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Proton magnetic resonance in paramagnetic and antiferromagnetic single crystals of CoCl₂.6H₂O Sawatzky, Erich 1960

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PROTON MAGNETIC RESONANCE IN PARAMAGNETIC AND ANTIFERROMAGNETIC SINGLE CRYSTALS OF CoCl « 6H 0 2  2  by Erich B.Sc,  Sawatzky  U n i v e r s i t y of B r i t i s h Columbia, 1958.  A t h e s i s submitted i n p a r t i a l f u l f i l m e n t of the requirements f o r the degree of Master of S c i e n c e i n the Department of Physics  We accept t h i s t h e s i s as conforming t o the required standard  The U n i v e r s i t y of B r i t i s h A p r i l , 1960  Columbia,  In the  presenting  requirements  f o r an  of  B r i t i s h Columbia,  it  freely available  agree that for  that  copying  gain  shall  advanced degree a t  or  not  his  Department o f  f o r reference  and  study.  I  for extensive  /3  be  #/>r/l;  copying of  granted  representatives.  the  It i s  of t h i s t h e s i s  a l l o w e d w i t h o u t my  VfiLfSICS  #60.  by  Columbia,  of  University shall  The U n i v e r s i t y o f B r i t i s h Vancouver Canada.  Date  the  Library  publication be  fulfilment  the  p u r p o s e s may  o r by  in partial  I agree that  permission  scholarly  Department  this thesis  make  further this  Head o f  thesis my  understood  for financial  written  permission.  ii  Abstract Standard r a d i o - f r e q u e n c y n u c l e a r resonance s p e c t r o s c o p y techniques have been a p p l i e d t o study the f i n e s t r u c t u r e of the  p r o t o n magnetic resonance a b s o r p t i o n l i n e i n s i n g l e  c r y s t a l s of CoC^'SI^O.  Cobaltous C h l o r i d e i s a paramagnetic  c r y s t a l a t h i g h temperatures and becomes a n t i f e r r o m a g n e t i c a t about 2.29°K.  The p o s i t i o n and number of "lines s t r o n g l y  depend on temperature and on the d i r e c t i o n of the e x t e r n a l l y a p p l i e d magnetic f i e l d .  Fewer l i n e s than the t h e o r e t i c a l  number o f twenty-four were always observed. At  room temperature the p r o t o n resonance a t 12 Mc/sec.  i n a f i e l d of 2.82 K gauss c o n s i s t s of a s i n g l e l i n e about six  gauss wide.  A s p l i t t i n g of t h i s l i n e i n t o a maximum of  six  components has been observed a t l i q u i d helium temperature.  The maximum o v e r a l l s e p a r a t i o n a t 4.2°K i s about 110 gauss. For  each d i r e c t i o n of the e x t e r n a l l y a p p l i e d magnetic  field  the  s e p a r a t i o n between the l i n e s i n c r e a s e s w i t h d e c r e a s i n g  temperature. The t r a n s i t i o n temperature i s measured and e f f e c t s due to  s h o r t - r a n g e order above the t r a n s i t i o n a r e observed. T h e o r e t i c a l formulae f o r the p o s i t i o n s o f the component  Tines a r e developed by c o n s i d e r i n g the two-proton s p i n system w i t h i n a water molecule o f h y d r a t i o n immersed i n the —>  homogeneous e x t e r n a l f i e l d H  and the inhomogeneous txme-  iii  averaged f i e l d of the cobalt ions* Measurements in the antiferromagnetic state have been partially completed.  iv  T a b l e of Contents Page Abstract  i i  L i s t of I l l u s t r a t i o n s  v  Acknowledgements  v i i  Chapter 1  -  Introduction  1  Chapter 2  -  Theory U n d e r l y i n g The Experiment  7  Chapter 3  -  Apparatus And E x p e r i m e n t a l Procedure  Chapter 4  -  Results  13  A.  -  The C o C l » 6 H 0 C r y s t a l  18  B.  -  D i s c u s s i o n Of E x p e r i m e n t a l O b s e r v a t i o n s  20  Introduction  20  Measurements In The Paramagnetic S t a t e  22  (a)  H  Q  In The Plane J- a - a x i s  22  (b)  H  Q  In a-c Plane  25  i. ii.  iii. iv.  2  2  T r a n s i t i o n Temperature  Measurements  Measurements In The A n t i f e r r o m a g n e t i c S t a t e  Bibliography  35 37 39  V  L i s t of  Illustrations  to f o l l o w Page Fig.  1  L e v e l of O s c i l l a t i o n  Fig.  2  A General  Fig.  3  B l o c k Diagram of the Apparatus  13  Fig. 4  Photographs of Apparatus  13  Fig.  5  Arrangement of C r y s t a l Mount  14  Fig.  6  Photograph of Three-Dimensional  L i n e and  at Frequency ( 0  o  i t s Derivative  2 3  C r y s t a l Model  19  Fig.  7  Proton Spectrum at 78°K  20  Fig.  8  Proton Spectrum at 4.2°K  20  Fig.  9  21  Fig.  10  R e l a t i o n s h i p between Recorded Spectrum and i t s I n t e g r a t e d L i n e Shape R o t a t i o n at 78°K w i t h Tf i n plane P e r p e n d i c u l a r t o the a - a x i s  22 25  0  Fig.  11  R o t a t i o n at 78°K w i t h H  Fig.  12  R o t a t i o n at 4.2°K w i t h I f i n a-c Plane  25  Fig.  13  F i e l d P a t t e r n of a D i p o l e  25  Fig. Fig.  14 15  T h e o r e t i c a l Line F i e l d Dependence of S p l i t t i n g at 78°K w i t h H" i n the Plane P e r p e n d i c u l a r t o the a - a x i s  27  Q  i n a-c Plane 0  q  at 78°K  27  Fig.  16  F i e l d Dependence of S p l i t t i n g w i t h H" i n the a-c Plane o  Fig.  17  Proton Spectrum at 2.5°K  35  Fig.  18  T r a n s i t i o n Temperature  35  27  vi to f o l l o w Page F i g . 19  Proton Spectrum a t 2.25°K  36  F i g . 20  Proton Spectrum a t 2.21°K  36  F i g . 21  Proton Spectrum at 1.52°K  37  F i g . 22  R o t a t i o n at 1.52°K w i t h H i n a-c Plane  37  vii  Acknowledgements The work d e s c r i b e d i n p a r t by r e s e a r c h  i n t h i s t h e s i s has been supported  grants  t o Dr. M. Bloom and Dr.  G.M.  V o l k o f f from the N a t i o n a l Research C o u n c i l of Canada and through the award of a N a t i o n a l Research C o u n c i l ship  Student-  (1959-60). To Dr. M. Bloom, who suggested and s u p e r v i s e d  research,  this  I wish t o express my s i n c e r e a p p r e c i a t i o n f o r  h i s constant  i n t e r e s t , many i l l u m i n a t i n g d i s c u s s i o n s , and  f o r h i s i n v a l u a b l e h e l p i n i n t e r p r e t i n g the r e s u l t s . I a l s o wish t o express my a p p r e c i a t i o n t o Dr. Volkoff f o r c r i t i c a l l y reading  the manuscript of t h i s  thesis. My thanks a r e a l s o due t o Mr. W. Morrison, constructed  the magnet  support.  G.M.  who  Chapter 1 Introduction The n u c l e a r magnetic resonance technique p r o v i d e s a powerful method of s t u d y i n g the i n t e r a c t i o n s between atomic n u c l e i and t h e i r magnetic environment atures.  both a t h i g h and low temper-  The work d e s c r i b e d i n t h i s t h e s i s r e p r e s e n t s a  p r e l i m i n a r y survey of the p r o t o n magnetic resonance i n s i n g l e c r y s t a l s o f CoClg'SHgO i n the paramagnetic and a n t i - f e r r o magnetic phases o f t h i s substance.  R e s u l t s o b t a i n e d here  s h a l l s e r v e as a guide f o r more d e t a i l e d i n v e s t i g a t i o n s of hydrated c o b a l t o u s c h l o r i d e planned f o r the near f u t u r e . Chapter 2 o f t h i s t h e s i s i s a summary of the t h e o r y u n d e r l y i n g the e x p e r i m e n t a l work t o be d e s c r i b e d and Chapter 3 i s a d e s c r i p t i o n of the e x p e r i m e n t a l apparatus.  Chapter 4 i s  a r e p o r t of the experimental o b s e r v a t i o n s on s i n g l e  crystals  of CoClg'BHgO i n e x t e r n a l magnetic f i e l d s up t o 3200 gauss and a t temperatures down t o 1.52°K.  The e x p e r i m e n t a l d a t a  so o b t a i n e d y i e l d i n f o r m a t i o n on both the c r y s t a l  structure  and the e l e c t r o n i c wave f u n c t i o n s of the atoms comprising the c r y s t a l , and b r i n g out some of the d i f f i c u l t i e s  encountered  i n paramagnetic c r y s t a l s w i t h a r e l a t i v e l y l a r g e number of water molecules o f h y d r a t i o n . In g e n e r a l , i f a sample c o n t a i n i n g n o n - i n t e r a c t i n g n u c l e i with spin I  0 and magnetic moment "j~T i s p l a c e d i n a  u n i f o r m magnetic f i e l d H , the n u c l e i can each assume a maximum Q  -2-  number of 21 + 1 o r i e n t a t i o n s w i t h r e s p e c t t o H . q  In t h i s  work o n l y the resonance spectrum of the protons i n the water molecules of h y d r a t i o n i s s t u d i e d . proton i s I = \  y  S i n c e the s p i n of a  o n l y two o r i e n t a t i o n s are p o s s i b l e ,  giving  r i s e t o two d i f f e r e n t energy l e v e l s s e p a r a t e d by 2p,pH . Q  T r a n s i t i o n s between these energy l e v e l s may externally applied r - f f i e l d angular frequency (jj  a t r i g h t angles to H  = 2u. H / h .  