o l e i n r e l a x a t i o n . B u t h e r e w a s a c u r i o u s t h i n g . T h e f a c t t h a t t h e Fe8 c r y s t a l s w e r e r e l a x i n g a t a l l i n t h e q u a n t u m r e g i m e w a s q u i t e s t r a n g e , f o r t h e f o l l o w i n g r e a s o n . T h e b a r e t u n n e l i n g a m p l i t u d e b e t w e e n t h e g r o u n d s t a t e s o f t h e F e 8 m o l e c u l e s w a s e s t i m a t e d ( b y e x a c t l y d i a g o n a l i z i n g p h e n o m e n o l o g i c a l s p i n H a m i l t o n i a n s w h o s e p a r a m e t e r s w e r e e x t r a c t e d f r o m v a r i o u s e x p e r i m e n t s ) t o b e A ~ 1 0 - 8 K [45]; a n d y e t t h e s c a l e o f t h e d i p o l a r i n t e r a c t i o n b e t w e e n d i f f e r e n t m o l e c u l e s c a n q u i t e e a s i l y b e e s t i m a t e d t o b e o f t h e o r d e r o f ~ 0.5 K. T h i s m e a n s t h a t i n a c r y s t a l o f m o l e c u l e s , o n l y a n e x t r e m e l y t i n y f r a c t i o n o f m o l e c u l e s c o u l d e v e r b e i n r e s o n a n c e , a n d t h e r e f o r e t h e i r d y n a m i c s s h o u l d b e f r o z e n . A r e s o l u t i o n o f t h i s d i f f i c u l t y w a s p r o p o s e d i n 1 9 9 7 b y P r o k o f i e v a n d S t a m p , a n d t h e p r o p o s e d m e c h a n i s m i n v o l v e d t h e n u c l e a r s p i n s p r e s e n t i n t h e Fe8 c r y s t a l [15]. A t t e m p e r a t u r e s l o w e r t h a n ~ 3 6 0 mK, t h e r e e x i s t s o n l y o n e s o u r c e o f d y n a m i c s i n t h e s y s t e m , a n d t h a t i s t h e n u c l e a r s p i n b a t h . T h e s e t y p i c a l l y d o n o t f r e e z e o u t u n t i l \iK t e m p e r a t u r e s a n d s o a t mK t h e y a r e e f f e c t i v e l y i n a h i g h - t e m p e r a t u r e l i m i t . T h e r a t e a t w h i c h t h e n u c l e i i n F e 8 p e r f o r m s o - c a l l e d T 2 f l i p s [64, 65, 6 6 ] , w h e r e t h e o v e r a l l m a g n e t i z a t i o n o f a p a i r d o e s n o t c h a n g e , w a s e s t i m a t e d b y t h e s e a u t h o r s t o b e ~ 1 kHz — 1 MHz. T h e e f f e c t o f t h e s e f l i p s i s t o c a u s e a t i m e - v a r y i n g m a g n e t i c f i e l d t o b e g e n e r a t e d a t e a c h c e n t r a l s p i n , w h i c h w a s p o s t u l a t e d t o i m m e n s e l y i n c r e a s e t h e " r e s o n a n c e w i n d o w " Chapter 1. Introduction and Overview 17 a n d t h e r e f o r e a l l o w t h e c r y s t a l t o r e l a x . T h e m e c h a n i s m w o r k s l i k e t h i s . A t t h e b e g i n n i n g o f r e l a x a t i o n , a l l m o l e c u l e s s i t i n a c o m b i n a t i o n o f t h e d i p o l a r fields c a u s e d b y a l l o t h e r m o l e c u l e s i n t h e c r y s t a l a n d t h e t i m e - v a r y i n g m a g n e t i c fields c a u s e d b y t h e n u c l e i p r e s e n t . T h e r e w i l l b e a s m a l l n u m b e r o f m o l e c u l e s t h a t c a n b e b r o u g h t t o r e s o n a n c e b y t h e t i m e - v a r y i n g m a g n e t i c fields g e n e r a t e d b y t h e T 2 f l i p p i n g n u c l e i . W h e n o n e o f t h e s e m o l e c u l e s t u n n e l s , i t r e a r r a n g e s t h e d i p o l a r field c o n f i g u r a t i o n i n t h e s a m p l e . T h i s c a n b r i n g o t h e r m o l e c u l e s t o r e s o n a n c e , a n d s o t h e c r y s t a l r e l a x e s . T h i s t h e o r y c o n t a i n e d c e r t a i n t e s t a b l e p r e d i c t i o n s a b o u t t h e r e l a x a t i o n c h a r a c t e r i s t i c s o n e s h o u l d s e e i f t h e h y p o t h e s e s w e r e c o r r e c t . O n e o f t h e s e w a s t h a t t h e r e l a x a t i o n o f Fe& i n i t s q u a n t u m r e g i m e s h o u l d b e s q u a r e r o o t i n t i m e . A l t h o u g h r e l a x a t i o n d a t a e x i s t e d u p t o t h i s p o i n t t h a t w a s t a k e n a s a f u n c t i o n o f t i m e , t h e p r e s e n c e o f t h e s q u a r e r o o t t e m p o r a l d e p e n d e n c e w a s n o t r e a l i z e d u n t i l a f t e r S t a m p a n d P r o k o f i e v l o o k e d f o r c o n f i r m a t i o n o f t h e i r t h e o r y . A t t h i s p o i n t i t w a s r e a l - i z e d t h a t e a r l i e r d a t a d i d i n f a c t f o l l o w a s q u a r e r o o t t e m p o r a l r e l a x a t i o n . S u b s e q u e n t m e a s u r e m e n t s v i n d i c a t e d t h e i d e a , n o t o n l y i n Fe% b u t a l s o i n M n i 2 . T h e o b s e r v a t i o n o f t h e s q u a r e r o o t t e m p o r a l d e p e n d e n c e i n M n i 2 w a s n o t u n d e r s t o o d u n t i l q u i t e r e c e n t l y [?]. T h e p r e d i c t i o n o f a s q u a r e r o o t r e l a x a t i o n r a t e d e p e n d s q u i t e c l e a r l y o n t h e s y s t e m t h a t i s r e l a x a t i n g b e i n g i n a q u a n t u m r e g i m e - t h a t i s , t h e p r e s e n c e o f t h e r m a l l y o c c u p i e d h i g h e r l e v e l s d e s t r o y s t h e s q u a r e r o o t . T h i s s e e m e d t o c o n t r a d i c t t h e f a c t t h a t t h e r e l a x a t i o n c h a r a c t e r i s t i c s i n M r i i 2 w e r e c l e a r l y t e m p e r a t u r e d e p e n d e n t d o w n t o a t l e a s t 6 0 mK. T h e r e s o l u t i o n o f t h i s d i f f i c u l t y i s t h a t Mnu c r y s t a l s c o n t a i n " r o g u e " s p e c i e s o f Mn-u t h a t r e l a x a t d i f f e r e n t r a t e s . B e c a u s e o f t h i s i t i s p o s s i b l e t h a t i n c e r t a i n t e m p e r a t u r e r e g i m e s o n e s p e c i e s o f M n i 2 i s i n i t s q u a n t u m r e g i m e a n d r e l a x e s a s a s q u a r e r o o t w h e r e a s t h e r e s t d o n ' t r e l a x a t a l l (see [63] f o r a d i s c u s s i o n o f t h i s p o i n t ) . I n t h e n e x t s e c t i o n w e s h a l l t a k e a c l o s e l o o k a t r e l a x a t i o n e x p e r i m e n t s , a s t h e y Chapter 1. Introduction and Overview 18 p r o v i d e t a n t a l i z i n g g l i m p s e s o f u n r e s o l v e d p u z z l e s , s o m e o f w h i c h w i l l b e r e s o l v e d i n l a t e r c h a p t e r s . 1.2 A n Introduction to Relaxation Experiments T h e g e n e r a l s t r a t e g y f o r p e r f o r m i n g a r e l a x a t i o n e x p e r i m e n t o n a m o l e c u l a r m a g n e t i c c r y s t a l i s q u i t e s t r a i g h t f o r w a r d [49]. O n e t a k e s a s a m p l e o f t h e m a t e r i a l a n d c o o l s i t d o w n t o s o m e t e m p e r a t u r e T i n s o m e s t a t i c field H o v e r s o m e t i m e i n - T h e i n i t i a l m a g n e t i z a t i o n o f t h e s a m p l e i s m e a s u r e d u s i n g a S Q U I D m a g n e t o m e t e r a r r a y . T h e s t a t i c —* field % i s t h e n a b r u p t l y c h a n g e d t o s o m e n e w field, w h i c h i s i n g e n e r a l t i m e d e p e n d e n t H(t), a n d t h e m a g n e t i z a t i o n o f t h e s a m p l e ( M ) i s t h e n m e a s u r e d a s a f u n c t i o n o f t i m e . —* —* T h i s g i v e s u s a q u a n t i t y M(T, t0, %, H(t), t) w h i c h t e l l s u s a b o u t h o w t h e c r y s t a l , i n i t i a l l y —* —* p r e p a r e d u s i n g \H, to}, r e l a x e s i n t h e p r e s e n c e o f t h e field H(t) a t t h e t e m p e r a t u r e T . S e v e r a l s u c h e x p e r i m e n t s h a v e b e e n p e r f o r m e d o f l a t e [49-63]. W e s h a l l f o c u s o u r a t - t e n t i o n o n t h o s e u s i n g t h e Fes s y s t e m , a s t h i s m a t e r i a l h a s g e n e r a t e d a w e a l t h o f e x c e l l e n t e x p e r i m e n t a l d a t a d e e p i n t o t h e s o - c a l l e d " q u a n t u m r e g i m e " - t h e r e g i o n w h e r e r e l a x a t i o n c h a r a c t e r i s t i c s b e c o m e c o m p l e t e l y i n d e p e n d e n t o f t e m p e r a t u r e ( t h i s h a p p e n s i n Fe$ f o r t e m p e r a t u r e s l o w e r t h a n Tc ~ 3 6 0 mK; see figure 1.11) [14]. T h e o t h e r h e a v i l y s t u d i e d m o l e c u l a r m a g n e t (M7112) i s s i m i l a r i n m a n y w a y s i n i t s r e l a x a t i o n c h a r a c t e r i s t i c s . H o w - e v e r , t h e r e i s a n i m p o r t a n t d i f f e r e n c e - t h e r e l a x a t i o n c h a r a c t e r i s t i c s o f M n i 2 s h o w c l e a r t e m p e r a t u r e d e p e n d e n c e d o w n t o t h e l o w e s t t e m p e r a t u r e s i n v e s t i g a t e d ( T ~ 6 0 mK) [17]. I n t h i s t h e s i s w e h a v e c h o s e n t o f o c u s o u r e f f o r t s o n u n d e r s t a n d i n g t h e p h y s i c s o f t h e q u a n t u m r e g i m e . N o w a s w e h a v e d e s c r i b e d t h e e x p e r i m e n t a b o v e , t h e r e a r e f o u r b a s i c p a r a m e t e r s t h a t w e c a n p l a y w i t h i n o r d e r t o c u s t o m i z e a p a r t i c u l a r r e l a x a t i o n e x p e r i m e n t - t h e t e m p e r a - —* t u r e a t w h i c h i t i s p e r f o r m e d ( T ) , t h e s t a t i c field i n w h i c h i t i s c o o l e d H, t h e t i m e o v e r Chapter 1. Introduction and Overview 19 w h i c h i t i s c o o l e d t0 a n d t h e t i m e - d e p e n d e n t field a p p l i e d d u r i n g r e l a x a t i o n H(t). O u r s t r a t e g y i n t h i s c h a p t e r w i l l b e t o s i m p l y p r e s e n t t h e e x p e r i m e n t a l s i t u a t i o n i n e a c h p a r - t i c u l a r c a s e t h a t w e w i l l r e v i e w ( t h a t i s , w e w i l l s t a t e w h a t t h e p a r a m e t e r s {T, to, H, H(t)} a r e ) a n d t h e n g i v e t h e r e s u l t s . N o t h e o r e t i c a l j u s t i f i c a t i o n o r e x p l a n a t i o n w i l l b e p r e s e n t e d h e r e - w e s h a l l m a k e t h i s t h e t a s k o f t h e r e m a i n d e r o f t h e t h e s i s . 1.2.1 D C Field Relaxation in Polarized Fe8 T h e first c l a s s o f e x p e r i m e n t t h a t w e w i l l r e v i e w w a s h i s t o r i c a l l y t h e first t o b e p e r f o r m e d , p e r h a p s b e c a u s e i t i s t h e s i m p l e s t [18, 50]. I n t h e s e e x p e r i m e n t s , t h e F e 8 s a m p l e s a r e c o o l e d s l o w l y f r o m r o o m t e m p e r a t u r e t o t h e q u a n t u m r e g i m e i n a l a r g e s t a t i c b i a s field H, a p p l i e d a l o n g t h e e a s y a x i s o f t h e c r y s t a l ( see c h a p t e r 2 f o r i n f o r m a t i o n a b o u t t h e c r y s t a l s y m m e t r y i n Fe8). T h i s h a s t h e e f f e c t o f p r e p a r i n g t h e c r y s t a l i n a n i n i t i a l l y p o l a r i z e d s t a t e . O n c e t h e i n i t i a l m a g n e t i z a t i o n h a s b e e n m e a s u r e d , t h e l a r g e s t a t i c b i a s field i s r e m o v e d , a n d a s m a l l e r D C b i a s field H i n a p p l i e d t o t h e c r y s t a l a l o n g i t s e a s y a x i s . T h e m a g n e t i z a t i o n a s a f u n c t i o n o f t i m e i s t h e n m e a s u r e d . S h o w n i n figures 1.12 a n d 1.13 a r e r e s u l t s f o r M(t) f o r t w o d i f f e r e n t c r y s t a l s i n d i f f e r e n t e x p e r i m e n t s o f t h i s t y p e . N o t e t h e u n u s u a l s q u a r e - r o o t r e l a x a t i o n c h a r a c t e r i s t i c . A l s o i n c l u d e d h e r e i s a s i m i l a r e x p e r i m e n t p e r f o r m e d o n t h e Mn\i m o l e c u l a r m a g n e t ( f i g u r e 1.14) [58]. T h e r e l a x a t i o n f o r t h i s m a t e r i a l i s a l s o s q u a r e - r o o t i n t i m e , a l b e i t w i t h a s t r o n g t e m p e r a t u r e d e p e n d e n c e . 1.2.2 D C Relaxation of Annealed Crystals A s e c o n d c l a s s o f e x p e r i m e n t s i n v o l v e a d i f f e r e n t m e t h o d o f p r e p a r a t i o n o f t h e s a m p l e u n d e r s t u d y . I n s t e a d o f s l o w l y l o w e r i n g t h e t e m p e r a t u r e d o w n i n t o t h e q u a n t u m r e g i m e w e m a y i n s t e a d r a p i d l y q u e n c h t h e t e m p e r a t u r e o v e r a t i m e to t h a t i s s o s m a l l t h a t t h e Chapter 1. Introduction and Overview 20 Figure 1.12: Relaxation of the magnetization measured at H = 0 after first saturating in a field of % — 3.5 T z. As indicated in figure 1.11, the curves superimpose for T < 360 mK. Shown in the inset are relaxation characteristics in the quantum regime for some H ^ 0, applied along the easy (z) axis. Figure from [50]. thermal distribution of molecular magnetization is frozen into the initial state of the crystal [49]. That is, the elevated temperature of the molecular ensemble before the quench enables a thermal distribution of the magnetic moments of the molecules. The state of the crystal after the quench retains this distribution initially, before it begins to relax. This procedure is called annealing. The initial magnetization of the sample can be arbitrarily chosen in this scenario, depending only on the pre-quench temperature and static field Ti. Note that the first type of experiment considered is a limiting case of this one. After the sample is quenched and the initial magnetization is measured, a longitudinal DC field is applied either against or in the direction of the magnetization, and the function M(t) is measured. Presented in figure 1.15 are results from this type of experiment. Chapter 1. Introduction and Overview 21 i i o a i « a sqrt(;j [sec"zj Figure 1.13: Short time relaxation of a single crystal of F e 8 , measured at 1 5 0 m A ' . Here several different DC bias fields H were applied along the easy axis of the crystal. Note that the data is plotted against square root t. The inset shows the slope of each of these lines as functions of the DC bias field. Figure from [55]. 1.2.3 H o l e D i g g i n g a n d t h e T i m e - D e p e n d e n t I n t e r n a l L o n g i t u d i n a l B i a s D i s - t r i b u t i o n —* It has been proposed that the D C relaxation of the magnetization M(H.t) can be related to the time-dependent distribution of internal bias fields in the sample [15, 51, 49]. In chapter 6 we shall review the current DC theory and supplant this with our A C results. The basic idea is that the relaxation characteristic M(H, t) is seen experimentally to be square root in time for short times. The "relaxation rate" Tsqrt(H), defined via M(H, t) ~ 1 — \JTsqrt(H)t can then be measured. The assumption then is that the internal distribution of longitudinal biases P(£, t) (where f is the longitudinal bias field) is proportional to the relaxation rate P(£,t) ~ Tsqrt(H) where the field H is applied at time t. Shown in figure 1.16 is the relaxation rate Tsqrt(Hz) of a sample that was prepared in Chapter 1. Introduction and Overview 22 11)0 150 20O 250 J0O Figure 1.14: Here we include some data from a different kind of molecular magnet, the Mn\2 system. Here we again see the clear square-root relaxation characteristic. However in this case the relaxation rates are temperature dependent. Figure from [57]. an initially polarized state. In figure 1.17 the evolution of these rates over time is shown. Note that as the sample evolves, there appears a "hole" in the relaxation rates near zero internal bias. Shown in figure 1.18 is the evolving relaxation rate spectrum of a sample that was initially annealed such that M(0) ~ 0.2. Again we see evidence of a hole being dug near zero bias in this distribution. This hole in the annealed samples has an interesting feature. If samples are annealed to |M(0) < 0.51 there is found an intrinsic hole width of approximately 0.8 mT (see figure 1.19). Stamp and Prokofiev suggested that this intrinsic linewidth was due to nuclear spins [20]. We present a framework in chapter 3 for calculating the linewidth due to the nuclei and demonstrate that our result agrees quantitatively with these experimental results, supporting the contention that nuclear broadening is responsible for this intrinsic hole. Chapter 1. Introduction and Overview 23 0.0005h 0.0015 M/M $ 0.001 h 0.002 0 _L_ 3 0 .^-2.2imT 0.56mT 1,12mT 2.80mT 3.36mT '3.92ml 1.68mT OmT 0 1 0 20 sqrt[t(s)] F i g u r e 1.15: H e r e i s d a t a f r o m a n e x p e r i m e n t o n a n Fe8 s a m p l e t h a t w a s a n n e a l e d i n z e r o f i e l d , g i v i n g i t z e r o i n i t i a l m a g n e t i z a t i o n . T h e s a m p l e w a s t h e n e x p o s e d t o l o n g i t u d i n a l D C fields o f v a r i o u s m a g n i t u d e s . W e see h e r e r e l a x a t i o n a w a y f r o m M = 0, i n t h e d i r e c t i o n o f t h e a p p l i e d field, w i t h t h e s a m e s q u a r e r o o t t e m p o r a l d e p e n d e n c e a s i n t h e i n i t i a l l y p o l a r i z e d c a s e . F r o m [49]. 1.2.4 A C Relaxation of Annealed Crystals T h e final k i n d o f r e l a x a t i o n e x p e r i m e n t t h a t w e s h a l l r e v i e w d i f f e r s f r o m t h e p r e v i o u s t y p e i n t h a t t h e field a p p l i e d d u r i n g r e l a x a t i o n c o n t a i n s a p e r i o d i c t i m e - d e p e n d e n t c o m p o n e n t , a p p l i e d i n t h e d i r e c t i o n o f t h e e a s y - a x i s o f t h e c r y s t a l [51]. T h e b e a u t y o f t h i s t y p e o f e x p e r i m e n t i s t h a t i t i s p o s s i b l e t o m e a s u r e e x t r e m e l y s m a l l r e l a x a t i o n r a t e s , w h i c h p r e s e n t s u s w i t h a u s e f u l p r o b e o f m u c h o f t h e p h y s i c s o f t h e s e s y s t e m s . I n a d d i t i o n , t h e r e n o w e x i s t s a q u a n t i t a t i v e t h e o r y o f h o w m o l e c u l a r m a g n e t s r e s p o n d t o t h i s k i n d o f p e r t u r b a t i o n , w h i c h w e s h a l l d e v e l o p i n t h e l a t e r c h a p t e r s . 1.2.5 Extraction of Tunneling Matr ix Elements S h o w n i n figures 1.20 a n d 1.21 a r e q u a n t i t i e s e x t r a c t e d f r o m A C r e l a x a t i o n m e a s u r e - m e n t s , w h i c h a r e r e l a t e d t o t h e t u n n e l i n g m a t r i x e l e m e n t s o f t h e s i n g l e - m o l e c u l e e f f e c t i v e H a m i l t o n i a n o f F e 8 ( w e s h a l l s h o w e x a c t l y h o w t h e y a r e r e l a t e d i n c h a p t e r 6 ) . W h a t i s Chapter 1. Introduction and Overview 24 M0H(T) F i g u r e 1.16: F i e l d d e p e n d e n c e o f s h o r t t i m e s q u a r e r o o t r e l a x a t i o n r a t e s Tsqrt(Hz). T h e i n i t i a l d i s t r i b u t i o n i s l a b e l l e d w i t h Min = — 0 . 9 9 8 Ms w h e r e a s t h e o t h e r s a r e d i s t r i b u t i o n s o b t a i n e d b y t h e r m a l a n n e a l i n g . T h e l a t t e r a r e d i s t o r t e d a t h i g h e r f i e l d s b y n e a r e s t n e i g h b o u r l a t t i c e e f f e c t s . F i g u r e f r o m [49]. -0.05 -0.025 0 0.025 0.05 f i 0 H(T) F i g u r e 1.17: Q u a n t u m h o l e - d i g g i n g . F o r e a c h p o i n t , t h e s a m p l e w a s first s a t u r a t e d i n a field o f -1.4 T a t a t e m p e r a t u r e o f T ~ 2 K a n d t h e n c o o l e d t o 4 0 m K . T h e s a m p l e w a s t h e n a l l o w e d t o r e l a x f o r t i m e s t0. A f t e r t h i s t i m e h a d e l a p s e d , a D C field Hz w a s a p p l i e d , a n d rsqrt w a s m e a s u r e d . N o t e t h e r a p i d d e c r e a s e i n r e l a x a t i o n r a t e n e a r Hz = 0. F i g u r e f r o m [49]. Chapter 1. Introduction and Overview 25 -0.04 -0 .02 0 0.02 0.04 0.06 0.08 H 0H(T) F i g u r e 1.18: Q u a n t u m h o l e d i g g i n g , a s i n figure 1.17, b u t n o w f o r a s a m p l e t h a t h a s b e e n a n n e a l e d t o Min = — 0.2 Ms. T h e r e s u l t i n g e v o l u t i o n s h o w s a v e r y n a r r o w h o l e (see i n s e t ) . N e a r z e r o b i a s t h e h o l e d e v e l o p s v e r y r a p i d l y a l t h o u g h t h e r e s t o f t h e d i s t r i b u t i o n h a r d l y c h a n g e s a t a l l . F i g u r e f r o m [49]. H0H(T> F i g u r e 1.19: H e r e i s p l o t t e d t h e d i f f e r e n c e b e t w e e n t h e r e l a x a t i o n r a t e s a t t = 0 (rinit) a n d a t t0 = 1 6 s (Tdig), f o r s e v e r a l d i f f e r e n t a m o u n t s o f a n n e a l i n g . N o t e t h a t f o r |M i n| < 0.5 t h e h o l e w i d t h b e c o m e s i n d e p e n d e n t o f |M« J , w i t h a n i n t r i n s i c w i d t h o f ~ 0.8 mT. F i g u r e f r o m [49]. Chapter 1. Introduction and Overview 20 actually measured here is simply the magnetization as a function of time, M(t) as per usual, in the presence of a longitudinal sawtooth A C field of amplitude A and frequency u>, a longitudinal DC bias field Hzz, and a static transverse DC field H = Hxx + Hyy. The authors find that M(t) ~ exp [—Ft], ie. the relaxation is exponential, with a rate F that is a function of the applied longitudinal A C field, the transverse DC field and the longitudinal DC field. They then define their quantity A in the figures shown via the following relation; Note that as advertised we are not going to try to justify this relation theoretically just yet-there will be much on this later. 11 t i i 1 1 1 1 0 0.2 0.4 0,6 0.8 1 1.2 1,4 r a n i ( T ) Figure 1.20: The quantity A here is related to the relaxation rate of the crystal's magne- tization via (1.3). Here it is shown as a function of the magnitude of the transverse DC field \H\ = JW* + H2 for several orientations of this field ip = tan'1 (Hy/Hx). In this case the longitudinal DC field was taken to be zero (Hz = 0) . Figure from [51]. Chapter 1. Introduction and Overview 27 r"""'"""r 11 i i r——-r 1 E I i f i r r i 1 1 -0.4*0.2 0 0.2 0.4 0.6 0,8 1 1.2 1.4 r u n s ( T ) Figure 1.21: The quantity A shown for ip = 0, as a function of \H\. Shown here are results for three different values of Hz. The lowest curve was obtained for Hz = 0; the middle curve for Hz = 0.22T, and the upper curve for Hz = 0.44T. In terms of the energy level structure of the Fe$ molecule's spin Hamiltonian presented in chapter 1, these applied fields correspond to resonance situations between | — 5 >+» | + 5 >, | — 5 >+»• 1 + 5—1 > and | 5 > o | + 5 — 2 > respectively. Notice that a parity effect is observed. Figure from [51]. 1.3 T h e s i s O v e r v i e w Our goal will be to work up to a quantitative theory of A C relaxation in F e 8 crystals. In order to do this we shall need to develop several key concepts. We begin in chapter two with an analysis of the problem of deriving an effective Hamiltonian for a single F e 8 molecule. In chapter three we calculate the hyperfine couplings between a central spin object and all the nuclei in the molecule. Using this information we calculate all the decoherence parameters introduced by Prokofiev and Stamp in their theory of the spin bath. We show Chapter 1. Introduction and Overview 2 8 t h a t t h e r e s h o u l d e x i s t m e a s u r a b l e i s o t o p e e f f e c t s i n Fe% a n d g i v e q u a n t i t a t i v e p r e d i c t i o n s o f t h e l i n e w i d t h d u e t o n u c l e a r s p i n s i n a n Fe& c r y s t a l w i t h a r b i t r a r y i s o t o p i c c o n t e n t . I n c h a p t e r f o u r w e i n t r o d u c e a n d d e v e l o p s o m e o f t h e m a c h i n e r y o f t h e L a n d a u - Z e n e r p r o b l e m . T h i s i n v o l v e s u s i n g a t i m e - d e p e n d e n t H a m i l t o n i a n t o e x t r a c t t r a n s i t i o n p r o b a b i l i t i e s b e t w e e n s t a t e s o f t h e c e n t r a l o b j e c t o f i n t e r e s t . C h a p t e r five i s t h e h e a r t o f t h e t h e s i s , a n d c o n t a i n s a n e x t e n s i o n o f t h e L a n d a u - Z e n e r p r o b l e m i n w h i c h n u c l e a r s p i n s a r e i n c l u d e d . W e u s e t h e r e s u l t s o f t h i s c a l c u l a t i o n t o find a s i n g l e - m o l e c u l e r e l a x a t i o n r a t e i n t h e p r e s e n c e o f a n e x t e r n a l m a g n e t i c field w i t h b o t h A C a n d D C c o m p o n e n t s . C h a p t e r s i x t h e n u s e s t h i s g e n e r a l s i n g l e - m o l e c u l e r e l a x a t i o n r a t e a s t h e i n p u t t o a m a s t e r e q u a t i o n s o a s t o m o d e l t h e t e m p o r a l e v o l u t i o n o f a c r y s t a l o f F e 8 m o l e c u l e s . W e e x t r a c t t i m e - d e p e n d e n t r e l a x a t i o n c h a r a c t e r i s t i c s f r o m o u r t h e o r y a n d c o m p a r e t h e s e t o e x p e r i m e n t a l r e s u l t s . W e c o n c l u d e o u r a n a l y s i s i n c h a p t e r s e v e n w i t h a s u m m a r y o f r e s u l t s a n d t h e c u r r e n t o u t l o o k f o r o u r t h e o r y o f A C r e l a x a t i o n . C h a p t e r 2 E f f e c t i v e H a m i l t o n i a n s I n t h i s c h a p t e r w e d e r i v e a l o w e n e r g y e f f e c t i v e H a m i l t o n i a n f o r a s i n g l e i s o l a t e d Feg m o l e c u l e . W e b e g i n b y l i s t i n g a l l t e r m s f o u n d i n t h e H a m i l t o n i a n o f a s i n g l e f r e e Fe3+ i o n . W e t h e n d e s c r i b e h o w t h e s e w i l l b e m o d i f i e d b y p l a c i n g t h e Fe3+ i o n s i n t o a c r y s t a l l i n e e n v i r o n m e n t , f o l l o w i n g t h e t r e a t m e n t o f A b r a g a m a n d P r y c e [67] ( s e e a l s o [68, 6 9 ] ) . T h i s l e a d s t o a s i n g l e i o n " s p i n H a m i l t o n i a n " . W e t h e n b u i l d u p t h e Fe$ H a m i l t o n i a n b y i n t r o d u c i n g e x c h a n g e / s u p e r e x c h a n g e t e r m s b e t w e e n t h e Fe3+ i o n s a n d t e r m s c o m i n g f r o m b o t h t h e n u c l e a r s p i n e n v i r o n m e n t [20] a n d p h o n o n [70] a n d p h o t o n [71] o s c i l l a t o r b a t h s . T h i s " b a r e " d e s c r i p t i o n o f t h e Fe% m o l e c u l e , c o n t a i n i n g e i g h t s i n g l e i o n F e 3 + t e r m s , e x c h a n g e / s u p e r e x c h a n g e c o u p l i n g s b e t w e e n t h e s e a n d t h e v a r i o u s e n v i r o n m e n t s i s t h e n i n v e s t i g a t e d . T h e e x c h a n g e / s u p e r e x c h a n g e c o u p l i n g e n e r g i e s a r e m u c h l a r g e r t h a n a l l o t h e r e n e r g y s c a l e s [72, 73]. T h i s s u g g e s t s t h e h y p o t h e s i s t h a t a t l o w e n e r g i e s t h e s e c o u p l i n g s l o c k t h e e l e c t r o n i c s p i n s t o g e t h e r i n t o a " g i a n t s p i n " [20]. W e a s s u m e t h a t t h i s i s t h e c a s e a n d w r i t e d o w n a " g i a n t s p i n H a m i l t o n i a n " t h a t w e p o s t u l a t e c o u l d i n p r i n c i p l e b e d e r i v e d f r o m t h e b a r e H a m i l t o n i a n i n a s i m i l a r m a n n e r t o h o w t h e s i n g l e i o n s p i n H a m i l t o n i a n s w e r e d e r i v e d f r o m t h e i r b a r e d e s c r i p t i o n s , i e . b y finding t h e " g i a n t s p i n " g r o u n d s t a t e o f t h e s y s t e m a n d p e r f o r m i n g p e r t u r b a t i o n t h e o r y a r o u n d i t t o e l i m i n a t e a l l t h e e l e c t r o n i c s p i n d e g r e e s o f f r e e d o m b u t o n e . W e t h e n p r o c e e d t o t h e i n v e s t i g a t i o n o f t h e p r o p e r t i e s o f g e n e r a l g i a n t s p i n H a m i l t o - n i a n s i n t h e a b s e n c e o f e n v i r o n m e n t a l c o u p l i n g s . W e c a l c u l a t e t u n n e l i n g m a t r i x e l e m e n t s 2 9 Chapter 2. Effective Hamiltonians 3 0 f o r H a m i l t o n i a n s w i t h v a r i o u s s y m m e t r i e s u s i n g i n s t a n t o n [74, 75], W K B [25, 26], p e r - t u r b a t i o n t h e o r y [70] a n d e x a c t d i a g o n a l i z a t i o n m e t h o d s . W e c o n c l u d e b y r e t u r n i n g t o t h e s p e c i f i c c a s e o f t h e F e 8 g i a n t s p i n H a m i l t o n i a n . T h e g i a n t s p i n p o s s e s s e s t w o p r e f e r r e d d i r e c t i o n s d u e t o c r y s t a l a n i s o t r o p y w h i c h a r e i d e n t i f i e d , i n z e r o e x t e r n a l field, w i t h t h e ± z d i r e c t i o n s ( s t a t e s \S = + 1 0 > a n d \S = — 1 0 > ) . A t t e m p e r a t u r e s m u c h l o w e r t h a n t h e d i f f e r e n c e i n e n e r g i e s b e t w e e n | + 9 > a n d | + 10 > s t a t e s ( ~ 5 K) o n l y t h e | ± 10 > s t a t e s h a v e s i g n i f i c a n t t h e r m a l p o p u l a t i o n s . T h i s a l l o w s u s t o d e r i v e a final e f f e c t i v e d e s c r i p t i o n w h e r e t h e c e n t r a l s p i n o b j e c t i s t r e a t e d a s a t w o l e v e l (| ± 10 > ) s y s t e m , f o l l o w i n g t h e t r e a t m e n t o f T u p i t s y n e t . a l . [74]. 2.1 The Fe3+ Free Ion Hamiltonian W e s h a l l b e g i n o u r a n a l y s i s o f t h e c o m p l i c a t e d Fes s y s t e m ( w h o s e s t r u c t u r e w a s s h o w n i n t h e i n t r o d u c t o r y c h a p t e r ) b y c o n c e n t r a t i n g o u r a t t e n t i o n o n t h e i r o n i o n s . W e s h a l l b e g i n b y s t u d y i n g a g e n e r a l H a m i l t o n i a n f o r a free Fe3+ i o n . T h i s t r e a t m e n t f o l l o w s t h a t o f [69]. T h e d o m i n a n t t e r m i n t h i s d e s c r i p t i o n i s t h e C o u l o m b i n t e r a c t i o n a m o n g s t t h e e l e c - t r o n s ( h e r e t h e r e a r e Ne o f t h e m ) a n d b e t w e e n t h e e l e c t r o n s a n d n u c l e a r c h a r g e Ze N e / » • 2 Z e 2 \ N e e2 V'-£&-T) +&£ (21) T h e n e x t m o s t i m p o r t a n t t e r m i s t h e m a g n e t i c i n t e r a c t i o n b e t w e e n t h e o r b i t a l a n g u l a r m o m e n t u m lj a n d t h e e l e c t r o n i c s p i n Sk VLS = ]C a3*h ' h + b3kl3 " $k + CjkSj • sk (2.2) 3,k w h e r e a ^ , bjk a n d Cjk a r e c o n s t a n t s . N e x t c o m e s t h e d i r e c t i n t e r a c t i o n b e t w e e n s p i n s Vss = y * j' *k - 3 ( f J f c ' fffX^i* • (2 3) jk Tjk rjk Chapter 2. Effective Hamiltonians 3 1 W e a k e r s t i l l a r e t h e t e r m s VN = 2gnjiBPn fc I rk rk \ o (2.4) w h e r e t h e t e r m i n c u r l y b r a c k e t s i s t h e d i p o l e - d i p o l e i n t e r a c t i o n b e t w e e n t h e n u c l e a r a n d e l e c t r o n i c m o m e n t s a n d t h e l a s t t e r m i s t h e s o - c a l l e d a n o m a l o u s h y p e r f i n e t e r m w h i c h c o m e s a b o u t f r o m t h e o v e r l a p o f t h e w a v e f u n c t i o n o f s e l e c t r o n s w i t h t h e n u c l e u s a n d Q e2Q 2 / ( 2 / - 1) 1(1 + 1) 3(rk-I)2 (2.5) 'fc 'fc w h i c h r e p r e s e n t s t h e e l e c t r o s t a t i c i n t e r a c t i o n b e t w e e n t h e n u c l e a r q u a d r u p o l e m o m e n t Q a n d t h e g r a d i e n t o f t h e e l e c t r i c field d u e t o t h e e l e c t r o n s . I n t e r a c t i o n w i t h a n e x t e r n a l m a g n e t i c field p r o d u c e s t h e t e r m s a n d VH = '£^B(lk + 2sk)-H k Vh = -9nPnH • I (2.6) (2.7) c o r r e s p o n d i n g t o t h e i n t e r a c t i o n s w i t h t h e e l e c t r o n s a n d n u c l e u s r e s p e c t i v e l y . T h e t o t a l f r e e i o n H a m i l t o n i a n i s n o w j u s t t h e s u m o f t h e s e ; H = VF + VLS + Vss + VN + VQ + VH + Vh (2.8) O r d e r s o f m a g n i t u d e o f t h e s e m a y b e o b t a i n e d f r o m o p t i c a l s p e c t r a a n d a r e , f o r Fe3+, VF ~ 5 - 1 0 5 K, VLS ~ 1 0 0 - 3 0 0 K, Vss ~1-2K,VN~ 1 - 2 0 0 mK a n d VQ ~ 1 - 2 mK [76]. W e s e e t h a t Vp i s b y f a r t h e d o m i n a n t t e r m i n t h i s e x p r e s s i o n . I f w e n e g l e c t a l l t e r m s b u t t h i s o n e , t h e n L a n d 5 ( t h e t o t a l a n g u l a r m o m e n t u m a n d s p i n o f t h e i o n ) c o m m u t e w i t h V p . T h i s m e a n s t h a t w e m a y i n t h i s a p p r o x i m a t i o n l a b e l t h e s t a t e s o f Chapter 2. Effective Hamiltonians 32 t h e f r e e i o n w i t h t h e q u a n t u m n u m b e r s L, Lz, 5, 5 2 , J a n d Jz. S i n c e t h e filled i n n e r s h e l l s h a v e 5 = L = J = 0 w e m a y d e s c r i b e t h e i o n b y r e f e r r i n g o n l y t o t h e s t a t e o f t h e p a r t i a l l y filled o u t e r 3 d s h e l l . I t i s k n o w n t h a t t h e Vp t e r m i n Fe3+ l e a d s t o a g r o u n d s t a t e t h a t i s a n o r b i t a l s i n g l e t 6<S 5/ 2 [77], w h e r e w e u s e t h e s t a n d a r d n o t a t i o n t h a t t h e s u p e r s c r i p t r e f e r s t o t h e s p i n m u l t i p l i c i t y 2 5 + 1, t h e c a p i t a l s c r i p t l e t t e r r e f e r s t o t h e t o t a l a n g u l a r m o m e n t u m o f t h e i o n (S -+ 0, V —> 1, V —> 2, et c . ) a n d t h e s u b s c r i p t r e f e r s t o t h e t o t a l a n g u l a r m o m e n t u m J. T h i s s t a t e m a y b e o b t a i n e d v i a t h e u s e o f H u n d ' s f i r s t t w o r u l e s [78]-we first m a x i m i z e t h e t o t a l s p i n b y filling u p five d o r b i t a l s w i t h s = + 1 / 2 e l e c t r o n s ( g i v i n g t o t a l s p i n 5/2) a n d t h e n m a x i m i z e t h e o r b i t a l a n g u l a r m o m e n t u m (L = 2 + 1 + 0 + ( - 1 ) + (- 2 ) = 0 ) . N o t e t h a t f o r a l l h a l f - f i l l e d s h e l l s ( h e r e w e h a v e 5 d e l e c t r o n s o u t o f a p o s s i b l e 10) w e g e t a n o r b i t a l s i n g l e t f o r t h e g r o u n d s t a t e . 2.2 T h e E f f e c t o f t h e C r y s t a l l i n e E n v i r o n m e n t I n g e n e r a l , w h e n a t r a n s i t i o n m e t a l i o n i s p l a c e d i n a c r y s t a l l i n e e n v i r o n m e n t , t h e first q u e s t i o n t h a t m u s t b e r e s o l v e d i s t h e q u e s t i o n o f t h e n a t u r e o f t h e b o n d i n g b e t w e e n t h e i o n a n d t h e l i g a n d s . T h i s i s b e c a u s e t h e d o m i n a n t n e w t e r m t h a t m u s t b e d e a l t w i t h c o m e s f r o m t h e e l e c t r o s t a t i c i n t e r a c t i o n b e t w e e n t h e i o n ' s d s h e l l e l e c t r o n s a n d a l l t h e c h a r g e d m a t t e r i n t h e m o l e c u l e . I f t h i s b o n d i n g i s m o s t l y i o n i c , t h e n o n e c a n m a k e t h e a p p r o x i m a t i o n t h a t t h e i o n s i t s i n a n e l e c t r o s t a t i c field c o m i n g p r e d o m i n a n t l y f r o m i t s n e a r e s t n e i g h b o u r s , w h i c h a r e t r e a t e d a s p o i n t c h a r g e s . T h i s i s t h e s o - c a l l e d crystal field [79] a p p r o x i m a t i o n . A l t h o u g h i t i s c r u d e , i t i s o f t e n a u s e f u l s t a r t i n g p o i n t f o r u n d e r - s t a n d i n g t h e e f f e c t o f t h e C o u l o m b i c i n t e r a c t i o n b e t w e e n t h e i o n a n d i t s e n v i r o n m e n t . A l i t t l e m o r e s o p h i s t i c a t e d i s t h e ligand field [80] a p p r o x i m a t i o n . I n t h i s t r e a t m e n t a l - l o w a n c e i s m a d e f o r t h e d i r e c t o v e r l a p o f t h e i o n ' s d s h e l l e l e c t r o n s w i t h t h e l i g a n d s , i e . a n a t t e m p t t o d e a l w i t h c o v a l e n c y i s p r e s e n t e d . B e t t e r y e t a r e molecular orbital [81] Chapter 2. Effective Hamiltonians 3 3 m e t h o d s , w h i c h u s e a s t h e i r s t a r t i n g p o i n t s t h e o r b i t a l s o f e a c h o f t h e l i g a n d s a n d i o n s a n d t h e i n t e r a c t i o n s b e t w e e n t h e s e . N o w i n o u r s p e c i f i c c a s e w e a r e i n t e r e s t e d o n l y i n t h e l o w e n e r g y p r o p e r t i e s o f t h e Fe3+ i o n . T h e f r e e i o n g r o u n d s t a t e i s , a s m e n t i o n e d , a n o r b i t a l s i n g l e t 6«S 5 / 2 - W h a t w i l l b e t h e e f f e c t o f t h e i n c l u s i o n o f t h e C o u l o m b i c e n v i r o n m e n t o n t h i s g r o u n d s t a t e a n d t h e l o w - l y i n g e x c i t e d s t a t e s ? I n o r d e r t o a n s w e r t h i s q u e s t i o n w e n e e d o n l y t o k n o w t h e r e l a t i v e s t r e n g t h s o f t h e " o n - s i t e " C o u l o m b t e r m s VF a n d t h e " o f f - s i t e " C o u l o m b t e r m s Vc- T h e s t r e n g t h o f t h i s c o u p l i n g i n s e v e r a l m a t e r i a l s c o n t a i n i n g Fe3+ i o n s h a s b e e n d e t e r m i n e d , w i t h t h e s e r a n g i n g f r o m Vc ~ 1 7 0 0 0 — 2 3 0 0 0 K [82]; h o w e v e r Vc h a s n o t b e e n m e a s u r e d i n F e 8 . F o r t u n a t e l y t h e r e a w a y t o k n o w w h a t t h e r e l a t i v e m a g n i t u d e o f t h e s e a r e w i t h o u t a d i r e c t m e a s u r e m e n t . I f VF ^> Vc t h e n t h e g r o u n d s t a t e o f t h e i o n w i l l r e m a i n 6<S 5/ 2, a s t h e e l e c t r i c field c a n n o t s p l i t a s i n g l e t . I f w e a r e i n t h e o p p o s i t e l i m i t Vc » VF t h e n t h e Fe3+ i o n s w i l l g o i n t o a " s p i n - p a i r e d " s t a t e w i t h s p i n 5 = 1/2 ( i e . H u n d ' s r u l e s a r e m o d i f i e d ) . S i n c e i t i s k n o w n e x p e r i m e n t a l l y t h a t t h e Fe3+ i o n s a r e i n f a c t i n a 5 — 5/2 s t a t e w e i n f e r t h a t w e a r e i n t h e l i m i t VF 3> Vc, w h i c h i s i n a c c o r d w i t h t h e c r y s t a l field s t r e n g t h s r e p o r t e d f o r o t h e r m a t e r i a l s w i t h Fe3+ c e n t e r s . W e se e t h a t t h e e x p e r i m e n t a l l y o b s e r v e d f a c t t h a t t h e Fe3+ i o n s h a v e s p i n 5 = 5/2 s i m p l i f i e s o u r t a s k t r e m e n d o u s l y . T h i s i s b e c a u s e t h i s i s prima facie e v i d e n c e t h a t t h e g r o u n d s t a t e o f t h e Fe3+ i o n s , e v e n i n t h e m o l e c u l a r e n v i r o n m e n t , i s 6«S 5 / 2 . M o r e p r e c i s e l y , t h i s i s e v i d e n c e t h a t t h e g r o u n d s t a t e o f t h e H a m i l t o n i a n H = VF + Vc, w h e r e Vc i n c l u d e s a l l C o u l o m b i c t e r m s c o m i n g f r o m t h e i n t e r a c t i o n o f t h e i o n w i t h t h e l i g a n d e n v i r o n m e n t , i s a n o r b i t a l s i n g l e t 6<5*5/2- N o w t h e t a c k t h a t w e s h a l l c h o o s e i n w h a t f o l l o w s i s t h i s . S i n c e t h e C o u l o m b i c e n v i r o n m e n t d o e s n o t s p l i t t h e g r o u n d s t a t e h e r e , b u t d o e s a f f e c t t h e e x c i t e d i o n i c s t a t e s ( t h e n e a r e s t s t a t e i s a AQ s t a t e [165] i n t h e f r e e i o n ) , a n d s i n c e w e a r e n o t i n t h e p o s i t i o n t o q u a n t i t a t i v e l y a c c o u n t f o r i t s e f f e c t s a n y w a y ( t h i s w o u l d r e q u i r e a m o l e c u l a r o r b i t a l Chapter 2. Effective Hamiltonians 3 4 a p p r o a c h , a n d e v e n t h e s e d o n o t a l w a y s w o r k [83]), w e s h a l l a d o p t t h e c r y s t a l field p a r a d i g m i n d e a l i n g w i t h c h a r g e s e x t e r n a l t o t h e i o n . I n t h i s p i c t u r e t h e e x c i t e d s t a t e s o f t h e Fe3+ f r e e i o n a r e s p l i t b y t h e c r y s t a l field, w i t h s o m e o r b i t a l s b e i n g f a v o u r e d a b o v e o t h e r s b e c a u s e o f t h e i r s p a t i a l d e p e n d e n c e a n d r e l a t i v e p o s i t i o n i n g i n t h e m o l e c u l e . I n o u r c a s e t h e Fe3+ i o n s a l l h a v e s i x n e a r e s t n e i g h b o u r s a r r a n g e d i n a d i s t o r t e d o c t a h e d r a l s h a p e . T h i s a l l o w s u s i n p r i n c i p l e t o c a l c u l a t e t h e s p l i t t i n g s o f t h e e x c i t e d s t a t e s . W e s h a l l n o t d o t h i s h o w e v e r - a s w e s h a l l see, t h e m a g n i t u d e o f t h e s e w i l l o n l y q u a n t i t a t i v e l y c h a n g e t h e r e s u l t s o f t h e a r g u m e n t s t o c o m e . 2.3 The Single Ion Effective Hamiltonian W e n o w w a n t t o c o n s i d e r t h e e f f e c t o f t h e f r e e i o n t e r m s VLS + Vss + VN + VQ + VH + Vh- W h a t w e s h a l l d o i s f o l l o w t h e t r e a t m e n t o f A b r a g a m a n d P r y c e [67], c a l c u l a t i n g t h e i r e f f e c t p e r t u r b a t i v e l y o n t h e 6<S 5/ 2 g r o u n d s t a t e o f H = Vp + Vc- T h e first s t e p i n t h i s t r e a t m e n t i s t o r e w r i t e t h e s e p e r t u r b a t i o n s i n t e r m s o f t o t a l s i n g l e i o n s p i n a n d a n g u l a r —* —* m o m e n t u m o p e r a t o r s 5 a n d L. T h i s w i l l b e p e r m i s s i b l e a s l o n g a s t h e r e i s n o s i g n i f i c a n t c h e m i c a l b o n d i n g b e t w e e n t h e i r o n i o n s a n d t h e i r s u r r o u n d i n g s , w h i c h a s w e s a w i n t h e p r e c e d i n g i s s u p p o r t e d b y t h e 5 = 5/2 n a t u r e o f t h e i o n s . T h i s h a s b e e n d o n e ; w e r e p e a t t h i s p r o c e s s h e r e ( f o r a d e t a i l e d e x p l a n a t i o n o f t h e s t e p s o u t l i n e d h e r e , s e e [68]). O u r t e r m s t r a n s f o r m a s f o l l o w s ; T h e e f f e c t i v e s p i n - o r b i t c o u p l i n g p a r a m e t e r A f o r f r e e Fe a t o m s i s m e a s u r e d t o b e \Z%K [84] a n d i s e x p e c t e d t o b e s l i g h t l y s m a l l e r f o r F e 3 + i o n s . T h e p a r a m e t e r p i s d i f f i c u l t t o VLS —> A L • 5 VSS -> P [(L • 5 ) 2 + l-L • S - l-L{L + 1 ) 5 ( 5 + 1) p \\{Lal/ + L^La)SQSp - l-L{L + 1 ) 5 ( 5 + 1) (2.9) Chapter 2. Effective Hamiltonians 3 5 c a l c u l a t e [85] b u t i s e x p e c t e d t o b e m u c h s m a l l e r t h a n A [86]. W e a l s o h a v e VN gntipixn ^ [ ( L • 5 ) + £L(L + 1 ) ( 5 • /) - ^(L • S)(L • i) (s-i) v , ^ Q 11 Q 2 / ( 2 7 - 1 ) \ r 3 3 , * 3 ( f - / ) 2 + | ( Z . i ) - L ( L + 1 ) 7 ( 7 + 1) w h e r e f o r t h e i r o n g r o u p (2/ - 1) - 4 5 , 77 = ±25e 5 ( 2 / - 1 ) ( 2 / + 3 ) ( 2 L - 1) ( t h e s i g n o f n d e p e n d s o n w h e t h e r t h e d s h e l l i s l e s s o r m o r e t h a n h a l f filled) a n d h) = l* .Jtf(*0l s (2.10) (2.11) (2.12) , r / J r° w h e r e ip (f) i s t h e s p i n d e n s i t y o f t h e o u t e r c o r e d e l e c t r o n s . T h e f a c t o r K c o m e s f r o m t h e p o l a r i z a t i o n o f t h e c o r e s - e l e c t r o n s d u e t o t h e d e l e c t r o n s a n d w i l l l e a d t o t h e c o n t a c t h y p e r f i n e i n t e r a c t i o n i n 57Fe. T h e t e r m s p r o p o r t i o n a l t o t h e e x t e r n a l field a r e VH = (i(,H-{L + 2§) Vh = gnixnH-I (2.13) P u t t i n g a l l t h e s e t o g e t h e r w e g e t t h e p e r t u r b a t i o n t e r m Hx = ( A - \p)L -S-p(L- 5 ) 2 + nPH • (L + 2S) +gnpnP | ( L .T)-K§- I- 1-{L •S)(L-T)-l-{L- !)