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Optical detection of spin-bath relaxation in some paramagnetic crystals Glattli, Hans 1966

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OPTICAL DETECTION OP SPIN-BATH RELAXATION IN SOME PARAMAGNETIC CRYSTALS f by HANS GLATTLI D i p l . Phys. ETH, Zurich, 1963 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n the Department of Physics We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA July, 1966 in presenting this thesis in partial fulfilment of the requirements f o r an advanced degree at the University of British Columbia, I agree that the Library shall make i t freely available for reference and study, 1 further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. It is understood that copying or publication of" this thesis for financial gain shall not be allowed without my written permission. Department The University of British Columbia Vancouver 8, Canada THE UNIVERSITY OF BRITISH COLUMBIA FACULTY OF GRADUATE STUDIES PROGRAMME OF THE FINAL ORAL EXAMINATION FOR THE DEGREE OF DOCTOR OF PHILOSOPHY of HANS GLATTLI Dipl.Phys., Swiss Federal I n s t i t u t e of Technology, (ETH), Z u r i c h , 1963 FRIDAY, AUGUST 19, 1966, at 1:30 P„M. IN ROOM 301, HENNINGS BUILDING COMMITTEE IN CHARGE Chairman; J . H. G. Smith M. Bloom C, F. Schwerdtfeger J . B. Farmer D. L„ W i l l i a m s W. Opechowski R. C„ W i l l i a m s E x t e r n a l Examiner: S. Geschwind B e l l Telephone L a b o r a t o r i e s , Incorporated Murray H i l l , New Jersey Research Supervisor: M. Bloom OPTICAL DETECTION OF SPIN-BATH RELAXATION IN SOME PARAMAGNETIC CRYSTALS ABSTRACT The magneto-optical Faraday effect has been used to observe the spin-bath relaxation at low tem-peratures in CeES and in .Eu-doped CaF2o The para-magnetic Faraday rotation § is an instantaneous measure of the magnetization M and i t is shown that in CeES and Eu 2 + ; CaF2» c> o° M at the light fre-quencies employed. The apparatus is the same as that previously described by Rieckoff and by Gr i f f i t h s . Pulsed MW-power at X-band has been used to disturb the equilibrium between spin system and bath. In CeES, the observed relaxation time T is of the order of a few msec, which is several orders of magnitude longer than the theoretical estimate of T^o This suggests a severe bottleneck in the energy transfer spin-bath. T~ is found to be environment-dependent'. In Hell, the relaxation is exponential. T is in good agreement with nonresonant relaxation measurements by Van den Broek and Van der Marel. I t , i s explained as arising from the Kapitza boun-dary resistance at the CeES-Hell interface. In Hel, the relaxation is non-exponential and is slower than in He gas at the same temperature. This sug-gests that in this case the thermal diffusion in the helium around the crystal is the bottleneck. The same relaxation behaviour is found when the crystal is heated d i e l e c t r i c a l l y with MW power far off re-sonance. This supports the assumption that the energy transfer spin-bath is limited by spatial dif-fusion. If the crystal is surrounded by a He film at a temperature below the X -point, T is found to be the same as in Hell up to a well defined average MW power level. For higher powers the relaxation be-haviour is similar to that of CeES immersed in Hel. In E u 2 + ; CaF;?} T^ is expected to have the form Ti = AT + BT5. The observed relaxation time T , however, is found to be concentration dependent. A l l measurements have been done on the +4- —> -\ transi-tion with H II [100] o For the three lowest concen-trations, the temperature dependence of T from 1.5 to 4.2°K can be fitte d with the expression r _ / ,=CT with C = 2.75 ( s e c O K ) -1 for 0.02% Eu, C = 3.5 for 0 , 8 % and C = 5 f o r 0 . 2 % . At a concentration of 27=,, r i s shorter and T~'~*> T% from l o 5 ° K to 7°K. The concentrations given correspond to the t o t a l Eu con-tent o The Eu^ + concentration has been i n f e r r e d from the magnitude of the saturation r o t a t i o n . T~ (T) seems to depend on both Ev£+ and Eu^" concentrations. It i s suggested that exchange coupled pa i r s of Eu^ + and c l u s t e r s i n v o l v i n g En*" may account f o r the con-ce n t r a t i o n dependence of T . Upper l i m i t s of A = 2 . 5 and B = 5 x 1 0 " ^ are found f o r T^ by extrapolating the lowest concentrations investigated. These values are somewhat lower than both measured and c a l c u l a t e d values found by Huang. GRADUATE STUDIES F i e l d of Study: Low Temperature S o l i d State Elementary Quantum Mechanics F. A„ Kaempffer Quantum Theory of Solids W. Opechowski Advanced Magnetism M, Bloom S t a t i s t i c a l Mechanics R. Barrie Group Theory Methods i n Quantum Mechanics W. Opechowski Applied Electromagnetic Theory M. M. Z. Kharadly AWARDS 1963-64 -Graduate Fellowships, The University of British Columbia. 1965 -National Research Council of Canada Studentship. PUBLICATIONS D. J. Griffiths and Hans Glattli, Optical Faraday Rotation Studies of Paramagnetic Resonance and Relaxation in Praseodymium Ethylsulphate. Can. J. Phys. 43, 2361, (1965). D. J. Griffiths and Hans Glattli, Imprisonment of Phonons by Pr3+ Ions in Praseodymium Ethyl-sulphate. Phys. Letters 21_, 275, (1966) „ (Prof. W. Opechowski) for the Supervisor Professor Myer Bloom ABSTRACT The magneto-optical Faraday e f f e c t has been used to observe the Bpin-bath relaxation at low temperatures l n cerium e t h y l s u l -phate (CeES) and l n europium doped calcium f l u o r i d e (EuT:CaP2)» The paramagnetic Faraday ro t a t i o n ^ Is an Instantaneous measure of the magnetisation M and It i s shown that i n CeES and Eu z +:CaF 2 <^ ~P M at the l i g h t frequencies employed. The apparatus i s the same as that previously described by Rieokoff and by G r i f f i t h s . Pulsed microwave power at X-band has been used to disturb the equilibrium between spin system and bath. In CeES, the observed relaxation time T i s of the order of a few mseo, which i s several orders of magnitude longer than th© t h e o r e t i c a l estimate of T^. This suggests a severe b o t t l e -neck l n the energy t r a n s f e r spin-bath. T" i s found to be envi-ronment dependent. In H e l l , the relaxation i s exponential* T Is i n good agreement with nonresonant relaxation measurements by Van den Broek and Van der Harel. I t Is explained as a r i s i n g from the Kapitza boundary resistance at the CeES-Hell i n t e r f a c e . In Hel, the relaxation i s non-exponential and i s slower than i n He gas at the same temperature. This suggests that l n t h i s ease the thermal d i f f u s i o n l n the helium around the c r y s t a l i s the bottleneck. The same relaxation behaviour Is found when the c r y s t a l i s heated d i e l e o t r i c a l l y with microwave power f a r o f f resonance. This supports the assumption that the energy trans-f e r spin-bath i s l i m i t e d by s p a t i a l d i f f u s i o n . I f the c r y s t a l 11 Is surrounded by a He f i l m at a temperature below the X -point, T Is found to be the same ae in H e l l up to a well defined aver-age microwave power l e v e l . For higher powers the relaxation be-haviour i s similar to that of CeES Immersed i n Hel. In Eu z + :CaF 2, T 1 ie expeoted to have the form Tj_ s AT+BT^. The observed relaxation time r , however, i s found to be con-centration dependent. A l l measurements have been done on the + -j ~+ ~~t t r a n s i t i o n with H II [100] . For the three lowest con-centrations, the temperature dependence of T from 1.5 to *+.2°K can be f i t t e d with the expression T " ' » CT with C « 2 .75(seo °K)"1 for 0,02% Eu, C a 3 .5 for 0.8# and C = 5 f o r 0 . 2# . At a concen-t r a t i o n of 2%, T i s shorter and r -V T 2 from 1.5 °K to 7°K. The concentrations given correspond to the t o t a l Eu oontent. The E u z + concentration has been i n f e r r e d from the magnitude of the saturation rotation. T (T) seems to depend on both E u 2 + and E u 3 + concentrations. It Is suggested that exchange coupled pairs of E u 2 + and d u s t e r s involving Eu^ + may account f o r the concentration dependence of T . Upper l i m i t s of A » 2.5 and B » 5 x 10~5 a r e found for by extrapolating the lowest con-centrations Investigated. These values are somewhat lower than both measured and calculated values found by Huang. I l l TABLE OF CONTENTS Page Abstract 1 L i s t of I l l u s t r a t i o n s and Tables v i Acknowledgements ix 1. INTRODUCTION 1 2. THEORETICAL BACKGROUND 2.1 Cerium Ethylsulphate 10 2.1.1 C r y s t a l Structure 10 2.1.2 Ground State and Paramagnetic Resonance 10 2.2 Spin-Lattice Relaxation 17 2.2.1 Introduction 17 2 . 2 . 2 The Direct Process 19 2 . 2 . 3 The Raman Process 22 2.2.4 Two Phonon Resonant Processes 26 2 . 3 The Phonon Bottleneck 33 2.3.1 Introduction 33 2 . 3 . 2 Thermodynpmical Treatment 34 2.3.3 Aooustic Mismatch at a Boundary 41 2.4 Europium Doped Calcium Fluoride 47 2.4.1 C r y s t a l Structure 47 2 . ^ . 2 Ground State end Paramagnetic Resonance 47 iv 2.4.3 Interactions Between Europium Ions .... 51 2.4.4 Spin-Lattloe Relaxation of Eu 2 + 53 2.4.5 Cross-Relaxation 56 2.5 The Magneto-Optioal Faraday E f f e c t 62 2.5.1 Introduction , 62 2.5.2 Faraday Rotation of Rare Earth lone ln Solids 63 3. EXPERIMENTAL ARRANGEMENT 3.1 The Apparatus 72 3.1.1 The Cryostat 75 3.1.2 The Magnet 75 3.1.3 The Microwave System 76 3 . l . k The Optical System 81 3.1.5 Signal Detection 82 3.2 Experimental Procedures 85 3.2.1 Preparations f o r the Measurements 85 3.2.2 Faraday Rotation Measurements 86 3.2.3 Determination of the Resonance Spectrum 88 3.2.4 Relaxation Time Measurements 90 3.2.5 Temperature Measurements 92 4. EXPERIMENTAL RESULTS AND DISCUSSIONS 4.1 Cerium Ethylsulphate 95 4.1.1 Faraday Rotation 95 V 4.1.2 Resonance Spectrum 98 4.1.3 Relaxation Times 101 4.1.4 Discussion 109 a) Thermodynamlcal Model 109 b) T below the X -point 114 o) r above the X -point 116 d) Application to PrES 124 4.2 Europium Doped Calcium Fluoride 130 4.2.1 Faraday Rotation 130 4.2.2 Resonance Spectrum 135 4.2.3 Relaxation Times 137 4.2.k Dlsoussion 142 4.3 Erbium Ethylsulphste 150 4.3.1 Introduction 150 4.3.2 The Polncare Representation 152 BIBLIOGRAPHY 158 v l LIST OF ILLUSTRATIONS AND TABLES Figure Page 2.1 Arrangement of H 20 and ES moleoules around the Ce Ion In CeES 11 2.2 Positions of the CeES groups l n the unit c e l l of the CeES l a t t l o e 12 2.3 Zeeman s p l i t t i n g of the two lowest doublets in CeES 16 2.4 Direct and Raman relaxation processes 18 2.5 Typioal energy l e v e l s and s p l i t t i n g s f o r two phonon resonant relaxation 28 2.6 Thermodynamlcal model f o r the phonon bottleneck 35 2.7 Crystal struoture of CaF 2 46 2.8 Paramagnetic resonance spectrum of Eu x +:CaF 2 ; calculated l i n e positions 50 2.9 Energy l e v e l s and cross-relaxations in "Eu i + :C aF 2 H>3100 Oe HH[100] 58 2.10 Typical l e v e l s of rare earth Ions in solids .. 66 2.11 Typical l e v e l s f o r rare earth ions i n solids? S-state ion i n a cublo f i e l d 70 3.1 Block diagram f o r the apparatus 73 3.2 The dewar cap 74 3.3 The current regulation 78 3.4 F i e l d configuration in the miorowave cavity .. 79 3.5 Crystal adjusting mechanism 79 3.6 Relaxation trace in CeES a) before averaging b) a f t e r averaging (CAT 400B) 84 v l l 3 . 7 F a r a d a y r o t a t i o n o f t h e g l a s s w a l l s o f t h e d e w a r 8 7 4 . 1 P a r a m a g n e t i c F a r a d a y r o t a t i o n o f C e E S 96 4 . 2 S a t u r a t i o n v e r s u s m a g n e t i c f i e l d i n C e E S 9 7 4 . 3 R e l a x a t i o n t i m e i n C e E S : M a g n e t i c f i e l d d e p e n d e n c e 103 4 . 4 R e l a x a t i o n t i m e i n C e E S : T e m p e r a t u r e d e p e n d e n c e 1 0 4 4 . 5 R e l a x a t i o n t i m e l n C e E S : T e m p e r a t u r e d e p e n d e n c e b e l o w t h e \ - p o i n t 105 4 . 6 R e l a x a t i o n t i m e d i s c o n t i n u i t i e s b e l o w t h e A - p o i n t 1 0 8 4 . 7 T h e r m o d y n n m i c a l m o d e l f o r C e E S - H e s y s t e m I l l 4 . 8 S p e c i f l o h e a t o f C e E S a s a f u n c t i o n o f t e m p e r a t u r e 1 1 7 4 . 9 S p e o i f l o h e a t o f C e E S a s a f u n c t i o n o f m a g n e t i c f i e l d 1 1 8 4 . 1 0 K a p i t z a r e s i s t a n c e o f C e E S - H e l l : T e m p e r a t u r e d e p e n d e n c e 119 4 . 1 1 K a p i t z a r e s i s t a n c e o f C e E S - H e l l : F i e l d d e p e n d e n c e 120 4 . 1 2 S p e o i f l o h e a t o f P r E S a s a f u n c t i o n o f t e m p e r a t u r e 1 2 7 4 . 1 3 K a p i t z a r e s i s t a n c e o f P r E S - H e l l : F i e l d d e p e n d e n c e 1 2 8 4 . 1 4 K a p i t z a r e s i s t a n c e o f P r E S - H e l l : T e m p e r a t u r e d e p e n d e n c e 129 4 . 1 5 F a r a d a y r o t a t i o n o f E u x + : C a F 2 c r y s t a l N o . 1 . . . 132 4 . 1 6 F a r a d a y r o t a t i o n o f E u * + : C a F ? c r y s t a l s N o . 2 - 5 f 133 4 . 1 7 R e s o n a n c e s p e c t r u m o f E u ^ : C a F 9 136 v i i i 4.18 Relaxation rate of Eu""" :CaF2 as a function of temperature for different concentrations .. 140 4.19 a) Typical relaxation trace of Eu :CaF2 141 b) Semilogarithmic plot of trace in a) 141 4.20 The Boinoare sphere - General relations 153 4.21 The Poincare sphere applied to relaxation measurements 155 Table I Relaxation time discontinuities below the X -point 106 II Saturation rotation and concentration of Eu :CaF2 specimens 134 III Coefficients for the spin-lattice relaxation in Eu :CaF9 144 ix ACKNOWLEDGEMENTS The research described in t h i s thesis was supported by the National Research Council of Canada through grants to Dr. M. Bloom and the award of a studentship to the author. The author i s indebted to the University of B r i t i s h Colum-bia f o r the award of two graduate fellowships. Among the many persons who have contributed to the comple-tion of t h i s work the author would l i k e to thank p a r t i c u l a r l y : Dr. M. Bloom f o r his continuous support throughout t h i s work; Dr. J . B. Brown fo r his active help and many int e r e s t i n g discussions during the experimental stages; Professor W. Opechowski f o r h i s interest and many h e l p f u l suggestions; Dr. c . F. Schwerdtfeger and Mr. B. J. Slagsvold f o r oheoklng the samples on th e i r Jh kMo/s spectrometer; Dr. D. J. G r i f f i t h s , from whom I learned most of the experi-mental know-how; Mr. T. E. Clarke for his assistance in taking the measure-ments; Mr. W. Brooks for the construction of the current regu-l a t i o n and the mounting of the dewar cap; Mr. R. Weissbaoh f o r h i s advice i n technical matters and for providing the l i q u i d helium; Mr. R. F. Trehearne and Mr. P h i l West from Nuolear Data X Ino. f o r lending us sn Enhancetron; Dr. J. A. Wad£ and Mr. Finn Bauok of the Neurological In-s t i t u t e st U.B.C. f o r the use of t h e i r Mnemotron CAT 400B; Mrs. R. A. Foreman for her expert typing an,d advioe i n questions of language; and, l a s t but most, my wife Josette, who not only contribu-ted to t h i s work through her continuous moral support, but also helped a c t i v e l y i n the time consuming prooese of evaluating the relaxation traces. 1 1 INTRODUCTION Spin-bath relaxation is the mechanism of energy exchange between a system of paramagnetic Ions ("spin system") of a cr y s t a l and. a constant temperature bath of phonons. Kronig (1939) suggested th^t a direct energy transfer from the spins to the l a t t i c e phonons oould take place due to the modulation of the cr y s t a l f i e l d by the l a t t i c e vibrations through the interplay of spin-orbit coupling. Van Vleck (1939, l°4o) calculated the time constant T^ characterising t h i s direot energy transfer by analysing the normal modes of v i b r a t i o n of an ootah^dral c l u s t e r surround-ing the paramagnetic ion. At the lowest temperatures, the one-phonon or di r e c t processes, where the spin makes a tran-s i t i o n with the simultaneous emission or absorption of a single phonon, Is dominant. At high temperatures, a two-phonon process i s dominant. Here, the spin t r a n s i t i o n i s accompanied by i n e l a s t i c scattering of a phonon into another of d i f f e r e n t frequency. The energy difference of the two pho-nons corresponds to the spin translti'on-frequency. This two-phonon process i s known as the Raman process. Finn, Orbaoh and Wolf (l96l) have shown that a resonant two-phonon process via excited states whioh are close to the ground state can account for the magnitude and temperature de-pendence of observed relaxation times which could not be ex-plained i n t^rms of d i r e c t and Raman processes. Orbach (1961) 2 described a phenornenological approach t o f i n d the o r b i t - l e t t i c e i n t e r a c t i o n and from there T^. This approach avoids the Van Vleck normal-mode expansion, which i s very complicated even f o r cubic symmetry. The c a l c u l a t i o n of the r e l a x a t i o n times from the Van Vleck-Orbach theory i s reviewed l n chapter 2.2. I t i s l n good agree-ment w i t h many experimental r e s u l t s . In many cases, however, b i g d i s c r e p a n c i e s between theory and experiment are observed and the basic mo^el has to be mod-i f i e d to account f o r the observed r e l a x a t i o n times. As e a r l y as 19^1, Van Vleck has pointed out a p o s s i b l e shortcoming of h i s model. At the loitf temperatures, f o r which the d i r e c t process i s u s u a l l y dominant, there e x i s t only very few phonons with energies corresponding to the Zeeman s p l i t -t i n g , the s o - c a l l e d phonons "on speaking terms" with the l a t -t i c e . The assumption of a phonon pystem i n I n t e r n a l e q u i l i b -rium at constant tempereture can be v i o l a t e d . The energy t r a n s f e r proceeds i n two steps, from the s p i n syetem v i a the phonons on speaking terms to the bath. Bath means that part of the phonon system which remains at a constant temperature during the whole r e l a x a t i o n process. The second step can be the l i m i t i n g process i n the energy exchange. This s i t u a t i o n i s known as the phonon b o t t l e n e c k . Two-step r e l a x a t i o n pro-cesses of t h i s k i n d show l n general nonexponential behaviour and only l n l i m i t i n g cases a s i n g l e r e l a x a t i o n time i s found. The observed r e l a x a t i o n time T i s longer than T^. (T^, 3 c a l l e d the s p i n - l a t t i c e r e l a x a t i o n time, Is used throughout t h i s t h e s i s to c h a r a c t e r i z e the time constant f o r the d i r e c t energy t r a n s f e r between the sp i n system and the l a t t i c e o s c i l -l a t o r s by the Van Vleck-Orbach mechanism.) T i s dependent on the co n c e n t r a t i o n of the paramagnetic ions and approaches T-^  i n the l i m i t of low co n c e n t r a t i o n . The phonon bot t l e n e c k i s s t u d i e d i n more d e t a i l i n chapter 2.3. Sp i n - s p i n i n t e r a c t i o n i s another complication l n r e a l o r y s t a l s which can l e a d to observed r e l a x a t i o n times q u i t e d i f f e r e n t from T^. This i n t e r a c t i o n can induoe cross--\ r e l a x a t i o n s . The term c r o s s - r e l a x a t i o n , introduced by Blosmbercren et a l ( 1 9 5 9 ) , i s used f o r processes i n whioh en-ergy i s t r a n s f e r r e d d i r e c t l y from one spin system to another spin system or to other l e v e l s of the same system. Such pro-cesses lead to r e l a x a t i o n paths In p a r a l l e l to the convention-a l process of d i r e c t energy exohange between the spin system i n question and the bath which i s c h a r a c t e r i z e d by T^ ( i n ab-sence of a phonon b o t t l e n e c k ) . The observed r e l a x a t i o n time i s f a s t e r than T-^  and the discrepancy increases with concen-t r a t i o n . More d e t a i l s on these c r o s s - r e l a x a t i o n processes are given l n chapter 2 . 4 . Several d i f f e r e n t methods have been used t o observe para-magnetic r e l a x a t i o n and to measure the r e l a x a t i o n times. In the nonresonant method, developed before World War I I and e x t e n s i v e l y a p p l i e d by the Leiden group, the absorption and d i s p e r s i o n p arts of the magnetic s u s c e p t i b i l i t y are meas-4 ured at audio and r f frequency and the value of deduced. With the a v a i l a b i l i t y of microwave techniques, a number of resonant methods have been developed, eg..the steady s t a t e s a t u r a t i o n measurements (Eschenfelder and Weidner, 1962), the pulse s a t u r a t i o n recovery method (Scott and J e f f r i e s , 1962), and the s p i n echo techniques (Mims, 1965). S e v e r a l o p t i c a l methods have a l s o been a p p l i e d success-f u l l y . Here, changes i n the o p t i c a l p r o p e r t i e s of the sub-stance are monitored. Paramagnetic resonance and r e l a x a t i o n i n exoited s t a t e s have been detected by Geschwind et, a l ( 1 9 6 1 ) using o p t i c a l methods. A d i s c u s s i o n of d i f f e r e n t o p t i c a l methods i s given by Geschwind et, a l , ( 1 9 6 5 ) . The r e s u l t s reported l n chapter 4 of t h i s t h e s i s have been obtained using the magneto-optical Faraday e f f e c t , Kramers ( 1 9 3 0 ) and Van Vleck and Hebb ( 1 Q 3 4 ) have shown th a t , under c e r t a i n oircumstances, the d i r e c t i o n of p o l a r i z a -t i o n of l i n e a r l y p o l a r i z e d l i g h t , a f t e r passing through a para-magnetic c r y s t a l , can change by an angle which i s p r o p o r t i o n a l to the magnetization. I t i s shown i n chapter 2 . 