Q  be induced by an Q  and of  The c o i l around the sample  i s p a r t of the resonant c i r c u i t of an o s c i l l a t i n g d e t e c t o r . T h i s c o i l produces the d e s i r e d r - f f i e l d H^, s o r p t i o n experiments  and i n ab-  i t u s u a l l y a l s o s e r v e s as the pick-up  coil. Resonance may  be observed by m o n i t o r i n g the l e v e l of  o s c i l l a t i o n of the r - f o s c i l l a t o r as i t s frequency i s v a r i e d . A d i p i n the l e v e l of o s c i l l a t i o n r e s u l t s when t r a n s i t i o n s are  induced between the n u c l e a r Zeeman l e v e l s , s i n c e then  energy i s absorbed from the r - f f i e l d .  As d e s c r i b e d l a t e r ,  t h i s resonance a b s o r p t i o n , though peaked a t the c l a s s i c a l Larmor frequency (n) = u.H /In, occurs over a range o f frequenQ  0  c i e s as i n d i c a t e d s c h e m a t i c a l l y i n f i g u r e It  i s o f t e n convenient, f o r s i g n a l - t o - n o i s e c o n s i d e r -  a t i o n s , t o modulate the magnetic sweeping  1.  f i e l d p e r i o d i c a l l y while  the o s c i l l a t o r frequency through the resonance,  and  to use the method of phase s e n s i t i v e d e t e c t i o n i n r e c o r d i n g the r - f l e v e l of o s c i l l a t i o n .  With a modulation  amplitude  s m a l l e r than the l i n e w i d t h the d e r i v a t i v e of the resonance  ^  Fig. I.  D i p in  the f r e q u e n c y fhrough  the  level of the Larmor  of  oscillation  oscillator frequency  60  as  passes 6Jo .  fo follow page 2.  -3-  l i n e i s observed as i n d i c a t e d s c h e m a t i c a l l y The  simple p i c t u r e of n o n - i n t e r a c t i n g  above i s never s t r i c t l y  true.  in figure nuclei  Interactions  2.  outlined  w i t h the  sur-  r o u n d i n g magnetic moments are always p r e s e n t , although i n l i q u i d s and ably,  the  l i n e widths u s u a l l y b e i n g determined by  geneities  i n the e x t e r n a l l y a p p l i e d magnetic  In the We  gases these i n t e r a c t i o n s are averaged  inhomo-  field.  case of s o l i d s the p i c t u r e changes  s h a l l consider only c r y s t a l l i n e s o l i d s .  consider-  considerably.  Here a l l n u c l e i ,  except f o r t h e i r thermal v i b r a t i o n s , are s i t u a t e d i n f i x e d p o s i t i o n s and  each nucleus e x p e r i e n c e s i n a d d i t i o n t o  externally applied f i e l d s H due  Q  and  a l o c a l magnetic  to the n e i g h b o u r i n g magnetic d i p o l e s .  Q c a  j |c=- 1 0 0 0 gauss;  n u c l e i i s u s u a l l y not n u c l e i and  the  be  l o c a l f i e l d due  l a r g e r than about 20 gauss.  paramagnetic i o n s are each o r i e n t e d  2S + 1 d i f f e r e n t ways r e s p e c t i v e l y , the  field  I f the c r y s t a l  c o n t a i n s paramagnetic i o n s , t h i s l o c a l f i e l d may order of J H ^  the  of  the  to other Since  the  i n 21 + 1  and  f i e l d produced  by  the s u r r o u n d i n g s at the s i t e s of d i f f e r e n t n u c l e i i n a u n i t c e l l may  v a r y between about + ( H 2  In the o n l y one  0 c a  j { and  - jH^  o c a  i| .  ease of non-paramagnetic s i n g l e c r y s t a l s w i t h  type of n u c l e a r magnetic moment present  (those of  the waters of c r y s t a l l i z a t i o n ) Pake* showed t h e o r e t i c a l l y and  observed e x p e r i m e n t a l l y t h a t the p r o t o n resonance  is split  i n t o two  components by  the d i p o l e - d i p o l e  line  interaction  between the p r o t o n - p a i r i n the water molecule. The  separation  of the  l i n e s i n any  given observation  (a)  Ahf Modulation amplitude  Fig. Z. F i e l d m o d u l a t i o n A H smaller t h a n line w i d t h ( a ) , derivative of line (b) .  t o follow p a g e 3  -4-  a l s o depends on the o r i e n t a t i o n of the c r y s t a l w i t h r e s p e c t —»>  to  the e x t e r n a l f i e l d H . Q  crystal, we^in  I f the sample i s a paramagnetic  a d d i t i o n t o the i n t e r a c t i o n s between the  n u c l e a r d i p o l e s and the e x t e r n a l f i e l d s and between the p r o t o n s themselves, the i n t e r a c t i o n s between t h e protons and the  e l e c t r o n i c magnetic moments of the paramagnetic  ions.  T h i s i n t e r a c t i o n g i v e s r i s e t o a d d i t i o n a l decomposition o f the  p r o t o n resonance l i n e .  Such f i n e s t r u c t u r e s of the  p r o t o n magnetic resonance l i n e were t r e a t e d  theoretically 2  and observed e x p e r i m e n t a l l y by N. Bloembergen and by N.J. P o u l i s ' 3  4  i n CuSO^'SHgO  i n CuCl «2H 0. 2  2  When o b s e r v i n g the p r o t o n resonance i n paramagnetic c r y s t a l s we would expect a v e r y broad l i n e because o f the l a r g e magnetic i n t e r a c t i o n s between t h e protons and the p a r a 5  magnetic i o n s .  But i t can be shown  t h a t i f exchange f o r c e s  between t h e magnetic i o n s a r e p r e s e n t , t h i s broadening a c t i o n of  the magnetic i o n s may be c o n s i d e r a b l y reduced.  of  C o C l * 6 H 0 such exchange i n t e r a c t i o n s a r e p r e s e n t and a t 2  In the case  2  room temperatures we observe a s i n g l e l i n e about s i x gauss wide. of  As a f i r s t approximation the temperature dependence  the resonance spectrum may be o b t a i n e d from t h e C u r i e law,  which s t a t e s t h a t a t h i g h temperatures the mean m a g n e t i z a t i o n of  a magnetic i o n i n a f i e l d I? a t temperature T i s g i v e n 0  by <H> - M. H /3kT 2  Q  where the average i s over time.  T h i s mean m a g n e t i z a t i o n g i v e s  -5-  r i s e to a time-average l o c a l f i e l d which depends s t r o n g l y on the  space c o o r d i n a t e s i n the c r y s t a l  of the magnetic moments.  and on the o r i e n t a t i o n  The energy l e v e l s  of the proton  magnetic moments are determined by the v e c t o r sum of t h i s l o c a l f i e l d with H .  However, i n paramagnetic resonance work,  q  the  local field  i s u s u a l l y much s m a l l e r than H ,  s i d e r o n l y the component i n the d i r e c t i o n field. the  Thus a d i f f e r e n t  different  sites  the  Q  of H  t o t a l magnetic f i e l d  Q  con-  of the l o c a l  i s produced at  i n a u n i t c e l l , and the d i f f e r e n t  i n the u n i t c e l l have d i f f e r e n t that H  and we  Q  Larmor f r e q u e n c i e s .  protons Assuming  i s c o n s t a n t over the sample, the l o c a l f i e l d w i l l  same f o r c o r r e s p o n d i n g protons i n d i f f e r e n t  S i n c e the l o c a l f i e l d  unit  be  cells.  i n c r e a s e s w i t h d e c r e a s i n g temperature,  the  resonance l i n e s p l i t s i n t o a number of component l i n e s  the  temperature i s lowered.  as  T h e r e f o r e , the number of com-  ponents observed depends on temperature, on the number of water molecules i n the u n i t c e l l and on the degree of symmetry possessed by the The i n t e r n a l  crystal. f i e l d at a p r o t o n due t o a magnetic i o n a  d i s t a n c e r away i s of order of magnitude (p.)/r . r-dependence will  i s an i n v e r s e cube, o n l y the near neighbours  have any profound i n f l u e n c e on the s p l i t t i n g and the  shape of the l i n e s . the  S i n c e the  Taking r = 2 x 10~  8  cm and H  Q  = 3000 gauss,  s p l i t t i n g at 300°K i s about 1 gauss, a t 78°K about 4 gauss,  and a t 4°K about 70 gauss.  We  t h e r e f o r e do not expect any  r e s o l u t i o n of the resonance l i n e at room temperature.  -6Theory predicts that CoCl2*6H20 has a possible maximum number of 24 component lines each of which should be several gauss wide, so that we cannot expect a complete resolution in a f i e l d of 3000 gauss even at liquid helium temperatures. The maximum number of 24 lines was never observed.  At 4.2°K  only six lines were found. In order to calculate the positions of the lines as a function of crystal parameters, temperature, and the external f i e l d H , degenerate perturbation theory is applied to the Q  Hamiltonian describing the system.  To simplify matters, some  approximations are made, since some of the terms in the Hamiltonian are much smaller than others. are summarized in Chapter 2.  