(L • 5 ) ] + gnlinH • 7(2.14) w h e r e P = 2fip (^). 2.3.1 First Order Perturbation Theory W e w a n t t o u s e (2.14) t o p e r t u r b o u r g r o u n d s t a t e o u t o f t h e 6 5 5 / 2 s t a t e . I n t h e first o r d e r p e r t u r b a t i o n w e find t h a t t h e o n l y t e r m s t h a t c o n t r i b u t e a r e o n e s t h a t d o n ' t c o n t a i n L, Chapter 2. Effective Hamiltonians 3 6 b e c a u s e < 0|L|0 > = 0 f o r a n o r b i t a l s i n g l e t , e x c e p t f o r t h e t e r m < 0\LiLj + LjLi\0 >= \L(L + l)Sij + kj w h e r e lu = 0. A s w e l l , w e h a v e i n t h i s c a s e t h a t £ = 2/21 a n d rj — 0. T h u s t h e t e r m s w e g e t f r o m first o r d e r p e r t u r b a t i o n f r o m &S§/2 a r e H2 = -gnPnP(^ + ^l^SJp + 2u.0H • S + gnlinH • I (2.15) 2.3.2 Second Order Perturbation Theory I n t h e s e c o n d o r d e r w e h a v e t o c a l c u l a t e a l l t h e t e r m s c o m i n g f r o m ^ < 0 | f f i | n > < n | # i | 0 > ,01_. £ E(n)-E(o) (2-16) w h e r e n l a b e l s t h e e x c i t e d o r b i t a l s t a t e s . W h e r e t h e s e e x c i t e d s t a t e s l i e , i e . t h e e x a c t v a l u e s o f E(n), a r e f u n c t i o n s o f t h e d e t a i l s o f t h e c r y s t a l field s p l i t t i n g a n d a s s u c h we s h a l l n o t a t t e m p t t o c a l c u l a t e t h e m e x a c t l y . W e m a y o b t a i n o r d e r o f m a g n i t u d e e s t i m a t e s f o r t h e s e b y c o m p a r i n g t o e x i s t i n g m a t e r i a l s t h a t c o n t a i n Fe3+ c e n t e r s . I n t h e s e c a s e s t h e e n e r g y o f t h e first e x c i t e d s t a t e E(l) i s i n t h e r a n g e 1 7 0 0 0 — 2 3 0 0 0 K [82], w h i c h i s e x t r e m e l y l a r g e c o m p a r e d t o t h e s c a l e i n w h i c h w e a r e i n t e r e s t e d . D e f i n i n g t h e t e n s o r s 0 <0\L*\nxn\I/l\0> E(n) - E(0) Q/3 _ _i€^_ v < 0\Ls\n >< n | L % + L 7 Z l | 0 > U ~ 2 & E(n)-E(0) ' U T ~*ho E(n)-E(0) ( 2 - 1 7 ) g i v e s , u p o n c o l l e c t i n g t e r m s , a n e f f e c t i v e H a m i l t o n i a n (2.18) Chapter 2. Effective Hamiltonians 3 7 w h e r e w e h a v e d e f i n e d gaP = 2(5al) - A Q / 3 ) , Da0 = - A 2 A Q / J - pla)3 Aa& - -P 7 7 , Rati = 1 - 2Pu.pAap (2.19) T h e m e a n i n g o f t h e s e t e r m s i s a s f o l l o w s [87]. ga@ i s t h e s o - c a l l e d " s p e c t r o s c o p i c s p l i t t i n g f a c t o r " . I t i s a n i s o t r o p i c i n g e n e r a l , c o n t a i n i n g r e f e r e n c e t o t h e h i g h e r l y i n g o r b i t a l s t a t e s . DaP i s a m e a s u r e o f t h e s p l i t t i n g o f t h e g r o u n d s t a t e a n d c o n t a i n s r e f e r e n c e t o b o t h t h e s p i n o r b i t c o u p l i n g a n d t h e s p i n - s p i n c o n t r i b u t i o n i n a n a s y m m e t r i c a l c r y s t a l field. Aa/} r e p r e s e n t s t h e h y p e r f i n e c o u p l i n g s b e t w e e n t h e n u c l e u s a n d t h e e l e c t r o n i c s p i n . T h i s t e r m i s m a d e u p o f c o n t r i b u t i o n s f r o m o v e r l a p o f s e l e c t r o n s w i t h t h e n u c l e u s ( « ) , a n o r b i t a l c o n t r i b u t i o n la/3 a n d a s p i n - o r b i t c o n t r i b u t i o n XAa/3. I n t h e e x p r e s s i o n f o r t h e c o u p l i n g b e t w e e n t h e n u c l e a r s p i n a n d a n e x t e r n a l f i e l d w e s e e t h a t t h e r e i s a n a n i s o t r o p i c c o m p o n e n t w h i c h c a n b e o f t h e s a m e o r d e r o f m a g n i t u d e a s t h e d i r e c t c o n t r i b u t i o n . 2.3.3 Higher Orders Perturbation Theory I t i s c l e a r t h a t p e r f o r m i n g t h i r d a n d h i g h e r o r d e r p e r t u r b a t i o n t h e o r y w i l l p r o d u c e t e r m s o f h i g h e r s p i n m u l t i p l i c i t y i n t h e s p i n H a m i l t o n i a n [25]. T e r m s i n t h e s p i n H a m i l t o n i a n u p t o 2Sth o r d e r i n t h e s p i n o p e r a t o r s Sx, Sy a n d Sz a r e i n g e n e r a l p o s s i b l e . I n t h e c a s e o f t h e F e 3 + i o n t h i s m e a n s t h a t a l l t e r m s u p t o fifth o r d e r m u s t b e c o n s i d e r e d i n a g e n e r a l t r e a t m e n t ; w e c a n w r i t e t h i s g e n e r a l fifth o r d e r s p i n H a m i l t o n i a n i n t h e f o r m HFe3+ = G Q 1 Q 2 Q 3 Q 4 Q 5 ( # ) SaiSa2Sa3Sa4Sa5 + g^^SaHp + gnu.n[Aal3SaIl3 + Raf3HaIp] (2.20) w h e r e t h e S{ai] c a n b e a n y o f Sx, Sy, Sz o r t h e i d e n t i t y 1, t h e p r e f a c t o r G c a n b e a f u n c t i o n o f m a g n e t i c field, a n d w e h a v e e x p l i c i t l y s e p a r a t e d o u t t h e Z e e m a n t e r m . W e Chapter 2. Effective Hamiltonians 3 8 —* o n l y k e e p t o first o r d e r i n t e r m s c o n t a i n i n g t h e n u c l e a r s p i n I. T h e t e r m s i n t h e s i n g l e i o n s p i n H a m i l t o n i a n m u s t i n h e r i t a n y s y m m e t r y t h a t t h e c r y s t a l l i n e e l e c t r i c field p o s s e s s e s , w h i c h r e d u c e s t h e t o t a l n u m b e r o f p o s s i b l e t e r m s . U n f o r t u n a t e l y i n o u r c a s e t h e Fe3+ i o n s d o n o t s i t i n p o s i t i o n s o f h i g h s y m m e t r y . H o w e v e r t h e s y m m e t r y i s c l o s e t o b e i n g c u b i c - t h e n e a r e s t n e i g h b o u r s o f t h e Fe3+ a r e d i s t r i b u t e d i n a d i s t o r t e d o c t a h e d r a l f a s h i o n . T o g i v e a c o n c r e t e e x a m p l e o f h o w t h e c r y s t a l field s y m m e t r y s e l e c t s s p e c i f i c t e r m s o u t o f t h e g e n e r a l 2Sth o r d e r s p i n H a m i l t o n i a n , c o n s i d e r t h e c a s e o f a n Fe3+ i o n i n a c u b i c c r y s t a l field. T h i s c a s e h a s b e e n p r e v i o u s l y t r e a t e d , w i t h r e s u l t i n g s p i n H a m i l t o n i a n [77] ( c o m p a r e t o (2.20). I t i s f o u n d t h a t t h e c o n s t a n t s gi, g2 a n d g3 a r e a l w a y s v e r y s m a l l a n d t h e r e f o r e t h e t e r m s p r o p o r t i o n a l t o t h e s e a r e u s u a l l y n e g l e c t e d [77]. T y p i c a l v a l u e s f o r a r a n g e f r o m 0.1 mK t o 3 mK [88]. A Useful Approximation M e a s u r e m e n t s o f ga/3 i n o t h e r i n s u l a t i n g m a t e r i a l s c o n t a i n i n g Fe3+ s u p p o r t t h e f o l l o w i n g a p p r o x i m a t i o n [89]. W e t a k e t h e t e n s o r ga^ t o b e i s o t r o p i c a n d f u r t h e r m o r e t h a t ga/3 = g = 2, t h e s p i n - o n l y v a l u e . T h e j u s t i f i c a t i o n f o r d o i n g t h i s c o m e s f r o m t h e L = 0 n a t u r e o f t h e g r o u n d s t a t e - p e r t u r b a t i o n s o f t h e s p i n l e v e l s c o m i n g f r o m c r y s t a l field a n d / o r s p i n - o r b i t c o u p l i n g w i l l b e v e r y s m a l l a n d t h e r e f o r e a s p i n - o n l y a p p r o x i m a t i o n f o r ga/3 i s j u s t i f i e d . H) rcubic Fe3+ — + + | (S4 + S4 + St) + ^HBSalp + 9i{S*Hx + S3Hy + S3HZ) g2(S5x + S5yHy + SlHz) + gnfxn [A^SJp + R^HJp] g3 [Sx(S4y + S4Z)HX + Sy(St + S$)Hy + SZ(S4X + S4y)Hy] (2.21) Chapter 2. Effective Hamiltonians 3 9 2.4 The Single Molecule Effective Hamiltonian W e n o w w a n t t o c o n s i d e r t h e H a m i l t o n i a n o f a n e n t i r e Fes m o l e c u l e . W e l a b e l t h e s p i n H a m i l t o n i a n (2.20) Hp f o r t h e pth i r o n i o n ( t h e r e a r e e i g h t o f t h e s e ) . N o w b e c a u s e e a c h s i t s i n a d i s t i n c t c r y s t a l field, w i t h d i f f e r e n t p r i n c i p l e a x e s , i t i s v e r y d i f f i c u l t ( a l t h o u g h i n p r i n c i p l e p o s s i b l e ) t o w r i t e d o w n t h e s p e c i f i c t e r m s f o r e a c h i o n . F o r e x a m p l e , i f w e w e r e t o m a k e t h e a p p r o x i m a t i o n t h a t t h e c r y s t a l field f o r e a c h i s e x a c t l y c u b i c a n d k e p t o n l y t h e t w o d o m i n a n t t e r m s i n (2.21), i e . H?& ~ ^(s4x + S4y + St)+gu.BH-S (2.22) t h e n w e i m m e d i a t e l y r u n i n t o t h e f o l l o w i n g t e c h n i c a l p r o b l e m . T h e cLXGS j clS d e f i n e d b y t h e l o c a l c r y s t a l fields f o r e a c h Fe3+, a r e d i f f e r e n t . T h i s m e a n s t h a t i f w e fix t h e a x e s s u c h t h a t f o r o n e o f t h e i o n s (2.22) i s c o r r e c t , t h e n i n w r i t i n g d o w n a l l t h e o t h e r s i n g l e i o n s p i n H a m i l t o n i a n s w e h a v e t o r o t a t e t h e a x e s t h r o u g h s o m e ( a l b e i t k n o w n ) a n g l e s f r o m t h e " n a t u r a l " b a s i s p i c k e d o u t b y t h e c r y s t a l field. T h i s h a s t h e e f f e c t o f h i d i n g t h e s y m m e t r y e x p l i c i t i n (2.22), a n d b r i n g i n g u s b a c k t o a g e n e r a l t y p e o f d e s c r i p t i o n l i k e ( 2.20). W e s h a l l t h e r e f o r e a t t h i s s t a g e w r i t e t h e c o l l e c t i o n o f e i g h t i r o n s i n g l e i o n s p i n H a m i l t o n i a n s a s H = £ Hk>+ = £ [GriQlQiai(#) S S ^ S S ^ S S , + 9PBSp -H + gnpu.n [A^s*!* P=I p—i + ErpHaI%]] (2.23) w h e r e f o r a l l q u a n t i t i e s t h e l a b e l p p o i n t s t o t h e pth i r o n i o n . Chapter 2. Effective Hamiltonians 4 0 2.4.1 Inclusion of Exchange and Superexchange Terms I t i s k n o w n t h a t t h e r e e x i s t e x c h a n g e a n d s u p e r e x c h a n g e c o u p l i n g s b e t w e e n t h e i r o n i o n s . I n g e n e r a l t h e s e c o u p l i n g s c a n b e a n i s o t r o p i c , l e a d i n g t o a g e n e r a l e x p r e s s i o n o f t h e f o r m Hex = £ JgS>S} (2.24) p<q w h e r e t h e s u m s o v e r p, q a r e o v e r t h e i o n s 1..8 a n d t h e l a b e l s a, j3 r e f e r t o s p a t i a l d i r e c t i o n s x, y, z. E x a c t d i a g o n a l i z a t i o n s t u d i e s o n t h i s t e r m h a v e b e e n p e r f o r m e d , w i t h r e s u l t s c o m p a r e d t o E P R a n d s u s c e p t i b i l i t y m e a s u r e m e n t s p e r f o r m e d o n F e 8 [18]. T h e m o d e l u s e d i n t h e s e c a l c u l a t i o n s a s s u m e s t h a t t h e c o u p l i n g s a r e i s o t r o p i c , a l t h o u g h i t i s n o t c l e a r t h a t t h i s h a s t o b e t h e c a s e ( D z y a l o s k i n s k i - M o r y a i n t e r a c t i o n s , f o r e x a m p l e , a r e e x c l u d e d i n t h e i s o t r o p i c c a s e [90]). T h e p a t h w a y s , a s w e l l a s t h e m a g n i t u d e o f t h e c o u p l i n g s t r e n g t h s e x t r a c t e d f r o m f i t s t o e x p e r i m e n t , a r e s h o w n i n f i g u r e 2.1. N o t e t h a t t h e c o u p l i n g e n e r g i e s e x t r a c t e d a r e f a r l a r g e r t h a n t h e s i n g l e - i o n a n i s o t r o p y t e r m s - a s we h a v e s e e n , t h e t e r m s i n t h e s i n g l e i o n s p i n H a m i l t o n i a n a r e t y p i c a l l y o n t h e o r d e r o f mK, w h i l e t h e e n e r g i e s e x t r a c t e d b y D e l f s e t . a l . [18] a r e i n t h e t e n s t o h u n d r e d s o f K e l v i n . 2.4.2 "Offsite" Dipolar and Quadrupolar Contributions T h e e l e c t r o n i c s t a t e o f e a c h Fe3+ i o n w i l l c o u p l e v i a d i p o l e - d i p o l e i n t e r a c t i o n s w i t h a l l t h e o t h e r d i p o l e s p r e s e n t i n t h e m o l e c u l e . T h i s i n c l u d e s t h e o t h e r Fe3+ e l e c t r o n i c s p i n s a n d a l l t h e n u c l e a r s p i n s i n t h e m o l e c u l e . A s w e l l , a l l n u c l e i i n t h e s y s t e m w i t h s p i n s I > 1/2 w i l l h a v e a n e l e c t r i c q u a d r u p o l e m o m e n t w h i c h c o u p l e s t o i n h o m o g e n e o u s e l e c t r i c fields i n t h e m o l e c u l e . I t w i l l t u r n o u t t h a t t h e s e c o n t r i b u t i o n s w i l l b e q u i t e i m p o r t a n t i n t h e c h a p t e r s t h a t f o l l o w . A s s u c h w e s h a l l g o o v e r t h e d e r i v a t i o n o f t h e s e t e r m s c a r e f u l l y . T h e g e n e r a l i n t e r a c t i o n b e t w e e n t h e e l e c t r o n i c a n d n u c l e a r d e g r e e s o f f r e e d o m m a y b e s p l i t i n t o t w o p a r t s [64]. T h e first o f t h e s e , t h e e l e c t r o s t a t i c i n t e r a c t i o n b e t w e e n n u c l e a r a n d e l e c t r o n i c c h a r g e s , w i l l b e c o n s i d e r e d first. T h e s e c o n d p a r t , t h e m a g n e t i c c o u p l i n g Chapter 2. Effective Hamiltonians 41 F i g u r e 2.1: E x c h a n g e p a t h w a y s i n Fe8 i n t h e i s o t r o p i c m o d e l o f D e l f s e t . a l . [18]. F i t s t o s u s c e p t i b i l i t y d a t a g i v e Ju ~ 3bK, J i 3 ~ 180K, J 1 5 ~ 22K a n d J 3 5 ~ 52K, w i t h a l l c o u p l i n g s a n t i f e r r o m a g n e t i c . b e t w e e n t h e m o m e n t s o f t h e n u c l e i a n d t h e m a g n e t i c f i e l d s g e n e r a t e d b y t h e s p i n a n d o r b i t a l c u r r e n t s o f t h e e l e c t r o n s , w i l l b e d e a l t w i t h l a t e r . T h e s t a n d a r d d e r i v a t i o n [64] o f t h e f o r m o f t h e e l e c t r o s t a t i c c o u p l i n g s b e g i n s b y d e s c r i b i n g t h e n u c l e i a n d e l e c t r o n i c c l o u d s a s c l a s s i c a l c h a r g e d i s t r i b u t i o n s pn{rn) a n d Peife) w i t h m u t u a l e l e c t r o s t a t i c e n e r g y Pe(re)Pn{rn)dredrn E (2.25) T h i s e x p r e s s i o n m a y b e s i m p l i f i e d b y e x p a n d i n g t h e C o u l o m b p o t e n t i a l i n t e r m s o f s p h e r - i c a l h a r m o n i c s . A f t e r s o m e a l g e b r a o n e c a n s h o w t h a t t h i s e n e r g y i s t h e e x p e c t a t i o n v a l u e Chapter 2. Effective Hamiltonians 4 2 o f t h e H a m i l t o n i a n Helec = £ A™B\ i m * (2.26) l,m w h e r e BY1 = - e A? = e 21 + lf 21 + lf i=l i-l (2.27) (2.28) H e r e Nn a n d Ne a r e t h e n u m b e r o f p r o t o n s p e r n u c l e u s a n d e l e c t r o n s r e s p e c t i v e l y , {Ri, Qi, <frj} a n d {r^, fy, 0j} a r e t h e p o l a r c o o r d i n a t e s o f t h e ith p r o t o n a n d ith e l e c t r o n r e s p e c t i v e l y a n d Y™ a r e t h e s t a n d a r d s p h e r i c a l h a r m o n i c s . T h e g e n e r a l e x p r e s s i o n (2.26) m a y b e f u r t h e r s i m p l i f i e d b y t h e f o l l o w i n g o b s e r v a t i o n s . I t i s w e l l - k n o w n t h a t s t a t i o n a r y n u c l e a r s t a t e s h a v e w e l l - d e f i n e d p a r i t i e s , i m p l y i n g t h a t t h e e x p e c t a t i o n v a l u e s o f t e r m s A™ o d d i n / a r e z e r o . T h a t t h i s i s i n f a c t t r u e h a s b e e n c o n f i r m e d e x p e r i m e n t a l l y t o a h i g h d e g r e e o f p r e c i s i o n w i t h t h e / = 1 ( e l e c t r i c d i p o l e ) t e r m [64]. T h i s l e a v e s u s w i t h a s u m o v e r e v e n /, w h i c h c a n b e m a n i p u l a t e d f u r t h e r . T h e I = 0 ( m o n o p o l e ) t e r m c a n b e s e e n t o b e c o n s t a n t , a n d c a n t h e r e f o r e b e o m i t t e d f r o m c o n s i d e r a t i o n . A s w e l l , i t i s s e e n e x p e r i m e n t a l l y t h a t t h e s t r e n g t h o f t h e t e r m s i n t h e s e r i e s d e c r e a s e r a p i d l y w i t h i n c r e a s i n g I. I n f a c t , d i r e c t c o n t r i b u t i o n s f r o m t h e / = 4 t e r m a r e s o w e a k t h a t t h e y h a v e n e v e r b e e n s e e n i n N M R s t u d i e s i n s o l i d s . W e m a y t h e r e f o r e o m i t a l l t e r m s w i t h I > 2, l e a v i n g o n l y t h e e l e c t r i c q u a d r u p o l a r t e r m HQ= J2 A?B™* (2.29) m = - 2 T h e t e n s o r s a p p e a r i n g i n (2.29) c a n b e r e c a s t i n a m o r e u s e f u l f o r m . O n e c a n s h o w [64, 91] t h a t t h e H a m i l t o n i a n (2.29) p r o d u c e s t h e s a m e m a t r i x e l e m e n t s i n t h e n u c l e a r Chapter 2. Effective Hamiltonians 4 3 s p i n s u b s p a c e a s t h e f o r m N eQk 6 / f c ( 2 / f c - l ) (2.30) w h e r e N i s t h e t o t a l n u m b e r o f n u c l e i , Q2yk (2.31) dxadxP w h e r e Vk i s t h e p o t e n t i a l a t t h e kth n u c l e u s d u e t o t h e c h a r g e d i s t r i b u t i o n i n t h e m o l e c u l e , a n d Qk i s t h e e l e c t r i c q u a d r u p o l e m o m e n t o f t h e kth n u c l e u s , w h i c h i s a m e a s u r e o f t h e n o n s p h e r i c a l d i s t r i b u t i o n o f c h a r g e i n s i d e t h e n u c l e u s . Qk i s z e r o f o r a l l n u c l e a r s p i n s w i t h Ik = 1/2 a n d i s t y p i c a l l y o n t h e o r d e r o f 1 - 10 • 1 0 - 2 4 cm2. I n Fe8, t h e n u c l e i t h a t h a v e n o n - z e r o e l e c t r i c q u a d r u p o l e m o m e n t s a r e t h e t w o s p e c i e s o f b r o m i n e 79Br a n d 81Br, 1 4 J V a n d 1 6 0 . T h e d e s c r i p t i o n (2.30) a c c o u n t s f o r a l l e l e c t r o s t a t i c e f f e c t s b e t w e e n e l e c t r o n i c a n d n u c l e a r c h a r g e d i s t r i b u t i o n s i n t h e m o l e c u l e . W e n o w t u r n t o t h e s e c o n d p a r t o f t h e e l e c t r o n - n u c l e u s i n t e r a c t i o n , t h e m a g n e t i c c o u p l i n g s . W e h a v e a l r e a d y u s e d t h e f a c t t h a t s t a t i o n a r y n u c l e a r s t a t e s h a v e a f i x e d p a r i t y t o e l i m i n a t e c e r t a i n t e r m s i n t h e m u l t i p o l e e x p a n s i o n o f t h e e l e c t r o s t a t i c c o u p l i n g . W e m a y p e r f o r m a s i m i l a r t r i c k h e r e b y n o t i n g t h a t t h e m a g n e t i c field f e e l s t h e o p p o s i t e p a r i t y e f f e c t o f t h e e l e c t r i c field ( s i n c e t h e y a r e a x i a l a n d p o l a r v e c t o r s r e s p e c t i v e l y ) . T h i s m e a n s t h a t a l l e v e n o r d e r s o f t h e m u l t i p o l e e x p a n s i o n o f t h e m a g n e t i c s t r u c t u r e o f t h e n u c l e u s h a v e t o b e z e r o . S i m i l a r l y t o t h e e l e c t r o s t a t i c c a s e , t h e s t r e n g t h o f t h e c o n t r i b u t i o n s o f h i g h e r o r d e r t e r m s i n t h e e x p a n s i o n f a l l o f f e x t r e m e l y r a p i d l y w i t h i n c r e a s i n g /; a s p r e v i o u s l y , n o e v i d e n c e o f / = 3 ( m a g n e t i c o c t o p o l a r ) t e r m s h a s e v e r b e e n d i r e c t l y s e e n b y N M R i n b u l k m a t t e r . T h i s m a k e s o u r j o b c o n s i d e r a b l y e a s i e r f r o m t h e s t a r t , a s w e c a n t r e a t t h e g e n e r a l n u c l e u s a s a m a g n e t i c d i p o l e w i t h o u t l o s s o f g e n e r a l i t y , a n d t h e i n t e r a c t i o n o f t h e m a g n e t i c (2.32) Chapter 2. Effective Hamiltonians 4 4 f i e l d g e n e r a t e d b y t h e e l e c t r o n s a n d a n y m a g n e t i c d i p o l e c a n e a s i l y b e d e a l t w i t h . O n e f i n d s a n i n t e r a c t i o n H a m i l t o n i a n o f t h e f o r m [64] .. _ 8 N+8 ^ ^ E E ^ F 1 [3 • h - 3 $ • rIfc)(/* • rlk)] (2.33) 4 7 r 1=1 fc=i rlk w h e r e w e h a v e u s e d o u r i s o t r o p i c g a p p r o x i m a t i o n , t h e s u m o v e r e l e c t r o n i c s p i n s i s o v e r t h e e i g h t Fe3+ s i t e s a n d t h e s u m o v e r n u c l e a r s i t e s i s o v e r a l l p o s s i b l e n u c l e a r s i t e s (N i s t h e n u m b e r o f s p i n s i n t h e l i g a n d b a t h ; a c c o u n t i n g f o r t h e p o s s i b i l i t y o f u p t o e i g h t 57Fe n u c l e i g i v e s k = l . . i V + 8 ) . W e d o n o t i n c l u d e t h e " s e l f - c o u p l i n g " t e r m h e r e w h e r e t h e e l e c t r o n i c s p i n i n t e r a c t s w i t h a 57Fe n u c l e u s o n t h e s a m e s i t e , a s t h i s t e r m i s i n c l u d e d i n t h e s i n g l e i o n s p i n H a m i l t o n i a n d e r i v e d e a r l i e r ( i t i s a " c o n t a c t " t e r m ) . gnic a n d u.n a r e t h e n u c l e a r p - f a c t o r o f t h e kth n u c l e u s a n d n u c l e a r m a g n e t o n r e s p e c t i v e l y . I t i s p o s s i b l e t o r e w r i t e t h i s t e r m b y d e f i n i n g a m a t r i x Mlk{rlk) = — — - 3 - ' Ik 1 ~ Sffkx -3flkxrlky -3flkxflkz -3rlkxrlky 1 - 3rfky -2>rikyrik z -3rikx?ikz -3fikyfikz 1 - 3ffkz T h i s a l l o w s u s t o w r i t e (2.33) i n t h e s i m p l e f o r m 8 N+8 Hmag = £ £ Qn^M^ SlJ* (2.35) 1=1 k=l w h e r e a a n d /3 a r e a g a i n s p a t i a l l a b e l s x, y o r z. N o t e t h a t o n e m a y c a l c u l a t e e x a c t l y t h e v a l u e s o f t h e s e t e r m s , a s t h e y d e p e n d o n l y o n t h e r e l a t i v e l o c a t i o n s o f t h e n u c l e i a n d t h e i r o n i o n s w h i c h a r e k n o w n f r o m c r y s t a l l o g r a p h i c d a t a ( t h i s w i l l t u r n o u t t o b e q u i t e i m p o r t a n t i n l a t e r c h a p t e r s , p a r t i c u l a r l y c h a p t e r s 3 a n d 5 ) . T o g e t h e r w i t h t h e t e r m d e r i v e d e a r l i e r (2.30) w e h a v e a c o m p l e t e d e s c r i p t i o n o f t h e " o f f s i t e " ( i e . n o t i n c l u d i n g o n - s i t e c o n t a c t h y p e r f i n e i n t e r a c t i o n s ) e l e c t r o m a g n e t i c i n t e r - a c t i o n o f a l l t h e n u c l e i w i t h a l l t h e e l e c t r o n i c s p i n s i n t h e Fe8 m o l e c u l e . W e m a y w r i t e (2.34) Chapter 2. Effective Hamiltonians 4 5 t h e final d e s c r i p t i o n o f t h i s n u c l e a r - e l e c t r o n i c c o u p l i n g i n t h e f o r m (2.36) 2.4.3 Intra-Nuclear Spin Couplings A s i m i l a r a n a l y s i s m a y b e p e r f o r m e d o n t h e e l e c t r o m a g n e t i c c o u p l i n g s b e t w e e n t h e n u c l e i t h e m s e l v e s . A s w a s i n d i c a t e d i n t h e p r e v i o u s s e c t i o n , b y f a r t h e d o m i n a n t c o n t r i b u t i o n t o t h i s e f f e c t c o m e s f r o m d i p o l e - d i p o l e i n t e r a c t i o n s o f t h e f o r m w h e r e gni a n d gnk a r e t h e n u c l e a r ^ - f a c t o r s f o r t h e I a n d k s p e c i e s o f n u c l e i . 2.4.4 Couplings of the Nuclear Bath to External Magnetic Fields W e s h a l l a s s u m e t h a t t h e n u c l e i i n t h e l i g a n d b a t h c o u p l e t o a n a p p l i e d e x t e r n a l m a g n e t i c field i n t h e s t a n d a r d Z e e m a n way; t h a t i s , 2.4.5 Coupling to Phonons B o t h t h e s u p e r - e x c h a n g e / e x c h a n g e a n d n u c l e a r s p i n c o u p l i n g s a r e l o c a l i n t h e s e n s e t h a t t h e i r r a n g e s d o n o t e x t e n d o u t s i d e o f a g i v e n m o l e c u l e ; i t i s e n o u g h t o t r e a t t h e i r e f f e c t s o n a " p e r m o l e c u l e " b a s i s . I n t h i s s e c t i o n w e s h a l l , f o l l o w i n g t h e t r e a t m e n t g i v e n i n [92], i n t r o d u c e c o u p l i n g s o f a s i n g l e m o l e c u l e w i t h p h o n o n fields w h i c h a t l o w e n e r g i e s h a v e l o n g w a v e l e n g t h s a n d a s s u c h o w e t h e i r p r o p e r t i e s t o t h e d e t a i l s o f t h e c r y s t a l l a t t i c e . T h e p r e s e n c e o f p h o n o n s i n t h e c r y s t a l c a n b e d e a l t w i t h b y i n t r o d u c i n g s t r a i n a n d r o t a t i o n fields w h i c h a r e w r i t t e n r e s p e c t i v e l y a s [92] (2.37) Hext = 2\^9nkPnIk ' H( k ext (2.38) UJ, = - [daUj - d 7 i t Q ) Chapter 2. Effective Hamiltonians 4 6 w h e r e M O = £ kX h 1 1/2 <4A) [4A + O*A] a r e t h e d i s p l a c e m e n t s a t f c a u s e d b y a p h o n o n field w h o s e c r e a t i o n / a n n i h i l a t i o n o p e r a t o r s a r e g i v e n b y {a\.x,ak\} a n d i s t h e m a g n i t u d e o f t h e p o l a r i z a t i o n v e c t o r . A l a b e l s t h e b r a n c h a n d i n c l u d e s i n g e n e r a l b o t h o p t i c a l a n d a c o u s t i c p h o n o n s . M i s t h e m a s s o f t h e u n i t c e l l a n d N i s t h e t o t a l n u m b e r o f u n i t c e l l s . T h e s e l o c a l s t r a i n s a n d r o t a t i o n s c a u s e t h e r e t o b e i n t r o d u c e d i n t o t h e H a m i l t o n i a n t e r m s m e d i a t e d b y t h e p h o n o n s . F o r e x a m p l e , i m a g i n e t h e e f f e c t o f a l o c a l r o t a t i o n o f a p a r t i c u l a r s p i n o n t h e DijSiSj t e r m i n (2.20), a n d i n p a r t i c u l a r o n t h e DZZSZSZ c o m p o n e n t ; SZ y SZ + U)ZXSX + U)ZySy (2.39) a n d DZZS2Z -> Dzz (s2 + u2zx + u 2 y u z x { S z , Sx} + u z y { S z , Sy}) (2.40) I n g e n e r a l o n e c a n s h o w [92] t h a t t h e s p i n o r b i t i n t e r a c t i o n l e a d s t o a n e f f e c t i v e c o u p l i n g b e t w e e n t h e i o n i c s p i n s a n d t h e p h o n o n fields o f t h e f o r m h 1/2 n 2NMuqX_ H (2.41) w h e r e VqX(St) = B%S?S? + (4 x q) • St (2.42) H e r e ̂  i s t h e p o l a r i z a t i o n v e c t o r o f t h e p h o n o n i n v o l v e d . T h e t e r m s c a n i n p r i n c i p l e b e c a l c u l a t e d f r o m k n o w l e d g e o f t h e c r y s t a l s y m m e t r y a n d h a v e b e e n e s t i m a t e d t o b e ~ 0.01 mK f o r t h e M n 1 2 m a t e r i a l [93]. O n e m u s t k e e p i n m i n d t h a t i n o r d e r t o d o Chapter 2. Effective Hamiltonians 4 7 t h i s o n e m u s t u s e t h e s y m m e t r i e s a p p r o p r i a t e f o r t h e i n d i v i d u a l i o n s a n d n o t t h e f u l l —# s y m m e t r y o f t h e c r y s t a l . T h e t e r m l i n e a r i n S a r i s e s f r o m t r e a t i n g t h e s p i n a s b e i n g l o c k e d t o t h e l a t t i c e d u e t o s p i n - o r b i t e f f e c t s . T h i s t e r m i s l i k e l y t o b e m u c h s m a l l e r t h a n t h e Bap t e r m s (see [92] f o r a d i s c u s s i o n o f t h i s p o i n t ) . 2.4.6 Coupling to Photons T h e c o u p l i n g t o t h e p h o t o n field i s t a k e n t o b e o f m a g n e t i c d i p o l a r o r i g i n [94]. T h e m a g n e t i c field d u e t o t h e p h o t o n field c a n b e w r i t t e n B 7 = V x l = E 2 7 ^ 1 / 2 Vukx ^ f ( c w + C y ( V x 4 ) (2.43) w h e r e V i s t h e v o l u m e o f t h e s a m p l e , {c\x, ck\} a r e t h e c r e a t i o n / a n n i h i l a t i o n o p e r a t o r s f o r t h e p h o t o n field a n d ̂ A i s t h e p o l a r i z a t i o n v e c t o r f o r t h e {k, A } m o d e . T h e c o u p l i n g t o t h e e l e c t r o n i c s p i n s i s t h e n s t r a i g h t f o r w a r d a n d i s g i v e n b y Hsl = J2gpBB1-Sl (2.44) 2.4.7 Bringing all the Terms Together-The Bare Fe8 Hamiltonian T h e d e s c r i p t i o n o f t h e s y s t e m s u n d e r s c r u t i n y g i v e n i n t h e p r e v i o u s s e c t i o n c o n t a i n s s e v e n t e r m s , n a m e l y (2.23), (2.24), (2.36), (2.37), (2.38), (2.41) a n d ( 2 . 4 4 ) . E x p l i c i t l y w e h a v e , f o r o u r e f f e c t i v e H a m i l t o n i a n , a n e x p r e s s i o n o f t h e f o r m H = J2[G?a>a>a*'*(ii) S^a2Spa3S^a5+gpBSp-H + 9npPn[Apa0S^ + R^HaI^}]+ £ <tfS>S} p<q=l N+8 + E fc=i Chapter 2. Effective Hamiltonians 4 8 4 7 r J<fc=l rife fc=l 8 9A l-l h 2NMugX 1/2 9 [4A^A(S<) - a*V&(3)] + £ g^B, • St (2.45) /=i L e t u s r e v i e w w h a t w e k n o w a b o u t e a c h t e r m i n t h i s e x p r e s s i o n . T h e first t e r m i s a s u m o v e r a l l t h e s i n g l e i o n Fe3+ s p i n H a m i l t o n i a n s . I n o r d e r t o e v a l u a t e t h e c o u p l i n g e n e r g i e s C 7 p l Q 2 Q 3 Q 4 Q 5 w e c o u l d d o t h e f o l l o w i n g . F i r s t , w e i d e n t i f y t h e s y m m e t r y o f t h e c r y s t a l l i n e field s u r r o u n d i n g e a c h i r o n i o n . T h i s r e d u c e s t h e n u m b e r o f n o n - z e r o c o u p l i n g s ( a s w e s a w i n t h e e x a c t l y c u b i c c a s e i n ( 2 . 2 2 ) ) . W e t h e n p i c k a s e t o f a x e s , a l i g n e d s o a s t o s i m p l i f y t h e s p i n H a m i l t o n i a n o f o n e o f t h e i o n s . N e x t w e d e t e r m i n e t h e a n g l e s n e c e s s a r y t o r o t a t e e a c h i r o n i o n f r o m t h e a x e s c h o s e n b y i t s l o c a l c r y s t a l field t o o u r c h o s e n b a s i s a n d a p p l y t h e s e r o t a t i o n s . T h i s p r o c e d u r e a l l o w s u s t o a p p r o x i m a t e l y e v a l u a t e t h e G ^ 1 0 2 0 ^ " 4 0 ^ for Fes- W e d o n o t k n o w f o r c e r t a i n w h a t t h e m a g n i t u d e s o f t h e s e c o u p l i n g s a r e , a l t h o u g h a s m e n t i o n e d e a r l i e r f o r Fe3+ i n a c u b i c field t h e y a r e t y p i c a l l y i n t h e mK r a n g e [77]. W e s h a l l n o t e x p l i c i t l y p e r f o r m t h i s t a s k f o r a r e a s o n t h a t w i l l s o o n b e m a d e c l e a r . T h i s t e r m a l s o c o n t a i n s c o u p l i n g s b e t w e e n t h e e l e c t r o n i c s p i n a n d e x t e r n a l field, w h i c h w e h a v e a p p r o x i m a t e d a s b e i n g i s o t r o p i c (ga^ ~ g = 2), t h e " c o n t a c t " i n t e r a c t i o n b e t w e e n t h e n u c l e a r a n d e l e c t r o n i c s p i n a n d t h e i n t e r a c t i o n b e t w e e n t h e e x t e r n a l field a n d t h e n u c l e a r s p i n . T h e s e t w o l a s t w i l l n o t i n g e n e r a l b e i s o t r o p i c . T h e s e c o n d t e r m i s t h e e x c h a n g e / s u p e r e x c h a n g e t e r m c o u p l i n g t h e e l e c t r o n i c w a v e - f u n c t i o n s o f t h e i r o n i o n s . E x t r a c t i n g q u a n t i t a t i v e p r e d i c t i o n s a b o u t h o w l a r g e t h e a r e i s a d i f f i c u l t t a s k a n d h a s y e t t o b e p e r f o r m e d s a t i s f a c t o r i l y [18]. I t i s n o t k n o w n w h e t h e r t h e a n i s o t r o p i c i t y h e r e i s i m p o r t a n t . A s w e m e n t i o n e d e a r l i e r , p r e l i m i n a r y i n - v e s t i g a t i o n s i n d i c a t e t h a t t h e s e c o u p l i n g s a r e i n t h e t e n s t o h u n d r e d s o f K e l v i n [18]. T h e t h i r d , f o u r t h a n d fifth t e r m s a r e t h e n o n - l o c a l c o u p l i n g b e t w e e n n u c l e a r s p i n s a n d Fe3+ e l e c t r o n i c s p i n s , n u c l e a r - n u c l e a r d i p o l e i n t e r a c t i o n s a n d n u c l e a r s p i n - e x t e r n a l Chapter 2. Effective Hamiltonians 4 9 f i e l d c o u p l i n g r e s p e c t i v e l y . A l l o f t h e q u a n t i t i e s i n t h e s e t e r m a r e e i t h e r k n o w n o r c a n b e c a l c u l a t e d ( w e d o t h i s e x p l i c i t l y f o r a l l t h e n u c l e i i n t h e m o l e c u l e i n c h a p t e r 3 ) . T h e c o u p l i n g e n e r g i e s h e r e a r e f o u n d t o b e b o u n d e d a b o v e b y ~ 5 mK. T h e s i x t h t e r m i s t h e p h o n o n - e l e c t r o n i c s p i n c o u p l i n g . H e r e t h e o n l y t e r m t h a t w e d o n o t k n o w i s t h e v a l u e o f t h e Bap t e r m s i n ( 1 . 2 1 ) , a l t h o u g h t h e s e h a v e b e e n e s t i m a t e d t o b e o n t h e o r d e r o f 0.01 mK i n Mn\2 [93]. T h e s e v e n t h a n d final t e r m i s t h e d i p o l a r m a g n e t o - o p t i c a l c o u p l i n g . H e r e t h e p h o t o n field i n t h e m a t e r i a l w i l l b e c h a n g e d t o a r e n o r m a l i z e d ( b y o p t i c a l p h o n o n s ) " p o l a r i t o n " field a n d t h e r e f o r e t h e f r e q u e n c i e s Cbkx a r e n o t k n o w n e x a c t l y , a l t h o u g h m e t h o d s t o a p p r o x i m a t e t h e s e a r e a v a i l a b l e [95]. 2.5 Exchange/Superexchange and the Giant Spin Picture A s w a s d i s c u s s e d i n t h e i n t r o d u c t o r y c h a p t e r , t h e l o w - e n e r g y p h e n o m e n o l o g y o f t h e F e 8 s y s t e m i n d i c a t e s t h a t s o m e h o w t h e e l e c t r o n i c s p i n s " l o c k t o g e t h e r " i n t o s o m e fixed-spin o b j e c t . B e c a u s e t h e e x c h a n g e / s u p e r e x c h a n g e c o u p l i n g e n e r g i e s a r e m u c h l a r g e r t h a n a l l t h e o t h e r e n e r g y s c a l e s i n o u r H a m i l t o n i a n (2.45), w e a r e p r e s e n t e d w i t h a m e c h a n i s m w h e r e b y w e c a n u n d e r s t a n d h o w t h i s c a n h a p p e n . I n a n u m e r i c a l d i a g o n a l i z a t i o n s t u d y p e r f o r m e d i n [18], i t w a s s u g g e s t e d t h a t t h e g r o u n d s t a t e o f t h e H a m i l t o n i a n (2.24) i s g i v e n b y a s t a t e w h e r e s i x o f t h e Fe3+ a l i g n p a r a l l e l t o e a c h o t h e r w h i l e t h e o t h e r t w o a l i g n t h e m s e l v e s a n t i - p a r a l l e l , g i v i n g a " q u a n t u m r o t a t o r " w i t h e x c e s s s p i n o f S = 6 • 5/2 — 2 • 5/2 = 10. B e c a u s e t h e jff a r e l a r g e , t h e r e e x i s t s a s i z e a b l e g a p t o e x c i t a t i o n s o u t o f t h i s g r o u n d s t a t e , w h o s e m a g n i t u d e t h e s e a u t h o r s s u g g e s t i s o n t h e o r d e r o f AE ~ 30K. T h i s l o c k i n g t o g e t h e r o f t h e e l e c t r o n i c s p i n s p r o f o u n d l y a f f e c t s t h e u l t i m a t e f o r m o f t h e l o w - e n e r g y e f f e c t i v e d e s c r i p t i o n . T o b e g i n w i t h , a s e x c i t a t i o n s f r o m t h e g r o u n d s t a t e a r e e n e r g e t i c a l l y i n a c c e s s i b l e a s l o n g a s kBT <C AE ( w e w i l l u l t i m a t e l y b e i n t e r e s t e d i n Chapter 2. Effective Hamiltonians 50 t h e mK r a n g e s o t h i s i s r e a s o n a b l e ) , w e m a y c o n s i d e r t h e t e r m i n o u r e f f e c t i v e d e s c r i p t i o n (2.24) t o b e s i m p l y a c o n s t a n t w h i c h w e h e n c e f o r t h r e m o v e f r o m c o n s i d e r a t i o n . N o t e t h a t t h i s d o e s n o t m e a n t h a t t h e d y n a m i c s o f t h e e l e c t r o n i c s p i n s a r e f r o z e n - i t i s s i m p l y t h a t t h e e f f e c t i v e d e g r e e o f f r e e d o m t h a t t h e y r e p r e s e n t i s , f o r ksT <^ AE, a s i n g l e c o l l e c t i v e " q u a n t u m r o t a t o r " o r " g i a n t s p i n " w h i c h i s s t i l l v e r y m u c h a d y n a m i c a l q u a n t i t y . N o w i t i s q u i t e a d i f f i c u l t m a t t e r t o a c t u a l l y derive a n e f f e c t i v e d e s c r i p t i o n i n t e r m s o f t h i s n e w c o l l e c t i v e d e g r e e o f f r e e d o m f r o m t h e H a m i l t o n i a n ( 2 . 4 5 ) . I n o r d e r t o d o t h i s o n e w o u l d h a v e t o first d e t e r m i n e t h e g r o u n d s t a t e o f t h e e l e c t r o n i c s p i n s a n d t h e n p e r f o r m p e r t u r b a t i o n s o u t o f t h i s g r o u n d s t a t e i n a s i m i l a r w a y d o n e f o r t h e s i n g l e i o n c a s e . I n s t e a d w h a t w e s h a l l d o i s , f o l l o w i n g [74], m a k e t h e f o l l o w i n g h y p o t h e s i s . W e s i m p l y a s s u m e t h a t t h e e x c h a n g e / s u p e r e x c h a n g e c o u p l i n g s l o c k t h e e l e c t r o n i c s p i n s t o g e t h e r i n t o a q u a n t u m r o t a t o r o r " g i a n t s p i n " S w i t h S = 10, w h e r e s i x ( t w o ) o f t h e e l e c t r o n i c s p i n s p o i n t p a r a l l e l ( a n t i p a r a l l e l ) t o t h e d i r e c t i o n o f 5, a s i n d i c a t e d b y [18]. T h i s w e r e f e r t o a s t h e giant spin hypothesis. W e t h e n , a s a c o r o l l a r y t o t h i s h y p o t h e s i s , r e w r i t e (2.45) i n t h e f o r m H = s a i i S ^ A . + S ^ ' E ^ + E ^ I ^ W + ^ M P=I p=i N+8 r 8 „ n 2 Af „ „ . . N 4 7 r l<k=l Tlk fc=l 1/2 qX 1=1 h °UV*&) - • St (2.46) /=i 2NMuqX_ w h e r e Sai. — Sx, Sy, Sz o r t h e i d e n t i t y . W h a t we h a v e d o n e h e r e i s r e p l a c e t h e s u m o v e r s i n g l e - i o n s p i n H a m i l t o n i a n s a n d t h e e x c h a n g e / s u p e r e x c h a n g e t e r m s w i t h a g i a n t s p i n H a m i l t o n i a n HGS — Gail-a{™SaiiSai2...SahQ. A s w e d i s c u s s e d i n t h e s i n g l e i o n c a s e , t h e s p i n H a m i l t o n i a n c a n c o n t a i n i n g e n e r a l t e r m s u p t o 2Sth o r d e r i n s p i n o p e r a t o r s , w h i c h Chapter 2. Effective Hamiltonians 51 i n t h i s c a s e i s 20 o r d e r . W e c a n h o w e v e r r e d u c e t h e t o t a l n u m b e r o f t e r m s s i g n i f i c a n t l y i n t h e c a s e o f Fe8, b e c a u s e t h e g i a n t s p i n H a m i l t o n i a n m u s t c o n t a i n t h e s y m m e t r y o f t h e f u l l c r y s t a l l i n e l a t t i c e , w h i c h i n t h e c a s e o f Fe8 i s t r i c l i n i c [23]. N o t e h o w e v e r t h a t w e a r e n o t g o i n g t o d e r i v e a n y o f t h e s e t e r m s . A s w e s a w i n t h e i n t r o d u c t o r y c h a p t e r m u c h e f f o r t h a s b e e n e x p e n d e d i n t r y i n g t o measure w h a t t h e r e l e v a n t c o u p l i n g s <5Qii -ai*> a r e . U n f o r t u n a t e l y b e c a u s e t h e r e a r e s o m a n y o f t h e s e , e v e n a f t e r t h e c r y s t a l s y m m e t r y h a s b e e n t a k e n i n t o a c c o u n t , m o s t h a v e b e e n i g n o r e d a s t h e i r m a g n i t u d e s d e c r e a s e q u i c k l y w i t h i n c r e a s i n g o r d e r i n Sj . T h i s i s h o w e v e r q u i t e d a n g e r o u s a s t h e s e s m a l l i g n o r e d t e r m s c a n c o n t r i b u t e g r e a t l y t o t h e p h y s i c s , i n p a r t i c u l a r t o t h e m a g n i t u d e o f t u n n e l i n g a m p l i t u d e s b e t w e e n d i f f e r e n t \ms > s t a t e s o f t h e g i a n t s p i n ( t w e n t y o n e o f w h i c h , r a n g i n g f o r S = 10 f r o m | - 10 > t o | + 10 >) [25]. T h e r e m a i n i n g t e r m s a r e i d e n t i c a l t o t h e i r f o r m s i n (2.45). T h e o n l y d i f f e r e n c e i s t h a t n o w w e m u s t r e m e m b e r t h a t e a c h i n d i v i d u a l e l e c t r o n i c s p i n i s l o c k e d t o t h e g i a n t s p i n . 2.6 Investigation of the Giant Spin Hamiltonian in the Absence of Environ- mental Couplings I f w e c o m p l e t e l y n e g l e c t a l l e n v i r o n m e n t s i n (2.46) a n d fix S = 53?= i Si w e o b t a i n H = r > i - ^ o SaiiSai2...Sai2Q + gpBH • S (2.47) W e a r e g o i n g t o d i g r e s s s o m e w h a t a t t h i s p o i n t f r o m o u r f o c u s o n F e 8 . W e w i l l i n w h a t f o l l o w s c o n s i d e r a l l p o s s i b l e c r y s t a l s y m m e t r i e s a n d n o t j u s t t h e F e 8 t r i c l i n i c s y m m e t r y . T h i s w e d o b e c a u s e t h e r e a r e o t h e r s i m i l a r m o l e c u l a r m a g n e t s ( M n i 2 , f o r e x a m p l e ) t h a t m a y h a v e g i a n t s p i n d e s c r i p t i o n s o f t h i s s o r t t h a t p o s s e s s d i f f e r e n t c r y s t a l s y m m e t r i e s ( f o r Mni2 t h i s i s t e t r a g o n a l [28]) a n d f o r t h i s r e a s o n i t i s w o r t h w h i l e t o s a y s o m e g e n e r a l t h i n g s a b o u t t h e d e s c r i p t i o n (2.47). Chapter 2. Effective Hamiltonians 52 N o w a s w e d i s c u s s e d e a r l i e r , t h e s p i n H a m i l t o n i a n c a n c o n t a i n t e r m s u p t o 2Sth o r d e r i n t h e s p i n v a r i a b l e s . I t i s h o w e v e r d i f f i c u l t t o k e e p a l l o f t h e s e t e r m s i n o u r d e s c r i p t i o n f o r l a r g e S, a n d f u r t h e r m o r e i t i s n o t c l e a r t h a t a l l o f t h e s e t e r m s c a n b e d i r e c t l y m e a s u r e d a n y w a y . W h a t i s u s u a l l y d o n e i s t h a t o n l y t h e l o w e s t o r d e r t e r m s i n S c o n s i s t e n t w i t h t h e c r y s t a l s y m m e t r y a r e k e p t , a n d a l l h i g h e r o r d e r t e r m s a r e t h r o w n away. I n w h a t f o l l o w s w e s h a l l f o l l o w t h i s t a c k . W e e m p h a s i z e h o w e v e r t h a t e v e n i f t h e h i g h e r o r d e r t e r m s a r e " s m a l l " t h e y c a n s t i l l s i g n i f i c a n t l y a f f e c t t h e p h y s i c s , i n p a r t i c u l a r t h e a m p l i t u d e s o f t h e t u n n e l i n g m a t r i x e l e m e n t s b e t w e e n s t a t e s o f t h e g i a n t s p i n . 