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Proton magnetic resonance in paramagnetic and antiferromagnetic CoCl₂·6H₂O Sawatzky, Erich 1962

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PROTON MAGNETIC RESONANCE IN PARAMAGNETIC AND ANTIFERROMAGNETIC CoCl «6H 0 2  2  by ERICH SAWATZKY B.Sc, University of B r i t i s h Columbia, 1958 M.Sc, U n i v e r s i t y of B r i t i s h Columbia, 1960  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n the Department of PHYSICS  We accept this thesis as conforming to the required standard  THE UNIVERSITY OF BRITISH COLUMBIA September, 1962  In p r e s e n t i n g  t h i s thesis i n p a r t i a l fulfilment of  the requirements f o r an advanced degree a t t h e U n i v e r s i t y British  Columbia, I agree t h a t the  a v a i l a b l e f o r reference  and  study.  of  L i b r a r y s h a l l make i t f r e e l y I f u r t h e r agree t h a t  permission  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 g r a n t e d by  the  Head o f my  It i s understood t h a t f i n a n c i a l gain  Department  representatives.  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  s h a l l not  be  a l l o w e d w i t h o u t my  of  The U n i v e r s i t y o f B r i t i s h Vancouver 8*. Canada. Date  Department o r by h i s  be  Columbia,  written  permission.  The University of B r i t i s h Columbia FACULTY OF GRADUATE STUDIES  PROGRAMME OF THE FINAL ORAL EXAMINATION FOR THE DEGREE OF DOCTOR OF PHILOSOPHY  of  ERICH SAWATZKY B.Sc, University of B r i t i s h Columbia, 1958 M.Sc, University of B r i t i s h Columbia, 1960 PUBLICATIONS 1_  2.  Proton. Resonance i n Paramagnetic and A n t i f erromagnetic CoC^^H^O. E. Sawatzky and M. Bloom. J . Phys. Soc. Japan, \J_ Sup B-I, 507 (1962). The Paramagnetic-Ant iferromagnetic Phase T r a n s i t i o n i n CoCl2*6H20. E. Sawatzky and M. Bloom. Physics Letters 2, 28 (1962).  MONDAY, OCTOBER 1, 1962, AT 2:30 P.M. l i t ROOM 301, PHYSICS BUILDING  COMMITTEE IN CHARGE Chairman: M. BLOOM J. GRINDLAY R. HOWARD  F.H. SOWARD  .  R. REEVES C.S. SAMIS J . TROTTER  G.M. VOLKOFF External Examiner: S.A. Friedberg Carnegie I n s t i t u t e of Technology Pittsburgh, Pennsylvania  PROTON MAGNETIC RESONANCE IN PARAMAGNETIC AND ANTIFERROMAGNETIC CoC 12*6^0 ABSTRACT The work, reported here is a detailed study of the proton magnetic resonance in single crystals of C o C l 2 * 6H2O. This substance is paramagnetic at high temperatures and becomes antiferromagnetic at about 2.25 K. The proton resonance frequency is a measure of the total magnetic f i e l d at the positions of the protons, which is the vector sum of the applied magnetic field with the internal field produced by the surrounding magnetic ions. At room temperature a single line about 6 gauss wide is observed. This line splits into a number of components at liquid helium temperatures. The position and number of lines strongly depend on temperature and on the direction of the externally applied magnetic field. The maximum overall splitting at 4.2°K is about 150 gauss in a field of 5000 gauss. At 2.1°K the maximum splitting observed is about 2500 gauss. From the resonance lines in the paramagnetic phase i t was possible to calculate the direction cosines of one proton-proton vector. The resonance spectra in both phases were found to agree well with the theory predicting the positions of the resonance lines and their dependence on crystal orientation. The transition temperature T^ was measured as a function of applied field and crystal orientation using the proton resonance lines, since they are very sensitive functions of temperature near T^j. T^ is found to be a complicated function of the applied field and crystal orientation, which cannot be described by TN(H,) = T(0) - const. H 2 , as predicted by the Weiss Molecular field theory. The transition takes place over a temperature region of about 10"2 °K, and effects due to short range order are observed just above T . N  The magnetic susceptibility in zero field was measured along the preferred axis of antiferromagnetic alignment. This, together with specific heat data from published literature, was used to show a mutual consistency between thermodynamic variables and TN obtained by NMR. The sublattice magnetization in the antiferromagnetic phase was measured as a function of temperature. It is found to depend logarithmically on Tjg - T, but is independent of applied field and crystal orientation. Further experiments are suggested, which would add greatly to the understanding of the magnetic behaviour of CoCl -6H 0. 9  9  GRADUATE STUDIES Field of Study:  Nuclear Magnetic Resonance  Quantum Mechanics Nuclear Physics Electromagnetic Theory Magnetism  F.A. Kaempffer J.B. Warren G.M. Volkoff M. Bloom  Related Studies: Crystallography Physical Metallurgy  K.B. Harvey E. Teghtsoonian  - ii -  ABSTRACT  The work reported here i s a detailed study of the proton magnetic resonance In single c r y s t a l s of CoClg'SHgO. This substance i s paramagnetic at high temperatures and becomes antiferromagnetic at about 2.25°K,  The proton  resonance frequency i s a measure of the t o t a l magnetic f i e l d at the positions of the protons, which l s the vector sum of the applied magnetic f i e l d with the i n t e r n a l f i e l d produced by the surrounding magnetic ions. At  room temperature a single l i n e about 6 gauss  wide i s observed.  This l i n e s p l i t s into a number of  components at l i q u i d helium temperatures.  The p o s i t i o n  and number of l i n e s strongly depend on temperature and on the  d i r e c t i o n of the externally applied magnetic f i e l d .  The maximum o v e r a l l s p l i t t i n g at 4.2°K i s about 150 gauss i n a f i e l d of 5000 gauss.  At, 2.1°K the maximum s p l i t t i n g  observed i s about 2500 gauss. the  From the resonance l i n e s i n  paramagnetic phase i t was possible to calculate the  d i r e c t i o n cosines of one proton-proton vector.  The reson-  ance spectra i n both phases were found to agree w e l l with the  theory predicting the positions of the resonance l i n e s  and t h e i r dependence on c r y s t a l orientation. The t r a n s i t i o n temperature TJJ was measured as a function of applied f i e l d and c r y s t a l orientation using the  - i i iproton resonance l i n e s , since they are very sensitive functions of temperature near T^.  i s found to he a  complicated function of the applied f i e l d and c r y s t a l orientation, which cannot be described by T (H,) = T(0) N  - const. H , 2  as predicted by the Weiss Molecular f i e l d  theory. The t r a n s i t i o n takes place over a region of about IO"  2  temperature  °K, and effects due to short range  order are observed just above  TJJ.  The magnetic s u s c e p t i b i l i t y i n zero f i e l d  was  measured along the preferred axis of antiferromagnetic alignment.  This, together with s p e c i f i c heat data from  published l i t e r a t u r e , was used to show a mutual consistency between thermodynamic variables and T^ obtained by  NMR.  The sublattlce magnetization i n the antiferromagnetic phase was measured as a function of temperature.  It  i s found to depend logarithmically on T^ - T, but i s independent of applied f i e l d and c r y s t a l orientation. Further experiments are suggested, which would add greatly to the understanding of the magnetic behaviour of CoCl .6H 0. 2  2  - vii -  ACKNOWLEDGMENT  My sincerest appreciations are due Dr. M. Bloom. Without his constant interest and guidance, many i l l u m i n ating discussions and invaluable help i n Interpreting the r e s u l t s , t h i s thesis would not have been possible. I also wish to express my appreciation to Dr. D. L. Williams f o r c r i t i c a l l y reading the manuscript of this thesis. This research has been supported f i n a n c i a l l y by grants to Dr. M. Bloom from the National Research Council and through the award of a National Research Council Studentship  (1959-1962).  The constant moral support from my wife has greatly helped i n the preparation of t h i s thesis.  - iv -  TABLE 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  Acknowledgment  vii  Chapter I  - Introduction  1  Chapter I I  - Apparatus and Experimental Procedure.  10  Chapter I I I  - Theory of the Positions of the Resonance Lines  24  - The Proton Resonance i n C o C l » 6 H 0 at  Chapter IV  2  2  38  Constant Temperature A - The C o C l - 6 H 0 C r y s t a l  38  B - Results and Discussions  39  2  Chapter V  2  (i) The proton resonance i n the paramagnetic phase at 4 . 2 ° K .... ( i i ) The proton resonance i n the antiferromagnetic phase at 2 , 1 ° K - The Paramagnetic-Antiferromagnetio  40 56 63  Phase Transition Chapter VT  - Thermodynamics of the Phase Transition  77  Chapter VII  - T h e Sublattice Magnetization below T^  88  Chapter VIII - Suggestions f o r Further Experiments .. Appendix A Appendix B  - Construction Hints and Operating Instructions - The Nagamiya, Yosida, and Kubo Theory f o r TJJH Applied to C o C l « 6 H 0 2  2  95 97 107  - V -  LIST OF ILLUSTRATIONS  Page  Fig. 1  - Block Diagram of Apparatus  11  Fig. 2  - Modified Tektronix 162 Waveform Generator ..  13  Fig. 3  - 30cps Source  15  F i g . 4a - C r y s t a l Growing Vessel  19  F i g . 4b - C r y s t a l Holder  19  Fig. 5  - Derivative of Proton Resonance Spectrum ....  41  Fig. 6  - Resonance Diagram i n the a-c Plane f o r T = 77°K (H = 5,000 gauss)  43  - Resonance Diagram i n the a-c Plane f o r T = 4.2°K (H = 3,000 gauss)  44  - Resonance Diagram i n the a-c Plane f o r T = 4.2°K (E = 5,000 gauss)  45  - Resonance Diagram i n the a'-b Plane f o r T = 4.2°K (H = 5,000 gauss)  46  Q  Fig. 7  Q  Fig. 8 Fig. 9  Q  F i g . 10 - Resonance Frequency Separation vs Applied Field  49  F i g . 11 - D e f i n i t i o n of Direction Angles F i g . 12 - Proton-Proton S p l i t t i n g as a Function of Orientation F i g . 13 - Resonance Diagram i n the a-c Plane f o r T = 2.1°K (H = 5,000 gauss) Q  51 53 57  F i g . 14 - Resonance Diagram i n the a'-b Plane f o r T = 2.1°K (H - 5,000 gauss)  58  F i g . 15 - Resonance Spectrum vs Applied F i e l d f o r T = 1.14°K  61  F i g . 16 - Relative Signal Amplitude vs Temperature f o r T > T , H = 5,000 gauss  65  F i g . 17 - The Region near T = 2.2°K of F i g . 16 Expanded  66  F i g . 18 - The Lie Width as a Function of Temperature .  69  Q  N  Q  - vi Page F i g . 19 - The Neel Temperature as a Function of Applied F i e l d F i g . 20 - P l o t of T  N  71  vs Orientation  73  F i g . 21 - The Anisotropy i n T^  74  F i g . 22 - The Molar S u s c e p t i b i l i t y as a Function of Temperature  82  F i g . 23 - Plot of (3M/3H)  S  VS C / T  84  H  F i g . 24 - O s c i l l a t o r Frequency vs Temperature  86  F i g . 25 - Resonance Frequency vs Temperature f o r T < T  89  N  F i g . 26 - P l o t of f j - f  x  vs l o g ( T  N  - T) f o r T  T  N  92  F i g . IA - O s c i l l a t i n g Detector  101  F i g . 2A - Williamson Power Amplifier  102  F i g . 3A - Audio Amplifier  103  F i g . 4A - Phase Sensitive Detector  104  F i g . 5A - Magnet Current Regulator  105  F i g . 6A - Low Temperature System  106  - 1 -  CHAPTER  I  INTRODUCTION  The nuclear magnetic resonance technique provides a powerful method of studying the interactions between atomic n u c l e i and their magnetic environment.  The steady state  branch of t h i s technique i s p a r t i c u l a r l y suited f o r the investigation of internal magnetic f i e l d s , because the nuclear resonance frequency i s a measure of the magnetic f i e l d at the nucleus under investigation.  In the f i e l d of  s o l i d state physics, the importance of nuclear magnetic resonance (NMR)  i s now well recognized, and valuable c o n t r i -  butions have been made to the understanding of many d i f f e r e n t classes of materials, among them paramagnetic and a n t i ferromagnetic substances. In  this work we are concerned with the proton  magnetic resonance i n single c r y s t a l s of CoClg . 6H 0 both i n g  the  paramagnetic and antiferromagnetic phases of this  substance.  The proton resonance i n both phases i s  profoundly influenced by the electronic magnetic moments of the  cobalt ions and by the protons themselves and may thus  be used to study the internal magnetic properties of the crystal. In general, i f a system of non-interacting protons with spin I  s  1/2 i s placed i n a uniform magnetic f i e l d  H,  - 2 each, proton can assume either of two orientations with respect  to this f i e l d H corresponding to two energy l e v e l s  ~/^joH  , where  yllp  i s the magnetic moment of the protons.  The difference i n energy between the two l e v e l s i s thus  2yUpH o  Transitions between the two l e v e l s may be induced  by applying  electromagnetic radiation, which i s c i r c u l a r l y  polarized with the magnetic vector rotating i n a plane perpendicular to the s t a t i c magnetic  field H .  The frequency  of t h i s radiation must be equal to the c l a s s i c a l Larmor frequency  co, = ZjupH/h. where ftp = 2.675 x 10 r a t i o f o r protons.  4  = <Tp W  rad/sec-gauss i s the gyromagnetio  In a t y p i c a l f i e l d of 5,GOO gauss, the  proton resonance frequency i s 21.297 Mc/sec. The simple picture of non-interacting  protons outlined  above i s never s t r i c t l y true, since interactions with neighbouring magnetic moments are always present, although i n l i q u i d s and gases these interactions are averaged considerably due to thermal motion of the atoms and molecules.  In  c r y s t a l l i n e s o l i d s the nuclei occupy, except f o r thermal vibrations, fixed positions, and each nucleus experiences, i n addition to the externally applied s t a t i c magnetic H  Q  field  and the small r - f f i e l d H]_ , an inhomogeneous i n t e r n a l  magnetic f i e l d due to the neighbouring magnetic dipoles.  The  proton resonance frequency i s a measure of the t o t a l magnetic f i e l d at the positions of the protons, which i s the vector  -  i.e.  _* loc H  -  —»  —»  sum of the applied f i e l d H  3  Q  with the i n t e r n a l f i e l d H ^ =  H  o  +  H  and the resonance frequency i s &)=  nt>  _^ int  <T i » H  00  I f the c r y s t a l contains paramagnetic ions, as i s the case i n CoCl *6H 0, the internal magnetic f i e l d Hj_  nt  may be  of the order of 1 0 0 0 gauss, whereas i n t e r n a l f i e l d s due to non-paramagnetic  ions are usually not larger than 20 gauss.  Since the magnetic moment of the paramagnetic ions may oriented i n ( 2 S + 1) d i f f e r e n t directions ( S  =  be  effective  spin of the paramagnetic i o n ) , the i n t e r n a l f i e l d produced by the  neighboring ions at the positions of d i f f e r e n t protons i n  a unit c e l l of the c r y s t a l may vary between approximately i  1 0 0 0 gauss.  This range of i n t e r n a l f i e l d s should give  r i s e to a spread of resonance frequencies, and we would expect very broad proton resonance l i n e s i n paramagnetic substances. However, i t can be shown ^ that i f exchange interactions 1  between the magnetic ions are present, t h i s broadening action of the magnetic ions i s considerably reduced.  From the  exchange interaction and the Zeeman energy of the cobalt ions i n the external f i e l d H  Q  the time averaged magnetic moment  <^jU>Q^ i s obtained to a f i r s t approximation ^ i n the para2  magnetic phase  - 4  where  ^  -  i s the Bohr magneton, ^  g-tensor, and  ©  i s the anisotropic  i s the Curie temperature  of the substance.  