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The effect of dynamic distortions on the magnetic behavior of transition metal clusters Jones, Donald H. 1984

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THE E F F E C T OF DYNAMIC D I S T O R T I O N S ON THE M A G N E T I C T R A N S I T I O N METAL CLUSTERS  BEHAVIOR  by DONALD H B.A.,  University  A T H E S I S SUBMITTED  JONES  Of C a m b r i d g e ,  1979  I N P A R T I A L F U L F I L M E N T OF  THE R E Q U I R E M E N T S  FOR  DOCTOR OF  THE DEGREE  OF  PHILOSOPHY in  THE F A C U L T Y OF GRADUATE S T U D I E S Department  We  accept to  this  thesis  ©  as conforming  the required.standard  THE U N I V E R S I T Y OF University  of Chemistry  Of B r i t i s h  BRITISH Columbia  Donald H Jones,  June  COLUMBIA June  1984  1984  OF  In p r e s e n t i n g  t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of  requirements f o r an advanced degree a t the  the  University  o f B r i t i s h Columbia, I agree t h a t the L i b r a r y s h a l l make it  f r e e l y a v a i l a b l e f o r reference  and  study.  I further  agree t h a t p e r m i s s i o n f o r e x t e n s i v e copying o f t h i s t h e s i s f o r s c h o l a r l y purposes may  be granted by  department o r by h i s or her  the head o f  representatives.  my  It i s  understood t h a t copying or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l gain  s h a l l not be  allowed without my  permission.  Department o f The U n i v e r s i t y of B r i t i s h 1956 Main Mall Vancouver, Canada V6T 1Y3  Columbia  written  i i Abstract The of  effect  of dynamic  c l u s t e r s which contain  coupled the mean may  spins  coupling cluster  i s studied constants  magnetic using  on t h e m a g n e t i c  ions  with  i n small  c o n f i g u r a t i o n . I t i s found  symmetry  equivalent  magnetic  energy  pronounced  magnetic  ions  states contains  with  different  Dynamic  distortions  special  type  assumption  distortions  that  effect  dynamic  f o r which  degeneracy  or  from t h e  distortions  the spectrum of i n addition to that spin.  c l u s t e r s c a n be r e g a r d e d  of v i b r a t i o n a l / m a g n e t i c  that  on c l u s t e r s w i t h  o r i e n t a t i o n s of the t o t a l  i n these  behavior  isotropically  the simplifying  are linear  have a p a r t i c u l a r l y  associated  distortions  as a  magnetostrictive  coupling.  A dynamic clusters  which contain  tetrahedra derived. are  distortion  model  equilateral  of i s o t r o p i c a l l y  Numerical  avoided  f o r the magnetic  coupled  complications  by t h e u s e o f g r o u p t h e o r y w h i c h c a n be u s e f u l l y  introduction  of intermediate  core  i n a basis  of S=l/2 with  of  regular t o S=5/2 i s matrix  methods  and ' f a c t o r a b l e '  simplified  by t h e  quantum numbers. D i s t o r t i o n s a r e  defined  by t h e n o r m a l modes  of the metal  of the c l u s t e r .  The and  ions  associated  Hamiltonians  considered  t r i a n g l e s and  behavior  dynamic  distortion  model  for equilateral  tetrahedral clusters i s tested against  susceptibility  data  base) and M 0(RCOO) 3  magnetic  f o r the c l u s t e r s Cu OX L a  + 6  (M = F e ( l I I )  trimetallic  6  a  or C r ( l l l ) ,  ( X=halide RCOO =  L=Lewis  carboxylate). the  most  Cu„OX L 6  a  The  dynamic  satisfactory which  a  f u n c t i o n of  excellent  but of  Infinite numerical  the  there  are  static  be  The  transition  an  3-D  with  results  of  the  numerical  an  expression  with  chain,  c h a r a c t e r i s e d by  many a s  1-D  this  distortions.  significant  Dynamic considered  as  even  can  It  be  then and  also  in terms  f o r the  ground  i s solved  Hamiltonian  A  as  a  to of  specific  heat  large  appropriate in  which  defined  f r e e z i n g out that,  imperfections  tendency  towards  dimerisation  arising  shown  phase  for  more  a well  the  s t a t e energy  lattice  distortions  of  Spin-Peierls transition,  regarded  using  the  e x t r a p o l a t i o n are  i s shown n u m e r i c a l l y  of  relative  examined  with  tens  is  the  a d i s c o n t i n u i t y i n the  The  for are  of  S=1/2, associated  in chains  of  atoms.  of  from  high their  as  the  numbered atoms of  becomes a l t e r n a t i n g a t  temperature,  model  = -2S(S+21n2-1).  0  makes t h e  systems.  chain  several  and  E  for  moment  systems  the  chains  Odd/Even H a m i l t o n i a n  spins;  are  i s defined  odd  effects associated a  finite  which  data  i n the  in evaluating  chains  the  than  transition  end  linear  and  provide  distortions.  on  infinite  dynamic  dynamic  Heisenberg  a p p e a r s when t h e numbers of  data  to  f o r the  trimetallic  difficulties  and  antiferromagnetic  for  the  spins  consistent  the  For  e x t r a p o l a t i o n s from  intermediate  i s shown  u n u s u a l maximum  experimental  'Odd/Even' H a m i l t o n i a n ,  chain.  rather  temperature.  between  model  i n t e r p r e t a t i o n to date  exhibit a  agreement  importance  distortion  symmetry c l u s t e r s can Jahn-Teller  activity  be and  for  as  i v S=l/2  the matrices  identical  to those  Interpretation terms  highlights  interactions terms  of  f o r the v i b r a t i o n a l / m a g n e t i c for paramagnetic the dynamic  the  which  provide  i n the dynamic  Jahn-Teller  distortion  importance the  distortion  of  coupling  model  systems. in  non-magnetic  symmetry-lowering Hamiltonian.  are  Jahn-Teller  intercluster distortion  V  Table  of  Contents  Abstract  i i  List  of  Tables  List  of  Figures  ix x  Notation  xi  Acknowledgements Chapter 1.1  1.  xiii  Introduction  Historical  Introduction  1.1.1  The  magnetic  1.1.2  Introduction  1.2  Outline  of  the  1.3  The  1.4  A Derivation  1  i n t e r a c t i o n Hamiltonian  1  to  4  c l u s t e r magnetochemistry  Thesis  Measurement of  8  and  Terminology  the  Heisenberg  of  Magnetism  Hamiltonian  1.4.1  The  two-electron  1.4.2  The  three-electron  1.4.3  The  Heisenberg-Dirac-VanVleck  1.4.4  L i m i t a t i o n s of  Chapter  2.  Methods of  12 17  system  the  18  system  HDW  Solution  21 model  25  model  of  Hamiltonian  the  25  Heisenberg  for  Clusters  2.1  Introduction  31  2.2  Matrix  34  2.3  The  2.4  Methods  Intermediate  Spin  A p p r o a c h and  Factorisation  2.3.1  Commutation  r e l a t i o n s and  intermediate  2.3.2  Examples of  clusters with  factorable Hamiltonians  Group Theory 2.4.1  Magnetic  and  Magnetic  groups  Clusters  spins  37 38 40 43 43  vi 2.4.2 Chapter 3.1  An e x a m p l e  of t h e use of group theory  3. D y n a m i c D i s t o r t i o n s  A Formalism  of Magnetic  fordescribing  50  Clusters;  Distortions  Trimers  of Magnetic Clusters....  3.1.1  The c o n f i g u r a t i o n  3.1.2  The d i s t o r t i o n H a m i l t o n i a n  3.1.3  An a p p l i c a t i o n  3.2 T h e J a h n - T e l l e r  57  Hamiltonian formalism of Magnetic C l u s t e r s  3.2.1  A p p l i c a t i o n of the Jahn-Teller with magnetic degeneracy  3.2.2  The m a g n i t u d e  of d i s t o r t i o n s  theorem  60 63  t o systems .. 63  i n magnetic  clusters  65  forTrimetallic Clusters  70  of the d i s t o r t i o n Hamiltonian  71  3.3.1  Solution  3.3.2  F l u x i o n a l i t y of magnetic  3.3.3  The t r i g o n a l b i p y r a m i d  Chapter  54  of d i s t o r t i o n  activity  3.3 A D i s t o r t i o n M o d e l  4.1  and d i s t o r t i o n spaces  54  4. D y n a m i c D i s t o r t i o n s  distortions  75 82  i n Tetrahedral  Clusters  Introduction  85  4.2 E-mode D i s t o r t i o n s  of Magnetic  Tetrahedra  4.2.1  T h e S=1  system  4.2.2  S > 1 systems  4.2.3  F l u x i o n a l i t y i n 'E' d i s t o r t e d  4.3 T-mode D i s t o r t i o n s  86 87 89  of Magnetic  tetrahedra  Tetrahedra.  100  4.3.1  The T  4.3.2  S > 1/2 s y s t e m s  109  4.3.3  General  112  2  d i s t o r t i o n Hamiltonian  97  d i s t o r t i o n s of tetrahedra  4.4 T e m p e r a t u r e D e p e n d e n t 4.5 D y n a m i c D i s t o r t i o n s  F l u x i o n a l i t y (TDF)  of Octahedra  100  113 117  vi i Chapter 5.1  5. A p p l i c a t i o n s o f t h e D y n a m i c  Introduction 5.1.1  t o Cu OX L„ a  6  The A n t i s y m m e t r i c  5.1.3 The s t a t i c 5.2 T h e d y n a m i c 5.2.1  clusters  exchange  5.1.2 T h e i n t e r c l u s t e r  exchange  distortion  distortion  D i s t o r t i o n Model  124  model  125  model  128  model  model  130  f o r Cu OX L„ a  136  6  The m a g n e t i c s p e c t r u m  136  5.2.2 T r e a t m e n t o f d a t a  138  5.2.3 D i s c u s s i o n  146  of r e s u l t s  5.2.4 C o n c l u s i o n s 5.3  122  [M 0(RCOO) ] 3  150  clusters  +  6  5.3.1  Chromium  acetate.  5.3.2  [Fe 0(RCOO) ] 3  5.3.3 D i s c u s s i o n 5.3.4 F a c t o r s  +  6  151 [Cr 0(CH COO) ]C1.6H 0 3  3  6  2  clusters  154  of r e s u l t s  158  a f f e c t i n g the importance  of dynamic distortions  5.4 T e t r a n u c l e a r 5.4.1  Fe(III) clusters  Chapter  6. L i n e a r  6.1  Introduction  6.2  The A n t i f e r r o m a g n e t i c 6.2.1  162  of r e s u l t s  Magnetic  165  Systems  to Linear  Chains Ground  168 State  Energy  Theoretical considerations  6.2.2 N u m e r i c a l  Extrapolation..  Approximations  6.3.2 R e s u l t s  180 190  to the Heisenberg  of t h e odd/even  172 175  6.3 T h e O d d / E v e n A p p r o x i m a t i o n 6.3.1  1 60 161  Treatment of data  5.4.2 D i s c u s s i o n  152  Hamiltonian  approximation  190 195  v i ii 6.4  Dynamic D i s t o r t i o n s and  7.2  Chains  205  6.4.1  D i m e t a l l i c systems  206  6.4.2  Three  211  6.4.3  Infinite  Chapter 7.1  Linear  7.  Other  atom c h a i n s chains  Aspects  214  of  Dynamic D i s t o r t i o n  Introduction  221  7.1.1  Fluxionality  221  7.1.2  Vibronic  225  coupling  Symmetry, Degeneracy properties  and  7.2.1  The  7.2.2  A p p l i c a t i o n to magnetic  7.3  Suggestions  7.4  Conclusion  References .  for  of  the  Jahn-Teller  Jahn-Teller  systems  clusters  further experiments  Effect  226 226 231 235 237 239  ix List  2.1  Branching diagram  2.2 C h a r a c t e r T a b l e s 3.1  Solutions  3.2 S o l u t i o n s  Solutions  for clusters  o f up t o 6 a t o m s  f o r the groups  o f H'0=O) f o r t h e  P4,P ,P 5  spectrum  4.2 T h e m a g n e t i c  spectrum  distortion  model  5.2  Summary o f d y n a m i c  distortion  spectra  Static data  5.5 F i t s  parameters  and Dynamic  Hamiltonian  75  model  81  96  f o r H' ( - 1 , - 1 , - 1 )  to the static  model  72  o f S>1  Fits  5.4  triangle  f o r the distortion  5.1  5.3 D i s t o r t i o n  49  equilateral  f o r H'(\//=0) f o r t e t r a h e d r a  33  6  of the t r i m e t a l l i c d i s t o r t i o n  3.3 T h e m a g n e t i c  4.1  of Tables  104  f o r Cu OX L„ a  f o r CUflOXgL,  f o r Cu„OX L 6  distortion  model  133  6  136  complexes  a  fits  to Fe 0(RCOO) 3  144 + 6  1 56  t o magnetic  moment  data  f or Fe O(CH COO) , t t  3  0  f o r E (S) f o r Heisenberg chains  162  6.1  Results  174  6.2  The g r o u n d  state  energy  of S=l/2  a n d S=1  rings  183  6.3 T h e g r o u n d  state  energy  of S=l/2  a n d S=1  chains.,  185  6.4  The g r o u n d  state  energy  o f S>1  6.5  Susceptibility  0  and s p e c i f i c  chains  heat r e s u l t s the Odd/Even  189 f o r chains in a p p r o x i m a t i o n ... 2 0 2  X  List  of  Figures  2.1. E x a m p l e s of f a c t o r a b l e c l u s t e r s 3.1.  Spectrum  f o r 3 atom,  4.1.  C o r r e l a t i o n diagrams  4.2.  Moments of  4.3.  C o r r e l a t i o n diagram T distortion  41  S=3/2 s y s t e m f o r S=1  'E' d i s t o r t e d  a s a f u n c t i o n o f r . . . . 77  tetrahedron,  S = 5/2  f o r S=1/2  E distortion..  tetrahedron  101  tetrahedron 107  2  4.4.  C o r r e l a t i o n diagrams  5.1.  Static copper  5.2.  f o r S=1  tetrahedron,  d i s t o r t i o n model c o u p l i n g c o n s t a n t s tetrahedron  T  2  distortion.111  f o r the 132  C a l c u l a t e d moments f o r J = 2 0 0 c m " w i t h , from t o p t o b o t t o m , J = - 1 0 0 , - 1 0 2 , - 1 0 6 , - 1 1 2 ,-1 60 a n d - 2 0 0 c m " 1  1  134  1  2  5.3.  90  The  p a r a m e t e r i s a t i o n of the magnetic  fit  moment d a t a  f o r Cu„OX L 6  spectrum  Dynamic  distortion  model  fits  t o Cu„OX L  5.5.  Dynamic  distortion  model  fits  t o Cu«OX L  5.6. 5.7.  S p e c t r a c o r r e s p o n d i n g t o F i g s . 5 . 4 a n d 5.5 I s o c e l e s and d i s t o r t i o n model f i t s t o data f o r Fe 0(CCl COO) (H 0)C1.H 0 3  6  2  model  6  a  data  ...141  a  data  142  :  6  2  fits  to data  to 139  a  5.4.  3  used  f o r Fe„  clusters  143 1 55  5.8.  Distortion  6.1.  Susceptibility  of Odd/Even  S=l/2 chains  196  6.2.  Susceptibility  of Odd/Even  S=1  197  6.3.  Susceptibility  of Odd/Even  S = 5/2  6.4.  Specific  heat  6.5.  Specific  6.6.  Specific  chains  164  chains  198  f o r Odd/Even  S=l/2 chains  199  heat  f o r Odd/Even  S=1  200  heat  f o r Odd/Even  S=5/2 c h a i n s  chains  201  xi Notat ion S  Spin  S'  Total  on an i n d i v i d u a l i o n . spin of  Intermediate S",S"'  cluster. spins.  S. +S . Number o f m a g n e t i c  n H  Spin 0  Space  T (n,S)  r P  of  s o l u t i o n s of Hamiltonian total  are  spin  found,  S'.  to the  R(S').  : Set  ' : Permutation  r e p r e s e n t a t i o n , I.R. group of order  : Permutation  P(r)  : Projection operator  C, c  : C o n f i g u r a t i o n space of c l u s t e r ,  d, d ^ :  Distortion  V  : Distortion  H' ( d )  : Vector  group  n.  P  D,  system.  s t a t e i n T.  : Irreducible n  f o r an u n d i s t o r t e d  of T of s t a t e s with  Dimension : Spin  }  in cluster.  Subspace of R of s t a t e s b e l o n g i n g i r r e d u c i b l e r e p r e s e n t a t i o n , r.  0(n,S,S'):  {  i n which  Subspace  R(S',T) Q(R,r)  | \p>  Hamiltonian  atoms  space,  element. onto  vector  Hamiltonian  i n V, d i s t o r t i o n  : Isotropic  I.R.  i n D,  r. vector  component  : Coulomb o r b i t a l  \jj,9,<f>  : Angular  Hamiltonian.  interaction  coordinates.  o f d.  space  exchange c o u p l i n g between  K  i n C.  term  atoms  i and j  xii radial  coordinates.  X  : Magnetic  susceptibility  M  : Magnetic  moment,  g  : g-factor,  h  : P l a n c k ' s c o n s t a n t , 6.6256  fi  : Planck's constant/27r  k  : Boltzman's 1kJ  mol"  per mole.  10"  1 .0545 1 0 "  constant,1.3805  10"  = 83.594 cm" ,  1  N  : Avogadro's  c o n s t a n t , 6.02205  e  : Electronic  charge,  M  : P r o t o n mass,  c  : Velocity  of l i g h t ,  1.609 10" 2.997  10  10"  1 9  2 7  =  1  2 3  C.  kg. 10  8  3 W  JK"  2 3  1cm"  1  1.6725  Js.  2 7  ms"  1  1  =  0.69503cm"  1.4388  K  1  xii i Acknowledgements I  would  like  t o t h a n k my  research  Dr.R.C.Thompson and Dr.J.R.Sams, guidance  during  to Dr.J.R.Coope I Walton  my  Fellowship.  Fund  for their  a t U.B.C.  f o r many h e l p f u l  am e x t r e m e l y Killam  studies  grateful  directors, assistance  Special  discussions  to the trustees  f o r t h e award  of a  and  t h a n k s a r e due and  suggestions.  of the Izaac  Predoctoral  1 CHAPTER 1.1  1  I N T R O D U C T I O N TO  Historical  1.1.1  The  first  causing  ISOTROPIC  Interaction  Hamiltonian  satisfactory explanation  magnetic  phenomena  1926 .  s u c h as  Heisenberg  in  reasonable  q u a l i t a t i v e explanation  magnetic  (Ewing)  or  fields'  into  of  the  agreement  i s the  the  but  a q  agreement w i t h  of  theory,  was of  qM  i s the  several  (and  Dirac ), 5  quantum mechanics,  requirement  that  satisfy  Pauli Exclusion  the  electrostatic  the  atoms,  with  electron  of  the  system  no  by a  phenomena  in  'molecular  explanation  to  of  or  bring  the  of  the  For  molecular q  =  i s necessary  the  theories  example,  form H  Classically,  using  then  that  in  the  + qM,  where  field  and  An/3 to  M  is obtain  Principle,  spins be  by  orbital  replaced  by  overlap,  an  of  of a  developed  the  system  must  an e s s e n t i a l l y  electrons  i n s u c h a way  recently  because  electronic wavefunction  caused  could  the  showed  i n t e r a c t i o n between  adjacent the  fields  induced  thousand  given  adjacent p a r t i c l e s  possible.  postulated.  forces  providing  many m a g n e t i c  but  field  was  theories  needed  was a  molecular  experiment*.  Heisenberg theory  experiment  field,  magnetisation,  expected,  of  been p r o p o s e d ,  field  applied  date  magnetic  forces  the  ferromagnetism  this  local  molecular  with  Weiss molecular H  of  had  3  of  i n t e r a c t i o n s between  i n terms  (Weiss)  magnitude  P r i o r to  1  t e r m s of 2  ( H E I S E N B E R G ) MODEL  Introduction  Magnetic  The  THE  that  associated would the  effective  be  real spin  with  correlated Hamiltonian  2 Hamiltonian:  (1.1) where J from the  H  is a  o v e r l a p and  certain  H  where J ^ j i s the Heisenberg,  approximations  theory  was of  In  two  the  Heisenberg  applying  regarding  Ising  had  8  since  H  take  Hamiltonian  though on  the  the  in real  Ising  Hamiltonian  Heisenberg vector,  spin  total  6  can  that, be  Hamiltonian:  a  s p i n s on  atoms  i  and  introducing further  number and  distribution  satisfactory  quantum  of  the  spin  mechanical  1928 . 7  proposed  = -2  a different  I J..S.  values  systems w i t h  S  spin Hamiltonian  to  S. jz  Hamiltonian  as  semi-classical,  i n v o l v e d commute, t h e y = S,S-1...-S; but  z  can  i t may 9  since  much e a s i e r t o  solve  i t involves scalar,  s p i n s . For  example,  than  rather  Ising's original  only  also  extreme a n i s o t r o p i c c o u p l i n g .  is typically  a d d i t i o n of  I Z  i s u s u a l l y regarded  spin operators  discrete  arise  the and  13  Ising  showed  of  interactions:  (1.3)  The  product  i  the  in  scalar  approach  the  arising  J jSj.Sj  (1.2)  to provide  ferromagnetism  magnetic  I  c o u p l i n g between  by  able  1925  describe  = -2  the  atoms. D i r a c  t o many e l e c t r o n s y s t e m s u s i n g  (1.2)  states,  represents  2  the  assumptions,  2  large) coupling constant  S, . S  spins associated with  extended  S,..S  (comparatively  orbital  under  j.  = -2J  The the  than paper  3  contained of  an  an e x a c t  infinite  expression for the z e r o - f i e l d  linear  c h a i n of m a g n e t i c a l l y c o u p l e d  w h e r e a s no c o r r e s p o n d i n g s o l u t i o n has y e t been  large  i t slimitations  interactions,  i n magnetic  perpendicular  mathematical  examples though  of  but  results  been  lattice  neither  are rare,  t h e quantum  applied  field,  to infinite  exchange which Phenomena  be m o d e l l e d  Magnetic  Heisenberg  become such  have  H a m i l t o n i a n s , but  the- H e i s e n b e r g  Hamiltonian  .  The  extremely  Good  nor the  experimental  from  serious  particularly  near  discrete  i t i s now  the  transition formation  approach.  clusters  by b o t h  with  to the  and domain  Hamiltonian  theory,  of, for  contributions  important  been m o d e l l e d  1 2  and  i n the  origins  lattices,  using the spin  "  the Ising  suffers  as h y s t e r e s i s  systems i n v o l v i n g  lattices  used  1 0  and  2-  while Heisenberg's  mechanical  t o the n e g l e c t of a n i s o t r o p i c  infinite  obtained  i s completely satisfactory. lattices  infinite  systems are often that  the Weiss molecular  temperature.  of  of  f o r phase  for the p a r a l l e l  of a v a r i e t y have  i t i s clear  i t explains  magnetic  cannot  lattices  infinite  model  defects.when regard  Ising  of I s i n g  instance,  lattices;  model  received a  as a model  t e c h n i q u e s and a p p r o x i m a t i o n s  sophisticated Heisenberg  atoms  Hamiltonian  mechanical  H a m i l t o n i a n has  particularly  susceptibilities  3- d i m e n s i o n a l  discussion  f o r the Heisenberg  as a quantum  the I s i n g  amount o f a t t e n t i o n ,  transitions  S=1/2  obtained.  Despite magnetic  susceptibility  rather  Ising  generally  i s more a p p r o p r i a t e  1 3  than  and  accepted " " , both 1  that  4 because and  i t can  because  be  derived  in a  i t gives better f i t s  anisotropic  Hamiltonians  (1.4)  H  = -2  L  of  J . .[S.  7=0  also  gives  been  Ising  considered  derivation  of  the  i n c l u d e d or  orbitally  degenerate.  electrons the  t h a t under  should  be  shown  caused  contributions midpoint  to  by  the  concerned  with  almost either  by  or a  J  exchange, in  coupling nearly  coupling  rank  or  tend  tensor  ferromagnetism exchange.  atoms  in infinite  (1.4)  of  ground including • a  orientation a-Fe203  was  Antisymmetric  vanish  i f the  i s a centre  linear  1 7  between  to cause  antiparallel,  c o u p l i n g must  two  Moriya  nearly.degenerate  parallel  have  the  tensor.  the  which  coupled  ]y  in equation  rank  second  weak  •) ]  atoms are  while  second  of  systems.  Magnetochemistry  a l l research the  Mixed,  c o u p l i n g between  elements  exchange  f o r example,  1960  the  antisymmetric  of  ly  arise  circumstances,  Introduction to Cluster  Until  a  S.  when s p i n - o r b i t  scalar,  by  s p i n s . The  between a p a i r  symmetry, as, 1.1.2  r a t h e r than  interacting t o be  a  represented  antisymmetric  ]x  IX  interacting  ions with degenerate  perpendicular, the  by  + S-  gives Heisenberg  approach,  certain  S.  a n i s o t r o p y may  i s represented  off-diagonal  of  ;  '  ]z  when t h e  i s represented  interacting states  1 6  Heisenberg's  coupling  showed  '  1 5  + 7(S.  spin Hamiltonian  terms are  Using  7=1  and  data.  way  form:  S.  I Z  satisfactory  to experimental  the  i]  where  theoretically  prediction  into of  magnetic the  phenomena  p r o p e r t i e s of  was  5 magnetically  isolated  ligand  theory  of  field  extended  usually number has  linear  the  interacting  The  acetate  ions  was  i n 1952.  properties  discrete  study of  crystallography  2 2  were d i s c o v e r e d adequately principal  transmitted.  ,  t o be  i n the  of  by  by  and  with  superexchange,  "  2 9  .  iron  and  the ions  also  the has  magnetically for iron  Bowers of  and  magnetic  copper  chromium  2 7  ;  acetate  systems,  X-ray  Many more d i m e r 2  and  f o r copper  2 1  the  shown, by  60s ""  systems  most were  found  to  be  s i m p l e H e i s e n b e r g H a m i l t o n i a n . The systems  mechanism,  been  the  the p r e c i s e  causing  the m e t a l atoms,  Magnetic  has  interactions  than d i r e c t  the v a r i a t i o n  of  structures;  later  interactions  to coordination  empirically  the  the dimer  between  2 8  the  were  found that  rather  dominant  of  and  the exchange  I t was  1960  of magnetic  i n 1950,  2 0  B l e a n e y and  correct.  the  atoms,  some o t h e r m e t a l s  clusters  Kambe  1950s and  the o r b i t a l  bridging  important  2 3  interest  which  usually  '  described  determination  ligands  systems  predictions  properties  Since  these systems, and  or  magnetochemistry.  l e d to predictions  These  the  above.  clusters  of copper  obtained  dimeric,  crystal  magnetic  In both cases c o n s i d e r a t i o n  e x p e c t e d t o be  trimeric.  introduced  evidence f o r discrete  chromium c a r b o x y l a t e  along  model  involving  enormously.  first  of  of m u t u a l l y i n t e r a c t i n g  i m p o r t a n t branch of  The  using  the c a l c u l a t i o n  chains characteristic  become a  was  or  1 9  Ising  systems  grown  * ,  1 8  lattices  using of  (paramagnetic) ions  spin  pathway  alignment are  involving  known  the  as  metal-metal interactions,  exchange  c h e m i s t s , who  systems were a b l e  i n the exchange  are  became to  constant,  investigate J, with  bond  6 lengths  and angles.  In  t h e 1960s s y s t e m s  coupled and  metal  Fisher's  chain  chains,  infinite  provided  exact  3 3  cannot  c a n be u s e f u l l y  with  t h e 2- a n d 3 - d i m e n s i o n a l  few  systems  until were  discussed  the trimers  involving  3 5  "  3 7  Development  e q u i l a t e r a l and i s o s c e l e s  bipyramidal"  1  Cluster the  ,  0  configurations  clusters  exchange  interactions  important  allows  2  atoms a r e thought catalysis" . 5  .  Bonner  of metal  and small  finite  above  T=0 ";  rather  3  than  comparatively  and four  atom  clusters  chemistry has  of three 3  5  3 6  ,  t o s i x atoms, rhombus  3 9  ,  trigonal  interest  because  in solids  study There  cluster  t o be be i m p o r t a n t  with  ions.  approximation a detailed  of  o r more a t o m s w e r e known  and p o s s i b l y  by v i b r a t i o n s .  clusters" "*"  ordering  triangle  systems a r e of p a r t i c u l a r  discrete  chains  clusters,  a r e many c l u s t e r s  octahedral" ,  use of the molecular  3 1  fordiscussion  of c l u s t e r  involving  3 8  /  results for  by Kambe,  three  a n d now t h e r e  3 7  3 0  systems.  of three  continued  tetrahedral  with  discussed  clusters  .  of exchange  f o r the case of  infinite  lattice  t h e m i d 6 0 ' s , when s e v e r a l synthesised  basis  undergo magnetic  they  from  solution  clusters,  thus  Apart  from computed  a theoretical  discrete  interactions  chains  of t h e s u s c e p t i b i l i t y of t h e S=l/2  and F i s h e r ' s  spin,  infinite  c o p p e r ) were p r o d u c e d  extrapolation  results. Like  range  (usually  calculation  3 2  by n u m e r i c a l  finite  the  atoms  involving  which  contain  of t h e modulation of a r e many b i o l o g i c a l l y like  a g g r e g a t i o n s of  i n heterogenous  7  When d i s c u s s i n g (i) (ii) (iii)  magnetic  Isotropic  clusters  exchange w i t h i n  I n t e r c l u s t e r exchange A l l coupling  assumption  of  isotropic coupling  the  derivation  of  the  i n t e r c l u s t e r chemical  interactions all  but  the  lowest  magnitude  are  c a l c u l a t i o n of  clusters  i s given  coupling  constants  excessively  experimental that  to the  section  1.4.  number  direct  The  of  For  many c l u s t e r s  susceptibility  data  6  '  3  7  explanation  is possible  i s removed. Removal  anisotropic  or  chains  of  copper  clusters Lines" type  9  J  have been  suggest  ions" "" ; 7  8  unless of  5  and  0  6  that,  applied Low  with  a  an  at  dipole/dipole  order  of  the  of between  equivalent introduction  of  fitting often  suggests  of  the  magnetic  of  the  above  assumption  extensively theory  for  (i) gives  systems  e x c h a n g e has 5 1  rise  to  Anisotropic  for Co(II)  some s u c c e s s  temperature  when t h e  one  Hamiltonians.  involving antisymmetric  Cu OX L„. a  used  negligible  equivalent.  no  antisymmetric  be  p a r a m e t e r s when  assumptions  Hamiltonians  avoid  s t r u c t u r a l evidence  i n t e r a c t i n g atoms are  exchange  interactions  assumption  with  1.4.  magnetic  smaller;  made t o  the  3  Direct  even  in  to  equal. in detail  involved,  dipole/dipole  is usually  large  data;  be  are  i s discussed  expected  6  expected  involved  bonds are  temperatures" .  assume:  cluster,  Hamiltonian  between c l u s t e r s are  interactions  an  Heisenberg  the  to  is negligible,  constants  The  Unless  i t i s usual  to  (<10K) m a g n e t i c  and  involving Cu(II)  been d e r i v e d c l u s t e r s of data  i n d i v i d u a l c l u s t e r s have a  by the  sometimes  magnetic  ground  state,  significant the as  very  effect  intercluster a  small,  surprising  that  within  the  '  5 6  distortions  1.2  of  the case  the  this that  to 6 metal  assumptions  has  of  been of  crystal phase  found  5 0  may  In  5 3  .  cases  Hamiltonian  and  i t is  expected  Although  5  a  Interactions  transitions ' ".  have  such  to the  lattice  non-equivalent  not  for  entirely  coupling constants  received comparatively l i t t l e caused  attention,  by  dynamic  clusters.  Thesis work  lower  i s concerned  with effects  the  of  symmetry  atoms w i t h s p i n s ,  are  1) T h a t an  not  i n h i g h symmetry  distortions  the  .  5 2  added  with regard to non-equivalence  O u t l i n e of Most  up  ,  has  properties  type, p e r t u r b a t i o n  more e v i d e n c e  cluster  particularly  field  interactions  i s usually  throughout  interactions 5 5  the magnetic  molecular extend  neglected  on  interaction  presumably  such  small intercluster  S,  of  of  clusters 1/2  dynamic  containing  t o 5/2.  Two  key  made:  magnetic  exchange  intracluster,  i s adequately  isotropic  described  by  (Heisenberg)  exchange  between  pair  Hamiltonian. 2)  That is  the  linear  coupling constant i n small changes  each  of  i n the c o n f i g u r a t i o n  atoms of  the  cluster.  Assumption approximation; Heisenberg  1)  i s conventional, at  least  i t i s d i s c u s s e d w i t h the d e r i v a t i o n  Hamiltonian given  i n 1.4.  Assumption  2)  as of  a  first the  involves  a  9 two-term Taylor distortion,  expansion  i j  J  Chapter  (  d  i j  for clusters  Hamiltonian  by  insight  time  matrix and  clusters  which e x h i b i t  are  lie  the  i t gives  be  effective  little  of  physical  which  even  for  clusters  i s introduced.  Distortions  in a vector  normal modes;  or  A  distortion the  space  high  amount  of  space  examined.  linearly  related  I t i s found  fluxionality,  similar  dynamic  symmetry.  spanned  spin Hamiltonian  C l u s t e r s c o n t a i n i n g an  s p i n s , though  are  by  the  distortions  e q u a t o r i a l plane  systems  but  only  o c c u r r i n g . A method  i s shown t o  behave m a g n e t i c a l l y  containing  S o l u t i o n of  is limited  are  Heisenberg  formalism  ions are  The  discussed.  the  3 a mathematical  metal  the  of  Chapter  (1.5).  triangular.  solution  of  by  coupled  clijOJij/adijio  +  f i r s t part  space  will  the  from  in a Hamiltonian  distortion,  in  distortion  considered  vibrational  i j <°>  J  are  effects  group theory  describing  c o u p l i n g , J^y  memory a v a i l a b l e ,  i f dynamic  the  =  methods  involves  In  }  2 methods of  Hamiltonian  computer  the  d^j, i.e.  (1.5)  In  of  i s expected  rather like  an  to  the  rapid  of  can  trigonal  distortion  t e t r a h e d r a of significantly  model  coupled  be  to occur isosceles  i s extended spins  distortion triangle  of  and  the  triangle be  system of  equilateral  applied to d i s t o r t i o n s  bipyramidal  in  clusters.  to  i n Chapter  more d i f f i c u l t  to  dynamic  i t s mean c o n f i g u r a t i o n w i l l approach  the  i s considered  equilateral  that  by  for  to treat  clusters 4.  Tetrahedral  than  three  10 atom and  systems because  both  because  of the larger  the distortion  space  number o f s p i n  of the tetrahedron  irreducible  r e p r e s e n t a t i o n s . The t r e a t m e n t  tetrahedral  systems  circumstances orientations states  S=l/2  model  a Lewis  copper  base,  i s found  intercluster  clusters,  are considered  development  of the  with a d i s c u s s i o n of the  or antisymmetric  found  though  3  that,  fitted  The  analytic  6  first;  data  6 i s devoted infinite  solutions  and even  the dynamic models  a r e then  distortion  the distortion  The and  distortion  involving have  been  proposed  considered;  i t  model  excellent  may  gives  be c a u s e d  r a t h e r than  for tetrametallic  the  linear  is  by  magnetic  Fe(III)  number  to 1-dimensional,  both  model  the l i t e r a t u r e .  exchange which  c h a i n problem  from  distortion  where X i s a h a l i d e  a  t h e i n t e r m e d i a t e quantum  amount o f a t t e n t i o n  obtained  a  clusters  data,  Finally  using  Chapter  from  Cu OX L ,  of t h e s o l v e n t m o l e c u l e s  interactions.  systems.  + 6  the dynamic  t o the magnetic  ordering  data  t o be s u p e r i o r t o o t h e r  M 0(RCOO)  no  the corresponding  of t h e dynamic  experimental  previously.  are  some  for certain  causes  i s concluded  5 applications  Chapter  tetranuclear  fits  as under  become e q u a l this  two  octahedron.  discussed using  model  may  of the d i s t o r t i o n ;  distortion  In  L  eigenvalues  spans  of f l u x i o n a l i t y i n  t o become e q u i v a l e n t . The t h e o r e t i c a l  dynamic  are  i s more c o m p l i c a t e d  states  clusters  approach.  chain,  has r e c e i v e d an enormous  physicists  f o r the thermal  and c h e m i s t s ,  p r o p e r t i e s have  the a n t i f e r r o m a g n e t i c ground  state  however, been  energy  i s  11 known e x a c t l y quantum used  only  numbers,  to develop  extrapolation, Though  some  concluded dynamic  as  Chapter  7 aspects  clusters  by l a t t i c e  strains,  c a n be u s e d  The  general  Exchange  in  high  2)  Distortions  t o improve Chapters  caused  vibrational/magnetic interactions  dramatic  effect  within  the origin of  physics,  a driving  using  model  such  of the d i s t o r t i o n  I t i s found such  that  as coupling  fluctuating  of  random  interpretation  include:  force  for distortion  clusters.  1) c a n u s e f u l l y  be c o n s i d e r e d  i t might  in clusters  as  be e x p e c t e d  that  since  both  have  m o d e s may  have  a  .  degenerate  on m a g n e t i c  of  non-magnetic  the physical  interactions;  _ 1  discussed  3 a n d 4.  are important  10-I00cm  3) D i s t o r t i o n s  by  The e f f e c t  distortion  of the thesis  provides  magnetic  hoped.  modes a n d s l o w l y  conclusions  symmetry  -  state  t h e model i s  i s then  are discussed.  phonon  coupling  first  model.  systems.  and the r o l e  solid  derived'in  1)  energies  was  of the f l u x i o n a l i t y ,  from  systems, i s  a n d , by  are obtained  of t h e dynamic  interactions,  intermediate  forcluster  to finite  chains  on f i n i t e  i n the Hamiltonian  model  than  on l i n e a r  of  known a s t h e ' o d d / e v e n '  results  useful  calculations  taken  such  chains  interesting  concepts  the  useful  an a p p r o x i m a t i o n  distortions  intercluster  The c o n c e p t  i s very  infinite  the nature  term  which  t o be l e s s  numerical  In  f o r S=l/2.  normal  behavior.  This  i s usually  most  of  12  apparent  for ferromagnetic clusters  dynamic  d i s t o r t i o n s may  a  decrease  rapid  produce  a  i n the magnetic  since  low  i n these  spin  ground  moment a t t h e  systems state  causing  lowest  temperatures. 1.3  The  Measurement  In  T e r m i n o l o g y of  magnetochemistry x.  susceptibility, the  and  This  i s a measure of  where M  which  M  H are not  M and  can  be  found  i s def ined  by  5  7  of  the change the  susceptibility applied  a magnetic  x may  susceptibility,  zero,  i s usually  independent  the  /  x  and  of  of x is  field,  is a  field.  second  on  a  In  rank  single  tensor  crystal.  be  If x >  0,  field  i . e . the  taken, x  i n systems  Thus x can  as a  between  field. <  9H  sample.  i n energy  i f x  field,  general  i s the a p p l i e d  measurements  = -9E  force  inhomogeneous magnetic  In  H  parallel,  M  d o n e by m e a s u r i n g  field  i s that  :  i s the energy  determining  field  concept  XH  by m a g n e t i c  (1.7) where E  =  i s the m a g n e t i s a t i o n and  general,  to  the  5 7  by:  (1.6)  to  fundamental  s u b s t a n c e t o m a g n e t i s a t i o n i n d u c e d by an  defined  M  t h e most  Magnetism  field  the 0,  measured  i s applied.  sample the  be  and  by  This  is  an  sample  i s attracted  i t i s repelled.  dependent; limit  as  i s usually  in this  case  the a p p l i e d  the  field  zero tends  at l e a s t approximately  of magnetochemical  interest.  13  Susceptibility or,  most  may  be  expressed per  commonly, p e r mole,  (1.8) where  N  x  i s Avogadro's  volume,  per  unit  mass,  x » M  = -N/H  M  unit  (9E/9H)  0  number.  P a r a m a g n e t i sm A paramagnetic associated to  an  applied  occurs state  this  o r more u n p a i r e d  magnetic  each  atom  which  does  not  Let  state  electron  be  spin  field. can  mix  be  the  equal  energy  which  excited  J=L+S, Let  system. can  electrons.  represented  contributions. of  independent  simplest  angular  J , in general  z-axis  The  with  the e l e c t r o n i c  unique  have the  one  when  applied.  the  with  substance contains  be  form  by  a  states  ions  each  It i s attracted of  paramagnetism  single  electronic  when  the  field  momentum a s s o c i a t e d the  sum  of  the a p p l i e d When H=0  taken as  with  the o r b i t a l field,  H,  is  and  define  a l l o r i e n t a t i o n s , of  E=0.  On  application  of  field:  (1.9)  where  'g'  E  = -g/3  H.J  i s the gyromagnetic  eh/47rmc. T h e  spin  i s now  ratio  quantised  and  /3 i s t h e B o h r  such  that  J =  where H  =  z  magneton,  - J , - J+1 , .... J ,  i.e. (1.10) The  J  mean e n e r g y  E of  = -g/3 H  the  J  system,  assuming  |H|  thermal equilibrium  reached i s : (1.11)  E M  = Z E  = Z  i  exp(-E /kT)  OEj/BH)  i  / Z  exp(-E /kT) i  exp(-E /kT) i  / Z  exp(-E /kT) i  is  14 Assuming  g/3HJ <<  kT, t h e e x p o n e n t i a l s  c a n be e x p a n d e d  using  e x p ( - x ) - 1-x, g i v i n g : (1.12)  M = - g /3 HJ(J+1 )/3kT 2  x =  and In  this  case  It terms  Ng 0 2  the system  i s often  (1.13)  M  J(J+l)/3kT  moment  constant  paramagnet d e s c r i b e d  as the temperature  Magnetons,  'quenched' results  with  given  g S(S+1). Deviations either by  The  systems c o n t a i n i n g  general  Van V l e c k x = 2  where t h e summation + H W 2  ( 2  and i s  2  ions  any o r b i t a l  symmetric  Law b e h a v i o r  angular  i s effectively  ligand field. formula,  M  This =  2  c a n be c a u s e d  by  atoms o r  the ions.  isolated  ions  are considered  using  formula": (Wj I  (1  = g J(J+l)  2  by t h e ' s p i n o n l y '  i n t e r a c t i o n s between  (1.14)  HW '  units)  the i n f l u e n c e of e x c i t e d s t a t e s of the i s o l a t e d  exchange  the  u  the electronic state  from C u r i e  2  above  metal  by t h e n o n - s p h e r i c a l l y  i n a moment  i n c.g.s.  v a r i e s . The u n i t s o f M a r e Bohr  j3. I n many t r a n s i t i o n  momentum a s s o c i a t e d  properties in  ju d e f i n e d b y :  ( x  2  the simple  magnetic  constant.  (3k/N)(xT)  = 2.828 (xT) For  Law, xT =  to discuss  magnetic  =  2  2  obeys C u r i e ' s  convenient  of the e f f e c t i v e  2  i s over  ', and W  (  1  1 ) 2  /kT  - 2Wl  2 )  )exp(-W^°»AT)  e x p ( - W ^ ° >/kT) a l l states  > e t c . a r e found  of the ion, W = W  (  using  0  perturbation  > +  15  ;."  theory.  If  only  departures second  interact  under  terms  'T.I.P', data,  term  l a w may  of various  exchange  the metal  orbital  1  '  terms w i l l  paramagnet,  metal  electron  ions  degeneracy  field  involved or near  spectroscopic state i s  be m o r e - c o m p l i c a t e d  and l a r g e  deviations  calculations  than  from  f o r both  of paramagnets  as  p a r a m e t e r s h a v e b e e n made f o r configurations  systems can u s u a l l y ligand  Paramagnetism', or  one o r b i t a l  properties field  The e f f e c t of  t o x; t h e  from  Theoretical  ligand  high  states  field.  term  Independent  and s p e c t r o s c o p i c  such d e t a i l e d  because  additive  I f more t h a n  be o b s e r v e d .  the t r a n s i t i o n  resort  that  (  are f o r the simple  Magnetic  and e x c i t e d  c a n s o m e t i m e s be e s t i m a t e d  the W  due t o t h e  known a s V a n V l e c k ' s  of the a p p l i e d  'Temperature  small.  i s occupied,  are entirely  as the ground  the influence  occupied,  magnetic  state  also  (2  i s t o add a c o n s t a n t  function  all  behavior  t e r m s , W >,  and i s u s u a l l y  Curie's  a  Law  arising  of t h i s  thermally  the  Zeeman  terms,  magnitude  they  from Curie  order  frequency  such  a s i n g l e , non-degenerate  be d e s c r i b e d  calculations,  have  5 8  such  degeneracy  '  5 9  .  without  presumably  low l o c a l  symmetry  does not occur.  D i a m a g n e t i sm The closed a  small  i n t e r a c t i o n between  electron  shells  negative  corrections  interpretation  and need  magnetic  i n which a l l spins  contribution  are usually  an a p p l i e d  are paired  t o x. S t a n d a r d  applied  and  leads  6 0  diamagnetic  to experimental  n o t be c o n s i d e r e d  field  data  further  before  here.  to  16 Magnetically It  Concentrated,  i s conventional  Dilute,  to  to  Condensed  systems  interactions  between  important  magnetically concentrated  the  as  i o n s can  be  magnetically systems  the  dilute.  In  there  important.  The  concentrated Cooperative which can  are  this are  and  the  work  will  phenomena a r e  the  be  systems  which spin  Those  are  known  an  those  as  t o as  magnetically  impossible  considered  in  extended which is  magnetically condensed.  in linear  t o be  which  between  within small clusters referred  in  as  throughout and  are  systems  distinction  transition,  latter  t h e r e f o r e be  .  interactions  localised  former  6 1  independently  l e a d i n g to a phase  interactions  in  ions c a r r y i n g unpaired  described  in which  lattice,  the  refer  and  chain  condensed,  systems  rather  than  concentrated.  Antiferromagnetism In  the  and  physics  usually  reserved  which a  spontaneous  occurs point The  at a  a  well  phase  definition  In  o r d e r i n g of  defined  of  term  spins  'ferromagnetism'  associated with  antiferromagnetism lattices  such of  of  as  equal  Curie the  and  ferro-  spin,  are  6 1  also  ,  spin  in  lattice  p o i n t . At  except  opposite  ferrimagnetism  the  ordering  is similar  v a r i a n t s of  unequal  magnetochemistry  the  is  substances  throughout  temperature,  More c o m p l i c a t e d  lattices  the  for magnetically concentrated  antiferromagnetism opposing  literature  transition  interpenetrating involved.  Ferromagnetism  this  occurs. that  two  are  and which  involves  defined.  i t i s common t o c o n s i d e r  condensed,  two  17 rather  than  concentrated  transitions of  ferro-  atoms  i s thus  "  exhibit  6 3  .  interactions  systems  behavior  i s well  magnetochemistry  of  sense  condensed  In J  systems  both  this  chains  f o r 1-D  work  i s largely  and chemistry  concerned  definition  and those  as  i n the  whatever  interactions  > 0 are described as ferromagnetic,  i n the  are often described  systems,  the magnetochemistry  physics  thus  and  chains;  (anti)ferromagnetism  i s impossible  the coupling. Since  i n M(T)  The d i f f e r e n c e i n  by t h e l i n e a r  such  though  6  temperature  between t h e  an extremum  at others.  literature  as the  ferromagnetic  a t some t e m p e r a t u r e s ,  illustrated  (anti)ferromagnetic " 'physics'  with  behavior  definitions  a r e o f t e n used. A  of exchange  ferromagnetic  terminology  r a t h e r weaker  M(T) r i s e s  Condensed  antiferromagnetic  of t h e l a c k of phase  one f o r w h i c h  as a r e s u l t 6 2  Because  and a s s o c i a t e d phenomena,  and a n t i f e r r o m a g n e t i s m  substance falls,  systems.  will  t h e form  with be  used.  c h a r a c t e r i s e d by with J < 0 as  antiferromagnetic. 1.4 D e r i v a t i o n o f t h e H e i s e n b e r g The  Heisenberg  many-electron Heisenberg  s y s t e m s by D i r a c  derivation spin  and a p p l i e d t o ferromagnets  Since  given  the two-electron  i s rather a special  i s also considered  case  case,  approach are  usually considered i n that  are separable,  i n some  detail.  by  below t h e  involved i n the spin Hamiltonian  p a r t s of the wavefunction  system  ( 1 . 1 ) was e x t e n d e d t o  i n 1928. I n t h e d e r i v a t i o n  approximations stressed.  Hamiltonian  Hamiltonian  the o r b i t a l  i n the and  the three-electron  18 1.4.1  The  two-electron  Consider  system  a system w i t h  are  localised  can  be  on d i f f e r e n t  a t o m s . The  H = H,  + H  w h e r e H,  i s the Hamiltonian  for  2, a n d V  atom  a t o m s . The H  Hamiltonian  f o r the  system  s o l u t i o n s o f H,  1H =  products  1, H  a small  l i ei n a  i s the  2  interaction (Hilbert)  1H!*1H , c o n t a i n i n g a l l l i n e a r  o f s t a t e s i n 1H, w i t h that  (ignoring  in  those  product of  2  s t a t e s f o r atoms  1 and  . Thus  2  they  can  be  i n D i r a c ' s n o t a t i o n , by:  1  |*^>  1'H,,  space  1H .  spin) are non-degenerate  H °|* °>  where  states  the ground  represented, (1.16)  between the  combinations  2  (i.) A s s u m e  Hamiltonian  s o l u t i o n s of H l i e i n the d i r e c t  2  space,  + V  2  f o r atom  represents  l i e i n 1H ,  2  the electrons  written:  (1.15)  of  two e l e c t r o n s ; assume  1  = E °|* °> 1  ;  1  H °|* °> 2  2  =  E °|* °> 2  i s the j - t h s t a t e above the ground  2  state for  atom ' i ' . (ii)  Assume  V does not c o n t a i n  (iii)  Assume  |* °>  if  V  = 0,  thus Under  these  function an  1  and  remain  <* °|* °> 1  2  assumptions  refers  orthogonal  =  |* °>, 2  the spin  which  orthogonal  explicitly,  are  orthogonal  when p e r t u r b e d  V,  0. |*, * > 0  0  2  and  |* °* °>, where 2  1  t o e l e c t r o n 1 and t h e second  basis  by  i n 1H w h i c h  i s complete  the  first  t o e l e c t r o n 2, enough  to allow  form an  19 accurate the  d e s c r i p t i o n of t h e low l y i n g  ground  states  states,  of t h e atoms c o n s i d e r e d  those  derived  separately.  In  from  this  basis: (1.17)  K J  H =  J w h e r e K = <* , ° * energy from and  0 2  | V | * , °4> 2 °> , J = <* , ° *  of the system  i s defined  the non-zero matrix 2 are interchanged  exchange  K  as zero  element  0 2  | V | * °* , °> , a n d t h e 2  when V  i s zero.  of V occurring  or 'exchanged'. J  parameter and t h e system  J arises  when e l e c t r o n s 1  i s known a s t h e  i s said to exhibit electronic  exchange. H has the s o l u t i o n s :  (1.18)  +  and  =  |* °'P >  +  |* °* °>,  Energy  = K+J;  |d>_>  =  |4,1°*2°>  -  |* °* °>,  Energy  = K-J.  +  Before |$ >  |$ >  i s symmetric 2, w h i l e  wavefunction exchange  For  with  fermions  1  2  1  spin  of the e l e c t r o n ,  t o t h e exchange  of e l e c t r o n s 1  But the e l e c t r o n s a r e  and so t h e t o t a l  electronic with  o f any p a i r o f them ( t h e P a u l i E x c l u s i o n case  o f two e l e c t r o n s i.e. the spin  the spin  and  respect  Principle). orbital  functions:  v// = { a a , a/3+j3a, j3/3}, a r e s y m m e t r i c +  while both  2  f o r t h e s y s t e m must be a n t i s y m m e t r i c  are separable,  (1.19)  thus  respect  |<i>_> i s a n t i s y m m e t r i c .  the particular  functions  2  i n c l u s i o n of the i n t r i n s i c  indistinguishable  to  0  1  \p_ = (af}-(ia)  eigenfunctions  i s  of H lead  antisymmetric to acceptable  wavefunctions.  20 (1.20)  !