5 that t h i s p r o p o r t i o n a l i t y holds f o r the two substances i n v e s t i g a t e d i n t h i s t h e s i s . Based on Kramers' theory, Becquerel, de Haas and Van den Handel ( 1 9 3 ? ) used the Faraday e f f e c t to measure the s u s c e p t i -b i l i t y of sever a l r a r e earth e t h y l sulphates at low temperatures. K a s t l e r ( 1 9 5 1 ) pointed out the p o s s i b i l i t y of an i n f l u e n c e of microwave r a d i a t i o n on the o p t i c a l Faraday r o t a t i o n . Opechowski ( 1 9 5 3 ) has given a quantum t h e o r e t i c a l c a l c u l a t i o n 5 of the o p t i c a l Faraday rotation i n presenoe of microwave res-onant r a d i a t i o n . The effeot has l a t e r been observed i n t h i s laboratory by Daniels and Wesemeyer (1958) and has been used to measure spin-bath relaxation times i n neodymium e t h y l s u l -phate, Nd(C2H5S0^)39H20 (Rleckoff, 1962), and i n praseodymium ethylsulphate ( G r i f f i t h s , 1965). The same technique as used by the above authors has been applied to study the spin-bath relaxation In undiluted oerium ethylsulphste (CeES) and i n europium doped calcium fluoride (Eu"1": CaF 2). Detailed descrip-tions of the apparatus have been given i n the references c i t e d above. A b r i e f review of the experimental set-up and a descrip-tion of the procedures employed i n the present investigations Is given In chapter 3» Both orystals used i n the present work have a large Fara-day rotation, X-band microwave tra n s i t i o n s within the range of our magnet and specimens of reasonable size and opti c a l quality are r e a d i l y a v a i l a b l e . They are therefore suitable for invest-igation with our apparatus. The relaxation behaviour of the two substances i s very d i f f e r e n t due to a d i f f e r e n t structure of the ground state of the paramagnetio ions. In neither substance can the observed relaxation times be explained In terms of the Van Vleok-Orbaoh mechanism. This i s due to the f a c t that T^_ has extreme values l n both cases. In CeES, T i s very short while in E u 1 + : CaF 2 T^ i s very long. CeES: C r y s t a l s t r u c t u r e and magnetic p r o p e r t i e s of the CeES are reviewed In chapter 2.1. The cerium ion i n an e t h y l s u l p h a t e l a t t i c e has as ground state a Kramers doublet. The c h a r a c t e r -i s t i c feature i s an e x c i t e d doublet ~5 cm"1 above the ground s t a t e . This c l o s e e x c i t e d doublet leads to an e x c e p t i o n a l l y strong s p i n - l a t t i c e coupling. T^_ of the Ce ions i s expected to be short due to e strong Orbach process. A t h e o r e t i c a l e s t -imate of T^ based on Orbach's phenomenological approach Is given l n chapter 2.2. This value i s s e v e r a l orders of magnitude shor-t e r than the observed r e l a x a t i o n times. I t seems l i k e l y , there-f o r e , t h a t a phonon bottleneck i s observed, even thoue-h the s p i n -l a t t i c e r e l a x a t i o n proceeds v i a the e x c i t e d doublet at 5 cm""1, thus I n v o l v i n g phonons with energies of a few times kT. The experimental r e s u l t s are described and discussed i n d e t a i l In chapter 4.1. I t i s seen that the observed r e l a x a t i o n times depend s t r o n g l y on the environment. The r a t e of energy t r a n s f e r spin-bath Is l i m i t e d by the thermal d i f f u s i o n from the c r y s t a l to I t s surroundings. The bath i n the sense defined e a r l i e r i s , i n t h i s case, the bulk l i q u i d or gaseous helium surrounding the c r y s t a l . I f the c r y s t a l i s Immersed i n H e l , the thermal d i f f u s i o n i n the l i q u i d seems to l i m i t the r a t e of energy t r a n s f e r . The observed r e l a x a t i o n behaviour Is slower than i n gaseous helium at the same temperature and i s nonexponential. This would be expected f o r thermal d i f f u s i o n i n three dimensions. In H e l l , the thermal d i f f u s i o n i n the bulk l i q u i d l e orders o f magnitude f a s t e r . The l i m i t i n g process here seems to be the K a p l t z a boundary r e s i s t a n c e at the i n t e r f a c e c r y s t a l — H e l l . The r e l a x a -t i o n i s e x p o n e n t i a l , i n agreement w i t h the assumption of a sur-faoe d i f f u s i o n process. The K a p l t z a r e s i s t a n c e has been c a l c u -l a t e d from the measured r e l a x a t i o n times and the known s p e c i f i c heat v a l u e s . The r e s u l t s agree i n magnitude, temperature and f i e l d dependence w i t h K a p l t z a r e s i s t a n c e measurements on other substances obtained with conventional heat conduction techniques. The r e l a x a t i o n behaviour of CeES does not depend on whether the c r y s t a l i s heated d l e l e c t r i c a l l y i n magnetic f i e l d s f a r from resonance or i f the energy i s t r a n s f e r r e d to the c r y s t a l mainly through the spins as Is the oase I f the magnetic f i e l d i s on resonance. This i n d i c a t e s that the d i s t r i b u t i o n of e x c i t a t i o n among the phonons of d i f f e r e n t frequencies l a the same l n both cases. A conventional heat conduction experiment would give a good t e s t f o r the v a l i d i t y of the explanations above. Such an experiment should be performed l n a simpler geometry such that the thermal d i f f u s i o n equation can be solved more e a s i l y . E u v + : CaF 2 The s i t u a t i o n l n the Eu i +-doped CaF 2 Is q u i t e d i f f e r e n t from the one i n CeES. The Eu 1 + f r e e i o n has a  S 3 - L ground s t a t e . The cubio f i e l d of the f l u o r i d e l a t t i o e can s p l i t the 8 - f o l d degenerate ground state only i n higher order through the i n t e r p l a y of s p i n - o r b i t 8 coupling. In magn^tlo f i e l d s o f a few kOe, the ground s t a t e i s s p l i t i n t o eight n e a r l y e q u i d i s t a n t l e v e l s which can be l a b e l l e d according to J 2 , the z-component of the t o t a l angular momentum. Magnetic d i p o l e t r a n s i t i o n s o f the form J 2-*J Z+1 can be induced by microwave r a d i a t i o n . The seven paramagnetlo resonance l i n e s r e s u l t i n g from these t r a n s i t i o n s are separated due to the c r y -s t a l f i e l d by an amount which depends upon the r e l a t i v e o r i e n -t a t i o n of the magnetic f i e l d w i t h respect to the c r y s t a l l o g r a -phic d i r e c t i o n s . As an a d d i t i o n a l c o m p l i c a t i o n , each l i n e cor-responding to a given e l e c t r o n i c t r a n s i t i o n i s s p l i t i n t o a t o t a l of twelve hy p e r f i n e l i n e s due to tv/o d i f f e r e n t isotopes each of which has a nuclear s p i n I = 5/2. In suoh a complicated m u l t i l e v e l system, c r o s s - r e l a x a t i o n s may i n f l u e n o e the r e l a x a t i o n behaviour, p a r t i c u l a r l y since the s p i n - l a t t i c e r e l a x a t i o n Is ex-pected to be slow f o r an S-state. A short review of the p e r t i -nent theory i s given i n chapter 2.4. A phonon bottleneck as l n CeES cannot occur i n E u x + : C P F 2 since the r a t e of energy t r a n s -f e r from the spins to the l a t t i c e i s very s m a l l . We have measured the spin-bath r e l a x a t i o n time of Eu ions i n CaF 2 by s a t u r a t i n g the J z a t r a n s i t i o n . The magnetio f i e l d was p a r a l l e l to the [lOO] d l r e o t l o n In the c r y s t a l . This c o n f i g u r a t i o n g i v e s minimum o r o s s - r e l a x a t i o n w i t h i n the s i n g l e -spin system since the c r y s t a l f i e l d s p l i t t i n g i s maximum and the i t r a n s i t i o n Is quite f a r removed from the other e l e c -t r o n i c t r a n s i t i o n s . The absence of c r o s s - r e l a x a t i o n to o ther l i n e s has been v e r i f i e d by Huang (1965) u s i n g the pulse s a t u r a -t l o n teohnique. The r e s u l t s obtained l n the present I n v e s t i g a -t i o n are shown and discussed l n chapter 4.2. The spin bath r e l a x a t i o n times are s<*en to d i f f e r c o n s i d -erably f o r d i f f e r e n t specimens, both i n magnitude and tempera-ture dependence. The r e l a x a t i o n times get s h o r t e r and t h e i r temperature dependence becomes steeper w i t h i n c r e a s i n g concen-t r a t i o n . At the lowest c o n c e n t r a t i o n , the r e l a x a t i o n times f o l l o w o l o s e l y a T~-*--law suggesting a d i r e o t process. At higher concentrations, some mechanism other than the Van Vleck-Orbach r e l a x a t i o n must provide a d d i t i o n a l channels through which the r e l a x a t i o n nan take place. Exchange coupled p a i r s or c l u s t e r s coupled to the s i n g l e spins by c r o s s - r e l a x a t i o n s could provide such mechanisms. This has been f i r s t suggested by Van Vleck (1959) and has since been used to e x p l a i n q u a l i t a t i v e l y c oncentration dependent r e l a x a -t i o n times of the k i n d observed l n E u l + : CsFg. I t Is impossible t o give more than q u a l i t a t i v e arguments i n t h i s case. The spectrum of the s i n g l e i o n i s already very complicated and the spectrum of exchange coupled p a i r s or c l u s t e r s i s n e c e s s a r i l y even more Involved. An unsuccessful attempt to measure the spin-bath r e l a x a t i o n i n u n d i l u t e d erbium ethylsulphnte Is described l n chapter ErES has a smaller Faraday r o t a t i o n and a l a r g e r b i r e f r i n g e n c e than the other rare earth e t h y l s u l p h a t e s and an alignment of the c r y s t a l - a x i s s u f f i c i e n t l y p a r a l l e l to the l i g h t could not be achieved with our apparatus. 1 0 2 . 1 Cerium E t h y l sulphate 2 . 1 . 1 C r y s t a l S t r u c t u r e The e t h y l sulphates (ES) form a croup of Isomorphous sub-stances. Their formula Is M ( C 1 2 H 5 8 0 ^ ) 3 x 9 H 2 0 where M stands f o r a l l t r i v a l e n t rare earth; ions from La to Yb and f o r Y. The c r y s -t a l s t r u o t u r e has been determined by K e t e l a a r ( 1 9 3 7 ) and more r e -ce n t l y by F i t z w a t e r and Rundle (1959). Figure 2 . 1 shows the arrangement of the 9 watermoleoules and of the e.thylsulphate r p d i c a l s around the cerium i o n . The c r y s t a l Is oomposed of two i n t e r l o c k i n g l a t t i c e s of these CeES groups (Figure 2.2). The u n i t c e l l contains two magnetically equivalent cerium i o n s . The point symmetry of the rare earth s i t e s i s 0 ^ n . Small departures from t h i s symmetry might ocour due to arrangement of the protons i n the water molecules according t o a lower symmetry. In f a c t , a small admixture of C ^ i s e b l e to aocount f o r the observed t r a n -s i t i o n s between the two lowest doublets (Devor and Hoskins, 1961). The volume of the u n i t c e l l i s 1.2 x 1 0 " * 2 * om^ which gives a spin concentration of 1 . 6 x 1 0 2 1 om""3 f o r the undiluted p a l t . The dens i t y i s «^  « 1 . 9 gom'^. Eaoh oerium Ion has two nearest neighbours at a dis t a n c e of 7.11 8 along the hexagonal a x i s and s i x next-nearest neighbours at 8.55 X at the edges of a t r i a n g u l a r prism surrounding the i o n . 2.1.2 G-roundstate and Paramagnetio Resonanoe Cerium Is the f i r s t Ion of the rare earth s e r i e s which Figure 2.1 Arrangement of H 20 and ES molecules around the Ce i o n i n CeES Figure 2.2 P o s i t i o n s of the CeES groups l n the u n i t c e l l of the CeES l a t t i c e 1 3 e x h i b i t s paramagnetism. I t has one 4f e l e c t r o n outside the closed s h e l l s . The fre e ion groundstate 2 F | Is s i x - f o l d degen-erate. I f placed i n a c r y s t a l , the f r e e ion i s subject to a orye-t a l l i n e e l e c t r i c f i e l d V o r of g i v e n symmetry. In p a r t i c u l a r , i n the e t h y l s u l p h a t e s , the rare e a r t h ion l e at a s i t e of sym-metry. The o r y s t a l f i e l d f o r t h i s symmetry i s given i n terms of the operator equivalents by E l l i o t t and Stevens ( 1 9 5 2 ) : 14 » «A[->0° *pA, ~*0l * rA°( PCf^'i* Ol <2-1) The CT operator equivalents and many of t h e i r matrix elements are t a b u l a t e d l n the book by Low ( i 9 6 0 ) . I f one neglects i n f i r s t approximation the admixture of the ZF T m u l t i p l e t , V act s only i n a manifold w i t h constant angular a o r momentum J • 5 / 2 and the terms i n A° and k\ v a n i s h . For a XF£ m u l t i p l e t , the values of the constants are (Low, I960): The operators 0° commute with J z . V c r s p l i t s the 2 F i m u l t i p l e t Into three Kramers doublets > <±ir/ > a n d <±£/ » w i t h the f o l l o w i n g energies: <±UVJ±\> = 3 ? Az ~ T T \  R" ~ IH ~~ 14 The values f o r A n r have been extrapolated from the values of the o t h e r rare e a r t h e t h y l s u l p h a t e s es given by Hiifner (1962). I t i s seen that the energy d i f f e r e n c e between the<±±/ * n d * n e doublet. Is very small: A E * E + £ - E ^ j . * 4 cm""1 A s l i g h t change i n A^r1 can therefore a f f e o t AE a p p r e c i a b l y and may even reverse I t s s i g n . This Is e x a c t l y what happens i n CeES. The concentrated s a l t has as lowest s t a t e the ^ i - f / doublet w i t h the <-T/ doublet l y i n g 4.8 cm""1 h i g h e r . I f Ce' +-ions are d i l u t e d i n t o the d i a -magnetic LaES l a t t i c e , the < ± i / doublet l i e s lowest w i t h A E • 3.94 0m""1 (Bogle, Cooke, and Whitley, 1951). I f Ce*+ i s d i -l u t e d l n YES, t h e ^ i x / doublet i s s t i l l l o w e s t , but with LE * 17.4 cm"1. E l l i o t t and Stevens (1952) c a r r i e d the p e r t u r b a t i o n c a l c u -l a t i o n s to second order. They obt a i n expressions f o r the r e l a t i v e energies of the three doublets and the g-values of the two low l y i n g doublets as a f u n c t i o n of the c r y s t a l f i e l d parameters A„ r- (n 35 2,4,6) and A6 F1 . These l a t t e r are unknown i n the case of Ce. The g-values and the s p l i t t i n g s between the doublets have been determined experimentally. This g i v e s 6 pieces of i n -formation to determine 4 unknowns, a l l o w i n g not only to determine the c r y s t a l f i e l d parameter but a l s o to check the theory. E l l i o t t and Stevens found t h a t the experimental g-values f o r the d i l u t e d CesLaES could not be f i t t e d e x a o t l y to the theory and suggested a small admixture of C^v symmetry. In the concentrated s a l t , the l i n e s are broad and the g-values 15 cannot be determined a c c u r a t e l y . Bogle ej^ a l observed paramag-n e t l o resonance absorption due to the lower doublet only at 2.5°K. Absorption due to the upper doublet has not yet" been found (Dweok and S e i d e l , 1966). The g-values found from s u s c e p t i b i l -i t y measurements by Bogle et_ a l are g ( r ) * 3.80 ± 0.0k, g ( f 0 ±0.4, g(|(±) s 1.0 ± 0 .2 and g x ( ± ) - 2.25 ±- 0.2. The s p l i t t i n g A between the two doublets has been deduced from the Schottky peak of the s p e o i f i c heat by Meyer and Smith (1959). They found A s 6.7 °K. A has a l s o been determined by s u s c e p t i b i l i t y measurements. Bogle et a l obtained 7.5 - 1.5 °K while Van d/en Broek and Van der Marel found A » 6.95 °K. Very r e c e n t l y , d l r e o t absorption of i n f r a r e d r a d i a t i o n between the two Kramers doublets has been observed by B u r g i e l and Meyer (1966) y i e l d i n g a value of A * 7.2°K. The s p l i t t i n g of the two doublets i n a magnetic f i e l d i s shown In Figure 2 . 3 . Figure 2 .3 Zeeraan s p l i t t i n g of the two lowest doublets l n CeES 17 2.2 S p i n - L a t t i c e R e l a x a t i o n 2.2.1 I n t r o d u c t i o n Since Finn, Orbaoh and Wolf (1961) introduced the two phonon resonant prooess to account f o r s p i n - l a t t l o e r e l a x a t i o n times which were unexpectedly short and f o l l o w e d an exponential temper-ature dependence, the mechanism of s p i n - l a t t i c e r e l a x a t i o n seems to be well understood. This Is t r u e , i f one r e s t r i c t s the term s p i n - l a t t l o e r e l a x a t i o n t o the process of energy t r a n s f e r from a s i n g l e paramagnetic spin t o the surrounding l a t t i c e phonons, through modulation of the o r y s t a l f i e l d by the l a t t i c e v i b r a t i o n , I.e. throuerh the o r b i t - l a t t i c e i n t e r a c t i o n Assume a paramagnetic ion i n a c r y s t a l of low enough sym-metry, to l i f t a l l degeneracies w i t h exception of the Kramers de-generacy f o r ions with an odd number of e l e c t r o n s . Assume nov; that only the ground Kramers doublet i s occupied and that the doublet l a s p i l t by a magnetic f i e l d to a separation o . A s p i n i n the upp»r l e v e l of t h i s doublet can r e l a x to the lower l e v e l by means of three d i f f e r e n t processes: - the d i r e c t process, where the spin makes a t r a n s i t i o n through the a c t i o n of the o r b i t - l a t t i c e i n t e r a c t i o n which creates simultaneously a phonon of energy 8 - the Raman process In which one phonon i s scattered i n e l a s -t i c a l l y i n t o another phonon x^hlle the s p i n makes simultan-eously a t r a n s i t i o n . The energy d i f f e r e n c e of the phonons corresponds to the s p i n t r a n s i t i o n frequency. Ii» I I I I I I /c > *= //> t n f > // > = l a , n i + / , \ > //> " A * T^OmQfn 7*o cess Figure 2.4 D i r e c t and Raman r e l a x a t i o n processes 19 - the Orbach prooess which Is actually a two-step resonant process v i a a close exolted state. In the case of satura-t i o n of an excited doublet, the relaxation proceeds v i a the ground state, thus making an "inverse" Orbach process which has a d i f f e r e n t temperature dependence. A de t a i l e d treatment of the three processes has been given by Orbach (1961), l n whloh an easy approach i s described to f i n d the matrix elements of the o r b i t - l a t t l o e i n t e r a c t i o n . In the following, a short review of Orbaoh's approach i s given and i s applied to the case of CeES. 2.2.2 The Direct Process The probability for a spin to make a t r a n s i t i o n from I by to | «.> (see Figure 2.4) under simultaneous emission of a pho-non Is (2.3) C(p i s the density of f i n a l states, whloh is the number of pho-non states with energy h . A Debye model gives 3 V S z (2.4) where V i s the volume of the c r y s t a l and v i s the v e l o c i t y of sound. The problem Is to fi n d the matrix elements of V ,. In the 20 same manner, as i s customary f o r the s t a t i c c r y s t a l f i e l d , we oan expand the o r b i t - l a t t i c e i n t e r a c t i o n i n terms of s p h e r l o a l harmonios. v.. = L Au~ r Y (<?,*) (2 -5) For small l a t t i c e s t r a i n s , one can expand the Atl and r e t a i n i n g only the l i n e a r term, one has \£ » Z . < C r~ YJS,1) = J -C Xf <2-«> where A^ are now the c o e f f i c i e n t s of the s t a t i c c r y s t a l f i e l d expansion. For not too b i g a n i s o t r o p i e s , the s t r a i n s C oan be replaced by an average i s o t r o p i c s t r a i n £ whose matrix elements are known to be (Abragam, 196l) / < * V , - ^ 7 7 ^ (2.7) Here M i s the c r y s t a l mass and n g the phonon occupation number, -which i s pqual to the Bo s e - E l n s t e i n f a c t o r JL The p r o b a b i l i t y f o r a d i r e c t process t r a n s i t i o n i n a c r y s t a l with Nfc n o n l n t e r a c t i n g ' s p i n s i n the upper state <4-l I s , t h e r e f o r e : 21 3 $ \ w i t h ( 2 . 1 0 ) Is the c r y s t a l d e n s i t y . Using «. from equation ( 2 . 9 ) and the corresponding expression f o r the reverse process W^ ^^ . , the r a t e equation f o r the population d i f f e r e n c e AN = N a -becomes C/(AA/) '3 " T c/t The s o l u t i o n of the above d i f f e r e n t i a l equation shows an exponen-t i a l time dependence &N(t)- =- ANCO) JL ^ ( 2 . 1 2 ) where - / 3 • -2-' » " T S V e'^ikTj ( 2 - 1 3 ) i s the r e l a x a t i o n r a t e f o r the d i r e c t process. Our measurements, performed at X-band frequencies, give S * 0.4°K and t h i s allows the approximation S « kT. Equation ( 2 . 1 3 ) can thus be w r i t t e n 22 approximately 7-= AT w l t h A = —<—7- (2 .1M A p p l i c a t i o n t o CeES For Kramers ions i n general Vol oan connect the Kramers con-jugate s t a t e s /a> and li>> only a f t e r admixture of higher s t a t e s through the Zeeman I n t e r a c t i o n . These matrix elements are pro-p o r t i o n a l to the magnetic f i e l d H and we have A z / 1 ^ o r , s i n c e % - g(* H T'' -V A / y T ~ (2.15) In CeES, admixture of the *F X m u l t i p l e t Is needed to obtain z nonvanishing matrix elements. Sinoe t h i s m u l t i p l e t Is about 2,000 am""1 higher than the groundstate, the d l r e o t process i s expeoted to be weak. Larson and J e f f r i e s ( 1 9 6 6 ) estimate r e l a x a -t i o n times of the order of seconds at helium temperatures. The Orbach process i s much f a s t e r i n t h i s temperature region and the d i r e c t process i s t h e r e f o r e not observed. 2.2.3 The Raman Prooess A h i g h e r order process becomes p o s s i b l e i f one considers terms of V 0 l b i l i n e a r i n £ , i . e . 2 3 (2.16) This operator gives r i s e to an i n e l a s t i o s c a t t e r i n g of a phonon of energy E 1 i n t o one of energy E 2 w i t h a simultaneous s p i n f l i p K>-*/«.