This mean magnetic moment gives r i s e to a time averaged i n t e r n a l f i e l d which depends strongly on the s p a t i a l coordinates and on the orientation of the c r y s t a l with respect to the external magnetic f i e l d  H. Q  The energy l e v e l s of  the proton magnetic moments are determined by the vector sum of t h i s i n t e r n a l f i e l d with H , Q  and non-equivalent protons  i n a unit c e l l have d i f f e r e n t resonance frequencies. Assuming  that H  Q  i s constant over the sample, the l o c a l f i e l d  w i l l be the same for equivalent protons i n the unit c e l l s and the proton resonance w i l l s p l i t into a number of component l i n e s , provided the l o c a l f i e l d s are s u f f i c i e n t l y d i f f e r e n t at non-equivalent protons.  To resolve a p a i r of l i n e s the  difference i n l o c a l f i e l d s must be greater than the proton l i n e widths which are t y p i c a l l y of order 5 to 10 gauss f o r hydrated s a l t s of the t r a n s i t i o n elements. f i e l d has roughly an inverse temperature  Since the i n t e r n a l  dependence, the  separation between component l i n e s increases with decreasing temperature.  Thus the number of component l i n e s observed i n  any p a r t i c u l a r applied f i e l d depends on the sample temperature, on the number of water molecules per unit c e l l , and on the degree of symmetry of the c r y s t a l . The i n t e r n a l f i e l d at a proton due to a ion at a distance  r  paramagnetic  i s of the order of magnitude  <£/6^/f •  The f i e l d therefore f a l l s o f f rapidly with increasing r  3  , so  - 5 that only the nearest neighbours make important contributions to H  l o o a l  .  Taking r = 2 x 10"  8  cm and  H  gauss, the s p l i t t i n g i n CoClg • 6HgO at 300°K  =  Q  5,000  i s about  2 gauss, at 78°K about 7 gauss, and at 4°K about 120 gauss. We therefore do not expect any resolution of the proton resonance l i n e at room temperature, and only p a r t i a l resolution at 78°K  and 4°K ,  Experimentally a single l i n e about  5 gauss wide i s observed at room temperature.  The maximum  s p l i t t i n g at 78 K is about 35 gauss and at 4.2° K about Q  170 gauss i n good agreement with the above order of magnitude o arguments,  since  r  was chosen a r b i t r a r i l y as  2 A,  The work to be described i n the following chapters of this thesis i s an extensive study of the proton magnetic resonance i n CoCl • 6H 0 as a function of temperature, c r y s t a l orientation and applied f i e l d .  A summary of the theory  describing the positions of the component l i n e s as a function of temperature, orientation, and applied f i e l d i s presented i n chapter I I I , netic phases.  f o r both the paramagnetic and antiferromagIn the paramagnetic phase the i n t e r n a l f i e l d  due to the cobalt ions i s found to be proportional to 3 COS^G^^ Vfa ^  , where r ^  tude of the radius vector between the i k  ttL  cobalt ion, and  Oik  t l L  i s the magni-  proton and the  i s the angle between  H  0  and  iTfo .  This implies that the resonance l i n e s i n the paramagnetic phase behave l i k e sine functions with a period of 180°.  In  - 6 addition to the i n t e r n a l f i e l d due to cobalt ions, a proton i n one water molecule experiences a f i e l d proportional to due to the other proton i n the same water molecule.  This f i e l d i s much smaller than that  due to the cobalt ions, but nevertheless gives r i s e to an additional s p l i t t i n g of component l i n e s which i s independent of the magnitude of the applied magnetic f i e l d .  The maximum  observed proton-proton s p l i t t i n g at 4.2°K was found to be 15 gauss.  From the proton-proton s p l i t t i n g as a function of  orientation i t was possible to f i n d the d i r e c t i o n cosines for  the l i n e joining one pair of protons.  The 180° p e r i o d i -  c i t y of the resonance curves i n the paramagnetic phase was v e r i f i e d experimentally.  The antiferromagnetic phase l s  characterized by a spontaneous alignment of the cobalt magnetic moments, independent of the applied f i e l d provided  H  0  i s not too large.  H, Q  This spontaneous a n t i p a r a l l e l  alignment gives r i s e to an i n t e r n a l f i e l d of the form  where  i s the angle between ^/^c^ n d a  H  Q  and OCQ i s  a phase angle depending on the orientation of ^/J^ with respect to the c r y s t a l axes and on the s p a t i a l coordinates. Thus the resonance curves are found to behave l i k e sine functions with a period of 360° instead of 180° as i n the paramagnetic phase.  The number of resonance l i n e s e f f e c t -  i v e l y doubles when we pass from the paramagnetic to the  -  7 -  antiferromagnetic phase due to the fact that we now have two sublattices, i . e . , ^JUc^ or i n a - d i r e c t i o n .  can point i n a  + direction  The resonance diagrams i n the a n t i -  ferromagnetic phase are thus symmetrical about a central frequency which i s determined by  H. Q  The s p l i t t i n g s i n  the antiferromagnetic phase are much larger than i n the paramagnetic phase.  S p l i t t i n g s of the order of 2,500 gauss  were observed at 2.1° K. that (jXc^ of  I t was v e r i f i e d experimentally  i n the antiferromagnetic phase i s independent  the applied f i e l d i n the range of f i e l d s used i n this  experiment. The phase t r a n s i t i o n or Neel temperature T^ i s treated i n chapter V.  I t i s found that the t r a n s i t i o n i s  not sharp, but takes place over a temperature range of a few millidegrees indicated by an overlap of both types of spectra i n this range.  The resonance l i n e amplitudes  decrease sharply as T  i s approached, whereas the l i n e  widths show an anomalous peak at T^ . This behaviour together with the anomalous peak i n the s p e c i f i c heat at T^  i s very suggestive of a X-type t r a n s i t i o n as observed  i n l i q u i d helium.  The t r a n s i t i o n temperature was studied  as a function of c r y s t a l orientation and applied f i e l d . The orientation dependence could be described by a funotion of the type  where  (p  i s the angle between  H  Q  and the preferred axis of  - 8 -  antiferromagnetic alignment. T^. H . 0  From the f i e l d dependence of  i t i s found that B(H ) i s nearly a l i n e a r function of 0  The dependence of T^  on H  with moleoular f i e l d theory.  Q  and  (p  i s inconsistent  Extrapolating to H  ~was found to be equal to 2.27°K«  When  H  0  Q  = 0, T^  i s applied  perpendicular to the preferred (c) axis, T^ i s found to have a broad minimum at about 3000 gauss and a maximum at 5000 gauss.  This type of behaviour i s not found i n other phase  t r a n s i t i o n s , where the l i n e separating two phases l s usually a monotonic function of the intensive thermodynamic coordinate.  No explanation has been given f o r t h i s  behaviour of ^  .  The magnetic s u s c e p t i b i l i t y p a r a l l e l to the preferred (c) axis i n zero applied f i e l d was measured as a function of temperature  by measuring the self-inductance of a c o i l  containing a sample of CoClg• 6HgO as a core.  These measure-  ments were found to be i n good agreement with previous 3)  s u s c e p t i b i l i t y measurements ' using an inductance bridge. The present measurements together with s p e c i f i c heat data are used to show the mutual consistency between the thermodynamic properties and the slope of the t r a n s i t i o n  temperature  near zero f i e l d as obtained with NMR. The temperature  dependence of the resonance  fre-  quencies i n the antiferromagnetic phase was studied i n various applied f i e l d s and a t several c r y s t a l orientations. In chapter VI we s h a l l see that the resonance frequency i s  - 9 -  a measure o f regardless  of  -»  t h e s u b l a t t l c e m a g n e t i z a t i o n M . I t was -> H  r i t h m i c a l l y on ( T studied. T^.  and o r i e n t a t i o n , t h a t  Q  N  - T) o v e r t h e w h o l e  that  M depends  loga-  temperature  range  This behaviour must, of course,  b r e a k down n e a r  The l o g a r i t h m i c b e h a v i o u r was f o u n d t o h o l d t o  a b o u t 20 m i l l i d e g r e e s o f  T  N  .  It  is  i n t e r e s t i n g to  t h i s behaviour i s q u a l i t a t i v e l y s i m i l a r to  behaviour of  the s p e c i f i c  r i t h m i c a l l y on (T  heat,  found,  within note  the  w h i c h a l s o depends  loga-  - T) n e a r t h e t r a n s i t i o n t e m p e r a t u r e .  -  10  CHAPTER  -  II  APPARATUS AND EXPERIMENTAL PROCEDURE The experimental data i n this thesis was obtained with a standard steady state nuclear resonance  spectrometer.  A block diagram of the apparatus i s shown i n figure 1 . C i r c u i t diagrams of the more common units are given i n Appendix A, together with some notes on the operation and construction of the various u n i t s .  D e t a i l s of the low  temperature system are also given i n Appendix A.  Only  modifications, o r i g i n a l designs, and the experimental procedure s h a l l be described i n t h i s chapter. The o s c i l l a t i n g detector was a s l i g h t l y modified version of a c i r c u i t by Watkins and Pound . 4  We have substi-  tuted a voltage sensitive capacitor,(varicap), f o r the mechanically variable capacitor i n the tuned c i r c u i t i n order to minimize i n c i d e n t a l noise and to eliminate sudden frequency changes during very slow sweeps due to i r r e g u l a r i t i e s i n the mechanical drive (see figure IA i n the Appendix). These modifications, together with the l i n e a r voltage sweep described below, proved very convenient and time saving. The sawtooth voltage required to sweep the frequency of the o s c i l l a t i n g detector by means of a varicap voltage sensitive capacitor i s derived from a modified Tektronix type  Oscil. Det.  Freq Counte]'  farrow Band Ampl.  Digita:. Recorde: •  Broad Band Ampl.  Phase Sens. Det.  Record Miliia  Scope  Phase Shift  (3  Sawtoot Sweep  Phase Shift  f  anostat  Vacuum Pump  30 cps, Source  Sample <3  Magnet  1  Fig. 1  Hagnet  Magnet Power Supply  Block Diagram of Apparatus  Oscil. Detector  2 Oscil. Detector f o r Field determination n d  - 12 162 Waveform Generator.  Part of t h i s generator i s shown i n  figure 2 (permission f o r reproduction of sohematic granted by Tektronix Inc., May 11, 1962), the modifications being indicated by dashed enclosures.  The basic operation of the  Tektronix 162 Waveform Generator i s f u l l y described i n the Tektronix 160 Series Instruction Manual.  The modifications  are described below. Changing the values of the timing r e s i s t o r s and timing capacitorsS) as indicated i n figure 2 provides a variable sawtooth voltage rundown from 6000 volts per hour to as slow as 8 volts per hour.  When SW6 Is closed, V2B i s at  cut-off and the plate of Y4 r i s e s to maximum voltage, causing the timing capacitor to quickly recharge and a new cycle may be started by opening SW6.  Thus, SW6 provides a means of  starting a new cycle during any part of the voltage rundown. The starting voltage i s preset by means of R7A and the cathode follower Y7A.  Any voltage between 50V and 150V may  be selected, producing a sawtooth output at the cathode of V5B decreasing uniformly from the preset voltage to p o s i t i v e 20V. The voltage sensitive capacitors used i n this experiment are rated at 100V maximum and are most sensitive at low voltages.  In order to obtain a sawtooth voltage varying  uniformly between 100V and 0V, the cathode of V5B i s connected to the g r i d of V7B, and the cathode of V7B i s returned to the -170 v o l t supply through a 50V Zener diode and a 33K  50K  LZ.170V  p i n  of  V4  |  |  i«-Wv\4>/\A-LwWv^ 6  Fig. 2  1.6  2  2.4 3.2 4  5  6.5  8  10  Modified Tektronix 162 Waveform  14  J o  lO^M  Generator  Arm of R21 SW3A«pin 1—>< of V4  +100 to 0 sawtooth output -•SW3B »<— pin 8 of V57  - 14 resistor.  The voltage appearing at the cathode of the Zener  diode i s 50V lower than that at the cathode of V5B.  Thus  we now have a sawtooth voltage of maximum, range between positive 100V and negative 50V.  The negative t a i l l s elim-  inated by means of diode F4 and R7B.  The f i n a l output i s  a uniformly decreasing voltage adjustable between 100V and 0V. Thus, with the modifications described above, any range and slope of the voltage rundown may be selected to sweep the frequency of the o s c i l l a t i n g detector.  The change  i n o s c i l l a t o r frequency corresponding to t h i s voltage sweep depends on the type and number of varlcaps used and on the inductance of the o s c i l l a t o r c o i l .  Various combinations of  varicaps were used, and i t was possible to obtain frequency sweep rates from a few hundred Kc/sec to a few cycles per second. A c i r c u i t diagram of the 30cps source and i t s accompanying  phase s h i f t i n g networks i s shown i n figure 3.  The frequency of t h i s source i s locked to the frequency of the  A.C. mains by means of a bistable multivibrator V2  which i s triggered by the 60cps mains voltage.  This  arrangement minimizes interference from the A.C. mains v i a f i e l d modulation and detector output amplification.  A 30cps  sinusoidal voltage i s obtained from the multivibrator square wave by means of two f i l t e r s tuned to 30cps. remainder of the network of the 30cps source i s s e l f explanatory from the schematic of figure 3.  The  Decoupling cathode follower  i—  1  2oo 1 0  Fig. 3  30 cps Source and Modulation Signal Output  Each i n 10 steps  F i g . 3 cont'd.  Phase S h i f t e r , Audio, and Reference Voltage Outputs  - 17 The magnet current was derived from a power supply adapted from a c i r c u i t by R. L. Garwin . ;  Since the magnet  used i n t h i s work has a considerably higher impedance than that used by Garwin, i t was necessary to Increase the r e f e r ence r e s i s t o r to 0.1 ohms.  Resistance variations i n the  reference r e s i s t o r were minimized by means of a water cooled o i l bath.  I t was also found that the performance of this  c i r c u i t improves greatly i f the s i g n a l transistors are water-cooled.  A c i r c u i t diagram of the existing regulator  i s shown i n figure 5A. The remainder of the apparatus i n figure 1 i s standard equipment and s h a l l not be described here, except for  i t s uses i n t h i s work.  not  mentioned here are shown i n Appendix A. The external f i e l d H  C i r c u i t diagrams of the units  Q  i s supplied by an air-cooled  iron-oore electromagnet (manufactured by Newport Instruments Ltd.,  Ser. No. 6010/3) with four inch diameter plane pole  t i p s and adjustable air-gap.  With an air-gap of 3.2 cm an  inhomogenelty of about 0.3 gauss per cm exists near the center of the pole faces at a f i e l d of 5.0 Kgauss.  The  magnet i s mounted on a rotating table provided with a graduated scale so that any f i e l d orientation r e l a t i v e to the sample i n the plane of r o t a t i o n can be obtained.  Field  modulation i s achieved by means of a pair of f l a t  coils  glued to the pole faces i n order to minimize modulation interference i n the magnet power supply and at the same time  - 18 to minimize the pole face separation.  