UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

AC relaxation in the Fe8 molecular magnet Rose, Geordie 2000

You don't seem to have a PDF reader installed, try download the pdf

Item Metadata

Download

Media
ubc_2000-487032.pdf [ 9.32MB ]
Metadata
JSON: 1.0085452.json
JSON-LD: 1.0085452+ld.json
RDF/XML (Pretty): 1.0085452.xml
RDF/JSON: 1.0085452+rdf.json
Turtle: 1.0085452+rdf-turtle.txt
N-Triples: 1.0085452+rdf-ntriples.txt
Original Record: 1.0085452 +original-record.json
Full Text
1.0085452.txt
Citation
1.0085452.ris

Full Text

A C R E L A X A T I O N IN T H E Fe  8  MOLECULAR MAGNET  By Geordie Rose B. Eng. (Engineering Physics), McMaster University, 1994  A THESIS 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 THE REQUIREMENTS FOR T H E DEGREE OF DOCTOR OF PHILOSOPHY  in T H E F A C U L T Y O F G R A D U A T E STUDIES DEPARTMENT OF PHYSICS A N D ASTRONOMY  We accept this thesis as conforming to the required standard  THE UNIVERSITY  OF BRITISH COLUMBIA  January 2000 © Geordie Rose, 2000  In presenting this thesis i n partial fulfilment of the requirements for an a d v a n c e d degree at the University of B r i t i s h C o l u m b i a , I agree that the L i b r a r y shall m a k e it freely available for reference a n d study.  I further agree that permission for extensive copying of this  thesis for scholarly purposes may or her representatively financial  be granted by the h e a d of my d e p a r t m e n t or by his  It isfc u n d e r s t o o d  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  D e p a r t m e n t of Physics and A s t r o n o m y The University of British C o l u m b i a 6224 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., V6T  Date:  that copying or p u b l i c a t i o n of this thesis for  1Z1  Canada  permission.  Abstract  We investigate the low energy magnetic relaxation characteristics of the "iron eight" (Fe ) 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 8  Fe  3+  ions surrounded by organic ligands. T h e molecules arrange themselves into a regular lattice w i t h triclinic symmetry. A t sufficiently low energies, the electronic spins of the Fe  3+  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.  We derive a low energy effective H a m i l t o n i a n for this system, valid for t e m p e r a t u r e s less t h a n T ~ 3 6 0 mK, w h e r e T 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 c  c  8  system crosses  over into a " q u a n t u m regime" where relaxation characteristics b e c o m e temperature i n d e p e n d e n t . W e s h o w t h a t i n this regime the 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 is t o the environmental s p i n b a t h i n the molecule. We show how to explicitly calculate these coup 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-  We  use this i n f o r m a t i o n to calculate the linewidth, topological 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 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 effect. 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 is significant a n d suppresses coherent 8  tunneling. We t h e n use our low energy effective H a m i l t o n i a n t o c a l c u l a t e the single-molecule relaxation rate i n the presence of an external magnetic field w i t h b o t h A C a n d  DC  components by solving the Landau-Zener problem i n the presence of a nuclear spin bath. 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 rate is t h e n u s e d as i n p u t into a m a s t e r e q u a t i o n i n o r d e r t o take i n t o a c c o u n t t h e m a n y molecule nature of the full system. O u r results are then c o m p a r e d to q u a n t u m regime relaxation experiments performed on the F e g system.  ii  Table of Contents  Abstract  ii  Table of Contents  iii  List of Tables  viii  List of Figures  ix  Acknowledgements 1  xix  Introduction and Overview  1.1  1.2  1.3  1  A n I n t r o d u c t i o n t o Fe  4  1.1.1  Giant Spins and Q u a n t u m Environments  8  1.1.2  A R e l a t e d S y s t e m is C h a r a c t e r i z e d  9  &  .1.1.3  Magnetic Characterization of F e  1.1.4  Results in the Q u a n t u m Regime  13  8  14  A n Introduction to Relaxation Experiments  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 Fe  19  1.2.2  D C Relaxation of Annealed Crystals  19  1.2.3  Hole Digging and the Time-Dependent Internal Longitudinal Bias  8  Distribution  21  1.2.4  A C Relaxation of Annealed Crystals  23  1.2.5  Extraction of Tunneling Matrix Elements  23  Thesis Overview  27 iii  2  Effective Hamiltonians  29  2.1  T h e Fe  Free Ion H a m i l t o n i a n  30  2.2  T h e Effect of the Crystalline Environment  32  2.3  T h e Single Ion Effective H a m i l t o n i a n  34  2.3.1  First Order Perturbation Theory  35  2.3.2  Second Order Perturbation Theory  36  2.3.3  Higher Orders Perturbation Theory  37  T h e Single Molecule Effective Hamiltonian  39  2.4  3+  2.4.1  Inclusion of E x c h a n g e and Superexchange  Terms  2.4.2  "Offsite" D i p o l a r and Quadrupolar Contributions  40  2.4.3  Intra-Nuclear Spin Couplings  45  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 ....  45  2.4.5  Coupling to Phonons  45  2.4.6  Coupling to Photons  47  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  Exchange/Superexchange  2.6  Investigation of the G i a n t Spin H a m i l t o n i a n i n the Absence of E n v i r o n -  and the Giant Spin Picture  mental Couplings  2.7 2.8  40  49 51  2.6.1  E x a c t Solution for Tunneling M a t r i x Elements v i a Diagonalization 5 2  2.6.2  Tunneling Matrix Elements via Perturbation Theory  59  2.6.3  Tunneling Matrix Elements via W K B  60  2.6.4  Tunneling Matrix Elements via Instanton Techniques  60  2.6.5  Comparison of Approximate M e t h o d s to E x a c t Solutions  60  Methods  B a c k to the Full Hamiltonian-Separation of Tunneling Energy Scale Using an Instanton Technique  62  Off-Diagonal T e r m s and the Instanton M e t h o d  68  iv  2.9 3  2.8.1  R e v i e w of the M e t h o d of T u p i t s y n et.al  69  2.8.2  The Tunneling Lagrangian  70  2.8.3  A n A s s u m p t i o n is M a d e  72  2.8.4  Solution for the Free Instanton T r a j e c t o r y  73  2.8.5  Inclusion of the E x t e r n a l Magnetic Field and the Nuclear Spins  .  75  T h e Final Single Molecule Effective H a m i l t o n i a n  77  Nuclear Spin Couplings in Fe$ and the Isotope Effect  78  3.1  Units and Constants  79  3.2  The Point Dipole Approximation  79  3.2.1  Magnetic Field at f due to a "Point Dipole" at 0  80  3.2.2  Magnetic Field at f due to E i g h t "Point Dipoles" at f  3.2.3  Isotopic Concentrations, Nuclear ^-factors and Quadrupolar  v  Fei  8  ....  80 Mo-  m e n t s i n Fe  80  8  3.3  3.2.4  Definition and E v a l u a t i o n of 7^,  7^,  3.2.5  Contact Hyperfine C o u p l i n g Energies for F e  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  Calculation of the Orthogonality B l o c k i n g Parameter K  85  3.2.8  Calculation of E  93  3.2.9  Calculation of Topological Decoherence Parameters A  u^. a n d ufc 5 7  81 82  3 +  0  U s i n g F r e e Fe  k  3+  ND  and A  .  Hartree-Fock Wavefunctions to M o d e l A c t u a l Spin Dis-  tributions 3.4  98  Tables of Nuclear Positions, Fields at N u c l e i a n d Hyperfine C o u p l i n g Energies  4  112  A n Introduction to the Generalized Landau-Zener Problem  4.1  97  Introduction to and Exact Solution of the Landau-Zener P r o b l e m v  122  .... 122  4.1.1  Alternate M e t h o d of Solution for the 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  4.1.2  126  A l t e r n a t e M e t h o d of Solution for the T r a n s i t i o n P r o b a b i l i t y II. D y c h n e ' s F o r m u l a  128  4.1.3  Analysis of the Transition F o r m u l a  130  4.1.4  Generalization of the Two-Level 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)  4.1.5  131  Generalization of the 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 II. 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  5  137  The Landau-Zener Problem in the Presence of a Spin Bath  5.1  T h e A d d i t i o n of an E n v i r o n m e n t to the L a n d a u - Z e n e r P r o b l e m : General Considerations  5.2 5.3  142  142  The Q u a n t u m Regime Effective Hamiltonian:  Inclusion of a Spin Envi-  ronment  143  General A C Field Solution in Fast Passage  147  5.3.1  A List of Approximations  Invoked in the Calculations T h a t Follow  5.3.2  General Strategy for C a l c u l a t i n g Relaxation Rates  151  5.3.3  Processing of the Transition Amplitude  155  5.3.4  Processing of the Transition 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  5.3.5  149  159  Processing of the Transition Probability (ii) 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  161  5.4  Solution Without Spin Bath  163  5.5  S o l u t i o n F o r a S p i n B a t h with no Q u a d r u p o l a r C o n t r i b u t i o n  169  5.5.1  169  Pure Orthogonality Blocking vi  5.5.2 5.6  T h e G e n e r a l Case; Inclusion of T o p o l o g i c a l D e c o h e r e n c e  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$ 5.6.1  Effect of the Nuclear Spin Environment 8  6  7  194  on the Large A Single  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 5.7  180  S u m m a r y and Discussion of Results  A C Relaxation in a C r y s t a l of M o l e c u l a r Magnets  202 203 206  6.1  Preamble  206  6.2  The Generalized Master Equation  209  6.3  Short T i m e Dynamics  210  6.3.1  Strongly Annealed Samples and the Large A L i m i t  212  6.3.2  General Solution Near the Nodes  214  S u m m a r y and Outlook  217  Appendices  221  A  Bias D i s t r i b u t i o n in a D i l u t e Solution of Dipoles  222  B  T i m e E v o l u t i o n of Nuclear Spin States  228  Bibliography  231  vii  List of  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-  81  3.2  Positions o f the i r o n ions, units i n A n g s t o m s  86  3.3  D a t a for Hydrogen  113  3.4  D a t a for Hydrogen  114  3.5  D a t a for Hydrogen  115  3.6  D a t a for Hydrogen  116  3.7  D a t a for Bromine  117  3.8  D a t a for Nitrogen  118  3.9  D a t a for C a r b o n  119  3.10  D a t a for Iron  120  3.11  D a t a for O x y g e n  121  5.1  Quantities coming from orthogonality and degeneracy blocking  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 a n d u 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  F r o m [48]  0  0  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 solely f r o m t o p o l o g i c a l decoherence effects  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$.  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  5.6  Quantities that come about due to interplay between orthogonality blocking, degeneracy b l o c k i n g a n d t o p o l o g i c a l decoherence effects  viii  .  197 201 201  List o f Figures  1.1  T h e n u m b e r o f atoms needed t o represent one bit o f information as a f u n c t i o n o f c a l e n d a r year. E x t r a p o l a t i o n o f t h e t r e n d suggests 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 is 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 ) vs. c a l e n d a r y e a r . A d a p t e d f r o m [1] 1.3  2  Energy (pico-Joules) dissipated p e r logical operation as a function o f cale n d a r y e a r . T h e 1 kT l e v e l is 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  from  [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 are i n Angstroms. Legend: R e d , iron; Purple, bromine; L i g h t Blue, oxygen; Green, nitrogen; Yellow, carbon; a n d D a r k B l u e Crosses, hydrogen. 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 species  5  1.5 A v i e w o f t h e F e u n i t c e l l i n t h e x — z p l a n e  6  8  1.6 A v i e w o f t h e Fe  &  u n i t cell i n t h e x — y plane. Here we a r e l o o k i n g right  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) 1.7 P r o j e c t i o n o f t h e M n i  u n i t cell onto the x  2  7  y plane. Here we are l o o k i n g  —  d o w n t h e easy axis o f t h e crystal. T h e axes scales a r e i n Angstroms. Legend:  R e d , manganese; Purple, oxygen; Yellow, carbon; D a r k B l u e  Crosses, hydrogen. N o t e t h e inner a n d outer "rings" o f m a n g a n e s e ions. . 1.8 P r o j e c t i o n o f t h e M n 1.9  i  2  u n i t cell onto t h e x — z plane  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  ix  —  z plane  10 11 12  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]  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 Fe . 8  temperatures t h e r m a l activation is observed, while for T < ~  15 A t high 360  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 of the 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 first 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 for T < 3 6 0 mK.  S h o w n i n the inset are relaxation characteristics i n the  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]  20  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 , m e a s u r e d a t 150mK. H e r e 8  —*  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 as 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 we 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 we a g a i n see 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 -  istic. H o w e v e r i n this case the r e l a x a t i o n rates are t e m p e r a t u r e dependent. 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 Fe  8  sample t h a t was annealed  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 exposed to longitudinal D C  fields  of various magnitudes.  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 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 case. F r o m [49]  23  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 T (H ). sqrt  The  z  i n i t i a l d i s t r i b u t i o n is l a b e l l e d w i t h M = — 0 . 9 9 8 Ms w h e r e a s t h e o t h e r s in  are distributions obtained by thermal annealing. T h e latter are distorted 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 effects. 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 40  mK.  T h e sample was t h e n allowed to r e l a x for times t . A f t e r t h i s t i m e h a d 0  e l a p s e d , a D C field H was a p p l i e d , a n d r z  sgrt  was m e a s u r e d . N o t e the  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 H — 0. F i g u r e f r o m [49]  24  z  1.18 Q u a n t u m h o l e d i g g i n g , as 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 M = —0.2 M . 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 in  s  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]. 25 1.19 H e r e is 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  (Ti ) nit  a n d a t t — 16s ( 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 0  t h a t f o r \Mi \ 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;„|, 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]  25  1.20 T h e q u a n t i t y A h e r e is 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 is s h o w n as a f u n c t i o n o f t h e m a g n i t u d e t h e t r a n s v e r s e D C field \H\ = field f = t a n (H /H ). - 1  y  x  of  + 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  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 (H = 0). F i g u r e f r o m [51] z  xi  26  1.21 T h e q u a n t i t y A s h o w n f o r ip = 0, as a f u n c t i o n o f \H\. 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 H.  T h e lowest curve was  z  for H  z  H  z  = 0; t h e m i d d l e c u r v e f o r H  S h o w n here are obtained  = 0.22T, a n d t h e u p p e r c u r v e f o r  z  = 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 Fe  8  molecule'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 resonance situations between | — 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 is o b s e r v e d . F i g u r e f r o m [51] 2.1  27  E x c h a n g e p a t h w a y s i n Fe  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 et.al. [18].  8  Fits to susceptibility d a t a give J J 3 5 ~ 52K, 2.2  ~ 35K,  i 2  J  i 3  ~ 180K,  J15 ~ 22K  and  with all couplings antiferromagnetic  41  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 , (iii) 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 (vii) T r i g o n a l  2.3  V a r i a t i o n o f As-s  54  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 left, \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 is p l o t t e d a±S /D 2  a n d o n t h e y a x i s l o g As-s1 0  H e r e we 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 2.4  V a r i a t i o nof A  s  55  _ s w i t h H /D  f o r a S /D  = 0.25 f o r f o u r d i f f e r e n t |5| v a l -  2  x  A  ues ( c l o c k w i s e f r o m t o p left, |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 is p l o t t e d H /DS  2  X  2.5  V a r i a t i o n o f As-s  w i t h a /D 2  and on the y axis l o g A ^ - s 1 0  55  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 left, \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 is p l o t t e d a /D 2  a n d o n t h e y a x i s l o g A s _ s . H e r e we h a v e t a k e n t h e 1 0  e x t e r n a l field t o b e z e r o  57 xii  2.6  V a r i a t i o n o f As-s w i t h Hx/D f o r a /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 2  ( c l o c k w i s e f r o m t o p left, |5| =2, 5, 10, a n d 15); o r t h o r h o m b i c O n t h e x a x i s i s p l o t t e d H /DS  2  a n d o n t h e y a x i s l o g As,-s  57  1 0  X  2.7  symmetry.  V a r i a t i o n o f A -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 S  e  t o p left, |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 is p l o t t e d a S /D  a n d o n t h e y a x i s l o g As-s-  4  H e r e we h a v e t a k e n t h e  1 0  e  e x t e r n a l field t o b e z e r o 2.8  V a r i a t i o n o f A ,-s s  58  w i t h H /D f o r a S /D  = 0.25 f o r f o u r d i f f e r e n t \S\  4  x  6  v a l u e s ( c l o c k w i s e f r o m t o p left, \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 H /DS  2  X  2.9  a n d o n t h e y a x i s l o g As-s  58  1 0  C o m p a r i s o n of p e r t u r b a t i o n theory, W K B  results and instanton results to  the exact s o l u t i o n for the t u n n e l i n g splitting between the two lowest levels of the H a m i l t o n i a n of orthorhombic  s y m m e t r y w i t h S=10.  Plotted on  t h e h o r i z o n t a l a x i s is a /D, a n d o n t h e v e r t i c a l a x i s l o g As,-s- L e g e n d : 1 0  2  Black, exact solution; Green, instanton solution; Red, perturbation theory and Blue, W K B  61  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  results to the exact solution  for 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 n i a n of tetragonal s y m m e t r y w i t h S=10.  Hamilto-  P l o t t e d on the horizontal axis  is 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 A s _ s . 1 0  Legend: Yellow, exact  solution; Red, perturbation theory and Green, W K B 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 H  GS  61  f o r t h e Fe s y s t e m . 8  The  region of validity of the m a p p i n g to a two-state system is the region w h e r e excited states are forbidden (this region i n s h a d e d grey i n the above).  xiii  . . 64  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 the easy-  a x i s . W e see 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 in f), a n d one where S tunnels from | + S  | — S > (off-diagonal i n f).  N o t e t h e s e p a r a t i o n o f scales; 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 greater than the tunneling time 3.1  (Jl f o r a l l n u c l e i i n Fesrepresent values for H 2  68  L a b e l i n g is as i n d i c a t e d i n t h e text. T h e d o t s ( l a b e l s 1..120), Br  ( l a b e l s 121..128), a n d  sl  N  l5  ( l a b e l s 129..146)  86  H, 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  87  3.2  1  3.3  1  H,  high end of the spectrum  87  3.4  2  H,  low end of s p e c t r u m  88  3.5  2  H,  entire s p e c t r u m  88  3.6  79  3.7  81  3.8  1 4  3.9  15  .Br, entire s p e c t r u m  89  Br,  89  entire s p e c t r u m  i V , entire s p e c t r u m  N,  90  entire s p e c t r u m  90  3.10 Fe, e n t i r e s p e c t r u m  91  3.11  91  57  1 3  C , entire s p e c t r u m  3.12 O, 17  3.13  1 7  low end of s p e c t r u m  92  0 , entire s p e c t r u m  92  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™ m a t e r i a l ax  93  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  x  material.  94  xiv  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 H f o r x  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 effect 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 is 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 together (no longer are they antiparallel). T h e curve s h o w n as "total" is the total result for a material containing 1 0 0 % o f t h e isotopes shown. . . 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 H f o r Fe *, Fe x  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 \A \ k  ND  a n d F e . ... 9 8 5 7  8D  8  for all nuclei, assuming  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 left) H, Br, 1  1 3  79  N,  U  1 7  0,  C a n d Fe, 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 57  is 0.0001; p l o t t e d o n t h e x a x i s i s  a n d on the y axis "number of  nuclei". Note that the contribution to  f r o m Fe i s a l m o s t e n t i r e l y 57  from the contact interaction  99  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 Fe  3+  wavefunction  101  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™ . 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 ax  102  3.21 H, 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  103  3.22 H, 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  103  3.23 H, 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  104  3.24 H, 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  104  3.25 . 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  105  3.26 Br, 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  105  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  106  3.28 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  106  3.29 Fe, 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  107  3.30  107  1  l  2  2  79  81  U  15  57  1 3  C , Hartree-Fock, entire spectrum xv  3.31  1 7  0 , Hartree-Fock, low e n d o f s p e c t r u m  108  3.32  1 7  0 , Hartree-Fock, entire s p e c t r u m  108  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 H i n t h e H a r t r e e x  Fock wavefunction  picture  109  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 H i n t h e H a r t r e e x  Fock wavefunction  picture, focusing o n small  fields  109  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 H f o r F e » , Fe$ a n d F e i n t h e 5 7  8  x  8  D  Hartree-Fock picture  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 ) H, Br, 1  1 3  C a n d F e , using the Hartree Fock approximation. 5 7  79  N,  14  1 7  0,  T h e bin width here  is 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 nuclei". 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 is a l m o s t e n t i r e l y  from t h e contact interaction 4.1  I l l  E n e r g y levels o f t h e L a n d a u - Z e n e r Hamiltonian.  Shown are both the  e i g e n s t a t e s o f a , 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), z  E (t) ±  4.2  = ±(A  + vH ) '  2  2  123  1 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 A /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 2  respectively  126  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  135  4.4  136  T r a n s i t i o n probabilities for t h e sinusoidal potential  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 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  xvi  ATT/U  140  4.6  for t h e r a m p scenario. P l o t t e d o n t h e x axis is  Transition probability A/u> a n d o n t h e y a x i s ^  141  5.1  Transition a m p l i t u d e as a train of blips  156  5.2  Transition probability normalized to the standard Landau-Zener transit 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 is for t h e sinusoidal (sawtooth) p e r t u r b a t i o n 5.3  Transition probability normalized to the standard Landau-Zener transit 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 is for t h e sinusoidal (sawtooth) p e r t u r b a t i o n 5.4  167  Transition probability normalized to the standard Landau-Zener transit 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 is for t h e sinusoidal (sawtooth) p e r t u r b a t i o n 5.5  166  Depiction o f t h e precession o f t h e k  th  168  nuclear spin during a "blip". T h e  central spin is s h o w n i n black, w i t h a s c h e m a t i c nuclear spin 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 feels 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 t t h e n u c l e a r s p i n x  feels 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 causes a precession o f t h e nuclear spin. A f t e r t h e central 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 less t h a n f u l l o v e r l a p w i t h its o r i g i n a l s t a t e  178  5.6  p  Xil  (dashed) a n d W ( 0 ) (solid) for p = H  197  5.7  p  lfl  ( d a s h e d ) a n d W ( ( L J ( s o l i d ) f o r p = Br  198  5.8  p  lfl  (dashed) a n d W ( 0 ) (solid) for p =  N  198  5.9  p  lfl  (dashed) a n d W ( 0 ) (solid) for p =  Fe  199  0  199  x  / i  79  5.10 p  llt  1 4  M  5 7  M  (dashed) a n d W ( 0 ) (solid) for p = M  xvii  1 7  5.11 p  Xil  (dashed) a n d W ( 0 ) M  ( s o l i d ) f o r (i = C  5.12 F u l l w i d t h W f o r Fe * ( d o t t e d ) , Fe s  200  n  ( d a s h e d ) a n d Fe 57  8D  8  (solid)  5.13 A (left) a n d A ( 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 * ( d o t t e d ) , Fe 3  and  4  5 7  Fe  8  8  (solid)  8D  200 (dashed) 202  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 | cos<I>| ( t h e l o w e r c u r v e ) a n d (2) 0  the large A result with nuclear spins A | c o s $ | (the middle curve) p l o t t e d 0  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 scale. T h e h o r i z o n t a l a x i s is H i n T e s l a - h e r e we h a v e ip = 0 (H = 0) x  6.1  204  y  T o p figure: Log A w  2  vs. H°, f o r 6 = s i n " (H%/H$) = 0. C u r v e s are, f r o m 1  b o t t o m t o t o p , WD — 0,10, 2 0 , 3 0 a n d 50 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 was d e r i v e d a s s u m i n g the i n s t a n t o n a p p r o x i m a t i o n for the t u n n e l i n g amplitude 6.2  215  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 ext 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 as 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 of 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 differ i n detail  B.l  216  H e r e we see 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)  xviii  230  Acknowledgements  T h e r e are m a n y people who have p r o v i d e d s u p p o r t d u r i n g the c o m p l e t i o n of this work. F i r s t l y I w o u l d like to t h a n k P h i l i p S t a m p , b o t h for i n t r o d u c i n g me t o the  complex  a n d i m p o r t a n t world of the nuclear s p i n b a t h a n d for a l l o w i n g me significant leeway in my a p p r o a c h to the difficult problems t r e a t e d i n this d o c u m e n t . T h e e d u c a t i o n I received from h i m i n the area of quantitative treatments of decoherence will be absolutely 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 we w i l l b e w o r k i n g together for m a n y years to come. I w o u l d also like to t h a n k the other great teachers I have h a d the o p p o r t u n i t y to learn from i n my time at U B C , and i n particular Ian Affleck, who has influenced my understanding of several aspects of condensed matter physics, and H a i g Farris, who  has  b e c o m e a g o o d friend, mentor and business p a r t n e r over the past year. A s well I w o u l d like to t h a n k colleagues 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 Duty, Sebastian Jaimungal, Alexandre Zagoskin, Jonathan O p p e n h e i m , Suresh Pillai, M i c h e l Olivier, M a r t i n Dube, Stephanie Curnoe, Igor T u p i t s y n , N i k o l a i Prokofiev a n d Jeff Sonier. A l s o I w o u l d like to t h a n k W o l f g a n g W e r n s d o r f e r for his a m a z i n g  Fe  8  data. T h e completion of this d o c u m e n t would not have been possible without the support 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 Dr.) 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 like to t h a n k C r a i g T h o m a s for t r y i n g to keep u p w i t h me o n the beach volleyball circuit, R A N D y for hours of revolting amusement, a n d S h a w n Kenney, 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. xix  F i n a l l y I w o u l d like t o t h a n k m y wife Valerie, 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 re-write I w a s d o i n g w o u l d b e t h e l a s t one.  xx  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 solutions such as the recent I B M S M A S H 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 capability 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 fashion 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  4  Chapter 1. Introduction and Overview  p r o c e s s k n o w n as 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 as 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 is 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 theoretical perspective), a n d yet how  (and interesting also from a purely  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 , is s t i l l n o t  satisfactorily understood. There exists a class of mesoscopic systems where an attempt can be m a d e at a quantitative theory of decoherence.  T h e s e are the so-called "molecular magnets". One  of  t h e s e m a t e r i a l s , w h i c h we w i l l r e f e r t o t h r o u g h o u t as " i r o n - e i g h t " (Fes), is p a r t i c u l a r l y well suited to a quantitative study of decoherence due to localized environmental  modes,  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 as a t o o l t o be used in future investigations of mesoscale systems i n the context of developing solid state q u a n t u m bits provides the m a i n m o t i v a t i o n for the w o r k presented i n this thesis. 1.1 Fe  s  A n Introduction to Fe  8  was first s y n t h e s i z e d i n 1984 b y W i e g h a r d t et.al. [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  imposing chemical  formula  { [ ( t o c n ) F e ( / i - 0) (/*2 6  8  3  -  2  OH) ]Br 12  7  • H 0}®[Br 2  • 8H 0f 2  (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 , was 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 ions per unit cell ever characterized. X-ray c r y s t a l l o g r a p h y studies performed 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 Fe  3 +  on this  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 FeN Os  and  the remaining two were surrounded by a distorted octahedral array of 6 O atoms.  The  3+  i r o n i o n s a r e c o u p l e d v i a 12 \x — 2  s  h y d r o x o bridges a n d two /x — oxo bridges. T h r e e views 3  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 here, 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 et.al. [23].  Chapter 1. Introduction and Overview  5  10  8+  +  +  6 + +  4+  2  +  +  +  0  - 2 -  +  + ++  +w + +  - 6 -  +  + 1  1  1  i 0  1 1  1  1  i  1  1  2  + +  + 1  ' i  1  • 4  1  1  i ' • • 6  1  i  1  '  1  1  8  i >  1  • 10  1  i ' > 12  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 Fe u n i t c e l l i n t h e x — z p l a n e . 8  Chapter 1. Introduction and Overview  7  10-  +  sG-  •1-  •  +  #  •»  +  42-  •  0-2-  +  +  +w  +  * m*  +  •  •  *  +  # + ** +  -6-  t  +  +  •  ,  +  <* * „ *  +  ••  +  +  +  -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 u n i t c e l l i n t h e x — y p l a n e . H e r e we 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 ) . 8  8  Chapter 1. Introduction and Overview  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 NH  e  a n d OH . 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 e  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 2250. 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 sites 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 Q u a n t u m  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  surpris-  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  1984  a n d 1993 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 derstanding the low-energy magnetic nature of this substance. T h e r e are two basic themes t h a t needed to be developed to u n d e r s t a n d the experim 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 first 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 of " g i a n t spins", t h a t is systems t h a t have large s p i n q u a n t u m n u m b e r . O n e of the 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 its 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 one c o u l d t h i n k o f this o b j e c t as a single degree of freedom, albeit w i t h s p i n q u a n t u m n u m b e r as large as S = 10 . T h e interesting t h i n g a b o u t these systems, a n d one of the 8  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 we 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 as i t is i n c r e a s e d s h o u l d cause the s y s t e m t o go 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 (say 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 ) i n a w a y t h a t we c a n s t u d y w i t h s o m e 8  intimacy. T w o papers that would prove to be i m p o r t a n t i n this respect were those of 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 1986. T h e s e d i s c u s s e d t h e dynamics of giant spins in the W K B  approximation. In particular, the p r o b l e m of how  Chapter 1. Introduction and Overview  9  large spin objects tunnel between energetic m i n i m a was treated. T h e second basic theme that needed t o be developed was a n understanding o f the effects 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 . It 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 effect 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 effects 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 bath. I n a solid a t l o w energies, i t is u s u a l l y t h e case t h a t d e l o c a l i z e d m o d e s , such as p h o n o n s a n d photons, c a n b e m a p p e d t o oscillator baths. T h i s is because l o w energy modes have long wavelengths a n d therefore their overlap w i t h t h e localized interesting degree of f r e e d o m is s m a l l . Various aspects o f the dynamics o f a central degree o f freedom coupled t o oscillator 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 et.al. 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 , the c e n t r a l degree o f f r e e d o m is a two-state s y s t e m (a spin). T h i s is t h e n c o u p l e d t o a n oscillator b a t h a n d t h e d y n a m i c s of the c e n t r a l s p i n are t h e n e x t r a c t e d . It is p o s s i b l e t o obtain analytic results for certain choices of parameters i n t h e spin-boson Hamiltonian, b u t i n general d y n a m i c a l solutions are difficult t o obtain. 1.1.2  A Related System is Characterized  In 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 u n d e r w e n t l o w t e m p e r a t u r e 8  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 [Mn {CH COO) (H 0) 0 ] 12  3  16  2  4  12  • 2CH COOH 3  • 4H 0 2  (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  10  Chapter 1. Introduction and Overview  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 is 2060. X - r a y c r y s t a l l o g r a p h y was p e r f o r m e d  [30]  on this substance, and the atomic positions stored i n the C a m b r i d g e Crystallographic D a t a b a s e . T h i s d a t a is p r e s e n t e d i n figures 1.7, 1.8 a n d 1.9.  We  see 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 Mn  4+  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 Mn  3+  ions with spin  i o n s w i t h s p i n 5 = 2. It was  proposed t h a t a l l these ions couple to each other p r i m a r i l y v i a superexchange through 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 we a r e l o o k i n g d o w n the easy axis of the crystal. T h e axes scales are i n A n g s t r o m s . Legend: Red, manganese; Purple, oxygen; Yellow, carbon; D a r k Blue Crosses, hydrogen. N o t e the inner a n d outer "rings" of m a n g a n e s e ions.  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  unit cell onto the x — z plane.  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  unit cell onto the y — z plane.  Chapter 1. Introduction and Overview  13  T h e 1991 p a p e r o f C a n e s c h i et.al. 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 magnetization Mn  and E P R  measurements on crystals of M n  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 .  1 2  1 2  which indicated that each  A m e c h a n i s m was d e s c r i b e d w h e r e b y t h e  twelve m a n g a n e s e ions i n each unit cell lock together v i a s u p e r e x c h a n g e at low tempera 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 was o b s e r v e d , a n d was  explained  as 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 effects. 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 , as 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 spondinging in a semi-classical picture to the giant spin p o i n t i n g parallel/antiparallel to the easy axis respectively) 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 effects of s o m e k i n d (as S is 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 was 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 was p e r f o r m e d i n 1993 b y S e s s o l i et.al. [31] a n d  provides  an excellent review. 1.1.3  Magnetic Characterization of  Fe  s  I n 1993 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%  at low tem-  p e r a t u r e s was p e r f o r m e d b y D e l f s et.al. [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 , magnetization  as 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  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  studies were performed which  groundstate.  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 Fe  3+  ions which couple to each other v i a exchange and superexchange v i a oxygen and  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 less 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 is 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 m = ± 1 0 s  b e i n g n e a r l y d e g e n e r a t e lowest 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  in a semi-classical picture  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 "up" 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 is g i v e n b y H = —DS , 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 2  i n t h e c r y s t a l [32, 33, 34]-see 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 effective H a m i l t o n i a n for this system). T h e p a r a m e t e r D was r e p o r t e d to be o n the order o f D~0.3K 1.1.4  [18].  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 results, b o t h f r o m theorists a n d f r o m the 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 were presented w i t h a truly mesoscopic problem with a growing n u m b e r of experimental results, a n d e x p e r i m e n t a l i s t s h a d access to systems where t h e y c o u l d directly 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 effects. 