<*>,> = (|* °* °> 1  |* >  2  + | * °*! °>) (a/3-/3a) E n e r g y  = ( ( * ° * °> - | * °* i °>) { a a , a 0 + /3a,/3j3}  2  n  2  d e f i n i n g £>' = ( S , + S ) ;  |$i>  i sassociated with  2  be r e p r e s e n t e d  S |*,> = 0 and S' |$ > l 2  2  2  s p i n 0, a n d \ $ > w i t h 2  by an e q u i v a l e n t H a m i l t o n i a n  S' , S'  2  energies  V does  as long  can is  thus  only  of the Heisenberg  not contain  as those  Hamiltonian  the electron spin,  o f H, a n d H  the operators  consequence  of  the Heisenberg  of  states having  depend S'  be  noted.  be e i g e n f u n c t i o n s o f  Similarly  t h e energy of  on t h e o r i e n t a t i o n  of t h e t o t a l  a n d S^_ a p p l i e d t o a s o l u t i o n  of t h i s  Hamiltonian  with  a r e , i . e .a s long as  by t h e o r i e n t a t i o n  t h e minimum  will  one v e r y  i t must commute  i n s t a t e s o f t h e same e n e r g y .  (2S'+1) d e g e n e r a t e  important  2  c o u p l i n g c a n be n e g l e c t e d .  result  unless  are required.  e i g e n f u n c t i o n s cannot  spin,  only the  w h i c h c a n be o m i t t e d  c o n s i d e r i n g t h e t h r e e - e l e c t r o n system,  property  spin-orbit the  involving  t h e r e f o r e t h e e i g e n f u n c t i o n s o f H must  2  1. T h u s V c a n  2  Before  Since  2  V = K - 2 J S,.S -J/2  The K a n d J / 2 t e r m s a r e c o n s t a n t s  useful  spin  = 2|$ >;  spin:  (1.21)  absolute  K-J.  2  But,  electron  = K+J;  2  result  Each  solution  of the total  i sthat  c a n be f o u n d  of H  spin.  of H One  a l l the eigenvalues  by s o l v i n g  z component o f t h e t o t a l  i na basis spin,  S'.  21 1.4.2 T h e t h r e e - e l e c t r o n The all  two e l e c t r o n  solutions  of H  system  system  i s rather  in orbital  s p a c e c a n be c o m b i n e d  functions  t o produce antisymmetric  solutions  of the o r b i t a l  requirement is  that  not the case  used  Hamiltonian  Using subspace low Thus  f o r the three-electron  t h e same a p p r o x i m a t i o n s  o f 1H =  energy  1H,*1H ....1H  states  arising  f o r n=3, u s i n g 3  (1.22) Define and  J  1  2  from  {|12  (1.23)  by t h e  which can a l s o  needed  >  This  1  3  1/2.  a s f o r n=2, a n n !  dimension  to describe  the i n t e r a c t i o n s of n  = <123|V|321>, J  of t h e form  overlap  i s small.  K  0  0  J  0  K  0  J l  0  0  K  J  H = J  1 2  J  2 3  J l 3  J l 3  J  2  3  2  the  electrons.  1 2  J  2 3  J l  3  J  1 2  J  2 3  J  1 2  J f 3  3  0  0  J l 3  0  K  0  J  0  0  K  2 3  1 2  3  = <123|V[132>  <123|V|231>  K  J  2  c a n be  The H a m i l t o n i a n i s :  3  be  t o extend the  3>,|231>,|312>,|213>,|132>,|321>}  = <123|V|213>, J  i f orbital  no  the basis i s :  K = <123|V|123>, t e r m s  ignored  spin  spin  "|123>" a s an a b b r e v i a t i o n f o r  |*,°(1)* °(2)* °(3)>, 2  system,  i s sufficient  2  with  i s antisymmetric.  approximations  t o atoms w i t h  i n that  thus  are excluded  wavefunction  to i l l u s t r a t e the extra  Heisenberg  wavefunctions,  Hamiltonian  the t o t a l  a s p e c i a l case  Though trivial can  H has  a n d S'_,  The  problem  including  reexpressing defined  H  spin,  new  from  functions  which  e v e n when t h e e l e c t r o n solved  basis.  Using  spin i s  by d e f i n i n g  functions  a  and  the antisymmetriser,  A,  by: A = /(1/n!)  functions:  A(123)(/3aa) satisfy  I  (-1)  v  f o r m an o r t h o g o n a l  the Exclusion  = A (1 23 ) ( aj3a) , a n d $ for solutions  of H  3  =  which  Explicitly:  =  ( 1 a2/33a+2/33a1 a+3a1 a2/3-2/31 a 3 a ~ 1 a3a2/3~3a2/31 a )  <t> =  ( 1 )32a3a+2a3a1 /3 + 3a1 /32a-2a1 /33a- 1 0 3 a 2 a - 3 a 2 a 1 j3 )  2  basis: K-J , 2  "Jl from  group P ,  ( 1a2a30 + 2a3/na+3/31a2a-2a1a30-1a302a-302a1a)  3  new  2  basis  Principle.  P  = $  this  x ( P )  of the permutation  <t> = A ( 1 23 ) ( aa/3) , *  (1.25)  the constant energies  matrix  obtained  by  the basis  H  =  2  3  "Jl  3  3  K-J,  3  "Jl  2  3  -Ji  2  2  absolute  ( 1 . 27 )  "J  K-J  K-(J, +J, +J  unless  to  three  efficiently  the P are the permutations  Apart  are, apart  of antisymmetrised  in this  (1.24) where  i s most  there  only  be c o r r e c t l y a n t i s y m m e t r i s e d ,  basis,  in  o f S'  operations  included.  the  6 eigenfunctions,  3  3  2  2 3  )/2,  are required,  this  w h i c h c a n be i s identical  applying:  —2Ji S>i.v5 2  — 2  {aa/3, a/3a, /3aa} a n d  2Ji S^i.S_ 3  ignoring  3  —  2J S_ .£> 23  orbital  2  and  3  ignored to the  23 antisymmetrisation  considerations.  Hamiltonian  i s equivalent  the  the d e r i v a t i o n given  clarity This  that  the electrons  between  for  state  coupling large  and  extension  •  J i  2  root  case  V i s  that the  i s the s t a t e of that  o n t h e same a t o m a r e a l w a y s  t o ions  system  described  on atom  of atom  2  with  1 be  1 and one , and *  3  and  of (1.27) f o r  are:  K - J 1 - J 1 3~J2 3 2  +  J = (J , +J 2  This  Hamiltonian  of two e l e c t r o n s  v/(Jl2  2  3  2  +  ,  Jl3  2  +  'l23  ) / 2 ; since  c a n be a p p r o x i m a t e d ,  - J .  rule,  electrons  2 be * . The e i g e n v a l u e s  , J i 3 , J 2 3 ,  K  Define  p o t e n t i a l s , while  t h e 3 atom  2, l e t t h e o r b i t a l s  on atom  (1.28)  using  the case  on a t o m  orbital  general  electrons  to  c a s e H, a n d  t o the statement  of t h e Heisenberg  be i l l u s t r a t e d  Consider  electron  i s equivalent  between  In this  Hund's f i r s t  unpaired  apply  positive.  S > l / 2 may above.  term.  o f an atom w i t h  constants  The  the  assumed f o r  on d i f f e r e n t atoms.  as the c e n t r a l f i e l d  maximum m u l t i p l i c i t y ,  3  Hamiltonian  i t has been  o n t h e same a t o m .  interelectron repulsion  ground  1  here  are localised  electrons  c a n be c o n s i d e r e d  the  J  to the real  the spin  i s not e s s e n t i a l and t h e r e s u l t s above a l s o  coupling 2  sense  system. In  H  (1.27)  In t h i s  J  the s o l u t i o n s of H a r e :  i f  —  1  using  r e s u l t i s exact  2  'Jl^'^IS  >> J  3  1  2  -  l  Jl3*l23 ^23^12) —  and J  the binomial J  1  2  =  J  2  3  ,  2  3  ,  the square  expansion,  fora l l  J i  3  ,  by i n which  24 ( 1 .29)  i//,  = ( 1, 0 , - 1 ) ,  4>  = ( 1 , - 2 , 1) ,  = K+J-J 13  = ( 1, 1, 1) ,  =  2  ^3  In  both  cases,  \p, w i l l  when c o n s i d e r i n g represented with  be o f v e r y  thermal  This  result  high  =  K+J  1 3  that  with  -J  K-2J-J 13  energy  a n d c a n be  p r o p e r t i e s , t h e system  by two s t a t e s ,  S'=1/2.  Energy  c a n be  S'=3/2 l y i n g  i s equivalent  to that  ignored  3 J above  that  f o r the  Hamiltonian: (1.30) describing The  high  H = - 2 J S".S coupling  spin  w h e r e S" = S , + S  2  between a s p i n  Hamiltonian  =  (J  1 2  +J  1/2 a t o m a n d a s p i n  2 3  as long  as the coupling  i s much  larger  electrons  o n t h e same a t o m  between  electrons  on d i f f e r e n t  between  electrons  on a g i v e n  atoms o r as l o n g  atom and e l e c t r o n s  than  )/2  1 atom.  i svalid  between  are  J  3  that  as a l l couplings on a n o t h e r  atom  equivalent.  Dirac's By  approach considering  the effect  of the operators  { | a a > , | a/3> , | 0a> , | /3/3>} i t c a n b e s e e n -2J  1 2  S .S 1  2  and (~Ji /2)(2P, -1 2  2  ), where P  on  electrons  1 a n d 2, a r e e q u i v a l e n t .  to  show t h a t  the Heisenberg  many e l e c t r o n highlights  using  the connection  permutations group  system  of electrons;  theoretical  between  1 2  Dirac  Hamiltonian (1.2).  that  the  used  6 5  methods o f s o l u t i o n  the labels this  property  be e x t e n d e d  to a  formulation  the coupling  i t i s useful  operators  permutes  could  The D i r a c  on t h e b a s i s  operator  and  when c o n s i d e r i n g t h e  of the Hamiltonian  which  25 are  introduced  1.4.3  The The  the  thesis  the  the  to as  system,  above  Hamiltonian  s t a t e s each  referred  of  which  the  'magnetic  atom,  simple  x  =  summation  The  factor  has  ( 2 S ' + 1)  A  system  (2S'+1) a r i s e s  >  1.  this  A l l the  Heisenberg  paramagnet, apply  Heisenberg-Dirac-Van from 1.4.4  these  Limitations 1) T h e  model  the  of  which  Vleck  now  HDW  r e p r e s e n t a t i o n of  interacting  In  2  be  formula  of  derived  states,  with  ions are  of  the  spin of  Hamiltonian,  total  different have  spin  S'  values  of  above  formula  definition x  S^,.  additional  f o r the  by  of  an  fi(S')  the  simple  the  m o d e l . The be  exp(-E/kT)  exp(-E/kT)  i s known a s  (HDW)  will the  i n the  derivation  (31)  assumptions  S' .  will  these  state  i n the  arising  the  to  of  s y m m e t r y may  i s represented  and  of  susceptibility  of  ( 2 S ' +1 )£2(S' )  associated with  assumptions  model,  to each  s i n c e each  with non-trivial  degeneracy;  The  a p p l y i n g the  a l l solutions  degeneracy  eigenvalues  S ' (S ' + 1 ) ( 2S ' + 1 )fl(S ' )  2  i s over  spectrum  weighting:  L The  of  by  in a  e i g e n f u n c t i o n of  spectrum'.  paramagnet  (N/3 g /3nkT) L 2  results  set  i s given  appropriate thermal  Model  i s an  corresponding  per  f o r the  (1.31)  2.  Heisenberg-Dirac-VanVleck Heisenberg  magnetic this  i n Chapter  limitations  arising  discussed.  Model the  interacting  represented  by  ions.  In  single,  the  HDW  26 non-degenerate  states  multiplicity. state  may  useful  of  principle  with  and  allowed  a  coupling  antisymmetric Order the an  2S'+1  rarely -  of  g and larger  100cm"  isotropic  For  of  i t s free than  coupling  orbitally  0.1,  and  expected at  or  into  the  both e f f e c t s spin  2.0  degenerate  two  i n which  must  be  suppose  ground  state,  an  atom has  enlarged basis  c o n s i d e r e d . I n g e n e r a l t h e r e i s no that  matrix elements  such  as:  is with from  1-2K.  electron, degenerate  clusters  departures  more c o m p l i c a t e d . Thus one  by  difference  . S i n c e Ag/g  significant  that  is split  i s the  i s considerably system  and  suggest  states  non-degenerate  nearly  be  spin Hamiltonian.  situation  atom  result  of  anisotropic  temperatures above  degenerate  state  occur as a  the e f f e c t s  and  v a l u e of  t o show  ground  c o n s t a n t i t can  2  orbitally  and  independent  ( A g / g ) J , w h e r e Ag  electron  It is  degenerate  model  t o x;  on  6 6  ground  separately.  the Heisenberg  the o r d e r of  are not  1  i s small  introduced  the  the exchange.  t o t h e HDW  calculations  degeneracy  amount  between  are  maximum  non-degenerate  a r e more c o m p l e x ,  terms  of magnitude  on  states  simple correction  of  other than  temperature  S i n c e T.I.P.  spin-orbit  spin  orbitally  orbitally  paramagnetism. f o r by  of  ground  an  coupling  states  effect  the cases  modifications  spin-orbit  associated  systems,  non-degenerate systems  an  significant  to consider  For the  In r e a l  have a  orbitally  with  of  clusters  for a a  the  reason  two  (nearly) form:  to  the  J  27 <*,°* °|V|* °*1°> 2  are  equal,  spin the  and  new  be  large  expected  to  clusters  rarely exist  and  between  very  ions the  rare  and  with the  any  Fortunately, in clusters,  orbitally  to  The  instability  theorem  caused  by  may  well  applied  small  be  cluster;  the  was  spin  with  degeneracy  to c l u s t e r s .  with  magnetic  by  considering  spectrum  contains  the  spectrum the  be  degeneracy  associated  with  Though  Heisenberg  Hamiltonian  is a  and  remove spin  that  small.  The  but  a  exchange  frequently  This  contain be  removed  i s most  cluster for  in addition  o r i e n t a t i o n of  unstable  include  i n mind,  Consider  coupling.  JT  is  proviso  s y m m e t r y , w h i c h may  degeneracy the  will  a Heisenberg  the  to  to  6 8  the  precise  within  3.  symmetry  with  expected  as  non-linear  extended  paramagnets  whatever  magnetic  associated  was  any  the  in  following  electronic state  lower  ions  i n t r o d u c t i o n to  i n the  that  later  dilute  d i s t o r t i o n s a f f e c t i n g the  illustrated  states  the  is  e f f e c t s both  An  to  become  degeneracy  in Chapter  Kramers degeneracy),  derived  Hamiltonian,  6 7  degenerate  t h e o r e m was  high-symmetry;  fields.  the  configurations  Jahn-Teller  d i s t o r t i o n s which  (except  equally  symmetry  of  m o d e l may  both because  i s continued  theorem  terms  contributions  orbital  c l u s t e r s i s given  discussion  degeneracy.  degeneracy  Heisenberg  interacting ligand  Jahn-Teller  respect  the  of  the  In  antisymmetric  because  paragraphs,  i n an  2  introduced.  high  i n magnetic  molecule  are  i n such  instability  The  1  2  and  inappropriate.  paramagnetic  1  a n i s o t r o p i c and  e x c h a n g e may  be  <*, * °|V j * ° * 1 >  parameters  Hamiltonian,  completely  and  2  the  to  easily  which  the  the  2S'+1  total  spin Hamiltonian,  spin. the  by  28 real  Hamiltonian  interactions applies.  f o r the system  degeneracy  will  degeneracy  associated  degeneracy,  lower  In  t o be  degenerate  i twill,  Heisenberg coupling  Chapter  The  orbitals notably multiple  they  open c h a i n  other  may  t h e symmetry and r e s o l v e t h e of the c l u s t e r . coupling  The  2S'+1  i s genuine  spin  be r e d u c e d t o  interactions, the effect i s  linear  molecules  be s t a b l e  linear  i n an  chains  a r e s h o w n t o be  orbitally  may,  under  some  resistant to distortion.  magnetic  are equal.  are considered  approximations  minor  spectra Because  Also,  are non-degenerate of these  separately  i f a l l  complications,  in this  thesis, in  inherent  i n t h e HDW  s i g n i f i c a n c e when a p p l i e d  to clusters.  of o r b i t a l  must  i n a l o s s o f o r t h o g o n a l i t y . Some  Slater  result 6 9  ,  have h e l d  exchange  terms,  Hamiltonian  disagree  7 0  '  7 1  .  that  neglect  undermines  f o r magnetic While  Any i n t e r a c t i o n  of overlap,  the usefulness problems.  overlap  i f the Hamiltonian  f o r small  overlap.  model a r e of  The n e g l e c t  modifications systems,  theorem  6.  Heisenberg authors  Jahn-Teller  in principle,  theorem  be s i m i l a r l y  constants  relatively 2).  orbital  significant.  s t a t e . Magnetic  chains  from  orbital  isotropic  by s p i n / s p i n  i n that  circumstances,  linear  with  the Jahn-Teller  exceptional  reduce  the energy  and though  Kramers doublets unlikely  results  and so t h e ' s t r o n g ' ,  D i s t o r t i o n s which  •;•  t e r m s may  authors, and hence  of the  Several  later  necessitate  i s t o be a p p l i e d  clusters inclusion  between  of o r b i t a l  to  infinite  overlap  only  29 causes  a second  systems,  order modification  defining  S=<*, |* 0  ( 1 .32)  The  energy  but  since  difference  remains  Hamiltonian;  negligible theory for  number,  the Heisenberg  spin.  effects  Direct  introduce  additional  molecular fields  magnetic  interactions  was t h e o r i g i n a l  spin/spin  cannot  interactions  terms  occur  are expected  difficulty  unpaired potential  direct  magnetic  forces  10A a p a r t ;  electrons  between  t o be  i n the c l a s s i c a l  clusters  classically,  into the  in clusters,  of f e r r o m a g n e t i s m ) . As an o r d e r o f magnitude  such  calculation  consider  their  two  mutual  energy i s :  (1.33) Taking  a good quantum  internal  (this  than 2 J ,  2  not contain  any d i r e c t  2  intact.  molecular f i e l d  while  t h e e n e r g i e s become:  i s now 2 ( J + K S ) / ( 1 - S " ) , r a t h e r  remains  2  3) V d o e s or  ,  E = K±J / 1 ± S  S'  formalism  0 > 2  i n J , e.g. f o r two-electron  MiM /r  the f i r s t  ( 1 .34)  -  3  2  term  3(MI.r)(M .r)/r  and using  E = -M (M ) /47rr 2  0  =  -  *  10"  Thus t h e n e g l e c t not  significant  the  derivation  5  2  S.I  units,  3  6  ( 4 7 T . 1 0 - / 4 7 T . 1 07  2 3  J  = 10"  of d i r e c t compared  3  2 7  ) ( 0 . 9 2 7 . 1 0-  cm"  2 3  )  2  1  intercluster  dipole/dipole  forces i s  to the other approximations involved i n  of t h eH a m i l t o n i a n .  30 4) arises  The  inequality  from  the  susceptibility valid  of  However  clusters  the  by  HDW  derivation cluster  of  the  systems.  well  model,  to  The  to derive  I t i s expected  low  as  additional  i n such  to  temperatures limit  spin states which  the  basic  both  w i t h i n and  be  sufficient  at  temperatures by  adequately  by  magnetic  exchange  Hamiltonian  single least  that  low  s t r u c t u r e of between  clusters  the be  and  high  the  size  to can  S'  may  be  associated with  the  that  the  clearly  satisfactory.  ligand  field and  interacting  by  valid  interacting  non-degenerate  clusters  the  are  of  the  more t h a n  a  observation  modelled  by  of  i f the  coupling.  effect  of  instability  be  which  explained  atoms  However  i t is  symmetry caused Jahn-Teller  ligand  This  Heisenberg  many c l u s t e r s  e x c e p t i o n a l can  K.  that almost  the  with Jahn-Teller  few  for  state is  fields,  t o make a n i s o t r o p i c / a n t i s y m m e t r i c t e r m s  behavior  considered  a  t o be  i t i s shown t h a t  associated  approximations  approximation  suggest  both  supported  the  Heisenberg  considered  reasonable  thesis  of  represented  generally  be  another  used  _  large.  Summary. M o s t  are  1 x,  for very  low-lying, high  the  inequality  paramagnet.  except set  This  exp(-x)  simple  i t does  with  represented very  approximation  for a l l clusters  fields. of  q)3HS'<< k T .  suggestion  effects, should  negligible is  a l l known c l u s t e r s Hamiltonian. dynamic  by  In  can  this  distortion  i s considered,  the  have p r e v i o u s l y been i n terms of  isotropic  31 CHAPTER 2  METHODS OF  S O L U T I O N OF FOR  2.1  THE  HEISENBERG  HAMILTONIAN  CLUSTERS  Introduct ion  The H e i s e n b e r g  Hamiltonian  f o r c l u s t e r s of exchange  coupled  ions: (2.1) where can  S^  introduction theoretical  exactly  of intermediate m e t h o d s . The  combined  to give  of  symmetry  quite  spectrum  spins  The HDW  r e s u l t i n g from  number  total  spin (2.2)  S',  c l u s t e r s , but they  equation  may  f o r some  be  systems  solvable  c a n be a p p l i e d  high to the  s o l u t i o n of the Hamiltonian  of M or x ( T ) , which can then  when  solving  i s the large  number  of spin  fi(n,S,S'),  i s given  states  /n i \ (2.3)  of s p i n  S  with  a l g e b r a i c a l l y by:  = w(n,S,S' ) - co(n,S,S' + 1 )  i s the c o e f f i c i e n t of x  / S . S~1 (x + x  be  involved.  S' where cj(n,S,S')  to  (2.1) f o r a l l but the  of a c l u s t e r of n atoms  fi(n,S,S')  useful for  data.  problem  of s t a t e s  group  a r e most  approximations  ions,  by t h e  o r by  by c o r r e l a t i o n b e t w e e n e x a c t l y  to experimental  The  numbers  two m e t h o d s  accurate  fitted  clusters  diagonalisat ion,  symmetry  theoretical predictions  smallest  on t h e i n t e r a c t i n g  quantum  obtain  The c e n t r a l  i  latter  symmetry c o n f i g u r a t i o n s . magnetic  i  by m a t r i x  c a l c u l a t i o n s on h i g h  lower  J jS .Sj  and S j a r e the t o t a l  be s o l v e d  exact  H = -2 L  . +...x  -  S~1  i n the  , -S%n +x )  expression:  32 For  S=l/2,  the c o e f f i c i e n t s  cj(n,l/2,S')  = n ! / [ ( n / 2 - S ' ) ! ( n / 2 + S ) ! ] , but  f o r S > l / 2 no  1  expression found  can  using  by  is  the  S' 6 11/2 5 9/2 4 7/2 3 5/2 2 3/2 1 1/2 0 n levels Branching in  Table  an  and  the 7 2  diagram  summing  i s begun  such  fl(n,S,S') .  The  i t e r a t i v e procedure  o b t a i n e d by  |S'-S"|<S. T h e S=l/2  derived  a branching diagram  constructed fi(n,S,S')  be  a r e g i v e n by  simple  a r e most  easily  branching diagram which  those by  cj(n,l/2,S')  is valid  fi(n-1,S,S")  placing  f o r a l l S;  f o r which  '1' a t  n=1  S'=S.  branching diagram i s : 1 1 1 1 1 1 1 1 1 1  5 4  3 2  1  2 3  4  5  6  7  8  1  2  3  6  10  20  35  70  2.1.  f o r S=1  t o S=5/2, up  297  90 1 32 42  14  2  diagrams  1 65  42  1  275  75  28  5  110  48  14  1 54  35  20  9  44  27  14  5  54  9  7 6  9  1 32 1 1  10  1 26 2 5 2 t o n=6  1 1  10  8  1 1  is  462  are  12 924  given  For  33  Table  2.1  n  3 1 3 2 1  Branching  }  4 3 6 6 3 1  ;  diagrams 5 6 15 15 10 4 1  for clusters  6 15 36 40 29 15 5 1  3 1 3 5 4 3 2 1  ;  S=1 total n  7  3  4 4  ;  2  51  141  5  6 34  20 9  4  34 1 1  3 2 1  4 5 12 16 17 15 10 6 3 1  S =2  19  ;  o f up t o 6  19  3  6 20 3 10 1  5  21  1 20  64  315 475  24 21  2 15 1  575 1 20  3 35  6 11 1  84  24 4  6 65 180 260 295 285 240 180 1 20 70 35 15 5 1 1 751  11 1  5 96  ;  45  6  4 1  4 6  ;  15  1 20 30  381  4  36. 10  85  2 90  5 16 45 65 70 64 51 35 20 10 4 1  atoms  609 11 5 581 1 00  15  505 79  10  405 56  5  6  300 35  1  3  204 20  1  126 10 70 4  S = 3/2  S = 5/2  35 1  total  12  44  1 55  580  27  1 46  780  15 5 1 4332  S' 0 1 2 3 4 5 6 7 8 9 10 1 1 12  S' 0 1/2 1 3/2 2 5/2 3 7/2 4 9/2 5 1 1/2 6 13/2 7 15/2 8 1 7/2 9 19/2 10 21/2 1 1 23/2 12 25/2 13 14 15  34 2.2  Matrix  Methods  The H a m i l t o n i a n using set  a basis  (2.1) can always  spanning  the t o t a l  of a l l p o s s i b l e p r o d u c t s  be e x p r e s s e d  spin  space  of i n d i v i d u a l  which a r e e i g e n f u n c t i o n s of S  forms  as a  matrix  of t h e problem. atom  spin  The  functions  such a b a s i s , e.g.  f o r n=3,  S=l/2: (2.4)  These  {aaa,aa(5,  (2S+1)  Hamiltonian; is  real  size  i.e.  All an  they  will  be  i n the Ising  H commutes w i t h  of the matrix  problem  eigenvalues S^ b l o c k ;  associated  of the I s i n g  manipulation  the basis  ^  basis with  This  found  to find  and m a n i p u l a t e  large matrices  to find  just  coupling  - j  = S.  1Z  S.  ]Z  + 1/2  solving the spin  direct i t i s very  the eigenvectors  eigenvalues.  elements are evaluated  S..S.  — 1  by  without  by e x p r e s s i n g  as:  (2.5)  separated  i s useful since  much more d i f f i c u l t  The m a t r i x  different  ( 2 . 4 ) c a n be  |S^| a r e o b t a i n e d  the  S^,,  {/3/3a,/3a/3, a/3/3} , {/3/3/3}.  c a n be  of the eigenvector.  spin,  b y b l o c k i n g by  t h e s o l u t i o n s of each b l o c k ,  each eigenvalue  than  of the t o t a l  reduced  [ a a a ] , {aa/3, a/3a, /3aa}  by c o m p a r i n g  with  c a n be  o f s t a t e s o f S'  b a s i s . (2.1)  basis.  t h e z-component  o f S^, s e p a r a t e l y . T h u s  the blocks  Ising  r e f e r r e d t o as the I s i n g  by c o n s i d e r i n g s u b s e t s  values into  s t a t e s a r e e i g e n f u n c t i o n s of the  n  and symmetric  Since the  a/3a,/3aa, |3/3a, /3a/3, a/3/3,/3j3/3}  (S.^S. i+  j -  +S. S. A  j+  i  -  )  the  of  35 where  S  and  +  angular  As  simplified  well  as  and  extrapolation.  1964  is  less  3 2  .  by  the  This  has  S>1/2  of  a  extrapolations  use  and  the  and  there  are  size  diagonalised the  matrix  Even  more  to  fundamental  Firstly,  the  smallest  clusters  experimental  set  data  {J}  {J}.  t o model  real  other and  the  s t a t e s and  the  data,  the  frequently gives  are  fitted,  accurate  long  ago  prohibits  i n Chapter size  one  the  of  6. the  f o r S=5/2,  longer of  matrix  n=5,  more atom  can  be  increases  practicable.  matrix  requiring  large  numbers  expensive.  There  are  direct  matrix  many p a r a m e t e r s  and  S=l/2  780X780 m a t r i x  i s no  prohibitively  i n v o l v e s too  by  Numerical  the  application  objections to  1-D,  obtain  problem  terms.  A d d i t i o n of solution  for  systems, e x t r a p o l a t i o n  matrix of  be  systems  p r o p e r t i e s , as  3-D  of  results  infinite  to  for  magnetic  to obtain  i n more d e t a i l  any  set  s u c c e s s f u l f o r the  experimental be  can  in  i m p o s s i b l e . Thus,  4332, and  may  sometimes  small clusters,  spin clusters  fits,  problem  used  number  cheaply.  in fitting  attempted  also  fairly  f o r lower  techniques of  S'=l/2 b a s i s  size  of  discussed  make s o l u t i o n 780  as  rules  the  symmetry  been  f o r 2-  sufficient  are  of  usual  f u n c t i o n of  F i s h e r were a b l e  well  E v e n when a p p l i e d t o p r o b l e m may  the  been most  Bonner  u s e f u l as  calculation  {J};  the  s t a t e p r o p e r t i e s of  f o r x ( T ) , as For  is a  used d i r e c t l y  ground  obeying  H,  methods have  for which  expressions as  operators  matrix,  being  matrix  thermal  system,  shift  coupling constants  clusters, the  are  momenta. The  n(n-l)/2 further  S.  rise  approach.  i n a l l but  to ambiguity  i n some s m a l l S=1/2  systems  the  when i t can  36 be  shown  that  any  spectrum  (and  therefore  in  principle,  take  into  dynamic thesis  attempts  result  account  that  the  {J}  from magnetic  in ambiguity.  the  distortions  to define  effects  using  of  Secondly,  {J}  magnetic data)  must,  i t is difficult  i n {J}  methods;  that  the  susceptibility  changes  matrix  assumption  from  caused  by  i t i s shown  i s constant  over  to  in  this  time  is  not  just ified. Despite with  these  considerable  structure  of  objections matrix success,  magnetic  to  test  clusters  7  3  -  7  methods have  5  ;  Hatfield  program which  c a l c u l a t e s the  clusters  of  f o r a l l { J } , and  I  have w r i t t e n a  s p i n .as w e l l a s magnetic  quite  only  severe  by  the  solution  the  this  thesis.  and  S=1,  f o r the of  with  impetus  distortions  matrix  time  minutes  Experiments of  {J},  on  properties  for a l l applied  only  uses a  the  written a  is useful for  which  gives  standard  of  fields.  general  zero  field  library  d i a g o n a l i s a t i o n . These programs memory c o n s t r a i n t s , h o w e v e r of  high  time  cluster  on  the  S = 2 , 5 / 2 , n=5.  In  each  n given  of  the  A m d a h l V8  takes  computer  the  role  p r o p e r t i e s of  of  are  limits  at  case order  U.B.C.  s o l v i n g program have p r o v i d e d  s t u d i e s of  magnetic  of  are  these  s p i n atoms; p r a c t i c a l  S=3/2 n=6;  largest value  f o r the the  program  n=8;  CPU  the  but  has  7 6  magnetic  program which  for clusters  S = 1 / 2 , N=12;  five  general  f o r the  are  of  similar  p r o p e r t i e s ; the  subroutine limited  ions  used,  hypotheses concerning  computer  S=l/2  been  much  structural  clusters  described  in  37 2.3  The In  several  Intermediate  Spin  1950 K a m b e  showed  iron  hypothesis metallic  ions  =  t h e compounds  i n an  behavior  consistent  contained  that  (S,+S ),  the  three  c o n f i g u r a t i o n . The  + S .S ) -  2  2  H c a n be  so t h a t  3  2J'S .S  3  solved,  H = -JS'  H c a n be  2  -  1  without  3  resort  formulation  is  'good quantum number',  +  2  i s u s e f u l because  in Dirac's  to  quantum  (J+2J')S  number  S"  2  commutes  2  with  and the e i g e n f u n c t i o n s  notation,  matrix  expressed:  (J'-J)S"  This  expressed,  of  with  c l u s t e r s of  by t h e i n t r o d u c t i o n o f t h e i n t e r m e d i a t e  (2.7)  a  was  isoceles triangle  H = -2J (S,.S  Kambe s h o w e d  S"  the magnetic  f o r such a system i s :  (2.6)  methods,  that  and chromium compounds  that  Hamiltonian  2 0  Approach and F a c t o r i s a t i o n  b y k'ets  |S'S">,  H,  S"  o f H c a n be  of energy  found  f rom: (2.8)  Not  S'  2  |S'S"> =  S'(S'+1)  |S'S">  S"  2  |S'S"> =  S " ( S " + 1)  |S'S">  a l l Hamiltonians  intermediate  spins  which  c a n be  c a n be e x p r e s s e d  solved  in this  way.  i n terms of  Thus  expression  of: (2.9) as  though  H = - 2 J (S, .S  -  2  S .S ) 2  3  H = - J ( S " - S " ' - S ) , where 2  2  2  S" = S + S - S ,  c o r r e c t , i s not h e l p f u l s i n c e  necessary  to consider  more  1  2  3  [S" ,S"' ] 2  2  S"'=S_.-S . 3  # 0  p r e c i s e l y the conditions  . I t i s now under  which  38 intermediate  spins may  u s e f u l l y be  introduced. This involves  c o n s i d e r a t i o n of the commutation r e l a t i o n s f o r the terms in the Heisenberg 2.3.1  Hamiltonian.  Commutation r e l a t i o n s and Consider  Hamiltonian, commutator  intermediate  the commutator between two [ . S ^ , . ] ,  is clearly  terms of the  where i ± j and  zero unless  spins  k * 1 . The  i or j equals  l o s s of g e n e r a l i t y , assume i = l , and  Heisenberg  k or 1 . Without  write:  (2.10) [S- .S . ,S. .S, ] = [S. S. ,S.„S,„] — l —] — i —k i a ] a ' i/3 k/3 where a and  /3 are dummy s u b s c r i p t s d e s c r i b i n g the  components of the spins and that  using the summation  i f a s u b s c r i p t occurs e x a c t l y twice  three  convention  i t i s summed over.  In atomic u n i t s (R = 1) the commutator for spins on the same atom i s , S S„ = e „ is a  tensor,  /3  apy  7'  , where e „  afiy  i s the  antisymmetric 1  thus:  (2.11)  [S. , S . J S . S. = i e . S. S. S.-, i a ' i/3 ]a k/3 a/37 1 D k/3  = i S..(S.AS.) —1 —] —k  a  a  1  by d e f i n i t i o n of the v e c t o r product. For general  i,j,k,l  the  expression i s : (2.12)  i  [Sj.Sj,^.^] =  { ( 6. ,-5. , )S . . (S, AS, ) + (6 .,-5 .. )S. . (S. AS, ) } il  The  result  Consider  ik —]  —k  —1  3I  ]k  — 1  —k  —1  (2.12) can be used to prove f o r m a l l y that  the commutator of a s i n g l e term in H,  S, .S,,  [H,S' ]=0. 2  with  S' : 2  39 (2.13)  S'  n = Z S..S. = n S i,j = 1  2  1  Ignoring  the t r i v i a l  (2.14) but  [  Z ( 6 ^ j ^ — 6^-^)  S  j f  = 0 since  over  i . Thus S'  therefore  with  the total  S"  i s some s p i n  contain  n S , t h e commutator i s : 2  ]  -2 i  2  other  Heisenberg  than  the f i r s t  S' =  (6  commutes w i t h  the conditions  n spins,  (2.15)  ^  11  [H,S" ];  (2.16) H  = -  a  {Z  third term  system term  of S" ,  spin.  m of which  The  second  total  there  m  n  + 2 Z  Z  i= 1  m  spin  X 1  term  J  l  b  i s zero,  1  3  )  to S" i.e.  i j  -  1  1  m Z i=1  n +  Z  that  S..S.}  J  i^j=m+1  i j  - .  1  term  1  describes  commutes w i t h t h e  i n common. C o n s i d e r  a  single  1 < b < m) : n Z. J . j S . S . , S j=m+1 ^ i  i  a  .S.]  1  S . . ( S A S . ) + (6., - 5 - ) S . . ( S A S . ) }  ~ J  since  S..S.  S " . I t also  (1 < a < m,  b  J  of H since  a r e no s p i n s  J..{(6.,-6. +  L e t t h e system  contribute  j=m+1  term  [ H , S . S , ] = -2 [  m n = "2i Z Z 1 j _  term, and  1  the f i r s t  a  (2.17)  each  i n each  2  the total  ^  1  s a y S_ .S_ ,  2  i =  i j  with  since  once  m S" = Z S . i=1  S..S.  J  commutes w i t h spin  ^.(S.A^)  n  H c a n be e x p r e s s e d a s :  2  i * j =1 2  -6 )  f o r [ H , S " ] = 0, w h e r e -  n Z S. i=1  m  S"  k  Hamiltonian.  1  Consider  i  i= k and i= l e x a c t l y  summation  Now c o n s i d e r  n Z S..Si * j  +  ]  constant,  S..S  2  a  _  a  ~  b  J  b  a,b < m < j , s o :  J  a  a  _  1  -  a  _  b  40 n [ H , S .S, ] = - 2 i Z -a -b .  (2.18)  =  For  = 0,  [H,S" ] 2  ( 2 . 1 8 ) must  addition  i s involved, each  Thus S"  commutes w i t h  in  2  i n S".  In t h i s  simplified will  spin  work  (JUA-J^A) +  term  S..(S AS,) -] - a -b  f o r a l l a,b. S i n c e  i n each  sum o v e r  vector  j must  from  n o t i n S" a r e e q u a l ,  a Hamiltonian  t o as  a i  3  i fthe couplings  by t h e i n t r o d u c t i o n  be r e f e r r e d  b  1  vanish  H only  S" t o a p a r t i c u l a r  not  m  be  every  spin  f o ra l l spins  w h i c h can u s e f u l l y  o f i n t e r m e d i a t e quantum  'factorable'.  zero.  be numbers  F o r example, t h e  Hamiltonian:  ( 2 . 1 9 ) H=. - 2 { J ( S i . S  can to  be p a r t i a l l y  +S .S +S .S 3  1  1  factorised  1 1  )+J23S2.S3+J ,S2.S +J3,S3.S, 2  }  a  b y t h e i n t r o d u c t i o n o f S"  =S +S +S 2  3  a  give:  (2.20)  The  H  =  —  J [ S ' — S" 2  3 spin problem  solved  term,  2.3.2  Examples  The  numbers  factorable  only  simpler  2  —  2J 3S_ .S_3 2  —  2  2 J 2 4S2 . S n  by t h e l a s t  3 terms  and the eigenvalues t o the value  of c l u s t e r s  expression  i s well  S ]  defined  according  isotropically  The  2 —  independently  first  of  2  with  spins  factorable  have  i n terms  appeared  factorable clusters  one n o n - z e r o  coupling  of (2.20)  of the  Hamiltonians  triangle  of intermediate  quantum  examples of  i n the l i t e r a t u r e  o f up t o 6 atoms w h i c h  constant  c a n be  t o those  f o ran i s o c e l e s  known, a n d s e v e r a l o t h e r  Hamiltonians  2j3uS3.Sn  o f S".  of the Hamiltonian  coupled  added  ~  a r e shown  4  0  '  7  7  involve  i n F i g . 2 . 1 . The  Fig.2.1.  Examples  of  small  factorable  clusters.  42 corresponding  Hamiltonians,  i n units  of - J , a r e l i s t e d  below.  S • •. d e n o t e s S . +S .+S, . —ilk — l — j —k  Dimer,  S'  - 2S ;  2  Tetrahedron, Trigonal Three  b) E q u i l a t e r a l  2  S'  S'  - 3S ;  2  2  - 4S ;  2  2  bipyramid,  membered  triangle,  10 e q u a l  chain  S'  c o u p l i n g c o n s t a n t s , S'  - S  2  - S  2 1  3  2  - 5S ; 2  ;  2  c ' 2 _ c 2 _ c2 32 3U I s  3  S' ~ S 3 « 5  ~~ S ;  2  2  2  C' 2 _ c  s  Trigonal Open  2  bipyramid,  trigonal  Square, S'  S*  - s, 3- 2  2  The clusters  atoms  list  bipyramid, - S,  S'  above  in  2 u  - S  2 2  unlikely  into  s i x spins  - S  3  „  - 2S  2  2  ;  5  2 6  ;  physically  reasonable  coupling constant.  interacting 2  2  Some  this  five  that  are possible,  clusters,  but p h y s i c a l l y s i xatoms i n  t h e c o u p l i n g s between a  a r e symmetry  given  e q u i v a l e n t . Example  d)  category.  of t h e above H a m i l t o n i a n s  the conditions f o r factorisation  of a l i n e a r  small  equally with a l l other  as i t i s impossible t o arrange  the Hamiltonian  terms  2 3  H = S' -6S ,  combinations  that  - S  includes only  i n a c o n f i g u r a t i o n such  provided  2  ;  2  2  i n the cluster,  Linear  thus  S'  5  r1  - S  2  as that w i t h  falls  - 3S  2 1  - S 2 4  2 3  atom and t h e r e m a i n i n g also  - S  2  d e s c r i b e d by a s i n g l e  extremely space  S'  ,  Octahedron,  such  2 _ c2 . ^ /  23«56  s  f o r an i s o c e l e s combination  triangle  of that  are allowed  are maintained; c a n be  expressed  f o rthe equilateral  43 triangle  and  complicated along c)  a  and  axis  that  f o r the  examples e x i s t ,  three-fold  an  a combination  factorisation  i s used  tetrahedral  a  discussed be  used  6  of  k)  directly  and  clusters  where  by  suggested  correlation  algebra  theory.  group  Group Theory  and  Group theory  be  of  the  carried  Magnetic  having  provides  the  most  satisfactory  non-crossing  rule  and  clusters.  quantum  results  2.4.1  Magnetic The  point  rotations, not  the  When u s e d  and  much more e a s i l y  four-fold  in  which  for k)  the  above  is  structural  data,  or  a l s o u s e f u l as  known  points  magnetic  spectrum  out  the  with  for  a i d of  to  can test  low  linear  Clusters when a p p l i e d t o h i g h  formalism  theory than  symmetry  r e p r e s e n t a t i o n s , but  i t also  f o r d i s c u s s i o n of  J a h n - T e l l e r theorem,  group  a  model  c)  in conjunction with  number a p p r o a c h ,  exact  along  of  q u a n t u m number H a m i l t o n i a n s  several irreducible  the  distorted  a combination  application  combining  i s most p o w e r f u l  clusters  all  An  by  They a r e  can  and  by  distorted  m).  more  tetrahedron  to define a d i s t o r t i o n  hypotheses.  which  a  represented  octahedron  symmetry c l u s t e r s  2.4  be  i n 5.1.3. I n t e r m e d i a t e  structural between  Cu OX L„  a t o m c h a i n . Many  f o r example,  a x i s can  f ) a b o v e , and  by  three  can  matrix  the  which apply  the  to  intermediate  frequently  provide  methods.  groups group  operate  symmetry  symmetry  elements,  i n o r d i n a r y , or  elements  such  'orbital',  as  reflections  space;  r e q u i r e d f o r d i s c u s s i o n of  they the  and  are spin  44 Hamiltonian. the  Instead,  spin Hamiltonian  operators  will  group'.  The  product  of  be  operators unchanged  referred  permutation  above  f u n c t i o n onto  i n the  considered  The  of  table  the  3  2  1 23 1 1 2  where s t a n d a r d i.e.  the  (12)3  of  the  definition which  to  model as  the  the  such  'magnetic  of  a  magnetic  b r i n g the  spin  definition  electrons  spin Hamiltonian  1.  The  and  in Dirac's  techniques  clusters  5=1/2  will  the  c o n t a i n i n g an  H  = -2J  =  -J/2(2P,  symmetry of  for this p A, A E  of  are  now  S=1  formalism,  for be  which  group  introduced  equilateral  equilateral  and  using  triangles.  triangle  of  S=l/2  exchange c o u p l i n g , J :  (2.21)  has  between  i n Chapter  a cluster  atoms w i t h  Heisenberg  the  set  s u b g r o u p of  is equivalent  8  The  as  is a  usual  i s made e x p l i c i t  treatment  examples  Consider  H  the  used.  thesis  operators  electrons leaving  localised.  groups  introduced  those 7  be  a cluster  g r o u p s . The  itself  relationship  theoretical the  of  t o be  permutation was  case  should in this  m a g n e t i c g r o u p of  space g r o u p e l e m e n t s as density  to  which permute  (S, .Sz+S,.S +S .S ) 3  2  the  + 2P,  3  +  2  2P  3  2 3  permutation  -  3)  group P . 3  The  character  group i s : 3 { (12)3} 1 -1 0  cyclic  changes  the  2{(123) } 1 1 -1  notation numbers  i s used 123  into  f o r the 213,  permutations,  while  (123)  changes  45 123 of  into  2 3 1 . The b r a c k e t s ,  elements  in a class  { }, d e n o t e  i s given  classes  by t h e number  a n d t h e number  outside the  brackets. Consider  the spin  representations,  basis  'I.R.s',  spanned  Similarly  t h e I.R. spanned  the  iaaa),  spanned  by s e t o f s t a t e s  i s A,. T h e g e n e r a l  by s t a t e s  (2.23)  r{s'}  having  a spin  = r{s'}  -  r{s'  {S'} i n c l u d e s z  Therefore  a l lbasis  with  expression  o f S' +U z  z where  by t h e b a s i s a r e :  r{aa/3, aj3a, (3aa} = A, + E  (2.22)  state  {aa|3, a/3a, /3aa}; t h e i r r e d u c i b l e  ,  this  s'  > o z  states  with  a given  contains which  from  the other i s split  the S=l/2 t o C^ , v  symmetry  two, t h e magnetic since  triangle  constants  group  with  i s made  becomes P  t h e I.R., E, c o r r e l a t e s  group  the magnetic  i s smaller elements  have no c o u n t e r p a r t s  magnetic  and  2  with  group.  I t i s also  the point  than  group;  the point  i n the magnetic  i s isomorphic  possible  group,  which i s  group,  ^^h'  which  such as t h e h o r i z o n t a l m i r r o r  convenient t o use the point  than  o f S'. z  2  isomorphic  is  value  3  in P .  2  For  the  f o r t h e I.R.s  r { S ' = 3 / 2 } = A, a n d r { S ' = l / 2 } = E , a n d t h e s t a t e s  degeneracy  A,+A  i . e .  i s :  S'=1/2 a r e d e g e n e r a t e . I f o n e o f t h e c o u p l i n g different  S'=3/2, 2  group.  to the point group  notation  f o r the magnetic  thus the t r i g o n a l  I n many  group;  plane, cases  i f so,  i t  f o r the magnetic  group  bipyramid  t o be l a r g e r with  10  equal  46 coupling 120  constants,  group  elements, whereas  to  P *P ) with  of  n, S=l/2  3  atoms  considered  of  in a full  of  group  electrons  group  t h e S=1  g r o u p s , f^s' equilateral  A, o r b i t a l s  is  T  ^  s  c  a  (isomorphic  group  of  clusters  o n t h e same a t o m m u s t  be  treatment. In this  product of the magnetic  system  with ^  n  triangle.  n  case group  intra-atomic  illustrated  e  with  complication of  theoretical  i s the direct  symmetry,  5  of P .  the additional  t h e c o r r e s p o n d i n g S=l/2  permutation  group  i s always a subgroup  exchanging  magnetic  i t s point  P  12 e l e m e n t s . T h e m a g n e t i c  S>1/2 c l u s t e r s  permutations  the  only  2  For  d) o f F i g . 2.1, h a s f u l l  Let orbitals  by t h e  example  1 a n d 2 be on a t o m  3 & 4 b e o n a t o m B, a n d 5 & 6 o n a t o m C. H c a n b e  expressed as:  (2.24)  where S  H  1 2  leaving  =  -2J (S,.S +S .S^+SB.S )-2J(S, .S „+S „.S 1  2  =S +S 1  2  and  J ' » ' J .  the electrons  (2.25)  3  6  There  on t h e i r  are eight  original  3  3  5 6  +S  5 6  .S, ) 2  permutations  atoms:  123456, ( 1 2 ) 3 4 5 6 , 1 2 ( 3 4 ) 5 6 , 1 2 3 4 ( 5 6 ) , ( 1 2 ) ( 3 4 ) 5 6 , ( 1 2 ) 3 4 ( 5 6 ) , 1 2 ( 3 4 X 5 6 ) , ( 1 2 ) ( 34 ) ( 56 ) .  For  each  of these permutations there  the  a t o m s on w h i c h  unchanged;  2  the electron  f o r the f i r s t  (2.26)  term  pairs  are 6 permutations changing are situated  of (2.23)  a l ltheir  by  H, t h a t  f  12(35X46),  a r e 48 p e r m u t a t i o n s . T h e r e q u i r e m e n t ,  the three  pairs  of e l e c t r o n s  H  these a r e :  1 2 3 4 5 6 , ( 1 5 3 ) ( 2 6 4 ) , (1 3 5 ) ( 2 4 6 ) (13) ( 2 4 ) 5 6 , ( 1 5 X 2 6 ) 3 4 .  In  but leaving  each  remain  imposed co-atomic i s  47 very  similar  octahedron symmetry  t o the requirement  remain  opposite  operator,  thus  octahedron,  0^. F o r e x a m p l e ,  the by  S=l/2 each  involve  the  than  derivation  t h e element  simplification  t h eatoms,  Hamiltonian  a s t h e 'elementary'  some i n f o r m a t i o n a  set  one  IS 1  than  i nt h i s Clearly  which  case t h e some  i st o be used  for  more  c a nbe a c h i e v e d by  theelectrons  particles  o f atoms r a t h e r  spin  ofthe spin  t o be permuted.  than  i nparticular, states  if P  t h egroup theory  S S z, z z  which  electrons are  a n d more m a n a g e a b l e g r o u p s a r e o b t a i n e d b u t  I.R. F o r example,  2  rather  i slost;  ofdegenerate  triangle  b u teven  Such a s i m p l i f i c a t i o n  smaller  spanned  states  S>1/2 c l u s t e r f o r  i f t h e method  considering  considered,  t h e I.R.s  t oexclude  results i stedious.  When p e r m u t a t i o n s  t o that f o r  S=1, a t o m s .  i snecessary  systems.  group of t h e  procedure  to find  must be t a k e n  group  (12)3456 corresponds t o  i n 0^. A s i m i l a r  i sn o n - t r i v i a l ,  ofuseful  of an  t h e magnetic  t othepoint  S=1 t r i a n g l e i s t h e s i m p l e s t  group theory  complex  plane  though care  S=0, r a t h e r  The  i sisomorphic  s y s t e m c a n now b e f o l l o w e d  spin,  corners  i t i sn o t s u r p r i s i n g that  discussion  horizontal mirror  opposite  on a p p l i c a t i o n o f a p o i n t  group under  a  that  3  will  rather  i t i sp o s s i b l e  be r e p r e s e n t e d  than 0^ i sused  a n a l y s i s , using  basis  that  b y more  than  f o r t h e S=1  kets  >i s :  3  r { S ^ = 3 } = r { | 1 1 1 > } = A,  r{s^=2} = r { | 1 i o > , | i o i > , | o i i > }  (2.27)  = A, + E  r{s^=i} = r { | i o o > , | o i o > , | o o i > , | 1 i - i > , | 1 - 1 1 > , | - 1 1 1 > } =  2A,+2E  r{s'=o} = r { | i o - i > , | i - i o > , | o i - i > , j o—11>,|-iio>,|-ioi>,|ooo>}  48 =  2A,  + A  2  + 2 E .  r{s'=3}=A,, r { s ' = 2 } = E , 3-fold  The  cannot  degeneracy  r{s'=1}=A!+E,  o f t h e S'=1  be r e p r e s e n t e d b y t h i s  r{s'=o}=A . 2  states  subgroup  under  this Hamiltonian  of the full  magnetic  group. The becomes groups  advantage feasible  which  the  full  The  magnetic  thus  magnetic  2.2  objects  t otreat  group  7  9  ;  groups w i l l  n  t h e I.R.s spanned  useful  f o rcalculations  tables  between  be  used  cluster  t ofind with  be u s e d  f o r any c l u s t e r  with  o f P«,P  i sthat i t  o f up t oS=5/2,  i n the rest  containing 5  and P  n=6, with  rather  of t h i s  than work.  n metal atoms 6  i s  are given i n  by a l l p e r m u t a t i o n s of sets o f  thecharacter  tables,  such as ( 2 . 2 7 ) ,  these permutation groups t h e I.R.s spanned  up t o 6 magnetic  group  o f S. The 'atom b a s e d '  ofP . Details  arelisted  magnetic  a l lclusters  a r e independent  a subgroup  Table  o f t h e 'atom b a s e d '  above.  Correlation  and their  by t h e s p i n  atoms.  these l i s t s a r e  subgroups can  states  of any  49 Table Pa A, A E T, T 2  2  2.2 C h a r a c t e r T a b l e s f o r t h e g r o u p s 1234 1 1 2 3 3  8{ 1 ( 2 3 4 ) } 1 1 1 0 0  -  6{12(34)} 1 1 0 -1 1  Pg,P  5  and P . 6  6{(1234)} 3{(12)(34 ) } 1 1 1 1 0 2 1 1 1 1  r { a a a a } = A,, T { a a a b } = A , + T T{aabb} = A,+E+T , r { a a b c } = A 1+E+T^+2T , r ( a b c d } = A,+A +2E+3T,+3T 2 f  2  2  1 2345 A, A G, G H, H I 2  2  2  1 1 4 4 5 5 6  2  2  15{1(23)(45)} 20{(12)(345)} 24{(12345)} 10{123(45)} 20(12(345)} 30{1(2345)} 1 1 1 -1 1 -1 2 0 0 -2 0 0 1 1 -1 -1 1 1 0 -2 0  r{aaaaa}=A,, r{aaaab}=A,+G,, r{aaabb}=A,+G,+H,, r{aaabc}=A,+2G +H,+I, T{aabbc}=A,+2G,+2H,+H +1, r{aabcd}=A +3G,+G +3H +2H +3I, r{abcde}=A +A +4G,+4G +5H +5H +6I. 1  2  1  2  1  2  1 23456  1  2  2  1  2  45(12(34)(56)}  15(1234(56)}  2  2  3  2  Mi  M,  1 1 5 5 5 5 9 9 1 0 10 1 6  T(aaaaaa} : Haaabbb}: T(aaaabc} : T(aabbcc}: r(aaabcd} : r(aabbcd};  1 •1 3 •3 1 •1 3 •3 2 2 0  144(1(23456)}  1 5 { ( 1 2 ) ( 3 4 ) ( 5 6 ) } 4 0 ( 1 2 3 ) ( 4 5 6 ) 120 ( 1 2 3 4 5 6 ) }  40(123(456)} A, A H, H H H» L, L  120{1 (23 ) ( 4 5 6 ) }  1 1 2 2 •1 •1 0 0 1 1 •2  90(12(3456)} 1 •1 •1 1 •3 3 3 •3 2 2 0  -2 -2 0  90((12)(3456)}  1 •1 0 0 1 -1 0 0 •1 1 0  1 1 •1 •1 2 2 0 0 1 •1 •2  1 1 -1 -1 -1 -1 1 1 0 0 0  1 1 0 0 0 0 -1 -1 0 0 1  1 -1 -1 1 0 0 0 0 1 -1 0  !, T ( a a a a a b } = A , + H , r(aaaabb}=A +H,+L , !+H,+H +L! , +2H,+L,+M , T ( a a a b b c } = A +2H +H +2L +M,+S, +2H +H +H«+3L +M +2S, +3H +H +3L +3M,+M +2S, +3H +2H +H,+4L +L +3M +M +4S r ( a a b c d e } = A + 4 H + H + 3 H + 2 H + 6 L + 3 L + 6M +4M +8S, T ( a b c d e f } = A + A + 5 ( H + H + H + H ) + 9 ( L , + L ) + lO(M,+M ) + 1 6S, l  1  3  1  1  1  1  3  1  1  3  1  1  1  1  1  2  1  1  1  1  3  1  1  2  3  2  3  1  2  2  I t  3  1  1  I (  2  2  1  2  2  2  50 2.4.2  An  example  The  group  above  d i d not  other  methods.  Hamiltonian all  theory lead  of group  f o r the three  t o any  results  I t i s obvious  of  from  t h e same t o t a l  example  together  give  will  be  now  t h e use  f o r the e q u i l a t e r a l  states  useful  of  an  exact  atom  which could  discussed  n o t be  the expression  of  as H = - J ( S  , 2  obtained the  and  -3),  without  factorable  resort  by  spin that  a r e d e g e n e r a t e . A more  theory  result  systems as  triangle  spin  i n which group  theory  clearly  Hamiltonians  to matrix  methods,  discussed.  Consider: (2.28) H(e)=-2[(S .S +S3.S )cose_+(S .S3+S .S )cos0+(S .S +S .S )cose ] l  2  0 ,8_  where  =  +  discussion For  6±2ir/3.  6,  2  1  This  of d i s t o r t i o n s  general  symmetry  a  a  n  d  ^  c  a  of  ^  n  1  Hamiltonian  symmetry  given D ,  2  3  +  i n the  i n Chapter  while  for  Consider  the  2  factorised.  e  4  i s important  tetrahedra  H has magnetic  i s ^>2d  u  4.  0=n7r/3  Hamiltonian:  (2.29)  where S  H(0)  1 3  =S +S 1  (2.30)  3  =  -2[S,.S +S .S  =  1/2  (S' -4S )  =  1/2  (S'  and  S  3  2  2  2 a  - 3S,  =S +S . 