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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 2«6H 20 by ERICH SAWATZKY B.Sc, University of British Columbia, 1958 M.Sc, University of Br i t i s h Columbia, 1960 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in the Department of PHYSICS We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA September, 1962 In presenting t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the requirements f o r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree that the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r reference and study. I f u r t h e r agree that permission f o r extensive copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s representatives. It i s understood that copying or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l gain s h a l l not be allowed without my written permission. Department of The U n i v e r s i t y of B r i t i s h Columbia, Vancouver 8*. Canada. Date PUBLICATIONS 1_ Proton. Resonance in Paramagnetic and Antif erromagnetic CoC^^H^O. E. Sawatzky and M. Bloom. J. Phys. Soc. Japan, \J_ Sup B-I, 507 (1962). 2. The Paramagnetic-Ant iferromagnetic Phase Transition in CoCl2*6H20. E. Sawatzky and M. Bloom. Physics Letters 2, 28 (1962). The University of British 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 British Columbia, 1958 M.Sc, University of British Columbia, 1960 MONDAY, OCTOBER 1, 1962, AT 2:30 P.M. lit ROOM 301, PHYSICS BUILDING COMMITTEE IN CHARGE Chairman: F.H. SOWARD M. BLOOM R. REEVES J. GRINDLAY . C.S. SAMIS R. HOWARD J. TROTTER G.M. VOLKOFF External Examiner: S.A. Friedberg Carnegie Institute 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 temp-eratures and becomes antiferromagnetic at about 2.25 K. The proton resonance frequency is a measure of the total magnetic field 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 it 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 antiferromag-netic 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 CoCl9-6H90. GRADUATE STUDIES Field of Study: Nuclear Magnetic Resonance Quantum Mechanics F.A. Kaempffer Nuclear Physics J.B. Warren Electromagnetic Theory G.M. Volkoff Magnetism M. Bloom Related Studies: Crystallography Physical Metallurgy K.B. Harvey E. Teghtsoonian - i i -ABSTRACT The work reported here i s a detailed study of the proton magnetic resonance In single crystals 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 total 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 internal f i e l d produced by the surrounding magnetic ions. At room temperature a single line about 6 gauss wide i s observed. This line s p l i t s 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 f i e l d . The maximum overall splitting at 4.2°K i s about 150 gauss in a f i e l d of 5000 gauss. At, 2.1°K the maximum splitting observed i s 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 reson-ance 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 TJJ was measured as a function of applied f i e l d and crystal orientation using the - i i i -proton resonance lines, 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 crystal orientation, which cannot be described by TN(H,) = T(0) - const. H2, as predicted by the Weiss Molecular f i e l d theory. The transition takes place over a temperature region of about IO" 2 °K, and effects due to short range order are observed just above TJJ. The magnetic susceptibility in zero f i e l d 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 T^ obtained by NMR. The sublattlce magnetization in the antiferromag-netic phase was measured as a function of temperature. It is found to depend logarithmically on T^ - T, but i s independent of applied f i e l d and crystal orientation. Further experiments are suggested, which would add greatly to the understanding of the magnetic behaviour of CoCl 2.6H 20. - v i i -ACKNOWLEDGMENT My sincerest appreciations are due Dr. M. Bloom. Without his constant interest and guidance, many illumin-ating discussions and invaluable help in Interpreting the results, this thesis would not have been possible. I also wish to express my appreciation to Dr. D. L. Williams for c r i t i c a l l y reading the manuscript of this thesis. This research has been supported financially 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 in the preparation of this thesis. - iv -TABLE OF CONTENTS Page Abstract i i List of Illustrations v Acknowledgment v i i Chapter I - Introduction 1 Chapter II - Apparatus and Experimental Procedure. 1 0 Chapter III - Theory of the Positions of the Resonance Lines 2 4 Chapter IV - The Proton Resonance in CoCl 2 » 6 H 2 0 at Constant Temperature 3 8 A - The C o C l 2 - 6 H 2 0 Crystal 3 8 B - Results and Discussions 3 9 (i) The proton resonance in the paramagnetic phase at 4 . 2 ° K .... 4 0 ( i i ) The proton resonance in the antiferromagnetic phase at 2 , 1 ° K 56 Chapter V - The Paramagnetic-Antiferromagnetio Phase Transition 6 3 Chapter VT - Thermodynamics of the Phase Transition 77 Chapter VII -The Sublattice Magnetization below T^ 8 8 Chapter VIII - Suggestions for Further Experiments .. 9 5 Appendix A - Construction Hints and Operating Instructions 97 Appendix B - The Nagamiya, Yosida, and Kubo Theory for TJJH Applied to CoCl 2 « 6 H 2 0 1 0 7 - 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 Fig. 4a - Crystal Growing Vessel 19 Fig. 4b - Crystal Holder 19 Fig. 5 - Derivative of Proton Resonance Spectrum .... 41 Fig. 6 - Resonance Diagram in the a-c Plane for T = 77°K (H Q = 5,000 gauss) 43 Fig. 7 - Resonance Diagram in the a-c Plane for T = 4.2°K (HQ = 3,000 gauss) 44 Fig. 8 - Resonance Diagram in the a-c Plane for T = 4.2°K (E = 5,000 gauss) 45 Fig. 9 - Resonance Diagram in the a'-b Plane for T = 4.2°K (H Q = 5,000 gauss) 46 Fig. 10 - Resonance Frequency Separation vs Applied Field 49 Fig. 11 - Definition of Direction Angles 51 Fig. 12 - Proton-Proton Splitting as a Function of Orientation 53 Fig. 13 - Resonance Diagram in the a-c Plane for T = 2.1°K (H Q = 5,000 gauss) 57 Fig. 14 - Resonance Diagram in the a'-b Plane for T = 2.1°K (HQ - 5,000 gauss) 58 Fig. 15 - Resonance Spectrum vs Applied Field for T = 1.14°K 61 Fig. 16 - Relative Signal Amplitude vs Temperature for T > T N, H Q = 5,000 gauss 65 Fig. 17 - The Region near T = 2.2°K of Fig. 16 Expanded 66 Fig. 18 - The Lie Width as a Function of Temperature . 69 - v i -Page Fig. 19 - The Neel Temperature as a Function of Applied Field 71 Fig. 20 - Plot of T N vs Orientation 73 Fig. 21 - The Anisotropy in T^ 74 Fig. 22 - The Molar Susceptibility as a Function of Temperature 82 Fig. 23 - Plot of (3M/3H) S VS C H / T 84 Fig. 24 - Oscillator Frequency vs Temperature 86 Fig. 25 - Resonance Frequency vs Temperature for T < T N 89 Fig. 26 - Plot of f j - f x vs log(T N - T) for T T N 92 Fig. IA - Oscillating Detector 101 Fig. 2A - Williamson Power Amplifier 102 Fig. 3A - Audio Amplifier 103 Fig. 4A - Phase Sensitive Detector 104 Fig. 5A - Magnet Current Regulator 105 Fig. 6A - Low Temperature System 106 - 1 -CHAPTER I  INTRODUCTION The nuclear magnetic resonance technique provides a powerful method of studying the interactions between atomic nuclei and their magnetic environment. The steady state branch of this technique i s particularly suited for the investigation of internal magnetic fie 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 solid state physics, the importance of nuclear magnetic resonance (NMR) i s now well recognized, and valuable contri-butions have been made to the understanding of many different classes of materials, among them paramagnetic and anti-ferromagnetic substances. In this work we are concerned with the proton magnetic resonance in single crystals of CoClg . 6Hg0 both in the paramagnetic and antiferromagnetic phases of this substance. The proton resonance in 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 in 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 levels ~/^joH , where yllp i s the magnetic moment of the protons. The difference in energy between the two levels is thus 2yUpH o Transitions between the two levels may be induced by applying electromagnetic radiation, which i s circularly polarized with the magnetic vector rotating in a plane perpendicular to the static magnetic f i e l d H . The frequency of this radiation must be equal to the classical Larmor frequency co, = ZjupH/h. = <Tp W where ftp = 2.675 x 10 4 rad/sec-gauss i s the gyromagnetio ratio for protons. In a typical 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 neigh-bouring magnetic moments are always present, although in liquids and gases these interactions are averaged consider-ably due to thermal motion of the atoms and molecules. In crystalline solids the nuclei occupy, except for thermal vibrations, fixed positions, and each nucleus experiences, in addition to the externally applied static magnetic f i e l d H Q and the small r - f f i e l d H]_ , an inhomogeneous internal magnetic f i e l d due to the neighbouring magnetic dipoles. The proton resonance frequency i s a measure of the total magnetic f i e l d at the positions of the protons, which i s the vector - 3 -—» —» sum of the applied f i e l d H Q with the internal f i e l d H^ n t > i.e. _* _^ H l o c = Ho + H i n t and the resonance frequency i s &)= <THi00» If the crystal contains paramagnetic ions, as is the case in CoCl *6H 0, the internal magnetic f i e l d Hj_ n t may be of the order of 1000 gauss, whereas internal fields due to non-paramagnetic ions are usually not larger than 20 gauss. Since the magnetic moment of the paramagnetic ions may be oriented in (2S + 1) different directions ( S = effective spin of the paramagnetic ion), the internal f i e l d produced by the neighboring ions at the positions of different protons in a unit c e l l of the crystal may vary between approximately i 1000 gauss. This range of internal fields should give rise to a spread of resonance frequencies, and we would expect very broad proton resonance lines in paramagnetic substances. However, i t can be shown1^ that i f exchange interactions between the magnetic ions are present, this broadening action of the magnetic ions is considerably reduced. From the exchange interaction and the Zeeman energy of the cobalt ions in the external f i e l d H Q the time averaged magnetic moment <^jU>Q^ is obtained to a f i r s t approximation2^ in the para-magnetic phase - 4 -where ^ i s the Bohr magneton, ^ i s the anisotropic g-tensor, and © i s the Curie temperature of the substance. This mean magnetic moment gives rise to a time averaged internal f i e l d which depends strongly on the spatial coordinates and on the orientation of the crystal with respect to the external magnetic f i e l d H Q. The energy levels of the proton magnetic moments are determined by the vector sum of this internal f i e l d with HQ, and non-equivalent protons in a unit c e l l have different resonance frequencies. Assum-ing that H Q i s constant over the sample, the local f i e l d w i l l be the same for equivalent protons in the unit cells and the proton resonance w i l l s p l i t into a number of component lines, provided the local fields are sufficiently different at non-equivalent protons. To resolve a pair of lines the difference in local fields must be greater than the proton line widths which are typically of order 5 to 10 gauss for hydrated salts of the transition elements. Since the internal f i e l d has roughly an inverse temperature dependence, the separation between component lines increases with decreasing temperature. Thus the number of component lines observed in any particular applied f i e l d depends on the sample tempera-ture, on the number of water molecules per unit c e l l , and on the degree of symmetry of the crystal. The internal f i e l d at a proton due to a paramagnetic ion at a distance r i s of the order of magnitude <£/6^/f3 • The f i e l d therefore f a l l s off rapidly with increasing r , so - 5 -that only the nearest neighbours make important contribu-tions to H l o o a l . Taking r = 2 x 10" 8 cm and H Q = 5,000 gauss, the spli t t i n g in CoClg • 6HgO at 300°K 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 reson-ance line at room temperature, and only