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 et.al. [17] i n 1996, which presented clear-cut evidence for incoherent macroscopic tunneling of the magnetization in 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 1997  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 figu 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  molecules in the crystals  measured. Results of magnetic relaxation measurements on these systems for the most part 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 the physics of relaxation i n molecular magnets. In the models of Prokofiev and Stamp, e a c h m o l e c u l a r m a g n e t is t r e a t e d as 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 F e . At high temperatures thermal activation is observed, while for T < ~ 360 rnK relaxation becomes temperature independent. Figure obtained from [14]. 8  Chapter 1. Introduction and Overview  16  the m o s t i m p o r t a n t o f w h i c h i t is a r g u e d are l o c a l i z e d m o d e s , s u c h as n u c l e a r s p i n s a n d m a g n e t i c impurities. N o t e t h a t localized m o d e s s u c h as these c a n n o t i n g e n e r a l be m a p p e d t o oscillator b a t h s as the c o u p l i n g s t r e n g t h s are n o t i n g e n e r a l s m a l l (for 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 show that the relaxation characteristics of giant spins i n condensed matter systems should be strongly influenced by these localized modes, a n d h a r d l y influenced at all by oscillator b a t h s (such as p h o n o n s ) .  In the q u a n t u m regime demonstrated to exist i n F e , the 8  t e m p e r a t u r e i n d e p e n d e n c e is evidence t h a t the p h o n o n b a t h is n o t p l a y i n g a significant role i n relaxation. 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 Fe 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 8  in the q u a n t u m regime was q u i t e strange, for the f o l l o w i n g reason. T h e bare t u n n e l i n g amplitude between the groundstates of the F e  molecules was e s t i m a t e d (by e x a c t l y  8  diagonalizing phenomenological spin Hamiltonians whose parameters were extracted from various experiments) to be 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 means that i n a crystal of molecules, only an extremely tiny fraction of molecules could ever be i n resonance, a n d therefore their d y n a m i c s s h o u l d be frozen. A resolution of this difficulty was p r o p o s e d i n 1997 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 Fe c r y s t a l [15]. A t 8  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 at which the nuclei i n F e  8  perform so-called T  2  f l i p s [64, 65, 66], 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 effect 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 at each central spin, w h i c h was p o s t u l a t e d t o i m m e n s e l y increase t h e "resonance w i n d o w "  Chapter 1. Introduction and Overview  17  and therefore allow the c r y s t a l to relax. T h e m e c h a n i s m w o r k s like this. A t the b e g i n n i n g of relaxation, a l l molecules sit 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 by the T f l i p p i n g nuclei. W h e n one of these molecules tunnels, i t rearranges the d i p o l a r 2  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 so the c r y s t a l relaxes. T h i s theory contained certain testable predictions a b o u t the relaxation characteristics o n e s h o u l d see 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 was t h a t t h e r e l a x a t i o n o f Fe&  i n its q u a n t u m regime should be square root i n time. 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 was t a k e n as 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 was 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 was r e a l ized that earlier data did in fact follow a square root temporal relaxation. Subsequent 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 was n o t 2  understood  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 is 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 is, t h e  presence  of thermally occupied higher levels destroys the square root. T h i s s e e m e d to contradict the fact that the relaxation characteristics in Mrii were clearly t e m p e r a t u r e dependent 2  d o w n t o a t l e a s t 60 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 is 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 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 as 2  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 ) . In the next section w e shall take a close look at r e l a x a t i o n experiments,  as t h e y  Chapter 1. Introduction and Overview  18  provide tantalizing glimpses o f unresolved puzzles, some of which will b e resolved i n later chapters. 1.2  A n Introduction to Relaxation Experiments  The general strategy for performing arelaxation experiment on a molecular magnetic 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 in- T h e i n i t i a l m a g n e t i z a t i o n of the s a m p l e is 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 array. T h e static —*  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 the s a m p l e ( M ) is t h e n m e a s u r e d as a f u n c t i o n o f time. —*  T h i s g i v e s u s a q u a n t i t y M(T,  —*  t , %, H(t), 0  t) w h i c h tells 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 of e x c e l l e n t experimental d a t a deep into the so-called " q u a n t u m regime"-the region where relaxation 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 T ~ 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 c  molecular magnet  (M7112)  is s i m i l a r i n m a n y w a y s i n its r e l a x a t i o n characteristics. How-  ever, 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 s h o w c l e a r 2  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 the q u a n t u m r e g i m e . N o w as we have d e s c r i b e d t h e experiment above, there a r e f o u r basic parameters t h a t we 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  19  Chapter 1. Introduction and Overview  w h i c h i t i s c o o l e d t 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 0  strategy i n this chapter will be t o simply present t h e experimental situation i n each part 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 is, 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)} are) a n d t h e n give the results. N o theoretical justification o r e x p l a n a t i o n will b e p r e s e n t e d here-we shall m a k e this t h e task o f the r e m a i n d e r o f the thesis. 1.2.1  D C F i e l d Relaxation in Polarized  Fe  8  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 s a m p l e s a r e 8  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 Fe ). T h i s h a s t h e effect 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 8  polarized state. O n c e t h e initial m a g n e t i z a t i o n h a s been measured, t h e large static bias field is 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 its 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 nfigures1.12 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 type. N o t e t h e unusual square-root r e l a x a t i o n characteristic. A l s o i n c l u d e d here is 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 relaxation for this m a t e r i a l is also square-root i n time, albeit w i t h a s t r o n g temperature dependence. 1.2.2  D C Relaxation of Annealed Crystals  A second class o f experiments involve a different m e t h o d o f p r e p a r a t i o n o f t h e sample under study. Instead o f slowly lowering t h e temperature d o w n into t h e q u a n t u m regime 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  o  a  i  i  «  sqrt(;j  a  [sec"zj  Figure 1.13: Short time relaxation of a single crystal of F e , measured at 1 5 0 m A ' . Here several different D C 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 D C bias field. Figure from [55]. 8  1.2.3  Hole Digging and the Time-Dependent Internal Longitudinal Bias Distribution —*  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 D C 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. M(H,  t) ~ 1 — \JT (H)t sqrt  The "relaxation rate" T (H), sqrt  can then be measured.  internal distribution of longitudinal biases P(£, is proportional to the relaxation rate P(£,t)  defined via  The assumption then is that the  t) (where f is the longitudinal bias field) ~ T (H) sqrt  time t. Shown in figure 1.16 is the relaxation rate T (H ) sqrt  z  where the field H is applied at 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  '3.92ml  0.002  M/M  $  3.36mT  0.0015  2.80mT .^-2.2imT  0.001  h  1.68mT 1,12mT  0.0005h 0  0  0.56mT  10  OmT  _L_  20  30  sqrt[t(s)]  F i g u r e 1.15: H e r e is d a t a f r o m a n e x p e r i m e n t o n a n Fe 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 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 case. F r o m [49]. 8  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 experiment is that it is possible to measure extremely  s m a l l r e l a x a t i o n rates, w h i c h  presents us with a useful p r o b e o f m u c h o f t h e physics o f these systems. In addition, there now exists a quantitative theory o f how molecular m a g n e t s respond t o this kind o f p e r t u r b a t i o n , w h i c h we 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  E x t r a c t i o n of Tunneling M a t r i x 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 ments, w h i c h are related t o t h e t u n n e l i n g m a t r i x elements o f the single-molecule effective H a m i l t o n i a n o f F e (we 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 8  Chapter 1. Introduction and Overview  24  M H(T) 0  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 T (H ). The i n i t i a l d i s t r i b u t i o n is l a b e l l e d w i t h M = — 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 latter are d i s t o r t e d a t h i g h e r fields b y nearest n e i g h b o u r l a t t i c e effects. F i g u r e f r o m [49]. sqrt  z  in  -0.05  -0.025  0  0.025  0.05  fi H(T) 0  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 t . 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 H w a s applied, a n d r 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 H = 0. F i g u r e f r o m [49]. 0  sqrt  z  z  Chapter 1. Introduction and Overview  -0.04  -0.02  0  25  0.02  H H(T)  0.04  0.06  0.08  0  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 , as 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 M = — 0.2 M . 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 inset). N e a r zero bias the hole develops very r a p i d l y a l t h o u g h the rest o f the distribution 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]. in  s  H H(T> 0  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 (r i ) a n d a t t = 1 6 s (Tdi ), 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 | < 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]. in t  0  g  in  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 D C bias field H z, and a static transverse D C field H = H x + H y. z  x  y  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 D C field and the longitudinal D C 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 0  t i i 1 1 0.2 0.4 0 , 6 0.8 1 rani  1 1.2  1  1,4  (T)  Figure 1.20: The quantity A here is related to the relaxation rate of the crystal's magnetization via (1.3). Here it is shown as a function of the magnitude of the transverse D C field \H\ = JW* + H for several orientations of this field ip = tan' (H /H ). In this case the longitudinal D C field was taken to be zero (H = 0 ) . Figure from [51]. 2  1  y  z  x  Chapter 1. Introduction and Overview  r"""'"""r  1  I  E  i  -0.4*0.2 0  27  1  i  f  i  1  i  r——-r  r  r  0.2 0.4 0.6 0,8  i  1  1  1  1.2 1.4  runs  (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 H . The lowest curve was obtained for H = 0; the middle curve for H = 0.22T, and the upper curve for H = 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]. z  z  1.3  z  z  Thesis Overview  Our goal will be to work up to a quantitative theory of A C relaxation in F e crystals. 8  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 molecule. 8  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  28  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 . In chapter four we introduce a n d develop s o m e o f t h e m a c h i n e r y Zener problem. T h i s involves using a time-dependent  of the Landau-  Hamiltonian to extract transition  probabilities between states o f t h e central object o f interest. 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 problem i n which nuclear spins are included. We use the results of this calculation to 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 both A C a n d D C components. C h a p t e r s i x t h e n uses 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 master equation so as t o m o d e l the temporal evolution of a crystal o f F e molecules. 8  extract time-dependent  We  relaxation characteristics from o u r theory a n d c o m p a r e these t o  experimental results. W e conclude o u r analysis i n chapter seven w i t h a s u m m a r y o f results a n d t h e current outlook for o u r theory o f A C relaxation.  Chapter 2 Effective Hamiltonians  In t h i s c h a p t e r we 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 Fe  3+  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 Fe  3+  ion.  ions into a crystalline  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] (see a l s o [68, 69]). 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 Fe  3+  ions and terms coming  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 baths. 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 +  terms,  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 is t h e n investigated. T h e exchange/superexchange c o u p l i n g energies are m u c h larger 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 is 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 we 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 be derived f r o m the bare 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 to how the single ion spin 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 " ground state of the system and performing perturbation theory around it to eliminate 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 one. We then proceed to the investigation of the properties of general giant spin Hamiltonians i n the absence of environmental couplings. We calculate t u n n e l i n g m a t r i x elements 29  30  Chapter 2. Effective Hamiltonians  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], per-  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 . We conclude by r e t u r n i n g to the specific case of the F e  giant spin Hamiltonian.  8  The  g i a n t s p i n possesses 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 >).  At  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 us 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 as 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 et.al. [74]. 2.1  The Fe  3+  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 was s h o w n in the i n t r o d u c t o r y chapter) by concentrating our a t t e n t i o n o n the i r o n ions. We 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 Fe  ion. T h i s treatment follows that  3+  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 is 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 elect r o n s ( h e r e t h e r e a r e N 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 e  /»•  Ne  V  Ze \  2  2  e  N e  2  '-£&-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 is 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 V S = ]C 3*h ' h + 3k 3 " $k + C Sj • s 3,k a  b  l  L  jk  k  (2.2)  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 V  ss  = y * ' * - ( J ' fffX^i* • jk jk jk j  T  k  3  f  fc  r  (2 3)  31  Chapter 2. Effective Hamiltonians  W e a k e r still are t h e terms V  N  =  2g ji Pn n  B  fc I  k  k  r  \  r  (2.4)  o  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 is 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 is 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  which  comes about from t h e overlap of the wavefunction o f s electrons with t h e nucleus a n d eQ  1(1 + 1)  2  Q  2 / ( 2 / - 1)  3(r -I)  2  (2.5)  k  'fc  'fc  which represents the electrostatic interaction between the nuclear quadrupole 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 V  = '£^B(lk  H  +  (2.6)  2s )-H k  k  and (2.7)  V = -9nPnH • I h  corresponding t o t h e interactions with t h e electrons a n d nucleus respectively. T h e total free i o n H a m i l t o n i a n is n o w j u s t t h e s u m o f these; H = V +V F  LS  +V  ss  (2.8)  +V +V +V + V N  Q  H  h  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 are, f o r V ~ 5 - 1 0 K, V 5  F  LS  ~ 1 0 0 - 3 0 0 K, V  ss  ~1-2K,V ~ N  Fe , 3+  1 - 2 0 0 mK a n d V ~ 1 - 2 mK Q  [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 terms b u t this one, then L a n d 5 (the t o t a l angular m o m e n t u m a n d spin o f t h e ion) c o m m u t e w i t h Vp. T h i s m e a n s t h a t we m a y i n this a p p r o x i m a t i o n l a b e l t h e states o f  32  Chapter 2. Effective Hamiltonians  t h e free 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, L , 5 , 5 , J a n d J . S i n c e t h e filled i n n e r z  2  z  shells have 5 = L = J = 0 w e m a y describe t h e i o n b y referring o n l y t o t h e state 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 Fe  3+  leads 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 <S / [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 6  5  2  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, etc.) a n d t h e s u b s c r i p t refers 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 first 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 Effect o fthe Crystalline 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 is 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 is 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 is 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 ion'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 is 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 sits 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 is c r u d e , i t is 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 effect 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 its 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 .  In this treatment al-  l o w a n c e is 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 ion'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 , ie. an attempt t o deal w i t h covalency is presented.  B e t t e r y e t a r e molecular orbital [81]  33  Chapter 2. Effective Hamiltonians  m e t h o d s , w h i c h use 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 these. Now Fe  3+  i n our specific case we are interested o n l y i n the low energy p r o p e r t i e s of the  ion. T h e free i o n g r o u n d s t a t e is, a s mentioned,  a n o r b i t a l s i n g l e t «S /26  5  What  will b e the effect of the i n c l u s i o n of the C o u l o m b i c environment o n t h i s groundstate  and  the low-lying excited states? In order to answer this question we need only to k n o w the relative strengths of the "on-site" C o u l o m b terms V a n d the "off-site" C o u l o m b terms F  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 Fe  ions has been  3+  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 been measured in F e . Fortunately there a w a y to know w h a t the relative m a g n i t u d e of 8  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 V ^> 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 F  r e m a i n <S / , 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 6  5  2  Vc » V t h e n t h e Fe  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 (ie.  3+  F  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 is 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 Fe  3+  ions are 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 V 3> Vc, w h i c h i s i n a c c o r d F  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 Fe  centers.  3+  W e see 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 Fe  i o n s h a v e s p i n 5 = 5/2  3+  simplifies our task tremendously. g r o u n d s t a t e o f t h e Fe  3+  T h i s i s b e c a u s e t h i s is prima facie e v i d e n c e t h a t t h e  ions, even i n the m o l e c u l a r e n v i r o n m e n t , is «S /2. M o r e precisely, 6  5  t h i s is 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 = V + Vc, w h e r e Vc i n c l u d e s F  all Coulombic terms c o m i n g from the interaction of the ion w i t h the l i g a n d environment, is a n o r b i t a l s i n g l e t <5*5/26  Now  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 is t h i s . S i n c e t h e  Coulombic  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 here, 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 Q 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 A  to q u a n t i t a t i v e l y account for its effects a n y w a y (this w o u l d require a m o l e c u l a r o r b i t a l  Chapter 2. Effective Hamiltonians  34  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]), we s h a l l a d o p t t h e c r y s t a l  field  paradigm i n dealing w i t h charges external to the ion. In this picture the excited states of t h e Fe  3+  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  others because of their spatial dependence and relative positioning i n the molecule. In o u r c a s e t h e Fe  3+  ions all have six nearest neighbours arranged i n a distorted octa h e d r a l  shape. T h i s allows us i n principle t o c a l c u l a t e the splittings o f t h e e x c i t e d states. 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 we 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 change the results of the arguments to come. 2.3  T h e 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 effect o f t h e f r e e i o n t e r m s VLS + Vss + VN + VQ + VH + VhW h a t we s h a l l d o is 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 effect p e r t u r b a t i v e l y o n t h e <S / g r o u n d s t a t e o f H = Vp + Vc6  5  2  T h e first s t e p i n t h i s  treatment is to rewrite these p e r t u r b a t i o n s i n terms of total single i o n s p i n a n d angular —*  —*  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 as l o n g as 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 as we 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 ; we 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 , see [68]). O u r t e r m s t r a n s f o r m as f o l l o w s ; VLS  —> A L • 5  V  -> P  SS  [(L • 5 ) 2 + -L • S - -L{L + 1 ) 5 ( 5 + 1)  p \\{L l/ a  l  l  + L^L )S S a  Q  - -L{L + 1 ) 5 ( 5 + 1) l  p  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 is m e a s u r e d t o b e [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 +  (2.9) \Z%K  ions. T h e p a r a m e t e r p is difficult to  35  Chapter 2. Effective Hamiltonians  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 V  gntipixn ^  N  [ ( L • 5 ) + £L(L + 1 ) ( 5 • /) - ^(L  • S)(L • i)  (s-i) v  ,  ^  Q  1  2/(27-1) \ r where for the iron group Q  3,* 3 ( f - / ) + | ( Z . i ) - L ( L + 1 ) 7 ( 7 + 1)  1  (2.10)  2  3  (2/ - 1) - 4 5 5 ( 2 / - 1)(2/ + 3 ) ( 2 L - 1)  ,  77 = ±25e  (2.11)  ( 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 less o r m o r e t h a n h a l ffilled)a n d , h) r / = Jl*  .Jtf(*0l  (2.12)  s  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 the p o l a r i z a t i o n o f t h e core s-electrons d u e to t h e d electrons a n d will lead t o t h e contact h y p e r f i n e i n t e r a c t i o n i n Fe. 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 57  V  H  = (i(,H-{L  + 2§)  (2.13)  V = g x H-I h  ni  n  P u t t i n g a l l t h e s e t o g e t h e r we g e t t h e p e r t u r b a t i o n t e r m H = (A - \p)L -S-p(L-  5)  x  +g p P n  n  | ( L .T)-K§-  2  + n H • (L + 2S) P  I- -{L •S)(L-T)- -{L1  !)(L •  l  5)] + g iH nl n  •  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 5 / s t a t e . I n t h e first o r d e r 6  5  2  p e r t u r b a t i o n we 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,  36  Chapter 2. Effective Hamiltonians  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 , we 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 we 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 &  + ^l^SJp  H = -g PnP(^ 2  2.3.2  n  + 2u. H • S + g i H 0  (2.15)  •I  nl n  Second Order Perturbation T h e o r y  I n t h e s e c o n d o r d e r we 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|ffi|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 , ie. 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 as s u c h we shall not a t t e m p t to calculate t h e m exactly. We may obtain order of m a g n i t u d e estimates 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 Fe  3+  c e n t e r s . I n t h e s e cases  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) is 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 is 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 we 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\L*\nxn\I/ \0> E(n) - E(0) l  0  Q/3 U  T  _ _i€^_ v < 0\L \n >< n | L % + L 7 Z l | 0 > ~ 2 & E(n)-E(0) s  ~*ho  E(n)-E(0)  '  U  ( 2  -  1 7 )  gives, 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)  37  Chapter 2. Effective Hamiltonians  where we have defined gP  = 2(5  A  -  a  al)  a&  - A )  ,  Q/3  -P  = -A A  D  2  a0  7  Q / J  - pl  a)3  ,  7  R  ati  = 1 -  2Pu.pA  ap  (2.19) T h e m e a n i n g o f t h e s e t e r m s is a s f o l l o w s [87]. g @ is 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 a  factor". It is anisotropic i n general, containing reference t o t h e higher l y i n g orbital states. D P is 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 state a n d contains reference t o a  both the spin orbit coupling a n d the spin-spin contribution i n a n asymmetrical field.  A  crystal  represents t h e hyperfine couplings between t h e nucleus a n d t h e electronic spin.  a/}  T h i s t e r m is 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 ( « ) , an orbital contribution l  a/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 XA . a/3  I n t h e expression for the  c o u p l i n g b e t w e e n t h e nuclear s p i n a n d a n external field we see t h a t there is 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 be o f the same order o f m a g n i t u d e as t h e direct contribution. 2.3.3  Higher Orders Perturbation Theory  It is 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  Hamiltonian  u p t o 2S o r d e r i n t h e s p i n o p e r a t o r s S , S a n d S 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 case o f th  x  the F e  3 +  y  z  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 H  Fe3+  = GQ1Q2Q3Q4Q5(#) SS S S S ai  +  g u. [A S I n  a  a3  a4  +R HI]  al3  n  a2  af3  l3  a  p  a5  +  g^^SaHp  (2.20)  w h e r e t h e S{ ] c a n b e a n y o f S , S , S 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 ai  x  y  z  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  38  Chapter 2. Effective Hamiltonians  —*  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 terms i n the single ion spin H a m i l t o n i a n m u s t inherit any s y m m e t r y that the c r y s t a l l i n e e l e c t r i c field possesses, 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 Fe  i o n s d o n o t sit 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  3+  t h e s y m m e t r y is 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 Fe  are distributed  3+  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 2S 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 th  t h e c a s e o f a n Fe  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 ,  3+  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] H)cubic Fe + r  3  | (S + S + St) + ^HBSalp 4  —  + 9i{S*H  4  + g (S + S H + SlH ) + g x [A^SJp 5  2  5  x  y  y  z  nf  n  + SH 3  x  +  + SH) 3  y  Z  R^HJp]  g [S (S + S )H + S (St + S$)H + S (S + S )H ]  +  4  3  x  4  y  4  Z  X  y  y  Z  4  X  y  y  (2.21)  ( c o m p a r e t o (2.20). It is f o u n d t h a t t h e c o n s t a n t s gi, g a n d g a r e a l w a y s v e r y s m a l l 2  3  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  Measurements of g  a/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 Fe  3+  support the following  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 g ^ 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 g a  a/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 spin-orbit c o u p l i n g will be very small and therefore a spin-only a p p r o x i m a t i o n for g  a/3  justified.  is  Chapter 2. Effective  2.4  39  Hamiltonians  T h e Single M o l e c u l e Effective H a m i l t o n i a n  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) H f o r t h e p 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 th  p  sits 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 axes, 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 in p r i n c i p l e possible) t o write d o w n t h e specific t e r m s for each ion. F o r e x a m p l e , i f we 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), ie. ~  H?&  ^(s  4  (2.22)  + S + St)+gu. H-S 4  x  y  B  then we immediately r u n into t h e following technical p r o b l e m . T h e t h e l o c a l c r y s t a l fields f o r e a c h  Fe , 3+  cLXGS j clS d e f i n e d  by  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 efixt 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) is 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 we have t o rotate t h e axes t h r o u g h s o m e ( a l b e i t k n o w n ) angles 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 shall therefore a t this stage write t h e collection o f eight i r o n single i o n spin H a m i l t o n i a n s as H  =£  P=I  +  k>  H  +  =  £ [Gr  p—i  iQlQiai  (#)  S S ^ S S ^ S S , + 9PBS  p  Er H I%]]  -H + g u. np  n  [A^s*!* (2.23)  p  a  where for a l l quantities t h e label p points t o t h e p iron ion. th  Chapter 2. Effective Hamiltonians  2.4.1  40  Inclusion of Exchange and Superexchange Terms  It 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 . In general these c o u p l i n g s can be anisotropic, l e a d i n g to a general expression of the f o r m H  ex  = £  (2.24)  JgS>S}  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.  Exact diagonalization studies on this term have been performed, with results  compared to E P R  and susceptibility measurements performed on 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 is n o t clear t h a t t h i s has to be the case ( D z y a l o s k i n s k i - M o r y a  interactions, for example, are  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 , as w e l l as 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 fits 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 seen, 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 et.al. [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 Fe  3+  ion will couple v i a dipole-dipole interactions with all the  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 Fe  3+  electronic spins and  all the nuclear spins i n the molecule. A s well, a l l nuclei i n the system w i t h spins 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 . It 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 we 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 general interaction between the electronic a n d nuclear degrees of f r e e d o m may  be  s p l i t i n t o t w o p a r t s [64]. T h e first o f these, 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 Fe 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 et.al. [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 ~ 180K, J ~ 22K a n d J ~ 52K, w i t h a l l couplings antiferromagnetic. 8  i 3  1 5  3 5  between t h e m o m e n t s o f t h e nuclei a n d t h e magnetic fields generated b y t h e s p i n a n d orbital currents o f the electrons, will be dealt with later. 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 p {r ) n  Peife)  n  and  with mutual electrostatic energy E  Pe(r )Pn{r )dr dr e  n  e  n  (2.25)  T h i s expression m a y b e simplified 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 terms of spheri 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 is t h e e x p e c t a t i o n v a l u e  42  Chapter 2. Effective Hamiltonians  of the Hamiltonian (2.26)  im*  Helec = £ l,m  A™B\  where A? = e  (2.27)  21 + lfi-l  BY = - e  (2.28)  1  21 + lfi=l  Here N a n d N are the n u m b e r o f protons p e r nucleus a n d electrons respectively, n  e  {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 i  th  proton and i  th  electron  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 . It 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 confirmed experimentally t o a high degree o f precision w i t h t h e / = 1 (electric dipole) 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 seen t o b e constant, a n d c a n therefore b e o m i t t e d f r o m consideration. A s well, i t is seen 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 terms i n t h e series 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 are so w e a k t h a t they have never b e e n seen i n N M R studies i n solids. W e m a y therefore 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 H= Q  term  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 .  One can show  [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  43  Chapter 2. Effective Hamiltonians  spin subspace as t h e f o r m N  eQk  (2.30)  6/ (2/ -l) f c  f c  where N is t h e t o t a l n u m b e r o f nuclei, Q2yk  (2.31)  dx dxP a  where V is the p o t e n t i a l a t t h e k nucleus d u e t o the charge d i s t r i b u t i o n i n t h e molecule, k  th  (2.32) a n d Qk is 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 k 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 th  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 I = 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 k  - 2 4  cm . I n Fe , t h e n u c l e i 2  8  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 Br a n d 79  Br,  81  1 4  JV and 0 . 1 6  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 nuclear charge distributions i n t h e molecule.  We now turn t o the second part of the  electron-nucleus interaction, the magnetic couplings. W e have already used the fact that stationary nuclear states have a fixed parity t o eliminate certain terms i n t h e multipole expansion o f t h e electrostatic coupling. W e m a y perform a s i m i l a r trick here b y n o t i n g t h a t t h e m a g n e t i c field feels 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 axial a n d polar vectors respectively). T h i s means that a l l even orders o f t h e multipole expansion o f t h e magnetic structure o f t h e nucleus have t o b e zero. 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 case, 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 ) terms h a s ever b e e n d i r e c t l y seen b y N M R i n b u l k matter. T h i s m a k e s o u r j o b considerably easier f r o m t h e start, as we c a n treat t h e general nucleus as a magnetic d i p o l e w i t h o u t loss of generality, a n d the interaction of the magnetic  Chapter 2. Effective Hamiltonians  44  field generated by the electrons a n d any m a g n e t i c d i p o l e c a n easily be d e a l t with.  One  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 47r  [3 • h  1  • r )(/* • r )]  - 3$  Ifc  (2.33)  lk  1=1 fc=i lk r  w h e r e we 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 is o v e r t h e e i g h t Fe  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 sites is 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 is  3+  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 Fe 57  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 Fe n u c l e u s o n t h e s a m e site, as t h i s t e r m is i n c l u d e d i n 57  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 is a " c o n t a c t " t e r m ) . g  nic  a n d u. a r e n  t h e n u c l e a r p - f a c t o r o f t h e k 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 . It is p o s s i b l e th  to rewrite this term by defining a m a t r i x 1  M {r ) lk  lk  =  ——-3-  ~ Sffkx  -3f r lkx  -3r r lkx  lkx  1 - 3rf  lky  ky  ' Ik  -3rikx?ikz  -3f f  lky  -3fi fikz ky  lkz  -2>ri rik z ky  (2.34)  1 - 3ff  kz  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  =£ £ 1=1  Qn^M^  (2.35)  S J* l  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 , as 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 the i r o n ions w h i c h are k n o w n f r o m crystallographic d a t a (this will t u r n out to be quite 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) we 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 " (ie. 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 Fe m o l e c u l e . W e m a y w r i t e 8  Chapter 2. Effective  45  Hamiltonians  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 S p i n Couplings  A similar analysis may be p e r f o r m e d on the electromagnetic couplings between the nuclei themselves.  A s was 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 effect 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 (2.37) where g a n d g are the nuclear ^-factors for the I a n d k species of nuclei. ni  2.4.4  nk  Couplings of the Nuclear B a t h to E x t e r n a l M a g n e t i c Fields  We shall a s s u m e that the nuclei i n the ligand bath couple to an a p p l i e d external 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 is, (2.38)  H xt = 2\^9n PnIk ' ext H k e  2.4.5  k  (  C o u p l i n g to Phonons  B o t h the super-exchange/exchange and nuclear spin couplings are l o c a l i n the sense that 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 is e n o u g h t o t r e a t t h e i r effects 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 we 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 as s u c h owe 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 presence of p h o n o n s i n the crystal can be dealt w i t h by i n t r o d u c i n g s t r a i n and 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 as [92] UJ,  = -  [d Uj a  -  d it ) 7  Q  46  Chapter 2. Effective Hamiltonians  where MO  =  1 1/2  h  £  <4 [4A A)  kX  +  O*A]  are 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 are g i v e n b y {a\. ,ak\} a n d  operators  is 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  x  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 is t h e m a s s o f t h e u n i t c e l l a n d N is t h e t o t a l n u m b e r o f u n i t cells. T h e s e local strains and rotations cause there to be introduced into the  Hamiltonian  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 effect o f a l o c a l r o t a t i o n of a particular spin on the  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  DijSiSj  D SS ZZ  Z  Z  component; S  Z  yS  Z  + U) S ZX  (2.39)  + U) ySy  X  Z  and D S  2  ZZ  Z  -> D  (s  2  zz  + u  2 zx  + u  2 y  u  z x  {S , S} z  x  + u {S , z y  z  S }) y  (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 1/2  h 2NMu _  (2.41)  n H  qX  where V (S ) qX  t  = B%S?S?  + (4  x q) • St  H e r e ^ is 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  (2.42) can in principle  be calculated f r o m k n o w l e d g e of the c r y s t a l s y m m e t r y and have b e e n estimated to be ~ 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  47  Chapter 2. Effective Hamiltonians  this one must use t h e symmetries appropriate for t h e i n d i v i d u a l ions a n d n o t t h e full — #  s y m m e t r y o f t h e crystal. T h e t e r m linear i n S arises f r o m t r e a t i n g t h e s p i n as 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 effects. T h i s t e r m is 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 B p 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 ) . a  2.4.6  C o u p l i n g 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  = Vxl=E  27^  1 / 2  ^ (c  + y ( V x 4 )  f  Vukx  w  C  (2.43)  w h e r e V is t h e v o l u m e o f t h e s a m p l e , {c\ , c \} 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 x  k  f o r t h e p h o t o n field a n d ^ is 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 A  t o t h e e l e c t r o n i c s p i n s is 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 H = J2gp B -S sl  2.4.7  B  1  (2.44)  l  B r i n g i n g all the Terms T o g e t h e r - T h e B a r e Fe  Hamiltonian  8  T h e description of the systems under scrutiny given in t h e previous section contains seven 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 (2.44). E x p l i c i t l y w e have, for o u r effective H a m i l t o n i a n , a n expression o f t h e f o r m H  = J2[G? > > *'*(ii) a  +  9 Pn[A S^ pa0  np  a  a  S^ S S^ +gp S -H p  a2  + R^H I^}]+ a  a3  £ p<q=l  +  N+8  E  fc=i  a5  B  p  <tfS>S}  48  Chapter 2. Effective Hamiltonians  47r  J<fc=l  ife  8  9A  l-l  fc=l  r  1/2  h  9 [4A^A(S<) - a*V&(3)]  2NMu  +£  g^B,  (2.45)  •S  t  /=i  gX  L e t us r e v i e w w h a t we 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 is a s u m o v e r a l l t h e s i n g l e i o n Fe  spin Hamiltonians. In order to evaluate the c o u p l i n g energies  3+  C7p  l Q 2 Q 3 Q 4 Q 5  we c o u l d d o t h e f o l l o w i n g . F i r s t , we 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 (as we 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 (2.22)). W e t h e n p i c k a s e t o f axes, a l i g n e d s o as 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 we 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 us 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 ^ Fes-  1 0 2 0  ^ " ^ for 4 0  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 are, a l t h o u g h  as m e n t i o n e d e a r l i e r f o r Fe  3+  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].  We shall not explicitly perform this task for a reason t h a t will soon be m a d e clear. 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  we  h a v e a p p r o x i m a t e d as b e i n g i s o t r o p i c (g ^ ~ 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 a  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 nuclear spin. T h e s e two last will not i n general be isotropic. T h e second t e r m is the exchange/superexchange  t e r m c o u p l i n g the e l e c t r o n i c wave-  functions of the i r o n ions. 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 how large the 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 is 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 is i m p o r t a n t . A s we 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 Fe  3+  electronic spins, nuclear-nuclear dipole interactions a n d nuclear spin-external  Chapter 2. Effective Hamiltonians  49  field c o u p l i n g respectively. A l l of the q u a n t i t i e s i n these t e r m are either k n o w n or c a n b e c a l c u l a t e d (we 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 is 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 we d o n o t k n o w i s t h e v a l u e o f t h e B p t e r m s i n (1.21), 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 a  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 is 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 G i a n t Spin P i c t u r e  A s was discussed i n the i n t r o d u c t o r y chapter, the low-energy p h e n o m e n o l o g y of the F e system indicates that somehow the electronic spins "lock together" into some object. Because the exchange/superexchange  8  fixed-spin  coupling energies are m u c h larger t h a n all  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), we 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 we 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 Fe  align parallel to each other while the other  3+  two align themselves anti-parallel, 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 excess s p i n of 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 is o n t h e o r d e r o f AE ~ 30K.  T h i s locking together of the electronic spins profoundly affects the ultimate form of 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 , as 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 as l o n g as k T <C AE (we 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 B  50  Chapter 2. Effective Hamiltonians  t h e mK r a n g e so t h i s i s r e a s o n a b l e ) , we 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 we 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 this does not m e a n t h a t the d y n a m i c s of the electronic spins are f r o z e n - i t is 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 is, 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 is s t i l l v e r y m u c h a d y n a m i c a l Now  quantity.  i t is 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  of this new collective degree of f r e e d o m f r o m the H a m i l t o n i a n  (2.45). 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  then  perform perturbations out of this g r o u n d state i n a similar way d o n e for the single ion case. I n s t e a d w h a t we s h a l l d o is, 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 . We  simply assume that the exchange/superexchange couplings lock the electronic  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 (two) 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, as i n d i c a t e d b y [18]. T h i s we r e f e r t o as t h e giant spin hypothesis. W e t h e n , as 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  S ^ A . + S ^ ' E ^+ E  i  P=I  N+8 r 8  2 47r  „  Af  „  l<k=l  „  I  .  lk  N  1/2  °U *&) V  /=i  qX  • St  w h e r e S . — S , S , S 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 is r e p l a c e t h e s u m x  y  z  single-ion spin Hamiltonians Hamiltonian  H  GS  —  W +  fc=l  T  1=1 2NMu _  ai  ^  n  .  h qX  ^  p=i  (2.46) over  and the exchange/superexchange terms with a giant spin  G - ™S S ...S . ail a{  aii  ai2  ahQ  A s we d i s c u s s e d i n t h e s i n g l e i o n case, 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 2S 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 th  ^  Chapter 2. Effective Hamiltonians  i n t h i s c a s e is 20  51  order. We can however reduce the t o t a l n u m b e r of t e r m s significantly  i n t h e c a s e o f Fe , 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 8  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 Fe 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 we  8  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 we saw 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 effort has b e e n e x p e n d e d i n t r y i n g to  measure  what the relevant couplings  <5 i - *> Qi  ai  are.  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 so m a n y o f these, 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 as 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 is h o w e v e r q u i t e d a n g e r o u s as t h e s e s m a l l i g n o r e d  terms can contribute greatly to the physics, i n particular to the m a g n i t u d e of tunneling 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 for 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 are identical to their 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 is t h a t  n o w we 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 is 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 H a m i l t o n i a n i n the Absence of E n v i r o n mental Couplings  If we 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 H  =r>i-^o  in  S S ...S  (2.46)  aii  ai2  ai2Q  a n dfixS = 53?= i + gp H B  Si  • S  we o b t a i n (2.47)  We are going to digress s o m e w h a t at this point f r o m our focus on F e . We will i n w h a t 8  follows consider all possible crystal symmetries and not just the F e T h i s we 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  triclinic symmetry.  8  (Mni , 2  for example) that  may have giant s p i n descriptions of this sort t h a t possess different c r y s t a l s y m m e t r i e s (for  Mni2  this is t e t r a g o n a l  things about the description  [28])  a n d for this reason i t is worthwhile to say s o m e general  (2.47).  Chapter 2. Effective  52  Hamiltonians  N o w as we 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  2S  th  order  i n t h e s p i n v a r i a b l e s . It is 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 is 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 is 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 we 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 still significantly affect the physics, i n p a r t i c u l a r the a m p l i t u d e s o f the tunneling matrix elements between states of the giant spin. 2.6.1  E x a c t Solution for Tunneling M a t r i x Elements v i a 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 as 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 is as 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 as 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}.  In the specific cases of the  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 we 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 we 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 we 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 we t h e n p l o t as 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  C u b i c System  A crystal 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 possesses a s p i n H a m i l t o n i a n o b e y i n g the symmetries [S —^Sy x  Sy —> — S ] x  ,  [S  x  S  z  S —y —S ] z  x  ,  [Sy  S  z  S —> ~Sy] z  53  Chapter 2. Effective Hamiltonians  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 we o n l y i n c l u d e t h e t w o lowest order terms o b e y i n g these symmetries the giant s p i n H a m i l t o n i a n c a n be w r i t t e n [96] H=-D  (S + S + St) + E [St + S« + S + WS S S ) + gu. S • H 4  x  6  y  y  2  Z  2  x  2  y  B  (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 is 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 we 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 ) is 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 we see 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 shall not say m o r e a b o u t this crystal symmetry. However, it is w o r t h n o t i n g t h a t the physics of cubic molecular magnets should be particularly entertaining because of the lack of an easy/hard axis. T h e Tetragonal System  In systems with tetragonal symmetry, the symmetries [S ^S x  S ^-S ]  y  y  x  ,[S ^-S ] Z  Z  (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 Hamiltonian of the form H = -DS + a (Si + Si) + gu. S • H 2  4  B  (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 is k n o w n as K r a m e r ' s d e g e n e r a c y [47, 97], a n d it arises here because there exists no p a t h by which our H a m i l t o n i a n can connect t h e s t a t e s \S > a n d | — S > i f S is o d d .  54  Chapter 2. Effective Hamiltonians  171 (i)  (ii) rtV  fl  a (iii)  J^b  (iv) (vi)  (vii)  O (v)  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 , (ii) 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 . T h e O r t h o r h o m b i c System  In systems w i t h orthorhombic symmetry, the [S  X  y  S  x  Sy  y  symmetries Sy  S  y  z  S  z  (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 tonian H = -DS  + a (5  2  2 Z  2  + S _) + x S 2  gf  B  •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 parameter  regimes.  Chapter 2. Effective Hamiltonians  55  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 left, |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±S /D a n d o n t h e y a x i s l o g &s,-s- H e r e we 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 zero. 2  1 0  -5.0  1  '  '  0.0 0.5  0.0  |  1.0 ,  '  1.5  '  '  -5.0  1  ,  ,  1  '  1  2.0 2.5 3.0  0.0 0.5 0.0  |  '  1.0  '  '  1  1  1.5 2.0 2.5 3.0 1  F i g u r e 2.4: V a r i a t i o n o f A , - s w i t h H /D f o r a S /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 left, |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 H /DS a n d o n t h e y a x i s l o g As,-s2  5  x  4  2  X  1 0  Chapter 2.  The  56  Effective Hamiltonians  M o n o c l i n i c 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 as 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 . Because of this the spin H a m i l t o n i a n for these systems m u s t c o n t a i n the orthorhombic s y m m e t r i e s . I n a d d i t i o n we 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 = -DS The  + a ( S + S _) + a^S* + a^St 2  2 Z  + gu. S • H  2  2  B  (2.53)  Triclinic System  Triclinic s y m m e t r y is o b t a i n e d v i a a distortion of m o n o c l i n i c symmetry. A s such the 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 case. I n a d d i t i o n , we 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 = -DS  2  The  - DS  + a (S + S _) + o j  4  Z  0  2  Z  2  + )  2  +  S j + a^Si  + gn. S • H B  (2.54)  Trigonal System  In systems w i t h t r i g o n a l symmetry, rotations a r o u n d the b o d y d i a g o n a l are three-fold symmetric. H = -DS  + a {S ,Sl  2  The  3  z  (2.55)  + S _}+ gp S • H 3  B  Hexagonal System  In systems w i t h hexagonal symmetry, the symmetries S -> e^ S ]  ,  3  x  x  [S -> e^ S ] 3  y  y  ,  [S -> -S ] z  z  (2.56)  m u s t be preserved. T h i s implies, keeping only the lowest order s p i n terms, a H a m i l t o n i a n H = -DS  2 Z  + a, (S« + S*) + g» S B  •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 parameter regimes.  Chapter 2. Effective Hamiltonians  ' 0.0  57  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 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 left, |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 a /D a n d o n t h e y a x i s l o g A$-s- H e r e we 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 zero. 2  2  1 0  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 H /D f o r a /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 left, \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 is p l o t t e d H /DS a n d o n t h e y a x i s l o g As -sx  2  2  X  1 0  58  Chapter 2. Effective Hamiltonians  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 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 left, \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 a S /D a n d o n t h e y a x i s l o g As,-s- H e r e we 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 zero. 6  i  6  1 0  F i g u r e 2.8: V a r i a t i o n o f A - w i t h H /D f o r a S /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 left, \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 is p l o t t e d H /DS a n d o n t h e y a x i s l o g As,-s4  S  S  x  6  2  X  1 0  Chapter 2. Effective  2.6.2  59  Hamiltonians  Tunneling M a t r i x Elements v i a Perturbation T h e o r y  The calculation of tunneling matrix elements v i a perturbation theory in the terms that 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 as l o n g as t h e r e is 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 cases we 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].  Hamiltonian  A -s S l  -DS  + a (S  -DS  + a (S  -DS  + a (S% + St)  2  2  2  «f(2S)!  + S _) 2  +  4S-i£>s-i[( _i)!]* S  af (2S)! /2  2  4  + S _) 4  +  4  S-2 S/2-l[( y _ )l]2 D  S  2  1  af (2S)! /3  2  4S-sz)S/3-i[(s/3-l)!]'  -DS  6  2  +H  x  2  H* (2S) S  ( 5 + + S-)  £,2S-l[(2S_i)!]  al (2S)\ S/3  -DS  2  + a {S ,(Sl 3  z  + S _)} 3  Z  )2S/3-l 2S/3-2[(5/ _ )!]2 6  3  1  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 the n u m b e r of s y m m e t r y b r e a k i n g t e r m s i n the H a m i l t o n i a n is increased, the solution for A using perturbation theory becomes a little m o r e complicated.  T h i s is  because of the c o m p e t i t i o n between these t e r m s i n d e c i d i n g w h i c h are the preferred paths between the low l y i n g states.  Chapter 2. Effective  2.6.3  60  Hamiltonians  T u n n e l i n g M a t r i x Elements via W K B M e t h o d s  It h a s b e e n s h o w n [25, 26] t h a t t h e  Hamiltonian  H = -DS + a (S + S _) 2  k  k  (2.58)  k  +  leads t o  ,_ ^p^!) "  „  2 S  A s  as l o n g a s a S <C k  k  2.6.4  DS  2  s  a n d the ambient energy  E  is s u c h t h a t  <C  E  5 9 )  DS . 2  Tunneling M a t r i x 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 of solution for tunneling m a t r i x elements i n spin Hamil-  tonians that we shall consider involves using instanton techniques.  W e will explicitly  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 thorhombic  (ie. 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 (2.52)) ^s  =  ^S^ (D-2a f a exp 2  4  2  (2.60)  1 /4 2  V  a  2  .  T h e same type o f procedure m a y be followed i n principle with a n y spin  Hamiltonian  t h a t possesses 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 practice o n e runs u p against p r o b l e m s w i t h a l l but the easiest q u a d r a t i c s p i n terms.  We  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 left 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  C o m p a r i s o n of A p p r o x i m a t e M e t h o d s to E x a c t Solutions  In order t o give s o m e idea about h o w effective these different a p p r o x i m a t i o n  schemes  a c t u a l l y are, 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 for t h e tetragonal a n d orthorhombic  crystal symmetries.  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 exact s o l u t i o n for the t u n n e l i n g splitting between the two lowest levels of the 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 is a /D, a n d o n t h e v e r t i c a l a x i s l o g 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 ; Red, perturbation theory and Blue, WKB. 2  1 0  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 for the 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 the two lowest levels of the H a m i l t o n i a n of tetragonal 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 is a S /D, a n d o n t h e v e r t i c a l a x i s l o g 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 . 2  4  1 0  Chapter 2. Effective  comparison  62  Hamiltonians  for the orthorhombic  case. H e r e we see t h a t , as we 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 a /D is s m a l l . T h i s i s s i m p l y b e c a u s e 2  w h e n we 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 we a s s u m e d t h a t t h e  fluctuations  around  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 a /D i s s m a l l . 2  a /D 2  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 is q u i t e g o o d . T h e W K B  the When  s o l u t i o n we see is q u i t e  b a d . It 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 a /D is c o r r e c t b u t t h e p r e f a c t o r is n o t . 2  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 a /D < 0.5 s t u d i e d . 2  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 case, s h o w n i n figure 2.10. H e r e we see a similar story. T h e W K B  s o l u t i o n gets 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  is 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 2.7  regime.  Back to the 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 of Tunneling E n e r g y Scale U s ing  an Instanton Technique  A s was 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 is s i m p l y n o t feasible to measure a l l of the non-zero c o m p o n e n t s of the tensor  <5 i Qi  - 2o ai  experimentally.  B e c a u s e o f t h i s w h a t is 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 ofite x p e r i m e n t a l  data.  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 we s h a l l a d o p t is H  = G i~ «>S ...Si ai  GS  T h i s form, with  ai  il  D =  = -DS  2  20  0.292A",  + E(S  2  Z  E = 0.046K  and  + S _) + C{Si 2  +  C =  +S  )  4 +  (2.61)  -2.1 • 10~ K, is g o o d e n o u g h t o 5  a c c u r a t e l yfitb 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 is c l e a r t h a t a  large n u m b e r of t e r m s have s i m p l y b e e n c h o p p e d off the "true" s p i n H a m i l t o n i a n (for example, even the quartic diagonal spin t e r m has not been included).  Chapter 2. Effective  63  Hamiltonians  T h e effects of 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 at low e n o u g h temperatures completely neglected  can be  [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  like t h e i r respective densities i n the crystal, w h i c h are v a n i s h i n g l y s m a l l at low t e m p e r a t u r e s . If we 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 et.al. [14] we a r e  a t t e m p e r a t u r e s less t h a n ~ 360 mK, a n d t h e r e f o r e k T <C DS [20]. W e t h e r e f o r e w r i t e B  the q u a n t u m regime Hamiltonian in the form H  =  -DS +E(S  +  £ 9n ii [A^SPJI  2  + S _) + C(Si  2  Z  P  + Sl)  2  +  n  + R/ PH q] a  +  gu. H-S B  + f;  Q  u. H  9nk  n  • i  k  *;=i  P=I  N+8  + £ A;=l  + ^ 47r  9n linM .1=1 k  E Kk=l  lk  ^0  +  6 I k  (  2 I k  -l)  '"P  V  (2-62)  ^[?i-?k-3{fi lk  T  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 we 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 DS 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 is o f t h e 2  o r d e r o f 29 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 see t h a t if all ambient energies ( p r i m a r i l y the lattice temperature  a n d e x t e r n a l field) a r e m u c h  less 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(S — (S — l ) ) t h e g i a n t s p i n w i l l o n l y b e 2  2  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 S = ± 5 . A t t h i s l o w e n e r g y s c a l e we s h a l l Z  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 to a two-level system parametrized by a P a u l i m a t r i x r , where f = ± 1 corresponds 2  to  S = ± 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 < 360 mK Z  [14]. T h e r e is 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 H is 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 z  Chapter 2. Effective Hamiltonians  64  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 H s f o r t h e Fe 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 is 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 forbidden (this region i n s h a d e d grey in the above). G  figure  8  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  original states and a higher-energy state on the other side of the barrier (for example, the application of a positive H could lead to a resonance c o n d i t i o n between | + 1 0 > and z  | — 9 >). I n t h i s case, 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 levels. 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 here, as 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 ,  65  Chapter 2. Effective Hamiltonians  i n w h a t f o l l o w s we 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 is s u c h t h a t H « C DS a n d o n l y t h e t w o l e v e l s | ± S > are  involved,  z  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 we 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 to larger values of H. z  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 S ( 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 ) z  a n d t e r m s that are not diagonal i n S (and therefore c o m i n g i n t o play only w h e n the z  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 is 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 (we w i l l s h o w t h i s ) that the time scale for tunneling physics = A  - 1  is m u c h smaller t h a n the t i m e between  t u n n e l i n g e v e n t s = QQ [20]. T o see 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 1  argument. 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 excess 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 is m u c h longer t h a n the time over w h i c h t u n n e l i n g occurs. We therefore expect the central 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 of scales" allows one to consider the effective d e s c r i p t i o n separately i n two different regimes; one i n the regions between t u n n e l i n g events a n d one d u r i n g the tunneling. 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 we s h a l l r e f e r t o as t h e " d i a g o n a l " r e g i o n (as 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 we 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 that the diagonal c o n t r i b u t i o n can be written JV+8  +  (±) 9n Vn P  z  p  N p=l  gp, SH f B  z z  fc-i  p=l  fc=l  N k=l  eQk L 6/ (2/ - 1) fc  fc  -\rkaf) jk Q/?  Chapter 2. Effective  +  (2.63)  J2°^[l I -S(I r )(I -r ) ttk lk r k  47r  r  lk  k  lk  r  where ( ± ) is shorthand  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 p i o n i c s p i n i s p o i n t i n g (see th  p  figure  66  Hamiltonians  2.1). O u r l a b e l l i n g s y s t e m h a s ( ± ) = 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 p  a n d ( ± ) = — 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 is, t — + 1 ) p  z  with signs reversed i f t h e central spin is "down". W e m a y rewrite this i n the form H  =  D  N  'l + f  z  T^E^-/* H Z ,E T T" •h +  +  P  fc=i  N  +  gu. SH f B  z z  'i-f  2  E7  N  2c) P  -4 + E 7 i - 4  p=l  2)  fc=l  eQ  k  + E k=l  "64(24 - 1)  «» (2.64)  4 7 r  Kk  lk  r  where (2.65) (2.66)  E -.(2)  -.(2)  5  g  7fc = 7fcs + 2^nH  p  5  = -g Vn nk  < - E < + ^  Lpe{pt>  pe{P4.}  • E  E  pe{p } T  M$ + HP  (2.67)  (2.68)  pe{p±}  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 f3 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 th  s p o n d s t o t h e (5 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 left. th  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  67  Chapter 2. Effective Hamiltonians  c o m p l e x flips. 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 feels 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 g o i n g f r o m 7 ^ —> 7 ^ a n d 7^ ^ —>  rf \ c  7 ^ ,  7 ^  k  and  7  2 c f c  ^.  T h e sign changes i n  come from the fact that the central spin object  lc  has reversed its d i r e c t i o n along t h e easy axis. 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 +T  k  (2.69)  4= 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 -.(1c) _ Ik  m =  A2c) Ik  (2.70)  k  f o r k l a b e l l i n g t h e p o s s i b l e Fe 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 57  (2.71) f o r l i g a n d s p i n s k = 1..N a n d i f I  II  I -.(lc)  u = IhlgnMTk k  -.(2c) \  ±  -7fcI ,  i-*(lc)  ,f\  u = \I \gn MTk k  k  k  ,  -,(2c),  +%  (2.72)  I  f o r Fe s p i n s k = N + 1..N + 8. W i t h t h e s e we c a n w r i t e 57  N+S  —I • m + —uU k  k  k  •l  k  +  gu. SH f B  z z  k=l N  E  r  eQ  k  T/fca/3 jk  +  POPN  9ni9n  k  An  h •I k  Kk  • r )(I lk  k  • f )] lk  'Ik  (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]. It d i f f e r s i n t w o r e s p e c t s . Firstly, it shows explicitly w h a t t h e energies  uJ ,ui k  k  a n d u n i t v e c t o r s m ,l k  k  are i n terms  of parameters i n the higher energy descriptions ( a n d allows us t o calculate these-we shall  68  Chapter 2. Effective Hamiltonians  |-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 excess s p i n S(t) a l o n g t h e e a s y - a x i s . W e see 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 scales; 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 is 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 . do this i n the next chapter). Secondly, i t includes the effect of q u a d r u p o l a r  couplings  between higher spin nuclei i n the ligand b a t h and electric field gradients i n the  molecule.  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 is s o m e w h a t h a r d e r t o extract, and a different a p p r o a c h will be required. 2.8  Off-Diagonal Terms and the Instanton M e t h o d  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 et.al. [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 , ie. 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 object tunnels. Two  objections have recently been raised w h i c h question the validity  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 we 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 objections.  69  Chapter 2. Effective Hamiltonians  2.8.1  Review of the M e t h o d of T u p i t s y n 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 is, 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 approach to the F e  m o l e c u l e t h a t we h a v e a d o p t e d , we h a v e c h o s e n a m o d e l w h e r e a l l  8  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 lies 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 as 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 out w h e n this description m u s t be modified because of the "true" eight spin nature of the object. —*  — +  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 >, ie. 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 these states can be defined to be Tapit) =  w h e r e L (T) 0  a n d LN{T)  1^ D(9, <j>) e x p  (-  j*  dr ( L 0 ( r ) + I ^ f r ) ) )  are the Lagrangians corresponding to the bare s p i n  (2.74) Hamiltonian  p l u s t h e e x t e r n a l field t e r m H = -DS] + E(S  + Si) + C(S* + Si) +gu, S-H  2  0  +  B  (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 so as 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 way; t h a t is, S = \S\ c o s <j) s i n 6 x  , S = \S\sin y  <f> s i n 9  , S = \S\ c o s 9  (2.76)  z  If we 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 afig i s m u c h s m a l l e r t h a n t h e 1  t i m e b e t w e e n t r a n s i t i o n s A g , the evolution operator connecting the two m i n i m a will be l  given by {H ) ND  afi  =  f (t)  t  afl  ,  (a^/?,Q - «i«A - ) 1  0  1  0  (2.77)  70  Chapter 2. Effective Hamiltonians  where 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 we a r e l o o k i n g f o r .  T h i s means that in order to calculate H  ND  i t s u f f i c e s t o c a l c u l a t e T g(t), a n d t h i s we a  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 (2.74). 2.8.2  T h e 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 we s h a l l e x p l i c i t l y use 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 tonian, vis. H — -(D - 2E)S + AES 2  0  (2.78)  2  Z  X  H o w e v e r , t h e t a c t i c s we 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  admitting  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 Fe i f t h e q u a r t i c 8  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 S = S + S + S we c a n 2  2  2  2  s h o w t h a t (2.78) i s e q u i v a l e n t t o H = -DS  2  0  + E(S  2  Z  + S _) 2  +  (2.79)  T h i s t r u n c a t i o n is 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 , as 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 we d i s c u s s e d e a r l i e r i n this chapter, the "true" spin H a m i l t o n i a n contains m a n y t e r m s that are inaccessible experimentally, and therefore dropping the quartic term simply emphasizes the point t h a t i n u s i n g a f o r m l i k e (2.78) we 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  description-the  effect 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 is 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 is 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  71  Chapter 2. Effective Hamiltonians  are 20sin0-M£Ssin 0sin2<£ = O  (2.80)  2  and + c o s 0) s i n 29 = 0  (j) s i n 9 - i4ES{  (2.81)  2  Solutions of the classical equations of m o t i o n are s i m p l y ^  =  J  7  h -i M E> _ ^— _ 2E  . ,  _7r _  t  s  i  n  (2.82)  and 0(t) = sind(t) = ——— w w  cosh Q t  ,  - 1 Q = 4£Ssinh 2 sinh 0  0  ID-2E AE  (2.83)  where n = ± labels rotations clockwise a n d counter-clockwise i n the easy plane. N o w i n the s t a n d a r d treatment, o n e assumes that the classical equations of m o t i o n are attractors i n t h e sense 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 is t r u e t h e n one c a n perform a gaussian integration over small  fluctuations  away from t h e classical  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 et.al. [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)  8(f) = -d>ESsm(2smh.-  1  \ V  OTP  ) c o s 9 5(j>  (2.84)  AE  N o w w e see 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> is 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 see 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 it  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 this is true it brings into question the validity of the gaussian i n t e g r a t i o n technique.  Chapter 2. Effective Hamiltonians  2.8.3  72  A n Assumption is M a d e  I n w h a t f o l l o w s we 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 we 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 .  As  U n r u h h a s p o i n t e d o u t [101] i t is 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 where  <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 et.al. [74]. F u r t h e r m o r e  have that  i n our case  we  T h a t t h i s n u m b e r is 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>  = 2.67.  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 steep, so t h a t t h e a s s u m p t i o n  that gaussian  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 is 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 solution to the instanton solution for the tunneling amplitude show g o o d agreement in t h i s c a s e - s e e figures 2.9, 6.1 a n d 6.2). T h e 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 L  0  =  -iS(j)6 sind+ (D-2E)S  sin 9+ 4ES sin 6 cos <f>  2  2  2  2  2  — S (H s i n 9 cos <f> + H s i n 9 s i n <j> + H c o s 9) x  y  z  and LN  = J2g fJin[A SlI pa0  +  p  np  p  R ^HJ^+J29n PnH-I p  k  p=l  k  k=l  N  + £  fe=i  +  ^ $ : ^ [ T i - T 4  7  r  l<k=\  k  - 3 ( I  _ 1 2  lk  k  (2.85)  • f )} lk  lk  T  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 T  • f ){I  t  is 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  1  is s u c h t h a t Q,Q flip-flop  1  <C T , _1  2  where  processes mediated by the  last t e r m i n the a b o v e expression occur. In this limit, the nuclear-nuclear t e r m provides  73  Chapter 2. Effective Hamiltonians  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 in the form =  LN  E 9n Pn p  k=l  P=I  AT  + E  (2.86)  k=l  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 is 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 some contribution from the nuclear-nuclear term. In o u r m o d e l , each electronic s p i n is locked t o t h e central spin. T h i s m e a n s t h a t —*  (2.87) and therefore 8  N  E 9n PnA^SlIl p  p=l  8  +  9n PnM^S I P  k  k p  fc=lp=l  N  4  |_pe{p} T  pe{pi}  E  + E E 9n pJ 3 k=\ Lpe{ptl k  dripPn  k  MS-  E  Kk  pe{P4.}  (2.88) 2.8.4  Solution for the Free Instanton Trajectory  We assume that  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  instanton solution for orthorhombic  of t h e exact solution with the  s y m m e t r y we see t h a t i n this r e g i m e t h e i n s t a n t o n  solution is off b y a b o u t a factor o f t w o f r o m t h e exact 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 case. 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  74  Chapter 2. Effective Hamiltonians  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 is the angle b e t w e e n S a n d the z axis  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 is 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 is n o e x t e r n a l m a g n e t i c field i t is 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 we 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  L  eff  (6)  2  w h e r e we h a v e d e f i n e d E = AEJ2 \S \  2  p  +D sin 9  — —9 AE P  (2.89)  2  a n d D = (D - 2E) £  |S | , f o l l o w i n g [75]. T h e 2  p  P  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 is r e a d i l y f o u n d f r o m (2.89) a n d is 9 = s i n 9{T)  , Vto = f \[hl5  = 1/ c o s h ( £ V )  (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 here. F i r s t is 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 the gaussian integrations have been performed w i t h r e n o r m a l i z e d Q, . 0  is t h e s a m e as t h e b a r e c l a s s i c a l f o r m  S e c o n d is t h a t the p a r a m e t e r s D a n d E d e p e n d on the fact  t h a t we 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 = 2ES  2  = 2 0 0 £ , w h i l e f o r u s E = 2E £  \S \ = E • 8 • 25/4 = 1 0 0 £ ; l i k e w i s e f o r D. 2  p  P  S u b s t i t u t i n g this extremal t r a j e c t o r y into the effective L a g r a n g i a n a n d i n t e g r a t i n g over r gives for the instanton a c t i o n Aefj = A + innS 0  ,  A = 0  ID - 2E 2S^—^—  where n = ± corresponds to clockwise and counterclockwise paths respectively. N o t e that neither the H a l d a n e phase c o l l e c t i v e c e n t r a l s p i n [75].  TJTTS  nor A depends on the eight spin nature of the 0  75  Chapter 2. Effective Hamiltonians  2.8.5  Inclusion of the E x t e r n a l Magnetic F i e l d and the Nuclear Spins  We now make the assumption  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  environmental spins are weak. Specifically we require that Q,o  «  l  \H\  Clo  «  (2.91)  1  It is 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 w h e r e £ i s o n e o f u[\  uifc  0  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 effects ( w h i c h w e are) 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 (as 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 ) . It 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 in particular the assumption order i n t h e external  [101]-  that the trajectory o f the central spin responds t o second  fields.  S u b s t i t u t i o n o f the extremal trajectory into t h e general effective lagrangian a n d integ r a t i n g over r yields t h e following effective action, N+8  A  E  F  F  = A  0  + irjTcS -  ir,A  •H  H  +n £ k=l  A  K  N  D  (2.92)  •I  K  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  terms  are t h e c o n t r i b u t i o n d u e t o t h e presence o f the environmental spins. E x p l i c i t l y these a r e irSgu, „ .S -ngn -y-%-x 2E 2  B  Sir  4O  0  L b>e{ } PT  E  <k  (2.93)  B  ID-2E 4E  E  Kk  E  Mf  k  P€{p.|.