2  u  2  -  2  2  -1/2(S,.S +S .S3+S3.S +S«.S,)]  f t  (S,  3/2  - 3S «  2 3  2 3  2  2  H  H(0)  = - H ( 0 + TT)  H(d)  = P[(13)(24)]  has  +S « -4S ) 2  H(0)  2  2  +  8S ) 2  the f o l l o w i n g  U(-6)  H ( 0 + 2TT/3) = P [ 1 ( 2 3 4 ) ]  r  2  symmetry:  the  51  S=l/2; f r o m ( 2 . 2 9 ) , t h e m a g n e t i c s p e c t r u m o f t h e S'=1  Consider  states i s : (2.31)  |111> = - 2 ,  where t h e e i g e n s t a t e s  |110> =  1,  are labelled  spectrum  for a l l 0=n  consider  t h e I.R.s spanned  7 r / 3  ' |101> =  by | S ' S  c a n be f o u n d  from  1 3  S  2 U  1.  >.  (2.30)  The m a g n e t i c a n d ( 2 . 3 1 ) . Now  by {S'=1}; t h e n e c e s s a r y g r o u p  tables  are: 2 A, B, B B  D  1234  3  2d A, A B, B E  (13)(24)  (14)(23)  1 1 -1 -1  1 -1 1 -1  1 -1 -1 1  1 1 1 1  2  D  (12) (34)  1 234  2{(1234)}  1 1 1 1 2  2  2  labelling  has  been chosen  1 1 1 1 -2  1 - 1• 1 -1 0  f o r t h e I •R.s, a s w e l l t o show c l e a r l y  The c o r r e l a t i o n  (2.32)  D  D For  2{(12)(34)}  1 1 -1 -1 0  The  groups.  (13) (24)  2  table  A,  B,+B  I  2 d  2{(13)24} 1 -1 -1 1 0  a s t h e names o f t h e isomorphism with p o i  the  f o r t h e two groups i s : B  3  2  I  I  E  A ,B , 2  2  A,,B,  D , the analysis i s : 2  (2.33)  r{S'=2}=T{|aaaa>}=A,  T{S^=1 } = T{ | aaa(3> , | aa(3a> , j a(3aa> , |/3aaa>} = A + B + B + B 1  r{S'=0} = r{aa/3/3,a/3aj3,ai3/3a,/3aa/3,/3a)3a,/3l3aa}  =  1  2  3  SA^B^Bj+Ba  'Zi Therefore of  r{S'=l}  = B +B +B , 1  2  3  H are constant as 8 varies.  f o r a l l 8,  and the eigenfunctions  The e i g e n f u n c t i o n s  c a n be f o u n d  52 explicitly  with  (2.34)  projection  operators,  denoted  =  aaafi-a(laa+aa(3a-fiaaa  P ( B ) (aaa/3) =  aaafi+afiaa-aafia-fiaaa  P(B )(aaa/3) 1  2  P(B )(aaa|3) 3  by P(r),  = aaa/3-a/3aa~aa/3a+/3aaa  H ( 0 ) + H ( 2 7 r / 3 ) + H ( - 2 7 r / 3 ) = 0; t h e r e f o r e , f o r e a c h the  eigenvalues  (2/3) [H(O)cos0  (2.35)  f o r these  3 Hamiltonians  + H(2TT/3)COS6»_  spectrum,  together  f o rgeneral  result  which  factorisation degenerate other to  cannot  + E ( 2TT/3 ) c o s 0 _  imply  that  form  transform  rule  t h e B,+B  a s A, i n D  to find  2  +  t h e S'=1  magnetic  n=1,2,3  by e i t h e r  t h e B, a n d B  2  or the B +B 2  some  non-crossing  eigenvalues  solutions  3  the group  3  theory or  s t a t e s become group,  ^2d'  ^  o  r  I.R.s combine  o f H; b o t h  and i t i s t h e r e f o r e  from  t o t h e magnetic  The  + E (-2TT/3 ) c o s 0 ]  a s 'E' i n t h e e n l a r g e d  t h e S'=0  eigenfunctions  possible  and so:  t h e d e g e n e r a t e I.R.  Now c o n s i d e r  the  be o b t a i n e d  and transform  0 = n7r/3, e i t h e r  Further, H(0)=  6, i s :  F o r 6=mt  alone.  i s zero. +  ( 2 . 3 6 ) E ( 0 ) = -2 cos(6» + 2n7r/3)  a  I . R . t h e sum o f  + H (-2TT/3 ) c o s f l ]  E(0)= (2/3) [E(O)cos0  (2.35) and (2.31)  thus:  group  theory  functions  impossible  alone;  however,  i n f o r m a t i o n by a p p l y i n g  to define i ti s s t i l l  the non-crossing  Hamiltonian.  rule  a p p l i e s t o the spectrum of  o f any H a m i l t o n i a n  and s t a t e s ,  simply,  that the  53 eigenvalues cannot  associated  cross  Consider  symmetry  and consider  H  H  1 l f  1 2  basis and H  a space  for this  2 2  variable  parameter  theprobability  variable never on  parameter  degenerate  As are  eigenfunctions on  cross.  next  using  have  i n E^^, where  thematrix =H  and H  2 2  elements  1 2  =0 a r e  t o choose t h e  of these  conditions  a t t h e same v a l u e  rule  6.  parameter  occur  8 0  they  transform  Further  algebraic  that  H  or  .  t h e same s y m m e t r y  o f t h e S'=0 e n e r g i e s  the linear  depends  i f non-adiabatic  i nD  2  a s A,+B,,  H ( 0 ) +H ( 2 it / 3 ) +H ( - 2 7r/3 ) = 0 , t h i s with  of the  and theassumption down  hold  the states are  The n o n - c r o s s i n g  i t may b r e a k  must v a r y  chapter.  1 1  o f H i n some  therefore  approximation  i n thevariable  Since  thevariation  obtained the  occurring  t h e S'=0 s t a t e s b o t h  cross.  Both H  one o r o t h e r  and cannot  non-degenerate  cannot  In general,  i sinfinitesimal,  changes a d i a b a t i c a l l y , changes  b y t w o s t a t e s o f t h e same  i t may b e p o s s i b l e  of both  symmetry  I t i sderived as  representation  space.  so that  t h e Born-Oppenheimer  'sudden'  spanned  a r ea l l non-zero.  f o r degeneracy;  o f t h e same  i nH i svaried.  a matrix  required  but  eigenvectors  as a parameter  follows.  arbitrary  with  and they  shows t h a t t h  information  with  results  8 i s most  easily  which a r ederived i n  54 CHAPTER 3 3.1  DYNAMIC D I S T O R T I O N S I N M A G N E T I C C L U S T E R S ;  A Formalism  A  formalism which  effects  of c o u p l i n g  coordinate,  expanded  smooth  be  (3.1) for  system The in  now  i  j  (  d  i  i s i n magnetic  detail;  )  j  of  which  of  be  shown  long  Taking the  clusters  magnetic on  a  distortion  a s J^_j i s a  the d i s t o r t i o n ,  i t can  first  be  two  by:  - J..(0)  a  +  (9J/9d)  well  the  depend  d e v e l o p e d . As  unless as  of magnetic  to discuss  in d—.  the d i s t o r t i o n  i t will  combination  be  series  distortions,  n a t u r e of  used  approximated  J  small  be  function  as a power  J ^ j can  can  distortions  constants,  d^_j, w i l l  sufficiently  terms,  for describing  TRIMERS.  as  0  i j  i j  = 0,  /3d  i j  )o  i n which  electronic/steric  coordinate, that  the v i b r a t i o n a l  oj  d-.,  i t i s best  normal  will  case  the  equilibrium. be  discussed  interpreted  c o o r d i n a t e s of  as  a  the  cluster. 3.1.1  The The  configuration configuration  described can  be  by  the  s e t of  interpreted  s p a c e , C, C can  a  be  bond  linearly  as  and of  distortion  a cluster  3m-6  a configuration  in a variety  space  (3.2)  C_  =  (C I 2  interatomic  1 3 (^2  of  of ways  l e n g t h s o r a n g l e s ; one independent  containing  m atoms can  orthogonal coordinates,  the c o n f i g u r a t i o n chosen  spaces  3 )  vector,  c,  the c l u s t e r . involving  {c},  in a The  which  vector basis  combinations  s i m p l e c h o i c e i s t o use distances,  be  e.g.  for  a  m=3:  for of  s e t of  55 In  this  work,  denoted  small distortions  c(0)=c ,  of the c l u s t e r  o  Distortions D, c l o s e l y  may  be c o n s i d e r e d  related  t o C by  The v i b r a t i o n a l of  basis  i n D;  distance  containing  as v e c t o r s  n o r m a l modes p r o v i d e  (3.3)  four  may b e s i m p l i f i e d distortions  atoms  i s represented  = (d  the point  1 2  C  ,d  1 3  ,d  l f l  ,d  2 3  ,d  2 a  ,d „)  group of the tetrahedron,  projection operators, i . e .  P(E)(d P(E)(d  =  3  found  2  cluster  (1,1,1,1,1,1)  =  2  f o r the normal  P(E)(d, )  of a  by  by:  expressions  coordinates  = ( 1 , 1 , 1 , 1 ,1 , 1 ),  2  choice  in a tetrahedral  C  2  P(A,)(d, )  o  an a l t e r n a t i v e  C = ( C'1 , C| 3, C ) j, C 3 r 2 4 » 3 4 )  using  d(0)=d =0.  In interatomic distance coordinates the  cluster  d  space,  the interatomic  consider  equivalent  in detail.  in a distortion  d = c - c ( 0 ) ; thus  As an e x a m p l e ,  just  undistorted  2  are considered  t r a n s f o r m a t i o n between  configuration.  T ,  t h e mean c o n f i g u r a t i o n ,  b a s i s and normal c o o r d i n a t e s  group theory.  In  from  (0,0,0,0,0,0) T{d} = A,+E+T , 2  in this  b a s i s c a n be  (3.4)  = ( 2 , - 1 , - 1 ,-1 , - 1 , 2 ) ,  1 3  )  = (-1,2,-1,-1,2,-1) = E  1 f t  )  = (-1,-1,2,2,-1,-1),  v  E =  E(d  y  1 2  -d  1 4  )  =  (1,0,-1,-1,0,1)  P(T )(d, )=(1,0,0,0,0,-1)=T  , T (d, )=(0,1,0,0,-1,0)=T ,  P(T )(d )=(0,0,1,-1,0,0)=T  .  2  2  2  3  y 2  1 4  The l a b e l s degenerate ^  x,y,z,  indicate  n o r m a l modes.  E  orthogonality within the  ,E , a n d T ,T ,T , t h u s x'y' x ' y ' z '  form a  56 Cartesian often  basis  be c o n v e n i e n t  coordinates; represented  t o work  d(E)  normal  =  r  ,cosi//, cosi^  (cos\p_  space  2  c l u s t e r of four using  energies  t h e above  associated  full  notation.  i svalid  with,  2  3  3m-6 d i m e n s i o n a l to discuss  Cu (CH COO)„,  +  ,cos\p,  However  only  cos^_  )  coordinates:  as a d i s t o r t e d  the concept of  f o rsmall  distortions for  as a harmonic  say,the T  2  distortion  indetail,  even  3m-6=84. H o w e v e r m o s t  i n t e r e s t when c o n s i d e r i n g  bending  of  , cos\p  oscillator;  modes, w i l l  be  equal  f o rsmall d i s t o r t i o n s .  too large  may  +  a t o m s c a n be e x p r e s s e d  v i b r a t i o n a l modes  The  no  o f D may be  by s p h e r i c a l p o l a r  w h i c h t h e c l u s t e r c a n be c o n s i d e r e d  is  Cartesian  2  tetrahedron  only  than  d(T )=p(sin0cos0,sin0sin0,cos0,-cos0,-sin0sin0,-sin#cos0)  Any  the  rather  d i s t o r t i o n s i n t h e E subspace  distortions i n the T  (3.6)  i n polar,  space. I t w i l l  by:  (3.5)  and  f o rsubspaces of t h e d i s t o r t i o n  be i g n o r e d .  magnetic  a l l t h e atoms  necessary  I t might  involved  t o provide  exchange;  distortion  on t h e m a g n e t i c  below  the formalism  that  non-metal  3  groups  exchange  modes a r e o f  i n copper  acetate  consideration  pathways  w o u l d be  description of the effects of  exchange;  however,  i smathematically  atoms a r e ignored.  dimer  f o r example, t h e  be t h o u g h t t h a t  i n possible  an adequate  f o rt h e small  of t h e normal  a n d s t r e t c h i n g modes o f t h e C H  clearly  space of a c l u s t e r  i t will  adequate  Thus f o r a l l c l u s t e r s  be shown  even  i f a l l  containing  57 four  equivalent  tetrahedron  in  have  as A  E, a n d T .  1 r  no e f f e c t  In real  2  indicate will,  importance This  behavior  which  important  directly  the magnetic distortion  Equation  0  H  0  the relative as t h i s  I.R.  possible to  data.  The e f f e c t  discussed  using  of  distortions  the concept of  Hamiltonian  (3.1) r e l a t e s  the magnetic  c a n be r e p r e s e n t e d  H(d) = H(0) + d  (9H/3d)  Hamiltonian,  s p a c e , V, w h i c h  0  interaction  The H a m i l t o n i a n  = H  linearly  has only  as the o r i g i n  0  as a vector.  independent by  components,  ( 3 . 1 ) , H', t h e  of a d i s t o r t i o n  The l i n e a r  f o rthe  + H'  3n-6 i n d e p e n d e n t  i s 3n-6 d i m e n s i o n a l  Hamiltonian  as:  of t h e c o n s t r a i n t s imposed  c a n be s p e c i f i e d  interpreted  may  Hamiltonian.  but because  distortion  behavior  a r e o c c u r r i n g , however i t  i t i s never  i s best  i s a f u n c t i o n of n ( n - l ) / 2  {J},  i n t h e same  modes t r a n s f o r m i n g  from magnetic  behavior  system  (3.7)  distortions  of the magnetic  the c o n f i g u r a t i o n of the c l u s t e r .  distorted  H  since  modes  of s e v e r a l p o s s i b l e exchange mechanisms i s  3.1.2 T h e d i s t o r t i o n  to  the magnetism  modes  o r T,  2  while  t o determine  of t h e normal  i s expected  determine  the  investigation  be i m p o s s i b l e  of each  6 distortion  behavior,  t h a t , s a y , E mode d i s t o r t i o n s  i n general,  only  of a  Distortions within A  on t h e m a g n e t i c  clusters  at the corners  to consider  any o f , s a y , t h e E modes, a f f e c t  way.  on  atoms a r r a n g e d  i t i s necessary  transforming can  metal  coordinates. Hamiltonian  a n d i n w h i c h H' c a n b e relationship  (3.1) a l s o  58 means t h a t point  the magnetic  group.  As mentioned  3m-6 d i m e n s i o n a l , in  V will,  When o n l y in  since  normal  as  distortion coordinate  to several  by t h e m e t a l  i s3n-6, a n d there  by c o n s i d e r i n g  by m a g n e t i c  only  distortions  i s necessary  of the c l u s t e r f o rdiscussion  core  degrees  system) degrees  From  no  of the metal  of the metal  but represent  core as  core  are not the  them a s a d e q u a t e l y  of the magnetic  Hamiltonian.  t h e number o f  the n(n-l)/2  o f H a n d t h e3n-6 ( 3 n - 5 f o r n=2, a  (3.5)and (3.6),  vectors  tetrahedron,  information  'linear'  o f f r e e d o m o f H' d o e s n o t a r i s e .  (3.1),  Hamiltonian  i n D.  one-to-one  that  o f {J}and t h e d i s t i n c t i o n between  of freedom  of a basis  are considered  When n<5 t h e n u m b e r o f n o r m a l m o d e s e q u a l s components  s p a c e D, i s  s u s c e p t i b i l i t y measurements i s  t o D. T h e d i s t o r t i o n s  distortions  toa  coordinates  i s a simple  b e t w e e n D a n d V . I t c a n now b e s e e n  contributing real  thef u l l  correspond  modes d e f i n e d  w h i c h c a n be o b t a i n e d lost  above,  be i s o m o r p h i c  3m-6 > 3 n - 6 e a c h  i ngeneral,  D i t s dimension  mapping  g r o u p o f H' m u s t  t h es e t of d i s t o r t i o n  i n V c a n be d e f i n e d .  ignoring  t h e A, d i s t o r t i o n  Thus which  for the c a n be t a k e n  into  H : 0  (3.8)  Hjj,(r,\J/) = - 2 J r  [ (S , . S + S .S« )cos<//_+ (S , . S + S 2  3  3  2  .S„  )cos\[/  +(S,.S«+S .S )cos(^ )] 2  H^{p,e,<j>)  = - 2 J p [ (S, . S - S 2  3  3  .S, J s i n f l c o s ^ + J S ! . S - S 3  +  2  .S )sin0sin0 4  + ( S , . S « - S , S )cos<9) ] 2  where  r,p=(d/J)(9d/9J)  orthonormal  basis  0  f o r t h erelevant  i n H' c a n b e d e f i n e d  normal  from  3  coordinate.  (3.1)and (3.4)  An and  59 Hamiltonians, basis. in  I t should  general, One  (9J/9d) normal  0  be n o t e d  immediate  mode. T h i s  vibrational  (9J/9d)  0  of using  i s because  energy  are tensors  0  i s also  vector,  D and V  i s that  degenerate normal  independent  mode  of the  t o be i s o t r o p i c . this  implies  force  Since  sensitivity  of the d i s t o r t i o n  that  0  coordinates  when e x p r e s s e d i n , s a y , results  from d i f f e r e n c e s i n t h e  Hamiltonian  vector  to  distortions  modes. ( 3 . 1 ) i s t h u s b e t t e r  expressed  as: (3.9)  where  The  {  =  i n t h e sense last  i n which  space w i l l by  i  J (0) i  + d  i  a n d d ^ a r e now g e n e r a l i s e d  cluster  space  J (d )  T. T  vector  defined space  (dJ /dd ) i  i  0  normal  coordinates  of the  above.  w h i c h must  be d i s c u s s e d  i s the spin  s o l u t i o n s of H a r e found. A n o t a t i o n  now b e d e f i n e d .  k  w i t h i n a d e g e n e r a t e mode. Any  i n (9J/9d)  normal  than  i . e . the vibrational  by s y m m e t r y  distance  different  rather  w i t h i n any degenerate  interatomic  within  in this  i n V do n o t ,  w i t h i n each  o f t h e same k i n d  isotropic  occurring  to describe  i s , by d e f i n i t i o n ,  k, i s c o n s t r a i n e d  anisotropy  vectors  normal,  t o be i s o t r o p i c  of the d i s t o r t i o n  (9J/9d)  orthogonal  coordinates  i s constrained  constant,  that  advantage  distance,  orientation  and  i n V c a n t h e n be e x p r e s s e d a s v e c t o r s  commute.  interatomic  the  H',  The t o t a l  spin  space w i l l  i s a)(n,S,0) o r c j ( n , S , l / 2 ) d i m e n s i o n a l  (2.1) and (2.2) ) and i s , i n g e n e r a l ,  (u  completely  for this be  denoted  as defined i n unrelated to  60 C,D it  and V  i n t r o d u c e d a b o v e . A s H c a n be s o l v e d b y b l o c k i n g b y S'  i s convenient  to define  subspaces,  R(S'),  of T by:  ( 3 . 1 0 ) R ( S ' ) = {|>//>eT: S' |^> = S' ( S ' +1 ) | ^> , f o r a l l |i//>eR(S')} 2  R(S')  c o n t a i n s a l l s t a t e s of t o t a l  dimensional. considering  T h e R ( S ' ) may  transforming  Consider  0  + H'  H  H'(7r/2)  2  distortion  group. in  they  H'{8)  occur  which Like  c a n be t a k e n any other  orthogonal  as zero  vector  3  common  t o H'(0) and  H . 0  in  group  Such  Heisenberg  i s a subgroup of the magnetic  as a vector  of c o n s t a n t  c a n be e x p r e s s e d w i t h i n each space,  magnitude  as a t r i v i a l  subspace  constant  R ( S ' ) o f T.  H' c a n b e e x p r e s s e d  i n an  R as:  (3.12) H'(0) = H'(O)cos0  Let  P  i n a 2-D  b a s i s spanning  of the magnetic  commutes w i t h  are relatively  0  8, ( i i )  vectors corresponding  whenever  o f V. H  +H'(0)  by p e r m u t a t i o n s  c a n be r e g a r d e d  an E subspace  by  formalism  i n the variable  a n d ( i v ) H'  Hamiltonians  reduced  form:  2  i n 2ir  the d i s t o r t i o n  are orthogonal  clusters;  of t h e  i s (i) periodic  , (iii)  0  Hamiltonian  = -J(S' -nS )  H' (±0±2n7r/3) a r e r e l a t e d of  be f u r t h e r  of s t a t e s w i t h i n R(S')  of d i s t o r t i o n  a Hamiltonian  (3.11) H = H  i t i s R(n,S,S')  r.  a s t h e I.R.,  3.1.3 An a p p l i c a t i o n  w h e r e H' (9)  sometimes  Q(S',D,  subspaces,  s p i n S';  + H'(7r/2)sin0  {X} b e t h e s e t o f e i g e n v a l u e s  o f H' a n d c o n s i d e r  a  matrix  61 representation given  R(S').  elements basis.  of this  an  o f H'(0)  Assume  that  Since  t h emagnetic  X(0)=X(-0)  the Hellman-Feynmann | \p>  is  degenerate,  of  R i szero  in  which H'(7r/2)  expressed  Now  elements  i se n t i r e l y  H ' ( 0 ) . . = a- •[  of  spectrum  o f H'(0)  i s  0 a t 9= 0,  dX/d0=  and (3.12),  but,  t a k i n g X=0 a n d  = «// | H' ( 9 + v/2 ) \ <//>  H'(7r/2)  always  = 0  a r ea l l zero.  thedegenerate  I f H'(0)  subspace  p o s s i b l e t o choose  o f f d i a g o n a l . H'(0)  a basis  c a n now b e  ( 1 - 6 . . ) s i n 0 + 5. . c o s 0 ]  theeigenvalues  I H — X I j =0 c a n b e e x p r e s s e d is  theorem  dX/d0 over  and i t i ss t i l l  consider  then  <^|H'|^>  t h e sum'of  as:  Off-diagonal  of H(0):  ( 3 . 1 3 ) d X / d 0 = 6/69 so thediagonal  8 1  a  proportional to sin 9 i n this  a r ec l e a r l y  first  eigenvector  and  i n a b a s i s spanning  Choose a b a s i s d i a g o n a l i s i n g H'(0).  non-degenerate. from  Hamiltonian  o f H'( 0) . T h e  as a c h a r a c t e r i s t i c  determinant  polynomial  which  a f u n c t i o n o f 0: (3.14) a a a  cos0-X sin0 sin0  1 1 1 2 1 3  • •••  where {X}  a a a  1 2 2 2 2 3  sin0 a s i n 0 .. cos0-X a sin0 sin0 a cos0-X 1 3  2 3  (3.14).  c,(0)  ...  ••••  The e i g e n v a l u e s  of thepolynomial.  symmetry  0 =  \  k  + c , ( 9 ) \  k  1  +c (0)X .? k  2  3 3  ••••  k=fi(S').  =  +c.(0).  o f H' a r e now t h e s e t  The form  i n 9 impose q u i t e s t r i c t The c o e f f i c i e n t s ,  +c, , ( 0 ) X  of t h ematrix  H(0)  of roots  and the  c o n d i t i o n s on t h e p o l y n o m i a l  c(0), will  now b e c o n s i d e r e d  = Tr[H(0)], but Tr[H(0)]=Tr[H(0)]cos0  i nturn.  and Tr[H(0)]  =  62 T r [H( 0 + 27r/3) ] , s i n c e  ZX  c  = 0 for a l l  only  ( 8 ) ;  2  this  therefore zero.  Z  c = 2  q=0;  Thus  result the  terms  dependent The  cos20  2  c  can  matrices  +q  c  2  obtained  An  by  X-X.  inserting  Z X-X. i<3  By  and  therefore  sin 0  Therefore  c,  contribute  2  f u n c t i o n can  be  =  i n 36  therefore  a l t e r n a t i v e expression  (IX) 8=0  in 8  i s even  and  9=0,  since  i s zero,  2  into  but  (3.14)  to  e x p r e s s e d as  the polynomial  considering  i s zero,  0.  similar.  and  2  i s periodic  eigenvalues, =  cos 0  but  i s constant.  be  on  resulting  sin20,  also  2  are  8.  coefficient.  c =const+p  these  (ZX)  i t can  this  the  =  2  p is  of  ZX,  and  sum  of  Z(X )+ 2  be  seen  that  that:  J  (3.15)  c (0);  only  3  c . 3  These  even. the 9=0,  c (6)  -  terms  of  2  can  be  c  3  the  since  sum  Z(X ) 2  form p c o s 0 + q s i n 8 c o s 6 c o n t r i b u t e 3  by  p'cos30+q'cosd,  unless  to  2  the polynomial  i s 9 dependent  i s the  = -1/2  represented  C l e a r l y q'=0  spectrum  constant  since  is periodic  p'=0.  I t can  of a l l p o s s i b l e d i f f e r e n t  be  c  is  3  i n 30 seen  products  and  that  of  for  three  eigenvalues.  C,  c ; 5  Following  c ( 8) = c o n s t a n t  c  6  { 6 ) .  the  as above  the r e s u l t s  5  6  and  higher  coefficients  becomes more c o m p l i c a t e d ;  thus  are  involved  6  a  2  2  u  and  for c : 6  c =pcos 0+qcos 0sin 0+rcos 0sin 0+tsin 0 = 6  are:  c ( 8) = p " c o s 3 0 .  I f 0(S')^6, c  situation  (3.16)  and  t h e same r e a s o n i n g  6  p'cos60+q'.  63 and  there It  are  can  be  determined greater  two  coefficients  seen  i f the  fi(S'),  known. F o r  that  spectrum  two  or  fl(S')<6  fi(S')=2.  X -const=0,  B(S')=3.  X  fi(S')=4. fi(S')=5.  2  6,  values case 3.2  X(0) ]X  2  X  -  1/2[Z  X(0) ]X  3  3.2.1  2  harmonic  Jahn-Teller  i s independent  the  - c cos30  X  - c cos30 =  - c«X  2  levels for a  this  of  i s now  Jahn-Teller Chapter  i s not  period of  Magnetic  Jahn-Teller  5  given  give  instability  of  c l u s t e r s using  theorem as  the  distortion  originally  stated  a  7 1  of  arise.  Clusters to  systems  modification  the  group  3.1.  The  of  theory  the  original  the of  Jahn-Teller  i s :  of  with  degeneracy  " a l l non-linear molecules are unstable for an o r b i t a l l y d e g e n e r a t e e l e c t r o n i c s t a t e "  Some s l i g h t  less  the  27r/3 may  detailed discussion  formalism  is  0  adjacent  necessarily  theorem  to  S'  between  l e s s than  possible  2 and  0  0  3  ft(S')  =  X - c „ =  magnetic It  be  8  of  3  3  activity  A p p l i c a t i o n of  must  for  - c cos30  number o f  terms  8;  of  spectra  change m o n o t o n i c a l l y  larger  be  are:  2  2  i f the  value  -[X,(0)X (0)X (0)]cos30  2  eigenvalues For  one  unrelated  results  spectrum  l/2[2  since The  the  -  8.  f o r any  can  (3.17)  X"  the of  explicit  X(0) ]X  that  a l l coefficients  i s known  1/2[I  is clear  than  fl(S')<6  -  5  determined.  for a l l 8  X =  3  be  more symmetry  the  0(S')=1.  It  0,  if  to  theorem  is  64 desirable This may  t o make  i s because  the application  the magnetic  group  because  functions orbital here  the states  involved  functions,  only  with  as d i s c u s s e d  the orbital  considered  as a whole.  orbitally  considered molecular  first  orbitals  the spin  place,  possible  resulting  by  matrix  since  orbital  theorem  elements  states  a n d H'  will  that  w h i c h may  include  of which  when  orbitals  produce  n  distortions be i m p o r t a n t i n  t o be v a l i d  i n Chapter  i n the  1.  |a> a n d  i s a perturbation  the degeneracy  and T e l l e r  e l e m e n t s may  the t o t a l l y  b y H',  be z e r o o n l y  be n o n - z e r o  only  elements are  and lowers t h e  showed u s i n g  I.R.s spanned  |b> a r e  of the H a m i l t o n i a n  Unless a l l such matrix  a l lpossible  system  i s p r o v e d by c o n s i d e r i n g a l l  splits  o c c u r . Jahn  <a|H'|b> may  integral  atoms each  degeneracy  <a|H'|b>, w h e r e  from d i s t o r t i o n .  when a  atoms i s  Jahn-Teller  fields,  and  c a s e , when a l l t h e a t o m s  the n atomic  ligand  and  are concerned  which a r i s e s  has n - f o l d  been d i s c u s s e d  a l l such matrix  molecule. the  have  considering  that  nor  1. We  Hamiltonian representation  a distortion  symmetry  spin  non-degenerate  of equal energy.  Jahn-Teller  degenerate  zero  as a whole,  i n Chapter  of n independent  non-degenerate  the individual  The  of the c l u s t e r ,  are neither  In the l i m i t i n g  independent, a system  causing  group  degeneracy  a number o f o r b i t a l l y  within  apparent.  b u t a n t i s y m m e t r i s e d sums o f p r o d u c t s o f s p i n  containing  are  clusters  of the d i s t o r t i o n Hamiltonian  n o t be i s o m o r p h i c t o t h e p o i n t  also  is  t o magnetic  group |a> a n d  for a  theory, |b>,  linear  i f the I.R.s spanned  symmetric  representation,  by  A,.  65 For group the  magnetic  should  clusters  be c o n s i d e r e d ,  symmetry  t o t h e (2S'+1)  magnetic  atoms need  t o degeneracy  'genuine'  atom  i s coupled  with  a maximum m a g n e t i c  t h e example  tetrahedron  of four  r(H')=E+T ;  t h e S'=1  2  isomorphic  lower  Only the instability  spectrum. A system two o t h e r  i n which atoms, i . e .  number o f t w o , c a n be  l i n e a r . To i l l u s t r a t e containing  S=1/2 a t o m s .  states transform group  the point  that  when c o n s i d e r i n g  of a c l u s t e r  to the point  degeneracy.  t o , a t most,  equivalent  than  c a n remove d e g e n e r a c y i n  coordination  as magnetically  rather  distortions  spin  i n the magnetic  magnetic  consider  only  be c o n s i d e r e d  each  described  since  of the spin Hamiltonian  addition  due  the magnetic,  as T  a  t h e theorem regular  As shown  above,  i n P„, which i s  2  T^, a n d t h e S'=0 s t a t e s  transform  a s E.  For only  E distortions  r<T |E|T > 2  2  can affect  and r<T |T |T > 2  representation,  2  2  i s unstable  an  S'=1  state and unstable  in  a n S'=0  of  method  distortions  method a p p l i e d Jahn-Teller  these states.  contain  with  respect with  F o r S'=1, b o t h  the totally  symmetric  by a n y d i s t o r t i o n . The  t o E and T respect  not, so  2  distortions  to E distortions  when i n when  state.  The m a g n i t u d e The  2  and the s t a t e s a r e s p l i t  cluster  3.2.2  A,, b u t r < E | T | E > d o e s  S'=0, T<E|E|E> c o n t a i n s  used  of distortions here  t o determine  due t o m a g n e t i c by T e l l e r  paper  o f 1938  i n magnetic  symmetry  f o rspin 6 7  .  clusters  the approximate i s adapted  degeneracy  magnitude  from t h e  i n t h e second  66 Suppose is  stable  when o n l y  interactions in k  a normal i s the  a cluster  now  that  addition  to that  of  this  energy  energy  It  can  estimated  be  will  an  the  v a l u e of  a distortion  [ kd /2 2  'soft' from  with  If v  shift  the o r i e n t a t i o n associated by  of  of  a root  g i v e n by  i s the v i b r a t i o n a l i n cm  - 1  = \x /v  i s i n A,  vibrational  2  and  ; and  AJ  AJ  =  2  - 1  that  addition  k.  mean s q u a r e E=kx .  x  2  can  2  d a t a , from  the  i n cm  ),  - 1  -  1  0  k can  /  k be  A distortion, be thermal and means  X is that  then:  X x A.  i s i n cm .  the above c a l c u l a t i o n  total  minimum  spectrum.  1  - 1  The  (3J/3d)  0  1cm  modes h a v i n g a  the  ( 9 J / 9 d ) = 1 cm" A  1  1A c h a n g e s J b y  with  and  0  frequency  A" ;  the  the  has  in  of  d'=  (infra-red)  2  state  by:  0  x ,  where  2  0  given  crystallographic  0  E=kd /2,  E'=d(9 J / 3 d ) •  the p o s i t i o n  E has  distortion  coupling  + d ( 9 J / 3 d ) ] = 0,  of energy  (3J/9d)  by  on  mode.  isotropic  energy  the v i b r a t i o n a l  state  ( 3 . 1 9 ) d'  w h e r e d'  from  amount, d',  from X-ray  ellipsoids.  approximated  approximated  a m p l i t u d e of v i b r a t i o n ,  estimated  be  dependence  necessary to estimate (3J/3d)  from  vibrational  energy  which  magnetic  otherwise non-degenerate  resulting  term  9/9d  i s now  this  i n magnetic  a l o n g d by  (3.18)  or  d,  can  associated  change  coordinate,  d,  configuration  other than  constant f o r the normal  degeneracy  The  interactions  coordinate,  magnetic  spin.  i n a symmetric  a r e c o n s i d e r e d . The  force  Suppose  exists  2  Distortion  large  any  effect  (9 J/9d ) 2  2  0  i s favoured for on  J. It i s clear  terms, neglected  67 in  t h e d i s c u s s i o n so f a r , merely  the  harmonic  the  magnetic Since  AJ  will  potential  X,j> a n d x  exchange pathway.  J  with  may  2  vary  consider  A"  be u s e d  than  that  clusters  changes,  spectrum. than  distortions  suggest  t h e Cu-O-Cu has been  who c o r r e l a t e d  8 2  systems. A value  o f v,  of X  frequency,  and x  c a n be t a k e n  2  as  indicated  However that  by  normal this  by s t r u c t u r a l  a n d i t may  small  order  the magnetic  Cu-O-Cu c o n t a i n i n g  Many c l u s t e r s  dramatic  perturbations, i n the  A J i n Cu-O-Cu c l u s t e r s (3.20)  since there  will  i s no  modes e x a c t l y c o r r e s p o n d large effect correlations  be an o v e r e s t i m a t e this  1  be a s s o c i a t e d w i t h  r a t h e r than  w h i c h have  X was o b t a i n e d  cluster.  two e s t i m a t e s o f  pathway  / 500 = 320 cm"  2  may  The r e a l  the vibrational  dimers  on  i t i s p o s s i b l e that AJ i s as large or  J. Distortions  qualitative  smaller  of  gives:  i n these  magnetic  effect  involving  by M e l n i k  as an e s t i m a t e  2  larger  range,  The Cu-0 s t r e t c h i n g  (3.20) A J = ( 2 0 0 0 ) ( 0 . 2 )  be  of t h i s  and reviewed  was o b t a i n e d .  1  will  1  0.2A. T h i s  Thus  clusters  The s e n s i t i v i t y  i n some d e t a i l  1  no q u a l i t a t i v e  i n a wide  t h e s t r u c t u r e o f many d i m e r i c  2000cm" 500cm"  and can have  the force constant  behavior.  be made. F i r s t  studied  modify  o n X. A l s o between  f o r 3J/9d  of d i s t o r t i o n  guarantee to the  the value of  a s e r i e s of  within a  of magnitude c a l c u l a t i o n  effects  probably  single  does  may b e l a r g e i n  clusters.  have a J of t h e o r d e r  o f a few 10s o f cm"  1  68 and  even  quite  trimetallic J=lOcnr energy  small  be s i g n i f i c a n t .  c l u s t e r chromium  and even  1  A J may  100cm"  i f X i s only  t h e n AJ=4cirr  1  acetate,  this  i s quite  sufficient  both  the s u s c e p t i b i l i t y  discussed  i n 3.3.3,  100cm* A~ , i f t h e r e 1  1  . In Chapter  1  Thus f o r t h e  to explain  i s a c  5 i t i s shown  with  that  the observed anomalies i n  and s p e c i f i c  heat  data  observed  for this  compound.  It cluster v  and  c a n be s e e n  the reduced  _  1  )=  (l/27r)  /(k/m);  reduced  the  constituent  mass o f t h e c l u s t e r w i l l atoms,  i t i s hardly  this.  The r e s o l u t i o n o f t h i s  modes  of the c l u s t e r contribute  distortion.  variation  of J with  i t i s clear  stretching the  modes  tend  that  likely  difficulty  fact  note  distortion  studied  that  map  distortion  onto  pronounced  several each  the v i b r a t i o n a l energies  e f f e c t s may  as w e l l  as  normal  of any  mode i n  system thus  be e x p e c t e d t o become  more  increases.  to  normal  increases,  a s t h e mass o f t h e c l u s t e r  dimers  vibrations contribute  t o d e c r e a s e a s t h e mass o f t h e s y s t e m  magnetic  i n copper  f o r bending  several  space  t o come  o f t h e c l u s t e r . The  i s large  that  square  are expected  2  as  a l l vibrational  2  to x  the masses of  t o be a s s m a l l  is^fhat  a n g l e has been  (3J/3d)  2  a b o v e , m =*  t o x , t h e mean  modes  E=kx  s u g g e s t s m-1 . T h o u g h  i s related to the r e s u l t that  i n the f u l l  V. F i n a l l y ,  bond  modes. The  distortion  bending  used  be l e s s t h a n  The l a r g e s t c o n t r i b u t i o n s  the lower energy  the equations:  i n the units 1  the  and  from  S u b s t i t u t i o n o f v=500cirr , x=0.2A  2  from  mass c o r r e s p o n d i n g t o t h e  v i b r a t i o n s c a n be c a l c u l a t e d  ( s  20/vx .  that  69 The d i s c u s s i o n However, thermal  i t does  problem  even  the o r b i t a l  separated of  favouring vibrations  making  Having  thermal  motion  average  phenomenon a  slight  (S,.^)  which  2  may  will  temperature 8 3  .  be  to give  0  much,  i t i s now  Neutron  normal  clusters  possible  mode. I n t h i s  Under  certain may  distort  to consider first  an H that  a  case the  configuration  i t i s clear  0  to a  about  the  which  spectrum  circumstances a  occur. This leads to  i n J and s m a l l  diffraction  the  behavior of  important. Consider  striction  dependence  perhaps  modes.  the equilibrium  for H .  t o be i m p o r t a n t .  between o r t h o g o n a l  normal  o c c u r ; however  known a s e x c h a n g e  terms  e n e r g i e s of the order  c a n be t a k e n a s r e d e f i n i n g  t o be t h a t  be  c a n be c o n s i d e r e d a s  w h e n , a n d b y how  i n a non-degenerate  position,  space.  i n general,  of the dynamic  relationships  shifts  at the  i n spin  or coupling,  forces  interactions,  term merely  be  atoms a r e c o u p l e d by t h e  be s h o w n b e l o w  types of d i s t o r t i o n s  magnetic  will  determined  may  reasonable to expect  up d e g e n e r a t e  of magnetic  distortion  new  phase  with  An  of l o o k i n g  involve  interactions,  The m a g n e t i c  certain  way  cannot,  reasonable picture  distortions.  'motions'  a n d by t h e s p i n s  functions  viewpoint w i l l  most p h y s i c a l l y  which  space  associated  effects.  the cluster  , i t i s entirely  - 1  latter  because  that  and s p i n  vibrational/magnetic  the  magnetic  a n d b o t h c o u p l i n g s may  10-100cm  This  distortions  more q u a l i t a t i v e ,  in orbital  be a c r u d e e s t i m a t e .  the zero point  significant  i s to consider  vibrations  can only  that  and even  t o cause  alternative,  As  seem c l e a r  vibrations  sufficient  g i v e n above  biquadratic,  experiments,  e.g.  i n which the  70 magnetic than  spectrum  i s d e t e r m i n e d d i r e c t l y , a r e more  s u s c e p t i b i l i t y measurements  striction.  Available  biquadratic For  8  exchange  direction,  The c l u s t e r  continuously  normal  may move passing  from  that  The e x i s t e n c e  with  linearly related  distortion  space  including  There  on d i s t o r t i o n  to V  have magnetic  model  forTrimetallic  i s different  normal  i nthe to the f u l l  modes o f t h e f o r degeneracy  spectra  which  exhibit  associated  with  different  This  i s because  only  the system  normal  Clusters  i n which  and l i e a t the c o r n e r s  that  degenerate  t o another  (as opposed  a l lvibrational  those t r i m e t a l l i c clusters  equivalent  equally  spectrum.  3.3 A D i s t o r t i o n Only  be  i sa possibility  i s a necessary and s u f f i c i e n t condition  the magnetic  I .R.  of 6 w i l l  and  o f a d e g e n e r a t e mode  space  are  both magnitude  one d i s t o r t i o n  spectrum  distortion  in  i s much m o r e  0  magnetic  cluster)  the situation  through H .  the average 0  for clusters  negligible.  from  that  the importance of  that  i f not a l lvalues  without  of H .  indicate  5  mode  d ' i s now a v e c t o r  a n d many  8  i s usually  a degenerate  interesting:  likely.  results ""  t o determine  appropriate  o f an e q u i l a t e r a l  degeneracy  orientations with  mode; n o s u b g r o u p s  the metal  full of P  3  3  triangle  i n addition  of the t o t a l P  atoms  symmetry  contain  a  to spin.  has a degenerate  71 S o l u t i o n of the d i s t o r t i o n  3.3.1  An  equilateral  distortions  lower  triangle  Hamiltonian  h a s n o r m a l modes  t h e symmetry  and a r e r e l a t e d  mode  E  AT+E.  to a  distortion  Hamiltonian: (3.21) H'(r,0) = ~ 2 J r ( S , . S c o s 0 _  + S,.S cos0  2  H' c a n b e s o l v e d in  Chapter  f o r d=nit/3  2. T h u s  +  3  u s i n g Kambe's m e t h o d  f o r 0=0, t a k i n g J r = 1 i n t h i s  as  S .S cos0 ) 2  3  +  introduced  theoretical  discussion: (3.22) H'(0) = (1/2) ( S ' The  solutions  the non-crossing  obtaining not  - 3S  of (3.22) a r e l i s t e d  H ' ( r , 0 ) h a s no n o n - t r i v i a l obey  2  the exact  symmetry  rule.  2 1 3  + 3S ) 2  i n Table and thus  with  interconversions  solution  remove  non-adiabatic between  the effects  of  the eigenvalues  I t i s therefore clear that a periodic  cause the s t a t e s t o i n t e r c o n v e r t as long  associated  3.1. F o r 0 * n7r/3,  changes occur,  symmetry-related  distortion.  even  change a s no  before  of 0  will  transitions  i . e . , dynamic  0 will  not i n general  72 Table  3.1  Solutions  Eigenvectors, eigenvalues.  |S',S  S=l/2  |3/2,1> , 0  S=1  |3,2>  o  | 1 ,2>  -5  |0, 1 >  1 3  f o r the equilateral  > are followed  ; | 1/2,1> |2,2>  triangle.  by t h e a s s o c i a t e d  ,- 3 / 2  ;  |l/2,0>  , 3/2.  "3  |2, 1>  3  ;  1  | 1 ,0>  4  ;  I 1 , 1>  0  S = 3/2 |9/2,3> , 0  ; |7/2,3> , - 9 / 2  |5/2,3> , -8  |5/2,2> , 1  |3/2,3> , " 2 1 / 2 ;  |3/2,2> , - 3 / 2  |3/2,0> , S=2  o f H'(fl=0)  15/2 ;  o  |6,4>  |l/2,2>  ;  ;  , "3  |7/2,2>  , 9/2  |5/2,1>  , 7  |3/2,1>  , 9/2  |1/2,1>  , 3  |5,4>  -6  |5,3>  6  |4,4>  -1 1  |4,3>  1  |4,2>  10  |3,4>  -15  |3,3>  "3  |3,2>  6  | 3, 1>  1 2  |2,4>  -18  |2,3>  -6  |2,2>  3  | 2 , 1>  9  ; |2,0>  12  | 1 ,2>  1  M,  7  | 1 ,3>  -8  |0,2>  ;  i>  0  S = 5/2 | 1 5 / 2 , 5 >  o  |13/2,5>  |11/2,5>  -14  |11/2,4>  |9/2,5>  -39/2 ;  |9/2,4>  |9/2,2>  33/2 ;  I 7/2,5> |7/2,2>  r  |5/2,4>  r-25/2  j 7/2,3>  3  |5/2,5>  -55/2 ;  1  1 5/2  |11/2,3>,  1 3  , 1 5/2  r-24  |7/2,4>  , -9  12  |7/2,1>  , 18  ;  |5/2,3>  , "1/2  r  29/2 ;  |5/2,0>  , 35/2  |3/2,3>  r  -3  |3/2,2>  ,  6  |1/2,3>  , -9/2 ; | l / 2 , 2 >  ,  9/2  , 17/2 ; |5/2,1>  I 3/2,4>  r  |3/2,1>  , 12  -9/2 ;  |13/2,4>,  |9/2,3>  |5/2,2>  "1 5  -15/2 ;  73 H'  i s of the form d i s c u s s e d  (3.17) a p p l y . for  This  means t h a t  s t a t e s i n R(S') w i t h a t 8=0  eigenvalues S=3/2 S'=3/2  (3.23)  c a n be  F o r 8=0  the polynomial  -089/2),X  the (X(0)}  (3.24)  X  4  -  -216X  2  (189/2)X  2  possible to obtain  without  matrix  methods  F o r i f S' (3.25)  = 3S-2  from the the  f o r {X} i s :  +  = 0  (81/16)105  = 0  (which  (81/16)105  solutions valid  become  (S>1/2) t h e r e  |3S-2,2S>,  energies  directly  the spectrum i s  - 216Xcos30 +  i s now  with  found  are the roots o f :  It  S).  results  f o r a l l eigenvalues  (X+21/2)(X+3/2)(X-9/2)(X-15/2) X"  the  a b o v e . As an e x a m p l e , c o n s i d e r  (0=4) e i g e n v a l u e s .  or Therefore  and thus  expressions  fi(S')<6  given  {-21/2,-3/2,9/2,15/2};  i n 3.1.3  |3S-2,2S-1>,  f o r general  rather unwieldy  are three  = 0  states  S  for large with  |3S-2,2S-2>,  under H'(0) o f :  (3.26) (1/2)[(3S-2)(3S-1)+3S(S+1) = L  X  2  = -  {1-6S,  {X(0)}  (3.27)  3.2  The  X  3  similar form  3{2S(2S+1),(2S-1)2S,(2S-2)(2S-1)}]  6S-2}  [1+(6S-2)(6S-1)],  therefore  All  1,  -  i s given  X,X X 2  3  =  -(6S-1)(6S-2)  by:  - 3(12S -6S+1) +  (36S -!8S+2) cos30  polynomials  fi(S')<6  2  for  2  of the s o l u t i o n s i n t h i s  are l i s t e d  = 0 i n Table  t a b l e shows t h a t f o r  7 4 fl(S')<6  the  eigenvalues 6=mr/3.  successive  This  correlation  diagram  eigenvalues  which  non-crossing,  The for An  5  result An  are  of  8  Table  3.2  f o r the  numerical  be  constitute  r e q u i r e s the  this  3  6  and  2  3  c =2048 6  The be  expressed  J  =0,  1 3  i n normal  =  example has  the  solution  S=5/2  solution  result  . 1 1 . 7/2  for  to  6  system.  some X^ =  f o r 0 = 0,  )(c +c  6  1550835/16,  Hamiltonian  in distortion  (3.29)  an  3  f o r any  vector  be  Jl X^ =  determined.  'cos60) =  6  0  c =694200, 5  c o o r d i n a t e s as  J  =  (J, +J, +J  r  = /[  2  3  2/3  2 3  trimetallic  n o t a t i o n . Thus a  i n i n t e r a t o m i c n o t a t i o n , of  $  a complete  6  Heisenberg  couplings,  up  c '=-4513.  expressed  As  =  3  and  a  smooth,  of- ( 3 . 2 8 )  6  5  w h e r e c = - 3 0 0 3 / 4 , c = 4 5 7 6 , c«  the  c '  ,  u  2  joining  f o r 0=7r/2 g i v e s n  with  (3.28) X +c X +c X cos30+c X -c cos36 + ( 5 4  with  by  S'=5/2 s t a t e s o f  6  2  obtained  solution  5". 1 1 . 1 7 . 2 9 . 7 / 2 , a l l o w s c  6  between  curves.  , comparing  6  6 varies  as  t o a good a p p r o x i m a t i o n ,  f o r 8=nir/3  known  f o r 0(S')>6  exact  . 1 1 .7 . 3 / 2  3  f o r a l l 6 can  a l l S<5/2 e x c e p t  0*n7r/3.  means t h a t ,  monotonic  results  exact  change m o n o t o n i c a l l y  J, ,J,  J,r,0  2  and  3  cluster  can  cluster  with  J  be  2  can  3  where:  )/3  {(J-J  1  2  )  2  +  (J-J  1  3  )  2  +  (J-J  ) } ] 2  2  3  cos" [J, -J)/r] 1  3  of  J=r=2J  (J,r,#) 1 2  /3,  notation, a  and  0=0.  linear  chain  with  Ji =J 2  2  3  ,  75 Table  3.2  Solutions  In  general,  of  0,  the  f o r e a c h S,  S'=3S+1-0 and  magnetic  0= 1 ; X =  0=2;  0=3;  0=4;  of the t r i m e t a l l i c  there = 0.  2S'+1  spectra  may  are given  value  (1/2) [ S ' - 3 S " + 3 S ] , 2  2  2  by t h e r o o t s o f :  X  2  - 9S  b ) S'=1/2  ,  X  2  -  a) S'=3S-2,  X  3  - 3(12S -6S+1)X  +  ( 3 6 S - 18S + 2 ) c o s 3 9  =  b ) S'=1  X  3  - 3  +  (9S +9S+2)cos30  0.  ,  a) S'=3S-3,  X"  2  = 0  (1/16)(6S+3)  X"  a) S'=3S-4, X  (3S +3S+1)X 2  = 0  - 9 ( 1 O S - 1OS + 3 ) X 2  -  2  2  2  2  3  (9/8)(20S +20S+9)X 2  +  =  2  = 0  2  +  (81/256)(144S*+288S +56S -88S-15) 3  2  0.  + 27(8S -8S+2)Xcos30  81.(9S"-18S +11S -2S)  - 9(20S -30S+13)X  5  2  2  2  2  = 0  + 27(28S -42S+16)X cos30  3  2  2  + 81 ( 6 4 S - 1 9 2 S + 2 0 0 S - 8 8 S + 1 6 ) X  -' 3 2• ( S - 1 ) ( 2 S - 1 ) c o s 3 0 = 0  b ) S'=2,  + 27(7S +7S+2)X cos36  3  3  X  2  - 9(5S +5S+3)X  5  2  81(2S)(S+1)(2S +2S-1)X 2  Fluxionality It  i s now  fluxionality, is  =  a given  a ) S'=3S-1 ,  (27/2)(2S+1) Xcos30  3.3.2  H'  with  Hamiltonian  0.  b) S'=3/2,  0=5,  t w o S'  Taking  + ^  be  distortion  expected  distortions coupling,  produced  by  2  2  +  0.  distortions  of the d i s t o r t i o n s  are  2  2  5  to consider  in equilateral  2  - 3 [2S(S+1)] cos30 =  of magnetic  necessary  3  5  the dynamics, or  described  triangular internal  s i n c e d i s t o r t i o n s must  above.  systems  Fluxionality  i n which  f o r c e s s u c h as  occur  within a  small  exchange  degenerate  76 n o r m a l mode. T h u s t h e s y m m e t r y - r e l a t e d t o ±8±2n/3  corresponding interconvert  i f there  The  f o r S=2  solution  rotation  pathway  i s a low energy i s shown  since  i f r goes  of the energy  the  symmetry  mean  symmetry  the  S=l/2  magnetic  energy  of the system.  at  i n the absence  to rotation  a pseudo-rotation.  which  t h e two z e r o  nuclear all  S' w i t h J2(S') = 2  (3.30))  H  along  If fluxionality  occurs,  will  be P , a n d t h e mean 3  simple.  The r e s u l t s anharmonic o f 8. vector  be r a p i d l y  given effects  There  i s no  i n the  fluxional  pathway  can'also  c a n be  ata l l  phase  states with  energies  relationships  I t i s clear  that  the matrix  cos#  sin#  sinfl  -cos0  regarded  be d e s c r i b e d i n  n o r m a l modes a s a v i b r o n i c  i n which  = ±E  a n d no  this  but d i f f e r e n t  wavefunctions.  than  i s no d i s t o r t i o n  of the d i s t o r t i o n  point vibrational  but constant  r, rather  a  a low energy  independent  The s i t u a t i o n  of the v i b r a t i o n a l  symmetry  such  that  of a c l a s s i c a l p i c t u r e i n which d i s a  terms  cylindrical  them.  I t c a n be s e e n  of s i g n i f i c a n t  i s entirely  I n terms  to  least  defined vector, motion  different  there  space and t h e system w i l l  temperatures.  as  provides  i s particularly  spectrum  barrier  distortion  well  system  show t h a t  t o zero  between  f o r constant  of the Hamiltonian  of the system  The  8,  and l i a b l e  pathway  i n Fig.3.1.  vector  of r f o r constant  lowering  above  are equally likely  of the d i s t o r t i o n  variation  configurations  S'=l/2  arising between  there  coupling i n have  out of the s p i n and  i s some b a s i s f o r  representation of H i s :  77  F i g . 3 . 1 . Spectrum f o r 3 atom, S=2 system as a f u n c t i o n of r . C = c h a i n , E = E q u i l a t e r a l t r i a n g l e and D = dimer  78 This  matrix  has the property  that  the eigenfunctions  <j> = ( c o s e / 2 , s i n 0 / 2 ) a n d 4>_= ( s i n 0 / 2 , - c o s 0 / 2 ) c h a n g e +  changes  b y 2ir. T h e e i g e n f u n c t i o n s  superpositions as r e a l  p orbitals  spin  1. T h e i m p o r t a n t momentum  difference i s that  +  rotation  about  momentum  i s zero  As  doublets  a f f e c t e d by r o t a t i o n  which  i s split  o f H.  value  of the  of t h e angular  useful  t o take  (i)  'small'  Small  S=5/2,  see Fig.3.1.)  8 dependent. states,  fluxional given  Resistance  system  J<0 i s a p a i r o f  by an amount  the non-degenerate, o f 8.  independent  r (r less  the ground  excited state with  and so f a s t  with  vector. For  independent non-magnetic  F o r S>1/2 i t  than  state lies an energy  to fluxionality  fluxionality  the appropriate  about  significantly  separation  occurs  i s expected. time  0 . 2 J f o r S=2  averaged  only  X.=  equation  [X^O) +  defines  For the rapidly spectrum  \ (n)]/2 i  the fast  fluxionality  which i s  in excited  by:  (3.31)  i s  r and ' l a r g e ' r s e p a r a t e l y .  r. For small  the f i r s t  of the d i s t o r t i o n  by d i s t o r t i o n  s t a t e h a s an e n e r g y  This  with  the functions of  s t a t e of t h e system  ground  be  1/2  i n each.  8 a s f o r S = 1 / 2 . F o r 2S e v e n ,  below  momentum  functions  i n the sense  r=0 a n d t h e e x p e c t a t i o n  2S o d d , t h e g r o u n d  and  from  as  S i n c r e a s e s , an i n c r e a s i n g number o f s t a t e s a r e  significantly all  angular  1/2 a r e n o t e i g e n f u n c t i o n s  f u n c t i o n s <p a n d 4>_ d o n o t d i f f e r  The  of  c a n be c o n s t r u c t e d  i f8  be c o n s i d e r e d  of functions associated with  just  angular  can each  sign  model f o r  {X} w i l l  79 equilateral spectrum Table that  triangles  of i s o t r o p i c a l l y  c a n be o b t a i n e d e x p l i c i t l y  3.1, a n d a r e l i s t e d though  i n Table  H'(0) averages  occurring It  excited  be  states  normal  that  with  as d r i v i n g  cos30  appropriate.  may  case  by  a  low-lying  rotation system  t o remain  slowly  at least  i n a state  r  susceptibility  temperatures  These  states,  a n g u l a r momentum,  may  with  of the e q u a t i o n s of Table  value,  zero,  between  i s negligible.  might  be more  this  approach  Thus f o r  o f ±9/2,0  ( r > 0.25J  from Table  vector  9.  may  of well case  on t h e t i m e  the ground  The e n e r g y  3.3  state  or  to  t o cause the  9 f o r thermal  the system  3.1  state  barrier  be s u f f i c i e n t  defined  scale  measurements. Table  f o r S=2,5/2) a  on 9 f o r m s  f o r some  to occur. In t h i s  fluxional  of the  i n 6 associated  spectrum  i s dependent  of the d i s t o r t i o n  equilibrium as  state,  parts  ±/21,0.  r. For large  the energy  do c a r r y  of (3.32)  would  which  i t s spectrum  for distortions  involved.  the difference  fluxionality  for  become  a solution  S=1,S' = 1 t h e f a s t  Large  possible  given i n  modes.  b y i t s mean  Fortunately,  the simple approach  (ii)  to zero,  the electronic  t h e change  replaced  be r e p l a c e d  time  The  point i s  n o a n g u l a r momentum. A t h i g h  states  In this  the results  3.3. The c r u c i a l  i s only  f o r a two-dimensional well  fluxionality.  and  carry  vibrational  pictured  3.2  degenerate  was shown a b o v e  vibronic  which  within  from  out over  does n o t . T h i s type of r e s u l t  coupled spins.  c a n be  described  of the magnetic shows t h a t  = 2n7r/3 a r e t h e r m o d y n a m i c a l l y t h e m o s t s t a b l e  f o r J r > 0, 9  states  of the  80 system.  