partial resolution at 78°K and 4°K , Experimentally a single line about 5 gauss wide i s observed at room temperature. The maximum splitt i n g at 78QK is about 35 gauss and at 4.2° K about 170 gauss in good agreement with the above order of magnitude o arguments, since r was chosen ar b i t r a r i l y as 2 A, The work to be described in the following chapters of this thesis i s an extensive study of the proton magnetic resonance in CoCl • 6H 0 as a function of temperature, crystal orientation and applied f i e l d . A summary of the theory describing the positions of the component lines as a function of temperature, orientation, and applied f i e l d i s presented in chapter III, for both the paramagnetic and antiferromag-netic phases. In the paramagnetic phase the internal f i e l d due to the cobalt ions i s found to be proportional to 3 COS^G^^ Vfa ^ , where r ^ is the magni-tude of the radius vector between the i t l L proton and the k t t L cobalt ion, and Oik i s the angle between H 0 and iTfo . This implies that the resonance lines in the paramagnetic phase behave like sine functions with a period of 180°. In - 6 -addition to the internal f i e l d due to cobalt ions, a proton in one water molecule experiences a f i e l d proportional to due to the other proton in the same water molecule. This f i e l d is much smaller than that due to the cobalt ions, but nevertheless gives rise to an additional splitting of component lines which i s independent of the magnitude of the applied magnetic f i e l d . The maximum observed proton-proton splitting at 4.2°K was found to be 15 gauss. From the proton-proton splitting as a function of orientation i t was possible to find the direction cosines for the line joining one pair of protons. The 180° periodi-city of the resonance curves in the paramagnetic phase was verified experimentally. The antiferromagnetic phase l s characterized by a spontaneous alignment of the cobalt mag-netic moments, independent of the applied f i e l d HQ, provided H 0 i s not too large. This spontaneous antiparallel alignment gives rise to an internal f i e l d of the form where i s the angle between ^/^c^ and H Q and OCQ is a phase angle depending on the orientation of ^ /J^ with respect to the crystal axes and on the spatial coordinates. Thus the resonance curves are found to behave like sine functions with a period of 360° instead of 180° as in the paramagnetic phase. The number of resonance lines effect-ively 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^ can point in a + direction or in a -direction. The resonance diagrams in the anti-ferromagnetic phase are thus symmetrical about a central frequency which i s determined by H Q. The splittings in the antiferromagnetic phase are much larger than in the paramagnetic phase. Splittings of the order of 2,500 gauss were observed at 2.1° K. It was verified experimentally that (jXc^ in the antiferromagnetic phase i s independent of the applied f i e l d in the range of fi e l d s used in this experiment. The phase transition or Neel temperature T^ i s treated in chapter V. It i s found that the transition i s not sharp, but takes place over a temperature range of a few millidegrees indicated by an overlap of both types of spectra in this range. The resonance line amplitudes decrease sharply as T is approached, whereas the line widths show an anomalous peak at T^ . This behaviour together with the anomalous peak in the specific heat at T^ i s very suggestive of a X-type transition as observed in liquid helium. The transition temperature was studied as a function of crystal 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. From the f i e l d dependence of T^ . i t i s found that B(H0) is nearly a linear function of H 0. The dependence of T^ on H Q and (p i s inconsistent with moleoular f i e l d theory. Extrapolating to H Q = 0, T^ ~was found to be equal to 2.27°K« When H 0 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 in other phase transitions, where the line separating two phases l s usually a monotonic function of the intensive thermodynamic coordinate. No explanation has been given for this behaviour of ^  . The magnetic susceptibility parallel to the preferred (c) axis in 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 in good agreement with previous 3) susceptibility measurements ' using an inductance bridge. The present measurements together with specific heat data are used to show the mutual consistency between the thermo-dynamic properties and the slope of the transition temperature near zero f i e l d as obtained with NMR. The temperature dependence of the resonance fre-quencies in the antiferromagnetic phase was studied in various applied fields and at several crystal orientations. In chapter VI we shall see that the resonance frequency is - 9 --» a measure o f 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 f o u n d , -> r e g a r d l e s s o f H Q and o r i e n t a t i o n , t h a t M depends l o g a -r i t h m i c a l l y on ( T N - T) o v e r t h e w h o l e t e m p e r a t u r e r a n g e s t u d i e d . T h i s b e h a v i o u r m u s t , o f c o u r s e , b r e a k down n e a r T ^ . 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 w i t h i n a b o u t 20 m i l l i d e g r e e s o f T N . I t i s i n t e r e s t i n g t o n o t e t h a t t h i s b e h a v i o u r i s q u a l i t a t i v e l y s i m i l a r t o t h e b e h a v i o u r o f t h e s p e c i f i c h e a t , w h i c h a l s o depends l o g a -r i t h m i c a l l y on (T - 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 in this thesis was obtained with a standard steady state nuclear resonance spectrometer. A block diagram of the apparatus i s shown in figure 1. Circuit diagrams of the more common units are given in Appendix A, together with some notes on the operation and construction of the various units. Details of the low temperature system are also given in Appendix A. Only modifications, original designs, and the experimental pro-cedure shall be described in this chapter. The oscillating detector was a slightly modified version of a circuit by Watkins and Pound4. We have substi-tuted a voltage sensitive capacitor,(varicap), for the mechanically variable capacitor in the tuned ci r c u i t in order to minimize incidental noise and to eliminate sudden frequency changes during very slow sweeps due to irregulari-ties in the mechanical drive (see figure IA in the Appendix). These modifications, together with the linear voltage sweep described below, proved very convenient and time saving. The sawtooth voltage required to sweep the frequency of the oscillating detector by means of a varicap voltage sensitive capacitor i s derived from a modified Tektronix type Freq Counte]' Digita:. Recorde: • Sawtoot Sweep f anostat Vacuum Pump Oscil. Det. Sample Magnet <3 1 farrow Band Ampl. Broad Band Ampl. Hagnet Magnet Power Supply Fig. 1 Block Diagram of Apparatus Phase Sens. Det. Record M i l i i a Scope Phase (3 Shift Phase Shift 30 cps, Source Oscil. Detector 2 n d Oscil. Detector for Field determination - 12 -162 Waveform Generator. Part of this generator i s shown in figure 2 (permission for 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 in the Tektronix 160 Series Instruction Manual. The modifications are described below. Changing the values of the timing resistors and timing capacitorsS) as indicated in 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 rises 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 is 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 positive 20V. The voltage sensitive capacitors used in this experi-ment 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 connec-ted to the grid of V7B, and the cathode of V7B i s returned to the -170 volt supply through a 50V Zener diode and a 33K 50K LZ.170V | p i n i«-Wv\4>/\A-LwWv^ J o of V4 6 1 . 6 2 2.4 3.2 4 5 6.5 8 10 14 lO^M Fig. 2 Modified Tektronix 162 Waveform Generator | +100 to 0 sawtooth Arm of output R21 SW3A«-pin 1—>< of V4 -•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 oscillating detector. The change in oscillator frequency corresponding to this voltage sweep depends on the type and number of varlcaps used and on the inductance of the oscillator 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 circuit diagram of the 30cps source and i t s accompanying phase shifting networks i s shown in figure 3. The frequency of this 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 via 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. The remainder of the network of the 30cps source i s self-explanatory from the schematic of figure 3. Decoupling cathode follower i—1 2oo Each in 1 0 10 steps Fig. 3 30 cps Source and Modulation Signal Output Fig. 3 cont'd. Phase Shifter, Audio, and Reference Voltage Outputs - 17 -The magnet current was derived from a power supply adapted from a circuit by R. L. Garwin ;. Since the magnet used in this work has a considerably higher impedance than that used by Garwin, i t was necessary to Increase the refer-ence resistor to 0.1 ohms. Resistance variations in the reference resistor were minimized by means of a water cooled o i l bath. It was also found that the performance of this circuit improves greatly i f the signal transistors are water-cooled. A ci r c u i t diagram of the existing regulator i s shown in figure 5A. The remainder of the apparatus in figure 1 i s standard equipment and shall not be described here, except for i t s uses in this work. Circuit diagrams of the units not mentioned here are shown i n Appendix A. The external f i e l d H Q is supplied by an air-cooled iron-oore electromagnet (manufactured by Newport Instruments Ltd., Ser. No. 6010/3) with four inch diameter plane pole tips 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 gradu-ated scale so that any f i e l d orientation relative to the sample in the plane of rotation can be obtained. Field modulation i s achieved by means of a pair of f l a t coils glued to the pole faces in order to minimize modulation interference in the magnet power supply and at the same time - 18 -to minimize the pole face separation. The modulation current is 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 line widths in CoCl2«6Hg0 are of the order of 5 gauss or larger. To observe the deriva-tive of the true line shape with a recording milliameter, the modulation amplitude should not exceed about 1/4 the line width and so a modulation of about 1 gauss was used when recording the derivatives of the resonance lines. 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 in 10 can be easily obtained. The magnetic f i e l d was measured before and after each run, and in some experiments several times during each run with the aid of a second oscillating detector by measuring the frequency of the proton resonance in 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 for protons in water. Single crystals of 00012*6H20 were grown from aqueous solutions and found to be of the same type described by P. Groth 7). CoClg.6Hg0 crystals grow very rapidly, but tend to form multi-oriented, polycrystalline blocks. Several growing methods were employed with varying degrees of success. The most successful arrangement i s shown in Fig.4a. - 19 -[^^\ Ground glass seal Saturated CoClo'6H 20 solution Seed Solid CoCl2'6H20 Heater Coil _ Kovar seal ^ Brass top to f i t !