These calculations  Chapter 2 Theory T h i s p a r t of the t h e s i s c o n s i s t s of a summary of the t h e o r y of  steady s t a t e n u c l e a r resonance s p e c t r o s c o p y as a p p l i e d t o  paramagnetic c r y s t a l s .  Although none of i t r e p r e s e n t s o r i g i n a l  work by the author, n e v e r t h e l e s s , i t s i n c l u s i o n i s r e q u i r e d t o i n t e r p r e t the e x p e r i m e n t a l work p r e s e n t e d i n Chapter.4. We  c o n s i d e r a paramagnetic c r y s t a l w i t h one or more water  molecules of h y d r a t i o n whose c r y s t a l s t r u c t u r e i s a t l e a s t p a r t i a l l y known.  From X-ray i n v e s t i g a t i o n s f o r i n s t a n c e , the  oxygen p o s i t i o n s can be determined, but not the p r o t o n p o s i t i o n s . Both the p r o t o n magnetic moments w i t h s p i n I = |- and the e l e c t r o n i c magnetic moments of the paramagnetic ions w i t h s p i n S produce l o c a l magnetic f i e l d s throughout the c r y s t a l .  The  magnitude and d i r e c t i o n of t h i s l o c a l f i e l d a t any g i v e n p o i n t depend on the o r i e n t a t i o n and s e p a r a t i o n of the moments at any g i v e n time, s i n c e both moments p r e c e s s about the e x t e r n a l magnetic f i e l d H . Q  Thus we have a system of protons immersed  i n the homogeneous e x t e r n a l f i e l d H  Q  and the r a p i d l y v a r y i n g  inhomogeneous f i e l d produced by the paramagnetic i o n s .  The  e n t i r e H a m i l t o n i a n d e s c r i b i n g t h i s system may be w r i t t e n i n the (1)  form H = - I p  k  . g . H k  Q  +  H  s g  + H  S e K  + H  S I  + H  n  -Z  Y  . h I.-"H  Q  -8The quantities g  k  and |3 represent the g-factor of cobalt written  as a tensor and the nuclear Bohr magneton respectively.  The  f i r s t term in H represents the Zeeman energy of the paramagnetic ions in the external f i e l d H , Hgs the magnetic interaction g Q  between the paramagnetic ions themselves, H interaction between them;  e x  the exchange  Hgj is the magnetic interaction be-  tween the paramagnetic ions and the proton moments, H J J the magnetic interaction between the protons themselves, and ^  I-JHQ  represents the magnetic interaction between the  proton moments and the external f i e l d H . The proton magnetic Q  moment is denoted by y f i f , where I is the spin operator and y the gyromagnetic ratio. magnetic resonance work spins clearly separated.  This notation is customary in nuclear and keeps the nuclear and electron The term in equation (1) connecting  the two spin systems is Hgj and may be written (2)  «  3 I  - ^ j r j f i l j - l l k _ SYitKfj'r^Xprk'rjk ) I r r ± k  i k  3  i k  5  where pt is the magnetic moment of the kth magnetic ion and fc  Tjjj. is the radius vector connecting the i*h proton and the kth i  o  n an(  j  r  ^ » |*"ik| •  B u  * Mlc varies rapidly in time due  to the exchange coupling between the magnetic ions represented g by H  e x  .  The exchange interaction causes a pair of antiparallel  spins to f l i p simultaneously, i . e . i f two spins are oriented as Tt|r and one reverses direction, the other also f l i p s due to the exchange coupling.  The exchange frequency is approxi-  -9-  mately g i v e n by h ^ For C o C l ' 6 H 0 , 2  2  e  —  x  = kT , N  where T  2°K, so t h a t \ >  i s the Neel  N  e x  -5 x 10  1 0  temperature.  cycles/sec.  The Larmor frequency f o r protons i n a f i e l d of 3000 gauss i s 12.8  Mc/sec. so t h a t the protons cannot f o l l o w the r a p i d  v a r i a t i o n s of the l o c a l f i e l d due t o the c o b a l t i o n s . the protons see o n l y the time-average We  Thus  f i e l d of the c o b a l t  ions.  t h e r e f o r e can use the time average ^ p t ^ of ji^. i n equation g  (2).  We  s h a l l a l s o n e g l e c t the term Hgg  compared w i t h H  ,  e x  s i n c e the exchange energy between a p a i r of c o b a l t i o n s i s about  100 times g r e a t e r than t h e i r magnetic  average m a g n e t i z a t i o n ( p T ^ of the kth  energy.  The  time-  c o b a l t i o n can now  be  c a l c u l a t e d i n p r i n c i p l e from the reduced H a m i l t o n i a n f o r the 7  c o b a l t system by the d i a g o n a l sum method d e s c r i b e d by Van V l e c k . The e f f e c t of the time-average p r o t o n resonance  magnetization  p.  on the  k  i s o b t a i n e d from the H a m i l t o n i a n f o r the  proton s p i n system, w i t h T^  k  r e p l a c e d by  in H .  Written  g I  out i n f u l l , (3)  H  the proton H a m i l t o n i a n becomes; :<?k>_ di^ik)^k> = -^<y.nl. . l L +S-W.n x'x i o xk) x rxk ik T  , i ?  3  r  3  r  ik)  5  S i n c e the d i p o l e - d i p o l e i n t e r a c t i o n i s p r o p o r t i o n a l t o  j-, r  the most important terms i n the p r o t o n - p r o t o n  iJ  inter-  a c t i o n are those r e p r e s e n t i n g the c o u p l i n g between n e a r e s t  -10-  neighbours.  T h e r e f o r e , o n l y i n t e r a c t i o n s between the p r o t o n  p a i r i n the same water molecule  are c o n s i d e r e d .  The  inter-  a c t i o n s w i t h other protons and the time dependent p a r t of the f i e l d due t o the c o b a l t i o n s c o n t r i b u t e o n l y t o the l i n e  widths  of  im-  the component l i n e s .  We  now  have a two-proton system  mersed i n the homogeneous f i e l d H f i e l d of the c o b a l t i o n s .  and the s t a t i c inhomogeneous  Q  To f i n d the p o s i t i o n of the  resonance  l i n e s , equation  (3) must be s o l v e d f o r i t s e i g e n v a l u e s which  g i v e the energy  l e v e l s of the system.  Using t h i s  o b t a i n s f o r the energy  l e v e l s to f i r s t  Hamiltonian,  2 N. Bloembergen  —yn H  *1 E  2  E  3  E  4  (4)  + a + d  Q  -d + V b  + d~  2  2  -d - fb  +  2  +yh H  Q  order:  d  2  - a + d  where  (5)  a  ^ y f i (H* +  H)  b  h*  H)  d  -I/4 Y f c ^<M*k> k  "4  -  0 £ 2  2  2  ( 1  2  (1 -  3cos 0  -  ° i k > T k \/  2  3 c o s 2  r  1 2  )r^| ( i  =  1  '  2 )  i  and Et and EZ are the z-components of the f i e l d produced  by  the  c o b a l d i o n s a t protons 1 and 2 r e s p e c t i v e l y when 1 i s chosen p a r a l l e l to H . 0  1  of  a and b are f u n c t i o n s of temperature  by  virtue  2  H£ and H^ which are c a l c u l a t e d from H$i w i t h p.y, r e p l a c e d by  -11 •  The energy levels of (4) give rise to four transition  frequencies as follows:  (6)  h  x  = yhH - a + 2d + ^ b + d  h  2  = yhH - a - 2d - ]f ^  h  3  = hH  0  - a + 2d - J / V + J  h  4  = hH  0  - a - 2d +  2  0  + cf*  0  Y  Y  2  l  j/b  x  with corresponding intensities (b-d- tyb  2  +  2  I, = I - - — 5 — ^ , ,,• 1 2 b + d - b/b*+cf 9  (7)  d) irr  2  (b-d + /b +ci ir  I s  1 4  and  2  z  >  2  b +d +b f ¥ W 2  2  We thus have a maximum of four lines for each proton pair in every water molecule in the unit c e l l .  The maximum number of  proton lines in CoClg'SB^O should therefore be 4 x 6 (number of H2O molecules per formula) -f- 2 (because of reflection symmetry of the unit cell) x 2 (number of formula units per unit c e l l ) , which gives 24 lines. In this chapter we have followed the usual description of magnetic fields at nuclear positions in paramagnetic crystals and neglected the contribution to average magnetic f i e l d due to the average magnetization M per unit volume. If contributions of this sort are important as we later see that they are, the external f i e l d H which appears in this Q  -12chapter should everywhere be replaced by B* =  HQ +  (4TJ—N) I f ,  where N, the demagnetization factor for the crystal being 8 studied, must be calculated from the geometry of the crystal • If the crystal is not an e l l i p s o i d , M w i l l be a function of position in the crystal and this non-uniformity w i l l be equivalent to an inhomogeneity in the applied f i e l d in that i t w i l l also contribute to the nuclear resonance line width.  -13-  Chapter 3 Apparatus and E x p e r i m e n t a l Procedure. A s t a n d a r d steady s t a t e n u c l e a r resonance spectrometer used f o r the work r e p o r t e d i n t h i s t h e s i s .  A b l o c k diagram of  the apparatus i s shown i n f i g u r e 3, and photographs The o s c i l l a t i n g d e t e c t o r used was  was  in figure  4.  a s l i g h t l y modified 9  v e r s i o n of a c i r c u i t of Watkins and Pound .  The o r i g i n a l  circuit  i s d e s c r i b e d i n d e t a i l i n r e f e r e n c e 9, and the m o d i f i c a t i o n s by H.H.  Waterman  10  and s h a l l not be d e a l t w i t h here.  