The modulation  current i s supplied by a Williamson type power amplifier driven by the 30cps source (see figures 1 and 3).  With  maximum input from the 30cps source, a modulation f i e l d of 100 gauss can be achieved. The proton resonance l i n e widths i n CoCl «6Hg0 are 2  of the order of 5 gauss or larger.  To observe the deriva-  tive of the true l i n e shape with a recording milliameter, the modulation amplitude should not exceed about 1/4 the l i n e width and so a modulation of about 1 gauss was used when recording the derivatives of the resonance l i n e s . With the above-mentioned  current regulator f i e l d 5  s t a b i l i t i e s of the order of 1 part i n 10 obtained.  can be e a s i l y  The magnetic f i e l d was measured before and a f t e r  each run, and i n some experiments several times during each run  with the a i d of a second o s c i l l a t i n g detector by  measuring the frequency of the proton resonance i n a sample of water doped with approximately 2% of CuSO^SHgO.  The  f i e l d was obtained from 0O = fiHOi where }( i s well known f o r protons i n water. Single c r y s t a l s of 00012*6H 0 were grown from 2  aqueous solutions and found to be of the same type described by P. Groth ). 7  CoCl .6H 0 c r y s t a l s grow very rapidly, but g  g  tend to form multi-oriented, p o l y c r y s t a l l i n e blocks. Several growing methods were employed with varying degrees of success.  The most successful arrangement i s shown i n Fig.4a.  -  19  -  _ Kovar seal  [ ^ ^ \ Ground glass seal  ^ !•  Saturated  Brass top to f i t dewar cap  Teflon spacer  CoClo'6H 0 2  solution  Seed  Solid CoCl2'6H 0 2  Heater Coil  Tapered plug 4a  F i g . 4a. C r y s t a l Growing Vessel F i g . 4b. C r y s t a l Holder  4b  - so  -  A temperature gradient of 1 or 2 degrees between the top and  the bottom of the solution i s maintained by  means of the heater c o i l .  A few i r r e g u l a r pieces of  c r y s t a l l i n e CoCl •6H„0 are placed i n the t a l l of the vessel, 2  <J  and the solution surrounding these pieces i s more concentrated and s l i g h t l y warmer than the top layers of the solution. centrated  Thus a continuous upward current of a more consolution i s maintained.  As t h i s solution current  reaches the upper layers It cools and some of the excess solute i s deposited thread.  on the seed suspended with a thin nylon  With this method i t was  possible to obtain good  single c r y s t a l s with dimensions roughly up to 3 x 2 x 2 Cylinders 8 mm  cm.  i n diameter and about 2 cm long were  cut from the grown c r y s t a l s , taking great care to insure that a p a r t i c u l a r c r y s t a l axis was axis.  p a r a l l e l to the cylinder  These cylinders were placed i n a thin-walled  "Teflon" tube which i s attached  to a 3/8  (0.5  mm)  inch stainless  s t e e l tube i n the manner shown i n figure 4b.  The s t a i n l e s s  s t e e l tube with the Teflon holder was mounted v e r t i c a l l y so that the f i e l d H axis (H  Q  Q  would be perpendicular  i s adjusted  to be h o r i z o n t a l ) .  to the cylinder I t i s estimated that  the error i n orientation did not exceed 5 degrees i n the various mountings.  With t h i s arrangement d i f f e r e n t c r y s t a l  cylinders could be investigated under i d e n t i c a l conditions since the same sample holder could be used f o r d i f f e r e n t crystals.  - 21 The low temperatures were achieved with the a i d of an ordinary double dewar glass oryostat.  A schematic of the  low temperature system i s shown i n figure 6A.  Temperatures  lower than 4.2°K were obtained by pumping on the helium vapour and were maintained constant by regulating the rate of pumping with a manostat.  The temperature was measured  by observing the vapour pressure with a meniscus type cathetometer.  The lowest temperature obtained was about  I t was feared that the Teflon holder would prevent rapid temperature equilibrium between the sample and the helium bath, but extensive investigations showed that t h i s was not the case.  A l l temperature sensitive properties of  CoClg*6HgO investigated i n t h i s work were rechecked against various rates of temperature change and were found to follow the temperature (vapour pressure) very c l o s e l y .  When  r a i s i n g the temperature, a strong l i g h t was directed into the l i q u i d helium bath and maintained there u n t i l the helium boiled v i o l e n t l y to insure proper s t i r r i n g of the l i q u i d . The t i p s of both dewars were unsilvered, providing a path f o r r a d i a t i o n into the helium.  I t was found that the temperature  of the sample could be raised by 0.1° K i n a few seconds by placing a common 100 watt l i g h t bulb near the dewar t i p s . Whenever measuring a temperature dependent quantity by continuously following the temperature (vapour pressure), the measurement was always made at decreasing temperatures,  - 22 and even then at a slower rate than previously found permissible.  When measuring the t r a n s i t i o n temperature by  following the s i g n a l amplitude as a function of vapour pressure (hence temperature), the temperature v a r i a t i o n was often stopped at some a r b i t r a r y temperature, and i t was found that the signal amplitude remained at whatever value the  temperature v a r i a t i o n was stopped, indicating that the  sample temperature followed c l o s e l y the vapour pressure. I f there were not a continuous equilibrium, the signal amplitude should continue to change, as i t i s a very sensitive function of temperature near the Neel temperature. I t was found that t h i s continuous temperature equilibrium was even true f o r increasing temperatures, because enough l i g h t from the room i n general could get into the dewars providing a continuous heat source.  However, a l l measurements were  made at decreasing temperatures. Frequency measurements were made with a Hewlett Packard Model 524C Electronic Counter coupled to a Model 561B D i g i t a l Recorder. the  In the antiferromagnetic phase where  separation between l i n e s i s very large (up to 15 Mc/sec.  at 2.1° K) the s i g n a l was merely brought onto an oscilloscope screen and the resonance frequency recorded.  In the para-  magnetic phase and i n experiments near the Neel temperature a l l signals were recorded on a recording milliameter.  A  marker on the recording milliameter was coupled to the d i g i t a l recorder i n order to correlate the numbers from the  \  \  - 23  -  d i g i t a l recorder to positions on the recording charts. The usual method of phase-sensitive detection  was  employed to record the derivatives of the actual resonance lines.  In a l l cases, the frequency of the o s c i l l a t o r rather  than the magnitude of the external f i e l d was rate of sweep was  varied.  The  adjusted i n each case to avoid d i s t o r t i o n  of the s i g n a l by Integration through the r e l a t i v e l y long time constant  i n the phase sensitive detector.  - 24 CHAPTER I I I THEORY OF THE POSITIONS OF THE RESONANCE LINES In  this chapter a b r i e f summary of the theory  describing the positions of the proton magnetic resonance l i n e s as a function of the externally applied magnetic f i e l d i n hydrated single crystals containing paramagnetic ions l s presented.  The calculations which follow below ON  were developed by N. Bloembergen ' i n connection with s i m i l a r measurements i n CuSo^»5HgO. The position and width of each resonance l i n e are complicated functions of the d i r e c t i o n and magnitude of the  applied magnetic f i e l d  H, 0  structure of the c r y s t a l .  and of the Internal  For each d i r e c t i o n of the applied  f i e l d the separation between the l i n e s increases with decreasing temperature.  The character of the resonance  spectrum changes d r a s t i c a l l y as we pass from the paramagnetic phase ( T > T ) to the ant i f erromagnetic phase (T < T^). N  Theo-  r e t i c a l formulae f o r the positions of the component l i n e s are  developed by considering the proton two-spin system  within a water molecule of hydration acted upon by the —>  externally applied f i e l d  H  Q  and the inhomogeneous time-  averaged f i e l d a r i s i n g from the electronic magnetic moments of the cobalt ions.  The dipolar Interactions with the pro-  tons i n neighbouring water molecules and with other nuclei  - 25  -  are assumed to contribute only to a broadening of the resonance l i n e s . We  consider a c r y s t a l containing paramagnetic ions  and one or more water molecules of hydration.  Except f o r  the positions of the protons, the c r y s t a l structure i s generally known.  At each point i n the c r y s t a l there  be, i n addition to the applied f i e l d , a  will  time--varying  inhomogeneous i n t e r n a l f i e l d contributed by the e l e c t r o n i c magnetic moments of the paramagnetic ions with spin S. There i s also a weaker f i e l d due to the neighbouring proton magnetic moments with spin I = 1/2.  The magnitude and  d i r e c t i o n of t h i s i n t e r n a l f i e l d at any given point depend on the orientation and separation of the magnetic moments at that time.  The Hamiltonian describing t h i s system may  be  written i n the form 1'  -JEp^-H,  +  z +%  (3.1)  5  si  where |3 i s the nuclear Bohr magneton and ^ tropic g- factor i n the tensor form.  i s the aniso-  The f i r s t and  last  terms i n (3.1) are the Zeeman energies of the cobalt ions and protons i n the externally applied f i e l d H  Q  respectively.  ^55  i s the magnetic Interaction between the cobalt ions,  and  $?ex  represents  ^  n e  exchange energy between them.  The term  the magnetic i n t e r a c t i o n between the cobalt ions  and proton moments, and  <^!J-J i s the magnetic i n t e r a c t i o n  - 26 -  between the protons  themselves*  Since we consider the proton spin system immersed i n the homogeneous applied f i e l d  H  and the inhomogeneous  Q  fluctuating f i e l d produced by the neighbouring magnetic moments, i t i s convenient to divide the Hamiltonian into two parts: calculated and  (a)  (3.1)  , from which the i n t e r n a l f i e l d i s  (b) the Hamiltonian f o r the proton spin  system subjected to the t o t a l l o c a l f i e l d at the proton s i t e s . In considering the cobalt ions we s h a l l neglect  c$?s£ , since  the e f f e c t of the protons on the system of cobalt spins i s very small as compared to al^ and the magnetic energy of the cobalt spins i n the f i e l d H . Q  great importance  This term, however, i s of  i n the Hamiltonian f o r the proton spin sys-  tem to be discussed below. neglected i n comparison with  The term S^&x  $ss  ^ay also safely be  , since the exchange  energy i s roughly 100 times greater than the magnetic energy between two cobalt ions.  Thus, the Hamiltonian of the cobalt  spin system, using the standard notation f o r the exchange interaction, i s :  K  where  J  >K  A]^ i s the so-called exchange i n t e g r a l between the  cobalt spins. The term i n equation (3.1) connecting the two spin systems i s  and may be written  - 27 -  where JJL^ i s the magnetic moment of the k u  / Ik  i s the radius vector connecting  cobalt ion and  th the i proton and the —>  k  tiL  cobalt ion.  As pointed out before, JJ^  varies r a p i d l y  with time due to the exchange coupling between the cobalt ions. The exchange i n t e r a c t i o n causes a p a i r of a n t i p a r a l l e l cobalt spins to change d i r e c t i o n simultaneously,  so that there i s  no energy involved, but the magnetic f i e l d due to JX^ changes d i r e c t i o n i n the neighbourhood of a p a i r of cobalt spins. The exchange frequency i s approximately given by where  k"T^  i s the paramagnetie-antiferromagnetic phase tran-  s i t i o n temperature. =  %CD&t*  5  x  10 cps. 10  For CoCl .6H 0 g  = 2°K, so that  2  The Larmor frequency of protons i n the 7  i n t e r n a l f i e l d s of magnetic c r y s t a l s never exceeds 10 cps, so that the protons cannot follow the rapid variations of the i n t e r n a l f i e l d .  The proton spins react only to the  time-averaged f i e l d of the cobalt ions. the time-average ^U^^ /X^  We therefore can use  of Jjt^ i n (3.3), i . e . , the operator  i n equation (3.3) i s replaced by a number, ^J*£y  The time-averaged magnetic moment  ty^-k)  °^  t  n  e  cobalt i o n  spins can now be calculated i n p r i n c i p l e from the reduced Hamiltonian (3.2) of the cobalt ion spin system by the  .  - 28 diagonal sum method described by Van V l e c k \ since the time 8  averaged  should be the same as the s t a t i s t i c a l mechan-  i c a l average f o r the equilibrium state of the system. ^JX^  Co"*  spin  i s proportional to the magnetization M i n the  paramagnetic phase  (T>TJJ)  and to the so-called sub-lattice  magnetization i n the a n t i f erromagnetic phase  (T < T^.).  The effect of the time-averaged magnetic moment  (/Mc)  OVl  t n e  Photon spin system i s now obtained from the  Hamiltonian of the protons:  V?  Since the dipole-dipole i n t e r a c t i o n i s proportional to 1 / r , only the nearest neighbours w i l l contribute appre55  ciably to t h i s i n t e r a c t i o n .  We therefore consider only  interactions between the two protons i n the same water molecule i n the l a s t term of  ( 3 . 4 ) .  The interactions with other  protons i n the c r y s t a l and with the time dependent part of the f i e l d produced by the cobalt ions contribute only to the width of the resonance l i n e s .  We now have the r e l a t i v e l y  simple system of two i n t e r a c t i n g protons Immersed i n the  homogeneous e x t e r n a l f i e l d internal field the protons  H  p l u s the s t a t i c  Q  produced by the neighbouring  a r e subjected E  loc  =  H  oobalt i o n s , i . e .  to a f i e l d  int  +  H  o*  To f i n d the resonance f r e q u e n c i e s , its  inhomogeneous  ( 3 . 4 ) must be s o l v e d f o r  eigenvalues which g i v e the energy l e v e l s o f the two-proton  system.  We here present  the r e s u l t s o f N. Bloembergen , who ;  o b t a i n s f o r the energy l e v e l s o f the two protons molecule to f i r s t o r d e r p e r t u r b a t i o n  E  z  i n one water  theory:  = - d + \ l b * + <&  (3.5)  where  a = Vz?A  (Hi + Hi) (3.6)  bd - -  Vzn(Hi~Hi) t M p k  2  ( i - 3 « o * 0  -3  ;  2  ) £  - 30 -  1 and H  2 and H^.  z  are the z-components of the f i e l d produced  by the cobalt ions at protons 1 and 2 respectively when the z-direction i s chosen p a r a l l e l to  EQ.  The quantities a and 1  b are functions of temperature by virtue of  and H  with /U^ replaoed by (jX^y  are calculated from  2  1  which  z  .  The  energy l e v e l s i n ,(3,.5) give r i s e to four t r a n s i t i o n frequencies as follows: loci  tlV  3  *rhti  -CL+ld-tf+d '  -  1  0  jfflHLoc  3  (3.7)  with corresponding i n t e n s i t i e s a r b i t r a r i l y normalized to 4 T 1  _  T 2  =  (b-«£ -te  +  dff  bW-b/t>%d ' 1  \  ,o  (S.8)  Thus, with the approximations made i n the preceding paragraphs, each water molecule of hydration i n p r i n c i p l e gives r i s e to four resonance l i n e s .  In the absence of the magnetic  ions a m b = 0, and the problem has the solution given by Pake  ;  f o r the water molecule i n gypsum.  