To a f i r s t  fluxionality values  approximation a system  c a n be c o n s i d e r e d a s b e i n g  of 6 and i t s magnetic  for  t h e Kambe m o d e l  J<0  the magnetic  slowly 'high  state  exhibits  moment w i l l  be l a r g e r  than a r a p i d l y  i s brought  closer  slow  fixed at the stable  will  b e t h e same a s  ( T a b l e 3 . 1 ) . I t i s t o be e x p e c t e d  f l u x i o n a l system spin'  spectrum  which  that i f  a t low temperatures f l u x i o n a l system  t o the ground  that  state.  fora  since  a  81 Table  3.3 T h e S p e c t r u m f o r t h e D y n a m i c As  quantum the  thus  In  only  the total  number. The c a l c u l a t i o n  e x a m p l e o f t h e S'=1 For  and  6 varies,  these  {X(7r)}  = {-4,-1,5},  the l i s t  Model  S', r e m a i n s a g o o d  o f {X} i s i l l u s t r a t e d  by  s t a t e s o f t h e S=1 t r i m e r .  s t a t e s (X(0)}  {X} = { ( - 5 - 4 ) / 2 ,  spin,  Distortion  = {-5,1,4} f r o m  using  (1-D/2,  t h e symmetry (4+5)/2}  below, each value  ( i ) above relation  (3.11).  ={-9/2,0,9/2}.  o f S' i s f o l l o w e d b y { X } :  S=l/2.  S'=3/2  {0}  ; S'=!/2  S=1  S' =3  {0}  ;  S'=2  {±3}.  ;  S'=0  {0}.  ;  S'=7/2  {±9/2}  ;  S'=3/2  {±9 ,±3}  S' =5  {±6}  S'=3  {±27/2  ,±9/2}  S' = 1  {±15/2  ,0}  {±15/2}  S' = 1 {±9/2 S = 3/2  S* = 1/2  {±3/2}.  ,0}  S'=9/2 S'=5/2  ,  {0} {±15/2  ,0}  {±3}. S = 2.  S=5/2.  S' =6  {0}  S' =4  {±21/2  S' =2  {±15  S* =0  {0}.  S'=15/2  {0}  S'=13/2,  S'=11/2  {±27/2,0}  S'=  9 / 2 , {±18,±6}  S'=7/2  {±21,±21/2,0}  S'=  5/2, {±45/2,±27/2,±9/2};  S'=3/2  {±27/2,±9/2}  S'=  1/2,  ,0}  ,±15/2  ,0}  {±9/2}.  82 3.3.3  The  The  trigonal  addition  triangle  results  Some a s p e c t s factorable  of  terms  useful atoms  the  o f the  i f the  atoms above  S=l/2 s y s t e m  Hamiltonians. o f the  There  as for and  two  present  and  equatorial  below  was  the  case  metal  spanned  b y {d})  to  there  atoms.  systems.  That  5  in P , i s 5  J.,  in  terms o f  i s more  useful  in  1  i s no i n t e r c o n v e r s i o n The  That  make a l l m e t a l  i n PC1 .  G  with  choices of atom  i n t e r n a l J , J , and  where  behavior.  offive  sufficient  halides  an e q u i l a t e r a l  magnetic  natural  (I.R.s  3  the  and  now b e i n v e s t i g a t e d  configuration  fluxionality  equatorial  will  are  ' n o r m a l modes'  equivalent,  axial,  o fa x i a l  i n much more c o m p l i c a t e d  parameterisat ion in  bipyramid  Hamiltonian  for  between  the  axial  undistorted  system i s : (3.32) H = - J  (S  2 2 3 1 )  -3S )  -J  2  6  =  -J s'  2  a  e  1 and  equatorial.  The  notation  (3.33),  by  two  for  2  - (J -J )s  where atoms  in  (S' - S  2 2 3 I t  -S, )  - J . (S, -2S  2  2  5  5  3 a  5 are  axial  solution  for  the  spins  .)  2  1  2 3  „  - ( j . - J  2  and  atoms  a  ) s  5  2,3 a n d  S=l/2 i s g i v e n and  2 1  energies  in have  4 are |S',S „,S, > 2 3  been  5  multiplied  convenience.  (3.33)  |5,3,2>  ,-6J  1  a , -3J  |3,3,0>  e  |1,3,2> '  ,10J  a  - 3 J - Je + 3J 1  i  - 3J -J. e I  ; |3,3,2> f  \  •  •  , 4J - 3J - J . a  e  , - 2J  ; I 1 , 1 , 2>  , 4J + 3J - J . a e I  1  a  + 3J  1  ; |3,1,2>  e  - J  I 1 , 1 , 0> , 3 J + 3 J • ' ' ' e I 1  Those  states  with  S 3«=l/2 have 2  a two-fold  degeneracy  which  is  i  83 split  by  (3.34) which  equilateral  (3.34) has  = -r(S  applies now  to  uniquely  and  the  (3.34)  good  ten  {J}  to  zero The  has  of  the  f o r S=l/2  independently.  f o u r quantum  numbers S ' , S  between  data  the  linear  this  normal  2  2 3 5  by  3 5  As i  , S  l  the 1  ±3r now  system  and  5  necessary  is  S „, 2  to  define  2A,'+2E'+A "+E", 2  the. e x i s t e n c e o f  be  +2S  3  = 3 / 2 and  trimetallics  modes,  of  the  two  distortion axial  distortion  can  |S',S,  be  2  2 3 U  values.  leaves  -r(S' -3S, in  J  combination  terms and  solutions  would  nine  mode w h i c h  S  for states with  2  one  =  )  2  theory  i s complicated  'axial'  (3.35) H This  = l/2.  a  p o s s i b l e t o d e f i n e an  contains  to d i s t o r t i o n  + 3S  2 2  four corresponding  i s just  also  of  experimental  mapping  belonging  3  S 3«=l/2 pair  each  the  The  is  2 3 ( |  d e s c r i b e d by  extremely  u -3S 2  2  eigenvalues with S  those  corresponds  triangle.  ( 3 . 3 4 ) H'  for  .  J  E'  modes.  Hamiltonians  unchanged.  Hamiltonian  It  which  factored:  2 1 5  -3S  2 2 4  +6S ) 2  ,S, ,S „> notation: 5  2  |5,3,2,2>  ,  0;  |3,3,2,2>  , -5;  |3,3,2,0>  , 1;  |3,1,2,2>  ,  |3,1,0,2>  ,  0;  |1,3,2,2>  , -8;  |1,1,2,2>  , 1;  |1,1,0,2>  ,-3;  | 1,1,2,0>  ,  7;  | 1 , 1,0,0>  ,  but  (3.35) commutes w i t h  equatorial treatment coupling  J are of  the  3.  (3.32) and  (3.34) o n l y  constrained to equal effect  requires matrix  of  4;  dynamic  methods.  and  in general  distortions For  the  i f the  E"  on  mode,  the  a  axial proper axial  which  and  84 corresponds  to a t i l t  axial  no  The  atoms  of the e q u a t o r i a l w i t h  factorable distortion  complications  in dealing with  be c o n s i d e r e d  as a r i s i n g  in  Hamiltonian  the s t a t i c  distortion bipyramid  method can only  from  Hamiltonian  the a x i a l  E'  in general  As a r e s u l t  defined.  a n d E" modes c a n of t h e atoms  the dynamic  be a p p l i e d t o t h e  distortions.  to the  c a n be  the non-equivalence  (3.32).  for equatorial  respect  trigonal  85 CHAPTER 4 4.1  DYNAMIC D I S T O R T I O N S I N  theory  regular  of  dynamic  t e t r a h e d r a of  developed.  The  distortions  tetrahedron  problem  triangle  because  n o r m a l modes s p a n  of  the  the  larger  equilateral along  a  The the in  only {J}  triangle,  by  a .  considered two  distortions  with  total  relationship  The  matrix  application  of  the  number  equal  to  to  of  the  f o r the  three  matrix  to  group t h e o r e t i c a l  and  introduced  in chapters  3.  distortion  model  discussion  i t is helpful,  distortions  be than  the  both  and  2  for  146  because the  must  2 and  number  occur  of  and  of  d  1 2  S>l/2 would  be  In  ,  and  chapter  f o r a l l S^5/2.  initially,  obtained  In  of  a/3a/3  f o r both  slow  and  reduced 3.  The  i t is  necessary  results  the  dynamic  theoretical E and  T  2  s o l u t i o n s which  both  there  extremely  the  to consider that  components  be  2 and  algebraic  this  are  modes. Thus  1,  linear  atoms  ( i g n o r i n g S'  order  s e p a r a t e l y . I t i s found  be  four  s t a t e s f o r S=5/2) and  i s derived  models can  or  s i x s p i n s t a t e s can  enough t o a l l o w d e f i n i t i o n  fluxionality  3  T ,  normal  aafifi  matrices  methods t o  the  states  say,  (there are  the  now  tetrahedron  three  that  spin  cumbersome  complete  the  number of  between,  problem S'  contain  mode.  1:1  use  of  property  b l o c k i n g by  E and  s p i n s t a t e s i n v o l v e d . As  systems the  atoms w i l l  in chapter  I.R.s,  systems w i t h  and  which  i s more c o m p l i c a t e d  n o n - l i n e a r S=1/2  degeneracy) are is  problem  number of  degenerate  in clusters  equivalent magnetic  equilateral  1 3  CLUSTERS.  Introduction A  d  TETRAHEDRAL  and  modes  are  fast  for a l l spin.  86 4.2  E-mode d i s t o r t i o n s The  Hamiltonian  was d e r i v e d  (4.1)  of magnetic  tetrahedra  f o r t h e E mode d i s t o r t i o n  of a  i n 3.1.2:  H'(r,^)  = - 2 J r [ (S , . S +S . S , ) cos\p . + ( S , . S +S . S « ) c o s t f 2  3  3  + (S, .S„+S  'Jr',  which  only  theoretical simplifies form  s c a l e s H', w i l l  d i s c u s s i o n . The D i t s solution.  introduced  (4.2) +  fi(S')=k, k  4  a  c  use of group theory.  It  k  Equation  Hamiltonian  greatly  -  2  c (X) X  k_3  3  cos3\£/  0  i n 2.4.2. a s an example o f  I t c a n now b e s e e n  f o r t h e S=l/2 case.  3  t h a t H' i s o f t h e  =  5  ( 4 . 1 ) was i n t r o d u c e d  thefactorisable  that  group theory i s  (4.2) and t h e r e s u l t s  f o r \p = n7r/3 d e f i n e  the exact  immediately: S=l/2;  X(S'=2)=0,  i su s e f u l a t t h i s  plane. defined As  2  .S )cos(i|/J]  2  ( 3 . 1 7 ) c a n be a p p l i e d :  {6)\ ~ cos3\p  the  (4.3)  I t c a n a l s o be seen  k  5  Hamiltonian  result  s y m m e t r y o f H'(\£)  k  2  as 1 i n t h e  {X}: X - (1/2L X ( 0 ) ) X ~  c (0)X " +  unnecessary  2  be t a k e n  i n 3.1.3. a n d so e q u a t i o n  The  for  tetrahedron  The t h r e e  point  velocity. convenience  The second  the idea  rotates with  dimension  of t h e energy  c a n be v i s u a l i s e d  o f an e q u i l a t e r a l  thetriangle  only  to introduce  S'=1 e n e r g i e s  by t h ec o r n e r s  \p c h a n g e s  { X (S ' = 1 ) } = - 2 c o s (V+n7r/3 ) , {X (S ' =0 )} =±3 .  triangle  t h e same  of t h e energy  a n d h a s no p h y s i c a l  as being o f r a d i u s 2.  angular  plane  significance.  i s added f o r  87 4.2.1  T h e S=1 s y s t e m It  for  c a n be seen  S=1  from  fi{S'=4,3,2,1,0}  solving  thebranching diagram, = {1,3,6,6,3}.  H' i s t o w r i t e  Table  The f i r s t  down t h e s o l u t i o n s  2.1, t h a t  step i n  forthe factorisable  H' ( 0 ) : ( 4 . 4 ) H ' ( 0 ) = 1/2 ( S ' - 3 S " - 3 S ' " 2  In  |S',S",S"'> n o t a t i o n  2  theeigenvectors,  2  + 8S ) 2  and associated  eigenvalues, a r e : (4.5) |2,2,1>,-1;  |4,2,2>,  0; | 3 , 2,2>,  - 4 ; | 3 , 2,1>,  |2,2,0>,  2; | 2 , 1 , 1 > ,  5; | l , 2 , 2 > , - 9 ;  | 1 , 1 , 1 > , 3; | 1 , 1 , 0 > , where each SVS"' be  eigenfunction  |2,2,2>,-7;  |1,2,1>,-3;  | 0 , 1 , 1 > , 2; | 0 , 0 , 0 > , 8.  i sfollowed  by i t s e n e r g y .  States  with  a r e d o u b l y d e g e n e r a t e . T h e s o l u t i o n s o f a l l H'(n7r/3) c a n  found  from  (4.4).  The n e x t states  6; | 0 , 2 , 2 > , - 1 0 ;  2;  stage  i s t o consider  i n t h e magnetic  t h e I.R.s spanned  group D ; t h egroup 2  table  by t h e s p i n  was g i v e n i n  2.4.2. The a n a l y s i s i s :  r{s =4} = T{|1111>} = A, ,  (4.6)  z  r{s'=3} =  r{|ino>}  =  A +B +B +B , 1  1  2  3  T{S =2} z  = r({|1100>}+{|111-1>}  = 4A +2B +2B +2B ,  T{S =1}  = r({|110-1>}+{|1000>}  = 4A +4B +4B +4B ,  Z  1  1  1  1  2  3  2  3  r{S'=0} = r ( { | 1 1 - 1 - 1 > + | 1 0 0 - 1 > } + | 0 0 0 0 > ) = 7 A + 4 B + 4 B + 4 B , 1  1  1  1  r(s'=4) = A , , T ( S ' = 3 ) = B + B + B , T ( S ' = 2 ) = 3 A + B + B + B , 1  2  3  1  1  2  3  88 T(S'=1)  2  result  the  S=1/2 system,  Since  states;  <  6,  Consider  a  single  spanning s t i l l B  for  2  X  are the roots -  3  34X  .  Though t h e t r a c e  eigenvalues -2cosi//  X ( B  with  2  )  = X( B  2  of the polynomial:  cos3\p  =  s u b s p a c e s Q(r,S') s e p a r a t e l y .  i n \p  +  for B  cosi//.  1  f  . i n 4/ f o r B  Inspection  possible solution.  ina  t o be z e r o  S'=2, { - 7 , - 1 , - 1 , 2 , 2 , 5 } ,  i s the only  i . e . s p a n n e d by  o f H' (\|>) e x p r e s s e d  constrained  , \jj=0)  0  one d i m e n s i o n a l ,  3  e a c h Q i s no l o n g e r  Thus  +1 6 0  )and Q ( B ) a r e each  be p e r i o d i c a n d e v e n 3  c a n be a p p l i e d t o t h e S ' = 0  S'=2, a n d c o n s i d e r  state.  f o rthe S'=1 states of  by two:  (3.16)  equation  (X(S'=0)},  Q ( B  = 3 A , .  = - 4 cos(\//+2n7r/3)  the eigenvalues  (4.8)  Q(B,),  multiplied  E(S'=3)  fl(0)  3  f o r S ' = 3 i s t h e same a s t h a t  The  (4.7)  r(S'=0)  = 2B,+2B +2B ,  shows  The ' B '  basis  i t must 2  a n d i n \{J.  of the s e t of that  X(BT)=  eigenvalues are  thus:  (4.9)  This  E (B)  leaves  only  = -2cos(i//+2n7r/3)  theA, states.  be a p p l i e d a n d t h e e i g e n v a l u e s (4.10)  ( X ( A , ) } ,  Now c o n s i d e r the =  problem  A +B +2E 2  2  X  S ' = 1 . Since i sslightly  which  3  -  39X  n= 0,1,2  Since  Tr[Q(A,,2)]  are the roots + 70cos3*//  =  = 0,  (4.2)  can  of the polynomial: 0  r ( S ' = l ) i n c l u d e s no I . R . s e x a c t l y o n c e more  complicated.  indicates that  those  In  D^^  (\//=0)  states which  form  T(S' = 1)  89 degenerate Therefore  pairs  of energy  -3 a n d +6 t r a n s f o r m  the s t a t e s of energy  -9,3 must  transform  Consider  H'(<//) i n a b a s i s s p a n n i n g Q ( B , 1 )  diagonal  and hence  eigenvalues  o f H' n e e d  change  i f \p c h a n g e s  sign  (4.11)  H'  X  sinxjj  2  At  y\i=2v/3  {-9,-3,-3,3,6,6} a n d have ZX=3; t h e o n l y  The  Though t h e must  cos\p  3  2  (4.12)  2  c sin\[/  s o l u t i o n s {X} f o r Q ( B , 1 ) .  From t h i s  as B .  b e p e r i o d i c i n 3\jj t h e y  + 6Xcos\// - 27cos \//  2  2  b y 7r:  -9cos<//,  Thus t h e e x p r e s s i o n the  i s of f - d i a g o n a l .  no l o n g e r  c  in D  3  i n which H'(0)i s  2  H'(7r/2)  a s B,+B  i t c a n be s e e n  that  the B  X + 6 X c o s \ / / - ( 2 1 + 6 c o s 2 ^ ) = 0, 2  s o l u t i o n s f o r B, a n d B  3  2  - c  2  s i n  2  \ p  = 0,  {X} m u s t  l i e  solution  i s  eigenvalues  provides in  {X}={-3,6}.  satisfy:  X=-3cos<//±v/( 3 6 c o s \ ^ + 1 5 s i n ^ ) 2  are identical  except  2  that  \p i s  r e p l a c e d , by ^±2^/3. A correlation tetrahedron, 4.2.2  f o r S'=1  The S >  and  before  f o r the eigenvalues  a n d S'=2, i s g i v e n  o f t h e S=1  i n Fig.4.1.  1 systems  Some g e n e r a l derived  diagram  results  which  continuing with  are valid explicit  f o ra l lS w i l l  results  be  f o r S=3/2,  S=5/2.  a)  S'=4S-1. The t h r e e  states of this  spin transform  as  S=2  90  0.0 Fig.4.1.  ~i 0.8  r  THETR/PI  i .6  C o r r e l a t i o n d i a g r a m s f o r E d i s t o r t i o n o f S=1 t e t r a h e d r o n . S'=2 ( a b o v e ) and S'=1 (below).  91 B,+B +B 2  2  |4S-1,2S,2S-1> Therefore  transform  as 3A +B +B +B  factorisable  Application 'B'  1  n=0,1,2 s i x states  s p e c t r u m a t v/>=0,  which  found  from  {1-8S,1-2S,1-2S,4S-2,4S-2,4S+1}  i s  in 4.1.1.  given  shows t h a t  the  three  energies:  (4.14)  (X(4S-2,B,*//)  leaves  t h e A,  found  =  states,  (4S-2)  cosU+2nir/3) , n= 0,1,2 {1-8S,4S-2,4S+1}  of energy  t o be  from:  (4.15)  (X(4S-2,A,i//) } X - ( 4 8 S - 1 2 S + 3 ) X + ( 8 S - 1 ) ( 4 S - 2 ) (4S+1 )cos3i//=0 3  S ' = 0 . There  c) states  are split  6S(2S+1) than  Hamiltonian,  contains  The  2  and 2 S .  H' i\p) a r e :  R(S,S')  in D .  3  of the argument  s t a t e s have  This  2  -4S,2S  energies  = - 4 S cos(i//+2n7r/3) ,  S>l/2. This  1  with  f o r general  {X(S' =4S-1  b) S ' = 4 S - 2 ,  the  |4S-1,2S-1,2S>  and  the energies  (4.13)  are |4S-1,2S,2S>,  At ^=0 t h e e i g e n s t a t e s  in D .  3  by  that  produce  'E'  Consider (c.f. (4.16)  mode d i s t o r t i o n .  Table  ground  cluster.  a s A,  [E(|0,2S,2S>)  by an amount  a singlet  transform  a r e 2S+1 s t a t e s o f t h i s  f o r any o t h e r  tetrahedral  2  in D  S',  2  this  a l a r g e enough  state  It will  Since  i n any  now  and that  t h e I.R.s spanned  be  they  by  spin. -  The S ' = 0  E(|0,0,0>)]  splitting 'E'  that  of four  will  coupled  a l lS'=0 states  therefore cannot  sets  i s more  distortion  isotropically  shown  =  cross.  objects  in  D . 2  2.2):  r{aaaa}=A , 1  T{aaab}=A,+B,+B +B , 2  3  T{aabb}=3A,+B,+B +B , 2  3  92  r{aabc}=3(A,+B +B +B ), 1  It  c a n be s e e n  will  be d e n o t e d  states. n(A),  i s equal  than  contains,  Now i n (S  ,S„  like  a s A,.  But there  there  excess  are S+1/2 sets  and  I f 2S i s even  o f I . R . A,  denoted  thus  then  the set  an e l e m e n t  i n excess  are also  (0,0,0,0)  therefore of the form  A,  states  and the r e s u l t  This  result  i s very relevant  i t means t h a t  they  o f t h e number  S sets  of the form  two e x t r a  states  2S+1 S ' = 0 states  than  states  B,  and thus  they  cross.  For 2S  cannot  {z,z,-z,-z},  and 2S+1  i s t h e same. to the magnetochemistry  clusters  since  magnetic  effects  of  The  spectra  f o r the S ' = 0 states  found  3  {S'=1};  Consider  a r e 2 S + 1 m o r e A,  a s A,,  be  B  4  are exactly  must a l l t r a n s f o r m  will  each of which c o n t r i b u t e  There  and  2  which  by an amount  o r {aabb} s e t s  a s B,. T h e r e  z integer,  transforming  Z  f o r any R(S'),  states  ) notation,  ,S„ 3  one s t a t e  transforming  {z,z,-z,-z}  states.  Z  3  {aaaa} and {aabb} spanned  2.23).  {S'=0}.  ,S„  Z 2  contributes  states  {aaaa}  consider  Z ]  which  no  2  s t a t e s , which  t h e n u m b e r o f B,  on t h e number o f s e t s  set can contain  states  1  t o t h e number o f B  t h e n u m b e r o f A,  i s greater  n(A)=n(B).  odd  1  {S'=S'}-{S'=S'+1}, (equation  this  of  r{abcd}=6(A +B +B +B )  3  t h e n u m b e r o f B,  n(B),  Further,  depending by  that  2  fluxionality  'E' mode d i s t o r t i o n  cannot  remove  on t h e S'=0  for S=3/2  of  the  states.  a n d 4 / 2 c a n be  from:  (4.17)  S= 3/2  X"  S =2  X  5  - 6  3 42X 2  2  3 3 X  3  +3  2  64Xcos3i//  3  +  3"105  +6 44X cos3<//+6" 132X 3  2  -  =  0  6 144cos3<// 5  =  0  93 Since  0(S=5/2,S'=0)=6,  determined The  by g r o u p  method  A, c a n a l s o  theory  used  Consideration  solution  sets  f o r S=5/2 c a n n o t  and f a c t o r i s a t i o n  t o show t h a t  be a p p l i e d  {aabb} a n d {aaaa}  the exact  to higher of basis  a l l S'=0 spins.  states  be  alone.  states  transform  as  {S^=even} c a n c o n t a i n while  o f t h e number o f ' e x c e s s '  {S'=odd} c a n n o t . z  A states  gives the  result: (4.18)  I f S'=4S-k, k e v e n S'=4S-k, k  The  I.R.s f o r t h e s t a t e s  written  down d i r e c t l y  , then  n(A)=n(B)+(k+2)/2;  odd , t h e n  n(B)=n(A)+(k+1)/2;  o f t h e S>1  given  the  tetrahedra  fi(S')  c a n now b e  from the branching  diagram. (4.19)  S = 3 / 2 . S'=6,A;  S=2.  S'=5,B;  S'=4,3A+B;  S'=3,3B+A;  S'=2,5A+2B;  S'=1,3B;  S'=0,4A.  S'=8,A;  S'=7,B;  S*=6,3A+B;  S'=5,3B+A;  S'=4,6A+3B;  S'=3,2A+5B;  S'=2,7A+3B;  S'=1,4B;  S'=9,B;  S'=8,3A+B;  S'=7,3B+A;  S'=6,6A+3B;  S'=5,6B+3A;  S'=4,9A+5B;  S'=3,3A+7B;  S'=2,9A+4B;  S'=1,5B;  S'=0,6A.  S'=0,5A. S = 5 / 2 . S'=10,A;  In  each case Finally  that  fl(S')  = n(A)+3n(B).  i t c a n be s e e n  i f S"+S"'  i s even  from t h e r e s u l t s  the state  already  obtained  | S ' , S " , S ' " > t r a n s f o r m s a s A, o r  94 B  2  B,  inD  2  and that  i f S"+S"'is  or B . This result 3  diagrams large  The  transforms as  i ta l l o w s  f o r H' (\p) e v e n  correlation  when 0 ( S ' ) i s t o o  solution.  solution  solutions  the state  i s important since  t o be c o n s t r u c t e d  f o rexact  odd then  f o r S=3/2 w i l l  now b e c o m p l e t e d , i . e .  f o r S'=3,2 a n d 1 o b t a i n e d . F r o m  (4.4)the spectrum f o r  \p=0 i s :  (4.20)  |3,3,3>,-15;  |3,3,2>,  - 6 ; |3,3,1>,  0;  |3,3,0>,  3;  |3,2,2>,  3,  |3,2,1>,  9;  |2,3,3>,-18;  |2,3,2>,  - 9 ; |2,3,1>, - 3 ;  |2,2,2>,  0;  |2,2,1>,  6;  |2,2,0>,  9;  |2,1,1>,  1 2 ; | 1,3 , 3 > , - 2 0 ;  |1,3,2>,-11;  |1,2,2>,  - 2 ; |1,2,1>,  4;  |1,1,1>,  10; |1,1,0>, 13;  Solutions  o f H' (\^=0) f o r a l l S>1 a r e l i s t e d  For  S'=3 n ( A ) = 1, n ( B ) = 3.  The  3 remaining pairs  which  shows t h a t  must  have  X=0 f o r a l l \p. •  states  must  be B , + B  T h e A, s t a t e  of degenerate  a t X=0 t h e B  2  spectrum  { - 6 , 3 , 9 } a t \//=27r/3. T h e c h a r a c t e r i s t i c of  i n T a b l e 4.1.'  i s {-15,0,3}  3  pairs  a t \jj=0 a n d  p o l y n o m i a l f o rQ(B ) i s 2  t h e form:  (4.21)  X  The  solution  only  For  3  + a c o s ^X  S'=2  2  + ( b + b'cos2<//) X + ( c c o s i ^ + c ' cos3»//) = 0  i s a=12, b=-45,  n(A)=5,  n(B)=2.  b'=0, C=108  The r e s u l t s  C'=-108.  f o r t h e A and B  states  are:  (4.22)  {X(A)}  : X  5  -  3 31 X  {X(B)}  : X  2  -  6X + ( 18cos2(\//+n7r/3)-45)  2  3  + 3 42X cos3\// 3  2  +. 3 " 7 2 X = 0 = 0  95 For  S'=1  (4.23)  the  9 eigenvalues  (X(B)}  X  +  3  12cosa  can  X  -  2  (656cos3a-1856cosa)/3  where  As using  an  effect  can  be  and  of  rapid  This  group  +  0  theory  and  the  solution  theory the  failure  dynamic  derived without group  from:  (428+112cos2a)X/3 =  increase exact  When 0 ( Q , S , S ' , D > 6  solution.  the  using  fi(S')  factorisation  exact  found  a =^+2n7r/3, n = 0,1,2.  S and  difficult.  be  becomes  method  i s not  distortions exact  of  can  spectrum  increasingly no  longer  provide  s e r i o u s s i n c e a model on  magnetic  eigenvalues  i n t r o d u c i n g the  the  by  concept  for  tetrahedra  correlation of  energy  bands.  96 Table  4.1  5 , 3/2 6 , 3 , 3> 4,3,2> 3, 3,2> 3,2,1> 2,2,2> 1 , 3 , 3> 1 , 1 , 1> 0,1 ,1> S , 8,4 6,4 5,4 5,3 4,4 4,3 3,4 3,3 2,4 2,3 2,2 1,3 1, 1 0,2  2 4> 3> 3> 2> 1> 1> 2> 1> 4> 2> 0> 3> 1> 2>  S , 5/2 10,5,5> 8,5,4> 7 4> 7 3> 6 2> 6 2> 5 3> 5 4> 5 3> 4 3> 4 3> 4 3> 3 5> 3 4> 3 3> 3 2> 2 3> 2 3> 2 1> 1 4> 1 2> 1 0> 0 2>  Solutions  o,  "2, -6 9 0 20 10 9 0 -3 -9 12 1 13 -9 9 33 0 18 1 1 19 6 0 -4 -12 15 2 17 •13 -10 14 -18 -3 9 -49 -19 5 23 -25 2 26 -39 9 33 17  of the E d i s t o r t i o n  Hamiltonian  5,3,3> 4,3,1> 3,3,1> 2,3,3> 2,2,1> 1,3,2>  , -6; , 4; , 0, , -18, , 6, , -11, 1 , 1 , o > , 13, 0,0,0> , 15  5,3,2> 4,2,2> 3,3,0> 2,3,2> 2,2,0> 1,2,2> 0,3,3>  7,4,4> , -8 6,4,2> , 6 5,4,2> , 0 4,4,4> , -26 4,4,0> , 4 4,2,2> , 16 3,4,1> , -3 3,3,0> , 12 2,4,3> , -21 2 , 3 , 1> , 6 2 , 1 , 1 > , 21 1,3,2> , -2 1,1,0> , 22 0 , 1 , 1 > , 18  7,4,3> , 4 6,3,3> , 9 5,4,1> , 6 4,4,3> , -14 4,3,3> , -2 3,4,4> , -30 3,3,3> , -6 3,2,2> , 12 2,4,2> , -12 2,2,2> , 9 1,4,4> , -35 1 , 2 , 2> , 7 0,4,4> , -36 0,0,0> , 24  ,5 ,5 ,5 ,5 ,5 ,3 ,5 ,4 ,3 ,5 ,4 ,3 ,5 ,4 ,3 ,2 ,4 ,3 ,2 ,4 ,2 ,5 , 1  5> 3> 3> 5> 1> 3> 2> 3> 2> 2> 2> 2> 4> 3> 2> 1> 4> 2> 0> 4> 2> 5> 1>  , , , , , , , , , , , , , , , , , , , , , , ,  -10; " 8; 0; -34; 8; 20; -4; 2; 23; -9; 6; 18; -34; "7; 14; 29; -22; 11; 29; -24; 18; -55; 29;  ,5 ,4 ,5 ,5 ,4 ,5 ,5 ,4 ,5 ,5 ,4 ,3 ,5 ,4 ,3 ,5 ,4 ,3 , 1 ,4 ,2 ,4 ,0  4> 4> 2> 4> 4> 5> 1> 2> 5> 1> 1> 1> 3> 2> 1> 5> 3> 1> 1> 3> 1> 4> 0>  , 3, , 7, , 3, , "9, , 9, , -2 , -21  , , , , , , , , , , , , , , , , , , , , , , ,  . 5, 1 1 9 -19 -4 -40 2 1 1 -45 -3 12 24 -22 2 20 -52 -10 17 32 -12 24 -25 35  H'(\ft,0)  4,3,3> 3,3,3> 3,2,2> 2,3,1> 2,1,1> 1,2,1> 0,2,2>  , -1 1 , -15 , 3 , -3 , 12 , 4 , -3  6,4,4> 5,4,4> 5,3,3> 4,4,2> 4,3,2> 3,4,3> 3,3,2> 3 , 2 , 1> 2,3,3> 2 , 2 , 1> 1 ,4,3> 1,2, 1> 0,3,3>  , -15 , -21 , 3 , -5 , 7 -18 , 3 18 -9 15 -23 13 -12  8 7 7 6 6 5 5 5 4 4 4 4 3 3 3 2 2 2 1 1 1 0  5> 5> 4> 3> 3> 4> 0> 1> 4> 4>  , , , , , , , , , ,  2> 1> 0> 4> 2> 2> 5> 3> 1> 3>  , , , , , , , , , ,  5 5 4 5 4 5 5 4 5 4 4 2 5 4 3 5 4 2 5 3 1 3  -19 -27 3 -7 8 -25 5 17 -30 -15 15 27 -13 8 23 -37 -1 20 -54 0 30 -1  o2>> , ,  97  4.2.3  Fluxionality Table  distorted  4.1  defines  model  on  the populations  of  states  a t some  ^=n7r/3,  fluxional then  their  populations  such that  cross  d i s c u s s i o n t h e number o f bands  between  and E(S,S',n)  rule;  As an example  this  and E(S,S',n)  o f <p. T h e  value  equal  now  depend  on t h e  and  In the  f o r each R(S,S')  will  the exact of t h i s  values  of  n=1....B(S,S').  by  group theory  (A(i//)}  notation consider the fast  correlation  S' 2  B  a rapidly  fluxionality  0  2  N {1} {3} {1,1}  and t h e  are not  system.  S=l/2  a s e t of  become  c a n be o b t a i n e d  s y s t e m . (4.24) d e f i n e s  (4.24)  to  into  will  where  t h e \/>=n7r/3 c o n f i g u r a t i o n s u s i n g  non-crossing  pair  a n d t h e n u m b e r a n d mean e n e r g y o f s t a t e s i n  a s N(S,S',n)  B(S,S')  S=1/2  be d i v i d e d  e n e r g y , and number o f s t a t e s o f e a c h b a n d .  band  If a  are expected  f o r some  and magnetic p r o p e r t i e s w i l l  be d e n o t e d B ( S , S ' ) ,  distortions  each band c o n t a i n s  the  each  f o r fast  s y s t e m b e c o m e d e g e n e r a t e a s ii  of a l l s t a t e s w i t h i n a band  following  for E  of r a p i d dynamic  populations  'mean'  a model  T h e s p e c t r u m , f o r e a c h S', may  f o r which the eigenvalues  thermal  model  o f t h e s t a t e s m u s t be c o n s i d e r e d .  o f s t a t e s o r bands  states  fluxionality  the the effect  in a rapidly  become e q u a l . sets  the slow  tetrahedra  m a g n e t i c t e t r a h e d r a . To d e f i n e  fluxionality  varies  i n 'E' d i s t o r t e d  E 0 0 {3,-3}  essential. fluxional  model f o r  98 The  corresponding  (4.25)  a l lS,  For  is  2S+1  bands  B 1 1 3 2 3  since  i s (see, f o r example,  N{11 {3} {1,4,1} {3,3} {1,1,1} states  i s rapid.  they cannot  I t becomes a p p a r e n t  makes a l l s t a t e s  for  S' 4 3 2 1 0  S=1  i f the f l u x i o n a l i t y  3(2S+1).  there  f o r S=1  t h e S'=4S a n d S ' = 4 S - 1  distortion form  table  equivalent,  a r e 2S b a n d s e a c h  cross,  i . e . B=1  E 0 0 {6,0,-6} {5,-5} {9,0,-9}  are unaffected T h e 2S+1  i s odd  unless  o f N=3. T h e f a s t  S'=0  and t h e band  i fS'  that  Fig.4.1):  S'=1,  by states separation  fluxionality i n which  fluxionality  case  spectrum  S=3/2 i s : (4.26) S=3/2  The  results  S' B 6 1 4 3 2 5 1 3 0 4  N {1} {1,4,1} {1,4,1,4,1} {3,3,3} {1,1,1,1}  E 0 {9,0,-9} {15,6.75,0,-6 {13,0,-13} {18,6,-6,-18}  t o S=5/2 a r e c o m p l e t e d b y :  (4.27) S=2  S=5/2  S' B 8 1 6 3 4 5 7 2 1 4 0 5 S' 1 100 8 6 4 2  B 1 3 3 5 9  1 0  5 6  N {1} {1,4,1} {1,4,1,4,1} {1,4,1,4,1,4,1} {3,3,3,3} {1,1,1,1,1}  E 0 {12,0,-12} {21,9.75,0,-9.75,-21} {27,16.5,9,0,-9,-16.5,-27} {24,8,-8,-24} {30,15,0,-15,-30}  N E {1} 0 {1,4,1} {15,0,-15} {1,13,1} {27,0,-27} {1,4,14,4,1} {36,15,0,-15,-36} {1,4,1,4,1,4,1,4,1} {42,29.25,21,6,0,-6,-21,-29.25,-42} {3,3,3,3,3} {38,19,0,-19,-38} {1,1,1,1,1,1} {45,27,9,-9,-27,-45}  99 All  states  except  those  o f R ( 5 / 2 , 4 ) a n d R ( 5 / 2 , 6 ) c a n be  found  from: (4.28) S' e v e n , S'=0 S'>0  B = ( 2 S + 1 ) N={1...}  S'=4S.  B=1 .  S ' = 4 S - 2 . B=3, N = { 1 , 4 1 }  AE = 6 S .  f  S ' = 4 S - 4 . B=5, N = { 1 , 4 , 1 , . } S ' = 4 S - 6 . B=7, N = { 1 , 4 , 1 , 4 S ' = 4 S - 8 . B=9, N = { 1 , 4 . . } S' o d d ,  with  E={12S-3,6S-0.75,0 ..}  E={18S-9,12S-7.5,6S-3,0,..}  N={3,...}  difficulties arise  large  classical  limit  infinity.  In this  S. T h e s e  i n which limit  , .. . }  E={24S-18,18S-15.75,12S-9,6S~9,0..}  S' = 1 B=2S  Significant systems  AE=6S+3  AE=6S +S-2, 2  i f (4.28) i s a p p l i e d  c a n be d e s c r i b e d  (4.1) i s d i v i d e d there  S'>1  successive  by S  2  fluxionality behave  like  distortion  there those  increasingly  bands,  j u s t one  with  S'>1,  regarded  and thus a l l s t a t e s  S' o d d i n ( 4 . 2 8 ) a n d t h e d y n a m i c  to systems with  large  For large small  band  possibility  S the large separation  behavior  as a crude  S c a n be slow  and f a s t  ( on a n S  2  scale)  implies  an  a n d t h u s an  t r a n s i t i o n s between  bands  i n t o o n e . The  of R(5/2,4) and R(5/2,6) representation  regarded  number o f bands  of non-adiabatic  and hence merging of s e v e r a l  exceptional  vector  o f 4. F o r f a s t  'band'  a s i t u a t i o n which i s between  fluxionality.  increasing  i s now  of each  h a s no e f f e c t o n t h e m a g n e t i c s p e c t r u m . T h e s p e c t r u m  (4.28) as a p p l i e d describing  configurations  the  and S tends t o  i s a continuum of states  stable  to  by c o n s i d e r i n g  S' a n d a n e n e r g y b a r r i e r t o r o t a t i o n o f t h e d i s t o r t i o n between  B=1.  of t h i s  noted  a b o v e c a n be  e f f e c t . The  100 modification of  of the magnetic  S=l/2 and other  therefore The not  finite  decrease  T  2  f o r S=5/2 a r e g i v e n  two n o n - t r i v i a l  because  no n o n - t r i v i a l  orientations  of the T  be l o n g  in detail.  4.3.1  2  defined  2  distortion  than  be c o n s i d e r e d  the three  i s somewhat  low energy  fold  and  the  procedure  and the four a n d an  f o r t h e space less  both  for a l l  be c o n s i d e r e d  diagram  spectrum of  f o r t h e E mode  v e c t o r . Though  could  correlation  distortion  i n chapter  2  does  f o r such  on t h e m a g n e t i c  i s maintained  distortion  between  considered  T  symmetry  f o l l o w e d below  interconversions  The T  2  fluxional  Tetrahedra  c o o r d i n a t e s must  axes w i t h i n T  complete  approach  The  of Magnetic  and i n v o l v e d , both  'rotation'  essentially The  are slowly  i n F i g 4.2.  i s more c o m p l i c a t e d  because  fold  which  c o n s i d e r a t i o n of the e f f e c t  mode d i s t o r t i o n  would  effect.  of E d i s t o r t i o n s  4.3 T-mode D i s t o r t i o n s  tetrahedra  fluxionality i s  as S i n c r e a s e s . T h e o r e t i c a l curves  distortions  The  of E d i s t o r t e d  s p i n s by f a s t  a p u r e l y quantum effect  behavior  complete;  configurations in T  obtained. only 2  are  Hamiltonian Hamiltonian  H'(p,0,0) = H'(x,y,z)  was  3.  (4.29) H,l.(p,0,0) = - 2 J p [ ( S , . S - S 2  3  .S, ) s i n 0 c o s 0 + ( S  .S -S  1  3  2  .S, ) s i n 0 s i n 0  + (S,.S„-S .S )cos0)] 2  3  = -2Jp [ x ( S , .S -S .S,)+y(S,.S -S .S„)+z(S,.S„-S .S )] 2  3  3  2  2  3  F i g . 4 . 2 . Moments o f E d i s t o r t e d S » 5 / 2 t e t r a h e d r o n ; f r o m t o p t o b o t t o m , X=2,3,4,5 and 6.  1 02 where  x + y + z = 1 . The J p t e r m ,  taken  as 1 i n the t h e o r e t i c a l discussion.  2  2  The H'(E);  result L X  and  =constant  2  i ti sproved  H'(x,y,z)  = xH, + y H  Since  permutations i.e.  H,,H  t o H'(T ) 2  be  just as t o i s expressed as  3  2  = x H, +y H  2  2  and H  2  2  2 2  +z H 2  2 3  are similar  3  which a r e elements = Tr[H, ]  2  H', w i l l  + z H , w h e r e H,=H' ( 1 , 0 , 0 ) , H = H ' ( 0 , 1 , 0 )  2  2  Tr[H' ( x , y , z ) ]  H'(x,y,z)  applies  scales  2  3  3  only  i n a s i m i l a r way. I f H ' ( T )  H =H' (0,0,1 ) , H'  zxH H,).  which  2  2  (they  i s similar to H'(x,-y,-z),  2  2  +  3  a r e r e l a t e d by  o f P « ) T r [ H,]  + 2 Tr[H,H  2  + 2(xyH,H +yzH H  = Tr[H ]  =  2  + H H 2  3  Tr[H ], 3  + H H, ] . But 3  H'(-x,y,-z) and H'(-x,-y,z)  thus Tr[H,H ] = 0 and Tr[H' (x,y,z)]  = Tr[H, ]  2  2  2  = L X  =  2  constant.  An  exact  obtained contain  solution  comparatively, thetotally  for  S=1/2 v a l i d  easily.  symmetric  2  representation  by T . d i s t o r t i o n .  polynomial  c a n be. e x p r e s s e d a s :  (4.30) where is  X  - c (r)X 2  proportional  to r . 2  2  i s zero  f(x,y,z)  = 0 as H ( T ) i s t r a c e l e s s and c 2  i s symmetry  and f(0,y,z)=0.  all  a and,  by p e r m u t a t i o n  all  a,/3, 7, x , y , z . T h e r e f o r e  Therefore  o f x,y,z  s o l u t i o n c a n now b e o b t a i n e d  f(x,y,z)  f(ax,y,z)=af(x,y,z) f o r  f ( ax , j3y , yz ) =aPy£  f(x,y,z)  2  related to  i . e . f ( x , y , z ) = f ( - x , - y , z ) = - f ( - x , - y , - z ) . Thus  -f(-x,y,z)  complete  and the states a r e  F o r S' = 1 t h e c h a r a c t e r i s t i c  - f(x,y,z)  the c o e f f i c i e n t of X  f(-x,-y,z), =  3  c a n be  F o r S'=0 < E | T | E > d o e s n o t  unaffected  2  f o r a l l x,y,z  i sproportional  (x , y , z )  t o xyz.  from any p a r t i c u l a r  The  for  1 03 solution.  Taking  factorisation -4,  (see  f(x,y,z) =  visualise  (4.31))  of  the  triangle  the  the  i n the  is  dimensional.  a l l S,  the  denoted  S"=S  elements  + 2  =  3  H'  0  state of  can  be  0  (S'  has  -  2  P  3  of the  produce of  = 8 =>  2  by =  2  to  0  f o r the to  in this  a  case as  this  E  mode,  rotation  of  the the  H'(x=y=z=-l//3), system,  c  is easier  ) =  +  i s more c o m p l i c a t e d  state  1//3  S +S„.  eigenvalues  Table  H'  plane,  obtained  expressed:  corresponds  rotation  a=0,  is  the  Hamiltonian  factorised:  2S"  +2S )  2  2  symmetry  with magnetic  (E,P(23),P(24),P(34),P(234),P(243)}  distortions The  vector  energy  ground  H'°.  ( 4 . 3 2 ) H'°  where  is  0  eigenvalues  p o s s i b l e energy  be  equation  X  result  X = 4/v/3. T h u s , a s  distortion  of  will  The  0  behavior  lowest  = -16.  3  o  rotation  For  solution,  ( X - X c o s 0 . ) (X-XQCOSC?) (X-X COS6>  1  three  the  i s 1 / (j/3 ) {4 ,-2 ,-2} , L  characteristic  0=cos" (-3/3xyz) and  S'=1  for which  -16/3»/3=C3/3v/3, c  i f the  (4.31)  where  x=y=z=1//3,  tetrahedra with H'(-1,-1,-1)  point  and  group  for a l l S <  5/2  group  i t s associated symmetry are  ^2^'  listed  in  4.2.  It  can  distortions, largest  be  seen  'T'  amount  from  the  table that,  mode d i s t o r t i o n s  and  leave  the  S'=0  split states  in contrast to the  'E'  S'=2S s t a t e s by  unchanged.  mode the  104 Table  4.2 The  total  S o l u t i o n s of the T d i s t o r t i o n degeneracy  of a s t a t e  |S',S">  Hamiltonian i s equal  s p i n = S" s t a t e s o f t h e c o r r e s p o n d i n g  fi(3,S,S").  The  energies  comparison  with other  should  be d i v i d e d  eigenvalues  t o t h e number o f  t h r e e atom  b y /3  of (4.29)  H'(-1,-1,-1)  system,  before  which  requires  x +y +z =1 . 2  2  2  S = 1/2. |2,3/2>  "4;  1,l/2> 3,2> 1 ,2>  o, -2, 6,  3,3> , 2, 1> 0, 1>  -8; 6; 0.  3/2, 9/2> 7/2> 5/2> 3/2> l/2>  0 -4 2 6 8  5,9/2> 4,5/2> 3,3/2> 2,l/2> 0,3/2>  , -12; , 10; , 12; , 12; , 0.  0 -6 2 -8 •16 •22 18 0  5/2. ),15/2> ( 0 1 3/2> -8 1 l/2> 2 1 l / 2 > -12 1 3/2> -50 5/2> 30 7/2> 6 9/2> -20 28 3 l/2> 2 3/2> 16 1 3/2> 12  1 8 7 6 5 5 4  1,3/2> ,  1 . 3> 2> 0>  2. 6> 5> 4> 4> 4> 4> 0> 2> s  o,  5,7/2> 3,9/2> 2,7/2> 1,5/2>  ,  2; 4; -6;  , 6; , -30; , -18; , -8;  0,1/2> , 2,3> 1 , 1>  4,9/2> 3,7/2> 2,5/2> 1,3/2>  0. -14; 2;  , -22; , -12; , -4; , 2;  7,6> 6,4> 5,3> 4,3> 3,3> 2, 3> 1 , 3>  •16 14 18 8 0 -6 -10  7, 5> 5,6> 4,6> 4,2> 3,2> 2,2> 1 , 2>  32 •42 •52 20 12 6 2  6,6> 5,5> 4,5> 3,5> 3, 1> 2, 1> 1 , 1>  •30 •18 -28 -60 18 1 4 10  9,15/2> 8,1l/2> 7,9/2> 6,9/2> 5,1l/2> 4,13/2> 4,5/2> 3,7/2> 2,9/2> 2,l/2> 0,5/2>  •20 18 24 10 •24 •60 20 -2 •26 22 0  9,13/2> 7,15/2> 6,15/2> 6,7/2> 5,9/2> 4,1l/2> 4,3/2> 3,5/2> 2,7/2> 1,7/2>  10 •54 •68 28 -2 •34 30 12 -8 •12  8,15/2> 7,13/2> 6,13/2> 5,15/2> 5,7/2> 4,9/2> 3,1l/2> 3,3/2> 2,5/2> 1,5/2>  •38 •24 •38 -80 16 -12 •42 22 6 2  1 05 The space and  path  followed  i s not uniquely defined  a more p r e c i s e  description behavior  considered Chapter  and f a s t  and t h e problem  first  most  configurations, {(-1,-1,-1)}.  time  the slowly  distortions  well a  2  and of  detailed  suggestions f o r the  fluxionality will  systems a r e  be d i s c u s s e d a g a i n i n  configurations  {(-1,-1,-1)} define  a cube  to interconvert  represented  The  system  low energy denoted  defines a tetrahedron i n the T  and (1,1,1), denoted  {(-1,-1,-1)}  cluster.  (- 1,- 1,-1) , ( - 1 , 1 , 1 ) , ( 1 , - 1 , 1 ) , ( 1 , 1 , - 1 ) ,  {(-1,-1,-1)}  opposes  (-1,1,1),  fluxional  i n one o f t h e f o u r  The h i g h e n e r g y  (1,-1,-1)  energy  Plausible  to obtain  i n the T  7.  spend  which  a s i t w a s f o r *E' mode  be n e c e s s a r y  t h e slow  below,  Consider  space.  would  of the motion.  of both  vector rotating  understanding of the p o t e n t i a l  time c o n s i d e r a t i o n s  will  by a d i s t o r t i o n  (-1 , - 1 , 1 ) , ( - 1 , 1 , - 1 ) ,  {(1,1,1)}, define  i . e . the points in T . 2  a tetrahedron  {(1,1,1)}  I f the system  +  has just  b e t w e e n members o f { ( - 1 , - 1 , - 1 ) }  as following  paths  such  (0,0,1), (1,-1,1), along  2  as (-1,-1,-1),  enough  i t may b e  (-1,0,0),  t h e edges o f t h e t e t r a h e d r o n  {-1,-1,-U. Consider  now t h e r a p i d l y  interconversion with in  i s rapid  fluxional  the distortion  a n a n g u l a r momentum m a k i n g  T . 2  great  Only circle  must a l s o  two p o i n t s  from  through  a point  I fthe  v e c t o r may b e  i t tend t o follow  the s e t {(-1,-1,-1)}  and a great c i r c l e  pass  cluster.  through a point i n {1,1,1}.  associated  a great  circle  c a n l i e on a i n {(-1,-1,-1)}  The s y s t e m  c a n be  1 06 pictured circles  as precessing defined  temperatures  equal  from  circles  free  {1,1,1}. At very  will  energy  between  be a s s o c i a t e d  and e s s e n t i a l l y  great  high with  free  precession  occur.  The  c o r r e l a t i o n of the spin  through can  by two p o i n t s  a l l great  approximately will  (perhaps d i s c o n t i n u o u s l y )  states  (1,1,1) and (1,-1,-1) w i l l  denoted  circle  now b e c o n s i d e r e d .  b y : x=cos/3 y = z = / 2  be p a r a m e t e r i s e d  f o r the great  sinj3 a n d w i l l  The p a t h be  P(/3) :  (4.34) =2 ( S , . S - S  H'(0)  2  3  .S« ) c o s j 3 + / 2 ( S !. S + S ! . S - S 3  H' c a n b e f a c t o r i s e d b o t h  when  a  2  .S -S u  . S ) sin/3  2  3  | x | = | y | = | z | ( g i v i n g H ' ° ) a n d when  y=z=0:  (4.33)  H*(-1,0,0) = 2  H'(-1,0,0) w i l l  T .  areassociated  S=1/2,  For  distortion. (4.30),  only  States  with  C  =  S  the corners 2 v  2  -  S  2 3  which a r e  of an octahedron i n  a r e a f f e c t e d by T  I f t h e s y s t e m h a s no  fluxional  between  only  that  part  with  an e n e r g y  I/(16/3) - 2 =0.31; i t i s t o be e x p e c t e d  will  rock  'slowly'  between  2  from angular  o ft h e  (1,-1,-1) and (1,1,1)  only  and f o r t h  U  distortions of the cluster.  i n { 1 , 1 , 1 } may i n t e r c o n v e r t  back  2 1  s o l u t i o n , w h i c h c a n be o b t a i n e d  i n Fig.4.3.  momentum a n d i s s l o w l y diagram  define  1  t h e S'=1 s t a t e s  The e x a c t  i splotted  correlation  3  1  r e l a t e d t o H'  They  2  be d e n o t e d H ' . T h e s i x H a m i l t o n i a n s  symmetry 2  (S, .S -S .S.i,)  that  {1,1,1}.  i s involved. b a r r i e r of t h e system The spectrum  1 07  1 1  -1.0  I  I  I  I  -0.6  -0.2  0.2  0.6  BETR/PI  F i g . 4 . 3 . C o r r e l a t i o n diagram T distortion. 2  f o r S=l/2 t e t r a h e d r o n  1  1.0  for  108  is the  divided other  i n t o two b a n d s , containing  exhibits  fast  (or near  Fig.4.3  that  i t will  to) a point  the ground  states.  state  and  I f the c l u s t e r  be u n a b l e  to avoid  passing  i n '{-1,-1,-1} . I t c a n be s e e n  the f l u x i o n a l i t y  will  cause  a l l the states  from  to  equivalent.  It state  t h e two e x c i t e d  fluxionality  through  become  one c o n t a i n i n g  c a n be  of a T  resistance relative  seen  from  t h e above d i s c u s s i o n  d i s t o r t e d tetrahedron  2  to fluxionality  since  to the undistorted  may  there  system  that  exhibit i s no  the ground  considerable  loss  of  the f l u x i o n a l i t y  energy becomes  rapid.  Group  theory  c a n be a p p l i e d  the  magnetic  for  t h e 'E' mode d i s t o r t i o n s s i n c e P(34),  element, group P .  expressed, transform  the  i n |S',S as A , 2  state  must  two A  2  (4.35)  Since At  , S  3 4  >  2  +  c a n be  c, ( 0 )  X  (4.33)  i s X +2X=0 2  0=cos" (1/V3) w i l l 1  obtained  which  that  this  which  found  +  c  2  gives  a  magnetic  c a n be  | 1,1,1>, | 1 , 0 , 1 > ,  as  transforms  than  symmetry  f o r j3=0  X=X(/3=0)cos/3 = 2cos/3.  a s A,. The  The  energies  which single of  by w r i t i n g :  (0)  c  =  2  0  c a n be e x p r e s s e d  i n d i c a t i n g that  be d e n o t e d  shows  a single  states  notation  Tr[R(S' = 1 ) ]=0 c , = 2 c o s 0 .  0=0,  S' = 1  |1,1,0>,  and  have energy  states  X  1 2  only  f o r a l l (3;  the three  of c o r r e l a t i n g  P ( 0 ) but i s l e s s u s e f u l  f o r the path  i s maintained  For S=l/2  2  A,  spectrum  t o the problem  0  O  .  the f u l l  At 0 = 0 , O  d=-e.  c =d+e  The  2  cos/3.  angle  X +v/(4/3) X+8/3 = 0 i s 2  solutioni s :  109 (4.36) 4.3.2  X  S  >  For atom the  +2cos0  2  1/2  S>l/2 problems  based magnetic first  time.  introduction  of  individual dimension  from  of  this  similar given  For states all  S  give  largest  that  for  S=1,  the  the  unaffected  of  the  2  from  'E'  be  factorised  O  the  greater  (E)  degeneracy  in P ;  one  3  non-crossing of  mode a n d  three  result  rule  at  detailed  than  of  S'=3  s t a t e s behave the  i n the  general  T  f o r the  distortion.  Apart  0  i s now  As  s t a t e s are  2S.  same way  result The 'E'  as  as  A .  {X(A,)}={-4,0,2}, will  1  (X(A,)}  : X : *  3  X  3  states  mode,  +  | 1 , 2 , 1>, |1,1,1> a n d  also give  and  for A . 2  2  2  2  -  /(4/3)  8X  =0  -20X/3 + / ( 6 4 / 3 )  =  0  which  Clearly  the  Writing:  (c cos3/3+c 3  Consider  which  2  solution  for  are  |1,1,0>,  {X(A )}={4,0,-2}.  the  S'=1  the  |1,0,1>,  + 2 X c o s / 3 + ( c , +c c o s 2 / 3 ) X +  +2X  3  will  the  f o r S'=4S-1  S'=0  | 1 , 2 , 2> , | 1 , 1 , 2>  transform  for A  very  calculation  f o r S'=1,S'=2 i s s o m e w h a t m o r e c o m p l i c a t e d .  2  is  S=1.  by  /3 = 0 t h e  the  this  /3=/3 .  factorisation a  for  0= 0 ,  atoms a t by  the  arise  to a  and  0 = 0o  three  2,  of  A,,  /3 = 0  0)  2  of  spin consisting  as  (4.37)  in chapter  (4.30) m u l t i p l i e d 2  incompleteness  the  transform  solution  the  of  application  S=l/2 system;  by  calculation  rise  I.R.  f o r the  only  to  intermediate  difficulty,  of  At  an  the  i s just  S' = 1.  Hamiltonian  breakdown  to  X =' -cos/3 ± / ( c o s /3 + 4 s i n  discussed  equivalence  s p i n s , may  apparent  0,  arising  groups,  The  the  be  2  systems  which allows  an  X -4s i n ^ =  cos0  fl  =  0  1 10 The  solutioni s : (4.37)  For  0 = 0,  A  and  1 f  the  X  +2X cos0 2  3  t h e S'=2  ( 7 + c o s 2 0 ) X +3 ( c o s 0 - c o s 3 0 )  -  states are  The  2  as those  '(4cos 0+6sin 0) 2  The  2  two A  A,  s t a t e s c a n be  2  f o r S = l / 2 S'=1. four  The  result  s t a t e s have  +'4X cos0 3  a  +(b+b'cos20)X  (d+d'cos20+d"cos40) 0=0  X"  0=0o  X"+4X cos0  The  + 4X  3  - 37X ~  3  o  2  -  144X  Writing:  (128/3)X  to define  d,d'  shown  eigenstates  with  Thus  d"=-63. The  solution for general  u  information  is  a r e r e l a t e d b y a C«  the point  0 c a n now  as  (1,0,0),(0,1,0 ) ,  (1,1,0) has  d+d'+d"=d-d'+d"=0, d'=0  and  axis i s  two  A  2  d=63,  be w r i t t e n :  2  f o r S=1  S'=2,1, i s shown  of the c o r r e l a t i o n  i s needed  fluxionality from  that  3  result  part  (1,1,1)  = 0  X +4X cos0+(5cos20-41)X -(6Ocos0+84cos3 0)X+63(1-cos40)=0  The that  112  I f the path  the points  i t c a n be  (4.39)  o  o f X° m o r e  considered,  X=0.  +  + c'cos30)X +  f o r c and c' a r e o b t a i n e d  and d".  (- 1,0 , 0 ) , ( 0 ,-1 , 0 ) , w h e r e  (c c o s 0  = 0  +8OXcos0  2  but f o r the c o e f f i c i e n t  necessary  +  = 0  s o l u t i o n s f o r b and b', and  before  2  ±  energies O  X  found i n  i s X = +2cos0  {X} = { 6 , 0 , - 2 , - 6 } a t 0 = 0 a n d i/3 {- 1 4 ,-2 , 6 , 6} a t 0 = 0 .  (4.38)  0.  | 2 , 2,2>, | 2 , 1,2> , | 2 , 0 , 2 > , | 2 , 2 , 0 > ,  |2,2,1>,|2,1,1>, A .  same way  =  f o r slow  diagram between  fluxionality.  i s rapid the effect  the spectrum.  i n Fig.4.4.,  of T  2  As  again  only  (1,-1,-1) and with  distortion  S=l/2, i f the i s removed  111  l l  I  -] 0  Fig  4.4.  -0.6  I  I  -0.2  0.2  THETfi/PI  I  0.6  C o r r e l a t i o n diagrams f o r T d i s t o r t i o n of S=1 t e t r a h e d r o n . S'=2 ( a b o v e ) and S'=1 (below). 2  I  1 .0  1 12 If the  this  effect  magnetic can  it  i s clear  by c o r r e l a t i n g  that  there  c a n be o n l y  i s never  dynamic  equal  rule  t o t h e number  distortion  H'=H'(- 1,- 1 , - 1 ) ,  f o r each model  a factorisable  H'=0.  4.3.3  distortions  from the i s rapid.  energy  of lowest  2  energy  distorted  by: ( i ) Slow  Hamiltonian,  t e t r a h e d r a of fluxionality,  ( i i ) Fast  of tetrahedra  i n 4.2 a n d 4.3 d o n o t a m o u n t  t o a complete  general  distortions.  would  example, a T  For  splits  can always  six  though,  i nthe  energies.  both  distortion;  the T  2  similarly  each w i l l  lowers  t h e symmetry of  mode a n d t h e d e g e n e r a c y significant  t h e E mode. T h u s t h e t w o t y p e s as competing,  even  be i g n o r e d  fluxionality, E distortion  2  results  solution for  involve a l l  be r a t h e r c o m p l i c a t e d  distortion  and s p l i t s  the T  envisaged and  2  o f t h e S'=0  fast  system  favours  Such a s o l u t i o n  o f {J} and would  calculation  the  states  o f S'.  derived  for  states of a  H ' ( E ) a n d H'(T) do n o t , i n g e n e r a l , commute, t h e  variables  This  applied fora l l states  value  forT  be s u m m a r i s e d  fluxionality,  that  between H(-1,-1,-1) and H(1,1,1) (see  one 'band'  s p i n can thus  As  removed  s i n c e t h e number o f h i g h e s t  spin  General  i t i s found  i f the f l u x i o n a l i t y  i f the non-crossing  given  The  f o r S>1  is entirely  s p e c t r u m of any s p i n  4 . 2 ) . Even  general  i s continued  of T d i s t o r t i o n  be s e e n  Table  approach  tend  T  2  of d i s t o r t i o n t o 'quench'  t h e r e f o r e c o n s i d e r a t i o n of the general  which  distortion c a n be the other,  distortion  113 Hamiltonian For retains  i sunlikely  both E and T  t o be n e c e s s a r y . slow  2  some d e g e n e r a c y  f l u x i o n a l i t y the magnetic  and hence  secondary  d i s t o r t i o n . The s l o w T  therefore  a n E mode. T h i s  derived  from  original  E distorted in this  occur. the  system case  Thus  triplet  4.4 T e m p e r a t u r e  that  i s also  f o r S=l/2,  low energy  The  T  might  such a secondary closer  mode. T h i s  the long  to the singlet  ground  move  state.  barrier  to rotation  symmetry magnetic energy  barrier  into  3 smaller  wells  are related  cluster  tends wells  forT distortion, by symmetry  of the within  a  to divide the forE i n each  operations  case  of the  cluster.  Two e x t r e m e barriers  d i s t o r t i o n would  a n d t h e p r e c e d i n g c h a p t e r h a s shown  of the c l u s t e r  and 4 smaller  distortion  2  mode a n d  F l u x i o n a l i t y (TDF)  normal  undistorted  a p p l y . T h e E mode o f t h e s l o w  with  in this  wells  of  associated  degenerate  secondary  such as t h e lack  2  of a high  the  mode o r t h o g o n a l t o t h e  2  the o r i g i n a l T  vector  distortion  still  i s t y p i c a l l y an e n e r g y  well  a x i s and  3  from  distortion  potential  a C  derived  Dependent  discussion  there  states  effects  retains  i s n o t t h e o r i g i n a l E mode b u t o n e  t h e two components o f t h e T  o f t h e S'=0  i s a p o s s i b i l i t y of  system  2  d i s t o r t i o n , and so r e s u l t s  splitting  so  there  spectrum  have  types of behavior associated  been d e f i n e d .  Thus  l i f e t i m e of the distorted and short  compared  with  i n slow and f a s t  configurations  t o the rate  a t which  are,  these  energy  fluxionality respectively,  the magnetic  114 spectrum  attains  distorted  thermal equilibrium.  configuration  may  depend  reasonable  to expect  that,  transition  from  to  increases,  i . e . temperature  occur.  As  such  slow  fast  between  somewhat  model as  rate  low  5.  determine  the  equilibrium,  2) T h e  constant  9J/3d) , since  and  potential system is  one  state.  spectrum  A  full  the magnetic which larger  treatment of  and  somewhat  important parameters,  states  of  not  constant  the  same s p i n ,  known. As of  S',  the motion  spectrum  energy  but  of  and no  a  the  since  between  derived.  