> to take up the energy d i f f e r e n c e E^ - E 2 a % (Figure 2.4) The t r a n s i t i o n p r o b a b i l i t y f o r t h i s process i s i n close analogy to equation (2.3) (2.1?) where fa means I n t e g r a t i o n over a l l p o s s i b l e f i n a l s t a t e s . A s i m i l a r procedure as f o r the d i r e c t process leads to (2.18) with the Debye c u t o f f energy k© as the upper l i m i t of the i n t e -g r a l . The main c o n t r i b u t i o n to the i n t e g r a l comes from the r e -gion where E ^  kT and sinoe S ^ < kT the i n t e g r a l gives a p p r o x i -mately f o r e » T (Ziman, 1954) £ <>T 24 Again, as f o r the d i r e c t process, the s o l u t i o n of the r a t e equa-t i o n f o r the population d i f f e r e n c e shows an exponential time de-pendence ir A A/ft) " v e ^* w i t h the r e l a x a t i o n r a t e -7--' = 3 6.' VJt, fi.-r)7 H 7T 3? Z£ 7 v'° yK 1 / (2.20) The comparison with the d i r e c t process i n CeES gives IK at X-band frequencies. The d i r e c t process i s seen to be stronger at helium temperatures. A second p o s s i b i l i t y f o r a Raman process a r i s e s from second order time dependent p e r t u r b a t i o n theory (Orbach, 1961; see a l s o Scott and J e f f r i e s , 1962). Here the o r b i t - l a t t i c e i n t e r a c t i o n of equation (2.6) acts twice through an Intermediate s t a t e / e > , The t r a n s i t i o n p r o b a b i l i t y f o r such a prooess i s 25 In the oase of a Kramers i o n w i t h one exoited doublet much c l o -ser to the ground s t a t e than a l l the others (as i s the case i n CeES) and using the same approximations as b e f o r e , we get from equation (2.22) ke (2.23) o c denotes here the lower l e v e l of the e x c i t e d doublet. For k 9 < < - A F the sum of the energy denominators i s always s m a l l . This Is known as Van Vleck c a n c e l l a t i o n . For kO>&, as Is the case In CeES, the Integrand has a s i n g u l a r i t y f o r E » A . This g i v e s r i s e to a resonant process which w i l l be t r e a t e d In chapter 2,2k below. I f one negleots the c o n t r i b u t i o n from the s i n g u l a r i t y , one obtains another c o n t r i b u t i o n to the Raman r e l a x a t i o n r a t e , which i s (2.24) The r a t i o of the two c o n t r i b u t i o n s to the Raman process i s : 2 6 ' z 77« vait A 7"" Vx I / 2 " ^(kT)x ( 2 . 2 5 ) 2. X.4 i v« and are of the same order of magnitude as A and the term i s t h e r e f o r e dominant f o r kT ^ ~ . 10 2 . 2 . 4 Two Phonon Resonant Processes The f i r s t e x c i t e d doublet i n CeES i s so c l o s e to the ground s t a t e , t h a t i t i s a p p r e c i a b l y populated even at helium tempera-t u r e s . We consider the rate equations f o r the whole f o u r - l e v e l system shown i n Figure 2 . 5 . I f the t r a n s i t i o n s o / -*<4-/ and < c / - » <°6/ are neglected, the following rate equations are obtained: ^ = w + u/ - (w + u/ clt = U/ + U / ^ -(LA/ + In exaot analogy to the d i r e c t process (equation (2.3)) the 2 7 t r a n s i t i o n p r o b a b i l i t i e s are —» fa , v 2 - 7 T t, p v-and correspondingly f o r the other t r a n s i t i o n s . The r a t e equations (2.26) have to s a t i s f y the o o n d l t i o n ct ct I f no t r a n s i t i o n s t o other l e v e l s occur. The s o l u t i o n s of the three remaining independent equations w i l l i n general be sums of three e x p o n e n t i a l s . I f , « A and I f ^ v«« ~ ^ ~ - V equations (2.26) s i m p l i f y to dt dt = c'(/V^ -2.cA£ (2.27) dt dA£ dt where c - 3 " A 2 7 r A *o t r • r and c » , ^ ^ Y ^ ^ O 1 7 7 " - t V ^ > V s " 28 A Figure 2.5 T y p i c a l energy l e v e l s and s p l i t t i n g s f o r two phonon resonant r e l a x a t i o n 29 In our experiments, e i t h e r one of the two doublets could be saturated by a microwave pulse. We assume t h a t the Boltzmann d i s t r i b u t i o n between the doublets i s not d i s t u r b e d , which means Trfa'AO = -£(w<)~o ( 2 . 2 8 ) In t h i s case, the equations f o r the population d i f f e r e n c e s A N I « N A - N D and A N 2 = N C - are ~ ( A A/,) = - 2 c A /V/ (2.29) Equation (2.29) describes the r e l a x a t i o n of the lower doublet by means of the two-phonon resonsnt r e l a x a t i o n known as the Orbach process. The r e l a x a t i o n r a t e is TT^3irr _ j ( 2 . 3 D which g i v e s the f a m i l i a r e x ponential temperature dependence f o r A » 7T > i O v (2.32) 30 The r e l a x a t i o n of the upper doublet v i s the lower doublet has been c a l l e d the Inverse Orbaoh process. I t s r e l a x a t i o n rate I s found from equation ( 2 , 3 0 ) to be 77. 3 A 5 v: 7 T * v y ( 2 . 3 3 ) This g i v e s f o r the case kT « A T. —i ( 2 . 3 4 ) I t i s seen that the r e l a x a t i o n r a t e f o r the inverse Orbaoh pro-cess approaches a lower l i m i t at low temperatures and does not decrease e x p o n e n t i a l l y as i s the case f o r the Orbaoh process. The approximations made i n the above d e r i v a t i o n of T-^ 0 and T ^ are r a t h e r rough f o r CeES. At the r e s o n a n c e - f i e l d f o r the e x o i -ted doublet, we have S, * 1.1 cm"*1 and A e 4.7 cm""1. This i s not s e r i o u s , however, si n c e the matrix elements Vq^  eto. are not known a c c u r a t e l y and the thus obtained values f o r the r e l a x a t i o n times can at best be regarded as an order of magnitude estimate. In order to estimate the r e l a x a t i o n times T^Q and T^, we have to know V 2 or more p r e c i s e l y VQ< , , vj" and vj^ . We have used the wave f u n c t i o n s given by E l l i o t t and Stevens f o r the d i l u t e Ce : LaES and the s t a t i c c r y s t a l f i e l d parameters have been extrapolated from values f o r other (undilu-ted rare earth e t h y l s u l p h a t e s . The values used are: 31 A^ r*- = 32. e ~ r ' A\ ~H - - 3 s- c~r' £ -< = _ 5-5. c _ - ' ( 2 . 3 5 ) C - / In the dynamic cane without the C^ n symmetry r e s t r i c t i o n s , other parameters occur (A^ _ , , A^ " , A* , A^ , A* and A^ ). They have been estimated using the Orbach (1961) r e c i p e : o j_ (2.36) V/e have neglected coherenoe between the d i f f e r e n t terms, i . e . we assumed that Vflth these approximations, the r e s u l t s are (2.37) V  1 - vj . -2. 3 0 0 C~"-V*- = V>- ^ H O O ^ ( 2 . 3 8 ) 3 2 Using the value V 2 *350 c a r 2 gives f o r the Orbach relaxation time at 1.5°K T l Q * 10-5 a e 0 (2.39) and for the inverse Orbaoh relaxation time T l l 1 0 ~ 7 8 8 0 (2.4o) Both estimates are several orders of magnitude fas t e r than the measured relaxation times, as w i l l be shown l a t e r and we measure therefore not s p i n - l a t t i c e relaxation times as defined at the beginning of this chapter. 33 2.3 The Phonon Bottleneck 2.3.1 I n t r o d u c t i o n In the preoeding treatment of s p i n - l a t t i c e r e l a x a t i o n , i t has been assumed throughout that the l a t t i o e has a oonstant temperature T^. This Is s t r i c t l y speaking only the case f o r a l a t t i c e w ith i n f i n i t e thermal c a p a c i t y . However, a r e a l l a t t i o e has, i n gpneral, a s m e l l e r s p e c i f i c heat than the spin system i n the low temperature region where r e l a x a t i o n experiments are usu-a l l y performed and the l a t t i c e temperature \ * i l l not n e c e s s a r i l y remain constant. Casimir (1939) has seen t h i s p o s s i b i l i t y and hps ext°nded the Caslmir-duFre formulae used i n nonresonant r e l a x a t i o n meas-urements f o r the case where T L i s not constant. Van Vleok (19^1) studied the energy exchange between the l a t t i o e o s c i l l a t o r s and postulated the Inadequacy of the heat c a p a c i t y of the l a t t i o e o s c i l l a t o r s to conduct the spin energy t o the bath i n a time shorter than the s p i n - l a t t i c e r e l a x a t i o n time T^. This e f f e c t has become known under the name of the phonon bottleneck. A l o t of data have been published, which give experimental evidence of a b o t t l e n e c k , namely by the Leiden group and by J e f f r i e s and h i s c o l l a b o r a t o r s . The most d i r e c t evidence of a b o t t l e n e c k has been obtained by Brya and Wacner (1965). They observed a phonon avalanche a f t e r i n v e r s i o n of a paramagnetic resonance l i n e by a d i a b a t i c passage i n a bottleneoked l a t t i o e . The bottleneck has been t r e a t e d i n two d i f f e r e n t ways. 34 Gorter £t s i ( 1 9 5 5 ) used, a thermodynamlcal approach to f i n d the time dependence of the s p i n temperature. Faughnan and Strandberg ( 1 9 6 1 ) used rate equations to determine the change of the sp i n population with time. Most of the numerous other p u b l i c a t i o n s on the subjeot use one or the other of the above approaches. In a very I n t e r e s t i n g and d e t a i l e d paper, G-lordmalne and Nash ( 1 9 6 5 ) showed the analogy between the phonon b o t t l e n e c k and the imprisonment of resonant photons i n gases. 2 . 3 . 2 Thermodynamlcal Treatment Consider three systems A, B, and C. Each system i s assumed to be i n i n t e r n a l e q u i l i b r i u m w i t h the r e s p e c t i v e temperatures T A, Tg, and TQ. Suppose system A has an i n f i n i t e s p e o i f i c heat and the i n t e r n a l energy and s p e c i f i c heats of B and C are U B , Cg and UQ, CQ. Energy i s fed i n t o B and C by an ex t e r n a l source at the r a t e Wg and WQ. Suppose that the thermal coupling of the three systems can be described by two time constants 7^S and T~AC as shown l n Figure 2 . 6 . Energy conservation leads to two coupled d i f f e r e n t i a l equations c B A 4 • y v u. rM cM TA* 4 i Figure 2 . 6 Thermodynamical model f o r the phonon b o t t l e n e c k 36 The systems A, B and 0 oan he I d e n t i f i e d w i t h bath, l a t t l o e and s p i n system r e s p e c t i v e l y . The above equations oan be l i n e -a r i z e d i f we assume that the temperature d i f f e r e n c e s are small: lr.-Tcl« TA IT* - T,l« TA such that CQ and Cg are Independent of time and can be taken at the bath temperature T A . In our experiments, the temperature d i f f e r e n c e s never exceeded a few m°K and the approximation i s reasonable i n t h i s case. Equations (2.40) and (2.41) have to be solved f o r two d i f -ferent i n i t i a l c o n d i t i o n s a p p l i c a b l e i n our experiments: a) The three systems A, B, and C are i n i t i a l l y at the same tem-perature. In t h i s oase, the s o l u t i o n s of equations (2.40) and ( 2 . 4 1 ) describe the warm-up of systems B and C due t o simultaneous h e a t i n g pulses to both systems. b) The systems B and C are I n i t i a l l y at the temperatures T B and TQ r e s p e c t i v e l y and Tg ^ T ^ due to a previous h e a t i n g pulse. The s o l u t i o n s of equations ( 2 . 4 0 ) and ( 2 . 4 l ) w i t h W B = W(j = 0 c h a r a c t e r i z e l n t h i s case the r e l a x a t i o n s of. T B and T C towards T A . a) T c(t = 0 ) = T B ( t a 0) a T A The d i f f e r e n t i a l equation f o r TQ i s obtained by e l i m i n a t i n g T B from equations (2.40) and ( 2 . 4 l ) and the s o l u t i o n i s 7^(&)- 7^(~) = ct e + 4- e 7 1 ( 2 . 4 2 ) 37 I f terms of the order 7 ^ - are negleoted oompared to u n i t y , the time oonstants are and The constants are a= 77 - 7 ^ r - ) (2.^5) 1 AS I , £ - 7 — U/f ( 2 . 4 6 ) ( 2 . 4 7 ) b) R e l a x a t i o n w i t h Tg(t • » 0) * T B, Tc(t« = 0) » T°c and wB = w c = 0 . The s o l u t i o n f o r T c ( t ' ) i s l n t h i s case with the same time oonstants TT and as found before i n 3 8 equations ( 2 . 4 3 ) and (2,kh) and w i t h and I t i s seen that f o r both cases a) and b) T t « TT and b « -a, b'«-a' due to the assumption O Q » C B » The longer time oonstent T. has been d e r i v e d i n t h i s manner by Stoneham (1965) and can be regarded as the e f f e c t i v e time constant f o r the thermal exchange of the combined system B + C and system A. The time TX which i s very much shorter than T, i f C B « C Q , oan be shown to c h a r a c t e r i z e the time needed f o r systems B and C to come i n t o e q u i l i b r i u m . Peterson (1965) has studied the r e l a x -a t i o n of two systems, both w i t h f i n i t e heat c a p a c i t y . T~z i s i d e n t i c a l to h i s time oonstant f o r small temperature d i f f e r e n c e s . For b i g disturbances, a whole d i s t r i b u t i o n of time constants are obtained depending upon the f u n c t i o n a l dependence c R ( T ) and c c(T) and TT , TZ are the asymptotic values f o r t -*• o°. This has been shown by K a l b f l e i s c h (1965) using the a l t e r n a t i v e rate equation approach f o r the d i r e c t process. The corresponding equations f o r the Orbaoh prooess are more Involved, but y i e l d 39 the same r e s u l t . From equation (2 .43) can be seen how a bottleneck oan a r i s e i n paramagnetio r e l a x a t i o n . I f the systems A, B and C are iden-t i f i e d w i t h the oonstant temperature both, w i t h the l a t t i o e os-c i l l a t o r s on speaking terms w i t h the spin system and with the s p i n system r e s p e c t i v e l y , then ~ca< i s the s p i n - l a t t i c e r e l a x a t i o n time T^, and TAa i s the l i f e - t i m e of the phonons with respeot to absorption by the bath. A b o t t l e n e c k a r i s e s i f •<3 and the observed r e l a x a t i o n time given by equation (2.^3) Is longer than T^, In severe b o t t l e n e c k cases c AH Be? a I t f o l l o w s t h s t 2~ <:< T~ and 7~ » 7~_ and the seoond r e l a x a t i o n time oan be oompletely neglected, l e a d i n g to a p e r f e c t l y exponential time dependence of the sys-tem C. This exponential behaviour i s based on the assumptions that: 1. the r e l a x a t i o n times d e s c r i b i n g the energy exchange be-tween the systems are Independent of temperature In the I n t e r v a l discussed, and that ho 2. the systems are In thermal e q u i l i b r i u m , I.e. each system i s described by a s i n g l e temperature. The nonexponentlal behaviour of the observed r e l a x a t i o n i n CeES above the X - p o i n t oan be explained as due to v i o l a t i o n of assumption 2. I t i s always p o s s i b l e to subdivide a system w i t h inhomo-geneous temperature i n t o more subsystems w i t h homogeneous temper-ature. Each new system l n s e r i e s between C and A c o n t r i b u t e s a new exponential term i n the time dependence of C. In the l i m i t corresponding to s p a t i a l d i f f u s i o n i n a medium wi t h a f i n i t e d i f -f u s i o n constant, a continuous d i s t r i b u t i o n of r e l a x a t i o n times Is needed to describe T ( t ) . I t i s I n t e r e s t i n g to know under what o o n d i t i o n the above r e -l a x a t i o n behaviour can be desoribed by means of two systems only. The a c t u a l l y observed r e l a x a t i o n time i s given by equation (2.43) the l a t t i c e can be lumped together w i t h the bath and the observed r e l a x a t i o n time of the spin system i s 7^ , No bottleneck occurs and i s the o r d i n a r y s p l n - l a t t l o e r e l a x a t i o n time, whloh Is u s u a l l y denoted as T . i f ^ » « ~?~r AQ £ It 41 the l a t t i o e and the spin system can be lumped together and the observed r e l a x a t i o n time Is c h a r a c t e r i s t i c of the energy exchange between the l a t t i o e system and the bath. This happens, to con-a l d e r a s p e c i a l case, i f the dominant r e s i s t a n c e to energy ex-change i s the boundary r e s i s t a n c e at the c r y s t a l i n t e r f a c e due to aoouetlo mismatch. This behaviour w i l l be t r e a t e d i n more d e t a i l below. 2.3.3 Aooustlo Mismatch at a Boundary The K a p l t z a Reslstanoe The phonons t r a n s p o r t i n g energy from the o r y s t a l Into the helium bath experience a boundary reslstanoe at the i n t e r f a o e . This i s due to the f a c t that the a c o u s t i c impedance Z = <j v chan-ges from the c r y s t a l to the helium. This e f f e c t has been observed by K a p l t z a (1941) between H e l l and copper. The p r o p e r t i e s of s u p e r f l u i d helium have been used to e x p l a i n t h i s K a p l t z a r e s i s t a n c e . A boundary r e s i s t a n c e i s , however, to be expected q u i t e g e n e r a l l y ot an I n t e r f a c e be-tween two media w i t h d i f f e r e n t c h a r a c t e r i s t i c Impedances. I t i s r e a d i l y observed between a metal and H e l l because there i s no temperature drop i n H e l l and a very small one i n the metal, such that the temperature d i f f e r e n c e across the i n t e r f a c e Is not masked by b i g gradients In the bulk s o l i d and l i q u i d . E x p e r i -mental data are a v a i l a b l e f o r the boundary r e s i s t a n c e between metals or very few d i e l e c t r i c s and H e l l or l i q u i d 3 H e . For small heat c u r r e n t s , the temperature drop A T across the 42 surface i s p r o p o r t i o n a l to the thermal f l u x f. Thus A T « Rj^f (2.53) and R K i s c a l l e d the Kapitza r e s i s t a n c e . ( K a p i t z a , 1941) Devia-t i o n from the l i n e a r dependence A T v f has been found by A n d r o n i k a s h v i l l (1956) f o r high f l u x e s . depends s t r o n g l y on the surface c o n d i t i o n of the s o l i d (Johnson and L i t t l e , 1963) and d i f f e r s only s l i g h t l y f o r d i f f e r -ent m a t e r i a l s . The known experimental data f o r R^ can be f i t t e d with the em p l r l o a l expression T> A —r- —<~> °K err,2" sec A ; = A T — 7 — (2.54) w i t h 10 < A < 40 and 2.6 < n < 4.15 . A h i g h l y p o l i s h e d surface has a higher R^ than a rough one. Surface inhomogeneitles which are b i g compared to the acoustio wave l e n g t h Increase the e f f e c t i v e area and give a sm a l l e r appar-ent Rg, while surface inhomogeneitles oomparable to the wavelength serve as impedanoe transformer by smoothing out the abrupt impe-dance mismatch at a. sharp boundary and hence lead to a decrease i n R^. The K a p i t z a r e s i s t a n c e was found to be the same f o r thermal currents l n both d i r e c t i o n s , i . e . with the s o l i d oooler or u h 6 t t e r than the surrounding H e l l . (Kuang Wey-Yen, 1962) C h a l l i s (1962) measured R^ between l e a d and H e l l and found 43 a s l i g h t magnetic f i e l d dependence. decreases by 7% from 1 kOe t o 7 kOe. Most of the measurements up to date have been made l n the temperature range from 1°K to 2°K. Abel et a l , however, found an abnormally low K a p l t z a r e s i s t a n c e between cerium magnesium n i t r a t e and He3 i n the m°K re g i o n , whioh does not f o l l o w a T"n law. Very l i t t l e headway has been made to c a l c u l a t e R^ theo-r e t i c a l l y . The mechanism proposed by Khalatnikov gives a p p r o x i -mately the r i g h t temperature dependence R^vT"^, but does not e x p l a i n the pressure dependence ( C h a l l i s , D r ansfeld, and W i l k s , 1961). No s i g n i f i c a n t progress has been made since (Anderson et a l , 1965). Energy Transport Through an I n t e r f a c e In order to examine the e f f e c t of the K a p l t z a r e s i s t a n c e on the r e l a x a t i o n process i n CeES, we consider the i d e a l i z e d system of a p e r f e o t l y conducting c r y s t a l at temperature T , separated from the bath at constant temperature T^ by a boundary re s l s t a n o e R^ . This model a p p l i e s to our measurements below the X- p o i n t , where the abnormally h i g h thermal c o n d u c t i v i t y of H e l l allows no temperature gradient i n the bulk l i q u i d . I f W Is the energy fed to the c r y s t a l per seoond by an e x t e r n a l source, we have f o r the time r a t e of change o f the I n t e r n a l energy VQR of the o r y s t a l kk where c^r l a the t o t a l heat capaolty of the c r y s t a l and f i s the thermal f l u x through the c r y s t a l surface s. For small f l u x e s 71- z; - * -r - *„ / (2-54) and equation (2.55) becomes C ' 4^ - W - ^ 3 (2.57) For the case of a heat pulse s t a r t i n g at t = 0 w i t h T Q r ( o ) = T A, the s o l u t i o n of equation (2.57) i s with the r e l a x a t i o n time Tk (2.59) I f the heat pulse i s shutv o f f a f t e r a t t a i n i n g the temperature T Q r i n the c r y s t a l , the r e l a x a t i o n i s given by The time constant TL i s the same i n both cases. I f o„„ i s the * c r s p e c i f i c heat of the c r y s t a l per g and. 9, i t s d e n s i t y , the time constant beoomes (2.60) U5 where (y - J 1B the volume - t o-surface r a t i o o f the c r y s t a l . The maximum temperature r i s e i n the c r y s t a l a f t e r a Ion?? he a t i n g pulse i e -r-r , A T^(°°) = (2.61) I t Is p r o p o r t i o n a l to the heating power W and i n v e r s e l y propor-t i o n a l to the t o t a l surface c o n d u c t i v i t y Kg = S/Rg. I f ^ i s the r a t e determining time oonstant, i t should be equal to T, as oal o u l a t e d w i t h the three system p i c t u r e i n the l i m i t i n g case of By i d e n t i f i c a t i o n of equations (2.43) and (2.60) one gets ~QA i s the l i f e t i m e of a phonon against absorption by the helium bath and i s given by where Cg i s the s p e o i f i o h ° a t of the l a t t i o e system end K g i s the t o t a l c o n d u c t i v i t y across the surface. A t y p i o a l value of = 10 cm 2deg. sec. J o u l e s ^ gives f o r our c r y s t a l ( V/S ^ 0 . 0 5 cm. , c B * 6 x lO"* 5 Joules(g°K)-1) T~A& ->s. 6 x 10"5 sec. Figure 2.7 C r y s t a l s t r u c t u r e of CsF 2 47 2.4 Europium Doped Calcium F l u o r i d e 2.4.1 C r y s t a l Struoture The p o s i t i o n of the caloium and f l u o r i n e ions l n the CaF 2 c r y s t a l l a t t i c e l a shown i n Figure 2 . 