case, a, b  I f , i n the present  d, which i s c e r t a i n l y true at room temperature,  - 31 then the influence of the cobalt ions may be p a r t l y ignored and Pake's solution s t i l l holds.  However, i n CoCl »6HgO the 2  number of protons per unit c e l l i s large and the cobalt magnetic moment i s not completely negligible even at room temperatures and we do not observe any resolved l i n e s at room temperature, but a single l i n e about s i x gauss wide. At helium temperatures, none of the terms i n (3.6) are necessarily n e g l i g i b l e , and each water molecule should-give r i s e to four l i n e s i n two p a i r s of two.  In CoCl '6HgO the  positions of the protons are not known.  However, i f we  2  assume that the protons i n each water molecule i n the unit c e l l are magnetically nonequivalent, then the s i x d i f f e r e n t water molecules of hydration should lead to 24 l i n e s i n s i x groups of four. that b ^>d,  At low temperatures we can safely assume  since the f i e l d of the cobalt ions i s much  larger than that of the protons.  Then we have four l i n e s of  equal i n t e n s i t y consisting of two p a i r s , the centres of which are separated by a distance 2b, and the separation between the two l i n e s i n a pair i s 4d.  From (3.6) we see  that f o r some p a r t i c u l a r configuration b may be equal to zero and only h a l f of the maximum number of l i n e s w i l l be observed. Also, when the- sums of the terms i n the resonance frequencies i n (3,7) containing the geometrical factors are equal, the resonance frequencies f o r the protons i n d i f f e r e n t water molecules w i l l coincide.  I t i s thus expected that the number  of resonance l i n e s observed depends on the orientation of the  - 32 c r y s t a l r e l a t i v e to the external magnetic f i e l d H  Q  and  on  the temperature of the system. The d i s t i n g u i s h i n g c h a r a c t e r i s t i c of the resonance diagrams i n the paramagnetic phase i s the p e r i o d i c i t y of the resonance curves.  From H^"  i n (3.6) we see that the z  resonance curves are sine functions with a period of 180 degrees.  In the next section we s h a l l see that t h i s period-  i c i t y changes to 360 degrees i n the antiferromagnetic phase. These points w i l l be discussed i n d e t a i l i n the next chapter i n connection with the proton resonance measurements i n paramagnetic and antiferromagnetic CoClg 6Hg0. #  A l l the above  arguments apply to the paramagnetic state of the c r y s t a l where \tik) <  &  ^  M /(T+  to a f i r s t approximation,  0  where  i s the Curie temperature. The antiferromagnetic phase (T < T ) i s characterized N  by a spontaneous alignment of the cobalt ion spins. exchange i n t e g r a l A.,  i n (3,2) may  The  be p o s i t i v e or negative.  I f i t i s p o s i t i v e , then we have an ordinary ferromagnet with all  the spins aligned i n one d i r e c t i o n .  I f A.,  i s negative,  adjacent spins w i l l a l i g n i n opposite d i r e c t i o n s giving r i s e i n the simplest case to two ferromagnetic ing  sublattices point-  i n opposite d i r e c t i o n s . The bulk magnetic moment of the  sample i s s t i l l zero i n the absence of an applied f i e l d , Inside the c r y s t a l there now  but  e x i s t s a strong i n t e r n a l f i e l d  alternating i n d i r e c t i o n from cobalt ion to cobalt ion. Recently i t has been suggested ^ that there may 10  exist more  - 33 than two directions of alignment i n some substances.  A  discussion of t h i s p o s s i b i l i t y i n CoCTg^SHgO i s postponed to a l a t e r chapter.  For the purpose of describing the main  features of the proton resonance curves i n antiferromagnetic C o C l » 6 H g O i t w i l l s u f f i c e to assume a two-sublattice 2  system.  The spontaneous alignment of the cobalt ion moments along some preferred d i r e c t i o n i n the c r y s t a l at a very low temperature i s an inherent property of the system and persists even i n the presence of an applied f i e l d .  There-  fore, i n c a l c u l a t i n g the positions of the resonance l i n e s i n the antiferromagnetic  phase, the arguments applied to the  paramagnetic phase must be s l i g h t l y modified. In considering the l o c a l f i e l d at the proton p o s i tions we must take into account the fact that the cobalt spins have f i x e d d i r e c t i o n s and that adjacent spins point i n opposite d i r e c t i o n s .  The t o t a l magnetic f i e l d  at a proton p o s i t i o n i s the vector sum of H f i e l d due to the cobalt ions.  cobalt ion  Q  with the i n t e r n a l  In order to calculate the  p o s i t i o n of each resonance l i n e as a function of the d i r e c t i o n  -> of the external f i e l d H , we need consider only the component Q  -* of the i n t e r n a l f i e l d p a r a l l e l to H . Q  component w i l l be discussed l a t e r .  The perpendicular  For s i m p l i c i t y  consider  one water molecule i n the v i c i n i t y of a single cobalt ion magnetic moment  JJ-Co a  n  d  assume that  jXcc i s  i  n  "tlie plane  —»  of r o t a t i o n of H . Q  The magnetic f i e l d due to the cobalt  ion at the position of proton i obtained from (3»3) by factoring out  , i s given by  -  34 -  (3.9)  The component of t h i s i n t e r n a l f i e l d p a r a l l e l to the applied field  H  Q  i s then  and may be written i n the form  MttlH, -  %  FcosoL-3COS  COS(oC+  'Ik  where  J  i s the f i x e d angle between  we assume that (JJ^  ( 3 , 1 0 )  ^U-k) an<3-  ^ik > since  i s coupled t i g h t l y to the c r y s t a l  l a t t i c e , and cL i s the angle between  ^/J^)  H Q * Equa-  tion (3.10) may be rewritten i n the more convenient form  where  d  k  = [(/-ieas^ft*)**  (5/2  sin  2&IK)J  - 35 and oCf. i s defined by  tan  Both  oC =• 3/g 0  A|< and  k  are constants of the c r y s t a l l a t t i c e , and  the v a r i a t i o n of i t U H o H  n  by aC alone.  $kZ&£  with orientation i s determined  Therefore, to determine the positions of the  proton resonance l i n e s i n the antiferromagnetic phase, the term H_  i n (3.6) must be replaced by (3.11).  noted that i n (3.11) ^$k} p o s i t i v e , while {J%£y negative.  I t should be  belonging to one sublattice i s  belonging to the other sublattice i s  Thus, the i n t e r n a l f i e l d i s oriented i n opposite  directions f o r corresponding protons i n adjacent unit c e l l s , and the number of resonance l i n e s should double as we pass from the paramagnetic to the antiferromagnetic phase.  Also,  f o r each resonance l i n e displaced i n one d i r e c t i o n due to Hint II Ho  there w i l l be another resonance l i n e displaced an —  -»  equal amount i n the opposite d i r e c t i o n due to H ^ ^ ^ ,  . The  resonance curve diagram w i l l therefore be symmetric about the centre of gravity of the l i n e s .  Tn addition to the  displacement of the resonance l i n e s due to the p a r a l l e l ponent of H^ , nt  com-  there i s also a contribution of the perpen-  d i c u l a r component, which causes a s l i g h t lack of symmetry of the resonance curves about the centre of gravity of the resonance diagram. The main difference between the paramagnetic phase  - 36 and the antiferromagnetic phase i s the p e r i o d i c i t y of the respective resonance curves.  In the paramagnetic phase the  resonance curves were sine functions with a period of 180 degrees r o t a t i o n of the applied f i e l d H .  Equation (3.11)  Q  shows that t h i s p e r i o d i c i t y has now been changed to 360 degrees i n the a n t i f erromagnetic phase due to the cos (pL + ^ factor i n H  5-  . Also, since i n this case the spins a l i g n spontaneously  i n their respective d i r e c t i o n s , the i n t e r n a l f i e l d s w i l l be much larger i n the antiferromagnetic phase than they are i n the paramagnetic phase.  This means that the resonance  lines  w i l l be displaced much greater distances from the centre of gravity.  In the next chapter we s h a l l see how well these  points are borne out by experiment. In  this chapter we have described the magnetic  at nuclear positions i n terms of the applied f i e l d  H  Q  field plus  the i n t e r n a l f i e l d produced by the surrounding magnetic moments at the nuclear p o s i t i o n s .  We have thus neglected the  contribution to the average magnetic f i e l d due to the average macroscopic magnetization M per u n i t volume.  In the next  chapter we s h a l l see that contributions of t h i s type amount to about 40 gauss at 4.2° K.  I f we want to take t h i s extra  f i e l d into account, the external f i e l d be replaced by  Bo =  H  0  H  Q  should everywhere  + ( 4 ^ - N)M, where N, the demag-  n e t i z a t i o n factor f o r the c r y s t a l , must be calculated from the geometry of the sample ^. 11  Only i f the sample i s i n the form  - 37 o f an e l l i p s o i d w i l l  M  be uniform over the whole sample.  For any o t h e r c o n f i g u r a t i o n ,  M  w i l l be a f u n c t i o n o f  p o s i t i o n i n the c r y s t a l and t h i s n o n - u n i f o r m i t y o f e q u i v a l e n t to an inhomogeneity  M  is  i n the a p p l i e d f i e l d i n that  i t w i l l c o n t r i b u t e t o the n u c l e a r resonance  line  width.  - 38 -  CHAPTER  17  THE PROTON RESONANCE IN CoCl «6H 0 2  2  AT CONSTANT TEMPERATURE A.  THE CoCl '6H 0 CRYSTAL 2  2  CoCl "6H 0 forms dark purple monoclinic c r y s t a l s 7) 2  2  of the type described by P. Groth  .  are a:b:c . 1.4788:1:0.9452 with  The u n i t c e l l edges  - 122°19'.  Perfect 12)  cleavage occurs along the C(OOl) face.  J . Mizuno et a l  ;  have analyzed single c r y s t a l s of CoCl '6H 0 by the X-ray 2  2  method and report two formula units per u n i t c e l l with space group C c = 6.67A . 0  3  - C /m 2  g l i  with a = 10.34A , b = 7.06A , and 0  0  The atomic positions, except f o r the protons,  are given i n the following table: of atom Position  X  Z  z  0  Co  origin  0  0  Cl  4(i)  .278  0  .175  °I .  8(3)  .0288  -.221  .255  °II  4(j)  .275  0  .700  The proton positions cannot be found by the X-ray d i f f r a c tion method. According to the authors of reference- - ) , two Cl"" 1  2  ions from 4 ( i ) and four water molecules from 8(j) form  - 39 an octahedron with the Co •f-t ion at the center to form the group  (CoCl *4HgO).  Co  ion and the four water molecules, and the two C l ~ ions  + +  are  2  The octahedral plane contains the  at the apexes of the octahedron.  The other two water  molecules of the formula unit are located at somewhat greater distances from the cobalt ions.  The groups seem to  be joined with one another by hydrogen bonds  0j«••H-OJJ-H*  •»0j  and 0j-H»»»01 i n the plane p a r a l l e l to (001), the a-b plane. B.  RESULTS AND DISCUSSIONS In t h i s chapter measurements at two fixed tempera-  tures, T = 4.2°K i n the paramagnetic phase and T = 2.1°K i n the antiferromagnetic phase are discussed.  The  temperature  dependence of the proton resonance, Including a detailed discussion of the results near the Neel temperature T  N  =  2.275°K, w i l l be presented i n the next chapter. In making these measurements the c r y s t a l was kept  fixed i n the laboratory reference frame while the orientation of  H  Q  was changed by rotating the magnet. ->  measurements  H  Q  In one set of  was oriented i n the a-c plane of the  c r y s t a l , while i n another set i t was oriented i n the  a'rfb  plane, where a' l i e s i n the a-c plane and i s mutually perpendicular to the b and c axes, i . e . , a 32°19  T  makes an angle of  with the a-axis. The angle between H and the s designated by y . In subsequent discussions 0  s h a l l r e f e r to rotations.  H  Q  p a r a l l e l to the a -axis i n both  The measurements were made with the magnitude of  - 40 H  Q  kept constant, while the o s c i l l a t o r frequency was swept  through the resonance spectrum at a rate slow enough to avoid d i s t o r t i o n of the l i n e shape. When the separation between the component l i n e s was small (1 Mc/sec or l e s s ) , as i s the oase i n the paramagnetic phase, the derivatives of the resonance l i n e s were recorded using a phase sensitive  detector and a recording milliameter.  The o s c i l l a t o r frequency was marked d i r e c t l y on the recording chart with an automatic "event marker" at approximately 5 Kc/sec i n t e r v a l s .  The resonance frequencies were then  established by observing the 1st, 3rd, 5th, etc. n u l l i n the derivative of the resonance spectrum.  In this way permanent  reoords of the spectra were obtained which could be used to study the l i n e shapes and i n t e n s i t i e s of the i n d i v i d u a l l i n e s . A sample recording i s shown i n Figure 5. In the case of large l i n e separations, as i n the antiferromagnetic phase, the l i n e s were not recorded on a recording chart.  The d i f f e r e n t resonance l i n e s were brought  into the centre of an oscilloscope screen by adjusting the o s c i l l a t o r frequency manually, and the resonance frequency was read d i r e c t l y from the frequency counter.  Derivatives  were recorded only when the l i n e widths were desired. (i)  The proton resonance i n the paramagnetic phase at 4.2°K. The results of the measurements i n the paramagnetic  phase are shown i n Figures 6 to 9.  Each set of points  20.8  20.9  21.0  21.1  21.2  21.3  21.4  21.5  Frequency i n Mc/sec Fig. 5  Derivative of the Resonance Spectrum with H i n the a-b plane. H =5000 gauss, f = 100°, T = 4,2 ° K Q  Q  21.6  21.7  - 43 p a r a l l e l to the frequency scale was obtained from a recording of the type shown i n Figure 5. In each of the Figures 6 to 9 the proton resonance frequency i n water i s indicated by a v e r t i c a l l i n e and the positions of the c r y s t a l axes are marked on the "orientation scale".  The measured resonance  frequencies are indicated by s o l i d points.  S o l i d l i n e s are  then drawn through the experimental points.  In places where  the resonance l i n e s overlap, the n u l l i n the derivative i s not a true measure of the peak resonance frequency, since the peak i s s l i g h t l y shifted due to the overlap.  No corrections  were made f o r such s h i f t s , since the l i n e shape i s not known. Figure 6 shows the resonance diagram f o r the H the a-c plane with H T s 78° K.  Q  =  Q  rotating i n  5000 gauss and the sample temperature  Figures 7 and 8 are the resonance diagrams f o r H Q  rotating i n the a-c plane at 4.3°K with H H  Q  = 4970 gauss respectively.  0  s  3005 and  Figure 9 shows the resonance  diagram f o r lf 0 rotating i n the a'-b plane with H gauss and T  s  Q  = 4990  4.2°K.  As pointed out i n chapters I and I I I , the separation between component l i n e s depends on the temperature of the sample and on the applied f i e l d .  The d i f f e r e n t resonance  l i n e s i n Figure 6 were never completely resolved and only a maximum number of s i x l i n e s could be observed.  However, the  resonance diagram of Figure 6 i s b a s i c a l l y similar to the diagrams of Figures 7 and 8, since a l l three diagrams were obtained with  H  rotating i n the a-c plane.  The resolution  180  H J.c G  ffl ©  u bO 0  Q  H ±a Q  90  H l|c o  Proton frequency i n water 21.300 Mc/sec a  o  •H +»  a  H  0  I'  a  +J  a  © •H  o  C T Ho l Resonance frequency Mc/sec 21.20  21.30  21.25  F i g . 