include  shape of itself.  to  1)  force  the Thus  a  triplet  state  interconversion from  would  of  The  against  the  be  ground  both  very  t h e v a l u e s o f many  states  of  provides a  thermal  relaxation  intermediate  for rotation  be  tested  removed  factors  the  now  low-lying  are well  such as  independent  the v i b r a t i o n a l  barrier  the above  the  fluxionality  spectrum  includes  redundant  be  may  forms.  attains  magnitudes  in.which the t r i p l e t s  complicated  are  3)  e x p e c t e d t o have a  than  the  a  and  will  these a f f e c t  0  well,  with  relative  will  type of  i t is  o r TDF,  model w h i c h  f o r TDF  a  temperature  a temperature  a new  tetrahedra  the magnetic  the  high temperature  in chapter  which  at which  and  model  t o S=l/2  data  Factors  the  as  fluxionality,  by  of  circumstances, a  modelled, using  approach,  tentative  applied  experimental  The  dependent be  lifetimes  i t s thermal energy  suitable  i t i s necessary to define  connection  A  on  fluxionality  behavior cannot  Heisenberg-Dirac-VanVleck spectrum  under  Since the  rates  between  different  quantum number,  the d i s t o r t i o n  of  spin, is a  vector,  1 1 5  the  relaxation  assumed  t o be much  different expected of  this  spin.  to  each  smoothly  slowly  i s now  (4.41)  as  i s required  i s a total  from  f  0  =  p f  0  +  of  f  0  , and a h i g h  an  on  to define  function,  f, which  to f, at high  'effective'  fraction in  (1~p)fi  a form  barrier  f o r p such  that  p=1  a t low  simple form i s :  of the exponential  corresponds  to e=0,  f=f,  corresponds  t o e>>6  since  f o r a l l T; in this  significantly  from  Fast  slow  case 1.  c a n be r e g a r d e d  of t h e system  o f h e i g h t e.  a  formalism  exp(-T/e)  an energy  p varies  depend  write:  c o r r e s p o n d i n g t o the thermal energy  before  S'  temperature  t o a p p l y t h e HDW  to define  t h e T i n t h e numerator  overcome  Because  of d i f f e r e n t  i t i s possible  a n d p = 0 a t h i g h t e m p e r a t u r e . One p =  relaxation i s  fluxionality.  partition  p, and  of s t a t e s of  to interconversion  a t low temperature  form,  sets  o f t h e same S'.  clusters  function,  I t i s possible  f  for states  a g i v e n S'  n e c e s s a r y t o choose  temperature  where  data  spin-lattice  the rates  partition  o f t h e same S' c a n b e  between  barriers  with  fluxional  (4.40)  It  will  cluster  f i tmagnetic  that  to consider  f , . A l l that  temperature. the  effective  As t h e e n e r g y  temperature  changes  than  In other words,  so, i n general,  function,  a s e t of s t a t e s  faster  i t i s possible  For low  within  t o be m o s t  separately. S'  rate  which  must  fluxionality  fluxionality  f , becomes c l o s e  to f  0  1 16 For  the  simplest  example  tetrahedron,  i n which  only  fluxionality, partition  function,  (4.42) where slow  35  application  f  the of  triplet  (4.40)  m u s t be  unique,  i t was  transition potential  triplet  splitting  originally  barrier, 8 6  .  due  which  I t may  suggested  be  has  an  that  +  by  triplet energy:  9(l-p) spectrum  for  case  even  though  large  reduce  tunnelling  concept and  fast  magnetic  n u m b e r , may magnetic  of  fluxionality,  data because be  be  as  effective involved  though  well  as  are very  of c l u s t e r s  function  more d i r e c t l y ,  fast  and  slow  satisfactory,  postponed  the next c h a p t e r .  until  such as  a decision  on  tend  small. which  Also exhibit  a  measure are  to  treatment  i s just  models  an  i n the present  for a  e.s.r.,  fluxionality  of  thermal  mass w i l l  satisfactory  the p a r t i t i o n  from  simple  i n the numerator  is significant  proportion  is far  through a  i n a d e q u a t e where e x p e r i m e n t s which  the  theoretically must  the d i s t a n c e s  'effective'  spectrum  Thus, while  the  for p  the e x p r e s s i o n f o r  tunnelling  barrier  since  by  energy  the energy  in  a TDF  form  to tunnelling  over  TDF  are affected  i n the magnetic  the above  activation  of  gives  S=l/2  r e s p e c t t o t h e mean t r i p l e t  admitted that  probability  exponential  slow  states  the  fluxionality. It  the  system,  = 3p{2exp+5/T +exp(-26/T)}  3  i s the  3  with  o f a TDF  simple the  involved.  are  the above model f o r  the experimental data are considered  117 4.5  Dynamic  The to T  0^,  2u^  and  a  magnetic  the n  d  t  *  indeed,  E  H  a  m  ^  pathways  H  =  0  in  T  t  o  n  ^  a  Octahedra  of  the  The  '  n  of  In  general  cluster  undistorted cluster  normal  o-  H  couplings, J  and  g  2  e  2 1 3  to J  -S  2 2 5  modes w h i c h  contains  can  be  an  reason  and  g  -  2  atom  n.  J (S i  As  to  +  T  2g  +  T  iu  +  'exterior' and  equal  2 1 f l  for  be  g  t o assume  of  degeneracies  expected  E  J^=0;  active central  are  3  +  isomorphic  adjacent  exchange  -S 6 )  i s opposite  spectrum  independent  i s no  is  g  between  contains  -J (S' -S  modes a r e  contains  there  corresponding  (4.43)  magnetic  and  group.  atoms.  A t o m n+3 the  i e  i f the  (4.43)  group  point  'interior'  opposite  the  Distortions  +S  2 2 5  the  atom  length.  +S  -6S )  2  2  3 6  tetrahedron,  associated with removed  by  the  dynamic  distortions.  As group  for  the  symmetry  trigonal (unless  factorable  distortion  aspects  the  of  The states  S=l/2  spectrum  may  of  denoted  bipyramid  J =J^) severely g  Hamiltonians will  (4.43)  f o r S=l/2  most  simply  now  by  discussion i t i s useful to  use  notation  parenthesis latter  |S'(S, „ S,„S  l i e i n an  notation gives  3  6  3 6  )S  unique  limits  full the  permutation  number  commute w i t h  of  H .  Some  0  be i n v e s t i g a t e d .  i s given  |S'S,„S  2 5  S  3 6  introduce 2 5  >,  e q u a t o r i a l plane a  l a c k of  which  system  present the  the  in >  the  d e s c r i p t i o n of  but  in  3  6  the  spins  octahedron. each  The  the  S , „ =S , „+S  in which of  (4.44).  state.  and  3 6  in This  1 18 | 3 ( 2 1 1 ) 1>  -6J e  | 2 ( 2 1 1 ) 1 > , |2(1 1 1 )1> |1(211)1>, |1(111)1>, |1(011)1>  ,  +4J  | 0 ( 1 1 1 ) 1>  + 6J  | 2 ( 2 1 1 ) 0 > , | 2 ( 1 1 0 ) 1 > , | 2 ( 1 0 1 ) 1>  e  -2J  | 1 ( 1 1 1 ) 0 > , | 1 ( 1 1 0 ) 1 > , | 1 ( 1 0 1 ) 1>  e  2J  | 0 ( 0 1 1 ) 0 > , | 0 ( 1 1 0 ) 1 > , | 0 ( 1 0 1 ) 1>  e  e  4J  -  3J /2  -  3J /2  -  3J /2  -  3J /2  +  J±/2  +  J./2  +  J /2  i  i  i  i  L  e  |1(000)1>, |1(110)0>, |1(101)0>  5J /2 i  9J /2  | 0(000)0>  Note  that  there  S1a=S 5=S 2  results  3 6  =1  a r e two s o u r c e s  the combination  i n a degeneracy  N=3, S = 1 ; i f S related  i a  ,S  2 5  and S  only  external  3 6  of degeneracy  of t h e three  found  to interchanges  The the  {  from  factorable  pairs  J a n d commutes w i t h  ( i nunits 3 ( 2 1 1 ) 1> 2(211)1>, 1(211)1>, 0 ( 1 1 1 ) 1> 2(211)0>, 1(111)0>, 0(011)0>, 1(000)1>, 0(000)0>  (4.45) axes  of  vectors.  The g e n e r a l  diagram f o r  there  i s degeneracy  spins.  Hamiltonian  which  modifies  0  2 1 3 t t 6  + 2S, „  2  + 2S  36  2  - S  25  2  )  r ) :  2( 1 1 1 ) 1> 1(111)1>, 1(011)1> 2(110)1>, 1(110)1>, 0(110)1>, 1(110)0>,  and t h e two H a m i l t o n i a n s  of the octahedron  of  momenta  H i s :  2  solutions  the branching  distortion  ( 4 . 4 5 ) H' = r / 2 ( S ' - 3 S with  angular  are not equal  of these  i n the system: i f  2(101)1> 1(101)1> 0(101)1> 1(101)0>  related  sum t o z e r o ;  distortion  they  , o , -3,+3 , " 5 , 1 ,4  , o , -2,1,1 , , , ,  2,-1, 4,-2, 0,0,0 0.  by p e r m u t a t i o n a r e E mode  Hamiltonian  forthis  of the  distortion mode c a n b e  1 19 written:  cosd.(S .S3+S3.S +S .S +S .S )  (4.47) H'(0) =  2  + cosd  5  5  6  6  2  (S,.S3+S3.S^+Sa.S +S .S , ) 6  6  + c o s 0 ( S , . S + S .Sn+Sj, . S + S .S , ) +  The  8 symmetry  more  symmetry  to I^h'  a  r  i s retained  states  with  e  m a  o n e o f {S , „ , S  with  two o f { S , „ , S 5 , S  shows  that  (4.46)  same  way It  states  c a n be s e e n leaves  with  the  ground  }  throughout  2 5  =S  3 6  that  t o zero  =0  except  2 5  =S  those  of three  form  as A  u  and  B^+B2^ B^^ +  . Analysis a s A^ a r e  0 independent  spectrum  For slow  like  those those  as  transforming  plane  group  tetrahedron).  with  eigenstates 9 i n the  tetrahedron.  E mode d i s t o r t i o n  behave  a  as  in  transform  of t h e S=l/2  triangle.  o f one o r  as B^+B^+B.^;  i n the energy  =1  of the  transforms  for (this)  3 6  (just  as  to zero,  the magnetic U  permutations  transform  equal  states  S, =S  o f t h e S'=1  S'=0,  3 6  rotate  a s t h e S'=1  the  =1  } equal  36  S,„=S  therefore  states  3 6  o f 0. T h e g r o u p s  fluxionality the  with  =S  a l l eigenstates  independent in  ,S  2  the state  with  5  ( 1 4 ) , ( 2 5 ) , ( 3 6 ) ,which  intainecl  2 5  with  and  5  for a l l E distortions  S , „=S 25  2  associated  of the p a i r s of spins  isomorphic  The  elements  2  unchanged the  exhibit  the largest s p l i t t i n g  state  i s S'=3  f o r r < 2 a n d S'=1  except  that  corresponding  fluxionality  states  fast  t h e S'=1,  and i f J >0, e  f o r r>2, never  other  internal  J c a n be v a r i e d  and the e x t e r n a l  J with  independently  the distortion  of both  J^=0  S'=0  S' =2.  The  not  each  Hamiltonian:  or  1 20 (4.48)  which H ,  H" =  xS,« +yS 2  commutes w i t h  f o r x + y + z = 0 H"  0  (4.49) The  H"  H  2 1 l t  assumption that  magnitude (4.45);  of  there  with  S  i a  =S  there  =S  (4.50)  3 6  .  2(x+y+z)S  I f x=y=z H h a s  + S  2 2 5  is linear  cos0  + S  one The  cosi9  (4.49) does  remaining  states  have  | 1(110)0>,|1(101)0>, |1(000)1>  removes  the e f f e c t  Where  and  consideration such  shows t h a t  of d i s t o r t i o n J  are at least  g  of d i s t o r t i o n  constrains  fast of  the  of  fact states  energies:  = =  -2,1,1 -2,1,1  =  -2,1,1  =  2,-1,-1  fluxionality  the form  completely  (4.49).  approximately  Hamiltonians  the  to that  from the  not a f f e c t  | 0(011)0>, |0(110)1>, | 0(101)1>  (3.17)  of  +  i s clear  | 2(211)0>, |2(110)1>, | 2(101)1>  of  t h e symmetry  t o be p r o p o r t i o n a l  | 1(111)0>,|1(110)1>, |1(101)1>  Application  2  i n the d i s t o r t i o n  (4.48)  E mode.  2  3 6  i s such a c o n s t r a i n t  i s only 2 5  cos0.  J  -  2 3 6  i n t h e E mode, i . e . :  the d i s t o r t i o n  that  that  a n d H'.  0  lies  = S  +zS  2 2 5  equal,  b a s e d on  the C  axis,  3  as: (4.51)  H = 2S'  - 5S  2  2 1  2  3  - 5S  2 t t 5 6  +  18S  2  and (4.52)  would  give  spectrum. a  H  =  S  2 1  some i d e a (4.51)  combination  2  3  -S„  of  results  o f T.  and  2 5  6  the e f f e c t s from T g 2  T„  o f T mode d i s t o r t i o n s  distortion  distortions.  and  I t can  (4.52) be  seen  on  from that  the  121 distortion and  may  i s associated  produce  = 0. T h i s  with  a s i n g l e t ground  of d i s t o r t e d high-spin  of  (6  even  state  approximate method would  treatment states  a large  6  i s blocked  splitting  i n a system with  by S .  J > g  0,  be e s s e n t i a l f o r  c l u s t e r s , as the large  f o r S=5/2) makes m a t r i x  i f the matrix  singlet  solution  number  impracticable  1 22 CHAPTER  5  A P P L I C A T I O N S OF THE DYNAMIC  Dynamic pronounced coupling least  in clusters  constants  a three  necessary normal  effects which  exhibit  relating  i n addition  effects  t o that  f o r large  magnetic  halide  has  been  bridges, shown  t o d i s t o r t i o n . At metal  since  susceptible  spin.  distortion  with  to small  degenerate  the total  that  associated  a  contains  i n complexes which  exchange  t o be v e r y  atoms i s  spans  with  suggests  0  symmetry and ( i i )  spectrum  associated  are particularly likely  or  sensitive equivalent  (3J/9d)  t o be most  (i) high  the d i s t o r t i o n Hamiltonian  mode a n d t h e H e i s e n b e r g  requirement  are expected  which a r e very  fold axis  so that  degeneracy The  distortion  D I S T O R T I O N MODEL.  contain these  oxide  bridges  changes i n  configuration. The is  clusters  a Lewis  Fe(III) above of  6  base),  3  clusters  distortion  model  An  compound  transition observed  The  1  1  7  complexes  one c a s e ,  3  '  1  2  8  magnetic  3  6  '  1 1 1  "  13 1  6  that 2  i s the l i k e l y and thermal  application  (where M  that  and L  i s .  s a t i s f y both the  of the magnetic  available  [Cr 0(CH COO) ]C1.6H 0, 3  "  ere X i s a halide  i n 5.2 a n d 5.3 a f t e r  i s the best  In at least  5  i n 5.1. I t i s found  f o r the t r i m e t a l l i c  clear.  3  interpretation  i s given  models  + 6  w n  a n d RCOO a c a r b o x y l a t e )  requirements.  these  (  37.87-110  and M 0 ( R C O O )  or Cr(III)  alternative  but  Cu„OX L„  behavior  a discussion  of  t h e dynamic  f o r the copper  the situation  i s much  of the i n t e n s i v e l y a non-magnetic  clusters, less  studied  phase  t o be t h e d o m i n a n t  cause  of the  behavior.  to tetrahedral  Fe(IIl)  clusters  discussed  1 23 in the  5.4  i s somewhat t e n t a t i v e  potential  treatment  of  of  large,  fluxionality the  i s expected  static,  H  = -J  0  In  2  principle,  distortions  slow. For  undistorted  triangle  H a m i l t o n i a n has  and  the  spin  2  lower  also  which  symmetry  with  clusters  1  3  2  "  1  3  8  ,  magnetic  l i e close  enough  into  antiferromagnetically  magnetic  distortion  presumably  larger  t e c h n i q u e of  spectrum  rather  ambiguity  at  effects  of  two  the  and  i n the d e r i v e d  may  be  be  which  interaction  a  since system,  s e p a r a t e d from  data  J values,  parameters.  frequently  contributions.  susceptibility  still  regarded  However,  J v a l u e s i n such  easily  useful  Even  to the  which as  the  a  s u c h an  singlet.  steric  to particular  i n 5.2,  t o be  of  ferromagnetic dimers  state  cannot  effects  states  favour the  c o u p l e d , as  least  electronic  than  the d i s c u s s i o n  a pair  the ground  fitting  in-energy  exhibit such  there are  interactions  are  a r e e x p e c t e d t o be  and  4  separation  of  may  i n the cubane-like Cu O  the magnetic  the energy  0  effects,  whenever  pronounced weakly  systems  character  as q u a s i - d e g e n e r a t e . Thus  in  the  H .  associated  same t o t a l  the  both  i n the  the  H a m i l t o n i a n s commute w i t h  show some d y n a m i c  there  i n which  indicate  (S' -nS )  a l ldistortion  lowers  to  H a m i l t o n i a n method  clusters  t o be  largely  form:  (5.1)  may  i s included  factorisable high spin,  t e t r a h e d r o n the  simple  and  the  and  is  the so,  magnetic  introduced  i t avoids  1 24 5.1  I n t r o d u c t i o n t o Cu^OXsLg c l u s t e r s  The  a  a  moment  y(T)  a  magnetic by  Cu OX L  simple  either  point  3 7  '  or almost '  8 8  group  copper  8 8  1 0 0  trigonal  '  9 0  s t r u c t u r e c o n s i s t s of  those  edge-bridging  from  data  been  with  discussed 1  0  1  '  1  0  5  maximum  i m p l i e s , assuming that  which  lies  ground  just  s t a t e . Such  on  chapter 5 1  1  0  '  1  are  or  discussed  below,  antisymmetric  static  and  as  i n t h e moment o f  About  '  8 9  ' ;  9 1  9 9  half 1  0  5  .  two  infra-red  of  them,  The  the  might  possibilities  w e l l as  dynamic  exchange  Cu^O exchange  a highly  or  distortions,  intercluster  position  independent  contains  spectrum  from  atoms  principally  magnetic  slightly conceivably  n o n - i s o t r o p i c c o n t r i b u t i o n s t o the exchange,  exchange,  centred  .  non-magnetic, a  oxygen  about  s t u d i e s  spectrum  above a  an  synthesised  temperature  the magnetic  in a tetrahedral  l i g a n d , L.  reported  Wong  i  group  the a x i a l  a  been  e.s.r.  and  fitted  h a l i d e s ; the copper  by  in this  be  indicate  show a c h a r a c t e r i s t i c  1  characteristic  constants,  magnetic,  which  500cm"" , h a v e  in conjunction  complexes  state  occupied  measurements have  of D i c k i n s o n  The  occur  bipyramidally coordinated,  at about  experimental  tetrahedral point  The  these complexes,  cannot  the complexes  .  being  in- the  j5o.5i,95-ioi,io«  some c o m p l e x e s  dozen  sometimes  one  and  with  a maximum  20-50K w h i c h  s e v e r a l of  exact  the oxygen  absorption  with  on  opposite  magnetic  region  exhibit  ,  tetrahedron  of  typically  i n the  determinations  exact  symmetry  complexes  H e i s e n b e r g model  Structural  are  6  intercluster  which  distortions. models,  arise  which  are The differ  1 25 strikingly ground as  from  state  opposed  5.1.1  arises  to  discussed  a  S'=1  Heisenberg  splitting  and  spin-spin)  S'=2  three  terms are  addition  of  (5.2)  H  coupling  -J(S  terms, D,  are  t e r m s was symmetry  system  to  by  parallel,  feature state,  =  than  (5.2)  of  the  the  non-magnetic  Heisenberg  Heisenberg  singlets,  quintet, are  i n T^.  In  , 2  -3)  which  i s the  symmetry  the  are  (and  dimension  spin-orbit  Hamiltonian  by  of  Cu  Moriya  1 7  ,  the of  S'  splitting  nor  2  who  to  6 6  by  the  be  even  also  cube the and  However, the of  for J  the  of  S'=2  spin-orbit derived  vectors,  the  a t o m s . As  4  neither  state  expected  representation  diagonals  D,  for  which  D  are the  the  is  not solutions  essential Heisenberg  >  0,  can  be  seen  whole can  be  regarded  by  a  argument.  C u O X L , c l u s t e r as 6  spin-orbit  of  model,  degeneracy,  terms:  which c o n s t r a i n  ground  by  I.Rs  spin  This  introduced  the  5-fold  ID^tS^xSj)  neglected.  model,  tetrahedral  and  no  Lines  exchange +  9-  reduced  are  i n the  S^r\jS_j,  rules  regular  be  somewhat c o m p l i c a t e d .  straightforward  a  the  the  included  tetrahedron  the  which  of  there  commutes w i t h  are  model  S=l/2 e x h i b i t  l i e along  the H  of  The  the  states  antisymmetric  Anisotropic  of  of  i n t e r a c t i o n s as  than  defined  in that  splitting  w h i c h must c l e a r l y  coupling  Cu„  model  exchange  cluster with  respectively,  certain  from a  Antisymmetric  The  smaller  distortion  first.  The  greater  the  a  as  defining  126 an  inverted  before the  transition  field  atoms  i s added,  E+T .  I t i s found  2  transition lowest.  state  that  D,  smaller  order  a quintet  degenerate  in this  states  c a n be d e n o t e d  5  S. If  case,  state  splits  as i n o r d i n a r y  the non-magnetic  into  tetrahedral  E doublet  lies  the s p l i t t i n g of the q u i n t e t i s than  to explain  that  of the t r i p l e t s as i t s o r b i t a l  D i s expected  energy Lines  model  orbitally  degenerate  strengths  ground  unusual  and there  Spectroscopic copper  i s also  true  state  would  hence  reduce  While  tend  trigonal 3 _ 5  dynamic  orbit  fairly  o f an  on t h e r e l a t i v e  a singlet  of about  Jahn-Teller  ground  8000cirr  1  state 1  3  9  "  fits  1  momentum a n d  to several  4  0  .  on an E  coupling.  good  very  coordinated  effects acting  angular  be  evidence.  bipyramidally  suggest  state  field (in  l i g a n d s ) , i t would  t o quench any o r b i t a l  i tprovides  i s required.  such a ground  supporting  E s t a t e a t an energy  the spin  model,  for orbital  2  in a trigonal  and e q u a t o r i a l  on o t h e r  that  this  the assumption  s t a t e , E. W h i l e  f o r copper  such as C u C l  the lowest  2  involves  i s no c o n v i n c i n g  data  species  (g-2) J/g  s t a t e may b e A o r E d e p e n d i n g  of the a x i a l  with  of wavenumbers  of only  therefore  by no means i m p o s s i b l e t h e ground  tens  data  t o make a c o n t r i b u t i o n t o t h e  of the order  singlets.  which  the experimental  s p l i t t i n g of several  parameter  magnetic  It  field  T h e S'=2  i s non-degenerate.  In  with  fold  complexes,  For small  field.  due t o t h e c o n f i g u r a t i o n o f t h e c o p p e r  the five  metal  considerably  is  crystal  the a d d i t i o n of the c r y s t a l  T^ c r y s t a l  The  metal  sets of  1 27 magnetic  data,  1/ I t d o e s anionic such at  model  has s e v e r a l  not f i ta l l a v a i l a b l e  clusters  which  have  parameters  which  i s difficult.  somewhat  s h a r p e r maximum  has  suggested that  paper  were  error  1  0  e.s.r.  5  .  3/ M o s t  studies  state  singlet, study  compound. earlier normal  In  fits  6  show t h a t  studies  g-factors, high  with  5  the zero f i e l d and that  and i t  samples,  the o r b i t a l l y  crystal  typical  splitting  a ground  The r e s u l t s  to reconcile  of the  state  of  this  for this  t h e r e s u l t s of which  indicated  d e g e n e r a t e model  ( e . g . 2.37 f o r C u n O C l ^ " " )  of t h e above d i f f i c u l t i e s ,  5 1  and i t s .  i t c a n be c o n c l u d e d  exchange does not c o n t r i b u t e  magnetic  a  original  single  i s a  3  on p o w d e r e d  g-values  predicts  detailed  untenable, at least  I t had been d i f f i c u l t  view  nature of  presented i n the  below t h e q u i n t e t . model  the derived  typically  recent  1  1  moment  of systematic experimental  ( a b o u t 0.5 c m " )  the Lines  antisymmetric unusual  t h e good  6  14cm"  clusters  have a v e r y h i g h  on C u „ O C l ( O P ( C H ) ) „ , w h i c h  e.s.r.  associated  a t 20K a n d  i s observed experimentally,  significantly,  i s small  lies  make  than  f o r both the  between  2/ The m o d e l  t o some e x t e n t a r t i f a c t s  compound o f t h e g r o u p , S'=2  correlation  and the c h e m i c a l and p h y s i c a l  complex  been  failing  a l o w moment  6  temperature. Also,  drawbacks:  data,  a s Cu„0C1 (3-quinuclidinone)„  this  the  Lines  significantly  b e h a v i o r o f CunOXgL,, c o m p l e x e s .  that  to the  1 28 5.1.2 T h e In  intercluster  the intercluster  state  quintet  field  which  weak  i s split  arises,  isotropic  standard  pair  intracluster average  .  of weakly  field  molecular  paramagnets law),  ferromagnetic  between  - z ' J ' S ^ . S j where clusters,  of C u „ O X L  The  c o u p l i n g s between  6  4  molecular  (5.3)  S..S. =  as long  S  i z  j z  orbitals  clusters,  by a n  ( c . f . chapter of the  o n t h e same  which  c a n t h e r e f o r e be  represented spins of  f o r the ground  w i t h J>0.  >  involves  atom  i s assumed t o  the total  a s z ' J ' < < J . S^=Sj=2  approximation  <S  to the  coordination  the role  a n d S j a r e now  clusters field  play  a  as i n v o l v e d i n  t o t h e S=2 d i m e r  c o u p l i n g s here  the l a t t i c e ,  Cu  each p o s s i b l e  by an i n t e r c l u s t e r  The c o u p l i n g b e t w e e n  throughout  separate  model  (exactly  i n v o l v e s severa.l  approximated  i s t h e same a p p r o x i m a t i o n  the Heisenberg  of the  tend t o  a r e a l l s m a l l compared  c a n be w e l l  from  t h e kT i n t h e  i s r e p l a c e d by kT-#,  a Curie  magnetic  approximation,  Each p a i r w i s e i n t e r a c t i o n  i f these  the ground  The d e r i v a t i o n  i n which  coupled  r a t h e r than  atoms;  z'. T h i s  a term  states  result,  c o u p l i n g they  the dimer.  extend  f o r CU„OXGL,,,  clusters.  equation  the intracluster  strong  by  field  coupling J' multiplied  applying 1.3),  5 3  model  i n v o l v e s 16 c o u p l i n g c o n s t a n t s , b e t w e e n  of metal  number  between  o f t h e HDW  approximations  in  exchange  a Curie-Weiss  clusters  exchange  by a s y m m e t r y - l o w e r i n g  the s u s c e p t i b i l i t y  follow  model  i n the molecular  molecular  denominator as  exchange  writing:  1 29 i.e.  each  clusters,  cluster <Sj >.  a  special  magnetic  weakness  direction phase  does produce and  a  phase  a  just  simple  of  be  at T-z'J'  is  precisely  the  magnetic  With  throughout  a  the  6  sometimes affect  model  complexes involve  complexes  6  phase  results  data with using a  the  care  lattice;  the  susceptibility range,  at which  5  ratio  3  i n a good  z'J'=0.3cm" , 1  this  model a r e  z'J':J  the v a l i d i t y  Cu„OCl (OP(C H ) ),, magnetic  of  any  temperature  range  spin  of  as  is which  interesting  6  seriously 6  temperature  feature  as i t  low  with  e x c e p t i o n of Cu„OCl (3-quinuc1idinone)„,  intercluster  Cu OX L„  in this  of  to  the  i s useful  used  3-D  rise  above  in a solution  essential  the  give  states  be  the  behavior occurs.  susceptibility  for  i s an  coupling  inaccurate  found  expected  of  from  cannot  into  Curie-Weiss type e x p r e s s i o n f o r the  often  rest  a molecular f i e l d ,  t h e model must  t o become  a  coupling  the q u i n t e t  expected  the  to the  t e m p e r a t u r e , the model  as would  involves  due  approximation suffers  hence  However,  transition  field  isotropic  z, and  splitting  problem.  model which  that  transition  high spin,  isotropic  Though t h i s  z  theoretical  experiences a  of  z'J'=-12K  transition  has  which  of  0  1  ever  been  As  no  observed that  data for  poor,  enough  and  to  (e.g. e v i d e n c e of i n Cu„OX L 6  a a  the v a l u e of z ' J '  r e p r e s e n t s an  exchange  which  the  to magnetic  i s large  J=40K).  intercluster  to  frequently  ,  5 0  3 - q u i n u c l i d i n o n e complex  the magnitude  fits  1  the d e r i v a t i o n  i t seems r e a s o n a b l e t o assume  f o r the  f i t  for  i n these  upper  bound  systems.  1 30 5.1.3  The  static distortion  The simple  purpose model,  certain  has  not  the  larger  this  in which  fixed,  some b u t  of  a l l of  clusters,  that  i t can  s u c h as  the  f l u x i o n a l i t y pathway because  slow  to  symmetry  are  too  The  model,  factorable  the  tetrahedron.  and  C^  of  P  be  for  good  to  simple  fits  approach  adapted  and  octahedron,  interpreted the  in X-ray is a  Hamiltonians  is as  a  permanent studies,  low  distortions  three  both  of  approach  static configuration  which  to  understanding  can  fold axis  or  as  energy  related  f l u x i o n a l i t y must  a  is solved  factorisation  i f there  between  taking  This  real  The  be in  since  distortion  Thus,  no  observed  then  a  give  bipyramid  that  energy  to  relatively  relatively easily  model can  small  can  data.  trigonal  lowest  interconversions  operations  Many  by  the  occur.  be (T  2  defined  for  distortion  symmetry):  (5.4)  and  the  a  Hamiltonian  is obtained.  f l u x i o n a l i t y model,  barrier  v  the  static distortion  distortions  be  disadvantage  factorisable.  genuine  a  the  show t h a t  distortions,  experimental  i t also  always  has  the  i s to  Heisenberg  factorisable  advantage  useful  section  the  but  is  model  hence,  H'  =  J'[S'  treating  the  2  -  2(S +S +S )  spins  1  S,,S  2  3  and  2  2  +  S  2S ] 2  3  as  an  equilateral  triangle:  (5.5)  can  be  H'  =  J , ( S T + S Z + S - , ) .S«  factorised.  However, g i v e n  +  J (S,+S ).S  the  2  2  3  s t r i k i n g resemblance  of  131 the  experimental  for  a pair  moment  data  of ferromagnetic  antiferromagnetically distortion  f o r t h e Cu^OXgL,, c o m p l e x e s dimers  coupled,  Hamiltonian  based  which  to  those  a r e weakly  i t i s natural to consider  on t h e two f o l d  axis  a  first.  The  Hamiltonian:  (5.6)  where  S  number  H = - J , ( S ' - 3 ) - ( j - J , ) (S, + S 2  =  1 3  2  2  S,+S  and S „=  3  S +S ,  2  of parameters  3  2  1 (  forthis  2 2 ( t  -3)  - xJ (S  which contains  kind  2  2  1 3  -S „ ) 2  2  t h e maximum  of f a c t o r i s a t i o n ,  has  energies:  (5.7)  |2,1 , 1 > , - 2 J - J ; 1  2  | 1 ,1 , 1 > , + 2 J , - J ;  |1,1,0>, J ( 1 - 2 X ) ; 2  The  arrangement  shown  of c o u p l i n g  i n F i g . 5.1. JT>0,  antiferromagnetic D  2d  s  Y  m  m  e  t  y ?  r  parameter a  T  2  distortion  ground  The  t  x affects  significant the  i f  e  constants  to a  i s reduced  the triplet  as f o r values  o f x=1  i s  2 v  »  and  The  states as i tcorresponds t o  a triplet  (J -J,). 2  i s brought  I ti s  close to  state.  of a l e a s t  i n Table  squares  5.1. t h e n o t a t i o n  f i t to this  model, a r e  ( X , L ) i s used  C u O X L , and the f o l l o w i n g a b b r e v i a t i o n s  are  introduced:  6  tetrahedron  to C  for  4  t o (5.6)  pairwise  of a ferromagnetic  symmetry  only  2  corresponding  2  ;  2  |0,0,0>,+3J .  2  i n addition to the E distortion  results  presented  n  |1,0,1>,J (1+2x);  J <0 corresponds  distortion  |0,1,1>,+4J,-J  2  a  as  shorthand  f o r the ligands  py - p y r i d i n e , p y o - p y r i d i n e - N - o x i d e ,  tetramethylurea, 3-quinuclidinone,  dmso - d i m e t h y l s u l p h o x i d e ,  3-quin  -  TPPO - t r i p h e n y l p h o s p h i n e  oxide.  The  tmu -  counter  L,  132  Fig.5.1.  Static d i s t o r t i o n coupling constants copper t e t r a h e d r o n .  for  the  133 ion  f o r both  Table  5.1  ionic  Fits  complexes,  to  Complex  the  static  J^cm  and  (C1,C1')  distortion  J /cm  x  2  (Cl,Br~),  J  g  FITf  200  -135  0.94  90  2. 10  0 .068  (CI,Br" )  220  -1 48  0.96  1 00  2. 07  0 .068  (Br,py)  45  0.94  21  2. 21  0 .038  (CI,pyo)  210  -106  =* 0  1 05  2. 1 4  0 .031  (CI,tmu)  1 90  -  *  0  95  2. 04  0 .09  (CI,dmso)  280  -141  ~  0  1 50  2. 1 1  0 .029  (Br,dmso)  1 95  -98.8  =* 0  1 00  2. 25  0 .027  -  -  30.7  -18.5  -2.0  43.2  (CI,3-quin) (CI,TPPO)  t  FIT  minimised No  Z(M  c a l c  -M  -  of  14  2. 10  0 .045  =* 0  13  2. 1 0  0 .045  the  i . e . the  calculated least  0 . 0 2 - 0 . 0 4 B.M.  adequate the  fits  to  shapes of  about  data.  some b u t the  routine  of  Fig.5.2  to  complex.  i s expected  be  a l l the  moment c u r v e s  exchange model,  curves  for this  I t can  not  are q u a l i t a t i v e l y ' s i m i l a r  antisymmetric theoretical  of  experimental fits  fitting  ) .  FIT  i n the  the  2  o b s  A  of  from  squares  obtained  value  -  0.895  f i t c o u l d be  general best  moments,  •  acceptable  scatter gives  97  = R.M.S. d e v i a t i o n  experimental  t  28  4  model  (ci,ci-)  .-  i s NMe  seen  that  compounds.  corresponding those  this  can  be  with  those  obtained  seen of  by  from  this  model  In to  the  on  the  comparing  r e f 50.  the  Thus  the  both  in (M  TEMPERRTURE/K Fig.5.2.  Moments c a l c u l a t e d from and, from t o p t o bottom,  s t a t i c d i s t o r t i o n model -J =100,102,106,112,160 2  w i t h J,=-l00cm and 200cm" . 1  1 35 models  give a  poor  f i t for  for  large  maximum moment, M>2.6B.M., w h i c h  the  ground  state  at  Hamiltonians  such  and  dynamic  explain low  the the  suggests  that  monomeric  The  fits  curve  for  range  20-60K. by  distortion  compounds  to  an  low  states,  spin  yield  intercluster the  have  a  a and  Introduction not  >  exceptionally  suggests  does  of  f i t (FIT  strongly  complexes  remaining  although  Two  yet  of better f i t  models  can  experimental curves (CI,CI")  been  and  at  (Cl,Br~)  contaminated  for  (Cl,TPPO) are  i.e. similar 5.1;  Ji+2J , in a  ground  quintet.  For  obtained  w i t h the  270,-136cm" . 1  of  1  f  for J  2  the  J  the 2  fits  fits  alone  singlet (Cl,pyo)  1  low  with  a  presented; these  can  lies  is  rotations  be to  negative  the  to  the  and  below  the  fits  are  210,-106;  Because  J=(4J,+2J )/6  of  extremely  sensitive  good  2  i t can  derived for a l l  150,-76;  case.  the  characteristic  just  comparably  i n every  d e r i v e d parameter  very  in  correspond  be  s m a l l and  sets  theoretical  obtained are  but  which  more  moment  i s associated with  parameter  J\+2J =-2cm" 2  of  is typically  state  example, J  and  that  the  This ambiguity  pairs  J , or  which  2  (5.7)  general  to either  compounds a r e  systematically  spectra.  v e c t o r . In  five  i t i s noticeable that  fits  identical  insensitive  magnitude  and  s a m p l e s may  the  in Table  distortion  quantity  lying  behavior ionic  poor  compound has  (5.5)  reference to equation  the model,  results  and  (Cl,TPPO) g i v e s a  essentially of  very  temperatures.  (5.4)  f o r the  these  low  a  species.  satisfactory,  seen  as  low  b e h a v i o r . The  temperatures  latter  w i t h no  moment d e c r e a s e s  only  This  and  0.10)  quintet  (Cl,3-quin).  (Cl,tmu)  of  must  this be  the  1 36 regarded  as only  a crude  extremely  insensitive  lies  the ground  near  The which for  the f i t with choice  interpret  this  distortion that  least  data.  and J  feature without  there  i s still  The chapter Table  magnetic  model with  that  there  together,  leads  which are at  the antisymmetric  i f some k i n d o f  and  exchange  slow with  e.s.r.  difficulties  data  t o make  providing a  magnetic  model  f o r which  i t c a n be  to fits  i s no c o n t r a d i c t i o n  f a r from  those f o r  t o t h e dynamic  to experimental  distortion  The m a g n e t i c  triplet  x=*1 ; i t i s d i f f i c u l t t o  a l l the results  f o r the observed  5.2 T h e d y n a m i c  a  x=0, a n d t h o s e  a r e enough q u e s t i o n s  out of the f i t s  explanation  case  two g r o u p s ,  reference  obtained  i s invoked  t h e model  5.2.1  gives  2  and has the advantage  arising that  of  into  gives  2  the static distortion  However  x= 1, i n which  fall  J,>0 a n d J < 0  model. Taking  fluxionality  the f i t s are  state.  a s good as those  model,  Similarly  to x unless  complexes apparently  this  said  estimate.  i t clear  complete  behavior.  f o r Cu OX L a  6  a  spectrum spectrum  for this  4. I t i s s u m m a r i s e d  5.2 Summary o f d y n a m i c  Distortion  Slow  E  s i n g l e t and splitting tr iplet splitting  i n Table  s y s t e m was d e r i v e d i n 5.2.  distortion  spectra Fast  triplet  singlet splitting no splitting  1 37 The  spectrum  degenerate For  fast  the  energy  of  triplets  well  plane' produces for static E  removed  of  the  equivalent  and  T  from  include  the  ground  state.  triangle  triplets.  exact  The  ( e q u a t i o n s (4.3)  in  and  distortion  H'(r,\//)  =  [ (S , . S + S  -2r  2  3  . S „ ) c o s f + (S , . S + S 3  2  .S  )cosi// )cosi/>  3  +  ]  E ( S ' = 1 ) = - 2 r c o s ( i | / + n 7 r / 3 ) , n = 0 , 1 , 2 ; E ( S ' = 2 ) = 0.  E(S'=0)=±3r;  T  .S,  2  + (S , . S „ + S  (5.9)  a  ' S' = 1 energy  distortions  2  to  ) are: E  (5.8)  pair  f l u x i o n a l i t y i s expected  fluxionality rotation  solutions (4.31)  f o r slow  distortion  2  R(p,e,<p)  =  -2p  [x(S , . S - S 2  3  .S  )+y(S, . S - S  t t  3  .S  2  )+z(S , . S y - S j .S  a  3  )]  E ( S ' = 0 ) = E ( S * =2)=0 E ( S ' = 1 )=-4/v/3cos(a+n7r/3) , cos3a=-3v/3xyz .  where  x = psinpcos</>,  Given attempt  the  result  rule  on  energy that  s i m p l i c i t y of  to define  must  y = p s i n 0 s i n 0 and  i n a m b i g u i t y . The  z e r o and  a t most  spectrum.  parameters  three  this  of  pure  number  for fast  are E  2  i t i s clear  the magnetic  'constraints' S',  of  of  i i ) an  two  of  that  or T  2  i) a  the  by  f l u x i o n a l i t y only  a J  Lande  arbitrary  distortion, t o two  any  spectrum  triplets,  necessary to describe  i s reduced T  5.2  from  e n e r g i e s f o r each  In the case  and  J  i i i ) the degeneracy  three  fluxionality, constraint,  the Table  s i x independent  the average  9cos6.  z=  or  the  fast  E  further i s defined.  mean  138 In  view  of  the  impossibility  alternative  procedure  parameters,  chosen  has  of  been used  spectrum,  5.2.  most c o n v e n i e n t c h o i c e of  one  based  always to  on  at  t h e most  which  energy  low  energy  reliable  The  spectrum  in Fig.5.3.  an J  and  for a l l small  static  distortion J but  only  and  f o r n=5  the  magnetic  may  be  must  possible  5.2.2  fitted  S=1/2  from two  be  least  the  a  h i g h symmetry. Thus,  spectrum  only  eight  becomes more c o m p l e x explicit  interpreted  is  this  even  rather  It is  the  spectrum For  f o r S=1  H a m i l t o n i a n , though  with  c a r e and  a  there are  parameters. even  than  of s p u r i o u s  those with  f o r n=3  and  leads  fitting  parameters.  necessary to define  t e n J and  which  typical  introduction  (n<6),  Table is  since  on  a  case  squares  i n the derived  clusters  t o f i t t o an  still  the  to define  in this  states,  an  Three  the q u i n t e t ,  to a magnetic  parameters  spectrum  examined  S>1 i t the  for  ambiguity.  Treatment The  of  six J  according to  is illustrated  ambiguity  there are  possible  results  between  the other spin  fitting  used  parameters  Hamiltonian avoids both  useful  been  interpreted  convergence  of  the a s s o c i a t e d  three  and  parameterisation  procedure  explicit  i s then  differences  routine.  The  to f i t the data.  f o r c o n v e n i e n c e , have  magnetic The  uniquely defining  of  data  e x p e r i m e n t a l data of D i c k i n s o n  to magnetic  distortion temperature  model  spectra  using  independent  a  of  least  the  and  Wong h a v e  type predicted  squares program.  spectrum  could  produce  by  the  Where a  been  no  satisfactory  1 39  o C\J _^  Fig.5.3.  P a r a m e t e r i s a t i o n of spectrum used f o r f i t t i n g d a t a t o t h e dynamic d i s t o r t i o n m o d e l .  1 40 fit  a temperature (5.10)  as  derived  good  fits The  dependent  triplet  f = 3p{2exp+6/T+exp(-25/T)} i n 4 . 5 , was  fits  are illustrated  parameters  spectra  the  low l y i n g  monomeric than which  those  in  impurity.  remain  temperature o r , more  In a l l cases  fluxionality,  were used,  a l l cases  (Cl,TPPO)  fast  except  poor  complexes  a t low temperatures,  dependence  i n the energy  simply, the presence the f i t s even  such  included  as  a r e as good where,  found  by e . s . r .  of  of or  better  f o r complexes  (Cl,pyo),  fewer  as a v a r i a b l e  parameter  two, t h e v a l u e s f o r ( C l , 3 - q u i n ) and  h a v i n g been  respectively.  'g' was  to give  and F i g . 5 . 5 , t h e  5.3. F o r t h e  g i v e n by a n y o t h e r m o d e l  exhibit  parameters  triplet,  found  i n F i g . 5 . 6 , and the r e s u l t i n g  i n Table  (C1,C1~) and ( B r , B r " ) t h e f i t s o f some  9(l-p)  o f D i c k i n s o n a n d Wong.  i n Fig.5.4  are given  are tabulated  because  +  function:  i n t r o d u c e d . The m o d e l was  t o a l l the experimental data  corresponding  either  partition  t o be 2.16 a n d  2.10  _ 0 1  j 50 Fig.5.4.  1 100  1 150  r 200  TEMPERATURE/K  1 250  1 300  D y n a m i c d i s t o r t i o n m o d e l f i t s t o Cu,,OX L,, d a t a . I n o r d e r o f i n c r e a s i n g t h e o r e t i c a l moment a t 2 0 0 K , s y m b o l s a r e -+ - ( C I , B r ) , O - ( C I , C I ) , & -(Br,py), O - (Cl,pyo), X -(Cl,dmso) and O -(Br.dmso). 6  i 350  1 50  i  100  r  150  200  TEMPERATURE/K  350  250  F i g . 5 . 5 . Dynamic d i s t o r t i o n model f i t s t o C u , O X L , d a t a . In o r d e r o f i n c r e a s i n g t h e o r e t i c a l moment a t 200K, symbols a r e A - ( C l , t m u ) , D - ( C l , T P P O ) , + - ( C l , 3 q u i n ) . 6  1 43  O C\J  o  CD CO  O O CO I o >-O0  CD  UJcD  O  CD  1  1 1  tCl.Cl)  (Br.py) (Cl.Br)  1  ]  1  (3)  (3)  121 1  (Cl.dmso) (Cl.tmu) (C1.TPP0) (CI,pyo) (Br.dmso) (C1.3quin)  F i g . 5 . 6 . Magnetic s p e c t r a of the Cu,OX L» complexes. P a r e n t h e s e s i n d i c a t e s t a t e s removed by f l u x i o n a l i t y . 6  144 Table  5.3 D i s t o r t i o n  Complex  model parameters  Spectrum parameters  f o r CunOXgLn  J  g  complexes  Fluxionality Distortion FIT  (cm" ) 1  (CI,CI")  Ai  = -135,  A  =-116  > 1 00  2. 08  E+T  2  Slow  0 .066  (CI,Br")  A1  =-145,  A =-132  > 1 00  2. 04  E+T  2  Slow  0 .066  (Br,py)  A,  =-22 A = - 1 8 . 5 = 1 40  22  2. 20  E  Slow  0 .037  =-5.0,  A =300  75  2. 1 3  E  Fast  0 .026  ( C I , d m s o ) A , =-3.8,  A =400  100  2. 09  E  Fast  0 .030  ( B r , d m s o ) A , =-5.4,  A =250  60  2. 27  E  Fast  0 .018  =95  25  2. 1 3  T  2  TDF  0 .020  25  2. 1 6  T  2  TDF  0 .035  1 0  2. 1 0  E  TDF  0 .028  A  (CI,pyo)  3  A,  (Cl,tmu)  3  A  3  3  3  3  3  A  3  (CI,3-quin)  3  3  =-16, e = 30  A  3 3  A =-1.4 1 00  e=2.6  3  =  3  ( C I , T P P O ) A , =-14.0, c=20, A = 3 8 , A =-10.3 3 3  3  A correspond  The p a r a m e t e r s Fig.5.3. These distortion  results  c a n be  model as  t o the energy  interpreted  lying quintet  physically  fluxionality  removes  singlet/quintet coupling  between  similar,  except  triplet  indicates  to a pair  state  that  The s i n g l e t  the system  to the  t h e low l y i n g  triplet  indicates  the dimers.  weak  In (Br,py)  ground  state  and  corresponds  of ferromagnetic dimers  separation  that  according  in  follows:-  (Br,dmso),(CI,pyo),(Cl,dmso). low  differences  i n which  states.  The  rapid small  antiferromagnetic  the s i t u a t i o n i s  f l u x i o n a l i t y i s slow,  i . e . a low l y i n g  i s 'frozen out'.  (Cl,Cl"),(Cl,Br").  In the anionic  clusters  a large  singlet  1 45 splitting lowest  (E d i s t o r t i o n )  triplet  distortion The  indicates that  represented  on  significantly  is  compared w i t h  fluxionality  magnetic  occurs  data;  probably  of  be  than  data. of  This  spectrum  large)  only  20K  the  I f the  decrease  In  below  lowtemperatures,  data,  thus  for  indicates  that T  considered  as  20K  antiferromagnetic The  temperatures  and  thus  cannot  be  fixed  these  (CI,3-quin) (unless g  would  found  triplet  be  of  to give  states  not by  but  fluxionality  are  represented  as  degenerate  fit  i s obtained  by  introducing a  the  experimental  for which is  the  a  i . e . , the  triplet  no  the  low  for  lying  copper  as  no  at 20K.  atom  state  the quintet be  weak in  the  low The  singlets  improvement  splitting  TDF  the  c o m p l e x e s may  fluxionality  singlet  to  trimers with  above about  the  ground  justification  s t a t e and  system  from e x c i t e d  excellent fits  in Fig.5.6  moment  unrealistically  A p p l i c a t i o n of  remaining  slow  temperature  independent,  ferromagnetic the  to  separated  i s dominant;  undergoes  fast  three  occupied,  occur.  ground  coupling to  system  not are  i . e . i n t r o d u c t i o n of  consisting  occurs.  they  temperature  distortion  2  also  that  fits  p r o v i d i n g some e x p e r i m e n t a l T<20K, t h e  mode  energy  c o m p l e x e s no  state well  were  was  model. At  cluster.  these  implies that  model d e r i v e d above, at  from  the  E  distortion  2  300K and  q u i n t e t s t a t e can  i n M(T)  f o r pure  excited states  gives acceptable  spectrum  of  1  must have a q u i n t e t g r o u n d states.  energy  100cm"" .  i s most marked  2.65B.M. a t  T  high  defined  (CI,tmu),(CI,3-quin). independent  such at  low  expected  three  even  J cannot  greater  The  are  populated  that  the  significant  i s slow.  Fig.5.6  but  in  parameter.  146 (C1,TPP0).  The  slow  E-fluxionality  this  data,  satisfactory  f i t to  satisfactory  f i t i s only  exhibit  TDF.  In  involve  5 parameters  meaningful this  It  result.  complex  reduced  the  is also  allowed  has  A  1  3  to  On  those  this  neutral  freely  found  by  be  the  coupled  or  The  1)  The  have  highest  complexes have  2)  The  ionic  a  too  are  to  1  acceptable  parameters  give  a  study  on  this  number,  : g=2.11,  three. A,  f i t i f g and  i.e. values  to  would  e.s.r.  A,=-14.Ocm" ;  best  allowed  model  many  recent  experimental as  are  e=14cm" , 1  f o r g and  data  indicating  e  very  for  the  that  tetrahedron, into  ( i . e . two  pairs  three  of  ferromagnetically  ferromagnetically  weak a n t i f e r r o m a g n e t i c  coupling  coupled  occurs.  results  with  are  noticeable  b r i d g i n g bromides,  g-values.  The  i n the  results:  (Br,py)  neutral chloride  and  (Br,dmso),  bridged  g=2.12±0.04.  l a r g e J , poor complexes  the  summarised  following trends  complexes  the  of  that  the  a monomer a n d  which  Discussion  t o an  the  e,  that  ferromagnetic  subclusters  atoms) between  and  nearly  e.s.r.  interpretation  atoms  ,g  A^-14.5;  =35cm" ,  splits  3  g=2.l0 and  yields 1  3 3  3  find  ferromagnetic  5.2.3  s t a t e TDF  parameters  complexes can  distortion  ground  3  a  entirely  singlet  determined  to vary  an  triplets  a  reassuring to  5.1.3, but  i f the  : A,,A ,A  of  gives  obtained  It i s fortunate  number  A =-10.9cm" , close  general  c.f.  model  f i t , and  suggests  that  lack these  of  the  might  moment m a x i m u m most  usefully  in be  the  147 treated  as  a  separate  fluxionality large  J  or  may  to  be  the  class  due  of  compounds. T h e i r  either  influence  to  of  a  the  large  A  counter  slow  associated  3  ions  in  with  the  the  crystal  latt ice. 3)  (Br,dmso),  ( C l , p y o ) and  oxygen  and  steric  halide  octahedron  fluxionality and  noted exhibit  the  apparent  the  donor  atom or  compound  unwise  ligand It  is  the  In  of  too  of  function  the  to  of at  into  i t s large  to  re-examine  (they  both) and  donor  and  single  mass and the  the  the  in  have  a  There  nature  is  of  of  the  However  P-0  available  been  other  small  result  high  which  ) a l l  rate  that  interactions.  this  and  temperatures.  (or  fast  (Cl,tmu),(Br,py)  significant  oxygen  steric  as  bond  it  would  this order.  structural  data  in  observations.  typically about  0.2A.  enough compared  the  i t is  in  make t o  either  is  the  a l l exhibit  (Cl,3quin)  low  atom  atoms and  significant  at  donor  is nitrogen  interactions,  an  the  contrast,  atom  least  between  much  ligand  ( B r , p y ) and  context  above  of  donor  steric  to  data  small  of  studies  read  In  definitely  significant  X-ray  3.OA  the  ( C l , T P P O ) has  vibrations  probably  are  this  interesting  The  about  which  the  i s unusual  light  with  in  correlation  possibility be  insignificant,  probably  fluxionality  fluxionality. TDF  are  X-ray  slow  which  between  interactions in  in  interactions  a l l temperatures.  (Cl,3quin),  steric  an  at  (Cl,dmso),  make t h e  distortion  show  r.m.s. d i s t o r t i o n s  Such d i s p l a c e m e n t s to  the  typical  assumption  valid.  Though  that there  are  associated very  Cu-Cu d i s t a n c e J  is a  is a  of  linear significant  1 48 problem said  in finding  that  large.  of  to  which  as  cases  t h e E mode, on  Cu OZ L 4  of  6  may  general,  may  give  which  the  of  than  the  as  relative  indirectly  by  system  perturbing  space a  a  group  as  which  rather  than  system  However  static  f o r T>0  effects  the  symmetry.  