• dewar cap Teflon spacer Tapered plug 4a 4b Fig. 4a. Crystal Growing Vessel Fig. 4b. Crystal Holder - s o -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 irregular pieces of crystalline CoCl •6H„0 are placed in the t a l l of the vessel, 2 <J and the solution surrounding these pieces i s more concen-trated and slightly warmer than the top layers of the solution. Thus a continuous upward current of a more con-centrated solution i s maintained. As this solution current reaches the upper layers It cools and some of the excess solute i s deposited on the seed suspended with a thin nylon thread. With this method i t was possible to obtain good single crystals with dimensions roughly up to 3 x 2 x 2 cm. Cylinders 8 mm in diameter and about 2 cm long were cut from the grown crystals, taking great care to insure that a particular crystal axis was parallel to the cylinder axis. These cylinders were placed in a thin-walled (0.5 mm) "Teflon" tube which i s attached to a 3/8 inch stainless steel tube in the manner shown in figure 4b. The stainless steel tube with the Teflon holder was mounted vertically so that the f i e l d H Q would be perpendicular to the cylinder axis (H Q i s adjusted to be horizontal). It i s estimated that the error in orientation did not exceed 5 degrees in the various mountings. With this arrangement different crystal cylinders could be investigated under identical conditions since the same sample holder could be used for different crystals. - 21 -The low temperatures were achieved with the aid of an ordinary double dewar glass oryostat. A schematic of the low temperature system i s shown in 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 It was feared that the Teflon holder would prevent rapid temperature equilibrium between the sample and the helium bath, but extensive investigations showed that this was not the case. A l l temperature sensitive properties of CoClg*6HgO investigated in this work were rechecked against various rates of temperature change and were found to follow the temperature (vapour pressure) very closely. When raising the temperature, a strong light was directed into the liquid helium bath and maintained there u n t i l the helium boiled violently to insure proper sti r r i n g of the liqu i d . The tips of both dewars were unsilvered, providing a path for radiation into the helium. It was found that the temperature of the sample could be raised by 0.1° K in a few seconds by placing a common 100 watt light bulb near the dewar tips. 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 transition temperature by following the signal amplitude as a function of vapour pressure (hence temperature), the temperature variation was often stopped at some arbitrary temperature, and i t was found that the signal amplitude remained at whatever value the temperature variation was stopped, indicating that the sample temperature followed closely the vapour pressure. If there were not a continuous equilibrium, the signal amplitude should continue to change, as i t i s a very sensi-tive function of temperature near the Neel temperature. It was found that this continuous temperature equilibrium was even true for increasing temperatures, because enough light from the room in 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 Digital Recorder. In the antiferromagnetic phase where the separation between lines is very large (up to 15 Mc/sec. at 2.1° K) the signal was merely brought onto an oscilloscope screen and the resonance frequency recorded. In the para-magnetic phase and in 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 di g i t a l recorder in 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 oscillator rather than the magnitude of the external f i e l d was varied. The rate of sweep was adjusted in each case to avoid distortion of the signal by Integration through the relatively long time constant in the phase sensitive detector. - 24 -CHAPTER III THEORY OF THE POSITIONS OF THE RESONANCE LINES In this chapter a brief summary of the theory describing the positions of the proton magnetic resonance lines as a function of the externally applied magnetic f i e l d in hydrated single crystals containing paramagnetic ions l s presented. The calculations which follow below O N were developed by N. Bloembergen ' in connection with similar measurements in CuSo^»5HgO. The position and width of each resonance line are complicated functions of the direction and magnitude of the applied magnetic f i e l d H0, and of the Internal structure of the crystal. For each direction of the applied f i e l d the separation between the lines increases with decreasing temperature. The character of the resonance spectrum changes drastically as we pass from the paramagnetic phase (T>T N) to the ant i f erromagnetic phase (T < T^). Theo-r e t i c a l formulae for the positions of the component lines 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 arising from the electronic magnetic moments of the cobalt ions. The dipolar Interactions with the pro-tons in neighbouring water molecules and with other nuclei - 25 -are assumed to contribute only to a broadening of the resonance lines. We consider a crystal containing paramagnetic ions and one or more water molecules of hydration. Except for the positions of the protons, the crystal structure i s generally known. At each point in the crystal there w i l l be, in addition to the applied f i e l d , a time--varying inhomogeneous internal f i e l d contributed by the electronic 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 direction of this internal 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 this system may be written in the form 5 (3.1) 1' - J E p ^ - H , + zsi+% where |3 i s the nuclear Bohr magneton and ^ i s the aniso-tropic g- factor in the tensor form. The f i r s t and last terms in (3.1) are the Zeeman energies of the cobalt ions and protons in the externally applied f i e l d H Q respectively. ^55 i s the magnetic Interaction between the cobalt ions, and $?ex ^ n e exchange energy between them. The term represents the magnetic interaction between the cobalt ions and proton moments, and <^ !J-J i s the magnetic interaction - 26 -between the protons themselves* Since we consider the proton spin system immersed in the homogeneous applied f i e l d H Q and the inhomogeneous fluctuating f i e l d produced by the neighbouring magnetic moments, i t i s convenient to divide the Hamiltonian (3.1) into two parts: (a) , from which the internal f i e l d i s calculated and (b) the Hamiltonian for the proton spin system subjected to the total local f i e l d at the proton sites. In considering the cobalt ions we shall neglect c$?s£ , since the effect 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 in the f i e l d H Q. This term, however, is of great importance in the Hamiltonian for the proton spin sys-tem to be discussed below. The term $ss ^ay also safely be neglected in comparison with S^&x , since the exchange energy is roughly 100 times greater than the magnetic energy between two cobalt ions. Thus, the Hamiltonian of the cobalt spin system, using the standard notation for the exchange interaction, i s : K J >K where A]^ i s the so-called exchange integral between the cobalt spins. The term in 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 cobalt ion and /u th Ik i s the radius vector connecting the i proton and the —> k t i L cobalt ion. As pointed out before, JJ^ varies rapidly with time due to the exchange coupling between the cobalt ions. The exchange interaction causes a pair of antiparallel cobalt spins to change direction simultaneously, so that there i s no energy involved, but the magnetic f i e l d due to JX^ changes direction in the neighbourhood of a pair of cobalt spins. The exchange frequency i s approximately given by %CD&t* k"T^ where i s the paramagnetie-antiferromagnetic phase tran-sition temperature. For CoClg.6H20 = 2°K, so that = 5 x 10 1 0cps. The Larmor frequency of protons in the 7 internal fields of magnetic crystals never exceeds 10 cps, so that the protons cannot follow the rapid variations of the internal f i e l d . The proton spins react only to the time-averaged f i e l d of the cobalt ions. We therefore can use the time-average ^U^^ of Jjt^ in (3.3), i.e., the operator /X^ in equation (3.3) i s replaced by a number, ^ J*£y . The time-averaged magnetic moment ty^-k) °^ t n e cobalt ion spins can now be calculated in principle 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 8 \ since the time averaged should be the same as the s t a t i s t i c a l mechan-i c a l average for the equilibrium state of the Co"* spin system. ^JX^ i s proportional to the magnetization M in the paramagnetic phase (T > T J J ) and to the so-called sub-lattice magnetization in the antif 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 interaction i s proportional to 1/r 5 5, only the nearest neighbours w i l l contribute appre-ciably to this interaction. We therefore consider only interactions between the two protons in the same water mole-cule in the last term of ( 3 . 4 ) . The interactions with other protons i n the crystal 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 lines. We now have the relatively simple system of two interacting protons Immersed in the homogeneous external f i e l d H Q plus the s t a t i c inhomogeneous i n t e r n a l f i e l d produced by the neighbouring oobalt ions, i . e . the protons are subjected to a f i e l d E l o c = H i n t + Ho* To f i n d the resonance frequencies, (3 .4 ) must be solved f o r i t s eigenvalues which give the energy l e v e l s of the two-proton system. We here present the r e s u l t s of N. Bloembergen ; , who obtains f o r the energy l e v e l s of the two protons i n one water molecule to f i r s t order perturbation theory: Ez = - d + \ l b * + <& (3.5) where a = Vz?A (Hi + Hi) b- Vzn(Hi~Hi) d - - t M p k 2 ( i - 3 « o * 0 ; 2 ) £ (3.6) -3 - 30 -1 2 and H z and H^ . 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 paral l e l to EQ. The quantities a and 1 1 2 b are functions of temperature by virtue of and H z which are calculated from with /U^ replaoed by (jX^y . The energy levels in ,(3,.5) give rise to four transition frequen-cies as follows: loci tlV3 *rhti0 -CL+ld-tf+d1' - jfflHLoc3 (3.7) with corresponding intensities a r b i t r a r i l y normalized to 4 T _ T = (b-«£ -te + dff 1 2 bW-b/t>%d1' \ , o (S.8) Thus, with the approximations made in the preceding para-graphs, each water molecule of hydration i n principle gives rise to four resonance lines. In the absence of the magnetic ions a m b = 0, and the problem has the solution given by Pake ; for the water molecule in gypsum. If, in the present case, a, b d, which i s certainly true at room temperature, - 31 -then the influence of the cobalt ions may be partly ignored and Pake's solution s t i l l holds. However, in CoCl2»6HgO the number of protons per unit c e l l i s large and the cobalt mag-netic moment i s not completely negligible even at room temperatures and we do not observe any resolved lines at room temperature, but a single line about six gauss wide. At helium temperatures, none of the terms in (3.6) are necessarily negligible, and each water molecule should-give rise to four lines in two pairs of two. In CoCl2'6HgO the positions of the protons are not known. However, i f we assume that the protons i n each water molecule in the unit c e l l are magnetically nonequivalent, then the six different water molecules of hydration should lead to 24 lines in six groups of four. At low temperatures we can safely assume that b ^ >d, since the f i e l d of the cobalt ions i s much larger than that of the protons. Then we have four lines of equal intensity consisting of two pairs, the centres of which are separated by a distance 2b, and the separation between the two lines in a pair i s 4d. From (3.6) we see that for some particular configuration b may be equal to zero and only half of the maximum number of lines w i l l be observed. Also, when the- sums of the terms in the resonance frequen-cies in (3,7) containing the geometrical factors are equal, the resonance frequencies for the protons in different water molecules w i l l coincide. It i s thus expected that the number of resonance lines observed depends on the orientation of the - 32 -crystal relative to the external magnetic f i e l d H Q and on the temperature of the system. The distinguishing characteristic of the resonance diagrams in the paramagnetic phase i s the periodicity of the resonance curves. From H^" in (3.6) we see that the z resonance curves are sine functions with a period of 180 degrees. In the next section we shall see that this period-i c i t y changes to 360 degrees in the antiferromagnetic phase. These points w i l l be discussed in detail in the next chapter in connection with the proton resonance measurements in paramagnetic and antiferromagnetic CoClg#6Hg0. A l l the above arguments apply to the paramagnetic state of the crystal where <\tik) ^ M0/(T+ to a f i r s t approximation, where & i s the Curie temperature. The antiferromagnetic phase (T < T N) i s characterized by a spontaneous alignment of the cobalt ion spins. The exchange integral A., in (3,2) may be positive or negative. If i t is positive, then we have an ordinary ferromagnet with a l l the spins aligned in one direction. If A., i s negative, adjacent spins w i l l align in opposite directions giving rise in the simplest case to two ferromagnetic sublattices point-ing in opposite directions. The bulk magnetic moment of the sample i s s t i l l zero in the absence of an applied f i e l d , but Inside the crystal there now exists a strong internal f i e l d alternating in direction from cobalt ion to cobalt ion. Recently i t has been suggested 1 0^ that there may exist more - 33 -than two directions of alignment i n some substances. A discussion of this possibility in CoCTg^SHgO i s postponed to a later chapter. For the purpose of describing the main features of the proton resonance curves in antiferromagnetic CoCl 2 »6HgO i t w i l l suffice to assume a two-sublattice system. The spontaneous alignment of the cobalt ion moments along some preferred direction in the crystal at a very low temperature i s an inherent property of the system and persists even in the presence of an applied f i e l d . There-fore, in calculating the positions of the resonance lines in the antiferromagnetic phase, the arguments applied to the paramagnetic phase must be slightly modified. In considering the local f i e l d at the proton posi-tions we must take into account the fact that the cobalt spins have fixed directions and that adjacent cobalt ion spins point in opposite directions. The total magnetic f i e l d at a proton position i s the vector sum of H Q with the internal f i e l d due to the cobalt ions. In order to calculate the position of each resonance line as a function of the direction -> of the external f i e l d HQ, we need consider only the component -* of the internal f i e l d parallel to H Q. The perpendicular component w i l l be discussed later. For simplicity consider one water molecule in 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 rotation 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 fac-toring out , i s given by - 3 4 -(3.9) The component of this internal f i e l d parallel to the applied f i e l d H Q i s then and may be written in the form MttlH, - FcosoL-3COS % COS(oC+ ( 3 , 1 0 ) 'Ik J where i s the fixed angle between ^U-k)  an<3- ^ik > since we assume that (JJ^ i s coupled tightly to the crystal l a t t i c e , and cL i s the angle between ^/J^) HQ* Equa-tion (3.10) may be rewritten in the more convenient form where d k = [ ( / - i e a s ^ f t * ) * * (5/2 sin 2&IK)J - 35 -and oCf. i s defined by tan oC0 =• 3/g $kZ&£k Both A|< and are constants of the crystal l a t t i c e , and the variation of H i n t U H o with orientation i s determined by aC alone. Therefore, to determine the positions of the proton resonance lines in the antiferromagnetic phase, the term H_ in (3.6) must be replaced by (3.11). It should be noted that in (3.11) ^$k} belonging to one sublattice i s positive, while {J%£y belonging to the other sublattice is negative. Thus, the internal f i e l d i s oriented in opposite directions for corresponding protons in adjacent unit ce l l s , and the number of resonance lines should double as we pass from the paramagnetic to the antiferromagnetic phase. Also, for each resonance line displaced in one direction due to Hint II Ho there w i l l be another resonance line displaced an — -» equal amount i n the opposite direction due to H^^^, . The resonance curve diagram w i l l therefore be symmetric about the centre of gravity of the lines. Tn addition to the displacement of the resonance lines due to the parallel com-ponent of H^ n t, there i s also a contribution of the perpen-dicular component, which causes a slight 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 periodicity of the respective resonance curves. In the paramagnetic phase the resonance curves were sine functions with a period of 180 degrees rotation of the applied f i e l d H Q. Equation (3.11) shows that this periodicity has now been changed to 360 degrees in the antif erromagnetic phase due to the cos (pL + ^ factor in H5- . Also, since in this case the spins align spontaneously in their respective directions, the internal fields w i l l be much larger in 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 shall see how well these points are borne out by experiment. In this chapter we have described the magnetic f i e l d at nuclear positions in terms of the applied f i e l d H Q plus the internal f i e l d produced by the surrounding magnetic moments at the nuclear positions. We have thus neglected the contribution to the average magnetic f i e l d due to the average macroscopic magnetization M per unit volume. In the next chapter we shall see that contributions of this type amount to about 40 gauss at 4.2° K. If we want to take this extra f i e l d into account, the external f i e l d H Q should everywhere be replaced by Bo = H 0 + (4^ - N)M, where N, the demag-netization factor for the crystal, must be calculated from the geometry of the sample 1 1^. Only i f the sample i s in the form - 37 -of an e l l i p s o i d w i l l M be uniform over the whole sample. For any other configuration, M w i l l be a function of p o s i t i o n i n the c r y s t a l and t h i s non-uniformity of M i s equivalent to an inhomogeneity i n the applied f i e l d i n that i t w i l l contribute to the nuclear resonance l i n e width. - 38 -CHAPTER 17 THE PROTON RESONANCE IN CoCl 2«6H 20 AT CONSTANT TEMPERATURE A. THE CoCl2'6H20 CRYSTAL CoCl2"6H20 forms dark purple monoclinic crystals 7) of the type described by P. Groth . The unit c e l l edges are a:b:c . 1.4788:1:0.9452 with - 122°19'. Perfect 12) cleavage occurs along the C(OOl) face. J. Mizuno et a l ; have analyzed single crystals of CoCl2'6H20 by the X-ray method and report two formula units per unit c e l l with space group C 3 g l i - C2/m with a = 10.34A0, b = 7.06A0, and c = 6.67A0. The atomic positions, except for the protons, are given in the following table: of atom Position X Z z Co origin 0 0 0 C l 4(i) .278 0 .175 °I . 8(3) .0288 -.221 .255 ° I I 4(j) .275 0 .700 The proton positions cannot be found by the X-ray diffrac-tion method. According to the authors of reference-1-2) , two Cl"" 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 (CoCl2*4HgO). The octahedral plane contains the Co + + ion and the four water molecules, and the two C l ~ ions are 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 0 j « • • H - O J J - H * •»0j and 0j-H»»»01 in the plane parallel to (001), the a-b plane. B. RESULTS AND DISCUSSIONS In this chapter measurements at two fixed tempera-tures, T = 4.2°K in the paramagnetic phase and T = 2.1°K in 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 in the next chapter. In making these measurements the crystal was kept fixed in the laboratory reference frame while the orientation of H Q was changed by rotating the magnet. In one set of -> measurements H Q was oriented in the a-c plane of the crystal, while in another set i t was oriented in the a ' r fb plane, where a' l i e s in the a-c plane and i s mutually per-pendicular to the b and c axes, i.e., a T makes an angle of 32°19 with the a-axis. The angle between H 0 and the rotations. The measurements were made with the magnitude of s designated by y . In subsequent discussions shall refer to H Q parallel to the a -axis in both - 40 -H Q kept constant, while the oscillator frequency was swept through the resonance spectrum at a rate slow enough to avoid distortion of the line shape. When the separation between the component lines was small (1 Mc/sec or less), as i s the oase in the paramagnetic phase, the derivatives of the resonance lines were recorded using a phase sensitive detector and a recording milliameter. The oscillator frequency was marked directly on the record-ing chart with an automatic "event marker" at approximately 5 Kc/sec intervals. The resonance frequencies were then established by observing the 1st, 3rd, 5th, etc. null in the derivative of the resonance spectrum. In this way permanent reoords of the spectra were obtained which could be used to study the line shapes and intensities of the individual lines. A sample recording i s shown in Figure 5. In the case of large line separations, as in the antiferromagnetic phase, the lines were not recorded on a recording chart. The different resonance lines were brought into the centre of an oscilloscope screen by adjusting the oscillator frequency manually, and the resonance frequency was read directly from the frequency counter. Derivatives were recorded only when the line widths were desired. (i) The proton resonance in the paramagnetic phase at 4.2°K. The results of the measurements in the paramagnetic phase are shown in Figures 6 to 9. Each set of points 20.8 20.9 21.0 21.1 21.2 21.3 21.4 21.5 21.6 21.7 Frequency in Mc/sec Fig. 5 Derivative of the Resonance Spectrum with H Q in the a-b plane. HQ=5000 gauss, f = 100°, T = 4,2 ° K - 43 -parallel to the frequency scale was obtained from a record-ing of the type shown in Figure 5. In each of the Figures 6 to 9 the proton resonance frequency in water i s indicated by a vertical line and the positions of the crystal axes are marked on the "orientation scale". The measured resonance frequencies are indicated by solid points. Solid lines are then drawn through the experimental points. In places where the resonance lines overlap, the null in the derivative i s not a true measure of the peak resonance frequency, since the peak i s slightly shifted due to the overlap. No corrections were made for such shifts, since the line shape i s not known. Figure 6 shows the resonance diagram for the H Q rotating in the a-c plane with H Q = 5000 gauss and the sample temperature T s 78° K. Figures 7 and 8 are the resonance diagrams for H Q rotating in the a-c plane at 4.3°K with H 0 s 3005 and H Q = 4970 gauss respectively. Figure 9 shows the resonance diagram for lf0 rotating in the a'-b plane with H Q = 4990 gauss and T s 4.2°K. As pointed out in chapters I and III, the separation between component lines depends on the temperature of the sample and on the applied f i e l d . The different resonance lines in Figure 6 were never completely resolved and only a maximum number of six lines could be observed. However, the resonance diagram of Figure 6 i s basically similar to the diagrams of Figures 7 and 8, since a l l three diagrams were obtained with H rotating in the a-c plane. The resolution 180 H G J.c ffl © u bO 0 Q 90 H Q ± a H o l | c a o •H +» a +J a © •H o H 0 I' a CT H o l Proton frequency i n water 21.300 Mc/sec Resonance frequency Mc/sec J I i _ 21.20 21.25 21.30 21.35 21.40 21.45 F i g . 6. Resonance Diagram for H Q i n a-c plane. H Q = 5000 gauss. T = 77 K 12.6 12.7 12.8 12.9 13.0 13.1 13.2 Fig. 7 Resonance Diagram for H Q in a-c plane. H Q = 3005 gauss. T - 4.2° K 180 Resonance frequency Mc/sec 20.9 21.0 21.1 21.2 21.3 21.4 21.5 21.6 Fig. 8 Resonance Diagram for H G in a-c plane. H Q - 4970 gauss. T - 4.2 K 20.8 21.0 21.2 21.4 21.6 21.8 Fig. 9 Resonance Diagram for H Q a'-b plane. H Q = 4990 gauss. T = 4.2 K - 4 7 -in Figures 7 to 9 i s much, better, and i n some orientations completely resolved lines could be observed. A l l four diagrams are characteristic of the paramagnetic phase as outlined in ohapter III. Each curve in the resonance dia-grams behaves to a f i r s t approximation like a sine function with a period of 180 degrees due to the factor (3 c o s 2 0 -1) in the internal f i e l d s . This i s in good agreement with the theory in chapter III. The resonance diagram for the a'-b rotation (Figure 9) i s symmetric with respect to the a'-axis and the b-axis, in agreement with the space group C2/m. The symmetry about the a'-axis i s due to a* lying in 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. gravity of the lines i s shifted from the proton frequency In water, indicating the need for the ideas put forth at the end of chapter III. The amount by which the lines are shifted from the proton frequency in water, due to the aver-resonance diagrams are available, but these can be obtained only i f the proton positions are known. However, approximate shifts can be obtained by assuming the protons are situated 13) at the oxygen positions. These ideas have been used ' in connection with earlier measurements to determine the ratios In each of the resonance diagrams the centre of age bulk magnetization i f . can be calculated i f theoretical The two ratios _ 48 -were found to be the same; as they should be, since M, the space average magnetization, and <^ /T^  > t n e time average magnetic moment of a given magnetic ion, should have the same temperature dependence. The theoretical number of 24 lines was never observed, At most, eight lines were present, suggesting that some of the protons have the same local f i e l d s , thus giving rise to identical resonance frequencies. To determine which of the protons are magnetically equivalent requires a knowledge of the proton positions. Figures 7 and 8 show that the eight lines appear in four pairs (indicated by l a , lb; etc.). The spectrum in the a-c rotation for = 115° at 4.2°K has been investigated as a function of the applied f i e l d H Q. In Figure 10 are shown plots of - f 4 and f"4b - f g a as a function of the magnitude of the applied f i e l d . 