6. Resonance Diagram f o r H  Q  i n a-c plane.  J  21.35  H  Q  I  21.40  = 5000 gauss.  i  _  21.45  T = 77  K  12.6  Fig. 7  12.7  12.8  Resonance Diagram f o r H  Q  12.9  i n a-c plane.  13.0  H  Q  13.1  = 3005 gauss.  13.2  T - 4.2° K  180  Resonance frequency Mc/sec 20.9  Fig. 8  21.0  21.2  21.1  Resonance Diagram for H  G  21.4  21.3  i n a-c plane.  H  Q  - 4970 gauss.  21.5  T - 4.2  21.6  K  20.8  Fig. 9  21.0  Resonance Diagram f o r H  21.2  Q  a'-b plane.  21.4  H  Q  = 4990 gauss.  21.6  T = 4.2  21.8  K  -  47  -  i n Figures 7 to 9 i s much, better, and i n some orientations completely resolved l i n e s could be observed.  A l l four  diagrams are c h a r a c t e r i s t i c of the paramagnetic  phase as  outlined i n ohapter I I I . Each curve i n the resonance  dia-  grams behaves to a f i r s t approximation l i k e a sine function with a period of 180 degrees due to the factor (3 c o s 0 2  i n the i n t e r n a l f i e l d s .  -1)  This i s i n good agreement with the  theory i n chapter I I I . The resonance diagram f o r the a'-b r o t a t i o n (Figure 9) i s symmetric with respect to the a'-axis and the b-axis, i n agreement with the space group C /m.  The  2  symmetry about the a'-axis i s due to a* l y i n g i n the mirror plane m, the a-c plane; and the symmetry about the b-axls i s due to b being a two-fold axis perpendicular to the mirror plane m. In each of the resonance diagrams the centre of gravity of the l i n e s i s shifted from the proton frequency In water, indicating the need f o r the ideas put f o r t h at the end of chapter I I I . The amount by which the l i n e s are shifted from the proton frequency i n water, due to the average bulk magnetization i f . can be calculated i f theoretical resonance diagrams are available, but these can be obtained only i f the proton positions are known. However, approximate s h i f t s can be obtained by assuming the protons are situated 13)  at the oxygen positions.  These ideas have been used  ' in  connection with e a r l i e r measurements to determine the r a t i o s The two r a t i o s  _ 48 were found to be the same; as they should be, since M, space average magnetization, and <^/T^  >  t n e  the  time average  magnetic moment of a given magnetic ion, should have the same temperature dependence. The  t h e o r e t i c a l number of 24 l i n e s was  At most, eight l i n e s were present,  suggesting  never observed, that some of  the protons have the same l o c a l f i e l d s , thus giving r i s e to i d e n t i c a l resonance frequencies. protons are magnetically the proton positions.  To determine which of the  equivalent requires a knowledge of  Figures 7 and 8 show that the eight  l i n e s appear i n four pairs (indicated by l a , l b ; e t c . ) . The spectrum i n the a-c r o t a t i o n f o r  = 115° at  4.2°K has been investigated as a function of the applied field H .  In Figure 10 are shown plots of  Q  f"4b - f  g a  field.  - f  4  and  as a function of the magnitude of the applied  The separation  ^ f = f ^ - f  4  i s seen to be inde-  pendent of the applied f i e l d and must thus be due to the proton dipole-dipole interaction i n one of the water molecules.  The  separation f  4 f e  - f  g  i s a l i n e a r function of  H  Q  and i s due to the i n t e r n a l f i e l d produced by the cobalt ions, which i s proportional to the applied f i e l d .  We  could  now  9)  apply the method of Pake ' to calculate the proton-proton distance  r  from the s p l i t t i n g of the l i n e pair 4. However,  the measured s p l i t t i n g s are r e l i a b l e only i n the v i c i n i t y of (p  = 115°.  For  (pg  70° and  (p£,  130° the s p l i t t i n g  i n l i n e p a i r 4 cannot be measured accurately since  F i g . 10 The Frequency Difference ( f 4 ~ f4b) and (±4& - f2a) vs applied F i e l d H .. The Orientation i s at 110° w.r.t. F i g . 8. T - 4.2°K a  Q  - 50 there Is too much overlapping of l i n e s . J. W. McGrath and A. A. S i l v i d i the proton-proton  14)  have found that  ;  distance r i n water molecules of hydration  i s nearly constant for a great variety of substances with an average value of  r = 1.595  i  0.003 A*, assuming that the o  proton-oxygen distance i s always 0.97 the average H-0-H  A.  With these  angle i s 100 ± 5 degrees.  values  I f we assume  that these values also hold f o r CoClg'6H 0, the d i r e c t i o n of 2  the proton-proton  vector r* due to the two protons giving  r i s e to l i n e p a i r 4 can be found. Due we  to the large magnetic moment of the cobalt ions  can safely assume that b ^>d  i n (3.7), and the separation  between l i n e s 4a and 4b i s  - 4d -  AH where  6^  (3cos 9 , l  i s the angle between  terms of frequency  H  tl  - l)  and  Q  2  (4.1) becomes  i s conveniently expressed  d i r e c t i o n cosines of ~r  and  and  H  Q  4 1  Expressed i n  Af = f J j ( W e , , - i ) The term c o s  <-'  _^  —»  r and  H. Q  <*•*) i n terms of the  The d i r e c t i o n cosines of  respectively are defined, using Figure 11 as  oC'- COS/9 , &'* COS B '  > Y'= COS, C U  (4.3)  - 51 -  Figure 11,  D e f i n i t i o n of the d i r e c t i o n angles of  r and  H. 0  Equation (4,2) can now be rewritten i n the form  df-  a 3(oC'cM$(p-  (4.4)  If'sinff ~ 1  where  when  r  i s chosen to be equal to 1.595  A .  Some simple algebraic manipulation i n (4.4) leads to  Af/a  (4.5)  = 3/2 (</-/'*}- 1  +  fy,(J -f )cos2((p+f ) l  l  0  - 52 where ^  i s defined hy  j.  ^ ^'  orf)  ^Tfi  tan Equation H  Q  (4.6)  (4.5) describes the proton-proton s p l i t t i n g when  rotates i n the a-c plane.  (Af/a) max occurs at  Since the maximum s p l i t t i n g  - 115° i n l i n e pair 4 i n the a-c  r o t a t i o n , we have that  cos Z(H5° and  +  therefore  -ii5°  or  % - n s * = &s*  so that, from (4.6)  cC'f  '  (r %-oL' 1)  0.5352  In Figure 12 equation  ( 4  '  7 )  (4.5) has been f i t t e d to the experi-  mental s p l i t t i n g Lt (solid points) i n the v i c i n i t y of  Cp = 115°.  The maximum s p l i t t i n g at (p - 115° i s  ' (M/o.)  mx  - Z. 00 * 0.06  which leads to  - 0 , so that  This implies that to the a-o plane. values of oC  "? i s i n a plane p a r a l l e l  From F i g u r e . 1 2 and equation  and Q  are found to be Z  J"  =  0 . 1 7 7 8  *  0 . 8 1 7 8 .  ( 4 . 7 ) the  -  2S  -  - 54 One set of solutions obtained from these values f o r the angles A and 0 i s eliminated using the f a c t that ~l£ must be 0  p a r a l l e l to the p-^ - p  l i n e when 4 f l s a maximum, and one  g  obtains A  =  65° and  C  = 25°  —^  Thus the d i r e c t i o n of r  f o r the two protons giving r i s e  to l i n e p a i r 4 has been f i x e d .  Since the two protons In  each water molecule should give r i s e to two pairs of l i n e s , each p a i r being s p l i t by an amount 4d, there should be a p a i r of l i n e s i n Figures 7 and 8 whose s p l i t t i n g i s i n phase with l i n e p a i r 4.  Line p a i r 2 s a t i s f i e s t h i s  requirement.  However, i t i s d i f f i c u l t to establish experimentally whether this pair of l i n e s i s r e a l l y the counterpart of pair 4, since l i n e p a i r 2 i s never completely separated from the r e s t of the spectrum. Applying equation (4.2) i n the a*- b rotation f o r the d i r e c t i o n of r At -29.6  obtained from l i n e p a i r 4, the s p l i t t i n g  i s -13.8 Kc/sec when Kc/sec when  H  Q  H^ i s p a r a l l e l to the a'-axis and  i s p a r a l l e l to the b-axis.  In Figure  9 the measured separation i n l i n e p a i r 1 i s about 25 Kc/sec when  ~R i s p a r a l l e l to the b-axis, and less when Q  any other d i r e c t i o n .  R  0  is in  When two l i n e s are so close to one  another that they overlap, as i s the case i n l i n e p a i r 1 i n Figure 9, the measured peak frequencies are not the true resonance frequencies, since the peaks are shifted towards  - 55 one another due to the overlap.  In l i n e pair 1 of Figure 9  the measured peak separation i s approximately l i n e width of each l i n e .  We may  equal to the  therefore assume that l i n e  p a i r 1 i n Figure 9 i s produced by the same protons as line p a i r 4 i n Figure 8, and l i n e p a i r 2 i n Figure 9 i s 'then the counterpart of l i n e p a i r 1. The i n t e n s i t y of l i n e p a i r 4 i n Figure 8 i s approximately 1/6  of the t o t a l i n t e n s i t y of the spectrum i n the a-o  rotation.  The i n t e n s i t y of l i n e p a i r 1 i n Figure 9 i s also  about 1/6  of the i n t e n s i t y of the spectrum i n the a'-b rota-  tion.  In the a-c r o t a t i o n the protons associated with the Oj  are magnetically equivalent, since plane of the c r y s t a l .  H  0  rotates i n the mirror  The two l i n e pairs discussed above  must thus originate from protons associated with the Oj^ , since one proton from each of the 0 comprises 1/6  of the t o t a l number of protons i n the u n i t c e l l .  In the a'-b observed.  water molecules  This may  r o t a t i o n (Figure 9) s i x l i n e p a i r s are now  be explained by the fact that i n this  r o t a t i o n only the Oj water molecules situated diagonally opposite from the cobalt ion are magnetically equivalent. There are thus a t o t a l of three magnetically equivalent groups of water molecules i n the u n i t c e l l , two associated with the octahedron about a cobalt ion, and the 0 ^  group.  Each group  gives r i s e to two p a i r s of l i n e s , thus accounting f o r the s i x pairs of l i n e s i n Figure 9, At present i t i s not possible to uniquely determine  - 56 the proton positions i n CoCl «6H 0 from the proton resonance & 2 measurements due to the r e l a t i v e l y large number of l i n e s and t h e i r extensive overlapping.  A solution to the problem should  become f e a s i b l e when the present measurements are repeated at much higher f i e l d s and a third rotation i n the b-c plane i s completed, made.  and accurate r e l a t i v e intensity measurements are  Since a crystallographic analysis was not the goal of  this work, these measurements were not pursued to the f u l l e s t possible extent. (ii)  The proton resonance  i n the antiferromagnetic phase at  2.1°K. Measurements similar to those mentioned i n the preceding section have been made i n the antiferromagnetic phase (T < T^) at T - 2.1°K.  The r e s u l t s are shown i n the  resonance diagrams of Figures 13 and 14 respectively f o r H rotating i n the a-c plane and f o r H plane.  Q  rotating i n the  Q  a'-b  These results exhibit some features s t r i k i n g l y  d i f f e r e n t from the measurements i n the paramagnetic  tempera-  ture region. The c h a r a c t e r i s t i c features of the rotations i n the antiferromagnetic phase are:  (l) Each curve i n Figures 13  and 14 behaves l i k e a sine function with a period of 360° instead of 180° as i n the paramagnetic  phase.  (2) The  resonance diagrams are symmetric with respect to a central frequency which i s about 200 Kc/sec higher than the proton frequency i n water, i.e. for each l i n e displaced by an amount  - 57 -  16  18  F i g . 13 H  0  20 22 24 Resonance frequency i n Mc/sec  Resonance Diagram f o r H  = 5000 gauss.  q  26  i n the a-c p l a n e .  T - 2.11° K  Proton frequency i n water a t 21.28 Mc/sec  27  - 58 -  Proton  frequency i n water at 21.28 Mc/sec  -  -f-^f  59  -  from the central frequency there i s an Identical l i n e  displaced by an equal and opposite amount — &>f from the central frequency.  ( 3 ) In general the l i n e s are much broader  than i n the paramagnetic phase.  ( 4 ) The maximum number of  l i n e s observed i s again 8, but there i s no evidence of a proton-proton s p l i t t i n g .  ( 5 ) The maximum separation between  l i n e s i s of the order of 10 Mc/sec Instead of 6 0 0 Kc/sec as i n the paramagnetic phase.  ( 6 ) In the a*-b r o t a t i o n (Figure  14) the resonance diagram i s again symmetric with respect to the a - a x i s and the b-axis i n agreement with space group C2/m. f  These points are i n good agreement with theory as outlined i n chapter I I I . In connection with  (3.11)  of chapter I I I i t was  pointed out that the number of l i n e s should double when passing from the paramagnetic to the antiferromagnetic phase. Experimentally only eight l i n e s are observed.  However, the  separation between any pair of l i n e s i s much too large to be attributed to a proton dipole-dipole i n t e r a c t i o n .  We there-  fore conclude that each of the curves i n Figures 13 and 1 4 consists of two resonance l i n e s , the s p l i t t i n g between them being too small to be observed since the l i n e s i n general are quite broad as compared to the l i n e widths i n the paramagnetic phase. f a c t has  Thus we can say that the number of l i n e s i n  doubled. The fact that s i x l i n e pairs were observed i n the  a'-b rotation i n the paramagnetic phase does not contradict  - 60 -  the above arguments i n regard to Figure 14, since i n the antiferromagnetic phase the cobalt spins are coupled to the c r y s t a l axes and l i e i n the a-c pendent of the applied f i e l d .  (mirror) plane inde-  The four 0^ water molecules  associated with the octahedron are thus magnetically equivalent even i n the a'-b r o t a t i o n i n the a n t i f e r r o magnetic phase. non-equivalent  Therefore, only two groups of magnetically  water molecules are present, and only eight  l i n e s are expected i n the antiferromagnetic phase In the a -b r o t a t i o n . f  The spectrum at (f = 133° i n the a-c r o t a t i o n has been investigated as a function of the applied f i e l d at T = 1.14° K.  The results are shown i n Figure 15, where we  have plotted the entire spectrum as a function of  H'. 0  These measurements show that the separation between the various l i n e s i s independent of the applied f i e l d . r e s u l t proves that the magnetic moment /U^of  This  the cobalt  ions i n the antiferromagnetic phase i s independent of the applied f i e l d as long as i t i s not too large.  The shape of  the resonance diagrams i s determined by the vector sum of  H  Q  of  with the i n t e r n a l f i e l d due to the antiferromagnetic  cobalt ions  ^//.^  •  1 1 1 6  small demagnetization  field  merely adds to the applied f i e l d . No attempt has been made to give t h e o r e t i c a l formulae f o r the resonance curves i n Figures 13 and  14  since the proton positions are not yet known and since i t  10  15  20  25  Resonance frequency i n Mc/sec F i g . 15  Plot of Resonance Frequencies 2 to 7 of F i g . 13 as a function of the  applied f i e l d .  f  = 133°. T - 1.14°K  30  - 62 i s not known what the sublattice arrangement of the cobalt ion magnetic moments i s i n the antiferromagnetic phase. In a l a t e r chapter we s h a l l see that there i s some evidence that more than two sublattices are present i n a n t i f e r r o magnetic CoCT »6H 0. 2  2  - 63 -  CHAPTER  THE  V  PARAMAGNETIC-ANTIFERROMAGNETIC PHASE TRANSITION  The proton magnetic resonance technique provides a simple but accurate method f o r investigating the paramagnetic-ant i f erromagnetic phase t r a n s i t i o n i n hydrated salts. the  In chapters I I I and IV i t was pointed out that  character of the resonance spectrum changes d r a s t i c a l l y  when the sample temperature i s lowered from the paramagnetic to the antiferromagnetic temperature region.  