reduction  or o r i e n t a t i o n  reduction  o f L,  the  even  in static  in this  distortion  field  on  singlet the  case will,  equivalent  remaining minima  such as  In  i n the  interconversions  those expected  cannot  importance Even  of so,  comparisons,  other systems.  the  i . e . excluding  susceptibility  above,  pathways.  such  will  between  them  splittings  static  model  and  triplets.  t r e a t m e n t of  exchange  with  t o dynamic  magnetic  o r t h o g o n a l modes  t h e o t h e r w i s e symmetry  nevertheless,  larger  formalism,  produces  o c c u r . The of  the  relationships,  of  be  anomalously  to p i c t u r e  as  have c u b i c  t h e r m a l p o p u l a t i o n s and  equivalence  A  still  are not  i t can  static.  the magnetism.  f a v o u r one  rise  are  or  structure  affect  may  distortions; have  fixed  the d i s t o r t i o n  to the  fluxionality  still  interactions  complexes  t t  phase  i n t h e two  t h e Cu-Cu l i n k a g e s ,  s y m m e t r y due distortion  comparison,  size  the v i b r a t i o n s  the magnetic  i n which  symmetry  in  vibrations  is essentially  Few  this  imposing c e r t a i n  phase  picture  of  fair  i s i t p r o b a b l y more c o r r e c t  interactions  comprise  for a  thermal e l l i p s o i d s  Thus  7r/2 o u t  molecules  lead  data  within  directly  the o x i d e and  Displacement  of  Heisenberg  t o a d e t e r m i n a t i o n of halide  some i n f o r m a t i o n both between  the  may  t h e Cu„  the c e n t r a l  magnetic be  obtained  clusters oxygen  and  atom  1 49 transforms to  T  as  or  T  Table  5.3  magnetic  to oxygen  associated  is linear  coupling  negative  the  preponderance  The  t o be  copper because  the  cause  atoms are of  their  and  the  ligands.  either  the  h a l i d e or Standard  to assess; would  (5.11)  2J  this  angle  0=109°. The  would  question which  positive  and  be  1  i s i n a very  from  The  to  observed to  be  mass and  or  strong  both  would  must  suggest  halide  that  bridge  ferromagnetic  Cu-O-Cu p a t h w a y be  formula  large 8 2  be  •  i s more  and  :  77.60  i s not  obtain  net  J  f o r the t e t r a h e d r a l  surprising  environment  the  formula.  J associated with  c o m m e n t s on  some o t h e r  in  Chloride  pathway  for  either  E distortions  the  s i n c e the  the  different  small negative  d i s c u s s i o n of  of  correlations  in a negative  discrepancy  were used  the  c o n t r i b u t i o n cannot  expected  result  of  larger  perfect angle  only  lead to  expected  oxide  p-orbitals.  the  possible. Further  after  the  (cm' ) = 7555 -  since  dimers  90°,  of  leads  distortion.  Given  w i t h a J>0.  as  i n the  c h l o r i n e a t o m s may  v i a orthogonal  difficult  and  and  both  i s close to  in  copper  Cu-X-Cu c o n t i b u t i o n i s p o s i t i v e  angle  are  i f J  likely  behavior. fixed  a displacement)  seem t h a t d i s t o r t i o n  i s most  Clearly  in  J  the  i t would  relatively bonds  of  of  distortions.  2  octahedron  the  ( i . e . as  2  distortions  2  Displacement E  T  this  M-O-M  topic  as  the  from Both the  are  containing  oxygen  that a  in  the  small  oxide  pathway  made i n clusters.  5.4.2.  1 50 5.2.4 C o n c l u s i o n s The  rival  exclusive, the  however  others.  the copper strong  For  each  interactions  isotropic  coupling fit  would  which i s  suppress  one o r o t h e r o f t h e  reasons I conclude that,  be r u l e d  though  and i n t e r c l u s t e r  u  6  small  exchange  o u t , the primary cause  behavior of Cu OX L„  magnetic  exchange,  constants.  One  of e q u i v a l e n t  as  distortions  in  t h e s e compounds.  intercluster  resulting  f o r the  complexes  exchange  ground  2) O n l y  the parameters  in clusters  intracluster  exchange  f o r Cu OX L„. a  coupling  i s  and  must  exchange  be r e m o v e d t o of the natural  c a n be e x p e c t e d  i s no o b v i o u s  little  e v i d e n c e f o r an  orbital  state.  with  obtained  the physical  using  to  equivalent  c o n s t a n t s i s t h e most  there  pathway.and  i t i s usual  The r e m o v a l  6  from magnetic  In contrast,  doublet  correlated  exchange  of these assumptions  the experimental data  assumption  be  while  In both cases the  state  sense  effects,  the degeneracy  distortion. ground  symmetry of  distortion.  1) When c o n s i d e r i n g assume  tend t o suppress  removes t h e h i g h  remove  from n o n - i s o t r o p i c cannot  mutually  i s expected t o dominate.  unusual magnetic  dynamic  would  In this  the following  interactions  mechanism does  s u p p r e s s i n g any o r b i t a l  f o r dynamic  exchange.  are not e n t i r e l y  4  distortion  of a non-magnetic  contributions  6  possible  coupling  force  intercluster possible  4  environment,  spin/orbit  production  for Cu OX L  Thus dynamic  the d r i v i n g  rather  models  the d i s t o r t i o n  model can  and c h e m i c a l n a t u r e of the  151 clusters. 3)  The model p r o v i d e s  behavior 4)  Only  of these  available  5.3  [M Q(RCOO) ]*  experimental  3  e  Clusters  i n Chapter  has been  e.s.r many  1 2 2  1 1 1  incomplete c l u s t e r s e.s.r.  3  "  3  6  '  1  2  0  directed  been  difference  fits to  the cases  spectrum  undergo as  o f M=Cr, t h e e x p e r i m e n t a l  towards a very  capacity  thorough  [Cr O(CH CO0) ]C1.6H 0, 3  1  1 1 8  investigated  1 2 0  arrangement of  circumstances  the magnetic  compound,  are only  triangular  suitable  spectroscopy  and M o s s b a u e r  1 2 7  6  affect  way; t h e m a g n e t i c 5  good  where M i s chromium o r  +  3  and heat  and o p t i c a l  systems have  of g i v i n g  [M 0(RCOO) ] ,  3. I n t h e c a s e  of a single  susceptibility  of the magnetic  data.  may, u n d e r  largely  investigation  i s capable  an e q u i l a t e r a l  d i s t o r t i o n s which  described work  o f t h e form  atoms a n d t h u s  dynamic  picture  clusters  typically exhibit  metal  physical  compounds.  t h e d i s t o r t i o n model  all  iron  a simple  1  2  '  '  1  1 2 9  3  1  7  .  6  2  m e a s u r e m e n t s a n d by In contrast,  but only  in a  s u s c e p t i b i l i t y data  f o r M=Fe,  rather f o r these  occasionally  s u p p l e m e n t e d by  data.  because  Partly  M=Cr a n d M=Fe w i l l  by  of t h i s  be c o n s i d e r e d  separately.  The  d i s t o r t i o n Hamiltonian  used  i n the following  discussion  is: (5.12) H = - J [ ( S ' - 3 S ) 2  The  corresponding  spectrum,  2  +  X/2(S' -3S" +3S )]  Table  2  2  2  3.1, i s m o d i f i e d  t o that  of  1 52 Table  3.3 b y f a s t  5.3.1  Chromium It  become since  acetate.  i s perhaps t h e most  with  model Uryu  suggested  heat the to  that  data  apparent  be d u e t o a s m a l l i s sensitive  S'=1/2 u n t i l  Significant disproved  S'=l/2  of the waters  studied  by Wucher  J=11cm~ ,  r=0.16  1  1 1 2  indicated  states  non-isotropic  interaction  other  than  may rather  . On t h e same J b u t r=0.08  i n the value  of r  t o the s p l i t t i n g of i s just  are thermally  that  due  occupied.  e x c h a n g e was c o n c l u s i v e l y  behavior " 5  t o 0.3K w h i c h  0=0.13K. G i v e n spin-orbit  exchange terms,  a direct  t o an  dependence; t h e s p e c i f i c  be due t o e i t h e r  introducing  fitted  a similar  inconsistency  magnetic  from C u r i e - W e i s s must  behavior  i n 1955. The  d o w n t o 2K w e r e  1 1 1  low temperatures  spin  intercluster  splitting  (n>2) c l u s t e r ,  of hydration  whereas t h e s u s c e p t i b i l i t y  higher  compound h a s  and thermal  by a s u s c e p t i b i l i t y e x p e r i m e n t  departures  this  polymetallic  temperature  a t very  S'=1/2 s t a t e s ,  that  magnetic  model w i t h  capacity  2  itself.  was f i r s t  triangle  t h e heat  could  studied  properties  cluster  3  6  unfortunate  susceptibility data  isosceles  3  interesting  The compound magnetic  3  rather  with  the M  [Cr 0(CH COO) ]C1.6H Q  intensively  i t srather  be a s s o c i a t e d than  fluxionality.  o r an  showed no this, the  coupling, intercluster  exchange i n t e r a c t i o n ,  or dynamic  distortion.  The Sorai  key experiments  et a l  1  1  7  ,  on t h i s  who r e d e t e r m i n e d  cluster  t o date a r e those of  the specific  heat  over  a wide  1 53 temperature optical differ  range,  and  (electronic)  Ferguson  spectrum.  s i g n i f i c a n t l y at  Wucher,  exhibit  reproduced presence  by  and  low  The  heat  arrangement  of more t h a n  two  of  3.5K  two  1 1 9  who  from  Kramers  of  interaction,  at  least  at very  t e m p e r a t u r e s . Measurements of  phase  temperature  transition  increase  at  211K,  i n t h e number o f  temperature. crystalline  The  phase  water,  an  idea  now  The  AH  f a r too  interactions.  assumption  that  transition  creates  obtained  f i t to the  X=0.1 by  a  and  the o p t i c a l  T=40K a n d obtained  As  Sorai  a result  t o be  of  1 1 9  two  parameters capacity  order  below t o an  an  this ordering  show a  data  can  with  f o r J=10.5,  was  with  and  supported below  those  measurements.  these experiments  i n these c l u s t e r s  the  reduction  symmetry and  any  more c o m p l i c a t e d e f f e c t s ,  seem t h a t  i s probably largely  ordering such  i t would  a t 211K  as dynamic  .  the  resolved  consistent  8  phase  heat  model be  2  reasonable  the  specific  1  of  transition.  types of c l u s t e r ,  cluster  , which  S'=l/2 s p l i t t i n g of  two  first  structural  with  two  heat  t h e r e was  associated  associated  temperature  spectroscopy  the heat  ascribed  made t h e e n t i r e l y  X=-0.03. The  assigned with from  was  e q u a l amounts of low  ;  - 1  transitions  s u p p o r t e d by  large  the to a  compound does not  the o r d e r i n g  J=10.5,  AH=3322 J m o l  transition  c o r r e s p o n d i n g anhydrous  magnetic  showed a peak due  infra-red  The  is clearly  intercluster  with  The  the e x i s t e n c e  and  t o room  which  be  doublets.  of c l u s t e r ,  capacity  data,  cannot  types  low  o b t a i n e d the  t h o s e o b t a i n e d by  which  doublets implies  h e n c e an  ,  capacity  temperatures  a b r o a d maximum a t  an  Gudel  rather  the  due  to  than  to  distortions  1 54 associated system were  with  have  magnetic  appeared  published,  intercluster seem  interactions.  since  invoking  exchange  t o be somewhat  the heat  between  1 2 5  redundant.  pairs  cluster  but, as i n the 'dimerised'  at  from  they  that  to the total  results  and  1 2 3  of c l u s t e r s ,  would  dynamic  distortion  Cu^O,, c u b a n e  non-magnetic  this  in this  clusters,  effects  i s not  feasible  present.  5.3.2  [Fe 0(RCOO) ]* 3  The  iron  clusters The  be v e r y c l o s e  parameter larger over  clusters  f i  measurements. to  significantly  on  and o p t i c a l  I t i s possible  contribute  of magnetic  capacity  antisymmetric exchange  effects  separation  Though p a p e r s  6  S  a much  ground  state  larger  energy  suitable  f o r magnetic  means t h a t  3 +  n o t be i n c l u d e d  t o magnetic  f o r t h e Cr system  susceptibility  of F e  t o 2.0 a n d n e e d  in fittting  than  are particularly  data; also,  J  and the magnetic  range. Mossbauer  m e a s u r e m e n t s c a n be u s e d  g i s likely  as a  variable  i s typically spectrum  as w e l l  to detect  extends  as magnetic  ordering.  The have  experimental data  been  using  fitted  a least  (isosceles) and  parameters compared  t o t h e dynamic  squares program.  and fast  essentially  illustrated  of Earnshaw  fits  ,  I t was  and f i t s  t o those  3  model  found  models  that  give  3 6  and  Long  defined  1 2 0  above  the slow  almost  identical,  to a l l the data. This i s  for Fe 0(CCl COO) (H 0) C1.H 0 3  Duncan  distortion  fluxionality  perfect,  3 5  6  2  3  obtained using  obtained using  1  2  0  2  i n F i g . 5 . 7 . The  the distortion  the isosceles  model a r e  model  i n Table  155  CD C O  ~  0  1 60  1 120  1 180  1 240  1  300  TEMPERRTURE/K Fig.5.7.  F i t s t o e x p e r i m e n t a l d a t a f o r complex L4. The t h e o r e t i c a l c u r v e s , i n o r d e r o f i n c r e a s i n g moment a t 20K, c o r r e s p o n d t o an i s o c e l e s t r i a n g l e w i t h X<0, f a s t f l u x i o n a l i t y , and an i s o s c e l e s t r i a n g l e w i t h X>0.  156 5.4. As rise was  indicated i n Table  t o two minima noted  by L o n g  (see Fig.3  the s i m i l a r i t y  Thus  f o r every  good  f i t with  sense is  superior  2  J i  = J 2 3 > J i 3 .  2  = J 2 3 < J i 3  The  to the s t a t i c  made t o d e f i n e  of r e f . 1 2 0 ) .  of the spectrum  f i t with 1  t h e i s o s c e l e s model  i n F I T as t h e two J v a l u e s  from  J  5.4,  8 and  this  fast  model  with  The X>0  there  are varied, ambiguity to that  ambiguity  is  case  model since  avoided.  gives as  arises  with  i s another  fluxionality in this  always  X<0,  similarly i s in a no  attempt  1 57 Table  5.4  Static  and dynamic d i s t o r t i o n  Fe (RCOO) * 3  t  Equilateral Triangle FIT J  6  FIT  X<0  -o.  fits  to  data Isoceles  J  model  Triangle J  X>0  FIT  Dynamic Distortion J X FIT  E1  23. 0 0 .119  25. 5  26 0. 0 0 9  26 .4  0. 38  E2  26. 9 0 .036  2 7 . 7 - 0 . 1 2 0. 020  27 .8  0. 1 5 0. 0 2 0  2 7 . 8 0. 1 4 0 .020  E3  28. 2 0 .043  2 9 . 6 - 0 . 1 4 0. 0 0 6  29 .9 0. 18 0. 0 0 4  2 9 . 7 0. 1 6 0 . 0 0 5  E4  29. 5 0 .053  31 . 3 - 0 . 1 5 0. 007  31 .7  E5  30. 8 0 .065  33. 7  E6  31 . 0 0 . 0 4 5  3 2 . 8 - 0 . 1 3 0. 0 0 3  33 .0  0. 1 6 0. 0 0 3  3 2 . 9 0. 1 5 0 .003  E7  3 2 . 6 0 .028  3 3 . 8 - 0 . 1 1 0. 01 1  33 .9  0. 12 0. 0 1 2  3 3 . 8 0. 1 1 0 .012  L1  2 7 . 9 0 .087  30. 2  1 4 0. 0 1 2  30 .4  0. 18 0. 0 1 2  3 0 . 2 0. 16 0 .012  L2  28. 7 0 .075  30. 4  1 2 0. 0 2 3  30 .8  0. 1 6 0. 0 2 2  3 0 . 6 0. 1 4 0 .022  L3  2 7 . 6 0 .057  2 8 . 9 - 0 . 1 1 0. 017  29 . 1 0. 1 4 0. 0 2 0  2 8 . 8 0. 1 2 0 .019  -o.  -o. -o. -o.  18 0. 0 0 5  0. 0 0 6  2 5 . 9 0. 32  0 .006  0. 20  0. 0 0 3  31 . 5 0. 18 0 . 0 0 5  34 .3 0. 23  0. 0 0 8  3 4 . 0 0. 21  0 .006  L4  29. 4  25 0. 027  31 .0  0. 38  0. 0 0 9  3 0 . 0 0. 30  0 .015  L5  2 4 . 2 - 0 . 22 0. 024  25 .2  0. 33  0. 0 0 8  2 4 . 6 0. 27  0 .015  L6  2 6 . 6 - 0 . 24  27 .6 0. 35  0. 0 1 7  2 6 . 8 0. 28  0 .026  0. 037  D1  2 6 . 8 0 .040  2 7 . 5 - 0 . 1 2 0. 0 2 0  D2  2 9 . 0 0 .047  30. 1  D3  2 7 . 2 0 .067  28. 7  D4  2 4 . 7 0 .034  25. 2  D5  26. 8 0 .036  27. 6  f  -o. -o. -o. -o.  27 .6 0. 1 5 0. 0 1 9  2 7 . 3 0. 1 4 0 . 0 1 9  1 3 0. 021  30 .3  0. 1 6 0. 021  3 0 . 3 0. 1 5 0 .021  1 7 0. 017  28 .8  0. 21  0. 0 1 5  2 8 . 7 0. 19 0 .016  1 3 0. 0 1 5  25 .3  0. 1 5 0. 0 1 3  2 5 . 2 0. 1 4 0 .014  1 1 0. 0 1 3  27 .6 0. 14 0. 0 1 3  2 7 . 6 0. 1 3 0 . 0 1 3  The f o r m u l a e a s g i v e n i n t h e o r i g i n a l L , r e f . 1 2 0 ; D, r e f . 3 6 ) a r e :  papers  ( E , r e f . 35;  1 58 E1 E2 E3 E4 E5 E6 E7 L1 L2 L3 L4 L5 L6 D1 D2 D3 D4 D6  : : : : : : : :  [Fe [Fe [Fe [Fe [Fe [Fe [Fe [Fe [Fe [Fe [Fe [Fe [Fe [Fe [Fe [Fe [Fe [Fe  3 3 3 3  : : : :  5.3.3  3 3  3 3 3 3 3 3 3 3 3 3 3 3  (C H C0 ) ].(C H C0 ) .3H 0 (CH C1C0 ) (OH) ].CIO,.4H 0 (C H C0 ) (OH) ].CIO,.3H 0 (CH C0 ) (OH) ]C1.4H 0 (C H C0 ) (OH) ].C H C0 .7H 0 (CCI3CO2) (OH)3](CCI3CO2) H 0 (C H C0 )5(OH) ].C H C0 . 0(HCOO) (H 0) ]OH.2H 0 0(CH COO) (H 0) ]C1.5H 0 0(CH C1C00) (H 0) ]C1.5H 0 0(CCl COO) (H 0) ]C1.H 0 0 ( C H C O O ) ( H 0 ) ] C H 0 . 2H 0 0 ( C H C O O ) ( H 0 ) ] C H 0 . 6H 0 (HCOO) (OH) ]OH.2HCOOH (CH3COO) (OH) ]N0 .5H 0 (C H COO) (OH) ]N0 .2H 0 (CH ClCOO) (OH)2]N0 .4H 0 (CCI3COO) (OH) ]N0 .3H 0 6  5  2  2  6  5  2  3  2  s  5  5  2  2  2  3  6  2  3  5  2  3  6  6  2  2  2  2  3  6  3  6  2  2  3  2  2  3  2  8  7  6  2  3  8  7  2  2  7  5  6  2  3  7  s  2  2  6  2  6  2  5  2  6  2  3  2  6  of  2  3  6  2  2  3  2  3  2  results  3  6  most  2  molecules which are not part that  symmetry  their ordering  may  On  ordering  i s about  complexes the ground rise  3  8cm  than - 1  of  were  hand  solvent  there and  X i s great state  3  and  makes  and  enough thus  t o i/5 t o w a r d s T=0.  distortion  3  to  80cm"  to bring  an  This  difference  molecules i s  i s no be  effect. Also  the s p l i t t i n g  about  a  the  the d i s t o r t i o n are  1  of the  in Fe . 3  In  exhibit a  close minimum  i n t h e amount  to suggest that  S'=l/2 several  S'=3/2 s t a t e  t h e moment m i g h t  i t reasonable  to the  X as might  the dominant  i n C r . Thus  in Cr  3  possible  solvent  i n t h e m a g n e t i c s p e c t r u m p r o d u c e d by in Fe  Fe 0  i n which the m o l e c u l e can have  the other  t h e amount  the  of the c l u s t e r  I n some c a s e s i t i s n o t e v e n  of three.  between  of  contribute  a x i s , a s t h e number o f  i f solvent  much l a r g e r  and  6  5  a multiple  changes  to  2  2  down a s t r u c t u r a l f o r m u l a  expected  3  2  2  distortion.  correlation  Fe  2  2  3  three-fold  states  3  i n the case of C r 0 ( C H C O O ) C 1 . 6 H 0  observed  not  2  2  6  i t i s possible  write  5  2  6  5  compounds c o n t a i n and  6  6  6  2  6  Discussion  As  6  2  of  the causes of  159. distortion  here  are  not  identical  to those  f o r the  chromium  acetate. One  of  the  distortion  and  six  18  of  the  similarly out  of  X >  0.2,  to  be  f e a t u r e s of isosceles  clusters  only  five  R  within  X w i t h the  the  number  numbers of  For  of  (L4,L5,L6) expected  on  indicate  a  variables: and  R,  of  the  the  is slightly fast  tendency  here. these  nature  compounds w i t h l a r g e  M(T)  simple  other molecules  fluxionality towards  slow  -30±3 c m  X.  and  - 1  0.16±0.04. I n the  size  five  five  effect  Only  with  seems  Attempts  to correlate  clusters  are  the anion  hampered and  J by  the  lattice.  data  low  model  J and  interactions  i n the  at  the  steric  of  X and  larger  of  i n f o u r of  Though a  that  significant  chemical nature  water  the  are  range  range  J and  group.  i t is unlikely  the c l u s t e r  and  o u t s i d e the  X o u t s i d e the  a phenyl  obtained w i t h both  i s the constancy  with unusual  involves  occurring  results  models  have J  have  six clusters  the  down  to  20K  temperatures  (Fig.5.7).  fluxionality  at  than  This  the  may  lowest  temperatures. In  conclusion,  Fe 0 clusters 3  results steric  and  300K and  the  this  the  excellent,  fits  magnetic, An  lower  dynamic  analysis  of  temperature  r e s p e c t . I would  isosceles  m o d e l be  and  and  the  the  static  heat  magnetic  suggest  used  to magnetic  the p h y s i c a l  p r e s e n t s great problems  determined.  in  are  though  data  for  intepretation  relative  has  between  in future  the dynamic, fits  the of  not  4.2  measurements would  that  of  importance  distortions  capacity  the  been  and  be  rather  to experimental  useful than  1 60 magnetic and  data  because  For energy that  may  a f f e c t i n g the importance  dynamic  distortion  change a s s o c i a t e d  associated  following be  with  with  non-magnetic  Cu  symmetry  The M  3  stronger  3  fl  must  the JT type  be g r e a t e r  of ordering.  3  core  than  The  3  has e q u i l a t e r a l t r i a n g u l a r symmetry, t h e low space  clusters typically  clusters considered  from  susceptibilities  that  suggest  might  and that  contain  impossible,  of c r y s t a l l i s a t i o n  non-cluster  molecules.  be e x p e c t e d  i n the neutral Cu  bonding t o be Cu„  systems.  systems behave  a  their  quite  magnetic  a number o f w a t e r ,  stoichiometrically  unlikely.  space  i s increased.  i n some o f t h e s e  arrangement  higher  above a r e - i o n i c . I o n i c  (RCOOH) m o l e c u l e s ;  molecules  rather  l a r g e r d i s t o r t i o n s which a r e not  removed as t h e t e m p e r a t u r e clusters typically  s y m m e t r y a t room  cubic.  t h e two a n i o n i c  the others  group  have  than Van d e r Waals c o u p l i n g  differently  3  types  a n d some a r e e x a c t l y  i s noticeable  M  distortion  t h e F e 0 , C r 0 a n d Cu„0 c l u s t e r s  between c l u s t e r s , v i a t h e a n i o n s ,  3/  'best f i t '  significant.  Though t h e M  group  of dynamic  distortion  d i f f e r e n c e s between  temperature.  It  a unique  e f f e c t s t o be o b s e r v e d ,  chromium c l u s t e r has a r a t h e r  2/  gives  r e s u l t s i n an unambiguous J .  5.3.4 F a c t o r s  1/  t h e new m o d e l  high  or a c i d  symmetry i s  i n the others  disorder  may make a h i g h  symmetry  T h e Cu„ c l u s t e r s t y p i c a l l y  of the  contain  no  161 4/  I t may  be  'inherently highly  symmetrical  The  3-  fluxionality  In  of  and  h e n c e may  magnitudes  to  pack  triangle  lattice,  tetrahedral  clusters  than  Cu^  of  copper(II)  be  particularly  the  to  pack  the  clusters.  i s known  for i t s  sensitive  to  1  of of  to  the  cluster magnetic  9J/9d  Tetranuclear  and  Fe  "cubane"-like  to  factors  effects, the  groups  which  the  transition  d i f f e r e n c e s i n the  force constants  may  be  relative relevent.  clusters  occur  are  important  in several 1  atoms a r r a n g e d  2  which  3  mixed  v a l e n c e . ( F e (111.) , F e (11 ) ) c o m p l e x e s a l s o o c c u r . i t is surprising  Fe„-carboxylate-methoxide c o m p l e x e s s t u d i e d . The  Fe 0(CH COO),o 4  The  3  interpretation  as  magnetic  is  hoped  the  , and  data  that  moment a t  the  f o r two  3  must  available  results  lower  have  will  6  be  3  6  regarded  tetrahedra  the not  such  Fe,(CH COO) (OCH ) ,  given  are  data  that  been  and  Given  the  more  compounds" , 1  are as  discussed somewhat  down t o a b o u t  have  some p r e d i c t i v e  The  a  simple  only  temperatures.  are  often contain  number  intensively  in linked  1  large  interest  areas;  in f e r r e d o x i n s " ~ " ; there  iron-tungsten clusters  metal  influence  undergo a phase  complexes  u  several.polynuclear of  above  Fe(lII)  Tetrahedral  biological  in a  " .  1  the  independent  4  3-dimensional  dimensional'  addition  tendency  Fe  equilateral  c o o r d i n a t i o n sphere  "plasticity"  5.4  more d i f f i c u l t  2-dimensional'  'inherently  5/  intrinsically  treatment  tentative  100K,  also  below.  but  value  i t  for  illustrates  1 62 how  the factorisable  interpretation Though 3  data  3  Fe  with  4  data  core, which  the Cu 0  f l  3  suggest  fluxional; systems  fast  H(E) = - J ( S  , 2  (5.14)  H(T) = - J ( S  , 2  Treatment  4  3  though  similar  the variation  decrease  behavior cannot  such  a model  this  range.  and  t o magnetic  might  i s very  -70)  data  be s t a t i c  presented  or  i n S=5/2  small.  -70)  -J'(S' -3(S  slowly tetranuclear  The a p p r o p r i a t e  2  -J'(S' -2S, 2  with  for this  2 1 3  2 2 3  +S  2 2 4  ) 70) +  +17.5)  temperature unusual  As might data,  a r e summarised  an  increasing  be e x p e c t e d  fits  i s small  cluster i s  Cu„0  species,  the behavior can  a s t h e r e i s a maximum, o r a t  be f i t t o a be  predicts  neutral  f o r the neutral  i n slope of M VS T with  This  temperature  centred i n the case of  distortions are:  to that  regarded as s i m i l a r l y  least  formulae,  an a p p r o x i m a t e l y  i s unlikely  T h e moment  6  qualitatively  on  of data  Fe 0(CH COO) •  be  2  (5.13)  5.4.1  which  fluxionality  f o r E and T  clusters.  the structural  6  oxygen  fits  unless the d i s t o r t i o n  Hamiltonians  studies  and  i s c a r b o x y l a t e b r i d g e d a n d , by  0  distortions  3  suggest  systems,  4  for larger  of X-ray  6  compound F e O ( C H C O O ) , . The below  feasible  and Fe„(CH COO) (OCH ) ,  0  and Mossbauer  tetrahedral analogy  of magnetic  method makes t h e f i t t i n g  t h e r e a r e no r e p o r t s  Fe„0(CH COO), infra-red  distortion  decreasing  f i t t o a model slope with  from  the lack  temperature. with  one J a s  decreasing T in of low  t o b o t h E a n d T m o d e l s c a n be o b t a i n e d ,  i n Table  5.5. The a d d i t i o n  of the  second  1 63 parameter, of  J ' , improves  10. A l l t h e f i t s  Table  5.5 F i t s  the least  squares  t o two p a r a m e t e r s  t o magnetic  moment  are essentially  data  a  1  The  T  J'>0  J*<0  J'>0  1.48 3.38 0.010  1.22 -2.24 0.008  1.35 10.55 0.011  e x i s t e n c e o f two f i t s  with  a rotation  space,  ratio  of the d i s t o r t i o n  the situation  isosceles  f o r both  triangle  associated ratio  f o r the T For  model  2  t h e chosen  singlet  stable  distortions;  the other  encouraging  thus  not.  J i s about  In Fig.5.8  and  fits  ,  models  i sa s s o c i a t e d  found  with the  -11/7 w h i c h i s state  energies; the  t o -8/3. i sthe  f o r both  E and T  have  been  even  different  (ueff=/30)  ground  extended as the T  state  i s a singlet.  2  whereas  The c u r v e s  2  c a n b e d i s r e g a r d e d . An  the theoretical  strikingly  predictions  - 1  0.072  fits  though  i s that  J i s quite  the d i s t o r t i o n  t w i c e t h e v a l u e o b t a i n e d on t h e one  model.  state  fits  feature of the remaining  d e f i n e d , J=1.15±0.1 c m  2  close  configuration  well  T  ground  c o n v e n t i o n , J'<0  thermodynamically  2  0.644  f o r the E d i s t o r t i o n the  to the value  i s similarly  sign  distortion  1.05 -4.68 0.008  E and T  Thus  3  with identical  0  J'<0  to that  f o r M 0.  o f J'<0:J'>0 i s c l o s e  perfect.  3  No  factor  vector i n the appropriate  i s analogous  model  a  f o rFe O(CH COO),  E J (cm" ) J' FIT  f i t by a b o u t  curves  mode i s  parameter  associated with the E  t o T=10K. The c u r v e s a r e distortion  results  for E.distortion c a n be r e g a r d e d  f o r t h e low temperature  moment.  as  i n an the  S'=5  ground  alternative  1 64  o  o CD  o in  o  CD  Lj  - oo J  '  o CNJ  -i  "T"  50  100  1  150  200  TEMPERRTURE/K  250  F i g . 5 . 8 . D i s t o r t i o n model f i t s t o d a t a f o r Fe,0(CH C00), (above) and F e , 0 ( C H C 0 0 ) , ( O C H ) « . 3  0  3  3  300  1 65 Feg(CH COO) (OCH ) . 3  have  6  3  6 carboxylate a  bridges  cannot  be  that  J  is significantly  with  the  as  c e n t r a l oxygen  E-model  obtained  Though  6  on  the  this  compound  i s expected  the  previous  cluster,  did  a t o m . The  fits  to  the  more a n t i f e r r o m a g n e t i c .  results  i n no  T-model;  improvement  taking  the  one  to  there  moment i n d i c a t e V a r i a t i o n of  i n f i t . Two  associated  fits  with  J' are  J'<0  the  result i s : (5.15) T - d i s t o r t i o n , J=-4.38, J'=-2.526 No  Thus  distortion,  For  J<0,  the  moment  towards 5.4.2  the  any  Discussion  the  other  several of  the  suggests  that  exchange  is  the  fully  oxygen  just  exchange  five  T=0,  S'=5  as  ground  state.  rise  s h o w n on  the  obtained  in this nature  be  and  i n the M-O-M  orbital  integral,  Fig.5.8.  that  f o r Fe„,  i n the two  Fe„  pathway  importance  polymetallic complexes  is quite  t h a t , as  integral found  25cm  strongly  Heisenberg  each atom, a J  <12|V|21>, of  and  suggests  relative  each atom. T h i s  exchange with  on  both  chapter  remembered h e r e  e l e c t r o n s on  comparable  J  centred  one  i n 6 . 5 . 1 . The  therefore  at  J values  i n s i g n of  I t should  involves  are  /30  p o s s i b l e exchange pathways  the  i m p l i e s an  more  of  an  results  between  change  ferromagnetic.  FIT=0.080  e x h i b i t a minimum and  clusters discussed  various The  there  value  to  comments c o n c e r n i n g  clusters.  Fe  of  -  3|J|/8 p r o d u c e s  i s expected  limiting  Comparison for  |J'| >  J=-3.85,  FIT=0.020  - 1  of  1cm  - 1  since  point  is  discussed  i n the  Fe„  clusters  i n some C u « 0  (while  in  1 66 that  i n the M 0  systems i s very  3  contribution centred  to J suggests  tetrahedra  orthogonal  carboxylate  contribution  to the  antiferromagnetic atom  avoids  pathway  2  orthogonal.  assertion  as  sp  the Fe  the  in Fe 0  and  an  both  dominant  i s a very  3  suggests  that  strongly the  oxygen  w h i c h w o u l d make s u c h a  i s some s u p p o r t i n g  t h e oxygen atom  oxygen  systems, which  that  This  hybridisation  i n oxygen  orthogonal.  suggests  pathway.  pathway  a n d Fe„  3  positive  hybridised  3  (J<0) e x c h a n g e  There  This  t h e M-O-M  pathway  bridges  M-O-M  the sp  an  pathway,  A c o m p a r i s o n between contain  that  involves  exchange  large).  i s rarely  evidence  found exactly  for this i n the  M  3  plane. With compound  these  formulated  conductivity, trigonal axial  J  a  results  the magnetic data  as F e 0 ( 0 C M e ) , ( 0 C M e ) , 5  solubility  bipyramid , an  i n mind, 2  2  f o r which  and mass s p e c t r u m s u g g e s t e d  o f Fe a t o m s ,  equatorial  2  f o r the  J  e  was  , and an  investigated. internal  Fitting  J. with 1  a  -  t o an  the  Hamiltonian: (5.16) H = - J ( S ' - 3 S ) e 2  though  leading  equatorial  2  - J  (S' -S' -S' ) 2  a  to considerable  J of about -15cm  obtained  to the t r i m e t a l l i c  J=-28.5,  r=-0.29,  - 1  2  2  ambiguity  . However  distortion  which are t y p i c a l  J.(S' -2S ) l 2  suggested  just model  B e c a u s e o f t h e more n o r m a l J  triangular  model  2  axial  as good a  fit  was  Fe 0 3  found i n the f i t  c a n be  and  with parameters  for equilateral  complexes.  the magnetic data  -  said  to support  to the a  167 trimetallic  rather  than  a pentametallic  structure  for  this  complex. The harder J>0,  r o l e of  to  the  define.  J might  be  M-X-M  As  of  the  pathway  i n Cu„  c e n t r a l oxygen  either  sign,  though  clusters  is sufficient a  large  contribution  is unlikely. A  small  ferromagnetic  J  with  angle  of  associated  consistent  with  the  the  M-X-M  is  about  c o r r e l a t i o n s observed  90°  somewhat to  cause  antiferromagnetic contribution would  in other  be  systems.  to  1 68 CHAPTER 6.1  6  L I N E A R MAGNETIC  SYSTEMS.  Introduction to Linear  The  linear  interest quantum  magnetic  effects  play  a vital  s o l u t i o n s have  counterparts  problem  problems  of i n t e r e s t  such  type  a s TCNQ  transition  5  0  ,  superconductors.  systems,  changes  coordination,  chemistry Both  sense of  fluids " 1  Magnetic because,  can give  and because  they  chains like  '  1 5  2  finite  chapter.  complicated  a n d 1-D 8  behavior insight  1  organic  of  high  great coupled  on s u b s t i t u t i o n  c a n be u s e d  The  9  becomes an  a r e of  into  5  bonding  of  and  to i n v e s t i g a t e the metal  . and i n f i n i t e  Only  i . e . those  7  a l l exchange  of p o l y m e r i s a t i o n i n t r a n s i t i o n 1 5 1  1  i n which a regular chain  i n the magnetic  chains  systems which  w i t h a maximum  two a r e d i s c u s s e d ,  solid  both  l i g a n d s and metals  other  1  1  no  1-D  (tetracyanoquinodimethane) " ' " .  temperature  i n chemistry  have  or  ferroelectrics " ,  i s an o b s t a c l e t o t h e p r o d u c t i o n  interest  exact  of the  t o t h a t o f many  realistic  6  i n which  complete  The m a t h e m a t i c s  related  models " , 1  systems  f o r S=l/2 which  1  chain  phenomenon  obtained  i n p h y s i c s * " : e . g . 2-D  alternating  this  theoretical  part. Essentially  been  i s closely  theoretical  conductors  both  i s of g e n e r a l  i n 2 or 3 dimensions.  magnetic  Peierls  chain  a s i t i s one o f t h e s i m p l e s t many-body  numerical  field  Chains  'Ladder'  s t r u c t u r e s which  are linear 'magnetic  type  would  s t a t e p h y s i c s , s i n c e they  are considered i n  may  i n the  c o o r d i n a t i o n number'  and s i m i l a r be c o n s i d e r e d be  'chemical'  infinite  more linear in i n only  one  169 dimension,  are excluded.  initially,  before  constant  will  There  can  be  treatment absence  of a phase  typically  of  have  that  cluster  degeneracy of  magnetic  higher  higher  anisotropic  (6.1)  XY  coupling  systems.  atoms would  atoms  than  atom  mathematical  i n (other)  i n chapter  low symmetry  clusters  chains clusters.  1, w h e r e  i t was  of the c r y s t a l  factor  fields  i n removing  the Heisenberg  e x c h a n g e more v a l i d . As a c o n s e q u e n c e  than  by t h e  range  i n magnetic  those  chain  length  i s simplified  s t a t e and making  systems  chains  systems are  for finite  be an i m p o r t a n t  i n the ground  The  chain  a t T>0  the metal  the i n t r i n s i c a l l y  linear  and, as the chain  to the-discussion  of  any model  their  exhibit  magnetic  anisotropy  and a r e o f t e n  modelled  with the  more  Hamiltonian:  H = -2J  ( a = 0 , 0= 1 )  1 5 3  Heisenberg  one  least  e.g. the three  magnetic  symmetry  i  The  the linear  clusters,  transition  symmetry, c h a i n  frequently  First,  lattice  Second,  i s relevant  by j u s t  for considering  as a c l u s t e r ,  of the i n f i n i t e  3  argued  true  the i n f i n i t e  interactions ".  This  reasons  thesis.  between  be c o n s i d e r e d  increases,  which a r e , at  considered.  in this  intermediate  chains  distortion, defined  are several  separately  Only  I j  =  +  * ", 1 5  (a=j3= 1 )  1 5 e  ~  16 2  1  aS- S- + /3(S- S- +S - S- ) i z Dz i x ]x l y ny the I s i n g ' 8  1  l i m i t s o f 6.1  5  5  ( a = 1 , j3=0) a n d t h e  are a l l of  theoretical  interest. The  third  reason  for considering  chains  separately  i s that  1 70 whatever  the  nature  exceptions  to  assumption  that  J  d(3J/3d) ,  only  the  the  affect  0  chains  the  of  many d e g e n e r a t e magnetically symmetry of no  in  of  distortions the  of  interest  a  magnetic  the  spin 2-  that  that which of  result chain  a=1,  3-D  useful, obtained  to  can  be  lower  of  0=0,  Hamiltonian developed  more t h a n  can  s y s t e m s as  particularly the  result  never  now  w e l l as  with  development  systems  are  respect  to  interest  before was  infinite  chains,  for  low  temperature  that  the  low  survey  which  are  experimental  chain  found  (1925) .  be  to  heat  The  8  be  properties.  temperature  the  Ising's solution  which can  was  the  given.  even  ,  degeneracy  historical  theoretical  1 6 3  with  the  experiment  be  studied  group  this  linear  and  magnetic  a  2  has  are  contain  brief  theory  chain  the  P ,  of  unstable  Bloch  the  s p i n . However  f o r the  by  are  +  modes, t h e s e  symmetry. A  first  = J(0)  no  be  chain  systems the  though  consequence  its intrinsic  k n o w n . The  and  predict that  was  J  of  group theory  total  not  the  magnetic  terms  of  chain  the  need  Hamiltonian  wave t h e o r y  and  terms  a direct  due  they  spectrum  to magnetochemists w i l l  examples were of  In  linear  l o n g i t u d i n a l v i b r a t i o n s of  vibrational  open m a g n e t i c  aspects  As  magnetic  as  In  in distortions,  theorem does not  only  of  the  I.R.s;  a d d i t i o n to  stable,  theorem.  non-degenerate  open c h a i n  the  Jahn-Teller  is linear  transverse  any  exchange c o u p l i n g ,  Jahn-Teller  irrelevant.  degenerate  spectrum  the  applied  to  very Bloch  capacity  of  3/2 the  Heisenberg  chain  antiferromagnetic 0 . 5 - 2 1 n 2 , was  i s p r o p o r t i o n a l to T  ground  state energy  c a l c u l a t e d by  Hulthen  in  of  '  .  the  1938  The S=l/2 c h a i n ,  1 6  ".  E  0  Comparatively  =  171 few  advances  properties On  of  the  contains  chains  2  as  chains  containing a  chains, metal  '  1 6 8  1 6 9  From  Co<, ( C H 0 5  7  point  2  of  in  1963,  who  of  numerical  e x t r a p o l a t i o n from  the  was  expressed  data  by  S=l/2  H a l l  expansions chain  infinite  1  7  3  in T  by  and  1/T  J<0  and  numerical  1 7  f o r the J>0  1 7 5  6  6  '  1  6  1  7  2  1 7 0  ;  .  are  extensive  c a l c u l a t i o n s on  including  those  axial  the  of  in  though  infinite  metal  again  ions  are  finite  copper  the  is  two  the  most  theory  exact  s o l u t i o n f o r x(T)  and  Fisher  systems  w e r e made  3 2  , who  to  Bonner  has  the  and  of  B l o t e  specific  anisotropy,  1  6  heat in  0  -  1  6  of  1975.  S=1/2  of  faster  to play  1  1 5 6  who S>l/2  of  series  the  19 6 8  result  fitting  development  in  the  x(T)  Fisher's  developed  systems  of  used  s u i t a b l e f o r the Baker  by  determine  susceptibility  S=1  those  now,  chain  With  be  .  1964.  ".  to  Haseda  and  7  c a l c u l a t i o n s have c o n t i n u e d  results  with  '  form  Weng made some c a l c u l a t i o n s on most a c c u r a t e  1  an  in  Jotham  1  transition  finite  chain  and  7  Bonner  in a polynomial  for both  computers  and  compound  magnetochemists  obtained  (S=°°) c h a i n ,  for  1  linear  classical  first  m o s t common,  known  8  view  1  systems c o n t a i n i n g  ) , are  involved  J=-5cm~ , w h i c h  2  Watanabe and  the  fewer  c o n t r i b u t i o n s to  3 3  by  followed  still  .H 0,  a  the  wide v a r i e t y of  frequently  the  important  are  Considerably  such as  most  Fisher  .  1960.  was  linear,  more e x a m p l e s chains  thermodynamic  3  magnetically  infinite  the  s i d e , C u ( N H ) (,SO  2  copper  known  of  w e r e made u n t i l  experimental  . Several  1 6 5  determination  -Cu-H 0-Cu-H 0- c h a i n s ,  recognised 1958  i n the  a  role.  , but  the  made chains,  Majumdar  1 5 8  and  172 later  Hatfield  exact  results  With  the  6  have p e r f o r m e d  2  f o r the  magnetic  comparatively  possible results  1  to of  One  improve this  of  some o f  from  difficult  unreliable.  Fisher's 6.3 is  an  the  this  work  by  are  dimerisation deal  of  finite  6.2  The  N,  both  has  so  a  ground  s t a t e of  of  simple  Ising  energy  and  f o r the  f o r S=l/2  in  been  work chains  accurate  the  i s that results  as  Bonner  f o r S>1/2.  true  spin  involves  which can  In  and  section  intermediate  be  easily  i s used.  solved  The  results  finite  has  and  Though  infinite the  tendency  recently received a instability  on  ground but  to great  long  ignored. state  ordered  1938  is  Hamiltonian  energy  antiferromagnetic Heisenberg  p r o p e r t i e s i s not  expression  -0.886294,  type  as  magnetic  f a r been  an  finite  i n 6.4.  chain  effect  i t has  chains.  6.3.  of  infinite  finite  obtain  e x t r a p o l a t i o n s . The  f o r a l l S,  in  to  6.2.  which  Hamiltonian,  of  UBC  numerical  i n which  atoms,  at  been o b t a i n e d  a n t i f e r r o m a g n e t i c ground  The  exact  have  the  in  results  a Hamiltonian  the  chains  the  i s considered  attention,  but  not  of  No  stability  chains  earlier  for small  discussed  the  of  approach  l a r g e numbers of  Heisenberg  its  1964  'odd/even'  Finally,  is  of  approximated  for of  result  results  alternative  spins,  the  conclusions  calculations  computer  discussed  extrapolation and  similar  susceptibility  powerful  work a r e  the  "  s t a t e and  trivial.  Hulthen  state energy, until  the  E = 0  chain  calculation obtained  an  0.5~21n2  =  of  numerical extrapolation  1 73 became  f e a s i b l e the  theory, Kubo  given  to  first  result for  and  second  S>1/2  order  was  that  ofspin  by A n d e r s o n  wave  and  1 7 6  .  1 7 7  Bonner finite  and  Fisher  S=l/2 s y s t e m s  atoms a g a i n s t for  best  n even  energy  1/n  and  demonstrated thermal  that  gave  2  n odd,  within  found  1% o f this  i n the  a p l o t o fE  rise  which  the  to  of  exact  the  two  of  very  value  nearly  obtained  went on t o  linear  their  offinite  0  i n t e r s e c t e d near  r e s u l t they  properties  course  chains  work on  rings  of n  straight  lines,  l/n =0 a t a n 2  by H u l t h e n .  extrapolate using  Having  for  the  a similar  technique. Numerical S>l/2 h a v e  extrapolations  been c a r r i e d out  some m o d i f i c a t i o n s necessary  a s the  intersect  near  believed  that  o f the  lines  for  original  1/n =0. D e s p i t e  for determining  the  I have  re-examined  numerical  results for  authors.  The  which  E (S) 0  i s given  the  spin  i s the  consistent  Blote  but  method  are  i t is  most  generally  reliable  o f S>l/2 c h a i n s .  extrapolations  larger  for  no longer  2  difficulty  f o r S>l/2  systems than with  some t h e o r e t i c a l b a s i s 0  v 1/n  energy  the  and  previous  expression:  = - 2S(S+21n2-l)  some r e s u l t s f o r E ( S ) to  0  properties  somewhat  r e s u l t s are  (6.2)  ofE (n)  extrapolation  method  obtained  state  Bonner-Fisher  this  2  the  ground  b y Weng, M a j u m d a r a n d  on a p l o t  numerical  the  i s given  wave e x p r e s s i o n  E  0  i n 6.2.1. A summary o f  in Table  6.1 w h e r e  'SW r e f e r s  = 2S(S+0.363+0.033/S)  obtained  1 74 by  Anderson  Table  6.1  and Kubo.  Results  8  16 1  1 6  The first  s p i n wave e x p a n s i o n  order  S=l/2  i s worse  expansion obtained  f o r low s p i n ;  may  o n S=1  extrapolation S=1/2  expression. 1%  accurate.  small  spins  and  chains  1 to S =  (6.2)  order  convergent.  Bonner-Fisher  result  Weng  type  to obtain differ  from  suggesting  large.  the others  that  this  results  for general  This  the results  the c o e f f i c i e n t s  f o r only  leaves  i nh i s  by more  technique  s y s t e m s a n d t h e e r r o r b o u n d s on h i s r e s u l t s ,  most a c c u r a t e  problems of  = 2S(S+0.424-0.019/S) from  Majumdar o b t a i n e d  the  S=1  than  to  and the  0  were u s e d  S=1, a r e c o r r e s p o n d i n g l y  =  result  t o o b t a i n E (S=1) and then  Weng'.s r e s u l t s  f o r the higher  very  0  rather  rings only;  was u s e d  a n d S=1  E  the second  order  14.43  i s variational  are significant  f o r example,  be a s y m p t o t i c  9.545  i n p o w e r s o f 1/S  the f i r s t  the expression  calculations  for  than  4/2 5/2 9.452 14.315 9.518 14.38 9.658 14.58 9.40±0.8 14.20±1.0 9.52±0.04 14.38±0.08  5.659  but not beyond. There  convergence for  3/2 5.589 5.655 5.734 5.70±0.06 5.67±0.02  1/2 0.863 0.929 0.886  1 5 6  chains  0  2/2 2.726 2.792 2.810 2.808±0.004 2.805910.0002 0.88629 2.772 0.886  S 1 s t o r d e r SW 2 n d o r d e r SW Weng Ma j umda r Blote Hulthen " (6.2) 1 5  f o r E ( S ) f o r Heisenberg  than  i s not  relatively except f o r  Blote's results  S. B l o t e c o n s i d e r e d  both  i n h i s e x t r a p o l a t i o n a n d made c a l c u l a t i o n s  as  rings for a l l S  5/2.  i s i n agreement  with  the previous  w h e r e t h e d i f f e r e n c e o f 1% i s w e l l  results  except f o r  outside Blote's error  175 bounds. and  A detailed discussion  a comparison  6.2.1  Theoretical  It (6.3)  will  that  formal  proof  has not been  Hamiltonian  symmetry  and boundary  set  state  ground  discussion  eigenfunction,  both  states  rest  to describe  S  n +  ^=S^;  two s i m i l a r but an n - f o l d  (b) a c h a i n  a x i s of o f n+1  energies  induced  by t h e  conditions  f o r t h e two systems  deviations  of E ( n ) from 0  of S ^S.^ +  + 1  \jj{0) . A l l o f f d i a g o n a l flips.  H0 =E 0 ,  problem  states  the f i r s t  tend  E (n=°°) 0  matrix  o  problem  with  where  <j> i s t h e 0  i n the basis  S =0. The z  basis  |S,-S,S,-S,S,-S...> a n d  of these and c a l l  c a n be o r d e r e d or ' f l i p s ' ,  o  o  as a matrix  the I s i n g ground  of the states  operations  spin  a  i+ 1  of the system  | - S , S , - S , S , - S , S . . .>; t a k e The  though  .S_  state  the eigenvalue  of the Ising  contains  reasonable,  interest.  Consider ground  i  conditions  i n t h e boundary  In this  o f no  n Z S i= 1  2  Hamiltonian  S^ = 0 f o r a l l i > n + 1 . A s n —> °° t h e d i f f e r e n c e s  differences  are  i n 6.2.2.  of the  (a) a r i n g of n atoms w i t h  the r i n g and chain  zero.  0  obtained.  ( 6 . 3 ) c a n be u s e d  systems:  to  i s given  0  (6.2) i s t h e o r e t i c a l l y  H - lim n — n  atoms w i t h  f o rE (S=l/2)  now be s h o w n b y c o n s i d e r a t i o n  distinct  in  that  f o r E (S=1)  considerations  equation  (6.3) The  with  of t h e e x t r a p o l a t i o n  by t h e number o f  needed  elements  i t i|/(0).  t o produce  of H a r i s e  them  from  from  such  176 n o t a t i o n i//(1,i),  The spin  flip  \p{0)  by two  be u s e d . for  i ; i//(2,i,j),  at position flips  for a state  related  i<j for a state  at p o s i t i o n s i and j , \//(3,i,j,k)  Thus t h e other  Ising  ground  state  S = l / 2 a n d ^ ( n S , 1 , 1 , 1 . . . , 3 , 3 , 3 . . ., . ..)  sets  elements in  M=0,1,2  {i//(M)},  of each  adjacent  are related  part  o f 4>  which  0  b(i)v//(1,i)  I  o  each  allowed;  takes  f o r S=l/2  nor  \p( 2 , i , i +1 ) , a n  for  the matrix  +  <xP(]  flips  only  The and t h e  to  i n v o l v e s s t a t e s near  1/2  into  those  to  written:  c ( i , j ) i/> ( 2 , i , j ) +  I  account  'adjacent'  elements  the fact  of spin  neither i//(2,i,i), flip,  , i ) |H|>//(1 , j ) >  =  flips a  exist.  c a n be o b t a i n e d ,  <i//(0) |H|<//(0)> = - 2 S  (6.5)  within the matrix,  0  a r e some c o m b i n a t i o n s  thus  S.  ...  i,j  t h e summation S there  to  etc., will  for general  </> c a n be  i where  related  i s \//(n/2 , 1 , 3 , 5 , 7 . . . )  by s p i n  M=0(1)}.  {{\p{M)},  <j> = \p(0)+  (6.4)  blocks  one  .  that  <//(0) , i . e . f r o m  e t c . form  block  blocks  Consider  t o \//(0) by  that for which  'multiple General  are not flip',  expressions  e.g:-  2  ( - 2 S + 0 ( S / n ) ) 6.^ 2  <^(2,i, j) |H|^(2,k,l)>  =  (-2S +0(S/n)  <<//(0) | H | ^ ( 1 , i ) > = <\p(l  , i ) |H|i//(2,i, j ) >  2  ja^e^ = 2S/n  for a l l  i , j .  thus: (6.6) + L i  H0  O  =  [-2S  2  +  ( 2 S / n ) L b ( i > ] 4>(0)  [ b ( i )(-2S +0(S/n) ) + 2  (2S/n)d+  + I [ c ( i , j ) (-2S +0(S/n) ) + 2S/n i ,j 2  Z c(i,j)) j  |^(1,i)>]  ( b ( i ) + L d ( i , j , k ) ) | <//( 2 , i , j ) > ] k  1 77 Premultipling (6.7)  gives:  <x//(0) | H | 0 ( O ) >  = E  «//(1,i)|H|0(O)> = b ( i ) E «//(2,i,j)|H|0(O)> =  0  = -2S  +  2  (2S/n)Zb(i)  = b(i)[-2S +0(S/n)]  +  2  0  c(i,j)E  (2S/n)(1+Zc(i,j))  = c(i,j)(-2S +0(S/n)) 2  0  + 2S/n ( b ( i ) + b ( j ) + Z d ( i , j , k ) )  etc.  k Quantities  which  a r e 0(S/n) w i l l  multiplied  by q u a n t i t i e s  intended  t o be v a l i d  infinite  n.  (6.8)  = -2S  E  0  Now ring all  now b e n e g l e c t e d  O ( n ) , i . e . though  f o r general  2  + (2S/n) Z b ( i ) = - 2 S S=l/2,  f o r which E  b ( i ) = b. Thus  two f l i p s ,  c(i,j)  the result i s  S, S i s n o t a s l a r g e a s t h e  b ( i ) = b (0) exp(-27ri/n) ; e i t h e r  with  where  Therefore:  consider  equal,  except  vary,  the f i r s t  but their  sum  0  +  2  (2S/nb(i)) Z c ( i , j )  = 21n2-0.5. F o r a c l o s e d  Z b ( i ) = 0 or the b ( i ) are  from  ( 6 . 