2.6.1 Exact Solution for Tunneling Matr ix Elements via Diagonalization W e s h a l l b e g i n o u r s t u d y o f (2.47) b y e x a c t l y d i a g o n a l i z i n g s o m e p a r t i c u l a r s u b s e t s o f i t a n d t h e r e b y e x t r a c t i n g t u n n e l i n g m a t r i x e l e m e n t s a s f u n c t i o n s o f {G}, \S\ a n d H. O u r p l a n o f a t t a c k i s a s f o l l o w s . W e b e g i n i n e a c h c a s e b y c h o o s i n g o n e o f t h e s e v e n c r y s t a l s y m m e t r i e s s o a s t o d e t e r m i n e t h e a l l o w e d f o r m o f {G}. I n t h e s p e c i f i c c a s e s o f t h e t e t r a g o n a l , o r t h o r h o m b i c a n d h e x a g o n a l s y s t e m s w e t h e n d i a g o n a l i z e a t r u n c a t e d v e r s i o n o f t h e r e s u l t a n t H a m i l t o n i a n f o r a r a n g e o f e x t e r n a l l y a p p l i e d D C fields, f o r c e n t r a l s p i n v a l u e s |5| = 1, 10 a n d 15. I n e a c h o f t h e s e c a s e s w e a s s u m e t h e e x i s t e n c e o f a n e a s y a x i s w h i c h w e i d e n t i f y w i t h t h e z a x i s . W e c a l c u l a t e t h e t u n n e l l i n g s p l i t t i n g s b e t w e e n t h e t w o l o w e s t l y i n g s t a t e s (| + S > a n d | — S > ) , c o r r e s p o n d i n g t o t h e g i a n t s p i n p o i n t i n g i n t h e ±z d i r e c t i o n s , w h i c h w e t h e n p l o t a s f u n c t i o n s o f t h e p a r a m e t e r s i n t h e b a r e H a m i l t o n i a n . The Cubic System A c r y s t a l w i t h u n d e r l y i n g c u b i c s y m m e t r y p o s s e s s e s a s p i n H a m i l t o n i a n o b e y i n g t h e s y m m e t r i e s [Sx —^Sy Sy —> — Sx] , [Sx Sz Sz —y —Sx] , [Sy Sz Sz —> ~Sy] Chapter 2. Effective Hamiltonians 5 3 T h i s r e s t r i c t s t h e a l l o w e d t e r m s i n t h e s p i n H a m i l t o n i a n . I f w e o n l y i n c l u d e t h e t w o l o w e s t o r d e r t e r m s o b e y i n g t h e s e s y m m e t r i e s t h e g i a n t s p i n H a m i l t o n i a n c a n b e w r i t t e n [96] H=-D (Sx + S4y + St) + E [St + S«y + S6Z + WS2xS2yS2) + gu.BS • H (2.48) T h e c a s e o f c u b i c c r y s t a l s y m m e t r y i s s o m e w h a t a n o m a l o u s i n t h a t i t i s t h e o n l y c a s e w e s h a l l e n c o u n t e r w h e r e a n a x i s ( e a s y o r h a r d ) i s n o t s i n g l e d o u t b y t h e c r y s t a l f i e l d - a s w e s e e f r o m t h e s y m m e t r y r e q u i r e m e n t s , a l l t h r e e a x e s i n t h e c r y s t a l a r e e q u i v a l e n t . W e s h a l l n o t s a y m o r e a b o u t t h i s c r y s t a l s y m m e t r y . H o w e v e r , i t i s w o r t h n o t i n g t h a t t h e p h y s i c s o f c u b i c m o l e c u l a r m a g n e t s s h o u l d b e p a r t i c u l a r l y e n t e r t a i n i n g b e c a u s e o f t h e l a c k o f a n e a s y / h a r d a x i s . The Tetragonal System I n s y s t e m s w i t h t e t r a g o n a l s y m m e t r y , t h e s y m m e t r i e s [Sx^Sy Sy^-Sx] ,[SZ^-SZ] (2.49) m u s t b e p r e s e r v e d (see f i g u r e 2.2). K e e p i n g o n l y t h e l o w e s t o r d e r t e r m s g i v e s a s p i n H a m i l t o n i a n o f t h e f o r m H = -DS2 + a4 (Si + Si) + gu.BS • H (2.50) S h o w n i n figures 2.3 a n d 2.4 a r e r e s u l t s o f e x a c t d i a g o n a l i z a t i o n o f (2.50) f o r a v a r i e t y o f p a r a m e t e r r e g i m e s . N o t e t h a t i n z e r o e x t e r n a l field, i f \S\ i s o d d t h e t u n n e l i n g s p l i t t i n g b e t w e e n t h e t w o l o w e s t l e v e l s h e r e i s z e r o . T h i s i s k n o w n a s K r a m e r ' s d e g e n e r a c y [47, 97], a n d i t a r i s e s h e r e b e c a u s e t h e r e e x i s t s n o p a t h b y w h i c h o u r H a m i l t o n i a n c a n c o n n e c t t h e s t a t e s \S > a n d | — S > if S i s o d d . Chapter 2. Effective Hamiltonians 5 4 (i) 171 rtVfl ( i i ) ( i v ) ( v ) ( v i ) O J ^ b a ( i i i ) ( v i i ) F i g u r e 2.2: A l l o f t h e u n i t c e l l s ( a f t e r A s h c r o f t a n d M e r m i n [138]). ( i ) C u b i c , ( i i ) T e t r a g - o n a l , ( i i i ) O r t h o r h o m b i c , ( i v ) M o n o c l i n i c , ( v ) T r i c l i n i c , ( v i ) H e x a g o n a l a n d ( v i i ) T r i g o n a l . The Orthorhombic System I n s y s t e m s w i t h o r t h o r h o m b i c s y m m e t r y , t h e s y m m e t r i e s [SX y Sx Sy y Sy Sz y Sz (2.51) m u s t b e p r e s e r v e d ( f i g u r e 2.2). K e e p i n g o n l y t h e l o w e s t o r d e r t e r m s l e a d s t o t h e H a m i l - t o n i a n H = -DS2Z + a2 (52 + S2_) + gfxBS • H (2.52) S h o w n i n figures 2.5 a n d 2.6 a r e r e s u l t s o f e x a c t d i a g o n a l i z a t i o n o f (2.52) f o r a v a r i e t y o f p a r a m e t e r r e g i m e s . Chapter 2. Effective Hamiltonians 5 5 F i g u r e 2.3: V a r i a t i o n o f A s , - s w i t h a^/D f o r f o u r d i f f e r e n t |S| v a l u e s ( c l o c k w i s e f r o m t o p l e f t , |SI = 2 , 6, 10, a n d 14); t e t r a g o n a l s y m m e t r y . O n t h e x a x i s i s p l o t t e d a±S2/D a n d o n t h e y a x i s l o g 1 0 &s,-s- H e r e w e h a v e t a k e n t h e e x t e r n a l field t o b e z e r o . -5.0 1 ' ' ' ' ' 1 -5.0 1 ' ' ' ' 1 1 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0 | , , , 1 0.0 | 1 F i g u r e 2.4: V a r i a t i o n o f A 5 , - s w i t h Hx/D f o r a4S2/D = 0.25 f o r f o u r d i f f e r e n t |S| v a l u e s ( c l o c k w i s e f r o m t o p l e f t , |S| = 2 , 5, 10, a n d 15); t e t r a g o n a l s y m m e t r y . O n t h e x a x i s i s p l o t t e d HX/DS2 a n d o n t h e y a x i s l o g 1 0 As,-s- Chapter 2. Effective Hamiltonians 56 The Monoclinic System S y s t e m s w i t h m o n o c l i n i c s y m m e t r y c o n t a i n a s a s p e c i a l c a s e t h e o r t h o r h o m b i c s y m m e t r y . B e c a u s e o f t h i s t h e s p i n H a m i l t o n i a n f o r t h e s e s y s t e m s m u s t c o n t a i n t h e o r t h o r h o m b i c s y m m e t r i e s . I n a d d i t i o n w e s h a l l h a v e q u a r t i c t e r m s , g i v i n g H = -DS2Z + a2 ( S 2 + S2_) + a^S* + a^St + gu.BS • H (2.53) The Triclinic System T r i c l i n i c s y m m e t r y i s o b t a i n e d v i a a d i s t o r t i o n o f m o n o c l i n i c s y m m e t r y . A s s u c h t h e d e s c r i p t i o n o f t h e t r i c l i n i c c a s e m u s t c o n t a i n t h e s y m m e t r y o f t h e m o n o c l i n i c c a s e . I n a d d i t i o n , w e p i c k u p a d i a g o n a l q u a r t i c s p i n t e r m ; H = -DS2Z - D0S4Z + a2 (S2+ + S2_) + o j + ) S j + a^Si + gn.BS • H (2.54) The Trigonal System I n s y s t e m s w i t h t r i g o n a l s y m m e t r y , r o t a t i o n s a r o u n d t h e b o d y d i a g o n a l a r e t h r e e - f o l d s y m m e t r i c . H = -DS2 + a3{Sz,Sl + S3_}+ gpBS • H (2.55) The Hexagonal System I n s y s t e m s w i t h h e x a g o n a l s y m m e t r y , t h e s y m m e t r i e s Sx -> e^3Sx] , [Sy -> e^3Sy] , [Sz -> -Sz] (2.56) m u s t b e p r e s e r v e d . T h i s i m p l i e s , k e e p i n g o n l y t h e l o w e s t o r d e r s p i n t e r m s , a H a m i l t o n i a n H = -DS2Z + a, (S« + S*) + g»BS • H (2.57) S h o w n i n figures 2.7 a n d 2.8 a r e r e s u l t s o f e x a c t d i a g o n a l i z a t i o n o f (2.57) f o r a v a r i e t y o f p a r a m e t e r r e g i m e s . Chapter 2. Effective Hamiltonians 5 7 ' 0.0 0.5 1.0" ' 0.0 0.5 1.0 F i g u r e 2.5: V a r i a t i o n o f As-s w i t h a2/D f o r f o u r d i f f e r e n t \S\ v a l u e s ( c l o c k w i s e f r o m t o p l e f t , |5| = 2 , 5, 10, a n d 15); o r t h o r h o m b i c s y m m e t r y . O n t h e x a x i s i s p l o t t e d a2/D a n d o n t h e y a x i s l o g 1 0 A$-s- H e r e w e h a v e t a k e n t h e e x t e r n a l field t o b e z e r o . 0.0 i 1 1 1 1 1 1 0.0 F i g u r e 2.6: V a r i a t i o n o f As-s w i t h Hx/D f o r a2/D = 0.25 f o r f o u r d i f f e r e n t |5| v a l u e s ( c l o c k w i s e f r o m t o p l e f t , \S\ = 2 , 5, 10, a n d 15); o r t h o r h o m b i c s y m m e t r y . O n t h e x a x i s i s p l o t t e d HX/DS2 a n d o n t h e y a x i s l o g 1 0 As -s- Chapter 2. Effective Hamiltonians 5 8 0.0 I 1 1 , , , 1 0.0 F i g u r e 2.7: V a r i a t i o n o f A s - s w i t h a6/D f o r f o u r d i f f e r e n t |5| v a l u e s ( c l o c k w i s e f r o m t o p l e f t , \S\ —2, 6, 10, a n d 14); h e x a g o n a l s y m m e t r y . O n t h e x a x i s i s p l o t t e d a6Si/D a n d o n t h e y a x i s l o g 1 0 As,-s- H e r e w e h a v e t a k e n t h e e x t e r n a l field t o b e z e r o . F i g u r e 2.8: V a r i a t i o n o f AS-S w i t h Hx/D f o r a6S4/D = 0.25 f o r f o u r d i f f e r e n t \S\ v a l u e s ( c l o c k w i s e f r o m t o p l e f t , \S\ —2, 5, 10, a n d 15); h e x a g o n a l s y m m e t r y . O n t h e x a x i s i s p l o t t e d HX/DS2 a n d o n t h e y a x i s l o g 1 0 As,-s- Chapter 2. Effective Hamiltonians 5 9 2.6.2 Tunneling Matr ix Elements via Perturbation Theory T h e c a l c u l a t i o n o f t u n n e l i n g m a t r i x e l e m e n t s v i a p e r t u r b a t i o n t h e o r y i n t h e t e r m s t h a t b r e a k t h e e a s y a x i s s y m m e t r y (z <r+ —z) i s q u i t e s t r a i g h t f o r w a r d a s l o n g a s t h e r e i s o n l y o n e s u c h t e r m i n t h e H a m i l t o n i a n . F o r t h e s e c a s e s w e s h a l l j u s t s t a t e t h e r e s u l t s , a l l o f w h i c h h a v e b e e n p r e v i o u s l y c a l c u l a t e d e l s e w h e r e [70]. H a m i l t o n i a n A S l - s -DS2 + a2 (S2+ + S2_) -DS2 + a4 (S+ + S4_) -DS2 + a6 (S% + St) -DS2 + Hx ( 5 + + S-) -DS2 + a3{Sz,(Sl + S3_)} «f(2S)! 4 S - i £ > s - i [ ( S _ i ) ! ] * af / 2 (2S)! 4 S - 2 D S / 2 - l [ ( S y 2 _ 1 ) l ] 2 af / 3 (2S)! 4 S - s z ) S / 3 - i [ ( s / 3 - l ) ! ] ' 2 H*S(2S) £,2S - l [ (2S_i) ! ] alS/3(2S)\ Z )2S/3 - l 6 2S/3-2 [ (5/ 3 _ 1 ) ! ]2 T a b l e 2.1: P e r t u r b a t i o n t h e o r y r e s u l t s f o r s o m e s i m p l e H a m i l t o n i a n s , f r o m [70]. W h e n t h e n u m b e r o f s y m m e t r y b r e a k i n g t e r m s i n t h e H a m i l t o n i a n i s i n c r e a s e d , t h e s o l u t i o n f o r A u s i n g p e r t u r b a t i o n t h e o r y b e c o m e s a l i t t l e m o r e c o m p l i c a t e d . T h i s i s b e c a u s e o f t h e c o m p e t i t i o n b e t w e e n t h e s e t e r m s i n d e c i d i n g w h i c h a r e t h e p r e f e r r e d p a t h s b e t w e e n t h e l o w l y i n g s t a t e s . Chapter 2. Effective Hamiltonians 6 0 2.6.3 Tunneling Matr ix Elements via W K B Methods I t h a s b e e n s h o w n [25, 26] t h a t t h e H a m i l t o n i a n H = -DS2 + ak (Sk+ + Sk_) (2.58) l e a d s t o A s , _ s ^ p ^ ! ) 2 S " „ 5 9 ) a s l o n g a s akSk <C DS2 a n d t h e a m b i e n t e n e r g y E i s s u c h t h a t E <C DS2. 2.6.4 Tunneling Matr ix Elements via Instanton Techniques T h e final a p p r o x i m a t e m e t h o d o f s o l u t i o n f o r t u n n e l i n g m a t r i x e l e m e n t s i n s p i n H a m i l - t o n i a n s t h a t w e s h a l l c o n s i d e r i n v o l v e s u s i n g i n s t a n t o n t e c h n i q u e s . W e w i l l e x p l i c i t l y p e r f o r m o n e s u c h c a l c u l a t i o n i n s e c t i o n (2.8), o b t a i n i n g t h e f o l l o w i n g f o r m f o r a n o r - t h o r h o m b i c ( i e . e a s y - a x i s e a s y - p l a n e ) H a m i l t o n i a n (see ( 2 . 5 2 ) ) ^s = ^S^2(D-2a2f4a12/4exp (2.60) V a2 . T h e s a m e t y p e o f p r o c e d u r e m a y b e f o l l o w e d i n p r i n c i p l e w i t h a n y s p i n H a m i l t o n i a n t h a t p o s s e s s e s w e l l - d e f i n e d s e m i - c l a s s i c a l t r a j e c t o r i e s b e t w e e n i t s m i n i m a . H o w e v e r i n p r a c t i c e o n e r u n s u p a g a i n s t p r o b l e m s w i t h a l l b u t t h e e a s i e s t q u a d r a t i c s p i n t e r m s . W e b e l i e v e t h a t i t i s p o s s i b l e t o d e r i v e f o r m s s i m i l a r t o (2.60) b u t h a v e l e f t t h i s t a s k t o f u t u r e i n v e s t i g a t i o n s [98]. 2.6.5 Comparison of Approximate Methods to Exact Solutions I n o r d e r t o g i v e s o m e i d e a a b o u t h o w e f f e c t i v e t h e s e d i f f e r e n t a p p r o x i m a t i o n s c h e m e s a c t u a l l y a r e , w e n o w c o m p a r e t h e r e s u l t s o f t h e p r e c e d i n g s e c t i o n s t o t h e e x a c t r e s u l t s f o r t h e t e t r a g o n a l a n d o r t h o r h o m b i c c r y s t a l s y m m e t r i e s . S h o w n i n figure 2.9 i s t h e Chapter 2. Effective Hamiltonians 61 F i g u r e 2.9: C o m p a r i s o n o f p e r t u r b a t i o n t h e o r y , W K B r e s u l t s a n d i n s t a n t o n r e s u l t s t o t h e e x a c t s o l u t i o n f o r t h e t u n n e l i n g s p l i t t i n g b e t w e e n t h e t w o l o w e s t l e v e l s o f t h e H a m i l t o n i a n o f o r t h o r h o m b i c s y m m e t r y w i t h S = 1 0 . P l o t t e d o n t h e h o r i z o n t a l a x i s i s a2/D, a n d o n t h e v e r t i c a l a x i s l o g 1 0 As-s- L e g e n d : B l a c k , e x a c t s o l u t i o n ; G r e e n , i n s t a n t o n s o l u t i o n ; R e d , p e r t u r b a t i o n t h e o r y a n d B l u e , W K B . F i g u r e 2.10: C o m p a r i s o n o f p e r t u r b a t i o n t h e o r y a n d W K B r e s u l t s t o t h e e x a c t s o l u t i o n f o r t h e t u n n e l i n g s p l i t t i n g b e t w e e n t h e t w o l o w e s t l e v e l s o f t h e H a m i l t o n i a n o f t e t r a g o n a l s y m m e t r y w i t h S = 1 0 . P l o t t e d o n t h e h o r i z o n t a l a x i s i s a4S2/D, a n d o n t h e v e r t i c a l a x i s l o g 1 0 As-s- L e g e n d : Y e l l o w , e x a c t s o l u t i o n ; R e d , p e r t u r b a t i o n t h e o r y a n d G r e e n , W K B . Chapter 2. Effective Hamiltonians 6 2 c o m p a r i s o n f o r t h e o r t h o r h o m b i c c a s e . H e r e we s e e t h a t , a s w e c o u l d h a v e e x p e c t e d , t h e i n s t a n t o n s o l u t i o n f a i l s q u i t e s p e c t a c u l a r l y i f a2/D i s s m a l l . T h i s i s s i m p l y b e c a u s e w h e n w e c a l c u l a t e d o u r i n s t a n t o n a c t i o n w e a s s u m e d t h a t t h e fluctuations a r o u n d t h e s e m i - c l a s s i c a l p a t h s w e r e s m a l l , w h i c h o f c o u r s e t h e y a r e n ' t i f a2/D i s s m a l l . W h e n a2/D i s l a r g e , t h e i n s t a n t o n s o l u t i o n i s q u i t e g o o d . T h e W K B s o l u t i o n w e s e e i s q u i t e b a d . I t t u r n s o u t t h a t t h e f u n c t i o n a l f o r m i n a2/D i s c o r r e c t b u t t h e p r e f a c t o r i s n o t . P e r t u r b a t i o n t h e o r y w o r k s q u i t e w e l l f o r t h e e n t i r e r a n g e o f a2/D < 0.5 s t u d i e d . W e n e x t t u r n o u r a t t e n t i o n t o t h e t e t r a g o n a l c a s e , s h o w n i n figure 2.10. H e r e w e s e e a s i m i l a r s t o r y . T h e W K B s o l u t i o n g e t s t h e f u n c t i o n a l f o r m c o r r e c t b u t a g a i n t h e p r e f a c t o r i s w r o n g . P e r t u r b a t i o n t h e o r y w o r k s q u i t e w e l l i n t h e s m a l l a±/D r e g i m e . 2.7 Back to the Ful l Hamiltonian-Separation of Tunneling Energy Scale Us- ing an Instanton Technique A s w a s d i s c u s s e d i n t h e i n t r o d u c t i o n a n d t o a l e s s e r d e g r e e i n s e c t i o n (2.6), i t i s s i m p l y n o t f e a s i b l e t o m e a s u r e a l l o f t h e n o n - z e r o c o m p o n e n t s o f t h e t e n s o r <5Qii -ai2o e x p e r i m e n t a l l y . B e c a u s e o f t h i s w h a t i s u s u a l l y d o n e i s t h a t t h e l o w e r o r d e r t e r m s i n Si a l l o w e d b y c r y s t a l s y m m e t r y a r e k e p t a n d t h e r e s t o f t h e t e r m s t h r o w n away. T h e c o e f f i c i e n t s o f t h e k e p t t e r m s a r e t h e n u s e d t o fit e x p e r i m e n t a l d a t a . I n t h e c a s e o f t h e Fe% s p i n H a m i l t o n i a n , t h e f o r m t h a t w e s h a l l a d o p t i s HGS = Gaii~ai«>Sil...Si20 = -DS2Z + E(S2+ + S2_) + C{Si + S4+) (2.61) T h i s f o r m , w i t h D = 0.292A", E = 0.046K a n d C = - 2 . 1 • 10~5K, i s g o o d e n o u g h t o a c c u r a t e l y fit b o t h t h e p e r i o d o f t h e " A h a r o n o v - B o h m " o s c i l l a t i o n s a n d m a g n i t u d e o f t h e t u n n e l i n g s p l i t t i n g [51] i n r e c e n t e x p e r i m e n t s . N o t e h o w e v e r t h a t i t i s c l e a r t h a t a l a r g e n u m b e r o f t e r m s h a v e s i m p l y b e e n c h o p p e d o f f t h e " t r u e " s p i n H a m i l t o n i a n ( f o r e x a m p l e , e v e n t h e q u a r t i c d i a g o n a l s p i n t e r m h a s n o t b e e n i n c l u d e d ) . Chapter 2. Effective Hamiltonians 6 3 T h e e f f e c t s o f t h e r m a l p h o n o n s a n d p o l a r i t o n s a t l o w e n o u g h t e m p e r a t u r e s can be completely neglected [20]. T h i s i s b e c a u s e p r o c e s s e s i n v o l v i n g t h e s e b o s o n i c m o d e s s c a l e l i k e t h e i r r e s p e c t i v e d e n s i t i e s i n t h e c r y s t a l , w h i c h a r e v a n i s h i n g l y s m a l l a t l o w t e m p e r a - t u r e s . I f w e a r e i n t h e " q u a n t u m r e g i m e " d e m o n s t r a t e d b y S a n g r e g o r i o e t . a l . [14] w e a r e a t t e m p e r a t u r e s l e s s t h a n ~ 3 6 0 mK, a n d t h e r e f o r e kBT <C DS [20]. W e t h e r e f o r e w r i t e t h e q u a n t u m r e g i m e H a m i l t o n i a n i n t h e f o r m H = -DS2Z+E(S2+ + S2_) + C(Si + Sl) + gu.BH-S + £ 9nPiin [A^SPJI + R/aPHQq] + f ; 9nku.nH • ik P = I *;=i N+8 + £ A;=l 9nklinMlk ^ 0 + 6 I k ( 2 I k - l ) V '"P .1=1 + ^ E ^[?i-?k-3{fi (2-62) 4 7 r Kk=l Tlk T h e r e s t i l l r e m a i n s o n e f e a t u r e o f (2.62) t h a t w e c a n t a k e a d v a n t a g e o f i n o r d e r t o s i m p l i f y i t . A n e x a m i n a t i o n o f t h e r e l a t i v e s t r e n g t h s o f t h e t e r m s i n (2.63) r e v e a l s t h a t t h e l a r g e s t t e r m i s t h e s i n g l e - m o l e c u l e a n i s o t r o p y t e r m DS2 i n t h e s p i n H a m i l t o n i a n w h i c h i s o f t h e o r d e r o f 2 9 K. A l l t h e o t h e r t e r m s a r e s m a l l c o m p a r e d t o t h i s . W e t h e r e f o r e s e e t h a t i f a l l a m b i e n t e n e r g i e s ( p r i m a r i l y t h e l a t t i c e t e m p e r a t u r e a n d e x t e r n a l field) a r e m u c h l e s s t h a n t h e g a p t o t h e first e x c i t e d l e v e l D(S2 — (S — l ) 2 ) t h e g i a n t s p i n w i l l o n l y b e a b l e t o a c c e s s t h e t w o l o w e s t e n e r g y l e v e l s SZ = ± 5 . A t t h i s l o w e n e r g y s c a l e w e s h a l l d e f i n e o u r final l o w e n e r g y e f f e c t i v e d e s c r i p t i o n , w i t h i n w h i c h t h e g i a n t s p i n i s m a p p e d t o a t w o - l e v e l s y s t e m p a r a m e t r i z e d b y a P a u l i m a t r i x r , w h e r e f 2 = ± 1 c o r r e s p o n d s t o SZ = ± 5 r e s p e c t i v e l y . T h i s d e s c r i p t i o n w i l l b e v a l i d i n t h e q u a n t u m r e g i m e T < 3 6 0 mK [14]. T h e r e i s a n o t h e r c a s e o f i n t e r e s t w h e r e t h i s m a p p i n g m a y b e p e r f o r m e d . I f a n e x t e r n a l l o n g i t u d i n a l field Hz i s a p p l i e d t o t h e s y s t e m , t h e e f f e c t i s t o bias t h e w e l l s d r a w n i n Chapter 2. Effective Hamiltonians 6 4 Energy A E = 0 •10-9-8-7-6 -5-4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 S . F i g u r e 2.11: Z - p r o j e c t i o n o f s p i n v e r s u s e n e r g y f r o m HGs f o r t h e Fe8 s y s t e m . T h e r e g i o n o f v a l i d i t y o f t h e m a p p i n g t o a t w o - s t a t e s y s t e m i s t h e r e g i o n w h e r e e x c i t e d s t a t e s a r e f o r b i d d e n ( t h i s r e g i o n i n s h a d e d g r e y i n t h e a b o v e ) . figure 2.11. I f t h e a p p l i e d field i s s t r o n g e n o u g h , i t c a n b r i n g t o r e s o n a n c e o n e o f o u r o r i g i n a l s t a t e s a n d a h i g h e r - e n e r g y s t a t e o n t h e o t h e r s i d e o f t h e b a r r i e r ( f o r e x a m p l e , t h e a p p l i c a t i o n o f a p o s i t i v e Hz c o u l d l e a d t o a r e s o n a n c e c o n d i t i o n b e t w e e n | + 1 0 > a n d | — 9 > ) . I n t h i s c a s e , t h e d o m i n a n t t u n n e l i n g d y n a m i c s i n t h e s y s t e m s t i l l i n v o l v e o n l y t w o l e v e l s . H o w e v e r , o n e m u s t e x e r c i s e c a u t i o n h e r e , a s t h e l o w e r - l y i n g s t a t e s ( i n t h i s e x a m p l e , | — 10 > ) a r e c o n n e c t e d t o t h e t w o p r i m a r y s t a t e s b y s e v e r a l m e c h a n i s m s , m o s t i m p o r t a n t l y p h o n o n e m i s s i o n t a k i n g | — 9 > t o | — 10 >. B e c a u s e o f t h i s c o m p l i c a t i o n , Chapter 2. Effective Hamiltonians 6 5 i n w h a t f o l l o w s w e s h a l l e x p l i c i t l y c o n s i d e r o n l y t h e c a s e w h e r e t h e e x t e r n a l l y a p p l i e d l o n g i t u d i n a l field i s s u c h t h a t Hz « C DS a n d o n l y t h e t w o l e v e l s | ± S > are i n v o l v e d , k e e p i n g i n m i n d t h a t i n c e r t a i n c i r c u m s t a n c e s w e m a y b e a b l e t o g e n e r a l i z e o u r r e s u l t s t o l a r g e r v a l u e s o f Hz. T h e d y n a m i c s o f f i n t h i s r e g i m e a r e s o l e l y t h e p r o d u c t o f t u n n e l i n g f r o m |5 >«-» | — S >. T h i s o b s e r v a t i o n l e a d s t o a n a t u r a l s e p a r a t i o n o f c o n t r i b u t i o n s c o m i n g f r o m t e r m s i n (2.63) t h a t a r e d i a g o n a l i n Sz ( a n d t h e r e f o r e a r e n o t i n v o l v e d i n t u n n e l i n g e v e n t s ) a n d t e r m s t h a t a r e n o t d i a g o n a l i n Sz ( a n d t h e r e f o r e c o m i n g i n t o p l a y o n l y w h e n t h e g i a n t s p i n t u n n e l s ) . T h i s s e p a r a t i o n i s h e l p f u l b e c a u s e i t w i l l t u r n o u t ( w e w i l l s h o w t h i s ) t h a t t h e t i m e s c a l e f o r t u n n e l i n g p h y s i c s = A - 1 i s m u c h s m a l l e r t h a n t h e t i m e b e t w e e n t u n n e l i n g e v e n t s = QQ 1 [20]. T o s e e w h y t h i s s e p a r a t i o n i s h e l p f u l , c o n s i d e r t h e f o l l o w i n g a r g u m e n t . L e t u s i m a g i n e a l i k e l y t r a j e c t o r y f o r t h e e x c e s s s p i n S(t), a s s u m i n g a n i n i t i a l c o n d i t i o n c o r r e s p o n d i n g t o S(0) — +S z. W e a s s u m e t h a t t h e t i m e b e t w e e n t u n n e l i n g e v e n t s i s m u c h l o n g e r t h a n t h e t i m e o v e r w h i c h t u n n e l i n g o c c u r s . W e t h e r e f o r e e x p e c t t h e c e n t r a l s p i n t o e v o l v e d y n a m i c a l l y i n a s i m i l a r w a y t o t h a t s h o w n i n figure 2.12. T h i s " s e p a r a t i o n o f s c a l e s " a l l o w s o n e t o c o n s i d e r t h e e f f e c t i v e d e s c r i p t i o n s e p a r a t e l y i n t w o d i f f e r e n t r e g i m e s ; o n e i n t h e r e g i o n s b e t w e e n t u n n e l i n g e v e n t s a n d o n e d u r i n g t h e t u n n e l i n g . T h e r e g i m e b e t w e e n t u n n e l i n g e v e n t s w e s h a l l r e f e r t o a s t h e " d i a g o n a l " r e g i o n ( a s o n l y t e r m s d i a g o n a l i n S e n t e r i n t o p l a y ) a n d t h e c o n t r i b u t i o n o f t h e s e t e r m s t o t h e final e f f e c t i v e d e s c r i p t i o n w e c a n s i m p l y r e a d o f f o u r e q u a t i o n (2.63). W e find u n d e r t h e s e c i r c u m s t a n c e s t h a t t h e d i a g o n a l c o n t r i b u t i o n c a n b e w r i t t e n JV+8 z p=l (±)P9npVn fc-i N N + gp,BSHzfz p=l fc=l k=l L eQk -\rkaf) jk 6 / f c ( 2 / f c - 1) Q/? Chapter 2. Effective Hamiltonians 6 6 + J2°^[lrIk-S(Irrlk)(Ik-rlk) 4 7 r ttk rlk (2.63) w h e r e ( ± ) p i s s h o r t h a n d f o r t h e d i r e c t i o n i n w h i c h t h e pth i o n i c s p i n i s p o i n t i n g (see figure 2.1). O u r l a b e l l i n g s y s t e m h a s ( ± ) p = 1 f o r i o n i c s p i n s {p-j-} = 1,2,4,5,6 a n d 8 a n d ( ± ) p = — 1 f o r s p i n s {p±} — 3 a n d 7 w h e n t h e c e n t r a l s p i n i s " u p " ( t h a t i s , tz — + 1 ) w i t h s i g n s r e v e r s e d i f t h e c e n t r a l s p i n i s " d o w n " . W e m a y r e w r i t e t h i s i n t h e f o r m 'l + fz H Z , TP" • h + fc=i HD = N E T T ^ E ^ - / * + ' i - f 2 N E7 P 2 c )-4 + E7i 2 ) -4 p=l fc=l N + gu.BSHzfz + E k=l eQk "64(24 - 1) « » 4 7 r Kk rlk w h e r e (2.64) (2.65) (2.66) E < - E < + ^ Lpe{pt> pe{P4.} (2.67) -.(2) -.(2) 5 g 5 7fc = 7fcs + 2^nHp = -gnkVn • E E M$ + HP pe{pT} pe{p±} (2.68) T h e n o t a t i o n h e r e i s s u c h t h a t t h e f3th c o m p o n e n t o f t h e e x p r e s s i o n s o n t h e r i g h t c o r r e - s p o n d s t o t h e (5th c o m p o n e n t o f t h e v e c t o r o n t h e l e f t . O n e c a n t h i n k o f t h e e x p r e s s i o n (2.64) i n t h e f o l l o w i n g way. T h e r e a r e t w o e l e c t r o n i c s p i n c o n f i g u r a t i o n s (1,2) t h a t c o r r e s p o n d t o b e f o r e (1) a n d a f t e r (2) t h e c e n t r a l s p i n Chapter 2. Effective Hamiltonians 6 7 c o m p l e x f l i p s . I n e a c h o f t h e s e c o n f i g u r a t i o n s , e a c h n u c l e a r s p i n i n t h e s y s t e m f e e l s a m a g n e t i c field c o m i n g f r o m t h e i o n i c s p i n s a n d t h e e x t e r n a l l y a p p l i e d field. I n t h e a b o v e e x p r e s s i o n , t h e s e fields a r e r e p r e s e n t e d b y 7 ^ , rfkc\ 7 ^ a n d 7 f c 2 c ^ . T h e s i g n c h a n g e s i n g o i n g f r o m 7 ^ —> 7 ^ a n d 7 ^ l c ^ —> c o m e f r o m t h e f a c t t h a t t h e c e n t r a l s p i n o b j e c t h a s r e v e r s e d i t s d i r e c t i o n a l o n g t h e e a s y a x i s . O n e m a y w r i t e (2.64) i n a m o r e t r a n s p a r e n t f a s h i o n b y d e f i n i n g u n i t v e c t o r s -(1) , -(2) Tk +Tk \Tk +Tk I f o r k l a b e l l i n g t h e l i g a n d s p i n s k — 1..N a n d 4 = mk = -.(1c) _ A2c) Ik Ik (2.69) (2.70) f o r k l a b e l l i n g t h e p o s s i b l e 57Fe n u c l e a r s p i n s k — N 4- l.-N + 8. W e a l s o d e f i n e e n e r g i e s (2.71) f o r l i g a n d s p i n s k = 1..N a n d II i f I I - . ( lc) -.(2c) \ ± ,f\ i-*(lc) , -,(2c), uk = IhlgnMTk - 7 f c I , uk = \Ik\gnkMTk +% I f o r 57Fe s p i n s k = N + 1..N + 8. W i t h t h e s e w e c a n w r i t e N+S —Ik • mk + —uUk • lk k=l N r E eQk T/fca/3 jk + + gu.BSHzfz POPN 9ni9nk (2.72) An Kk 'Ik h • Ik - • rlk)(Ik • flk)] (2.73) T h i s f o r m i s s i m i l a r t o t h a t d e r i v e d b y P r o k o f i e v a n d S t a m p [20]. I t d i f f e r s i n t w o r e s p e c t s . F i r s t l y , i t s h o w s e x p l i c i t l y w h a t t h e e n e r g i e s uJk,uik a n d u n i t v e c t o r s mk,lk a r e i n t e r m s o f p a r a m e t e r s i n t h e h i g h e r e n e r g y d e s c r i p t i o n s ( a n d a l l o w s u s t o c a l c u l a t e t h e s e - w e s h a l l Chapter 2. Effective Hamiltonians 6 8 |-S> |+S> |+S> F i g u r e 2.12: T y p i c a l e v o l u t i o n o f t h e p r o j e c t i o n o f t h e e x c e s s s p i n S(t) a l o n g t h e e a s y - a x i s . W e s e e t w o r e g i m e s ; o n e w h e r e S e v o l v e s w i t h o u t t u n n e l i n g ( d i a g o n a l i n f ) , a n d o n e w h e r e S t u n n e l s f r o m | + S ><-*• | — S > ( o f f - d i a g o n a l i n f ) . N o t e t h e s e p a r a t i o n o f s c a l e s ; t h e t i m e b e t w e e n t u n n e l i n g e v e n t s i s m u c h g r e a t e r t h a n t h e t u n n e l i n g t i m e . d o t h i s i n t h e n e x t c h a p t e r ) . S e c o n d l y , i t i n c l u d e s t h e e f f e c t o f q u a d r u p o l a r c o u p l i n g s b e t w e e n h i g h e r s p i n n u c l e i i n t h e l i g a n d b a t h a n d e l e c t r i c f i e l d g r a d i e n t s i n t h e m o l e c u l e . T h e " o f f - d i a g o n a l " c o n t r i b u t i o n t o t h e e f f e c t i v e H a m i l t o n i a n i s s o m e w h a t h a r d e r t o e x t r a c t , a n d a d i f f e r e n t a p p r o a c h w i l l b e r e q u i r e d . 2.8 Off-Diagonal Terms and the Instanton Method A m e t h o d h a s b e e n d e v e l o p e d b y T u p i t s y n e t . a l . [74] t h a t a l l o w s t h e e x t r a c t i o n o f t h e " o f f - d i a g o n a l " t e r m s i n t h e e f f e c t i v e H a m i l t o n i a n , i e . t h o s e t h a t a c t w h e n t h e c e n t r a l s p i n o b j e c t t u n n e l s . T w o o b j e c t i o n s h a v e r e c e n t l y b e e n r a i s e d w h i c h q u e s t i o n t h e v a l i d i t y o f t h i s m e t h o d [101]). I n t h i s s e c t i o n w e s h a l l r e v i e w t h e m e t h o d a n d p o i n t o u t t h e o b j e c t i o n s . Chapter 2. Effective Hamiltonians 6 9 2.8.1 Review of the Method of Tupitsyn et.al. I n t h e e f f e c t i v e d e s c r i p t i o n (2.63) we h a v e r e d u c e d t h e H i l b e r t s p a c e o f t h e c e n t r a l d e g r e e o f f r e e d o m d o w n t o d i m e n s i o n D — 2. T h e s e s t a t e s a r e c o l l e c t i v e o b j e c t s o f t h e f o r m —* —• S = Yli Si; t h a t i s , t h e y a r e f o r m e d o f e l e c t r o n i c s p i n s t h a t h a v e l o c k e d t o g e t h e r . I n t h e a p p r o a c h t o t h e F e 8 m o l e c u l e t h a t w e h a v e a d o p t e d , w e h a v e c h o s e n a m o d e l w h e r e a l l t h e e l e c t r o n i c s p i n s l o c k t o g e t h e r s u c h t h a t e a c h l i e s p a r a l l e l o r a n t i p a r a l l e l t o t h e e a s y a x i s . W e s h a l l i n w h a t f o l l o w s t r e a t t h i s c o l l e c t i v e s t a t e a s a s p i n 10 q u a n t u m r o t a t o r , p o i n t i n g o u t w h e n t h i s d e s c r i p t i o n m u s t b e m o d i f i e d b e c a u s e o f t h e " t r u e " e i g h t s p i n n a t u r e o f t h e o b j e c t . —* —+ T h e s t a t e s \a > a n d |/? > a r e c o l l e c t i v e s t a t e s w i t h a = \ + S > a n d /3 — \ — S >, i e . r e f e r r i n g t o t h e c e n t r a l s p i n " p o i n t i n g i n t h e u p / d o w n d i r e c t i o n s " . T h e t r a n s i t i o n a m p l i t u d e b e t w e e n t h e s e s t a t e s c a n b e d e f i n e d t o b e Tapit) = 1^ D(9, <j>) e x p ( - j* dr ( L 0 ( r ) + I ^ f r ) ) ) (2.74) w h e r e L0(T) a n d LN{T) a r e t h e L a g r a n g i a n s c o r r e s p o n d i n g t o t h e b a r e s p i n H a m i l t o n i a n p l u s t h e e x t e r n a l field t e r m H0 = -DS] + E(S2+ + Si) + C(S* + Si) +gu,BS-H (2.75) a n d t h e c o n t r i b u t i o n s f r o m t h e n u c l e a r s p i n s r e s p e c t i v e l y . T h e s p h e r i c a l a n g l e s 9 a n d <f> a r e i n t r o d u c e d s o a s t o c h a r a c t e r i z e S i n t h e s t a n d a r d w a y ; t h a t i s , Sx = \S\ c o s <j) s i n 6 , Sy = \S\sin <f> s i n 9 , Sz = \S\ c o s 9 (2.76) I f w e a s s u m e t h a t t h e " b o u n c e t i m e " b e t w e e n m i n i m a fig 1 i s m u c h s m a l l e r t h a n t h e t i m e b e t w e e n t r a n s i t i o n s A g l , t h e e v o l u t i o n o p e r a t o r c o n n e c t i n g t h e t w o m i n i m a w i l l b e g i v e n b y {HND)afi = tfafl(t) , ( a ^ / ? , Q 0 - 1 « i « A 0 - 1 ) (2.77) Chapter 2. Effective Hamiltonians 7 0 w h e r e H i s t h e n o n - d i a g o n a l p a r t o f t h e e f f e c t i v e H a m i l t o n i a n t h a t w e a r e l o o k i n g f o r . T h i s m e a n s t h a t i n o r d e r t o c a l c u l a t e HND i t s u f f i c e s t o c a l c u l a t e Tag(t), a n d t h i s w e c a n a t t e m p t t o d o b y s o l v i n g f o r t h e i n s t a n t o n ( s e m i c l a s s i c a l ) s o l u t i o n s o f ( 2 . 7 4 ) . 2.8.2 The Tunneling Lagrangian I n t h e c a l c u l a t i o n t h a t f o l l o w s w e s h a l l e x p l i c i t l y u s e t h e e a s y - a x i s e a s y - p l a n e s p i n H a m i l - t o n i a n , v i s . H0 — -(D - 2E)S2Z + AES2X (2.78) H o w e v e r , t h e t a c t i c s w e e m p l o y h e r e c a n b e u s e d f o r a n y s p i n H a m i l t o n i a n a d m i t t i n g c l e a r l y d e f i n e d s e m i c l a s s i c a l p a t h s b e t w e e n t h e m i n i m a \a > a n d \/3 >. N o t e t h a t t h i s s p i n H a m i l t o n i a n i s e q u i v a l e n t t o t h e o n e e x p e r i m e n t a l l y o b t a i n e d f o r Fe8 i f t h e q u a r t i c s p i n t e r m s i n t h i s l a t t e r a r e n e g l e c t e d . U s i n g t h e r e l a t i o n s h i p S2 = S2 + S2 + S2 w e c a n s h o w t h a t (2.78) i s e q u i v a l e n t t o H0 = -DS2Z + E(S2+ + S2_) (2.79) T h i s t r u n c a t i o n i s p e r f o r m e d s i m p l y f o r c o n v e n i e n c e , a s i n c l u s i o n o f t h e q u a r t i c t e r m s m a k e s t h e a n a l y t i c c a l c u l a t i o n s t h a t f o l l o w v e r y d i f f i c u l t . A s w e d i s c u s s e d e a r l i e r i n t h i s c h a p t e r , t h e " t r u e " s p i n H a m i l t o n i a n c o n t a i n s m a n y t e r m s t h a t a r e i n a c c e s s i b l e e x p e r i m e n t a l l y , a n d t h e r e f o r e d r o p p i n g t h e q u a r t i c t e r m s i m p l y e m p h a s i z e s t h e p o i n t t h a t i n u s i n g a f o r m l i k e (2.78) w e r e a l l y a r e u s i n g a p h e n o m e n o l o g i c a l d e s c r i p t i o n - t h e e f f e c t o f d r o p p i n g h i g h e r o r d e r t e r m s o n t h e a m p l i t u d e o f t u n n e l i n g m a t r i x e l e m e n t s c a n b e d r a s t i c [164]. T h i s b e i n g s a i d , w h a t i s i m p o r t a n t i n t h i s c a s e i s t h e i n s t a n t o n t r a j e c t o r y , a n d t h i s i s n o t e x p e c t e d t o b e s t r o n g l y a f f e c t e d b y h i g h e r o r d e r t e r m s [74]. F o r t h e e a s y - a x i s e a s y - p l a n e m o d e l (2.78) o n e c a n s h o w t h a t t h e e q u a t i o n s o f m o t i o n Chapter 2. Effective Hamiltonians 71 a r e 2 0 s i n 0 - M £ S s i n 2 0 s i n 2 < £ = O (2.80) a n d (j) s i n 9 - i4ES{ + c o s 2 0) s i n 29 = 0 (2.81) S o l u t i o n s o f t h e c l a s s i c a l e q u a t i o n s o f m o t i o n a r e s i m p l y 7r . , i E> — 2E ^ = J 7 _ _ t s i n h - M _ ^ _ (2.82) a n d 0 ( t ) = sind(t) = ——— , Q0 = 4 £ S s i n h w w c o s h Q0t 2 s i n h - 1 ID-2E AE (2.83) w h e r e n = ± l a b e l s r o t a t i o n s c l o c k w i s e a n d c o u n t e r - c l o c k w i s e i n t h e e a s y p l a n e . N o w i n t h e s t a n d a r d t r e a t m e n t , o n e a s s u m e s t h a t t h e c l a s s i c a l e q u a t i o n s o f m o t i o n a r e a t t r a c t o r s i n t h e s e n s e t h a t a s m a l l p e r t u r b a t i o n a w a y f r o m t h e s e c o s t s a c t i o n . I f t h i s i s t r u e t h e n o n e c a n p e r f o r m a g a u s s i a n i n t e g r a t i o n o v e r s m a l l fluctuations a w a y f r o m t h e c l a s s i c a l e q u a t i o n s o f m o t i o n i n t h e m a n n e r s u g g e s t e d b y T u p i t s y n e t . a l . [74]. H o w e v e r i t w a s r e c e n t l y p o i n t e d o u t b y U n r u h [101] t h a t i f w e p e r t u r b t h e v a r i a b l e <f> i n t h e e q u a t i o n o f m o t i o n (2.81) w e o b t a i n I r) OTP 8(f) = -d>ESsm(2smh.-1 \ ) c o s 9 5(j> (2.84) V AE N o w w e se e t h a t a s l o n g a s c o s # > 0 a n y p e r t u r b a t i o n o f d> i s a t t r a c t e d t o t h e c l a s s i c a l s o l u t i o n . H o w e v e r w e se e f r o m o u r s o l u t i o n s t h a t c o s # c h a n g e s s i g n a t 9 = ir/2. T h u s i t w o u l d a p p e a r t h a t a t t h i s p o i n t t h e v a r i a b l e d> i s p u s h e d a w a y f r o m i t s s t a b l e p o i n t . I f t h i s i s t r u e i t b r i n g s i n t o q u e s t i o n t h e v a l i d i t y o f t h e g a u s s i a n i n t e g r a t i o n t e c h n i q u e . Chapter 2. Effective Hamiltonians 72 2.8.3 A n Assumption is Made I n w h a t f o l l o w s w e s h a l l m a k e t h e a s s u m p t i o n t h a t w e m a y p e r f o r m g a u s s i a n i n t e g r a t i o n s o v e r s m a l l f l u c t u a t i o n s i n <f> a r o u n d t h e s o l u t i o n s o f t h e c l a s s i c a l e q u a t i o n s o f m o t i o n . A s U n r u h h a s p o i n t e d o u t [101] i t i s n o t c l e a r t h a t t h i s a s s u m p t i o n h o l d s e v e n i n t h e l i m i t w h e r e <C 1 - t h e c a s e t r e a t e d b y T u p i t s y n e t . a l . [74]. F u r t h e r m o r e i n o u r c a s e w e h a v e t h a t = 2.67. T h a t t h i s n u m b e r i s l a r g e m e a n s t h e p o t e n t i a l t h a t h o l d s (j> c l o s e t o t h e c l a s s i c a l s o l u t i o n s i s n o t v e r y s t e e p , s o t h a t t h e a s s u m p t i o n t h a t g a u s s i a n i n t e g r a t i o n s c a n b e p e r f o r m e d i s q u e s t i o n a b l e ( h o w e v e r o u r r e s u l t s o f c o m p a r i n g t h e e x a c t s o l u t i o n t o t h e i n s t a n t o n s o l u t i o n f o r t h e t u n n e l i n g a m p l i t u d e s h o w g o o d a g r e e m e n t i n t h i s c a s e - s e e figures 2.9, 6.1 a n d 6.2). The Formal Calculation E x p l i c i t l y t h e L a g r a n g i a n s a p p e a r i n g i n (2.74) a r e L0 = -iS(j)6 sind+ (D-2E)S2 sin2 9+ 4ES2 sin2 6 cos2 <f> — S (Hx s i n 9 c o s <f> + Hy s i n 9 s i n <j> + Hz c o s 9) a n d LN = J2gnpfJin[Apa0SlIpp + Rp^HJ^+J29nkPnH-Ik p=l k=l N + £ fe=i + ^ $ : ^ [ T i - T k - 3 ( I t • flk){Ik • flk)} (2.85) 4 7 r l<k=\ Tlk W e s h a l l a s s u m e t h a t t h e i n s t a n t o n b o u n c e t i m e Q.Q 1 i s s u c h t h a t Q,Q 1 <C T 2 _ 1 , w h e r e T 2 _ 1 i s t h e t i m e s c a l e o v e r w h i c h t h e n u c l e a r - n u c l e a r flip-flop p r o c e s s e s m e d i a t e d b y t h e l a s t t e r m i n t h e a b o v e e x p r e s s i o n o c c u r . I n t h i s l i m i t , t h e n u c l e a r - n u c l e a r t e r m p r o v i d e s Chapter 2. Effective Hamiltonians 7 3 a s t a t i c b i a s field a c t i n g o n e a c h n u c l e u s a n d t h e r e f o r e w e c a n r e w r i t e o u r L a n g r a n g i a n i n t h e f o r m LN = E 9npPn P = I AT + E k=l k=l (2.86) w h e r e n o w t h e s t a t i c field a t e a c h n u c l e u s i s g i v e n b y t h e s u m o f t h e e x t e r n a l field p l u s s o m e c o n t r i b u t i o n f r o m t h e n u c l e a r - n u c l e a r t e r m . I n o u r m o d e l , e a c h e l e c t r o n i c s p i n i s l o c k e d t o t h e c e n t r a l s p i n . T h i s m e a n s t h a t —* (2.87) a n d t h e r e f o r e 8 N 8 E 9npPnA^SlIl + 9nkPnM^SPIkp p=l fc=lp=l 4 dripPn |_pe{pT} pe{pi} N E k=\ + 9nkpJk3 E MS- E Kk Lpe{ptl pe{P4.} (2.88) 2.8.4 Solution for the Free Instanton Trajectory W e a s s u m e t h a t fluctuations i n t h e v a r i a b l e <j> a r e s m a l l i n o u r m o d e l . T h i s w i l l o n l y s t r i c t l y b e t r u e f o r 4E/(D — 2E) l a r g e (see figure 2.9). I n o u r c a s e w e h a v e s e e n t h a t 4E/(D — 2E) ~ 0.37. L o o k i n g b a c k t o o u r c o m p a r i s o n o f t h e e x a c t s o l u t i o n w i t h t h e i n s t a n t o n s o l u t i o n f o r o r t h o r h o m b i c s y m m e t r y w e s e e t h a t i n t h i s r e g i m e t h e i n s t a n t o n s o l u t i o n i s o f f b y a b o u t a f a c t o r o f t w o f r o m t h e e x a c t o n e i n t h e d e t e r m i n a t i o n o f A 0 . N e v e r t h e l e s s w e s h a l l a d o p t t h i s m e t h o d i n t h i s c a s e . D o i n g t h i s a l l o w s u s t o p e r f o r m a G a u s s i a n i n t e g r a t i o n o v e r <j>. T h i s w i l l l e a v e u s w i t h a n e f f e c t i v e d e s c r i p t i o n i n t e r m s Chapter 2. Effective Hamiltonians 7 4 o f o n l y o n e p a t h - v a l u e d p a r a m e t e r 9(T) w h i c h i s t h e a n g l e b e t w e e n S a n d t h e z a x i s d u r i n g t h e i n s t a n t o n t r a j e c t o r y . I n t h e c a s e w h e r e t h e g i a n t s p i n i s n o t c o u p l e d t o t h e e n v i r o n m e n t a l s p i n s a n d t h e r e i s n o e x t e r n a l m a g n e t i c field i t i s k n o w n t h a t t h i s e f f e c t i v e d e s c r i p t i o n r e d u c e s t o ( h e r e w e i n c l u d e o n l y t e r m s t h a t a f f e c t t h e e q u a t i o n s o f m o t i o n ) [75] <?2 . ~ Leff (6) — —92 + D s i n 2 9 (2.89) AE w h e r e w e h a v e d e f i n e d E = AEJ2p\SP\2 a n d D = (D - 2E) £ p |S P| 2, f o l l o w i n g [75]. T h e c l a s s i c a l e q u a t i o n o f m o t i o n i s r e a d i l y f o u n d f r o m (2.89) a n d i s 9 = s i n 9{T) = 1/ c o s h ( £ V ) , Vto = f \[hl5 (2.90) T h e r e a r e t w o t h i n g s w o r t h n o t i n g h e r e . F i r s t i s t h a t t h e f o r m o f t h e i n s t a n t o n a f t e r t h e g a u s s i a n i n t e g r a t i o n s h a v e b e e n p e r f o r m e d i s t h e s a m e a s t h e b a r e c l a s s i c a l f o r m w i t h r e n o r m a l i z e d Q,0. S e c o n d i s t h a t t h e p a r a m e t e r s D a n d E d e p e n d o n t h e f a c t t h a t w e a r e r e a l l y d e a l i n g w i t h a n e i g h t s p i n o b j e c t . I n t h e c a s e o f a s p i n 10 o b j e c t , E = 2ES2 = 2 0 0 £ , w h i l e f o r u s E = 2E £ p \SP\2 = E • 8 • 2 5 / 4 = 1 0 0 £ ; l i k e w i s e f o r D. S u b s t i t u t i n g t h i s e x t r e m a l t r a j e c t o r y i n t o t h e e f f e c t i v e L a g r a n g i a n a n d i n t e g r a t i n g o v e r r g i v e s f o r t h e i n s t a n t o n a c t i o n ID - 2E Aefj = A0 + innS , A0 = 2S^—^— w h e r e n = ± c o r r e s p o n d s t o c l o c k w i s e a n d c o u n t e r c l o c k w i s e p a t h s r e s p e c t i v e l y . N o t e t h a t n e i t h e r t h e H a l d a n e p h a s e TJTTS n o r A0 d e p e n d s o n t h e e i g h t s p i n n a t u r e o f t h e c o l l e c t i v e c e n t r a l s p i n [75]. Chapter 2. Effective Hamiltonians 7 5 2.8.5 Inclusion of the External Magnetic Field and the Nuclear Spins W e n o w m a k e t h e a s s u m p t i o n t h a t t h e a p p l i e d m a g n e t i c field a n d t h e c o u p l i n g s t o t h e e n v i r o n m e n t a l s p i n s a r e w e a k . S p e c i f i c a l l y w e r e q u i r e t h a t « l \H\ « 1 (2.91) Q,o Clo I t i s k n o w n [74] t h a t t h e m o d i f i c a t i o n s t o t h e i n s t a n t o n t r a j e c t o r y c a l c u l a t e d a b o v e (2.90) c o m i n g f r o m t h e e x t e r n a l field a n d t h e s p i n b a t h first a p p e a r t o s e c o n d o r d e r i n a n e x p a n s i o n i n p o w e r s o f £ / f 2 0 w h e r e £ i s o n e o f u[\ uifc o r t h e e x t e r n a l field m a g n i t u d e —* T h i s m e a n s t h a t i f w e a r e o n l y i n t e r e s t e d i n first-order c o r r e c t i o n s d u e t o t h e s e e f f e c t s ( w h i c h w e a r e ) w e c a n n e g l e c t t h e s e a n d u s e t h e t r a j e c t o r y (2.90) i n t h e p r e s e n c e o f t h e e x t e r n a l field a n d t h e s p i n b a t h ( a s l o n g , o f c o u r s e , a s t h e c o n d i t i o n s (2.91) h o l d ) . I t m u s t b e n o t e d h e r e t h a t t h e v a l i d i t y o f t h i s a p p r o a c h h a s b e e n q u e s t i o n e d [101]- i n p a r t i c u l a r t h e a s s u m p t i o n t h a t t h e t r a j e c t o r y o f t h e c e n t r a l s p i n r e s p o n d s t o s e c o n d o r d e r i n t h e e x t e r n a l fields. S u b s t i t u t i o n o f t h e e x t r e m a l t r a j e c t o r y i n t o t h e g e n e r a l e f f e c t i v e l a g r a n g i a n a n d i n t e - g r a t i n g o v e r r y i e l d s t h e f o l l o w i n g e f f e c t i v e a c t i o n , N+8 A E F F = A0 + irjTcS - ir,AH • H + n £ A K N D • IK (2.92) k=l w h e r e AH i s t h e c o n t r i b u t i o n d u e t o t h e e x t e r n a l m a g n e t i c field H a n d t h e A N D t e r m s a r e t h e c o n t r i b u t i o n d u e t o t h e p r e s e n c e o f t h e e n v i r o n m e n t a l s p i n s . E x p l i c i t l y t h e s e a r e irSgu,B„ .S2-ngnB -y-%- 2E -x (2.93) Sir 4 O 0 L b>e{ P T} E <k ID-2E 4E E Kk E P€{p.