}  (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 7»fc  In  /_|_\  V  '4^o f o r A; = N + I..N  + 8  - OF  .1 (2.95)  4E  r e p r e s e n t i n g t h e Fe i o n s . I n b o t h c a s e s t h e f3 c o m p o n e n t o f 57  th  t h e t e n s o r s o n t h e left 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 (3 c o m p o n e n t o f t h e v e c t o r s th  on t h e right h a n d side. T h e s e expressions s h o u l d be c o m p a r e d t o equations (2.28)  (2.27)  and  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 is 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 to the presence o f t h e nuclear spins have been explicitly written i n terms o f parameters in a higher energy H a m i l t o n i a n a n d the nuclear spins c a n have a r b i t r a r y spin numbers. 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 is 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 0  A  0  =  ^ y|^exp(-Ao)  (2.96)  0  A s we have seen earlier, this expression for A is off 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 0  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  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 sfig < < t « 1  (figure AQ ,  2.9).  the relationship between the  1  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 is H  ND  =  * ( _r (t) + h.c.) f  (2.97)  4t  where T ( r ) = itA i t  0  £ exp(-A  e / /  )  (2.98)  a n d f _ is 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 our expression  for  (2.92)  A ff e  a n d the criteria  (2.91)  allows us t o write t h e off-diagonal  part o f the effective H a m i l t o n i a n as N+8  H  ND  = 2 A f _ c o s ( $ - i Y, 0  fc=i  AN,D  • h) + h.c.  (2.99)  Chapter 2. Effective Hamiltonians  77  where (2.100)  $ = nS - An • H  2.9  T h e F i n a l Single Molecule Effective H a m i l t o n i a n  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 we 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 effective H a m i l t o n i a n is N+8  _L  H = £ !~/ • m + Y^l'/fc • Ik+ k=l  fc  Qk  + E  6/ (2J -ir fc  B  ^  yka0 jk  e  N  gu. SH f  z z  k  fc  a P  4?r  %|n.  + 2A0f_ cos($ -i £ A k=l  k  ^ ./ _ j;. fc  3 (  f  hk){  k  N>D  . ] fjfc)  • h) + h.c.  ( 2  1 Q 1 )  ra /fc r  A t t h i s jt=i p o i n t we r e i t e r a t e t h a t o b j e c t i o n s as 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 is a f f e c t e d is 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 .  We  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 we 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 we 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 as 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 approximations m a d e i n the instanton calculation are valid.  Chapter 3 N u c l e a r S p i n C o u p l i n g s i n Fe a n d t h e I s o t o p e E f f e c t 8  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 j g a n d %g k  ( 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 Fe  3+  ions at the k  th  nuclear  spin before a n d after t h e central spin c o m p l e x tunnels respectively. W e are going 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 first w i l l t r e a t e a c h Fe  3+  i o n i n t h e molecule as a  point m a g n e t i c dipole ( a n d as such we call this t h e "point dipole approximation").  The  s e c o n d m e t h o d we s h a l l u s e is 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 ions b y using previously calculated Hartree-Fock wavefunctions  f o r free Fe  3+  ions. T h i s  "spreading out" o f t h e m a g n e t i c dipole changes t h e values for t h e fields a t t h e nuclei. T h e s e fields (and therefore 7 ^ a n d 7 ^ ) are then used t o c a l c u l a t e the 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 u . 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 k  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 Fe  3+  g  f a c t o r is 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) } 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 k  t o t h e p r e s e n c e o f Fe 57  3+  ions, 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,  the  topological decoherence parameter A a n d the full linewidth o f t h e nuclear spins W for arbitrary isotopic concentration.  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 effect.  78  79  Chapter 3. Nuclear Spin Couplings in Fe and the Isotope Effect 8  3.1  Units and  Constants  We choose to w o r k i n the SI system of units. We therefore h a v e t h a t  T- =  1  0  "  W  7  A  2  47T  T h e B o h r M a g n e t o n is  HB = 0.9271203 • 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 is fx = 0.505038 • l O "  ^ = 7.622462  2 6  n  (3.3)  T h e p r o t o n a n d electronic i r o n g-factors are g = 5.58510  , g =2  pr  (3.4)  Fe  H e r e we 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  Fe  3+  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 we s h a l l use a r e 20.837 GHz = 1 K = 1.3807 • 1 0 ~ 3.2  The Point Dipole  2 3  J = 0.695045 c m  - 1  = 8.617 • 1 0 "  5  eV  (3.5)  Approximation  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 we 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 as 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 which will o n l y be useful if the s p a t i a l extent of the i r o n wavefunctions is m u c h less 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 careful treatment in the following section and compare its results to those obtained using the point dipole approximation.  Chapter 3. Nuclear Spin Couplings in Fe%  3.2.1  80  and the Isotope Effect  M a g n e t i c F i e l d 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 is 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 is m = QFePBS = 9FePBSS  (3.7)  T h e r e f o r e w e find t h a t l(r) = -^-^g PBs[s-S(s-f)r}  (3.8)  Fe  w h i c h is 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 is m e a s u r e d i n A n g s t r o m s . 3.2.2  M a g n e t i c F i e l d at f  due to E i g h t "Point Dipoles" at f  p  Fei  Define the vector joining the p nucleus a n d j th  th  8  i r o n i o n t o b e f j = f — fj. T h e n t h e P  p  t o t a l field a t t h e p 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 th  is 7 ( r ) = 4.636 £ p  j=i \ pj\ r  [3(S • f )f i  pj  pj  -  T  (3.10)  where again distances are measured i n Angstroms. 3.2.3  Isotopic Concentrations, Nuclear ^-factors a n d Q u a d r u p o l a r M o m e n t s in  Fe  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 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 gn \I\ a n d a r e 8  listed i n units o f nuclear magnetons.  n  Chapter 3. Nuclear Spin Couplings in F e  Species H H  1  2  13  C  N N  U  15  16  0  0  1 7  1SQ  5 7  Fe  si  B r  Concentration  \f\  99.98 0.02 98.88 1.12 99.62 0.38 99.757 0.039 0.204 91.068 2.20 50.56 49.47  1/2 1 0 1/2 1 1/2 0 5/2 0 0 1/2 3/2 3/2  8  81  and the Isotope Effect  N u c l e a r M o m e n t [// ] Q u a d r u p o l e M o m e n t Q [10 n  2.79255 0.857354 0 0.70225 0.40365 -0.2830 0 -1.8935 0 0 0.05 2.10576 2.2696  2 4  cm ]  0 0.00273 0 0 0.02 0 0 -0.005 0 0 0 0.335 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 trations. It is quite possible t o alter these concentrations a n d as s u c h i n w h a t follows we shall treat t h e general case where t h e p a r t i c u l a r isotopic concentrations i n t h e material may be varied. 3.2.4  Definition and Evaluation of  ,  ui' and u£ l  k  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 f 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 k  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 (see (2.67) a n d (2.68)). 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 we 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 ) .  2  82  Chapter 3. Nuclear Spin Couplings in Fe$ and the Isotope Effect  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 r w i t h t h e e x t e r n a l f i e l d is k  Uk  -  9nPnh  • Ik  l+f  z  [g pJk•  ik ]  QnPri  + ^-y^  ]  n  [gnPnh  •  fi ] 2)  [^•(#+€ )+U-(7i -#)] )  (3-11)  1 )  A s before we take -  -(1)  -(2)  _ 7fc  -,(1) _ ^(2)  + Ik  f _ 7fc  * l7i"+#l  '  _  4'=  I^WTI '-  € ' I  1  Ik  fi-\o\  ' " i T i " " ^ !  '  1  4 = l4l9«M»l7i + 7 f l  (3-13)  1)  .  '  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 (2.73)) Uk = ^ik-m  +  k  3.2.5  ^4h-i  (3.14)  k  Contact Hyperfine Coupling Energies for  5 7  Fe  3 +  T h e r e is 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 Fe 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 is o f t h e 57  f o r m (see (2.61)) U  c p  = gn^A^SVl  (3.15)  W e are going 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 follows t h a t t h e off-diagonal elements of the tensor A  a r e zero. T h i s g i v e s t h e f o r m  pal3  (3.16)  U; = u I -S c  p  p  p  where S a n d I are the electronic a n d nuclear spin of t h e p  th  p  p  57  Fe i o n r e s p e c t i v e l y .  Similarly t o w h a t we d i d with t h e dipolar term we write, w i t h S ^ a n d 5 p  2 ) p  the p  th  Chapter 3. Nuclear Spin Couplings in F e  8  83  and the Isotope Effect  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;i -s p  f  [T  p  • (S^ + S f ) + f I  P  z  • (SM - Si ))]  (3.17)  2  p  Define 5(1) + 5(2) P ~ I 5(1) , 5(2) i  5(1) _ 5(2)  „ I  I  Op  ^-,5(1)  '  5(2)  I^P  4 =\IPK\SU-SW\ c  ,  u,}*  I  I  = \T \ ;\%» P U  '  (3.19)  +  T h e n we 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 U  c p  = ^I  •m  c  p  p  + ^J\% • %  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 free  5 7  Fe  (3.20)  ion due to polarization of the s  3 +  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 was f o u n d t o b e H ~ 63 T [99]. I f we 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 c  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;  | c p  ~ 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 ^ k=N+i..N+8^ an< u)  as 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 300 — 800 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 is 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  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  + 8 are involved i n  F e n u c l e i k = N + 1..N  + 8 also  are involved i n contact hyperfine interactions w h i c h absolutely s w a m p the dipole-dipole  84  Chapter 3. Nuclear Spin Couplings in Fe$ and the Isotope Effect  i n t e r a c t i o n s o f t h e Fe 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 57  N+8  (3.22)  k=l  w h e r e i t is 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 t h e t e r m s f o r k = N + 1..N 3.2.6  + 8 a r e c o n t a c t t e r m s f r o m (3.20).  Calculation of u  k  I n t h e Fes  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  and u)  k  from Knowledge of A t o m i c Positions  m o l e c u l e , we 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 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 w a n d u)^ k  T h i s a l l o w s us  are, f o r k = I..N, v i a t h e use 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 is c a l c u l a t e d as f o l l o w s . T h e central spin H a m i l t o n i a n for F e , w h e n truncated to t e r m s of quartic order or 8  — +  less 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 H = -DS  2  0  +E(S 2  Z  X  S ) + C(S* + Si) + 9ii H 2  B  (3.23)  •S  as d i s c u s s e d i n c h a p t e r 2. H e r e we t a k e D = 0.292/C, E = 0 . 0 4 6 ^ a n d C = -2.9 • 10~ K 5  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 et.al. [51]. I f we 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 S —>• S s i n 9 s i n 0, S —> S cos <j> s i n 9 a n d S —> S c o s 9 a n d s u b s t i t u t e t h e s e x  y  z  i n t o (3.23) i t is 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 as f u n c t i o n s o f H. W e  assume that the  i n d i v i d u a l electronic spins are locked to the d i r e c t i o n i n w h i c h the effective central spin is 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 to the F e  3 +  s p i n s b e i n g { t , 1\ 1,1\  the labelling 1 - 4 Angstroms.  8 corresponds  correspond  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 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 positions are i n  Chapter 3. Nuclear Spin Couplings in Fe%  85  and the Isotope Effect  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 .  We  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 the zero  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 H, Br, 1  79  N,  U  Fe,  57  1 3  C and  0.  17  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) we assume a 1 0 0 % concentration of //). 1  3.2.7  Calculation of the Orthogonality B l o c k i n g 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 K  n ^  (3.24)  = - i -7i  (3-25)  = — In  AT+8c  os  Lfc=l  where cos2A  1)  2)  7  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 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 57  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 cos2& = -s£ .Sj 1 )  2 )  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 field.  (3.26) magnetic  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  86  Chapter 3. Nuclear Spin Couplings in Fe$ and the Isotope Effect  Nuclear Label  X  y  z  Fel Fe2 Fe3 Fe4 Fe5 Fe6 Fe7 Fe8  5.737857 6.912112 6.656427 5.327499 5.198078 3.616621 3.888089 4.848748  .064911 -2.210346 3.323106 2.408451 -2.412666 2.196296 -3.321701 -.043695  1.58115 3.04665 .33915 3.17265 -3.16665 -2.98815 -.29265 -1.54365  Table 3.2: Positions of the iron ions, units in Angstoms.  120.0  Figure 3.1: cJ for all nuclei in F e . Labeling is as indicated in the text. The dots represent values for H (labels 1..120), Br (labels 121..128), and A (labels 129..146). k  8  2  81  1 5  7  87  Chapter 3. Nuclear Spin Couplings in Fe and the Isotope Effect 8  6  |  —  ,  —  —  ,  ,  |  ,  ,  ,  ,  ,  ,  r - ,  ,  1  5 -  4 <D  E 3 z  I  I  2 - I I  I  °0.0  I  5.0 10.0 w_k|| [MHz], Binned in 0.1 MHz  15.0 steps  20.0  F i g u r e 3.2: H, 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  6 i  •  •  1  •  •  ,  •  •  1  • •1  5 -  4f  0.0  30.0 60.0 w_k|| [MHz], Binned in 0.1 MHz  90.0 steps  F i g u r e 3.3: H, h i g h e n d o f t h e s p e c t r u m . 1  120.0  Chapter 3. Nuclear Spin Couplings in Fe and the Isotope Effect 8  6 i-  5 -  4I  10.0 w_k|| [MHz], Binned in 0.1 MHz  steps  F i g u r e 3.4: H, l o w e n d o f s p e c t r u m . 2  6 |  .  .  .  ,  .  .  54h  w_k|| [MHz], Binned in 0.1 MHz  steps  F i g u r e 3.5: H, e n t i r e s p e c t r u m . 2  88  Chapter 3. Nuclear Spin Couplings in Fe$ and the Isotope Effect  0.0  5.0 w_k|| [MHz], Binned in 0.1 MHz  10.0 steps  F i g u r e 3.6: Br, e n t i r e s p e c t r u m . 79  <D  E  0.0  5.0 w_k|| [MHz], Binned in 0.1 MHz  F i g u r e 3.7: Br, sl  10.0 steps  entire spectrum.  89  Chapter 3. Nuclear Spin Couplings in Fe$ and the Isotope Effect  5.0 w_k|| [MHz], Binned in 0.1 MHz  10.0 steps  F i g u r e 3.8: N, e n t i r e s p e c t r u m . 14  10.0 w_k|| [MHz], Binned in 0.1 MHz  steps  F i g u r e 3.9: N, e n t i r e s p e c t r u m . 15  90  Chapter 3. Nuclear Spin Couplings in Fe$ and the Isotope Effect  0.00  0.10 0.20 0.30 w_k|| [MHz], Binned in 0.001 MHz  0.40 steps  0.50  F i g u r e 3.10: Fe, e n t i r e s p e c t r u m . 57  <D  £  1  r  0.0  5.0 w_k|| [MHz], Binned in 0.1 MHz  F i g u r e 3.11:  1 3  10.0 steps  C , entire spectrum.  91  Chapter 3. Nuclear Spin Couplings in Fe$ and the Isotope Effect  5.0 10.0 w_k||, [MHz], Binned in 0.1 MHz  F i g u r e 3.12:  1 7  15.0 steps  20.0  0 , low end of spectrum.  20.0 40.0 w_k||, [MHz], Binned in 0.1 MHz steps  F i g u r e 3.13: 0, e n t i r e s p e c t r u m . 17  60.0  92  93  Chapter 3. Nuclear Spin Couplings in Fe and the Isotope Effect 8  d i r e c t i o n f o r a n Fe 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 H, Br, N, Fe, 1  81  U  8  57  1 3  C and  0. W e s h a l l c a l l t h i s m a t e r i a l Fe™ . 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  17  ax  o f 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 don't c h a n g e m u c h a s f u n c t i o n s o f field. 5 7  500.0 Total  400.0  300.0 CO  o. o.  200.0  100.0 0.0 0.0  0.5 H_x [Tesla]  1.0  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 applied i n t h e x direction for t h e F e ™ material.  fields  a i  3.2.8  Calculation of E  0  T h e q u a n t i t y E is 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 0  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 k n u c l e a r s p i n h a s 21 + 1 e q u a l l y th  spaced energy levels between  iwf/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 is  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 is d e f i n e d t o b e a =< E > — < E > 2  2  2  w h e r e E is t h e energy o f t h e nucleus. I f we m a k e t h e  a p p r o x i m a t i o n that t h e p r o b a b i l i t y o f each level b e i n g o c c u p i e d is i d e n t i c a l (effectively  Chapter 3. Nuclear Spin Couplings in Fe$ and the Isotope Effect  94  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  material.  an infinite spin temperature approximation) then < E > = 0 and |2  2/+1  cr  2  =  < E  2  >= £ i=i  PiE  2  2(2-1)'  i <4 = 21+ 1 4  2/  7 + 1 |,2 12/ *  (3.27)  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 i 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 . th  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 is t h e n g i v e n b y t h e c e n t r a l limit theorem  as N  -° =fe=i zZ°.  (3.28)  2  w h e r e N is 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 Fe t h i s g i v e s 8  = £(<K£(4/H£(4l.J) *2  ^79.  Chapter 3. Nuclear Spin Couplings in Fe$ and the Isotope Effect  +  ^ « : ) ksi  B  5 E «  +  2  )  2  )  ki5N  kn  t  s «  +  95  N  + E ( 4 L / ) + £ E M + E( kl3c  +  k57Fe  E UJ)  (3-29)  k57Fe  T h e r e l a t i o n b e t w e e n t h e h a l f w i d t h o f t h e d i s t r i b u t i o n W, E  a n d o is f o u n d v i a  0  or  El  = Aa  2  = 2W  (3.31)  2  T h e f u l l w i d t h o f t h e d i s t r i b u t i o n is  W = 2W = V2E  (3.32)  0  T h e w i d t h W is o f course a f u n c t i o n o f t h e i s o t o p i c c o n c e n t r a t i o n i n a p a r t i c u l a r  Fe  8  s a m p l e . F o r e x a m p l e , i f we p i c k t h e easiest case w h e r e w e have 1 0 0 % c o n c e n t r a t i o n s o f  H, Br,  1  79  N,  U  1  6  0 ,  Fe a n d C (we s h a l l c a l l t h i s m a t e r i a l F e * ) t h e n we find, i n zero  56  l2  8  2 e x t e r n a l field i n t h e p o i n t d i Y p o l e co a pl p\ r o x=i m ation, that 7 1 8 8 7 . 5 4 5 [MH z\ k H  120  ki =l H  \ E  4  7  J =  70.27  [MHz]  2  79 =l  fc  Br  o *J U  "L  E .  w = 181.8184  [MHz]  2  (3.33)  1  w h i c h gives  E  0  = 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 /x , we find that n  (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 H we are assuming that all the hydrogens are H; if the isotope is Br l  1  assume all the bromines are The addition of Fe  8 1  then we  Br, etc.).  to the mix significantly changes the value of the linewidth. This  57  is because the contact hyperfine coupling energies b l  81  are large. Let us define the material  Fe% to be identical to Fe * in every way except that every iron ion is a Fe 57  8  100% concentrations of H, Br, 1  79  15  JV, Fe, 57  C and  l2  1 6  ion; ie.  0 . Then the contribution to the  zero-field linewidth coming from the contact terms is, via (3.29), 8  8  £ 4' k=i  c2  ~ £ 2304 MHz k=i  2  = 18432 MHz  2  (3.37)  which is a significant fraction of the contribution from the protons (see (3.33)). Addition  97  Chapter 3. Nuclear Spin Couplings in Fe% and the Isotope Effect  500.0  400.0 x  —  300.0  |  •1 BramWe-81 Nitrogen-14 Caroon-13 Oxyg.n-17 lron-67  - Total 5 200.0  100.0  0.0 0.0  0.5 H_x [Tesia]  1.0  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 H f o r 1 0 0 % concentrations o f these isotopes. N o t e that wf a n d therefore W drops slowly with field. T h i s effect 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 the central spin c o m p l e x are forced closer together (no longer are they antiparallel). T h e c u r v e s h o w n a s " t o t a l " is 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 shown. x  of this t e r m gives 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 A  and A  k ND  If we 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 Fe 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 57  c a n see 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  A . k  ND  I n a d d i t i o n , i f we 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 we c a n calculate the general form for A . k  ND  F u r t h e r m o r e , w e see 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 . c o u p l i n g energies  a r e m u c h less 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 A  k ND  Because the turn out to  98  Chapter 3. Nuclear Spin Couplings in Fe$ and the Isotope Effect  500.0  400.0  I  —  57-Fe Fe_8C Fe_8'  300.0  200.0  100.0  0.0 0.0  0.5 H_x [Tesla]  1.0  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 H f o r F e * , x  8  Fe D 8  and  Fe .  57  8  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] is g i v e n i n t h i s l i m i t b y (3-39)  * = ^E\An\  2  T h i s is f o u n d to be  A =  4.45 • 1 0 ~ , 5  A  = 8.87-  H T  5  and  A  = 1.38 • 1 0 " f o r t h e Fe „ 5  a n d Fe o 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 ! 8  8  A  Fe  57  8  is 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 event. 3.3  U s i n g Free Fe  3+  Hartree-Fock Wavefunctions to M o d e l A c t u a l Spin Dis-  tributions  I n t h i s s e c t i o n we 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 . w h a t we s h a l l d o is 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 Fe  3+  Here  i o n as a  p o i n t we s h a l l a s s u m e t h a t i t is " 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 Fe  3+  wavefunction.  99  Chapter 3. Nuclear Spi  0.000  0.001  0.002  0.001  0.003 0  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 \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 left) H, Br, N, O, C a n d Fe, 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% \ 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 Fe 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 . k  ND  1  79  U  17  13  57  k  D  NtD  57  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 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 ). I n a free F e 5  3 +  3 +  i o n (its electronic  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 -  aligned d u e t o t h e Hund's rule w h i c h asks for m a x i m i z e d spin a n g u l a r m o m e n t u m giving 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 +  ion  at the origin; i t is  =~  E /  *T2-(™i ' [r-r]){f-r  3  \r  j=i  — T \  2  ) (3.40)  where 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 electrons, a n d the integration r is over a l l space. W e assume that t h e wavefunctions ^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 Fe  8  and the Isotope Effect  Since the total angular momentum of the Fe  100  in its ground state is zero, the spin  3+  of the d electrons is distributed spherically and therefore we can approximate (3.40) by the expression J  4rr  \r — r  d  3  m  \f  —  r  I  2  (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 Fe  3+  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 + Cr )r 2  exp (-r/d)  2  (3.43)  1  where A = -60.786097  , B = 68.94202-1  ,C = -22.48757-L  A  A  , d = 0.282745A  (3.44)  2  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 analytically, we shall do them numerically using the following technique. We pick P points out of the \il>(r)\ distribution to represent one iron ion wavefunction. This will be exact 2  as P —> oo. Then n  }  4rrP tt p  \r-r \  3  p  3  fh T  _  (rh • [f-  f ])(fp  f) p  (3.45)  Tp |  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)  101  Chapter 3. Nuclear Spin Couplings in Fe and the Isotope Effect s  24  15  / \ I  \  05  T-  I  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 Fe  wavefunction.  3+  is 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 " m a t e r i a l ( 1 0 0 % H, Br, N, 1  Fe).  57  1  1  81  U  1 7  0, C and 1 3  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 wavefunction. O n l y t h e higher energy nuclei are affected significantly. T h i s however c o u l d b e m e a n i n g f u l for several quantities of interest, p r i m a r i l y t h e intrinsic l i n e w i d t h W w h i c h is 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) are t h e binned hyperfine values o b t a i n e d using t h e Hartree-Fock wavefunction-these are 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 (3.15), u s i n g t h e H a r t r e e - F o c k wavefunctions instead of the point dipole approximation.  W e m a y repeat our calcula-  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 Fe * 8  Chapter 3. Nuclear Spin Couplings in F e  120.0  8  i  and the Isotope Effect  i.•  102  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 ™ " . 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 as d o t s . 1  are, 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  UJI = 7 1 5 1 2 . 2 2 [ M # z ]  £ k =i  2  H  H  I £ y  4?  =  B  n.2l[MHzf  fc =i Br  o  18  2  | £ 4  N  6  = imm[MHzf  (3.46)  k =l N  w h i c h gives E = 258.9 MHz , W = 366.2 MHz = 17.57 mK Q  (3.47)  N o t e that the values obtained are quite close to those o b t a i n e d using the point dipole 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 case, 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 approximations.  Chapter 3. Nuclear Spin Couplings in Fes, a n d t i e Isotope Effect  0.0  4.0 8.0 12.0 16.0 w_kll, MHz, Binned in 0.1 MHz steps  20.0  Figure 3.21: H, Hartree-Fock, emphasizing low end of the spectrum.  40.0  80.0  120.0  w_kll MHz, Binned in 0.1 MHz steps  Figure 3.22: H, Hartree-Fock, high end of the spectrum. 1  103  Chapter 3. Nuclear Spin Couplings in Fes a n d the Isotope Effect  E  0  0.0  4.0 w_kll,  F i g u r e 3.23:  2  8.0  12.0  16.0  20.0  M H z , B i n n e d in 0.1 M H z s t e p s  i f , Hartree-Fock, low end of spectrum.  .Q  E  0.0  10.0 w_kll,  F i g u r e 3.24: H, 2  20.0  30.0  40.0  50.0  M H z , B i n n e d in 0.1 M H z s t e p s  H a r t r e e - F o c k , entire s p e c t r u m .  104  Chapter 3. Nuclear Spin Couplings in Fe&  0.0  2.0  4.0  and the Isotope Effect  6.0  8.0  10.0  w_kll, MHz, Binned in 0.1 MHz steps  F i g u r e 3.25 :  0.0  Br,  79  2.0  Hartree-Fock, entire spectrum.  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: Br, 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 . 81  105  Chapter 3. Nuclear Spin Couplings in Fe and the Isotope Effect 8  2.0  4.0  6.0  8.0  10.0  w_kll, MHz, Binned in 0.1 MHz steps  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 .  F i g u r e 3.27:  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: 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 . 15  106  Chapter 3. Nuclear Spin Couplings in Fe$ and the Isotope Effect  0.0  0.1  0.2  0.3  0.4  107  0.5  w_kll, MHz, Binned in 0.001 MHz steps  Figure 3.29 : Fe, 57  2.0  Hartree-Fock, entire spectrum.  4.0  6.0  8.0  10.0  w_kll, MHz, Binned in 0.1 MHz steps  Figure 3.30:  1 3  C , Hartree-Fock, entire spectrum.  Chapter 3. Nuclear Spin Couplings in Fe$ and the Isotope Effect  0.0  2.0  4.0  6.0  8.0  10.0  w_kll, MHz, Binned in 0.1 MHz steps  Figure 3.31: O, Hartree-Fock, low end of spectrum. 17  10.0  20.0  30.0  40.0  50.0  w_kll, MHz, Binned in 0.1 MHz steps  Figure 3.32: 0, 17  Hartree-Fock, entire spectrum.  108  109  Chapter 3. Nuclear Spin Couplings in Fe% and the Isotope Effect  500.0  400.0  300.0  Total Hydrogen Bromine Nitrogen Carbon Oxygen lron-57  200.0  100.0  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 Hartree-Fock wavefunction picture.  0.00  0.01  0.02  function of H in the x  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 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.  function of H in the x  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  | 2 2 2 ^™ II || 22 /0.857354\ /0.857354\ X>1'„ ^ E4'„ ( - j ^ J 120  5  22  (3.48)  k =i D  which gives £  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 H for the three materials Fe *, x  Fe D  8  8  a n  d Fe 57  8  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~ , 8.73 • 10~ , and 1.35 • 10~ for the F e „ , 5  5  5  8  and  Fe D 8  materials respectively. Values for l A ^ ^ j  are shown in figure 3.36.  Fe  57  8  111  Chapter 3. Nuclear Spin Couplings in Fe$ and the Isotope Effect  =•  300 0  S7-F« e Fa_8D Fe 8-  1 S  200.0  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 a s a f u n c t i o n o f H f o r F e s * , Hartree-Fock picture. x  0 000  0.001  0.002  0.003 0  0.001  0 002  Fe$D  and F e 5 7  8  in the  0 003  F i g u r e 3.36: B i n n e d t o p o l o g i c a l d e c o h e r e n c e p a r a m e t e r s | A ^ | f o r a l l n u c l e i , a s s u m i n g 1 0 0 % c o n c e n t r a t i o n s o f ( c l o c k w i s e f r o m b o t t o m left) H, Br, A T , ( 9 , C a n d Fe, u s i n g t h e H a r t r e e F o c k a p p r o x i m a t i o n . T h e b i n w i d t h h e r e i s 0.0001; p l o t t e d o n t h e x axis is |A^£,| a n d o n t h e y axis "number o f nuclei". N o t e t h a t t h e c o n t r i b u t i o n t o |A^£,| f r o m Fe i s a l m o s t e n t i r e l y f r o m t h e c o n t a c t i n t e r a c t i o n . D  1  57  79  1 4  1 7  1  3  57  Chapter 3. Nuclear Spin Couplings in Fe$ and the Isotope Effect  3.4  Tables of Nuclear Positions, Fields at N u c l e i and Hyperfine  112  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) for both point dipole and Hartree-Fock k  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: H, Br, X  79  N,  14  Fe,  57  C,  n  and  1 7  0.  Chapter 3. Nuclear Spin Couplings in Fe and the Isotope Effect  113  8  Nuclear Label  X  HI H2 H3 H4 H5 H6 H7 H8 H9 H10 Hll H12 H13 H14 H15 H16 H17 H18 H19 H20 H21 H22 H23 H24 H25 H26 H27 H28 H29 H30  10.8 9.38 9.6 10.3 9.07 7.29 8.03 9.54 9.52 8.24 9.94 10.6 3.4 3.68 5.91 5.79 7.62 5.88 10.8 10.2 11 10.3 6.01 7.02 2.87 3.72 1.63 2.01 6.79 7.63  y  3.58 2.12 2.61 4.3 5.17 4.78 6.8 6.61 7.18 6.26 5.34 5.61 5.03 4.34 5.96 5.65 4.27 4.07 -.193 -1.76 -2.19 -1.09 1.8 1.88 .67 2.35 2.12 .981 -2.16 -.505  z  1.92 1.38 -.66 -.173 -1.88 -2.37 -1.13 -.0732 2.03 2.61 3.62 2.36 3.38 4.85 5.19 3.57 7.01 6.59 5.82 6.24 4.4 3.45 7.08 6.14 5.39 5.84 3.77 2.96 -5.9 -5.25  17 IP1 (1)  (P.D.)  a; II [MHz] (P.D.)  IT I m (H.F.)  J l [MHz] (H.F.)  .0509 .161 .112 .0422 .0978 .284 .0556 .0242 .0277 .0988 .054 .0347 .15 .256 .106 .2 .0927 .179 .0807 .109 .0924 .137 .186 .249 .195 .321 .11 .109 .307 .195  2.167 6.854 4.757 1.797 4.168 12.055 2.365 1.026 1.175 4.209 2.301 1.477 6.383 10.939 4.532 8.519 3.945 7.632 3.436 4.626 3.938 5.832 7.915 10.634 8.311 13.67 4.68 4.656 13.088 8.328  .0521 .152 .112 .0415 .0943 .298 .0577 .0246 .0274 .101 .054 .0345 .149 .251 .104 .2 .0924 .183 .0821 .109 .0924 .136 .189 .249 .201 .326 .108 .107 .326 .192  2.22 6.476 4.76 1.772 4.016 12.676 2.46 1.052 1.172 4.292 2.304 1.472 6.344 10.708 4.444 8.532 3.944 7.824 3.504 4.628 3.944 5.776 8.056 10.628 8.58 13.88 4.592 4.572 13.94 8.196  Table 3.3: Data for Hydrogen.  (1)  Chapter 3. Nuclear Spin Couplings in Fe and the Isotope Effect  114  8  Nuclear Label  X  H31 H32 H33 H34 H35 H36 H37 H38 H39 H40 H41 H42 H43 H44 H45 H46 H47 H48 H49 H50 H51 H52 H53 H54 H55 H56 H57 H58 H59 H60  8.61 8.83 6.52 6.88 4.57 4.24 4.52 2.79 3.43 4.47 4.21 3.44 .303 -.515 .385 -.312 1.19 .406 .0679 1.72 1.75 3.39 6.48 8.07 8.47 6.89 3.61 2.01 2.3 3.84  y  z  -1.06 -2.96 -2.19 -3.93 -4.3 -5.18 -5.19 -3.84 -5.64 -3.71 -5.96 -5.37 -3.93 -6.69 -4.09 -7.02 -1.79 -6.2 -1.73 -7.13 5.46 -3 5.64 -4.65 .862 -3.45 -4.2 1.95 1.99 -6 .338 -5.82 -7.11 -2.18 -6.44 -1.32 -5.37 -3.3 -5.16 -3.29 -6.92 .851 -5.77 1.73 -3.54 6.51 -3.5 6.48 -1.16 7.11 -1.24 5.94 1.38 -5.96 1.26 -7.1 3.54 -6.3 3.65 -6.53  |7 I [T] J l [MHz] (P.D.)  (P.D.)  l 7 l [T] (H.F.)  .104 .118 .277 .153 .194 .0985 .174 .0938 .234 .179 .163 .131 .145 .0924 .116 .0798 .0331 .0274 .042 .124 .0333 .193 .21 .186 .14 .364 .361 .137 .188 .195  4.423 5.023 11.756 6.517 8.255 4.201 7.418 4.004 9.961 7.632 6.943 5.581 6.198 3.938 4.964 3.4 1.409 1.169 1.785 5.287 1.423 8.218 8.967 7.915 5.949 15.498 15.383 5.862 8.014 8.302  .101 .118 .277 .148 .19 .0971 .178 .0962 .232 .186 .156 .127 .144 .0919 .115 .0807 .0326 .0272 .0425 .118 .0321 .18 .221 .199 .138 .368 .387 .14 .184 .192  (1)  Table 3.4: Data for Hydrogen.  (1)  J l [MHz] (H.F.) 4.316 5.032 11.784 6.312 8.1 4.136 7.568 4.1 9.884 7.924 6.664 5.404 6.136 3.916 4.916 3.44 1.392 1.16 1.812 5.048 1.372 7.676 9.436 8.492 5.904 15.716 16.488 5.96 7.856 8.188  Chapter 3. Nuclear Spin Couplings in F e a n d the Isotope Effect 8  Nuclear Label H61 H62 H63 H64 H65 H66 H67 H68 H69 H70 H71 H72 H73 H74 H75 H76 H77 H78 H79 H80 H81 H82 H83 H84 H85 H86 H87 H88 H89 H90  X  .623 1.95 1.75 .617 8.72 9.24 6.92 6.26 1.1 1.72 .272 -.208 8.05 7.98 5.76 7.86 7.11 6.67 3.8 7.07 4.83 5.68 3.59 6.76 2.49 4.9 2.54 3.73 2.7 3.58  y -2.88 -1.71 -4.88 -5.23 -5.09 -4.12 -5.64 -5.47 4.07 4.94 -2.64 -4.22 -.671 -2.98 -3.43 3.23 2.81 5.16 2.86 3.95 .935 -.979 -3.76 -3.06 .701 3.6 2.85 -5.2 -2.96 -2.99  z  .611 .255 2.25 .629 3.6 5.09 4.83 3.17 -4.8 -3.24 -1.92 -1.48 4.23 2.43 3.78 1.98 -1.02 .275 2.4 4.32 4.13 -4.05 -4.2 -2.37 -4.22 -3.77 -2.31 -.498 -1.82 1.29  i7 i m (P.D.)  J l [MHz] (P.D.)  .08 .251 .129 .0481 .137 .165 .131 .163 .16 .142 .0859 .0453 .622 1.82 1.08 1.12 2.44 .758 1.12 .432 .877 .943 .579 .934 .647 .765 2.09 .72 1.24 2.06  3.407 10.708 5.505 2.05 5.85 7.021 5.57 6.965 6.83 6.053 3.662 1.927 26.468 77.829 46.004 47.703 103.95 32.286 47.649 18.397 37.415 40.203 24.676 39.825 27.551 32.557 89.109 30.655 53.006 87.885  (l)  Table 3.5: Data for Hydrogen.  i7rm  a;I' [MHz]  (H.F.)  (H.F.)  .0718 .232 .132 .0439 .131 .171 .129 .157 .154 .139 .0828 .0446 .629 1.87 1.04 1.16 2.65 .779 1.02 .439 .823 .833 .547 .903 .64 .772 1.55 .631 1.34 2.74  3.056 9.884 5.612 1.868 5.604 7.284 5.52 6.708 6.572 5.916 3.532 1.904 26.844 79.864 44.464 49.468 113.492 33.22 43.636 18.688 35.084 35.512 23.344 38.492 27.288 32.876 66.084 26.856 57.224 116.536  (l)  Chapter 3. Nuclear Spin Couplings in Fe$ and the Isotope Effect  Nuclear Label  X  H91 H92 H93 H94 H95 H96 H97 H98 H99 H100 H101 H102 H103 H104 H105 H106 H107 H108 H109 H110 Hill H112 H113 H114 H115 H116 H117 H118 H119 H120  8.78 7.92 5.34 6.2 8.76 3.29 7.21 7.9 7.15 5.05 2.81 2.66 5.85 2.57 5.99 7.21 2.99 5.45 7.68 4.29 1.72 7.97 2.53 6.65 3.35 7.55 1.73 2.95 3.85 9.69  y  .0423 2.02 -1.03 -4.09 3.34 .719 3.45 -.733 6.11 3.95 3.16 .832 1.45 -2.02 -.381 -.719 -4.75 -3.95 -3.16 4.05 -.0423 7.92 6.18 6.66 -6.11 5.74 -3.34 8.36 7.44 -.874  z  5.01 3.9 3.83 1.14 2.78 1.14 -1.76 1.32 .465 1.35 2.39 -1.47 -3.24 -3.9 -4.76 -1.14 -5.18 -1.35 -2.34 -1.14 -5.01 7.83 7.14 9.57 -.465 10.6 -2.78 4.44 5.43 .24  |7 I [T] (P.D.) (1)  .231 .403 .542 .371 .303 .187 .934 .432 .181 1.16 .242 .556 .638 .392 .396 .185 .15 1.17 .249 .378 .227 .0219 .0427 .0265 .182 .0242 .291 .0227 .0389 .0877  Jl  [MHz] (P.D.) 9.837 17.188 23.124 15.776 12.891 7.979 39.767 18.425 7.7 49.5 10.279 23.686 27.153 16.747 16.94 7.888 6.398 49.671 10.567 16.101 9.663 .934 1.817 1.133 7.755 1.033 12.401 .969 1.662 3.742  Table 3.6: Data for Hydrogen.  |7 I [T] (H.F.)  J l [MHz]  .242 .406 .575 .354 .293 .182 1.16 .415 .172 1.23 .246 .544 .622 .38 .42 .193 .152 1.19 .23 .387 .226 .0222 .0427 .027 .187 .0246 .3 .0229 .0396 .0891  10.316 17.348 24.52 15.064 12.54 7.756 49.388 17.656 7.32 52.368 10.492 23.176 26.464 16.208 17.892 8.208 6.488 50.76 9.792 16.488 9.652 .948 1.824 1.152 7.992 1.048 12.828 .976 1.688 3.804  (1)  (H.F.)  Chapter 3. Nuclear Spin Couplings in Fe and the Isotope Effect  117  s  Nuclear  X  y  z  |7 | [T] J l [MHz] (1)  (P.D.)  Label  |7  (1)  I  [T]  Jl  [MHz]  (P.D.)  (H.F.)  (H.F.)  Brl  10.2  2.65  5.67  .0521  2.215  .0537  2.293  Br2  3.37  -2.3  4.91  .137  5.856  .142  6.063  Br3  5.2  7.11  7.43  .0289  1.23  .0298  1.273  Br4  .265  -2.64  -5.66  .0516  2.201  .0535  2.279  Br5  7.09  2.28  -4.91  .14  5.957  .145  6.167  Br6  7.64  -4.75  .0737  .091  3.878  .0943  4.015  Br7  1.06  1.08  -.26  .0816  3.481  .0844  3.604  Br8  3.7  5.53  .2  .0734  3.132  .076  3.243  T a b l e 3.7: D a t a for B r o m i n e .  Chapter 3. Nuclear Spin Couplings in Fe and the Isotope Effect  118  8  Nuclear Label NI N2 N3 N4 N5 N6 N7 N8 N9 . N10 Nil N12 N13 N14 N15 N16 N17 N18  X  8.52 8.73 6.48 8.87 8.09 7.75 3.45 6.55 4.83 5.7 3.96 7 2.01 4.07 1.77 2.73 1.7 2.48  y  -1.05 -3.03 -3.78 3.76 3.55 5.71 3.02 4.24 1.44 -1.4 -4.13 -3.14 1.12 3.86 2.92 -5.67 -3.67 -3.44  z  5.1 3.35 4.49 1.97 -.923 .515 3.18 4.89 4.98 -4.92 -4.86 -3.23 -5.06 -4.4 -3.2 -.426 -1.94 .983  |7 I [T] (P.D.)  J l [MHz] (P.D.)  .0547 .0891 .112 .0486 .138 .0333 .0917 .0525 .13 .13 .0582 .0929 .0568 .1 .0927 .0342 .0532 .146  2.332 3.796 4.765 2.073 5.892 1.419 3.908 2.242 5.526 5.533 2.476 3.964 2.423 4.283 3.947 1.459 2.268 6.228  (1)  Table 3.8: Data for Nitrogen.  i7 im (i)  (H.F.)  J l [MHz] (H.F.)  .053 .0849 .108 .0479 .14 .0392 .0807 .0577 .132 .136 .0525 .0884 .0638 .0896 .083 .045 .0605 .145  2.26 3.62 4.61 2.04 5.95 1.67 3.44 2.46 5.63 5.78 2.24 3.77 2.72 3.82 3.54 1.92 2.58 6.18  Chapter 3. Nuclear Spin Couplings in Fe and the Isotope Effect 8  Nuclear Label  X  CI C2 C3 C4 C5 C6 C7 C8 C9 CIO Cll C12 C13 C14 C15 C16 C17 C18 C19 C20 C21 C22 C23 C24 C25 C26 C27 C28 C29 C30 C31 C32 C33 C34 C35 C36  10 10.2 8.52 6.95 7.21 7.76 9.72 9.51 8.22 8.46 8.82 9.69 3.94 5.59 6.6 6.18 3.43 2.45 7.07 8.09 6.36 4.7 3.83 4.26 2.73 3.2 3.5 .382 1.88 .49 1.25 1.13 .645 1.4 1.68 2.44  y  -1.15 -1.83 -4.37 -4.98 -3.19 -1.57 3.16 3.41 4.94 6.13 6.22 5.27 4.46 5.22 3.79 2.13 1.57 1.85 -1.45 -1.88 -4.53 -5.22 -3.64 -2.07 1.66 3.26 4.96 1.66 4.27 1.26 -6.23 -5.13 -3.34 -2.74 -4.91 -5.92  z  5.49 4.1 4.1 4.11 6.03 6.15 1.34 -.168 -1.55 -.569 2 2.58 3.98 4.4 6.3 6.23 5.12 3.65 -5.07 -3.72 -4.28 -4.55 -6.35 -6.27 -6.12 -5.96 -3.95 -4.01 -3.84 -5.37 -1.68 -2.69 -1.37 .183 1.25 .938  |7 I [T] (P.D.)  a; II [MHz] (P.D.)  i7 i m (H.F.)  [MHz] (H.F.)  .0352 .0422 .0659 .0638 .0833 .0762 .035 .035 .0633 .0217 .0206 .0194 .0837 .0598 .0507 .0812 .0936 .0521 .0919 .0521 .0851 .0568 .0507 .0758 .0753 .0805 .0643 .0432 .0666 .0361 .0153 .0253 .0298 .0551 .0425 .0333  1.499 1.797 2.807 2.72 3.553 3.254 1.487 1.491 2.703 .925 .878 .829 3.572 2.553 2.162 3.456 3.993 2.224 3.915 2.216 3.631 2.416 2.162 3.233 3.205 3.432 2.739 1.835 2.837 1.536 .652 1.081 1.273 2.348 1.814 1.424  .0357 .0441 .069 .0615 .0805 .0758 .035 .0342 .0615 .0221 .0213 .0198 .0819 .0615 .0518 .0821 .0919 .0495 .088 .0528 .0821 .0577 .0488 .0795 .0748 .0823 .0643 .0418 .0687 .038 .0147 .0249 .0307 .0558 .0434 .0331  1.52 1.88 2.94 2.62 3.43 3.23 1.49 1.46 2.62 .941 .908 .844 3.49 2.62 2.21 3.5 3.92 2.11 3.75 2.25 3.5 2.46 2.08 3.39 3.19 3.51 2.74 1.78 2.93 1.62 .627 1.06 1.31 2.38 1.85 1.41  (1)  Table 3.9: Data for Carbon.  (i)  •  Chapter 3. Nuclear Spin Couplings in Fe and the Isotope Effect  120  8  Nuclear Label  X  FE1 FE2 FE3 FE4 FE5 FE6 FE7 FE8  5.72 6.9 6.65 5.31 5.19 3.61 3.89 4.84  y  .0651 -2.21 3.34 2.41 -2.43 2.2 -3.33 -.0439  z  1.58 3.05 .339 3.18 -3.17 -2.99 -.293 -1.55  |7 I [T] (P.D.)  J l [MHz]  .00729 .00467 .003 .00793 .00779 .00467 .00293 .00737  (1)  J ' [MHz]  (P.D.)  |7 I [T] (H.F.)  .311 .199 .128 .338 .332 .199 .125 .314  .00729 .00467 .003 .00793 .00779 .00467 .00293 .00737  .311 .199 .128 .338 .332 .199 .125 .314  Table 3.10: Data for Iron.  (1)  (H.F.)  Chapter 3. Nuclear Spin Couplings in Fe% and the Isotope Effect  Nuclear  X  y  z  Label  |7 I [T] (1)  Jl  Jl [MHz]  [MHz]  (P.D.)  (P.D.)  (H.F.)  (H.F.)  01  6.9  1.68  3.29  .746  31.831  .699  29.8  02  5.4  -1.44  2.99  .798  34.036  .877  37.4  03  5.57  -3.6  1.31  .253  10.783  .249  10.6  04  4.14  .682  1.88  .544  23.202  .518  22.1  05  7.59  -.53  2.12  .608  25.926  .671  28.6  06  5.8  3.68  1.76  1.2  51.071  1.06  45  07  6.14  1.37  .185  .969  41.303  .931  39.7  08  4.5  -1.34  -.115  .973  41.493  .978  41.7  09  3  .503  -2.03  .633  26.96  .612  26.1  .887  37.8  .877  37.4  010  5.13  1.44  -2.94  .783  33.367  Oil  3.65  -1.68  -3.24  .798  33.975  012  6.5  -.649  -1.85  .471  20.113  .483  20.6  013  4.74  -3.71  -1.76  1.14  48.459  1.15  49.2  •  014  4.94  3.55  -1.29  .239  10.154  .251  10.7  O100  8.98  8.16  7.79  .0121  .516  .012  .512  020  1.58  6.01  7.14  .0246  1.046  .0258  1.1  030  -1.02  6.85  5.61  .012  .51  .0116  .493  040  11.4  7.23  9.35  .00868  .37  .00901  .384  050  6.85  6.23  10.6  .0155  .662  .0153  .651  060  3.56  8.15  4.43  .0182  .775  .0174  .741  070  9.32  5.56  10.5  .0139  .593  .0134  .571  4.67  .0115  .492  .0116  .496  .101  4.302  .0973  4.15  080  1.11  8.53  090  8.96  -1.17  .278  T a b l e 3.11: D a t a for O x y g e n .  Chapter 4 A n Introduction to the Generalized Landau-Zener Problem  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  Introduction to and Exact Solution of the Landau-Zener Problem  Consider a two-level system (TLS) with time-dependent Hamiltonian H(t) = vto  z  where {a} are the Pauli matrices.  + AJ  x  (4.1)  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 eigenstates of o , which are linear in time, and the eigenstates of H(t), E±(t) = ± ( A + v t ) / . z  2  2  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  t h a t h = l)  .d i— dt  H(t)  (4.2)  [ A )  yields  I——  = Vtlpa  + Alpb  dt  Atp - Vtlp a  .dip dt  (4.3)  b  b  e l i m i n a t i n g ip a l l o w s us t o w r i t e a  ib\ + (y t + A - iv) ip = 0 2  2  2  b  (4.4)  w h e r e o v e r d o t s d e n o t e d e r i v a t i v e s w i t h r e s p e c t t o t. T h i s is t h e e q u a t i o n for a p a r a b o l i c c y l i n d e r f u n c t i o n . It h a s t w o s o l u t i o n s w h i c h m a y b e w r i t t e n (i)  _  - F H I _ , A £  (ivt ) 2  I  /  (2)  (4.5) /T  4  4u  '  5  4  *  '  w h e r e W is t h e W h i t t a k e r f u n c t i o n [113]. I f w e c o n s i d e r t h e a s y m p t o t i c f o r m s o f these s o l u t i o n s as t —> — oo, we find t h a t (1) 2  6  (2) |2  t—y —  cx>  (4.6)  0  T h i s i n d i c a t e s t h a t t h e choice o f o n e o f t h e t w o s o l u t i o n s is e q u i v a l e n t t o t h e choice o f a n i n i t i a l c o n d i t i o n o n t h e w a v e f u n c t i o n ip . T h e a n a l y s i s o f t h e o t h e r w a v e f u n c t i o n ip b  a  proceeds i n a n identical manner; s o l v i n g  4> + (y t + A + iv) ip = 0 2  a  2  2  a  (4.7)  Chapter 4. An Introduction to the Generalized Landau-Zener Problem  125  gives two solutions  (4.8)  y/t  4  4u  '4  7  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 tp at time a  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 ip at time t given that it started in state ip = | t > at a  a  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 ->• oo) = 1 - e~^ t  (4.11)  and therefore the probability to make a transition is P (t n  -> 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 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 we have t a k e n A /v  = 0 . 6 3 , 1 a n d 5 for t h e s o l i d , d o t t e d a n d d a s h e d lines r e s p e c t i v e l y .  2  4.1.1  Alternate Method of Solution for the Transition Probability I. A l l Orders Perturbation Expansion  It is also p o s s i b l e t o solve for t h e t r a n s i t i o n p r o b a b i l i t i e s w i t h o u t first f i n d i n g t h e wavef u n c t i o n s [104]. C o n s i d e r t h e a m p l i t u d e  A -> =<b\U(t ,ti)\a> a  b  (4.13)  f  where  U{t ,t ) f  = Te tt * i  i  H{T)  (4.14)  r  is t h e t i m e e v o l u t i o n o p e r a t o r ( T m e a n i n g " t i m e o r d e r e d " ) a n d a, b c a n be e i t h e r " u p " or " d o w n " (<T = ± 1 , r e s p e c t i v e l y ) . S p l i t t i n g t h e H a m i l t o n i a n i n t o d i a g o n a l a n d o f f - d i a g o n a l z  (in o) p a r t s H(t)  = Hd(t) +  a l l o w s us t o r e w r i t e t h e e v o l u t i o n o p e r a t o r i n t h e m o r e  c o n v e n i e n t f o r m [71]  Ufo^)  = e- % l  Te- %"*  Hd{T)dT  l  dT  (4.15)  where 7-7  (+\  H^(t)  J It  = e * Jf  d(T)dT  H  JJ  He A  -i  f'  i  Jt  H (r)dT d  .  .  (4-16)  Chapter 4. An Introduction to the Generalized Landau-Zener Problem  127  Expansion of the time-ordered exponential yields A^  b  = f)(-t)  f  B  dt f *  71=0  n  x  f  2  *  , d,TVTd  dt, <  i f '  2  d,TVT&  i  drvra  x  . .  z  Ao ...e i  i f*  1  Ao e -°°  Jt  n-i  Jt  b\e ^ 'A&  * „  z  e  dt _ ...  n  n  J  x  x  drvru  i  z  ^  \a >  / ,,  M  (4-17)  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  f) A " ^ 2  N  =  Q  rt2n  d t I dtzn 2n  J—ti  T*2  c f t ^ - ! •. .•. • dtm-\  Jti  dt - ?T-il-W iv  ie  (4.18)  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 P as before.  = 1 - \A \ = 2  n  n  (4.20)  Chapter 4. An Introduction to the Generalized Landau-Zener Problem  128  i  4.1.2  Alternate M e t h o d of Solution for the Transition Probability II. Dychne's F o r m u l a  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  R  sin 9/2  (4.21)  cos (9/2  which, if 9 is a function of time, can diagonalize the general Hamiltonian  H = V\\{t)cr + A(t)o z  =  x  Vji(t)  A(t)  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  + vt 2  2  (4.25)  The result of Landau and Dychne states that the probability for making a transition from one eigenstate of o  z  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  P^ a  b  = ~ e  2  I  m  ^  (4.26)  where  f ( t ) = 2 j E{r)dT  (4.27)  1  and  t  c  is t h e zero o f E(t)  i n the u p p e r h a l f p l a n e t h a t is closest t o t h e r e a l a x i s . N o t e  t h a t t h i s r e q u i r e s t h a t we a n a l y t i c a l l y c o n t i n u e t h e t i m e v a r i a b l e t h r o u g h o u t t h e c o m p l e x p l a n e . T h e e r r o r i n t h i s e x p r e s s i o n c o m e s f r o m t h e neglect o f t h e c o n t r i b u t i o n o f a l l t h e o t h e r zeroes o f E(t)  i n t h e u p p e r h a l f c o m p l e x p l a n e w h i c h are o m i t t e d here. G e n e r a l l y  s p e a k i n g , one c a n t e l l w h e t h e r or n o t t h i s m e t h o d w i l l give useful r e s u l t s b y l o o k i n g at t h e s t r u c t u r e o f t h e zeroes o f E(t).  I f t h e r e are m a n y c l o s e l y s p a c e d zeroes i n E(t)  off t h e r e a l  axis, then the c o n t r i b u t i o n s from s u b d o m i n a n t terms w i l l grow a n d this m e t h o d w i l l fail. F o r e x a m p l e , i n t h e case o f the s i m p l e L a n d a u Z e n e r m o d e l , there are o n l y t w o p o i n t s i n the c o m p l e x t i m e p l a n e w h e r e E(t)  vanishes, n a m e l y t  c  = izA/y^,  a n d o n l y o n of these  is i n the u p p e r h a l f p l a n e . B e c a u s e of t h i s we e x p e c t t h a t i n t h i s case t h e results o b t a i n e d u s i n g D y c h n e ' s f o r m u l a s h o u l d be e x a c t . T h i s r e s u l t is i n t e r e s t i n g as i t p r o v i d e s i n s i g h t i n t o the reasons w h y t h i s m o d e l is e x a c t l y s o l v a b l e a n d m a n y o t h e r s i m i l a r m o d e l s are n o t - f o r e x a m p l e , c h a n g i n g the t i m e d e p e n d e n c e f r o m l i n e a r t o say c u b i c i n Vj|(t) causes the m e t h o d s u s e d i n the p r e c e d i n g c h a p t e r s t o f a i l t o p r o d u c e e x a c t results. T h i s is m o s t p r o b a b l y r e l a t e d t o the existence of n o w two zeroes o f E(t)  i n the u p p e r h a l f p l a n e w h i c h  i n t r o d u c e errors i n t o the f o r m u l a (4.26). It is q u i t e l i k e l y t h a t t h e r e is a deep c o n n e c t i o n here b e t w e e n the t h e o r y of s p e c i a l f u n c t i o n s a n d issues i n c o m p l e x a n a l y s i s . H o w e v e r , we d o n o t choose t o p u r s u e t h i s avenue at the present t i m e - w e s h a l l o n l y m e n t i o n i t i n passing.  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* VA  2  + v r dr 2  2  = WA  2  +vt + ^ 2  2  In (yt + VA  +vt)  2  2  2  (4.28)  and therefore A 7T 2  Im ate) = ^rj  (4.29)  and P^ a  b  = e'^T  (4.30)  which agrees with our previous exact results.  4.1.3  Analysis of the Transition F o r m u l a  We now wish to step back from the preceding technical exercise and draw some conclusions from this analysis. From the outset it should have been clear that the presence of the tunneling term Ao  x  ipajt,-  in the Hamiltonian would mix the two states that we are calling  Furthermore one expects that the dimensionless parameter A /v should be impor2  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 equations 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 equation. 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 transition 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  Generalization of the Two-Level Landau-Zener Problem I.  Let  E x a c t S o l u t i o n f o r A(t) ~ Vj|(t)  us now consider a more general case of (4.1), namely [102] H(t) = V\\(t)a + A{t)a  We  (4.31)  x  z  may perform the same type of analysis as we did in section 5.1. The Schrodinger  equation  .d ( dt  = H(t)  i—-  (4.32)  yields  ^ = (t)^ -  t  A  a  vm {l  (4.33)  b  eliminating x/j allows us to write a  A -  + (v~(\ + A  2  -i Mi - Mi  A  ^6  = 0  (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) {t= - 0 0 ) | = 1 2  b  (4.35)  Chapter 4. An Introduction to the Generalized Landau-Zener Problem  132  ie. that the system has been prepared in state tp in the deep past. b  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 7  ~  Vn + A  A A"  i  2  ^ii-Mix  2  ^6  = 0  (4.37)  We now choose the mapping such that i = A(t)  (4.38)  We then find that 1 +  v  2  ILL  i  /A  A  A  A  2  ^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 =  (4.40)  KA  with K constant, then (4.39) reduces to ib'l + ( l + K ) ^ 2  = 0  (4.41)  + c i e i-yz  (4.42)  Define 7 = 1 + K, ; then 2  2  ip (z) = c e b  0  1 7 2  Chapter 4. An Introduction to the Generalized Landau-Zener Problem  133  is t h e g e n e r a l s o l u t i o n o f (4.41). I n o r d e r t o d e t e r m i n e t h e n o r m a l i z a t i o n c o n s t a n t s we require the following.  F i r s t l y , we w a n t t h e s o l u t i o n for t h e p r o b a b i l i t y t o be b o u n d e d  e v e r y w h e r e b y 1; e x p l i c i t l y ,  |^ | 6  2  = |  | + | 2  C o  | + *  e  2  C l  c  C l  + c  2 i Z 7  l C o  e-  < 1  2 j Z 7  (4.43)  A s w e l l , we k n o w t h e s o l u t i o n for V\\ = 0 w i t h o u r i n i t i a l c o n d i t i o n is s i m p l y  |V; | = | c | + | C ! | 2  2  fc  0  2  c* e  2iz  +  + c^cne"  oCl  = cos (z)  212  2  (4.44)  T h e s e f i x t h e c o n s t a n t s t o b e CQ = c\ = 1/2, a n d  \Mz)\  2  = cos ( z)  (4.45)  2  7  In t e r m s o f o u r o r i g i n a l p a r a m e t e r s t h i s 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  f A(r)dr\  \^ (t)\ = c o s [(1 + K ) ' 2  2 1 2  2  b  (4.46)  T h i s r e s u l t has b e e n d e r i v e d p r e v i o u s l y b y different m e t h o d s [115]. W e m a y pause n o w and  ask h o w t h i s e x a c t s o l u t i o n c o m p a r e s t o t h a t o b t a i n e d u s i n g D y c h n e ' s f o r m u l a . I n  t h i s p a r t i c u l a r case, we f i n d t h a t  E±(t) = ± /A (t) 2  ]  and  t h a t t h e zeroes o f E±(t)  + V (t) = ± v T + ^ | A ( t ) | 2  (4.47)  2  are s i m p l y t h e zeroes o f A(t)  (where, o f course, we a n a l y t i -  c a l l y c o n t i n u e t h e t i m e t t h r o u g h o u t t h e entire c o m p l e x p l a n e ) . 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 r o m D y c h n e ' s f o r m u l a is t h e n  P where t  c  =<i \U(-oo, + o o ) | t > ~ -  I  4VWlm  n  is t h e zero o f A(t)  e  tc  ^  d  T  closest t o t h e r e a l a x i s i n t h e u p p e r h a l f p l a n e .  (4.48)  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)  (4.49)  cosh cot  The indefinite integral of A(t) is A  ft  / A(r)dr  J  (4.50)  = —tan (sinha;£) + c _1  2  UJ  To be consistent with our initial condition (4.35) we take c = AIT/2UJ. 2  The transition  probability from our exact result is then \Mt)\  12  - cos  , A  (1 + « : ) — tan (sinhwt) +  2  2  1/2  _1  7T  UJ  (4.51)  The t —» oo asymptotic of (4.51) is P  n  = cos  An  2  L  i  i  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 t . This breakdown of Dychne's formula in this regime has been noted previously [102]. c  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  135"  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 one; A{t) = 0 A(t) = Asmu(t + t ) 0  ,  |*| > t  ,  0  \t\<t  0  (4.53)  w h e r e wefixt = mr/2u. H e r e we find t h a t f o r \t\ < t 0  0  |<M*)| = cos (1 + 2  2  ^1/2-  [cosu;(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  we 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 effect, 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 we 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  4.1.5  137  Generalization of the T w o - L e v e l Landau-Zener P r o b l e m II. E x a c t Solution by M a p p i n g to Riemann's Differential E q u a t i o n  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  z(t) = - [tanhwi + 1]  (4.55)  Li  Then (4.31) becomes  A  +  l  l  A  z +z-1 -  A  Vn + A  +  z(z-l)  A  2  ~ A  W\\  4to z(z - 1)  2u  2  W  V  A  A = 0(4.56)  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) = A t a n h w i  ,  1 A)' cosh cot  A(t) = (A + 2  2  (4.57)  1 2  As functions of z, these are V\\(z) = A(2z - 1)  ,  A(z) = (A +  A ) ^z(l-z)  2  2  1  (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  n A  +  !  -  .Z  where  1  I +  7 Z —  1.  A  l  z(z-l)  k  z  k  z —  1  A  =  o  (4.59)  Chapter 4. An Introduction to the Generalized Landau-Zener Problem  138  This is a specific case of the R D E . Its solutions are ^  = z (za  b  + a - c, 1 + b - c; 2 - c; z]\  iy [CF[a, b; c; z] + Dz ~ F[l l  c  (4.61)  where C and D are constants chosen such that \ipb(z = 0 ) | = 0. In order to determine 2  what the constants a,  a, b and c are, we refer to Abramowitz and Stegun [113]. Using  7,  their nomenclature, 1 - a - a ' =  ,  1/2  1 - 7 - 7 = 1 / 2  aa = k 77  =  ,  /?/?' =  a+ b= \ + a —a +  ,  -A\  7 — 7  c= 1+ a —a  (4.62)  In terms of our original parameters we find that a=^(l  ,  + 2A i) 0  a=  y = b=\  Ai  2  1 +  -A i 0  +  A  1  1 + AAQI  (4.63)  and our general solution becomes ^  (l+2A i)/2^_ yA i  =  z  0  1  0  CF[^-A ,^  + A ;^  1  +  Dz-  F[-2A i  - A  {l+AAol)l2  Q  u  1  -2A i 0  +  + 2A i;z] 0  2A i; z] 0  (4.64)  The condition | ^  = 0)| = 0  (4.65)  2  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 ^ ( 2 = l)  = ^°C e  r ( | + 2Aoz)r(| + 2A 2) 0  r ( l + 2A i + 4 i ) r ( l + 2 A i - A ) 0  0  x  (4.66)  Chapter 4. An Introduction to the Generalized Landau-Zener Problem  139  U s i n g the reflection formulae  we find t h a t  C e * ° (1 + 16 A ) 2  2  \AY =  2  A  COS 7rAi  2  2  0  4(4A  2  + A ) 2  cosh  2  (4.68)  2A TY 0  A s we r e q u i r e t h a t t h e p r o b a b i l i t y be b o u n d e d a b o v e b y one we t a k e  C= 2  -  ( ^ + ^> ^ ° ( l + 16A ) 4  2 e  2  1  (4 69) '  w h i c h gives, i n t e r m s o f o u r o r i g i n a l p a r a m e t e r s , t h e f i n a l t r a n s i t i o n p r o b a b i l i t y cos  |^(t-+oo)| = l - — 2  ^  2  (4.70)  ^  W e e m p h a s i z e t h a t t h i s r e s u l t is exact. T o the best o f o u r k n o w l e d g e , t h i s r e s u l t has n o t b e e n p u b l i s h e d p r e v i o u s l y . T h i s s o l u t i o n is s k e t c h e d i n figure (4.5). If we t r y t o c o m p a r e t h i s e x a c t r e s u l t t o t h a t o b t a i n e d f r o m D y c h n e ' s f o r m u l a , we encounter  another  interesting conundrum.  In order to o b t a i n closed form  solutions  for D y c h n e ' s f o r m u l a , one has t o be a b l e t o o b t a i n t h e i n d e f i n i t e i n t e g r a l o f E(t) \JV\\(t)  =  + A ( t ) . N o w i t is e v i d e n t t h a t t h e presence o f t h e s q u a r e r o o t w i l l m a k e t h e set 2  of V\\(t) a n d A ( t ) t h a t c a n be d e a l t w i t h q u i t e s m a l l .  T h e present case is a n e x a m p l e  of a H a m i l t o n i a n t h a t p r o d u c e s t i m e - d e p e n d e n t e n e r g y eigenstates t h a t d o n o t p r o d u c e c l o s e d f o r m s o l u t i o n s t o t h i s i n t e g r a t i o n . T h i s l i n e o f t h o u g h t is q u i t e i n t e r e s t i n g , as it seems t o relate the i n t e g r a b i l i t y o f square r o o t f u n c t i o n s t o t h e s o l u t i o n of s e c o n d o r d e r differential equations.  It is p l a u s i b l e t h a t progress c o u l d be m a d e i n the s t u d y o f these  k i n d s of i n d e f i n i t e i n t e g r a l s v i a the c o n n e c t i o n t h a t has b e e n e s t a b l i s h e d here t o hypergeometric equation.  the  Chapter 4. An Introduction to the Generalized Landau-Zener Problem  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 is 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 is a n o t h e r c a s e w h e r e we c a n m a p  exactly to the R D E ,  14P  ATT/UJ  on  a n d t h a t is f o r t h e  p a r a m e t e r set V[|(<) = A t a n h u / t  ,  A(t) = A  (4.71)  T h i s is s i m i l a r to the previous example, except t h a t the t u n n e l i n g t e r m is constant for a l l t i m e . T h e p r o c e d u r e is 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 ; we 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 | 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 T  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 4* h ^ ; , . / 2  l  N o t e that for A/A «  ^ ^ ° °  )  |  2  = s  i  n  A  )  <- > 4  2  72  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  is 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 is 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 Fe molecular magnet 8  can be described by a two level system. In the presence of an externally applied timedependent 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 f eigenstates of the central spin object in the presence of z  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 onemolecule 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 Environment  The form of the low energy effective Hamiltonian for the F e molecular magnet was 8  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^rh ) ; one due to the field acting on the nuclear spins coming from k  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  t o t h e i n t e r a c t i o n o f t h e field w i t h t h e c e n t r a l s p i n ( t h e d i a g o n a l t e r m ~  fH z  a n d the  z  off-diagonal terms i n the phase $ ) . T h e first o f these is  uim  = \I \g p {^  k  k  nk  n  + o f ) = \I \g ix {f  l) k  k  nk  n  l)  H(t) + H(t +  + 7 $+  k s  (5.1)  fin" )) 1  w h e r e we have d e f i n e d t h e fields -(1,2) _ IkS —  E  9n Pn k  sf'  2 )  -  E  (5.2)  M°SS«W  t o b e those d u e t o t h e c e n t r a l s p i n 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 a t t h e k  th  n u c l e u s , t is t h e t i m e a t w h i c h t h e flip o c c u r e d a n d fin"  11 S  c e n t r a l s p i n c o m p l e x . I f we a s s u m e t h a t fig  1  t h e t u n n e l i n g t i m e for t h e  is m u c h less t h a n t h e t i m e s c a l e over w h i c h  the e x t e r n a l l y a p p l i e d field changes, t h e n we find t h a t  u^rhkit)  « Ihlgn^nd^  T h i s a s s u m p t i o n a l s o leads t o uj l k  u> l = \I \g p (t  ]  K K  k  nk  -  n  k  + IkS +  being time independent,  o f ) = \h\9nMikS  (5.3)  2H{t)) since  ~ iks + H{t) - H(t + fi^ )) « u l (5.4) 1  K  k  A l s o we find t h a t i n g e n e r a l t h e phase $ b e c o m e s t i m e d e p e n d e n t i f t h e e x t e r n a l  field  c o n t a i n s t r a n s v e r s e c o m p o n e n t s t h a t v a r y w i t h t i m e . I n t h e p a r t i c u l a r case o f t h e easyaxis easy-plane H a m i l t o n i a n i n the l i m i t studied i n chapter 2 (that is, the D C bias  field  \H\ - C fi a n d AE j(D — 2E) n o t t o o s m a l l (so t h a t t h e i n s t a n t o n c a l c u l a t i o n is v a l i d , see 0  c h a p t e r 2)) we f i n d t h a t  nS gn 2  $ = 7rS-  B  -l  —  2 E  t l  T  H  nSg^B —  fin  „ t i n .  ->• $ ( t ) = TTS + 1  2  ~  H (t) x  ^—H {t) y  (5.5)  Chapter 5. The Landau-Zener Problem in the Presence of a Spin Bath  145  T o g e t h e r w i t h t h e o b v i o u s d i a g o n a l c o u p l i n g t o t h e e x t e r n a l field we find t h a t i n t h e presence o f a g e n e r a l e x t e r n a l t i m e - d e p e n d e n t field t h e o n e - m o l e c u l e effective H a m i l t o n i a n is N+8  H(t)  =  ^ - 4 • m {t) + ^J %  £  k  k  •l  k  + gn SH (t)f B  z  z  fc=l  ( +  N  2 A f _ cos \^(t) 0  N +  S  +  - i £  8  _  [6/ (2/ fc  fc  - 1) "  + h.c.  N  _yko.fi jk l a 0  _\  A , D • hj  \ +  Mo/^n 47r  9ni9n  E rZk  k  ik Ti-T -s(Ti-  r  k  rik) (h • hk) (5.6)  w h e r e t h e n o t a t i o n is t h e s a m e as t h a t i n t r o d u c e d i n c h a p t e r 2. F r o m t h i s p o i n t o n w a r d s we s h a l l d r o p a l l g-factors a n d m a g n e t o n s for reasons o f 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 c o n f u s i o n m a y arise w e s h a l l e x p l i c i t l y i n c l u d e t h e m . T h i s e x p r e s s i o n i s s t i l l q u i t e c o m p l i c a t e d ; however, o n e notes t h a t i t h a s t h e f o r m o f a generalized L a n d a u - Z e n e r H a m i l t o n i a n which has been coupled to a spin environment. N o t e t h a t t h e p h o n o n b a t h does n o t enter i n t o t h e q u a n t u m r e g i m e effective d e s c r i p t i o n , for reasons d e t a i l e d i n c h a p t e r 2 ( a l t h o u g h o f course t h e o s c i l l a t o r b a t h is present w h e n relaxation becomes temperature dependent-a  p r o p e r t r e a t m e n t o f these b a t h s is u n d e r  c o n s t r u c t i o n . N o t e t h a t t h e r e are several (often c o n f l i c t i n g ) t r e a t m e n t s of o s c i l l a t o r b a t h s i n t h i s c o n t e x t [116, 117]). W e s h a l l n o w s i m p l i f y t h e e x p r e s s i o n (5.6) b y r e s t r i c t i n g o u r a t t e n t i o n t o t h e f o l l o w i n g two kinds o f externally applied  field.  I n a l l t h a t follows we shall be considering this  e x t e r n a l field t o b e e i t h e r a l o n g i t u d i n a l s i n u s o i d a l field p l u s a s t a t i c c o m p o n e n t i n a n arbitrary direction  H (t) cos  = (A cos cot + 0  z +H  x  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 l u s a s t a t i c c o m p o n e n t i n a n a r b i t r a r y d i r e c t i o n  H (t)  2Aco  =  saw  +  £  7T  H  x  2n+ 1  e  cu  t)e  2n-l t-  x +H y  (5.8)  y  N o t e t h a t w e have c h o s e n t o use t h e n o t a t i o n t h a t t h e l o n g i t u d i n a l c o m p o n e n t o f t h e e x t e r n a l l y a p p l i e d field i s d e n o t e d b y H  z  have t i m e - d e p e n d e n t  —> £. N o w b e c a u s e n e i t h e r o f these t w o fields  t r a n s v e r s e c o m p o n e n t s , we find t h a t t h e phase $ r e m a i n s t i m e  independent. T h i s simplifies o u r expression somewhat. T o emphasize the relation of o u r effective H a m i l t o n i a n t o t h e s i m p l e r L a n d a u - Z e n e r m o d e l s s t u d i e d p r e v i o u s l y , we r e w r i t e it i n the form  H(t) = g(t) + V\\(t)f + A f _ + h.c.  (5.9)  z  where  m  = at) +  ^Y,^[ rh-3(l ^f )(r .f )} r  l  4 7 F  Kk  lk  lk  k  (5.10)  lk  r  Vj|(t) = f + H (t)  (5.11)  z  /  N+8  \  (5.12) A = 2A cos ( $ - t E  A\  0  > D  • Ij k  a n d we have defined N+8  N  N+8  Qk  yka/3  e  k=l  k=l  k=\  Z  6/ (2/ fc  fc  - 1)  jk a  \  (5.13)  and N+8  E  f -  k=l  Note that subspaces.  g(t), V\\(t), A a n d A  k  r  ND  N+8  T fc  E  k=l  (5.14) -ylk  • h  Z  are a l l o p e r a t o r s i n t h e  {I } ( e n v i r o n m e n t a l s p i n ) k  Chapter 5. The Landau-Zener Problem in the Presence of a Spin Bath  5.3  147  General A CField Solution in Fast Passage  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 / v , which is 2  small if the time it takes the external field to sweep through the resonance is much smaller than the "bounce time" A " . We see explicitly in this case that if the sweep velocity is 1  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 amplitude from our initial time (say, t = 0) to our final time t or we can choose a "coarse graining" time t < t and only evolve the system from t = 0 to t . After this shorter-time c  c  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. Because 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 , where To is the energy range over 0  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 where T  _ 1 2  0  is roughly r  0  ~  \fNT2~  1  for an ensemble of N nuclei,  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 is much larger than u, then the spin bath does have time to equilibrate 0  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 t = c  TT/LU for  both the sinusoidal and sawtooth fields). If the timescale for the readjustment of the density matrix of the system (which is t « l / r ) is such that tb/t < 1 then this coarse b  0  c  graining procedure is justified. Typically in magnetic insulators at mK T  _1 2  temperatures  ~ 10 — 10 Hz [48, 64]. In the specific case of Fe% we can calculate r 4  (see table  6  5.2); we find that r  0  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, t /t « b  c  TTTQ/UJ,  which gives co <  TITO  (5.15)  <~ 10 MHz  In the experiments that have been performed to date on the Fe and M n 8  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 approximation 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 t will be difficult to maintain, as the nuclear spin bath has c  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 List of Approximations Invoked in the Calculations T h a t Follow  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 nuclearnuclear 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 a polarization  group M are in thermal equilibrium  at  group temperature 0M = 0. The "polarization group" of our set of N + 8  nuclear spins is defined to be [20] N+8  M  =£  I  kz  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 z = 7 ^ ; that is, the z direction for the k  th  k  nuclear spin  corresponds to the direction of the field at the spin when the central spin complex is in its +T eigenstate. Because the internal field due to the iron spins is quite inhomogeneous, Z  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  5.3.2  151  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 f eigenstates of the central spin complex during one z  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 onemolecule 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 f eigenstates, we also have to explicitly include the effects of the other N + 8 systems. z  Formally we can see how this is done by noting the following. For the Hamiltonian (5.6) the amplitude to go from some initial f eigenstate \a > to some final f eigenstate \/3 > z  z  during a single sweep of the A C field can be written A% =< 0\ < lf\e^o ^ \p CH  dT  >\ > a  (5.16)  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 (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 k  th  the subset N£ is labeled  spin of  = l..Njf. The total number of spins in the spin bath is  N = J2fj.N^. As an example, consider the Fe * material. Here we have that 8  A^  2  = 120  ,  i V f = 18  ,  N% = 8  (5.18)  3  With this notation the spin bath states \P > and \P > can be written in the form  \r> = n »  f  >  N  1  nK>kK>®i4>-«^>} j  (^=i  i^> = n(ii 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 > to be all the allowed spin bath states with polarization M  group  in the p  th  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  p=  I.-CM^  153  states w i t h  w h i c h we l a b e l  i i>l 7  \K>=\i\ U/x =  ( - °) 5  2  J  l  B e c a u s e t h i s n o t a t i o n m a y at first s i g h t a p p e a r c o n f u s i n g , l e t us e x p l a i n here a g a i n w h a t each o f t h e s y m b o l s r e p r e s e n t . T h e s t a t e | M £ > is a o f t h e set o f s t a t e s \M^ > . fixed  for each species.  T h e states | M  M  particular  s t a t e w h i c h is a m e m b e r  > have p a r t i c u l a r p o l a r i z a t i o n g r o u p s  F o r example, imagine a fictitious material w i t h 2 protons a n d 2  n i t r o g e n a t o m s . W e b e g i n b y fixing (say) MH = 0 a n d MN — 1- T h e n t h e states | M are t h e set o f p a r t i c u l a r s t a t e s  with polarization group M  M  M  >  . S o i n t h i s e x a m p l e we  c o u l d l a b e l o u r states  \M  H  >= | + 1/2,-1/2 >  ,  |i\4>=|i,o>  ,  \M  > = | - 1/2,+1/2 >  2 H  and  | M £ > = | O , I >  the f o r m e r o f w h i c h c o n s t i t u t e t h e set | M # > a n d t h e l a t t e r t h e set \M_y >. B e c a u s e o f o u r a p p r o x i m a t i o n (iii) i n t h e p r e v i o u s s e c t i o n we m a y define a d e n s i t y m a t r i x for the s p i n s t a t e set \M^ > o f t h e f o r m 1  -e-e^Ml  X  M?\  )  9  Q  AMI  >< MP\  (5.21)  with partition function  Z  M  ^ M , )  = j : e - ^ i=i  E  P  ^ C  M  ^  (5.22)  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 APG  K^t]  (5-23)  / 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^  APG  = i  ii-  For a single species spin 1/2 bath,  M  W  = ^ \  .„ .„  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  < a\e- ^ ^ \(3 l  M M  H  )dT  >< ftjfc  ^ \a  H  dT  >}  (5.26)  Chapter 5. The Landau-Zener Problem in the Presence of a Spin Bath  155  I n o r d e r t o finish o u r j o b , we n e e d t o t r a n s l a t e t h i s t r a n s i t i o n p r o b a b i l i t y i n t o a r e l a x a t i o n rate.  W e c a n d o t h i s b y n o t i n g t h a t s i n c e t h i s is a " p e r sweep" t r a n s i t i o n p r o b a b i l i t y ,  t h e r e l a x a t i o n r a t e c a n be o b t a i n e d b y d i v i d i n g t h i s q u a n t i t y b y t h e t i m e for one sweep ( w h i c h is o f course s i m p l y t h e c o a r s e - g r a i n i n g t i m e t ). c  loss o f g e n e r a l i t y here choose \ct >=  \ t > a n d \(3 >=  F u r t h e r m o r e , we m a y w i t h o u t  \ 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 group state i n i t i a l l y  = l {pM» Tr  <t | e - * J > ( ^ |  > , is  i><! | ' J > M * - | > } C  (5.27)  T  a n d the f i n a l e n s e m b l e a v e r a g e d r e l a x a t i o n r a t e is, f r o m (5.23),  ^=EI1 APG  [WMSMI]  (5-28)  f-  W e n o w t u r n t o the e x p l i c i t e v a l u a t i o n o f t h i s q u a n t i t y .  5.3.3  Processing of the Transition A m p l i t u d e  I n t h i s s u b s e c t i o n we s h a l l recast t h e t r a n s i t i o n a m p l i t u d e (5.16) i n a n e w f o r m t h a t is easier t o d e a l w i t h . W e b e g i n w i t h o u r o r i g i n a l d e f i n i t i o n  A% =< p\ < I \e^o f  C  W*\r >\a>  H  (5.29)  T h i s f o r m is c o r r e c t t o a l l o r d e r s i n A . A s i t is o u r w i s h t o d o p e r t u r b a t i o n t h e o r y i n t h i s f l i p p i n g t e r m we m a y r e w r i t e t h i s i n the f o r m  A%  n=0  * where  r «*.-.. r  . E f *. H  J  A  e  ° J  J  ^  d  r  ° H  J  i  [  T  )  *. < a <  I  '  ^  ^  H  J  ^  °  ...H^& ^\P d  > \a >  (5.30)  H_t(t) c o n t a i n s a l l t e r m s i n t h e H a m i l t o n i a n d i a g o n a l i n t h e f r e p r e s e n t a t i o n H {t) d  =g{t)  + V\\{t)T  z  (5.31)  Chapter 5. The Landau-Zener Problem in the Presence of a Spin Bath  and  156  contains all off-diagonal terms #  = A r _ + h.c.  A  (5.32)  This amplitude can be thought of as a sum over diagrams containing n "blips" (see figure (5.1a)).  r~~Ln_  n 'i  '« h h  h  >i  (!>)  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 4i = i/ dt f e  <l\<lf\e fi* i  1  H*e £* <V\I >\l>  Hd{T)  i  H  (5.33)  i  J0  Now it turns out that the presence of the nuclear dipole-dipole term in H {t) introduces d  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 l a t t e r are m u c h l a r g e r t h a n the a m p l i t u d e o f t h e n u c l e a r - n u c l e a r noise ~  r  0  we m a y neglect t h e m - w e s h a l l a s s u m e t h i s is the case for n o w a n d t h e n e x p l i c i t l y s h o w i n s e c t i o n 5.6 t h a t t h i s h o l d s . T h e l o n g i t u d i n a l t e r m s , h o w e v e r , m a y n o t be so a b s o r b e d . T h i s is because the e n e r g y scale t h a t we m u s t c o m p a r e these t o is A , w h i c h is m u c h 0  smaller than  r. 0  T h e r e f o r e we s h a l l r e p l a c e o u r m i c r o s c o p i c a n d d e t e r m i n i s t i c d e s c r i p t i o n w i t h equivalent  an  stochastic f o r m , f o l l o w i n g [20]. I n t h i s s t o c h a s t i c v e r s i o n t h e i n t e r n a l d y n a m i c s  of the s p i n b a t h are r e p l a c e d b y a r a n d o m l y v a r y i n g t i m e - d e p e n d e n t field a c t i n g o n t h e c e n t r a l s p i n c o m p l e x w h i c h m o d e l s the d y n a m i c s g e n e r a t e d b y t h e o r i g i n a l n u c l e a r d i p o l e d i p o l e t e r m a n d each e n v i r o n m e n t a l s p i n s u b s p a c e is a s s u m e d t o be d e c o u p l e d f r o m a l l o t h e r s . I n t e r m s o f o u r H a m i l t o n i a n t h i s e n t a i l s r e p l a c i n g t h e i n t r a - n u c l e a r t e r m i n (5.10) w i t h a t e r m 8£(t)f , z  w h e r e 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 t h e f l u c t u a t i n g  n u c l e a r - n u c l e a r b i a s field. E x p l i c i t l y we have t h a t  H{t)  = $(t) + Vj|(t)f + Af_ + h.c. ->  Hit)  = C(£) + (V\\(t) + 6£(t)) f + Af-+  2  z  h.c.  (5.34)  W e have d e n o t e d t h i s n e w s t o c h a s t i c H a m i l t o n i a n v i a a n o v e r b a r ; t h a t is, we w r i t e for t h e o r i g i n a l H a m i l t o n i a n a n d H(t)  Hit)  for the s t o c h a s t i c a p p r o x i m a t i o n .  B e c a u s e o u r n e w H a m i l t o n i a n n o w c o n t a i n s t h i s s t o c h a s t i c t e r m , we s h a l l use a s l i g h t l y 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  > | t> .  <1 | < lf\ fi* H e f? <V\I i  e  SdiT)  i  dTR  i  A  w i t h the o v e r b a r r e m i n d i n g us t h a t we are n o w u s i n g H(t)  t o e v o l v e the s y s t e m , a n d n o t  Hit). W i t h t h i s choice, the i n n e r p r o d u c t i n the f s u b s p a c e c a n be t a k e n , y i e l d i n g  A\{ = i fj  dh < I \e ^ f  1  ^ ' ^ A ^ f o  1  «MC(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 + AcOSUJTl  (5.37)  zk  k=l  where N  (f£ + H x + H y) x  y  eQk \rkafi 6I (2I - 1) k  jk  (5.38)  k  N+8  c =E G  (5.39)  fc=i  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,  Jo  n  £. *[tf e  + t +  «* e^^+Zo i +  1 )  r  >  ( 5 > 4 0 )  We have defined the initial and final polarization groups of the \i  th  N^  = E W  ,  L  M  fll  = E  species to be  S  i,  (5.41)  Because we are always in the limit that SgHB Ng pn  » 1  n  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  B e c a u s e a l l t h e e n v i r o n m e n t a l s p i n spaces are i n d e p e n d e n t i n H(t)  159  we m a y s e p a r a t e  each c o n t r i b u t i o n . R e w r i t i n g t h e cosine c o m i n g f r o m t h e flip t e r m as a s u m o f e x p o n e n tials \  W+8  /  A = 2 A cos U - i £ A \ 0  \  k=l  N+8  •I) = A  iD  k  1  £ <?  m  0  h=±l  II  ''  EHAKN D H  k=l  we f i n d t h a t A*f  A •  /i^  — £±01J  A*  4^  oX\e  L  S i n W t l  -X?  d T  ^/W /o +  l d r  ^/W ^-* )l +  1  J  1  N+8  Y  *  EIH  h=±l w h e r e \P > (\I k  II  <//|e ^" i {  T f c } ( t c  "  i l )  e ^.°' * e ^ / l  /  i {  + t A : } t l  |4  >  (5.43)  k=l  >) is t h e i n i t i a l (final) s t a t e o f the k  th  k  :  n u c l e a r s p i n a n d r5£j/ is the ( t i m e  d e p e n d e n t ) n u c l e a r - n u c l e a r l o n g i t u d i n a l b i a s field a c t i n g d u r i n g t h e t r a n s i t i o n f r o m \P > t o \P  >.  T h i s is the f o r m o f t h e t r a n s i t i o n a m p l i t u d e t h a t is b e s t s u i t e d for u s i n g as  i n p u t i n t o 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 r o b a b i l i t y .  5.3.4  Processing of the Transition  Probability  (i) T h e F o r m a l E x p r e s s i o n T h e n e x t s t e p o n o u r j o u r n e y t o w a r d s the final o n e - m o l e c u l e r e l a x a t i o n r a t e is the c o m p u t a t i o n o f the t r a n s i t i o n p r o b a b i l i t y f r o m the r e s u l t (5.43). A s we n o t e d i n o u r i n t r o 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 r o m the s t a t e | j > <S)\P > t o the s t a t e  | l> <g>\lf >  with Hamiltonian  H(t)  is s i m p l y  PH = A\"A\{ Now  i n o u r case we are w o r k i n g w i t h t h e t r a n s f o r m e d H a m i l t o n i a n H(t).  this expression? it  (5.44) D o e s t h i s change  L o o k i n g at t h e f o r m o f o u r t r a n s i t i o n a m p l i t u d e (5.43) we r e a l i z e t h a t  c o n t a i n s e x p l i c i t m e n t i o n o f 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 t o be  Chapter 5. The Landau-Zener Problem in the Presence of a Spin Bath  160  a v e r a g e d over i n t h e final e x p r e s s i o n for t h e 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 m e a n s t h a t i f we use H(t),  t h e e x p r e s s i o n for t h e t r a n s i t i o n p r o b a b i l i t y b e c o m e s  P;( = J V[S^ (t)}V[S^ (t)}A^A\{ f  (5.45)  f  w h e r e we have e x p l i c i t l y i n c l u d e d a f u n c t i o n a l average over t h e  fluctuating  bias  field.  U s i n g o u r r e s u l t (5.43) we find t h a t t h i s t r a n s i t i o n p r o b a b i l i t y i s  p;{ = Al £ Nj~*  e  *  tofvmit)]  fj  i(W-m**) J J  h,m=±l  PK„(i)]e'[-  ( , i  """- "' A  ^ -i{a+r }t2 rnA%^ J -i{C -r }(t -t )^jf  <  e  k  e  D  ke  |ji  >  k  k  c  2  ! |  - ^ 2  >  k=l  < lJ_\e ^k-^k}(tc-n) hA% T i{i +y }t i  e  D  ke  k  k  i  To translate this into the form P ^  (5.46)  we s u m over a l l final s t a t e s o f t h e s p i n b a t h ( u s i n g  t h e c o m p l e t e n e s s r e l a t i o n ) a n d r e s t r i c t t h e set o f i n i t i a l s p i n b a t h s t a t e s t o be those w i t h polarization group M ^ , giving  P^f  = A fj  dt, fj  2  0  £  e  i(fc  ^ ^( "-^- >"^)+«^-^)] si  s  2e  *- **> I " ^ [ ^ ( t ) ] P [ ^ ( t ) ] e m  h,m=±\  2  l  ^  d  ^  (  r  £  )  ~  peMj  7  K Y[ < M  p f c  |e~ ^ l {  + T k } t 2  e  m A  ~ "' ' e~ ^~ '^-^ k  D  h  l {  T  »' '  hA e  e^  D Ik  l{  k+r  k } t l  \M  pk  > (5.47)  w h e r e \M  pk  > is t h e i n i t i a l s t a t e of t h e k^  polarization group M  ;  i  nuclear spin i n the p  th  spin b a t h state h a v i n g  a n d S^M^ i s t h e f l u c t u a t i n g l o n g i t u d i n a l b i a s t e r m c o m i n g f r o m  t h e n u c l e a r s p i n s i n the M  M  polarization group.  Chapter 5. The Landau-Zener Problem in the Presence of a Spin Bath  5.3.5  161  Processing of the Transition Probability (ii) 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 T N o i s e 2  We  see f r o m the p r e c e d i n g t h a t t h e c o n t r i b u t i o n s f r o m t h e effective n u c l e a r - n u c l e a r t e r m  i n o u r H a m i l t o n i a n H(t)  c o m i n g from the p  species are w h o l l y d e s c r i b e d b y the expres-  th  sion  /V In  V[S^(t)]e- K ^ 2l  dT5  {T)  (5.48)  o r d e r t o e x p l i c i t l y e v a l u a t e t h i s c o n t r i b u t i o n we first n e e d t o s p e c i f y the p r o b a b i l i t y  f u n c t i o n a l P[8£,M^(t)]- W e a s s u m e t h a t t h i s r a n d o m process is g a u s s i a n a n d  therefore  take  7> [<5£ (*)] =  ^  M/i  d s i  ^  ^ ^  d S 2 S  M  S l ) K M  ^  S l  ~  S 2 ) 5  ^ ^ M  (5.49)  S 2 )  w h e r e t h e q u a n t i t y KM,,, c a n be u n d e r s t o o d i n t e r m s o f the f o l l o w i n g .  