7 ) , b = 1-2ln2.  flip  destroys  For the states  t h e symmetry and so t h e  i s a f u n c t i o n of the b ( i ) only,  from  (6.7): (6.9) and  (1/n) Z c ( i , j ) j  =  (l-21n2)  (1/n ) Z c (i , j) = (1-2ln2)/n i, j 2  To  make  differences different  between  boundary  Coefficients prime,  further progress  2  fora l l i Z b(i) =  (ring).  (1-2ln2)  i t i s necessary  the c o e f f i c i e n t s  b,c...  2  (chain)  to consider the  induced  by t h e  c o n d i t i o n s f o r t h e r i n g s and c h a i n s .  and f u n c t i o n s f o r chains  e.g. b ' ( 2 , i , j ) .  Consider  will  be w r i t t e n a s a  S=l/2 and w r i t e :  1 78 (6.10) |<//'(0)>= \\p{0)>  =  | aPapaPaPaP  =  |  point  having  point  a  a n d no f l i p  vanish  in  a ring.  of  the remaining  It  (6.11)  spin  a s no f l i p  Expressed  flips,  these  for  a ring  the c ( i , j )  i s known  that  satisfy  as found  above.  Though  imposed  by t h e end e f f e c t s / s p i n f l i p  in  the equations  of  a typical  of  t h e number  on t h e c o e f f i c i e n t s  are identical,i.e.:  c'(i,i)=c'(i,i±1)=0  Zb'(i)  =  E b(i)  the boundary  = E  0  respect  =  0  conditions e f f e c t on t h e  t h e sum o f  t o the end or f l i p  basis  sum - b u t i t i s t h e sum w h i c h such  0  state  of f l i p s  in 0  as (6.7). O  will  be  Therefore  i . e .  spin i t  i s important  the coefficient  'b' r a i s e d  i n the state. This  these  i s unchanged,  a f f e c t s the c o e f f i c i e n t s of the next  forE  and  n(l-21n2)  have a l a r g e  c o e f f i c i e n t s b ' ( i ) and c ' ( i , j ) ,  not a f f e c t their  flip  that:  exactly  one f l i p  t h e ends of  b'(i)=b'(i±1)=0  satisfy  =  with  point  to the  to a first  L b'(i) = I b ( i ) , . s i n c e E'  i t i s expected  |i//'(0)> a n d  At t h i s  between  conditions  / b(i)  does  apaPaP>  paaPaPa.p>  at or adjacent  a t or adjacent  Zc(i,j)  although  i s defined.  conditions  the b'(i)  coefficients  \ \  | aPaPaPaPap  end. For both  flip  as boundary  a chain  individual  \ appaaPapaP  i s possible  i s possible  For  therefore  =  i n the system  one e x t r a  initial chain  |^(1,i)>  || r e f e r s t o a c h a i n  a unique  states  =  aPapPaaPaPPaapapaP>  t h e symbol  \\p(],i)> the  |<//'(l,i)>  | aPaPapapaPaPapaPaP>  |\(>(2,i,j)> where  | | aPaPap>  argument  t o t h e power c a n be  179 extended  t o a s many f l i p s  (6.12) L c ( i , j ) and  will  only  which point Thus f o r  Now basis  break  has been  flipped  setting the  on  B, d e r i v e d  from t h e  i t was shown t o breaking  0  less  disturbance.  i.e.  using  a partial Therefore  thebasis  have  involves  spin.  Thus E ( S ) 0  states  t o zero. In  w h i c h h a s no e f f e c t  this  and w i l l  produce  boundary  condition,  section  0  i n (6.5).  even '  expressed as  of the matrix  {i//(M)} w h e r e M<<n f o r a n y  i sthes o l u t i o n of a matrix  e l e m e n t s have been g i v e n  i.e.  f o r S=l/2 t h i s was  no e f f e c t on E  large  i n the sets  states,  | . . .S,-S+1,S-1,S..> i s  imposing  s o l u t i o n o f an a r b i t r a r i l y  which  that  r i n g opening  B, w i l l  these  and c(i,i+1)  thering,  E . F o r S>l/2 t h e f l i p p e d s t a t e t o only  1 , S ,-S , S ,-S>  S>l/2; i t i s n o t c o m p l e t e .  of ignoring  0  equivalent  ,-S>  | S ,-S , S , ~S , S- 1 ,-S+  which, do e x i s t f o r  above d i s c u s s i o n  matrix  not i n the Ising  t h em u l t i p l y and a d j a c e n t l y  t h e e f f e c t on E ( S )  equivalent  the  with  c o e f f i c i e n t s such as c ( i , i )  formally  f o r an  | S ,-S , S ,-S , S ,-S , S ,-S , S  B does n o t i n c l u d e  states  Consider  lengths.  by r e p l a c i n g :  | a0a0aa0/3a/3aj3a/3a|3> basis  -at  {\//(M)}.  t o S but i n thebasis  | a/3a0a/3a/3a/3a|3a/3a/3> w i t h  The  / c(i ,j)  i s 0(n)  into finite  s o l v i n g t h e S>l/2 p r o b l e m ,  consider  L d(i,j,k)  c a n b e made t r u e  number o f s e t s  corresponding  and  / b'(i) =  'divided'  n theexpression  large  S=l/2 b a s i s  £ c'(i,j)  down when t h e n u m b e r o f f l i p s  thechain  large  arbitrarily  / b =  as necessary, e.g.  for which t h e  I t c a n be seen by  180 adding  2S  times  2  ignoring  (S/n)  independent  The  S.  E (S)  the  S>l/2,  However  upper  on  E .  for E  i n the  with  In  view  . The  to of  E  study  of  6.2.2  Numerical As  S  for  E (n) 0  of  -2/n  which  would  with  a l l extra effect  the  M=0(n). have  difficulty  (6.2)  would  an  incomplete  important  | . . . S ,-S+1  decrease  0  states  flips  terms could  E (S=1) provides  on  as  O(n)  are  difficulties,  several  increases  then  elements might  type  has  {^(M)} w h e r e  from  \//(l,i),  that:  with  (6.2)  an  can (6.2)  be  no be  an  basis. the  spin  significant.  , S-1 ,-S . . . > ,  contribute  a  S. only  be  and  numerical  incentive  for  regarded  further  extrapolation.  Extrapolation  increases very  matrix  no  c o n t r a d i c t i o n between  numerical  increases sequence  0  these  t e n t a t i v e . The  extrapolations  flips  element  2  only  and  is  and  states  have  states  the  S  i t follows  rigourous,  assumes t h a t  i t arises  diagonal  S=l/2  by  (S+2ln2-1)  in sets  that  O(n)  for  from  dividing  eigenvector  number of  occur  were  matrix,  -2S  majority,  since  0  -2S +(8S-2)/n term  vast  this  diagonal  negative  =  is far  proved  i f states  Thus the  2Sb  total  they  If  0  bound  variation  as  since  this  lowest  above approach  i t i s not  However  is  the  i . e . the  eigenvector  the  -  2  above approach  (2S+1)° a n d  to  i t i s known  = -2S  0  First,  effect  that  Since  weaknesses.  for  identity  terms,  of  (6.13)  the  the  size  r a p i d l y and can  be  of  the  basis  correspondingly  obtained.  As  (6.2)  of  the  fewer  matrix terms  contradicts  problem in  the  Blote's  181 result  for  For and  do  S=1,  S=1  not  this  the  lines  for  for  odd  n against  The  estimate  the  assumption  behavior  even  of  on  a  be  considered  Bonner-Fisher  l/n =0.  Blote  2  1  n were p l o t t e d a g a i n s t 1/n  6  obtained  0  this  was  in  plot  are  t h e o r e t i c a l l y rather  if  the  n=°° was  "critically"  straight  that  and  3  not  the  results  obtained. dependent  on  unexpected  continued.  to  Blote,  obtain  advantage  of  'mismatch  I have used  series  rings  greatly  the  seam'  for  both  numerical  open c h a i n s  i s that  the  cyclic  symmetry  s i z e of  the  matrix  to  e f f e c t s due  to  the  be  p r o b l e m may  sometimes g i v e  misleading  n.  For  open c h a i n s  convergence  n  monotonic. reducing  The  to  solution.  matrix  for  some e x t e n t  Calculations  the  the on  S=l/2 and  those  for  S>1  are  equal  for  these  elements positions  The  of  elements  since  r e s u l t s f o r S=l/2 and  S=1  of  E/n  6.2  replace  plots  6.2  for  S=l/2 F1(2)=0.75  the  points  and  against  corresponding  6.3.  be  CPU  S=1  spins, to  in Tables  can  be  However conditions small  banded,  required  for  significantly  therefore  matrix only  the  stored.  rings The  1/n,  and  chains  columns  1/n  2  and  i s the  intercept  to  and  n=3  time  to  frequently  easily  are  and  used  results for  a l l off-diagonal  need  presented  the  chains  be  boundary  = °° i s m o r e  memory and  than  can  closed  The  solved.  periodic  the  to  and  extrapolation.  of  simpler  detail.  found  1  1/n  i n t e r s e c t i o n near  2  E (S) that  Following  reduce  will  i n t e r s e c t near  results  rings  spin  n=1.  E  'F1','F2'  1/n . 3  on 0  are  Thus  and in  'F3' Table  l/n=0 drawn  from  i s considered  as  a  182 positive  The  q u a n t i t y , i . e . |E |,  results  Classically, spin  not  an  energy  Bonner-Fisher  n;  = E  0  divided  too small.  S=l/2  for rings  E(n)  i s evenly  causing  throughout.  0  the sequence  F2  into  n  considered but  of - 2 J S  2  {1 - c o s ( 7 r / n ) }  = -JS 7r /n 2  b a s i s of  as e x p e c t e d  i s approximately  the mismatch  of m i s a l i g n m e n t ,  I n s p e c t i o n of T a b l e  i s exactly  first.  f o r n odd  intervals  i s the t h e o r e t i c a l  method.  the behavior  be  f o r n even  change  This  will  6.2  7r/n, 2  for n  the  shows t h a t f o r  even  constant  2  in  for quite  for  n>3.  small  183 Table  6.2  The  ground  state  energy  o f S = l / 2 a n d S=1  rings  S=!/2 n E 2 3. 0 0 0 0 0 0 3 1 .5 0 0 0 0 0 4 4. 0 0 0 0 0 0 5 3. 7 3 6 0 6 8 6 5. 6 0 5 5 5 1 7 5. 7 1 0 3 5 9 7. 3 0 2 1 8 7 8 9 7. 5 9 4 6 0 0 10 9. 0 3 0 8 9 3 1 1 9. 4 3 7 8 7 3 12 10. 7 7 4 7 8 2 13 1 1 .2 5 9 1 6 9 14 12. 5 2 7 0 9 9 15 13. 0 6 7 3 3 5 16 14. 2 8 4 5 9 3  E/n 1 .500000 0 .500000 1 .000000 0 .747214 0 .934258 0 .815766 0 .912773 0 .843844 0 .903089 0 .857988 0 .897898 0 .866090 0 .894793 0 .871156 0 .892787  F1 ( n ) 0. 7 5 0 0 0 0 0. 5 0 0 0 0 0 1 .1 1 8 0 3 4 0. 8 0 2 7 7 5 0. 9 8 7 1 4 6 0. 8 4 8 3 1 8 0. 9 4 2 1 2 0 0. 8 6 4 3 5 3 0. 9 2 1 6 3 6 0. 8 7 1 9 4 4 0. 9 1 0 6 4 8 0. 8 7 6 1 5 8 0. 9 0 4 0 8 3 0. 8 7 8 7 4 7  F2(n) 0. 5 6 2 5 0 0 0. 8 3 3 3 3 3 0. 8 8 6 2 7 1 0. 8 8 1 6 6 5 0. 8 8 7 1 7 4 0. 8 8 5 1 5 0 0. 8 8 6 8 4 0 0. 8 8 5 8 7 3 0. 8 8 6 6 3 0 0. 8 8 6 1 0 1 0. 8 8 6 5 1 2 0. 8 8 6 1 9 2 0. 8 8 6 4 4 3 0. 8 8 6 2 3 5  F3(n) 0 .519231 0 .928571 0 .815323 0 .906578 0 .855073 0 .897095 0 .868795 0 .892929 0 .875116 0 .890768 0 .878541 0 .889511 0 .880603 0 .888716  S=1 n 2 3 4 5 6 7 8 9 10  E/n 4.000000 2.000000 3.000000 2.612452 2.872474 2.734914 2.834239 2.773342 2.818826  F1 ( n ) 3. 0 0 0 0 0 0 2. 0 0 0 0 0 0 3. 5 3 1 1 2 9 2. 6 1 7 4 2 3 3. 0 4 1 0 7 0 2. 7 1 9 5 3 3 2. 9 0 7 8 4 1 2. 7 5 7 1 7 4  F2(n) 2. 2 5 0 0 0 0 2. 6 6 6 6 6 7 2. 9 5 6 9 5 6 2. 7 7 0 4 5 4 2. 8 6 2 4 7 9 2. 78.5079 2. 8 3 2 1 8 5 2. 7 9 1 4 2 5  F3(n) 2 .076923 2 .857143 2 .781188 2 .818779 2 .805133 2 .806338 2 .807489 2 .802655  S = 1/2 nS e v e n o n l y . n E E/n 4 4. 0 0 0 0 0 0 1 .000000 7. 3 0 2 1 8 4 8 0.912773 12 10. 7 7 4 7 7 6 0.897898 16 14. 2 8 4 5 9 3 0.892787 20 17. 8 0 8 7 7 3 0.890439  F1 . F2 0. 8 2 5 5 4 6 0. 8 8 3 6 9 7 0. 8 6 8 1 4 8 0. 8 8 5 9 9 8 0. 8 7 7 4 5 4 0. 8 8 6 2 1 6 0. 8 8 1 0 4 5 0. 8 8 6 2 6 4  F3 0 .900312 0 .891635 0 .889057 0 .887975  S=1 nS e v e n o n l y . n E E/n 2 8. 0 0 0 0 0 0 4.000000 4 12. 0 0 0 0 0 0 3.000000 17. 2 3 4 8 4 4 6 2.872474 8 22. 673912 2.834239 10 28. 188260 2.818826 12 33. 739104 2.811592  F1 2. 0 0 0 0 0 0 2. 6 1 7 4 2 2 2. 7 1 9 5 3 4 2. 7 5 7 1 7 4 2. 7 7 5 4 2 2  F3 2 .857143 2 .818779 2 .806338 2 .802655 2 .801655  E 8. 6. 12. 13. 17. 19. 22. 24. 28.  000000 000000 000000 062258 234846 144397 673912 960079 188260  F2 2. 6 6 6 6 6 7 2. 7 7 0 4 5 3 2. 7 8 5 0 8 0 2. 7 9 1 4 2 5 2. 7 9 5 1 5 1  184 Table  6.3  The  S=l/2 n E 2 1 .500000 3 2 .000000 4 3 .232051 5 3 .855773 6 4 .987154 7 5 .672479 8 6 .749865 9 7 .472643 10 8 .516070 1 1 9 .264187 12 1 0. 2 8 4 1 8 1 1 3 1 1. 0 5 0 6 4 4 1 4 1 2 .053449 15 1 2.833841 1 6 1 3. 8 2 3 4 7 4 1 7 1 4 .61481 7 S=1 n 2 3 4 5 6 7 8 9 1 0 1 1  E 4. 0 0 0 0 0 0 6. 0 0 0 0 0 0 9. 2 9 1 5 0 3 1 1 .6 6 0 4 2 5 14. 7 4 0 5 5 0 17. 2 6 9 0 6 4 20. 249274 22. 865863 25. 789120 28. 460718  ground  state  energy  o f S = l / 2 a n d S=1  chains.  E/n 0 .750000 0 .666667 0 .808013 0 .771155 0 .831192 0 .810354 0 .843733 0 .830294 0 .851607 0 .842199 0 .857015 0 .850050 0 .860961 0 .855589 0 .863967 0 .859695  E/(n-1) 1 .500000 1 .000000 1 .077350 0 .963943 0 .997431 0 .945413 0 .964266 0 .934080 0 .946230 0 .926419 0 .934926 0 .920887 0 .927188 0 .916703 0 .921565 0 .913426  F1 ( n ) 1 .000000 0 .866025 0 .927886 0 .877552 0 .908353 0 .881355 0 .900082 0 .883103 0 .895772 0 .884055 0 .893229 0 .884634 0 .891598 0 .885012 0 .890488  F2(n) 0 .750000 0 .827350 0 .829929 0 .849736 0 .851187 0 .859857 0 .860826 0 .865605 0 .866307 0 .869306 0 .869840 0 .871887 0 .872308 0 .873788 0 .874129  F3(n) 0 .692308 0 .816300 0 .799942 0 .840952 0 .832831 0 .852885 0 .848012 0 .859868 0 .856615 0 .864444 0 .862116 0 .867671 0 .865921 0 .870069 0 .868705  E /h 2. 0 0 0 0 0 0 2. 0 0 0 0 0 0 2. 3 2 2 8 7 6 2. 3 3 2 0 8 5 2. 4 5 6 7 5 8 2. 4 6 7 0 0 9 2. 5 3 1 1 5 9 .2. 540651 2. 5 7 8 9 1 2 2. 5 8 7 3 3 8  E/(n-1) 4 .000000 3 .000000 3 .097168 2 .915106 2 .948110 2 .878177 2 .892753 2 .858233 2 .865458 2 .846072  F1 ( n ) 3 .000000 2 .645751 2 .830213 2 .724524 2 .804319 2 .754362 2 .798400 2 .769923 2 .797427  F2(n) 2 .250000 2 .430501 2 .518883 2 .563864 2 .607555 2 .626818 2 .653416 2 .663806 2 .681878  F3(n) 2 .076923 2 .369001 2 .423578 2 .513130 2 .544374 2 .585452 2 .606090 2 .629013 2 .643874  185 It  appears that  F2  forms  n w h i c h a p p r o a c h t h e known above and is  f o r odd  F2(15)  and  imply  accurate  even  the r e s u l t  S=l/2,  F1  f o r F1  With occurs are  also  and  change  better  F2,  i n the t a b l e ,  closer  t h r o u g h n=16  to the exact  converging and  For  below  2 and  as  useful;  i n the opposite  i . e . from above and  n=20  value.  less  and  which  i n t h e same way  the r e s u l t  converge but  and  even  F2(14)  o  f o r some a b e t w e e n  the a v a i l a b l e 2.016  data  sense  f o r odd  3, F a m i g h t  and  i t i s found that  2.017. F o r a=2.0l66  but converge to a  i s incorrect  as  result.  It i s clear  exactly  that  assumed  that  unless  i n any  the  0  by v a r y i n g  consider  S=1.  a  fail  and  F2 the  to  and F2  even and  give  an  2.775<E <2.907  sequence  i s not as c o n s t a n t  suggests  2.795<E <2.832. Once a g a i n  also  reversed  f o r F3  nor  i s larger  sub-sequences <  0  0.886299  than the  at large  exact  n is  the  the attempt to  as F l ( S = l / 2 )  refine  the sequences  and  behaves, as F2(S=1/2) but  i s convergence as  0  and  crossover  i s unreliable.  F1(S=1) behaves . F2  both  extrapolation  suggests  0  the  0.886296 < E  the behavior  given  must e v e n t u a l l y  of E  result  t h e l o w e r bound  extrapolation  Now  from below  i n the sense of the c o n v e r g e n c e between  between  value  and  result.  monotonic  which  values  f o r odd  E =0.88634±0.0001, a r e s u l t  i s much s l o w e r  suggests that  even  monotonically  final  sub-sequences  sub-sequences  n. T h i s F3  forms  convergence  those  sub-sequences  t o 0.005%. E x t r a p o l a t i o n  t h e l o w e r bound even  F3  limit  n. The  raises  but  two  rapid.  the  F2  the sense of convergence i s  s u g g e s t 2.8074<E <2.8027, o  a  1 86 nonsensical that  result.  f o r a=2.0l66  s e q u e n c e s must apparently for  obtained is  the  what  since a  might  fail  Thus,  though  certainly Now  both are  too  i s the  is entirely by  the  upper  c o r r e c t the consider  included  i t might  as  interactions  was  sub-sequences S=1/2  to E  be  and  'best'  of  chains  for S=l/2;  the  i f a  the thus  i s too  large. of  a?  than  The  I t would be  as  the  1/n.  ruled  column  in this  monotonic  of  E (S=1/2,n=4)  value  with i s in  E/n  no  plots one. almost  by  the  6.3.  is  number  of  The increasing for  for E/(n-l).  can  the  i n Table E/(n-l)  case.  decreasing  result  type  seem out  correct  listed  that division  as  by  suspect.  The  n are  E  question  d e f i n e d above are  0  for  obtained  0  a l l Bonner-Fisher  for chains,  S and  estimate  for E  terms cannot  for E  convergence 0  values  larger  0  even  refined  a.  exponents,  the  monotonic  j3 v a r i e s w i t h  basis  S=1  more a p p r o p r i a t e  S=1  the  'refined'  thought  and  of  most  bounds are  0  the  the  bounds  tends  reversal  arbitrary  /3 i n E/(n-/3) a m o r e r e f i n e d  correctness  than  E  of  to  c o n t r a d i c t i o n s are  results  f o r odd  and  an  one  undermines  'safe' value  lower  the  This  seriously  s a f e and  indicating  least  analogous  f o r S=1,  large  largest  E(n)  For  data  S=l/2 and  Classically  the  n.  logarithmic, l/log(n),  of  varying  sequences  f o r a l l a>2.7 w h e r e  value  both  i n that at  for larger  available  plotting  arises, that  reverse  h e r e m u s t be  extrapolation using  2.797. F o r  simple  situation  f o r S=l/2  monotonic  numerical With  The  be  rings  By  obtained  but  the  doubt.  i s much s l o w e r  i s 91.4%  of  the  for  exact  S=1 E  0  0  187 w h e r e a s n=8 i s n e e d e d appears  that  important,  0(S/n)  creating  all  S>1/2. O n l y  not  form  F1  f o r S=1 t o o b t a i n  terms a s d i s c u s s e d an added  F1 i s u s e f u l  lower  than  converge  the  be g i v e n .  value  t o the upper  second  from  below  Classically,  column  o fTable  i s significantly  apparently  the best  and below.  nearer  to the  limit. ofnumerical  E(n)=E -2S /n. 2  0  6.3 a s e q u e n c e  f o r odd and even  S = 1 . F o r S=1 t h e s e q u e n c e  from above 0  One m o r e e x a m p l e o f t h e d a n g e r s will  a s F2 and F3 do  f o r S=l/2 a n d 2 . 7 6 9 9 < E < 2 . 7 9 7 4 f o r  o  limit  may b e  in extrapolations t o  in extrapolation  0.8850<E <0.8905  S = 1 . F o r S=l/2 t h e e x a c t  i n 6.2.1  difficulty  two sub-sequences w h i c h  suggests  s i m i l a r accuracy.I t  suggests  I f 2S /n 2  which  n i s obtained  this  0  i s added t o  i s monotonic  f o r both  2.7692 < E  result yet. Testing  extrapolation  S=1/2  and  < 2.7789,  r e s u l t o n S=l/2  gives  0.8891<E <0.8952, which  i s i n c o r r e c t and once again  high.  However  i s genuine,  o  t h e upper  bound  A v a r i e t y of f u r t h e r and this  work  dependent behavior detail.  has been  i s worse  I n general  series  be  important  with  than  useless  both  tried.  limit  applied,  on t h e raw  data  A feature o f  i s strongly  and s i n c e  this  kind of  the r e s u l t s are not presented i n  the d i f f i c u l t y  transformations  suggests  ofobtaining that  reliable  logarithmic  results  t e r m s may  f o r large n.  therefore  a useful  been  the apparent  on the t r a n s f o r m a t i o n  by  It  that  f o r S=l/2 a t l e a s t .  transformations,  on t h e sequences F1,F2,F3 have  too  appears  that  r e s u l t should  the lowest  be used  power  ofn consistent  i n the extrapolation.  Since  188 F1  has a t h e o r e t i c a l b a s i s  but  n o t f o r r i n g s , F1 o f c h a i n s  reliable  2.7699 < E ( S = 1 ) <  are  previous  that  extrapolations Cautious  has been attempts  states  Table  are given  6.4. New  to E  0  between  a s t h e most  but i t does  b y Weng, M a j u m d a r (6.2) and  Rings S=3/2,.n=8 S=2, n=6 S = 5 / 2 , n=8  and  Blote  numerical  bounds  f o r E (S>1) 0  for finite  rings  i n (6.16),  results f o r chains  r e s u l t s , f o r both  with  totally  5.732511; 9.721564; 14.52762  symmetric are listed  rings  and c h a i n s , a r e  n=8  9.630467.  underlined. (6.16)  seem  removed. to define  been made. R e s u l t s  ground  (6.2) i s c o r r e c t  extrapolations  i n c o r r e c t . The c o n f l i c t  have  be t a k e n  approximation)  2.7974  0  (6.15) does not imply that  will  (classical  r e s u l t f o r S=1. T h u s :  (6.15)  in  f o r chains  189 Table  6.4  The  ground  state  S = 3/2 n E 2 7. 5 0 0 0 0 0 3 12. 0 0 0 0 0 0 4 18. 3 6 2 3 6 2 5 23 . 4 3 6 5 3 4 6 29. 514154 7 34. 794436 8 40. 74661 2 9 46. 129158  E/n 3. 7 5 0 0 0 0 4. 0 0 0 0 0 0 4. 5 9 0 5 9 1 4. 6 8 7 3 0 7 4. 9 1 9 0 2 6 4. 9 7 0 6 3 4 5. 0 9 3 3 2 6 5. 1 2 5 4 6 2  S= 2 2 3 4 5 6 7  energy  o f S>1  F1 ( n ) 6. 0 0 0 0 0 0 5. 4 3 1 1 8 1 5. 7 1 8 2 6 7 5. 5 7 5 8 9 6 5. 6 7 8 9 5 1 5. 6 1 6 2 2 9 5. 6 6 7 3 6 1  chains  F2(n) 4. 5 0 0 0 0 0 4. 8 7 0 7 8 7 5. 0 7 3 9 1 7 5. 1 8 1 7 7 4 5. 2 6 5 7 6 6 5. 3 1 7 4 2 8 5. 3 6 2 5 4 3  F3(n) 4 . 153846 4 .710675 4 .876667 5 .057314 5 . 1 33092 5 .220519 5 .263043  6 7 8 8 8  12. 20. 30. 39. 49. 58.  000000 000000 442143 206223 285120 309994  6. 0 0 0 0 0 0 6. 6 6 6 6 6 7 7. 6 1 0 5 3 6 7. 8 4 1 2 4 5 8. 2 1 4 1 8 7 8. 3 2 9 9 9 9  10. 9. 9. 9. 9.  000000 221072 6031 1 2 421488 551885  7. 5 0 0 0 0 0 8. 147381 8. 5 0 1 9 4 5 8. 6 9 7 1 0 7 8. 8 3 9 1 1 8  S = 5/2 2 17. 3 30. 4 45. 5 58. 6 74. 7 87.  500000 000000 521040 953800 054960 821137  8. 7 5 0 0 0 0 10. 0 0 0 0 0 0 1 1 .3 8 0 2 6 0 1 1 .7 9 0 7 6 0 12. 3 4 2 4 9 3 12. 54587.7  15. 14. 14. 14. 14.  000000 010520 476900 266960 433668  1 1 .2 5 0 0 0 0 12. 2 5 7 0 1 3 12. 7 9 8 0 6 2 13. 1 1 2 2 8 0 13. 3 3 2 4 5 7  For to  the l i m i t e d data  available,  behave as f o r the lower  spins.  The  The  (6.17)  results S  2  estimates  (6.18) E : 0  For  S=2  and upper  3/2  F1 E+2S /n The  f o r lower  of E  0  from  s e q u e n c e F1  bounds  for E  2  5.616,5.679 5.586,5.669  (6.16)  i n an a t t e m p t  2  0  0  10 . 3 8 4 6 1 5 1 1. 7 5 6 0 1 1 1 2 .284133 1 2 .747644 12 . 9 7 8 8 5 6  the r e s u l t s  sequence E ( n ) + 2 S / n were c o n s i d e r e d E .  .923077 .840612 .164853 .468355 .610248  0  appear  and the to  estimate  are: 5/2  9.220,9.599 9.439,9.610,  14.01,14.48, 14.29,14.50,  (6.17) a r e :  S = 3 / 2 , 5 . 6 4 7 ± 0 . 0 3 1 ; S=2,  a n d S=5/2 t h e e s t i m a t e  S=3/2 i t i s s o m e w h a t l o w e r .  9.52±0.08; S=5/2,  i s t h e same a s B l o t e ' s ,  A l lnumerical  14.38±0.10 for  estimates are  190 consistent In the  with  (6.2).  conclusion,  result  E  Numerical e x t r a p o l a t i o n  = 2S(S+21n2-l) suggested  0  considerations,  however  associated  the  regarded  with  as  exactly both  anyway; of  members of  the  for  finite  n.  6.3  The  accurate  cannot  i s most,  for which  E  and  because  terms  be  even  i s known  0  in obtaining  increased d i f f i c u l t y  sequences  as  S increases sufficient  0(S/n) are  important  Approximation shown  S>1/2  the Hamiltonian  result  the  i t was  of  uncertainty  rapidly  that  i t is difficult  numerical extrapolations  properties  extreme  e s t i m a t e s the  f o r S=1/2,  with  theoretical  i t s usefulness decreases  Odd/Even 6.2  the  by  numerical extrapolation  satisfactory  because  In  numerical  proved,  anomalously,  i n v i e w of  is consistent  c h a i n s . An  f o r the  low  alternative  i s approximated  before  to obtain temperature  approach  solution  in  will  which  now  be  introduced. 6.3.1  Approximations The  first for  quantum m e c h a n i c a l  applied  to  real  ferromagnetism  certain  to the Heisenberg theory  systems  by  in infinite  approximations  about  Hamiltonian  of magnetic  Heisenberg  systems  was  interactions  i n 1928 . A 7  d e r i v e d by  the d i s t r i b u t i o n  of  the  model  making spin  states.  Heisenberg  c o n s i d e r e d an  infinite  lattice  of  atoms  was  each  191 coupled per  t o a number o f n e a r e s t  neighbours,  atom, o f a l l s t a t e s w i t h a g i v e n  W(S')=-(zJ/n)S' .  F o r S=l/2  2  energies the  value  the standard  d e v i a t i o n , A, o f Heisenberg  the  obtained  result:  2  The  total  number  calculated  =  J z(n -4S' )(3n -4S' )/(8n ) 2  2  2  of states with  from  2  total  fl(S')=w(S')-w(S'+1),  2  fi(S')  where w(S')  =  as i n Chapter  low  states Stirling's  energy  used.  ferromagnetic  Assuming  a Gaussian  function  and hence  function  of temperature  obtained  an e x p r e s s i o n  k(l-[l-8/z]  z=4,  have  point  shown  that  given  approach  i ssurprisingly successful.  total  s p i n S' c a n n o t  p r o p e r t i e s as a  0  Heisenberg = 2J/  i sp r e d i c t e d only though  t h e number o f a p p r o x i m a t i o n s  S>l/2 t h e s t a n d a r d  be c a l c u l a t e d  involved  and there  expression  for  fi(S').  assumption  that  a l l s t a t e s o f t h e same s p i n h a v e  energy.  This  2S(S+l)/3k  made t h e ' s t i l l  l e d t o an e x p r e s s i o n  . This  approximation  S=l/2, a n d i n c o r r e c t l y  for theCurie  i s much l e s s  i f z >  1-D, z = 2 , a n d 2-D,  deviation of theenergies  Heisenberg  c a n be  the partition  p o i n t , systems with  However,  with  of states,  point, T  Curie  systems do n o t have a C u r i e  For  S' i s l a r g e f o r  be c a l c u l a t e d .  for theCurie  c a n be  approximation  and magnetic  can then  ) . Thus a r e a l  Later calculations  2. S i n c e  distribution  the thermal  2  s p i n S',  n!/[(n/2-S')!(n/2+S')!],  for  energy,  o f S' i s g i v e n b y  a b o u t W(S') c a n a l s o be c a l c u l a t e d ;  (6.19) A ( S ' )  8.  z . T h e mean  n=6 m a y . this  of s t a t e s  i s no s i m p l e cruder' t h e same  point T  adequate  predicts ferromagnetic  0  =2Jz  that  Curie  that  points  1 92 for  a l l lattices, In  1966,  Earnshaw  approximation, model  the  which  were  to  valid  be  irrespective  t h a t a l l s t a t e s of  finite then  being  synthesised  f o r S>1/2  linear  available.  application  The  thesis  approximation system  as  way.  i s now  the  z=2(n-l)/n  (6.20) can Earnshaw  Hamiltonian clusters  finite  chain  and  no  systems was  hoped  Fisher's result  systems w i l l  the  to  numerical  be  were  referred  to  Earnshaw's  correct Hamiltonian  for  the  that  chains the  (6.20)  2  -(2J/n)  into  the  table of  2  HDW  less  the  In  The  2  formula  giving  n=3-l0,  i s even  a  i n the  u(T)  S=1/2-5/2. of  the  exact  satisfactory  spin  for chains  i s only  e q u i v a l e n t as  i n the  equilateral  systems the  the  exact  difference  same s p i n may  states.of different results  of  usual  for  (6.20)  other  linear  eigenvalues  systems.  states having  the  (S' -nS ).  simplification  for infinite  atoms are  of  2  H=  published a  between  inspection  thus  inserted  t e t r a h e d r o n . For of  [S' -nS ]  'average c o o r d i n a t i o n number'.  and  be  by  than  coupled  energies as  to  degenerate,  approximation  for which  to Bonner  = -[zJ/(n-l)]  Unfortunately  or  . The  and  Earnshaw's approximation.  antiferromagnetic  all  3 0  are  by:  where z  of  same S'  chains,  involves replacing  (6.20) H  chain  the  'cruder'  antiferromagnetic cluster  corresponding  this  z.  a p p l i e d Heisenberg's  extrapolations  in  of  spin, 6.2.  as  For  be  at  can  and  accurate i f triangle  between least be  example  the as  large  verified the  by  Earnshaw  193 model a  suggests  ground  the  state  Another  ground  serious  failing  p e r atom  f o r l a r g e n. The o n l y ground  the splitting  from  (6.19) t h a t  will  results  have  show  that  of t h e Earnshaw  approximation  systems  as n increases, the  tends also  i n the antiferromagnetic  from  chains  i s S.  at a l l temperatures;  antiferromagnetic  of  1/2 o r 0 w h e r e a s e x a c t  to antiferromagnetic  vanishes  zero  spin  s t a t e energy  maximum  a l l odd n a n t i f e r r o m a g n e t i c  state of spin  ground  applied  that  i s that  t o zero  as  and the s p e c i f i c  the temperature  susceptibility  heat  a t which t h e  occurs  tends t o  contribution to the  s t a t e energy  f o rlarge  systems  comes  o f s t a t e s o f t h e same S'. I t c a n b e s e e n  a s S' d e c r e a s e s  Earnshaw's approach  fi(S')  was p o i n t e d  increases.  The i n a d e q u a c y  o u t by B a r r a c l o u g h  1 7 8  in  1968.  Since  neither  extrapolation S>l/2  considered  satisfactory,  t o these  systems.  I have  the spins  chains of  investigated a  The atoms o f t h e c h a i n a r e and two  intermediate  on t h e o d d numbered a n d t h e e v e n  numbered atoms a r e i n t r o d u c e d .  The t r u e  spin Hamiltonian  approximated by:  (6.21) H = - c J ( S' -S"' -S" ) = - 2 c J 2  w h e r e S" = S + S + S + 1  3  5  simple  quantum numbers a s an  t o be n u m b e r e d c o n s e c u t i v e l y  representing  nor numerical  properties of l i n e a r  involving intermediate  approximation  spins  f o rt h e magnetic  i sentirely  Hamiltonian  Earnshaw's approximation  2  S".S"'  2  a n d S'" = S + S , + S + ... 2  6  i s  1 94 where c i s a s c a l i n g Earnshaw's  approximation,  interactions The  worst  States  aspects  increases,  thought which  worthwhile  (6.23)  the  S'  split  state In view  chains  spin  of these  to infinity tends  to -2S  a r e now  has  non-zero  i s now  improvements  i t was (6.21),  approximation, factor,  , n  2  i s exact  approx  i n some  c, i s g i v e n  p e r atom,  by:  even. only  i s exact  f o r the 2 f o r n = 2 a n d 3.  the a n t i f e r r o m a g n e t i c ground 2  removed.  i n c r e a s e s a s S'  the approximation  n odd; 4 ( n - 1 ) / n  approximation  state  number o f  (6.21) w i t h c=1.  f o r odd c h a i n s  the scaling  c h a i n , t h e odd/even  chain  by  by an amount w h i c h  to investigate  4/(n+l),  Earnshaw  n tends  to the real  implied  be known a s t h e o d d / e v e n  For linear  membered  i t i s equal  of Earnshaw's approximation  correctly.  The  t o the z / ( n - l ) of  the a n t i f e r r o m a g n e t i c ground  will  detail.  analogous  by t h e number  and t h e ground  predicted  As  divided  o f t h e same  energy  factor  state for  t h e same a s t h e c l a s s i c a l  value. Eigenstates as  kets  of t h e odd/even H a m i l t o n i a n  |S',S",S">  degeneracies  of degeneracy  associated with the half  n ( n / 2 , S , S " ) . 0 ( n / 2 , S , S " ' ) where branching and  diagram.  specific  energies  from  heats  o f OE  chains  fi(n/2,S,S')  A program which  systems u s i n g these  of the  fl(n,S,S',S",S"')  i s found  calculates  R e s u l t s f o r a l l n,S  expressed  to the product  ( 6 . 2 1 ) a n d t h e HDDV f o r m u l a  straightforward. obtained.  equal  c a n be  the  =  from t h e susceptibility  degeneracies,  i s therefore  u p t o S'=250 h a v e  been  the  195 6.3.2 R e s u l t s o f t h e o d d / e v e n The S=l/2,  approximation  s u s c e p t i b i l i t y and s p e c i f i c  are plotted  Some d a t a below.  relevant  in Figs.  6.1  for  to the discussion  The ' n a t u r a l '  units,  g=2 f o r x ; a n d R ( t h e a r e used  linear  transformed is and by  systems f o r  of these  a n d /3 i s t h e B o h r  The r e s u l t s  t o 2-D o r 3-D  magneton,  are presented they  systems.  s y s t e m s c a n i n n o way b e o b t a i n e d  scaling.  are listed  f o r the specific  one of t h e weaknesses o f t h i s a p p r o a c h 3-D  results  6.5.  2  c h a i n s ; as e x p l a i n e d above, t o apply  i n Table  The  (N/3 /3k) = 1 , where n i s A v o g a d r o ' s  gas constant)=1  throughout.  calculated.  t o 6.6 a n d l i s t e d  number, k i s B o l t z m a n n ' s c o n s t a n t  heat,  o f OE  S=1 a n d S=5/2 a n t i f e r r o m a g n e t s h a v e b e e n  .results  and  heat  normalised  may e a s i l y This  'adaptability'  as s o l u t i o n s from  be  f o r 2-  solutions  f o r 1-D  in  T/J F i g . 6 . 1 . S u s c e p t i b i l i t y o f Odd/Even S=1/2 c h a i n s . From b o t t o m t o t o p a t T/J=0.5 N=2,4,8,16,32,64,128,256 a n d 512.  F i g . 6 . 2 . S u s c e p t i b i l i t y o f Odd/Even S=1 c h a i n s . From b o t t o m to t o p a t T/J=0.5 N=2,4,8,16,32,64,128 and 256  LO  U3 00  0 0  6.0  12.0  T/J  18.0  F i g . 6 . 3 . S u s c e p t i b i l i t y of Odd/Even S=5/2 c h a i n s . From t o p t o t o t o p a t T/J=3.0 N=2,4,8,16,32 a n d 64.  24.0  F i g . 6 . 4 . S p e c i f i c Heat o f Odd/Even S=l/2 c h a i n s . I n o r d e r o f i n c r e a s i n g maxima, N=2,4,8,16,32,64,128,256 and 512.  Fig.6.5.  S p e c i f i c Heat o f Odd/Even S=1 c h a i n s . In o r d e r o f i n c r e a s i n g maxima, N=2,4,8,16,32,64,128 a n d 256.  o Cvl  CD  0.0  6.0  12.0  T/J  18.0  F i g . 6 . 6 . S p e c i f i c Heat o f Odd/Even S=5/2 c h a i n s . In o r d e r o f i n c r e a s i n g maxima, N=2,4,8,16,32 a n d 64.  24.0  202 Table  6.5 S u s c e p t i b i l i t y a n d S p e c i f i c  Odd/Even  approximation  Heat  f o rantiferromagnetic  Susceptibility  S=l/2  S=1  S=5/2  results  f o r the  chains  Spec i fi c Heat  < max =J  y max A  T(Y max  n 2 4 8 16 32 64 128 256 512  1 .2071 1.1139 1.1827 1.2634 1.3294 1 .3785 1.4138 1.4390 1.4568  1.247 1.173 1.034 0.967 0.946 0.947 0.955 0.964 0.973  0.5117 0.4636 0.4825 0.5764 0.7001 0.8299 0.9518 1.0588 1.1486  0.703 0.714 0.743 0.752 0.772 0.803 0.836 0.867 0.894  n 2 4 8 16 32 64 128 256  1.5375 1.3262 1.3141 1.3431 1.3775 1.4077 1.4320 1.4505  2.048 2.293 2.347 2.396 2.449 2.498 2.540 2.573  0.5486 0.5696 0.7105 0.8864 1.0678 . 1.2370 1.3852 1.5098  0.805 1 .591 1 .766 1 .892 2.015 2.132 2.236 2.326  n 2 4 8 16 32 64  1.8615 1.4632 1.3888 1 .3857 1.4024 1.4227  0.5488 0.7217 0.9625 1.1947 1.4053 1.5886  0.808 5.79 6.96 6.87 8.62 9.25  T  V A  5.76 8.17 9.38 10.10 10.56 10.88  c  m  E(1.0)  - o . 601 - o . 597 - o . 4487 05 -- 0o .. 341 - o . 22 83 66 - o . 1 92 - o . 1 54 -o.  203 The  x,  susceptibility,  The the  a/  A smooth,  at  a temperature  b/  x tends  rounded  maximum  which  by  :-  f o r which  varies  with  x-1.5  t o a p p r o x i m a t e l y 1.0 a s T g o e s x  smooth,  i s qualitatively  Comparing  rounded  chain.  maximum  However  t h e OE a n d e x a c t  chain  t o 0,  i s proportional  the result results  for a l lS  S.  S. A t h i g h t e m p e r a t u r e s  linear  S=l/2  i s characterised  for a l l  t o S(S+1)/T. correct for  i s quantitatively  poor.  f o r the antiferromagnetic  :-  (6.22) *max Exact OE S=oo  .The because  ••  T (  *mai  )  *(T=0)=Xo  0.8815 1 .282 = 1.5 =1.0 1.204 0.2382 susceptibility  0.6079 =1.0 1.0  an  o r d e r e d a/3a/3a0a/3 s t a t e  OE  energy  representation with  ( E = - 0 . 8 8 ) a n d more m a g n e t i c o  If this  1.38  into  by all  i s taken  no means spin  energy  suggests that  states  compression  account  accurate.  °  1 .450 =1.5 1.204  The  a/  of  The  than  Low  0.962 =1.0 0  similarity  i s improved  i s characterised  temperature, C tends  as  the  spectrum  to  by a f a c t o r but i s s t i l l  of the x vs T curves f o r  may  m  f o r S=l/2  the exact  of the spectrum  the result  Xo a n d x  state  a r e o c c u p i e d a t low  be r e l a t i v e l y  0  heat  / R  of the ground  E .  specific  max  0.350 1.0  ITlcL X  functions  C  -0.5. Thus  i s more c o m p r e s s e d  temperatures.  / x  i s o v e r e s t i m a t e d a t low t e m p e r a t u r e s  of the inadequate  spectrum  *max  by:  1/n.  simple  of  204 b/  I n t e r m e d i a t e T,  forms  were  T.  increased  phase  as  1/n  further  large  and  transition;  unbounded  increases which  C proportional  decrease  increasingly of  n  at a temperature  c/ High The  As  n.  temperatures. I t i s clear  quantum numbers a r e incorrectly  does not may  be  that  a  would  show  heat  investigation  of  the  anomaly,  though  d i s c o n t i n u o u s at or near  even  specific  kind  be  unlike  that heat  of at  essential  i f intermediate numbers  interaction.  susceptibility  t h e maximum  some  such quantum  range  i f n  an  with may  the  lost  because  long  with  smooth  some o f  Hamiltonian are  of  show a n y  heat  exhibits  introduced,  associated  suggests that  specific  imply a degree  Careful temperature  2  This behavior i s quite  which  maximum  spin,  s h a r p maximum a s s o c i a t e d  all  the chain  shaped  1/nT .  the s p e c i f i c  real  of  with  the extremes  the  features  chain  lambda  varies  to  t h e maximum  for infinite linear  at  a  i n the  near  specific  i t i s possible  t h e maximum  the  in x  heat  that  the  slope  for  infinite  n. It a s an  can  any  seen  that  t h e OE  approx  approximation f o r the c a l c u l a t i o n  properties this  be  of H e i s e n b e r g  respect since model  without particle  1 7 9  .  mathematical  linear  Despite this  interest size  the  linear  on  as  system  failure  i t gives  magnetic  chains.  some  phase  i s less of  the  provides a  t h e OE  unusual  in  severe test  for  H a m i l t o n i a n i s not  indication  infinite  satisfactory  thermodynamic  I t i s not  transitions  t r e a t m e n t s assume an  than  of  the e f f e c t  f o r which number o f  most atoms.  of  205 The  magnetic  limited  by m i c r o c r y s t a l b o u n d a r i e s  it  has been  1%  in linear  expected heat  i n t e r a c t i o n s i n any r e a l  suggested chains  finite  that ,  1 8 0  size  specific range  heat  with  decreases  rarely  in this  observed  There  context  logarithmic  n appears that  i s a connection chains  couplings  susceptibility  t o a mean c h a i n  1 8 2  .  here  with  of ' i n f i n i t e '  rather  impurities  and i m p e r f e c t i o n s  characteristics  than  with  linear  2R a r e  of pure  1-dimensional  with  Second  between magnetic  order  (3J/3d)  t o as s t r i c t i o n  be due t o  spectrum  than  to  1  .  8  ""  0  may  occur  with  small  i n the distortion with  coupling 0  8  6  Heisenberg  distortions  i.e. (3J/3d) , 1  in  of t h e  of the f i n i t e  linear  forces,  effects  1 8 3  As a r e s u l t  effects associated  and e l a s t i c  i n the  Sensitivity  systems"  non-zero  the changes a s s o c i a t e d rather  tail  random  Chains  linear  t o be q u a d r a t i c  or with  1 8 1  i s "one o f t h e b a s i c t o p o l o g i c a l  of the magnetic  referred  I ti s  o f more t h a n  impurity.  systems as w e l l as i n c l u s t e r s .  coordinate.  considerable  slower.  heats  lengths  non-degeneracy  tend  of the s p e c i f i c  s y s t e m s may  paramagnetic  associated  chains,  100. As  c a l c u l a t i o n s on  random  6.4 D y n a m i c D i s t o r t i o n s a n d L i n e a r  linear  length  fora  The f r e q u e n t l y o b s e r v e d  randomness  Effects  i s =  o f t h e maximum  t o be e v e n  specific  defects  and  inpractice.  antiferromagnetic exchange  of such  the temperatures  n i s nearly  be  imperfections  maxima. The v a r i a t i o n  of n but f o r large  relevant  and other  the proportion  leading  and s u s c e p t i b i l i t y  sample w i l l  ,  and have  are often  been  observed  206 in  several In  occur  3-dimensional  infinite  as  the  infinite  chains  system  ring.  The  of  such  dynamic  rather  6.4.1  than  vibrational  products function, such  term.  , can  The  have  magnetic  The  as  typically  the  ring  defined  been an  area  "freezing  of  out'  t e c h n i q u e s used  in  much more  distortion  field  may  an  a well  a  mathematical  phonon  dimers.  spectrum  i n which  recently  dynamic  in  equivalent to  regarded  normal  be  must  modes o f  a  model,  as  be  considered  small  cluster.  the  treated  |n> by  found  H a m i l t o n i a n by  Later work  1 9 0  potential  the  showed  static  on  energy can  minimum of  a s s o c i a t e d w i t h the S'  and  levels.  be  thus As  written  is a vibrational  perturbation  that  to the  the  i n the H a m i l t o n i a n can t h e r e f o r e  dependent  of  property that  proportional  oscillator  |n>|S>, w h e r e  unique  i n the  the wavefunctions  and  the  term  shift  is typically  and  has  be  f o r the  local  i n the  d i m e r i s a t i o n ) at  theory are  systems  harmonic  a model spin  9  as  systems  rearrangement  degenerate  8  lattice  the  associated with a  shift  1  H a m i l t o n i a n must  Hamiltonian.  slight  *  those  merely  Dimetallic  this  7  3-dimensional  distortion  the  8  well  transition,  which  distortions.  than  Dimetallic  be  1  and  transition  sophisticated full  (incipient  as  changes  c o n s i d e r e d as  spin-Peierls  research  spin-Peierls  the  be  temperature,  vigourous  systems  first-order  can  becomes a l t e r n a t i n g transition  lattice  striction addition that,  are  |S>  theory. L i n e s can  a  spin developed  8 3  be  included  of a b i q u a d r a t i c  because  the  a  Born-Oppenheimer  and  effects  distortion;  leads to  they  as  the  distortion  (S,.^) terms  in 2  207 commute w i t h exact  s o l u t i o n , which  (ST.SJ). the  H, t h e L i n e s  Hamiltonian  in increasing  (assuming  simple  powers of  J i s a f u n c t i o n of  p o s i t i o n b u t n o t t h e momentum o f t h e n u c l e i ) i s :  where the  H = TT /2 + u £ / 2 2  2  normal 1  mode, A = ( 1 / J ) 9 J / 9 £ ) , 0  similar  2  linear  f o rquite  convention 221,  a large  range  that  2  theory  H  2  i n the usual  that  r i s a constant terms  2 0  )9 J/9£ ) 2  factor  and  2  indicate that  away a r a t h e r  J i s nearly  sharp  small. t o that  0  turning  The  sign  used  i n r e f s 83  of 2 i s included  i n r. In the  - J  2  0  T V = -J  way. The e x a c t f o r each  (A£ + B £ ) 2  0  s o l u t i o n makes u s e o f t h e  S' a n d t h a t  striction  o f d i f f e r e n t S'. H c a n t h e r e f o r e  be  H = TT /2 + J 2 U - 6 )  where  fi =cj -2J0B7 2  2  2  2  2  , 6= - ( J A / 0 ) T 2  0  + C  and C = - J  Therefore:  (6.27)  the  result,  E(n,S')  = RO(v+1/2) + C  up t o s e c o n d  order  in r i s :  r-(J A/0) r /2 . 2  0  0  2  does  expressed  as:  (6.26)  point  H i s expressed as:  = 7T /2 + w £ / 2  solved  couple  B=(1/J  i s opposite  approach  2  0  momentum a n d f r e q u e n c y o f  B i susually  and a c o n v e n t i o n a l  (6.25)  fact  ( 1 +A£+B£ ) r  0  typically  f o rJ i n (6.24)  perturbation  and  0  dimers  maximum J a n d t h u s  and  - J  2  £, rr a n d u> a r e t h e c o o r d i n a t e ,  r=2S .S .  not  has a r e l a t i v e l y  c a n be e x p a n d e d  The s t r i c t i o n  (6.24)  of  Hamiltonian  208 (6.28)  E(n,S')=  RCJ(V+1/2) - J  0  [ 1+ (RB/w) ( v + 1 / 2 ) ] T -  ( J O / 2 O J ) [ A + ( R B / w ) (v+1/2) ] r 2  Thus  the Lines  2  model  2  gives  2  rise  t o both biquadratic  temperature  dependence  of J . E x p e r i m e n t a l l y ,  temperature  dependence  of J f o r d i n u c l e a r  been  explained  elasticity'  which  non-rigidity"  The may  of such  i s enhanced  on  (6.28)  on t h e s t r a i n  consider  S=l/2;  striction  cavity  1  9  1  of i n c l u d i n g coordinate  referring  terms  and a  the large  copper or  by t h e " q u i t e  of the c r y p t a t e  effect  depend  First  i n terms  2  cryptates  has  'exchange specific  .  terms  will  now  t o equation  other be  than J T which  considered.  (1.21),  the  full  Hamiltonian i s :  (6.29)  In  general  H = K - J/2 - 2JS .S 1  K, a s w e l l  as J , w i l l  2  = K -  J(T+1/2)  be a f u n c t i o n  of the  distortion:  (6.30)  K = K  A'  may,  to  e x p e c t them  of  orbital  V  t o be s i m i l a r  = TT /2 + u £ / 2 =  change  2  £[K A' 0  will  i n energy  -  from A, though  as they both a r i s e  Following  2  0  B a n d B' t e r m s  order  be d i f f e r e n t  overlap. H  w h e r e A' = ( 1 / K ) 3 K / 3 £ )  2  i n general  (6.31)  The  (1 + A ' £ + B ' £ )  0  2  Lines' + K  0  i t i s reasonable from  t r e a t m e n t H may -  0  some be  form  written:  J(T+1/2)  J A(T+1/2)] 0  be r e i n t r o d u c e d i s now:  i n (6.35).  The  second  209 (6.32)  (T+1/2) If J  AE  = +1  2  A /CJ 2  splitting. K A'>>J A 0  energy  i s no  0  levels  i s needed  0  effect  on  For  the  2  a  S,  most  exchange  interaction  of  between  atoms a r e c o u p l e d  with  an  by  S « Z  expressing  eigenfunction consider  { A ( 1 a2a. into  two  and  each  two  low  splitting  be  t o have  to  a  significant  dimers.  (6.29)  demonstrated  with  is H  = K  -  by  pairs one  of  pair  exchange  H as  n e l e c t r o n s , (S=n/2). Let e l e c t r o n s on of o r b i t a l s  integral  a matrix  i n an  on  Ising  and  the S'=n  the basis  blocks  of  size  problem  basis  1 and  and  E = K  - n(n-l)J'  of  2n  eigenvalue  states with  n.  Each  diagonal  i n each  solutions for this <//(S'=n-l)  =  S',  2,  S'=n-1, s u c h z  element  block  i s an - J".  as  can  be  separated  is E +(n-l)J' 0  i s - J ' . There  \//(S'=n) =  can  i s the  atoms  an  atom  blocked  s u b s c r i p t s denote  o f f - d i a g o n a l element  and  same  the  different  2  with  energy  the  J " . The  2  2  the  that  i n S=1/2  equivalent  but  seen  . n a ) 1 ( 1 a.2/3. . . n a ) } . I f J " = 0 t h e m a t r i x  (1,1,...1) >  by  If  { A ( 1 a 2 a . . . n a ) , ( 1 a 2 a . . . n a ) } , where A  Clearly  antisymmetriser  Now  reduced  i n energy  for striction  atoms each  just  by  i s obtained.  I t can  conveniently  are  singlet-triplet  shift  constant.  that  solved  result a  f o r S=1/2.  energies  i n the  suffer  suppose  be  singlet  theory:  a pair  J ' and  and  triplet  the equation  Consider  be  again  in order  result  perturbation  and  singlet-triplet  general  J(T+2S ),  triplet  Lines'  0  2  0  change  d i f f e r e n c e remains  J A=K A' 0  singlet  If K A'=J A/2 the  0  f o r the  i . e . there  2  0  -1  the  O  [K A'-J A(r+l/2) ]  2  and  K A'=0 both  0  = -0/2w )  are  | (1 , 1 , . . . 1 ) ,  | ( 1 , 1 , . . 1 ) , (-1 ,-1 , . .-1 ) > ; 2  i f J "= 0  210 these  have  energy  E .  I f J " i s now  0  a p p l i e d as  a perturbation  one  obtains: (6.33)  «MS'=n)|V|^(S'=n)>  = 0,  Having  defined  energy  rest  of  the  In  terms  the  S'=n  and  The  energies  n J,  spin of  S'=n-1  states  now  be  result  i s not  on  the J"  two  dependent  atoms;  and  positive  couplings  a measured J of  exchange  integral  It  can  each  be  between  atom  seen  30cm"  1  the  -  i s ignored  of  the  as  the case rise  the spins  J=J"/n  i n an  orbitals  2  The  the J'  t o Hund's S on  as  are  first  the when a l l J ' ,  shows t h a t , f o r  of 750cm"  r e s t of  are  i s useful  as  S=5/2 s y s t e m  i n the  coupling  integrals.  give  that  orbitals  i s involved  that  t o be  result  2  result i s :  good quantum numbers o n l y  The  example,  on  in J'  p a i r of  such  which  to note  atoms remain  a l l J " , are equal.  orbital  one  of  i s expected  It is interesting  constituent  the d e t a i l s  a sum  J=J"/n .  E(S'=n) = K  hence the H e i s e n b e r g , treatment  as J'>>J". T h i s strong  by  between  J[S'(S'+1)-2S]  i f more t h a n  i s replaced  rule.  splitting  Thus the g e n e r a l  = K -  2  interval  expressed as the term  2J"/n  d i f f e r e n c e , the  the Lande  = - J T , the  where  striction.  energy  = 2J"/n; therefore  H = K - J(r+2S )  perturbation,  rule.  H  i s 2Jn  found above can  by  an  f o l l o w s from  (6.34)  coupled,  long  and  the Hamiltonian,  i s unaffected  between  and  spectrum  2  This  the  absolute  E(s'=n-1)= K - J ( n - 2 n ) ,  2  it  an  $ ( S ' = n-1)|V|<//(S'=n-1)> =  1  implies  i f just  an  one  exchange.  the s o l u t i o n f o r the  Lines  21 1 model for  can  now  simplicity,  (6.35)  has in  be  the  H  followed except T must  = rr /2  T new  terms  replaced  + w £ /2  2  solution  be  2  by  assuming  to  and  0  B=B'  Thus,  2  0  +B£ ) (r+ 2 S - K / J 2  0  corresponding  A=A'  (T+2S -K /J )•  - J ( 1 + A£  2  a r e added  that,  2  0  (6.27);  when  this  is  0  )  expanded  to J :  (6.36) J  = J  Thus a  [ 1 + { (<n>+l/2)hB/w} + ( J / C J ) [ A 2  0  the  first  change  order  between  the  exchange  static  Three  In  to  atom  the  symmetric with  striction  i n the magnetic  difference  6.4.2  + hB  2  0  spectrum which  integral  0  A,  as  well  results  J  0  as  in a  B,  small  value  atom c h a i n  of J  and  .  effects.  