7 (from VTykoff: C r y s t a l S t r u c t u r e ) . Each u n i t c e l l contains 4 Ca t + and 8 F" i o n s . The f l u o r i n e form a simple cubic l a t t i o e with a l a t t i c e constant of « 2 . 7 2 8 . The caloium ions can be found at every other cen-t e r of the f l u o r i n e cubic l a t t i c e , thus forming a faoe centered oubio l a t t i c e w ith the l a t t i c e oonstant a<, . Each Ca*+ has 8 F~ nearest neighbours at a distance of 2 . 3 6 8 and 12 Ca t + nearest neighbours at 3« 86A. Eaoh F*~ has 4 Ca nearest neighbours at and 6 F~ nearest neighbours at 2. 722. D i v a l e n t europium ions are r e p l a c i n g the calcium ions i n CaF 2 without causing no-t i c e a b l e d i s t o r t i o n . The c r y s t a l f i e l d symmetry remains c u b i c This Is not the case w i t h t r i v a l e n t rare e a r t h Ions which oan be found at s i t e s of lower than cubic symmetry. 2.4.2 Ground State and Paramagnetic Resonanoe The f r e e - i o n ground s t a t e of d i v a l e n t europium Is S x . The c r y s t a l f i e l d V cannot l i f t the e i g h t - f o l d degeneracy im-or mediately. The s p i n - o r b i t c o u p l i n g admixes, however, i n f i r s t order the 6 P i m u l t i p l e t and In seoond order the m u l t i p l e t s fD T and 4D 7 . A oublc f i e l d has only o f f - d i a g o n a l matrix elements between s t a t e s c o n t a i n i n g the seoond order p e r t u r b a t i o n i n s p i n -48 o r b i t c o u p l i n g . The matrix elements are, t h e r e f o r e , of the order of V Q r where j Is the s p i n - o r b i t coupling parameter. The e f -fe c t i s a decomposition of the ground s t a t e i n t o two doublets and one quartet w i t h an o v e r a l l s p l i t t i n g of 0.178 o n f 1 ( R y t e r , 1957)* This i s of the order of magnitude of the Zeeman p e r t u r b a t i o n <JCi . L a c r o i x (1957) has d i a g o n a l i z e d the Hamiltonian c o n s i d e r i n g V Q r and <Kt simultaneously. The r e s u l t i n g 8 by 8 s e c u l a r determin-ant i s r a t h e r oomplloated. An a l t e r n a t i v e procedure i s t o desc r i b e the ground s t a t e by a phenomenologioal spin-Hamiltonian as has been done by Baker, Bleaney and Hayes (1958): S « 7/2 i s the e f f e o t l v e spin and the oJ are the operator equi-v a l e n t s corresponding to the Legendre-polynomials In the c r y s t a l f i e l d expansion and are tabulated by Baker et. a l . F , Bv , B6 snd A are parameters determined by f i t t i n g the s p i n Hamiltonian to the observed ground state s p l i t t i n g s . We neglect f o r the moment the hyperflne i n t e r a c t i o n A(S»I). In magnetic f i e l d s of a few kOe, the ground s t a t e c o n s i s t s then of 8 l e v e l s which can be desoribed approximately by the magnetic quantum number M » + £ ) ~ ... There are seven allov;ed magnetic d i p o l e t r a n s i t i o n s AM » 1 with the f o l l o w i n g frequencies: + 2. — z V- — — 2. 2. 2. ~ 2 2 H (2.65) l,m,n are the d i r e c t i o n cosines of the magnetio f i e l d w i t h r e -spect to the f o u r - f o l d cubio axes. The experimental values are (Ryter, 1957): | £ J - 55.75 x lCT^cm- 1 , U J = 0.25 x lO-^cm" 1 , h- < o and g • 1.9927 There are two st a b l e Isotopes, Eu and Eu , of approximately the Same n a t u r a l abundanoe. Both have a n u c l e a r spin I = 5/2. The h y p e r f i n e i n t e r a c t i o n t h e r e f o r e s p l i t s each e l e c t r o n i c t r a n -s i t i o n i n t o twice s i x hyperfine l i n e s . The hyperfine s p l i t t i n g constants are k' ri * 34.1 x lO'^cm""1 and k' S* = 15.1 x lO^om" 1. The resonanoe f i e l d s f o r the d i f f e r e n t t r a n s i t i o n s have been c a l c u l a t e d at the constant frequency of 8500 Mo/s and. are p l o t -ted i n Figure 2.8 f o r the two d i r e c t i o n s [100] and [ i l l ] . The s i g n of b^ has been assumed negative l n a n t i c i p a t i o n of our own r e s u l t s . where £ v and p f/ooj On] 2 . r 3.5-Figure 2 . 8 Paramagnetic resonance spectrum of Eu +:CaF ? c a l c u l a t e d l i n e p o s i t i o n s d 51 2.4,3 I n t e r a c t i o n s Between Europium Ions In the preceding paragraph, we have t r e a t e d the problem of a s i n g l e Eu*"+ i o n , s u b s t i t u t e d f o r a Ca l + i o n i n the CaF 2 l a t -t i c e . With i n c r e a s i n g concentration the p r o b a b i l i t y f o r one Eu"+ i o n to have another E u 1 + i n the close neighbourhood becomes apprec i a b l e and the i n t e r a c t i o n between the e l e c t r o n i c spins of such p a i r s has to be taken i n t o account. The Hamiltonian JC^ f o r a p a i r of paramagnetic ions with spins S, and separated by a distance r i a_ can be w r i t t e n J £ R ~ <£, + < (2.66) where <£, snd JC^ are the s i n g l e - i o n Hamiltonlans, which are, i n our case < ^ c = + X * (2.67) Note th a t JCer w i l l i n general be d i f f e r e n t from the s i n g l e - i o n case, since the c r y s t a l f i e l d at one ion s i t e might be d i s t o r t e d due to t h e neighbouring i o n . As an example, the c r y s t a l f i e l d f o r a nearest neighbour E u 1 + p a i r l n CaFg w i l l have a x i a l r a t h e r than cubic symmetry. The I n t e r a c t i o n Hamiltonian oan be w r i t t e n approximately as A, •£) * - 3&r;!fA-rJ] (2.68) The f i r s t term i s the i s o t r o p i c exchange I n t e r a c t i o n and J i e 52 the exohange I n t e g r a l due to the overlap of the e l e o t r o n wave fun c t i o n s of the two Ions. The seoond terra contains c l a s s i c a l d i p o l a r i n t e r a c t i o n (- -j^r ) and a pseudo-dipolar term (D^ ) known as the a n i s o t r o p i c exohange i n t e r a c t i o n (Van Kranendonk and Van Vleok, 1958). For spins S >± , higher order terms are p o s s i b l e (quadrupole, octupole, e t c . ) . The p a l r - H a m i l t o n l s n can now be writ t e n 1 (Owen, 1961) ^ = ^  ((f-S) + J(S, ) + *Z - & + A, fcz) L ^ J where J = lgl + ^ I f the i s o t r o p i c exohange i n t e r a c t i o n Is the dominant term i n the Hamiltonian, the p a i r - s t a t e s can be grouped i n t o m u l t i -p l e t s of t o t a l s p i n S (3 = S, + Sx , S,+ S t - l , 0) w i t h energies J '• . The e f f e c t of the d i p o l a r and c r y s t a l f i e l d terms i s to l i f t the ( 2 S + l ) - f o l d degeneracy (even i n zero f i e l d ) . For an ant 1 ferromagnetic exohange I n t e r a c t i o n , ( J > 0 ) , the lowest p a i r state i s a s i n g l e t and. the paramagnetic a b s o r p t i o n l i n e s due to p a i r s w i l l disappear at low temperatures. This has been observed i n copper acetate by Bleaney and Bowers (1952). I f the remaining terms i n the Hamiltonian are small compared to the Zeeman i n t e r -a c t i o n , the l i n e s of the p a i r spectrum w i l l be olose to the l i n e s of the s i n g l e - i o n spectrum. For Eu* + p a i r s 3, = SZ = 7/2 and t h i s g i v e s a t o t a l of 6 4 l i n e s without considering the hyperfine I n t e r a c t i o n . EuF 2 has been observed to remain paramagnetic down to 1.6°K (Lee ©t, a l , 1 9 6 5 ) . A molecular f i e l d model f o r nearest neighbour (nn) i n t e r a c t i o n gives nearest neighbours. This gives l J n n | < 0 . 0 8 ° K There i s the p o s s i b i l i t y t h a t J n n i a not very d i f f e r e n t i n magnitude and opposite i n sign as compared to next nearest neighbour i n t e r a c t i o n J n n n . This would r a i s e the upper l i m i t f o r the exchange I n t e g r a l . I t i s , however, l i k e l y that the over-a l l s p l i t t i n g of the p a i r l i n e s does not exoeed a few °K. Due to the complicated s i n g l e ion spectrum, i t seems hopeless to f i n d " and i d e n t i f y p a i r t r a n s i t i o n s i n Eu*"+ : CsF 2 as has been p o s s i b l e i n ruby by 3tatz et a l ( 1 9 6 1 ) and by G i l l ( 1 9 6 D . 2.4.4 S p i n - L a t t i c e R e l a x a t i o n of Eu*"+ The c a l c u l a t i o n of the s p i n - l a t t i o e r e l a x a t i o n time T^ as reviewed i n chapter 2.2 cannot be applied to the case of Eu 1 + i n a cubic f i e l d without m o d i f i c a t i o n s . The t r a n s i t i o n p r o b a b i l i t y f o r a one phonon process between two l e v e l s of the ground stat e manifold has s t i l l the form of equation (2.9) (2.70) where T Q i s the t r a n s i t i o n temperature and z i s the number of ( 2 . 7 1 ) 54 The matrix elements of the o r b i t - l a t t i c e i n t e r a c t i o n contained i n A v a n i s h '.between the states of the 83X m u l t i p l e t i n f i r s t approximation and only combined a c t i o n of V 0^ and s p i n - o r b i t coupling give non-zero matrix elements. The d i r e o t process Is therefore weaker f o r S-state ions than f o r ions w i t h J > 0 i n the ground s t a t e . The two-phonon Raman process has to be modified. Equation (2.23) f o r the t r a n s i t i o n p r o b a b i l i t y i s s t i l l v a l i d but care must be taken i n i n t e g r a t i n g . P r e v i o u s l y , equation (2.23) was in t e g r a t e d approximately by assuming Av>kT. The two energy denominators were almost equal i n magnitude but opposite i n si g n , thus l e a d i n g to the Van Vleck c a n c e l l a t i o n . I n the case of a m u l t i l e v e l ground s t a t e , however, the opposite approximation A « kT a p p l i e s i n the Raman r e g i o n , I f the intermediate s t a t e /c> belongs to the ground state m u l t i p l e t . The two energy denomin-ators l n equation (2.23) are s t i l l approximately equal l n magni-tude, but of the same sign and no c a n c e l l a t i o n occurs. The t r a n s i t i o n p r o b a b i l i t y becomes w - / ^ 4 / — ^ •3^2. + 7 10 (2.72) The i n t e g r a l gives f o r T « e 55 Such a temperature dependence has f i r s t been suggested by Orbach and Blume (1962). In order to f i n d the relaxation time characterizing the return to equilibrium of two l e v e l s , one has to solve the rate equations f o r the whole 8-level multiplet. + 1 - Y uy _ . . / f : - , ^ , \ (2.74) oft 1* K . - W . . f ( = 2 f _ 2 This leads, i n general, to seven time constants (since ^ 7? = 0 )• It is easy t o show that the relaxation i s exponential i f a l l nonvanishing t r a n s i t i o n p r o b a b i l i t i e s per electron are equal. In this case, the relaxation time can be written approxi-mately T^ 1 AT •+• BT5" (2.75) Orbaoh's approach to calculate the matrix elements of the o r b i t - l a t t i c e i n t e r a c t i o n does not work f o r oubic c r y s t a l s . A cubio f i e l d has only terms i n A^ and A™ (equation 2.64) and thus the A™ terms of the dynamic c r y s t a l f i e l d expansion cannot be estimated from the c o e f f i c i e n t s f o r the s t a t i c c r y s t a l f i e l d . Huang (1964) obtained an expression f o r V o l by examining the normal modes of vi b r a t i o n of a cluster containing the Eu ion at the center of a oube formed by eight F~ ions. This ap-proach has been used f i r s t by Van Vleok (193°, 1940). From the normal mode expansion of V Q^, Huang obtained the relaxation time T J 1 * 3T + 4 x 1 0 " S 5 (seo^ 1) (2.76) 5 6 2.4.5 C r o s s - r e l a x a t i o n The Van Vleok-Orbaoh theory used to c a l c u l a t e T^ In CeES and i n Eu* + : CaF 2 assumes n o n l n t e r a c t l n g s p i n s . At high concen-t r a t i o n s s p i n - s p i n I n t e r a c t i o n s ( d i p o l a r and exchange i n t e r a c -t i o n s ) can play an important r o l e l n r e d i s t r i b u t i n g the popula-t i o n s of the d i f f e r e n t l e v e l s l n the spin system thus l e a d i n g to cros6*-relaxation e f f e c t s . Three d i f f e r e n t types of c r o s s r e l a x a t i o n s may a f f e c t the r e l a x a t i o n behaviour of Eu1"1" : CaF 2. These processes w i l l be examined i n the f o l l o w i n g : a ) , c r o s s - r e l a x a t i o n between d i f f e r e n t hyperfine components of a given e l e c t r o n i c t r a n s i t i o n . I f . the hyperfine s p l i t t i n g of a given e l e c t r o n i o t r a n s i t i o n i s not l a r g e compared to the d i p o l a r width of the s i n g l e compo-nents, the hyperfine s t r u c t u r e i s not r e s o l v e d . This i s the case l n an inhomogeneously broadened resonanoe l i n e . Innsubh a l i n e , a short microwave pulse may saturate o n l y parts of the l i n e . Bloembergen et. a l (1959) have used a random walk model to estimate the time i t takes f o r the e x o i t a t i o n to d i f f u s e through the whole l i n e . I f A*' Is the width of the whole resonanoe l i n e and the width of i t s homogeneously broadened components, the d i f f u -sion time i s given by 57 (2 .77) In Eu : CaF 2 the l i n e width of one hyperfine component i s ap-proximately 5 gauss, as measured from the spectrum at 34 Gc/s and the t o t a l s p l i t t i n g i s 180 gauss. This gives ^ 2 msec. This value should be too low, sinoe the h y p e r f i n e components are p a r t l y resolved and the ov e r l a p i s s m a l l . Huang has de-tec t e d the in f l u e n c e of c r o s s r e l a x a t i o n s due t o inhoraogeneous s a t u r a t i o n of the l i n e up to pulse lengths of 0 .5 msec. This time i s somewhat shorter than the estimate from the random walk model. A s h o r t e r time could be due to increased i n t e n s i t y of the wings of the s i n g l e components or to m u l t i p l e spin f l i p s as proposed by Bloembergen el; a l (1959). b) c r o s s ^ e l a x a t l o n between d i f f e r e n t e l e c t r o n i c t r a n s i t i o n s There are many p o s s i b i l i t i e s f o r m u l t i p l e s p i n f l i p s be-tween the d i f f e r e n t e l e c t r o n i c l e v e l s of E u z + : CaF 2. Consider c r o s s r e l a x a t i o n processes which spread the e x c i t a t i o n from the center +{ t r a n s i t i o n to the other l i n e s . One process which conserves energy i s shown i n Figure 2 . 9 a . (The l e v e l d i s -tances i n Figure 2 .9 correspond to the a c t u a l s p l i t t i n g s o f the 5, m u l t i p l e t w i t h H 11 [100J .) Processes of lower order than t h i s quadruple spin f l i p are p o s s i b l e only through the f a r wings of the resonance l i n e s and are th e r e f o r e weak. As an example, a double s p i n f l i p Is shown l n Figure 2 . 9 h . Prooesses a) and b) z s_ Z 2. j_ Z 4-z -2. z z 4> r 2 > Flgure 2.9 Energy l e v e l s and c r o s s - r e l a x a t i o n s l n Eu :Ca.F2 H H 3100 Oe [1003 co 59 oonserve angular momentum. Figure 2 . 9 o and 2 . 9 d show the t r i p l e s p i n - f l i p s whloh do not conserve angular momentum. A l l prooesses except a) can take plaoe only through the wings of the l i n e s , I f H II [100] , and oan be expected to be weak. The prooess 2.9d i n v o l v e s two nuclear s p i n f l i p s together w i t h an e l e c t r o n i c t r a n s i t i o n . The a c t u a l c a l c u l a t i o n of the t r a n s i t i o n p r o b a b i l i t i e s due to a l l the above mentioned processes i s not p o s s i b l e without the knowledge of the l i n e shapes and would be extremely Involved i n our oase due to the complicated hyperfine a t r u o t u r e . c) c r o s s - r e l a x a t i o n to other spin systems Consider two d i f f e r e n t spin systems 1 and 2 with two l e v e l s of almost equal spaoing. Let the c r o s s r e l a x a t i o n r a t e between the two systems be w 1 2 • l / T 1 2 * I f W l * 1 / / T 1 W 2 = 1 / / < r i are the s p i n - l a t t i c e r e l a x a t i o n rates i n the re s p e c t i v e systems, the r a t e equations f o r the population d i f f e r e n c e s n^ and n 2 are: and N 2 are the populations i n the two l e v e l s of each system and h° * *1 pre the e q u i l i b r i u m population d i f f e r e n c e s . Rannestad and Wagner (1963) have di s c u s s e d the s o l u t i o n s of the above rate equations i n d e t a i l . In g e n e r a l , two deoay oon-stants are found which reduce to one i n the l i m i t of N 2. 60 This Is the observed r e l a x a t i o n r a t e r~' I f system 1 r e l a x e s through s p i n - l a t t l o e r e l a x a t i o n and c r o s s - r e l a x a t i o n w i t h sys-tem 2. r - < _ U/ + . J * , 2 . W , I f e i t h e r one of the r e l a x a t i o n rates i s dominant we o b t a i n three s p e o i a l cases: a) WX> W2, W12 The s p i n - l a t t i c e r e l a x a t i o n i s dominant and c r o s s - r e l a x a -t i o n can he neglected: 7"~'« Vf^  b) w 1 2 > > W l w 2 The o r o e s - r e l a x a t I o n i s very f a s t compared to both s p i n -l a t t i c e r e l a x a t i o n r a t e s . We o b t a i n - , A i (2.80) 2 /V O W 2 > > W ] L. W 1 2 System 2 r e l a x e s very r a p i d l y to the bath and the observed r e l a x a t i o n rate Is r"'= W,+W,x^ (2.81) I t i s seen that the energy exohange of system 1 v i a 2 w i t h the bath i s l i m i t e d by the slower of the two r e l a x a t i o n r a t e s i n s e r i e s and i t s e f f i c i e n c y i s diminished by the r a t i o of the populations l n both systems -JJ- . From equation (2.79), a great v a r i e t y o f temperature and concent r a t i o n dependences are p o s s i b l e . W^  and W2 can have any of the temperature dependences of the s p i n - l a t t i c e r e l a x a t i o n rates and a l s o might depend on temperature through the Boltzman f a c t o r , i f the c r o s s - r e l a x a t i o n takes place to an e x c i t e d s t a t e of system 2. — depends on the r e l a t i v e concen-t r a t i o n of the two systems and W 1 2 oan a l s o be concentration de-pendent through the l i n e shapes. I f there are s e v e r a l l e v e l s o f the same system o r s e v e r a l systems to which c r o s s - r e l a x a t i o n s are p o s s i b l e , I t does not seem p o s s i b l e to solve the rate equations exoept i n l i m i t i n g cases. Rannestad and Wagner f i n d f o r the time constant T of system 1 l n presenoe o f c r o s s - r e l a x a t i o n to (ra-1) other systems: f o r the oase of dominant s p i n - l a t t i c e r e l a x a t i o n s and /V, W, + ^_ /V, U(-R = + ZM- ( 2 - 8 3 ) 2 f o r the oase of dominant c r o s s - r e l a x a t i o n . Equation (2.83) r e -duoes f o r JEL K A/, to z w> The i n f l u e n c e o f the above discussed processes on the ob-served spin-bath r e l a x a t i o n times i n Eu a" +: CaF 2 w i l l be d i s -cussed l n chapter 4. 62 2 . 5 The Magneto-optical Faraday E f f e c t 2.5.1 I n t r o d u c t i o n The d i r e c t i o n of p o l a r i z a t i o n of l i n e a r l y p o l a r i z e d liarht a f t e r passing through a medium, can change ss a f u n c t i o n of an e x t e r n a l l y a p p l i e d magnetic f i e l d . This i s known as the magneto-optical Faraday e f f e c t . Kramers (1930) showed that the r o t a t i o n of the plane of p o l a r i z a t i o n , c a l l e d the Faraday r o t a t i o n <^  , can be propor-t i o n a l to the magnetization M of the medium under o e r t a l n gen-e r a l c o n d i t i o n s . This p r o p o r t i o n a l i t y <^  >v> M leads to a simple method of measuring changes i n magnetization. One can measure the time dependence of the l i g h t i n t e n s i t y I ( t ) t r a n s -mitted by an a n a l y s e r and i f the transmis s i o n law of the ana-l y s e r I(<?) i s known, the magnetization can be c a l c u l a t e d . The time r e s o l u t i o n of t h i s method i s only l i m i t e d by the response time of the l i g h t d e t e c t i o n system. Under s u i t a b l e c o n d i t i o n s , microwave r a d i a t i o n can induce magnetic d i p o l e t r a n s i t i o n s between the energy l e v e l s of para-magnetic ions thus causing a r e d i s t r i b u t i o n of the populations i n the various l e v e l s . A change i n population w i l l a f f e o t the magnetization end consequently the Faraday r o t a t i o n . Opeohowskl ( 1 9 5 4 ) has given a quantum mechanical treatment of the Influence of microwave r a d i a t i o n on the Faraday r o t a t i o n . No attempt has been made to apply t h i s r i g o r o u s treatment to the o r y s t a l s studied i n t h i s t h e s i s . We are mainly I n t e r e s t e d l n the r e l a x a t i o n process, i . e . l n the recovery of the magnetization from a n o n - e q u l l l b r l u m value. The microwave r a d i a t i o n I s only used to d i s t u r b the spin system i n a w e l l defined manner by e q u a l i z i n g the population d i f f e r e n -ces l n o e r t a i n l e v e l s . Microwave r a d i a t i o n i s not the only way o f e s t a b l i s h i n g non-equilibrium p o p u l a t i o n s . K a l b f l e i s o h ( 1 9 6 4 ) pulsed the DO magnetic f i e l d from a low to a higher v a l u e . I f t h i s i s done l n a time short enough such that the populations i n the Zeeman l e v e l s do not change a p p r e c i a b l y , the e f f e c t i v e tempera-ture Tfl of the s p i n system w i l l be r a i s e d . This can be seen from the d e f i n i t i o n of T g, the s o - c a l l e d s p i n temperature Here n^ and n 2 are the populations i n two l e v e l s w i t h an energy d i f f e r e n c e of &12- T h © two methods e s t a b l i s h d i f f e r e n t i n i t i a l o onditions f o r the r e l a x a t i o n which f o l l o w s the microwave pulse or the DC magnetic f i e l d step. In the f o l l o w i n g , the c o n d i t i o n s f o r the p r o p o r t i o n a l i t y o -v M are examined and i t i s shown that f o r the substances i n -v e s t i g a t e d i n t h i s t h e s i s , t h i s p r o p o r t i o n a l i t y holds at the l i g h t frequencies employed. 