The probability  f u n c t i o n a l is defined so t h a t t h e average o f a n y o p e r a t o r over i t is s i m p l y  < A^MM  > = /V[5^(t)]A[8^(t)]V[5^(t)}  (5.50)  It f o l l o w s therefore t h a t the a u t o c o r r e l a t i o n f u n c t i o n o f t h e noise is  < s^( )s^( ) Sl  > = j  S2  2>[(&£M (t)]<w*i)<y^ M  (5.51)  The  f o r m a l s o l u t i o n t o t h i s e q u a t i o n is [119]  < 6ZMM) ZM,{S ) S  2  where K  l M  >=  MM  K  ~ )  (- )  S2  5  52  is defined b y  Jds'K (si Mll  -  s)K ^s l  M  - s ) =6(si2  s) 2  (5.53)  Chapter 5. The Landau-Zener Problem in the Presence of a Spin Bath  162  I n o u r case, we are i n t e r e s t e d i n a noise f u n c t i o n a l S£,M^(t) w h i c h changes b y r o u g h l y SUJM^ i n a t i m e T /N£  ( t h i s q u a n t i t y is r e l a t e d t o the s p r e a d i n e n e r g y space of t h e  polarization group  -simply Y .  2  = VNs5tu ^)-  It t h e n diffuses i n energy space w i t h  M  My  Mff  diffusion c o n s t a n t  = r w h e r e we have defined A  t  ^  KJ  2  = -^ .  T h i s n o w a l l o w s us t o find o u t w h a t K (s\  L  M  -  —s)  l  M  2  is  for o u r p r o b l e m . T h i s is because <  ( < S £ M „ ( S I )  5i ,{s ))  >  2  -  M  2  < (5£MM)) 2  = =  > + < (^M,(S ))  Si ^\si-s \  = A  2D  M  > - 2 < ^ ( )S^ (s )  2  2  2  M i i  Mfi  Sl  Mfi  2  \si-s \  (5.55)  2  a n d therefore A  <  ^ M , ( S I ) ^ M , ( S  2  )  >  =  K \{ M  Sl  -) S2  3  = -^-  (\ \ + \s \ - \ Sl  2  - s \)  S l  2  + 5f^(0)  (5.56)  W e n o w i n t r o d u c e the c h a r a c t e r i s t i c f u n c t i o n a l $ [ Q M J d u a l t o o u r p r o b a b i l i t y f u n c t i o n a l  V[KMM  = I^O^We"*  F  *  ^  0  W  ^  (  T  )  <  &  [  Q  M  |  1  (  *  )  ]  (5.57)  F o r a g a u s s i a n p r o b a b i l i t y f u n c t i o n a l i t follows t h a t  $[QM  (t)] = e ^ ^ ^ ^ M ^ e ' ^ *  8  1  ^  8  2  ®  (5.58)  T h e q u a n t i t y t h a t we are t r y i n g t o e v a l u a t e (5.48) is s i m p l y $[—2]. B e c a u s e we k n o w w h a t K (s\ l  M  —s) 2  fv  is f r o m p h y s i c a l g r o u n d s , we m a y n o w w r i t e  V[6£ (t)]e-*% '> dT6iM  Mll  lT)  >  = $[-2]  Chapter 5. The Landau-Zener Problem in the Presence of a Spin Bath  163  Performing the s integrals gives 3  /V  V i S U M ^  [S^(t)\  ^ ^ ^  1  = e-^  ( t 2  -  t l ) 3  - -, r  ( i 2  -  (5.60)  t l ) 2  We find therefore that pM»  =  A  A  f  tc  2  /"*  df  °Jo  n  ^2  c  3  d t  Jo  gj(/i*-"i**)  >p  h,m=±l  3  i[^(sin * -sin t2)+«* -*2)] --^(t2-ti) -r^(t2-t ) U  1  W  1  1  e  2  ^  peM? ^ f> M  Nt  JJ  < M  p f c  |e  _ i {  ^  + i } t 2  e ^V^ ^^^  >  m  e  (5.61) Note that since  A  «C T  M F I  MF  (that is,  _  T  MFI  for the systems under consideration,  ^> T ~ ) 2  L  we can always drop the cubic term in (ti — t ) in the above expression. We therefore find 2  that P^  AQ  =  dti °  rft 4^( ~ ) ^* )] ~ ^^ ^ SINWTL SINWI2 +  A*  0  1_I2  R  2E  2-IL  £  E  °  J  e  _ 1 _  i(/.*-m*-)  h,m=±l  J  V  ZMI » CM  Nt  JJ  < j\^P | - {Cfc+t }i2 mA^ -4 -i{C -f }(t -t ) /i^^ f c  i  e  f c  D  e  e  f c  f c  1  2  e  >  (5.62)  5.4  Solution Without Spin Bath  The expression (5.62) is still quite opaque. In order to see how one extracts meaningful results, we shall see how to resolve it in the situation where the spin bath is absent. Here the Hamiltonians H(t) and H(t) are identical and (k = T  K  Uk=i  < M k\Ml >= l  P  n  and P$ -> P ,  1  = A  n  dt  2  r  x  °  J  h  C r f t 2 e  =A  N  D  = TM = 0. Since  we may write (5.62) as  ^(sin-ti-sina; )+ r(« - )] t 2  £  1  i 2  Vh,m=±l  j[h*-m**)  (5  gg)  Chapter 5. The Landau-Zener Problem in the Presence of a Spin Bath  T h e s u m s over {h,m}  P  c o l l a p s e , a n d we have  = 2 A |cos$| 2  n  164  f Jo  2  [ ^ 4^( Jo  tc  tc  s i n  2 E  ^-  s i n  ^)+C(ti-*2)]  (  5  6  4  )  T h e s e i n t e g r a l s c a n n o w be p e r f o r m e d , w i t h r e s u l t A 7T 2  2  ^ ^ H c o s c & l where J  (5.65)  2  a n d £ are t h e A n g e r a n d W e b e r f u n c t i o n s r e s p e c t i v e l y [113].  T h i s result,  n o r m a l i z e d t o t h e t r a n s i t i o n p r o b a b i l i t y for t h e s t a n d a r d L a n d a u - Z e n e r process  P  (o) _  A  2  ?  _  R  AO|COS$| TT 2  n  (5.66)  Aui  IS n  2A\  PA  p(0)  LO  u,  \  ui  (5.67)  T h i s q u a n t i t y is a f u n c t i o n o n l y of A/ui a n d ^/ui. W e have p l o t t e d i t as a f u n c t i o n oi^/ui for t h r e e fixed values o f A/u> i n the figures t h a t f o l l o w . N o t e t h a t s i m i l a r t r a n s i t i o n a m p l i t u d e s have b e e n c a l c u l a t e d p r e v i o u s l y for r e l a t e d p r o b l e m s [115, 120, 121]. P e r h a p s t h e m o s t c l o s e l y r e l a t e d 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 t h e H a m i l t o n i a n [115]  H =  £ —A cos uit  A  n  (5.68)  -e  Ao  T h e p u b l i s h e d r e s u l t is t h e t r a n s i t i o n p r o b a b i l i t y over one e n t i r e c y c l e of t h e field, w h i c h is a t i m e w h i c h is d o u b l e t h a t o f the c o a r s e - g r a i n i n g t i m e we are u s i n g , a n d i t is g i v e n b y 1 2 A'in  2  P w h e r e J^(x)  is a fj,  LO  1  order Bessel function.  ^  /2A u  V  ui  (5.69)  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 P  = 4A |cos$| / 2  n  2  W  2  dt, f '  J—t /2 c  dfee'IW-^fc-*)]  C 2  (5.70)  J—t /2 c  These integrals may be performed, giving error functions [113] which can be easily plotted. This result, again normalized to P$, 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  :  \  0.10.08 0.06^  \  0.04 0.02  \ 2  0  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  i  lA  i 0.8 0.6 0.4  02 0  2  4  6  8  1 0  12  14  16  18  2 0  2 n 1.81.61.4  :  /  1.2^  \ \  1-  0.8  \  0.6 0.4^  v  ~ \  0.2^  0  2  4  6  8  10 12 14 16 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-  321 i-..^»,,-..,w,-»» «'V: l  100  200  300  400  500  600  100  200  300  400  500  600  12.A  0.6  0.4 A  02. A  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  5.5  Solution For a Spin B a t h w i t h no Quadrupolar  169  Contribution  If t h e s p i n b a t h c o n t a i n s o n l y s p i n 1/2 n u c l e i , or i f t h e c o n t r i b u t i o n due t o the e l e c t r i c q u a d r u p o l a r t e r m is o m i t t e d f r o m t h e H a m i l t o n i a n , t h e n t h e e x p r e s s i o n (5.62) s i m p l i f i e s c o n s i d e r a b l y . I n t h i s s e c t i o n we s h a l l c a l c u l a t e t h e f u l l g e n e r a l r e l a x a t i o n r a t e p r o d u c e d by t h e effective H a m i l t o n i a n (5.6) i n the l i m i t w h e r e t h e t e r m s r e p r e s e n t i n g t h e e l e c t r i c q u a d r u p o l a r effects are t a k e n t o z e r o . F o r m a l l y w h a t we s h a l l d o is t a k e  It s h a l l t u r n o u t i n w h a t f o l l o w s t h a t o u r choice o f a s i n u s o i d a l s w e e p i n g field i n t r o duces t e c h n i c a l d i f f i c u l t i e s i n e v a l u a t i n g i n t e g r a l s w h e n t h e s p i n b a t h is i n c l u d e d . F o r t h i s r e a s o n we s h a l l i n t h i s s e c t i o n c o n s i d e r o n l y t h e 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 t h i s a p p l i e d field (5.62) c a n be w r i t t e n  = A  £  2  r  dt, dt e [^ dti /r T/2U J-K/2U, J—7T/2UI J—TT/2UJ /2w  II  /2u  i  ( t i  2  <  -  t 2 ) 2 +  «  t i  -* )] - ^ 2  r  e  ( t i  -  t 2 ) 2  E  e  ^*-m*-)  uh,m=±X I 1  M \e~ ^'' » e »'^ pk  l  k  t2  mA  (5.71)  5.5.1  P u r e Orthogonality Blocking  In o r d e r t o get a feel for h o w t h i s c a l c u l a t i o n w i l l go i n the g e n e r a l case, l e t ' s s t a r t off w i t h a p a r t i c u l a r l y i n t e r e s t i n g l i m i t - t h a t o f 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 (for the d e f i n i t i o n s o f o r t h o g o n a 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 d e g e n e r a c y b l o c k i n g we refer reader t o [20]). I n t h i s case we take the 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 A ^ k  D  The  the  t o be zero.  r e a s o n t h a t t h i s l i m i t is p a r t i c u l a r l y s i m p l e is t h a t w i t h o u t these t e r m s , we m a y t a k e  the axes of q u a n t i z a t i o n o f each o f the n u c l e a r s p i n s t o be s u c h t h a t 7 ^  = z without  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 Atf . We D  find in this limit that  p^f"=A ,i cos $i 2  J _ »  n  £  r  /2uj  dtl  ^ e4^^-^ ^-^] - ^^^) 2  r  2  2  e  J-TT/2UJ J-TT/2UJ  < l ^^-^)<Mf|e-< ^ )  e  r  l7u  2  /  )  ( t l  -  t 2 )  |Mf >  (5.72)  eMl fc =i  Cm  M  P  where we have used notation such that I  means the z state of the k ^ spin, normalized 1  kfiZ  such that —1 < Ikz < 1- Note that we must be careful here, as the conversion from field to energy units contains a factor of \I \. We now choose, without loss of generality, k  7^ = c z + x Q  (5.73)  Cl  where co = - | 7 £ | c o s 2 ^ )  ,  = |7g |sin2^  (5.74)  )  C l  wiith  (5.75)  cos 2/3 , = - 7 ^ . 7 ^ fc|  Changing variables to X = u>{t -t ) x  2  ,  Y = J-±±^  (5.76)  allows us to write |2 CO  U  l  —  J-00  fl* J-^^l "*** < Ml \e- ~^-'^ \Ml  N  E  J-n/2 r/2  2  1  >i  1  k  ^l  -  x  k  >  (5.77)  Here we have extended the limits on the X integral. This will be permissible as long as r „ 3> to (which we assume here), as the gaussian term cuts off the large X contributions. M  Chapter 5. The Landau-Zener Problem in the Presence of a Spin Bath  Similarly if  <  171  (which we shall see is also the case-see table 5.2 and tables  ^/NTTM^  2.3-2.11) we may expand the inner product to quadratic order in X, giving < Mf \e- ^" ^ \MP i  I  x  >t  k  -(2)  l-iX<  Mf\^  • I \M?  >  k  —  —  <  '-(2) Tk [ ^ • I k A  Mf\  \Mf  > (5.78)  1^(2) 12 x2 \f  IT1 I 1 + %XI J-^cos 2ft, 2)  k  2  k  to  x l7< l 2  2 )  + cos 2/? / 2  a ^ M  2ui  l  2  1  2  fc/i  2  • sin 2/3,  1  2  4 2  (5.79)  W  = e Insertion of this into (5.72) yields i%»  = ° A  c  °f  r  { 2  CO'  dY r  d X e ^ x ^ A e - ^  T/2 l^[ l 2zXI ^cos*p 2)  n A?  E  M,  l c  kaZ  where we have used the fact that |7^|  =  J—  k  ITI^I  sin 2/3, 2  1  (5.80)  ( i - the magnitude of the field acting on e  the k ^ nuclear spin before/after the central spin complex flips is the same). Defining the 1  quantities j2  TV," 1  Vlf  = 2 £  1 J f c  |cos ft 2  £  l7i | sin 2^ 1 1 }  2  2  •  z  (5.81)  k»=l  we can write this as A |coscfr| 2  p M i i  =  CJ  2  2 r  /2  /  7-7T/2 J-n/2  d  y  /  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 p ^ with its average value; that is, we take 2  l7i | sin 2^ 1)  P\^\>Z  2  )  2  kit — 1  \  (5.83)  /  Since  ^ ^ ^ m - ' M ^ the average value of  (5.84)  is simply  I^Psin^^  P?, = % ^ E  (5.85)  The approximation of neglecting the width of the distribution of numbers {pi } is justified M  whenever this width is much less than T  M/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 FesIn this situation, there are an enormous number of possible states because N  H  (and  therefore C ) is large. In F e there are 120 protons per molecule. This means that there M[]  are 2  1 2 0  ~ 10  contains N  H  8  36  states in 2N + 1 = 241 polarization groups. The Mjj polarization group  choose \M  H  H  — N \/2 H  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 ) . In this limit we find that / i  ^ ^ £ l # W &  (5.87)  Now the width W ( M ) is a little trickier to dealt with. In general it is apparent that this M  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 spin  (which, for  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 ) for | M | > VNf will be irrelevant, simply because A1  M  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 ) ~ ( l - J ^ l j W((VJ M  ,  | 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  174  i t is s i m p l y 2  N  '  -+ - ? = £ 1 ^ | cos A 2  N o t e t h a t t o b e c o n s i s t e n t here we m u s t also t a k e we t o o k W(M )  r  M ( i  -> r  for t h e s a m e r e a s o n t h a t  0/J  —»• W ( O ^ ) . W e s h a l l h o l d off o n u s i n g these a p p r o x i m a t i o n s for t h e t i m e  fl  being.  (5-89)  W e c a n m a k e e x c e l l e n t progress i n e v a l u a t i n g o u r expressions w i t h u n s p e c i f i e d  , e f ^ a n d W ( M ) . U n t i l w e r e a c h t h e p o i n t w h e r e w e need t o have e x p l i c i t f o r m s for / t  these we s h a l l j u s t leave t h e m as f u n c t i o n s o f t h e p o l a r i z a t i o n g r o u p . U s i n g t h e g a u s s i a n w e i g h t i n g a l l o w s us t o r e w r i t e o u r s u m i n t h e f o r m  l  ^  y  l A  M  r°° , „  1  - /  1 > -tjl^Hl  tlfX  ™W  e ^r  t1  /-oo  = e ~ ^ y - = —  g  de^e  e  2  {  2  2  ^"e  l  flu.  X  e ^r  W (M )/2+  «M  = e  'fx l  P  ^  (5.90)  x  S u b s t i t u t i o n o f (5.90) i n t o t h e e x p r e s s i o n for t h e t r a n s i t i o n p r o b a b i l i t y (5.82) y i e l d s p M t  ^  =  A^OS$| CO  l  2 r  /2  ^  J-K/2 J-IT/2  2  dY r r  a  v  A  L ^'Z.d x A ^ ^ e ^ e  ^  w  X  Y  +  x  ]  2  2  ^  +  f  ^  W  x  ,  J-OO  (5.91)  Evaluation of the Ensemble Average Over Polarization Groups Examination of P ^  reveals t h e f o l l o w i n g useful fact. B e c a u s e t h e e n e r g y spreads i n t h e  different p o l a r i z a t i o n g r o u p s o v e r l a p c o n s i d e r a b l y , i e .  «  ^  (  M  ^ + ^  + r ^  (5.92)  we m a y a p p r o x i m a t e t h e ensemble average over a l l p o l a r i z a t i o n g r o u p s b y i n t e g r a t i o n s . F o r m a l l y t h i s is d o n e b y t a k i n g  = APG  fi  \i  r  KN? J-oo  dM,e- V ° M  "  2N  (5.93)  Chapter 5. The Landau-Zener Problem in the Presence of a Spin Bath  175  w h e r e we have e x t e n d e d t h e i n t e g r a t i o n l i m i t s (the g a u s s i a n c u t s off the l a r g e M  contri-  b u t i o n s ) . F r o m (5.28) we have  APG  n  A^|cos$|  r/2  2  J-ix/2  J-OO M„  n  J=  f  00  W(M )  dM,e- ll^eV<  A:  X  M  »  M  w (M^)/2+^ +r2 2  M  X  2  (5.94) w h e r e we have w r i t t e n e f ^ e x p l i c i t l y as a f u n c t i o n o f t h e i n t e g r a l s over t h e set {M^}  u s i n g (5.87). I n o r d e r t o p e r f o r m  we n o w i n v o k e o u r W ( M  / 1  ) —» W ( O ^ ) a p p r o x i m a t i o n . T h e  i n t e g r a l s over the set { M ^ } are t h e n e a s i l y p e r f o r m e d , g i v i n g A |cos$| 2  2  W2  2  f'  •Pf-I- ~  J-n/2T / 2  dY r  dxA% +Me-$ ' XY  x  (5.95)  J-OO  w h e r e we have d e f i n e d the f u l l e n e r g y w i d t h  (5.96) w h e r e a l l these q u a n t i t i e s are e v a l u a t e d for the zero p o l a r i z a t i o n g r o u p s p i n set for each species.  Evaluation of the X and Y Integrals T h e X i n t e g r a l c a n n o w be p e r f o r m e d , y i e l d i n g A 1 cos $ 2  U  P  -  I  2  r/  2  yfUloW J-n/2  m  2  dY e x p  '4W  2  (2AY \  TTLU  t  f' LU  (5.97)  C h a n g i n g variables to  Z =  LU  (2AY  2W V vro;  C  + LU  (5.98)  Chapter 5. The Landau-Zener Problem in the Presence of a Spin Bath  176  a l l o w s us t o w r i t e o u r r e l a x a t i o n r a t e i n t h e final f o r m ,  o>  A n | cos $ |  2  r + ,„ z  2  ,  7  where  ^  = ^  ("<»)  Evaluation of the Relaxation Rate in Various Interesting Limits I n t h e l i m i t w h e r e \A\ 3> y / ^ + ~ W 2  we recover o u r o l d l a r g e A r e s u l t , w h i c h is o f c o u r s e  2  independent of , r"  1  A^|cos$| = —^— L 2  5.101  A  If \A\ <C \ / £ + W 2  2  v  '  then the relaxation rate becomes A |cos$| 2  2  2  414/2  (5.102)  Discussion of Results: I. Pure Orthogonality Blocking T h e g e n e r a l r e s u l t (5.99) p r o d u c e s t w o l i m i t i n g cases (5.101, 5.102) t h a t are b o t h q u i t e i n t e r e s t i n g . T h e first o f these (5.101), v a l i d for l a r g e A, t e l l s us t h a t i n t h i s l i m i t o r t h o g o n a l i t y b l o c k i n g effects do not affect the relaxation  rate at all. H o w c a n we e x p l a i n t h i s  physically? In order to understand  t h e r e s u l t , we s h a l l present  orthogonality blocking works i n a molecular magnet. level s y s t e m , b e g i n s i t s e v o l u t i o n i n some f  z  a graphical depiction of how  T h e central spin complex, a two-  e i g e n s t a t e w h i c h we choose w i t h o u t loss o f  g e n e r a l i t y t o b e | y > . N o w for a l l t h e t i m e t h a t i t r e m a i n s i n t h i s s t a t e t h e n u c l e a r s p i n s  Chapter 5. The Landau-Zener Problem in the Presence of a Spin Bath  feel i t s d i p o l a r field. T h e k  177  n u c l e a r s p i n i n i t i a l l y is e x p o s e d t o t h i s field w h i c h we have  called 7 ^ . W e saw p r e v i o u s l y t h a t b e c a u s e o f t h e n a t u r e o f 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 we c o u l d choose t h e a x i s o f q u a n t i z a t i o n o f t h e k  th  n u c l e a r s p i n freely a n d for c o n v e n i e n c e chose  t h e z a x i s t o be p a r a l l e l t o t h e d i r e c t i o n o f 7 ^ . t h i s choice o f basis t h e k  ih  P h y s i c a l l y w h a t t h i s m e a n s is t h a t for  n u c l e a r s p i n feels o n l y a l o n g i t u d i n a l field as l o n g as the c e n t r a l  spin c o m p l e x remains i n its i n i t i a l state | y > . N o w w h e n the 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 t o t h e o t h e r f n u c l e a r s p i n feels a different field 7 ^ of course, t h a t t h e The  k  th  z  eigenstate  | i>,  the  th  which contains transverse components. T h i s m e a n s ,  nuclear spin w i l l  precess i n the n e w field 7 ^ .  d i a g r a m t h a t we are e v a l u a t i n g i n o r d e r t o solve for t h e t r a n s i t i o n p r o b a b i l i t y is  the one s h o w n i n figure 5.5. W e see t h a t t h i s process c o n t a i n s a l e n g t h o f t i m e t  x  w h e r e t h e c e n t r a l s p i n c o m p l e x is i n the s t a t e | \ > . n u c l e a r s p i n precesses i n the Now  k  field  D u r i n g t h i s l e n g t h o f t i m e the  —i  2  k  th  7^.  w h a t does t h i s have t o d o w i t h o u r large A r e s u l t (5.101)?  L e t us go b a c k t o  a n e a r l i e r e x p r e s s i o n (5.95) a n d l o o k c l o s e l y at the i n t e g r a l s i n v o l v e d . I n p a r t i c u l a r , let us e x a m i n e the Y i n t e g r a t i o n . N o t i c e t h a t there is o n l y one t e r m u n d e r t h e i n t e g r a t i o n t h a t is a f u n c t i o n of Y.  T h i s t e r m m a y be i s o l a t e d a n d is  (5.103)  Now  i f A/to  is "large e n o u g h " , we see t h a t t h i s e x p r e s s i o n w i l l b e g i n to a p p r o x i m a t e a  d e l t a f u n c t i o n i n A ' . H o w large is l a r g e e n o u g h ? L o o k i n g b a c k t o the e x p r e s s i o n (5.95) we see t h a t t h e X i n t e g r a l is n o w o f the f o r m  (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 k 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 is 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 feels a f i e l d y± f o r t i m e s t < ti, w h i c h we 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 feels 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 in general transverse components. T h i s causes a precession of the nuclear spin. A f t e r the 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 less t h a n f u l l o v e r l a p with its original state. th  Changing variables to Z =  AX/LJ  a l l o w s us t o w r i t e (5.105)  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 is 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 the exponentials grow slowly c o m p a r e d to the  t e r m a n d we c a n n e g l e c t t h e m , a l l o w i n g  f o r e a s y s o l u t i o n o f (5.105). 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 the expression all references to the nuclear spins disappear! W h a t has h a p p e n e d ? W e see t h a t t h i s e x p r e s s i o n is 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  — t ). 2  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\ — t < ~ 2  ^. 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 "  Chapter 5. The Landau-Zener Problem in the Presence of a Spin Bath  179  a n d therefore t h e l e n g t h o f t i m e t h a t t h e n u c l e a r s p i n s c a n precess i n t h e field j  . This  k  is t h e p h y s i c a l r e a s o n w h y o r t h o g o n a l i t y b l o c k i n g does n o t affect o u r l a r g e A r e l a x a t i o n r a t e - t h e processes i n v o l v e d a r e c o m p l e t e l y d o m i n a t e d b y s h o r t b l i p s a n d therefore t h e n u c l e a r s p i n s d o n ' t have t i m e t o precess o u t o f t h e i r i n i t i a l states. L e t u s n o w t u r n o u r a t t e n t i o n t o t h e o t h e r l i m i t i n g case t h a t w e h a v e w o r k e d o u t , t h a t for s m a l l A (5.102). W h a t w e have f o u n d i s t h a t t h e r e l a x a t i o n r a t e falls off l i k e a gaussian w i t h a p p l i e d bias £. T h i s seemingly contradicts results o b t a i n e d for A = 0 w h e r e t h e r e l a x a t i o n r a t e was f o u n d t o b e e x p o n e n t i a l w i t h b i a s [20]  T  - i  =  ±L -|£l/to £0  (5.106)  e  w h e r e £0 is a n e n e r g y scale i n the P r o k o f i ' e v a n d S t a m p t h e o r y [20] s u c h t h a t r  0  < £0 < W  (W is t h e f u l l e n e r g y w i d t h o f the n u c l e a r s p i n d i s t r i b u t i o n - i n o u r case we s h a l l f i n d t h a t i t is W ~ 100 — 400MHz  d e p e n d i n g o n the c h o i c e o f i s o t o p e s i n t h e Fe$). W h y i t is t h a t  o u r r e s u l t s e e m i n g l y disagrees w i t h t h i s e a r l i e r r e s u l t c a n b e t r a c e d t o a s u b t l e difference i n t h e w a y t h a t o u r e x p r e s s i o n (5.102) a n d t h e e x p r e s s i o n (5.106) were c a l c u l a t e d . W h e n A = 0 t h e r e is a n a d d i t i o n a l c o n s t r a i n t w h e n we c a l c u l a t e t h e t r a n s i t i o n p r o b a b i l i t y for t h e c e n t r a l s p i n , a n d t h a t is t h a t w h e n 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 I t , M >—>• I I, M' > i t is necessary i n o r d e r t o conserve e n e r g y t h a t M' = — M .  That  is, t h e t o t a l e n e r g y o f the c e n t r a l s p i n p l u s t h e e n v i r o n m e n t a l s p i n s h a s t o b e t h e s a m e before a n d after a flip a n d therefore t h e e n v i r o n m e n t a l s p i n s c a n n o t g o i n t o a n y M' t h e y want.  N o w i n t h e c a l c u l a t i o n o f (5.106) t h i s c o n s t r a i n t , w h i c h is e x p l i c i t l y used i n t h e  c a l c u l a t i o n 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 , is responsible for the exponential dependence on  external bias. In o u r case we have n o t e x p l i c i t l y p u t i n a n y s u c h c o n s t r a i n t .  I n s t e a d we have a l -  l o w e d i n o u r e x p r e s s i o n for t h e 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 a n y M'. T h e energy c o n s t r a i n t here o f course s t i l l e x i s t s ( i t is i m p l i c i t i n t h e e x p r e s s i o n for t h e  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 A = 0 because we are in the fast passage limit AQ <§C  (5.102) is not  valid for  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 We  T h e General Case; Inclusion of Topological Decoherence  now turn our attention to the more general case, including the contributions due  to the topological decoherence terms A . k  ND  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  reason.  T h e basis i n w h i c h t h e A  are c a l c u l a t e d is t h a t o f the s p i n H a m i l t o n i a n for  k  the p r o b l e m .  N  181  D  T h a t is, t h e axes o f q u a n t i z a t i o n o f t h e n u c l e a r s p i n s c a n n o t be c h o s e n  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 t h e A j)  i n t o t h i s n e w basis.  k  N  I n o r d e r t o get a  c o n c r e t e feel for w h y t h i s presents a p r o b l e m , let us c o n s i d e r t h e specific case o f 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 d i s c u s s e d at l e n g t h i n c h a p t e r 2. F o r m o l e c u l e s w i t h t h i s s y m m e t r y we s h o w e d t h a t (2.73)  nS  Tk  9n h'n  N,D  k  £  Lpe{p }  D —E  Mfk - E  MJS-4 2E '  pe{pj.}  t  E  |_pe{pi-}  M  $  -  £  M  t  (5.107)  for l i g a n d n u c l e a r s p i n s k = 1..N  and  xk _ 4 g7r  for a n y  Fe  57  |c  y— \ %  ID- E 2E -x  (5.108)  i n t h e m a t e r i a l . B y s p l i t t i n g t h e r e a l a n d i m a g i n a r y p a r t s o f these we c a n  write  A  N  ~(l)~  a  h + ia  -~(2)  \  4  =  —  D  k  .  k  l  k  ~k  h  (5.109)  2  where  a  B-eA  k N  D  2)  \ImA \ k  (5.110)  and R A%,D e  k  l  for a l l n u c l e a r s p i n s k = 1..N  + 8.  q u a n t i z a t i o n t o be s u c h t h a t 7 ^  =  ImA n =  k  K  2  k  N  D  •  (D.lll)  a  N o w as we d i d p r e v i o u s l y , we choose t h e axes of  z a n d 7 ^ = —cos2(l z + s i n 2 / ? ^ x for a l l k. T h e n k  we see t h a t since these n e w axes d o n o t i n g e n e r a l c o r r e s p o n d t o those o f t h e s p i n  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 i s n e c e s s a r y t o r o t a t e 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 i n t o these n e w bases. T h e s e r o t a t i o n s y i e l d i n g e n e r a l n e w t o p o l o g i c a l decoherence t e r m s  —k  r,N,D  a  n  l ^  l  a  k  (5.112)  2  n  in t e r m s o f w h i c h w e m a y w r i t e  n / ^ A ^ c ^ J—Tr/2w  ,,  J—n/2u  E  rL,(ti-«2) _ ± 2  n  £ ei{h$—m<&*)  CM,  / i , m =L1± l  u  n  ih^\l \{ti-t2) { X)  e  k  "f-  \j^pk  < M \e k ^ ^~ k 2-h^ -i(h-t )1^ pk  ma  L>n  Ik  ima  Z>A  e  2  >  (5.113) N o t e t h a t n o w t h a t w e have t o p o l o g i c a l decoherence effects i t is necessary t o i n c l u d e a l l t h e n u c l e i ( a n d therefore t h e p r o d u c t over species a) here because t h e s u m over m, h d o e s n ' t c o m m u t e w i t h t h e p r o d u c t over n u c l e i . C h a n g i n g v a r i a b l e s t o  = u(h  x  Y =  -1 ) 2  A  (i  +1 ) 2  x  (5.114)  2ir  a l l o w s us t o w r i t e TTpffa  =  f  A,2uJ d  Y  J-A/2.  U  n  C  A hx] XY+  dXe  II  y  e  i(W-m**  , t ± l  7-00  E  M,  r  e -^r- ^ l  h  x  < M \e k ^' *~ k *' »e~ ^ pk  ma  l)  rk  ima  )h  Tk  l1  where we have t a k e n t h e l i m i t s o n t h e X i n t e g r a l t o i n f i n i t y as before.  \M  pk  >  (5.115)  Chapter 5. The Landau-Zener Problem in the Presence of a Spin Bath  183  Evaluation of the Inner P r o d u c t over Spin B a t h States A s we saw p r e v i o u s l y i n t h e case o f 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 , because t h e g a u s s i a n t e r m ( i n X) X  c o m i n g f r o m t h e average over the n u c l e a r - n u c l e a r b i a s field c u t s off l a r g e  c o n t r i b u t i o n s t o t h e e x p r e s s i o n (5.115), we m a y e x p a n d t h e o r t h o g o n a l i t y b l o c k i n g  t e r m i n s i d e the s p i n i n n e r p r o d u c t t o q u a d r a t i c o r d e r i n X.  A s w e l l , we s h a l l r e s t r i c t  o u r a t t e n t i o n t o the case w h e n a l l the |o4 '  1^1,  w h i c h is a g a i n t h e p h y s i c a l l i m i t  for t h e m a t e r i a l s i n w h i c h we are i n t e r e s t e d .  T h i s a l l o w s us t o e x p a n d t h e t o p o l o g i c a l  decoherence t e r m s i n s i d e t h e i n n e r p r o d u c t t o q u a d r a t i c o r d e r i n  (1 2)  '  as w e l l . K e e p i n g  (1 2)  o n l y u p t o q u a d r a t i c i n X,  04 '  or p r o d u c t s t h e r e o f a l l o w s us t o w r i t e the e n v i r o n m e n t a l  spin inner product i n the form < M£  f c  |e  m a  *  1 }  "i'^  - i m a  '-(2)  r  D  •f])  k  k  A*  r  • f,  k/1  •"•r.N.D  2  D  k  A:,  r N D  k  1  |Mf>  -(2)  -iX < M f | m 4 ^ *  -(2)  • J,  k  1  r,N,D  >  \Mf >  •i]  k  k  |Mf  UJ  k  •I }\[A ^  tD  Mf\  •I) | M f >  k  k  + - < M f | ([%$ M fI  f e  • I + hA ^  tD  +mh <  X jMf > - — < 2  M f| ^ •/  + < M f | (m£$  |Mf >  2  -(2)  l-iX<  _(2)  * ^  + h  UJ  •™r,N,D  UJ  M f  k  i  >  (5.116) We  s h a l l present r e s u l t s for these t e r m s one b y one a n d t h e n c o m b i n e the results  we are  finished.  after  W e have t h a t  < M f | ( m l ^ n • Ik + hA y  • h) | M f >  h  r  =<  M f | ( m + h) a ^ n ^  = (m + h) a ^h I {  k  lkiiZ  kiiZ  •4  !D  - » ( m - h)a h ^  - i(m -  { 2) k  2k  hja^^Jk^  •/  f c  jMf  > (5.117)  Chapter 5. The Landau-Zener Problem in the Presence of a Spin Bath  184  T h e next t e r m is  4f K • 4) - «lffa*,• 4)1 2  2  2 < Mf,  k  wf >  (2/, 2  a  -a  ^  {2Y  \ k p Z  kfj,Z  (1 - n\ J  (5.118)  k  Next we have >  < --< M f  • 4)  + 4f  2  (oT - T 2  fa*,  • 4)  + ictfta^  2  [n  lkfl  • I ^n k, k  2  • 4]]  \Mf  >  \  2  a (or  - T  2  ,  (2)  )  2  2  •vp.  .  And  T  (1)  ~  (2) / -  .  .  -  \  (5.119)  finally -(2)  <Mf\  (m  h  [Al^-h]  v  LO  l7£ |cos2/3^ )  T  a  kz  LO  n  )„ + h) - ia (jn (m  - h)  2  {  k/  2kfiZ  u  l7g|sin2/?  hi  -1  2  fc)i  { k»(  m  )  2  "  LU  a  i^ ik,z(m  2  1  z  + h)h  4  ~ )hik» h  y  - ictklim +  - ia J{m [  lkpiX  h)h ) 2kiiy  k  h)n ) 2kiiX  (5.120)  Chapter 5. The Landau-Zener Problem in the Presence of a Spin Bath  185  W e n o w s u b s t i t u t e these r e s u l t s i n t o t h e e x p r e s s i o n for t h e t r a n s i t i o n p r o b a b i l i t y , g i v i n g M  1  1  t  n  =  2A|r M A -  d  roc  y  J-A/2U) V-yl/aj  ;  X  [  x  y  +  i  x  ]  i ( f t #  . J/ -oo -co «" e  1  A  i  m #  .,  _f^,,  m,h=±\  XX o? p 2  "  2  _  ^^g(^+^)[P31^-i^p4i ]-l(m-/l)[p32 -iXp42 ]+(l-t-m/l)/921 -(l-m/l)p22 i M  M  M  M  /  PGM? (5.121)  w h e r e we have defined T  „  P21/x  (1)  2  /or2_r2  2  / i  (!  -  \V V Z  P32/x  ^ ', 2  AT]  1  1  (24 -4lJ . l 2  E4f(i--U =  ,  T  j - ^ -+ 9 £  ik,z)  n  JV,"  JV"  J  \  a  (2)/2kiiX  -  XKy  \  -  n  _  ^2/c  2 M  n  l f c ( i X  )  JV,"  + 2 £  4  (!)  h zOi\la\l[n n ^ %  a  l ( " ° t | . « H v  M 2 /  ni  f c M X  )  (i).  £  J  N'.  E  ^4?^*  E  —  tk^zJ  (l 2I  *  (2).  i n  k u - l  P42fi  E  ^  ^  2  v-;-  ^  ,  (1)-  (5.122)  and  P<*P = E A In  (5.123)  o t h e r w o r d s , i f t h e r e is a / J s u b s c r i p t o n one o f these q u a n t i t i e s t h e n i t refers t o  t h e species \i.  For example, p  2 1 / t  of n u c l e a r s p i n s i n species ji.  is t h e q u a n t i t y d e f i n e d a b o v e for t h e specific subset  I f t h i s s u b s c r i p t is n o t t h e r e t h e n i t refers t o t h e s u m  over a l l s p e c i e s - f o r e x a m p l e , p i is t h e s u m over t h e c o n t r i b u t i o n s f r o m a l l t h e species 2  P21 =  E / i P2Ui-  Chapter 5. The Landau-Zener Problem in the Presence of a Spin Bath  186  The Large A Limit Now  t h e r e is a l i m i t here w h e r e t h i s s i m p l i f i e s c o n s i d e r a b l y , a n d t h a t i s w h e n A is 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 scales i n t h i s e x p r e s s i o n .  I n t h i s case, t h e i n t e g r a l  over Y gives a d e l t a f u n c t i o n i n X, f o r reasons i d e n t i c a l t o t h o s e d i s c u s s e d i n t h e 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 case. T h i s a l l o w s us t o w r i t e 2A 7T 2  1 1  T  1_  LOA  ;  ^ m,h=±l  1  ^  \ {rn+h)pzi -i{m-h)pz2^+{l+mh)p2\^-{l-mh)p22A e  pt  1  (5.124) T h e S u m O v e r States i n t h e M* All  four q u a n t i t i e s p i ,  Polarization Group  h  P22p,, P3ip a n d P32p. d e p e n d e x p l i c i t l y o n t h e s t a t e o f t h e s p i n  2 tl  b a t h v i a t h e i r d e p e n d e n c e o n t h e set {h^z}- A t t h i s stage o f t h e c a l c u l a t i o n we w i s h t o p e r f o r m t h e s u m over t h e s t a t e s i n t h e Mjj p o l a r i z a t i o n g r o u p . I n o r d e r t o d o t h i s we 1  define t h e q u a n t i t i e s V" P2\p  =  2  Z, kJ  ^  l  =  P  T  - 1)  ^  2iVf ^  +  ffi  (1)2,  n  \  2  u„J  fc„=i  (2) /. 2  Z.  1 Mp,  (1 - n  )  ^  7  V  _  ffi  T~2  X  I  Z,  \ ~ ~ *  I  (1)  T m L l fc a  s  ^xni ,y -  n ,yn ^ )\ 2k  k  1  x  .  (2),„  (1)  n  -  n  2 k i i y  n  \ I  l k i i X  )\  r  I (1)  T - ^ ^  (2)/-  \oi\y l{n k  2 k i i X  n  x k t i y  -  -  M  n ^ n )\ 2k  y  lkfiX  k^ — l  ^ ( 5 / ^ - 1)  •"'• /  „  fc„=i  s  ,  j-^—  2kiiZ  (2)2 a  fcp. —  P3lp -  n 2  Z, 1%<.K* it,»  kfi=l  =  a  I  +^  ^Il~lt»z)  2  (2)/.  \ ^ kJ( a  — 1  Z, < U -  TT, 1 1  22  , I M j , v - - I (1)  k .z)  1  fcfi  I^Ip  \^  V  n  1  =  =  p ^  Z  \ - ik,z)  a  hp  2  (!) ft  ~ n M  (1  + 2 iVf  n  X  I  fc =l  >  -  P32/X  kfi —  _  )  (2)  I  (2)  1  ~  -T7JI 2_, |ttfc 2A 2 fc„=l N  -  M  Y  V  s  „  Z, K>1("2*^n  1  l^l B  1 M ^  2  l ( ~ 2^z)  M  ..  lfc)1  j, - n fc ri 2  M?/  lfcMX  )|  Chapter 5. The Landau-Zener Problem in the Presence of a Spin Bath  w  E  =  3lli  JV,"  l  187  P £ l4 W.  (1)-  2  (5.125)  F o r reasons i d e n t i c a l t o t h o s e g i v e n i n t h e s e c t i o n o n 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 we m a y n o w c h a n g e t h e s u m over s t a t e s t o a n i n t e g r a t i o n ; e x p l i c i t l y we t a k e 1  ( m + / i ) p 3 i / i —i{m-h)p32^  +(l+mh)p i -(l-mh)p22n 2  ll  -(P31u.-P31ii)  1  dp3i^e  2irW31p  2  ~(P32^~P32/i)  ^  W  27rW J-oo  e  dP3211  32  ( 21  ~ 21  p  p  M  M  )  ~  e e x p [(m + / i ) p i 3  2  22 ~P22^) M  e  M  - « ( m - /i)/o 2/i + (1 +  M  ( p  2W2  2W"2  3  mh)p  21fl  - (1 -  ^  2  mh)p } 22)M  (5.126)  N o t e t h a t t h e r e a s o n t h a t these i n t e g r a t i o n s c a n b e t a k e n t o b e i n d e p e n d e n t is because we a r e o n l y k e e p i n g t o q u a d r a t i c o r d e r i n | a ^ | . T h e s e g a u s s i a n i n t e g r a t i o n s a r e e a s i l y performed, giving 1  E  CM,  3  ( m + / i ) p 3 i  (1 + mh) [W  exp  M  - i ( m - / i ) p 3 2  2  M  + ( l + m / i ) p  2  i  M  - ( l - m / i ) p  + p i J - (1 - mh) [W 2  2  , ,  +p  2  3 ltl  2  2/1  2 2  J + (m + h)p  3lll  - i(m - h)p  32lx  (5.127)  I n s e r t i n g these r e s u l t s i n t o (5.124) gives us M,\i  IP1 M  _  2A ,7T 2  ~  0 " "  g  i ( / i * - m * * )  m,/i=±l  ^2,-1  /1  ITT/2  e x p I (1 + m / i ) \W'£ + P211 - (1 - mh) \ W% + p 1 + ( m + / i ) A n - « ( m - / i ) p X  2  22  3 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 i , 2  P22, P31 and p . In order to perform the ensemble average over the set {M^} we perform 32  the same calculation as was introduced in the section on pure orthogonality blocking; namely we take  En%->n  APG  A*  L=  r  dM e~ H ' M  2N  (5.129)  ti  A»  Substitution of this into the expression for the relaxation rate (5.130) APG  Ii  gives  r  -l  _  9A Z  A  2  A  0  A  e  i(/i*-m**)  m,/i=±l  exp [(1 + mh) [w \ + p ] - (1 - mh) [\W + p ] + (m + h)p 2  3  2i  2  22  31  - i(m - /i)p ] 32  (5.131)  Performing the {M^} integrals gives r  -i  =  2A| A  ^  e i  ( *- *. h  m  ) e x p  [1 + m/i)  12  m,/i=±l  [1 — mh) W,  2 9  +  4(5^-1)  exp  12  1  [(m + /i)W i - i ( m - / i ) W 3  3 2  ]  (5.132)  We can now perform the {m, h} sums. The result can be written  T  ~  l  =  [e cosh 2cp! Al  + e~ ' cos 2<p x 2  0  (5.133)  Chapter 5. The Landau-Zener Problem in the Presence of a Spin Bath  189  w h e r e we have defined  A  = <  l  A = 2  + W -  4iy  2 2 3+  ^  1  6  ( 5 J  )  £ £ 4f(l A« fc„=i  g"  1  £  }  £ fc =i  4f  - n? ) M  (1 -  (5-134)  (5.135)  M  D e f i n i n g n e w phases  (/>o = ^ c o s  (e~  - 1  A2  cos2(/? ) 0  and  (f>i = - c o s h  - 1  (e  A l  c o s h 2ipx)  we w r i t e  $ = <> /o + t(f>i  (5.136)  i n t e r m s o f w h i c h t h e e x p r e s s i o n (5.133) c a n b e w r i t t e n •, r"  1  An|cOS$| =  0 1  1 A  2  (5.137)  w h i c h is i d e n t i c a l t o t h e r e s u l t w i t h o u t t o p o l o g i c a l decoherence except that the phase $ has been renormalized  by the interaction  with the nuclei. T h i s is e x a c t l y w h a t we w o u l d  e x p e c t t o have h a p p e n here. N o t e t h a t t h e i n c l u s i o n o f t o p o l o g i c a l decoherence does n o t change t h e r e s u l t t h a t i n t h e l a r g e A l i m i t o r t h o g o n a l i t y b l o c k i n g effects d i s a p p e a r .  G e n e r a l Solution for A r b i t r a r y A N o w l e t us a t t a c k t h e 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 r a t e g y t h a t we w i l l use i n t h i s m o s t g e n e r a l case is as f o l l o w s .  Chapter 5. The Landau-Zener Problem in the Presence of a Spin Bath  1. F i r s t , we p e r f o r m t h e s u m over s t a t e s i n M , p  190  b y c o n v e r t i n g t h i s s u m t o a set o f  i n t e g r a l s over q u a n t i t i e s t h a t are f u n c t i o n s o f t h e set {p G M };  e x p l i c i t l y we t a k e  p  V-  1  (P31,-P31,)  dpzi„e  2  -)  2  /  -oo „2 (P 1,-^21,) 2  \  \ / 2 Z7T 7r ^ '--oo o o  ^  1  ^  roo  7=1  1  w*  2  Z"  1  r  -==—  "  41  '  0 0  /  V^rW(M^)  41lJ  1  de^e  J-oo  2  dp  3 2 /  V 7 T V Z2 7 T  ./-oo J-oo  2  2Vf2  ,e  32,  -^22,-P22,) 2W.  00  2  /-oo  dp 5(p , - p ^)—= / 41fl  -(P32,-P32,)  J —CO  \/ ZlTvVii,.  f°°  1  rOO  W  M  ^  dp ^(P42n ~ 42  '—  " ™"  •  PA2U)-7==  V27T X/Z7T  roo  /  dpi^ipi^ - p ) lfl  J-oo  1  w  2  W  (5.138)  ^  1  ;  A s we have seen p r e v i o u s l y , t a k i n g a l l these i n t e g r a t i o n s t o be i n d e p e n d e n t is j u s t i f i e d as l o n g as we o n l y keep t o q u a d r a t i c o r d e r i n o u r s m a l l p a r a m e t e r s . N o t e t h a t t h e w i d t h s o f the three d i s t r i b u t i o n s  { p i ^ } ,  { p A i p \  and { p A 2 i i \  are a l l q u a d r a t i c i n s m a l l p a r a m e t e r s a n d  therefore have b e e n d r o p p e d (because these a p p e a r i n q u a r t i c o r d e r i n s m a l l p a r a m e t e r s after t h e i n t e g r a t i o n s ) , effectively r e p l a c i n g i n t e g r a t i o n s over g a u s s i a n s w i t h i n t e g r a t i o n s over d e l t a f u n c t i o n s . 2. N e x t we p e r f o r m t h e e n s e m b l e average over p o l a r i z a t i o n g r o u p s M^. E x p l i c i t l y we take  L= APG  V•  V  r  dNLe- H ° M  L \irN? J-OO  2N  (5.139)  "  3. N e x t we i n t e g r a t e over t h e v a r i a b l e X, w h i c h represents t h e " b l i p l e n g t h " 4- W e t h e n p e r f o r m t h e s u m s over {m, h}. 5. F i n a l l y , we change v a r i a b l e s f r o m Y —> Z. T h i s gives us o u r final answer. L e t us p e r f o r m a l l five o f these steps e x p l i c i t l y , s t a r t i n g w i t h (5.121);  I  2A TT rAI* 2  TTA^ =  dY  T dXei  ^  XY+  V *(**-™**) e  Chapter 5. The Landau-Zener Problem in the Presence of a Spin Bath  1  n  < ^2  x  191  " l  2  e "  jr  lX  £  e  ( + m  f t  )[P3i -^/'4i ]-i(m-/i)[p32 -iXp42 ]+(l+m/i)p2i -(l-m/i)/922 M  M  M  M  M  M  (5.140)  In o r d e r t o c o m p l e t e step 1. we m u s t c a l c u l a t e t h e f o l l o w i n g ; S  1 l  U  M  f  X  X p\ 2  u  J2^g( + )[P31 -J^/541 ]-J(m-/l)[p32 -iA'p42 ]+(l+m/l)p21^-(l-m'l)P22 m  f t  M  M  M  M  A l  —>•  PGM? -(P32M~/'32M)  (P31H~P31M)  e  f  3 1  e  -^21^-^21^)  - ( P |  1  2  M  ^ - P 2 2 M )  2f  2nW< \  3 2  1^ roo - P41/i)-^= / dp ^S(p42^ V^TT •/-co  roo  ?= / dp i S(p4i Z7T J—oo 4 lt  ll  oo -g  /  ]_  ( f-e,")  2  t  2VV (Mj )  /•oo  ~ p 2n)-7= ^Pl/APl/i v27T ./-co  42  4  p) lp  v2„2 g  2  l  -co  V2TTW(MA y-oo e x p [ ( m + h) [ p 3 1 / i  - i A T p i ^ ] - z ( m - /i) [/9 4  32Ai  -  + (1 + mh)p i  iXp ] i2fi  2 IJL  mh)p ]  - (1 -  22ll  (5.141)  E v a l u a t i n g these i n t e g r a l s a n d i n s e r t i n g t h e r e s u l t s i n t o (5.140) gives /i/zuj  t4-  *  i  / J-oo  -A/2u  X  2  n  r  roo  exp  f^  2  2  W  ... 