there  a r e two  stretch,  The  normal  w h i c h may  striction  modes,  be  the  associated  Hamiltonian  corresponding  (6.24) i s :  (6.37)  H  = 7T /2  + w  2  0  *, /2  + o)  2  where  c=(2S -K /J )  totally  O  symmetric  £o /2  " J (1+A£ ) (T,+T +C)  2  2 1  0  ^, /2 -  vibration.  0  +  3  J (1+A{,)(T,-T )  2  0  i s a constant  2  0  2 0  3  which  Applying  appears  (6.28)  only  (and  f o r the  i g n o r i n g c)  gives:  (6.38) =- ( j  2 0  A  2  / 2  2  AE  = -(J  ) [ ( l /  2 W  o  2 0  A /2) 2  +l/o;  2 l  )(r  [ ( r , +r ) /u 2  3  2 1  +r  2 3  )  +  2 0  +  ( r , - T ) /a>, ) ] 2  2  3  ( 1 /u>  0  2  - 1 /co, ) ( r , r + r r , ) ] 2  3  0  )  causes  chains  antisymmetric  striction  2  the e x p e r i m e n t a l l y measurable  three  and  parameter  (v+ 1 / 2 ) /u ] ( 2 S - K / J  2  3  212 (If  the  i.e., the is  two  the  pairwise  two  first  pairwise  term  three  The being  In  case  of  simplest  has  the  the  first  the  metal  special term  neighbour  exchange  the  =  where J ' = ( J A / 2 )  E =-J-J' a  The and  has  r , which 2  simply  ( 1 /CJ  0  2  doublets  of at  J<0  this  It obtained  by  simply and  the  distortion  dynamic  scale  J;  the  is  coupling  first.  As  spin  (T,T  + T  3  well  states, i t 3  r i )  adds a as  a  as  =r  . Thus  2  trivial next-nearest  t o h a v e J>0;  i.e. H  can  2  i s expected  t o be  at  Defining  E„=-J", E = 3 J ' / 2 2  striction  that  and  a quartet  2J-J'/2.  as  small  of  decreases  and  the  the  static  chain  above  result  with  while  increased.  to compare  averaging  asymmetric  procedure  2  and  i s expected  i f J>0  solving H  distortions  this  only  0  same e f f e c t  re-expressed  is interesting  dimers,  = 3CJ .  2 1  and  scales H  compared w i t h  splitting  = 3/4-T  the  separately,  vibrational  number of  2  shows t h a t  splitting  be  of  E =3J'/2 and  2J"-3J'/2,  if  2  and  CO^CJQ  (6.39) c o n s i s t s of  can  doublet  T  treated  t r e a t e d as  mass, W  the  2  J"=J+J'/2 t h e s e this  of  - 1 /CJ, )  spectrum  are  3  considered  -J'T  3  2  r  as:  -J(r,+r )  0  positive.  that  second  most  (6.39) H  be  (6.38) merely  and  expressed  case  equal  i n terms  property  constant,  be  idealised  S=l/2 w i l l  in  and  in general  atoms of  case  T,  i n t e r a c t i o n s are  i s obtained;  incorrect).  between  couplings  the  model.  the  f o r the  range of  resulting Clearly  distortion  possible  magnetic  'A'  that  s p e c t r u m , as  in  mode v i b r a t i o n s m e r e l y  terms can  be  included  by  213 writing:  -Ax (6.40)  H  =  -J(T,+T )  -xAJ(r -r )  3  1  1+Ax, -1  = J  3  ,1+Ax, 0  0  where  the matrix  distortion  basis  parameter.  E(S'=1/2)=3(AJX) /2, as t h a t obtained  replaces  t h e term  distortion for  degeneracy  separated energy and so  the idea  may  data.  energy  be e x t r e m e l y  This  interest several exchange  S=l/2 system between  difficult  have  Hamiltonian.  been  2  and dynamic  there  spectrum)  results only  a separation  given  nickel  be i s only  only,  chains  one  energies  t o d e f i n e J , and  interval  to distinguish  fitted  cannot  o f t h e S=1/2  i s not of merely  a s s e v e r a l 3 - a t o m S=1 of these  <x >  contains  effects  t h e mean  such  exchange  s t a t e of a f f a i r s  that  equivalent  as again  a Lande  f o r S>1/2  neighbour  h a s t h e same  except  spectrum  exchange  i n the magnetic  In general,  asyymetric  spin.  of d e v i a t i o n s from  nearest  model  to yield  the magnetic  and b i l i n e a r  f o r t h e 3 atom  t h e S=3/2  next  i n which  d i f f e r e n c e (that  meaning. it  are expected  due t o t h e t o t a l  Biquadratic  spectrum  i n t h e f r e q u e n c i e s . The L i n e s  approaches  a l l systems  on t h e s t r i c t i o n  Ax  E(S'=3/2)=J,  This  2  ,1-Ax,  and x i s an  H has eigenvalues -2J-3(AJx) /2.  2  form  i s { a a B , a B a , Baa}  ,1-Ax  rule  h a s no  i s p o s s i b l e but  biquadratic  from  say, s u s c e p t i b i l i t y theoretical e x i s t  to a next-nearest  1  9  2  '  1  9  3  and  neighbour  214 6.4.3  Infinite  As  the chain  spectrum  phonon  modes w h i c h 3-D  with  lattice  In  this  will  from  a slightly  1 9  *'  +  i+ 1  H = -2J 2[(1+a)S .S i i  a s a c a n now  the spectrum  Alternating characterised  by a  low energy  wavelength  throughout  vibrations associated  transition  8  aS^.S.^)  notation:  +  i + l  (1-a)S .S _ i  i  t o t h e even  3  "  1  8  9  0  ]  to a shift  numbered  2  1  9  2  independent  (SP) systems  t e m p e r a t u r e , below  a^O  5  i n (6.42)  exhibit which  separates  from  the the continuous spectrum  both  system,  i.e. creates  an e n e r g y  a l t e r n a t i n g a n d SP c h a i n s .  In general  g a p , a n d n=0  A noticeable  9  6  .  In a  a increases to  0 . 2 , a t T=0.  infinite  1  a=0 a b o v e  than  state  ones  , are  a maximum, u s u a l l y l e s s a n S'=0  of  i n a=0.  e.g. C u ( N 0 ) 2 . 5 H 0  7  discussed  :  symmetric  by a t e m p e r a t u r e 1  1 9 5  J are often  respect  i s now  chains,  spin-Peierls  i s replaced  be r e l a t e d v i a ( 3 J / 3 d )  and  of H  vibrational  contains  energy  different  odd numbered atoms w i t h  defined  The s p e c t r u m  Hamiltonian  the  well  molecules  the equivalent  i  be u s e d  contrast  the d i s c r e t e  distortions.  H = - 2 J S(S_ .S i  work  (6.42)  spectrum.  local  of the  (6.41)  of small  as w e l l as h i g h e r  significant  terms  increases  c o r r e s p o n d t o waves o f l o n g  Deviations in  length  characteristic  continuous  the  chains  'knee'  a*0 of the  a t T=0 i n i n the  215 susceptibility in  appearance  Figs.6.1  at the t r a n s i t i o n to that  f o r long  causes  of a l t e r n a t i o n  understood.  associated  with a non-magnetic,  thought  t o be c a u s e d  couplings In  Alternation  the  freezing  not  along  spins  drive  of t h i s  a degenerate  there  varies  ( t o produce  from  thesis  above  normal  transition.  some c a s e s  phenomenon  between  static  ordering)  the behavior distortion  phonon  as  which,  because  temperature.  i n t h e ground .state  zero.  must  i t  i s  the  For c l a s s i c a l as a  energy  b e some d e c r e a s e i s thus  field.  c a n be d e s c r i b e d a s  mode, d o e s n o t a f f e c t  SP b e h a v i o r  such  be  the magnetic  i s no d e c r e a s e  the transition,  are not yet  b u t t h e SP t r a n s i t i o n i s  the t r a n s i t i o n  As t h e r e  similar  chains,  i n at least  structural  units,  out of a dynamic  susceptibility  may  by i n t e r a c t i o n  a n d t h e 3-D  the language  o f t h e SP  a n d SP b e h a v i o r  completely  s t a c k i n g of the polymer  somewhat  a n d even Odd/Even  t o 6.3, i s c h a r a c t e r i s t i c  The  the  temperature,  i n energy t o  an e s s e n t i a l l y  quantum  phenomenon. Solitons to  travel  area in  9  6  along  have a l s o been studies.  1  This  "  1  9  7  ,  or non-linear excitations  chains observed  without,  in principle,  i n chains  has aroused  a great  i n neutron deal  o f n o n - l i n e a r p h y s i c s has been  recent  formalism conditions around  years.  I t might  to cluster  the.normal  loss  using  of i n t e r e s t  the subject  rotation  periodic  of energy,  diffraction  be p o s s i b l e t o a d a p t  systems,  associated with  which a r e able  as the whole  of intense the soliton  boundary  of the d i s t o r t i o n  mode, b u t i t i s u n l i k e l y  study  that  the  vector concept  216 would the  be  useful  travelling  metal  i n c l u s t e r s as wave  soliton  assuming few  periodic  numerical  where E  per  i s zero  0  0  atom,  chains  will  dimerisation at  least  delicate  by  chains  of  would  the  SP  be  tetrahedra  across  as  several  of  with  tens  of  as  the  a  non-zero atoms.  of  hundreds  proportion  of  atoms w h i c h  view  are  8  rings  state  energy  the  about  for  chains  half  a  dozen  s y s t e m s ) and  lattice  e f f e c t s may  to  extremely  expected  of  on  for  remains  0  in  . For  tendency  linear  atoms because  9  that  of  (only  of  1  or  calculations  show  3E/3a)  In  known o u t  end  " ~  9  significant  transition  these  1  infinite,  ground  .These  SP  seem t h a t  in either r i n g s  the  100  investigated  Some n u m e r i c a l  discussed.  S=l/2 chains  l e s s than  e f f e c t s have  different low  contexts;  with  the  occur  in  defects  1 8 0  significantly  1 8 2  cause and  an  previously  increase  lower  the  ,  influence  the  (6.43) H  =  distortion  -2J  to  a  8  in  rather  ordering 3  at  temperature  .  Hamiltonian:  [(\+a/2)S,.S  a corresponds  1  only  in susceptibity  3-D  interchain interactions  Consider  which  been c o n s i d e r e d  they  temperatures  associated  in  or  transition.  End  very  extends  conditions  symmetry.  now  several  examples are  it  trimers  typically  i s here defined  0  associated  nature  significant  phenomena a r e  boundary  antiferromagnetic  of  typically  calculations, finite  3E /3a) ,  open  as  atoms.  SP/alternation  a  small  2  shift  +  of  (l-a)S .S 2  the  3  +  (1+a/2)S .S,]  c e n t r a l atoms  3  with  217 respect  t o fixed outer  chosen  t o be o r t h o g o n a l  of  the chain.  to the t o t a l l y  The c o n d i t i o n  examples ensures  that  limits  large  of either  notation  a t o m s . The c o e f f i c i e n t s o f (6.43) a r e  (9E /3a) 0  0  p e r atom tends  n or c l a s s i c a l  H={J,2/^,^23/^,^3n/J-••},  i s independent  are,  of - J , and t o f i r s t  and  0  S=1/2  i n the  Introducing  the  state  as  quintet  i s no t e n d e n c y t o  f o r J<0 t h e s i n g l e t order  i s c l e a r l y favoured.  c a l c u l a t i o n s a r e needed.  Hamiltonians  obtained  mode  energies  i n a:  = - 1 . 5 ± 1/3(1+0/2)  dimerisation  numerical  rings;  t o zero  ( 6 . 4 3 ) c a n be w r i t t e n  of a and there  as i n closed  (6.44) E  spin.  J>0, t h e ground  dimerisation, in units  stretching  I J ^ =0 i n t h i s and subsequent  {1+a/2,1-a,1+a/2}. F o r S=1/2, energy  symmetric  f o r a = ±10"  6  chains  Some r e s u l t s  of t h e form H =  by s o l v i n g  For longer  f o r 9E /9a f o r 0  {1+a,1-2a,1+2a,....1+2a,1-a} and a = 0 a r e :  ( 6 . 4 5 ) n=2, 0.0; n=4, 0 . 2 1 6 5 1 ; n=6, 0 . 2 4 1 3 2 ; n=8, 0 . 2 3 9 9 5 ; n=l0 Though that  0.23182; n=12,0.22288;  (n-2) orthogonal  leading  d i s t o r t i o n Hamiltonians  t o a l t e r n a t i o n has t h e l a r g e s t  others  will  (6.45)  i s that  limit,  zero,  given  n=14,0.21410;  n o t be c o n s i d e r e d . (9E /9a) 0  0  decreases  as n increases;  above suggests  that  The most only  inspection  t h e decay  n=16,0.20587;  c a n be  (9E /9a) 0  0  defined, and t h e  i n t e r e s t i n g feature very  slowly  of t h e l a s t  i s even  slower  of  towards the few t e r m s than  that of  l/log(n).  The  spin  v a r i a t i o n o f t h i s phenomenon  i s also  of i n t e r e s t  218 as  i n the classical  for  limit  0  n=4, a r e i n u n i t s o f  (6.46)  The  high  0.30430;  S=3/2,  0.37581;  S = 2,  0.45670;  S=5/2,  0.54142;  S=3,  0.62766;  S=7/2,  0.71460;  S = 4,  0.80195.  results  c a n be o b t a i n e d  related  t o (6.43)  the  same g r o u n d  for  t h e four-atom that  most  simply  H={2+2a,1-2a,0} w h i c h ,  by t h e a d d i t i o n o f t h e ' T ' 2  {1+a/2,0,- 1-a/2}, w h i c h  2  - J :  S=1,  atom H a m i l t o n i a n  1/S  results  0.21651;  spin  S and thus  = 0 f o r a l l n . Some  0  S=l/2,  'three'  a  (3E /3a)  leaves  state energy. chain  (3E /3a) 0  the limit  since  0  i t  i s  distortion  t h e S'=0 e n e r g i e s The sequence  by s o l v i n g t h e  unchanged, has  (6.46)  indicates that  i sproportional to S f o r large  as S tends  to infinity  i s zero  only  on  scale.  For  S=1 t h e s e q u e n c e  corresponding  t o (6.45) i s :  ( 6 . 4 7 ) N=4, 0 . 3 0 4 3 0 ; N=6, 0 . 2 5 5 2 6 ; N=8, 0 . 1 9 9 5 1 ; N = 1 0 , 0 . 1 5 6 7 4  Now c o n s i d e r interactions. {l-a,1+a,1-a contain  odd c h a i n s ;  Consider  the  centre  one  atom  Hamiltonians  the centre  Z J = 0 the chain  o f 1+a. T h e s e  f o r n=4m+1  number o f  of t h e form  t o maintain  couplings  of the chain  from  have an even  the Hamiltonian  ,1+a,1-a};  two a d j a c e n t  these  will  must  be p l a c e d i n  (m i n t e g e r ) a n d d i s p l a c e d b y  f o r n=4m+3. F o r e x a m p l e t h e  f o r n = 7 a n d n = 9 a r e {1 - a , .1 +a, 1 - a , 1 +a, 1 +a, 1 -a} a n d  {1-a,1+a,1-a,1+a,1+a,1-a,1+a,1-a}. The s e q u e n c e s (6.45) a n d (6.47) a r e :  equivalent to  219 (6.48)  S=l/2  n=3,  n=11 S=1 (There  0.0;  n=5,  is alternation of  n=4m+1, m  first  order  finite  regarded  Though E the said for  0  S=l/2 In  I t can  as  i s larger  and  have that  with  favoured. chains  (9E /9a) 0  0  number o f a t o m s c a n alternation. clusters  be  said  In comparing retain  of magnetic faces.  energetically  remain  (6.48)  S=l/2  (A f r u s t r a t e d  favourable  i s no  remaining  sense  finite  spin  the can  ends be  with  rapidly.  9  9  6.2  which  a l i g n m e n t of  a  be  chains  detailed  However  contain  a  innate  tendency  clusters  i t can  even  associated  system  I t can  for long  given.  to d i s t o r t 1  f o r a l l but  f o r S>l/2.  i s not  chains  chains  frustration  the  In t h i s  important  t o have an  a tendency  there  strongly  of  t h e c o m m e n t s made i n s e c t i o n  'infinite'  real  only  alternation,  i s smaller  become u n i m p o r t a n t  seem t h a t  the  dimerisation.  but  to  0.16391.  as  though  d e c r e a s e s much more  (6.45)  0.16553  symmetry  2  simple  of  effects  of  P  become more  quantum end rapidly  n=9,  (6.48)  at the expense  even,  f o r S=1,  chains  view  triangular  seen  incipient  of  because  be  pairs  coupled  interpretation  that  system can  t h e end  as w e l l  0.18070;  the sequences  change a s s o c i a t e d  s o u r c e s of  shortest that  0.15059.  are energetically  odd, as  0 . 1 5 7 6 6 ; n=!5  0 . 2 1 8 4 9 ; n=7,  integer).  antiferromagnetically  of  0 . 1 5 0 1 6 ; n=9,  within  i n which  interactions  0 . 1 5 7 1 6 ; n=7,  the d i s t o r t e d  energy  distortions  n = 5,  0 . 1 5 7 0 0 ; n=13  n=3,  Hamiltonian for  0.0;  spins  is  finite towards be  said  for high  spin  with  i s one  i t does  their  i n which  no  possible.)  220 Frustration order sum  {J}  can  excited the  arise  only  occurs  at  states could limit  S and  of n.  high The  and  thus  a l s o be  for which which  temperature  development  f o r t r a n s m i s s i o n of  lattice  defined  throughout  the  interactions.  the  as of  a  effects  well  preserves  the  of  (3E/3a) as  theory  transition  the  would  SP  distortion 0  tends  i n the  from  to  limits  explaining  p o i n t , and  alternation  lattice  first  effects.  considered.  mechanism  chains  a well  T>0  of  3-D  a distortion  from quantum  existence  short  systems  above a n a l y s i s i s f a r from e x h a u s t i v e ;  transition  large  in linear  changes a s s o c i a t e d with  The  in  i s impossible  require a  zero of  the  hence  the  on  some  comparatively treatment  of  221 CHAPTER 7  OTHER A S P E C T S OF THE D Y N A M I C D I S T O R T I O N  Most high  of t h i s  chapter i s concerned with  symmetry magnetic  closer  to solid  thesis.  state  The J T i d e a ,  clusters physics  which  a re-examination of  from a p e r s p e c t i v e  than  that  applies  to a l lclusters  i n t h e magnetic  discussion  of concepts such as f l u x i o n a l i t y ,  The  dynamic  distortion  because  non-magnetic  intercluster  any decrease  the  active  conclusions  t u n n e l l i n g and  data  which  of the spin  a fuller  fluctuating  tothe  are the real  treatment,  random  strains with  involving to explain  the Jahn-Teller  t o t h e model a l r e a d y  f o rthese c l u s t e r s  derived.  magnetic  a r e made.in  a r e summarised  cause  Hamiltonian and  experiments t o supplement  of the thesis  i t i s open  of the d i s t o r t i o n  associated  add l i t t l e  As such  7.3, a n d t h e  i n 7.4.  Introduction Fluxionality  The 1963  would  f o rfuture  susceptibility  7.1.1  suitable  by a  reference  the rotation  of the degeneracy  clusters,  Suggestions  7. 1  fluctuation,  such as s l o w l y  reduction  of s o l i d s .  interactions  I n 7.2 i t i s s h o w n t h a t  features  with  as p r e s e n t e d i n c h a p t e r s 3 and  i t makes no e x p l i c i t  i n t h e symmetry  by t h e i r  vector.  model  on a ' m o l e c u l a r ' v i e w  to c r i t i c i s m  hence,  i s introduced  of the  coupling.  4 i s based  of  spectrum,  somewhat  of the rest  degeneracy  vibronic  MODEL.  term  fluxionality  to describe  was c o i n e d  the magnetic  by D o e r i n g a n d R o t h i n  equivalence observed  i n t h e n.m.r  222 spectrum  of  the  hydrocarbon  bullvalene  applied  in cluster chemistry  ligands  bonded  applied  in  the  to  the  inorganic  with  molecular  can  minima  has  been  states  be  of  a  in a  the  as  one  in which  since  the  excited  of  factors  a  state  state  long and  as  or H  second i s the  which  i s time  i t is clear  Tunnelling  temperature  the  geometrically fluxional  exist  low ligands  dependent these  low-lying  at  between  ,  and  symmetry  or related  f o r p a r t i c l e s of  low  thin potential barriers; tunnelling  which a molecule one  a  tunnelling  the  t u n n e l l i n g . The  independent, the  i n terms of  either  works a g a i n s t  favours rate  2 0 7  localised within  from  of  i s favoured  i s l o c a l i s e d with  states  in which  equivalent  discussed  states  the  these  .  Doering's  one  there  of  .  been  barriers. A  occupation  often  in ammonia  2 0 6  was  2 0 3  in  temperature.  mass, as  fluxionality  to  changes  have  several  energy  effective  the  fluxionality  of  recently,  temperature  low  minima  clusters,  with  particular configuration via  more  apparent  by  potential  a  describe  widely  equivalence  and,  2  p o t e n t i a l has  2 0 5  systems are  interconverting  first  0  been  a p p l i c a t i o n s . Thus,  application  arises  Fluxional  in  to  2  the  which a permutation  The  d e p e n d s on  "  1  I t has  f l u x i o n a l system  separated defined  effected.  molecule  *  2 0  through  stereochemistry states  0  fields  these  Born-Oppenheimer  equivalent  lying  each of  definition  cluster  ligand  2  d i f f e r e n t d e f i n i t i o n s of  associated implied  of  .  describe  core  chemistry  stereochemistry  Slightly  metal  to  2 0 0  rate  stationary  magnetic  of  initially  minimum  other  in  prepared  i s transformed  minima. However, state  v a r i a t i o n a l theorem  solutions  that  a  to  as exist  molecule  223 which  is localised  fluxionality that  of  the  near  involves ground  one  real  and  chemistry  most quantum  as  eigenstates real  states  dynamic  Fluxionality wavefunctions Hamiltonian Partial  which  tunnelling  discussion  minima  the  are  full  and  consider  a  square  non-degenerate  2 & 4 or from  configurations vibrations  of  state.  degeneracy  of  full  f o c u s on  the  there  no  is  ligands. the  e i g e n f u n c t i o n s of the  potential  has  a  system.  several  i t i s only  these. the molecule  and i t s  surface with  since  Such a molecule atom  1 can  These c o n f i g u r a t i o n s of  tetrahedral  correlate the parent  several  w i t h 4 e q u i v a l e n t atoms  E mode d i s t o r t i o n  configuration  of  a potential  molecule  3 & 4.  quantum  important.  electronic  configurational  arising  of  the  of  when  of  into  definition,  f o r many p u r p o s e s  i n t e r a c t i o n s 'between  example  awkwardly  than  used:  (b) where  an  greater  of m o l e c u l e s  by  which  the concepts  H a m i l t o n i a n of  t o c o n s i d e r one  As  energy  only arise  necessary  are  an  for  or permutation  the Born-Oppenheimer  surroundings  3,  treatments  Hamiltonians are  well-defined  has sense  in which,  i s not  minimum,  f i t rather  distortion  and  under  (a) where  &  In t h i s  tunnelling  o f H,  motion,  motion,  state.  fluxionality  potential  w i t h the molecule.  be  can  has  s y m m e t r y . The  The  3-fold  state  molecule  t o atoms  pictured  a parent, higher  ground  in a  bonded be  minima,  energy  three and  as  square  two  i s only  lowest square  2  224 to  the extent  that  configurations of  the f u l l  Hamiltonian  the degenerate  the  full T^.  atoms and l e s s included  always the  mode  symmetry It  lowest.  than  symmetry  n  i f only  i f the inclusion  magnetic  clusters  spectrum  attains  Interactions  that  t o be u s e f u l ,  they  molecules  the time  chemi s t r y .  proper  n equivalent rotations are  Such a molecule  h a s an  rotations enlarges  s c a l e on  the parent  For example, scale  fluxionality  high  which  symmetry  i n the application  i s t h a t on w h i c h  must  to  the magnetic  thermal e q u i l i b r i u m . between a molecule  provides  important  state.  a meeting  typically  treats  the 'molecular  and i t s surroundings  are a p p l i e d , perhaps  a r e an e s s e n t i a l  i n the solid  which  considering  eigenstate of  t o the time  from  o f some m e a s u r e m e n t . M o r e  physics,  with  t h e word  by t h e r e f e r e n c e  when e x t e r n a l f i e l d s  interactions  i n P„  group.  i s t o be m e a s u r e d .  discussion,  a s A,+E  of the c o n f i g u r a t i o n  of improper  c o n f i g u r a t i o n , or d i s t o r t i o n  course  the three  eigenfunctions  In each  sense any molecule  be q u a l i f i e d  important  three  the expectation value  P  between  transform  extreme  i s clear  system,  the  f o r the system  c a n be d e s c r i b e d a s f l u x i o n a l .  inversion the  elements  The l o w e s t  E state lies  Hamiltonian  In t h i s  matrix  c a n be i g n o r e d .  and  is  tunnelling  i n the  i n the present  f e a t u r e of any treatment The t r e a t m e n t  ground  between  atomic  lattices  problem'  2 0 8  ,  may  of solid  such state  before  and c o o r d i n a t i o n  of  be  225 The d i s t i n c t i o n an  important  occur  only  between  i n case  fluxionality  compounds In  both  motion  with  ( b ) . In terms  in solid  exist  intrinsic  property  7.1.2 V i b r o n i c  of the  labelled lie  in H .  interactions product  between  space,  functions  in H  eigenfunction expressed  Born  20 9  eigenvalue  than  molecules' with  no  i s  real  i s n o t an  e  w h i c h c a n be  of a n u c l e a r which  only  Hamiltonian involves  e l e c t r o n s and n u c l e i l i e i n the d i r e c t  n  with  involves  l i e i n a space  H *H , which contains e  which  of a Hamiltonian  functions  in H  of the Hamiltonian  n  a l l sums o f p r o d u c t s . A vibronic state  which  cannot  be  of  i s an  adequately  o f an e l e c t r o n i c and a  nuclear  . 2 1 0  showed  the eigenfunction  that  state  i s u s e f u l because of the n u c l e i ;  that  the error  corresponding  by a s i n g l e p r o d u c t  approximation less  on  inorganic  fluxionality  the eigenstates  and O p p e n h e i m e r  representing  effect  molecule.  as a s i n g l e product  wavefunction  eigenstates  such  a n d momenta  The e i g e n s t a t e s  n  such as  to 'fluxional  of a H a m i l t o n i a n  similarly  g  i t i s likely  coupling  coordinates  H ;  can  stereochemistry.  f o r (a) because  The e i g e n s t a t e s electronic  dependent  and f o r (b) because  eigenvalues  have a dominant  s t a t e compounds  (a) and (b) r e f e r e n c e  misleading,  (a) and (b) i s  of f l u x i o n a l i t y ,  interactions will  temperature  cases  somewhat  Hamiltonians  one. V a r i a t i o n i n t h e molecular  that. intermolecular any  partial  to a  involved in non-degenerate  i s negligible.  The  BO  t h e e l e c t r o n mass i s v e r y  the nuclear  much  momenta t h e r e f o r e  have  226 a  n e g l i g i b l e e f f e c t on t h e e l e c t r o n i c p a r t s  and  the nuclei  electrons.  only  respond  I n t h e BO a p p r o x i m a t i o n  nuclear  momenta  nuclear  p o s i t i o n s appear  for a  each  states  the in  to yield  as JT type  nuclear  as parameters.  terms  b r e a k s down which  c o n f i g u r a t i o n cause  i s overcome  which a r e l i n e a r  i n which the  The e l e c t r o n i c be c o n s i d e r e d to yield  energies to define  the vibrational  therefore  combination  be r e g a r d e d  approximation  7.2 S y m m e t r y , D e g e n e r a c y  The  most  transition  T, i s J T - a c t i v e  large  In Jahn-Teller  of several eigenvalue  2 1 2  '  2 1 3  .  small  states  theory  this  respect  states  states,  problem.  2  1  1  obtained  JT theory  can  .  and t h e J a h n - T e l l e r  studied  changes  i n the electronic  product  Effect  systems  JT systems a r e those orbital  i n which a  state, either E or  t o a n o r m a l mode o f i t s l i g a n d  e . g . a n E o r T mode o f a n o c t a h e d r a l  complex  t h e r e s t of  a c o r r e c t i o n t o t h e BO  i o n i n a degenerate with  situation  changes  of Jahn-Teller  frequently  metal  in this  as providing  f o rdegenerate  The p r o p e r t i e s  f o rdegenerate e l e c t r o n i c  by t h e i n t r o d u c t i o n o f v i b r o n i c  s o l u t i o n o f an a p p r o p r i a t e  field,  without  do n o t commute w i t h  are introduced,  of the eigenfunctions.  7.2.1  eigenstates  c o n f i g u r a t i o n can then  BO a p p r o x i m a t i o n  difficulty  by  due t o t h e  of the system.  Hamiltonian  parts  field  the Hamiltonian  p o t e n t i a l w h i c h c a n be s o l v e d  The states  i s solved  nuclear  nuclear  t o the average  of the wavefunction  S i g n i f i c a n t extension  or tetrahedral  of the o r i g i n a l  JT theorem  227 is  necessary  aspects  of  to provide  the  wavefunctions  coupling which  Phenomena degeneracy dynamic weak o r the  2 1 5  ,  centres  are  to a  be  external  the  of  Ham  can  point  take  place.  form 2  must This  intramolecular  2  1  '  2  of 2  2  and 2 2 3  .  No  situation  fluxionality  planar  point the  the  and  are  no  effects of  according  JT in  JT  effects  of  in  transform  of  to  that  commutes w i t h  the  still  for  general 2 2 0  .  no  be  momentum,  forbidden  because a l l  the  original  in  i n which  does not  the  essential  occur  according  and  angular  I.Rs  i n ammonia,  to  formalism  however,  can  which  and-may  similarity  absorption  to  of  between  important  orbital  i s analogous  part  ,  context  removal  distortion  as  ",  systems,  The  can,  (Vibronic coupling  'distortion'  JT  1  JT-active centre  increased  transform  group.  one  a quenching  state eigenfunctions  as  In  2  2 1 7  spin Hamiltonian  just  JT  ground  case  nuclear  static  cooperative  i s d e l i b e r a t e and in a  dynamic  electronic  of  effects.  'distortion'  transitions  group.  i n terms  transition.  containing no  e f f e c t ,  eigenstates  state  concentrated  phase  discussed  i n the  infra-red  ground  9  and  i n t e r a c t i o n s between  and  magnetism  field,  observed  1  and  complexes.  between J T - a c t i v e c e n t r e s  system  degeneracy  2  i n which  cooperative  a  electronic  dynamic  local  negligible,  usefully  In  ,  a l l static  i t i s most u s e f u l t o d i s t i n g u i s h  systems,  terminology can  and  2 1 8  2 1 6  the  i n such  associated with  work  interactions lead  occurs  cooperative  strong  JT  between  have been d e s c r i b e d  present  dilute  a d e s c r i p t i o n of  to  both  of  the  I.R.s  of  the  occur  in  r e s t of  this the  228 Hamiltonian.) because  the  No  electronic  degeneracies, in  this  are an  accidental  The  reduced  JT  potential  the  by  2  1  .  8  permanent  magnetic  i.e.  If  a  static  characterised of and  transitions  can  transitions, distortion and  the  as  is a  lattice  from c r y s t a l  the  In  the  the  free  driven be  by  the  in  with  characterised,  of  packing  for  effects  each  by  ""  2  2  5  or  related  of  second  an  order Such  magnetic  . As  static  the  separate to  a  the  to  between to  and  interacting  occur.  analogy 2  lines  centre,  transition,  to  may  2  other,  of  first  respect  difficult not  the  a  such  i s even  phase  effect,  to  the  with  JT-a.ctive  number  the  interactions  i t is often  one  only  shape of  associated  There  (anti)ferrodistortive result  be  sensitive  the  in  2  approximation.  of  may  E  an  result  as  JT  (AT+E+A is  with  i n an  a  cooperative  the  quantum  1+2+1  systems along  energy  molecule  s y s t e m s can  occur.  discontinuities  states  JT  fields  the.limit  degeneracy  dilute  more t h a n  may  vibronic  of  i s very  JT  but  occurs  several  Thus a  harmonic  As  external  when  degeneracy  stress  system  a  E*E,  by  Essential  vibrational  4-fold  small  infinite  by  below.  one  fields. a  replaced  same I.R.  symmetry  'condensed'  becomes  parameter,  and  distortion  clusters.  derivatives  with  applying  provide  of  the  The  are  degeneracies  which a r i s e s  system c o n t a i n s  may  possibility  centres  of  essential  degeneracy,  surface,  centres  span  and  external  strain,  fields  that  degeneracy.  degeneracy  be  large  and  4-fold  result  the  discussed  to  state  mode h a s  essential  be  refers  related  in  degeneracies  will  electronic  normal 0^)  as  context  symmetry E  reduction  the  metal JT  phase phase JT atoms  effects  electronic  229 2 26  degeneracy  Coupling  between  conventionally is  denoted  The  denoted  E*e,  T*e, and t h a t  matrices  which  independent  that  define  H(x)  =  t h e BO s u r f a c e  1  states  H(x ) =  •1  0  0 - 1 0  T*T;  H(X)  The relevant  =  E*e  (7.2)  the  • 3  systems a r e and a r e :  0 0  2  0  0  H(y)  =  0 0 1  0  , H(z) =  0  1 0  0 0 1  0 0 0  1 0 0  0  1 0 0  0 0 0  1 0  system  has been  studied  copper(II).  t o the potential cos0  sine?  sine?  -cose  ±1 w h i c h  hat'  2 2 7  .  i n most d e t a i l ,  and i s  The e i g e n f u n c t i o n s  of the JT  surface: <// = ( c o s e / 2 ,  sine/2)  \//. = ( s i n e / 2 ,  -cose/2)  +  a r e independent  p o t e n t i a l has c y l i n d r i c a l  'Mexican  0  0 -v/3  0 0 0  eigenvalues,  electronic  =  0  to octahedral  contribution  have  0  f o r these  1 0  H(y)  0  a n d a T mode, T * T .  0 1  0 -1  T*e  a n d a n E mode  a n d modes i n v o l v e d  H(y) =  0  a n d a n E mode i s  between a T s t a t e  between a T s t a t e  of the p a r t i c u l a r  E*e,  (7.1)  an E e l e c t r o n i c s t a t e  o f e.  symmetry  I t c a n be shown t h a t  The  full  a n d i s known a s  the solutions to  230 the  total  states  Hamiltonian  c o n s i s t i n g of just  (7.3)  *  =  +  \p. a r e now  exp(±ip0/2)•  e l e c t r o n i c momentum  The e l e c t r o n i c p a r t s  characteristic the  total  undergo  that  they  f u n c t i o n must  change  of systems  with  angular  states In  with  other  nuclear  words,  remains. between fold  with a is  three  T*e  spin/spin  leading  and thus  a l l vibronic  i s removed.  e f f e c t s destroy  correspondence molecule  i s rather  less  the  fold  to a  conditions  three  associated  t h e ground  state i s  effects are included.  There  the configurational  discussed  interesting  the eigenstates  axis  distortion.  b y 2n,  with  when  tunneling  i n the limit  a static  when a n h a r m o n i c  coupling,  associated  hat but the three  as 8 changes  of the square  coupling  spin  number j . F o r e a c h  Anharmonic  with  must  i n sign i s  i n (7.2) i s removed  p e c u l i a r boundary  of sign  E, even  parts  t h e s y s t e m c a n be p i c t u r e d a s  associated  an i n t e r e s t i n g  degeneracy  of  case  +  b y 27r; s i n c e  the nuclear  nij<0  splitting  and  the unusual  as 6 changes  quantum  of the Mexican  of the rather  doublet,  of * have  half-integral  p o t e n t i a l minima,  the change  of t h e form  i s one w i t h  the apparent  In this  states  a n d no e s s e n t i a l d e g e n e r a c y  symmetry  degeneracy  Because  with  effects are included.  cylindrical  functions  sign  momentum  IIK>0 t h e r e  are doublets  nuclear  i n s i g n . The change  characteristic  solution  states:  be s i n g l e v a l u e d  t h e same c h a n g e  a nuclear  by v i b r o n i c  \p_<i>_  +  +  represented  t w o BO p r o d u c t  <j> a n d 0. a r e n o n - o r t h o g o n a l  where and  c a n be a d e q u a t e l y  i n 7.1.1.  a s , i n the absence  o f H ( J T ) commute  with  231 the  rest  of  different  2-D  degeneracy  T*r  the Hamiltonian. harmonic  i s not  coupling  Each  oscillator  complete  interaction  the three-D harmonic  been p e r f o r m e d  symmetry and  momentum J=0  electronic  potential  degeneracy,  vibrational  functions  quanta,  and  configurational degeneracy 7.2.2  as  long  once.  high as  and of  (T+T)*7  The  with  2 2 9  R(S')  In general  f o r m . The  potential  degeneracy For  large  separate  four.  a  2-D  n+1  The  for small  with  excited  states  JT  have  multiplied  interaction  wells.  The  by  the  lowest  vibrational by  the  ground  state  k.  the matrices  S=l/2  contains forms  This  each  least  at  apply most  including some I . R . s  of c o u p l i n g  would  (7.1)  I.R.  systems, at  such as  correspond, i n the  i n which  electronic  between  has  three-fold  ( s e e 3.1.1) c o n t a i n s  R(S')  JT  potential  2J+1  T^  f o r the  i s no  multiplied  of  has  clusters  to a s i t u a t i o n  relationship  If there  lowest  for a l lvalues  introduced.  each  calculations  .  and  degeneracy  more c o m p l i c a t e d  states,  fold  oscillator  symmetry m e t a l c l u s t e r s  are  the three  h a v e a number o f  to magnetic  JT paramagnets  low-lying  four  i n each  i s the case  and  with  three.  degeneracy,  each  This  once  into  i s retained  octahedron. than  n=1  Application For  J=1  "  2 2 8  the ground  and  i s divided  n=0  and  numerical  have  the  with  potential  i s t h e most complex  eigenvalues  angular  associated  removed.  symmetry. R e l a t i v e l y  spherical  i s therefore  there.are  the more (E+E)*e theory  several  degeneracy.  the above approach,  which  involves  232 the  magnetic  used  group  i n Chapters  specific which  example  transform  of the u n d i s t o r t e d  3 and 4 w i l l  now  of C u O X L .  Consider f i r s t  u  6  shifted  affects  singlet  to a  t h e S'=1  their  a n d S'=0  energies will  i n a breakdown  states  energy  i t i s likely  spin.  Such  f o r Cu„OCl (TPPO)«  singlets  model  The  by a  down  rule  state  with  and the t r i p l e t s  with  independent.  well-defined  i d e n t i c a l coppers  degeneracy  which  applies  t o note  cannot  that  a Hamiltonian rule  both singlets  a t 85cm" . T h i s 1  1  0  5  10cm"  1  approach  with  e.s.r.  14cm" . 1  operator which  has P  a  Cu„0 symmetry  and e i g e n v a l u e s w i t h the  w h i c h , by d e f i n i t i o n , a r e t i m e  Thus c o n c e p t s such as r o t a t i o n  i n 4.1  shift  would  s p e a k i n g , t h e H a m i l t o n i a n f o r an i s o l a t e d  i s a single  corresponding  at  are  the JT  a t low temperature and i s not c o n s i s t e n t  Strictly  introduced  that  m a i n t a i n e d and w i t h o u t t h e Lande  which puts the singlets  associated  cluster  a  using  6  the ground  cluster  may  function.  I t i s interesting  above  7  i s not  a difference  interval  a reasonable f i tt o a spectrum  0  distortion  6  of t h e Lande  f o r each  the degeneracy  1  of the  of the Cu„OX L  gives  data  states  the levels  the partition  be d i f f e r e n t .  treatment of the data  breaks  above  i n the dynamic  d i s t i n c t JT systems  t h e mean  with  t h e S'=0  degeneracy  i n energy  by i n c r e a s i n g  i s obtained  essentially  result  as d i s c u s s e d  by t a k i n g t h e  splitting.  As  in  their  i n energy. Decrease  t h e magnetism  same e f f e c t  u  and the approach  be e x p l o r e d  a s E i n P„. Though  removed by t h e JT e f f e c t become  cluster,  o f t h e S'=1  a p p l y . The d i s t o r t i o n  triangle  Hamiltonians H  233 and  H  cannot  fc  those  arise  non-magnetic  essential  from  internal  interactions  feature  of  the  solid  effects;  t h e y must a r i s e  between c l u s t e r s state.  Thus H  which  and  interpreted internal  basic  of  occurs.  A  T'.  Below  ground  exhibit  situation  may  be  the  effects"  2  3  considerably  modify  discontinuities the  e.s.r.  FeCr O 2  f t  and  classic,  significant magnetic  only  discontinuities  degeneracy)  above a  static  distortion  must a l w a y s  systems.  excited  state  as  cluster  transition  to lower exist  Systems JT  of a  such as  activity  symmetry  in  and  i s i n a d o u b l e t a t =44K h a v e s u c h as  "dynamic  fluctuating  Cu„OX L 6  a  the  As  another  cubic  state  JT  range  Mossbauer  splittings,  o r d e r a b o v e T'.  studies  2 3 2  arise  even  have ever  behave  similar  observe  as  close  would  i t s limiting  be  T', in  be  that  the as  no  difficult  to  b e e n o b s e r v e d . Any  at high temperature to  in a  be  on  from  above  t h e r e may  I t may  of d i s t o r t i o n be  2 3 1  sharp  which  are observed  removal  M might  strains"  removing  transitions  o f Cu<,0 c l u s t e r s i n M(T)  JT  t o z e r o a t h i g h t e m p e r a t u r e . Even phase  in  dependence o f , f o r example  example,  crystal,  resulted  random  random  t h e b e h a v i o r a b o v e T'  slowly  short  spectra  i s that  the quadrupole  in this  ground  JT  "slowly  spectrum.  decrease  approach  i n the temperature  show t h a t  distortion  a  of concepts  i n which  0  better  m u c h m o r e c o m p l i c a t e d . S t u d i e s o f TmPOi, , i n  activity  development  T'  state  J>0  the JT  the JT  temperature  which  which  on  an  cluster.  (and has  transition  (undiluted)  c o m p o n e n t s o f p h o n o n modes t h a n the  picture  i s undistorted  temperature,  for  resolved  vibrations  The which  as  are  H. a r e t  e  from  way slow  v a l u e /3(g/2)  before  234 significant The  changes  occur.  theory of slowly  beyond magnetochemistry, encountered e.g.  idea  However  resolves  treatment  i t c a n be s e e n  the apparent  Cu„0 a n d t h e r e m o v a l  at  a l l temperatures, which  This  treatment  parameter,  the  but  suggests  the JT  this  activity clusters  f o r the dynamic  an a l t e r n a t i v e  which  activation  transition  might  simple molecular  and hence  e x p l a n a t i o n of  e, f o r m e r l y a n  as a phase  ordering,  parameter  p = { 1 - T/e  expansion  of T « e  temperature  be o b s e r v a b l e by field  type  the 'proportion  4. F i t t i n g  be e x p e c t e d  phase  }  of  treatment distorted  1 / 2  g i v e s a T i n t h e numerator  o f t h e ad hoc e x p r e s s i o n e x p ( - T / e )  qualitatively  hardly the  a static  expansion  chapter  between  of  ( s e e 5.2.2) i s g i v e n b y :  (7.4)  as  ordering, are  i n h i g h symmetry  i s the basis  as a r e  transitions,  the introduction  contradiction  c a n be i n t e r p r e t e d  ordering  Series  that  fluxionality.  o c c u r s . On a v e r y  triplet'  of s p i n - P e i e r l s  of degeneracy  also  dependent  which  X-ray,  goes f a r  model.  temperature  below  strain  o f phonon modes a n d p a r t i a l  of  distortion  random  a n d t h e same k i n d s o f p r o b l e m s  i n a thorough  the continuum  involved.  fluctuating  the data using good  on s u c h  transition  interpretation  fits.  may  o f TDF.  just  introduced i n  (7.4) g i v e s s l i g h t l y  Since perfect  a s i m p l e model  i n some c a s e s  agreement  poorer  could  i t c a n be s a i d  p r o v i d e an  that  alternative  235 In  conclusion,  lattice  would  be  t r e a t m e n t of  interpretation  and  similar  clusters.  add  little  of  4.  (most  o f ) the magnetic  distortion  model  7.3  intercluster  thesis  has  provide on  As cases,  of magnetic additional  and  temperature the absence  in detail,  more u s e f u l ,  6  as  even  i t m a k e s no  a  would  in chapters independence  of  explicit  the  dynamic  though  i t might  explicit  reference  experiments  the biggest  distortion  model  importance  of  clear  studies,  i n the course of  ions.  Mossbauer  information even  where be  single  cannot  interactions  on  lead  and  e.s.r.  effects  exchange  problem  can  which  and  dynamic  To  studies  state  genuine  be  in applying  distortions  range.  unambiguous  symmetry  and  techniques are  i s i n t h e d e t e r m i n a t i o n of  static  supplemented  t o an  in high  the ground  these  when  this  may perhaps  possible  necessary.  i n t e r c l u s t e r magnetic  temperature  Cu„OX L  such a treatment  s u s c e p t i b i l i t y d a t a , even  e x p e r i m e n t s may  non-magnetic  the  effects  'purist'  of magnetic  f l u x i o n a l i t y but  further  the  strain  room t e m p e r a t u r e X - r a y  clusters  as  increasingly  magnetic  interpretation  b e h a v i o r of  interactions.  become  that  unusual  spectra  Suggestions for further Is  by  such  is actually  a Jahn-Teller  the  the  satisfactory  d i s t o r t i o n model developed  I n some r e s p e c t s ,  describing  to  of  throughout  entirely  For magnetochemists  t o the dynamic  parameters  offend  interactions  n e c e s s a r y t o p r o v i d e an  theoretical  3 and  3-D  t h i s end  more  may  ruled the  the arise  out  dynamic  relative from  distortions low  i n most  throughout  temperature  X-ray  236 studies,  particularly  Specific phase  heat  transition,  observed  would  magnetically sensitive  below  clusters  measurements c o u l d as  in Cr 0,  and  3  indicate  whether  i n d u c e d . The  than  e o f TDF  not  specific  susceptibility t o 6.6  the  heat  t o phase  Figs.  6.1  Clusters  which  have v e r y h i g h symmetry a t  particularly  temperature symmetry  Specific information magnetic the  as  the  spectrum.  spins Thus  the  the  model and  distortion  and  low  lying  might  useful  magnetic  spectrum  was  give  I n f o r m a t i o n about  well  resolved.  clusters,  a  co-workers,  Optical  simplicity  neutron  technique which would  much  be  more  see, f o r  indeed JT-active  in principle low-lying  triplet  above a  above  has  give  quintet  To  of  of  the  Fluxionality  those  date 2 3 3  "  scattering  2 3 5  are  2 3  been p i o n e e r e d by  such  " has  not  (INS)  i n which  two  the t r a n s i t i o n s  properties  the  experiment.  be  spectroscopy  i n the  a doublet of  distinguishable.  but  some  states  singlet  directly.  would  T=0.  experiments would  the magnetic  Inelastic  a  low  c o m p l i c a t e s u c h an  measured  experiments are a v a i l a b l e .  and  the b l u r r e d  s h o u l d be  t h e most  of comparative  could  room t e m p e r a t u r e  also,  e n e r g i e s of  the q u i n t e t  triplet  of  AH  transitions,  occur above  model,  Perhaps  advantage  must  f o r Cu„0, t h e  model  any  transition  heat  i f they are  measurements can  a n t i s y m m e t r i c exchange  dynanic  for specific  transition  heat  on  intercluster  of  studies  lowering  useful.  f o r t h e Odd/Even a p p r o x i m a t i o n .  interesting  X-ray  of  is typically  example,  be  be  f o r the presence  the magnitude  or  the  test  would  the  which always  from  Gudel  and  seem more p r o m i s i n g f o r , s a y , t h e C u „ O X L „ 6  and  237 Fe 0(RCOO) 3  systems.  + 6  cubane  type c l u s t e r s  purely  isotropic  experiment,  and found  exchange  INS m i g h t  fluxionality As  2 3 6  give a  clusters  a r e doped  lattice,  perhaps  Dilution  in  paramagnetic  JT s y s t e m s  ;  into  clusters  2 3 8  "  as i n the pure  the  likely,  7.4  a reversal  of a cross  with  i n which the  2 3 7 6  out  ,  would  routinely have  linked  for Cu OCl (TPPO) 4  6  temperature  distortion, possibly  was  6  a n d t h e 'hump'  indicates  either  associated  i n the very dissimilar  i n the sign  to the  host  with  or, less  of J .  Conclusion (1) H i g h  highly exhibit  symmetry c l u s t e r s  sensitive  to small  dynamic  distortion  with  changes  3  3  6  distortions exhibit  2  associated  either  dynamic  ground  with  which  be  may  pathways which a r e configuration modify,  as i n  a s i n Cu^OXgL,,, a n y  non-magnetic  or excited  d i s t o r t i o n may  exchange  in cluster  effects  Cr 0(CH COO) C 1 . 6H 0, o r d o m i n a t e ,  the  timescale  clusters  6  c o m p o u n d was a b s e n t . T h i s  p a c k i n g of the c l u s t e r  with  triplet  such as Mg„OBr  a  moment  found  static,  u  non-JT-active host  Cu«0X L  the pores  the e f f e c t i v e  perhaps  .  2 3 9  to decrease monotonically  larger,  about  experiments are c a r r i e d  found  a  short  d i l u t i o n experiments,  of s i m i l a r  Cu„O  consistent  are essential  i n a non-magnetic,  been doped  results  t o some  systems.  between c l u s t e r s  of i n t e r e s t .  1 0 5  to give  new i n f o r m a t i o n  be  polystyrene  been a p p l i e d  As a r e l a t i v e l y  6  d i s t o r t i o n model,  already  .  i n the Cu OX L„  interactions  dynamic  INS h a s a l r e a d y  state  'frozen  effects.  Jahn-Teller out*  Such  static clusters  activity  and  a t low temperature t o  238 yield  a symmetry-reduced (2) G r o u p t h e o r y  produce a powerful of  technique  distortions  (3)  considerably Heisenberg  reduces  The  3  exchange and magnetic  chains  dynamic approach may  exhibit  formalism  t o exchange  transition.  an  remaining,  adequate  such  as  intercluster  temperature.  linear  systems  i s complicated  ( i i )the systems.  difficulties These  by a p p r o x i m a t i o n s  f o r S>l/2.  such  extrapolation  The a p p l i c a t i o n  to linear chains  striction  be a s i g n i f i c a n t f a c t o r  or  f o r which  to provide  and even n u m e r i c a l  i s unreliable  static  theory  ferromagnetic  of i n f i n i t e  are not circumvented  distortion  spin-Peierls  fails  of J a h n - T e l l e r a c t i v i t y ,  Odd/Even a p p r o x i m a t i o n  either  o f known c l u s t e r s  in infinite  i n the mathematics  difficulties  distortion  o r d e r i n g a t low  situation  ( i ) the lack  5 2 4  Hamiltonians  symmetry.  t h e number  3  the spin exhibit  few a n o m a l o u s c l u s t e r s 4  (4) The  which  of the dynamic  Ni„(OCH )„(acac) (CH OH)  finite  high  i n t r a c l u s t e r exchange  explanation.  inherent  for solving  clusters  from  Application  structure.  a n d f a c t o r i s a t i o n c a n be c o m b i n e d t o  i s o t r o p i c a l l y coupled  dynamic  by  (anti)ferrodistorted  leads  and a s u g g e s t i o n  in alternation  from of  to a  t h a t end  phenomena  as the  such  new effects as the  239  REFERENCES 1.  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