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 in one of the water mole-cules. The separation f 4 f e - f g i s a linear function of H Q and i s due to the internal 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 splitting of the line pair 4. However, the measured splittings are reliable only in the v i c i n i t y of (p = 115°. For (pg 70° and (p£, 130° the splitting in line pair 4 cannot be measured accurately since Fig. 10 The Frequency Difference ( f 4 a~ f4b) and (±4& - f2a) vs applied Field HQ.. The Orientation is at 110° w.r.t. Fig. 8. T - 4.2°K - 50 -there Is too much overlapping of lines. 14) J. W. McGrath and A. A. S i l v i d i ; have found that the proton-proton distance r in 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 A. With these values the average H-0-H angle i s 100 ± 5 degrees. If we assume that these values also hold for CoClg'6H20, the direction of the proton-proton vector r* due to the two protons giving rise to line pair 4 can be found. Due to the large magnetic moment of the cobalt ions we can safely assume that b ^>d in (3.7), and the separation between lines 4a and 4b i s AH - 4d - (3cos l9tl, - l) <4-1' where 6^  i s the angle between H Q and Expressed in terms of frequency (4.1) becomes Af = f J j ( W e , , - i ) <*•*) The term cos 2 i s conveniently expressed in terms of the _ ^ — » direction cosines of r and H Q. The direction cosines of ~r and H Q respectively are defined, using Figure 11 as and oC'- COS/9 , &'* COS B > Y'= COS, C ' U (4.3) - 51 -Figure 11, Definition of the direction angles of r and H 0. Equation (4,2) can now be rewritten in the form d f - a where 3(oC'cM$(p- If'sinff ~ 1 (4.4) when r i s chosen to be equal to 1.595 A . Some simple algebraic manipulation in (4.4) leads to Af/a = 3/2 (</-/'*}- 1 + fy,(Jl-fl)cos2((p+f0) (4.5) - 52 -where ^  i s defined hy j. orf) ^ ^' (4.6) tan ^Tfi Equation (4.5) describes the proton-proton splitting when H Q rotates in the a-c plane. Since the maximum splitting (Af/a) max occurs at - 115° in line pair 4 in the a-c rotation, we have that cos Z(H5° + and therefore -ii5° or % - n s * = &s* so that, from (4.6) cC'f ' 0.5352 (r%-oL'1)  ( 4 ' 7 ) In Figure 12 equation (4.5) has been f i t t e d to the experi-mental splitting Lt (solid points) in the vic i n i t y of Cp = 115°. The maximum splitting at (p - 115° i s ' (M/o.)mx - Z. 00 * 0.06 which leads to This implies that - 0 , so that "? i s in a plane paral l e l to the a-o plane. From Figure.12 and equation ( 4 . 7 ) the values of oC and Q are found to be Z = 0 . 1 7 7 8 J " * 0 . 8 1 7 8 . - 2S -- 54 -One set of solutions obtained from these values for the angles A and 0 i s eliminated using the fact that ~l£0 must be parallel to the p-^  - p g line when 4 f l s a maximum, and one obtains A = 65° and C = 25° —^ Thus the direction of r for the two protons giving rise to line pair 4 has been fixed. Since the two protons In each water molecule should give rise to two pairs of lines, each pair being s p l i t by an amount 4d, there should be a pair of lines in Figures 7 and 8 whose splitting i s in phase with line pair 4. Line pair 2 satisfies this requirement. However, i t i s d i f f i c u l t to establish experimentally whether this pair of lines i s really the counterpart of pair 4, since line pair 2 is never completely separated from the rest of the spectrum. Applying equation (4.2) in the a*- b rotation for the direction of r obtained from line pair 4, the split t i n g At i s -13.8 Kc/sec when H^ i s parallel to the a'-axis and -29.6 Kc/sec when H Q is parallel to the b-axis. In Figure 9 the measured separation in line pair 1 i s about 25 Kc/sec when ~RQ is parallel to the b-axis, and less when R0 i s in any other direction. When two lines are so close to one another that they overlap, as is the case in line pair 1 in 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 line pair 1 of Figure 9 the measured peak separation i s approximately equal to the line width of each line. We may therefore assume that line pair 1 in Figure 9 is produced by the same protons as line pair 4 in Figure 8, and line pair 2 in Figure 9 i s 'then the counterpart of line pair 1. The intensity of line pair 4 in Figure 8 i s approxi-mately 1/6 of the total intensity of the spectrum in the a-o rotation. The intensity of line pair 1 in Figure 9 i s also about 1/6 of the intensity of the spectrum in the a'-b rota-tion. In the a-c rotation the protons associated with the Oj are magnetically equivalent, since H 0 rotates in the mirror plane of the crystal. The two line pairs discussed above must thus originate from protons associated with the Oj^ , since one proton from each of the 0 water molecules comprises 1/6 of the total number of protons in the unit c e l l . In the a'-b rotation (Figure 9) six line pairs are observed. This may now be explained by the fact that in this rotation only the Oj water molecules situated diagonally opposite from the cobalt ion are magnetically equivalent. There are thus a total of three magnetically equivalent groups of water molecules in the unit c e l l , two associated with the octahedron about a cobalt ion, and the 0 ^ group. Each group gives rise to two pairs of lines, thus accounting for the six pairs of lines in Figure 9, At present i t i s not possible to uniquely determine - 56 -the proton positions in CoCl «6H 0 from the proton resonance & 2 measurements due to the relatively large number of lines and their extensive overlapping. A solution to the problem should become feasible when the present measurements are repeated at much higher fields and a third rotation in the b-c plane is completed, and accurate relative intensity measurements are made. Since a crystallographic analysis was not the goal of this work, these measurements were not pursued to the ful l e s t possible extent. (i i ) The proton resonance in the antiferromagnetic phase at 2.1°K. Measurements similar to those mentioned in the preceding section have been made in the antiferromagnetic phase (T < T^) at T - 2.1°K. The results are shown in the resonance diagrams of Figures 13 and 14 respectively for H Q rotating in the a-c plane and for H Q rotating in the a'-b plane. These results exhibit some features strikingly different from the measurements in the paramagnetic tempera-ture region. The characteristic features of the rotations i n the antiferromagnetic phase are: (l) Each curve in Figures 13 and 14 behaves like a sine function with a period of 360° instead of 180° as in 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 in water, i.e. for each line displaced by an amount - 57 -16 18 20 22 24 26 27 Resonance frequency i n Mc/sec F i g . 13 Resonance Diagram for H q i n the a-c plane. H 0 = 5000 gauss. T - 2.11° K Proton frequency i n water at 21.28 Mc/sec - 58 -Proton frequency in water at 21.28 Mc/sec - 59 --f-^f from the central frequency there i s an Identical line displaced by an equal and opposite amount — &>f from the central frequency. (3) In general the lines are much broader than in the paramagnetic phase. (4) The maximum number of lines observed is again 8, but there i s no evidence of a proton-proton sp l i t t i n g . (5) The maximum separation between lines i s of the order of 10 Mc/sec Instead of 600 Kc/sec as in the paramagnetic phase. (6) In the a*-b rotation (Figure 14) the resonance diagram i s again symmetric with respect to the a f-axis and the b-axis in agreement with space group C2/m. These points are in good agreement with theory as outlined in chapter III. In connection with (3.11) of chapter III i t was pointed out that the number of lines should double when passing from the paramagnetic to the antiferromagnetic phase. Experimentally only eight lines are observed. However, the separation between any pair of lines i s much too large to be attributed to a proton dipole-dipole interaction. We there-fore conclude that each of the curves in Figures 13 and 14 consists of two resonance lines, the spli t t i n g between them being too small to be observed since the lines in general are quite broad as compared to the line widths in the para-magnetic phase. Thus we can say that the number of lines in fact has doubled. The fact that six line pairs were observed in the a'-b rotation in the paramagnetic phase does not contradict - 60 -the above arguments in regard to Figure 14, since in the antiferromagnetic phase the cobalt spins are coupled to the crystal axes and l i e in the a-c (mirror) plane inde-pendent of the applied f i e l d . The four 0^ water molecules associated with the octahedron are thus magnetically equivalent even in the a'-b rotation i n the antiferro-magnetic phase. Therefore, only two groups of magnetically non-equivalent water molecules are present, and only eight lines are expected in the antiferromagnetic phase In the a f-b rotation. The spectrum at (f = 133° in the a-c rotation has been investigated as a function of the applied f i e l d at T = 1.14° K. The results are shown in Figure 15, where we have plotted the entire spectrum as a function of H'0. These measurements show that the separation between the various lines i s independent of the applied f i e l d . This result proves that the magnetic moment /U^of the cobalt ions in the antiferromagnetic phase is independent of the applied f i e l d as long as i t is not too large. The shape of the resonance diagrams is determined by the vector sum of of H Q with the internal f i e l d due to the antiferromagnetic cobalt ions ^//.