The narrow  band a m p l i f i e r and phase s e n s i t i v e d e t e c t o r were b u i l t and d e s c r i b e d by H.H.  Waterman . 10  fully  The remainder of the apparatus i s  s t a n d a r d equipment and s h a l l not be d e s c r i b e d , except f o r i t s use i n t h i s work. The l a r g e magnetic f i e l d H^, was  s u p p l i e d by an a i r - c o o l e d  i r o n - c o r e electromagnet manufactured by Newport instruments Co., which has f o u r i n c h diameter plane p o l e t i p s w i t h a d j u s t a b l e air-gap.  The c r y o s t a t was  c o u l d not be l e s s than 3.2  of such dimensions t h a t the a i r - g a p cm.  With t h i s a i r - g a p and a f i e l d  of 3000 gauss the homogeneity was measured t o be about 0.3 per cm.  gauss  The magnet i s mounted on a r o t a t i n g t a b l e p r o v i d e d w i t h  a c i r c u l a r b r a s s s c a l e graduated i n degrees and w i t h a v e r n i e r arranged t o measure the magnet o r i e n t a t i o n t o one t e n t h of a degree.  The whole support was  c o n s t r u c t e d i n such a way  the magnet can c o n v e n i e n t l y be r o t a t e d through 360  that  degrees  Freq. Meter  iKl  Oscill. D etector  Narro w Band Amplifier  Radio Received  Phase < Sen sitive Detector  ^ Recording ^Milliameter  Phase Shifter  Audio  c  Broad Band Amplifier  Motor  Oscill. D e l  Sample  Oscillator  Scope Field  Modulqtion  Williamson Amplifier  El e c t r o M agnet Modulation Coils  Fig.3. B L O C K  M agnet Power  Vacuum Pump  DIAGRAM OF APPARATUS.  Figure 4 Photographs of Apparatus to follow page 13  14-  without i n t e r f e r i n g w i t h the r e s t of the apparatus* C y l i n d e r s about cut  from f a i r l y  1.5  cm long and 0.8  cm i n diameter were  l a r g e c r y s t a l s , which were of the same type  d e s c r i b e d by P. G r o t h . 1 1  During t h i s o p e r a t i o n g r e a t care  was  taken t o make one of the c r y s t a l axes p a r a l l e l t o the a x i s of the c y l i n d e r . difficult  The p r e p a r a t i o n of the c r y s t a l s proved t o be a  t a s k , mainly because CoClg'SI^O c r y s t a l s have a low  m e l t i n g p o i n t (86°C) and are q u i t e b r i t t l e .  Two  such  cylinders  were c u t , one w i t h the a - a x i s and the other w i t h the b - a x i s p a r a l l e l t o the c y l i n d e r a x i s .  In each case the c r y s t a l  was  a t t a c h e d t o the end of a German s i l v e r tube 1 cm i n diameter and 80 cm l o n g .  The c y l i n d e r s were mounted i n such a way  t h e i r axes were p a r a l l e l t o the a x i s of the German s i l v e r The r - f c o i l was a filling was  that tube.  wound d i r e c t l y on the c r y s t a l t o g i v e as h i g h  f a c t o r as p o s s i b l e .  The number of t u r n s i n the  coil  chosen t o g i v e the r - f o s c i l l a t o r a frequency range of  about 7.5  Mc/sec. (7 Mc/sec. - 14.5  Mc/sec.).  When a l i g n e d ,  the c r y s t a l w i t h c o i l and the end of the German s i l v e r  tube  were imbedded i n " P l a s t i c Wood" and then coated w i t h g l u e t o i n s u r e a permanent mount (see f i g . 5 ) .  During the mounting  the d i r e c t i o n of the a x i s p e r p e n d i c u l a r t o the c y l i n d e r (hence t o one of the c r y s t a l axes) was  axis  marked on the b r a s s  p i e c e a t the top end of the German s i l v e r tube, so t h a t the o r i e n t a t i o n of the c r y s t a l i s known w i t h r e s p e c t t o the p o r t i n g frame, and hence the magnetic  —^  field H . Q  The  sup-  angular  Co-ax  connector  Koi/ar seal  Brass holder fftfing into  dewar cap.  O-ring  Lucite spacer  der man silver tube. Central lead  I n n e r dew/ar  Crystal  with coil  "Plastic Wood" Fig. 5 . A r r a n g e m e n t of c r y s t a l mount,  to follow p a g e M-  -15-  o r i e n t a t i o n of the c r y s t a l c y l i n d e r i n the plane of r o t a t i o n of the  magnet i s not v e r y c r i t i c a l , but the d i r e c t i o n of the  c y l i n d e r a x i s s h o u l d be as n e a r l y p e r p e n d i c u l a r t o H A d e v i a t i o n of t h i s a x i s from the i d e a l p o s i t i o n may s p u r i o u s l i n e s i n the resonance spectrum. the  I t was  as p o s s i b l e .  Q  introduce  estimated that  c y l i n d e r a x i s was v e r t i c a l t o w i t h i n about two degrees. The c r y o s t a t was  a common double dewar system.  The  sup-  p o r t i n g frame of the dewar cap was p r o v i d e d w i t h s e t screws so t h a t the German s i l v e r tube c o u l d be a d j u s t e d t o be and hence the c y l i n d e r a x i s  (a- or b - a x i s of the c r y s t a l )  always p e r p e n d i c u l a r t o the f i e l d H a d j u s t e d t o be h o r i z o n t a l ) .  vertical  Thus  Q  (the d i r e c t i o n of H  Q  was is  and H*-^ are always perpen-  d i c u l a r t o each o t h e r . Temperatures  lower than the b o i l i n g p o i n t of helium are  o b t a i n e d by pumping on the helium vapour. a t u r e o b t a i n e d was of  about 1.52°K.  The lowest temper-  The i n n e r dewar has a c a p a c i t y  about two l i t e r s and without pumping kept l i q u i d helium f o r  approximately twelve hours.  The temperature i s c o n t r o l l e d by  r e g u l a t i n g the pumping speed and i t i s measured by o b s e r v i n g the  vapour p r e s s u r e w i t h a meniscus type eathetometer. A p a i r of c o i l s are mounted on the magnet p o l e p i e c e s and  are  s u p p l i e d from a W i l l i a m s o n type power a m p l i f i e r .  The sweep  frequency i s p r o v i d e d by an a u d i o - o s c i l l a t o r which f e e d s i n t o a phase s h i f t i n g network. the  To t h i s network are a l s o connected  h o r i z o n t a l sweep of the o s c i l l o s c o p e and the r e f e r e n c e  -16-  v o l t a g e input of the phase s e n s i t i v e d e t e c t o r , so t h a t any phase r e l a t i o n s h i p between the t h r e e u n i t s i s p o s s i b l e .  The  o s c i l l o s c o p e i s used mainly f o r adjustment purposes but may be used f o r a c t u a l measurements i f o n l y the frequency of the resonance l i n e s i s sought.  When so used, the modulation  amplitude i s a c t u a l l y l a r g e r than t h e l i n e width. the  To observe  d e r i v a t i v e of the t r u e l i n e shape w i t h a r e c o r d i n g  milli-  ameter the modulation amplitude s h o u l d be l e s s than about  1/4  the  l i n e width.  Observations a t such low modulation amplitudes  are  made w i t h t h e a i d of a narrow band a m p l i f i e r and the phase  sensitive detector. oscillator a fairly  To decrease the n o i s e generated i n the long time constant i s i n s e r t e d between  phase s e n s i t i v e d e t e c t o r and r e c o r d e r r e d u c i n g the n o i s e band pass.  T h i s , however, r e q u i r e s t h a t the time of sweeping  a s i g n a l be a t l e a s t s e v e r a l times t h e time c o n s t a n t . way  through  In t h i s  i t i s p o s s i b l e t o r u n through a l i n e spectrum c o n t i n u o u s l y  and r e c o r d l i n e s s e p a r a t e d by many gauss on the same c h a r t . To o b t a i n t h e frequency a t which resonance o c c u r s , frequency markers a r e made on the c h a r t a t r e g u l a r i n t e r v a l s by mixing the  r a d i a t i o n from t h e o s c i l l a t o r w i t h t h a t of a B22-CA  frequency meter and an o r d i n a r y r a d i o r e c e i v e r . r e c e i v e r and frequency meter a r e f i r s t d e s i r e d frequency.  The r a d i o  s e t t o zero-beat a t the  As the o s c i l l a t o r frequency passes through  t h i s z e r o - b e a t , one t e r m i n a l of the r e c o r d e r i s momentarily grounded  c a u s i n g the needle suddenly t o swing t o one s i d e .  -17-  The f i e l d H is determined by placing a water sample as Q  close to the crystal as possible and measuring i t s resonance frequency with the aid of a second oscillating detector and the oscilloscope.  The f i e l d is obtained from 00  where y is well: known for protons in water.  0  = yB.  0f  The magnet current  was regulated with a highly stable Varian magnet current power supply, which kept the f i e l d constant throughout an entire helium run.  -18-  Chapter  4  Results A.  The C o C l « 6 H 0 C r y s t a l . 