'M  2  e x p [(1 + mh)(W \ z  +p) 2l  ,-,  - ) + i — - e f ^ — iXp i (m /  4 ll  LO  2  - (1 - m / i ) ( W |  2  + p  2 2  + h) — (m — h)Xp, '42/x  ) + ( m + h)p  3l  - i(m -  h)p  32  (5.142)  W e n o w m o v e o n t o s t e p 2. T h e r e l a x a t i o n r a t e is w r i t t e n (5.28)  T~ = L  V APG  n I*  [WM»T I M  Chapter 5. The Landau-Zener Problem in the Presence of a Spin Bath  2A ,  r^  2  /  A  A  /  dY  XY+  J-A/2u  n  J-oo  1= X  e  <("*-™*-)  .. , , , m,/i=±l  r  KM  exp  y  dxA ^  192  dM,e~ V ° M  J-OO  2N  "  2  ,,,2 LO  — ^ Z  ^  +  e x p [(1 + m / i ) ^ + p  2 x  ) + z — e f " - iXp (m I LO  + h) - (m -  41ll  ) - (1 - m / i ) ( T y  2 3  2  + p  2 2  ) + (ra + / * ) p  31  - i(m -  h)Xp » 42  /i)p ]] 3 2  (5.143) P e r f o r m i n g t h e i n t e g r a t i o n s over { M ^ } gives _! _ 2 A  2  j  dY T  7-/l/2i  A  J-Al2w  y  dxA » \ XY+  x  J-oo  m  m h ) ^ - ( l - m h ) ^  . .. L , , m,/i=±l  (m + h)  exp  e^~ ^exp (l +  h z ( r a — h)  LO  (5.144)  LU  w h e r e W, \ \ a n d A are as p r e v i o u s l y a n d 2  A  -  A  4  »W  (5.145)  ^ F  (5.146)  WO  W  o  ~  ~W  W32  W e n o w p e r f o r m s t e p 3., t h e i n t e g r a t i o n over t h e b l i p l e n g t h X. 2 A j 7y r / u ; 2  .-1  1  2  1  WA  r  AI2w e  J-AI2w -A/2w  »(h*-m**)  e x p  T h i s gives  1 + mh) — — (1 — mh) —  2  ... , m,/i=±l  2  2  LO  exp  H  LU  h ( m + h)  LO  i(m  —  h)  (5.147)  LO  W e n e x t p e r f o r m s t e p 4-, t h e s u m s over { r a , / i } . T h i s gives 2 A T T / O ; M/2w 2  T  =  1  r  —  /  J-A/2u  WA  + exp - A  2  ,2  2  + A  dY  2  AW'  cosh  exp  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 t e p 5. is a change o f v a r i a b l e s  w h i c h a l l o w s us t o w r i t e t h i s i n the f o r m  r~ =  A  l  o  2  °  dZe~  z2  /  2y7rA Jz~  [e  A l _ A 2  L  - c o s h [2(p, + 2\SZ]  + e"  A 2 + A 2  cos [2<p0 + 2A Z]1 4 J  (5.150)  where  <"«>  * -(^) D e f i n i n g n e w phases  ^ ( Z ) = ^ cos" [e" 1  A 2 + A 2  cos [2<p + 2X4Z}]  (5.152)  c o s h [2<pi + 2 A Z ] ]  (5.153)  0  and  ^ ( Z ) = ]- c o s h  - 1  [e  A l _ A  3  3  we w r i t e  $ ( Z ) = 0o(Z) + ^ i ( Z )  (5.154)  i n t e r m s o f w h i c h o u r final r e l a x a t i o n r a t e m a y be w r i t t e n  T  - i  o y/nA A  =  /^ r f Z e +  +  Jz-  _ z 2  |cos$(zT)|  2  (5.155)  Evaluation of the General Relaxation Rate in the Small A Limit W h e n A is l a r g e the g e n e r a l r e l a x a t i o n r a t e reduces t o t h a t for p u r e t o p o l o g i c a l decoherence (5.137). W h e n A b e c o m e s s m a l l , however, we e x p e c t t o see s o m e i n t e r p l a y b e t w e e n  Chapter 5. The Landau-Zener Problem in the Presence of a Spin Bath  194  o r t h o g o n a l i t y b l o c k i n g a n d t o p o l o g i c a l d e c o h e r e n c e effects. I n t h i s l i m i t we m a y r e w r i t e the r e l a x a t i o n r a t e i n t h e f o r m  A  2  exp  _e_  cos$ '  4W  2  ^  (5.156)  2W  J  We  see f r o m (5.156) t h a t t h e r e is a s e e m i n g l y u n u s u a l d e p e n d e n c e  u p o n the external  DC  b i a s field w h e n b o t h o r t h o g o n a 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 are  present. H o w e v e r t h e r e a s o n for t h i s f o r m is e v i d e n t . A s i n t h e case o f 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 , we see t h a t there exists a g a u s s i a n p r o f i l e w i t h e x t e r n a l b i a s . t o p o l o g i c a l p h a s e t e r m has a c q u i r e d a l o n g i t u d i n a l b i a s d e p e n d e n c e .  A s well,  the  T h i s last occurs  because the axes o f q u a n t i z a t i o n of each o f the k n u c l e a r s p i n s t h a t we have chosen, t h a t is  z  k  —  and x  = ** ^ ^ p ^ k  k  ' ^°  n  o  t  m  g  e  n  e  r  a  l c o r r e s p o n d w i t h t h e axes of the  c e n t r a l s p i n H a m i l t o n i a n , w h i c h are also t h e axes w i t h w h i c h the t o p o l o g i c a l decoherence t e r m s are d e s c r i b e d . W h e n we c a l c u l a t e the v a l u e s for the t o p o l o g i c a l decoherence A  k N  D  terms  , these are b y d e f i n i t i o n o f f - d i a g o n a l i n t h e axes o f the s p i n H a m i l t o n i a n . B u t w h e n  these are r o t a t e d i n t o the axes defined b y 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  they  a c q u i r e , i n g e n e r a l , l o n g i t u d i n a l c o m p o n e n t s . It is these l o n g i t u d i n a l c o m p o n e n t s , c o m i n g f r o m t h i s m i s m a t c h o f preferred axes, t h a t gives rise t o the presence of a l o n g i t u d i n a l bias t e r m i n t h e 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 t h o g o n a l i t y b l o c k i n g effects d i s a p p e a r , s u c h as we f o u n d h a p p e n s i n the large A l i m i t , t h i s r o t a t i o n o f axes is n o t necessary a n d t h i s effect is n o t present  5.6 We  (5.137).  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  8  n o w w i s h t o e v a l u a t e the q u a n t i t i e s e n t e r i n g i n t o (5.155) for the specific case of the  Fe$ m o l e c u l a r m a g n e t .  W e w i l l b e g i n b y e v a l u a t i n g the c o n t r i b u t i o n s t h a t arise f r o m  Chapter 5. The Landau-Zener Problem in the Presence of a Spin Bath  Quantity  195  Definition  1  LOQ  1  W (0)  v  v  + -,ll 8  7 V + 8 (, .11  Ef=>*  2  i Y  k  =  1  \  k  , , \  2  °>  w  2  T a b l e 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 .  r + p + w (o) 2  orthogonality and degeneracy  b l o c k i n g effects,  2  2  namely p ,  q u a n t i t i e s are d e f i n e d , i n t e r m s o f t h e m o r e f u n d a m e n t a l r e p e a t e d for c o n v e n i e n c e i n t a b l e 5.1. a n d LO for t h e t h r e e species -Fe *, Fe D 0  8  8  T , and W ( 0 ) .  2  2  H o w these  q u a n t i t i e s i n t h e t h e o r y , is  I n t a b l e 5.2 we present zero field v a l u e s for T and  5 7  0  F e s - T h e q u a n t i t i e s p\ a n d W ( 0 ) w i l l be  f u n c t i o n s o f a n y e x t e r n a l b i a s field p r e s e n t , b o t h b e c a u s e of t h e field m a g n i t u d e s and their changing directions, i m b o d i e d i n the orthogonality 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 ) a n d T  0  \j J k  .  as f u n c t i o n s o f a n e x t e r n a l s t a t i c  t r a n s v e r s e m a g n e t i c field o r i e n t e d a l o n g t h e x d i r e c t i o n (as defined b y 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 ) i n t h e F e s m o l e c u l a r m a g n e t , for v a r i o u s species. I n figure 5.12 is s h o w n t h e f u l l w i d t h W for t h e t h r e e v a r i e t i e s Fe$*, Fe^o  and  5 7  Fe 8  Chapter 5. The Landau-Zener Problem in the Presence of a Spin Bath  Fe  Fe  8  r  0  UJ  0  196  Fe  57 8D  8  10.64  3.22  12.96  8.32  2.93  10.04  T a b l e 5.2: Z e r o e x t e r n a l field values for T a n d u> for t h e t h r e e species s h o w n . U n i t s are 0  in  0  MHz.  W e n o w c o n s i d e r t h e t e r m s o w i n g t h e i r e x i s t e n c e o n l y t o t o p o l o g i c a l decoherence effects. T h e s e t e r m s , u n l i k e t h e ones d e a l t w i t h i n t h e p r e c e d i n g , are n o t s t r o n g f u n c t i o n s of a n e x t e r n a l l y a p p l i e d transverse field ( a l t h o u g h t h e r e is a s m a l l d e p e n d e n c e , since t h e application of an external  field  shifts t h e m i n i m a o f t h e c e n t r a l s p i n o b j e c t ,  c h a n g i n g t h e p h a s e t h a t i t a c c u m u l a t e s i n t u n n e l i n g f r o m | y > t o | i> T h e r e f o r e i t is e n o u g h t o c a l c u l a t e t h e i r v a l u e s i n zero transverse  field.  thereby  or viceversa).  T h e definitions of  A i a n d A are r e p e a t e d i n t a b l e 5.3. C o n t r i b u t i o n s t o these f r o m different n u c l e a r species 2  are s h o w n i n t a b l e 5.4. V a l u e s for A i a n d A for t h e t h r e e v a r i e t i e s F e * , 2  8  Fe  and  8D  are s h o w n i n t a b l e 5.5.  Quantity  Definition  Ai  ^  A  ^  2  \  4  t—'l  [ 1  4  Nt  \r °  \n f,  \T '  la  N  N  [^k^l  ,  W  n  ( 2 )  1 '*(5/*.-l) v - A t f  2  ll 1 W 2  t  2ku.z\\  n  ll  "I"  6  1  (l)  ) ? " ' Lku. = l  o a  k  ( 2 ) 2  u  2 n  fl \  l  T a b l e 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 decoherence  ~  n  J  2  )}  2  2k z)  n  a  effects.  Fe  57  8  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 External Field Hx[T]  F i g u r e 5.6: pi^ (dashed) a n d W ( 0 ) M  79  V =  Ai^  Br  0.2  ( s o l i d ) for p =  l  13^  i47v  0  Fe  17  1.45 • 1 ( T  6  0.06 • 1 0 "  6  0.03 • 1 0 "  6  0.66 • 1 0 ~  12.8 • 1 0 "  6  0.13 • 1 0 "  6  0.13-  6  0.61 • l O "  10"  H.  6  6  57  0.11 • 1 0 "  6  0.18 • 1 0 ~  0.31 • 1 0 "  6  0.60 • l O "  6  6  T a b l e 5.4: Z e r o field values o f t h e t o p o l o g i c a l decoherence t e r m s for species i n Fe%.  F i n a l l y , we c o n s i d e r t e r m s t h a t arise f r o m t h e i n t e r p l a y o f 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 decoherence. 5.6.  T h e s e q u a n t i t i e s are l i s t e d i n t a b l e  T h e s e o f course w i l l be f u n c t i o n s of a n y e x t e r n a l t r a n s v e r s e  presence o f the o r t h o g o n a l i t y b l o c k i n g t e r m s . functions of The  H for F e * , Fe x  8  and  8D  field  because o f the  S h o w n i n figure 5.13 are A  3  and A  4  as  Fe .  57  8  "bare" p h a s e $ = op + i<pi is u n i q u e l y defined b y the choice o f a c e n t r a l s p i n 0  Chapter 5. The Landau-Zener Problem in the Presence of a Spin Bath  F i g u r e 5.8: p  1(l  ( d a s h e d ) a n d W ( O ^ ) ( s o l i d ) for p. =  1 4  N.  198  Chapter 5. The Landau-Zener Problem in the Presence of a Spin Bath  300.0  200.0  100.0  0.1  External Field Hx [ T ]  F i g u r e 5.9: p  (dashed) a n d W(O^) (solid) for p =  lfl  F i g u r e 5.10: p  Fe.  5 7  (dashed) a n d W(O^) (solid) for p =  17  lfl  O.  199  Chapter 5. The Landau-Zener Problem in the Presence of a Spin Bath  200  200.0  0.1  0.2  External Field Hx [ T ]  Figure  5.11: p  lfl  (dashed) a n d  W(0 )  (solid) for  M  0.1  External Field Hx [ T ]  Figure  5.12:  Full width  W for F e * ( d o t t e d ) , Fe 8  8D  p. =  C.  13  0.2  (dashed) a n d  5 7  Fe  8  (solid).  Chapter 5. The Landau-Zener Problem in the Presence of a Spin Bath  Fe *  Fe  8  1.54 • l O  A  13.06- 1 0 "  2  Fe  57 8D  Ai  -  8  6  0.53 • 1 0 "  6  6  4.16 • 1 0 "  6  1.72-10"  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 t e r m s for t h r e e v a r i e t i e s o f  Quantity  3  A  4  Fe . 8  Definition  v  A  201  v  V  Z^ti  JV "  i  W ,  S  0  Z^/c = l M  Wo,  .  y/~^  w  (1)-  |  (2)-  1"*:,  ,  ^z\  n  T a b l e 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 , degeneracy b l o c k i n g a n d t o p o l o g i c a l decoherence  effects.  H a m i l t o n i a n . I n o u r case we have chosen t o use the s p i n H a m i l t o n i a n  H = -DS  2  + E{S% + S _) + C(S 2  Z  4 +  + St)  where D = 0.292A", E = 0.046AT a n d C = - 2 . 9 • 1 0 " A " . 5  (5.157)  T h e bare 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 n e a r l y d e g e n e r a t e g r o u n d states | ± 10 > c a n be f o u n d v i a e x a c t d i a g o n a l i z a t i o n (see c h a p t e r 2) a n d is  A  0  « 3.9 • \Q~*K  (5.158)  T r u n c a t i n g t h e q u a r t i c s p i n t e r m gives us a n "easy-axis e a s y - p l a n e " 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  0.0005  202  0.001  0.0000 0.0  0.000 0.0  0.2  0.1 Hx[T]  0.2  F i g u r e 5.13: A (left) a n d A (right) for t h e t h r e e v a r i e t i e s Fes* ( d o t t e d ) , Fe8D 3  and  5 7  Fe  8  4  (dashed)  (solid).  w h i c h gives b a r e phases  cpo = TTS  TrSgn H B  S ivgiJ, H 2  y  B  '  '~  x  (5.159)  2E  L o o k i n g b a c k at o u r g e n e r a l e x p r e s s i o n for 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 (5.155), we see t h a t we have n o w c a l c u l a t e d a l l o f t h e p a r a m e t e r s i n t h i s e x p r e s s i o n . F i x i n g the s p i n H a m i l t o n i a n gives a single m o l e c u l e A C r e l a x a t i o n r a t e w i t h no free  parameters.  Effect of the Nuclear Spin Environment on the Large A Single Molecule  5.6.1  R e l a x a t i o n R a t e i n Fe  8  S h o w n i n figure (5.14) are p l o t s o f t w o q u a n t i t i e s . T h e first is the q u a n t i t y A =  V ' A T  -  1  c a l c u l a t e d w i t h o u t the a d d i t i o n of a n y n u c l e a r spins; t h a t is,  A The  = An I cos $1  (5.160)  s e c o n d is t h e q u a n t i t y A , w i t h t h e a d d i t i o n o f n u c l e i t o t h e effective H a m i l t o n i a n ;  t h a t is,  A  = A |cos<l>| n  (5.161)  Chapter 5. The Landau-Zener Problem in the Presence of a Spin Bath  We  203  see t h a t t h e p h a s e 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 t h e n u c l e i causes a t i n y  s m o o t h i n g o f t h e "nodes" seen i n A c a l c u l a t e d u s i n g t h e b a r e s p i n H a m i l t o n i a n . F u r t h e r m o r e , because we k n o w h o w t h i s r e n o r m a l i z a t i o n d e p e n d s o n t h e n u c l e a r s p i n s w e can  p r e d i c t t h e s h a p e o f t h i s s m o o t h i n g effect.  T h i s b e i n g s a i d , i t is o b v i o u s i n t h i s  p a r t i c u l a r m a t e r i a l t h a t t h i s effect is e x t r e m e l y w e a k , because o f t h e s m a l l values o f t h e t o p o l o g i c a l decoherence One rate.  parameters.  s h o u l d n o t e t h a t w h a t w e a r e c a l c u l a t i n g here is t h e single molecule r e l a x a t i o n I f w e were t o i n c l u d e m a n y m o l e c u l e s , as is t h e case i n a r e a l c r y s t a l , we w o u l d  have t o i n c l u d e t h e effects o f t h e t r a n s v e r s e d i p o l a r fields i n $ (ie. H  x  andH  y  don't only  c o m e f r o m t h e e x t e r n a l field; t h e r e are also i n t e r n a l d i p o l a r fields o f t h i s k i n d ) . T h i s is t h e b a s i c r e a s o n w h y t h e d a t a s h o w n i n figures 1.20 a n d 1.21 is so " s m o o t h e d " - t h i s effect c o m e s a b o u t because o f n o n - z e r o H  y  i n t h e p h a s e $ c o m i n g f r o m i n t e r n a l d i p o l a r fields  (we s h a l l t r e a t t h e m u l t i - m o l e c u l e case i n t h e n e x t c h a p t e r ) .  5.7 In  S u m m a r y and Discussion of Results t h i s c h a p t e r we b e g a n w i t h t h e f u l l effective H a m i l t o n i a n (5.6) for a g e n e r a l m o l e c u l a r  magnet i n a n external time dependent  field.  W e d e r i v e d a f o r m a l e x p r e s s i o n for t h e one-  m o l e c u l e r e l a x a t i o n r a t e i n s u c h a s y s t e m , a n d s o l v e d i t f o r t h e s p e c i a l case o f a s a w t o o t h e x t e r n a l field i n t h e absence o f q u a d r u p o l a r c o u p l i n g s t o t h e n u c l e i .  T h e g e n e r a l one-  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 i s case w a s f o u n d t o b e (5.162) w h e r e needed d e f i n i t i o n s m a y b e f o u n d i n (5.151) a n d t h e e q u a t i o n s t h a t d i r e c t l y f o l l o w , (5.134), (5.135), (5.145) a n d (5.146). T h i s f o r m for t h e r e l a x a t i o n r a t e s i m p l i f i e s i n t h e l i m i t t h a t t h e s w e e p i n g a m p l i t u d e is  g r e a t e r t h a n b o t h t h e energy w i d t h W a n d a n y e x t e r n a l b i a s £ . I n t h i s l i m i t t h e  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 | c o s $ | (the lower curve) and (2) the large A result with nuclear spins A | c o s $ | (the middle curve) plotted in units of Kelvin. Note the logarithmic vertical scale. The horizontal axis is H in Tesla-here we have (p = 0(H = 0). 0  0  x  y  relaxation rate can be written A ,| c o s $ | A 2  2  (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  205  b e i n g s a i d , we have s h o w n q u a n t i t a t i v e l y h e r e i n j u s t h o w i t h a p p e n s a n d h o w t o c a l c u l a t e t h e effects o f a s p i n e n v i r o n m e n t o n t h i s k i n d o f t o p o l o g i c a l d e c o h e r e n c e . W h e n t h e a m p l i t u d e o f t h e s w e e p i n g field is m u c h s m a l l e r t h a n t h e t o t a l e n e r g y s p r e a d +  we find t h a t  (5.164)  T h e r e are t w o t h i n g s t o n o t e here. O n e is t h a t t h e f o r m for t h e o n e - m o l e c u l e r e l a x a t i o n r a t e is  gaussian, n o t  e x p o n e n t i a l as is r e p o r t e d i n t h e l i t e r a t u r e . W e have e x p l a i n e d t h e  reasons b e h i n d t h i s d i s c r e p a n c y a n d t h e reasons w h y we b e l i e v e t h e g a u s s i a n f o r m is m o r e appropriate to the physics of molecular magnets.  T h e s e c o n d t h i n g here is t h a t t h e r e is  a n e x p l i c i t i n t e r p l a y b e t w e e n 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 m a n i f e s t s i t s e l f i n a £ d e p e n d e n t B e r r y phase. for t h e specific case o f Fe ,  H o w e v e r , we have s h o w n t h a t  t h e o s c i l l a t i o n s i n £ c o m i n g f r o m t h i s t e r m are far t o o s m a l l  8  t o be seen. A g a i n , we d o n o t e x p e c t t h i s t o h o l d i n t h e g e n e r a l c a s e - i t is q u i t e p o s s i b l e t h a t i n s y s t e m s t h a t have l a r g e a m o u n t s of t o p o l o g i c a l d e c o h e r e n c e c o m i n g f r o m a s p i n b a t h the oscillations i n T  _ 1  ( £ ) w i l l be a p p a r e n t (these c o u l d be m e a s u r e d i n a s i m i l a r  m a n n e r t o recent e x p e r i m e n t s p e r f o r m e d o n Fe  8  which extract just this quantity).  W e c o m p a r e d our general large A r e l a x a t i o n rate w i t h t h a t of the simple no-environment s p i n H a m i l t o n i a n (5.157). W e f o u n d t h a t t h e n u c l e a r s p i n s s m o o t h o u t t h e cusps n e a r the nodes i n the r e l a x a t i o n rate. Fe ; 8  T h e effect o f t h e n u c l e i c o n s i d e r e d is r a t h e r s m a l l i n  however, t h i s need n o t be t h e case i n g e n e r a l .  Chapter 6  A C Relaxation in a C r y s t a l of Molecular Magnets  I n t h i s c h a p t e r we use t h e r e s u l t s o b t a i n e d i n the p r e v i o u s c h a p t e r t o i n v e s t i g a t e  what  f o r m t h e 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 i n a c r y s t a l o f m o l e c u l a r m a g n e t s takes.  In  essence w h a t we s h a l l d o is i n s e r t t h e o n e - m o l e c u l e r e l a x a t i o n r a t e c a l c u l a t e d i n c h a p t e r 5 (5.155) i n t o a m a s t e r e q u a t i o n [15, 167] a n d solve i t i n v a r i o u s l i m i t s .  In particular  we s h a l l be c o n c e r n e d w i t h s h o r t - t i m e r e l a x a t i o n i n t h e presence o f a n e x t e r n a 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 d e r i v e d i n t h e p r e v i o u s s e c t i o n are s i n g l e - m o l e c u l e r e l a x a t i o n rates. I n o t h e r w o r d s , i n t h e i r d e r i v a t i o n we have o m i t t e d c o m p l e t e l y f r o m c o n s i d e r a t i o n a l l effects c o m i n g f r o m i n t r a - m o l e c u l a r t e r m s i n the f u l l c r y s t a l H a m i l t o n i a n . T h e f u l l H a m i l t o n i a n o f a c r y s t a l of m o l e c u l a r m a g n e t s m a y be w r i t t e n i n t h e f o r m  ff = £ t t i + £ V y i  (6.1)  i<j  w h e r e Hi is the s i n g l e - m o l e c u l e H a m i l t o n i a n for the i  th  a n d Vij gives the t o t a l i n t e r a c t i o n between the i  th  m o l e c u l e ( i n c l u d i n g e x t e r n a l fields) and j  t h  molecules.  w i l l be d o m i n a t e d b y m a g n e t i c d i p o l a r i n t e r a c t i o n s b e t w e e n  T h i s interaction  off-site m a g n e t i c  atoms.  N o w w h e n we a t t e m p t t o c a l c u l a t e the 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 the entire c r y s t a l , we see t h a t it is n o t e n o u g h t o k n o w 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 s - t h e r e are o t h e r i m p o r t a n t t e r m s i n the c r y 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  207  In o r d e r t o get a feel for t h e p h y s i c s g o i n g o n i n s i d e one o f these c r y s t a l s d u r i n g a r e l a x a t i o n e x p e r i m e n t , l e t ' s a t t e m p t t o q u a l i t a t i v e l y d e s c r i b e t h i s r e l a x a t i o n first before d e a l i n g 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 r e a t m e n t w i l l f o l l o w t h e course o f t h e  s t a n d a r d e x p e r i m e n t s o f t h i s t y p e t h a t are c u r r e n t l y b e i n g p e r f o r m e d o n these s y s t e m s (see c h a p t e r 1). T h e r e are t w o s t a n d a r d w a y s o f i n i t i a l l y p r e p a r i n g these m a t e r i a l s before t h e i r 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 are m e a s u r e d .  T h e first o f these i n v o l v e s p l a c i n g t h e s a m p l e i n a  l a r g e (5-8 T ) D C m a g n e t i c field a l i g n e d a l o n g the z a x i s o f t h e c r y s t a l at s o m e h i g h t e m p e r a t u r e , a n d t h e n c o o l i n g t h e s a m p l e d o w n s l o w l y t o the mK  regime. T h i s technique  has t h e effect o f p r e p a r i n g t h e m a t e r i 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 - a l l the m a g n e t i c i o n s w i l l be s u c h t h a t each m o l e c u l e w i l l be i n the " u p " s t a t e at t h e b e g i n n i n g o f the exp e r i m e n t . T h e s e c o n d w a y t o p r e p a r e t h e s a m p l e is t o p l a c e t h e c r y s t a l i n a s m a l l e r (0-5 T)  z a x i s m a g n e t i c field at a h i g h t e m p e r a t u r e , a n d t h e n quench t h e s y s t e m ' s t e m p e r a -  t u r e v e r y q u i c k l y d o w n t o t h e mK  r e g i m e . T h i s has the effect o f g i v i n g t h e c r y s t a l s o m e  less t h a n f u l l i n i t i a l m a g n e t i z a t i o n ( w h i c h is a f u n c t i o n o f t h e s t r e n g t h of the o r i g i n a l D C field)  w h i l e m a k i n g sure t h a t t h e r e are n o s i g n i f i c a n t off-site c o r r e l a t i o n s at the b e g i n n i n g  of the e x p e r i m e n t ( w h y t h i s c o n s i d e r a t i o n is i m p o r t a n t w i l l s o o n b e c o m e  apparent!).  L e t ' s b e g i n o u r q u a l i t a t i v e p o n d e r i n g s w i t h some g e n e r a l l y a p p l i c a b l e r e m a r k s .  In  e i t h e r of t h e a b o v e cases, each m o l e c u l e i n the s a m p l e w i l l feel s o m e b i a s field c o m i n g f r o m the s u m over the c o n t r i b u t i o n s o f a l l the o t h e r m o l e c u l e s .  T h e s e b i a s fields w i l l  have s o m e d i s t r i b u t i o n over the c r y s t a l , w h o s e d e t a i l s d e p e n d o n the s a m p l e shape. N o w i f the l o n g i t u d i n a l (ie., a l o n g the S-axis) b i a s o n a n y p a r t i c u l a r m o l e c u l e is m u c h b i g g e r t h a n the "resonance w i n d o w " g i v e n b y o u r 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 (5.155), t h i s m o l e c u l e w i l l be  frozen i n i t s o r i g i n a l s t a t e . It is c o m p l e t e l y u n a b l e t o r e l a x ! W e n o t e i n  p a s s i n g t h a t i f we r e m o v e the effects o f n u c l e a r s p i n s f r o m o u r 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 , the resonance w i n d o w s h r i n k s d o w n t o e n c o m p a s s o n l y e x t e r n a l biases £ < AQ.  Chapter 6. AC Relaxation in a Crystal of Molecular Magnets  208  B e c a u s e t h e v a l u e s of the b a r e 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 these m a t e r i a l s are s m a l l (typically < crystal. of ~  10~ K), a n y s i g n i f i c a n t 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 w i l l freeze the entire e  T y p i c a l l y t h e i n i t i a l b i a s d i s t r i b u t i o n i n these m a t e r i a l s varies over a r a n g e  0.1 — 1 K,  orders o f m a g n i t u d e l a r g e r t h a n A ! B e c a u s e s i g n i f i c a n t  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  0  short-time  a n d Mn^,  we infer t h a t t h e n u c l e a r s p i n s  are a c t u a l l y crucial t o t h e p h y s i c s here [20, 5 1 , 49]. Now  let's i m a g i n e w h a t w i l l h a p p e n  as t h e c r y s t a l s t a r t s t o r e l a x .  t o specify t h e t y p e of field a p p l i e d d u r i n g t h e r e l a x a t i o n process.  W e n o w have  T h e r e are t w o m a i n  p o s s i b i l i t i e s for w h i c h e x p e r i m e n t a l r e s u l t s have been o b t a i n e d ; e i t h e r we a p p l y a D C field i n a n y d i r e c t i o n , 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 o f the f o r m of the a p p l i e d field, the f o l l o w i n g c o n s i d e r a t i o n s a p p l y . W e have i m a g i n e d t h a t t h e r e 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 t h e r e w i l l be s o m e f r a c t i o n o f the c r y s t a l for w h o m the bias is s m a l l e r t h a n t h e i r r e s o n a n c e w i n d o w ( w h i c h is a f u n c t i o n of t h e a p p l i e d field t h r o u g h (5.155)). T h i s f r a c t i o n is able t o r e l a x . W h e n one o f these m o l e c u l e s does r e l a x (by t u n n e l i n g t o the of bias fields over the crystal.  " d o w n " state) i t rearranges the  distribution  T h i s is b e c a u s e i n s t e a d of c o n t r i b u t i n g a n " u p " t o the  t o t a l i n t e r n a l field it n o w c o n t r i b u t e s a " d o w n " . W e see therefore t h a t t h e i n t e r n a 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 p e c u l i a r w a y - a l l m o l e c u l e s t h a t are i n s i d e the b i a s w i n d o w w i l l b e g i n t o flip, a n d w i l l " d i g a h o l e " i n the b i a s d i s t r i b u t i o n n e a r zero bias, s e n d i n g t h e i r w e i g h t o u t i n t o the d i s t r i b u t i o n ' s w i n g s . T h i s w i l l s l o w d o w n t h e r e l a x a t i o n , as t h e r e are less a n d less m o l e c u l e s a v a i l a b l e i n the r e s o n a n c e w i n d o w as t i m e progresses. A s we have seen i n c h a p t e r 1, t h i s effect has r e c e n t l y been o b s e r v e d i n Fe  8  t e r m s o f a q u a n t i t a t i v e theory, w h a t does t h i s m e a n ?  [51]. N o w i n  Chapter 6. AC Relaxation in a Crystal of Molecular Magnets  6.2  209  T h e Generalized Master E q u a t i o n  T h e r e is one c h a r a c t e r i s t i c here t h a t is a p p a r e n t , a n d t h a t is t h a t as t h e c r y s t a l s t a r t s to relax,  correlations b e t w e e n different sites b e g i n t o d e v e l o p . T h a t i s , t h e r e l a x a t i o n o f  t h e c r y s t a l is s t r o n g l y d e p e n d e n t o n i t s h i s t o r y .  It has b e e n suggested  [15] t h a t these  c o r r e l a t i o n s m a y be d e a l t w i t h s i m i l a r l y t o t h e w a y t h a t c o r r e l a t i o n s i n s p i n glasses are t r e a t e d , n a m e l y b y the d e f i n i t i o n o f a series o f m a n y - m o l e c u l e d i s t r i b u t i o n w h i c h are r e l a t e d v i a a n e x p r e s s i o n o f t h e B B G K Y t y p e .  functions  W e s h a l l i n w h a t follows use  t h i s t e c h n i q u e a n d see w h a t r e s u l t s 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 l u s i v e . W e b e g i n b y w r i t i n g t h e m a g n e t i z a t i o n o f a c r y s t a l o f m o l e c u l a r m a g n e t s i n the f o r m  M(t) = £ j dHM{r, n,t)  = yj  r  a"H{2P {r, H, t) - 1) t  (6.2)  r  where P f ( r , ri, t) is t h e n o r m a l i z e d p r o b a b i l i t y o f the c e n t r a l s p i n c o m p l e x at site f t o be " u p " (ie. i n s t a t e \S  Z  = +S))  a n d i n a D C b i a s field ri a t t i m e t.  T o o b t a i n a s o l u t i o n for P f (f, ri, t) we s h a l l p r o c e e d i n the f o l l o w i n g m a n n e r . W e n o t e d i n the p r e v i o u s c h a p t e r t h a t for frequencies lower t h a n the n u c l e a r T  2  energy scale > ~  T  0  each pass o f the A C field t h r o u g h r e s o n a n c e is d e c o r r e l a t e d f r o m a l l o t h e r s u c h passes. T h i s m e a n s t h a t we m a y c a l c u l a t e 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 i n g l e pass a n d b y s i m p l y d i v i d i n g t h i s b y t h e p e r i o d o f t h e ac field o b t a i n 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 T~ (A,UJ;'H). 1  K n o w l e d g e of t h i s r a t e a l l o w s us t o w r i t e a k i n e t i c or " m a s t e r " e q u a t i o n  of the f o r m [15, 20]  Pa(r,il,t) = -T-\A,u-%){P {r,n,t)  -  a  -  a  /dH'T-\A,u H')[P™,(? T &  £ -J  P_ (?,H,t)}  4  f  t  i ^  r ,a  (6.  where P^  is a 2 - m o l e c u l e d i s t r i b u t i o n f u n c t i o n , ie. the j o i n t p r o b a b i l i t y o f t w o m o l e c u l e s  Chapter 6. AC Relaxation in a Crystal of Molecular Magnets  h a v i n g the characteristics i m p l i e d by their arguments.  Vi)(r — r)  210  represents the D C bias  at f c o m i n g f r o m t h e m o l e c u l e at site r , w h i c h as we have m e n t i o n e d is d u e t o m a g n e t i c d i p o l a r i n t e r a c t i o n s . W e have n o t e x p l i c i t l y w r i t t e n o u t the h i g h e r o r d e r m u l t i m o l e c u l a r t e r m s P^ \P^ \ Z  etc., r e p r e s e n t i n g these b y a n e l l i p s i s (...).  A  A s the c r y s t a l relaxes, the  influence o f these h i g h e r o r d e r t e r m s w i l l b e g i n t o be s i g n i f i c a n t . H o w e v e r , i n t h e i n i t i a l stages o f r e l a x a t i o n , we s h a l l assume t h a t these c a n be n e g l e c t e d . 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 s o m e l i m i t e d t i m e f r o m t h e p r e p a r a t i o n o f t h e i n i t i a l s t a t e a n d i t r e m a i n s t o be s h o w n t h a t t h i s t i m e is l o n g e n o u g h t o be s i g n i f i c a n t .  6.3  Short T i m e Dynamics  In the e a r l y stages o f t h e r e l a x a t i o n o f a c r y s t a l o f m o l e c u l a r m a g n e t s , t h e r e w i l l be t w o d o m i n a n t c o n t r i b u t i o n s t o the p h y s i c s . T h e first o f these is g i v e n b y t h e first t e r m o n the r i g h t - h a n d side o f  (6.3),  a n d c o r r e s p o n d s t o " l o c a l " r e l a x a t i o n . I f we were t o neglect  a l l t e r m s b u t t h i s first one we w o u l d find t h a t the r e l a x a t i o n w o u l d be l i n e a r i n t i m e . A n o t h e r w a y of s a y i n g t h i s is t h a t i f a l l o f the m o l e c u l e s r e l a x e d i n d e p e n d e n t l y o f each o t h e r , t h e n t h e t o t a l r e l a x a t i o n o f the c r y s t a l has t o be e x p o n e n t i a l ( a n d therefore l i n e a r at s h o r t t i m e s ) . D e f i n i n g the q u a n t i t i e s  M(f, H, t) = P ( f , H, t) - P^f, %, t) t  ,  M(H, t) = £ M(f, H, t)  ,  M(t) = J dHM(H, t)  f  a l l o w s us t o w r i t e  P ( f , li, t) = Q  -T~\A,  to; U){P (r, a  U, t) - P. (r, a  H, t)}  (6.4)  i n the f o r m  M(H,t)  = -r- (A,w;H)M(??,t) 1  (6.5)  Chapter 6. AC Relaxation in a Crystal of Molecular Magnets  211  N o w i f t h i s is t h e w h o l e s t o r y (ie. we n e g l e c t a l l c o r r e c t i o n s t o t h i s ) t h i s is e a s i l y s o l v e d , giving  M{H,t)  = M(ft, 0 ) e -  t / r ( A  '^  (6.6)  )  a n d therefore  M(t) = J driM(n, o)e- ' '  (6.7)  2/T{A UJ il)  w h i c h has s h o r t t i m e b e h a v i o u r  M(t) ~ \ - t j  dfi M(H, 0) T~ (A, LO; it)  (6.8)  1  w h i c h , as we have c l a i m e d , is l i n e a r i n t i m e . N o w we t u r n t o t h e l e a d i n g s u b - d o m i n a n t t e r m i n (6.3). H o w d o we i n c o r p o r a t e the l e a d i n g c o r r e c t i o n s t o the l o c a l t e r m , a n d w h a t effect w i l l t h i s have o n t h e r e l a x a t i o n characteristics? A s t h e c r y s t a l b e g i n s t o r e l a x , t h e i n t e r n a l b i a s d i s t r i b u t i o n ( a n d therefore  M{ri,t))  w i l l c h a n g e i n t i m e i n a w a y t h a t one c a n c a l c u l a t e a n a l y t i c a l l y i n a n e l l i p s o i d a l c r y s 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 o f biases is e x a c t l y the effect t h a t the l e a d i n g c o r r e c t i o n t e r m i n (6.3) has o n t h e r e l a x a t i o n . W e c a n therefore t a k e i n t o a c c o u n t  the  M(ri,t)  l e a d i n g c o r r e c t i o n s b y s i m p l y i n s e r t i n g the f o r m d e r i v e d i n A p p e n d i x A for  i n t o the l o c a l t e r m (6.5). T h i s gives us the f o l l o w i n g e q u a t i o n 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 o c a l r e l a x a t i o n a n d the first c o r r e c t i o n s t o i t ;  M{t) = w h e r e M(it,  -j  dH r " (A, LO] it) M(H, t)  (6.9)  1  t) is the ( e v o l v i n g ) d i s t r i b u t i o n o f biases d e r i v e d i n A p p e n d i x A a n d r~  l  (A, LO; it)  is the 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 d e r i v e d i n the p r e v i o u s c h a p t e r (5.155).  Chapter 6. AC Relaxation in a Crystal of Molecular Magnets  6.3.1  212  Strongly A n n e a l e d Samples and the Large A L i m i t  A s we have seen e a r l i e r w h e n the e x t e r n a l s w e e p i n g a m p l i t u d e is l a r g e e n o u g h t o sweep a l l m o l e c u l e s t h r o u g h resonance, ie. w h e n A > H  + W,  z  c o n s i d e r a b l y . W e have also seen t h a t i n t h e F e are m i n i m a l .  8  D  the relaxation rate simplifies  m a t e r i a l t o p o l o g i c a l decoherence  effects  I f we i g n o r e these we f i n d t h a t t h e r e l a x a t i o n r a t e c a n b e w r i t t e n i n the  form  A \cos^{H ,H )\ A 2  r- (A,H ,H )  =  1  x  y  2  0  x  y  (6.10)  L e t us n o w c o n s i d e r the e x p e r i m e n t a l l y r e l e v a n t case o f s t r o n g a n n e a l i n g , w h e r e t h e i n i t i a l m a g n e t i z a t i o n M(t ) 0  <C M o , the s a t u r a t e d m a g n e t i z a t i o n .  I n t h 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 g a u s s i a n (see [200] a n d refs.  therein)  3/2  M(H,t ) 0  =  M(t ) 0  nW (M) 2  £  exp  (Hi - H?)  (6.11)  /  T h e h a l f - w i d t h WD(M) for Fe$.  is d i r e c t l y m e a s u r e d i n e x p e r i m e n t s [] a n d is m u c h l a r g e r t h a n E  0  N o t e t h a t the o b s e r v a t i o n t h a t Wr> is a f u n c t i o n of M a n d is d i r e c t l y e x t r a c t a b l e  f r o m e x p e r i m e n t is d u e t o w o r k b y S t a m p a n d T u p i t s y n [198]. P l u g g i n g (6.10) a n d (6.11) i n t o (6.9) a l l o w s us t o solve for M(t).  M(t) = M(t ) exp [-T (H, 0  W e find that  M)t  AC  (6.12)  w h e r e the r e l a x a t i o n r a t e is  T,c(H,M)  =  ^ exp  2A§ w  h  {  M  )  A  J dH j dH x  y  |cos  X  {H - H° ) + (H - H\0\2 2  Wl(M)  $(H ,  x  x  y  H  y  (6.13)  N o w as we have stressed t h r o u g h o u t t h i s d o c u m e n t , o u r a n a l y s i s d e p e n d s o n t h e choice of a p a r t i c u l a r s p i n H a m i l t o n i a n (for Fe$ t h i s was t h 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 ) .  Chapter 6. AC Relaxation in a Crystal of Molecular Magnets  The  213  c h o i c e o f t h e p a r a m e t e r s i n t h e s p i n H a m i l t o n i a n s t r o n g l y affects the  of the t u n n e l i n g m a t r i x e l e m e n t o n e x t e r n a l t r a n s v e r s e fields. the dependence on H  x  and H  y  dependence  T h i s is r e l e v a n t here as  i n the r e l a x a t i o n r a t e changes i f we use a different s p i n  H a m i l t o n i a n as we s a w i n c h a p t e r 2. I n a d d i t i o n , i n d e r i v i n g o u r r e l a x a t i o n r a t e (6.9), we have a s s u m e d t h e i n s t a n t o n c a l c u l a t i o n w h i c h o f course is n o t e x a c t .  In order to  e x t r a c t the c o r r e c t t u n n e l i n g a m p l i t u d e as a f u n c t i o n o f e x t e r n a l field, we have t o e x a c t l y d i a g o n a l i z e t h 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 h a t is, t h e c o r r e c t e x p r e s s i o n  cos$(H ,  takes o u r A |  x  0  H )\ t o \A(H ,H )\ y  x  extracted by exact d i a g o n a l i z a t i o n .  y  T h e r e is a specific case w h e r e we c a n get a r o u n d t h e  first  difficulty,  a n d t h a t is  near t h e nodes o f t h e t u n n e l i n g a m p l i t u d e - t h i s case has b e e n t r e a t e d i n d e p e n d e n t l y b y S t a m p a n d T u p i t s y n [198, 200] a n d we w i l l r e v i e w t h e i r r e s u l t s i n t h e f o l l o w i n g s e c t i o n . In  t h i s case t h e t u n n e l i n g a m p l i t u d e w i l l be 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 a n d  therefore c o m p a r i n g o u r r e s u l t s t o t h e i r s a l l o w s us t o gauge the a c c u r a c y o f o u r i n s t a n t o n approximation. In t h e case w h e r e t h e s p i n H a m i l t o n i a n is 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 we have seen t h a t , i n t h e l i m i t b e i n g c o n s i d e r e d ,  cos  ${H ,H )\ x  cos  y  2TTS  2irS t (H gr B  + H° )  y  S Kgn (H  + Hl)  2  y  + cosh  On  B  x  E  (6.14) w h e r e we have s p l i t the transverse fields i n t o t w o c o n t r i b u t i o n s , one f r o m e x t e r n a l l y applied  fields  H  i n t e r n a l fields H  xy  xy  a n d one f r o m i n t e r n a l t r a n s v e r s e gives us o u r i n t e r n a l d i p o l a r  T (H,M)  =  AC  fields  H .  I n t e g r a t i n g over the  x>y  field-averaged  relaxation rate  ^\cos$(H° ,H° ) x  (6.15)  y  where 2n  2  cos  S g n wl(M) 2  2  2  2irSg^ H  0  B  COS  2irS-  B  O  0  y  +  Chapter 6. AC Relaxation in a Crystal of Molecular Magnets  s 7r 4  2  2 g  2 M  w  (M)  2  2£2  214  cosh  (6.16)  E  The relaxation rates (6.15) are shown in figure 6.1 for several values of  6.3.2  W (M). D  General Solution Near the Nodes  Stamp and Tupitsyn [200, 199] have demonstrated that near the nodes of \A(H , X  one can expand \A(H ,H )\ X  y  around its zeroes, as \A(H ,H )\ X  H )\  2  y  = A c o s h Z , where for  y  0  the Fe biaxial symmetry 8  Z = 4 +ilby= X  [(Hy/Fy) + I {H ~ x  H^)/F  (6.17)  X  (see [200] for complete details of this calculation). The "periods" F and F vary slowly x  y  with H and H , and so we expand as Fi{H ) — F;(0) + Fi(0)"Hf/2 + .... We can thus x  y  t  solve for the A C relaxation rate around the n  "nodal field" H^;  ih  r^(Hl,M)  =  ^ C (M)  + (rft +  n  C (M) = *-Wl{M) \F-\H ^) n  X  tfj  + F- (0) 2  (6.18)  (6.19)  This is a parabolic increase around minima with values exactly proportional to the independently measurable  Wp(M).  Choosing a specific form for the F e spin Hamiltonian (such as the easy-axis easy8  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 (i7°/77°) and H , A C relaxation measurements ought to obtain _1  0  information about the giant spin Hamiltonian inaccessible to E S R experiments [200].  Chapter 6. AC Relaxation in a Crystal of Molecular Magnets  Figure  6.1:  Top  figure:  b o t t o m to top, W  D  Log  w  A  2  vs.  215  Hy, for 6 = sin" (Hll/Hy) = 0. C u r v e s are, f r o m  == 0 , 1 0 , 20, 30 a n d 50 mT.  1  B o t t o m figure: S a m e , b u t w i t h 9 =  1°.  