^ • 1 1 1 6 small demagnetization f i e l d merely adds to the applied f i e l d . No attempt has been made to give theoretical formulae for the resonance curves in Figures 13 and 14 since the proton positions are not yet known and since i t 10 15 20 25 30 Resonance frequency in Mc/sec Fig. 15 Plot of Resonance Frequencies 2 to 7 of Fig. 13 as a function of the applied f i e l d . f = 133°. T - 1.14°K - 62 -i s not known what the sublattice arrangement of the cobalt ion magnetic moments i s in the antiferromagnetic phase. In a later chapter we shall see that there is some evidence that more than two sublattices are present in antiferro-magnetic CoCT 2»6H 20. - 63 -CHAPTER V THE PARAMAGNETIC-ANTIFERROMAGNETIC  PHASE TRANSITION The proton magnetic resonance technique provides a simple but accurate method for investigating the para-magnetic-ant i f erromagnetic phase transition in hydrated salts. In chapters III and IV i t was pointed out that the character of the resonance spectrum changes drastically when the sample temperature i s lowered from the paramagnetic to the antiferromagnetic temperature region. This change in character of the resonance spectra can be used to deter-mine the transition temperature T^ as a function of the applied f i e l d , and the crystal orientation relative to the applied f i e l d . The transition temperature of CoClg*6HgO in 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 Friedberg 1 6) (T N = 2.29°K) ; and by Miss Voorhoeve and Dokoupil 1 7^ (T = 2.28°K) from the X-transition in the specific heat, and by Van der Lugt and Pou l i s 1 8 ) (T K = 2.275 i ,01°K) from proton magnetic reson-ance experiments similar to those to be described in the following paragraphs. We have essentially duplicated and extended the measurements of Van der Lugt and Poulis with - 64 -Improved accuracy. Our results are in good agreement with theirs, hut additional details have been brought out not reported by these authors. The transition from the paramagnetic phase into the antiferromagnetic phase i s characterized by a broadening and decrease in amplitude of the resonance lines in the para-magnetic phase. In Figure 16 a plot of the signal amplitude of the resonance lines l a , 2a, and 4b of Figure 8 for <^ = 130° and H Q = 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 derivative. Figure 16 shows that the amplitudes of the resonance lines increase slightly as the Neel temperature i s approached and drop very rapidly near the transition temperature. However, the rate of change of the amplitude i s different for different lines. Similar graphs were obtained for other values of (f a n d l l 0 , and in each case the general behaviour i s similar to that of Figure 16. The region near the transition temperature Is of particular interest. Figure 17 is a magnification of the region near T = 2.2°K, including part of the antiferromag-netic temperature region. Over a small temperature region, approximately 7 millidegrees in Figure 17, lines from both phases are present simultaneously. As we shall see in a later paragraph, the lines associated with the paramagnetic phase broaden rapidly as the transition temperature i s - 67 -approached. Thus very broad lines of low amplitude begin to overlap and eventually i t becomes impossible to identify any single line. Also, the lines associated with the anti-ferromagnetic phase f i r s t appear near the proton frequency in water so that these lines also overlap with one another and with the lines from the paramagnetic phase very near the transition temperature. It i s thus impossible to analyze this 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 . If there i s any variation in this range with , then i t i s beyond the accuracy of this experiment. Measurements for 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 millidegrees. Such regions of coexistence have also been observed in- Azurlte 1 9) (about 160 millidegrees) and in MnFg 2 0 ^ (about 25 millidegrees). On f i r s t sight one might attribute this coexistence region to lattice imperfections in the crystal or to "supercooling" and "superheating", i.e. hysterisis. Lattice imperfections are ruled out by the fact that the same range was always present for different samples and that the range i s f i e l d dependent. We have eliminated the possibility of hysterisis by varying the temperature within the coexistence range and were able to reproduce any particular set of lines - 68 -over several hours. We thus tentatively attribute the region of coexistence of both types of spectra to short range order. It would be interesting to test this conjec-ture experimentally. The line widths also show an anomalous change near the transition temperature. In Figure 18 a plot of the line width of line 5 of Figure 9 (paramagnetic) and of line 8 of Figure 14 (antiferromagnetic) is presented as a function of temperature with H 0 = 5000 gauss at s 120°. This graph i s very suggestive of a A-type transition. It has been pointed out by Nakamura (private communication) that super-exchange interactions between the protons via the cobalt ions, or between protons and paramagnetic impurities at the cobalt ion positions, could produce a line broadening of this type. S t r i c t l y speaking, the two branches of the line widths in Figure 18 should cross over near 2.24°K, since some lines associated with the antiferromagnetic phase appear before those of the paramagnetic phase have vanished. No attempt has been made to measure line widths throughout the transition region, since here most of the lines overlap, making accurate line-width determination impossible, especially in view of the very small amplitudes. As pointed out previously, hysterisis i s absent in this transition. Indeed, the fact that some of the ohange occurs even before the transition region i s reached shows that there i s no choice 1 I I I I I I I I I 1.8 2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6°K Fig. 18 The line width of line 5 in Fig. 9 and line 8 in Fig. 13 vs temperature H - 5000 gauss, $ = 120°. •paramagnetic, A antif erromagnetic - 70 -of paths for the transition. The new phase i s established gradually rather than at one single temperature. Figure 17 points out some ambiguity in the defini-tion of the transition temperature. St r i c t l y speaking, there i s a small range of temperature over which the tran-sition takes place. We have investigated the transition temperature as a function of the applied "field and crystal orientation. These measurements were performed by continuously monitoring the amplitude of one of the lines associated with the paramagnetic phase, and T^ was chosen as that temperature at which this amplitude had decreased to the noise l e v e l . Thus the lower values of the transition range were obtained. These measurements are possible, since each resonance frequency i s independent of temperature in the paramagnetic phase, except for the last 10 or 15 m i l l i -degrees where the frequencies are shifted towards the proton frequency in water. These shifts, however, are at most 50 Kc/sec whereas the line widths in the transition region are usually much greater, so that the amplitude measure-ments are only slightly affected. A graph of T^ as a function of the applied f i e l d in the a-c plane for three values of dp i s shown in Figure 19. T^ i s independent of orientation in the a*-b plane. The ^P-dependence of T^ has been inves-tigated for several values of H Q and T N(^p) i s found to behave like a sine function with a period of 180 degrees. This behaviour of T K r((^) can be expressed in the form 2.30 -o 2.25 -55 H <a u ft 2 ® d o •rl -P •H 2 2. rt u H 20 -15 -H_ in Kgauss Fig. 19 Paramagnetic-antiferromagnetic phase to= transition temperature as a function of the applied f i e l d for <jP = 0 ° , 45°, and 90°. - 72 -TN(<P)]H = A(U0) + B(Hc)cos*<(p (5.1) where A(H0) and B(HQ) are functions of H Q. In Figure 20 a plot of T N ( ^ ) Is shown for H Q = 5000 gauss and rotating In the a-c plane. For this value of H Q the hest f i t for (5.1) i s obtained with A(H_) = 2.162°K B(HQ) = 0.078°K and from Figure 19, B(H0) i s found to be nearly linear in H 0 as shown in Figure 21 with B(HQ) = (1.45.10"5 x H 0)°K. We cannot attribute this anisotropy of T N to the anisotropic g-factor in CoCl 2"6H 20. The principal values of the g-factor have been measured by Based!'21) in connection with paramag-netic susceptibility measurements, and by Date 2 2) in connec-tion with paramagnetic and antiferromagnetic resonance experiments. The values given by Haseda are g . = 2.7, gb o 4.9, and g Q = 4.9, Date*s values are only slightly different. Thus, the g-tensor varies identically in the a'-c plane as in the a'-b plane, yet T^ Is independent of orientation when H Q rotates in the a r-b plane. Therefore, the variation of T^ with in the a-c plane cannot be a direct consequence of the variation in the g-tensor. 2 . 1 5 9 0 1 8 0 Relative orientation in 2 7 0 degrees Fig. 2 0 Plot of T J J vs Cp for H Q in the a-c plane and H Q • 5 0 0 0 gauss. .Experiment — [ T n < ? )] ^ _ 5 - 2 . 1 6 2 « + 0 . 0 7 8 ° cos* f . - 75 -Nagamiya, Yosida and Kubo25^ have established a 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 this theory with minor modifications to conform with CoCT2'6HgO i s presented in Appendix B. The results by Nagamiya et a l show that T^ may be expressed in the form TN(H,) -T M(o)- C«H* +••• (5,3> to a f i r s t approximation in HQ, when the exchange inter-actions are assumed to be isotropic. The quantity C depends on the orientation of B 0^. Date 2 4^ has found i t necessary to introduce additional anisotropic exchange interactions to explain his antiferromagnetic microwave resonance experiments in CoCl2»6HgO. These additional terms, however, only change the f i e l d independent quantity C in (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 N/dH Q - 0 for HQ= 0. Experimentally i t i s found that T N(H 0) has a f i n i t e slope at H Q = 0, and none of the curves in Figure 19 can be approximated by a quadratic function In H Q. In particular, the term B(HQ) in (5.1) was found to be 2 linear in H Q , which completely rules out an H Q dependence of T N(H 0) on E Q . Equation (5.3) implies that ! I ^ ( E o 1O ) decreases monotonically with increasing HQ, whereas experimentally we 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 transition temperature in zero f i e l d i s always greater than that in a f i n i t e f i e l d . Similar measurements have been performed in CuClg^HgO 2 5). However, the available data i s not s u f f l -sufficiently accurate to really test the present theory. It i s not surprising that this theory does not agree with the present measurements, since these were obtained very accurately, and the theory i s based on a crude approxi-mation, not Including any details of the antiferromagnetic sublattice arrangements. - 77 -CHAPTER VI THERMODYNAMICS OF THE PHASE TRANSITION In this chapter we shall discuss some thermodynamic aspects of the paramagnetic-antiferromagnetic phase trans-itions. As i s seen in Figure 18, the proton resonance line widths exhibit a lambda-shaped anomalous peak near the transition temperature TJ, i.e. they increase anomalously as T-*TN either from below or above T^. This type of behaviour has also been observed in M ^ g 2 0 ) . Also, the specific heat in CoCl -6H 0 has been measured by Robinson and Friedbergi°) as a funotion of temperature between about 1°K and 20°K in zero applied f i e l d . These measurements show that the speoific heat also has an anomalous peak at 2«29°K. The peak i s identified with a cooperative order-disorder transition and the temperature at which the peak occurs i s the transition temperature. Robinson.and Friedberg found that the behaviour of the magnetic contribution to the speoific heat near the transition temperature could be described by a logarithmic function of the form log (T-T N), which i s also characteristic of the anomaly in the X-transition of liquid helium. Such a singularity 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 Fairbank 2 6^ in their analysis of the liquid helium A-transitlon to discuss some - 78 -thermodynamic aspects of the magnetic phase transition in GoClg'SHgO. We confine ourselves to the case [TN(H0)]H „ c > since here the slope ^ of the X-line T N(H 0) i s always f i n i t e . The case where a H beoomes in f i n i t e w i l l be discussed later. Using the identity [dxjy ' l a x / ? wwx 157/2 where W, X, Y, Z are functions of state with two independent variables; and the Maxwell relations, we have (di) 9fdT) _/9J\ (dT) (6.2) and (Si) ./5T) -/ar) tea.) (68, Since C H >oo as T — * (^CH « f ( | i p ) H ) we have from (6.2) that /dr) /<2I) . ^ at T=TM ( 6 - 4 ) - 79 -and from (6,3) that (dT) 'dl\ _ dTN ( 6 . 5 ) The latter can easily be shown by using the relation (6.6) (dL\ = -(dT) (dT) The left-hand side in (6.6) vanishes on the X-line and (SI)S i s f i n i t e and equal to the slope of the X-line at T = T K, so that ^ s t vanish. Using this fact in (6.3) immediately gives (6.5). In the above relations H denotes the applied magnetic f i e l d . In order to study the variation of thermodynamic properties near the transition temperature Buckingham and Fairbank introduce a new variable, the "neighbourhood temperature" t, a function of state defined by t s T-T N(H). (6.7) The line t = 0 i s the X - l i n e and. a l i n e t = t 0(constant) Is a line parallel to the X - l i n e in the phase diagram. Using (6.1) and the new variable we have ( 6 . 