|.} Mfk (2.94) Chapter 2. Effective Hamiltonians 76 f o r k = 1..N r e p r e s e n t i n g t h e l i g a n d n u c l e a r s p i n s a n d In - OF .1 (2.95) 7»fc /_|_\ V 4E '4^o f o r A; = N + I..N + 8 r e p r e s e n t i n g t h e 57Fe i o n s . I n b o t h c a s e s t h e f3th c o m p o n e n t o f t h e t e n s o r s o n t h e l e f t h a n d s i d e s a r e i d e n t i f i e d w i t h t h e (3th c o m p o n e n t o f t h e v e c t o r s o n t h e r i g h t h a n d s i d e . T h e s e e x p r e s s i o n s s h o u l d b e c o m p a r e d t o e q u a t i o n s (2.27) a n d (2.28) i n [20], n o t i n g o f c o u r s e t h a t h e r e t h e h a r d d i r e c t i o n i s t h e x d i r e c t i o n w h i l s t i n [20] i t i s t h e y d i r e c t i o n . A s i d e f r o m t h i s t h e o n l y d i f f e r e n c e s h e r e a r e t h a t t h e t e r m s d u e t o t h e p r e s e n c e o f t h e n u c l e a r s p i n s h a v e b e e n e x p l i c i t l y w r i t t e n i n t e r m s o f p a r a m e t e r s i n a h i g h e r e n e r g y H a m i l t o n i a n a n d t h e n u c l e a r s p i n s c a n h a v e a r b i t r a r y s p i n n u m b e r s . T h e t u n n e l i n g s p l i t t i n g i n z e r o field A 0 i s g i v e n i n t h i s i n s t a n t o n p i c t u r e b y A 0 = ^ 0 y | ^ e x p ( - A o ) (2.96) A s w e h a v e s e e n e a r l i e r , t h i s e x p r e s s i o n f o r A 0 i s o f f b y a p p r o x i m a t e l y a f a c t o r o f t w o f r o m t h e e x a c t s o l u t i o n f o r (D - 2E)/4E ~ 0.37 ( f i g u r e 2.9). A s d i s c u s s e d p r e v i o u s l y , f o r t i m e s fig 1 < < t « AQ 1, t h e r e l a t i o n s h i p b e t w e e n t h e t r a n s i t i o n a m p l i t u d e a n d t h e o f f - d i a g o n a l p a r t o f t h e e f f e c t i v e H a m i l t o n i a n i s HND = * ( f_r4t(t) + h.c.) (2.97) w h e r e T i t ( r ) = itA0 £ e x p ( - A e / / ) (2.98) a n d f _ i s a P a u l i l o w e r i n g o p e r a t o r i n t h e s u b s p a c e o f t h e t w o - l e v e l N e e l v e c t o r . U s i n g o u r e x p r e s s i o n (2.92) f o r Aeff a n d t h e c r i t e r i a (2.91) a l l o w s u s t o w r i t e t h e o f f - d i a g o n a l p a r t o f t h e e f f e c t i v e H a m i l t o n i a n a s N+8 HND = 2 A 0 f _ c o s ( $ - i Y, AN,D • h) + h.c. (2.99) fc=i Chapter 2. Effective Hamiltonians 77 w h e r e $ = nS - An • H (2.100) 2.9 The Final Single Molecule Effective Hamiltonian A s a r e s u l t o f a l l t h e a b o v e c o n s i d e r a t i o n s w e find t h a t t h e f o r m o f t h e s i n g l e m o l e c u l e e f f e c t i v e H a m i l t o n i a n i s N+8 H = £ k=l N + E jt=i _L !~/fc • mk + Y^l'/fc • Ik eQk yka0 jk 6 / f c ( 2 J f c - i r a P + gu.BSHzfz + 2 A 0 f _ c o s ( $ - i £ AkN>D • h) + h.c. k=l ^ % | n . ^ . /fc _ 3 ( j ; . hk){fk . fjfc)] ( 2 1 Q 1 ) 4 ? r ra r/fc A t t h i s p o i n t w e r e i t e r a t e t h a t o b j e c t i o n s a s t o t h e v a l i d i t y o f t h e i n s t a n t o n c a l c u l a t i o n h a v e b e e n r a i s e d [101]. T h e t e r m t h a t i s a f f e c t e d i s t h e o f f - d i a g o n a l c o n s t r i b u t i o n . W e s h a l l find l a t e r o n t h a t t h e p r e d i c t i o n s t h a t w e o b t a i n f r o m t h e u s e o f t h i s t e r m m a t c h b o t h e x a c t d i a g o n a l i z a t i o n a n d e x p e r i m e n t a l r e s u l t s e x t r e m e l y w e l l i n t h e l o w field r e g i m e t h a t w e a r e c o n s i d e r i n g . W e t r e a t t h i s a s e v i d e n c e ( b u t c e r t a i n l y n o t p r o o f ) t h a t t h e a p p r o x i m a t i o n s m a d e i n t h e i n s t a n t o n c a l c u l a t i o n a r e v a l i d . C h a p t e r 3 N u c l e a r S p i n C o u p l i n g s i n Fe8 a n d t h e I s o t o p e E f f e c t H e r e w e p r e s e n t o u r r e s u l t s f o r t h e q u a n t i t i e s jkg a n d %g ( f o r d e f i n i t i o n s s e e (2.67) a n d (2.68)) i n Fes, w h i c h r e p r e s e n t t h e d i p o l a r f i e l d s d u e t o t h e Fe3+ i o n s a t t h e kth n u c l e a r s p i n b e f o r e a n d a f t e r t h e c e n t r a l s p i n c o m p l e x t u n n e l s r e s p e c t i v e l y . W e a r e g o i n g t o d o t h i s u s i n g t w o d i f f e r e n t m e t h o d s . T h e f i r s t w i l l t r e a t e a c h Fe3+ i o n i n t h e m o l e c u l e a s a p o i n t m a g n e t i c d i p o l e ( a n d a s s u c h w e c a l l t h i s t h e " p o i n t d i p o l e a p p r o x i m a t i o n " ) . T h e s e c o n d m e t h o d w e s h a l l u s e i s t o m o d e l t h e a c t u a l s p a t i a l s p i n d i s t r i b u t i o n n e a r t h e i r o n i o n s b y u s i n g p r e v i o u s l y c a l c u l a t e d H a r t r e e - F o c k w a v e f u n c t i o n s f o r f r e e Fe3+ i o n s . T h i s " s p r e a d i n g o u t " o f t h e m a g n e t i c d i p o l e c h a n g e s t h e v a l u e s f o r t h e f i e l d s a t t h e n u c l e i . T h e s e f i e l d s ( a n d t h e r e f o r e 7 ^ a n d 7 ^ ) a r e t h e n u s e d t o c a l c u l a t e t h e d i p o l a r c o u p l i n g e n e r g i e s u>[' a n d uk. T h e c o n v e r s i o n o f field u n i t s t o e n e r g y u n i t s i s c a l c u l a t e d u s i n g t h e d i p o l e - d i p o l e i n t e r a c t i o n u s i n g k n o w n n u c l e a r ^ - f a c t o r s a n d a s s u m i n g t h a t t h e Fe3+ g f a c t o r i s i s o t r o p i c a n d e q u a l t o 2. W e t h e n u s e {u)k} a n d {ui^}, t o g e t h e r w i t h k n o w n c o n t a c t h y p e r f i n e c o u p l i n g s d u e t o t h e p r e s e n c e o f 57Fe3+ i o n s , t o c a l c u l a t e t h e o r t h o g o n a l i t y b l o c k i n g p a r a m e t e r K, t h e t o p o l o g i c a l d e c o h e r e n c e p a r a m e t e r A a n d t h e f u l l l i n e w i d t h o f t h e n u c l e a r s p i n s W f o r a r b i t r a r y i s o t o p i c c o n c e n t r a t i o n . W e find t h a t a l l t h e s e q u a n t i t i e s s h o w a s i g n i f i c a n t i s o t o p e e f f e c t . 7 8 Chapter 3. Nuclear Spin Couplings in Fe8 and the Isotope Effect 7 9 3.1 U n i t s a n d C o n s t a n t s W e c h o o s e t o w o r k i n t h e S I s y s t e m o f u n i t s . W e t h e r e f o r e h a v e t h a t T h e B o h r M a g n e t o n i s T- = 1 0 " 7 W 47T A2 HB = 0 . 9 2 7 1 2 0 3 • 1 0 - 2 3 ^ (3.2) a n d t h e n u c l e a r m a g n e t o n i s fxn = 0.505038 • l O " 2 6 ^ = 7.622462 (3.3) T h e p r o t o n a n d e l e c t r o n i c i r o n g - f a c t o r s a r e gpr = 5.58510 , gFe = 2 (3.4) H e r e w e h a v e a s s u m e d a n i s o t r o p i c s p i n - o n l y g - f a c t o r f o r t h e e l e c t r o n i c s p i n o f t h e Fe3+ i o n s [?]. T h e u n i t c o n v e r s i o n f a c t o r s w e s h a l l u s e a r e 2 0 . 8 37 GHz = 1 K = 1.3807 • 1 0 ~ 2 3 J = 0.695045 c m - 1 = 8.617 • 1 0 " 5 eV (3.5) 3.2 T h e P o i n t D i p o l e A p p r o x i m a t i o n H e r e we b e g i n t h e p r o b l e m o f c a l c u l a t i n g t h e m a g n e t i c fields c r e a t e d b y t h e i r o n i o n s . T h e t a c k w e u s e h e r e i s t o t r e a t t h e i r o n i o n s a s p o i n t d i p o l e s . T h i s i s a " f i r s t o r d e r " a p p r o a c h w h i c h w i l l o n l y b e u s e f u l i f t h e s p a t i a l e x t e n t o f t h e i r o n w a v e f u n c t i o n s i s m u c h l e s s t h a n t h e d i s t a n c e b e t w e e n t h e i r o n i o n s a n d t h e p r o t o n s . W e w i l l a t t e m p t a m o r e c a r e f u l t r e a t m e n t i n t h e f o l l o w i n g s e c t i o n a n d c o m p a r e i t s r e s u l t s t o t h o s e o b t a i n e d u s i n g t h e p o i n t d i p o l e a p p r o x i m a t i o n . Chapter 3. Nuclear Spin Couplings in Fe% and the Isotope Effect 8 0 3.2.1 Magnetic Field at r due to a "Point Dipole" at 0 T h e field a t f d u e t o a p o i n t d i p o l e a t t h e o r i g i n i s 7 (0 = - ^ ^ l [ ^ - 3 ( m - f ) f ] (3.6) w h e r e m i s t h e m a g n e t i c d i p o l e m o m e n t o f t h e d i p o l e a t t h e o r i g i n , w h i c h i n o u r c a s e i s m = QFePBS = 9FePBSS (3.7) T h e r e f o r e w e find t h a t l(r) = -^-^gFePBs[s-S(s-f)r} (3.8) w h i c h i s e q u i v a l e n t t o 7 ( f ) = 4 . 6 3 6 | ^ [ 3 ( 5 - r ) r - s ] T (3.9) w h e r e r i s m e a s u r e d i n A n g s t r o m s . 3.2.2 Magnetic Field at fp due to Eight "Point Dipoles" at fFei 8 D e f i n e t h e v e c t o r j o i n i n g t h e pth n u c l e u s a n d jth i r o n i o n t o b e fPj = fp — fj. T h e n t h e t o t a l field a t t h e pth n u c l e u s d u e t o t h e e i g h t i r o n i o n s i n t h e p o i n t d i p o l e a p p r o x i m a t i o n i s 7 ( r p ) = 4.636 £ [ 3 ( S i • fpj)fpj - T (3.10) j=i \rpj\ w h e r e a g a i n d i s t a n c e s a r e m e a s u r e d i n A n g s t r o m s . 3.2.3 Isotopic Concentrations, Nuclear ^-factors and Quadrupolar Moments in F e 8 I n t a b l e 3.1 w e p r e s e n t i n f o r m a t i o n o n t h e p r o p e r t i e s o f t h e v a r i o u s n u c l e i t h a t c a n b e f o u n d i n t h e F e 8 m o l e c u l e . T h e n u c l e a r m a g n e t i c m o m e n t s a r e e q u a l t o gnn\I\ a n d a r e l i s t e d i n u n i t s o f n u c l e a r m a g n e t o n s . Chapter 3. Nuclear Spin Couplings in F e 8 and the Isotope Effect 8 1 S p e c i e s C o n c e n t r a t i o n \f\ N u c l e a r M o m e n t [//n] Q u a d r u p o l e M o m e n t Q [10 2 4 c m 2 ] 1H 99.98 1/2 2.79255 0 2H 0.02 1 0.857354 0.00273 98.88 0 0 0 1 3 C 1.12 1/2 0.70225 0 UN 99.62 1 0.40365 0.02 15N 0.38 1/2 -0.2830 0 1 6 0 99.757 0 0 0 1 7 0 0.039 5/2 -1.8935 -0.005 1SQ 0.204 0 0 0 91.068 0 0 0 5 7 F e 2.20 1/2 0.05 0 50.56 3/2 2.10576 0.335 s i B r 49.47 3/2 2.2696 0.280 T a b l e 3.1: N u c l e a r s p i n i n f o r m a t i o n f o r n u c l e i o c c u r i n g i n F e s - F r o m [48]. T h e i s o t o p i c c o n c e n t r a t i o n s s h o w n i n t a b l e 3.1 a r e t h e " n a t u r a l l y o c c u r i n g " c o n c e n - t r a t i o n s . I t i s q u i t e p o s s i b l e t o a l t e r t h e s e c o n c e n t r a t i o n s a n d a s s u c h i n w h a t f o l l o w s w e s h a l l t r e a t t h e g e n e r a l c a s e w h e r e t h e p a r t i c u l a r i s o t o p i c c o n c e n t r a t i o n s i n t h e m a t e r i a l m a y b e v a r i e d . 3.2.4 Definition and Evaluation of , ui'kl and u£ 7 ^ a n d 7 ^ a r e t h e d i p o l a r fields ( i n T e s l a ) a t t h e p o i n t fk b e f o r e / a f t e r t h e c e n t r a l s p i n flips r e s p e c t i v e l y , f o r a l l n u c l e i k = 1..N + 8 ( s e e (2.67) a n d ( 2 . 6 8 ) ) . N o t e t h a t i n t h e a b s e n c e o f a n e x t e r n a l l y a p p l i e d m a g n e t i c field w e h a v e 7 ^ = — 7 ^ ( t h e field d u e t o t h e c e n t r a l s p i n c l u s t e r j u s t flips i t s d i r e c t i o n w h e n t h e c e n t r a l o b j e c t t u n n e l s ) . Chapter 3. Nuclear Spin Couplings in Fe$ and the Isotope Effect 8 2 T h e e n e r g y o f i n t e r a c t i o n o f t h e n u c l e a r d i p o l e a t rk w i t h t h e e x t e r n a l f i e l d i s Uk - 9nPnh • Ik l+fz QnPri [gnpJk• ik]] + ^-y^ [gnPnh • fi2)] [^ • (#+€ ) )+U - ( 7 i 1 ) - # ) ] (3-11) A s b e f o r e w e t a k e -(1) -(2) -,(1) _ (̂2) - _ 7fc + Ik f _ 7fc Ik fi-\o\ * _ l 7 i " + # l ' ' " i T i " " ^ ! ' 1 ' 4 ' = I ^ W T I 1 ' - € ' I . 4 = l4l9«M»l7i 1 ) + 7 f l (3-13) w h i c h g i v e s a n i n t e r a c t i o n t e r m o f t h e f o r m ( c o m p a r e t o (2.7 3 ) ) Uk = ^ik-mk + ̂ 4h-ik (3.14) 3.2.5 C o n t a c t H y p e r f i n e C o u p l i n g E n e r g i e s f o r 5 7 F e 3 + T h e r e i s a n o t h e r c o n t r i b u t i o n d u e t o c o n t a c t h y p e r f i n e i n t e r a c t i o n s d u e t o t h e p r e s e n c e o f a n y 5 7Fe n u c l e i i n t h e m a t e r i a l . A s w a s d i s c u s s e d i n c h a p t e r 2, t h i s c o u p l i n g i s o f t h e f o r m (see (2.61)) Ucp = gn^A^SVl (3.15) W e a r e g o i n g t o m a k e t h e a p p r o x i m a t i o n i n w h a t f o l l o w s t h a t t h e o f f - d i a g o n a l e l e m e n t s o f t h e t e n s o r Apal3 a r e z e r o . T h i s g i v e s t h e f o r m U; = ucpIp-Sp (3.16) w h e r e Sp a n d Ip a r e t h e e l e c t r o n i c a n d n u c l e a r s p i n o f t h e pth 57 Fe i o n r e s p e c t i v e l y . S i m i l a r l y t o w h a t w e d i d w i t h t h e d i p o l a r t e r m w e w r i t e , w i t h Sp^ a n d 5 p 2 ) t h e pth Chapter 3. Nuclear Spin Couplings in F e 8 and the Isotope Effect 8 3 e l e c t r o n i c s p i n b e f o r e a n d a f t e r t h e c e n t r a l s p i n flips r e s p e c t i v e l y , u; = u;ip-sp f [TP • (S^ + S f ) + fzIp • (SM - Si2))] (3.17) D e f i n e 5(1) + 5(2) „ 5(1) _ 5(2) P ~ I 5(1) , 5(2) I ' ^ - , 5 ( 1 ) 5(2) I ' i Op I I^P I 4c=\IPK\SU-SW\ , u,}* = \TP\U;\%» + (3.19) T h e n w e m a y w r i t e t h e i n t e r a c t i o n t e r m i n t h e f o r m Ucp = ^ I p • mcp + ^J\% • % (3.20) T h e v a l u e o f t h e field a t t h e n u c l e u s o f a f r e e 5 7 F e 3 + i o n d u e t o p o l a r i z a t i o n o f t h e s e l e c t r o n s b y t h e o u t e r s h e l l 3d e l e c t r o n s h a s b e e n p r e v i o u s l y c a l c u l a t e d a n d w a s f o u n d t o b e Hc ~ 63 T [99]. I f w e t a k e t h e n u c l e a r ^ - f a c t o r o f t h e 5 7 F e i o n t o b e g = 0.05 [166], t h i s g i v e s a z e r o field l o n g i t u d i n a l c o n t a c t h y p e r f i n e c o u p l i n g o f u ; p | c ~ 4 8 MHz V p (3.21) T h e s e c o m p l e t e l y o v e r w h e l m t h e d i p o l a r c o u p l i n g e n e r g i e s (^k=N+i..N+s a n <^ u)k=N+i..N+8^ a s t h e d i p o l a r fields a t t h e i r o n n u c l e i a r e o n t h e o r d e r o f 3 0 0 — 8 0 0 g a u s s (see t a b l e 3.10). B e c a u s e o f t h i s , i t i s e a s i e s t t o t h i n k a b o u t t h e " d i a g o n a l " e f f e c t s o f t h e n u c l e a r s p i n s i n t h e f o l l o w i n g way. A l l n u c l e a r s p i n s i n t h e m o l e c u l e k = 1..N + 8 a r e i n v o l v e d i n d i p o l e - d i p o l e i n t e r a c t i o n s v i a (3.14). H o w e v e r , t h e 5 7 F e n u c l e i k = N + 1..N + 8 a l s o a r e i n v o l v e d i n c o n t a c t h y p e r f i n e i n t e r a c t i o n s w h i c h a b s o l u t e l y s w a m p t h e d i p o l e - d i p o l e Chapter 3. Nuclear Spin Couplings in Fe$ and the Isotope Effect 8 4 i n t e r a c t i o n s o f t h e 57Fe n u c l e i . T h e r e f o r e t h e i n t e r a c t i o n t e r m i n o u r H a m i l t o n i a n i s (3.22) N+8 k=l w h e r e i t i s u n d e r s t o o d t h a t t h e t e r m s f o r k = 1..N a r e t h e d i p o l e - d i p o l e t e r m s (3.14) b u t t h e t e r m s f o r k = N + 1..N + 8 a r e c o n t a c t t e r m s f r o m (3 . 2 0 ) . 3.2.6 Calculation of uk and u)k from Knowledge of Atomic Positions I n t h e Fes m o l e c u l e , w e k n o w w h e r e a l l t h e a t o m s a r e i n t h e m o l e c u l e . T h i s a l l o w s u s t o c a l c u l a t e w h a t t h e p a r a m e t e r s wk a n d u)^ a r e , f o r k = I..N, v i a t h e u s e o f (3.10) a n d (3.13) t o g e t h e r w i t h t h e k n o w l e d g e o f t h e d i r e c t i o n s i n w h i c h t h e c e n t r a l s p i n o b j e c t p o i n t s i n t h e t w o l o w e s t l y i n g e n e r g y l e v e l s | + 1 0 > a n d | — 10 >. T h i s l a s t i s c a l c u l a t e d a s f o l l o w s . T h e c e n t r a l s p i n H a m i l t o n i a n f o r F e 8 , w h e n t r u n c a t e d t o t e r m s o f q u a r t i c o r d e r o r —+ l e s s i n S, c a n b e w r i t t e n i n t h e p r e s e n c e o f a n e x t e r n a l field H i n t h e f o r m H0 = -DS2Z +E(S2X- S2) + C(S* + Si) + 9iiBH • S (3.23) a s d i s c u s s e d i n c h a p t e r 2. H e r e w e t a k e D = 0.292/C, E = 0 . 0 4 6 ^ a n d C = - 2 . 9 • 10~5K i n k e e p i n g w i t h t h e findings o f W e r n s d o r f e r e t . a l . [51]. I f w e w r i t e S i n s p h e r i c a l c o o r d i n a t e s Sx —>• S s i n 9 s i n 0, Sy —> S c o s <j> s i n 9 a n d Sz —> S c o s 9 a n d s u b s t i t u t e t h e s e i n t o (3.23) i t i s t h e n a s i m p l e e x e r c i s e t o find t h e a n g l e s (9, <j>) a n d t h e r e f o r e t h e q u a n t i t i e s —• Sj f o r a l l t h e e l e c t r o n i c s p i n s i n t h e m o l e c u l e a s f u n c t i o n s o f H. W e a s s u m e t h a t t h e i n d i v i d u a l e l e c t r o n i c s p i n s a r e l o c k e d t o t h e d i r e c t i o n i n w h i c h t h e e f f e c t i v e c e n t r a l s p i n i s p o i n t i n g , w i t h r e l a t i v e s i g n s g i v e n b y t a k i n g | + 10 > a n d | — 10 > t o c o r r e s p o n d t o t h e F e 3 + s p i n s b e i n g { t , 1\ 1,1\ Ti t> I, T} a n d {I, I, t> l> l> 4> t> 4} r e s p e c t i v e l y , w h e r e t h e l a b e l l i n g 1 - 4 8 c o r r e s p o n d s t o t h a t i n t a b l e 3.2. A l l c o o r d i n a t e p o s i t i o n s a r e i n A n g s t r o m s . Chapter 3. Nuclear Spin Couplings in Fe% and the Isotope Effect 8 5 W e p r e s e n t i n f o r m a t i o n o n t h e z e r o field v a l u e s o f t h e u>{! i n t h e f o l l o w i n g m a n n e r . W e l a b e l a l l t h e n u c l e i i n t h e m o l e c u l e s u c h t h a t t h e h y d r o g e n a t o m s a r e t a g g e d 1..120, t h e b r o m i n e s 121..128, t h e n i t r o g e n s 129..146, t h e i r o n s 147..154, t h e c a r b o n s 155..190 a n d t h e o x y g e n s 191..213 ( t h e p o s i t i o n s o f e a c h o f t h e s e a r e g i v e n i n s e c t i o n 2.5). F o r e a c h n u c l e u s t h e r e a r e a v a r i e t y o f p o s s i b l e i s o t o p e s . S h o w n i n figure (3.1) a r e t h e v a l u e s f o r t h e z e r o field s h o w n a s f u n c t i o n s o f n u c l e u s l a b e l f o r 1H, 79Br, UN, 57Fe, 1 3 C a n d 170. T h i s i n f o r m a t i o n i s p r e s e n t e d i n a d i f f e r e n t m a n n e r i n figures (3.2) t h r o u g h (3.13). I n t h e s e figures w e b i n t h e h y p e r f i n e v a l u e s , i n e a c h c a s e a s s u m i n g t h a t t h e i s o t o p e i n q u e s t i o n r e p r e s e n t s 1 0 0 % o f t h e e l e m e n t i n q u e s t i o n ( f o r e x a m p l e , i n figure (3.2) w e a s s u m e a 1 0 0 % c o n c e n t r a t i o n o f 1 / / ) . 3.2.7 Calculation of the Orthogonality Blocking Parameter K T h e P r o k o f ' e v a n d S t a m p t h e o r y [20] c o n t a i n s a p a r a m e t e r K w h i c h i s d e f i n e d t o b e AT+8 n c o s ^ Lfc=l (3.24) K = — In w h e r e c o s 2 A = - 7 i 1 ) -7i 2 ) (3-25) S i n c e w e k n o w w h a t t h e fields 7^ a n d 7^ a r e , w e m a y c a l c u l a t e t h e fa a n d t h e n K. I f t h e n u c l e a r s p i n i n q u e s t i o n i s a 5 7 F e t h e n b e c a u s e t h e c o n t a c t h y p e r f i n e field i s s o m u c h l a r g e r t h a n t h e d i p o l a r field ( 6 3 T a n d ~ 0.3 — 0.8 T r e s p e c t i v e l y ) w e c a n t a k e c o s 2 & = - s £ 1 ) . S j 2 ) (3.26) N o t e t h a t f o r a l l n u c l e i fa i s a s t r o n g f u n c t i o n o f t h e e x t e r n a l l y a p p l i e d D C m a g n e t i c field. S h o w n i n figures 3.14 a n d 3.15 i s K a s a f u n c t i o n o f a D C field a p p l i e d i n t h e x Chapter 3. Nuclear Spin Couplings in Fe$ and the Isotope Effect 86 Nuclear Label X y z Fel 5.737857 .064911 1.58115 Fe2 6.912112 -2.210346 3.04665 Fe3 6.656427 3.323106 .33915 Fe4 5.327499 2.408451 3.17265 Fe5 5.198078 -2.412666 -3.16665 Fe6 3.616621 2.196296 -2.98815 Fe7 3.888089 -3.321701 -.29265 Fe8 4.848748 -.043695 -1.54365 Table 3.2: Positions of the iron ions, units in Angstoms. 120.0 Figure 3.1: cJk for all nuclei in F e 8 . Labeling is as indicated in the text. The dots represent values for 2H (labels 1..120), 81Br (labels 121..128), and 1 5 A 7 (labels 129..146). Chapter 3. Nuclear Spin Couplings in Fe8 and the Isotope Effect 8 7 6 | — , — — , , | , , , , , , r - , , 1 5 - 4 - <D E 3 - I I z 2 - I I I I °0.0 5.0 10.0 15.0 20.0 w_k|| [MHz], Binned in 0.1 MHz steps F i g u r e 3.2: 1H, e m p h a s i z i n g l o w e n d o f t h e s p e c t r u m . 6 i • • 1 • • , • • 1 • • 1 5 - 4 f 0.0 30.0 60.0 90.0 120.0 w_k|| [MHz], Binned in 0.1 MHz steps F i g u r e 3.3: 1H, h i g h e n d o f t h e s p e c t r u m . Chapter 3. Nuclear Spin Couplings in Fe8 and the Isotope Effect 8 8 6 i - 5 - 4 I 10.0 w_k|| [MHz], Binned in 0.1 MHz steps F i g u r e 3.4: 2H, l o w e n d o f s p e c t r u m . 6 | . . . , . . 5 - 4 h w_k|| [MHz], Binned in 0.1 MHz steps F i g u r e 3.5: 2H, e n t i r e s p e c t r u m . Chapter 3. Nuclear Spin Couplings in Fe$ and the Isotope Effect 89 0.0 5.0 10.0 w_k|| [MHz], Binned in 0.1 MHz steps F i g u r e 3.6: 79Br, e n t i r e s p e c t r u m . <D E 0.0 5.0 10.0 w_k|| [MHz], Binned in 0.1 MHz steps F i g u r e 3.7: slBr, e n t i r e s p e c t r u m . Chapter 3. Nuclear Spin Couplings in Fe$ and the Isotope Effect 9 0 5.0 10.0 w_k|| [MHz], Binned in 0.1 MHz steps F i g u r e 3.8: 14N, e n t i r e s p e c t r u m . 10.0 w_k|| [MHz], Binned in 0.1 MHz steps F i g u r e 3.9: 15N, e n t i r e s p e c t r u m . Chapter 3. Nuclear Spin Couplings in Fe$ and the Isotope Effect 91 0.00 0.10 0.20 0.30 0.40 0.50 w_k|| [MHz], Binned in 0.001 MHz steps F i g u r e 3.10: 57Fe, e n t i r e s p e c t r u m . <D £ 1 r 0.0 5.0 10.0 w_k|| [MHz], Binned in 0.1 MHz steps F i g u r e 3.11: 1 3 C , e n t i r e s p e c t r u m . Chapter 3. Nuclear Spin Couplings in Fe$ and the Isotope Effect 9 2 5.0 10.0 15.0 20.0 w_k||, [MHz], Binned in 0.1 MHz steps F i g u r e 3.12: 1 7 0 , l o w e n d o f s p e c t r u m . 20.0 40.0 60.0 w_k||, [MHz], Binned in 0.1 MHz steps F i g u r e 3.13: 170, e n t i r e s p e c t r u m . Chapter 3. Nuclear Spin Couplings in Fe8 and the Isotope Effect 9 3 d i r e c t i o n f o r a n Fe8 m o l e c u l e w i t h 1 0 0 % c o n c e n t r a t i o n s o f 1H, 81Br, UN, 57Fe, 1 3 C a n d 170. W e s h a l l c a l l t h i s m a t e r i a l Fe™ax. N o t e t h a t t h e c o n t r i b u t i o n t o K f r o m t h e p r e s e n c e o f 5 7 F e i s m i n i m a l b e c a u s e t h e m i n i m a o f S d o n ' t c h a n g e m u c h a s f u n c t i o n s o f f i e l d . 500.0 400.0 300.0 CO o. o. 200.0 100.0 0.0 0.0 0.5 1.0 H_x [Tesla] F i g u r e 3.14: T h e o r t h o g o n a l i t y b l o c k i n g p a r a m e t e r K o v e r a l a r g e r a n g e o f e x t e r n a l fields a p p l i e d i n t h e x d i r e c t i o n f o r t h e F e ™ a i m a t e r i a l . 3.2.8 Calculation of E0 T h e q u a n t i t y E0 i s r e l a t e d t o t h e s p r e a d i n e n e r g y s p a c e d u e t o t h e p r e s e n c e o f m a n y n u c l e a r s p i n s . F r o m (3.14) w e s e e t h a t i n t h e a b s e n c e o f q u a d r u p o l a r o r c o n t a c t i n - t e r a c t i o n s a n d i n z e r o e x t e r n a l m a g n e t i c field t h e kth n u c l e a r s p i n h a s 21 + 1 e q u a l l y s p a c e d e n e r g y l e v e l s b e t w e e n i w f / 2 . I n t h e p r e s e n c e o f a n a p p l i e d field t h e s i t u a t i o n i s s i m i l a r . T h e v a r i a n c e i n t h e d i s t r i b u t i o n o f t h e e n e r g y l e v e l s f o r t h i s n u c l e u s i s d e f i n e d t o b e a2 =< E2 > — < E > 2 w h e r e E i s t h e e n e r g y o f t h e n u c l e u s . I f w e m a k e t h e a p p r o x i m a t i o n t h a t t h e p r o b a b i l i t y o f e a c h l e v e l b e i n g o c c u p i e d i s i d e n t i c a l ( e f f e c t i v e l y T o t a l Chapter 3. Nuclear Spin Couplings in Fe$ and the Isotope Effect 9 4 10.0 cd Q. a 0.03 H_x [Tesla] F i g u r e 3.15: T h e p a r a m e t e r K f o r s m a l l v a l u e s o f e x t e r n a l f i e l d a p p l i e d i n t h e x d i r e c t i o n f o r t h e FeT'ax m a t e r i a l . a n i n f i n i t e s p i n t e m p e r a t u r e a p p r o x i m a t i o n ) t h e n < E > = 0 a n d |2 2/+1 cr2 = < E2 >= £ PiE2 = i=i i <4 21+ 1 4 2 ( 2 - 1 ) ' 2 / 7 + 1 |,2 (3.27) 1 2 / * H e r e £>i = ^f+i i s t h e p r o b a b i l i t y o f t h e ith e n e r g y l e v e l o f t h e n u c l e u s b e i n g o c c u p i e d . T h e t o t a l v a r i a n c e o f t h e d i s t r i b u t i o n o f a l l n u c l e a r l e v e l s i s t h e n g i v e n b y t h e c e n t r a l l i m i t t h e o r e m a s N -°2 = zZ°. (3.28) fe=i w h e r e N i s t h e t o t a l n u m b e r o f n u c l e a r s p i n s . I n t h e c a s e o f Fe8 t h i s g i v e s = £ ( < K £ ( 4 / H £ ( 4 l . J ) *2 ^79. Chapter 3. Nuclear Spin Couplings in Fe$ and the Isotope Effect 9 5 + ^ « : ) + 5 E « 2 ) + s « 2 ) ksi B t knN ki5N + E ( 4 L / ) + £ E M + E( kl3c k57Fe + E UJ) (3-29) k57Fe T h e re l a t ion between the h a l f w i d t h of the d i s t r i b u t i o n W, E0 a n d o is found v i a or El = Aa2 = 2W2 ( 3 . 31 ) T h e fu l l w i d t h of the d i s t r i b u t i o n is W = 2W = V2E0 ( 3 . 32 ) T h e w i d t h W is of course a func t ion of the i so top ic concen t r a t ion i n a p a r t i c u l a r Fe8 sample . F o r example , i f we p ick the easiest case where we have 1 0 0 % concent ra t ions of 1H, 79Br, UN, 1 6 0 , 56Fe a n d l2C (we sha l l c a l l th i s m a t e r i a l F e 8 * ) then we find, i n zero ex te rna l field i n the po in t d ipo l e a p p r o x i m a t i o n , tha t 120 Y colk\H = 7 1 8 8 7 . 5 4 5 [MH kiH=l z\2 \ E 4 7 J = 7 0 . 2 7 [MHz]2 fc79Br=l o E "Lw = 1 8 1 . 8 1 8 4 [MHz]2 ( 3 . 33 ) *J U . 1 w h i c h gives E0 = 2 6 8 . 6 MHz , W = 3 7 9 . 8 MHz = 1 8 . 2 3 mK ( 3 . 34 ) Chapter 3. Nuclear Spin Couplings in Fe$ and the Isotope Effect 96 What we have calculated here is an intrinsic linewidth due to spreading caused by the presence of the nuclear spins in the molecule. Note that these numbers are sensitive to which isotopic concentrations we choose for our molecule. This should lead to a clear isotope effect in this intrinsic linewidth which is easy to calculate using our formalism (it just amounts to changing the spin and nuclear moments for the new isotopes). For example, if we replace all the hydrogen nuclei by deuterium (we shall call this material Fesr)), which has 1 = 1 and g = 0.857354 /xn, we find that (3.35) which gives E 0 = 69.06 MHz , W = 97.67 MHz = 4.687 mK (3.36) Note as well that because wjj is a function of external magnetic field (because the minima of S are), the linewidth EQ is also a function of external magnetic field. Shown in figure 3.16 is the intrinsic linewidth as a function of a field applied in the x direction due to specific isotopes for 100% concentrations of these isotopes (in other words, if the isotope is lH we are assuming that all the hydrogens are 1H; if the isotope is 81Br then we assume all the bromines are 8 1 Br, etc.). The addition of 57Fe to the mix significantly changes the value of the linewidth. This is because the contact hyperfine coupling energies are large. Let us define the material b l Fe% to be identical to Fe 8 * in every way except that every iron ion is a 57Fe ion; ie. 100% concentrations of 1H, 79Br, 1 5JV, 57Fe, l2C and 1 6 0 . Then the contribution to the zero-field linewidth coming from the contact terms is, via (3.29), 8 8 £ 4'c2 ~ £ 2304 MHz2 = 18432 MHz2 (3.37) k=i k=i which is a significant fraction of the contribution from the protons (see (3.33)). Addition Chapter 3. Nuclear Spin Couplings in Fe% and the Isotope Effect 9 7 0.5 1.0 H_x [Tesia] F i g u r e 3.16: I n t r i n s i c l i n e w i d t h W d u e t o p a r t i c u l a r i s o t o p e s a s a f u n c t i o n o f Hx f o r 1 0 0 % c o n c e n t r a t i o n s o f t h e s e i s o t o p e s . N o t e t h a t w f a n d t h e r e f o r e W d r o p s s l o w l y w i t h field. T h i s e f f e c t c o m e s a b o u t b e c a u s e a s t h e e x t e r n a l field i s r a i s e d , t h e t w o m i n i m a o f t h e c e n t r a l s p i n c o m p l e x a r e f o r c e d c l o s e r t o g e t h e r ( n o l o n g e r a r e t h e y a n t i p a r a l l e l ) . T h e c u r v e s h o w n a s " t o t a l " i s t h e t o t a l r e s u l t f o r a m a t e r i a l c o n t a i n i n g 1 0 0 % o f t h e i s o t o p e s s h o w n . o f t h i s t e r m g i v e s t h e l i n e w i d t h £ 0 = 301.0 MHz , W = 425.6 MHz = 20.43 mK (3.38) w h i c h i s 1.12 t i m e s t h e l i n e w i d t h f o r F e 8 * . 3.2.9 Calculation of Topological Decoherence Parameters AkND and A I f w e a s s u m e f o r t h e m o m e n t t h a t t h e r e a r e n o 57Fe n u c l e i i n o u r m o l e c u l e , t h e n w e c a n s e e f r o m (2.89) t h a t a l l p a r a m e t e r s c a n b e c a l c u l a t e d i n t h e e x p r e s s i o n f o r AkND. I n a d d i t i o n , i f w e a s s u m e t h a t t h e c o n t a c t i n t e r a c t i o n i s o f t h e f o r m (3.16) t h e n w e c a n c a l c u l a t e t h e g e n e r a l f o r m f o r AkND. F u r t h e r m o r e , w e s e e t h a t t h e s e a r e n o t f u n c t i o n s o f t h e e x t e r n a l m a g n e t i c field, u n l i k e t h e o r t h o g o n a l i t y b l o c k i n g p a r a m e t e r s . B e c a u s e t h e c o u p l i n g e n e r g i e s a r e m u c h l e s s t h a n t h e e n e r g y s c a l e f i n , a l l o f t h e AkND t u r n o u t t o 500.0 400.0 x — 300.0 5 200.0 100.0 •1 BramWe-81 Nitrogen-14 | Caroon-13 Oxyg.n-17 lron-67 - Total 0.0 0.0 Chapter 3. Nuclear Spin Couplings in Fe$ and the Isotope Effect 9 8 0.5 1.0 H_x [Tesla] F i g u r e 3.17: I n t r i n s i c l i n e w i d t h W as a f u n c t i o n o f Hx f o r F e 8 * , Fe8D a n d 57Fe8. b e s m a l l (see figure 3.18). T h e t o p o l o g i c a l d e c o h e r e n c e p a r a m e t e r A [20] i s g i v e n i n t h i s l i m i t b y * = ^E\An\2 (3-39) T h i s i s f o u n d t o b e A = 4.45 • 1 0 ~ 5 , A = 8.87- H T 5 a n d A = 1.38 • 1 0 " 5 f o r t h e Fe8„ 57Fe8 a n d Fe8o m a t e r i a l s r e s p e c t i v e l y . N o t e t h a t t h e s e a r e e x t r e m e l y s m a l l ! A i s r o u g h l y t h e n u m b e r o f n u c l e a r s p i n s flipped p e r c e n t r a l s p i n t u n n e l i n g e v e n t . 3.3 Using Free Fe3+ Hartree-Fock Wavefunctions to M o d e l Actua l Spin Dis- tributions I n t h i s s e c t i o n w e a t t e m p t t o d o a l i t t l e b e t t e r t h a n t h e p o i n t d i p o l e a p p r o x i m a t i o n . H e r e w h a t we s h a l l d o i s i n s t e a d o f t r e a t i n g t h e m a g n e t i c d i p o l e n a t u r e o f t h e Fe3+ i o n a s a p o i n t w e s h a l l a s s u m e t h a t i t i s " s p r e a d o u t " i n a w a y d i c t a t e d b y t h e s p a t i a l s p r e a d o f t h e Fe3+ w a v e f u n c t i o n . 500.0 400.0 I — 300.0 200.0 100.0 0.0 0.0 57-Fe Fe_8C Fe_8' Chapter 3. Nuclear Spi 99 0.000 0.001 0.002 0.003 0 0.001 0.002 0.003 F i g u r e 3.18: B i n n e d t o p o l o g i c a l d e c o h e r e n c e p a r a m e t e r s \AkN D\ f o r a l l n u c l e i , a s s u m i n g 1 0 0 % c o n c e n t r a t i o n s o f ( c l o c k w i s e f r o m b o t t o m l e f t ) 1H, 79Br, UN, 17O, 13C a n d 57Fe, u s i n g t h e p o i n t d i p o l e a p p r o x i m a t i o n . T h e b i n w i d t h h e r e i s 0.0001; p l o t t e d o n t h e x a x i s i s \A%D\ a n d o n t h e y a x i s " n u m b e r o f n u c l e i " . N o t e t h a t t h e c o n t r i b u t i o n t o \AkNtD\ f r o m 57Fe i s a l m o s t e n t i r e l y f r o m t h e c o n t a c t i n t e r a c t i o n . S p e c i f i c a l l y , w e a r e i n t e r e s t e d i n t h e five 3d e l e c t r o n s i n t h e F e 3 + i o n ( i t s e l e c t r o n i c c o n f i g u r a t i o n i s o f c o u r s e [Ar]3d 5). I n a f r e e F e 3 + i o n , t h e s e five d e l e c t r o n s a r e s p i n - a l i g n e d d u e t o t h e H u n d ' s r u l e w h i c h a s k s f o r m a x i m i z e d s p i n a n g u l a r m o m e n t u m g i v i n g a t o t a l s p i n o f 5/2. W e c a n w r i t e d o w n w h a t t h e field a t a p o i n t r i s d u e t o a F e 3 + i o n a t t h e o r i g i n ; i t i s = ~ E / j=i 3*T2-(™i ' [r-r]){f-r ) \r — T \ 2 (3.40) w h e r e m = gFel^BS = gFe^B^S (3.41) b e c a u s e t h e s p i n o f e a c h d e l e c t r o n i s o n e h a l f . H e r e t h e s u m o v e r j i s o v e r t h e five d e l e c t r o n s , a n d t h e i n t e g r a t i o n r i s o v e r a l l s p a c e . W e a s s u m e t h a t t h e w a v e f u n c t i o n s ^j{r) a r e p r o p e r l y n o r m a l i z e d . Chapter 3. Nuclear Spin Couplings in Fe8 and the Isotope Effect 100 Since the total angular momentum of the Fe3+ in its ground state is zero, the spin of the d electrons is distributed spherically and therefore we can approximate (3.40) by the expression 3 4rr J \r — r d m \f — r I2 (m • [f — r])(f' — r) (3.42) where ip(r) represents the spin distribution around the iron ion. The wavefunction ip(r) for a free Fe3+ ion has been previously calculated using a Hartree-Fock approach [100]. We can fit the numerical results of this calculation using the form ib(r) = (A + Br + Cr2)r2 exp (-r/d)1 (3.43) where A = - 6 0 . 7 8 6 0 9 7 , B = 68.94202-1 , C = -22 .48757 -L , d = 0 .282745A (3.44) A A2 This is. the form we shall use in the following. Note that it is not exactly correct as immersion in the crystal will change the electronic distributions and therefore the spin distribution. However it is clear that using the free Hartree-Fock wavefunction here will give more realistic results than the point dipole approximation (it remains to see how different these are). The integrations in (3.42) are handled as follows. Instead of trying to do these ana- lytically, we shall do them numerically using the following technique. We pick P points out of the \il>(r)\2 distribution to represent one iron ion wavefunction. This will be exact as P —> oo. Then 3 n } 4 r r P p t t \r-rp\3 fh _ (rh • [f- fp])(f- fp) T Tp | (3.45) Convergence is reached for P ~ 80 for all nuclei (the closer to an iron nucleus a proton is, the larger number of points are required for convergence). If the point dipole ap- proximation were exact, then P = 1 would suffice (one point). Shown in figure (3.20) Chapter 3. Nuclear Spin Couplings in Fes and the Isotope Effect 101 24 15 05 / \ I \ I T-05 15 F i g u r e 3.19: H a r t r e e - F o c k r e s u l t s f o r t h e f r e e Fe3+ w a v e f u n c t i o n . i s a c o m p a r i s o n o f t h e Jjj. v a l u e s o b t a i n e d u s i n g t h e p o i n t d i p o l e a p p r o x i m a t i o n a n d t h e H a r t r e e - F o c k m e t h o d , u s i n g t h e F e g 1 " 1 m a t e r i a l ( 1 0 0 % 1H, 81Br, UN, 1 7 0 , 1 3 C a n d 57Fe). W e find t h a t t h e l o w e r e n e r g y n u c l e i a r e n o t a f f e c t e d b y t h e c h a n g e t o t h e H a r t r e e - F o c k w a v e f u n c t i o n . O n l y t h e h i g h e r e n e r g y n u c l e i a r e a f f e c t e d s i g n i f i c a n t l y . T h i s h o w e v e r c o u l d b e m e a n i n g f u l f o r s e v e r a l q u a n t i t i e s o f i n t e r e s t , p r i m a r i l y t h e i n t r i n s i c l i n e w i d t h W w h i c h i s s e n s i t i v e t o t h e h i g h e r e n e r g y c o u p l i n g s . S h o w n i n figures (3.21) t h r o u g h (3.32) a r e t h e b i n n e d h y p e r f i n e v a l u e s o b t a i n e d u s i n g t h e H a r t r e e - F o c k w a v e f u n c t i o n - t h e s e a r e t h e H F a n a l o g u e s o f figures (3.2) t h r o u g h (3.13). W e m a y r e c a l c u l a t e K f o r t h e n e w fields g e n e r a t e d i n t h i s a p p r o a c h . S h o w n i n figures (3.33) a n d (3.34) a r e t h e a n a l o g u e s o f figures (3.14) a n d (3 . 1 5 ) , u s i n g t h e H a r t r e e - F o c k w a v e f u n c t i o n s i n s t e a d o f t h e p o i n t d i p o l e a p p r o x i m a t i o n . W e m a y r e p e a t o u r c a l c u l a - t i o n s f o r t h e i n t r i n s i c l i n e w i d t h a s w e l l . W i t h o u r n e w field v a l u e s t h e n u m b e r s f o r Fe8* Chapter 3. Nuclear Spin Couplings in F e 8 and the Isotope Effect 102 120.0 i i . • i 110.0 100.0 - 90.0 80.0 Nucleus Label F i g u r e 3.20: C o m p a r i s o n o f p o i n t d i p o l e a n d H a r t r e e - F o c k m e t h o d s ; z e r o f i e l d wjl v a l u e s i n F e ™ " 1 . T h e H a r t r e e - F o c k r e s u l t s a r e s h o w n a s d o t s . a r e , i n z e r o e x t e r n a l field u s i n g t h e H a r t r e e - F o c k w a v e f u n c t i o n a p p r o x i m a t i o n , 120 2 £ UJIH = 7 1 5 1 2 . 2 2 [ M # z ] 2 kH=i I £ 4B? = n.2l[MHzf y fcBr=i o 18 2 | £ 4N = imm[MHzf (3.46) 6 kN=l w h i c h g i v e s EQ = 258.9 MHz , W = 366.2 MHz = 17.57 mK (3.47) N o t e t h a t t h e v a l u e s o b t a i n e d a r e q u i t e c l o s e t o t h o s e o b t a i n e d u s i n g t h e p o i n t d i p o l e a p p r o x i m a t i o n . N o t e h o w e v e r t h a t t h i s d i d n o t h a v e t o b e t h e c a s e , a s s o m e o f t h e l a r g e r w| i n c r e a s e d a n d s o m e d e c r e a s e d i n g o i n g f r o m t h e p o i n t d i p o l e t o H a r t r e e - F o c k a p p r o x i m a t i o n s . Chapter 3. Nuclear Spin Couplings in Fes, a n d t i e Isotope Effect 103 0.0 4.0 8.0 12.0 16.0 20.0 w_kll, MHz, Binned in 0.1 MHz steps Figure 3.21: H, Hartree-Fock, emphasizing low end of the spectrum. 40.0 80.0 w_kll MHz, Binned in 0.1 MHz steps 120.0 Figure 3.22: 1H, Hartree-Fock, high end of the spectrum. Chapter 3. Nuclear Spin Couplings in Fes a n d the Isotope Effect 104 E 0 0.0 4 . 0 8 .0 1 2 . 0 1 6 . 0 2 0 . 0 w _ k l l , M H z , B i n n e d in 0.1 M H z s t e p s F i g u r e 3.23: 2 i f , Ha r t r ee -Fock , low end of spec t rum. .Q E 0.0 10 .0 2 0 . 0 3 0 . 0 4 0 . 0 w _ k l l , M H z , B i n n e d in 0.1 M H z s t e p s 5 0 . 0 F i g u r e 3.24: 2H, Ha r t r ee -Fock , entire spec t rum. Chapter 3. Nuclear Spin Couplings in Fe& and the Isotope Effect 1 0 5 0.0 2.0 4.0 6.0 8.0 10.0 w_kll, MHz, Binned in 0.1 MHz steps F i g u r e 3.25 : 79Br, H a r t r e e - F o c k , e n t i r e s p e c t r u m . 0.0 2.0 4.0 6.0 8.0 10.0 w_kll, MHz, Binned in 0.1 MHz steps F i g u r e 3.26: 81Br, H a r t r e e - F o c k , e n t i r e s p e c t r u m . Chapter 3. Nuclear Spin Couplings in Fe8 and the Isotope Effect 106 2.0 4.0 6.0 8.0 w_kll, MHz, Binned in 0.1 MHz steps 10.0 F i g u r e 3.27: N, H a r t r e e - F o c k , e n t i r e s p e c t r u m . 0.0 2.0 4.0 6.0 8.0 10.0 w_kll, MHz, Binned in 0.1 MHz steps F i g u r e 3.28: 1 5 J V , H a r t r e e - F o c k , e n t i r e s p e c t r u m . Chapter 3. Nuclear Spin Couplings in Fe$ and the Isotope Effect 107 0.0 0.1 0.2 0.3 0.4 0.5 w_kll, MHz, Binned in 0.001 MHz steps Figure 3.29 : 57Fe, Hartree-Fock, entire spectrum. 2.0 4.0 6.0 8.0 w_kll, MHz, Binned in 0.1 MHz steps 10.0 Figure 3.30: 1 3 C , Hartree-Fock, entire spectrum. Chapter 3. Nuclear Spin Couplings in Fe$ and the Isotope Effect 108 0.0 2.0 4.0 6.0 8.0 10.0 w_kll, MHz, Binned in 0.1 MHz steps Figure 3.31: 1 7 O, Hartree-Fock, low end of spectrum. 10.0 20.0 30.0 40.0 w_kll, MHz, Binned in 0.1 MHz steps 50.0 Figure 3.32: 170, Hartree-Fock, entire spectrum. Chapter 3. Nuclear Spin Couplings in Fe% and the Isotope Effect 1 0 9 500.0 400.0 300.0 200.0 100.0 Total Hydrogen Bromine Nitrogen Carbon Oxygen lron-57 F i g u r e 3.33: T h e o r t h o g o n a l i t y b l o c k i n g p a r a m e t e r f u n c t i o n o f Hx i n t h e H a r t r e e - F o c k w a v e f u n c t i o n p i c t u r e . 0.00 0.01 0.02 0.03 H_x [Tesla] F i g u r e 3.34: T h e o r t h o g o n a l i t y b l o c k i n g p a r a m e t e r f u n c t i o n o f Hx i n t h e H a r t r e e - F o c k w a v e f u n c t i o n p i c t u r e , f o c u s i n g o n s m a l l fields. Chapter 3. Nuclear Spin Couplings in Fe% and the Isotope Effect 110 As before, we can redo this calculation with any isotopic concentration. Here we treat the case where all protons are replaced by deuterium. This gives 1 2 0 2 2 ^ II 2 /0.857354\ 2 kD=i which gives || ™ ||  / 0 . 