2 . 5 . 2 Faraday R o t a t i o n of Rare Earth Ions l n S o l i d s Shen (1964) has g e n e r a l i z e d the Kramers formula f o r the Faraday r o t a t i o n t o Include not only e l e c t r i c - d i p o l e t r a n s l -6k t i o n s . He pets the f o l l o w i n g expression f o r the oomplex Faraday-r o t a t i o n : Z — *fW - " -C ^ 7 <2-85) n Is the average r e f r a c t i v e index of the medium; N i s the number of atoms per u n i t volume; w i s the l i g h t frequency; uu ~ —1 i s the t r a n s i t i o n frequenoy between stetes /6> and /«> ; Tba + i s the associated damping term and / i t t are the o s c i l l a t o r strengths f o r r i g h t and l e f t c i r c u l a r l y p o l a r i z e d l i g h t . For e l e c t r i c d i p o l e t r a n s i t i o n s , we can w r i t e (2.86) where r+ = x ± l y and i s the Boltzmann f a c t o r e k 7~ • I f the l i g h t frequency Is f a r from resonance, the damping terms can be neglected and equation (2.85) becomes %~ JTTn /j£Zj(l<*<^*>/ -l<t>l'-/«>l J (2.87) In order to s i m p l i f y the above expression, we have t o s p e c i f y the s t a t e s . For f r e e Ions, the stat e s can be s p e c i f i e d by / n > T i rns> In the Russell-Saunders approximation, which Is quite good f o r rare e a r t h s . For the ground s t a t e of r a r e earth Ions i n s o l i d s , J remains a good quantum number, but mj has to be replaced by the symmetric quantum number /*. . In the e x c i t e d s t a t e s , the e f f e c t of the c r y s t a l f i e l d oan be strong. J i s no longer a good quantum number and i s replaced by (j . For i l l u s -t r a t i o n , a part of the l e v e l diagram f o r a h y p o t h e t i c a l Ion i s shown l n Figure 2.10. I f the c r y s t a l f i e l d and Zeeraan s p l i t t i n g s are small compared to the d i f f e r e n c e between the average t r a n s i -t i o n frequency and the l i g h t frequency, I.e. i f where the frequency f a c t o r l n equation (2.87) oan be expanded i n terms of AU> : -2. (oo , _ +• Aco , , \X~ co 1" — Z> rco) + AW , D (u,)+... C o ! . t o - CO (2.89) 66 free /on Crys/b/ fate Zee/r,on Figure 2.10 Typioal l e v e l s of r a r e e a r t h lone i n s o l i d s 6 7 I f one r e t a i n s only the f i r s t terra of expansion ( 2 . 8 9 ) , equa-t i o n ( 2 . 8 7 ) f o r the Faraday r o t a t i o n gives % = C£D"V)£ ( 2 . 9 0 ) with _ 2 T r e*d/fo t+i) v W 5 t c « F o l l o w i n g Van Vleok and Hebb (193*0, we change from the l*'f'y> and I n J~/u.> to the \n Tm> r e p r e s e n t a t i o n and apply the Kronlg-Honl formula f o r the e l e c t r i c d i p o l e moment operator. The r e s u l t i s (2.91) 7' At the low temperatures employed i n our experiments, only the lowest J - m u l t l p l e t Is oocupied and equation (2.91) can be w r i t -ten % - AT - C . " A < > ( 2 - 9 2 ) w i t h J 1 I t i s seen that „^ i s p r o p o r t i o n a l to the magnetization. Shen ( 1 9 6 3 ) has c a l c u l a t e d the c o n t r i b u t i o n t o the Faraday r o t a -t l o n from the -£><>,_.,rlu'J term, whioh a r i s e s from the Zeeman 68 p e r t u r b a t i o n on the frequencies _ . He gets two a d d i t i o n a l terms to the r o t a t i o n , one diamagnetic and one para-magnetic. They are small t o the extent that approximation (2.88) h o l d s . F i n a l l y , both the c r y s t a l f i e l d and Zeeman i n t e r -a c t i o n s perturb the ground J m u l t i p l e t such that J Is no longer a good quantum number. The former gives a sm a l l paramagnetic c o n t r i b u t i o n while the l a t t e r causes a dlamagnetio r o t a t i o n , which i s the analogue t o the s o - c a l l e d Van Vleok paramagnetism. A p p l i c a t i o n to CeES CeES c r y s t a l s are completely transparent down to a wave-length of 27008 and the absorption bands due to the strong e l e c -t r i o d i p o l e t r a n s i t i o n s s t a r t at 2500$. The l i g h t frequencies used are around s 20,000 cm"1. We have, th e r e f o r e C o — » 2.0 O O O c m " ' Since the c r y s t a l f i e l d s p l i t t i n g i s r e l a t i v e l y small (Van Vleck and Hebb, 1934) A<^ > becomes ^ C o • » = • / a O O cm'1 Therefore, the c o e f f i c i e n t s f o r the dominant terms i n equa-t i o n (2.87) due to the strong e l e c t r i c d i p o l e t r a n s i t i o n s (4f) -» (5d) are Z> j and ^ ' " ^ * & . The approximation (2.88) thus holds to w i t h i n one peroent. A p p l i c a t i o n t o Eu*+ : CeF 2 1.+ The strong a b s o r p t i o n bands f o r Eu are l n the v i s i b l e region and the approximation (2.88) i s not v a l i d . However, the t r a n s i t i o n s which c o n t r i b u t e mainly to the sum i n (2.87) are the allowed e l e c t r i c d l p o l e t r a n s i t i o n s S -* P. I f the Eu* ion i s at a oublo s i t e , the c r y s t a l f i e l d s p l i t t i n g s of the P mult-i p l e t s are only of the order of 10 om"*1 (Van Vleolc and Penney, 1934) and J remains a good quantum number even i n the ex o l t e d P s t a t e s (Figure 2.11). I t Is then p o s s i b l e to expand the frequency d i s p e r s i o n f a c -t o r s u s i n g a much l e s s s t r i n g e n t requirement, namely (2.93) with This y i e l d s the expansion Co - C o Co — co - ; / - 2 A C O , r f c 3 l (2.94) I f only the ground J m u l t i p l e t i s occupied and i f only the f i r s t term i n expansion (2.94) i s r e t a i n e d , the same procedure as used before gives the Faraday r o t a t i o n (2.95) Figure 2.11 T y p i c a l l e v e l s f o r rare e a r t h ions l n s o l i d s : S-state ion i n a cubic f i e l d 71 w i t h The Faraday r o t a t i o n l e again p r o p o r t i o n a l to the magnetization M, whloh, i n t h i s approximation, Is given by the B r i l l o u l n f u n c t i o n B co - J ^ J - _ J _ ^ / J L ; , 2 . 9 « , with X = 9£r?H and J a 7/2 f o r Eu"" J Is a good quantum number and departure of the magnetization from the B r i l l o u l n curve Is s m a l l . In f a c t , higher J - s t a t e s are admixed to the ground s t a t e only i n second order as mentioned i n ohapter 2.4. As mentioned above, the s p l i t t i n g of the e x c i t e d P-states by a oubio c r y s t a l f i e l d i s of the order of 10 cm""1. Henoe, the frequenoy d i f f e r e n c e AC*>„, 7 > /_ m J >_ i e s m a l l , cor-responding to 3$ at the l i g h t frequencies used. The a p p r o x i -mation (2.Q3) i s v a l i d even f o r l i g h t frequencies very c l o s e to the a b s o r p t i o n bands due to the e l e c t r i o d i p o l e S -* P t r a n s i -t i o n s whloh begin below 4200& ( F e o f i l o v and K a p l y a n s k i i , 1963)• The frequency f a c t o r P^,,^, enhances, i n t h i s case, the Faraday r o t a t i o n considerably due t o I t s small denominator, without s p o i l i n g the p r o p o r t i o n a l i t y between Faraday r o t a t i o n and mag-n e t i z a t i o n . 7 2 3 EXPERIMENTAL ARRANGEMENT 3 . 1 Apparatus The b a s i c apparatus required f o r the o p t i c a l d e t e c t i o n of paramagnetic resonance and r e l a x a t i o n d escribed In t h i s t h e s i s inolude - a helium c r y o s t a t to maintain the sample at the low temperatures needed to ob t a i n long enough s p i n - l a t t l o e r e l a x a t i o n times and appreciable paramagnetic r o t a t i o n . - a magnet to provide strong enough f i e l d s to s p l i t the ground l e v e l s of the paramagnetic ions by an amount corresponding t o convenient microwave frequencies. • a mlorowave system oapable of s a t u r a t i n g t r a n s i t i o n s between the ground l e v e l s . - an o p t i c a l system to produce a c o l l i m a t e d beam of l i n e a r l y p o l a r i z e d monochromatic l i g h t . - a d e t e c t i o n syrtem to analyse the s t a t e of p o l a r i z a t i o n of the l i g h t a f t e r passing through the c r y s t a l . The apparatus used In our experiments I s b a s i c a l l y the same as that described i n great d e t a i l by R i e c k o f f ( 1 9 6 2 ) and G r i f f i t h s ( 1 Q 6 5 ) . We review, t h e r e f o r e , only b r i e f l y the gen-e r a l f e a t u r e s , d e s c r i b i n g at the same time i n more d e t a i l the few adaptions whioh have been made f o r our experiments. A block diagram of the apparatus i s shown i n Figure 3 . 1 . A N finite 7~ X.K 33 6tps/ron tr,/rr l U fitter polarise/- A* v  /'w J-cjrq /ar Figure 3.1 Block diagram f o r the apparatus Figure 3.2 The dewar cap 75 3.1.1 The Cryostat The helium dewar, whose diameter i s Just l a r g e enough f o r the microwave c a v i t y and the a d j u s t i n g meohanisms, i s placed Into an ove r s i z e n i t r o g e n dewar. Both dewars are made of pyrex glass and s t r i p s i l v e r e d . Two windows, cut out of s e v e r a l l a y e r s of p l a s t i c tape and held to the Inner dewar by a copper sleeve, kept the l i q u i d n i t r o g e n out of the l i g h t path. This e l i m i n a t e s noise i n the l i g h t s i g n a l a r i s i n g from b o i l i n g n i t r o g e n . A new dewar oap was used, which i s of the same basio design as those used c u r r e n t l y i n our low temperature l a b o r a t o r y . The d e t a i l s are shown i n Figure 3»2. The pumping l i n e and the con-nections to the helium return l i n e and manometers enter the brass cap from the s i d e . The cover Is screwed on top of the cap and contains the feedthrough f o r waveguide, t r a n s f e r syphon and a d j u s t i n g rods. A combination of forepump and H g - d l f f u s l o n pump was used to pump out the t r a n s f e r syphon Jacket, the manometers and the w a l l of the helium dewar to a hard vacuum. A Kinney high-speed pump was employed to reduce the pressure over the l i q u i d helium surface i n order to obtain temperatures down to 1.4°K. 3.1.2 The Magnet The low-Impedance watercooled electromagnet was custom made to s p e c i f i c a t i o n s from Dr. J.M. Daniels by the Tatena E l e c -t r i c Mfg. Co. i n Tokyo. An a x i a l hole through the magnet per-mite o p t i o a l i n v e s t i g a t i o n s w i t h a l i g h t beam p a r a l l e l t o the magnetic f i e l d as r e q u i r e d f o r our experiments. The magnet produces a f i e l d of 12 kOe at ho amperes. This i s l a r g e enough at X-band frequencies to I n v e s t i g a t e substances with g-values as low as 0.5. The current through the magnet was provided by a DC generator and regulated by means of water cooled r h e o s t a t s . The generator r i p p l e was reduced through a f i l t e r of two 5600 Mfd oapsoltors i n p a r a l l e l . The mpgnet ourrent was s t a -b i l i z e d by power t r a n s i s t o r s , whose base curren t was regulated by combined ourrent and voltage feedback l o o p s . This s t a b i l i -z a t i o n has been described by G-arwln et a l (1959). A diagram of the o i r c u l t Is shown l n Figure 3 0 . A l o n g term s t a b i l i t y of at l e a s t ±1 gauss was achieved, whioh i s c e r t a i n l y l e s s than the l i n e width of the h y p e r f i n e components of the Eu 1* l i n e s and Is b e t t e r than the r e s o l u t i o n l i m i t set by the i n -horaogeneity of the magnetic f i e l d . The value of the magnetic f i e l d was determined by measuring the current through the mag-net. The f i e l d values oould then be taken from a c a l i b r a t i o n ourve H vs I . 3.1.3 The Microwave System A 2 K 3 9 r e f l e x k l y s t r o n produced 250 mW microwave power at X-band frequencies. The k l y s t r o n heater c u r r e n t was taken from a 6 V o l t c a r b a t t e r y and the h i g h voltages were provided by a home msde regulated power supply. The k l y s t r o n was matched to the r e c t a n g u l a r waveguide tr a n s m i s s i o n l i n e by a v a r i a b l e sue-77 ceptance. A f e r r l t e I s o l a t o r protected the k l y s t r o n from r e -f l e c t e d power. The microwave power was then passed through a 30 db f l a p a t t e n u a t o r , two v a r i a b l e p r e o l s l o n attenuators and the magic T, before reaching the sample c a v i t y . The r e f l e c t e d power from the c a v i t y was monitored l n the f o u r t h branch of the magic T, which contained a stub tuner and a o r y s t a l d e t e c t o r (1N23). A o a v i t y wavemeter followed by a o r y s t a l d e t e c t o r was coupled to the microwave system w i t h a 20 db d i r e c t i o n a l oou-p l e r i n s e r t e d Just a f t e r the f e r r l t e i s o l a t o r (see Figure 3»l)« I t served to measure the resonance frequency of the sample o a v i t y . The c y l i n d r i c a l sample o a v i t y was coupled to the waveguide by a c i r c u l a r i r i s . Two screws extending i n t o the waveguide were used t o raatoh the loaded c a v i t y to the transmis s i o n l i n e at room temperature. U s u a l l y the matching d e t e r i o r a t e d con-s i d e r a b l y at l i q u i d N 2 temperature, but at helium temperatures, the matoh was again s a t i s f a c t o r y . The c a v i t y was operated i n the TE 112 mode. The f i e l d con-f i g u r a t i o n i n a h o r i z o n t a l plane through the sample I s shown i n Figure 3 . 4 . This p a r t i c u l a r mode permits a simple s o l u t i o n to the problem o f a d j u s t i n g the o r y s t a l at helium temperatures. A t e f l o n plug f i l l s the lower h a l f of the o a v i t y . A c y l i n d r i -c a l t e f l o n rod i s i n s e r t e d i n a h o r i z o n t a l s l o t l n the upper part of the plug. The plug was turned through a gear d r i v e while the rod was t i l t e d by p u l l i n g on one end of the nyl o n thread passing through the rod. This allowed the c r y s t a l to Figure 3.3 The current r e g u l a t i o n 00 Figure 3 . 4 F i e l d c o n f i g u r a t i o n l n the microwave c a v i t y F igure 3 . 5 C r y s t a l a d j u s t i n g mechanism r o t a t e about two perpendioular axes. The s t r e n g t h of the nylon Is not v e r y great at helium temperatures and I t breaks v e r y e a s i l y . I t has been found that imperfections i n the n y l o n thread ( o r d i n a r y 1 2 pound f i s h i n g l i n e ) occur f a i r l y o f t e n . Breaking has ooourrefl l e s s o f t e n when the nylon thread was examined under a low power microscope to f i n d a perfeot p i e c e . A new pieoe of nylon thread was used f o r each experimental run since repeated use was found to weaken the thread. A schema-t i c drawing of the o a v i t y w i t h the a d j u s t i n g mechanism i s shown l n Figure 3 . 5 . For more d e t a i l s , we r e f e r to R i e c k o f f and Weissbaoh ( 1 9 6 2 ) . For that s e c t i o n of the t r a n s m i s s i o n l i n e , which extends i n t o the dewar, a t h i n - w a l l m a t e r i a l w i t h low thermal conduction i s r e q u i r e d . The home made german-silver waveguide used pre-v i o u s l y was replaced by a s t a i n l e s s s t e e l waveguide obtained from S u p e r i o r Tube Co., Norrlstown, Pa. The heat leak was found t o be s m a l l e r and thus the helium l a s t e d several hours longer. Since In a l l substances I n v e s t i g a t e d l n t h i s t h e s i s , magnetic-dipole t r a n s i t i o n s are induced by a microwave magnetic f i e l d perpendicular to the DC magnetic f i e l d , the H-plane of the wave guide was set p e r p e n d i c u l a r tn the DC f i e l d . Pulse modulation of the microwaves was achieved by apply-ing a pulse from a Tektronix 1 6 1 pulse generator to the r e f l e c -t o r e l e c t r o d e of the k l y s t r o n v i a a 1 Mfd condensor. The pulse width used v a r i e d between 1 msec and 1 0 0 mseo and the r e p e t i -t i o n r a t e was i n a l l cases at l e a s t ten times the r e l a x a t i o n 81 time l n order t o achieve complete recovery before a p p l y i n g a new p u l s e . To tune the microwave system, a sawtooth v o l t a g e from a T e k t r o n i x 162 waveform generator was a p p l i e d to the r e -f l e c t o r . This permitted the k l y s t r o n frequency to be swept through the o a v i t y resonanoe. 3.1.4 The O p t i c a l System The l i g h t source was a General E l e c t r i c H100 A4/T mercury-a r c lamp cooled by f o r c e d a i r . The current was taken from a DC generator and passed through a choke and a current r e g u l a t o r tube Amperite 10-4C l n order to keep the l i g h t i n t e n s i t y as oonstant and fre e from r i p p l e as p o s s i b l e . A s e r i e s of lenses o o l l i m a t e d the l i g h t t o a narrow and nearly p a r a l l e l beam at the sample s i t e . Corning g l a s s f i l t e r s were used t o s e l e c t s i n g l e l i n e s from the Hg-speotrum. Both analyser and p o l a r i z e r x>/ere of the Glan-Thompson type and mounted i n a d i v i d e d c i r c l e which allowed angle readings of f i v e minutes of a r c In many cases, the l i g h t i n t e n s i t y at the d e t e c t i n g end was too high f o r the p h o t o m u l t i p l i e r to operate i n i t s l i n e a r region. An i r i s was improvised u s i n g p l a s t i c tape i n these cases and was i n s e r t e d between the l i g h t source and the condenser l e n s . Even w i t h optimum alignment, a halo due to s c a t t e r e d l i g h t oould be seen around the image of the c a v i t y hole at the de t e c t -ing side of the o p t i c a l bench. This s c a t t e r e d l i g h t was e l i m i -nated by another i r i s narrow enough to pass only the l i g h t 82 ooming d i r e c t l y from the c a v i t y . The c o n t r i b u t i o n of soattered l i g h t to the photo s i g n a l was thus minimized. 3 . 1 . 5 S i g n a l Deteotion The l i g h t passing through the analyser was detected by an RCA 621? p h o t o m u l t i p l i e r . A bank of twelve 90-volt r a d i o bat-t e r i e s d e l i v e r e d the necessary s t a b l e v o l t a g e s . The photoour-rent was passed through a 10 k l l r e B i s t o r . The r e s u l t i n g photo s i g n a l V p h was found t o be p r o p o r t i o n a l to the l i g h t i n -t e n s i t y up to V p h « 700 mV. The r e l a x a t i o n s i g n a l s were f i l -t ered w i t h a v a r i a b l e RC network. A time constant long enough not t o d i s t o r t the s i g n a l s , but as short as p o s s i b l e to cut down the noise was chosen. The s i g n a l s were u s u a l l y d i s p l a y e d d i r e c t l y on a Tektronix 502 o s o l l l o s o o p e u s i n g the b u i l t - i n d i f f e r e n t i a l D C -amplifier. The r e q u i r e d opposite b i a s to cancel the DC-component of the l i g h t was taken from a No. 6 dry o e l l r e g u l a t e d by a 10 k l l h e l i p o t . I n the case of CeES, the s i g n a l s were very weak, es-p e c i a l l y at high temperatures, and use was made of s i g n a l aver-aging devioes. An enhancetron was k i n d l y l e n t t o us f o r an overnight run by Mr. R.F. Trehearne and Mr. P h i l West from Nuclear Data Inc. Subsequently, a CAT kOOB ( T e c h n i c a l Measurement Corp, Mnemotron Dlv.) was used f o r which thanks are given to Dr. J.A. Wada and Mr. Finn Bauck from the n e u r o l o g i c a l I n s t i t u t e at the U n i v e r s i t y of B r i t i s h Columbia. With both of these averaging d e v i c e s , vre 83 could Increase the s l g n a l - t o - n o i s e r a t i o c o n s i d e r a b l y . S i g n a l s completely Immersed l n the noise y i e l d e d reasonable traces a f t e r I n t e g r a t i o n f o r only a few minutes (Figure 3 . 6 ) . The box-car i n t e g r a t o r used i n previous work by G r i f f i t h s (1965) i s u n s u i t a b l e i n the case of CeES beoause of i t s depend-ence on z e r o - l i n e s h i f t s . Appreciable z e r o - l i n e s h i f t s over the l ong time periods necessary f o r i n t e g r a t i o n by the box-car could not be avoided due to the very l a r g e Faraday r o t a t i o n , which i s extremely s e n s i t i v e t o temperature changes. a) before averaging b) a f t e r averaging (CA? UOQB) Figure 3.6 R e l a x a t i o n t r a c e i n CeES 85 3.2 Experimental Procedure 3.2.1 Preparations f o r the Measurements The e t h y l s u l p h a t e samples were grown i n t h i s l a b o r a t o r y from aqueous s o l u t i o n by slow evaporation at i c e temperature. The growing, o r i e n t i n g and p o l i s h i n g prooedure has been desoribed i n d e t a i l by G r i f f i t h s (1965). The europium-doped CaF 2 samples were purchased ready out and o r i e n t e d from Optovac Inc., North B r o o k f i e l d and were p o l i s h e d i n our l a b o r a t o r y . The samples were cooled to helium temperatures by standard low temperature teohniques described i n d e t a i l by G r i f f i t h s (1965). The e t h y l s u l p h a t e samples dehydrate under reduced pressure or i n a dry atmosphere. At the pressure of a few mm Hg, the surfaces of the o r y s t a l s dehydrate i n seconds s u f f i c i e n t l y to d e p o l a r i z e the l i g h t and thus reduce the s i g n a l - t o - n o i s e r a t i o a p p r e c i a b l y . I t Is t h e r e f o r e important t o pump the a i r q u i c k l y out of the inner dewar and re p l a c e I t immediately w i t h dry helium at atmos-pheric pressure. A f t e r t h a t , the precoolincr should be s t a r t e d without delay. A f t e r having reached the d e s i r e d low temperature, the o r y s t a l s are aligned f o r maximum l i g h t t r a n s m i s s i o n . In the e t h y l s u l p h a t e s , the alignment i s very c r i t i c a l . I f the l i g h t d i r e c t i o n and the hexagonal a x i s of the c r y s t a l are not p a r a l l e l , b i r e f r i n g e n c e occurs whloh may d i m i n i s h the Faraday r o t a t i o n ap-p r e c i a b l y . This e f f e c t i s Investigated i n some d e t a i l l n chap-t e r 4.3. A good cheok of the proper alignment i s the Faraday 8 6 r o t a t i o n at low magnetic f i e l d s , which should be l i n e a r w i t h f i e l d . 