This change  i n character of the resonance spectra can be used to determine the t r a n s i t i o n temperature T^ as a function of the applied f i e l d , and the c r y s t a l orientation r e l a t i v e to the applied  field. The t r a n s i t i o n temperature of CoClg*6HgO i n zero 15)  applied f i e l d has previously been determined by Sugawara  '  (T^ a 2.35"* .05°K) from proton magnetic resonance experiments; by Robinson and F r i e d b e r g ) ( T = 2.29°K) ; and by Miss 16  N  Voorhoeve and D o k o u p i l ^ (T = 2.28°K) from the 17  X - t r a n s i t i o n i n the s p e c i f i c heat, and by Van der Lugt and P o u l i s ) ( T = 2.275 i ,01°K) from proton magnetic reson1 8  K  ance experiments similar to those to be described i n the following paragraphs.  We have e s s e n t i a l l y duplicated and  extended the measurements of Van der Lugt and Poulis with  - 64 Improved accuracy.  Our results are i n good agreement with  t h e i r s , hut additional d e t a i l s have been brought out not reported by these authors. The t r a n s i t i o n from the paramagnetic phase into the antiferromagnetic phase i s characterized by a broadening and decrease i n amplitude of the resonance l i n e s i n the paramagnetic phase.  In Figure 16 a p l o t of the signal amplitude  of the resonance l i n e s l a , 2a, and 4b of Figure 8 f o r <^= and H  Q  130°  = 5000 gauss i s shown as a function of temperatures.  The amplitudes given here are differences between maximum and minimum values of the f i r s t d e r i v a t i v e .  Figure 16 shows that  the amplitudes of the resonance l i n e s increase s l i g h t l y as the Neel temperature i s approached and drop very r a p i d l y near the t r a n s i t i o n temperature.  However, the rate of change of  the amplitude i s d i f f e r e n t f o r d i f f e r e n t l i n e s .  Similar  graphs were obtained f o r other values of (f a n d l l , and i n 0  each case the general behaviour i s similar to that of Figure 16. The region near the t r a n s i t i o n temperature Is of particular interest.  Figure 17 i s a magnification of the  region near T = 2.2°K, including part of the antiferromagnetic temperature region.  Over a small temperature region,  approximately 7 millidegrees i n Figure 17, l i n e s from both phases are present simultaneously.  As we s h a l l see i n a  l a t e r paragraph, the l i n e s associated with the paramagnetic phase broaden rapidly as the t r a n s i t i o n temperature i s  - 67 approached.  Thus very broad l i n e s of low amplitude  begin  to overlap and eventually i t becomes impossible to i d e n t i f y any single l i n e .  Also, the l i n e s associated with the a n t i -  ferromagnetic phase f i r s t appear near the proton  frequency  in water so that these l i n e s also overlap with one  another  and with the l i n e s from the paramagnetic phase very near the t r a n s i t i o n temperature.  I t i s thus impossible to analyze  t h i s region quantitatively from the existing data and to determine the exact range of coexistence.  The range of co-  existence seems to be independent of (fi  I f there i s any  v a r i a t i o n i n this range with accuracy of t h i s experiment.  .  , then i t i s beyond the Measurements f o r H  Q  = 3000  gauss indicate that the coexistence range increases with decreasing H . 0  At 3000 gauss the temperature range over  which both spectra exist simultaneously i s of the order of 15 m i l l i d e g r e e s . Such regions of coexistence have also been observed MnFg  2 0  ^  in- A z u r l t e ) (about 160 millidegrees) and i n 1 9  (about 25 m i l l i d e g r e e s ) . On f i r s t sight one might attribute t h i s coexistence  region to l a t t i c e imperfections i n the c r y s t a l or to "supercooling" and "superheating", i . e . h y s t e r i s i s .  Lattice  imperfections are ruled out by the fact that the same range was always present f o r d i f f e r e n t samples and that the range is field  dependent.  We have eliminated the p o s s i b i l i t y of  h y s t e r i s i s by varying the temperature within the coexistence range and were able to reproduce any p a r t i c u l a r set of l i n e s  - 68 over several hours.  We  thus tentatively a t t r i b u t e the  region of coexistence of both types of spectra to short range order.  I t would be interesting to test t h i s conjec-  ture experimentally. The l i n e widths also show an anomalous change near the t r a n s i t i o n temperature.  In Figure 18 a plot of the l i n e  width of l i n e 5 of Figure 9 (paramagnetic) and of l i n e 8 of Figure 14 (antiferromagnetic) i s presented as a function of temperature with H  0  = 5000 gauss at  s  120°.  i s very suggestive of a A-type t r a n s i t i o n .  This graph  I t has been  pointed out by Nakamura (private communication) that superexchange interactions between the protons v i a the cobalt ions, or between protons and paramagnetic impurities at the cobalt ion p o s i t i o n s , could produce a l i n e broadening of this type. S t r i c t l y speaking,  the two branches of the l i n e  widths i n Figure 18 should cross over near 2.24°K, since some l i n e s associated with the antiferromagnetic phase appear before those of the paramagnetic phase have vanished.  No  attempt has been made to measure l i n e widths throughout the t r a n s i t i o n region, since here most of the l i n e s overlap, making accurate line-width determination  impossible,  e s p e c i a l l y i n view of the very small amplitudes.  As pointed  out previously, h y s t e r i s i s i s absent i n this t r a n s i t i o n . Indeed, the fact that some of the ohange occurs even before the t r a n s i t i o n region i s reached shows that there i s no choice  1 1.8 F i g . 18 H  I  I  I  2.0  2.2  2.4  I  I  I  I  I  2.6  2.8  3.0  3.2  3.4  I 3.6°K  The l i n e width of l i n e 5 i n F i g . 9 and l i n e 8 i n F i g . 13 vs temperature  - 5000 gauss,  $  = 120°.  •paramagnetic,  A a n t i f erromagnetic  - 70 of paths f o r the t r a n s i t i o n .  The new phase i s established  gradually rather than at one single temperature. Figure 17 points out some ambiguity i n the d e f i n i t i o n of the t r a n s i t i o n temperature.  S t r i c t l y speaking,  there i s a small range of temperature over which the trans i t i o n takes place.  We have investigated the t r a n s i t i o n  temperature as a function of the applied "field and c r y s t a l orientation.  These measurements were performed by  continuously monitoring the amplitude of one of the l i n e s associated with the paramagnetic phase, and T^ was chosen as that temperature at which t h i s amplitude had decreased to the  noise l e v e l .  Thus the lower values of the t r a n s i t i o n  range were obtained.  These measurements are possible, since  each resonance frequency i s independent of temperature i n the  paramagnetic phase, except f o r the l a s t 10 or 15 m i l l i -  degrees where the frequencies are shifted towards the proton frequency i n water.  These s h i f t s , however, are at most  50 Kc/sec whereas the l i n e widths i n the t r a n s i t i o n region are  usually much greater, so that the amplitude measure-  ments are only s l i g h t l y affected. of dp  A graph of T^ as a function  the applied f i e l d i n the a-c plane f o r three values of i s shown i n Figure 19.  i n the a*-b plane.  The  T^ i s independent of orientation  ^P-dependence of T^ has been inves-  tigated f o r several values of H  Q  and  T (^p) i s found to N  behave l i k e a sine function with a period of 180 degrees. This behaviour of T ((^) can be expressed i n the form Kr  2.30  -  2.25  -  o  55  H  <a u  ft 2 20 ®  -  d o  •rl -P •H  2 2. 15  -  rt  u  H  H_ i n Kgauss F i g . 19  Paramagnetic-antiferromagnetic phase to= t r a n s i t i o n temperature  a function of the applied f i e l d for  <jP = 0 ° , 45°, and 90°.  as  - 72 -  = A(U ) +  T (<P)] N  where A(H ) 0  B(Hc)cos*<(p  0  H  and B(H )  are functions of H .  Q  Q  p l o t of T ( ^ ) Is shown f o r H N  the a-c plane.  Q  (5.1)  In Figure 20 a  = 5000 gauss and rotating In  For this value of H  the hest f i t f o r (5.1)  Q  i s obtained with A(H_)  =  B(H )  = 0.078°K  Q  and from Figure 19, B(H ) 0  H  0  i s found to be nearly l i n e a r i n  as shown i n Figure 21 with B(H )  = (1.45.10" x H )°K. 5  Q  We  2.162°K  0  cannot attribute this anisotropy of T  g-factor i n CoCl "6H 0. 2  2  N  to the anisotropic  The p r i n c i p a l values of the g-factor  have been measured by Based!' ) i n connection with paramag21  netic s u s c e p t i b i l i t y measurements, and by D a t e ) i n connec22  tion with paramagnetic experiments.  The values given by Haseda are g . = 2.7,  gb o 4.9, and g different.  and antiferromagnetic resonance  Q  = 4.9,  Date*s values are only s l i g h t l y  Thus, the g-tensor varies i d e n t i c a l l y i n the  a'-c plane as i n the a'-b plane, yet T^ Is independent orientation when H  rotates i n the a -b plane. r  Q  the v a r i a t i o n of T^ with  of  Therefore,  i n the a-c plane cannot be a  d i r e c t consequence of the v a r i a t i o n i n the g-tensor.  2.15 180  90  Relative orientation i n Fig.  20  Plot of  TJJ  .Experiment  vs Cp for —  H  Q  i n the a-c plane and  [ T < ? )] ^ n  270  degrees  _  5  -  H  2.162«  Q  +  •  5000  gauss.  0.078°  cos* f  .  - 75 Nagamiya, Yosida and Kubo ^ have established a 25  theory describing the variation of T^ with applied f i e l d on the basis of the Weiss molecular f i e l d approximation.  A  summary of t h i s theory with minor modifications to conform with CoCT '6HgO i s presented i n Appendix B. 2  The r e s u l t s by  Nagamiya et a l show that T^ may be expressed i n the form  T (H,) - T ( o ) - C«H* +••• N  >  (5,3  M  to a f i r s t approximation i n H , when the exchange i n t e r Q  actions are assumed to be i s o t r o p i c . on the orientation of  B^ . 0  The quantity C depends  D a t e ^ has found i t necessary to 24  introduce a d d i t i o n a l anisotropic exchange interactions to explain h i s antiferromagnetic microwave resonance i n CoCl »6HgO. 2  experiments  These additional terms, however, only change  the f i e l d independent quantity C i n (5.3), but not the nature of the f i e l d dependence.  Such a behaviour, however, does not  agree with the present measurements, since this implies that dT /dH N  Q  - 0 for H = Q  0.  Experimentally i t i s found that  T ( H ) has a f i n i t e slope at H N  0  = 0, and none of the curves  Q  i n Figure 19 can be approximated by a quadratic function In H.  In p a r t i c u l a r , the term B(H )  Q  Q  i n (5.1) was found to be 2  l i n e a r i n H Q , which completely rules out an H T ( H ) on E Q . N  0  Q  dependence of  Equation (5.3) implies that ! I ^ ( E 1 O ) decreases o  monotonically with increasing H , whereas experimentally we Q  observe a minimum at about 3000 gauss and a maximum at 5000 gauss.  This cannot be explained by any existing theories. The  -  76  -  present theory does, however, agree with the general trend of this experiment, namely, that the t r a n s i t i o n temperature i n zero f i e l d i s always greater than that i n a f i n i t e  field.  Similar measurements have been performed i n CuClg^HgO ). 25  However, the available data i s not  suffl-  s u f f i c i e n t l y accurate to r e a l l y test the present theory. I t i s not surprising  that t h i s theory does not agree  with the present measurements, since these were obtained very accurately, and the theory i s based on a crude approximation, not Including any d e t a i l s of the antiferromagnetic sublattice arrangements.  - 77 CHAPTER VI THERMODYNAMICS OF THE PHASE TRANSITION In this chapter we s h a l l discuss some thermodynamic aspects of the paramagnetic-antiferromagnetic phase transitions. As i s seen i n Figure 18, the proton resonance l i n e widths exhibit a lambda-shaped anomalous peak near the t r a n s i t i o n temperature TJ,  i . e . they increase anomalously as  T-*T either from below or above T^. N  has also been observed i n M ^ g  2 0  ).  This type of behaviour  Also, the s p e c i f i c heat  i n CoCl -6H 0 has been measured by Robinson and Friedbergi°) as a funotion of temperature between about 1°K and 20°K i n zero applied f i e l d .  These measurements show that the s p e o i f i c  heat also has an anomalous peak at 2«29°K.  The peak i s  i d e n t i f i e d with a cooperative order-disorder t r a n s i t i o n and the temperature at which the peak occurs i s the t r a n s i t i o n temperature.  Robinson.and  Friedberg found that the  behaviour of the magnetic contribution to the speoific heat near the t r a n s i t i o n temperature could be described by a logarithmic function of the form log (T-T ), which i s also N  c h a r a c t e r i s t i c of the anomaly i n the X - t r a n s i t i o n of l i q u i d helium.  Such a s i n g u l a r i t y i s integrable,giving a f i n i t e  enthalpy and entropy and no latent heat.  We can therefore  follow the method used by Buckingham and F a i r b a n k ^ i n their 26  analysis of the l i q u i d helium  A - t r a n s i t l o n to discuss some  - 78 thermodynamic aspects of the magnetic phase t r a n s i t i o n i n GoClg'SHgO. [T (H )] N  0  „  H  We confine ourselves to the case > since here the slope ^  c  T ( H ) i s always f i n i t e . N  of the X-line  The case where a H  0  beoomes  i n f i n i t e w i l l be discussed l a t e r . Using the i d e n t i t y  [dxjy  '  lax/?  wwx  157/2  where W, X, Y, Z are functions of state with two independent variables; and the Maxwell r e l a t i o n s , we have  (di)  fdT)  _/9J\  ./5T)  -/ar)  9  (6.2)  (dT)  and  (Si) Since  C  H  >oo  as  T—*  tea.) (^C  H  «  f(|ip)  (68  H  ,  )  we have from (6.2) that  /dr)  /<2I) . ^  at T=T  (6-4) M  - 79 and from (6,3) that  'dl\ (dT)  dT  _  (6.5)  N  The l a t t e r can e a s i l y be shown by using the r e l a t i o n  (dL\  =  (6.6)  (dT)  -(dT)  The left-hand side i n (6.6) vanishes on the (SI)  S  X - l i n e and  i s f i n i t e and equal to the slope of the  at T = T , K  so that  ^ s t vanish.  (6.3) immediately gives (6.5).  X-line  Using this f a c t i n  In the above r e l a t i o n s H  denotes the applied magnetic f i e l d . In order to study the v a r i a t i o n of thermodynamic properties near the t r a n s i t i o n temperature Buckingham and Fairbank introduce a new v a r i a b l e , the "neighbourhood temperature" t, a function of state defined by t The l i n e t = 0 i s the a l i n e p a r a l l e l to the Using  s  T-T (H).  X-li  (6.7)  N  n e  and.  a  li  n  e  t = t (constant) Is 0  X - l i n e i n the phase diagram.  (6.1) and the new variable we have  (6.8)  _ 80 and (6.9)  Combining  (6.8) and (6.9) gives  9a ^fiaffdM)  T  UTJ  +  W I T  fds\  lar/t  _fdH\fdtL\  ( 6 , 1 0 )  {3jNJ[dTjt  Buckingham and Fairbank argue that the quantity (^pj^ relatively  varies  l i t t l e over a small temperature range i n the  neighbourhood of the t r a n s i t i o n , although i t has an i n f i n i t e temperature derivative at the t r a n s i t i o n temperature i t s e l f . This can be seen from the following argument:  Near the  t r a n s i t i o n l i n e we could write  S = S(r ) +/»(h)fft) N  where A(H) i s a function o f the applied f i e l d H and the function f ( t ) has an i n f i n i t e derivative at t = 0. Then  so that t  f&L)  depends on t i n a manner similar to the  dependence of S i t s e l f .  