8 5 7 3 2 X>1'„ ^ 5 E 4 ' „ ( - j ^ J (3.48) £ 0 = 68.32 MHz , W = 96.62 MHz = 4.676 (3.49) which is very close to the point dipole result. Similarly to what we did in the point dipole case we compute the intrinsic linewidth W as a function of Hx for the three materials Fe 8*, Fe8D a n d 57Fe8 and show the results in figure (3.35). We conclude by recalculating the topological decoherence parameters. Using our new field values we find that A = 4.23 • 10~5 , 8.73 • 10~5 , and 1.35 • 10~5 for the Fe 8 „ , 57Fe8 and Fe8D materials respectively. Values for l A ^ ^ j are shown in figure 3.36. Chapter 3. Nuclear Spin Couplings in Fe$ and the Isotope Effect 111 =• 300 0 S 7 - F « e Fa_8D 1 Fe 8- S 2 0 0 . 0 0 . 5 H_x [Testal F i g u r e 3.35: I n t r i n s i c l i n e w i d t h W as a f u n c t i o n o f Hx f o r F e s * , Fe$D a n d 5 7 F e 8 i n t h e H a r t r e e - F o c k p i c t u r e . 0 000 0.001 0.002 0.003 0 0.001 0 002 0 003 F i g u r e 3.36: B i n n e d t o p o l o g i c a l d e c o h e r e n c e p a r a m e t e r s | A ^ D | f o r a l l n u c l e i , a s s u m i n g 1 0 0 % c o n c e n t r a t i o n s o f ( c l o c k w i s e f r o m b o t t o m l e f t ) 1H, 79Br, 1 4 A T , 1 7 ( 9 , 1 3 C a n d 57Fe, u s i n g t h e H a r t r e e F o c k a p p r o x i m a t i o n . T h e b i n w i d t h h e r e i s 0.0001; p l o t t e d o n t h e x a x i s i s | A ^ £ , | a n d o n t h e y a x i s " n u m b e r o f n u c l e i " . N o t e t h a t t h e c o n t r i b u t i o n t o | A ^ £ , | f r o m 57Fe i s a l m o s t e n t i r e l y f r o m t h e c o n t a c t i n t e r a c t i o n . Chapter 3. Nuclear Spin Couplings in Fe$ and the Isotope Effect 112 3.4 Tables of Nuclear Positions, Fields at Nuclei and Hyperfine Coupling Energies In what follows we shall be considering only the case where the external magnetic field is zero for clarity of presentation. In these tables we indicate the locations of each ion in the molecule, presented in Cartesian coordinates (x,y,z) in Angstroms. As well we present the magnitude of the field at each nucleus due to the eight iron spins (note that in zero external field this magnitude is the same for both configurations of the central spin complex) both for the point dipole approximation and the Hartree-Fock approximation. We also present the hyperfine coupling energies u)k for both point dipole and Hartree-Fock cases (u^ is zero when the external field is zero). Here we have chosen the following isotopes in order to convert from field to energy units: XH, 79Br, 14N, 57Fe, nC, and 1 7 0 . Chapter 3. Nuclear Spin Couplings in Fe8 and the Isotope Effect 113 Nuclear X y z 17(1)IP1 a; II [MHz] IT(1)I m J l [MHz] Label (P.D.) (P.D.) (H.F.) (H.F.) HI 10.8 3.58 1.92 .0509 2.167 .0521 2.22 H2 9.38 2.12 1.38 .161 6.854 .152 6.476 H3 9.6 2.61 -.66 .112 4.757 .112 4.76 H4 10.3 4.3 -.173 .0422 1.797 .0415 1.772 H5 9.07 5.17 -1.88 .0978 4.168 .0943 4.016 H6 7.29 4.78 -2.37 .284 12.055 .298 12.676 H7 8.03 6.8 -1.13 .0556 2.365 .0577 2.46 H8 9.54 6.61 -.0732 .0242 1.026 .0246 1.052 H9 9.52 7.18 2.03 .0277 1.175 .0274 1.172 H10 8.24 6.26 2.61 .0988 4.209 .101 4.292 H l l 9.94 5.34 3.62 .054 2.301 .054 2.304 H12 10.6 5.61 2.36 .0347 1.477 .0345 1.472 H13 3.4 5.03 3.38 .15 6.383 .149 6.344 H14 3.68 4.34 4.85 .256 10.939 .251 10.708 H15 5.91 5.96 5.19 .106 4.532 .104 4.444 H16 5.79 5.65 3.57 .2 8.519 .2 8.532 H17 7.62 4.27 7.01 .0927 3.945 .0924 3.944 H18 5.88 4.07 6.59 .179 7.632 .183 7.824 H19 10.8 -.193 5.82 .0807 3.436 .0821 3.504 H20 10.2 -1.76 6.24 .109 4.626 .109 4.628 H21 11 -2.19 4.4 .0924 3.938 .0924 3.944 H22 10.3 -1.09 3.45 .137 5.832 .136 5.776 H23 6.01 1.8 7.08 .186 7.915 .189 8.056 H24 7.02 1.88 6.14 .249 10.634 .249 10.628 H25 2.87 .67 5.39 .195 8.311 .201 8.58 H26 3.72 2.35 5.84 .321 13.67 .326 13.88 H27 1.63 2.12 3.77 .11 4.68 .108 4.592 H28 2.01 .981 2.96 .109 4.656 .107 4.572 H29 6.79 -2.16 -5.9 .307 13.088 .326 13.94 H30 7.63 -.505 -5.25 .195 8.328 .192 8.196 Table 3.3: Data for Hydrogen. Chapter 3. Nuclear Spin Couplings in Fe8 and the Isotope Effect 114 Nuclear X y z |7(1)I [T] J l [MHz] l 7 ( 1 ) l [T] J l [MHz] Label (P.D.) (P.D.) (H.F.) (H.F.) H31 8.61 -1.06 -2.96 .104 4.423 .101 4.316 H32 8.83 -2.19 -3.93 .118 5.023 .118 5.032 H33 6.52 -4.3 -5.18 .277 11.756 .277 11.784 H34 6.88 -5.19 -3.84 .153 6.517 .148 6.312 H35 4.57 -5.64 -3.71 .194 8.255 .19 8.1 H36 4.24 -5.96 -5.37 .0985 4.201 .0971 4.136 H37 4.52 -3.93 -6.69 .174 7.418 .178 7.568 H38 2.79 -4.09 -7.02 .0938 4.004 .0962 4.1 H39 3.43 -1.79 -6.2 .234 9.961 .232 9.884 H40 4.47 -1.73 -7.13 .179 7.632 .186 7.924 H41 4.21 5.46 -3 .163 6.943 .156 6.664 H42 3.44 5.64 -4.65 .131 5.581 .127 5.404 H43 .303 .862 -3.45 .145 6.198 .144 6.136 H44 -.515 1.95 -4.2 .0924 3.938 .0919 3.916 H45 .385 1.99 -6 .116 4.964 .115 4.916 H46 -.312 .338 -5.82 .0798 3.4 .0807 3.44 H47 1.19 -7.11 -2.18 .0331 1.409 .0326 1.392 H48 .406 -6.44 -1.32 .0274 1.169 .0272 1.16 H49 .0679 -5.37 -3.3 .042 1.785 .0425 1.812 H50 1.72 -5.16 -3.29 .124 5.287 .118 5.048 H51 1.75 -6.92 .851 .0333 1.423 .0321 1.372 H52 3.39 -5.77 1.73 .193 8.218 .18 7.676 H53 6.48 -3.54 6.51 .21 8.967 .221 9.436 H54 8.07 -3.5 6.48 .186 7.915 .199 8.492 H55 8.47 -1.16 7.11 .14 5.949 .138 5.904 H56 6.89 -1.24 5.94 .364 15.498 .368 15.716 H57 3.61 1.38 -5.96 .361 15.383 .387 16.488 H58 2.01 1.26 -7.1 .137 5.862 .14 5.96 H59 2.3 3.54 -6.3 .188 8.014 .184 7.856 H60 3.84 3.65 -6.53 .195 8.302 .192 8.188 Table 3.4: Data for Hydrogen. Chapter 3. Nuclear Spin Couplings in F e 8 a n d the Isotope Effect Nuclear X y z i 7 ( l ) i m J l [MHz] i7(l)rm a;I' [MHz] Label (P.D.) (P.D.) (H.F.) (H.F.) H61 .623 -2.88 .611 .08 3.407 .0718 3.056 H62 1.95 -1.71 .255 .251 10.708 .232 9.884 H63 1.75 -4.88 2.25 .129 5.505 .132 5.612 H64 .617 -5.23 .629 .0481 2.05 .0439 1.868 H65 8.72 -5.09 3.6 .137 5.85 .131 5.604 H66 9.24 -4.12 5.09 .165 7.021 .171 7.284 H67 6.92 -5.64 4.83 .131 5.57 .129 5.52 H68 6.26 -5.47 3.17 .163 6.965 .157 6.708 H69 1.1 4.07 -4.8 .16 6.83 .154 6.572 H70 1.72 4.94 -3.24 .142 6.053 .139 5.916 H71 .272 -2.64 -1.92 .0859 3.662 .0828 3.532 H72 -.208 -4.22 -1.48 .0453 1.927 .0446 1.904 H73 8.05 -.671 4.23 .622 26.468 .629 26.844 H74 7.98 -2.98 2.43 1.82 77.829 1.87 79.864 H75 5.76 -3.43 3.78 1.08 46.004 1.04 44.464 H76 7.86 3.23 1.98 1.12 47.703 1.16 49.468 H77 7.11 2.81 -1.02 2.44 103.95 2.65 113.492 H78 6.67 5.16 .275 .758 32.286 .779 33.22 H79 3.8 2.86 2.4 1.12 47.649 1.02 43.636 H80 7.07 3.95 4.32 .432 18.397 .439 18.688 H81 4.83 .935 4.13 .877 37.415 .823 35.084 H82 5.68 -.979 -4.05 .943 40.203 .833 35.512 H83 3.59 -3.76 -4.2 .579 24.676 .547 23.344 H84 6.76 -3.06 -2.37 .934 39.825 .903 38.492 H85 2.49 .701 -4.22 .647 27.551 .64 27.288 H86 4.9 3.6 -3.77 .765 32.557 .772 32.876 H87 2.54 2.85 -2.31 2.09 89.109 1.55 66.084 H88 3.73 -5.2 -.498 .72 30.655 .631 26.856 H89 2.7 -2.96 -1.82 1.24 53.006 1.34 57.224 H90 3.58 -2.99 1.29 2.06 87.885 2.74 116.536 Table 3.5: Data for Hydrogen. Chapter 3. Nuclear Spin Couplings in Fe$ and the Isotope Effect Nuclear X y z |7 ( 1 )I [T] J l [MHz] |7 ( 1 ) I [T] J l [MHz] Label (P.D.) (P.D.) (H.F.) (H.F.) H91 8.78 .0423 5.01 .231 9.837 .242 10.316 H92 7.92 2.02 3.9 .403 17.188 .406 17.348 H93 5.34 -1.03 3.83 .542 23.124 .575 24.52 H94 6.2 -4.09 1.14 .371 15.776 .354 15.064 H95 8.76 3.34 2.78 .303 12.891 .293 12.54 H96 3.29 .719 1.14 .187 7.979 .182 7.756 H97 7.21 3.45 -1.76 .934 39.767 1.16 49.388 H98 7.9 -.733 1.32 .432 18.425 .415 17.656 H99 7.15 6.11 .465 .181 7.7 .172 7.32 H100 5.05 3.95 1.35 1.16 49.5 1.23 52.368 H101 2.81 3.16 2.39 .242 10.279 .246 10.492 H102 2.66 .832 -1.47 .556 23.686 .544 23.176 H103 5.85 1.45 -3.24 .638 27.153 .622 26.464 H104 2.57 -2.02 -3.9 .392 16.747 .38 16.208 H105 5.99 -.381 -4.76 .396 16.94 .42 17.892 H106 7.21 -.719 -1.14 .185 7.888 .193 8.208 H107 2.99 -4.75 -5.18 .15 6.398 .152 6.488 H108 5.45 -3.95 -1.35 1.17 49.671 1.19 50.76 H109 7.68 -3.16 -2.34 .249 10.567 .23 9.792 H110 4.29 4.05 -1.14 .378 16.101 .387 16.488 H i l l 1.72 -.0423 -5.01 .227 9.663 .226 9.652 H112 7.97 7.92 7.83 .0219 .934 .0222 .948 H113 2.53 6.18 7.14 .0427 1.817 .0427 1.824 H114 6.65 6.66 9.57 .0265 1.133 .027 1.152 H115 3.35 -6.11 -.465 .182 7.755 .187 7.992 H116 7.55 5.74 10.6 .0242 1.033 .0246 1.048 H117 1.73 -3.34 -2.78 .291 12.401 .3 12.828 H118 2.95 8.36 4.44 .0227 .969 .0229 .976 H119 3.85 7.44 5.43 .0389 1.662 .0396 1.688 H120 9.69 -.874 .24 .0877 3.742 .0891 3.804 Table 3.6: Data for Hydrogen. Chapter 3. Nuclear Spin Couplings in Fes and the Isotope Effect 117 N u c l e a r X y z |7(1)| [T] J l [MHz] |7 ( 1 )I [T] J l [MHz] L a b e l ( P . D . ) ( P . D . ) ( H . F . ) ( H . F . ) B r l 10.2 2.65 5.67 .0521 2.215 .0537 2.293 B r 2 3.37 -2.3 4.91 .137 5.856 .142 6.063 B r 3 5.2 7.11 7.43 .0289 1.23 .0298 1.273 B r 4 .265 -2.64 -5.66 .0516 2.201 .0535 2.279 B r 5 7.09 2.28 -4.91 .14 5.957 .145 6.167 B r 6 7.64 -4.75 .0737 .091 3.878 .0943 4.015 B r 7 1.06 1.08 -.26 .0816 3.481 .0844 3.604 B r 8 3.7 5.53 .2 .0734 3.132 .076 3.243 Tab le 3.7: D a t a for B r o m i n e . Chapter 3. Nuclear Spin Couplings in Fe8 and the Isotope Effect 118 Nuclear X y z |7 ( 1 )I [T] J l [MHz] i7(i)im J l [MHz] Label (P.D.) (P.D.) (H.F.) (H.F.) NI 8.52 -1.05 5.1 .0547 2.332 .053 2.26 N2 8.73 -3.03 3.35 .0891 3.796 .0849 3.62 N3 6.48 -3.78 4.49 .112 4.765 .108 4.61 N4 8.87 3.76 1.97 .0486 2.073 .0479 2.04 N5 8.09 3.55 -.923 .138 5.892 .14 5.95 N6 7.75 5.71 .515 .0333 1.419 .0392 1.67 N7 3.45 3.02 3.18 .0917 3.908 .0807 3.44 N8 6.55 4.24 4.89 .0525 2.242 .0577 2.46 N9 4.83 1.44 4.98 .13 5.526 .132 5.63 . N10 5.7 -1.4 -4.92 .13 5.533 .136 5.78 N i l 3.96 -4.13 -4.86 .0582 2.476 .0525 2.24 N12 7 -3.14 -3.23 .0929 3.964 .0884 3.77 N13 2.01 1.12 -5.06 .0568 2.423 .0638 2.72 N14 4.07 3.86 -4.4 .1 4.283 .0896 3.82 N15 1.77 2.92 -3.2 .0927 3.947 .083 3.54 N16 2.73 -5.67 -.426 .0342 1.459 .045 1.92 N17 1.7 -3.67 -1.94 .0532 2.268 .0605 2.58 N18 2.48 -3.44 .983 .146 6.228 .145 6.18 Table 3.8: Data for Nitrogen. Chapter 3. Nuclear Spin Couplings in Fe8 and the Isotope Effect Nuclear X y z |7(1)I [T] a; II [MHz] i7 ( i ) i m [MHz] Label (P.D.) (P.D.) (H.F.) (H.F.) CI 10 -1.15 5.49 .0352 1.499 .0357 1.52 C2 10.2 -1.83 4.1 .0422 1.797 .0441 1.88 C3 8.52 -4.37 4.1 .0659 2.807 .069 2.94 C4 6.95 -4.98 4.11 .0638 2.72 .0615 2.62 C5 7.21 -3.19 6.03 .0833 3.553 .0805 3.43 C6 7.76 -1.57 6.15 .0762 3.254 .0758 3.23 C7 9.72 3.16 1.34 .035 1.487 .035 1.49 C8 9.51 3.41 -.168 .035 1.491 .0342 1.46 C9 8.22 4.94 -1.55 .0633 2.703 .0615 2.62 CIO 8.46 6.13 -.569 .0217 .925 .0221 • .941 C l l 8.82 6.22 2 .0206 .878 .0213 .908 C12 9.69 5.27 2.58 .0194 .829 .0198 .844 C13 3.94 4.46 3.98 .0837 3.572 .0819 3.49 C14 5.59 5.22 4.4 .0598 2.553 .0615 2.62 C15 6.6 3.79 6.3 .0507 2.162 .0518 2.21 C16 6.18 2.13 6.23 .0812 3.456 .0821 3.5 C17 3.43 1.57 5.12 .0936 3.993 .0919 3.92 C18 2.45 1.85 3.65 .0521 2.224 .0495 2.11 C19 7.07 -1.45 -5.07 .0919 3.915 .088 3.75 C20 8.09 -1.88 -3.72 .0521 2.216 .0528 2.25 C21 6.36 -4.53 -4.28 .0851 3.631 .0821 3.5 C22 4.7 -5.22 -4.55 .0568 2.416 .0577 2.46 C23 3.83 -3.64 -6.35 .0507 2.162 .0488 2.08 C24 4.26 -2.07 -6.27 .0758 3.233 .0795 3.39 C25 2.73 1.66 -6.12 .0753 3.205 .0748 3.19 C26 3.2 3.26 -5.96 .0805 3.432 .0823 3.51 C27 3.5 4.96 -3.95 .0643 2.739 .0643 2.74 C28 .382 1.66 -4.01 .0432 1.835 .0418 1.78 C29 1.88 4.27 -3.84 .0666 2.837 .0687 2.93 C30 .49 1.26 -5.37 .0361 1.536 .038 1.62 C31 1.25 -6.23 -1.68 .0153 .652 .0147 .627 C32 1.13 -5.13 -2.69 .0253 1.081 .0249 1.06 C33 .645 -3.34 -1.37 .0298 1.273 .0307 1.31 C34 1.4 -2.74 .183 .0551 2.348 .0558 2.38 C35 1.68 -4.91 1.25 .0425 1.814 .0434 1.85 C36 2.44 -5.92 .938 .0333 1.424 .0331 1.41 Table 3.9: Data for Carbon. Chapter 3. Nuclear Spin Couplings in Fe8 and the Isotope Effect 120 Nuclear X y z |7 ( 1 ) I [T] J l [MHz] |7 ( 1 ) I [T] J ' [MHz] Label (P.D.) (P.D.) (H.F.) (H.F.) FE1 5.72 .0651 1.58 .00729 .311 .00729 .311 FE2 6.9 -2.21 3.05 .00467 .199 .00467 .199 FE3 6.65 3.34 .339 .003 .128 .003 .128 FE4 5.31 2.41 3.18 .00793 .338 .00793 .338 FE5 5.19 -2.43 -3.17 .00779 .332 .00779 .332 FE6 3.61 2.2 -2.99 .00467 .199 .00467 .199 FE7 3.89 -3.33 -.293 .00293 .125 .00293 .125 FE8 4.84 -.0439 -1.55 .00737 .314 .00737 .314 Table 3.10: Data for Iron. Chapter 3. Nuclear Spin Couplings in Fe% and the Isotope Effect N u c l e a r X y z |7(1)I [T] J l [MHz] Jl [MHz] L a b e l ( P . D . ) ( P . D . ) ( H . F . ) ( H . F . ) 0 1 6.9 1.68 3.29 .746 31.831 .699 29.8 0 2 5.4 -1.44 2.99 .798 34.036 .877 37.4 0 3 5.57 -3.6 1.31 .253 10.783 .249 10.6 0 4 4.14 .682 1.88 .544 23.202 .518 22.1 0 5 7.59 -.53 2.12 .608 25.926 .671 28.6 0 6 5.8 3.68 1.76 1.2 51.071 1.06 45 0 7 6.14 1.37 .185 .969 41.303 .931 39.7 0 8 4.5 -1.34 -.115 .973 41.493 .978 41.7 0 9 3 .503 -2.03 .633 26.96 .612 26.1 0 1 0 5.13 1.44 -2.94 .783 33.367 .887 37.8 O i l 3.65 -1.68 -3.24 .798 33.975 • .877 37.4 0 1 2 6.5 -.649 -1.85 .471 20.113 .483 20.6 0 1 3 4.74 -3.71 -1.76 1.14 48.459 1.15 49.2 0 1 4 4.94 3.55 -1.29 .239 10.154 .251 10.7 O 1 0 0 8.98 8.16 7.79 .0121 .516 .012 .512 0 2 0 1.58 6.01 7.14 .0246 1.046 .0258 1.1 0 3 0 -1.02 6.85 5.61 .012 .51 .0116 .493 0 4 0 11.4 7.23 9.35 .00868 .37 .00901 .384 0 5 0 6.85 6.23 10.6 .0155 .662 .0153 .651 0 6 0 3.56 8.15 4.43 .0182 .775 .0174 .741 0 7 0 9.32 5.56 10.5 .0139 .593 .0134 .571 0 8 0 1.11 8.53 4.67 .0115 .492 .0116 .496 0 9 0 8.96 -1.17 .278 .101 4.302 .0973 4.15 T a b l e 3.11: D a t a for O x y g e n . C h a p t e r 4 A n I n t r o d u c t i o n t o t h e G e n e r a l i z e d L a n d a u - Z e n e r P r o b l e m An introduction to the standard Landau-Zener problem is presented. Three methods of solution are considered; one involves solving directly for the wavefunctions of the two-level system, the second is a perturbation expansion in the tunneling term and the third uses a complex analysis technique usually referred to as Dychne's formula. We demonstrate that it is possible to solve a more general version of the simple Landau-Zener Hamiltonian and present the solutions (see [102] for a complementary analysis). 4.1 I n t r o d u c t i o n t o a n d E x a c t S o l u t i o n o f t h e L a n d a u - Z e n e r P r o b l e m Consider a two-level system (TLS) with time-dependent Hamiltonian H(t) = vtoz + AJx (4.1) where {a} are the Pauli matrices. This rather generic effective description was first considered by Landau [103], Zener [104] and Stuckelberg [105]. Landau and Stuckelberg used this Hamiltonian to model the evolution of two atoms scattering off each other, while Zener used it to model the evolution of the electronic states of a bi-atomic molecule. It has since been used in a large number of different contexts; chemical reaction kinetics [106], biophysics [107], examination of the solar neutrino puzzle [108], aspects of nuclear magnetic resonance [109], behaviour of atoms in photon fields [110], surface scattering [111], electric breakdown in solids [112] and many more. The reason for its wide usage is evident. There are many real physical systems that for one reason or another can be 122 Chapter 4. An Introduction to the Generalized Landau-Zener Problem 123 modeled as two level systems-either they really are two level systems (for example, the spin Hilbert space of a spin 1/2 particle) or they can be mapped to one (for example, the O(DS) low energy effective Hamiltonian derived in chapter 2). It is often useful to know how these systems respond to an externally applied time-dependent perturbation. It is clear that the simplest effective description of a two level system coupled to a time dependent perturbation is that given in (4.1). We have sketched the energy levels of this Hamiltonian as functions of time in figure 4.1. Figure 4.1: Energy levels of the Landau-Zener Hamiltonian. Shown are both the eigen- states of oz, which are linear in time, and the eigenstates of H(t), E±(t) = ± ( A 2 + v 2 t 2 ) 1 / 2 . One of the most useful features of the description (4.1) is that one can solve it exactly, in the sense that one can solve for the wavefunctions explicitly as functions of time. In order get a feel for the model, we shall outline in the following how these wavefunctions are extracted. Insertion of (4.1) into the Schrodinger equation (we choose a system of units such Chapter 4. An Introduction to the Generalized Landau-Zener Problem 124 tha t h = l) .d i— dt [ A ) H(t) (4.2) y ie lds I—— = Vtlpa + Alpb dt .dipb dt Atpa - Vtlpb e l i m i n a t i n g ipa a l lows us to wr i t e ib\ + (y2t2 + A2 - iv) ipb = 0 (4.3) (4.4) where overdots denote der ivat ives w i t h respect to t. T h i s is the equa t ion for a p a r a b o l i c c y l i n d e r func t ion . It has two solu t ions w h i c h m a y be w r i t t e n ( i ) _ - F H / I _ , A £ I (ivt2) (2) / T 4 ' 4 u 5 4 * ' (4.5) where W is the W h i t t a k e r func t ion [113]. I f we consider the a s y m p t o t i c forms of these so lu t ions as t —> — oo, we find tha t (1) 2 6 (2) |2 t—y — cx> 0 (4.6) T h i s indica tes tha t the choice of one of the two solu t ions is equivalent to the choice of an i n i t i a l c o n d i t i o n on the wavefunct ion ipb. T h e analys is of the other wavefunct ion ipa proceeds i n an iden t i ca l manner ; s o l v i n g 4>a + (y2t2 + A2 + iv) ipa = 0 (4.7) Chapter 4. An Introduction to the Generalized Landau-Zener Problem 125 gives two solutions y/t 4 4u ' 4 7 (4.8) which have deep past asymptotics (1)|2 (4.9) This is consistent with our interpretation of the choice of one of these being equivalent to the choice of initial conditions. That is, preparation of the system in state tpa at time t = — oo requires that we use solutions ip^ and Similarly, preparation in the state ipb requires the use of ip^ and ip^. Finding the transition probabilities is now a straightforward exercise. The probability of finding the system in state ipa at time t given that it started in state ipa = | t> at time t = —oo is simply |2 Pt(t) = \^(t)\2 = \ft 4 + 4v '4 (4.10) This solution is plotted in figure 4.2. As t —>• +oo (4.10) asymptotically approaches P t ( t ->• oo) = 1 - e~^ (4.11) and therefore the probability to make a transition is Pn(t -> oo) = e'^ (4.12) Chapter 4. An Introduction to the Generalized Landau-Zener Problem 126 F i g u r e 4.2: T r a n s i t i o n p r o b a b i l i t y (4.10) as a func t ion of t ( in un i t s of A ) . Here we have taken A2/v = 0.63, 1 and 5 for the s o l i d , do t t ed a n d dashed l ines respect ively. 4.1.1 Alternate Method of Solution for the Transition Probability I. Al l Orders Perturbation Expansion It is also poss ible to solve for the t r a n s i t i o n p robab i l i t i e s w i t h o u t first f i nd ing the wave- funct ions [104]. C o n s i d e r the a m p l i t u d e Aa->b=<b\U(tf,ti)\a> (4.13) where U{tf,ti) = TeittH{T)*r (4.14) is the t i m e e v o l u t i o n opera tor ( T m e a n i n g " t ime ordered") and a, b c an be ei ther "up" or "down" (<Tz = ± 1 , respec t ive ly) . S p l i t t i n g the H a m i l t o n i a n in to d i agona l and off-diagonal ( in o) par t s H(t) = Hd(t) + a l lows us to rewr i te the evo lu t i on opera to r i n the more convenient f o r m [71] Ufo^) = e-l%Hd{T)dTTe-l%"*dT (4.15) where 7-7 (+\ J It Hd(T)dT JJ -i f' Hd(r)dT . . H^(t) = e J f * HAe Jti (4-16) Chapter 4. An Introduction to the Generalized Landau-Zener Problem 127 Expansion of the time-ordered exponential yields A^b = f ) ( - t ) B f dtn fn dtn_x... f2 dt, < b\ei^drvra'A&x 71=0 * * * , d,TVTdz „ i f ' 2 d,TVT&z . . i f * 1 drvruz i ^ / , , M e Jtn-i Aox...eJti Aoxe J-°° \a > (4-17) f) A 2 " ^ dt 2 n c f t^ - ! . . . dtie- iv?T-il-W (4.18) Let us assume that a = b, and furthermore that \a >= | t> (the solution for \a >= | 1> is similar, differing only in an overall phase factor). Inspection of (4.17) shows that in this case only paths with n even contribute, and furthermore that the inner product in (T-space is easy to perform. Explicitly we find ftf rt2n T*2 tz  I dtm-\ • • • N = Q J—ti Jti Jti where <p is an uninteresting phase. One sees that all of the time integrals may now be performed if we take U = — oo and tj = +oo; otherwise we are stuck. That is, it seems as though this approach gives us less information than the solution for the wavefunctions performed in the preceding section; this is somewhat strange, as usually if a problem is solvable in terms of known special functions in one representation it is usually solvable in all of them. In any case, we shall now take ti = —oo and tj = +oo. In this case the integrations over the time set {tj} can be performed explicitly, giving with subsequent probability Pn = 1 - \An\2 = (4.20) as before. Chapter 4. An Introduction to the Generalized Landau-Zener Problem i 128 4.1.2 Alternate Method of Solution for the Transition Probability II. Dychne's Formula The final method that we shall review for extracting results from (4.1) was first suggested by Landau [114] but later became known as Dychne's formula [?]. This method uses arguments from the theory of complex analysis in order to extract transition amplitudes from two level time dependent Hamiltonians such as the Landau Zener model (4.1). We shall not give a detailed analysis of this method, but just present its basic result. We begin by defining the standard rotation matrix cos6>/2 - s i n 9/2 sin 9/2 cos (9/2 which, if 9 is a function of time, can diagonalize the general Hamiltonian Vji(t) A(t) R (4.21) H = V\\{t)crz + A(t)ox = A(t) -Vj|(t) (4.22) giving RHRj E-{t) 0 0 E+(t) (4.23) where E±(t) = ±JA*(t) + W(t) (4.24) are the adiabatic energy levels. In the case of the simple Landau Zener Hamiltonian, we find that E±{t) = ± V A 2 + v2t2 (4.25) The result of Landau and Dychne states that the probability for making a transition from one eigenstate of oz to the other if the Hamiltonian is evolved over the range Chapter 4. An Introduction to the Generalized Landau-Zener Problem 129 —oo < t < oo is a p p r o x i m a t e l y Pa^b = e ~ 2 I m ^ (4.26) where f ( t ) = 2 j1E{r)dT (4.27) and tc is the zero of E(t) i n the upper h a l f p lane tha t is closest to the rea l ax is . N o t e tha t th i s requires tha t we a n a l y t i c a l l y cont inue the t i m e var iab le t h r o u g h o u t the c o m p l e x plane . T h e error i n th is expression comes f rom the neglect of the c o n t r i b u t i o n of a l l the other zeroes o f E(t) i n the upper h a l f c o m p l e x p lane w h i c h are o m i t t e d here. G e n e r a l l y speaking , one can t e l l whe ther or not th i s m e t h o d w i l l give useful results by l o o k i n g at the s t ruc ture of the zeroes of E(t). I f there are m a n y closely spaced zeroes i n E(t) off the rea l axis , then the con t r i bu t i ons f rom s u b d o m i n a n t t e rms w i l l grow a n d th is m e t h o d w i l l f a i l . Fo r example , i n the case of the s imple L a n d a u Zener m o d e l , there are o n l y two po in t s i n the c o m p l e x t i m e p lane where E(t) vanishes, n a m e l y tc = i z A / y ^ , a n d o n l y on of these is i n the upper h a l f p lane. Because of th is we expect tha t i n th is case the results ob t a ined us ing D y c h n e ' s f o r m u l a shou ld be exact . T h i s resul t is in te res t ing as i t provides ins ight in to the reasons w h y th is m o d e l is exac t ly solvable and m a n y other s i m i l a r mode ls are no t - fo r example , chang ing the t i m e dependence f rom l inear to say cub ic i n Vj|(t) causes the me thods used i n the preceding chapters to fa i l to p roduce exact results . T h i s is most p r o b a b l y re la ted to the existence of now two zeroes of E(t) i n the upper h a l f p lane w h i c h in t roduce errors in to the f o r m u l a (4.26). It is qui te l i ke ly tha t there is a deep connec t ion here between the theory of spec ia l funct ions and issues i n c o m p l e x analys is . However , we do not choose to pursue th is avenue at the present t i m e - w e sha l l o n l y m e n t i o n i t i n pass ing. Chapter 4. An Introduction to the Generalized Landau-Zener Problem 130 Let us now apply Dychne's formula to the Landau Zener Hamiltonian. We find that f (*) = 21* VA2 + v2r2dr = WA2 + v2t2 + ^ In (yt + VA2 + v2t2) (4.28) and therefore A 2 7 T Im ate) = ^rj (4.29) and Pa^b = e'^T (4.30) which agrees with our previous exact results. 4.1.3 A n a l y s i s o f t h e T r a n s i t i o n F o r m u l a We now wish to step back from the preceding technical exercise and draw some conclu- sions from this analysis. From the outset it should have been clear that the presence of the tunneling term Aox in the Hamiltonian would mix the two states that we are calling ipajt,- Furthermore one expects that the dimensionless parameter A2/v should be impor- tant in the final transition expression. Both these suspicions, as we have seen (4.12), turn out to be justified. What else can we say about this solution? Perhaps the key point here is that as we are dealing with a two-level system the equa- tions for the wavefunctions have to be second order homogeneous differential equations. For the specific Hamiltonian that we were working with (4.1) this differential equation turned out to be one that is a well-studied specific case of the hypergeometric equa- tion. This allowed us to write down general solutions. Supplanting these solutions with information about the asymptotic behaviour of the system then gave us the specific tran- sition probabilities that we were after. This realization suggests that there exists a class of two level time dependent Hamiltonians whose wavefunctions are obtainable in terms Chapter 4. An Introduction to the Generalized Landau-Zener Problem 131 of known special functions, and that this class is that whose Schrodinger equation can be mapped to the hypergeometric equation. The full determination of what is required of the Hamiltonian in order for it to belong to this class is a difficult task. We shall in the following demonstrate the method for some specific cases. 4.1.4 G e n e r a l i z a t i o n o f t h e T w o - L e v e l L a n d a u - Z e n e r P r o b l e m I. E x a c t S o l u t i o n f o r A(t) ~ Vj|(t) Let us now consider a more general case of (4.1), namely [102] H(t) = V\\(t)az + A{t)ax (4.31) We may perform the same type of analysis as we did in section 5.1. The Schrodinger equation . d i—-dt ( = H(t) yields t^ = A(t)^a - v{lmb eliminating x/ja allows us to write A - + (v~(\ + A2 -i Mi - Mi A ^6 = 0 (4.32) (4.33) (4.34) where again the overdot represents a time derivative. As before, we have to supplant this equation with a specification of the initial conditions; here we shall assume that \i)b{t= - 0 0 ) | 2 = 1 (4.35) Chapter 4. An Introduction to the Generalized Landau-Zener Problem 132 ie. that the system has been prepared in state tpb in the deep past. The equation (4.34) cannot be solved with known special functions for general V\\(i) and A(t). However, it is possible to recast it in a form that allows solution for quite a few interesting specific cases. We map t - r z(t) (4.36) where the only requirement we have at this stage being that the map is onto, ie. 1 —> 1. With differentiation with respect to z denoted by primes, (4.34) becomes A + z A 7 ~ A" Vn + A 2 i 2 ^ i i - M i x ^6 = 0 We now choose the mapping such that (4.37) i = A(t) (4.38) We then find that 1 + v2 ILL A 2 i A /A A ^6 = 0 (4.39) Writing our equation in this form highlights the fact that if V\\ and A are constant multiples of each other, ie. Mi = KA (4.40) with K constant, then (4.39) reduces to ib'l + ( l + K2) ^ = 0 (4.41) Define 7 2 = 1 + K,2; then ipb(z) = c0e 1 7 2 + c i e i-yz (4.42) Chapter 4. An Introduction to the Generalized Landau-Zener Problem 133 is the general s o l u t i o n o f (4.41). In order to de te rmine the n o r m a l i z a t i o n constants we require the fo l lowing . F i r s t l y , we want the s o l u t i o n for the p r o b a b i l i t y to be b o u n d e d everywhere by 1; e x p l i c i t l y , | ^ 6 | 2 = | C o | 2 + | C l | 2 + c * C l e 2 i Z 7 + c l C o e - 2 j Z 7 < 1 (4.43) A s w e l l , we k n o w the s o l u t i o n for V\\ = 0 w i t h our i n i t i a l c o n d i t i o n is s i m p l y |V; f c | 2 = | c 0 | 2 + | C ! | 2 + c*oCle2iz + c^cne" 2 1 2 = c o s 2 ( z ) (4.44) These fix the constants to be CQ = c\ = 1/2, and \Mz)\2 = c o s 2 ( 7 z ) (4.45) In te rms of our o r i g i n a l parameters th is t r a n s i t i o n p r o b a b i l i t y is w r i t t e n \^b(t)\2 = cos 2 [(1 + K2)1'2 f A(r)dr\ (4.46) T h i s result has been der ived p rev ious ly by different me thods [115]. W e m a y pause now and ask how th is exact s o l u t i o n compares to tha t ob t a ined us ing Dychne ' s fo rmula . In this p a r t i c u l a r case, we f ind tha t E±(t) = ±]/A2(t) + V2(t) = ± v T + ^ 2 | A ( t ) | (4.47) and tha t the zeroes of E±(t) are s i m p l y the zeroes of A(t) (where, of course, we a n a l y t i - ca l l y cont inue the t i m e t t h roughou t the entire c o m p l e x p lane) . T h e t r a n s i t i o n p r o b a b i l i t y f rom D y c h n e ' s f o r m u l a is then Pn =<i \U(-oo, + o o ) | t > ~ e-4VWlmItc ^ d T (4.48) where tc is the zero of A(t) closest to the real ax is i n the upper ha l f p lane. Chapter 4. An Introduction to the Generalized Landau-Zener Problem 134 Let us solve for the transition probabilities for some specific potentials using our exact result. One that occurs quite regularly in this type of problem is a "pulse" potential that looks like A A(t) cosh cot The indefinite integral of A(t) is ft A / A(r)dr = —tan _ 1 (s inha;£) + c2 J UJ (4.49) (4.50) To be consistent with our initial condition (4.35) we take c 2 = AIT/2UJ. The transition probability from our exact result is then 1 2 - cos2 \Mt)\ The t —» oo asymptotic of (4.51) is Pn = cos2 , A (1 + « : 2 ) 1 / 2 — tan _ 1(sinhwt) + 7T UJ (4.51) An i i L u) (4.52) If we were to try to use Dychne's formula for this case, we would find that it fails. This is because since our energy levels are asymptotically approaching zero as t —>• oo, there is no zero of E(t) that is closest to the real axis and therefore one cannot find a unique tc. This breakdown of Dychne's formula in this regime has been noted previously [102]. Our exact solution is quite interesting. We see that the t —> oo transition probability may be varied between zero and one by altering an external parameter (A or u). A train of such pulses with these varied could be used as "quantum logic gates". The reason that this particular application is interesting is that there is absolutely no source of decoherence present in this system-phase coherence is sustained throughout the evolution of the system simply because there is nothing coupled to the two level system that can remove it. We shall not say anything further about this potential application; some embellishment of this basic idea may be found in recent reviews [102]. Chapter 4. An Introduction to the Generalized Landau-Zener Problem 1 3 5 " F i g u r e 4.3: T r a n s i t i o n p r o b a b i l i t i e s f o r t h e p u l s e p o t e n t i a l . A n o t h e r p o t e n t i a l o f i n t e r e s t i s t h e f o l l o w i n g o n e ; A{t) = 0 , |*| > t 0 A(t) = Asmu(t + t0) , \t\<t0 w h e r e w e fix t0 = mr/2u. H e r e w e find t h a t f o r \t\ < t0 (4.53) |<M*)|2 = cos2 (1 + ^ 1 / 2 - [ c o s u ; ( t + to) — 1] (4.54) w i t h d y n a m i c s f r o z e n f o r \t\ > to- T h i s s o l u t i o n i s s h o w n i n f i g u r e (4.4). N o t e t h a t h e r e Chapter 4. An Introduction to the Generalized Landau-Zener Problem 136 w e a l s o h a v e a t u n a b l e s w i t c h i n g e f f e c t , t h i s t i m e a l s o d e p e n d i n g o n t h e v a l u e o f n t h a t w e p i c k . F i g u r e 4.4: T r a n s i t i o n p r o b a b i l i t i e s f o r t h e s i n u s o i d a l p o t e n t i a l . Chapter 4. An Introduction to the Generalized Landau-Zener Problem 137 4.1.5 Generalization of the Two-Level Landau-Zener Problem II. Exact Solution by Mapping to Riemann's Differential Equation There is a similar approach to the solution of (4.31) that generates solutions for a different class of functions V\\(t) and A(t). Let us take the mapping Then (4.31) becomes A + l l - + A z z - 1 A A + z(t) = - [tanhwi + 1] Li Vn + A2 (4.55) z(z-l) W\\ ~ VWA A A = 0(4.56) 4to2z(z - 1) 2u which is suspiciously close to Riemann's differential equation (RDE) [113]. We have been able to map this equation into the R D E in two specific cases, which we shall review. Let us take V\\(t) = Atanhwi , A(t) = (A2 + A2)1'2 1 cosh cot (4.57) As functions of z, these are V\\(z) = A(2z - 1) , A(z) = (A2 + A2)1^z(l-z) (4.58) This models some scenario where the tunneling between the two states is externally enhanced near t = 0 and killed for \t\ >> 1 while the two levels are crossed. Insertion of these into (4.56) yields A + n I 1 ! - + 7 .Z Z — 1. A l z(z-l) k k z z — 1 A = o (4.59) where Chapter 4. An Introduction to the Generalized Landau-Zener Problem 138 This is a specific case of the R D E . Its solutions are ^b = za(z- iy [CF[a, b; c; z] + Dzl~cF[l + a - c, 1 + b - c; 2 - c; z]\ (4.61) where C and D are constants chosen such that \ipb(z = 0 ) | 2 = 0. In order to determine what the constants a, 7 , a, b and c are, we refer to Abramowitz and Stegun [113]. Using their nomenclature, 1 - a - a ' = 1/2 , 1 - 7 - 7 = 1 / 2 aa = k , /?/?' = -A\ 7 7 = , a + b = \ + a — a + 7 — 7 c = 1 + a — a In terms of our original parameters we find that a=^(l + 2A0i) , y = -A0i a= A i 2 and our general solution becomes ^ = z(l+2A0i)/2^_1yA0i b=\ + A1 1 + 1 + AAQI CF[^-A1,^ + A1;^ + 2A0i;z] + Dz-{l+AAol)l2F[-2AQi - Au -2A0i + 2A0i; z] (4.62) (4.63) (4.64) The condition | ^ = 0 ) | 2 = 0 (4.65) gives D = 0. The solution that we require is the t —» 00 asymptotic; this corresponds to the z —> 1 limit. Near z = 1 we find that r ( | + 2 A o z ) r ( | + 2 A 02) ^ ( 2 = l) = e ^ ° C r ( l + 2A0i + 4 i ) r ( l + 2 A 0 i - A x ) (4.66) Chapter 4. An Introduction to the Generalized Landau-Zener Problem 139 U s i n g the ref lect ion formulae we find tha t \AY = 2 C2e2*A° (1 + 16 A20) 4 ( 4 A 2 + A 2 ) COS2 7rAi c o s h 2 2A0TY (4.68) A s we require tha t the p r o b a b i l i t y be b o u n d e d above by one we take C2= 4 ( ^ + ^ > (4 69) - e 2 ^ ° ( l + 1 6 A 2 ) 1 ' w h i c h gives, i n te rms of our o r i g i n a l parameters , the f ina l t r a n s i t i o n p r o b a b i l i t y cos 2 ^ | ^ ( t - + o o ) | 2 = l - — ^ (4.70) W e emphasize tha t th i s result is exact. T o the best of our knowledge , th i s result has not been pub l i shed prev ious ly . T h i s so lu t ion is sketched i n figure (4.5). If we t r y to compare th is exact result to tha t o b t a i n e d f rom D y c h n e ' s fo rmula , we encounter another in te res t ing c o n u n d r u m . In order to o b t a i n closed fo rm so lu t ions for D y c h n e ' s fo rmu la , one has to be able to o b t a i n the indef in i te in tegra l of E(t) = \JV\\(t) + A 2 ( t ) . N o w i t is evident tha t the presence of the square root w i l l make the set of V\\(t) and A ( t ) tha t can be deal t w i t h qui te s m a l l . T h e present case is an example of a H a m i l t o n i a n tha t produces t ime-dependent energy eigenstates tha t do not p roduce closed fo rm so lu t ions to th is in teg ra t ion . T h i s l ine of though t is qui te in teres t ing , as i t seems to relate the i n t e g r a b i l i t y of square root funct ions to the so lu t ion of second order differential equat ions . It is p laus ib le tha t progress c o u l d be m a d e i n the s t udy of these k inds of indef ini te in tegra ls v i a the connec t ion tha t has been es tabl ished here to the hypergeomet r ic equa t ion . Chapter 4. An Introduction to the Generalized Landau-Zener Problem 1 4 P F i g u r e 4.5: T r a n s i t i o n p r o b a b i l i t y P\± f o r t h e p u l s e / r a m p s c e n a r i o . P l o t t e d i s ATT/UJ o n t h e x a x i s a n d ATT/U o n t h e y a x i s . T h e r e i s a n o t h e r c a s e w h e r e w e c a n m a p e x a c t l y t o t h e R D E , a n d t h a t i s f o r t h e p a r a m e t e r s e t V[|(<) = A t a n h u / t , A(t) = A (4.71) T h i s i s s i m i l a r t o t h e p r e v i o u s e x a m p l e , e x c e p t t h a t t h e t u n n e l i n g t e r m i s c o n s t a n t f o r a l l t i m e . T h e p r o c e d u r e i s i d e n t i c a l t o t h a t d o n e e x p l i c i t l y i n t h e p r e c e e d i n g ; w e s h a l l Chapter 4. An Introduction to the Generalized Landau-Zener Problem 14 j F i g u r e 4.6: T r a n s i t i o n p r o b a b i l i t y P T | f o r t h e r a m p s c e n a r i o . P l o t t e d o n t h e x a x i s i s A/ui a n d o n t h e y a x i s - j u s t s t a t e t h e r e s u l t . W e find t h a t s i n h 2 4* l ^ ^ ° ° ) | 2 = s i n h ^ ; A , ) . / 2 < 4- 7 2> N o t e t h a t f o r A / A « 1, t h i s r e d u c e s t o t h e s t a n d a r d L a n d a u - Z e n e r r e s u l t (4.12). T h i s i s a n o t h e r n e w e x a c t r e s u l t . T h i s s o l u t i o n i s s h o w n i n g r a p h i c a l f o r m i n figure (4.6). Chapter 5 The Landau-Zener Problem in the Presence of a Spin Bath At temperatures T < 360 mK thermal occupation of all but the two lowest lying levels in the Feg molecular magnet is vanishingly small (see figure 1.11 for experimental evidence of this and chapter 2 for theoretical justification). This means, as discussed in chapter 2, that in this "quantum regime" the central spin complex of the Fe8 molecular magnet can be described by a two level system. In the presence of an externally applied time- dependent magnetic field, the Hamiltonian of an isolated molecule is of the generalized Landau-Zener form (4.31) with the added complication that the central two level degree of freedom is coupled to an environmental spin bath. In this chapter we solve for the transition probability between fz eigenstates of the central spin object in the presence of an external A C field. We use this general result to calculate the one-molecule relaxation rate for any system with a Hamiltonian of the form (5 .6) . We then calculate the one- molecule relaxation rates for the specific case of the Fe% system. 5.1 The Addition of an Environment to the Landau-Zener Problem: General Considerations Whenever we study a condensed matter system, there is of necessity a distinction made between a (perhaps collective) degree of freedom and "everything else". When we write down an evolution operator (such as (4.31)) for this interesting degree of freedom without considering the "everything else" of the system we run the risk of misunderstanding what is actually happening inside the material. While the exact solution of toy models such 142 Chapter 5. The Landau-Zener Problem in the Presence of a Spin Bath 143 as (4.31) may perhaps be pleasing to the aesthetic sense, it does little to further physical understanding. It is not even clear that questions of principle in quantum mechanics may be settled via the use of toy models such as these. As we discussed at length in chapter 2, there exist standard ways of obtaining realistic effective Hamiltonians for some condensed matter systems. These effective descriptions, no matter what the material under study, always require us to specify a subset of the total information contained in them to be the "degrees of freedom of interest", while relegating all the "uninteresting degrees of freedom" to an environment [20, 21, 22, 74, 75]. This split, while not necessary in principle, is usually necessary to make the study of the system tractable (for, as we have seen, realistic effective Hamiltonians are complicated objects!). The consequences of this (artificial) distinction between system and environment are still not fully understood. Some of the ramifications, both physical and philosophical, were addressed in previous chapters and in earlier classic works [74, 20, 21, 22]. In this section, however, we shall consider some of the technical problems that arise due to the inclusion of an environment. 5.2 The Quantum Regime Effective Hamiltonian: Inclusion of a Spin Envi- ronment The form of the low energy effective Hamiltonian for the F e 8 molecular magnet was derived in chapter 2. We may ask how this Hamiltonian is changed by the application of an external time-dependent magnetic field. Looking back to (2.101) we see that there are four terms that will be affected; one due to direct interaction of the field with the nuclear spins (uj^rhk) ; one due to the field acting on the nuclear spins coming from the central spin whose minima are functions of the external field (w['4); and two due Chapter 5. The Landau-Zener Problem in the Presence of a Spin Bath 144 to the in t e rac t ion of the field w i t h the cen t ra l s p i n (the d i a g o n a l t e r m ~ fzHz a n d the off-diagonal t e rms i n the phase $ ) . T h e first o f these is uimk = \Ik\gnkpn{^kl) + o f ) = \Ik\gnkixn{fkl)s + 7 $ + H(t) + H(t + fin"1)) (5.1) where we have defined the fields -(1,2) _ IkS — 9nkPn E s f ' 2 ) - E M°SS«W (5.2) to be those due to the cen t ra l sp in before/af ter the cen t ra l sp in flips respec t ive ly at the kth nucleus, t is the t i m e at w h i c h the flip occured a n d fin"1 1 S the t u n n e l i n g t i m e for the cent ra l sp in c o m p l e x . I f we assume tha t fig 1 is m u c h less t h a n the t imescale over w h i c h the ex te rna l ly a p p l i e d f ield changes, then we find t ha t u^rhkit) « Ihlgn^nd^ + IkS + 2H{t)) (5.3) T h i s a s s u m p t i o n also leads to ujklk b e ing t i m e independent , since u>KlK = \Ik\gnkpn(t] - o f ) = \h\9nMikS ~ iks + H{t) - H(t + fi^1)) « uKlk (5.4) A l s o we find tha t i n general the phase $ becomes t i m e dependent i f the ex te rna l field conta ins transverse componen t s tha t va ry w i t h t i m e . In the p a r t i c u l a r case of the easy- axis easy-plane H a m i l t o n i a n i n the l i m i t s tud ied i n chapter 2 ( that is, the D C bias field \H\ - C fi0 and AE j(D — 2E) not too s m a l l (so tha t the in s t an ton c a l c u l a t i o n is v a l i d , see chapter 2)) we f ind tha t $ = 7 r S - nS2gnB nSg^B „ -l — t l T H — t i n . 