2  2  Large c r y s t a l s  (3 x 1.5 x 1 cm) were grown from a s a t u r a t e d  aqueous s o l u t i o n of CoClg'SHgO by s l o w l y e v a p o r a t i n g i t a t room temperature. P. G r o t h  1 2  CoC^'SI^O i s of the m o n o c l i n i c p r i s m a t i c type.  g i v e s f o r a : b : c = 1.4788 : 1 : 0.9452 w i t h  = 122°19*.  P e r f e c t cleavage occurs along the c | 0 0 l j f a c e .  X-ray s t u d i e s of the s i n g l e c r y s t a l were c a r r i e d out by J . Miguno 12 et a l .  .  These authors r e p o r t two formula u n i t s per u n i t  w i t h space group determined  as c |  - C /m.  The atomic p o s i t i o n s  2  h  cell  are g i v e n i n the f o l l o w i n g t a b l e : Kind of Atom  Position  X  y  z  Co  origin  0  0  0  CI  4(i)  .278  0  8(j)  .0288  4(j)  .275  T  °II  0 + .221 0  .175 .255 .700  A photograph of a t h r e e - d i m e n s i o n a l model of the c r y s t a l s t r u c t u r e i s shown i n f i g u r e 6. A c c o r d i n g t o the authors of r e f e r e n c e 12, the two C l ~ i o n s and f o u r water molecules Co  + +  a r e arranged o c t a h e d r a l l y about the  i o n s t o form the group  GoC^^HgO,, and the other two  -19-  waters of the formula u n i t are l o c a t e d at somewhat g r e a t e r d i s t a n c e s from the c o b a l t i o n s . " r e l a t i v e l y f r e e " waters.  These s h a l l be  termed  Hydrogen bonds of the type  Oj ••• H - O J J - H ••• Oj and Oj ••• H ••• CI seem t o form the group l i n k a g e s i n the plane p a r a l l e l t o (001), which would l e a d t o the p e r f e c t cleavage along (001) as r e p o r t e d by P. G r o t h . 1 1  Figure 6.  Photograph of three-  dimensional model of crystalstructure of CoCl »6H 0. 2  White:  Co  ++  2  ions  Small Black:  Cl~ ions  Large Black:  H 0 molecules. 2  to follow page 19  -20-  B.  D i s c u s s i o n s of Experimental Observations,  i.  Introduction. A complete  a n a l y s i s of the magnetic behaviour of CoC^'SI^O  i s beyond the scope of t h i s t h e s i s .  The work r e p o r t e d here i s  a p r e l i m i n a r y survey of the p r o t o n resonance v a r i o u s temperatures.  i n CoClg'SHgO a t  I t i s hoped t h a t the r e s u l t s o b t a i n e d  w i l l s e r v e as a guide f o r more d e t a i l e d i n v e s t i g a t i o n s f o r the near f u t u r e .  planned  These s h a l l be a c o n t i n u a t i o n and  ex-  t e n s i o n of the work r e p o r t e d here. The experimental r e s u l t s f a l l  i n t o t h r e e groups:  (a) measurements i n the paramagnetic of the phase t r a n s i t i o n ,  and  state,  (b) o b s e r v a t i o n  (c) p a r t i a l l y completed  measure-  ments i n the a n t i f e r r o m a g n e t i c s t a t e . In making the measurements the c r y s t a l was w h i l e the o r i e n t a t i o n of H  0  In one s e t of measurements H  was Q  s h a l l r e f e r to H  Q  fixed  changed by r o t a t i n g the magnet.  was  o r i e n t e d i n the a-c plane of  the c r y s t a l , w h i l e i n another s e t i t was d i c u l a r to the a - a x i s .  kept  i n the plane  In the subsequent  perpen-  d i s c u s s i o n z e r o angle  p e r p e n d i c u l a r to the a - a x i s f o r the a-c  r o t a t i o n and t o l f p a r a l l e l t o the a-c plane f o r the r o t a t i o n 0  i n the plane p e r p e n d i c u l a r to the a - a x i s . A r e c o r d i n g of the spectrum plane and a t 160° spectrum  a t 78°K w i t h H  i s shown i n f i g u r e 7.  a t 4.2°K i s shown i n f i g u r e 8.  The  Q  i n the a-c  corresponding  Similar recordings  Fig.7. Proton r e s o n a n c e s p e c t r u m in C o C I - 6 H 0 a t T = 7 8 ° K . H = 3 0 2 0 g a u s s . Ho at 160° and r o t a t i n g in a - c plane. F r e q . m a r k e d in 2 0 K c / s e c . steps. 2  2  o  to follow page 20  Fig.8. S a m e d a t a as in f i g . 8 . T = 4 . 2 ° K . Freq.in 4 0 K c / s e c steps. to follow page 2.0  -21-  were o b t a i n e d i n each case f o r angular o r i e n t a t i o n s between 0 and 180 degrees  a t 10 degree i n t e r v a l s .  The r e s u l t s of the  measurements of the angular dependence of the resonance p o s i t i o n s a r e shown i n f i g u r e s 10, 11, and 12. spond t o r o t a t i o n s w i t h H  Q  These c o r r e -  i n the plane p e r p e n d i c u l a r t o the  a - a x i s a t 78°K, and t o r o t a t i o n s w i t h H 78°K and 4.2°K r e s p e c t i v e l y .  Q  i n the a-c plane a t  The p o s i t i o n s of the l i n e s a r e  g i v e n i n the frequency s c a l e . f i g u r e s corresponds  line  Each c r y s t a l p o s i t i o n i n these  t o a c h a r t of the types i n f i g u r e s 7 and 8.  The p o s i t i o n s o f the p r o t o n l i n e s correspond t o maxima i n the a b s o r p t i o n s p e c t r a and hence t o z e r o s i n t h e i r  derivatives.  S i n c e most of the l i n e s o v e r l a p the a b s o r p t i o n curve w i l l  also  c o n t a i n minima, but o n l y the 1 s t , 3 r d , 5 t h , e t c . , z e r o s i n the d e r i v a t i v e curves r e p r e s e n t p r o t o n l i n e s  (see f i g u r e 9 ) .  Due  t o t h i s o v e r l a p p i n g of n e i g h b o u r i n g l i n e s the observed maxima are s l i g h t l y s h i f t e d from t h e i r t r u e p o s i t i o n s . have been made f o r such s h i f t s .  No c o r r e c t i o n s  Each p o i n t i n the graphs of  f i g u r e s 10, 11, and 12 corresponds  t o such a p r o t o n l i n e .  In  each of these f i g u r e s the s o l i d v e r t i c a l l i n e r e p r e s e n t s the frequency of the protons i n water  (from which H  Q  i scalculated)  and s h a l l subsequently be termed the " f r e e p r o t o n " frequency.  resonance  r e c o r d i n g milliameter. Ho in a-c plane at 160° . T * 4 . 2 ° K.  ^  a)  (b) Spectrum of (a) redrawn on rectangular scale  sec R e s o n a n c e lines c o r r e s p o n d to  "zeros" in derivative  (c)  Fi<j. 9.  Graph  (b)  integrated  Derivative of resonance line and actual line obtained by integrating (b) graphically . t o follow page 21  -22-  ii.  Measurements i n the Paramagnetic  State.  The p r o t o n magnetic resonance spectrum was s t u d i e d i n s i n g l e c r y s t a l s of CoClg'SHgO i n a f i e l d H gauss.  Q  of about 3100  The maximum number of 24 l i n e s p r e d i c t e d by theory  was never observed.  In the paramagnetic r e g i o n two complete  r o t a t i o n s were made a t 78° K, one w i t h H  Q  i n t h e plane perpen-  d i c u l a r t o the a - a x i s , t h e other w i t h H  Q  i n the a-c p l a n e .  At 4.2°K one complete r o t a t i o n was made w i t h I? i n the a-c 0  plane.  The c r y s t a l remained f i x e d i n space a t a l l times,  w h i l e the magnet was r o t a t e d about i t i n 10 degree  intervals.  From each of these graphs we can see d i r e c t l y t h a t the s p e c t r a r e p e a t themselves a f t e r a 180-degree  r o t a t i o n of H . Q  S e v e r a l checks were made a t angles between 180 and 360 degrees, and these confirmed the 180-degree  symmetry;  consistent with  paramagnetic measurements.  T h i s r e p e t i t i o n of the resonance  spectrum a f t e r a 180-degree  r o t a t i o n i s due t o the 180-degree  p e r i o d i c i t y of ( 3 c o s © - l ) . 2  (a)  H  Q  i n t h e Plane P e r p e n d i c u l a r t o the a - a x i s .  F i g u r e 10 r e p r e s e n t s t h e o n l y r o t a t i o n w i t h H plane p e r p e n d i c u l a r t o the a - a x i s .  Q  i n the  A maximum number of t h r e e  l i n e s was observed, and they a l l o v e r l a p p e d s t r o n g l y .  It i s  thus i m p o s s i b l e t o f o l l o w any i n d i v i d u a l l i n e through the whole rotation. b-axis.  F i g u r e 10 e x h i b i t s a g e n e r a l symmetry about the T h i s i n d i c a t e s t h a t t h e protons a r e s i t u a t e d  symmetri-  ©  ©  200  © ©  ©  180  Free p r o t o n frequency ^  ©  ©  ©  160  © ©  ©  0  140  ©  © CO  120  0  0>  o> a>  -o  \  ®  tu  A  I  800  .& /O  .820  ©^  c. o IS 8 0 c a>  T O  a> >  -860  .880  92 O  .900  Freq. in Mc/sec  »  f  ©  60  ©  \  ©  c? 4 0  © ©  \ ©*  20  /  ©  /  0  /  © 0 -10  ©  © © Fig. 10. Paramagnetic 'CoC4  2  6 H 0 at 7 8 ° K 2  H o r o t a t i n g in plane perpendicular to a-axis t o follow page22  -23-  c a l l y w i t h r e s p e c t t o the b - a x i s which i s c o n s i s t e n t w i t h the crystal  structure.  When H  i s p a r a l l e l t o the b - a x i s , a l l the l i n e s occur  Q  above the f r e e proton frequency.  In t h i s p o s i t i o n the l o c a l  f i e l d produced by the c e n t r a l c o b a l t i o n i n an octahedron i s the same a t a l l f o u r surrounding oxygens O j , and i s i n the same d i r e c t i o n as H f i g u r e 13a);  Q  (the c r y s t a l s t r u c t u r e r e v e a l s t h i s (see  and the g - f a c t o r of Co i s p o s i t i v e ) .  