Therefore,  even though (dS\ V8T/t  has an i n f i n i t e temperature derivative at T^, i t s t o t a l variation over a small temperature range near T^. w i l l nevertheless be small.  Similar arguments may be applied to the  - 81 quantity /2l£iV  .  Therefore, a plot of  should he l i n e a r near T  N  The s p e c i f i c heat  versus f ^ - j  fcLHfdTw'  with a slope  Cg of CoCl "6H 0 as a function 2  2  of temperature has been measured i n zero f i e l d by Robinson and  Friedberg ).  the  adiabatic s u s c e p t i b i l i t y  16  Flippen and Friedberg ) have measured 3  /«2£lV  "near" H  n  = 0 (the  measurements Involve a small o s c i l l a t o r y f i e l d ) .  We have  repeated some of the s u s c e p t i b i l i t y measurements with an improved accuracy. the  self-inductance of a c o i l containing a sample of  CoCl »6H 0. 2  the  Our measurements were made by measuring  2  The c o i l formed part of the tuned c i r c u i t of  same Pound-Knight-Watkins o s c i l l a t o r used to perform the  NMR measurements discussed i n the preceding chapters, and the  changes In self-inductance were calculated from the  changes i n the o s c i l l a t o r frequency.  The o s c i l l a t o r frequency  at 4.2°K was set to 21 Mo/sec and the t o t a l change i n frequency was found to be approximately 60 Kc/sec as the temperature changed from 4.2°K to about 1 3°E. 0  We have  calibrated our measurements against those of Flippen and Friedberg at 1.5°K.  The results of these measurements for  zero f i e l d are shown i n Figure 22, where the crosses (x) indicate the points published by Flippen and Friedberg.  The  two sets of measurements are i n good agreement, but i n the present measurements many more points were obtained, e s p e c i a l l y near the t r a n s i t i o n temperature.  The t r a n s i t i o n  temperature i s c l e a r l y indicated by a discontinuity i n the  1.2 F i g . 22  1.6  2.0  The molar s u s c e p t i b i l i t y  2.4  2.8  3.2  3.6  4.0  °K  along the c-axis as a function of temp.  - 63 slope of the s u s c e p t i b i l i t y , and t h i s method could a c t u a l l y be used to measure the t r a n s i t i o n temperature. These r e s u l t s , together with Figure 19, can now  be  used to show the mutual consistency of the heat capacity Cg as a function of temperature and f i e l d derivatives of the magnetization  i n the v i c i n i t y of T^ i n zero applied f i e l d .  In doing so we assume that the difference between the adiabatic and isothermal s u s c e p t i b i l i t i e s i s n e g l i g i b l e i n very small f i e l d s which i s reasonable i n the absence of an applied field. in  We  therefore use the adiabatic s u s c e p t i b i l i t y  (p-j^  (6.10) instead of the isothermal s u s c e p t i b i l i t y ( d U V  obtain  /  to  j , since the error introduced i n t h i s change  w i l l be n e g l i g i b l e .  A p l o t of  °W vversus ersus  Figure 23 f o r both T < T_ and T>T. . T  ature t = T -  '  [5777c i s shown i n  The neighborhood temper-  V T  i s Indicated by the arrows.  A l i n e a r plot i s obtained over the entire temperature The value of thus obtained range s tudied f o r T ^ T .. _ ~j i s 1.71 x 10 gauss/°K and i s shown as a dotted l i n e i n Figure W  N  WA  [B  4  19, and i s seen to be i n agreement with the T (H) N  obtained using proton resonance f o r H c-axis.  Q  curve  p a r a l l e l to the  The proton resonance data at low f i e l d s are rather  poor, because at such low frequencies the o s c i l l a t o r sensit i v i t y has greatly decreased, so that i t would be interesting to make plots similar to those of Figure 23 f o r large applied fields.  Unfortunately, no heat capacity measurements have  been made so f a r i n the presence of large d.c. f i e l d s .  0.07  0.10  Molar s u s c e p t i b i l i t y {dM/dH)c 0.15  0.20  (T< T ) N  0.25  CD  F i g . 23  Plot of (dtA/dti )s  v  s C  H  /T i n CoCl «6H 0. 2  2  - 85 For T > T between  we do not get a linear relationship  N  (^/d^  and  except f o r the region t£0„l°K.  I f one Identified the magnitude of the slope of the curve i n (HL^  t h i s region with  , one obtains  41  - l * ^ x 10 8  4  gauss /°K which i s i n agreement with the value f o r T< within experimental error.  However, there i s a d i f f i c u l t y i n  t h i s assignment, because the slope of the f^|)$  curve f o r t>0.1°K Is negative.  impossible since (dJt^  must be p o s i t i v e .  Qi  versus  This Is c l e a r l y Since no way  out  \dTfi) of this dilemma has been found, i t appears that the numerical agreement mentioned above may anticipate (^-j, temperature  be fortuitous and that one  may  going through a minimum as the Neel  i s approached from above f o r t  0,06°K.  There are two pieces of evidence which support this conjecture. to zero  F i r s t l y , extrapolation of  ^g)s  T£T,_, namely (6.10).  versus ^ £ 1 ) 5  should give the same value f o r T > T (S§\r- -  dJL (^M)^.  and  as seen from equation  The straight l i n e obtained f o r T<  to approximately -0.125 c a l / ( ° K )  K  2  extrapolates  - mole f o r t h i s quantity.  In order to obtain a negative value f o r t h i s quantity f o r T>T^,  C^]  s  must go through a minimum or  £&.  through a  maximum. More conclusive evidence that  (g$")s S °  e s  through a  minimum i n zero f i e l d i s that we have observed that t h i s does happen f o r H  Q  2000 gauss.  This i s seen i n Figure 24,  which i s a plot of o s c i l l a t o r frequency versus  temperature  - 86 H =0 Kgauss Q  21.185H  21.180f—  21.175f  .170\  o  Q> CQ \  S .165| a •H  >» O  a a>  a" . 160f u  «H  o +> ci  H  .155}  o CO  o  21.150  21.145 1.6  J  F i g . 24  I  L  2.0  J  1  T°K  1  O s c i l l a t o r frequency vs  L  2.5  J  1  Temperature  U  - 87 -  for various applied f i e l d s .  The curve for II  :  0 i s the  one from which the s u s c e p t i b i l i t i e s i n Figure 23 were obtained.  In order to make use of careful measurements of for t<0,06°K, the s p e c i f i c heat measurements  would also have to be made closer to T^.  Since there are no  s p e c i f i c heat measurements available f o r H  Q  j- 0, we are  unable to make quantitative use of our data of Figure 24 except f o r the H  Q  = 0 curve.  I t would also be desirable to do measurements of s u s c e p t i b i l i t y f o r the case of H preferred axis since  4=  0  perpendicular to the  i s i n f i n i t e f o r two values of  f i e l d i n that o r i e n t a t i o n . These measurements have not been carried  out yet, but one may anticipate that the nature of  the phase t r a n s i t i o n may change d r a s t i c a l l y above 3000 gauss. F i n a l l y , the anomalous increase of of frequency) a t low temperatures f o r H H  0  = 7,000 gauss i n Figure 24  0  =  (J^ffjs  (decrease  6,700, and  i s a manifestation of the  lowering of the c r i t i c a l f i e l d with decreasing temperature f o r the "flop-over" antiferromagnetic phase t r a n s i t i o n as IB)  discussed by Van der Lugt and P o u l l s  \  - 88 -  CHAPTER VII  THE SUBLATTICE MAGNETIZATION BELOW T  N  In this chapter sublattice magnet 125a tion measurements as a function of temperature f o r T<1 T^ are discussed. I t w i l l be interesting to note an empirical r e l a t i o n between the sublattice magnetization i n the antiferromagnetic state and the experimentally observed logarithmic dependence of the heat capacity on | T -  .  The results were obtained by measuring the proton resonance frequencies as a function of temperature at constant external f i e l d and a given o r i e n t a t i o n . Measurements were performed at two f i e l d s and about a dozen d i f f e r e n t orientations. A representative set of measurements i s shown i n Figure 35, where a-o plane at (p  B  H  Q  = 5,260 gauss was In the  0. Each set of points p a r a l l e l to the  frequency scale was obtained by holding the temperature constant and recording the frequency at which resonance occurs.  The measurements were usually terminated when the  t r a n s i t i o n region was approached, since here too many l i n e s overlap, making accurate measurements impossible. As stated before, the resonance frequencies are determined by the l o c a l f i e l d at the positions of the d i f f e r e n t protons, where' H B^  nf  l o c  = H  + B^ f ® r  Q  n  ie  quantity  denotes the i n t e r n a l magnetic f i e l d at the proton  'K -T = N  2.24°K  /  2.2  1  i  2.0  \  •\.  \  *8  V  1.8  00  to  1.6  1.4 •5  |  A  6  1.2h J_  _L  16  17  _L  19  18  20  21  22  1  23  1  24  J_  25  26  Resonance frequency i n Mc/sec F i g . 25  Proton resonance i n a n t i f e r r o m a g n e t i c CoC^'SB^O as a f u n c t i o n o f temperature, H  Q  = 5.26  Kgauss,  H  Q  c-axis.  -  90 -  positions produced by the neighbouring magnetic  ions. I t  i s now convenient to define the scalar quantity  k(^ T) For H 5> Q  H i n t  =|nW«fHffJ  -\jTiJ-\rZ\  }  , a condition s a t i s f i e d i n most of the experi-  ments, h Is the component of In  along the applied f i e l d  chapter I I I we saw that the component of the  i n t e r n a l f i e l d along  H  i s proportional to the sum of the  Q  time-averaged magnetic moments  ^ • In the a n t i f erro-  magnetic phase the magnetic moments of one sublattice a l l point i n one d i r e c t i o n , so that the sum of the magnetic moments over one sublattice gives the sublattice magnetization.  Thus a measure of h ( H , T) gives a measure of the 0  sublattice magnetization as a function of temperature. Since &) = Z'TfJ  = ^H  l o 0  j ( 7 . 1 ) may be rewritten i n terms of  frequency: f(«o,T) - f  where  2#Tfn )fK  and  r e s  (H  l 0 O  T) - f  o  -ffgS i s the measured resonance  frequency of a given l i n e at a temperature  T, and j i s the  proton resonance frequency i n water at an external f i e l d To avoid the contribution to  H. Q  from the bulk magnetization  of the sample, i t i s convenient to measure the difference i n frequency of a p a i r of resonance l i n e s , rather than the actual frequency.  This difference i s then roughly propor-  t i o n a l to the difference i n sublattice magnetization.  Thus,  -  91 -  the temperature dependence of the sublattice magnetization may be obtained by measuring  —¥  which i s independent of H  Q  and the bulk magnetization of  the sample. The rapid v a r i a t i o n of ^ f g  near T^ (Figure 25)  suggests a logarithmic dependence of Plots of  Af-j  Afjj  on  |T  - T^| .  versus l o g ( T - T) were prepared, and i t was N  found that a l l the proton resonance l i n e s give the following r e s u l t , which Is independent of H  Afij(fe) _ „ fa ( - f r - r )  where of  Q  and c r y s t a l orientation  + const  i n~  0,13±io%  ( 7  -pj (H ,0) was obtained by extrapolating the plots L  0  /\f;j versus l o g ( T - T) to ( T - T) = T  A plot  N  A^j  N  i . e . to T = 0.  versus l o g (T^ - T) corresponding to the  results of Figure 25 i s shown i n Figure 26. values of n  N >  The i n d i v i d u a l  are indicated f o r each p a i r of l i n e s ,  similar plot (not shown) was prepared f o r H  Q  k  = 0,using the  curve published by Van der Lugt and Poulis ^ , and the same relationship was found to hold. The above results suggest that the sublattice magnetization has the form  -  2 )  I  0.02  I  I  0.04  I  i  i i i I  0.1  L  0.2  I  I  0.4  I  I  i  i  i I  1.0  U  2.0  - 93 -  Mj^j.  -  n  in  T -T N  T  +  x  .  (7.3)  N  Equation (7,3) must, of course, break down as T approaches T. N  No evidence of this was observed to within about 20  millidegrees of T^.  We therefore conclude that the break-  down occurs within the t r a n s i t i o n region. Similar measurements have been performed and there I t was found that M(T) < = • < (T - T ) / 1  N  3  i n Mj^Fg ^ , 2  near T . I t N  i s thus not at a l l clear whether the sublattice magnetization i n d i f f e r e n t substances can be described by the same function. The shape of the curve given by (7,3) d i f f e r s considerably from the dependence of M(T) given by the Weiss molecular f i e l d approximation ). 18  No explanation i s given  here f o r the behaviour of M(T) i n CoCl »6H 0. g  2  An attempt to  understand I t could be made following D z i a l o s h i n s k i i ' s 27) method  , but the magnetic space groups of CoCl »6Hg0 are 2  not known. Neutron d i f f r a c t i o n experiments would, therefore, be desirable, but so f a r there are very few neutron d i f f r a c tion data on hydrated s a l t s of the t r a n s i t i o n elements.  A  group t h e o r e t i c a l a n a l y s i s ) of the proton resonances i n 2 8  the antiferromagnetic state may be successful however, and i s being pursued i n this department. In connection with the possible sublattice structure of CoCl »6H 0 2  2  two experimental r e s u l t s should be mentioned:  1)) Daniels and G r i f f i t h  (to be published) have studied the  -  angular dependence of  94  -  < f - r a d i a t i o n from single c r y s t a l s of  CoClg 6Hc>0, where the cobalt ions In the surface layers of e  the c r y s t a l were replaced by M n the d i s t r i b u t i o n of  54  ions.  I t was found that  <T-radiation could not be explained on  the basis of a two-sublattice model.  Although no sublattice  structure could be given from these measurements, i n d i c a tions were that a more complicated sublattice structure than the two-sublattice model exists i n CoCl »6H 0. g  2))  Measurements by Flippen and Friedberg  2  show that the  s u s c e p t i b i l i t y p a r a l l e l to the preferred axis does not approach zero at T = 0, which i s incompatible  with a two-  sublattice model, since such a model requires that ")C^ = 0 at T » 0.  - 95 -  CHAPTER YIII  SUGGESTIONS FOR FURTHER EXPERIMENTS The present measurements and others mentioned i n the foregoing chapters indicate the need f o r more extensive and varied experiments In CoClg*6H 0 to f u l l y explain the g  physical properties of t h i s substance.  The following experi-  ments would be f r u i t f u l : 1)  Measurements i n the paramagnetic phase should be  repeated at much higher f i e l d s than those employed present experiments.  i n the  Such measurements would greatly improve  the resolution of resonance l i n e s and enable a complete crystallographic analysis. 2)  A thorough study of the proton resonance l i n e shapes  and l i n e i n t e n s i t i e s would provide information as to how many protons contribute to a p a r t i c u l a r resonance l i n e .  This  information w i l l be p a r t i c u l a r l y useful i n the proposed magnetic space group analysis using the resonance diagrams of the antiferromagnetic phase. 3) detail.  The t r a n s i t i o n region should be investigated i n An exact knowledge of the range of the t r a n s i t i o n  and i t s f i e l d and orientation dependence could lead to a f u l l e r understanding of the nature of the t r a n s i t i o n .  In  this region double-resonance experiments would be of p a r t i cular i n t e r e s t .  The e f f e c t , i f any, on resonance l i n e s i n  - 96 the antiferromagnetic phase when a l i n e of the paramagnetic phase i s saturated would immediately  t e l l i f the t r a n s i t i o n  takes place by a gradual build-up of domains, or i f i n d i v i d u a l magnetic moments fluctuate between the two phases. 4)  NMR experiments on deuterated samples of CoCl *6H 0 2  might prove very u s e f u l .  