3.2.2 Faraday R o t a t i o n To determine the Faraday r o t a t i o n , the analyser was adjus-ted to minimum l i g h t t r a n s m i s s i o n f o r a given f i e l d . I f the angle of the analyser Is ^>^IN (H) , then the t o t a l Faraday r o t a -t i o n i s given by Each value of ^ was averaged from several readings. The mean d e v i a t i o n of the average v a r i e d between 0.3 deg and 1 deg de-pending on the degree of p o l a r i z a t i o n of the l i g h t . $,w has to be c o r r e c t e d f o r the Faraday r o t a t i o n of the g l a s s w a l l s of the dewar <?c pnd f o r the diamagnetic r o t a t i o n of the sample 0^  to o b t a i n the paramagnetic c o n t r i b u t i o n <^ p . Since <^ c and are of opposite s i g n xtfith respect to <j , we have - I M + i s . 1 H<M ( 3- 2 ) <k Is independent of temperature and has been determined a t room temperatures, where i s n e g l i g i b l e . <^>c turned out to be a l s o independent of temperature down to 1 .5°K. The Faraday r o -t a t i o n of our dewar i s shown l n Figure 3.7. F i g u r e 3.7 Faraday r o t a t i o n of the g l a s s w a l l s of the dewar 88 3.2.3 Determination of the Resonance Speotrum The r e l a t i v e change i n the population d i f f e r e n c e of a two l e v e l s p i n system due to a mlorowave r a d i a t i o n f i e l d i s u s u a l l y c a l l e d the s a t u r a t i o n s. A n ° - A » (3.3) An and An0 are the population d i f f e r e n c e s between the two l e v e l s i n presenoe and i n absence of microwaves r e s p e c t i v e l y . The p o p u l a t i o n d i f f e r e n c e of a two l e v e l s p i n system Is propor-t i o n a l to the magnetization M and we oan w r i t e S = (3.4) I f the magnetization i s i n turn p r o p o r t i o n a l to the Faraday r o -t a t i o n , the s a t u r a t i o n becomes S * 9 a " ? (3.5) where <^  and <^ 0 are the paramagnetic Faraday r o t a t i o n s i n pres-ence and i n absence of the microwave r a d i a t i o n r e s p e c t i v e l y . In a m u l t i l e v e l system, equation f3»4) oan be used to define the s a t u r a t i o n s, sinoe equation (3.3) does, i n ge n e r a l , not hold. The s a t u r a t i o n of a spi n system i s t h e r e f o r e d e f i n e d as the r e l a t i v e change i n magnetization due to microwave r a d i a t i o n , s i s a measure f o r the miorowave absorption and a p l o t of s v e r -sus the a p p l i e d DC magnetlo f i e l d H y i e l d s , t h e r e f o r e , the para-89 magnetlo resonance speotrum. Equation ( 3 . 5 ) has been used to determine the resonance spectrum f o r CeES and E u r + : CaF 2-- The experimental procedure Is very t e d i o u s . I t requires reading the angles of minimum l i g h t t r a n s m i s s i o n w i t h and without microwaves f o r eaoh f i e l d value. This step-by-step prooedure i s incapable of g i v i n g de-t a i l s of a spectrum w i t h c l o s e l y spaced l i n e s as i n the E u 4 + spectrum, since the minimum step p o s s i b l e w i t h our rheostat Is 3 0 gauss at medium f i e l d s . I t Is s u f f i c i e n t , however, to det e r -mine the spectra found i n u n d i l u t e d rare e a r t h s a l t s which ex-h i b i t broad resonance l i n e s . I f the microwave power r e q u i r e d to produce a measurable s a t u r a t i o n heats the sample a pulse method oan be used to deter-mine the s a t u r a t i o n . The l i g h t i n t e n s i t y I tra n s m i t t e d by an ana l y s e r i s 2 . Z i s the angle of the analyser w i t h respect to the plane of p o l a r i z a t i o n of the l i g h t . 1^ i s the maximum tran s m i t t e d l i g h t i n t e n s i t y - 0 ) and I„. i s the minimum ( ^ f = z : ) . A small ohange a. A^j around the angle ^ > ^ ...) gives from equation ( 3 * 6 ) | A / | = (la -lz ) A-f « (?<> - A j a ^ ( 3 . 7 ) since |A<-f| = IAC^I i f i s the change i n Faraday r o t a t i o n , 90 eg. due to a microwave pulse. The s a t u r a t i o n due to a mlcrov/ave pulse oan now he dete r -mined from equation (3*7) Equation (3.8) hoe been used to determine the s a t u r a t i o n l n CeES. The prooedure Is the f o l l o w i n g : The DC photoslgnal V ^  was measured f o r maximum and mlmimum l i g h t t ransmission and the angle corresponding to minimum l i g h t i n t e n s i t y was recorded. Then the microwaves were pulsed i n t o the o a v i t y at long r e p e t i t i o n r a t e s ( g e n e r a l l y 2 sec) i n order to avoid heating. The s a t u r a t i o n s i g n a l h e i g h t A V ^  was deter-mined from the o s c i l l o s c o p e or from a photo of the s a t u r a t i o n t r a c e . Care was taken t o stay l n the l i n e a r region of the pho-t o m u l t i p l i e r such that V ^ v l . For a l l measurements In CeES A c > was small enough to keep equation (3.7) v a l i d . 3.2.4 R e l a x a t i o n Time Measurements For the substances I n v e s t i g a t e d l n t h i s t h e s i s , the Faraday r o t a t i o n i s p r o p o r t i o n a l to the magnetization. Henoe, we have £<^(t) -v&M(t) and the decay of the photoslgnal ^ v p n observed a f t e r a microwave pulse Is a l i n e a r measure of the recovery of the magnetization. This i s true i f A V ^ ^ - V A I and A I -v> A q The f i r s t p r o p o r t i o n a l i t y Is assured by the p h o t o m u l t i p l i e r c h a r a c t e r i s t i c i f V P H < 700 mV. The d e v i a t i o n from the propor-S = (3.8) t l o n a l l t y A I ( O y^> i g smallest i f the analyser Is set at -f ~ - \ A<^w where ^ i a the angle of minimum l i g h t t r ansmission without the microwave s i g n a l . A mlorowave pulse r o t a t e s the d i r e c t i o n of p o l a r i z a t i o n by an angle A<^ m from ^ ~ x ^<?M *° ^oo "ir A ? W a n < a ' t 1 1 6 decay w i l l proceed i n the reverse d i r e c t i o n a f t e r s h u t t i n g o f f the mlorowaves. For &cjw < 20 deg, we have A Iff) -v> A<^(f; w i t h i n the measuring accuracy of the r e l a x a t i o n t r a c e . For l a r g e r values A < ^ M , R l e c k o f f has computed t a b l e s to c o r r e c t A<^  f o r the d e v i a t i o n from the p r o p o r t i o n a l i t y A 1(f) ' V A ^ ( t ) . The c o r r e c t i o n pro-cedure i s , however, time consuming and values o f A<^M > 20 deg di d not give a b e t t e r s i g n a l - t o - n o i n e r a t i o . Care was taken, t h e r e f o r e , not t o exceed A q M = 2 0 deg by a t t e n u a t i n g the mic-rowave power. This was only needed i n Eu z + : CaF 2, the s i g n a l s l n CeES being always s m a l l . The r e l a x a t i o n t races were recorded from the o s c i l l o s c o p e screen on p o l a r o i d or 35 mm f i l m . The traces were measured point f o r p o i n t and p l o t t e d on semllogar-lthmic paper. The slopes of the obtained s t r a i g h t l i n e s g i v e the r e l a x a t i o n time. I f no s t r a i g h t l i n e I s obtained, the r e -l a x a t i o n i s nonexponential and cannot be c h a r a c t e r i z e d by a s i n g l e r e l a x a t i o n time. This occurred only i n CeES above the X-point as w i l l be mentioned i n chapter 4.1. A second scope (Tektronix 545A) was used to monitor the pulse response of c a v i t y and wavemeter. This allowed t o ob-serve c o n s t a n t l y the tuning of the k l y s t r o n during the pulse and to oorrect i t s frequency f o r small s h i f t s In the c a v i t y resonance during warm-up by a d j u s t i n g the height of the pulse 92 applied to the r e f l e c t o r e l e c t r o d e . The 545A o s c i l l o s c o p e provided at the same time a delayed t r i g g e r which was used to t r i g g e r the 502 o s c i l l o s o o p e a second time a f t e r the pulse had died out i n order to obtain the zero-l i n e of the exponential t r a c e . 3.2.5 Temperature Measurements From 1.4°K t o 4.2 °K, the c r y s t a l was Immersed l n l i q u i d helium and the temperature could be measured by monitoring the helium vapour pressure. Up to 2.3°K» an o i l manometer and above t h i s temperature a mercury manometer were used. In order to reach temperatures above 4.2°K, the l i q u i d helium l e v e l had to drop below the c a v i t y . I n t h i s case, the Faraday r o t a t i o n was used to measure the o r y s t a l temperature. This was only done l n the Eu r + : CaFg samples at higher concen-t r a t i o n s . The paramagnetic Faraday r o t a t i o n can be w r i t t e n as was used t o obtain temperatures T > h.2°K from the known and H v a l u e s . The accuracy of t h i s method to f i n d T depends on the magnitude of the Faraday r o t a t i o n of the sample. The de-crease l n Faraday r o t a t i o n towards hie-her temperatures decreases the accuracy of the T value. At the same time, however, the (3.9) has been determined f o r each specimen as a f u n c t i o n of H at K.2\ as shown i n Figure k.15 and it. 16 and the o p vs y curve 9 3 s i g n a l - t o - n o l s e r a t i o of the r e l a x a t i o n t r a c e d e t e r i o r a t e s and the Inaccuracy i n T Is of the same magnitude as the tempera-ture inaccuraoy. An estimate of these i n a c c u r a c i e s i s shown i n Fiffure 4.18 as e r r o r f l a g s at the high temperature point of c r y s t a l No. 2. This i s an extreme case. The s l g n a l - t o - n o i s e r a t i o i s e x c e p t i o n a l l y -good f o r such a high temperature owing to the long r e l a x a t i o n time. On the other hand, the Faraday r o t a t i o n i s r e l a t i v e l y small due to the low Eu z + content of sample No. 2. A s l i g h t temperature dependence of the f u n c t i o n yf » has been found f o r E u l + : CaF 2 between 1 . 5°K and 4.2°K. This i s presumably due to the fa c t that J i s not a p e r f e c t l y good quan-tum number, a f a c t which also gave r i s e to a d e v i a t i o n of from the B r i l l o u l n curve. In t h i s case, f would change w i t h g the r e l a t i v e p o p ulation i n the d i f f e r e n t l e v e l s of the m u l t i p l e t . The o v e r a l l s p l i t t i n g Is 2°K at the f i e l d employed and hence the populations change s t r o n g l y between 1 . 5 ° K and 4.2°K. At 4.2°K, the l e v e l s are almost e q u a l l y populated and the func-t i o n f should not change any more w i t h i n c r e a s i n g temperature. We checked the e r r o r Introduced i n the T-values f o r the u n l i k e l y case that f s t i l l changes w i t h temperature above 4.2°K at the same r a t e as between 1 . 5 ° K and 4.2 °K. I t was found that t h i s e r r o r was below the measuring inaccuracy f o r a l l values. In some cases, the temperature of the c r y s t a l between the X-point and 4.2 °K was a l s o meastired by means of the Faraday r o t a t i o n . This allowed a check of the r e l i a b i l i t y of the me-thod by monitoring simultaneously helium vapour pressure and Faraday r o t a t i o n . Once t h i s r e l i a b i l i t y was e s t a b l i s h e d , the c r y s t a l was warmed up slower than the helium surface. In t h i s way,no bubbles are formed l n the lower part of the c r y o s t a t even above the X - p o i n t . Such bubbles Introduce a huge noise i n the l i g h t s i g n a l which makes r e l a x a t i o n measurements impos-s i b l e . They do not always appear and i n some cases, we vrere able to avoid bubbling even when the c r y s t a l was at the same temperature as the helium s u r f a c e . However, t h i s seemed to be a matter of luck and was i n no way s y s t e m a t i c a l l y r e p r o d u c i b l e . 95 4 . EXPERIMENTAL RESULTS 4 . 1 Cerium E t h y l s u l p h a t e 4 . 1 . 1 Faraday R o t a t i o n In order to make sure the c r y s t a l was p r o p e r l y a l i g n e d , the Faraday r o t a t i o n was measured i n each run. The Faraday r o t a t i o n o f CeES has been s t u d i e d l n d e t a i l by Beoquerel, DeHaas and VanDenHandel ( 1 Q 3 8 ) u s i n g the green Hg-line (5 4 6 0 & ) . In Figure 4 . 1 , our measurements at 546o2 and 4 3 5 0 $ are p l o t t e d along w i t h Becquerel's r e s u l t s as a f u n c t i o n of H/T. The r o -t a t i o n a t the highest f i e l d s used l n our measurements have been normalized to f i t the Becquerel values. The very high r o t a t i o n of CeES can be used to deteot small temperature changes with great aocuraoy. The Faraday r o t a t i o n at a given temperature i s 0 = A tonk •^L ' k T with A » - 4 3 0 deg mm"1. This gives f o r our c r y s t a l s , whloh have a t y p i c a l l e n g t h of 8 mm : at 2 kOe and 1 . 5 °K ~ « 4 0 0 deg ( " K ) * * 1 and at 2 kOe and 4 . 2 ° K " 5 0 a e & ( ° K ^ " " 1 The angle of minimum l i g h t t r a n s m i s s i o n can e a s i l y be d e t e r -mined w i t h an accuraoy of 1 deg. This corresponds to an ac-curacy i n determining the temperature of 2 m°K at 1 . 5 ° K and of 2 0 m°K at 4 . 2 ° K . Figure 4.1 Paramagnetic Faraday r o t a t i o n of CeES J S [%i o o 0 , o .10 .OS .06 .oy i ? 1 I i i I o ? t • o o - i r Figure 4.2 Saturation versus magnetic f i e l d i n CeES —r~ f o - 4 9 8 4 . 1 . 2 Resonance Spectrum The paramagnetic resonance spectrum oan be determined b y measuring the s a t u r a t i o n of the spi n system due t o a microwave f i e l d as a funotion of the DC magnetlo f i e l d at a given temper-ature. I f Is the paramagnetic r o t a t i o n i n presence of the microwave f i e l d and the r o t a t i o n i n absenoe of miorowave, the s a t u r a t i o n Is given by equation ( 3.5 ) deriv e d i n chapter 3 . 2 : In CeES, the miorowave power required t o obta i n a measur-able change In r o t a t i o n i s very high and heats both the sample and the helium bath at the rate of about 2 m°K per seoond Inde-pendent of magnetic f i e l d . This temperature change a f f e c t s the paramagnetic Faraday r o t a t i o n so s t r o n g l y that i t was not pos-s i b l e to obta i n the paramagnetic s a t u r a t i o n by means of Faraday r o t a t i o n measurements. I t I s , however, p o s s i b l e to pulse the microwave power with long enough r e p e t i t i o n rates to keep the temperature of\the l i q -u i d helium oonstant, while s t i l l a t t a i n i n g the same s a t u r a t i o n values i n the pulse as with DC-microwavea. The s a t u r a t i o n v a l -ues oan be c a l c u l a t e d from the height of the p h o t o m u l t i p l i e r s i g n a l as described i n chapter 3*2, The obtained s a t u r a t i o n i s p l o t t e d i n Figure 4 . 2 as a f u n c t i o n of magnetic f i e l d together w i t h the expeoted resonance l i n e centers corresponding t o g | l ( r ) » 3 . " 8 g (j)m 1*0, A d e f i n i t e resonance i s seen to coinoide quite w e l l w i t h g * 3 . 8 and i s i d e n t i f i e d as due to the s a t u -99 r a t i o n of the lower doublet l±4> • The l i n e width Is about 1 , 0 0 0 gauss. This Is 50% h i g h e r than the value given by Bogle, Cooke and Whitley ( 1 9 5 1 ) meas-ured by conventional paramagnetic resonance absorption. S a t u r a t i o n of the higher doublet has not been observed with c e r t a i n t y . The few higher l y i n g points around 6.5 kgauss c o i n -cide w i t h the background w i t h i n measuring accuraoy. The most remarkable feature of the spectrum Is the almost constant "background s a t u r a t i o n " a t t a i n i n g roughly 1 / 3 of the resonanoe s a t u r a t i o n . I t does not seem po s s i b l e to e x p l a i n t h i s broad, s t r u c t u r e -l e s s background i n terms of paramagnetic s a t u r a t i o n . A s m a l l c r y s t a l from the same growing s o l u t i o n was used to check the paramagnetic resonance spectrum w i t h a conventional 3 4 G-c/s spectrometer. No microwave absorption was deteoted except f o r a few l i n e s around g * 2 due to i m p u r i t i e s . These could y i e l d to some cross s a t u r a t i o n , but t h i s would show a marked f i e l d dependence c h a r a c t e r i s t i c of the impurity l e v e l s and can not give a broad s t r u c t u r e l e s s band. I t i s , however, p o s s i b l e to e x p l a i n the observed back-ground as due to d i e l e o t r i c h e a t i n g of the c r y s t a l l a t t i c e . Suoh a heating would a f f e c t both Kramers doublets and only one spin-temperature Is r e q u i r e d i n t h i s case to describe the whole spin system as opposed to paramagnetio resonance s a t u r a t i o n of one doublet, where the sp i n temperatures i n both doublets may be d i f f e r e n t . R e f e r r i n g to chapter 2 . 3 , we can i d e n t i f y system A as the 100 l i q u i d helium surrounding the o r y s t a l , B as the o r y s t a l l a t t i c e and 0 as the spin system. The temperature increase A T q i n the spin system due to a long pulse i s given be equation ( 2.47) where W0 •» o o f f resonance. A ?7too) =-p*"U£ ^AB * 8 t n e t i m e constant associated with the thermal exohange be-tween o r y s t a l and helium-bath, Og Is the s p e o i f l o heat o f the c r y s t a l l a t t l o e and Wg l a the power fed to the l a t t i c e through d i e l e c t r i c heating. We can estimate the energy A U d i s s i p a t e d l n the o r y s t a l : U » o v A T T y p i c a l values f o r our o r y s t a l are T « 2 x 10-3 °K o v - 5 x 103 erg Ct)' 1 This g i v e s f o r a pulse of 10 mseo an average d i s s i p a t i o n of 0.1 mW. I f one uses the Kap i t z a r e s i s t a n c e p i c t u r e , assuming that spin and l a t t i c e are i n e q u i l i b r i u m at a l l times, equation (2.$D) gives the temperature r i s e a f t e r a long pulse RkW With 10 om 2degseo and 3 0.5 ora2 the power needed to cre-Joules ate the above temperature r i s e i s W - 3 A T « 0.1 Watt . Two experimental observations support the idea of s p i n heating v i a the l a t t i c e ; a) As w i l l be discussed i n more d e t a i l below, the l a t t i c e -bath r e l a x a t i o n time inoreases by approximately a f a c t o r of 15 l n passing from H e l l to Hel. From equation (2.47) f o l l o w s that AT b(°° ) should increase by the same amount. The microwave pulses a v a i l a b l e v/ere too short to determine A T q ( o ° ) a c c u r a t e l y above the X-p o i n t . E x t r a p o l a t i o n of the short heating ourves to t —» co shows, however, a minimum inoreaee of A T ( o o ) by a f a c t o r 10 over the value belov/ the X-p o i n t . b) Further evidence of a strong heat d i s s i p a t i o n l n the c r y s t a l due to miorowave r a d i a t i o n i s given by the f a c t that the bath temperature increases under the i n f l u e n c e of DC-mlorowave by an amount of about 10 m°K o f f resonance. This temperature increase i s very slow, i n i t i a l l y about 2 m°K per second and does not a f f e o t the pulsed measurements. The time constant associated with t h i s slow temperature increase i s a measure f o r the energy transport through the helium surface due to evaporation and Is thus a f u n c t i o n of the pumping speed. This slow heating e f f e c t oan be c l e a r l y observed on the o i l manometer as an inorease i n the helium vapour-pressure. i i.1.3 R e l a x a t i o n Times Figure 4,3 shows the dependence of the observed r e l a x a t i o n time T on the magnetic f i e l d at 1.48°K, Figure 4.4 shows the temperature dependence o f T from 1.4 to 4.2°K at the resonance f i e l d of the lower doublet. A sharp Inorease i n T i s observed i n passing from the H e l l region through the X - p o i n t . Above t h i s p o i n t , the r e l a x a t i o n traces can he r e s o l v e d i n t o two time constants. Both stay approximately constant from 2.2 to 4.2°K. I t has to be pointed out that the i n t e r p r e t a t i o n of the measured r e l a x a t i o n traces as composed of two exponentials i s not q u i t e unique. Some d i s t r i b u t i o n of r e l a x a t i o n times would probably f i t the data as w e l l . Also i n Figure 4.4, a point i s shown whloh represents the r e l a x a t i o n time of the c r y s t a l surrounded by helium gas at 4.2°K.r i s about three times s m a l l e r than the smaller of the two values which are found when the o r y s t a l Is immersed l n l i q -u i d helium at the same temperature. Only one r e l a x a t i o n time Is found i n d i c a t i n g that the r e l a x a t i o n process Is ex p o n e n t i a l . Figure 4.5 shows the temperature dependence of the r e l a x a -t i o n time below the X-point f o r two d i f f e r e n t f i e l d v a l u e s , together with some values obtained from non-resonant d i s p e r s i o n -absorption measurements by Van den Broek and Van der. Marel (l°63)» The l a t t e r seem to e x h i b i t a s l i g h t l y stronger temperature de-pendence at low magnetic f i e l d s than our measurements. In a l l cases, our r e l a x a t i o n traces show a very good exponential behav-i o u r from l.k°X up to the sudden r i s e at the X - p o i n t . D i s c o n t i n u i t i e s of the same type RP those at the \ - p o i n t have been observed at 1.5°K, when the l i q u i d helium l e v e l was allowed to drop below the c r y s t a l . With small microwave power, the c r y s t a l stays pt the same temperature and e x h i b i t s the same r e l a x a t i o n behaviour as when i t i s immersed l n H e l l . As soon as the microwave power exceeds /msec J I o 1 o o I I o I ? o | A ° o * o relaxation A / . e a / ; ^ i / 2 3 V 5" 6 /V {AOs J Figure 4.3 Relaxation time i n CeES: Magnetic f i e l d dependence o o [rn secj o o o o o o o o o o o o T o o o 2 U I • ' • He .OJ 3< 1 r - 1 — — 1 r-•2. \-f>oir>t 3 ^ Figure h.k R e l a x a t i o n time In CeES: Temperature dependence Figure h.5 R e l a x a t i o n time In CeES: Temperature dependence below the A - p o i n t a w e l l defined t h r e s h o l d vnlue, a very sharp Increase l n T i s observed, very rauch s i m i l a r to the one seen l n passing through the X - p o l n t . The e f f e c t Is r e v e r s i b l e . The f a s t r e l a x a t i o n Is again e s t a b l i s h e d , I f the average microwave power fed i n t o the c a v i t y i s decreased. This can be done by decreasing e i t h e r one of pulse power, r e p e t i t i o n r a t e or pulse l e n g t h . The t h r e s h o l d value of the average power seems to decrease w i t h time l n one s i n g l e run, i . e . w i t h i n c r e a s i n g distance between the c r y s t a l and the l i q u i d helium l e v e l . Table I R e l a x a t i o n time d i s c o n t i n u i t i e s below the X - p o i n t Observation No. MW pulse rep. r a t e mseo MW pulse-l e n g t h mseo MW_power atten u a t i o n db Average MW power at d i s c o n t i n u i t y i n db (0 d b = f u l l DC-MW) 1 100 8 0 - 11 2 200 8 0 -14 3 200 7 - 2 0 3.7 ( - 1 3 . 