8 ) _ 80 -and (6.9) Combining (6.8) and (6.9) gives 9a ^fiaffdM) +fds\ _fdH\fdtL\ ( 6 , 1 0 ) T UTJ W I T lar/t {3jNJ[dTjt Buckingham and Fairbank argue that the quantity (^pj^ varies relatively l i t t l e over a small temperature range in the neighbourhood of the transition, although i t has an inf i n i t e temperature derivative at the transition temperature i t s e l f . This can be seen from the following argument: Near the transition line we could write S = S(rN) +/»(h)fft) where A(H) i s a function of the applied f i e l d H and the function f(t) has an infinite derivative at t = 0. Then so that f&L) depends on t in a manner similar to the t dependence of S i t s e l f . Therefore, even though (dS\ V 8 T /t has an inf i n i t e temperature derivative at T^, i t s total variation over a small temperature range near T^ . w i l l never-theless be small. Similar arguments may be applied to the - 81 -quantity /2l£iV . Therefore, a plot of versus f ^ - j should he linear near T N with a slope fcLHfdTw' The specific heat Cg of CoCl2"6H20 as a function of temperature has been measured in zero f i e l d by Robinson and Friedberg 1 6). Flippen and Friedberg 3 ) have measured the adiabatic susceptibility /«2£lV "near" H n = 0 (the measurements Involve a small oscillatory f i e l d ) . We have repeated some of the susceptibility measurements with an improved accuracy. Our measurements were made by measuring the self-inductance of a c o i l containing a sample of CoCl 2»6H 20. The c o i l formed part of the tuned ci r c u i t of the same Pound-Knight-Watkins oscillator used to perform the NMR measurements discussed in the preceding chapters, and the changes In self-inductance were calculated from the changes in the oscillator frequency. The oscillator frequency at 4.2°K was set to 21 Mo/sec and the total change in frequency was found to be approximately 60 Kc/sec as the temperature changed from 4.2°K to about 1 03°E. 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 in Figure 22, where the crosses (x) indicate the points published by Flippen and Friedberg. The two sets of measurements are in good agreement, but in the present measurements many more points were obtained, especially near the transition temperature. The transition temperature is clearly indicated by a discontinuity in the 1.2 1.6 2.0 2.4 2.8 3.2 3.6 4.0 °K Fig. 22 The molar susceptibility along the c-axis as a function of temp. - 63 -slope of the susceptibility, and this method could actually be used to measure the transition temperature. These results, 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 in the v i c i n i t y of T^ in zero applied f i e l d . In doing so we assume that the difference between the adia-batic and isothermal susceptibilities i s negligible in very small fields which i s reasonable in the absence of an applied f i e l d . We therefore use the adiabatic susceptibility (p-j^  in (6.10) instead of the isothermal susceptibility(dUV to obtain / j , since the error introduced in this change w i l l be negligible. A plot of °W versus [5777c i s shown in Figure 23 for both T < T_T and T>T.T. The neighborhood temper-' V ature t = T - i s Indicated by the arrows. A linear plot i s obtained over the entire temperature range s tudied for T^T W. The value of thus obtained i s 1.71 x 10 4 gauss/°K and i s shown as a dotted line in Figure 19, and is seen to be in agreement with the TN(H) curve obtained using proton resonance for H Q parallel to the c-axis. The proton resonance data at low fields are rather poor, because at such low frequencies the oscillator sensi-t i v i t y has greatly decreased, so that i t would be interesting to make plots similar to those of Figure 23 for large applied f i e l d s . Unfortunately, no heat capacity measurements have been made so far in the presence of large d.c. f i e l d s . N . _ WA [B~j Molar susceptibility {dM/dH)c (T< T N) 0.07 0.10 0.15 0.20 0.25 CD Fig. 23 Plot of (dtA/dti )s v s C H /T in CoCl 2«6H 20. - 85 -For T>T N we do not get a linear relationship between and (^/d^ except for the region t£0„l°K. If one Identified the magnitude of the slope of the curve in this region with (HL^ , one obtains 41 - l * 8 ^ x 10 4 gauss /°K which i s in agreement with the value for T< within experimental error. However, there is a d i f f i c u l t y in this assignment, because the slope of the Qi versus f^ |)$ curve for t>0.1°K Is negative. This Is clearly impossible since (dJt^ must be positive. Since no way out \dTfi) of this dilemma has been found, i t appears that the numerical agreement mentioned above may be fortuitous and that one may anticipate (^-j, going through a minimum as the Neel temperature i s approached from above for t 0,06°K. There are two pieces of evidence which support this conjecture. F i r s t l y , extrapolation of versus ^£1)5 to zero ^ g ) s should give the same value for T > T K and T£T,_, namely (S§\r- - dJL (^M)^. as seen from equation (6.10). The straight line obtained for T< extrapolates to approximately -0.125 cal /(°K) 2 - mole for this quantity. In order to obtain a negative value for this quantity for 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 in zero f i e l d i s that we have observed that this does happen for H Q 2000 gauss. This is seen in Figure 24, which i s a plot of oscillator frequency versus temperature - 86 -21.185H HQ=0 Kgauss 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 J I L J 1 1 L J 1 U 1.6 2.0 T°K 2.5 Fig. 24 Oscillator frequency vs Temperature - 87 -for various applied fields. The curve for II : 0 i s the one from which the susceptibilities in Figure 23 were obtained. In order to make use of careful measurements of for t<0,06°K, the specific heat measurements would also have to be made closer to T^. Since there are no specific heat measurements available for H Q j- 0, we are unable to make quantitative use of our data of Figure 24 except for the H Q = 0 curve. It would also be desirable to do measurements of susceptibility for the case of H 0 perpendicular to the preferred axis since 4= i s infin i t e for two values of f i e l d in that orientation. These measurements have not been carried out yet, but one may anticipate that the nature of the phase transition may change drastically above 3000 gauss. Finally, the anomalous increase of (J^ffjs (decrease of frequency) at low temperatures for H 0 = 6,700, and H 0 = 7,000 gauss in Figure 24 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 for the "flop-over" antiferromagnetic phase transition as I B ) discussed by Van der Lugt and Poulls \ - 88 -CHAPTER VII THE SUBLATTICE MAGNETIZATION BELOW T N In this chapter sublattice magnet 125a tion measure-ments as a function of temperature for T<1 T^ are discussed. It w i l l be interesting to note an empirical relation between the sublattice magnetization in the antiferromag-netic 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 orientation. Measure-ments were performed at two fields and about a dozen different orientations. A representative set of measurements is shown in Figure 35, where H Q = 5,260 gauss was In the a-o plane at (p B 0. Each set of points parallel 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 transition region was approached, since here too many lines overlap, making accurate measurements impossible. As stated before, the resonance frequencies are determined by the local f i e l d at the positions of the different protons, where' H l o c = H Q + B ^ n f r® i e quantity B^ n f denotes the internal magnetic f i e l d at the proton 'K 2.2 2.0 1.8 1.6 1.4 1.2h -TN= 2.24°K J_ _L _L / 1 i \ \ 1 1 •5 | A6 J_ •\. *8 V 00 to 16 17 18 24 25 26 19 20 21 22 23 Resonance frequency i n Mc/sec F i g . 25 Proton resonance i n antiferromagnetic CoC^'SB^O as a function of temperature, H Q = 5.26 Kgauss, H Q c-axis. - 90 -positions produced by the neighbouring magnetic ions. It is now convenient to define the scalar quantity k(^}T) -\jTiJ-\rZ\ =|nW«fHffJ For HQ5> H i n t , a condition satisfied in most of the experi-ments, h Is the component of along the applied f i e l d In chapter III we saw that the component of the internal f i e l d along H Q i s proportional to the sum of the time-averaged magnetic moments ^ • In the antif erro-magnetic phase the magnetic moments of one sublattice a l l point in one direction, so that the sum of the magnetic moments over one sublattice gives the sublattice magneti-zation. Thus a measure of h(H 0 , T) gives a measure of the sublattice magnetization as a function of temperature. Since &) = Z'TfJ = ^H l o 0 j(7.1) may be rewritten in terms of frequency: f(«o,T) - f r e s ( H l 0 O T ) - f o where 2#Tfn )fK and -ffgS i s the measured resonance frequency of a given line at a temperature T, and j i s the proton resonance frequency in water at an external f i e l d H Q. To avoid the contribution to from the bulk magnetization of the sample, i t i s convenient to measure the difference in frequency of a pair of resonance lines, rather than the actual frequency. This difference i s then roughly propor-tional to the difference in sublattice magnetization. Thus, - 91 -the temperature dependence of the sublattice magnetiza-tion may be obtained by measuring —¥ which i s independent of H Q and the bulk magnetization of the sample. The rapid variation of ^ fg near T^ (Figure 25) suggests a logarithmic dependence of Afj j on | T - T^| . Plots of Af-j versus log(T N - T) were prepared, and i t was found that a l l the proton resonance lines give the following result, which Is independent of H Q and crystal orientation Afij(fe) _ „ fa ( - f r - r ) + const i n~ 0,13±io% ( 7 - 2 ) where -pLj (H0,0) was obtained by extrapolating the plots of /\f;j versus log(T N - T) to (T N - T) = T N > i.e. to T = 0. A plot A ^ j versus log (T^ - T) corresponding to the results of Figure 25 i s shown in Figure 26. The individual values of n are indicated for each pair of lines, k similar plot (not shown) was prepared for H Q = 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 I I I I i i i i I L I I I I i i i I U 0.02 0.04 0.1 0.2 0.4 1.0 2.0 - 93 -M j ^ j . - n in TN-T + x . (7.3) TN 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 transition region. Similar measurements have been performed in Mj^ Fg 2 ^ , and there It was found that M(T) <=•< (T N - T ) 1 / 3 near T N. It is thus not at a l l clear whether the sublattice magnetization in different substances can be described by the same function. The shape of the curve given by (7,3) differs considerably from the dependence of M(T) given by the Weiss molecular f i e l d approximation 1 8). No explanation i s given here for the behaviour of M(T) in CoCl g»6H 20. An attempt to understand It could be made following Dzialoshinskii's 27) method , but the magnetic space groups of CoCl2»6Hg0 are not known. Neutron diffraction experiments would, therefore, be desirable, but so far there are very few neutron diffrac-tion data on hydrated salts of the transition elements. A group theoretical analysis 2 8) of the proton resonances in the antiferromagnetic state may be successful however, and is being pursued in this department. In connection with the possible sublattice structure of CoCl 2»6H 20 two experimental results should be mentioned: 1)) Daniels and G r i f f i t h (to be published) have studied the - 9 4 -angular dependence of <f-radiation from single crystals of CoClge6Hc>0, where the cobalt ions In the surface layers of the crystal were replaced by Mn 5 4 ions. It was found that the distribution of <T-radiation could not be explained on the basis of a two-sublattice model. Although no sublattice structure could be given from these measurements, indica-tions were that a more complicated sublattice structure than the two-sublattice model exists in CoCl g»6H 20. 2)) Measurements by Flippen and Friedberg show that the susceptibility parallel 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 in the foregoing chapters indicate the need for more extensive and varied experiments In CoClg*6Hg0 to f u l l y explain the physical properties of this substance. The following experi-ments would be f r u i t f u l : 1) Measurements in the paramagnetic phase should be repeated at much higher fields than those employed in the present experiments. Such measurements would greatly improve the resolution of resonance lines and enable a complete crystallographic analysis. 