2 E fin ->• $ ( t ) = TTS + 1 2 ~ Hx(t) ^—Hy{t) (5.5) Chapter 5. The Landau-Zener Problem in the Presence of a Spin Bath 1 4 5 Together w i t h the obv ious d i a g o n a l c o u p l i n g t o the ex te rna l field we find tha t i n the presence of a general ex t e rna l t ime-dependent field the one-molecule effective H a m i l t o n i a n is N+8 H(t) = £ fc=l ^ - 4 • mk{t) + ^Jk% • lk + gnBSHz(t)fz ( N + 8 _ _ \ + 2 A 0 f _ cos \^(t) - i £ A N , D • hj + h.c. N _yko.fi jk + S [ 6 / f c ( 2 / f c - 1) " l a 0\ + Mo/̂ n 9ni9nk E 47r rZk rik Ti-Tk-s(Ti- rik) (h • hk) (5.6) where the n o t a t i o n is the same as tha t i n t r o d u c e d i n chapter 2. F r o m th is po in t onwards we sha l l d r o p a l l g-factors a n d magnetons for reasons of n o t a t i o n a l s i m p l i c i t y . W h e n e v e r confusion m a y arise we sha l l e x p l i c i t l y i nc lude t h e m . T h i s expression is s t i l l qu i te c o m p l i c a t e d ; however, one notes tha t i t has the f o r m of a general ized L a n d a u - Z e n e r H a m i l t o n i a n w h i c h has been coup led to a sp in env i ronment . N o t e tha t the p h o n o n b a t h does not enter in to the q u a n t u m regime effective desc r ip t ion , for reasons de ta i led i n chapter 2 (a l though of course the osc i l l a to r b a t h is present when r e l axa t i on becomes t empera tu re dependen t - a p roper t r ea tment of these baths is under cons t ruc t ion . N o t e tha t there are several (often conf l ic t ing) t rea tments of osc i l l a to r ba ths i n th i s context [116, 117]). W e sha l l now s i m p l i f y the expression (5.6) by r e s t r i c t i ng our a t t en t ion to the fo l l owing two k inds of ex te rna l ly a p p l i e d field. In a l l tha t fol lows we sha l l be cons ider ing th is ex te rna l field to be ei ther a l o n g i t u d i n a l s inuso ida l field p lus a s ta t ic componen t i n an a r b i t r a r y d i r ec t ion Hcos(t) = (A cos cot + 0 z + Hx x + Hy y (5.7) Chapter 5. The Landau-Zener Problem in the Presence of a Spin Bath 146 or a l o n g i t u d i n a l l y a p p l i e d s a w t o o t h field p lus a s ta t ic c o m p o n e n t i n an a r b i t r a r y d i r e c t i o n Hsaw(t) = 2Aco 7T £ e 2 n + 1 cu t)e t- 2n-l + Hx x + Hy y (5.8) N o t e tha t we have chosen to use the n o t a t i o n tha t the l o n g i t u d i n a l componen t of the ex te rna l ly a p p l i e d field is denoted by Hz —> £. N o w because ne i ther of these two fields have t ime-dependent t ransverse components , we find t ha t the phase $ remains t i m e independent . T h i s s impl i f ies ou r expression somewhat . T o emphas ize the r e l a t ion of our effective H a m i l t o n i a n to the s i m p l e r L a n d a u - Z e n e r mode l s s t ud i ed prev ious ly , we rewr i te i t i n the fo rm H(t) = g(t) + V\\(t)fz + A f _ + h.c. (5.9) where m = at) + ^Y,^[rrh-3(ll^flk)(rk.flk)} 4 7 F Kk  rlk (5.10) Vj|(t) = f + Hz(t) / N+8 \ A = 2 A 0 c o s ( $ - t E A \ > D • Ikj and we have defined N+8 N+8 k=l k=l  Z k=\ N eQk yka/3 jk 6 / f c (2 / f c - 1) a \ and N+8 N+8 f - E T f c - E -ylk • h k=l k=l  Z (5.11) (5.12) (5.13) (5.14) N o t e tha t g(t), V\\(t), A a n d AkND are a l l operators i n the {Ik} ( env i ronmenta l spin) subspaces. r Chapter 5. The Landau-Zener Problem in the Presence of a Spin Bath 147 5.3 G e n e r a l A C F i e l d S o l u t i o n i n F a s t P a s s a g e We understand from looking at the effective description (5.9) that it will not be possible to compute the transition probabilities from the solution of a simple second-order differential equation. This is because all the environmental spins are dynamic and coupled to the central degree of freedom, vastly increasing the Hilbert space, and requiring the solution of (for a bath of N spin / nuclei) a (21 + \)Nth order differential equation which proves to be quite impossible in practice. We therefore must look for a different method of solution. Consider the perturbation expansion presented in section 4.1.1. We saw in the simple case presented there that there exists a natural perturbation parameter A 2 / v , which is small if the time it takes the external field to sweep through the resonance is much smaller than the "bounce time" A " 1 . We see explicitly in this case that if the sweep velocity is large enough, it is sufficient to calculate only to first order in this quantity. Let us set up this perturbation expansion for our effective description. As we indicated earlier, our externally applied field will be either sinusoidal or a sawtooth function. Often we shall find that analytic results are easier to obtain with the latter. Whenever possible we shall solve for interesting quantities for both shapes. Before we proceed with our technical investigation, let us pause and consider what the dynamics of the central spin complex should look like. When calculating the time evolution of this collective degree of freedom we can either calculate the evolution am- plitude from our initial time (say, t = 0) to our final time t or we can choose a "coarse graining" time tc < t and only evolve the system from t = 0 to tc. After this shorter-time evolution, we calculate probabilities and piece together a series of these, assuming that each piece is independent of the other pieces. Now in the usual case of a model like (4.31) it is clearly not permissible to do this, as the dynamics of the central degree of freedom Chapter 5. The Landau-Zener Problem in the Presence of a Spin Bath 148 at any particular time interfere (in the quantum mechanical sense) with dynamics at all other times. Another way of saying this is that it is necessary to calculate the evolution amplitude over the entire time range of interest-there is no decoherence (loss of phase information) in this system. However, the coupling to the environmental spin bath in (5.9) changes this story. Be- cause the spin bath is dynamically active, both because of its internal dynamics (coming from the nuclear dipole-dipole term in the Hamiltonian) and its response to the dynamics of the central spin moments, it absorbs phase information from the central spin complex [64, 167]. This has the effect of decorrelating successive sweeps of the A C field through resonance-while the evolution of the entire system is of course unitary, the evolution of the central spin degree of freedom is not. One can understand why this should be so from the following argument. During each sweep of the A C field through resonance, the central spin configuration can flip, and in so doing it can flip or rearrange the distribution of the environmental spins. This rearranges the density matrix of the spin environment, which can in general contain off-diagonal terms. Now as the sweeping field moves off-resonance, the nuclear spin-spin relaxation mechanism tries to equilibrate the environmental spin set. If the sweeping frequency is larger than ~ r 0 , where To is the energy range over which the nuclei sweep due to nuclear dipole-dipole interactions, by the time the sweeping field comes back to a resonance the off-diagonal spin bath elements will still be present (note that the energy scale r 0 is roughly r 0 ~ \fNT2~1 for an ensemble of N nuclei, where T 2 _ 1 is the standard spin-spin relaxation time). This means that for high enough sweeping frequencies one cannot neglect quantum correlations between successive passes of the field. In this case one could invoke methods from the so-called Floquet theory [118]. However, if T 0 is much larger than u, then the spin bath does have time to equilibrate itself between passes of the field, and we can therefore neglect all quantum correlations between successive passes, treating each sweep as being "decorrelated from its history". Chapter 5. The Landau-Zener Problem in the Presence of a Spin Bath 149 This consideration presents us with a natural "coarse graining time", the time it takes for one full sweep through a resonance by the AC field (which is tc = TT/LU for both the sinusoidal and sawtooth fields). If the timescale for the readjustment of the density matrix of the system (which is tb « l / r 0 ) is such that tb/tc < 1 then this coarse graining procedure is justified. Typically in magnetic insulators at mK temperatures T 2 _ 1 ~ 104 — 106Hz [48, 64]. In the specific case of Fe% we can calculate r 0 (see table 5.2); we find that r 0 ~ 3 — 13MHz, depending on the nuclear spin isotopes present. This gives us an approximate gauge of the highest frequencies that we can apply before the coarse graining approximation breaks down. For a sinusoidal sweep, tb/tc « TTTQ/UJ, which gives co < TITO <~ 10 MHz (5.15) In the experiments that have been performed to date on the Fe8 and M n 1 2 materials, the sweeping frequencies are much less than this; in the Fes experiments these frequencies were in the range 0.01 — 5 Hz [51]. We shall therefore use the coarse graining approx- imation in what follows. It is worth noting here that this point gives ample warning that temporal quantum coherence of the central degree of freedom in this system on timescales of the order of tc will be difficult to maintain, as the nuclear spin bath has ample time to both absorb phase information from the central spin and bias the system between sweeps of the AC field-and therefore processes mediated by the AC field will most likely be incoherent in the sense that each sweep through resonance is decorrelated from all other such sweeps. 5.3.1 A L i s t o f A p p r o x i m a t i o n s I n v o k e d i n t h e C a l c u l a t i o n s T h a t F o l l o w For the sake of clarity we shall list in this subsection all the approximations that we shall be making in the sections that follow. Whenever one of these is invoked, we shall refer Chapter 5. The Landau-Zener Problem in the Presence of a Spin Bath 150 the reader to the following list. (i) The coarse graining approximation. This approximation is described in some detail in the preceding section. To recap, whenever the condition UJ < -KTQ is met, one can neglect quantum mechanical correlations between successive sweeps of the A C field. This is because the internal dynamics of the spin bath mediated by nuclear- nuclear terms in the Hamiltonian carry phase information away from the central spin. (ii) The fast passage limit. The justification for taking this limit was presented in the preceding section. Formally in this limit we consider only the regime where A 2 AUJ where A and UJ are the amplitude and frequency of the sweeping field respectively. (iii) Nuclear spins with a given polarization group M are in thermal equilibrium at a polarization group temperature 0M = 0. The "polarization group" of our set of N + 8 nuclear spins is defined to be [20] N+8 M = £ Ikz k=\ We note that this quantity explicitly depends on which axes of quantization we pick for the nuclear spins. What we choose to do in this work is to pick the axes of quantization of the nuclear spins to be such that zk = 7 ^ ; that is, the z direction for the kth nuclear spin corresponds to the direction of the field at the spin when the central spin complex is in its +TZ eigenstate. Because the internal field due to the iron spins is quite inhomogeneous, these axes will only be mutually aligned in the case where the external field H is much larger than the internal fields (which we have seen in chapter 3 are in the 0.01 — 0.26 Tesla range). Chapter 5. The Landau-Zener Problem in the Presence of a Spin Bath 151 5.3.2 General Strategy for Calculating Relaxation Rates The core of the calculation of the one-molecule relaxation rate is the calculation of the transition probability between the fz eigenstates of the central spin complex during one sweep of the A C field. Before we begin to attack this problem, we shall explicitly describe the general strategy that shall be used to calculate these probabilities and hence the one- molecule relaxation rates. Qualitatively the situation that we have to deal with is similar to that examined in the previous sections dealing with the simple Landau-Zener transition. The big difference here is that the Hilbert space of the system now contains not one two level system but a two level system plus the full Hilbert space of the environment. This means that when we calculate the transition amplitudes (and then the probabilities) between the fz eigenstates, we also have to explicitly include the effects of the other N + 8 systems. Formally we can see how this is done by noting the following. For the Hamiltonian (5.6) the amplitude to go from some initial fz eigenstate \a > to some final fz eigenstate \/3 > during a single sweep of the A C field can be written where we have defined \P > and \P > to be the initial and final states of the spin bath respectively. Let us take a moment and explicitly detail the formalism that we shall use in order to describe the spin bath states. In a general molecular magnet there can be many species of environmental spins included in the effective Hamiltonian. For example in Fe$ we have 120 protons, 18 nitrogens, 8 bromines, 8 irons, 36 carbons and 23 oxygens. Our formalism must be able to handle these different species. To this end we shall define the following notation. The symbol A% =< 0\ < lf\e^oCH^dT\p >\a> (5.16) (5.17) Chapter 5. The Landau-Zener Problem in the Presence of a Spin Bath 152 represents the number of environmental spins of spin s and species JJL. The kth spin of the subset N£ is labeled = l..Njf. The total number of spins in the spin bath is N = J2fj.N^. As an example, consider the Fe 8 * material. Here we have that A ^ 2 = 120 , i V f = 18 , N3% = 8 (5.18) With this notation the spin bath states \P > and \P > can be written in the form f N> 1 \r> = n n K > k K > ® i 4 > - « ^ > } » ( ^ = i j i ^ > = n ( i i K > U n K > < > - ® i / i r > } The transition probability is then simply P% = A^A% (5.19) This expression for the "per sweep" transition probability contains explicit reference to the initial and final states of the spin bath. Now in any actual experiment performed on these materials, what is actually measured is a relaxation rate. In the calculation of this rate we will have to specify the distribution and characteristics of the initial and final spin bath states during a sweep of the external A C field. Now because of reasons elaborated in the preceding section, the strategy that we shall use in specifying the form of the initial states of the spin bath is the following. We shall define the set of states | M M > to be all the allowed spin bath states with polarization group in the pth subset of nuclear spins. That is, we explicitly treat each species of nuclear spin separately, defining a polarization group for each. We shall see later that in this calculation all the environmental spin subspaces separate; this means that it is permissible here to treat each species as being independent of all the other species. We define the number of states in the polarization group to be CM^- We have then Chapter 5. The Landau-Zener Problem in the Presence of a Spin Bath 153 p = I.-CM^ states w i t h w h i c h we l abe l \K>=\i\ i 7 i > l ( 5 - 2 ° ) U/x = l J Because th is n o t a t i o n m a y at first s ight appear confus ing, let us e x p l a i n here aga in w h a t each of the s y m b o l s represent. T h e state | M £ > is a particular state w h i c h is a m e m b e r of the set of states \M^ > . T h e states | M M > have p a r t i c u l a r p o l a r i z a t i o n groups fixed for each species. F o r example , imag ine a f ic t i t ious m a t e r i a l w i t h 2 pro tons a n d 2 n i t rogen a toms. W e beg in by fixing (say) MH = 0 a n d MN — 1- T h e n the states | M M > are the set of p a r t i c u l a r states w i t h p o l a r i z a t i o n g roup M M . So i n th is example we c o u l d l abe l our states \MH > = | + 1 / 2 , - 1 / 2 > , \M2H > = | - 1 / 2 , + 1 / 2 > and | i \ 4 > = | i , o > , | M £ > = | O , I > the former of w h i c h cons t i tu te the set | M # > and the la t t e r the set \M_y >. Because of our a p p r o x i m a t i o n (iii) i n the prev ious sec t ion we m a y define a densi ty m a t r i x for the sp in state set \M^ > of the fo rm 1 -e-e^Ml X M?\ 9 )Q AMI >< MP\ (5.21) w i t h p a r t i t i o n func t ion Z M ^ M , ) = j : e - ^ E P ^ C M ^ (5.22) i= i Chapter 5. The Landau-Zener Problem in the Presence of a Spin Bath 154 These definitions allow us to write down the transition probability for the central spin complex in the form Pa3=J2U K^t] (5-23) APG / i where the sum is over all possible configurations of A l l Polarization Groups (APG) and P $ = T r ^ Y * % } (5-24) where we have summed over all final spin bath states for our transition probability (any state of the spin bath is acceptable as a final state). Here the ensemble average over all possible configurations of all polarization group states APG is normalized such that £ n ^ = i APG ii- For a single species spin 1/2 bath, WM = ^ \ . „ . „ I (5-25) 2 The way this works is as follows. Each particular molecule begins its quantum mechanical evolution in one of the nuclear spin states f l ^ 1-̂  >• The probability of its making a transition from central spin state \a > to central spin state |/3 > is then W^P^p. The relaxation rate which we are after will involve contributions from all molecules in the crystal, which will of course be in different polarization groups. Therefore in order to extract the quantity of interest we need to perform an ensemble average over all these contributions. This ensemble average is given by (5.23). We can write the expression (5.24) in a simpler form by explicitly summing over all final spin bath states, using the completeness relation J2\if> \I^ >< = 1. Performing the sum gives p$ = Tr | p M M < a\e-l^H^)dT\(3 >< ftjfc H^dT\a >} (5.26) Chapter 5. The Landau-Zener Problem in the Presence of a Spin Bath 155 In order to finish our j o b , we need to t rans la te th is t r a n s i t i o n p r o b a b i l i t y i n to a r e l a x a t i o n rate. W e can do th is by n o t i n g tha t since th is is a "per sweep" t r a n s i t i o n p robab i l i t y , the r e l a x a t i o n rate can be o b t a i n e d by d i v i d i n g th is q u a n t i t y by the t i m e for one sweep (wh ich is of course s i m p l y the coarse-gra in ing t i m e tc). F u r t h e r m o r e , we m a y w i t h o u t loss of genera l i ty here choose \ct >= \ t > a n d \(3 >= \ i>. Therefore the one-molecule r e l a x a t i o n rate, w i t h the molecule ' s p o l a r i z a t i o n g roup state i n i t i a l l y > , is = lTr{pM» < t | e - * J > ( ^ | i><! | C ' J > M * - | T > } (5.27) a n d the f inal ensemble averaged r e l a x a t i o n rate is, f rom (5.23), ^ = E I 1 [WMSMI] (5-28) APG f- W e now t u r n to the exp l i c i t eva lua t i on of th is quant i ty . 5.3.3 P r o c e s s i n g o f t h e T r a n s i t i o n A m p l i t u d e In th is subsect ion we sha l l recast the t r a n s i t i o n a m p l i t u d e (5.16) i n a new fo rm tha t is easier to dea l w i t h . W e beg in w i t h our o r i g i n a l def in i t ion A% =< p\ < If\e^o C HW*\r >\a> (5.29) T h i s fo rm is correct to a l l orders i n A . A s i t is our w i s h to do p e r t u r b a t i o n theory i n th i s f l i p p i n g t e r m we m a y rewr i te th i s i n the fo rm A% . E f *. r «*.-.. r * . < a < I ' ^ ^ H J ^ n=0 J ° J ° J ° * H A e J ^ d r H i [ T ) . . . H ^ & d ^ \ P > \a > (5.30) where H_t(t) conta ins a l l te rms i n the H a m i l t o n i a n d i a g o n a l i n the f representa t ion Hd{t) =g{t) + V\\{t)Tz (5.31) Chapter 5. The Landau-Zener Problem in the Presence of a Spin Bath 156 and contains all off-diagonal terms # A = A r _ + h.c. (5.32) This amplitude can be thought of as a sum over diagrams containing n "blips" (see figure (5.1a)). n r ~ ~ L n _ 'i >i h '« h h (!>) Figure 5.1: Transition amplitude as a train of blips. Taking \a >= | t> and \P >— I i> we find that the leading order diagram contains only one blip (see figure (5.1b). This leading order term is simply 4 i = i / f e d t 1 <l\<lf\eifi*Hd{T)H*ei£*H<V\Ii>\l> (5.33) J 0 Now it turns out that the presence of the nuclear dipole-dipole term in Hd{t) introduces unnecessary complications into our calculation. This is because it couples the subspaces of all the environmental spins; that is, one cannot consider each environmental spin subspace separately if this term is explicitly present. However we may use the following physical argument to rewrite this term in a different way. The nuclear dipole-dipole term has the effect of causing a time-varying random field acting on the central spin, coming from the fast nuclear-nuclear dynamics in the bath. The transverse components of this field may be absorbed into the already existing transverse terms in the Hamiltonian. As Chapter 5. The Landau-Zener Problem in the Presence of a Spin Bath 157 l o n g as these la t te r are m u c h larger t h a n the a m p l i t u d e o f the nuclear -nuclear noise ~ r 0 we m a y neglect t h e m - w e sha l l assume th is is the case for now a n d then e x p l i c i t l y show i n sec t ion 5.6 tha t th is holds . T h e l o n g i t u d i n a l te rms, however, m a y not be so absorbed . T h i s is because the energy scale tha t we must compare these to is A 0 , w h i c h is m u c h smal l e r t h a n r 0. Therefore we sha l l replace our mic roscop ic a n d d e t e r m i n i s t i c desc r ip t ion w i t h an equivalent stochastic f o rm, f o l l o w i n g [20]. In th is s tochas t ic ve rs ion the in t e rna l d y n a m i c s of the sp in b a t h are rep laced by a r a n d o m l y v a r y i n g t ime-dependent field a c t i ng o n the cen t ra l sp in c o m p l e x w h i c h mode l s the d y n a m i c s generated by the o r i g i n a l nuclear d ipo le - d ipo l e t e r m and each e n v i r o n m e n t a l sp in subspace is assumed to be decoupled f rom a l l others. In te rms of our H a m i l t o n i a n th is entai ls r ep l ac ing the in t ra -nuc lear t e r m i n (5.10) w i t h a t e r m 8£(t)fz, where 8£(t) represents the l o n g i t u d i n a l c o m p o n e n t of the f luc tua t ing nuclear-nuclear bias f ie ld. E x p l i c i t l y we have tha t H{t) = $(t) + Vj|(t)f2 + Af_ + h.c. -> Hit) = C(£) + (V\\(t) + 6£(t)) fz + Af-+ h.c. (5.34) W e have denoted th is new s tochas t ic H a m i l t o n i a n v i a an overbar; tha t is, we wr i t e Hit) for the o r i g i n a l H a m i l t o n i a n a n d H(t) for the s tochast ic a p p r o x i m a t i o n . Because our new H a m i l t o n i a n now conta ins th is s tochas t ic t e r m , we sha l l use a s l i gh t ly different n o t a t i o n for the a m p l i t u d e , w r i t i n g A\{ = ifjdh <1 | < lf\eifi*SdiT)HAeif?dTR<V\Ii > | t> . w i t h the overbar r e m i n d i n g us tha t we are now us ing H(t) to evolve the sys tem, and not Hit). W i t h th is choice, the inner p r o d u c t i n the f subspace can be taken , y i e l d i n g A\{ = i fj dh < If\e1^ ^ ' ^ A ^ f o 1 « M C ( r ) + i * ( T ) } | r > ( 5 3 5 ) Chapter 5. The Landau-Zener Problem in the Presence of a Spin Bath 158 where we have defined F(r) = V»(T)+5Z(T) = t + AcosujT + £ + 6Z(T) (5.36) Here we have chosen for the sake of concreteness the sinusoidal form for the sweeping field. For the sake of notational simplicity we shall separate terms in £ that are time-dependent from those that aren't, writing N+8 C(r) = £ [ 4 + AcOSUJTlzk (5.37) k=l where (f£ + Hxx + Hyy) N eQk \rkafi jk 6Ik(2Ik - 1) N+8 c = E G fc=i (5.38) (5.39) and the time-dependent piece is the Zeeman interaction between the external A C field and the nuclear spin. We find that, after performing the r integrals in the exponents, n Jo £ . e * [ t f + t + « * i + e ^ ^ + Z o 1 ) r > ( 5 > 4 0 ) We have defined the initial and final polarization groups of the \ith species to be N^ = E LW , Mfll = E i, S (5.41) Because we are always in the limit that SgHB » 1 Ngnpn we can drop the direct field-nucleus interaction ~ A term from our expression. (5.42) Chapter 5. The Landau-Zener Problem in the Presence of a Spin Bath 159 Because a l l the e n v i r o n m e n t a l sp in spaces are independent i n H(t) we m a y separate each c o n t r i b u t i o n . R e w r i t i n g the cosine c o m i n g f rom the f l ip t e r m as a s u m of exponen- t ia l s / W+8 \ N+8 A = 2 A 0 cos U - i £ A \ i D • Ik) = A 0 £ <?m II EHAKN'D'H \ k=l 1 h=±l k=l we f ind tha t A*f A • A* 4 ^ S i n W t l - X ? d T ^ / W + / o l d r ^ / W + ^ - * 1 ) l / i ^ — £±01J oX\e L 1 J N+8 Y EIH* II < / / | e i { ^ " T f c } ( t c " i l ) e / l ^ . ° ' / * : e i { ^ + t A : } t l | 4 > (5.43) h=±l k=l where \Pk > (\Ik >) is the i n i t i a l (final) s tate of the kth nuclear sp in a n d r5£j/ is the ( t ime dependent) nuclear-nuclear l o n g i t u d i n a l bias field a c t i n g d u r i n g the t r a n s i t i o n f rom \P > to \P >. T h i s is the f o r m of the t r a n s i t i o n a m p l i t u d e tha t is best su i ted for us ing as i n p u t i n to the c a l c u l a t i o n of the t r a n s i t i o n p robab i l i t y . 5.3.4 P r o c e s s i n g o f t h e T r a n s i t i o n P r o b a b i l i t y ( i ) T h e F o r m a l E x p r e s s i o n T h e next step on our j ou rney towards the final one-molecule r e l a x a t i o n rate is the c o m - p u t a t i o n of the t r ans i t i on p r o b a b i l i t y f rom the result (5.43). A s we no ted i n our in t ro - d u c t i o n of the p r o b l e m , the t r a n s i t i o n p r o b a b i l i t y f rom the state | j > <S)\P > to the state | l> <g>\lf > w i t h H a m i l t o n i a n H(t) is s i m p l y PH = A\"A\{ (5.44) N o w i n our case we are w o r k i n g w i t h the t rans formed H a m i l t o n i a n H(t). Does th is change th is expression? L o o k i n g at the fo rm of our t r a n s i t i o n a m p l i t u d e (5.43) we real ize tha t it conta ins exp l i c i t m e n t i o n of the t i m e - v a r y i n g r a n d o m field 5£(i). T h i s field has to be Chapter 5. The Landau-Zener Problem in the Presence of a Spin Bath 160 averaged over i n the final express ion for the t r a n s i t i o n p r o b a b i l i t y . T h i s means tha t i f we use H(t), the express ion for the t r a n s i t i o n p r o b a b i l i t y becomes P;( = J V[S^f(t)}V[S^f(t)}A^A\{ (5.45) where we have e x p l i c i t l y i n c l u d e d a func t iona l average over the fluctuating bias field. U s i n g our resul t (5.43) we find t h a t th i s t r a n s i t i o n p r o b a b i l i t y is p;{ = Al £ * fj tofvmit)] P K „ ( i ) ] e ' [ - ( , i " " " - A " ' ! | - 2 ^ Nj~* ei(W-m**) J J < ^e-i{a+rk}t2ernA%^DJke-i{Ck-rk}(tc-t2)^jf > h,m=±l k=l < lJ_\ei^k-^k}(tc-n)ehA%DTkei{ik+yk}ti | j i > (5.46) T o t rans la te th is in to the f o r m P ^ we s u m over a l l final states of the sp in b a t h (using the completeness re la t ion) a n d res t r ic t the set of i n i t i a l sp in b a t h states to be those w i t h p o l a r i z a t i o n g roup M ^ , g i v i n g P^f = A20 fj dt, fj ^2e^(si"-^-s>"^)+«^-^)] £ e i ( f c *- m **> I " ^ [ ^ ( t ) ] P [ ^ ( t ) ] e - 2 l ^ d ^ ( r ) £ ~ h,m=±\ 7 peMj K Y[ < M p f c | e ~ l { ^ + T k } t 2 e m A ~ k " ' D ' h e ~ l { ^ ~ T ' ^ - ^ ehA»'D'Ik el{^k+r k } t l \Mpk > (5.47) where \Mpk > is the i n i t i a l state of the k^ nuclear sp in i n the pth sp in b a t h state h a v i n g p o l a r i z a t i o n group M ; i a n d S^M^ is the f l uc tua t i ng l o n g i t u d i n a l b ias t e r m c o m i n g f rom the nuclear spins i n the M M p o l a r i z a t i o n group. Chapter 5. The Landau-Zener Problem in the Presence of a Spin Bath 161 5.3.5 P r o c e s s i n g o f t h e T r a n s i t i o n P r o b a b i l i t y ( i i ) A v e r a g i n g o v e r t h e R a n d o m l y F l u c t u a t i n g T2 N o i s e W e see f rom the p reced ing tha t the c o n t r i b u t i o n s f rom the effective nuclear -nuclear t e r m i n our H a m i l t o n i a n H(t) c o m i n g f r o m the pth species are w h o l l y desc r ibed by the expres- s ion /V V[S^(t)]e-2lKdT5^{T) (5.48) In order to e x p l i c i t l y evaluate th i s c o n t r i b u t i o n we first need to specify the p r o b a b i l i t y func t iona l P[8£,M^(t)]- W e assume tha t th i s r a n d o m process is gauss ian and therefore take 7> [<5£M / i(*)] = ^ d s i ^ d S 2 S ^ M ^ S l ) K M ^ S l ~ S 2 ) 5 ^ M ^ S 2 ) (5.49) where the q u a n t i t y KM,,, can be unde r s tood i n te rms of the fo l lowing . T h e p r o b a b i l i t y func t iona l is defined so tha t the average of any opera tor over i t is s i m p l y < A^MM > = / V [ 5 ^ ( t ) ] A [ 8 ^ ( t ) ] V [ 5 ^ ( t ) } (5.50) It fol lows therefore tha t the a u t o c o r r e l a t i o n func t ion of the noise is < s^(Sl)s^(S2) > = j 2>[(&£MM(t)]<w*i)<ŷ (5.51) T h e f o r m a l so lu t ion to th is equa t ion is [119] < 6ZMM)SZM,{S2) > = KMM ~ S2) (5-52) where KMl is defined by Jds'KMll(si - s)KMl^s - s2) =6(si- s2) (5.53) Chapter 5. The Landau-Zener Problem in the Presence of a Spin Bath 162 In our case, we are in teres ted i n a noise func t iona l S£,M^(t) w h i c h changes by r o u g h l y SUJM^ i n a t i m e T2/N£ ( this q u a n t i t y is re la ted to the spread i n energy space of the Mff p o l a r i z a t i o n g roup - s i m p l y YMy. = VNs5tuM^)- It t hen diffuses i n energy space w i t h diffusion constant = - KJt ^ r2 where we have defined AM = -^L. T h i s now al lows us to find out w h a t KMl(s\ — s2) is for our p r o b l e m . T h i s is because < ( < S £ M „ ( S I ) - 5iM,{s2))2 > = < (5£MM))2 > + < (^M,(S2))2 > - 2 < ^Mfi(Sl)S^Mfi(s2) > = 2DSiM^\si-s2\ = A M i i \ s i - s 2 \ (5.55) and therefore A 3 < ^ M , ( S I ) ^ M , ( S 2 ) > = KM\{Sl -S2) = -^- (\Sl\ + \s2\ - \ S l - s2\) + 5 f ^ ( 0 ) (5.56) W e now in t roduce the charac te r i s t ic func t iona l $ [ Q M J d u a l to our p r o b a b i l i t y func t iona l V[KMM = I^O^We"* F * 0 ^ W ^ ( T ) < & [ Q M | 1 ( * ) ] (5.57) F o r a gaussian p r o b a b i l i t y func t iona l i t fol lows tha t $[QM (t)] = e ^ ^ ^ ^ M ^ e ' ^ * 8 1 ^ 8 2 ® (5.58) T h e quan t i t y tha t we are t r y i n g to evaluate (5.48) is s i m p l y $[—2]. Because we k n o w w h a t KMl(s\ — s2) is f rom p h y s i c a l grounds , we m a y now w r i t e fv V[6£Mll(t)]e-*%dT6iM'>lT) = $ [ - 2 ] Chapter 5. The Landau-Zener Problem in the Presence of a Spin Bath 163 Performing the s integrals gives 3 /V [S^(t)\ V i S U M ^ 1 ^ ^ ^ = e - ^ ( t 2 - t l ) 3 - r - , ( i 2 - t l ) 2 (5.60) We find therefore that A 3 pM» = A 2 ftc df / " * c d t i [ ^ ( s i n U * 1 - s i n W t 2 ) + « * 1 - * 2 ) ] e - - ^ ( t 2 - t i ) 3 - r ^ ( t 2 - t 1 ) 2 n °Jo Jo ^2 g j(/i*-"i**) >p ^ h,m=±l peM? ^Mf> Nt J J < M p f c | e _ i { ^ + i } t 2 e m ^ V ^ e ^ ^ ^ > (5.61) Note that since A M F I «C TMF_ (that is, TMFI ^> T2~L) for the systems under consideration, we can always drop the cubic term in (ti — t 2) in the above expression. We therefore find that P^ = A Q dti A*0 rft2E4^(SINWTL~SINWI2)+^*1_I2)]E~R^^2-IL^ £ ei(/.*-m*-) _ 1 _ J° J° h,m=±l VZMICM» Nt J J < j\^P f c | e - i { C f c + t f c } i 2 emA ^ D - 4 e - i { C f c - f f c } ( t 1 - t 2) e/i^^ > (5.62) 5.4 S o l u t i o n W i t h o u t S p i n B a t h The expression (5.62) is still quite opaque. In order to see how one extracts meaningful results, we shall see how to resolve it in the situation where the spin bath is absent. Here the Hamiltonians H(t) and H(t) are identical and (k = TK = A N D = TM = 0. Since Uk=i < Mlk\Ml >= 1 and P$ -> Pn, we may write (5.62) as P n = A 2 dtx r C r f t 2 e ^ ( s i n - t i - s i n a ; t 2 ) + £ r ( « 1 - i 2 ) ] V - j[h*-m**) ( 5 gg) J° h h,m=±l Chapter 5. The Landau-Zener Problem in the Presence of a Spin Bath 164 T h e sums over {h,m} col lapse , a n d we have Pn = 2 A 2 | c o s $ | 2 ftc [tc ^ 2 E 4 ^ ( s i n ^ - s i n ^ ) + C ( t i - * 2 ) ] ( 5 6 4 ) Jo Jo These integrals c a n n o w be per formed, w i t h resul t A 2 7 T 2 ^ ^ H c o s c & l 2 (5.65) where J and £ are the A n g e r and W e b e r funct ions respec t ive ly [113]. T h i s resul t , n o r m a l i z e d to the t r a n s i t i o n p r o b a b i l i t y for the s t a n d a r d L a n d a u - Z e n e r process P(o) _ A 2 ? R _ AO|COS$| 2TT IS PA n p ( 0 ) n LO Aui (5.66) u, \ ui 2A\ (5.67) T h i s quan t i t y is a func t ion o n l y of A/ui and ^/ui. W e have p lo t t ed i t as a func t ion oi^/ui for three fixed values of A/u> i n the figures tha t fo l low. N o t e tha t s i m i l a r t r a n s i t i o n ampl i tudes have been ca l cu l a t ed p rev ious ly for re la ted p rob lems [115, 120, 121]. Pe rhaps the most c losely re la ted p r o b l e m for w h i c h a p u b l i s h e d s o l u t i o n exists is for the H a m i l t o n i a n [115] H = £ —A cos uit A n A o -e (5.68) T h e pub l i shed resul t is the t r a n s i t i o n p r o b a b i l i t y over one entire cycle of the field, w h i c h is a t i m e w h i c h is double tha t of the coarse-gra in ing t i m e we are us ing , and i t is g iven by P A'in2 ^ / 2 A 1 2 LO1 u V ui (5.69) where J^(x) is a fj, order Bessel func t ion . Chapter 5. The Landau-Zener Problem in the Presence of a Spin Bath 165 We may also solve this problem for the case of a sawtooth field. In this case we have that Pn = 4 A 2 | c o s $ | 2 / W 2 dt, fC'2 dfee 'IW-^fc-*)] (5.70) J—tc/2 J—tc/2 These integrals may be performed, giving error functions [113] which can be easily plotted. This result, again normalized to P$, is compared to the results obtained from the sinusoidal perturbation in the figures that follow. Note that what we have calculated here is the transition probability for one sweep. Because we do not have the spin bath to absorb phase information from the central spin we cannot really claim to have calculated a "relaxation rate", as each sweep of the field here is correlated (in the quantum mechanical sense) and this calculation implicitly assumes a decorrelation. Chapter 5. The Landau-Zener Problem in the Presence of a Spin Bath 166 0.2 n 0.18-; 0.16-: 0.14 \ 0.12: \ 0.1- 0.08 0.06̂  \ 0.04 0.02 \ 0 2 4 6 8 10 0.1-, 0.08- 0.06- 0.04- 0.02- 0 2 4 6 8 10 12 14 16 18 20 Figure 5.2: Transition probability normalized to the standard Landau-Zener transition probability P^/Pff plotted against 2^/ui for 2A/to = 0.1. The top (bottom) graph is for the sinusoidal (sawtooth) perturbation. Chapter 5. The Landau-Zener Problem in the Presence of a Spin Bath 167 3 2 . 8 2 . 6 2 . 4 22. 2 1 . 8 1 . 6 lAi i 0 . 8 0 . 6 0 . 4 02 0 2 4 6 8 1 0 1 2 1 4 1 6 1 8 2 0 2 n 1 . 8 - 1 . 6 - 1 . 4 : 1 . 2 ^ / \ 1 - \ 0 . 8 0 . 6 \ 0 . 4 ^ v ~ \ 0 . 2 ^ 0 2 4 6 8 1 0 1 2 1 4 1 6 1 8 2 0 2 2 2 4 2 6 2 8 3 0 Figure 5 .3 : Transition probability normalized to the standard Landau-Zener transition probability P^/pff plotted against 2£/tu for 2A/u> = 10. The top (bottom) graph is for the sinusoidal (sawtooth) perturbation. Chapter 5. The Landau-Zener Problem in the Presence of a Spin Bath 168 5- 3- 2- 1 i-..^»,,-..,w,-»»l«'V: 100 200 300 400 500 600 12.A 0.6 0.4 A 02. A 100 200 300 400 500 600 Figure 5.4: Transition probability normalized to the standard Landau-Zener transition probability P^/pff plotted against 2£/UJ for 2A/u = 500. The top (bottom) graph is for the sinusoidal (sawtooth) perturbation. Chapter 5. The Landau-Zener Problem in the Presence of a Spin Bath 169 5.5 Solution For a Spin Bath with no Quadrupolar Contribution If the sp in b a t h conta ins o n l y s p i n 1/2 nuc le i , or i f the c o n t r i b u t i o n due to the e lec t r ic q u a d r u p o l a r t e r m is o m i t t e d f rom the H a m i l t o n i a n , t h e n the express ion (5.62) s impl i f ies considerably . In th i s sec t ion we sha l l ca lcu la te the fu l l general r e l a x a t i o n rate p r o d u c e d by the effective H a m i l t o n i a n (5.6) i n the l i m i t where the te rms represent ing the e lec t r ic q u a d r u p o l a r effects are t aken to zero. F o r m a l l y w h a t we sha l l do is take It sha l l t u r n out i n w h a t fol lows tha t our choice of a s inuso ida l sweeping field i n t ro - duces t echn ica l diff icul t ies i n eva lua t ing integrals w h e n the sp in b a t h is i n c l u d e d . F o r th i s reason we sha l l i n th is sec t ion consider o n l y the s a w t o o t h p e r t u r b a t i o n (5.8). W i t h th i s a p p l i e d field (5.62) can be w r i t t e n dti / T / 2 U J-K/2U, h,m=±X = A 2 r/2w t, r/2u d t 2 e i [ ^ ( t i - t 2 ) 2 + « t i - * 2 ) ] e - r ^ ( t i - t 2 ) 2 E e ^ * - m * - ) J—7T/2UI J—TT/2UJ u I 1 £ II < Mpk\e~l^''k»t2emA»'^ (5.71) 5.5.1 Pure Orthogonality Blocking In order to get a feel for how th is c a l c u l a t i o n w i l l go i n the general case, let 's s tar t off w i t h a p a r t i c u l a r l y in te res t ing l i m i t - t h a t of pure o r t hogona l i t y b l o c k i n g (for the def ini t ions of o r thogona l i t y b l o c k i n g , t o p o l o g i c a l decoherence a n d degeneracy b l o c k i n g we refer the reader to [20]). In th is case we take the t opo log i ca l decoherence te rms Ak^D to be zero. T h e reason tha t th is l i m i t is p a r t i c u l a r l y s imp le is tha t w i t h o u t these terms, we m a y take the axes of q u a n t i z a t i o n of each of the nuclear spins to be such tha t 7 ^ = z w i t h o u t Chapter 5. The Landau-Zener Problem in the Presence of a Spin Bath 170 having to worry about the effect that these rotations have on the form of the AtfD. We find in this limit that p^f"=A2,i cos $i 2 rl7u dtl r/2uj ^ 2 e4^^-^ 2 ^-^] e - r ^^^) 2 J-TT/2UJ J-TT/2UJ J _ £ n e < ) l / ^ ^ - ^ ) < M f | e - < ) ^ ( t l - t 2 ) | M f > (5.72) Cm» PeMl fcM=i where we have used notation such that IkfiZ means the z state of the k1^ spin, normalized such that —1 < Ikz < 1- Note that we must be careful here, as the conversion from field to energy units contains a factor of \Ik\. We now choose, without loss of generality, 7 ^ = cQz + Clx (5.73) where co = - | 7 £ ) | c o s 2 ^ , C l = | 7 g ) | s i n 2 ^ (5.74) wi ith Changing variables to cos 2/3fc|, = - 7 ^ . 7 ^ (5.75) X = u>{tx-t2) , Y = J-±±^ (5.76) allows us to write |2 U CO 2 J-n/2 J-00 r/2 l N* ^l1 >i ^ - — E fl J-^1"*** < Mlk\e-l~^-'^x\Mlk > (5.77) Here we have extended the limits on the X integral. This will be permissible as long as r M „ 3> to (which we assume here), as the gaussian term cuts off the large X contributions. Chapter 5. The Landau-Zener Problem in the Presence of a Spin Bath 171 Similarly if < ^/NTTM^ (which we shall see is also the case-see table 5.2 and tables 2.3-2.11) we may expand the inner product to quadratic order in X, giving < Mf \e-i^"I^x\MPk >t -(2) '-(2) Tk l-iX< Mf\^ • Ik\M? > — — < Mf\ [ ^ • I k A \Mf > (5.78) I T 1 2 ) I x2\fk 1 + %XIkJ-^- cos 2ft, - to 2uil 2 1̂ (2) 12 a ^ M 1 2 + cos 22/? f c / i/ 2 x 2 l 7 < 2 ) l 2 4 W 2 • sin 2 2/3, 1 = e (5.79) Insertion of this into (5.72) yields i%» = A ° l c ° f { 2 r dY r d X e ^ x ^ A e - ^ CO' T/2 l^[2)l c M, E n A? 2zXIkaZ^cos*pk J— sin 2 2/3, 1 (5.80) where we have used the fact that |7^| = I T I ^ I ( i e - the magnitude of the field acting on the k1^ nuclear spin before/after the central spin complex flips is the same). Defining the quantities TV," = 2 £ V l f f c 1 J | c o s 2 f t k»=l we can write this as 1 £ l 7 i 1 } | 2 s i n 2 2 ^ 1 j2 • z (5.81) p M i i = A 2 |coscfr| 2 r / 2 C J 2 7 - 7 T / 2 / d y / J-n/2 J- CM, (5.82) Chapter 5. The Landau-Zener Problem in the Presence of a Spin Bath 172 The Sum Over Initial States I M * > We are now going to perform the sum over the initial states \M? >. Before we begin, let us take a close look at the expression (5.82). We see that both e% and pi^ are functions of the particular state that the spin bath is in. Now what we are going to do now is to replace p2^ with its average value; that is, we take P\^\>Z l7 i 1 ) | 2 s in 2 2^ ) (5.83) kit — 1 \ / Since ^ ^ ^ m - ' M ^ (5.84) the average value of is simply P ? , = % ^ E I^Ps in^^ (5.85) The approximation of neglecting the width of the distribution of numbers {piM} is justified whenever this width is much less than TM/i (which is certainly the case here). With this approximation the sum in question becomes — e " ^ £ (5.86) In order to clarify the procedure that we shall adopt in what follows, let us imagine how we would proceed in a specific case. As a typical example, let us consider how to deal with the protons in Fes- In this situation, there are an enormous number of possible states because NH (and therefore CM[]) is large. In F e 8 there are 120 protons per molecule. This means that there are 2 1 2 0 ~ 1 0 3 6 states in 2NH + 1 = 241 polarization groups. The Mjj polarization group contains NH choose \MH — NH\/2 states, each of which contribute to the sum (5.86). In Chapter 5. The Landau-Zener Problem in the Presence of a Spin Bath 173 addition, we note that the values that e% can take are bounded. So the situation that we must deal with is one where we have an extraordinary number of states squashed into a bounded energy range. It is natural in this situation to convert the sum over initial states into an integral over e,. Now because of the exceedingly large number of states available to the spin bath, we may invoke the central limit theorem in order to supply an appropriate weighting function for the integration. That is, as we increase Ng, the numbers that we get for for each particular state of the spin bath will begin to approximate a gaussian distribution centered at some value ef^ with some width W ( M / i ) . In this limit we find that ^ ^ £ l # W & (5.87) Now the width W ( M M ) is a little trickier to dealt with. In general it is apparent that this quantity has its maximum for M M = 0 and monotonically decreases to zero for \M^\ = Ng. How it does this will in general be a function of how the fields | T ! ^ | are distributed. It turns out, however, that to a good approximation we can consider this width to be independent of M^. There are two factors that make this approximation reasonable. The first is that the number of states available in a given polarization group (which, for spin 1/2 nuclei, is simply N choose |JV - M | / 2 ) , fall off sharply for l A f ^ j > ~ v W - This means that the behaviour of W(M A 1 ) for | M M | > VNf will be irrelevant, simply because there aren't enough states in these higher polarization groups to make any difference in the final relaxation rate. The second factor is that for \M^\ < y/~Ng, the width W ( M / i ) will be roughly of the form W ( M M ) ~ ( l - J ^ l j W((VJ , | M , | < yfw < Ng (5.88) for Ng large. The zero polarization group width we get from the central limit theorem; Chapter 5. The Landau-Zener Problem in the Presence of a Spin Bath 1 7 4 i t is s i m p l y 2 N ' -+ - ? = £ 1 ^ | cos2 A (5-89) N o t e tha t t o be consis tent here we mus t also take -> r 0 / J for the same reason tha t we took W(Mfl) —»• W ( O ^ ) . W e sha l l h o l d off o n u s ing these a p p r o x i m a t i o n s for the t i m e be ing . W e c a n make excellent progress i n eva lua t i ng our expressions w i t h unspecif ied r M ( i , e f^ a n d W ( M / t ) . U n t i l we reach the p o i n t where we need to have e x p l i c i t forms for these we sha l l j u s t leave t h e m as funct ions of the p o l a r i z a t i o n group . U s i n g the gauss ian we igh t ing a l lows us to rewr i te our s u m i n the f o r m l ^ l r°° , „ -tjl^Hl 'fx X flu. 1 /-oo tg1 e1 > tlfX y el^r = e ~ ^ y - = — - / de^e ™2W e{^r «M W 2(M M ) / 2 + P 2 = e l A ^ " e ^ x (5.90) S u b s t i t u t i o n of (5.90) in to the express ion for the t r a n s i t i o n p r o b a b i l i t y (5.82) y ie lds dY r d x A ^ ^ e ^ e 2 J-IT/2 J-OO (5.91) p M ^ = A ^ O S $ | 2 r / 2  ̂ r a v A ^ X Y + L x ] ^'Z.