7 M - 1 8 . 2 ) 4 250 8 0.8 - 1.7 ( - 1 5 . 7 M - 1 6 . 7 ) 5 250 40 11.0 - 11 .9 19.0 - 19 .9 6 320 40 8.4 - 8.8 17.5 - 17 .9 1 0 7 In Tablf -'I . the microwave data f o r the d i f f e r e n t d i s c o n -t i n u i t i e s observed during the same run are assembled end. the corresponding r e l a x a t i o n times are p l o t t e d i n Figure 4.5 as a f u n c t i o n of average microtirave power ( i n db at t e n u a t i o n from f u l l •DC-microwave power). An attempt has been made to detect any change of T w i t h pulse l e n g t h or microwave power, while the c r y s t a l was immersed i n l i q u i d helium. A v a r i a t i o n of both miorowave pulse power and pulse l e n g t h by a f a c t o r 10 d i d not produce any change i n T below the \ - p o i n t . At 4.2 0K, a change i n power by 6 db and a change i n pulse length by a f a c t o r 10 gave the same r e l a x a t i o n times. Due to g e n e r a l l y weak s i g n a l s , i t was Impossible to extend these rather narrow l i m i t s and s t i l l pet measurable t r a c e s . For s e v e r a l t r a c e s , the time constant f o r the approach to s a t u r a t i o n , which i s b e t t e r c a l l e d the heating time constant, has been measured. Within the accuracy of our measurements, t h i s time oonstant i s the same as the corresponding T f o r r e l a x a t i o n . This i s l n agreement w i t h the s o l u t i o n of the rate equations treated i n ohapter 2.3. [msec] Figure 4.6 R e l a x a t i o n time d i s c o n t i n u i t i e s below the X-point 109 4 , 1.4- D i s c u s s i o n a) Thermodynamloal model In order to understand the r e l a x a t i o n behaviour i n CeES the model shown i n Figure 4.7 i s used to desoribe the c r y s t a l and i t s surroundings. The b a s i c assumption i s that there are at most f o u r systems i n s e r i e s , and that eaoh of these systems i s i n i n t e r n a l e q u i l i b r i u m desorlbed by a temperature. T]_ i s the usual s p i n - l a t t i c e r e l a x a t i o n time, i . e . the time constant measured i f the phonon system of the o r y s t a l which we c a l l the l a t t i c e i s a p e r f e c t h ^ a t r e s e r v o i r remaining always a t the bath temperature. 7^ i s the time constant a s s o c i a t e d w i t h the spreading out of the e x c i t a t i o n from the phonons "on speaking terms" (Van Vleck, 1941) to the other l a t t i o e modes. ^ " ^ J J and ^"pji a r e time constants associated w i t h the s p a t i a l d i f f u s i o n of the l a t t i o e e x c i t a t i o n away from the c r y s t a l . This time may be q u i t e d i f f e r -ent f o r the phonon modes on speaking terms w i t h the s p i n s , f o r which the l a t t i c e i s "opaque r t,than f o r the other l a t t i c e modes, slnoe the modes i n the v l o l n i t y of the paramagnetic resonanoe l i n e may have a very much sma l l e r group v e l o c i t y (Persioo et a l , 1 9 6 3 ) . The phonons i n t e r a c t i n g w i t h the spins are e f -f e c t i v e l y imprisoned by the paramagnetio i o n s . This problem has been considered l n great d e t a i l by Q-lordraaine and Nash (1Q65) and found to be analogous to the imprisonment of resonant pho-tons i n gases. Wg and W-^  are the energies fed to the s p i n and the l a t t i c e system r e s p e c t i v e l y during a miorowave pulse. The hot phonon system o o n s l s t s of the excess number of phonons over the e q u i l i b r i u m value at the l a t t i c e temperature. We have now to consider two d i f f e r e n t h eating and r e l a x a -t i o n processes, depending on whether the magnetic f i e l d i s at the resonance value (W g > V 7 - L ) or o f f resonance (W = 0 ) . These con s i d e r a t i o n s w i l l show that the model of Figure 4.7 can be s i m p l i f i e d by e l i m i n a t i o n of the hot phonon system. At resonance, the two doublets of the s p i n system are not i n e q u i l i b r i u m , the lower doublet being s a t u r a t e d and t h e r e f o r e at a higher temperature than the upper doublet. I t r e l a x e s v i a an Orbach process, thus forming two phonon spikes around A and A+S . The hot phonon system would, t h e r e f o r e , contain a nega-t i v e occupation number of phonons l n a band around A and a p o s i t i v e number of the same macnitude i n a band around A+S . Tp i s determined by the d i f f e r e n c e of the two occupation numbers and Tp = T^ f o r v a n i s h i n g d i f f e r e n c e , i . e . f o r the case, that no hot phonons are produced. The bottleneoked Orbaoh process has been s t u d i e d i n a s i m i l a r way by Stonehara ( 1 9 6 5 ) . The r e l a x a t i o n w i l l proceed v i a c r y s t a l l a t t i c e through frequency d i f f u s i o n , since i s long due to resonant reab-s o r p t i o n of the hot phonons as pointed out e a r l i e r . Knowing that T^ i s very s h o r t , we assume T g * T p and equation (2.43) gives f o r the observed r e l a x a t i o n time sinoe Cy y> c( and Tt « T . Figure 4.7 Thermodynamlcal model f o r CeES-He system 1 1 2 o s here i s the s p e c i f i c h°at of the doublet which i s at reso-nance. ' f l t h N • 1 . 6 x 1 0 2 1 spins cra""3 (lower doublet) <S = 0.4°K and T s - 1 . 5 ° K , we have f o r a c r y s t a l of 0 . 0 5 ora^ o s - 2 x 1 0 2 e r g ( " K ) ' 1 Off resonance, the hot phonon system cannot e x i s t , sinoe no part of the s p i n system i s sat u r a t e d . The population of a l l f o u r l e v e l s are rearranged by means of the s p i n - l a t t i c e i n t e r a c t i o n to f o l l o w the l a t t i c e temperature. This l a t t e r i s increased by l a t t i c e heating during the microwave pul s e . The r e l a x a t i o n time i s given by as before. But now, o g has to be taken as the s p e c i f i c heat of the whole spin system, not only the doublet which i s being saturated r s i n the previous case. o 8 can be taken from the s p e c i f i c heat measurements of Meyer and Smith ( 1 Q 5 9 ) snd i s at 1 . 5 ° K f o r our c r y s t a l c B = 2 x 1 0 3 erg(°K)- 1 This i s ten times l a r g e r than the value used p r e v i o u s l y f o r the f i e l d on resonance. VJe expect, t h e r e f o r e , a d i s t i n c t l y d i f f e r -ent value of the r e l a x a t i o n time at resonance than o f f resonanoe. tH 113 The experimental r e s u l t s show, however, a smooth f i e l d dependence of T . This i n d i c a t e s that the enerary d i s t r i b u t i o n i n the s p i n and l a t t i c e system i s the same f o r heating as f o r s a t u r a t i o n and that i n no case are hot phonons produoed. This conclusion i s confirmed by experiments of Dransfeld (1958), Shiren and Tucker (1959) and Feughnan and Strandberg (1961). These authors have t r i e d without success to detect hot phonons produced i n b o t t l e n e c k s i t u a t i o n s . We conclude, t h e r e f o r e , from the i n s e n s i t i v i t y of the r e -l a x a t i o n times to magnetic f i e l d , that at a l l times, a l l f o u r l e v e l s of the s p i n systems as w e l l as the systems of l a t t i c e o s c i l l a t o r s are i n e q u i l i b r i u m , i . e . determined by a s i n g l e temperature and the r e l a x a t i o n time f o r the spin system v i a l a t t i o e to the bath Is given by T = ±- T where Z~ 1 H i s the time oonstant f o r absorption of a l a t t i c e phonon by the bath. The r e l a x a t i o n problem Is now reduoed to the problem of s p a t i a l d i f f u s i o n of thermal energy through the c r y s t a l l a t -t i o e , across the boundary and i n t o the surrounding helium. The observed strong dependence of ~T on the environment sug-gests t h a t the l i m i t i n g f a c t o r i n t h i s energy exchange i s not the s p a t i a l d i f f u s i o n i n the c r y s t a l . Cracks l n the c r y s t a l , i n t o which H e l l might penetrate, are able to e x p l a i n o n l y the d i s c o n t i n u i t i e s i n r e l a x a t i o n time at or below the X - p o i n t . In that region s u p e r f l u i d h e l -114 ium might shorten out a thermal d i f f u s i o n b o t tleneck i n s i d e the c r y s t a l . However, cracks cannot e x p l a i n the decrease of the r e l a x a t i o n time i n gaseous helium at 4.2 °K to nearly the value l n H e l l . We suggest, t h e r e f o r e , t h a t the l i m i t i n g f a c t o r i n thermal exchange i s e i t h e r the boundary r e s i s t a n c e between the c r y s t a l and the surrounding helium bath or the l i m i t e d d i f f u s i o n i n the helium i t s e l f . This i s supported by the f a c t that K. C e , the thermal d i f f u s i o n oonstant l n CeES, i s orders of magnitude higher than the d i f f u s i o n constants of helium gas or H e l . b) T below the X -point Since the thermal c o n d u c t i v i t y of H e l l i s abnormally h i g h (the d i f f u s i o n constant K- IS of the order of 1 0 3 om 2seo~^), we can regard the temperature i n the helium bath below the \ -point as constant. We have, t h e r e f o r e , the s i t u a t i o n t r e ated i n chapter 2.3 with the Kap i t z a boundary r e s i s t a n c e as l i m i t i n g f a c t o r . The r e l a x a t i o n i n t h i s case i s exponential and T K i s independent of pulse l e n g t h or power, as confirmed by our meas-urements. The observed time oonstant Ty- i s according to equation I t should be the same f o r heating and r e l a x a t i o n , which i s con-s i s t e n t v/ith our experimental r e s u l t s . Equation (2.62) can be used to determine the K a p i t z a boun-(2.62) 1 1 5 dary r e s i s t a n c e Rj. from our measured r e l a x a t i o n times, i f the s p e o i f i c heat i s known. We have c a l c u l a t e d c as the sum of or the Schottky s p e c i f i c heat c g and the l a t t i c e s p e o i f i c heat c-^ o g has been c a l c u l a t e d from the p a r t i t i o n f u n c t i o n Z by means of the formula K where R » 1 . 9 8 7 oal(Mole T O " 1 , Z • 2— exp(-kT and the sum Is taken over a l l f o u r populated l e v e l s w i t h A^ determined, by g u ( £ ) = 3 . 8 , g S 1 . 0 and 6 . 9 5 °K. c-^  has been taken as the d i f f e r e n c e between the e x p e r i -mentally determined t o t a l s p e c i f i c h°at i n zero f i e l d as meas-ured by Meyer and Smith ( 1 9 5 9 ) and the t h e o r e t i c a l Schottky value at zero f i e l d , o^ gives o n l y a small c o n t r i b u t i o n to o at l i q u i d helium temperatures, cr Figure 4 . 8 shows the temperature dependence of o Q R / R f o r two d i f f e r e n t f i e l d s and Figure' 4 . 9 Is a p l o t of the f i e l d dependence of o Q R / R at two d i f f e r e n t temperatures. The K a p i t z a r e s i s t a n c e Rjr has been obtained from these values f o r c c r and from the experimental r e l a x a t i o n times. The r e s u l t s are p l o t t e d i n Figure 4 . 1 0 f o r the temperature dependence and i n Figure 4 . 1 1 f o r the f i e l d dependence. The obtained R J J can be described by the r e l a t i o n R, s AT""n crc^degseo * Joule with A - 30 and n » 2 . 4 . Rg i s independent of f i e l d , The maximum d e v i a t i o n of R^ . from the mean value i n Figure 4.11 l a 10%. The accuracy of the values i s l i m i t e d by the inaccuracy of the r e l a x a t i o n time measurements. The obtained r e s u l t s agree q u i t e w e l l with known values from other substances. c) T above the X -point Non-exponential r e l a x a t i o n above the X - p o i n t suggests that s p a t i a l conduction r a t h e r than t r a n s f e r through the i n t e r -face i s the l i m i t i n g process. This i s a l s o supported by the f o l -lowing argument: The r e s l s t a n o e due to a boundary d i s c o n t i n u i t y can be r e -garded as an a c o u s t i c mismatch f o r the phonons i n c i d e n t on the boundary. The a c o u s t i c impedance Z In a m a t e r i a l i s gi v e n by the product of d e n s i t y <^  and v e l o c i t y of sound v The transmi s s i o n c o e f f i c i e n t T^ 2 at a d i s c o n t i n u i t y i n t e r f a c e i s given i n analogy to transmis s i o n l i n e theory by T = 2 Z 2 T 1 2 = y _T7 Z 2 + Z l The a c o u s t i c impedances f o r the d i f f e r e n t media are CeES: Z = 3.7 x 10 5 gem""2 s e c " 1 a H e l l , 1.4°K: Z a « 3-4 x lo3 » » H e l , 4.2°K: Z R = 2.2 x 1C-3 « » He gas: Z e «. 1 0 2 which g i v e s the f o l l o w i n g t r a n s m i s s i o n c o e f f i c i e n t s : F i g u r e 4.9 S p e c i f i c heat of CeES as a f u n c t i o n of magnetic f i e l d 120 K L jouce J O O T~° ~8 ° ° o o o o o —r 1 ( 1 ' ' / z 3 V S 6 7 Figure 4.11 Kapltza resistance of CeES-Hell: F i e l d dependence CeEs Hel : T 1 2 = 1.2 x 10~ 2 CeES He gas : T 1 2 = 5.5 x 10~** From these t r a n s m i s s i o n c o e f f i c i e n t s I t i s seen that the boundary r e s i s t a n c e and therefore the r e l a x a t i o n time should be considerably longer f o r the cr y s t a l - H e gas system as com-pared to the c r y s t a l - l i q u i d He system. This i s i n di s a g r e e -ment w i t h the measurements at 4.2 °K which show a f a s t e r r e l a x a -t i o n f o r the crystal-He gas system. I t has been shown by M i l l s (1964) that the tra n s m i s s i o n can be considerably enhanced i n t h e presence of a porous l a y e r on the surface of the c r y s t a l , whloh w i l l serve as an Impedance transformer. This e f f e c t Is c e r t a i n l y present l n the e t h y l s u l -phates, which dehydrate very e a s i l y . I t would, however, only reduce the magnitude f o r the boundary r e s i s t a n c e without chang-in g the s i g n of ( T c e - * H e l ) " (TCe-*He gas) a n d therefore cannot give a smaller boundary r e s i s t a n c e f o r the CeES - He gas I n t e r -face as compared to the CeES - Hel boundary. The s p a t i a l d i f f u s i o n i s governed by the d i f f u s i o n equation m K V T dT The r e l e v a n t parameter i s the d i f f u s i o n constant K where K i s the thermal o o n d u o t i v i t y 122 The values f o r K_ are: He gss : K. » 9 x l O " 2 P m seo Hel : vc • 4 .5 x 1 0 " ^ seo CeES : * ~ 70 cm£ seo For the simplest oase of l i n e a r heat flow l n x > 0 , a p o s s i b l e s o l u t i o n of the d i f f u s i o n equation Is T ( x , t ) - A f ( x ) with f ( x ) s a t i s f y i n g the equation In t h i s oase, the thermal r e l a x a t i o n time i s given by r = . In g e n e r a l , the boundary c o n d i t i o n s are not harmonic and the r e l a x a t i o n i s not exponential but has some complicated time dependence. In a l l cases, however, the thermal r e l a x a t i o n i s r e l a t e d t o the d i f f u s i o n constant i n such a way that a higher K_ means f a s t e r r e l a x a t i o n . The slow r e l a x a t i o n i n Hel can there-fore be a t t r i b u t e d to a small value of . In He gas, K- i s considerably higher and at the same time the mismatch at the boundary i s increased, such that the boun-dary r e s i s t a n c e oould again play a r o l e . This would be i n d i c a -ted by a s i n g l e e x p o n e n t i a l . The r e l a x a t i o n t r a c e s obtained at 4 . 2 °K i n the gas are not good enough to deoide t h i s question d e f i n i t e l y . An attempt to draw q u a n t i t a t i v e conclusions from the ex-perimental r e l a x a t i o n behaviour would Involve the s o l u t i o n of the d i f f u s i o n equation f o r the c r y s t a l - b a t h system, t a k i n g into acoount the boundary r e s i s t a n c e . The unknown or not w e l l known parameters of t h i s s o l u t i o n would have to be adjusted i n order to f i t the experimental r e l a x a t i o n t r a c e s . The e f f o r t i n v o l v e d i n such c a l c u l a t i o n s i s c o n s i d e r a b l e , even f o r a. r a t h e r s i m p l i f i e d model. No general r e s u l t s are ob-tained and numerical methods have to be introduced at an e a r l y stage of the c a l c u l a t i o n . Moreover, i t i s doubtful i f a unique f i t t o the experimental data could be obtained even from a s o l u -t i o n c o n t a i n i n g a l l r e l e v a n t parameters, since too many are un-known w i t h i n a wide range of v a l u e s . The c a l c u l a t i o n s have, t h e r e f o r e , not, been c a r r i e d through. A few words might be said -with regard to the d i s c o n t i n u -i t i e s i n r below the X-point when the 1'quld H e l l l e v e l was below the c r y s t a l . A l l the parts i n the c r y o s t a t with a temper-ature below the X - p o i n t are covered by a f i l m of l i q u i d H e l l . The s u p e r f l u l d property of H e l l 8llows a considerable amount of heat to be conducted and t h e r e f o r e keeps the c r y s t a l at the temperature of the b u l k l i q u i d . I f , however, the heat, pulses are too b i g such t h a t the thermal flow i n the f i l m exceeds a c r i t i c a l v a l u e , the f i l m i s not able to conduct the heat away because the thermal behaviour of the f i l m becomes that of a normal l i q u i d and the r e l a x a t i o n behaviour i s the same as that observed l n H e l . 124 d) A p p l i c a t i o n to PrES The r e l a x a t i o n behaviour of PrES shows e f f o o t s s i m i l a r to the ones observed l n CeES. We have, t h e r e f o r e , re-examined the r e l a x a t i o n times obtained i n PrES. PrES has a very strong e p i n - l a t t i o e I n t e r a c t i o n and i t s s p i n - l a t t i c e r e l a x a t i o n times ere very short. Larson and J e f f r i e s ( 1 9 ^ 6 ) have measured T-j_ i n d i l u t e Pr:LaES and f i n d a bottlenecked d i r e c t process and an Orbaoh process v i a the e x c i t e d s i n g l e t at 12 cm"1. Tj_ i s below 100 yw-sec f o r temper-atures down to 1.4 °K. The observed r e l a x a t i o n times i n the concentrated PrES are much longer and e x h i b i t a completely d i f f e r e n t temperature and f i e l d dependence than would be expected from s p i n - l a t t i o e r e -l a x a t i o n theory ( G r i f f i t h s , 1965). A d i s c o n t i n u i t y was observed at the A-point very s i m i l a r to the one i n CeES. G r i f f i t h s suggested that above the X - p o i n t , the l i m i t e d heat c o n d u c t i v i t y of the surrounding helium i s r e s p o n s i b l e f o r the b o t t l e n e c k . Below the X-point, G r i f f i t h s proposed an explanation of the observed magnetic f i e l d dependence using the thermal c o n d u c t i -v i t y of the c r y s t a l which i s s t r o n g l y f i e l d dependent as a r e -s u l t of phonon s c a t t e r i n g by the paramagnetic ions. We xrould l i k e to show that temperature and f i e l d dependence of the r e l a x a t i o n times In PrES below the X - p o i n t can a l s o be explained as a r i s i n g from the K a p l t z a boundary r e s i s t a n c e . We b e l i e v e t h i s e xplanation t o be more l i k e l y , s ince s p a t i a l con-duction leads to non-exponential time dependence and i n both cases (CeES and PrES) no d e v i a t i o n from the exponential behaviour of the r e l a x a t i o n t r a c e s below the X - p o i n t has been observed. As i n the case of cerium, we c a l o u l a t e the Kap i t z a r e s i s t -ance R^ from the formula The s p e o i f i c heat o f PrES between 1.4°K and 2 . 1 ^ i s the sum of three c o n t r i b u t i o n s : the l a t t i c e s p e c i f i c heat, which i s close to the s p e o i f i c heat of NdES; the low temperature t a l l of the Schottky anomaly a r i s i n g from the ex c i t e d s i n g l e t s t a t e at 16 °K and the h i g h temperature t a i l of the Sohottky peak a r i s i n g from the ground s t a t e s p l i t t i n s r . The two former are independent of magnetic f i e l d , w h i l e the l a t t e r i s very s e n s i t i v e to an ex t e r -na l magnetio f i e l d . Such a f i e l d inoreases the s p l i t t i n g of the ground s t a t e and s h i f t s the Schottky peak towards h i g h e r temperatures, the s p e c i f i c heat Increases s t r o n g l y with mag-n e t i c f i e l d as shown i n Figure 4.12. A s i m i l a r increase i n ~c i s observed, l e a v i n g the K a p i t z a r e s i s t a n c e f i e l d - i n d e p e n d e n t . Just below the X - p o i n t , the s p e o i f l o heat i s mainly due to the higher Schottky peak which i s p r a c t i c a l l y f i e l d Independent. In f a c t , T loses i t s f i e l d dependence a l s o at 2.15°K ( G r i f f i t h s , 1965). Figure 4.13 shows as a f u n c t i o n of H, as c a l c u l a t e d from the r -values i n Figure 3 of G r i f f i t h s and G l a t t l i (1065). There seems to be a weak f i e l d dependence i n R R. This e f f e c t i s not n e c e s s a r i l y genuine and might be due t o a s l i g h t e r r o r i n c a l c u l a t i n g c e r . There I s , l n f a c t , an u n c e r t a i n t y about 126 the values of c c r s i n c e n e i t h e r the a c t u a l d i s t r i b u t i o n of the c r y s t a l f i e l d d i s t o r t i o n s nor an accurate g-value are known f o r PrES. In accordance w i t h the s t r u c t u r e l e s s f l a t resonance found by G r i f f i t h s , we have assumed a r e c t a n g u l a r l i n e shape i n the c a l c u l a t i o n s f o r c o r # Figure 4.14 shows R^ at 6 kgauss as a f u n c t i o n of tempera-tu r e . Reasonably c l o s e agreement w i t h the va l u e s f o r CeES i s obtained. This i s t o be expected, since the ac o u s t i c mismatch should not change much from one ES t o the other. 