2) A thorough study of the proton resonance line shapes and line intensities would provide information as to how many protons contribute to a particular resonance li n e . This information w i l l be particularly useful in the proposed magnetic space group analysis using the resonance diagrams of the antiferromagnetic phase. 3) The transition region should be investigated in detail. An exact knowledge of the range of the transition 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 transition. In this region double-resonance experiments would be of parti-cular interest. The effect, i f any, on resonance lines in - 96 -the antiferromagnetic phase when a line of the paramagnetic phase is saturated would immediately t e l l i f the transition takes place by a gradual build-up of domains, or i f individual magnetic moments fluctuate between the two phases. 4) NMR experiments on deuterated samples of CoCl2*6H20 might prove very useful. For instance, i t would be very interesting to see what effect the replacement of protons by deuterons would have on the transition temperature. Since deuterated samples should give resonance diagrams very simi-la r to those presented in chapter IV, such samples could be used for the high f i e l d experiments without the d i f f i c u l t y of obtaining high enough oscillator frequencies. Some other very useful experiments would be: 5) A study of the specific 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 forth in chapter VI. It would be most interesting to see i f such measurements give the same phase diagram as obtained with NMR measurements. 6) Susceptibility measurements should be made in the presence of an applied f i e l d for the three principal orientations. 7) As pointed out before, neutron diffraction experi-ments would be most useful. These would determine the magnetic space group of antiferromagnetic CoGl2'6H20 and probably the nature of the sublattice structure. - 97 -APPENDIX A CONSTRUCTION HINTS  AND OPERATING- INSTRUCTIONS 1) Oscillating detector (Figure IA). The oscillating detector was constructed in a box made of 1/8 inch brass sheet. The H.T. and filament supplies and their accompanying components are housed in a "false bottom", and the leads from this compartment to the tubes are kept as short as possible. Faraday shields are placed between the oscillator (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 tubes (2 x 6C4). This type of construction minimizes microphonic pick-up and undesirable r-f leakage between the various sections. 2) Williamson Power Amplifier (Figure 2A) . Best results are obtained with this c i r c u i t i f the following adjustments are made when the input i s grounded: a) Adjust Rr>5 to equal 1200*Vload Impedance". b) Connect a suitable ma meter in the lead to the center-tap of the output transformer primary and set the total current to 125 ma by means of Rp,. - 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 in this adjustment since they are due to the A.C. mains and fluctuations in 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 for water cooling. The reference resistor is made from a 1 inch manganin strip and immersed in an o i l bath which is maintained at a constant temperature by means of a cold water jacket. The reference voltage supply i s imbedded in styrofoam to avoid voltage fluctuations due to temperature variations. Operation: Switch on the reference voltage and set the helipot to some arbitrary value. The meter on the Honeywell Null 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 is approximately 4 volts. The null detector w i l l "hunt" momentarily, but should soon settle 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 volts. - 99 -4) Low temperature system (Figure 6A). Except for the manometers and dewars, which are made of glass, the low temperature system Is made of copper and stainless steel 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 i t with He gas to 1 atm. pressure. Evacuate the inner dewar jacket to approximately 4 cm Hg, and then f i l l the outer dewar with liquid nitrogen. From time to time add He gas to the inner dewar to keep the pressure at approximately 1 atm. Keep the nitrogen level in the outer dewar as high as pos-sible at a l l times. When the pressure in the inner dewar has settled to a constant reading (approximately 90 min.), the system has reached liquid nitrogen temperature and i s ready for a He transfer. To transfer: Completely evacuate the inner dewar jacket and be sure to close stopcock A. Open the free end of the syphon and insert i t into the liquid 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 liquid He can by means of the compressed He gas cylinder, effecting a transfer of liquid He into the inner dewar. Temperatures lower than 4.2°K: Open valves l c , 2, 3, 4, and 14 f u l l y . Close 5 and immediately begin to open la and then lb i f necessary. When the desired pressure - 100 -has been reached, close l c and 4. The pressure i n the bellows B i s now fix e d and any change i n the rate of pumping w i l l be corrected by the expansion or contraction of the bellows. To change the temperature, valve 4 i s opened and the pressure i n the bellows i s adjusted to the desired value. Fig. IA Oscillating Detector Twin "T" network. When t h i s i s removed, we have the broad band amplifier =r.003 From Osc. Det, ' Lii U L Wvv .ooa_ >COK „ 006=5 1 Metal can 6SJ7 17K 1-250K _ J Audio Gain 50 .IK >15K 8z±z :47K 6C5 8 >250 > K 3 K=b 25 250V — r - * to centre of primary of H448 i n phase s e n s i t i v e detector to scope O ~T~ .1 4 •250K 6SJ7 150 F i g . 2A Narrow Band and Broad Band Amplifier - 103 -+250V I O O K : 4.7M 4.7M 6SN7 LOOK O Q-Record. rtz Milliam. zZZ. \ZV\A/\Ar Zero meter Close when 20K balancing etc Balance 4.7M C jif .02 4.05 .1 4wl5 .2 •.3 .4 etc •4.7M Tsec .1 .25 .5 .75 1.0 1.5 2.0 [ 50K 30K Plate of l a s t ampl. O stage 1/26SN7 ( r±I__ N / V Vv-to scope "^or balance Ref. voltage input o H448 - V W 33K TL 2 = 4.7M T 1/2 6SN7 F i g . 3A Phase Sensitive Detector Fig. 4A Williamson Power Amplifier ±A(b Honeywell Null Detector Fig. 5A Transistor Magnet Current Regulator syphon to He gas storage Oil bubbler to l i q . He ^vessel 8 He 4 7 « * o Hg 13 -» 11 lo syphon 12 ^ jacket Manostat To main pump B I) C Outer Dewar Hg «>D o i l sample Temperature measuring manometers Fig. 6A Schematic of low temperature system 107 APPENDIX B THE NAGAMIYA, YOSIDA AND KUBO THEORY FOR %(H) APPLIED TO 0 o 0 1 g « 6 H 2 0 The exchange interaction between two ionic spins, responsible for the spontaneous alignment of neighboring spins in ferromagnetism and antiferromagnetism, i s equivalent to an interatomic potential VLi - - '/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 integral, which i s negative for antiferromagnetic substances. In the simplest form of the molecular f i e l d model, one assumes that J i s the same for 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 i belongs. The potential energy of atom i in the f i e l d produced by the sublattice j i s , apart from an additive constant, given by Vi - I - - Z T(tj) - t f - jTej' A (B.i) where /M[ i s the magnetic moment of an individual atom of spin and He^ i s the exchange magnetic f i e l d produced by the sublattice j and acting on atom i . We now make the - 108 -basic assumption that the whole antiferromagnetic crystal i s divided into two similar sublattlces, one with i t s spins pointing in the positive direction and the other with i t s spins pointing in the negative direction. The exchange f i e l d acting on each spin of the positive sub-lattice results from the spins of the negative sublattice and also from the other spins of the positive sublattice. If we define M + = H/Z (/T f) and M*~ = N/fc where N i s the number of magnetic Ions per unit volume, then the exchange fields on the positive and negative sub-lattice are, respectively, from (B.l) rTg - - AM'- r / ? + _> _^ (B.2) WI « ~AM* - P M~ where A and P , containing the exchange integral J, are the isotropic molecular f i e l d constants of the crystal. Date 2 4) has found i t necessary to introduce additional anisotropic molecular f i e l d tensors A and V to explain his antiferromagnetic microwave resonance experiments in CoCl g*6H 20. A and V are 3x3 matrices having zero traces and identical principal axes which determine the principal axes of the antiferromagnetic state. Equations analogous to (B.2) may now be written as - 109 -He (A-tA')M'- ( r + P')fA+ (B.3) The total effective 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 H', which is defined in terms of the principal values of g by i t s components as follows: Since the principal axes of g' coincide with the principal axes of the antiferromagnetic state in CoClg'6HG0, the total effective f i e l d may now be written 7?, . 7?± (B.5) He« ' H' * He where H~ i s given in (B.3). H"^ ^ i s the total f i e l d at the position of an ion belonging to the positive ore nega-tive sublattice, depending on whether the plus or minus sign Is used. We may now proceed in the usual way to find the equilibrium values of M - . After some algebraic manipula-tion we obtain the familiar expression (B.6) - 110 -where Bjfy1) i s the B r i l l o u i n function f o r a spin S with y t = | j?+ fTe*| Sq(3/feT I t has been found appropriate to 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 values of A and P be, respectively, A'x)Aj; A'i a n d P/ j Py'j Pi » where x, y, z corres-pond to the c r y s t a l axes c, b, and a'. In the absence of an applied f i e l d we may write M+c |*]-« H 0 and (B.6) becomes where M 0 o = In the l i m i t of T-+T N(0), the t r a n s i t i o n temperature In zero f i e l d , M o - * 0 and tank ((jo)-* % where (j0 i s the quantity i n the square brackets i n (B.7). Substituting these values i n (B.7) and solving f o r T we obtain Vo;- M«tf[(A + A 0 - ( r ( B - 8 ) the t r a n s i t i o n temperature i n zero applied f i e l d . In order to f i n d the paramagnetic-antiferromagnetic phase t r a n s i t i o n temperature i n the presence of an applied f i e l d , we may proceed as follows: - I l l -a) H Q parallel to the preferred (x) axis. Let A x = A + A * and Pt = T+ , then M 1 becomes 2 k T (B.9) Rewriting (B.9), we have tonk'^yUn.k" 1^ - (Arrx)(W-M~)W ( B . I O ) In the limit of T-^T^, the magnetization M + -» M"~, so that M, (B.ll) oo When T = T N, we have that M + = M , so that (B.9) becomes M+ = Moo tank '(H'-(A* + r*)M+tefi] (B.12) - 112 -Substituting ( B . 1 2 ) into (B.ll) we obtain ZkT (^-rx)M„qp> rN(o) >^,*g ) M*)q|S] ( B . 1 3 ; ZkT The solution of ( B . 1 3 ) for T gives T N(H Q). Since T N(H Q) varies only by a relatively small amount with H0, we are interested in small corrections to T N ( 0 ) , and can thus use M + T N ( 0 ) in ( B . 1 3 ) as a f i r s t approximation to find TjjfH). We may also note that M+(TN(o))- iXA(o>)H< i ( B . 1 4 ) so that ( B . 1 3 ) becomes & to/id TM(0) .9 ^ XpTJ Zt\(o) ( B . 1 5 ) Equation ( B . 1 5 ) gives T^ as a function of the applied f i e l d H Q in a f i r s t approximation and becomes exact as Tjjj(H0) -> T J J(O), I.e. when H Q - > 0 . For small values of H Q ( B . 1 5 ) - 113 -becomes TM'TU(0) * TN(0) 9 b) H Q perpendicular to the preferred axis;(i.e. H Q parallel to the y axis). In this case the antiferromagnetic spins are no longer parallel with H . In general, M"1" —>_ —> and M l i e in the x-y plane and make equal angles with HQ, and M = M~\ The derivation of T N(H 0) in this case i s more complicated than in the case discussed above, because the angle between H 0 and M must be taken into account. The calculations give the following result: TH(Ho) - T N ( 0 ) s 1 rN(o)  3 j s gs(Ax-r„) ( B . 1 7 ) - 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. Te 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 for 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 Brit. 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. Heller 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 in Physics 4, 1 (1955). 24) M. Date, Proc. Int. Conf. on Magnet, and Crystall. 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. Dzialoshinskii, Soviet Physics, JETP 5, 1259 (1957). 28) E. P. Riedel and R. D. Spence, Physica 26, 1174 (1961). 

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