-  w 2 ^ + f ^ W x , t l CO 2 J-K/2 Evaluation of the Ensemble Average Over Polarization Groups E x a m i n a t i o n of P ^ reveals the fo l lowing useful fact. Because the energy spreads i n the different p o l a r i z a t i o n groups over lap considerably , ie. « ^ ( M ^ + ^ + r ^ (5.92) we m a y a p p r o x i m a t e the ensemble average over a l l p o l a r i z a t i o n groups by in tegra t ions . F o r m a l l y th is is done by t a k i n g APG fi \i = r dM,e-MV2N° KN? J-oo " (5.93) Chapter 5. The Landau-Zener Problem in the Presence of a Spin Bath 175 where we have ex tended the in t eg ra t ion l i m i t s (the gauss ian cuts off the large M c o n t r i - bu t ions ) . F r o m (5.28) we have APG n A ^ | c o s $ | 2 r/2 J-ix/2 J-OO n M„ W(MM) J= f00 dM,e-Mll^eV< » X A : w 2(M^)/2+^ M+r2 X 2 (5.94) where we have w r i t t e n ef^ e x p l i c i t l y as a func t ion of u s ing (5.87). In order to pe r fo rm the integrals over the set {M^} we now invoke our W ( M / 1 ) —» W ( O ^ ) a p p r o x i m a t i o n . T h e integrals over the set { M ^ } are then easi ly per formed, g i v i n g A 2 | c o s $ | 2 W 2 T / 2 •Pf-I- ~ f ' 2 dY r dxA%XY+Me-$x' J-n/2 J-OO (5.95) where we have defined the fu l l energy w i d t h (5.96) where a l l these quant i t i es are eva lua ted for the zero p o l a r i z a t i o n g roup sp in set for each species. Evaluation of the X and Y Integrals T h e X in tegra l can now be per formed, y i e l d i n g PU - C h a n g i n g var iables to A 2 1 cos $ I2 r/2 yfUloW J-n/2 dY exp m2 (2AY t f' '4W2 \ TTLU LU (5.97) LU (2AY C Z = + - 2W V vro; LU (5.98) Chapter 5. The Landau-Zener Problem in the Presence of a Spin Bath 176 a l lows us to wr i t e ou r r e l a x a t i o n rate i n the final f o r m , o> A n | cos $ | 2 rz+ ,„ 72 , where ^ = ^ ( " < » ) Evaluation of the Relaxation Rate in Various Interesting Limits In the l i m i t where \A\ 3> y / ^ 2 + ~ W 2 we recover our o l d large A resul t , w h i c h is of course independent of , A ^ | c o s $ | 2 r " 1 = — ^ — L 5.101 A v ' If \A\ <C \ / £ 2 + W2 t hen the r e l a x a t i o n rate becomes A 2 | c o s $ | 2 2 414/2 (5.102) Discussion of Results: I. Pure Orthogonality Blocking T h e general result (5.99) produces two l i m i t i n g cases (5.101, 5.102) tha t are b o t h qu i te in teres t ing . T h e first of these (5.101), v a l i d for large A, te l ls us tha t i n th is l i m i t o r thog- ona l i t y b l o c k i n g effects do not affect the relaxation rate at all. H o w can we e x p l a i n th is phys i ca l l y? In order to unde r s t and the result , we sha l l present a g r a p h i c a l dep i c t i on of how or thogona l i t y b l o c k i n g works i n a molecu la r magnet . T h e cen t ra l sp in complex , a two- level sys tem, begins i ts evo lu t i on i n some fz eigenstate w h i c h we choose w i t h o u t loss of general i ty to be | y > . N o w for a l l the t i m e tha t i t r emains i n th is state the nuclear spins Chapter 5. The Landau-Zener Problem in the Presence of a Spin Bath 177 feel i ts d i p o l a r field. T h e k nuc lear s p i n i n i t i a l l y is exposed to th i s field w h i c h we have ca l l ed 7 ^ . W e saw p rev ious ly tha t because of the na ture of pure o r t h o g o n a l i t y b l o c k i n g we c o u l d choose the axis of q u a n t i z a t i o n of the kth nuclear sp in freely a n d for convenience chose the z axis to be pa ra l l e l to the d i r e c t i o n of 7 ^ . P h y s i c a l l y w h a t th is means is tha t for th i s choice of basis the kih nuclear sp in feels o n l y a l o n g i t u d i n a l f ield as l o n g as the cen t ra l s p i n c o m p l e x remains i n i ts i n i t i a l s tate | y > . N o w w h e n the cent ra l s p i n c o m p l e x tunnels to the o ther f z eigenstate | i>, the kth nuclear sp in feels a different field 7 ^ which contains transverse components. T h i s means, of course, tha t the kth nuclear sp in w i l l precess i n the new field 7 ^ . T h e d i a g r a m tha t we are e v a l u a t i n g i n order to solve for the t r a n s i t i o n p r o b a b i l i t y is the one shown i n figure 5.5. W e see tha t th is process conta ins a l eng th of t i m e tx — i 2 where the cent ra l sp in c o m p l e x is i n the state | \ > . D u r i n g th i s l eng th of t i m e the kth nuclear sp in precesses i n the field 7 ^ . N o w w h a t does th i s have to do w i t h our large A resul t (5.101)? L e t us go back to an ear l ier expression (5.95) a n d l ook closely at the integrals i nvo lved . In pa r t i cu la r , let us examine the Y i n t eg ra t ion . N o t i c e tha t there is on ly one t e r m under the in teg ra t ion t ha t is a func t ion of Y. T h i s t e r m m a y be i so la ted a n d is N o w i f A/to is "large enough" , we see tha t th is expression w i l l beg in to app rox ima te a de l t a func t ion i n A ' . H o w large is large enough? L o o k i n g back to the express ion (5.95) we see tha t the X in tegra l is now of the fo rm (5.103) (5.104) Chapter 5. The Landau-Zener Problem in the Presence of a Spin Bath 17& t = 0 F i g u r e 5.5: D e p i c t i o n o f t h e p r e c e s s i o n o f t h e kth n u c l e a r s p i n d u r i n g a " b l i p " . T h e c e n t r a l s p i n i s s h o w n i n b l a c k , w i t h a s c h e m a t i c n u c l e a r s p i n u n d e r n e a t h . T h i s n u c l e a r s p i n f e e l s a f i e l d y± f o r t i m e s t < ti, w h i c h w e c h o o s e t o b e t h e a x i s o f q u a n t i z a t i o n . A f t e r t h e c e n t r a l s p i n flips a t ti t h e n u c l e a r s p i n f e e l s a d i f f e r e n t field 7̂  w h i c h c o n t a i n s i n g e n e r a l t r a n s v e r s e c o m p o n e n t s . T h i s c a u s e s a p r e c e s s i o n o f t h e n u c l e a r s p i n . A f t e r t h e c e n t r a l s p i n flips b a c k , t h e n u c l e a r s p i n w i l l b e i n a s t a t e t h a t h a s l e s s t h a n f u l l o v e r l a p w i t h i t s o r i g i n a l s t a t e . C h a n g i n g v a r i a b l e s t o Z = AX/LJ a l l o w s u s t o w r i t e N o w h e r e i s t h e c r u x o f t h e m a t t e r . I f A i s m u c h b i g g e r t h a n £ a n d W, t h e n t h e t e r m s i n f o r e a s y s o l u t i o n o f ( 5 . 1 0 5 ) . M o r e o v e r , we see t h a t b e c a u s e t h e s e q u a n t i t i e s d r o p o u t o f t h e e x p r e s s i o n a l l r e f e r e n c e s t o t h e n u c l e a r s p i n s d i s a p p e a r ! W h a t h a s h a p p e n e d ? W e s e e t h a t t h i s e x p r e s s i o n i s d o m i n a t e d b y s m a l l v a l u e s o f Z. B u t Z i s n o t h i n g b u t Z = AX/LO — A(ti — t2). T h e r e f o r e c o n t r i b u t i o n s o n l y c o m e f r o m t\ — t2 < ~ .̂ W e see t h e n t h a t l a r g e A r e d u c e s t h e l e n g t h o f t i m e t h a t t h e c e n t r a l s p i n s t a y s i n t h e " b l i p " (5.105) t h e e x p o n e n t i a l s g r o w s l o w l y c o m p a r e d t o t h e t e r m a n d w e c a n n e g l e c t t h e m , a l l o w i n g Chapter 5. The Landau-Zener Problem in the Presence of a Spin Bath 1 7 9 a n d therefore the l eng th of t i m e tha t the nuclear spins can precess i n the field jk . T h i s is the p h y s i c a l reason w h y o r thogona l i t y b l o c k i n g does not affect our large A r e l a x a t i o n r a t e - the processes i nvo l ve d are c o m p l e t e l y d o m i n a t e d by short b l ip s a n d therefore the nuclear spins don ' t have t i m e to precess out of the i r i n i t i a l states. L e t us now t u r n ou r a t t en t ion to the other l i m i t i n g case tha t we have w o r k e d out , tha t for s m a l l A (5.102). W h a t we have found is tha t the r e l a x a t i o n rate falls off l ike a gauss ian w i t h a p p l i e d bias £. T h i s seeming ly con t rad ic t s results o b t a i n e d for A = 0 where the r e l a x a t i o n rate was found to be exponen t i a l w i t h bias [20] where £0 is an energy scale i n the P roko f i ' ev a n d S t a m p theory [20] such tha t r 0 < £0 < W (W is the fu l l energy w i d t h of the nuclear sp in d i s t r i b u t i o n - i n our case we sha l l f ind tha t i t is W ~ 100 — 400MHz depend ing on the choice of isotopes i n the Fe$). W h y i t is tha t our resul t seeming ly disagrees w i t h th is ear l ier result can be t raced to a subt le difference i n the way tha t our express ion (5.102) a n d the expression (5.106) were ca l cu la t ed . W h e n A = 0 there is an a d d i t i o n a l cons t ra in t w h e n we ca lcu la te the t r a n s i t i o n prob- a b i l i t y for the cen t ra l sp in , and tha t is tha t w h e n the cen t ra l sp in c o m p l e x tunnels I t , M >—>• I I, M' > i t is necessary i n order to conserve energy tha t M' = — M . T h a t is, the t o t a l energy of the cent ra l s p i n p lus the env i ronmen ta l spins has to be the same before a n d after a fl ip and therefore the e n v i r o n m e n t a l spins cannot go in to any M' they want . N o w i n the c a l c u l a t i o n of (5.106) t h i s cons t ra in t , w h i c h is e x p l i c i t l y used i n the c a l c u l a t i o n of the t r a n s i t i o n p robab i l i t y , is responsible for the exponential dependence on external bias. In our case we have not e x p l i c i t l y pu t i n any such cons t ra in t . Ins tead we have a l - lowed i n our express ion for the t r a n s i t i o n p r o b a b i l i t y | t , M >—> | I, M' > for any M'. T h e energy cons t ra in t here of course s t i l l exists (it is i m p l i c i t i n the express ion for the T - i = ±L e - | £ l / t o £0 (5.106) Chapter 5. The Landau-Zener Problem in the Presence of a Spin Bath 180 transition probability) but it is of a slightly different nature than when A = 0. This is because when A / 0 the Hamiltonian is explicitly time-dependent and therefore the system of central spin plus nuclear spins does not in general conserve energy-energy may be exchanged with the external field. Moreover, our expression (5.102) is not valid for A = 0 because we are in the fast passage limit AQ <§C Au. So it appears that the exponential dependence on bias (5.106) arises solely because of the explicit addition of an energy constraint. Now in a molecular magnet, one can ask if this constraint is realistic. The answer here is that it is not. In these materials, there are dynamic external dipolar fields coming from all the other molecules in the crystal (in addition to any externally applied fields of the type that we are discussing). In terms of the structure of the transition probabilities we see that each individual molecular magnet will have an explicitly time-dependent field acting on it from a external source. This means that in calculating the one-molecule relaxation rates it is crucial to not put in any artificial energy constraints on the central spin plus nuclear spin system. Let us now summarize our point of view on the discrepancy between our result and the A = 0 result. The exponential dependence on bias found previously [74, 20] is a limiting case that is not relevant for a real molecular magnet, arising because of the addition of an explicit energy constraint. Our result, arising from an explicitly time-dependent Hamiltonian, is the description that is relevant for molecular magnets even in the absence of an externally applied time-varying field because of the presence of time-varying internal dipolar fields which can exchange energy with the one-molecule system. 5.5.2 T h e G e n e r a l C a s e ; I n c l u s i o n o f T o p o l o g i c a l D e c o h e r e n c e We now turn our attention to the more general case, including the contributions due to the topological decoherence terms AkND. The inclusion of these terms increases the technical difficulties involved in evaluating the trace over the spin bath for the following Chapter 5. The Landau-Zener Problem in the Presence of a Spin Bath 181 reason. T h e basis i n w h i c h the AkN D are c a l c u l a t e d is tha t of the sp in H a m i l t o n i a n for the p r o b l e m . T h a t is , the axes of q u a n t i z a t i o n o f the nuclear spins cannot be chosen a r b i t r a r i l y w i t h o u t p r o p e r l y r o t a t i n g the AkNj) i n to th i s new basis. In order to get a concrete feel for w h y th i s presents a p r o b l e m , let us consider the specific case o f the easy-axis easy-plane sp in H a m i l t o n i a n discussed at l eng th i n chapter 2. F o r molecules w i t h th is s y m m e t r y we showed tha t (2.73) nS 9nkh'n Lpe{pt} pe{pj.} ' |_pe{pi-} Tk N,D £ Mfk - E MJS-4 D — E 2E E M $ - £ M t (5.107) for l i g a n d nuclear spins k = 1..N and xk _ g7r4|c y — %\ ID- E 2E -x (5.108) for any 57Fe i n the m a t e r i a l . B y s p l i t t i n g the rea l a n d i m a g i n a r y par ts of these we can wr i t e ~(l)~ k . -~(2) ~k AN D — ak hl + iak h2 (5.109) where a B-eA k N D\ 42) = \ImAk\ (5.110) and ReA%,D -K ImA k N D n2 = • (D. l l l ) lk ak for a l l nuclear spins k = 1..N + 8. N o w as we d i d previous ly , we choose the axes of q u a n t i z a t i o n to be such tha t 7 ^ = z and 7 ^ = —cos2(lkz + s in2/?^x for a l l k. T h e n we see tha t since these new axes do not i n general cor respond to those of the sp in Chapter 5. The Landau-Zener Problem in the Presence of a Spin Bath 182 H a m i l t o n i a n , i t is necessary to rota te the t o p o l o g i c a l decoherence terms in to these new bases. These ro ta t ions y i e l d i n general new t o p o l o g i c a l decoherence terms r,N,D —  ak  n l ^  l a k  n2 (5.112) i n t e rms of w h i c h we m a y wr i t e n / ^ A ^ c ^ £ e ,, J—Tr/2w J—n/2u u L1 n rL, ( t i-«2) 2 _ ± CM, i{h$—m<&*) / i , m = ± l E n ih^\l{kX)\{ti-t2) e "f- < Mpk\emakL>n^Ik^~imakZ>A2-h^e-i(h-t2)1^ \j^pk > (5.113) N o t e tha t now tha t we have t opo log i ca l decoherence effects i t is necessary to inc lude a l l the nucle i (and therefore the p roduc t over species a) here because the s u m over m, h doesn ' t c o m m u t e w i t h the p r o d u c t over nuc le i . C h a n g i n g var iab les to A (ix +12) x = u(h -12) Y = 2ir (5.114) al lows us to wr i t e TTpff- = fA,2uJ d Y r dXeAXY+hx] y e i ( W - m * * a  U J-A/2. 7-00 , t ± l n C M, E II el-^r-h^x < Mpk\emakl)^'rk*~imak)h*'Tk»e~l1^ \Mpk > (5.115) where we have t aken the l i m i t s on the X in tegra l to in f in i ty as before. Chapter 5. The Landau-Zener Problem in the Presence of a Spin Bath 183 Evaluation of the Inner Product over Spin Bath States A s we saw p rev ious ly i n the case of pure o r t hogona l i t y b l o c k i n g , because the gauss ian t e r m ( in X) c o m i n g f rom the average over the nuc lear -nuc lear b ias field cuts off large X con t r i bu t ions to the express ion (5.115), we m a y e x p a n d the o r t hogona l i t y b l o c k i n g t e r m ins ide the sp in inner p r o d u c t to quadra t i c order i n X. A s w e l l , we sha l l res t r ic t our a t t en t ion to the case w h e n a l l the |o4 ' 1 ^ 1 , w h i c h is aga in the phys i ca l l i m i t for the mate r i a l s i n w h i c h we are interested. T h i s a l lows us to e x p a n d the t o p o l o g i c a l (1 2) decoherence te rms ins ide the inner p r o d u c t to q u a d r a t i c order i n ' as we l l . K e e p i n g (1 2) o n l y up to quadra t i c i n X, 04 ' or p roduc t s thereof a l lows us to wr i t e the e n v i r o n m e n t a l s p i n inner p roduc t i n the f o r m _(2) < M £ f c | e m a * 1 } " i ' ^ - i m a * 2 ^ | M f > - ( 2 ) X2 l-iX< M f | ^ • / f e j M f > - — < Mf\ ' - ( 2 ) UJ | M f > + < M f | (m£$tD • Ik + hAkr^D • Ik) | M f > + - < M f | ([%$tD •Ik}\[Akr^D • fk]2) | M f > +mh < M f I Ak/1* • f, •"•r.N.D 1k Akr:N,D • ik] \Mf > -iX < M f | m 4 ^ * • J, r,N,D  1k - (2 ) UJ + h - ( 2 ) UJ •™r,N,D ik M f > (5.116) W e sha l l present results for these te rms one by one a n d then combine the results after we are finished. W e have tha t < M f | ( m l ^ n • Ik + hAhry!D • h) | M f > =< M f | ( m + h) a ^ n ^ • 4 - » ( m - h)a{k2)h2k^ • / f c j M f > = (m + h) a{k^hlkiiZIkiiZ - i(m - hja^^Jk^ (5.117) Chapter 5. The Landau-Zener Problem in the Presence of a Spin Bath 1 8 4 T h e n e x t t e r m i s 2 < Mf,k 4f K • 4)2 - «lf fa*, • 4)21 wf > a ( 2 / , 2 - ^ \ k p Z kfj,Z -a {2Y (1 - n\kJ (5.118) N e x t w e h a v e < > --< M f a , ( 2 ) 2 •vp. (or2 - T2 ) . T (1) (2) / - ~ . . - \ • 4)2 + 4f fa*, • 4)2 + ictfta^ [nlkfl • Ik^n2k, • 4]] \Mf > (oT2 - T2 \ (5.119) A n d finally <Mf\ (m [Al^-h] v l 7£ )|cos2/3^ - ( 2 ) h LO LO l7g|sin2/?fc)i LU T2 1kuz ai^nik,z(m + h) - ia{k/jn2kfiZ(m - h) ( 2)„ hi2 -12 z) " 4 + h)hlkpiX - ia[kJ{m - h)n2kiiX) {ak»(m ~ h)hik»y - ictklim + h)h2kiiy) (5.120) Chapter 5. The Landau-Zener Problem in the Presence of a Spin Bath 185 W e now subs t i tu te these resul ts in to the expression for the t r a n s i t i o n p robab i l i t y , g i v i n g M = 2 A | r M A - d y roc i [ x y + i x ] i ( f t # _ m # . , 1 1 t ; V-yl/aj . / - c o _ f ^ , , n A J A/2U) J-oo «" X2o? m,h=±\ 1 e X p X 2 " ^^g(^+^)[P31^-i^p4iM]-l(m-/l)[p32M-iXp42M]+(l-t-m/l)/921M-(l-m/l)p22 /i PGM? (5.121) where we have defined P21/x P32/x P42fi and T „ / o r 2 _ r 2 \ , AT]1 (1) 2 / i 2 \ V Z V J ^ 2 ' , 1 T (!) (2)/- - \ (! -  nik,z) j - ^ - + 9 £ hazOi\la\l[n2kiiXnXKy - n 2 M n l f c ( i X ) JV," JV" E 4 f ( i - - U (24 2-4lJ . l JV," 4 + 2 £ ^ % a l ( " ° t | . « H v _ ^ 2 / c M 2 / n i f c M X ) = £ J ( i ) . E ^4?^* E — * i n N'. k u - l (2Il tk^zJ (2). E ^ ^ 2 ^ v - ; - , (1)- (5.122) P<*P = E A (5.123) In other words , i f there is a / J subscr ip t on one of these quant i t i es then i t refers to the species \i. F o r example , p 2 1 / t is the quan t i t y defined above for the specific subset of nuclear spins i n species ji. If th is subscr ip t is not there then i t refers to the s u m over a l l species-for example , p 2 i is the s u m over the con t r i bu t i ons f rom a l l the species P21 = E / i P2Ui- Chapter 5. The Landau-Zener Problem in the Presence of a Spin Bath 1 8 6 T h e L a r g e A L i m i t N o w there is a l i m i t here where th is s impl i f ies cons iderably , a n d tha t is when A is m u c h larger t h a n a l l the other energy scales i n th is express ion. In th i s case, the in tegra l over Y gives a de l t a func t ion i n X, for reasons i den t i c a l to those discussed i n the pure o r t hogona l i t y b l o c k i n g case. T h i s a l lows us to wr i t e 2 A 2 7 T m,h=±l 1 1 T ; LOA ^ 1 1 1_ ^ \e{rn+h)pzipt-i{m-h)pz2^+{l+mh)p2\^-{l-mh)p22A (5.124) T h e S u m O v e r S t a t e s i n t h e M * h P o l a r i z a t i o n G r o u p A l l four quant i t ies p2itl, P22p,, P3ip a n d P32p. depend e x p l i c i t l y on the state of the sp in b a t h v i a the i r dependence on the set {h^z}- A t this stage of the c a l c u l a t i o n we w i s h to pe r fo rm the s u m over the states i n the Mjj1 p o l a r i z a t i o n g roup . In order to do th is we define the quant i t ies V " (!)2 ft 2 ^ Z V 1kP.z) , I M j , v-- I (1) ( 2 ) / . „ P2\p = Z, akJ \ l - n i k , z ) T ^ + 2 i V f ^ \ a ^ a k J ( n 2 ^ x n i k , y - n2k,yn1^x)\ hp = 1 fcfi — 1 I^Ip - 1) ffi (1)2, 2 \ 1 Mp, I (1) ( 2 ) , „ . \ I = TT, Z, < U - n u „ J + ̂  Z, 1%<.K* nit,» - n 2 k i i y n l k i i X ) \ 1 1 fc„=i ^ 7 V s fc„=i \ ^ ( 2 ) 2 / . 2 ^Il~lt»z) , r I (1) (2)/- - - - M p 2 2 ^ = Z. (1 - n2kiiZ) j-^— T - ^ ^ \ o i \ y k l { n 2 k i i X n x k t i y - n2k^ynlkfiX)\ kfi=l k^ — l ^ ( 5 / ^ - 1) ffi (2)2 2 1 M ^ ( 1) (2) „ . . = T~2 Z, a l (X ~ n 2 ^ z ) + 2 iVf Z, K>1("2*^n l f c ) 1j, - n2fcM?/rilfcMX)| fcp. — 1 kfi — 1 _ • " ' • X / I \ ~ ~ * I (1) ~ I - _ I (2) ~ P3lp - T m L l a f c M n l ^ l > P32/X - -T7JI 2_, |ttfcMN2AM2 s fcB=l Y V s fc„=l Chapter 5. The Landau-Zener Problem in the Presence of a Spin Bath 187 w3lli = E l ( 1 ) - JV," P £ l42W. (5.125) F o r reasons i den t i ca l to those g iven i n the sec t ion on pure o r t h o g o n a l i t y b l o c k i n g we m a y now change the s u m over states to an in tegra t ion ; e x p l i c i t l y we take 1 ( m + / i ) p 3 i / i —i{m-h)p32^ +(l+mh)p2ill-(l-mh)p22n 1 2irW-3 1 p -(P31u.-P31ii) dp3i^e 2 W ^ 27rW32J-oo dP3211 ~ ( P 3 2 ^ ~ P 3 2 / i ) e ( p 2 1 M ~ p 2 1 M ) 2W"2 e 2 M ~ ( p 2 2 M ~ P 2 2 ^ ) 2 W 2 e 2 ^ exp [(m + / i ) p 3 i M - « ( m - /i)/o32/i + (1 + mh)p21fl - (1 - mh)p22)M} (5.126) N o t e tha t the reason tha t these in tegra t ions can be taken to be independent is because we are o n l y keeping to q u a d r a t i c order i n | a ^ | . These gauss ian in tegra t ions are easi ly per formed, g i v i n g 1 CM, E 3 ( m + / i ) p 3 i M - i ( m - / i ) p 3 2 M + ( l + m / i ) p 2 i M - ( l - m / i ) p 2 2 , , exp (1 + mh) [W32ltl + p 2 i J - (1 - mh) [W22/1 + p 2 2 J + (m + h)p3lll - i(m - h)p32lx (5.127) Inser t ing these results in to (5.124) gives us 2 A 2 ,7T IP1 M, \i _ 0 " g i ( / i * - m * * ) exp I (1 + m/ i ) \W'£X + P211 - (1 - mh) \ W%2 + p221 + ( m + / i ) A n - « ( m - / i ) p 3 2 M ~ " m , / i = ± l ^2,-1 /1 ITT/2 (5.128) Chapter 5. The Landau-Zener Problem in the Presence of a Spin Bath 188 Evaluation of the Ensemble Average Over Polarization Groups Our expression for Yl^P^ depends explicitly on the set {M^} via the quantities p 2 i , P22, P31 and p 3 2 . In order to perform the ensemble average over the set {M^} we perform the same calculation as was introduced in the section on pure orthogonality blocking; namely we take E n % - > n APG A* A» L= r dMtie~MH2N' Substitution of this into the expression for the relaxation rate (5.129) (5.130) APG Ii gives 9 A 2 r - l _ Z A A 0 e i ( / i * - m * * ) A m , / i = ± l exp [(1 + mh) [w3\ + p 2 i ] - (1 - mh) [\W22 + p 2 2] + (m + h)p31 - i(m - /i)p 3 2] (5.131) Performing the {M^} integrals gives r - i = 2 A | ^ e i ( h * - m * . ) e x p A m , / i = ± l [1 + m/i) [1 — mh) W,2 9 + 4 ( 5 ^ - 1 ) 12 12 exp 1 [(m + / i )W 3 i - i ( m - / i ) W 3 2 ] (5.132) We can now perform the {m, h} sums. The result can be written T ~ l = [eAl cosh 2cp! + e~x'2 cos 2<p0 (5.133) Chapter 5. The Landau-Zener Problem in the Presence of a Spin Bath 189 where we have defined 6 A l = < + W - 1 ) £ £ 4 f ( l - n?M) (5-134) A« fc„=i A 2 = 4iy322 + ^ ( 5 J g " 1 } £ £ 4 f (1 - (5.135) fcM=i D e f i n i n g new phases (/>o = ̂  c o s - 1 ( e ~ A 2 c o s 2 ( / ? 0 ) a n d (f>i = - c o s h - 1 ( e A l cosh 2ipx) we wr i t e $ = </>o + t(f>i (5.136) i n t e rms of w h i c h the expression (5.133) c an be w r i t t e n •, An | cOS$| 2 r" 1 = 0 1 A 1 (5.137) w h i c h is iden t i ca l to the resul t w i t h o u t t opo log i ca l decoherence except that the phase $ has been renormalized by the interaction with the nuclei. T h i s is exac t ly wha t we w o u l d expect to have h a p p e n here. N o t e tha t the i n c l u s i o n of t o p o l o g i c a l decoherence does not change the resul t tha t i n the large A l i m i t o r thogona l i t y b l o c k i n g effects d isappear . General Solution for Arbi trary A N o w let us a t tack the a r b i t r a r y A case, s t a r t i n g w i t h (5.121). T h e s t ra tegy tha t we w i l l use i n th is most general case is as fol lows. Chapter 5. The Landau-Zener Problem in the Presence of a Spin Bath 190 1. F i r s t , we pe r fo rm the s u m over states i n Mp, by conve r t i ng th i s s u m to a set of integrals over quant i t i es tha t are funct ions o f the set {p G Mp}; e x p l i c i t l y we take 1 V- „2 (P31 , -P31 , ) 2 -) rOO - ( P 3 2 , - P 3 2 , ) 2 2Vf2 e 32, dpzi„e  2 M W / d p 3 2 / , - o o \/ ZlTvVii,. J — C O (P21,-^21,) 1 ^ -^22,-P22,) 2W.2 1 f°° 2w* 1 Z" 0 0 \ roo ^ /-oo ^ roo 7=1 dp41fl5(p41lJ, - p41^)—= / dp42^(P42n ~ PA2U)-7== / dpi^ipi^ - plfl) Z7T - ' - o o V Z 7 T J - o o X/Z7T J - o o \ /27r ^ - o o " ' V 2 7 T ./-oo ' — " ™" • V27T 1 r 0 0 1 1 -==— / de^e 2 w 2 W (5.138) V^rW(M^) J-oo 1 ^ ; A s we have seen previous ly , t a k i n g a l l these in tegra t ions to be independen t is jus t i f i ed as l o n g as we o n l y keep to quad ra t i c order i n our s m a l l parameters . N o t e t ha t the w i d t h s of the three d i s t r i b u t i o n s { p i ^ } , { p A i p \ a n d { p A 2 i i \ are a l l quadra t i c i n s m a l l parameters and therefore have been d r o p p e d (because these appear i n qua r t i c order i n s m a l l parameters after the in tegra t ions) , effectively r ep l ac ing in tegra t ions over gaussians w i t h in tegra t ions over d e l t a funct ions . 2. N e x t we per fo rm the ensemble average over p o l a r i z a t i o n groups M^. E x p l i c i t l y we take APG V V• L L= r dNLe-MH2N° \irN? J-OO " 3. N e x t we integrate over the var iab le X, w h i c h represents the " b l i p l eng th" 4- W e then per fo rm the sums over {m, h}. 5. F i n a l l y , we change var iables f rom Y —> Z. T h i s gives us our final answer. L e t us pe r fo rm a l l five of these steps e x p l i c i t l y , s t a r t ing w i t h (5.121); 2A 2TT rAI* (5.139) T T A ^ = I dY T dXeiXY+^ V e*(**-™**) Chapter 5. The Landau-Zener Problem in the Presence of a Spin Bath 191 n 1 < x 2 " l ^2 elX" j r £ e ( m + f t ) [ P 3 i M - ^ / ' 4 i M ] - i ( m - / i ) [ p 3 2 M - i X p 4 2 M ] + ( l + m / i ) p 2 i M - ( l - m / i ) / 9 2 2 M (5.140) In order to comple te step 1. we mus t ca lcu la te the fo l lowing ; 1 S X2p\ l X u J 2 ^ g ( m + f t ) [ P 3 1 M - J ^ / 5 4 1 M ] - J ( m - / l ) [ p 3 2 M - i A ' p 4 2 M ] + ( l + m / l ) p 2 1 ^ - ( l - m ' l ) P 2 2 A l U M f PGM? — > • ( P 3 1 H ~ P 3 1 M ) e 3 1 f - ( P 3 2 M ~ / ' 3 2 M ) e 3 2 M - ^ 2 1 ^ - ^ 2 1 ^ ) 2f 2nW< 1 - ( P | 2 ^ - P 2 2 M ) \ roo 1̂  roo ]_ /•oo ?= / dp4iltS(p4ill - P 4 1 / i ) - ^ = / dp42^S(p42^ ~ p42n)-7= ^ P l / A P l/i - plp) Z7T J—oo V^TT •/-co v27T ./-co / oo -co ( tf-e,")2 v2„2 - g 2 V V 2 ( M j l ) g V2TTW(MA y - o o exp [ ( m + h) [ p 3 1 / i - i A T p 4 i ^ ] - z ( m - /i) [/9 3 2 A i - iXpi2fi] + (1 + mh)p2iIJL - (1 - mh)p22ll] (5.141) E v a l u a t i n g these integrals and inse r t ing the results in to (5.140) gives t4- / /i/zuj roo r * i -A/2u J-oo ... ,-, n exp X2 f^2 2 W ' M- ) + i — - e f ^ — iXp4ill(m + h) — (m — h)Xp, 2 / LO 2 '42/x exp [(1 + mh)(Wz\ + p2l) - (1 - m / i ) ( W | 2 + p 2 2 ) + ( m + h)p3l - i(m - h)p32 (5.142) W e now move on to step 2. T h e r e l a x a t i o n rate is w r i t t e n (5.28) T~L= V APG I* n [WM»TMI Chapter 5. The Landau-Zener Problem in the Presence of a Spin Bath 192 2 A 2 , rA^ / / dY dxAXY+^ y e<("*-™*-) A J-A/2u J-oo .. , , , m,/i=±l n exp 1 = r dM,e~MV2N° KM J-OO " X 2 ,,,2 + ^ LO — ^ ) + z — e f " - iXp41ll(m + h) - (m - h)Xp42» Z I LO exp [(1 + m / i ) ^ + p 2 x ) - (1 - m / i ) (Ty 3 2 2 + p 2 2 ) + (ra + /*)p 3 1 - i(m - / i ) p 3 2 ] ] (5.143) P e r f o r m i n g the in tegra t ions over { M ^ } gives exp _! _ 2 A 2 A 7-/l/2i j dY T dxAXY+»x\ y e^~m^exp J-Al2w J-oo ... L , , m,/i=±l (m + h) h z ( r a — h) LO LU (l + m h ) ^ - ( l - m h ) ^ (5.144) where W, \ \ a n d A 2 are as p rev ious ly a n d A - WO»W A - W o ^ F 4 ~ ~WW32 (5.145) (5.146) W e now per fo rm step 3., the in t eg ra t ion over the b l i p l eng th X. T h i s gives 2 A 2 7 r 1 / 2 u ; -A/2w . -1 exp LO 1 j y WA 2 rAI2w J-AI2w ... , e » ( h * - m * * ) e x p m,/i=±l 1 + mh) — — (1 — mh) — 2 2 H h (m + h) i(m — h) LU LO LO W e next pe r fo rm step 4-, the sums over { ra , / i } . T h i s gives ,2 2A 2 TT 1 / 2 O; M/2w T = r — / dY WA J-A/2u exp cosh (5.147) + exp - A 2 + A 2 AW' LO COS 2 * * + ^ ( y + £ ) W LO (5.148) Chapter 5. The Landau-Zener Problem in the Presence of a Spin Bath 193 F i n a l l y , s tep 5. is a change of var iables w h i c h a l lows us to wr i t e th is i n the f o r m A 2 r~l = o ° / dZe~z2 [ e A l _ A 2 - cosh [2(p, + 2\SZ] + e " A 2 + A 2 cos [2<p0 + 2A4Z]1 2y7rA Jz~ L J (5.150) where * - (^ ) <"«> D e f i n i n g new phases ^ ( Z ) = ^ c o s " 1 [ e " A 2 + A 2 cos [2<p0 + 2X4Z}] (5.152) a n d ^ ( Z ) = ]- c o s h - 1 [ e A l _ A 3 cosh [2<pi + 2 A 3 Z ] ] (5.153) we wr i t e $ ( Z ) = 0o(Z) + ^ i ( Z ) (5.154) i n te rms of w h i c h our f inal r e l axa t i on rate m a y be w r i t t e n T - i = A o ^ + / + r f Z e _ z 2 | c o s $ ( z T ) | 2 (5.155) y/nA Jz- Evaluation of the General Relaxation Rate in the Small A Limit W h e n A is large the general r e l a x a t i o n rate reduces to tha t for pure t o p o l o g i c a l decoher- ence (5.137). W h e n A becomes s m a l l , however, we expect to see some in te rp lay between Chapter 5. The Landau-Zener Problem in the Presence of a Spin Bath 194 o r thogona l i t y b l o c k i n g a n d t o p o l o g i c a l decoherence effects. In th is l i m i t we m a y rewr i te the r e l a x a t i o n rate i n the f o r m (5.156) A 2 exp _e_ 4W2 c o s $ ' ^ 2WJ W e see f rom (5.156) t ha t there is a seemingly unusua l dependence u p o n the ex te rna l D C bias field w h e n b o t h o r thogona l i t y b l o c k i n g a n d t opo log i ca l decoherence effects are present. However the reason for th is f o r m is evident . A s i n the case of pure o r t h o g o n a l i t y b l o c k i n g , we see tha t there exists a gauss ian profi le w i t h ex te rna l bias . A s w e l l , the t o p o l o g i c a l phase t e r m has acqui red a l o n g i t u d i n a l bias dependence. T h i s last occurs because the axes of q u a n t i z a t i o n of each of the k nuclear spins tha t we have chosen, tha t is zk — and xk = **k ^ ^ p ^ ' ^ ° n o t m g e n e r a l cor respond w i t h the axes of the cent ra l sp in H a m i l t o n i a n , w h i c h are also the axes w i t h w h i c h the t o p o l o g i c a l decoherence terms are descr ibed . W h e n we ca lcu la te the values for the t o p o l o g i c a l decoherence terms AkN D , these are by def in i t ion off-diagonal i n the axes of the sp in H a m i l t o n i a n . B u t when these are ro t a t ed in to the axes defined by the o r thogona l i t y b l o c k i n g parameters they acquire , i n general , l o n g i t u d i n a l components . It is these l o n g i t u d i n a l components , c o m i n g f rom this m i s m a t c h of preferred axes, tha t gives rise to the presence of a l o n g i t u d i n a l bias t e r m i n the t o p o l o g i c a l phase i n (5.155) a n d (5.156). W h e n o r thogona l i t y b l o c k i n g effects d isappear , such as we found happens i n the large A l i m i t , th i s r o t a t i o n of axes is not necessary a n d th is effect is not present (5.137). 5.6 T h e G e n e r a l S i n g l e M o l e c u l e R e l a x a t i o n R a t e i n Fe8 W e now w i s h to evaluate the quant i t ies en te r ing in to (5.155) for the specific case of the Fe$ mo lecu l a r magnet . W e w i l l beg in by eva lua t ing the con t r i bu t ions tha t arise f rom Chapter 5. The Landau-Zener Problem in the Presence of a Spin Bath 195 Q u a n t i t y D e f i n i t i o n LOQ W 2 ( 0 ) w2 1 v i Y + 8 - , l l 1 v 7 V + 8 (, .11 , , \ 2 E f = > * k = 1 \ k °> r 2 + p2 + w 2(o) T a b l e 5.1: Q u a n t i t i e s c o m i n g f rom o r t h o g o n a l i t y and degeneracy b l o c k i n g . o r t hogona l i t y and degeneracy b l o c k i n g effects, n a m e l y p2, T2, and W ( 0 ) . H o w these quant i t ies are defined, i n terms of the more fundamen ta l quant i t ies i n the theory, is repeated for convenience i n table 5.1. In t ab le 5.2 we present zero field values for T 0 and LO0 for the three species -Fe 8 *, Fe8D and 5 7 F e s - T h e quant i t ies p\ a n d W ( 0 ) w i l l be funct ions o f any ex te rna l bias field present, b o t h because of the field magn i tudes \jkJ and the i r chang ing d i rec t ions , i m b o d i e d i n the o r thogona l i t y b l o c k i n g parameters . S h o w n i n figures 5.6 t h r o u g h 5.11 are pi, W ( 0 ) and T0 as funct ions of an ex te rna l s ta t ic transverse magne t i c field or iented a l o n g the x d i r ec t i on (as defined by the cen t ra l sp in H a m i l t o n i a n ) i n the F e s molecu la r magnet , for var ious species. In figure 5.12 is shown the fu l l w i d t h W for the three variet ies Fe$*, Fe^o and 5 7 F e 8 - Chapter 5. The Landau-Zener Problem in the Presence of a Spin Bath 1 9 6 Fe8 Fe8D 57Fe8 r 0 10.64 3.22 12.96 UJ0 8.32 2.93 10.04 T a b l e 5.2: Zero ex te rna l field values for T 0 a n d u>0 for the three species shown. U n i t s are i n MHz. W e now consider the terms o w i n g the i r existence o n l y to t o p o l o g i c a l decoherence effects. These terms, un l ike the ones deal t w i t h i n the preceding, are not s t rong funct ions of an ex te rna l ly a p p l i e d transverse field ( a l though there is a s m a l l dependence, since the a p p l i c a t i o n of an ex te rna l field shifts the m i n i m a of the cent ra l sp in object , thereby chang ing the phase tha t i t accumula tes i n t u n n e l i n g f rom | y > to | i> or viceversa) . Therefore i t is enough to ca lcu la te the i r values i n zero transverse field. T h e def in i t ions of A i and A 2 are repeated i n table 5.3. C o n t r i b u t i o n s to these f rom different nuclear species are shown i n tab le 5.4. Va lues for A i a n d A 2 for the three variet ies F e 8 * , Fe8D and 57Fe8 are shown i n tab le 5.5. Q u a n t i t y D e f i n i t i o n A i A 2 ^ \ 4 \rN° \nWf, , l l 2 1 '*(5/*.-l) v-Atf ( l ) 2 n ~ 2 J ^ [ 4 \TN' l a ( 2 ) n t l l 2 1 W - 1 ) ? " ' o ( 2 ) 2 f l n 2 )} t—'l1 Nt [^k^l n2ku.z\\ "I" 6 Lku. = l a k u \ l n2kaz) T a b l e 5.3: Q u a n t i t i e s c o m i n g solely f rom t o p o l o g i c a l decoherence effects. Chapter 5. The Landau-Zener Problem in the Presence of a Spin Bath 197 400.0 300.0 I 200.0 100.0 0.0 0.0 0.1 0.2 External Field Hx[T] F i g u r e 5.6: pi^ (dashed) a n d W ( 0 M ) (solid) for p =l H. V = 79 Br i47v 1 3 ^ 170 57Fe A i ^ 1.45 • 1 ( T 6 0.06 • 1 0 " 6 0.03 • 1 0 " 6 0.66 • 1 0 ~ 6 0.11 • 1 0 " 6 0.18 • 1 0 ~ 6 12.8 • 1 0 " 6 0.13 • 1 0 " 6 0.13- 1 0 " 6 0.61 • l O " 6 0.31 • 1 0 " 6 0.60 • l O " 6 T a b l e 5.4: Zero field values of the t o p o l o g i c a l decoherence te rms for species i n Fe%. F i n a l l y , we consider te rms tha t arise f rom the in te rp lay of o r t h o g o n a l i t y b l o c k i n g , degeneracy b l o c k i n g and t opo log i ca l decoherence. These quant i t ies are l i s t ed i n table 5.6. These of course w i l l be funct ions of any ex te rna l transverse field because of the presence of the o r thogona l i t y b l o c k i n g te rms. Shown i n figure 5.13 are A 3 and A 4 as funct ions of Hx for F e 8 * , Fe8D and 57Fe8. T h e "bare" phase $ = op0 + i<pi is u n i q u e l y defined by the choice of a cen t ra l sp in Chapter 5. The Landau-Zener Problem in the Presence of a Spin Bath 1 9 8 F i g u r e 5.8: p1(l ( d a s h e d ) a n d W ( O ^ ) ( s o l i d ) f o r p. = 1 4 N. Chapter 5. The Landau-Zener Problem in the Presence of a Spin Bath 1 9 9 300.0 200.0 100.0 0.1 External Field Hx [ T ] F i g u r e 5.9: plfl ( d a s h e d ) a n d W ( O ^ ) ( s o l i d ) f o r p = 5 7 Fe. F i g u r e 5.10: plfl ( d a s h e d ) a n d W ( O ^ ) ( s o l i d ) f o r p =17 O. Chapter 5. The Landau-Zener Problem in the Presence of a Spin Bath 2 0 0 200.0 0.1 External Field Hx [ T ] 0.2 F i g u r e 5.11: plfl (dashed) a n d W ( 0 M ) (solid) for p. =13 C. 0.1 External Field Hx [ T ] 0.2 F i g u r e 5.12: F u l l w i d t h W for F e 8 * (dot ted) , Fe8D (dashed) and 5 7 F e 8 ( sol id) . Chapter 5. The Landau-Zener Problem in the Presence of a Spin Bath 201 Fe8* Fe8D 57Fe8 A i 1.54 • l O - 6 0.53 • 1 0 " 6 1 . 7 2 - 1 0 " 6 A 2 13.06- 1 0 " 6 4.16 • 1 0 " 6 13.66 • 1 0 " 6 T a b l e 5.5: T o p o l o g i c a l decoherence te rms for three variet ies of Fe8. Q u a n t i t y D e f i n i t i o n A 3 A 4 v V J V S " W 0 , i (1)- | v Wo, . (2)- , Z^ti Z^/cM = l y/~^w 1"*:, n^z\ T a b l e 5.6: Q u a n t i t i e s tha t come abou t due to in t e rp lay between o r t h o g o n a l i t y b l o c k i n g , degeneracy b l o c k i n g a n d topo log ica l decoherence effects. H a m i l t o n i a n . In our case we have chosen to use the sp in H a m i l t o n i a n H = -DS2Z + E{S% + S2_) + C(S4+ + St) (5.157) where D = 0.292A", E = 0.046AT and C = - 2 . 9 • 1 0 " 5 A " . T h e bare t u n n e l i n g s p l i t t i n g between near ly degenerate g round states | ± 10 > can be found v i a exact d i a g o n a l i z a t i o n (see chapter 2) a n d is A 0 « 3.9 • \Q~*K (5.158) T r u n c a t i n g the qua r t i c sp in t e r m gives us an "easy-axis easy-plane" s p i n H a m i l t o n i a n Chapter 5. The Landau-Zener Problem in the Presence of a Spin Bath 2 0 2 0.0005 0.0000 0.0 0.1 Hx[T] 0.001 0.000 0.2 0.0 0.2 F i g u r e 5.13: A 3 (left) a n d A 4 ( r ight) for the three variet ies Fes* (dot ted) , Fe8D (dashed) and 5 7 F e 8 ( so l id) . w h i c h gives bare phases cpo = TTS TrSgnBHy S2ivgiJ,BHx (5.159) ' ' ~ 2E L o o k i n g back at our general expression for the single molecu le r e l a x a t i o n rate (5.155), we see tha t we have now ca l cu la t ed a l l of the parameters i n th is express ion. F i x i n g the sp in H a m i l t o n i a n gives a single molecu le A C r e l a x a t i o n rate w i t h no free parameters . 5.6.1 E f f e c t o f t h e N u c l e a r S p i n E n v i r o n m e n t o n t h e L a r g e A S i n g l e M o l e c u l e R e l a x a t i o n R a t e i n Fe8 S h o w n i n figure (5.14) are p lo ts of two quant i t ies . T h e first is the q u a n t i t y A = V ' A T - 1 ca l cu la t ed w i t h o u t the a d d i t i o n of any nuclear spins; tha t is, A = An I cos $1 (5.160) T h e second is the quan t i t y A , w i t h the a d d i t i o n of nucle i to the effective H a m i l t o n i a n ; tha t is, A = A n |cos<l>| (5.161) Chapter 5. The Landau-Zener Problem in the Presence of a Spin Bath 203 W e see tha t the phase r e n o r m a l i z a t i o n $ —>• $ c o m i n g f r o m the nuc le i causes a t i n y s m o o t h i n g of the "nodes" seen i n A ca l cu l a t ed us ing the bare s p i n H a m i l t o n i a n . F u r - the rmore , because we k n o w how th i s r e n o r m a l i z a t i o n depends o n the nuclear spins we can p red ic t the shape of th is s m o o t h i n g effect. T h i s b e i n g sa id , i t is obvious i n th is p a r t i c u l a r m a t e r i a l tha t th i s effect is ex t r eme ly weak, because of the s m a l l values of the t o p o l o g i c a l decoherence parameters . O n e s h o u l d note tha t w h a t we are c a l c u l a t i n g here is the single molecule r e l axa t i on rate. I f we were to inc lude m a n y molecules , as is the case i n a rea l c r y s t a l , we w o u l d have to inc lude the effects of the transverse d i p o l a r fields i n $ (ie. Hx and Hy don ' t o n l y come f rom the ex te rna l field; there are also i n t e rna l d i p o l a r fields of th is k i n d ) . T h i s is the basic reason w h y the d a t a shown i n figures 1.20 and 1.21 is so " s m o o t h e d " - t h i s effect comes abou t because of non-zero Hy i n the phase $ c o m i n g f rom in t e rna l d i p o l a r fields (we sha l l t reat the m u l t i - m o l e c u l e case i n the next chapter ) . 