Therefore,  i f we d i s r e g a r d the r e l a t i v e l y f r e e waters and the neighbouring octahedra, the protons o f the i s o l a t e d octahedron s h o u l d have a resonance at  frequency g r e a t e r than the f r e e p r o t o n frequency  this crystal position.  The minimum and maximum resonance  f r e q u e n c i e s occur s i m u l t a n e o u s l y a t 45° on e i t h e r s i d e of the b-axis.  When HQ i s i n t h i s p o s i t i o n the l o c a l f i e l d  i s almost  —>  i n the same d i r e c t i o n as H opposite t o H to resonance  Q  Q  f o r two of the f o u r waters, and  f o r the other two (see f i g u r e 13b), g i v i n g  l i n e s above and below the f r e e proton  r e s p e c t i v e l y as e x h i b i t e d by f i g u r e 10. 10 not f a l l i n g  on curves  rise  line  The p o i n t s i n f i g u r e  (1) or (2) may then be a t t r i b u t e d t o  the r e l a t i v e l y f r e e waters. It  i s of i n t e r e s t t o examine some aspects of f i g u r e 10  q u a n t i t a t i v e l y , s i n c e the plane p e r p e n d i c u l a r t o the a - a x i s i s w i t h i n about 20 degrees o f the r e f l e c t i o n plane of the octahedron.  For the angle 45°, H  Q  i s almost p a r a l l e l t o the  v e c t o r j o i n i n g the c e n t r a l c o b a l t i o n w i t h two of the Oj atoms  -24-  ( a c t u a l l y , cos© = 0.97 i f © i s the angle between H  and the  Q  v e c t o r j o i n i n g the oxygen atoms and the c o b a l t i o n ) , and e x a c t l y p e r p e n d i c u l a r to the v e c t o r from the c o b a l t i o n to the other two Oj atoms i n the octahedron.  Taking the protons t o  be near the Oj atoms and n o t i n g t h a t the f i e l d due t o the c o b a l t i o n i s p r o p o r t i o n a l t o (3cos ©-l)r"" , we expect the 2  3  s h i f t s of the proton f r e q u e n c i e s due t o these two groups of water molecules from the frequency c o r r e s p o n d i n g t o the 3cos © 2  average t o t a l i n t e r n a l f i e l d t o be i n the r a t i o 18 -j-  .  or  C l e a r l y , i n order t h a t t h i s be so, the average i n -  t e r n a l f i e l d must be taken t o correspond t o a proton resonance frequency approximately 19 Kc higher than the f r e e proton v a l u e (see f i g u r e 10). of  We  a t t r i b u t e t h i s t o the c o n t r i b u t i o n  (4TT - N ) M "to the average f i e l d d i s c u s s e d a t the end of  chapter  2.  —*  S i n c e H was a p p l i e d p e r p e n d i c u l a r t o the c y l i n d e r - a x i s our sample which i s r o u g h l y a c y l i n d e r of l e n g t h 1.6 times Q  of  the diameter, N i s approximately e q u a l t o 1.6JT CH that M — — 2 T  .  Assuming  , where C i s the C u r i e constant f o r our  and T 2 s 7 8 ° K , we  crystal  c a l c u l a t e C-0.014.  The C u r i e constant i s g i v e n by  _ N g /3 2  2  S(S + 1)  3 k where N i s the number of paramagnetic  i o n s per cm , g , f t , and  -25-  S have been d e f i n e d i n chapter 2, and k i s Boltzman's c o n s t a n t . 13  Putting g ^ 4 The  and g u e s s i n g t h a t S = -|, we o b t a i n C ~ 0.016.  c l o s e agreement of these two c a l c u l a t i o n s of C i s  p r o b a b l y f o r t u i t o u s , but i t p r o b a b l y a l s o i n d i c a t e s t h a t our general i n t e r p r e t a t i o n i s correct. I t w i l l now be i n t e r e s t i n g i n f u t u r e s t u d i e s t o measure the temperature  dependence o f M by t h i s method, and w i t h the  same geometry measure the temperature s p l i t t i n g of the l i n e s . "space  The f i r s t  dependence of the maximum  i s p r o p o r t i o n a l t o the  averaged" m a g n e t i z a t i o n per u n i t volume w h i l e the  second s h o u l d be p r o p o r t i o n a l t o the time average moment of an i n d i v i d u a l c o b a l t i o n . if  magnetic  I t would be s u r p r i s i n g  they d i d not have the same temperature  dependence.  Never-  t h e l e s s , an experimental check i s of i n t e r e s t w i t h r e s p e c t t o some fundamental i d e a s concerning i n t e r n a l magnetic f i e l d s . In the next s e c t i o n concerning the r o t a t i o n of H  Q  i n the a-c  plane approximate v e r i f i c a t i o n of the above i d e a s i s observed, s i n c e t h e r e we a l s o have d a t a i n the paramagnetic l i q u i d helium (b)  H  Q  state at  temperatures. i n the a-c p l a n e .  F i g u r e 11 r e p r e s e n t s the r o t a t i o n w i t h I? i n the a-c 0  plane a t 78°K and f i g u r e 12 the same r o t a t i o n a t 4.2°K.  In  f i g u r e 11 the number of l i n e s i s again s m a l l and the l i n e s are never completely r e s o l v e d , so t h a t l i n e i d e n t i f i c a t i o n i s  Relative  o r i e n t a t i o n in  degrees  no Q  Q  O  —  *"  TJJ Q  X  0  II  X 0  -\  OJ 0  —•-  Ol  3  0  gne  o  0  ne.  o  ro  TI Q -\ Q  Q  —•-  0 0  0  —  0  5  ro •  Q C  cn  CD X  ro  O  Fig.ll. Pgramagnetic CoCl2-6H 0 Ho in a-c plane 2  at 7 8 ° K. t o follow page 2 5  Fig. 13.  Field pattern of a dipole. to follow page Z5  -26-  difficult. Again the spectra are repeated after a 180-degree rotation due to the periodicity of (3cos 0-l), as explained 2  in the previous section.  Comparing figures 11 and 12 we can  recognize a general similarity in the two plots.  In figure  11 the maximum frequencies occur at 0 degrees and 180 degrees, whereas in figure 12 they occur at -19 degrees and 161 degrees. At these orientations H is in the plane of the four oxygens Q  O j forming the reflection plane of the octahedron, and maximum frequencies are expected.  However, these maximum fre-  quencies should occur at the same crystal orientations regardless of temperature.  Since different crystals were used for  these rotations, the above discrepancy is probably due to faulty alignment of the sample for the rotation at 78°K. There must also be a slight misalignment of this sample relative to figure 10, since the spectrum at 0 degrees in figure 11 does not quite agree with the 0-degree spectrum of figure 10. In figure 11 the majority of lines occur at frequencies below the free proton frequency, whereas in figure 12 the majority of lines occur above the free proton l i n e .  The ratio  of the maximum frequency below the free proton line to that above in figure 11 is about 85 : 55, whereas in figure 12 the same ratio is about 165 : 305.  In other words, at the lower  temperatures the whole system of lines is shifted to higher frequencies.  -27-  S i n c e the time averaged  magnetic moment of the c o b a l t  i o n s i s p r o p o r t i o n a l t o H , the s p l i t t i n g caused by the c o b a l t Q  i o n s s h o u l d be a l i n e a r f u n c t i o n of H . Q  The s e p a r a t i o n o f  extreme l i n e s was measured as a f u n c t i o n of H  Q  a t 78°K f o r H  Q  i n the a-c plane and w i t h 160 degrees o r i e n t a t i o n , and a l s o for H  Q  i n the plane p e r p e n d i c u l a r t o the a - a x i s a t 0 degrees  orientation. respectively.  The r e s u l t s a r e shown i n f i g u r e s 16 and 15 In both cases the s p l i t t i n g i s found t o be  l i n e a r i n H , but when e x t r a p o l a t e d f o r H Q  does not become z e r o . proton-proton For H  Q  Q  = 0 the s p l i t t i n g  T h i s i s t o be expected,  s i n c e the  i n t e r a c t i o n i s independent of the a p p l i e d f i e l d .  i n the a-c plane the s p l i t t i n g e x t r a p o l a t e s t o about  83 Kc or 19 gauss, and f o r E  Q  i n the p l a n e p e r p e n d i c u l a r t o  the a - a x i s i t e x t r a p o l a t e s t o about 53.5 Kc or 12 gauss. second of these v a l u e s i s almost  The  e x a c t l y the u s u a l p r o t o n -  p r o t o n s p l i t t i n g i n waters of h y d r a t i o n .  The f i r s t v a l u e i s  somewhat h i g h t o r e p r e s e n t pure proton-proton  interactions,  but s i n c e we do not know the r e l a t i v e amplitudes  of the  q u a n t i t i e s a and b i n formula 6, the s l o p e of the l i n e i n f i g u r e 15 may a c t u a l l y change as H  Q  approaches z e r o .  In view of the above r e s u l t s the i d e a s put forward a t the end of the l a s t s e c t i o n concerning the r o t a t i o n i n the plane p e r p e n d i c u l a r t o the a - a x i s can now be checked r o u g h l y . For the angle -19° i n t h i s r o t a t i o n H" i s i n the r e f l e c t i o n plane of the octahedron  and f o r the angle 71° l l  0  is  •9 *6 -7 -6 -5" Fig. 14.  