2  For instance, i t would be very  i n t e r e s t i n g to see what e f f e c t the replacement  of protons by  deuterons would have on the t r a n s i t i o n temperature.  Since  deuterated samples should give resonance diagrams very simil a r to those presented i n chapter IV, such samples could be used f o r the high f i e l d experiments without the d i f f i c u l t y of obtaining high enough o s c i l l a t o r frequencies.  Some other  very u s e f u l experiments would be: 5)  A study of the s p e c i f i c heat as a function of an  externally applied f i e l d including the region very close to TJ.  As pointed out before, such experiments could  immediately  be used to check some of the ideas put f o r t h i n chapter VI. I t would be most interesting to see i f such measurements give the same phase diagram as obtained with NMR measurements. 6)  S u s c e p t i b i l i t y measurements should be made i n the  presence of an applied f i e l d f o r the three p r i n c i p a l orientations. 7)  As pointed out before, neutron d i f f r a c t i o n experi-  ments would be most u s e f u l .  These would determine the  magnetic space group of antiferromagnetic CoGl '6H 0 and 2  probably the nature of the sublattice structure.  2  - 97 -  APPENDIX  A  CONSTRUCTION HINTS AND OPERATING- INSTRUCTIONS  1)  O s c i l l a t i n g detector (Figure IA). The o s c i l l a t i n g detector was constructed i n a box  made of 1/8 inch brass sheet.  The H.T. and filament  supplies and their accompanying components are housed i n a "false bottom", and the leads from t h i s compartment to the tubes are kept as short as possible.  Faraday shields are  placed between the o s c i l l a t o r (6J6) and the r - f amplifier (3 x 40B3), between the r - f amplifier and the detector (6AL5), and between the detector and the audi-output (2 x 6C4).  This type of construction minimizes  tubes  microphonic  pick-up and undesirable r - f leakage between the various sections. 2)  Williamson Power Amplifier  (Figure 2A) .  Best r e s u l t s are obtained with t h i s c i r c u i t i f the following adjustments  are made when the input i s grounded:  a)  Adjust Rr> to equal  b)  Connect a suitable ma meter i n the lead to the  5  1200*Vload Impedance".  center-tap of the output transformer primary and set the t o t a l current to 125 ma by means of R ,. p  - 98 c)  Connect a moving c o l l voltmeter (0 - 10V) across  the whole of the output transformer primary and adjust R^^ u n t i l the meter reads zero volts indicating proper balance. Ignore small random fluctuations i n this adjustment  since  they are due to the A.C. mains and fluctuations i n tube characteristics. 3)  Magnet power supply (Figure 4A) . The power transistors (2N278*s) are a l l mounted on  a 1/4 inch copper plate soldered to a 1/2 inch O.D. copper tube f o r water cooling.  The reference r e s i s t o r i s made  from a 1 inch manganin s t r i p and immersed i n an o i l bath which i s maintained at a constant temperature cold water jacket.  by means of a  The reference voltage supply i s  imbedded i n styrofoam to avoid voltage fluctuations due to temperature  variations.  Operation:  Switch on the reference voltage and set the  helipot to some a r b i t r a r y value.  The meter on the Honeywell  N u l l Detector w i l l indicate a negative reading.  Now  increase the A.C. voltage by means of the variac u n t i l the detector meter reads zero and the voltage drop across the power transistors i s approximately 4 v o l t s .  The n u l l  detector w i l l "hunt" momentarily, but should soon s e t t l e at zero.  To obtain the desired magnet current, slowly vary  the reference voltage and the A.C. supply voltage u n t i l the desired magnet current i s reached, always keeping the drop across the power transistors at approximately 4 v o l t s .  - 99 4)  Low temperature  system (Figure 6A).  Except f o r the manometers and dewars, which are made of glass, the low temperature  system Is made of copper  and s t a i n l e s s s t e e l tubing. To "pre-cool":  Insert the syphon into the dewar,  close the free end, and evacuate the syphon jacket. Evacuate  the inner dewar and then f i l l  to 1 atm. pressure.  Evacuate  the inner dewar jacket to  approximately 4 cm Hg, and then f i l l l i q u i d nitrogen.  i t with He gas  the outer dewar with  From time to time add He gas to the  inner dewar to keep the pressure at approximately 1 atm. Keep the nitrogen l e v e l i n the outer dewar as high as poss i b l e at a l l times.  When the pressure i n the inner dewar  has settled to a constant reading (approximately 90 min.), the system has reached l i q u i d nitrogen temperature  and i s  ready f o r a He transfer. To transfer:  Completely evacuate  jacket and be sure to close stopcock A.  the inner dewar Open the free end  of the syphon and insert i t into the l i q u i d He vessel. Open valve 5 to allow the He gas to return to the storage vessel.  Apply 4 to 6 cm Hg pressure to the l i q u i d He can  by means of the compressed He gas cylinder, e f f e c t i n g a transfer of l i q u i d He into the inner dewar. Temperatures lower than 4.2°K: 3, 4, and 14 f u l l y . la  Open valves l c , 2,  Close 5 and immediately begin to open  and then l b i f necessary.  When the desired pressure  - 100 c l o s e l c and 4.  has been reached,  The pressure i n the  bellows B i s now f i x e d and any change i n the r a t e o f pumping w i l l be c o r r e c t e d by the expansion o f the b e l l o w s .  To change the temperature,  opened and the p r e s s u r e i n the bellows the d e s i r e d v a l u e .  or contraction valve 4 i s  i s a d j u s t e d to  F i g . IA  O s c i l l a t i n g Detector  250V —r-*  Twin "T" network. When t h i s i s removed, we have the broad band amplifier  .ooa_  =r.003  >15K  >COK  1-  „ 006=5 1  to centre of primary o f H448 i n phase s e n s i t i v e detector  17K  250K  8z±z  t o scope O :47K  _J  Metal can  ~T~ .1  From Osc. Det, U ' Lii  L  6C5  Wvv  6SJ7  Audio Gain  50  F i g . 2A  .IK  Narrow Band and Broad Band  8 >250 > K  3 K  Amplifier  =b  25  6SJ7  4 •250K  150  - 103 +250V  LOOK  6SN7  IOOK:  O  rtz  Q-  Record. M i l l i a m . zZZ. 4.7M  4.7M  \ZV\A/\Ar Zero meter C l o s e when balancing  20K  Tsec .1 .25 .5 .75 1.0 1.5 2.0  C jif .02  4.05 .1  4wl5  .2 •.3 .4  etc  etc  4.7M  •4.7M  Balance  [  50K  30K P l a t e of last ampl. O stage 1/26SN7 ( r ± I _ _  N / V V  v-  2  to scope "^or balance - V W  33K  H448  Ref. v o l t a g e input o  F i g . 3A  TL =  Phase S e n s i t i v e D e t e c t o r  T  4.7M 1/2 6SN7  F i g . 4A  Williamson Power Amplifier  ± A(b  Honeywell N u l l Detector  F i g . 5A  Transistor Magnet Current Regulator  To main pump  syphon to He gas storage B I) C Oil bubbler  «>D  4 7  to l i q . He ^vessel  «*o  8  Hg  oil  11  lo syphon 12 ^ jacket 13 -» Hg  He  sample Manostat F i g . 6A  Schematic of low temperature system  Outer Dewar  Temperature measuring manometers  107  APPENDIX  B  THE NAGAMIYA, YOSIDA AND KUBO THEORY FOR % ( H ) APPLIED TO  0o01g«6H 0 2  The exchange i n t e r a c t i o n between two ionic spins, responsible f o r the spontaneous alignment of neighboring spins i n ferromagnetism  and antiferromagnetism, i s  equivalent to an interatomic p o t e n t i a l  V -Li  '/2 J ( l +  where Sj_ and S j are the spin angular, momentum vectors of atoms i and j respectively, and J i s the exchange i n t e g r a l , which i s negative f o r antiferromagnetic substances.  In the  simplest form of the molecular f i e l d model, one assumes that J i s the same f o r a l l interacting neighbors and we replace J Sj  by 2  )  , where (S^}  i s the s t a t i s t i c a l  average of S-j over the sublattice to which The p o t e n t i a l energy of atom the sublattice  j  i  i  belongs.  i n the f i e l d produced by  i s , apart from an additive constant,  given by  - - Z T(tj)  Vi - I  - t f - jT ' A ej  (B.i)  where /M[ i s the magnetic moment of an i n d i v i d u a l atom of spin  and H ^ i s the exchange magnetic f i e l d produced by  the sublattice  e  j and acting on atom  i . We now make the  - 108  -  basic assumption that the whole antiferromagnetic c r y s t a l i s divided  into two s i m i l a r sublattlces, one with i t s  spins pointing i n the positive d i r e c t i o n and  the other  with i t s spins pointing i n the negative d i r e c t i o n .  The  exchange f i e l d acting on each spin of the positive  sub-  l a t t i c e results from the spins of the negative sublattice and also from the other spins of the positive s u b l a t t i c e . I f we  M + = H/Z (/T f)  define  M*~ =  and  N/fc  where N i s the number of magnetic Ions per unit volume, then the exchange f i e l d s on the positive and negative l a t t i c e are, respectively, from  rTg - - AM'- r / ? _>  WI where  A  _^  ~AM*  «  P  and  -  sub-  (B.l)  +  (B.2)  P M~  , containing the exchange i n t e g r a l J,  are the i s o t r o p i c molecular f i e l d constants of the c r y s t a l . D a t e ) has found i t necessary to introduce a d d i t i o n a l 24  anisotropic molecular f i e l d tensors  A  and  V  to  explain  his antiferromagnetic microwave resonance experiments i n CoCl *6H 0. A g  and  2  and V  are 3x3 matrices having zero  traces  i d e n t i c a l p r i n c i p a l axes which determine the p r i n c i p a l  axes of the antiferromagnetic state. to (B.2) may  now  be written  as  Equations analogous  -  He  109  -  (r +  (A-tA')M'-  P')fA  +  (B.3)  The t o t a l e f f e c t i v e f i e l d acting on a sublattice i s found by adding  H", 0  the external f i e l d , to  ^  9  .  To take into  account the anisotropy of the g-factor of the magnetic ions, i t i s convenient  to introduce the f i c t i t i o u s f i e l d  i s defined i n terms of the p r i n c i p a l values of  g  H',  which  by i t s  components as follows:  Since the p r i n c i p a l axes of  g' coincide with the p r i n c i p a l  axes of the antiferromagnetic state i n CoClg'6H 0, the t o t a l G  e f f e c t i v e f i e l d may  H« e  where  H~  '  now 7?,  be written .  (B.5)  7?±  H' * He  i s given i n (B.3).  H"^^  i s the t o t a l f i e l d at  the p o s i t i o n of an ion belonging to the positive ore negative s u b l a t t i c e , depending on whether the plus or minus sign Is used.  We may  now proceed i n the usual way  equilibrium values of  M - .  to f i n d the  A f t e r some algebraic manipula-  tion we obtain the f a m i l i a r expression  (B.6)  -  Bjfy )  where  -  i s the B r i l l o u i n f u n c t i o n f o r a s p i n S w i t h  1  fTe*| Sq(3/feT  = | j?+  yt  110  I t has been found a p p r o p r i a t e t o use 8-1/2  f o r CoClg'SHgO,  i n which case  [Bs(y- )]s= 1/2 " +  Let  the p r i n c i p a l v a l u e s o f A  A'x)Aj;  a n d  A'i  P/ j Py'j  P  and  Pi  be, r e s p e c t i v e l y ,  » where x, y, z  corres-  pond to the c r y s t a l axes c, b, and a'. In Mc  |*]-« H  +  where In  M  0o  the absence o f an a p p l i e d f i e l d we may w r i t e 0  and (B.6) becomes  =  the l i m i t o f T - + T ( 0 ) ,  the t r a n s i t i o n temperature  N  f i e l d , M - * 0 and o  tank  in  the square  in  (B.7) and s o l v i n g f o r  Vo;-  brackets i n (B.7).  M«tf[(A  0  i s the q u a n t i t y  S u b s t i t u t i n g these v a l u e s  we o b t a i n  +  the t r a n s i t i o n temperature In  T  where (j  %  ((jo)-*  A  0  -  (  i n zero a p p l i e d  r  (  B  we may proceed  i n the presence  as f o l l o w s :  -  8  )  field.  o r d e r t o f i n d the 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  t r a n s i t i o n temperature  In zero  o f an a p p l i e d  phase field,  - Ill a)  H  p a r a l l e l to the preferred  Q  A = A+A * x  and  P = T+  (x) a x i s . Let  , then M  t  1  becomes  (B.9) 2kT Rewriting (B.9), we have  tonk'^yUn.k" 1^  (Arrx)(W-M~)W  -  ( B . I O )  In the l i m i t of T-^T^, the magnetization M -» M"~, so that +  M,  (B.ll)  oo  When T = T , we have that M N  + =  M , so that (B.9) becomes  M+ = Moo tank '(H'-(A* +  r*)M+tefi]  (B.12)  -  Substituting  112  -  into (B.ll) we obtain  ( B . 1 2 )  ZkT  >^,*g)M*)q|S]  (^-r )M„qp>  r (o)  The solution of  ( B . 1 3 )  .  1 3  ;  ZkT  N  x  ( B  f o r T gives T ( H ) . N  Q  Since  T (H ) N  Q  varies only by a r e l a t i v e l y small amount with H , we are 0  interested i n small corrections to M  +  in  T (0) N  TjjfH).  ( B . 1 3 )  T  N  ( 0 ) ,  and can thus use  as a f i r s t approximation to f i n d  We may also note that  iXA(o>) < H  M+(T (o))N  ( B . 1 4 )  i so that  ( B . 1 3 )  becomes  (B.15)  T (0)  &  to/id  M  Equation H  Q  ( B . 1 5 )  .9  ^  XpTJ  Zt\(o)  gives T^ as a function o f the applied  field  i n a f i r s t approximation and becomes exact as Tjjj(H ) ->  TJJ(O),  0  I.e. when  H  Q  - >  0 .  For small values of  H  Q  ( B . 1 5 )  -  113 -  becomes  TM'T (0)  *  U  TN(0)  b)  H  Q  9  perpendicular to the preferred a x i s ; ( i . e . H  Q  p a r a l l e l to the y a x i s ) . In t h i s case the antiferromagnetic spins are no longer p a r a l l e l with  H .  In general,  M"" 1  —>_  and  —>  M  and M  l i e i n the x-y plane and make equal angles with = M~\  H, Q  The derivation of T ( H ) i n this case i s more N  0  complicated than i n the case discussed above, because the angle between calculations  TH(Ho) - T ( 0 )  r (o) N  N  H  and  0  M  must be taken into account.  The  give the following r e s u l t :  s  1 3  j s  g (A -r„) s  x  (B.17)  - 114 -  BIBLIOGRAPHY 1)  J " . H. Van Vleck, Phys. Rev. 74, 1168 (1948).  2)  N. Bloembergen, Physica 16, 95 (1950).  5)  R. B. Flippen and' S. A. Fried'berg, J". Appl. Phys. (Supp) 31, 338S (1960).  4)  D. G. Watkins, Ph.D. Thesis, Harvard University (1952), Cambridge, Mass, U.S.A.  5)  R. J". Blume, RSI 32, 743 (1961).  6)  R. L. Garwin, RSI 30, 105 (1959).  7)  P. Groth, Chemische Krystallographie, 1. T e i l , p. 248 Wilhelm Engelman Verlag, Leipzig (1906).  8)  J . H. Van Vleck, J". Chem. Physics 5, 320 (1937).  9)  G. E. Pake, J". Chem. Phys. 16, 327 (1948).  10) See f o r example: A. J". Heeger et a l , Phys. Rev. 125, 1652  (1961).  11) American Inst, of Physics Handbook, Chapter 5, p. 240. 12) J". Mizuno, K. Ukai and T. Sugawara, J". Phys. Soc. Japan 14, 583 (1959). 13) E. Sawatzky, M.Sc. Thesis, Univ. of B r i t . Col., (1960). 14) J . W. McGrath and A. A. S i l v i d i , J". Chem, Phys. 54, 522 (1961). 15) T. Sugawara, J . Phys. Soc. Japan 14, 1248 (1959). 16) ,w,. K. Robinson and S. A. Friedberg, Phys. Rev. 117, 402 (1960) . 17) Miss W. H. M. Voorhoeve and Z. Dokoupil, Physica, (to be published). 18) W. Van der Lugt and N. J". Poulis, Physica 26, 917 (1960). 19) W. Van der Lugt, N. J". Poulis, T. W. J*. Van Agt and C. J". Gorter, Physica 28_, 195 (1962).  - 115 -  20) P. H e l l e r and. G, B. Benedeck, Phys. Rev. Letters 8, 428, (1962). 21) T. Haseda, J . Phys. Soc. Japan 15, 483 (1960). 22) M. Date, J . Phys, Soc. Japan 14, 1244 (1959). 23) T. Nagamiya, K, Yoslda, and R. Kubo, Advances i n Physics 4, 1 (1955). 24) M. Date, Proc. I n t . Conf. on Magnet, and C r y s t a l l . J". Phys. Soc. Japan, 17, Sup. B-I, 422 (1961). 25) No J . Poulis and G. E. G. Hardeman, Physica 18, 201 (1952) . 26) M. J . Buckingham and W . M. Fairbank, Prog. Low Temp. Physics 3, 80 (1961). 27) J . E. D z i a l o s h i n s k i i , Soviet Physics, JETP 5, 1259 (1957). 28) E. P. Riedel and R. D. Spence, Physica 26, 1174 (1961).  

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