5.7 S u m m a r y a n d D i s c u s s i o n o f R e s u l t s In th i s chapter we began w i t h the fu l l effective H a m i l t o n i a n (5.6) for a general mo lecu la r magne t i n an ex te rna l t i m e dependent field. W e der ived a f o r m a l express ion for the one- molecu le r e l axa t i on rate i n such a sys tem, and solved i t for the spec ia l case of a sawtoo th ex te rna l f ield i n the absence of q u a d r u p o l a r coupl ings to the nuc le i . T h e general one- molecu le r e l axa t i on rate i n this case was found to be where needed def ini t ions m a y be found i n (5.151) and the equat ions tha t d i r ec t l y fol low, (5.134), (5.135), (5.145) and (5.146). T h i s fo rm for the r e l axa t i on rate s impl i f ies i n the l i m i t t ha t the sweeping a m p l i t u d e is greater t h a n b o t h the energy w i d t h W and any ex te rna l bias £. In th is l i m i t the (5.162) Chapter 5. The Landau-Zener Problem in the Presence of a Spin Bath 204 -7.0 -11.0 Figure 5.14: Presented here are (1) the bare result A 0 | c o s $ | (the lower curve) and (2) the large A result with nuclear spins A 0 | c o s $ | (the middle curve) plotted in units of Kelvin. Note the logarithmic vertical scale. The horizontal axis is Hx in Tesla-here we have (p = 0(Hy = 0). relaxation rate can be written A 2 , | cos$ | 2 A (5.163) where the nuclear spins have caused a renormalization of the Berry phase $ —> $ in a way that we can calculate with no free parameters (once the positions of the nuclei are fixed). Note that in this case orthogonality blocking does not affect the relaxation rate at all. We saw in the case of the Fe$ system that this phase renormalization coming from the nuclear spins was slight. However this need not be the case in general. We may draw some far reaching conclusions from this result- basically that the presence of spin environments that couple to magnetic degrees of freedom "randomize" the central Berry phase term, destroying Aharonov-Bohm type oscillations. This is hardly surprising. This Chapter 5. The Landau-Zener Problem in the Presence of a Spin Bath 2 0 5 be ing sa id , we have shown q u a n t i t a t i v e l y here in ju s t how i t happens and how to ca lcu la te the effects of a sp in env i ronment on th is k i n d of t o p o l o g i c a l decoherence. W h e n the a m p l i t u d e of the sweeping field is m u c h sma l l e r t h a n the t o t a l energy spread + we find tha t T h e r e are two th ings to note here. O n e is tha t the f o r m for the one-molecule r e l a x a t i o n rate is gaussian, not exponen t i a l as is repor ted i n the l i t e ra tu re . W e have exp la ined the reasons b e h i n d th is d i sc repancy and the reasons w h y we bel ieve the gauss ian fo rm is more appropr i a t e to the phys ics of mo lecu la r magnets . T h e second t h i n g here is tha t there is an e x p l i c i t in te rp lay between t o p o l o g i c a l decoherence a n d o r t h o g o n a l i t y b l o c k i n g effects here w h i c h manifests i t se l f i n a £ dependent B e r r y phase. However , we have shown tha t for the specific case of Fe8, the osc i l l a t ions i n £ c o m i n g f r o m th is t e r m are far too s m a l l to be seen. A g a i n , we do not expect th i s to h o l d i n the general case- i t is qui te poss ible tha t i n systems tha t have large amounts of t o p o l o g i c a l decoherence c o m i n g f rom a sp in b a t h the osc i l l a t ions i n T _ 1 ( £ ) w i l l be apparent (these c o u l d be measured i n a s i m i l a r manne r to recent exper iments per formed on Fe8 w h i c h ex t rac t jus t th is quan t i t y ) . W e compared our general large A r e l axa t i on rate w i t h tha t of the s imple no-env i ronment sp in H a m i l t o n i a n (5.157). W e found tha t the nuclear spins s m o o t h out the cusps near the nodes i n the r e l a x a t i o n rate. T h e effect of the nuc le i cons idered is ra ther s m a l l i n Fe8; however, th i s need not be the case i n general . (5.164) Chapter 6 A C Relaxation in a Crystal of Molecular Magnets In th is chapter we use the results ob t a ined i n the prev ious chapter to invest igate w h a t f o r m the r e l axa t i on of the m a g n e t i z a t i o n i n a c r y s t a l o f m o l e c u l a r magnets takes. In essence wha t we sha l l do is insert the one-molecule r e l a x a t i o n rate ca lcu la t ed i n chapter 5 (5.155) in to a mas ter equa t ion [15, 167] and solve i t i n var ious l i m i t s . In p a r t i c u l a r we sha l l be concerned w i t h shor t - t ime r e l axa t i on i n the presence of an ex te rna l s a w t o o t h field of a r b i t r a r y a m p l i t u d e . 6.1 Preamble T h e results der ived i n the prev ious sect ion are s ingle-molecule r e l a x a t i o n rates. In o ther words , i n the i r de r iva t ion we have o m i t t e d comple t e ly f rom cons ide ra t ion a l l effects c o m - i n g f rom in t r a -molecu la r t e rms i n the fu l l c rys t a l H a m i l t o n i a n . T h e fu l l H a m i l t o n i a n of a c ry s t a l of mo lecu l a r magnets m a y be w r i t t e n i n the f o r m ff = £ t t i + £ V y (6.1) i i<j where Hi is the s ingle-molecule H a m i l t o n i a n for the ith molecu le ( i n c l u d i n g ex te rna l fields) and Vij gives the t o t a l i n t e r a c t i o n between the ith a n d j t h molecules . T h i s in t e rac t ion w i l l be d o m i n a t e d by magne t i c d i p o l a r in terac t ions between off-site magne t i c a toms. N o w when we a t t empt to ca lcu la te the r e l axa t i on charac ter i s t ics of the entire c ry s t a l , we see tha t i t is not enough to k n o w the s ingle-molecule r e l a x a t i o n ra tes- there are other i m p o r t a n t te rms i n the c ry s t a l H a m i l t o n i a n . 206 Chapter 6. AC Relaxation in a Crystal of Molecular Magnets 2 0 7 In order to get a feel for the phys ics go ing o n ins ide one of these c rys ta l s d u r i n g a r e l a x a t i o n exper iment , let 's a t t e m p t to q u a l i t a t i v e l y descr ibe th is r e l a x a t i o n first before dea l ing d i r e c t l y w i t h (6.1). T h i s q u a l i t a t i v e t rea tment w i l l fo l low the course of the s t a n d a r d exper iments of th i s t ype tha t are cu r ren t ly b e i n g pe r fo rmed on these systems (see chapter 1). T h e r e are two s t a n d a r d ways of i n i t i a l l y p r epa r ing these ma te r i a l s before the i r re lax- a t i o n character is t ics are measured . T h e first of these involves p l a c i n g the sample i n a large (5-8 T ) D C magne t i c field a l igned a long the z axis of the c r y s t a l at some h i g h t e m - pera ture , and then c o o l i n g the sample d o w n s lowly to the mK regime. T h i s technique has the effect of p r epa r ing the m a t e r i a l i n an i n i t i a l l y p o l a r i z e d s t a t e - a l l the magne t i c ions w i l l be such tha t each molecu le w i l l be i n the "up" state at the b e g i n n i n g of the ex- per iment . T h e second way to prepare the sample is to place the c ry s t a l i n a smal le r (0-5 T ) z ax is magne t i c field at a h i g h tempera ture , and then quench the sys tem's t empera - ture very q u i c k l y d o w n to the mK regime. T h i s has the effect of g i v i n g the c rys t a l some less t h a n fu l l i n i t i a l m a g n e t i z a t i o n (which is a func t ion of the s t rength of the o r i g i n a l D C field) wh i l e m a k i n g sure tha t there are no signif icant off-site cor re la t ions at the b e g i n n i n g of the exper iment (why th is cons ide ra t ion is i m p o r t a n t w i l l soon become apparent!) . L e t ' s beg in our qua l i t a t i ve ponder ings w i t h some genera l ly app l i cab le remarks . In ei ther of the above cases, each molecu le i n the sample w i l l feel some bias field c o m i n g f rom the s u m over the con t r i bu t ions of a l l the other molecules . These bias fields w i l l have some d i s t r i b u t i o n over the c r y s t a l , whose detai ls depend on the sample shape. N o w i f the l o n g i t u d i n a l (ie., a long the S-axis) bias on any p a r t i c u l a r molecu le is m u c h bigger t h a n the "resonance w i n d o w " g iven by our s ingle-molecule r e l a x a t i o n rate (5.155), th i s molecu le w i l l be frozen i n i ts o r i g i n a l state. It is comple t e ly unab le to relax! W e note i n pass ing tha t i f we remove the effects of nuclear spins f rom our s ingle-molecule r e l axa t i on rate, the resonance w i n d o w shr inks d o w n to encompass o n l y ex te rna l biases £ < AQ. Chapter 6. AC Relaxation in a Crystal of Molecular Magnets 208 Because the values of the bare t u n n e l i n g m a t r i x elements i n these ma te r i a l s are s m a l l ( t y p i c a l l y < 10~eK), any s ignif icant l o n g i t u d i n a l bias d i s t r i b u t i o n w i l l freeze the entire c r y s t a l . T y p i c a l l y the i n i t i a l b ias d i s t r i b u t i o n i n these ma te r i a l s varies over a range of ~ 0.1 — 1 K, orders of m a g n i t u d e larger t h a n A 0 ! Because s ignif icant shor t - t ime r e l a x a t i o n is seen e x p e r i m e n t a l l y i n b o t h F e 8 a n d Mn^, we infer tha t the nuclear spins are a c t u a l l y crucial to the phys ics here [20, 51, 49]. N o w let 's imag ine wha t w i l l h a p p e n as the c r y s t a l s tar ts to re lax . W e now have to specify the type of field a p p l i e d d u r i n g the r e l axa t i on process. T h e r e are two m a i n poss ib i l i t i e s for w h i c h expe r imen ta l results have been ob ta ined ; e i ther we a p p l y a D C field i n any d i r ec t ion , or we a p p l y a l o n g i t u d i n a l A C field. N o w regardless of the fo rm of the a p p l i e d f ie ld , the fo l lowing cons idera t ions apply . W e have i m a g i n e d t ha t there exists a d i s t r i b u t i o n of l o n g i t u d i n a l biases i n the c r y s t a l . N o w there w i l l be some f rac t ion of the c ry s t a l for w h o m the bias is sma l l e r t h a n the i r resonance w i n d o w (wh ich is a func t ion of the a p p l i e d field t h r o u g h (5.155)). T h i s f rac t ion is able to re lax . W h e n one of these molecules does re lax (by t u n n e l i n g to the "down" state) i t rearranges the distribution of bias fields over the crystal. T h i s is because ins tead of c o n t r i b u t i n g an "up" to the t o t a l i n t e rna l field i t now cont r ibu tes a " d o w n " . W e see therefore tha t the i n t e rna l field d i s t r i b u t i o n w i l l evolve i n t i m e i n a pecu l i a r w a y - a l l molecules tha t are ins ide the bias w i n d o w w i l l beg in to f l ip , and w i l l " d i g a hole" i n the bias d i s t r i b u t i o n near zero bias, send ing the i r weight out in to the d i s t r i b u t i o n ' s wings . T h i s w i l l s low d o w n the r e l axa t ion , as there are less and less molecules ava i lab le i n the resonance w i n d o w as t i m e progresses. A s we have seen i n chapter 1, th i s effect has recent ly been observed i n Fe8 [51]. N o w i n te rms of a quan t i t a t i ve theory, w h a t does th is mean? Chapter 6. AC Relaxation in a Crystal of Molecular Magnets 209 6.2 The Generalized Master Equation T h e r e is one charac te r i s t i c here tha t is apparent , a n d tha t is tha t as the c r y s t a l s tar ts to re lax , correlations between different sites b e g i n to develop. T h a t is , the r e l a x a t i o n o f the c ry s t a l is s t rong ly dependent on i t s h is tory . It has been suggested [15] tha t these cor re la t ions m a y be deal t w i t h s i m i l a r l y to the way tha t cor re la t ions i n s p i n glasses are t rea ted, n a m e l y by the def in i t ion of a series of many-mo lecu l e d i s t r i b u t i o n funct ions w h i c h are re la ted v i a an expression of the B B G K Y type . W e sha l l i n w h a t follows use th i s technique a n d see w h a t resul ts we c a n o b t a i n a n d w h i c h r e m a i n e lus ive . W e beg in by w r i t i n g the m a g n e t i z a t i o n of a c r y s t a l of mo lecu l a r magne ts i n the f o r m M(t) = £ j dHM{r, n,t) = yj a"H{2Pt{r, H, t) - 1) (6.2) r r where P f ( r , ri, t) is the n o r m a l i z e d p r o b a b i l i t y of the cen t ra l sp in c o m p l e x at site f to be "up" (ie. i n s tate \SZ = +S)) a n d i n a D C bias f ie ld ri a t t i m e t. T o o b t a i n a so lu t i on for P f (f, ri, t) we sha l l proceed i n the fo l l owing manner . W e noted i n the prev ious chapter tha t for frequencies lower t h a n the nuclear T2 energy scale > ~ T0 each pass of the A C field t h r o u g h resonance is decorre la ted f rom a l l o ther such passes. T h i s means tha t we m a y ca lcu la te the t r a n s i t i o n p r o b a b i l i t y for a s ingle pass and by s i m p l y d i v i d i n g th is by the p e r i o d of the ac f ie ld o b t a i n a s ingle-molecule r e l a x a t i o n ra te T~1(A,UJ;'H). K n o w l e d g e of th is rate a l lows us to wr i t e a k ine t i c or "master" equa t ion of the fo rm [15, 20] Pa(r,il,t) = -T-\A,u-%){Pa{r,n,t) - P_a(?,H,t)} - £ /dH'T-\A,ufH')[P™,(?tT 4& -J i ^ r ,a (6. where P^ is a 2-molecule d i s t r i b u t i o n func t ion , ie. the j o i n t p r o b a b i l i t y of two molecules Chapter 6. AC Relaxation in a Crystal of Molecular Magnets 2 1 0 h a v i n g the charac te r i s t ics i m p l i e d by the i r a rguments . Vi)(r — r) represents the D C bias at f c o m i n g f r o m the molecu le at site r , w h i c h as we have men t ioned is due to magne t i c d i p o l a r in te rac t ions . W e have not e x p l i c i t l y w r i t t e n out the h igher order m u l t i m o l e c u l a r te rms P^Z\P^A\ etc., represent ing these by an e l l ips i s (...). A s the c r y s t a l relaxes, the influence o f these h igher order te rms w i l l b eg in to be s ignif icant . However , i n the i n i t i a l stages of r e l a x a t i o n , we sha l l assume tha t these can be neglected. T h i s a p p r o x i m a t i o n w i l l o n l y h o l d for some l i m i t e d t i m e f rom the p r e p a r a t i o n of the i n i t i a l s tate a n d i t r emains to be shown tha t th is t i m e is l ong enough to be s ignif icant . 6.3 S h o r t T i m e D y n a m i c s In the ear ly stages of the r e l axa t i on of a c ry s t a l of mo lecu l a r magnets , there w i l l be two d o m i n a n t con t r i bu t i ons to the physics . T h e first of these is g iven by the first t e r m on the r i g h t - h a n d side of (6.3), and corresponds to " l o c a l " r e l axa t i on . If we were to neglect a l l t e rms bu t th i s first one we w o u l d find tha t the r e l a x a t i o n w o u l d be l inear i n t ime . A n o t h e r way of s ay ing th is is tha t i f a l l of the molecules re laxed independen t ly of each other, then the t o t a l r e l a x a t i o n of the c r y s t a l has to be exponen t i a l (and therefore l inear at short t imes ) . D e f i n i n g the quant i t ies M(f, H, t) = P t ( f , H, t) - P^f, %, t) , M(H, t) = £ M(f, H, t) , M(t) = J dHM(H, t) f al lows us to wr i t e P Q ( f , li, t) = -T~\A, to; U){Pa(r, U, t) - P.a(r, H, t)} (6.4) i n the fo rm M(H,t) = - r - 1 ( A , w ; H ) M ( ? ? , t ) (6.5) Chapter 6. AC Relaxation in a Crystal of Molecular Magnets 211 N o w i f th i s is the whole s to ry (ie. we neglect a l l correc t ions to th is) th i s is eas i ly solved, g i v i n g M{H,t) = M ( f t , 0 ) e - t / r ( A ' ^ ) (6.6) a n d therefore M(t) = J driM(n, o)e-2/T{A'UJ'il) (6.7) w h i c h has short t i m e behav iour M(t) ~ \ - t j dfi M(H, 0) T~1 (A, LO; it) (6.8) w h i c h , as we have c l a i m e d , is l inear i n t i m e . N o w we t u r n to the l ead ing s u b - d o m i n a n t t e r m i n (6.3). H o w do we incorpora te the l ead ing correc t ions to the l o c a l t e r m , a n d w h a t effect w i l l th i s have on the r e l axa t i on charac ter i s t ics? A s the c ry s t a l begins to re lax, the i n t e rna l bias d i s t r i b u t i o n (and therefore M{ri,t)) w i l l change i n t i m e i n a way tha t one can ca lcu la te a n a l y t i c a l l y i n an e l l i p so ida l c rys t a l (see A p p e n d i x A ) . T h i s r e d i s t r i b u t i o n of biases is exac t ly the effect tha t the l ead ing cor rec t ion t e r m i n (6.3) has on the r e l a x a t i o n . W e can therefore take in to account the l ead ing correc t ions by s i m p l y in se r t ing the f o r m der ived i n A p p e n d i x A for M(ri,t) in to the l o c a l t e r m (6.5). T h i s gives us the fo l l owing equa t ion for the r e l a x a t i o n rate, i n c o r p o r a t i n g b o t h the l oca l r e l a x a t i o n a n d the first correct ions to i t ; M{t) = - j dH r " 1 (A, LO] it) M(H, t) (6.9) where M(it, t) is the (evolving) d i s t r i b u t i o n of biases der ived i n A p p e n d i x A and r~l (A, LO; it) is the general s ingle-molecule r e l a x a t i o n rate der ived i n the previous chapter (5.155). Chapter 6. AC Relaxation in a Crystal of Molecular Magnets 212 6.3.1 Strongly Annealed Samples and the Large A Limit A s we have seen ear l ier when the ex te rna l sweeping a m p l i t u d e is large enough to sweep a l l molecules t h r o u g h resonance, ie. w h e n A > Hz + WD, the r e l a x a t i o n rate s impl i f ies cons iderably . W e have also seen tha t i n the F e 8 m a t e r i a l t o p o l o g i c a l decoherence effects are m i n i m a l . If we ignore these we f ind tha t the r e l a x a t i o n rate can be w r i t t e n i n the fo rm A20\cos^{Hx,Hy)\2 r-1(A,Hx,Hy) = A (6.10) L e t us now consider the expe r imen ta l l y relevant case of s t rong annea l ing , where the i n i t i a l m a g n e t i z a t i o n M(t0) <C M o , the sa tu ra ted m a g n e t i z a t i o n . In th i s s i t u a t i o n the i n i t i a l m a g n e t i z a t i o n d i s t r i b u t i o n over the a p p l i e d field is gauss ian (see [200] a n d refs. therein) 3/2 M(H,t0) = M(t0) exp £ (Hi - H?) (6.11) nW2(M)/ T h e h a l f - w i d t h WD(M) is d i r ec t ly measured i n exper iments [] and is m u c h larger t h a n E 0 for Fe$. N o t e tha t the observa t ion tha t Wr> is a func t ion of M and is d i r e c t l y ex t rac tab le f rom exper imen t is due to work by S t a m p and T u p i t s y n [198]. P l u g g i n g (6.10) and (6.11) in to (6.9) a l lows us to solve for M(t). W e f ind tha t M(t) = M(t0) exp [-TAC(H, M)t (6.12) where the r e l a x a t i o n rate is 2A§ T,c(H,M) = ^ w h { M ) A exp J dHx j dHy |cos $(HX, Hy Wl(M) {Hx - H°x)2 + (Hy - H\ 0\2 (6.13) N o w as we have stressed th roughou t th is documen t , our analys is depends on the choice of a p a r t i c u l a r sp in H a m i l t o n i a n (for Fe$ th i s was the easy-axis easy-plane H a m i l t o n i a n ) . Chapter 6. AC Relaxation in a Crystal of Molecular Magnets 213 T h e choice o f the parameters i n the sp in H a m i l t o n i a n s t rong ly affects the dependence of the t u n n e l i n g m a t r i x element o n ex te rna l t ransverse fields. T h i s is relevant here as the dependence on Hx a n d Hy i n the r e l a x a t i o n rate changes i f we use a different s p i n H a m i l t o n i a n as we saw i n chapter 2. In a d d i t i o n , i n d e r i v i n g our r e l a x a t i o n rate (6.9), we have assumed the i n s t an ton c a l c u l a t i o n w h i c h of course is not exact . In order to ext rac t the correct t u n n e l i n g a m p l i t u d e as a func t i on of ex te rna l f ield, we have to e x a c t l y d iagona l ize the sp in H a m i l t o n i a n plus the ex te rna l field. T h a t is , the correct express ion takes our A 0 | cos$(Hx, Hy)\ to \A(Hx,Hy)\ ex t r ac ted by exact d i a g o n a l i z a t i o n . T h e r e is a specific case where we can get a r o u n d the first diff icul ty, a n d t ha t is near the nodes of the t u n n e l i n g a m p l i t u d e - t h i s case has been t rea ted i ndependen t ly by S t a m p a n d T u p i t s y n [198, 200] and we w i l l r ev iew the i r results i n the fo l l owing sect ion. In th is case the t u n n e l i n g a m p l i t u d e w i l l be ex t r ac ted by exact d i a g o n a l i z a t i o n and therefore c o m p a r i n g our results to theirs a l lows us to gauge the accuracy of our i n s t an ton a p p r o x i m a t i o n . In the case where the sp in H a m i l t o n i a n is the easy-axis easy-plane m o d e l we have seen tha t , i n the l i m i t be ing considered, cos ${Hx,Hy)\ cos 2TTS 2irSgrtB(Hy + H°y) On + cosh S2KgnB(Hx + Hl) E (6.14) where we have sp l i t the transverse fields in to two con t r ibu t ions , one f rom ex te rna l ly app l i ed fields Hxy a n d one f rom in t e rna l t ransverse fields Hx>y. In tegra t ing over the in te rna l fields Hxy gives us our in t e rna l d i p o l a r field-averaged r e l a x a t i o n rate TAC(H,M) = ^\cos$(H°x,H°y) where c o s 2n2 S2g2n2Bwl(M) COS 2irS- 2irSg^BH0y O 0 + (6.15) Chapter 6. AC Relaxation in a Crystal of Molecular Magnets 214 s 4 7 r 2 g 2 M 2 w 2 (M) 2£2 cosh E (6.16) The relaxation rates (6.15) are shown in figure 6.1 for several values of WD(M). 6.3.2 General Solution Near the Nodes Stamp and Tupitsyn [200, 199] have demonstrated that near the nodes of \A(HX, Hy)\2 one can expand \A(HX,Hy)\ around its zeroes, as \A(HX,Hy)\ = A 0 c o s h Z , where for the Fe8 biaxial symmetry Z = 4X+ilby= [(Hy/Fy) + I {Hx ~ H^)/FX (6.17) (see [200] for complete details of this calculation). The "periods" Fx and Fy vary slowly with Hx and Hy, and so we expand as Fi{Ht) — F;(0) + Fi(0)"Hf/2 + .... We can thus solve for the A C relaxation rate around the nih "nodal field" H^; r^(Hl,M) = ^ Cn(M) + (rft + tfj (6.18) Cn(M) = *-Wl{M) \F-\HX^) + F - 2 (0) (6.19) This is a parabolic increase around minima with values exactly proportional to the inde- pendently measurable Wp(M). Choosing a specific form for the F e 8 spin Hamiltonian (such as the easy-axis easy- plane model we have been considering) allows us to use (6.13) to plot r^c ( -^ j ° \ M) over the whole range of Hj_ for different values of M (see figure 6.2). By varying M, the field angle 9 = sin _ 1 ( i7°/77°) and H0 , A C relaxation measurements ought to obtain information about the giant spin Hamiltonian inaccessible to ESR experiments [200]. Chapter 6. AC Relaxation in a Crystal of Molecular Magnets 215 F i g u r e 6.1: T o p figure: Logw A 2 vs. Hy, for 6 = sin"1 (Hll/Hy) = 0. Cu rves are, f rom b o t t o m to top, WD == 0 ,10 , 20, 30 and 50 mT. B o t t o m figure: Same, bu t w i t h 9 = 1°. N o t e tha t these results are ob t a ined f rom our r e l a x a t i o n rate w h i c h was der ived a s suming the in s t an ton a p p r o x i m a t i o n for the t u n n e l i n g a m p l i t u d e . ERROR^undefined 4 ^ Relaxation in a Crystal of Molecular Magnets OFFENDING COMMAND: STACK: C h a p t e r 7 S u m m a r y a n d O u t l o o k In t h i s thesis we have presented a quan t i t a t i ve theory of the phys ics o f m a g n e t i c r e l a x a t i o n i n the presence of t ime-dependent ex te rna l fields i n c rys ta l s o f m o l e c u l a r magnets . In so d o i n g we have achieved considerable ins igh t in to the effect o f sp in env i ronments on the q u a n t u m d y n a m i c s of mesoscale magne t i c moments . T h i s research p r o d u c e d several in te res t ing results , w h i c h we reproduce here. W e used the s t a n d a r d P r o k o f ' e v a n d S t a m p theory [15, 20, 122], together w i t h the c ry s t a l l og r a ph i c s t ruc ture of the Fe8 m o l e c u l a r magnet , to ex t rac t quan t i t a t i ve values for the t o p o l o g i c a l decoherence {ajj.1'2^} a n d o r thogona l i t y b l o c k i n g {j3k} parameters for th is m a t e r i a l . W e found tha t t opo log i ca l decoherence effects were m i n i m a l (lajj.1'2^ ~ 10~3) bu t tha t o r t hogona l i t y b l o c k i n g was s ignif icant . W e found tha t the p r i m a r y source of the o r t h o g o n a l i t y b l o c k i n g came f rom the large number of p ro tons i n the o rgan ic ma t t e r i n the Fe8 molecu le . In terms of the value K = — In [T[k cos (3).] m the P r o k o f ' e v a n d S t a m p theory, we f ind tha t K q u i c k l y rises w i t h transverse field, r each ing K ~ 1 for transverse field m a g n i t u d e of \H\ ~ 110 gauss i n Fe8* (see figures 3.14 and 3.15). A l l quant i t ies tha t are funct ions of the hyperfine coup l ings were found to e x h i b i t a s ignif icant isotope effect. In pa r t i cu l a r , A, K, and the nuclear sp in l i n e w i d t h W a l l depend on the i so top ic concen t ra t ion i n an Fe8 sample . F i t t i n g the shape of osc i l l a t ions i n t u n n e l i n g a m p l i t u d e to our de r ived r e l a x a t i o n rate, we found tha t i n t e rna l d i p o l a r transverse fields i n F e s a r e ° n the order of yjH2 + H2 ~ 0.03^ ~ 30 gauss. T h i s means tha t even i n the absence of any ex te rna l transverse fields 217 Chapter 7. Summary and Outlook 218 K ~ 0.09 i n Fes* , w h i c h is jus t large enough to cause s ignif icant decoherence [15, 20, 122], e ras ing any hope of seeing macroscop ic q u a n t u m coherence i n F e 8 * . W e de r ived an express ion for the s ing le -molecule r e l a x a t i o n rate i n any m o l e c u l a r magne t i c substance w i t h an effective H a m i l t o n i a n of the fo rm (5.6) b e i n g ac ted u p o n by a s a w t o o t h A C field, w h i c h is g iven by where 2W (7.2) w i t h £ the l o n g i t u d i n a l bias, W the t o t a l energy spread avai lab le to the molecu le , and A the a m p l i t u d e of the field. A 0 is the bare t u n n e l i n g m a t r i x element of the sp in H a m i l t o - n i a n pa r t of (5.6) and <&(Z) is the B e r r y phase of the cent ra l sp in complex , r eno rma l i zed by the i n t e r ac t i on w i t h the nuclear spins. T h i s express ion m a y be in te rpre ted i n the fo l lowing way. T h e effect of the nuclear spins is b o t h to b roaden the l i n e w i d t h W of the molecule and to r eno rma l i ze the B e r r y phase $ —>• $(Z). T h e first of these effects comes abou t f rom o r t h o g o n a l i t y b l o c k i n g and degeneracy b l o c k i n g . T h e second comes abou t p r i m a r i l y f rom t o p o l o g i c a l decoherence, a l t hough the Z dependence of $ is due to an in t e rp lay between t o p o l o g i c a l decoherence and o r t h o g o n a l i t y b l o c k i n g . In the l i m i t where A is m u c h bigger t h a n \W + £ | , the l i n e w i d t h W becomes i r re levant , a n d a l l o r t hogona l i t y b l o c k i n g effects d isappear , g i v i n g , A ^ | c o s $ | 2 , N r " 1 = 0 1 A 1 (7.3) In the oppos i te l i m i t A <C \W + £ | we f ind tha t - i A 2 2 AW2 cos$(c;/2W0| 2 (7.4) Chapter 7. Summary and Outlook 219 ie. the r e l a x a t i o n is gauss ian, w i t h an u n u s u a l b ias dependent B e r r y phase. W e showed tha t i n Fe8, because the t o p o l o g i c a l decoherence effects are so s m a l l , th i s pecu l i a r feature is overpowered by the gauss ian a n d i t is pe rmiss ib le to take <5>(£/2W) —>• $ . T h i s gauss ian dependence disagrees w i t h p r ev ious ly quo ted results w h i c h find the r e l a x a t i o n to be exponen t i a l i n bias [15, 20, 122]. W e bel ieve tha t the reason for th i s d i s p a r i t y is tha t i n the c a l c u l a t i o n l ead ing to the exponen t i a l b ias dependence the au thors added an energy cons t ra in t w h i c h is u n p h y s i c a l for these systems (see chapter 6 for a d i scuss ion of th is p o i n t ) . W e used our single molecu le r e l a x a t i o n rates as i npu t i n to a mas te r equa t ion i n order to take in to account the c rys t a l l i ne na ture of the real m a t e r i a l [15, 167]. W e s tud ied several specific cases a n d found the fo l lowing general observat ions . If the sweeping a m p l i t u d e o f the field is larger t h a n the t o t a l l o n g i t u d i n a l bias ava i lab le to the molecules A > \W + £ + ED\ the r e l a x a t i o n is l inear at short t imes . W h e n the sweeping a m p l i t u d e A <C \W + £ + ED\ the r e l a x a t i o n is i n i t i a l l y l inear , t u r n i n g over to square root at la ter t imes . T h i s occurs for b o t h L o r e n t z i a n a n d gauss ian i n i t i a l l o n g i t u d i n a l b ias profiles. W e used these general results i n the specific case of A C measurements per formed on Fe8 [39]. In th is case we are i n the large A l i m i t , g i v i n g l inear r e l a x a t i o n A?I ros <I>I2 M ( t ) ~ M ( 0 ) ( l - ^ & t ) (7.5) where <§ is the r eno rma l i zed B e r r y phase, averaged over the t ransverse in t e rna l d i p o l a r fields (7.11). W e then fit th is r e l a x a t i o n rate to the expe r imen ta l da t a . W e found the fo l l owing in te res t ing results. O u r de r ived expression for the t u n n e l i n g a m p l i t u d e produces results w h i c h quan t i t a - t i ve ly m a t c h those fund us ing exact d i a g o n a l i z a t i o n techniques for t ransverse fields less t h a n ~ 0.5 T . T h i s provides evidence tha t the ins tan ton c a l c u l a t i o n is v a l i d here. O u r express ion also q u a l i t a t i v e l y agrees w i t h expe r imen ta l results . Chapter 7. Summary and Outlook 220 In addition to these results, our research produced several new results related to the central problem investigated here. In chapter 2 we showed how the W K B , perturbation theory and instanton methods for calculating tunneling splittings in spin Hamiltonians for selected Bravais lattice symmetries compare to exact results obtained via diagonalization. We found that the standard W K B result's prefactor [25, 26] (corresponding to the first correction ~ h in the W K B expansion) disagrees with the exact result in each case studied. In chapter 3 we extended the work of Tupitsyn, Prokofiev and Stamp on the effective Hamiltonian of a central spin system [74, 75, 15, 20, 122] to allow for the calculation of the parameters {ak ' '} and {(5k) in real systems. As well, we included in our derivations the effect of electric quadrupolar terms coming from nuclei with spins greater than 1/2. In chapter 4 we proposed a classification scheme for all exactly solvable time depen- dent generalized Landau-Zener Hamiltonians. Our proposal is that if the equation for the wavefunction can be mapped to Riemann's equation then the problem is solvable. We presented two such mappings, along with exact results for the transition probabilities in each case. This concludes the presentation of our results. In conclusion, we would like to point out some avenues for future research related to this work. It will be quite straightforward at this point to perform investigations on molecular magnets other than the seminal Fe8, using the methods developed herein. This will give us a new quantitative tool with which to study decoherence and relaxation in these materials. In terms of the mathematical physics side of this work, there were many tantalizing relations between the theory of special functions and integral and differential equations glimpsed which may be of interest; we have noted these in the text whenever one occured. The problem of how to treat multiple crossings coherently in the presence of a spin bath Chapter 7. Summary and Outlook 221 (ie. w h e n our coarse g r a i n i n g a p p r o x i m a t i o n breaks down) , or the re la ted p r o b l e m of how to go b e y o n d the fast-passage a p p r o x i m a t i o n , r e m a i n works i n progress. T h e re so lu t ion of the con f l i c t i ng results ob t a ined w i t h the i n c l u s i o n of an osc i l l a to r b a t h , necessary i n the 0(DS2) effective desc r ip t ion , have to be addressed i n order to unde r s t and the t h e r m a l / q u a n t u m crossover. T h e i n c l u s i o n o f the q u a d r u p o l a r t e rms i n the t r a n s i t i o n p r o b a b i l i t y m a y p rov ide m a n y hours of gruesome enjoyment . Appendix A Bias Distribution in a Dilute Solution of Dipoles In th i s a p p e n d i x , we der ive a general express ion for the bias d i s t r i b u t i o n ins ide a c r y s t a l of mo lecu l a r magnets . T h e general a rguments tha t we w i l l use have been p u b l i s h e d p rev ious ly [123]. It is however wor thwh i l e to unde r s t and how the formulae quo ted i n the text arise, and tha t sha l l be the purpose of th is expos i t i on . W e beg in by present ing the fo l lowing ques t ion . G i v e n an ensemble of d ipo les i n a c rys t a l of a r b i t r a r y shape, w h a t w i l l be the p r o b a b i l i t y tha t the t o t a l d i p o l a r field at some site r has the value ril B a s i c a l l y we can jus t wr i t e d o w n the general s o l u t i o n to th is p r o b l e m ; i t is a s u m over a l l possible ways of o b t a i n i n g th is value w i t h the spins we've got, Ti Here each site fi ^ f i n the c ry s t a l can ei ther have a sp in p o i n t i n g "up" or a s p i n p o i n t i n g " d o w n " , ie. i n the ±z d i rec t ions . T h e d i p o l a r field is t aken to be H(?{ - f ) = p r ^ T a (4 - 3f ( 4 • r ) ) ( A . 2 ) where v is a un i t ce l l vo lume , ED is the d i p o l a r field scale, and d^ is the d i r ec t i on tha t the ith sp in is p o i n t i n g . W e now specia l ize our t rea tment to the case of an e l l i p so ida l sample . In th is case the f dependence drops out of P(f,rl), and we can wr i t e n 222 Appendix A. Bias Distribution in a Dilute Solution of Dipoles 2 2 3 W r i t i n g the de l t a func t ion out as an e x p o n e n t i a l gives pM = <k>f^ff*"%'-U**>f (A'4) T h i s can be r e w r i t t e n as W e note tha t the c o n t r i b u t i o n f rom H(fi) f l ips s ign depend ing on the s ign of . The re - fore we can w r i t e the p reced ing as p M = ^ / *w* n / f 9 n / g ^ « > * ( A.6) where we have s u b d i v i d e d the t o t a l number of spins TV i n to two subsets, those tha t are up ( { t } ) and those tha t are d o w n ( { ! } ) . W e now note tha t because we are o n l y a l l o w i n g the spins i n our so lu t i on to po in t i n the ±z d i rec t ions we have a s y m m e t r y here tha t we can exp lo i t ; the so lu t i on for P^H) has to be of the fo rm P(H) = 6{HX = 0)5(Hy = 0)P(Hz) (A.7) T h a t is, the f ina l so lu t ion i n th is idea l i zed case has the p r o b a b i l i t y for the transverse fields to be zero. W e therefore recast our express ion (A.6) in to an express ion for P(HZ), ie. the p r o b a b i l i t y of be ing i n a l o n g i t u d i n a l bias Hz; (.4.8) W e can pe r fo rm these integrat ions; / r w - ' " / ? ( ' , y ) \l Jo JO ' Appendix A. Bias Distribution in a Dilute Solution of Dipoles 224 Changing variables to x = 1 / r 3 gives 27r r ,„ . „ f°° dx 1 _ | L r d 0 s m e f ° ^ ( l - e ± i Y E ^ l - z ™ 2 e > ) (A.10) This can be broken up into two parts, like this 27T r ,n . „ r°° dx 1 30 y dBsindJ ^[{l-cosYEDv{l-3cos 29)x)±i[s\nYEDv{l-3cos29)x)) ( A . l l ) The leftmost integral is easy to do. We find that 27T r .„ . „ r°° dx Z T fn 0 0 cLx / \ — d9 sin 9 — (l-cosYEDv(l-3cos29)x) 30 Jo Jo XZ V ' r d9 sin 0 | 1 - 3 cos2 9\ HI Jo (A.12) ir2EDv\Y\ r 30 8TT2EDV\Y 9V3Q The second one is a little trickier. We write it as follows; —— / d9sm9 — sinYEDv(l - 3cos29)x 30 Vo 7o x 2 = ^ 7 ^ " I d6sind I — (sin f y / W l - 3cos 2 0)z) - YEDv(l - 3cos 2 6)x) oil Jo Jo xl v v ' ' (A.13) The reason we can do this is that the term we have added is equal to zero, as d9sm6(l-3cos2 9) = 0 (A.14) J 0 and the apparently divergent integral over x contains a cutoff which we've suppressed here (besides, the divergence of this term is only logarithmic). With this new expression we find, defining b = YEDv(l-3cos29) (A.15) Appendix A. Bias Distribution in a Dilute Solution of Dipoles 225 that rA dx [ (sin (bx) — bx) Jo xz = 6 ( C i ( A 6 ) - l n ( A ) + l - l n ( 6 ) - 7 ) (A.16) and the whole expression is ±iYEDv2n „ ̂  / d9 sin 9(1 - 3 cos2 9) oil Jo (Ci(AYEDv(l - 3 cos2 9)) - ln(A) + 1 - ln YEDv(l - 3 cos2 9) - 7 ) ±iYEDv2ir [V d9 sin 9(1 - 3 cos2 9) Jo 3Q (Ci(AYEDv(l -3cos29)) - ln (YEDv(l - 3 cos2 9))) (A.17) The leftmost term is ±iYEDv27r r* 3Q j* d9 sin 0(1 - 3 cos2 9) (Ci(AYEDv(l - 3 cos2 9)) I\Y\EDVTT28 / A The rightmost term is ±iYEDv27r r« 3ft Collecting our results, we find that dr 3Q lo d° S l n 9 i y l ~ 3 C°s2 ^ {1u(AYEdV(1 ~ 3 c o s 2 e)) ±iYEDv2ir (A 8 1 , T 1—. 4 1— \ V3 ~ 9 V ^ i ? e ( t a n h ~ ^ ) + 9V^TriJ (A.19) J n = 1 - pY [1 ± 1 ± i =F iK] ,{Y>0} = l + pY[l^l±i±iK] , {Y < 0} (A.20) Appendix A. Bias Distribution in a Dilute Solution of Dipoles 226 where we have defined G o i n g back to our o r i g i n a l express ion for P ( £ ) we see tha t 1 r°° dYeinzl m m P(HZ) = — / dYeiH*Y II [1 " PY H + iK]} JJ [1 - pY [2 + i - iK}] + -!- [° dYelH*Y Y[[i + p Y [ 2 - i - iK]} [1 + pY [+i + iK]] 2 7 r J ~ ° ° m m (A .22) T h i s can be w r i t t e n P(HZ) = — f°° dYeiY^Hz+{1~{N^~N^}p{K~l)IN)e~2^[N^~~Ni}YplN z 2ix Jo 1 roo _|_ _ / d y e - i y ( ^ + { l - ( i V t - i V 1 ) } p ( A - - l ) / i V ) e - 2 T ( l - { i V t - i V A } y p / J V 27T 7o (A .23) where iVf , iVj. are the numbers of u p / d o w n spins respect ively. A s w e l l , we have assumed tha t Q is large enough so tha t we can rewr i te the sums as exponent ia l s . P e r f o r m i n g the integrals gives TM 1 F ™ = l T ( g , + g)» + P ( A ' 2 4 ) where we have defined E(t) = p(K - 1)JV(1 - M) , T(t) = 2npN(l - M) (A .25) where M = ^Njt~N^ is the m a g n e t i z a t i o n of the sample . N o w i n order to wr i t e our fu l l P{ri), we have to inc lude the Hx and Hy dependence w h i c h we've seen is i n the idea l case very s imple . In a rea l m a t e r i a l , the de l t a funct ions i n th is dependence w i l l be spread somewhat . F o r th is reason we are go ing to replace the Appendix A. Bias Distribution in a Dilute Solution of Dipoles 227 delta functions with gaussians, whose width will be much smaller than the energy scale ED of the longitudinal distribution. Because we have a x -H- y symmetry here, we shall choose the transverse widths to be the same. This gives, for our final result, TM 1 T2  e XP Hi + Hi ™2t (A.26) 7T (Hz + E)2 + T 2 2irWjt where, as we have mentioned, W^t <C ED is the width of the transverse field distribu- tion. Appendix B T i m e Evolution of Nuclear Spin States In th is a p p e n d i x we sha l l show how the nuclear sp in states evolve over t i m e due to the i r c o u p l i n g w i t h the cen t ra l sp in c o m p l e x i n a m o l e c u l a r magnet . W e sha l l assume the same cond i t ions as were presented i n chapter 6, and fur thermore we sha l l assume tha t we are i n the large A l i m i t . W e m a y wr i t e d o w n a master equa t ion for the flow of the one-molecule sys tem i n the space \S,M > where \S > is the two-level state of the cen t ra l sp in and \M > is the state of the sp in env i ronment of the fo rm Ps,M = -PS,M £ TJ-is',M->M' + £ T s ' \ S , M ' ^ M P S ' , M ' ( B - 1 ) s' ,M' S' ,M' where P S , M is the n o r m a l i z e d p r o b a b i l i t y of be ing i n state \S,M > and T~^S, M M> is the t r ans i t i on rate f r o m state \S, M > to state \S', M' >. N o w as we e x p l i c i t l y showed i n chapter 6, these r e l a x a t i o n rates do not select for any p a r t i c u l a r state of the cent ra l sp in sys tem. T h a t is, the rate of flow | t>—> \ I > is i d e n t i c a l to the rate of flow | J,>—>• | y > i n the cent ra l sp in space. T h i s means tha t we can take Ts->s',M^>M' ~ TS'->S,M->M' — TM->M' (B-2) and wr i t e our mas te r equa t ion i n the form PM = -2PM £ r - ^ M , + 2 £ r-)^MPM, (B .3) M' M' where we have defined PM = Y,pSM (B .4) s 228 Appendix B. Time Evolution of Nuclear Spin States 229 W e can fur ther s imp l i fy th i s express ion by l o o k i n g c losely at the s t ruc ture of the t r a n s i t i o n p robab i l i t i e s i n chapter 6. W e see tha t these also do not select for specific sp in b a t h states. T h a t is , for a l l { M , M'}. T h i s means tha t the master equa t ion s impl i f ies even more . W e can wr i t e i t i n the fo rm PM = -2PME r M l M , + 2 £ r M l M , PM, (B .6) M' M' K n o w i n g the func t iona l f o r m of T~^^M, for a r b i t r a r y { M , M'} a l lows us to solve th is equa t ion n u m e r i c a l l y (it is s i m p l y a sys tem of M l inear equat ions) . If we s i m p l y assume tha t r~J-_^M, is not a s t rong func t i on of M' t hen our mas te r equa t ion becomes PM = -2PMrMl + 2 ^ (B .7) where TM1 = E M ' TM\M' a n c ^ ^ ^ s ^ n e t o t a l number of sp in b a t h states. T h i s is r ead i ly solved; i n the l i m i t t ha t C ^> 1 the so lu t ion is PM(t) - - P M ( 0 ) e- 2 TM* (B .8) together w i t h the cons t ra in t tha t E M Pu{t) = 1. W h a t th is means is tha t the different sp in b a t h states w i l l reach a s teady state so lu t ion where PM = PM> for a l l {M, M'} i n a t i m e on the order of TM w h i c h is s i m p l y the to t a l large A t r a n s i t i o n rate d i v i d e d by the average number of nuclear s p i n flips tha t occur per sweep. S h o w n i n figure B . l is a n u m e r i c a l s o l u t i o n of (B.3) showing the convergence of the spec t ra l weights of the var ious nuclear sp in states for a general sys tem w i t h , = Appendix B. Time Evolution of Nuclear Spin States 2 3 0 F i g u r e B . l : Here we see a sys tem w i t h seven e n v i r o n m e n t a l sp in states i n i t i a l l y prepared i n one o f t h e m e v o l v i n g v i a ( B . 3 ) . Bibliography [1] Colin P. Williams and Scott H. Clearwater, Explorations in Quantum Computing, Springer-Verlag (1998). [2] R. W. 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