See  -4  -3 -2  -I  O  I  2 3  4  5  text.  to follow page 27  Fig. 16.  S a m e a s f i g . 15. H at 160° in a-c plane. T = 7 8 °K e  +o follow page 2 7  -28-  perpendicular to t h i s plane.  Again, t a k i n g the protons t o be  near the Oj atoms, a l l the protons a s s o c i a t e d w i t h the Oj atoms are m a g n e t i c a l l y e q u i v a l e n t f o r the a-c r o t a t i o n .  Cal-  c u l a t i n g the p o s i t i o n of t h i s l i n e as a f u n c t i o n of o r i e n t a t i o n of  H  Q  i n the a-c plane, we  o b t a i n curve  (1) of f i g u r e 14.  extreme p o s i t i o n s of the l i n e above and below the f i e l d are i n the r a t i o of 1 : 2 r e s p e c t i v e l y . the average  internal f i e l d  average  T h i s means t h a t  i n f i g u r e 11 s h o u l d be taken t o  correspond t o a p r o t o n resonance than the f r e e p r o t o n v a l u e .  approximately 20 Kc higher  Within experimental error  this  i s i n agreement w i t h the r e s u l t s f o r the r o t a t i o n w i t h H the plane p e r p e n d i c u l a r to the a - a x i s . i n curve  The  (1) of f i g u r e 11 are about  in  The extreme f r e q u e n c i e s  12,427 Mc/sec. and 12,536  Mc/sec. which comes t o a d i f f e r e n c e of 109 Kc. t a t i o n a t 4.2°K, c o n s i d e r i n g curve  Q  For the r o -  ( l a ) i n f i g u r e 12,  the  average f i e l d s h o u l d be taken t o correspond t o a p r o t o n frequency about  197 Kc higher than the f r e e proton frequency.  The extreme f r e q u e n c i e s f o r l i n e l a i n t h i s f i g u r e are about 13,183 Mc/sec. and 13,657 Mc/sec.}  a d i f f e r e n c e of 372  Therefore M - (4.2°K) „ M (77.3°K)  197 20  1  Q  and LL (4.2°K) Li (77.3°K)  ~  372 - 83 109 - 83  ~  11  Kc.  -29-  From these r e s u l t s we see t h a t the "space  averaged"  magnetization  M and the average v a l u e of the i n d i v i d u a l c o b a l t moment the same temperature  dependence w i t h i n experimental  The dependence s h o u l d not be expected  have  accuracy.  t o f o l l o w a simple C u r i e  law. U s u a l l y one can approximate the temperature magnetic s u s c e p t i b i l i t i e s by a Curie-Weiss t h a t M and < pl> are p r o p o r t i o n a l t o temperature  of the substance,  '  10.5  n  — g  law.  , where  dependence of I f we ©  then our r e s u l t s g i v e  or  Q  t o be approximately TJJ — 2«28°K.  n o r m a l l y g r e a t e r than T , N  i s the C u r i e approximately  ^3.4°K  As d e s c r i b e d l a t e r , we have measured the Neel of t h i s substance  assume  temperature Since Q i s  t h i s r e s u l t seems r e a s o n a b l e .  s h o u l d be emphasized, however, t h a t we  It  have not t r i e d t o c o r r e c t  f o r the s h i f t i n the p o s i t i o n of the l i n e s due  to overlap.  With w e l l r e s o l v e d l i n e s and the proton p o s i t i o n s known, the r a t i o of extreme f r e q u e n c i e s f o r a p a r t i c u l a r l i n e below 2 and above z e r o s h i f t due  —3  t o (3cos 0 - l ) r  c o u l d be  obtained.  In such a case the above ideas would, i n f a c t , p r o v i d e a method f o r measuring the average  magnetization  If.  A c c o r d i n g t o theory the s i x d i f f e r e n t water molecules  of  h y d r a t i o n i n CoCl^BHgO s h o u l d l e a d t o 24 l i n e s i n s i x groups of f o u r .  Each group of f o u r s h o u l d c o n s i s t of two p a i r s ,  c e n t e r s of which are s e p a r a t e d by a d i s t a n c e 2b  the  (see formula 5 ) ,  -30-  and the separation between these two lines is 4 d . should be of equal intensity (see formula 7).  They  However,  experiments on C o C l g ^ ^ O never give 24 lines in a f i e l d of 3 K gauss.  Thus, i t may be concluded that certain lines  overlap even at liquid helium temperatures.  At 4.2°K a  system of six lines consisting of three pairs i s observed. One of the major goals in this work is to find the positions of the protons in the unit c e l l .  Since these are  not yet known, we cannot give theoretical curves of the type of figure 12.  However, an angular dependence of the  form of figure 14 is obtained i f we make the following assumptions: molecules,  (i) disregard the two relatively free water ( i i ) assume that only the cobalt at the centre  of the octahedron influences the four surrounding waters, ( i i i ) neglect the effects of cobalt ions in neighbouring octahedra, and  (iv) assume that the two protons in a water  molecule do not interact and are situated at the Oj positions. Curve 1 in figure 14 represents H =» (3cos ©-l)r""3 = z  2  (3sin y3 c o s G - l ) r ~ ^ as a function of ce, where y3 is the angle 2  2  between the Co*"** - O j vector and the projection of this line on the plane of rotation, and .a is the angle between H and Q  the plane of the Ox atoms.  For the a-c rotation /3 = 7r/4  and r i s , of course, a constant. Choosing an arbitrary amplitude, this curve can be f i t t e d exactly to curve l a in figure 12 when the angles of  -31-  rotation are chosen to coincide in the two figures. H  Q  When  is perpendicular to the reflect ion plane of the octa-  hedron, lines of lowest frequency are observed, since in this position the local f i e l d due to the central cobalt is opposite to the direction of H because cobalt has a positive Q  g-factor (3 unpaired electrons in the 3d s h e l l ) . perpendicular to the  When I? is 0  axis of the octahedron, i . e . parallel  to the reflection plane, lines of maximum frequency are observed.  This is in agreement with figure 14.  In reality, of course, the two protons are not at the oxygen positions and they do interact, but the fact that curve 1 in figure 14 and curves l a and lb in figure 12 are almost exactly in phase means that one of the protons is located in the reflection plane of the octahedron.  If we  consider the octahedron as an isolated system, potential energy and symmetry considerations should cause the other proton also to be in this plane.  But in the crystal one  such octahedron is surrounded by many others, and the second proton in the waters may be twisted slightly out of this plane.  Such a position would produce a pair of curves  slightly out of phase with curves la and lb in addition to having different frequencies.  Curves 2a and 2b are out of  phase by about 30° with curves la and l b . The splitting in the pair 1 and 2 is of the order of 10-15 gauss, which is the usual splitting in a water molecule due to proton-proton interaction.  It may be, then, that  -32  the s e p a r a t i o n i n p a i r s 1 and 2 i s due  t o the proton d i p o l e -  dipole interaction.  Curves  near <a = 71° when  i s p e r p e n d i c u l a r to the r e f l e c t i o n  of  the octahedron.  l a and l b i n f i g u r e 12 c o a l e s c e  T h i s i m p l i e s that the proton-proton  s p l i t t i n g i n curve-pair 1 i s zero at t h i s p o s i t i o n . fore,  plane  There-  (3cos ©j2~]i) = 0 i n t h i s o r i e n t a t i o n , where 9 j 2 *2  s  t  h  e  angle between the v e c t o r c o n n e c t i n g the two protons and i t s p r o j e c t i o n on the plane of r o t a t i o n of the magnet.  I f the  s p l i t t i n g i s t r u l y z e r o , and more d e t a i l e d s t u d i e s may  reveal  t h a t i t i s not, then the d i r e c t i o n of the l i n e c o n n e c t i n g the two protons c o u l d be  determined.  The s p l i t t i n g between the c e n t r e s of p a i r s 1 and 2 i s t h e r e f o r e assumed t o be caused by the c e n t r a l c o b a l t .  In  t h i s argument the e f f e c t of the neighbouring c o b a l t s was neglected.  S i n c e the nearest c o b a l t - n e i g h b o u r s t o any of the  protons are about  twice the d i s t a n c e of the c e n t r a l c o b a l t t o  any of i t s surrounding p r o t o n s , t h e i r e f f e c t i s reduced by a f a c t o r 8 at l e a s t due t o the  -1^ r  behaviour of a d i p o l e  field,  3  and so s h o u l d produce  o n l y a s l i g h t change i n the curves of  figure r . Curve 2 i n f i g u r e 14 r e p r e s e n t s the l o c a l f i e l d at the p o s i t i o n of an O J J due t o i t s f o u r n e a r e s t c o b a l t drawn t o the s c a l e of curve 1.  neighbours  T h i s curve, however, i s not  i n phase w i t h c u r v e - p a i r 3 i n f i g u r e 12, which merely that the protons of the r e l a t i v e l y f r e e waters  are not  indicates located  -33-  at the O J J positions.  