N o t e t h a t these r e s u l t s are 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 was d e r i v e d a s s u m i n g the i n s t a n t o n 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 ^ Relaxation in a Crystal of Molecular Magnets 4  OFFENDING COMMAND: STACK:  Chapter 7 Summary and Outlook  In  t h i s thesis we have p r e s e n t e d a q u a n t i t a t i v e t h e o r y o f t h e p h y s i c s o f m a g n e t i c r e l a x a t i o n  in  t h e presence o f t i m e - d e p e n d e n t e x t e r n a l fields i n c r y s t a l s o f m o l e c u l a r m a g n e t s .  In  so d o i n g we have a c h i e v e d c o n s i d e r a b l e i n s i g h t i n t o the effect o f s p i n e n v i r o n m e n t s o n t h e q u a n t u m d y n a m i c s o f mesoscale m a g n e t i c m o m e n t s . T h i s research p r o d u c e d s e v e r a l i n t e r e s t i n g r e s u l t s , w h i c h we r e p r o d u c e here. We  u s e d t h e s t a n d a r d P r o k o f ' e v a n d S t a m p t h e o r y [15, 20, 122], t o g e t h e r w i t h the  c r y s t a l l o g r a p h i c s t r u c t u r e o f t h e Fe  8  m o l e c u l a r m a g n e t , t o e x t r a c t q u a n t i t a t i v e values for  t h e t o p o l o g i c a l decoherence {ajj. ' ^} a n d o r t h o g o n a l i t y b l o c k i n g {j3 } 1 2  k  p a r a m e t e r s for t h i s  m a t e r i a l . W e f o u n d t h a t t o p o l o g i c a l decoherence effects were m i n i m a l  (lajj. ' ^ ~ 10~ ) 1 2  3  b u t 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 was s i g n i f i c a n t . W e f o u n d t h a t t h e 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 c a m e f r o m the l a r g e n u m b e r o f p r o t o n s i n t h e o r g a n i c m a t t e r i n the Fe  8  m o l e c u l e . I n t e r m s o f the v a l u e K = — In [T[ cos (3).]  t h e o r y , we f i n d t h a t K q u i c k l y rises w i t h t r a n s v e r s e m a g n i t u d e o f \H\  field  m  k  field,  the P r o k o f ' e v a n d S t a m p  r e a c h i n g K ~ 1 for t r a n s v e r s e  ~ 110 gauss i n Fe * (see figures 3.14 a n d 3.15). A l l  quantities  8  t h a t are f u n c t i o n s o f the h y p e r f i n e c o u p l i n g s were f o u n d t o e x h i b i t a s i g n i f i c a n t i s o t o p e effect.  I n p a r t i c u l a r , A, K, a n d t h e n u c l e a r s p i n l i n e w i d t h W a l l d e p e n d o n the i s o t o p i c  c o n c e n t r a t i o n i n a n Fe  8  sample.  F i t t i n g the s h a p e o f o s c i l l a t i o n s i n t u n n e l i n g a m p l i t u d e t o o u r d e r i v e d r e l a x a t i o n rate, we  f o u n d t h a t i n t e r n a l d i p o l a r t r a n s v e r s e fields i n F e s  0.03^  a  r  e  °  n  t h e o r d e r o f yjH  2  + H  ~ 30 gauss. T h i s m e a n s t h a t even i n the absence o f a n y e x t e r n a l t r a n s v e r s e  217  2  ~  fields  Chapter 7. Summary and Outlook  218  K ~ 0.09 i n F e s * , w h i c h is j u s t large e n o u g h t o cause s i g n i f i c a n t d e c o h e r e n c e [15, 20, 122], e r a s i n g a n y h o p e o f seeing m a c r o s c o p i c q u a n t u m coherence i n F e * . 8  W e d e r i v e d a n e x p r e s s i o n for the 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 a n y m o l e c u l a r m a g n e t i c s u b s t a n c e w i t h a n effective H a m i l t o n i a n o f t h e f o r m (5.6) b e i n g a c t e d u p o n b y a s a w t o o t h A C field, w h i c h is g i v e n b y  where  (7.2)  2W  w i t h £ t h e l o n g i t u d i n a l b i a s , W the t o t a l e n e r g y s p r e a d a v a i l a b l e t o t h e m o l e c u l e , a n d A the a m p l i t u d e o f t h e  field.  A  0  is t h e b a r e t u n n e l i n g m a t r i x e l e m e n t o f t h e s p i n H a m i l t o -  n i a n p a r t o f (5.6) a n d <&(Z) is the B e r r y p h a s e o f the c e n t r a l s p i n c o m p l e x , r e n o r m a l i z e d b y the i n t e r a c t i o n w i t h the n u c l e a r s p i n s . T h i s e x p r e s s i o n m a y be i n t e r p r e t e d i n the f o l l o w i n g way. T h e effect o f the n u c l e a r s p i n s is b o t h t o b r o a d e n the l i n e w i d t h W o f t h e m o l e c u l e a n d t o r e n o r m a l i z e the B e r r y phase $ —>• $(Z).  T h e first o f these effects c o m e s a b o u t f r o m o r t h o g o n a l i t y b l o c k i n g a n d  d e g e n e r a c y b l o c k i n g . T h e s e c o n d c o m e s a b o u t p r i m a r i l y f r o m t o p o l o g i c a l decoherence, a l t h o u g h the Z d e p e n d e n c e o f $ is d u e t o a n i n t e r p l a y b e t w e e n t o p o l o g i c a l decoherence and orthogonality blocking.  I n the l i m i t w h e r e A is m u c h b i g g e r t h a n  \W + £ | , t h e  l i n e w i d t h W b e c o m e s i r r e l e v a n t , a n d a l l o r t h o g o n a l i t y b l o c k i n g effects d i s a p p e a r , g i v i n g  r"  ,  1  =  A^|cos$| 0 1  , (7.3)  2  1  A  N  In the o p p o s i t e l i m i t A <C \W + £ | we f i n d t h a t -i  A  2  2  AW  2  cos$(c;/2W0|  2  (7.4)  Chapter 7. Summary and Outlook  219  ie. t h e r e l a x a t i o n is g a u s s i a n , w i t h a n u n u s u a l b i a s d e p e n d e n t B e r r y p h a s e . W e s h o w e d t h a t i n Fe , 8  because t h e t o p o l o g i c a l decoherence effects are so s m a l l , t h i s p e c u l i a r feature  is o v e r p o w e r e d b y t h e g a u s s i a n a n d i t is p e r m i s s i b l e t o t a k e d e p e n d e n c e disagrees  <5>(£/2W)  —>• $ . T h i s g a u s s i a n  w i t h p r e v i o u s l y q u o t e d results w h i c h find t h e r e l a x a t i o n t o  be  e x p o n e n t i a l i n b i a s [15, 20, 122]. W e b e l i e v e t h a t t h e reason for t h i s d i s p a r i t y is t h a t i n t h e c a l c u l a t i o n l e a d i n g t o t h e e x p o n e n t i a l b i a s d e p e n d e n c e t h e a u t h o r s a d d e d a n energy c o n s t r a i n t w h i c h is u n p h y s i c a l for these s y s t e m s (see c h a p t e r 6 for a d i s c u s s i o n o f t h i s point). W e used o u r s i n g l e m o l e c u l e r e l a x a t i o n rates as 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 c r y s t a l l i n e n a t u r e o f the r e a l m a t e r i a l [15, 167].  We  s t u d i e d s e v e r a l specific cases a n d f o u n d t h e f o l l o w i n g g e n e r a l o b s e r v a t i o n s . I f the s w e e p i n g a m p l i t u d e o f the field is l a r g e r t h a n t h e t o t a l l o n g i t u d i n a l b i a s a v a i l a b l e t o the m o l e c u l e s A > \W + £ + ED\ the r e l a x a t i o n is l i n e a r at s h o r t t i m e s . W h e n t h e s w e e p i n g 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 i n e a r , t u r n i n g over t o s q u a r e r o o t at l a t e r t i m e s . T h i s o c c u r s for b o t h L o r e n t z i a n a n d g a u s s i a n i n i t i a l l o n g i t u d i n a l b i a s profiles. W e used these g e n e r a l results i n t h e specific case o f A C m e a s u r e m e n t s p e r f o r m e d o n Fe  [39]. I n t h i s case we are i n the l a r g e A l i m i t , g i v i n g l i n e a r r e l a x a t i o n  8  A?I ros <I>I  2  M ( t ) ~ M ( 0 ) ( l - ^ & t )  (7.5)  w h e r e <§ is the r e n o r m a l i z e d B e r r y p h a s e , averaged over the t r a n s v e r s e i n t e r n a l d i p o l a r fields  (7.11).  W e t h e n fit t h i s r e l a x a t i o n r a t e to the e x p e r i m e n t a l d a t a .  W e f o u n d the  f o l l o w i n g i n t e r e s t i n g results. O u r d e r i v e d e x p r e s s i o n for the t u n n e l i n g a m p l i t u d e p r o d u c e s r e s u l t s w h i c h q u a n t i t a t i v e l y m a t c h those f u n d u s i n g e x a c t d i a g o n a l i z a t i o n t e c h n i q u e s for t r a n s v e r s e fields less than ~  0.5 T . T h i s p r o v i d e s e v i d e n c e t h a t t h e i n s t a n t o n c a l c u l a t i o n is v a l i d here. O u r  e x p r e s s i o n also q u a l i t a t i v e l y agrees w i t h e x p e r i m e n t a 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 {a ' '} and {(5k) in real systems. As well, we included in our derivations k  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 dependent 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 Fe , using the methods developed herein. This will 8  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 o u r coarse g r a i n i n g a p p r o x i m a t i o n b r e a k s d o w n ) , or the r e l a t e d p r o b l e m o f h o w t o go b e y o n d t h e fast-passage a p p r o x i m a t i o n , r e m a i n w o r k s i n progress. T h e r e s o l u t i o n of t h e c o n f l i c t i n g r e s u l t s o b t a i n e d w i t h t h e i n c l u s i o n o f a n o s c i l l a t o r b a t h , i n the  0(DS ) 2  necessary  effective d e s c r i p t i o n , have t o b e a d d r e s s e d i n o r d e r t o u n d e r s t a n d  t h e r m a l / q u a n t u m crossover.  the  T h e inclusion of the q u a d r u p o l a r terms 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 provide m a n y hours of gruesome  enjoyment.  Appendix A  Bias Distribution in a Dilute Solution of Dipoles  In t h i s a p p e n d i x , we d e r i v e a g e n e r a l e x p r e s s i o n for t h e b i a s d i s t r i b u t i o n i n s i d e a c r y s t a l of m o l e c u l a r m a g n e t s .  T h e g e n e r a l a r g u m e n t s t h a t we w i l l use have b e e n  published  p r e v i o u s l y [123]. It is however w o r t h w h i l e t o u n d e r s t a n d h o w the f o r m u l a e q u o t e d i n t h e t e x t arise, a n d t h a t s h a l l be the p u r p o s e o f t h i s e x p o s i t i o n . We  b e g i n b y p r e s e n t i n g the f o l l o w i n g q u e s t i o n .  G i v e n a n ensemble o f d i p o l e s i n a  c r y s t a l o f a r b i t r a r y s h a p e , w h a t w i l l be t h e p r o b a b i l i t y t h a t the t o t a l d i p o l a r some site r has t h e v a l u e ril  field  at  B a s i c a l l y we c a n j u s t w r i t e d o w n the g e n e r a l s o l u t i o n t o  t h i s p r o b l e m ; i t is a s u m over a l l p o s s i b l e w a y s o f o b t a i n i n g t h i s v a l u e w i t h the  spins  we've got,  Ti  H e r e each site fi ^ f i n t h e c r y s t a l c a n e i t h e r have a s p i n p o i n t i n g " u p " or a s p i n p o i n t i n g " d o w n " , ie. i n t h e ±z  d i r e c t i o n s . T h e d i p o l a r field is t a k e n t o be  H(? - f ) = p r ^ T a (4 - 3 f ( 4 • r ) ) {  (A.2)  where v is a u n i t c e l l v o l u m e , ED is the d i p o l a r field scale, a n d d^ is the d i r e c t i o n t h a t the i  th  s p i n is p o i n t i n g .  W e n o w s p e c i a l i z e o u r t r e a t m e n t t o the case o f a n e l l i p s o i d a l s a m p l e . I n t h i s case the f d e p e n d e n c e d r o p s o u t o f P(f,rl),  a n d we c a n w r i t e  n 222  Appendix A. Bias Distribution in a Dilute Solution of Dipoles  223  W r i t i n g t h e d e l t a f u n c t i o n o u t as a n e x p o n e n t i a l gives  M = <k>f^ f*"%'- **  p  f  U  '  >f  (A 4)  T h i s c a n be r e w r i t t e n as  W e n o t e t h a t t h e c o n t r i b u t i o n f r o m H(fi)  flips s i g n d e p e n d i n g o n t h e s i g n o f  . There-  fore we c a n w r i t e t h e p r e c e d i n g as  p  M  = ^  / *w*  n / f  9  n / g ^ « > *  ( A.6)  w h e r e we have s u b d i v i d e d the t o t a l n u m b e r o f s p i n s TV i n t o t w o subsets, t h o s e t h a t are u p ( { t } ) a n d those t h a t are d o w n ( { ! } ) . W e n o w n o t e t h a t because we are o n l y a l l o w i n g the s p i n s i n o u r s o l u t i o n t o p o i n t i n the ±z c a n e x p l o i t ; t h e s o l u t i o n for P^H)  d i r e c t i o n s we have a s y m m e t r y here t h a t we  has t o be o f t h e f o r m  P(H) = 6{H = 0)5(H = 0)P(H ) X  y  (A.7)  z  T h a t is, the f i n a l s o l u t i o n i n t h i s i d e a l i z e d case has the p r o b a b i l i t y for the t r a n s v e r s e  (A.6) i n t o a n e x p r e s s i o n for  fields t o be zero. W e therefore recast o u r e x p r e s s i o n ie. t h e p r o b a b i l i t y o f b e i n g i n a l o n g i t u d i n a l b i a s  H; z  (.4.8) W e c a n p e r f o r m these i n t e g r a t i o n s ;  / r  w  - '"/?(' ) , y  \l  Jo  JO  '  P(H ), Z  Appendix A. Bias Distribution in a Dilute Solution of Dipoles  224  Changing variables to x = 1 / r gives 3  r  27r  1_ | L r  ,„ . „ f°° dx 0 s m e f ° ^ (l -  d  ± i Y E e  ^ - ™ l  z  2 e  >)  (A.10)  This can be broken up into two parts, like this 27T  30  1  r , .  y  n  „  dBsindJ  r°° dx  ^[{l-cosYE v{l-3cos 9)x)±i[s\nYE v{l-3cos 9)x)) 2  2  D  D  (A.ll) The leftmost integral is easy to do. We find that 27T r ZTT f .„ . „ rr°° cLx dx — d9 sin 9 — 30 Jo Jo X n  0 0  Z  ir E v\Y\ 2  /  \  (l-cosYE v(l-3cos 9)x) 2  D  V  '  r r d9 sin 0 | 1 - 3 cos 9\ 2  D  HI 30  Jo  8TT E V\Y 2  (A.12)  D  9V3Q  The second one is a little trickier. We write it as follows;  =  —— / d9sm9 30 Vo 7o  — sinYE v(l x  ^ 7 ^ " I d6sind I  — (sin f y / W l - 3cos 0)z) - YE v(l x '  oil  Jo  Jo  -  D  2  3cos 9)x 2  2  l  v  D  v  - 3cos 6)x) ' 2  (A.13) The reason we can do this is that the term we have added is equal to zero, as d9sm6(l-3cos  9) = 0  2  J0  (A.14)  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 = YE v(l-3cos 9) 2  D  (A.15)  Appendix A. Bias Distribution in a Dilute Solution of Dipoles  225  that r dx [ (sin (bx) — bx) Jo x A  z  =  6(Ci(A6)-ln(A) + l-ln(6)-7)  (A.16)  and the whole expression is ±iYE v2n „^ oil D  / d9 sin 9(1 - 3 cos 9) Jo 2  (Ci(AYE v(l  - 3 cos 9)) - ln(A) + 1 - ln YE v(l  ±iYE v2ir 3Q D  - 3 cos 9) - 7 )  2  D  2  D  [ d9 sin 9(1 - 3 cos 9) Jo V  (Ci(AYE v(l  2  -3cos 9))  - ln (YE v(l  2  D  - 3 cos 9)))  (A.17)  2  D  The leftmost term is ±iYE v27r 3Q  r* j* d9 sin 0(1 - 3 cos 9) (Ci(AYE v(l  D  - 3 cos 9))  2  2  D  I\Y\E VTT 8 2  D  /  A  The rightmost term is ±iYE v27r 3Q  r« lo °  ±iYE v2ir  (A 8 1 V3 ~ 9 ^  D  d  D  S l n 9 i y l  V  3ft  ~ ° 3 C  i ? e  , (  s2  t a n h  ^ { ( 1u  (  AYEdV  1  ~  T 1—. 4 1— \ ~ ^ ) + 9V^TriJ  3 c o s 2  ))  e  (A.19)  Collecting our results, we find that  J  dr  n  =  1 - pY [1 ± 1 ± i =F iK]  =  l + pY[l^l±i±iK]  ,{Y>0} , {Y < 0} (A.20)  Appendix A.  Bias Distribution in a Dilute Solution of Dipoles  226  w h e r e we h a v e d e f i n e d  G o i n g b a c k t o o u r o r i g i n a l e x p r e s s i o n for P ( £ ) we see t h a t  P(H )  1 r°° — / dYe * dYe  =  Z  iH  II [1 " PY H  Y  inzl  + iK]} JJ [1 - pY [2 + i - iK}]  m  +  -!- [° 2 7 r  J  dYe * lH  Y  ~°°  m  Y[[i  +  p  Y [ 2 - i - iK]}  m  [1 + pY [+i + iK]] m (A.22)  T h i s c a n be w r i t t e n  P(H ) Z  z  =  — f°° 2ix Jo  _|_  _  1  dYe ^ iY  ~ ^~ ^ ~ e~ ^ ^~~  Hz+{1  {N  N  }p{K  l)IN)  2  [N  Ni}YplN  roo  /  d  y -iy(^+{l-(iV -iV )}p(A--l)/iV) -2 (l-{iV -iV }yp/JV t  e  1  e  t  T  A  27T 7o  (A.23)  w h e r e iVf, iVj. are t h e n u m b e r s o f u p / d o w n s p i n s r e s p e c t i v e l y . A s w e l l , we h a v e a s s u m e d t h a t Q is l a r g e e n o u g h so t h a t we c a n r e w r i t e t h e s u m s as e x p o n e n t i a l s . P e r f o r m i n g t h e i n t e g r a l s gives  TM F  1  ™ = l T ( , )» P g  +  g  ( A  +  '  2 4 )  w h e r e we h a v e d e f i n e d  E(t)  where M = ^ ~ ^ Njt  N  = p(K  - 1)JV(1 - M)  ,  T(t)  = 2npN(l  - M)  (A.25)  is 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 .  N o w i n o r d e r t o w r i t e o u r f u l l P{ri),  we h a v e t o i n c l u d e t h e H  x  and H  y  dependence  w h i c h w e ' v e seen is i n t h e i d e a l case v e r y s i m p l e . I n a r e a l m a t e r i a l , t h e d e l t a f u n c t i o n s i n t h i s d e p e n d e n c e w i l l b e s p r e a d s o m e w h a t . F o r t h i s r e a s o n we are g o i n g t o r e p l a c e t h e  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 E  D  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 7T  Hi + Hi  1 (H  + E)  2  z  + T  2  2irWj  T2  t  e X  P  ™  2  (A.26)  t  where, as we have mentioned, W^t <C ED is the width of the transverse field distribution.  Appendix B  T i m e E v o l u t i o n of Nuclear Spin States  I n t h i s a p p e n d i x w e s h a l l s h o w h o w t h e n u c l e a r s p i n s t a t e s e v o l v e over t i m e d u e t o t h e i r coupling w i t h the central spin c o m p l e x i n a molecular magnet. W e shall assume the same c o n d i t i o n s as were p r e s e n t e d i n c h a p t e r 6, a n d f u r t h e r m o r e we s h a l l a s s u m e t h a t w e are i n the large A l i m i t . W e m a y w r i t e d o w n a m a s t e r e q u a t i o n for t h e flow o f t h e o n e - m o l e c u l e s y s t e m i n t h e space \S,M > w h e r e \S > is t h e t w o - l e v e l s t a t e o f t h e c e n t r a l s p i n a n d \M > is t h e s t a t e of the s p i n e n v i r o n m e n t o f the form  Ps,M  = -P ,M S  £  J-is',M->M'  T  +  s' ,M'  £  s ' \  T  S  , M ' ^ M  P  (B- ) 1  S ' , M '  S' ,M'  w h e r e P S , M is t h e n o r m a l i z e d p r o b a b i l i t y o f b e i n g i n s t a t e \S,M  > a n d T~^ , S  M  > is  M  the t r a n s i t i o n r a t e f r o m s t a t e \S, M > t o s t a t e \S', M' >. N o w as w e e x p l i c i t l y s h o w e d i n c h a p t e r 6, these r e l a x a t i o n r a t e s d o n o t select for a n y p a r t i c u l a r s t a t e o f t h e c e n t r a l s p i n s y s t e m . T h a t i s , t h e r a t e o f flow | t>—> \ I > is i d e n t i c a l t o t h e r a t e o f flow | J,>—>• | y > i n t h e c e n t r a l s p i n space. T h i s m e a n s t h a t we c a n t a k e s->s',M^>M'  ~  T  S'->S,M->M'  T  — M->M'  (B-2)  T  and write our master equation i n the form P  M  = -2P  £  M  r - ^  M  , + 2£  M'  r-)^ P , M  M  (B.3)  M'  w h e r e we have d e f i n e d P  M  = Y, SM s p  228  (B.4)  Appendix B. Time Evolution of Nuclear Spin States  229  W e c a n f u r t h e r s i m p l i f y t h i s e x p r e s s i o n b y l o o k i n g c l o s e l y at t h e s t r u c t u r e 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 i e s i n c h a p t e r 6. W e see t h a t these a l s o d o n o t select for specific s p i n b a t h s t a t e s . T h a t is,  for a l l { M , M'}.  T h i s m e a n s t h a t t h e m a s t e r e q u a t i o n s i m p l i f i e s even m o r e .  W ecan  write it i n the form  r l , +2 £ r l , P ,  = -2PME  PM  M  M  M  M' K n o w i n g t h e f u n c t i o n a l f o r m o f T~^^ , M  M  (B.6)  M  M' for a r b i t r a r y { M , M'}  a l l o w s us t o solve t h i s  e q u a t i o n n u m e r i c a l l y ( i t is s i m p l y a s y s t e m o f M l i n e a r e q u a t i o n s ) . I f we s i m p l y a s s u m e that  r~J-_^ , is n o t a s t r o n g f u n c t i o n o f M' t h e n o u r m a s t e r e q u a t i o n b e c o m e s M  P  M  where T  1 M  = E M '  M\M'  T  a  n  c  ^^ ^ ^ s  = -2P r M  n  e  +2  l M  ^  (B.7)  t o t a l n u m b e r o f s p i n b a t h states. T h i s is r e a d i l y  s o l v e d ; i n t h e l i m i t t h a t C ^> 1 t h e s o l u t i o n is  P (t) M  together w i t h the constraint that E M  -  -P (0)eM  Pu{t)  2 T  (B.8)  M*  = 1. W h a t t h i s m e a n s is t h a t t h e different  s p i n b a t h states w i l l r e a c h a s t e a d y s t a t e s o l u t i o n w h e r e P  M  t i m e o n the order of T  M  = P > for a l l {M, M'} i n a M  w h i c h is s i m p l y t h e t o t a l l a r g e A t r a n s i t i o n r a t e d i v i d e d b y t h e  average n u m b e r o f n u c l e a r s p i n flips t h a t o c c u r p e r 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 o f ( B . 3 ) s h o w i n g t h e convergence o f t h e s p e c t r a l w e i g h t s o f t h e v a r i o u s n u c l e a r s p i n states for a g e n e r a l s y s t e m w i t h  , =  Appendix B. Time Evolution of Nuclear Spin States  230  F i g u r e B . l : H e r e we see a s y s t e m w i t h seven e n v i r o n m e n t a l s p i n states i n i t i a l l y p r e p a r e d 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 . Keyes, "Miniaturization of Electronics and its Limits", IBM Journal of Research and Development, Vol. 32, January (1988) pp. 24-28. [3] Malone, "The Microprocessor: A Biography", T E L O S , Santa Clara, (1995). [4] R. W . Keyes, "The Future of the Transistor", Scientific American, June (1993), pp.70-78. [5] New Scientist, December (1999). [6] R.P. Feynman, "There's Plenty of Room at the Bottom", Engineering and Science, Vol. 23, pp.22-36 (1960). [7] C. Bennett, "Logical Reversibility bf Computation", IBM Journal of Research and Development, Vol. 17, (1973) pp. 525-532. [8] D. Deutsch, Proc. Roy. Soc. London, Vol. A400, pp.97-117 (1985). [9] R.P. Feynman, "Quantum Mechanical Computers", Optics News, Vol. 11, pp.11-20 (1985) [10] A . Turing, On Computable Numbers with an Application to the Entscheidungsproblem", Proc. Lond. Math. Soc, Vol.42 (1937), pp.230-265, erratum in 43 (1937), pp.544-546. [11] S. Jaimungal, M.H.S. Amin and G . Rose, Persistent Currents and Boundary Conformal Field Theory, Int. J. Mod. Phys. B, in press; and references therein. [12] R. Rouse, Siyuan Han, J.E. Lukens, Phys. Rev. Lett. 75 8 1614 (1995). [13] Y . Nakamura, Yu. Pashkin and J. S. Tsai, cond-mat/9904003 (1999). [14] C. Sangregorio et.al., Phys. Rev. Lett. 78 24 4645-4648 (1997). [15] N.V. Prokofiev and P.C.E. Stamp, Phys. Rev. Lett. 80 26 5794-5797 (1998). [16] D. Gatteschi et.al., Science 265 1054 (1994). 231  Bibliography  232  [17] L . Thomas et.al. Nature 3 8 3 145-147 (1996). [18] C. Delfs et.al, Inorg. Chem. 3 2 3099-3103 (1993), and references therein. [19] P. Shor, "Algorithms for Quantum Computation: Discrete Logarithms and Factoring" , Proceedings 35 Annual Symposium on Foundations of Computer Science (1994), pp.124-134. th  [20] N . V . Prokofiev and P. C. E . Stamp, Theory of the Spin Bath, cond-mat/0001080, and references therein. [21] R.P. Feynman, F.L. Vernon, Ann. Phys. 2 4 , 118 (1963). [22] A . O . Caldeira, A . J . Leggett, Ann. Phys. 1 4 9 , 374 (1983). [23] K . Wieghardt, Klaus Pohl, Ibrahim Jibril and Gottfried Huttner, Angew. Chem. Int. Ed. Engl. 23, 77-78 (1984). Also see K . Weighardt et.al., Angew. Chem. 95 739 (1983) and Angew. Chem. Int. Ed. Engl. 22 727 (1983) for earlier compounds. [24] Information from Cambridge Crystallographic Database obtained from Jean Reid, reid@ccdc.cam.ac.uk, and Eugene Cheung of the Chemistry Dept. of U B C . RefCode for Fe is C O C N A J . 8  [25] J.L. van Hemmen and A . Suto, Europhys. Lett., 1, 481-490 (1986); J.L. van Hemmen, A . Suto, Physica 141B, 37-75 (1986). [26] M Enz, R Schilling, J. Phys C 1 9 , L711-5; ibid., 1765 (1986). [27] A . J . Leggett et al., Rev. Mod. Phys. 5 9 , 1 (1987). [28] A . Caneschi et.al., J. A m . Chem. Soc. 113, 5873-5874 (1991). [29] T. Lis, Acta Crystallogr. Sect. B 3 6 2042-2046 (1980). [30] P.D.W. Boyd et.al., J. Am.Chem.Soc. 1 1 0 (1988). [31] R. Sessoli et.al., Nature 3 6 5 141 (1993). [32] W . Low, Paramagnetic Resonance in Solids, p. 113, Academic Press 1960. [33] Van Vleck, J.H., J, Chem. Phys. 3 807 (1935). [34] Pryce, M.H.L., Phys. Rev. 8 0 1107 (1950). [35] C. Paulsen et ai, J. Mag. Magn. Mat. 1 4 0 , 1891 (1995), and references therein. [36] J.R. Friedman et al, Phys. Rev. Lett., 7 6 , 3830-3833 (1996).  Bibliography  233  A . - L . B a r r a et.al., Phys. Rev. Lett. 5 6 , 8192 (1997).  R. C a c i u f f o  e t . a l , Phys. Rev. Lett. 8 1 21 4744-4747 (1998).  D . Loss, D . P. diVincenzo, G . Grinstein, Phys. Rev. Lett. 6 9 , 3232 (1992). J . V o n Delft, C L . Henley, Phys. Rev.Lett. 6 9 3236 (1992).  P.C.E.Stamp,  E . M . C h u d n o v s k y , B . B a r b a r a , Int. J . M o d . Phys.  B6,  1355 (1992).  E . Chudnovsky, J . A p p l . Phys. 7 3 (10), (1993). B . Barbara, L . C . Sampaio, J . E . Wegrowe, B . A . R a t n a m , A . Marchand, C.Paulsen, M . A . Novak, J . L . Tholence, M . Uehara and D . Fruchart, J . A p p l . Phys. 7 3 (10), (1993). N . V . Prokof'ev and P . C . E . Stamp, J . Phys. C M 5, L663-670 (1993). N . V . Prokofiev and P . C . E . Stamp, J L T P 1 0 4 143-209 (1996). P . C . E . Stamp, Unconventional Environments, N A T O workshop on Tunneling of Magnetization, June-July (1994). N . V . Prokofiev and P . C . E . Stamp, J . Phys. C o n d . M a t . 5 L663-L670 (1993). H . Kopfermann, Nuclear Moments, Academic Press Inc. 1958. W . Wernsdorfer et.al. Phys. Rev. Lett. 8 2 19 3903-3906 (1999), and references therein. C. Sangregorio, T . O h m , C . Paulsen, R . Sessoli, and D . Gatteschi, Phys. Rev. Lett., 7 8 , 4645 (1997). W . Wernsdorfer and R . Sessoli, Science 2 8 4 133 (1999). J . M . Hernandez, X . X . Zhang, F . Luis, J . Bartolome, J . Tejada and R . Ziolo, Europhys. Lett. 3 5 (4), pp.301-306, (1996). J . F . Fernandez, F . Luis and J . Bartolome, Phys. Rev. Lett. 8 0 , 25 (1998). J . R. Friedman, M . P . Sarachik, J . Tejada and R . Ziolo, Phys. Rev. Lett. 7 6 20 (1996). T . O h m , C . Sangregorio, C . Paulsen, Europhys. J . B 6 195 (1998). J . R. Freidman, M . P . Sarachik, R . Ziolo, cond-mat/9807411 (1998).  Bibliography  234  [57] L . Thomas and B . Barbara, J L T P 113 1055 (1998). [58] J. Tejada, X . X . Zhang, LI. Balcells, J. Appl. Phys. 73 (10), (1993). [59] J . Tejada et.al, Phys. Rev. Lett. 79 9 1754 (1997). [60] W . Wernsdorfer et.al. J . Magn. Mag. Mat. 145 1 (1995). [61] F . Fominaya et.al, Heat Capacity Anomalies induced by magnetization quantum tunneling in a Mni 0i - acetate single crystal, preprint. 2  2  [62] A . M . Gomes et.al, Phys. Rev. B 57 9 5021 (1998). [63] W . Wernsdorfer, R. Sessoli and D. Gatteschi, Europhys. Lett. 47 (2), 254 (1999). [64] A . Abragam, Principles of Nuclear Magnetism, Oxford Science Publications, 1960. [65] A . Abragam, B. Bleaney, "Electron Paramagnetic Resonance of Transition Ions" (Clarendon, 1970). [66] W . J. Caspers, "Theory of Spin Relaxation", Interscience Publishers 1964. [67] A . Abragam and M . H . L . Pryce, Proc. Roy. Soc. A 135 (1951). [68] H . J. Zeiger and G. W . Pratt, Magnetic Interactions in Solids, p.79, Clarendon Press, Oxford 1973. [69] W . Low, Paramagnetic Resonance in Solids, pp.9-10, Academic Press 1960. [70] F. Hartmann-Boutron, P. Politi and J. Villain, Int. J. Mod. Phys. B, Vol. 10, No. 21, pp. 2577-2637 (1996). [71] G . D. Mahan, Many Particle Physics, Plenum Press, 1993. [72] A . Bencini and D. Gatteschi, E P R of Exchange Coupled Systems, Springer-Verlag 1990. [73] O. Kahn, "Molecular magnetism", V C H publishers (1993). [74] I S . Tupitsyn, N . V . Prokof'ev, and P.C.E. Stamp, Int. J. Mod. Phys. B, 11, 2901 (1997). [75] I S . Tupitsyn, J E T P Lett., 67, 28 (1998). [76] W . Low, Paramagnetic Resonance in Solids, p. 10, Academic Press 1960. [77] H . J. Zeiger and G. W. Pratt, Magnetic Interactions in Solids, p.155, Clarendon Press, Oxford 1973.  Bibliography  235  [78] Ashcroft and Mermin, Solid State Physics. [79] K . W . H . Stevens, Magnetic Ions in Crystals, Princeton University Press 1997. [80] B. N . Figgis, Introduction to Ligand Fields, Interscience Publishers, 1966. [81] L . E. Orgel, A n Introduction to Transition Metal Chemistry: Ligand-Field Theory, Butler and Tanner Ltd., 1966. [82] W . Low, Paramagnetic Resonance in Solids, p. 114-115, Academic Press 1960. [83] L . E. Orgel, A n Introduction to Transition Metal Chemistry: Ligand Field Theory, pp. 11-18, Butler and Tanner Ltd., 1966. [84] H . J. Zeiger and G . W. Pratt, Magnetic Interactions in Solids, p.85, Clarendon Press, Oxford 1973. [85] H . J. Zeiger and G . W. Pratt, Magnetic Interactions in Solids, p.132, Clarendon Press, Oxford 1973. [86] H . J. Zeiger and G . W . Pratt, Magnetic Interactions in Solids, p. 135, Clarendon Press, Oxford 1973. [87] W . Low, Paramagnetic Resonance in Solids, p. 45, Academic Press 1960. [88] H . J. Zeiger and G. W . Pratt, Magnetic Interactions in Solids, p.156, Clarendon Press, Oxford 1973. [89] W . Low, Paramagnetic Resonance in Solids, p. 119, Academic Press 1960. [90] B . Barbara et.al., J. Mag. Magn. Mat. 177-181, 1324 (1998). [91] H . J. Zeiger and G . W. Pratt, Magnetic Interactions in Solids, p.99, Clarendon Press, Oxford 1973. [92] J. Villain et al., Europhys. Lett. 27, 159 (1994) [93] F. Hartmann-Boutron, P. Politi and J. Villain, Int. J. Mod. Phys. B, Vol. 10, No. 21, p. 2630 (1996). [94] N . V . Prokofiev and Magnetisation-QTM'94"  P.C.E. Stamp, pp 347-371 in "Quantum Tunneling of (ed. L . Gunther and B. Barbara), Kluwer Publ. (1995)  [95] G . D. Mahan, Many Particle ences within.  Physics, pp. 355-374, Plenum Press, 1993, and refer-  Bibliography  236  [96] V . A . Kalatsky, E . Muller-Hartmann, V . L . Pokrovski and G.S. Uhrig, Berry's phase for large spins in external fields, preprint, Dec.5th, 1997. [97] A . Garg, Europhys. Lett. 2 2 (3), pp. 205-210 (1993), and references therein. [98] We are pursuing treatments of this technical problem with Igor Tupitsyn from Moscow. [99] A . J. Freeman and R. E. Watson, Hyperfine Interactions in Magnetic Materials, p.249, in Rado and Suhl, Magnetism II (1963). [100] A . J. Freeman and R. E. Watson, Hyperfine Interactions in Magnetic Materials, p.291, in Rado and Suhl, Magnetism II (1963). [101 Personal communication, William Unruh, University of British Columbia. [102 M . Grifoni and P. Hanggi, Physics Reports 3 0 4 , 219 (1998). [103 L. D. Landau, Phys. Z. Sowjetunion 1, (1932) 89. [104 C. Zener, Proc. R. Soc. London, Ser. A 137, (1932) 696. [105 E. G. C. Stuckelberg, Helv. Phys. Acta 5, (1932) 369. [106 P. Wolynes, J. Chem. Phys. 8 6 , 1957 (1987). [107 H. Frauenfelder and P. Wolynes, Science 2 2 9 , 337 (1985); A . Garg, J. N . Onuchic, and V . Ambegoakar, J. Chem. Phys. 8 3 4491 (1985). [108 C. P. Slichter, Principles of Magnetic Resonance, 2 1978).  nd  ed. (Springer-Verlag, Berlin,  [109 Y . Kayanuma, Phys. Rev. Lett. 5 8 , 1934 (1987). [110 M . Tsukada, J. Phys. Soc. Jpn. 51, 2927 (1982). [111 C. Zener, Proc. Roy. Soc. London, Ser. A 145, 523 (1934). [112 Y . Gefen and D . J . Thouless, Phys. Rev. Lett. 59, 1752 (1987). [113 Abramowitz and Stegun, Handbook of Mathematical Functions, (1968). [114 L. D. Landau and E . M . Lifshitz, "Quantum Mechanics". [115 J. M . Lopez-Castillo, A . Filali and J. P. Jay-Gerin, J. Chem. Phys. 97, 1905 (1992). [116 P. Ao and J. Rammer, Phys. Rev. B 4 3 7 (1991).  237  Bibliography  [117] Y . Gefen, E . Ben-Jacob, A . O. Caldeira, Phys. Rev. B . 3 6 , 2770 (1987). [118] G. Floquet, A n . , de L'ecole Norm. Sup. 12 (1883) 47. Also see [102] for a list of references for Floquet theory. [119] S. Coleman, Aspects of Symmetry, Cambridge University Press (1989). [120] C . . H . Henry and D. V . Lang, Phys. Rev. B 1 5 989 (1977). [121] M . A . Kmetic and W . J. Meath, Phys. Lett. A 1 0 8 , 340 (1985). [122] N . V . Prokof'ev and P.C.E. Stamp, J. Low Temp. Phys., 1 0 4 , 143 (1996). [123] The reference usually given here is P. W . Anderson, Phys. Rev. 8 2 342 (1951). However, this reference is nothing but a cryptic paragraph. Therefore we felt it best to rederive this expression for ourselves (see Appendix A ) . [124] J. Von Delft, C L . Henley, Phys. Rev. B 4 8 , 965 (1993). [125] A . K . Zvezdin, V.V.Dobrovitski, B.N.Harmon, M.I. Katsnelson, cond-mat/9807096 (1998). [126] Personal communication of unpublished data on tunneling quenching via transverse field application in Fe , W . Wernsdorfer, Nov. 16th, 1998. 8  [127] S. Hill, J. Perenboom, N.S.Dalai, T. Hathaway, T. Stalcup and J.S. Brooks, Phys. Rev. Lett. 8 0 11 (1998). [128] W.T.Coffey, D.S.F. Crothers, J.L. Dormann, Y u . Kalmykov, E . C Kennedy, W . Wernsdorfer, Phys. Rev. Lett. 8 0 25 (1998). [129] N . V . Prokofiev and P.C.E. Stamp, Decoherence in the Quantum Dynamics of a "Central Spin" Coupled to a Spin Environment, preprint, April 12th, (1999). [130] A . Chiolero, D. Loss, Phys. Rev. B 5 6 2 (1997). [131] E . N . Bogachek and I.V. Krive, Phs. Rev. B 4 6 22 (1992). [132] R. Sessoli, H-L Tsai, A . R . Schake, et.al. J. A m . Chem. Soc. 115, pp.1804-1816, (1993). [133] P.C.E. Stamp, Physica B 1 9 7 , 133 (1994) [Proc. LT-20, Aug. 1993]. See also P.C.E. Stamp, Nature 3 5 9 , 365 (1992). [134] N . V . Prokofiev, B . Svistunov and I. Tupitsyn, Tunneling Problems by Monte Carlo, preprint, (1999).  Bibliography  238  [135] Yunbo Zhang et.al., cond-mat/9901325 (1999). [136] M . N . Leuenberger and D. Loss, cond-mat/9810156 (1998). [137] Jiushu Shao and P. Hanggi, Phys. Rev. Lett. 81, 26 (1998). [138] A . Shimshoni, Y . Gefen, Ann. Phys. 210, 16 (1991). [139] M . Murao, C. Uchiyama, F. Shibata, Physica A 209 pp.444-456 (1994). [140] V . Y u . Golyshev and A.F.PopKov, Europhys. Lett. 29 (4). pp.327-332 (1995). [141] David Awschalom, David P. DiVincenzo, Joseph F. Symth, Science 258 p.414 (1992). [142] S. Gider et.al., Science 268 77 (1995); replies by J. Tejada and A . Garg with counter from S. Gider and D. Awschalom Science 272 424-426 (1996). [143] E. Dan Dahlberg and Jian-Gang Zhu, Physics Today p.34 April (1995). [144] John L . Simonds, Physics Today p^26 April (1995). [145] Jacques Villain, From Perturbation Theory to Instantons, preprint February 2nd (1998). [146] Seiji Miyashita, J. Phys. Soc. Japan 65 8 pp.2734-2735 (1996). [147] S.L. Heath and A . Powell, Angew. Chem. Int. Engl. 31 191 (1992). [148] K . L . Taft and S.J. Lippard, J. A m . Chem. Soc. 9629 (1990). [149] K . L . Taft et.al., J.Am.Chem.Soc. 116 823 (1990). [150] A . Caneschi et.al. J.Am.Chem.Soc. 113 5873-5874 (1991). [151] R. Sessoli et.al. J. Am.Chem.Soc. 115 1804-1816 (1993). [152] A . Fort et.al., Phys. Rev. Lett. 80 612 (1998). [153] E. M . Chudnovsky and L. Gunther, Phys. Rev. Lett. 60 661 (1988). [154] Yicheng Zhong et.al., Inelastic Neutron Scattering Study of Mnyi Acetate, condmat/9809133 (1998). [155] B. Bleaney and K . W . H . Stevens, Proc. Phys. Soc. 61 108 (1950). [156] V . V . Dobrovitski, M.I. Katsnelson and B . N . Harmon, Mechanisms of decoherence in weakly anisotropic molecular magnets, cond-mat/9906375, June 24th (1999).  Bibliography  239  [157] S.E. Barnes, Manifestation of intermediate spin for Fe , cond-mat/9907257, July 19th (1999); A . Garg, Large transverse field tunnel splittings in the Fe spin Hamiltonian, cond-mat/9906203, June 14th (1999). 8  8  [158 J. Villain et.al, Europhys. Lett. 27 159 (1994).  [159 H. J. Zeiger and G . W . Pratt, Magnetic Interactions in Solids, p.136, Clarendon Press, Oxford 1973. M . A . Novak and R. Sessoli, in Quantum Tunneling of the Magnetization, Vol.301 [160 of Nato Adavnced Study Institute, Series E: Applied Science, ed. L . Gunther and B . Barbara (Kluwer, Dordrecht, 1995), p. 171. See A . Garg, Phys. Rev Lett. 70, C2198 (1993), and reply of D. D. Awschalom [161 et a l , ibid., C2199 (1993); or A . Garg, Phys. Rev. Lett. 71, 4241 (1993), and the associated Comment of D. D. Awschalom et a l , ibid., C4276 (1993); or A . Garg, Science 272, 425 (1996), with the reply of D. D. Awschalom et a l , ibid., 425 (1996). [162 D.D. Awschalom et a l , Phys. Rev. Lett. 68, 3092 (1992). [163 This derivation is in the spirit of [74] and [75]. [164 I. S. Tupitsyn and B . Barbara, Quantum Tunneling of the Magnetization in Molecular Complexes with Large Spins. Effect of the Environment, preprint, p.12 . [165 L. E. Orgel, " A n Introduction to Transition Metal Chemistry: Ligand Field Theory", p. 92, Butler and Tanner Ltd., 1966. [166 W. Gordy, W . Smith and R. Trambarulo, "Microwave Spectroscopy", John Wiley and Sons, 1953. [167 G. Rose, P.C.E. Stamp, J. Low Temp. Phys. 113 1054 (1998). [168 P.C.E. Stamp, Phys. Rev. Lett., 66, 2802 (1991). [169 G. Tatara and H . Fukuyama, Phys. Rev. Lett., 72, 772 (1994); J. Phys. Soc. Jap., 63, 2538 (1994). [170 M . Dube' and P.C.E. Stamp, J. Low Temp. Phys., 110, 779 (1998). [171 K.Hong, N . Giordano, Europhys. Lett. 36, 147 (1996); W . Wernsdorfer et a l , Phys. Rev. B55, 11552 (1997); and refs. therein. [172 P.C.E. Stamp, Phys. Rev. Lett., 61, 2905 (1988). [173 A. Garg, Phys. Rev. Lett., 74, 1458 (1995).  Bibliography  240  [174] For Quantum Spin Glasses, see M . J . Thill, D . A . Huse, Physica A 2 1 4 , 321 (1995), and refs. therein. [175] N . Nagaosa, A . Furusaki, M . Sigrist, H . Fukuyama, J. Phys. Soc. Jap. 6 5 , 3724 (1996). [176] G.Scharf, W.F.Wreszinski, J.L. van Hemmen, J. Phys. A 2 0 , 4309 (1987). [177] I.Y. Korenblit, E . F . Shender, J E T P 4 8 , 937 (1978). [178] A . J. Leggett, pp 396-507 in "Matter and Chance: Proc. 1986 les Houches Summer School", ed. J. Souletie, J. Vannimenus, R. Stora, North-Holland, Amsterdam (1987); and A . J. Leggett, in "Frontiers and Borderlines in Many-Particle Physics", ed. R . A . Broglia, J.R. Schrieffer, North-Holland, Amsterdam (1988). [179] P.C.E.Stamp, Phys. Rev. Lett. 6 6 , 2802 (1991). [180] G. Tatara, H . Fukuyama, J. Phys. Soc. Jap. 6 3 , 2538 (1994); G. Tatara, H . Fukuyama, Phys. Rev. Lett. 7 2 , 772 (1994). [181] References on bosonic oscillator models of quantum environments; F. Bloch, Zeit. fur Physik 8 1 , 363 (1933), for a bosonic representation of spin waves; S. Tomonaga, Prog. Theor. Phys. 5, 349 (1950), and D. Bohm and E.P. Gross, Phys. Rev. 7 5 , 1851 (1949), for a representation of the low-energy excitation of Fermi systems by "collective mode" oscillators (the beginning of "bosonization"); and other similar models for phonons, photons, etc. Important early discussions of the effect of such environments on the quantum dynamics of a system coupled to them are in G . W . Ford, M . Kae, P. Mazur, J. Math. Phys. 6, 504 (1965); J. Schwinger, J. Math. Phys. 2, 407 (1961); and I.R. Senitzky, Phys. Rev. 1 1 9 , 670 (1960). [182] For response function theory, see eg., L . D . Landau and E . M . Lifshitz, "Statistical Physics" (Pergamon); or D.Pines, P. Nozieres, "Theory of Quantum Liquids", Ch. 2 and Ch. 5 (Benjamin, 1965). [183] R.P. Feynman, A . R . Hibbs, "Quantum Mechanics and Path Integrals" (McGrawHill, 1965). [184] U . Eckern, G . Schon, V . Ambegaokar, Phys. Rev. B 3 0 , 6419 (1984); G. Schon, Phys. Rev. B 3 2 , 4469 (1985). [185] Y. Kagan, N . V . Prokof'ev, J E T P 6 9 , 1250 (1989). [186] "Magnetostriction: Theory and Applications of Magnetoelasticity", by E. du Tremolet de Laicheisserie (CRC Press, 1993).  Bibliography  241  [187] L.D. Landau, E . M . Lifshitz, "Electrodynamics of Continuous Media", (Pergamon). [188] P. Politi et a l , Phys. Rev. Lett. 75, 537 (1995); and Int. J . Mod. Phys. 10, 2577 (1996). See also A . Burin et al., Phys. Rev. Lett. 76, 3040 (1996), for a Comment on this paper; and the reply of Politi et al., immediately following. [189] Y . Kagan, N . V . Prokof'ev, Ch. 2 in ref. [190]. [190] Quantum Tunneling in Condensed Matter, Elsevier Science Publishers B . V . , eds. Yu. Kagan and A . J . Leggett (1992). [191] A . Schmid, Ann. Phys. 170, 336 (1986). [192] U . Weiss, "Quantum Dissipative Systems", (World Sci., 1993). [193] A . J . Leggett, Phys. Rev. B 3 0 , 1208 (1984). [194] B . A . Jones, C.M.Varma, J.W. Wilkins, Phys. Rev. Lett. 61, 125 (1988), and refs. therein. [195] P. Hanggi, P. Talkner, M . Borkovec, Rev. Mod. Phys. 62, 251 (1990). [196] V . Hakim, V . Ambegaokar, Phys. Rev. Phys. Rev. A 3 2 , 423 (1985). [197] A . J . Leggett, Prog. T h . Phys. Supp. 69, 80 (1980). [198] Personal communication, Philip Stamp, University of British Columbia and Spinoza Institute, Utrecht. [199] I. Tupitsyn and B . Barbara, Quantum Tunneling of the Magnetization in Molecular Complexes with Large Spins. Effect of the Environment; to be published in "Magnetoscience - From Molecules to Materials", Miller, Drillon (Eds.) W I L E Y V C H Verlag GmbH, 2000. [200] G. Rose, I. Tupitsyn and P.C.E. Stamp, submitted to P R L .  

Cite

Citation Scheme:

    

Usage Statistics

Country Views Downloads
United States 21 1
Poland 8 0
China 6 27
Czech Republic 2 0
United Kingdom 2 0
France 2 0
Canada 2 0
Ukraine 1 0
City Views Downloads
Mountain View 14 0
Unknown 13 9
Ashburn 5 0
Beijing 3 0
Vancouver 2 0
Shenzhen 2 27
Boardman 2 0
Brno 1 0
Fuzhou 1 0
Ostrov 1 0

{[{ mDataHeader[type] }]} {[{ month[type] }]} {[{ tData[type] }]}
Download Stats

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
http://iiif.library.ubc.ca/presentation/dsp.831.1-0085452/manifest

Comment

Related Items