The splitting in this pair of lines  cannot definitely be accounted for.  It is probably due to  a proton-cobalt interaction, since i t is too large for a proton-proton interaction (about 30 gauss), unless these water molecules are greatly distorted. Quantitative results cannot be given at this time, since not sufficient data have been recorded.  The work  planned for the immediate future w i l l incorporate these qualitative arguments, and i t is hoped that i t w i l l be possible to establish the proton positions. The f i r s t obvious extension of the work reported here is to repeat the measurements at much higher fields so that a better resolution is obtained.  Probably more lines w i l l  then appear and definite identification should be possible. A study of the splitting between certain pairs of lines as a function of f i e l d at given positions should reveal which splittings are caused by proton-proton interactions and which are due to the cobalt ions. It is also planned to perform double resonance experiments.  With the aid of these i t should be possible to  study line shapes even i f the lines normally s t i l l slightly overlap.  Crystals containing different concentrations of  DgO should reveal some interesting phenomena, as i t should be possible to eliminate certain proton lines and to observe the lower frequency deuteron resonance lines.  With the  -34-  results from these measurements and the completion of the observations in the antiferromagnetic state i t should be possible to completely describe the magnetic behaviour of  CoCl *6H 0. o  o  -35-  iii.  T r a n s i t i o n Temperature Measurements.  T. Haseda and E. Kanda  14  , and M. Leblanc  15  independently  found t h a t CoClg^SHgO e x h i b i t s an a n t i f e r r o m a g n e t i c behaviour below about 3°K. W.K. Robinson and S.A. F r i e d b e r g  observed  a lambda-type anomaly i n the s p e c i f i c heat o f CoClg'SHgO a t 2.29°K and assumed t h i s anomaly t o be a s s o c i a t e d w i t h a p a r a m a g n e t i c - a n t i f e r r o m a g n e t i c phase t r a n s i t i o n . the t r a n s i t i o n temperature  In t h i s work  was measured by o b s e r v i n g the  change i n the p r o t o n spectrum. F i g u r e 1? shows the proton spectrum 160°  i n the a-c p l a n e .  a t 2.5°K w i t h H a t Q  The frequency of the o s c i l l a t o r was  a d j u s t e d so t h a t the r e c o r d e r pen rode on the f i r s t maximum s l o p e of the spectrum which corresponds in figure  17 as i n d i c a t e d .  With the frequency h e l d constant  at t h i s p o i n t , t h e temperature r e s u l t the graph of f i g u r e  t o the f i r s t maximum  was s l o w l y decreased.  18 was o b t a i n e d .  As a  The temperature  check p o i n t s a r e marked on the graph by s m a l l p i p s i n the curve, and the corresponding temperatures  are l i s t e d i n t h i s  figure. F i g u r e 18 shows that the onset o f the t r a n s i t i o n  occurs  at s l i g h t l y above 2.28°K i n agreement w i t h the s p e c i f i c heat measurements by Robinson and F r i e d b e r g . the l i n e of the paramagnetic i s not abrupt.  At t h i s  temperature  s t a t e d i s a p p e a r s , but the change  The t r a n s i t i o n takes p l a c e over a  temperature  Fig. 17. Proton  spectrum  at  at T=2.5°K. Ho at 160° and in a - c plane.  Freq.in 4 0 K c / s e c . steps  fofollowpage 3 5  -36-  range of about 0.07° (pips 7 t o 11).  T h i s range i s not caused  by time e f f e c t s i n the r e c o r d i n g system.  S e v e r a l such graphs  were o b t a i n e d w i t h d i f f e r e n t r a t e s of temperature change, and the range through which the t r a n s i t i o n takes p l a c e was 0.07° i n each case.  about  The e r r o r i n the temperature measurements  may be as l a r g e as +0.03°, but temperature d i f f e r e n c e s c o u l d be measured t o a much h i g h e r degree of a c c u r a c y .  The  fact  t h a t the t r a n s i t i o n i s not sudden but takes p l a c e over a c e r t a i n temperature spread i n d i c a t e s t h a t s h o r t - r a n g e magnetic order e f f e c t s are p r e s e n t . I t was p o s s i b l e t o keep the temperature constant at any p o i n t , and s e v e r a l r e c o r d i n g s of the spectrum were made w i t h the t r a n s i t i o n p a r t i a l l y completed. 2.25°K i s shown i n f i g u r e 19.  One such r e c o r d i n g at  T h i s spectrum s t i l l  resembles  t h a t of f i g u r e 17 (recorded at 2.5°K), but a change i s e a s i l y recognizable.  Recordings at lower temperatures w i t h i n the  t r a n s i t i o n range f u r t h e r d e v i a t e from f i g u r e 17, u n t i l the t r a n s i t i o n t o the a n t i - f e r r o m a g n e t i c s t a t e i s complete. F i g u r e 20 shows p a r t of a spectrum a t 2.21°K ( j u s t below the t r a n s i t i o n temperature).  Other lines occur i n t h i s spectrum  s e v e r a l Mc/sec. on e i t h e r s i d e of the p a r t shown.  T = 2 . 2 5 ° K . Ho a t 1 6 0 ° a n d in a-c p l a n e . F r e q . in 40 Kc/sec. steps. to rolioW paqe  3S  -37-  iv.  Measurements i n the A n t i f e r r o m a g n e t i c S t a t e .  As s t a t e d b e f o r e , C o C l 2 * 6 H 0 becomes a n t i f e r r o m a g n e t i c 2  at about 2.28°K.  Measurements i n the a n t i f e r r o m a g n e t i c s t a t e  are o n l y p a r t i a l l y complete. 1.52°K i n a f i e l d of 3100  These were c a r r i e d out a t  gauss w i t h H  Q  i n the a-c  A r e c o r d i n g of the spectrum w i t h I? a t 20° 0  f i g u r e 21.  i s shown i n  The r e s u l t s are shown i n f i g u r e 22.  i s of the same type as f i g u r e 12, b e i n g the same i n both  plane.  the angular  Figure  22  orientation  graphs.  These r e s u l t s e x h i b i t some f e a t u r e s s t r i k i n g l y  different  from the measurements i n the paramagnetic temperature r e g i o n . The  spectrum and the number of l i n e s change e s s e n t i a l l y i n  p a s s i n g from the paramagnetic t o the a n t i f e r r o m a g n e t i c r e g i o n . L i n e s h i f t s of 7.5 Probably  Mc/sec. have so f a r been r e c o r d e d .  these w i l l i n c r e a s e t o about 15 Mc/sec. when the  r o t a t i o n i s completed. g e n e r a l l y much broader 28 g a u s s ) .  The  l i n e s are w e l l r e s o l v e d and  (of the order of 120 Kc/sec. or about  F i g u r e 21 shows a t y p i c a l r e c o r d i n g of the  resonance a t 1.52°K f o r a g i v e n o r i e n t a t i o n of H .  The  q  proton data  i n f i g u r e 22 are d e r i v e d from such r e c o r d i n g s . One  s t r i k i n g and as y e t unexplained  feature i n a l l re-  c o r d i n g s i n the a n t i f e r r o m a g n e t i c s t a t e i s the v e r y s t r o n g l i n e approximately  30 Kc/sec. above the f r e e proton  T h i s l i n e i s not a f f e c t e d by the o r i e n t a t i o n of ~E  Q  frequency. i n the  R<j. 21. Proton resonance speirum. in Anti-ferromagnetic state at l.52°K. No in a-c plane at ZO°. to.follow  paqe 37  -38-  a-c plane* When the rotation in the antiferromagnetic state is completed, i t appears that the spectrum w i l l be symmetric only with respect to 360° rotation instead of 180° as in the paramagnetic state.  This result is expected, since  in the antiferromagnetic state the magnetic ions are oriented with respect to the crystal axes rather than the — »  external magnetic f i e l d H . Q  -39-  Bibliography 1.  G.E. Pake, J. Chem. Phys. 16, 327 (1948).  2.  N. Bloerabergen, Physica 16, 95 (1950).  3.  N.J. Poulis, Physica 17, 392 (1951).  4.  N.J. Poulis, Physica 18, 201 (1952).  5.  J.H. Van Vleck, Phys. Rev. 74, 1168 (1948).  6.  N. Bloembergen, E.M. P u r c e l l , and R.V. Pound, Phys. Rev. 73, 679 (1948).  7.  J.H. Van Vleck, J . Chem. Phys. 5, 320 (1937).  8.  American Inst, of Physics Handbook, Chapter 5, p. 240.  9.  D.G. Watkins, Ph.D. Thesis, Harvard University, Cambridge, Mass., u;S.A; (1952).  10.  H.H. Waterman, Ph.D. Thesis, University of B.C. (1954).  11.  P. Groth, Chemische Krystallographie, 1. T e i l , p. 248,  12.  Wilhelm Enzelmann Verlag, Leipzig (1906). J . Mizuno, K. Ukai, T. Sugawara, J . Phys. Soc. Japan 14, 383 (1959).  13.  M. Date, J . Phys. Soc. Japan 14, 1244 (1959).  14.  T. Haseda, E. Kanda, j ; Phys. Soc. Japan JL2, 1051 (1957).  15.  M. Leblanc, Ph.D. Thesis, University of B.C. (1958).  16.  W.K. Robinson, S.A. Friedberg, Tech. Report No. 5, Carnegie Inst, of Tech., Dept. of Physics (1959).  

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