12? Figure 4.12 S p e c i f i c heat of PrES as a f u n c t i o n of temperature ° o o ° o o o 0 o o o I ' I I I I I I V ' 7 8 3 /o // Figure 4.13 Kapltza reelstance of PrE3-HeII: F i e l d dependence 130 4.2 Europium Doped Calcium F l u o r i d e 4.2.1 Faraday R o t a t i o n The ohange of the angle of p o l a r i z a t i o n w i t h magnetic f i e l d has been measured i n order to cheok that the Faraday r o t a t i o n was p r o p o r t i o n a l to the magnetization. The measured r o t a t i o n has been correoted i n each case f o r the Faraday r o t a t i o n due to the g l a s s w a l l s of the dewar and f o r the diamagnetic r o t a t i o n of the o r y s t a l . The r e s u l t i n g paramagnetic r o t a t i o n has been nor-malized to match the B r i l l o u l n f u n c t i o n B x at the highest z values of H/T. Figure 4.15 shows the normalized paramagnetic r o t a t i o n of c r y s t a l No. 1 as a f u n c t i o n of H/T using the purple, green and yellow l i n e of the H g - l l g h t . No systematic d i f f e r e n c e exceeding the measuring accuracy i s observed and t h i s i n d i c a t e s that even the purple l i g h t i s s t i l l f a r enough from the opticr-1 t r a n s i t i o n causing the Faraday r o t a t i o n to maintain p r o p o r t i o n a l i t y between r o t a t i o n and magnetization. The s l i g h t d e v i a t i o n from the B r i l l o u l n curve observed above 3 k0e/°K i s probably not due to a v i o l a t i o n of t h i s proportional i t y but a r i s e s r a t h e r from a d e v i a t i o n of the magnetization from the f r e e - i o n value. The measurements from the other c r y s t a l s , p l o t t e d i n Figure 4 . 1 6 , confirm the r e s u l t s from c r y s t a l No. 1. The s a t u r a t i o n r o t a t i o n per mm. thickness ^ Is shox^n i n Table I I f o r the d i f f e r e n t o r y s t a l specimens. The <^ oo are pro-p o r t i o n a l to the Eu7""*" co n c e n t r a t i o n , i f the e f f e c t of concentra-131 t l o n dependence of the o p t i c a l speotrum on the Faraday r o t a t i o n i s n e g l i g i b l e . Suoh an e f f e o t would be strongest f o r l i g h t frequencies olose to the o p t i c a l t r a n s i t i o n s , as can be seen from equation (2.87). We found t h a t the r a t i o of <j„ f o r d i f -f e r e nt concentrations does not depend on the l i g h t frequency. This supports the assumption t h a t the ^ are p r o p o r t i o n a l to the E u x + concentration and, t h e r e f o r e , a d i r e c t measure f o r the r e l a t i v e c o n c e n t r a t i o n of the d i f f e r e n t specimens. The molar oonoentration of europium ions added to the CaF2~melt i n the growing prooess i s given i n Table I I . There Is a marked disagreement between the added Eu oonoentration and the r e l a t i v e Eu2""1" concentrations obtained from r o t a t i o n measure-ments. I t i s known that europium enters the f l u o r i d e l a t t i c e i n both the t r i v a l e n t and the d i v a l e n t s t a t e . (Shen, 196k),. The E u , + ion has a 7 F Q s i n g l e t groundstate and gives no paramagne-t l o c o n t r i b u t i o n to the Faraday r o t a t i o n . The r a t i o of E u 2 + to Eu^ "*" i s very s e n s i t i v e to the crowing c o n d i t i o n s . I n c r y s t a l No. 1, europium was added i n the form of Eu 2 0^ , while i n No. 3 the same molar amount was added i n the form of EuF^. As a r e -s u l t , No. 1 has a ten times h i g h e r Eu*"+ c o n c e n t r a t i o n than No.3« Assuming that the lowest concentration ( o r y s t a l No. k) i s mostly Eu , we have ca l o u l a t e d values f o r both the Eu and Eu* + oonoentrations. These values are a l s o given i n Table I I and have to be regarded as r a t h e r rough approximations. I t Is seen that two of the h e a v i l y doped c r y s t a l s have a very small 132 Figure 4.15 Faraday r o t a t i o n of Eu :CaF c r y s t a l No. 1 133 c / / + 5) -| 1 1 1 1 1 r I 2. 3 V 6 7 Figure 4 . 1 6 Faraday r o t a t i o n of Eu x +:CaF 2 o r y e t a l s No. 2 - 5 Specimen No. Satur a t i o n r o t a t i o n ^ [deg mm""^7 Eu To t a l % % 3 + Eu % 4 3 5 0 X 5 4 6 0 t 5790 A* 1 1 3 7 24.4 18.8 2 0 . 3 1 . 7 2 27.8 0.2 0 . 0 6 0.14 3 16.2 2 . 5 2 0 . 0 3 5 2 4 9.6 0.02 0 . 0 2 5 8.8 0.8 0 . 0 1 5 0.8 Table I I Sa t u r a t i o n r o t a t i o n and con c e n t r a t i o n of Eu:CaF2 speolmens Eu2""*" content. The pood f i t of t h e i r paramagnetic r o t a t i o n to 3+ the B r i l l o u l n ourve supports the assumption that the Eu Ion does not c o n t r i b u t e t o the paramagnetic r o t a t i o n . From the s a t u r a t i o n r o t a t i o n s and f o r two d i f f e r -ent l i g h t frequencies co, and c o 2 , we oan c a l c u l a t e the average t r a n s i t i o n frequenoy causing the Faraday r o t a t i o n . From equa-t i o n (2.87) and (2.89) we get c3^ X c o 1_ 1 c o - c o / 0 x 2. = ',2 The values f o r 9 ^ . from Table I I give £3" ^ 2 3 900 cm - 1 or 41808 . This agrees w e l l w i t h the o p t i c a l spectrum, which shows a strong absorption band at t i o n s . (Low, i 9 6 0 ) 24 200 cm"*1 a t t r i b u t e d to 8S — 6P t r a n s i -4.2.2 Resonanoe Spectrum The s a t u r a t i o n s of the s p i n system as a funotlon of the e x t e r n a l magnetic f i e l d was determined i n the same way as i n the case of CeES by measuring the paramagnetic r o t a t i o n f o r a given f i e l d i n presenoe of microwaves c> and l n absenoe of microwaves 137 Since the minimum step p o s s i b l e xvith our rheostat I s about 30 gauss, t h i s point-by-point procedure i s not capable of r e s o l -v i n g the hyperfine s t r u c t u r e . I t was, however, p o s s i b l e to ob-t a i n an envelope of the hyperfine l i n e s corresponding t o a given e l e c t r o n i c t r a n s i t i o n . The r e s u l t s obtained from c r y s t a l No, 1 at 1.5°K and 4.2°K are shown i n Figure 4.17. In both cases, the magnetlo f i e l d was along a cube edge [lOOj. The t r a n s i t i o n s ±-|-*±£ and ±T-*-T are too c l o s e and ere not resolv e d . The o v e r a l l width of any one e l e o t r o n i o t r a n s i t i o n agrees w i t h the t o t a l h y p e r f i n e s p l i t t i n g as measured by Ryt e r (1957). The o v e r a l l s p l i t t i n g of the S x groundstate i s a few at the mapnetic f i e l d s a v a i l a b l e t o us. The upper l e v e l s should, t h e r e f o r e , be ap p r e c i a b l y depopulated as one cools down from 4.2°K to 1.5 °K and the corresponding t r a n s i t i o n should become weaker. This i s observed f o r the l i n e at the h i g h - f i e l d side of the spectrum. We I d e n t i f y , t h e r e f o r e , t h i s l i n e as being the + f -» + -f t r a n s i t i o n . This corresponds t o a negative value of b^ i n equation (2.64). The s l i g h t d i f f e r e n c e l n the resonance oenter-values be-tween the two sp e c t r a I s not genuine. I t can be acoounted f o r by a s h i f t i n the resonance frequency of the oavit y and by a s l i g h t d i f f e r e n c e i n alignment of the c r y s t a l with respect to the e x t e r n a l magnetic f i e l d . 4.2.3 R e l a x a t i o n Times The temperature dependence of the spin-bath r e l a x a t i o n 1 3 8 time T has been measured In the speolmens No. 1 to 5 from 1.5°K up t o the highest temperatures which s t i l l gave a r e a -sonable s i g n a l - t o noise r a t i o . The r e s u l t s are shown i n F i g -ure 4.18, Most p o i n t s shown are averages from sev e r a l measure-ments. The measurements have been done on the c e n t e r l i n e x and w i t h both DC magnetic f i e l d and l i g h t d i r e o t l o n p a r a l l e l to the f o u r - f o l d cubic a x i s . These conditions assure maximum d i s -tance between the i n v e s t i g a t e d l i n e and other e l e o t r o n i o t r a n s i -t i o n s and therefore minimize c r o s s - r e l a x a t i o n s between d i f f e r e n t e l e c t r o n i c t r a n s i t i o n s . I t i s seen that the absolute value as w e l l as the f \ i n c t l o n a l dependence of the r e l a x a t i o n r a t e T~' depend? on the specimen. The d i f f e r e n t specimens have a l l n e a r l y the same shape and dimen-sions but they d i f f e r i n t h e i r europium concentrations as shown i n Table I I . The experimental points at the lowest temperatures can i n each case be f i t t e d w ith a s t r a i g h t l i n e w i t h the f o l l o w i n g slopes: C r y s t a l No. 1 2 3 4 5 Slope 1 . 9 6 1.22 1 . 8 8 1 . 0 7 1.11 Max. Dev i a t i o n 0.12 0 . 0 3 0 . 1 1 0 . 1 0 0 . 1 0 The mean value of the slope and I t s p o s s i b l e d e v i a t i o n have been-determined g r a p h i c a l l y . At h i g h e r temperatures, d e v i a t i o n from the s t r a i g h t l i n e occurs and a smooth ourve has been drawn 139 through t h e experimental p o i n t s . The e r r o r f l a g s attached to the high-temperature point of specimen No. 2 are the maximum e r r o r s due to measuring in a c c u ^ racy. At lower temperatures, t h e accuracy i s considerably higher and, i n g e n e r a l , does not exceed the s i z e of the c i r c l e s as drawn around the po i n t s i n Figure 4.18. The r e l a x a t i o n time has been found to be constant across the whole +T-»-f t r a n s i t i o n . This i s Important since the cavity-resonance s h i f t s by about 20Mc/s d u r i n g the warm-up from 1.5°K to 4.2°K, and i t i s impossible w i t h our rough f i e l d r e g u l a t i o n t o c o r r e c t f o r t h i s change. A s h i f t of t h i s s i z e i s , on the other hand, comparable to the spacing between the hyperfine components and t h e r e f o r e we cannot avoid s i t t i n g at d i f f e r e n t spots on the -f -* ~i U n e f o r d i f f e r e n t temperatures during the same run. I t i s al s o impossible to induce the microwave t r a n s i t i o n e x a c t l y at the same spot i n two d i f f e r e n t Puns. No change l n absolute value or temperature dependence of tr has been •"'eteoted l n d i f f e r e n t runs on the same c r y s t a l and the srood r e p r o d u c i b i -l i t y of the data are thus a f u r t h e r c o n f i r m a t i o n that the temper-ature dependences of 7" are genuine p r o p e r t i e s of the l i n e . I t has f u r t h e r been checked as to whether T was dependent on microwave power or pulse l e n g t h . No ohange In T has been observed by var y i n g the miorowave power up t o 40 db and the pulse l e n g t h by a f a o t o r of 30. In no case has c l e a r evidence been found f o r nonexponential behaviour of the r e l a x a t i o n traoe. 140 ' ' — 1 1 1 1 1 1 1 1 1 . ' 2 . S IQ a o Figure 4.18 R e l a x a t i o n r a t e of Eu 2 >:CaF 2 as a f u n c t i o n o f temperature f o r d i f f e r e n t concentrations 142 A t y p i c a l r e l a x a t i o n t r a c e i a shown i n Figure 4 . 19 together w i t h i t s aemllogarithralo p l o t . 4.2.4 Disousaion The oonoentration dependence of the r e l a x a t i o n times cannot be explained by a phonon-bottleneok, although the temperature dependence f o r the specimens 1 and 3 Is very n e a r l y T'1* DT2. This temperature dependence can r e s u l t from the l i n e a r i z e d rate equations f o r the d i r e c t process i n the presence of a weak phonon-bott l e n e c k (Scott and J e f f r i e s , 1962). The oonstant D should, however, decrease w i t h i n c r e a s i n g c o n c e n t r a t i o n i n c o n t r a d i c t i o n w i t h our r e s u l t s . No bottleneck would be expected l n t h i s case, anyway. The c o n d i t i o n f o r a phonon bottleneck has been found e a r l i e r to be (equation (2.51)) C a AB ^~ L sc where o^ and Cg are the heat c a p a c i t i e s of s p i n system and l a t -t i c e o s c i l l a t o r s r e s p e c t i v e l y , TAB i s the phonon l i f e t i m e and T a c i s the s p i n - l a t t i o e r e l a x a t i o n time T-^ . I n our case « 103 i n the d i r e c t prooess region. I f one assumes a phonon l i f e t i m e of r~AB -a* lCT^sec. (Faughnan and Strandberg, I 9 6 I ) then 77 ~£~An ^ 10" J. This i s two orders of magnitude smaller than our observed r e l a x a t i o n times. In our measurements, there seem to be, i n a d d i t i o n to the conventional s p i n - l a t t l o e r e l a x a t i o n , one or sev e r a l oompetlng 143 processes whioh become i n c r e a s i n g l y important at high concentra-t i o n s . I f the t o t a l r e l a x a t i o n r a t e due to these processes i s X, we can w r i t e f o r the observed r e l a x a t i o n r a t e approximately The s p l n - l a t t l c e r e l a x a t i o n rate has the form T~' = AT + BTr as shown i n Chapter 2.4. We do not know anything about A . The only reasonable assumption i s that A does not decrease w i t h i n c r e a s i n g temperature. This y i e l d s the f o l l o w i n g upper l i m i t s f o r the c o e f f i c i e n t s A and B of the s p i n - l a t t l o e r e l a x a -t i o n : A -2.5 B » 5 x io-5 The r e s u l t i n g temperature dependence of T~' Is shown i n F i g -ure 4.18. Table I I I i s a compilation of r e s u l t s obtained by d i f f e r e n t workers. In the f i r s t column, the r e s u l t s obtained by Huang (1965) are shown. He used the p u l s e - s a t u r a t i o n recovery technique. The measurements were done at X-band on the 7 t r a n s i t i o n w i t h H II [100]. The speolmens used had 0.0047$, 0.0074$ and 0.18$ E u x + c o n c e n t r a t i o n . The second oolumn shows the t h e o r e t l o a l estimate obtained f o r T^ by Huang. The f o u r t h oolumn shows the estimate of the s t r e n g t h of the d i r e c t process deduced from u l t r a -sonio absorption measurements by Dobrow and Browne (1962). 144 Huang Huang Our measurements Dobrow (Pulae s a t u r a t i o n ) ( t h e o r e t i c a l estimate) (Faraday r o t a t i o n ) ( u l t r a s o n l o absorption) A 12 3 2.5 0.1 B 5.3 x l o - 1 1 4 x 10'k 5 x 10~ 5 Table I I I C o e f f i c i e n t s f o r the s p i n - l a t t l o e r e l a x a t i o n In Eu :CaF2 The r e s u l t s are seen to disagree q u i t e c o n s i d e r a b l y . I t has to be noted that the three experimental methods measure d i f f e r e n t p h y s l o a l events. With the pdlse s a t u r a t i o n method, the recovery of the para-magnetic absorption l i n e a f t e r a strong s a t u r a t i n g pulse i s ob-served u s i n g a very low monitoring microwave poiirer l e v e l . The mlorowave absorption i s p r o p o r t i o n a l to the population d i f f e r -ence of the two l e v e l s between whloh the resonance takes place. The measured q u a n t i t y I s , t h e r e f o r e , a population d i f f e r e n c e be-tween two l e v e l s . S p i n - l a t t i c e r e l a x a t i o n i s not the o n l y pos-s i b i l i t y f o r reoovery of the absorption s i g n a l . Other p o s s i b i l -i t i e s are ( c f . ohapter 2.4.5): a) c r o s s - r e l a x a t i o n between d i f f e r e n t h y p e r f i n e l i n e s w i t h i n the same e l e c t r o n i c t r a n s i t i o n , or frequenoy d i f f u s i o n w i t h i n an inhomogeneously broadened resonance l i n e i n cases where the s a t -u r a t i n g pulse "PburHB a hol e " i n the l i n e . b) } c r o s s - r e l a x a t i o n between d i f f e r e n t e l e c t r o n i c t r a n s i t i o n s . c) .. c r o s s - r e l a x a t i o n to other systems l i k e i m p u r i t i e s or 1 4 5 exchange coupled c l u s t e r s of two or more paramagnetio Ions. Our method of monitoring the change i n paramagnetic r o t a -t i o n a f t e r a microwave s a t u r a t i n g pulse measures the time depen-denoe of the magnetization due to the Eu -Ions (to the extent that they only c o n t r i b u t e to the paramagnetio Faraday r o t a t i o n ) . The processes a) and b) above do not c o n t r i b u t e to the decay, I f they conserve angular momentum. The measured r e l a x a t i o n r a t e s are, however, enhanced by prooess o) to the same extent as the p u l s e - s a t u r a t i o n measurements. The u l t r a s o n i c absorption measurements are not a f f e c t e d by any of the above mentioned processes. Here, the strength of the o r b i t - l a t t i c e i n t e r a c t i o n i s determined by measuring the absorp-t i o n c o e f f i c i e n t of microwave-phonons. Processes b) have been observed by Huang to i n f l u e n c e the r e l a x a t i o n times f o r pulses s h o r t e r than 0.5 msec. For longer pulses, the whole e l e c t r o n i c t r a n s i t i o n can be considered as homogeneously saturated and the i n f l u e n c e of prooesses b) can be neglected. The processes a) have a l s o been detected by Huang f o r d i f -ferent d i r e c t i o n s and d i f f e r e n t e l e c t r o n i c t r a n s i t i o n s . They are r e a d i l y recognized by t h e i r temperature independence. For the -» - f t r a n s i t i o n and with the magnetic f i e l d along a cube edge, the distance to a l l other t r a n s i t i o n s i s maximum. Cr o s s r e l a x a t i o n s of the type a) can, In t h i s oase, be expected to be weak. In f a c t , Huang's r e l a x a t i o n times beoome temperature 146 dependent f o r t h i s c o n f i g u r a t i o n . The Influence of processes c) on both r e l a x a t i o n methods but not on the u l t r a s o n i c absorption measurements suggests that they might be r e s p o n s i b l e f o r the considerable enhancement of the r e l a x a t i o n r a t e s measured by Huang and i n the present inves-t i g a t i o n as oompared to Dobrow's values. v/e do not b e l i e v e other paramagnetic Ions as i m p u r i t i e s are able to account f o r the whole discrepancy. The spectrum of paramagnetic i o n s , which could be present only i n very high d i l u t i o n , would have a r e l a t i v e l y small number of sharp l i n e s and the c r o s s r e l a x a t i o n would show a marked f i e l d dependence. In a d d i t i o n , the i m p u r i t y content would be s t r o n g l y sample de-pendent. I t seems to be q u i t e g e n e r a l , that s p i n - l a t t i c e r e l a x a t i o n times measured by pulse s a t u r a t i o n techniques are found to be s h o r t e r than the estimates fromacoustlo absorption measurements. (For a c o m p i l a t i o n of data and references see Dobrow (1966), t a b l e 1«) This i n d i c a t e s that we have to look f o r a more gen-e r a l e xplanation. I t has been shown e a r l i e r t h a t p a i r s of ions might be coupled together through the exchange i n t e r a c t i o n to form para-magnetic systems w i t h a d i f f e r e n t l e v e l scheme than the s i n g l e i o n s . C r o s s r e l a x a t i o n from s i n g l e Ions to exchange coupled p a i r s ( o r l a r g e r c l u s t e r s ) has been proposed by s e v e r a l authors to e x p l a i n the c o n c e n t r a t i o n dependence and enhancement of r e -l a x a t i o n rates (Van Vleok, 1959 and I 9 6 I ; ( J i l l , 1961 and G i l l and E l l i o t t , 1961; Bloembergen and Pershan, 1961; Statz et a l , 1961; Rameatad and Wagner, 1963; Schulz and J e f f r i e s , 1966). The f o l l o w i n g i s only a very q u a l i t a t i v e i n v e s t i g a t i o n o f the f e a s i b i l i t y of suoh a mechanism to enhance considerably the spin-bath r e l a x a t i o n r a t e s . From equation (2.80) f o l l o w s the o o n d l t i o n f o r a p a i r or c l u s t e r to have an a p p r e c i a b l e e f f e c t on the s i n g l e ion r e l a x a -t i o n : Here N 2 i s the population of the l e v e l s of the p a i r speotrum which have a strong s p i n - s p i n ooupling to the s i n g l e i o n l e v e l , % i s the population of the s i n g l e ion l e v e l and W-^  and W2 are the r e s p e c t i v e s p i n - l a t t l o e r e l a x a t i o n r a t e s . S t a t z et. a l (1961) and G i l l (1961) have measured r e l a x a t i o n times of t r a n s i t i o n s i n the G r 3 + p a i r spectrum i n ruby. The p a i r r e l a x a t i o n r a t e s are found to be about three orders of magnitude f a s t e r than the s i n g l e - i o n r e l a x a t i o n r a t e s . This e f f i c i e n t r e l a x a t i o n mechanism cannot be due to the o r d i n a r y Van VlecSc-Orbaoh r e l a x a t i o n through modulation of the c r y s t a l f i e l d . Modulation of the exchange i n t e r a c t i o n by the l a t t i c e v i b r a t i o n s has been used by the above authors to e x p l a i n q u a l -i t a t i v e l y the strong s p i n - l a t t i c e coupling of p a i r s . I f we adopt the value w 2 / w i 1°^, we have as the l i m i t f o r o r o s s r e l a x a t i o n v i a p a i r s t o become n o t i c e a b l e 148 I f N 2 l a taken aa the number of nearest paramagnetic neighbours i n a o r y s t a l with randomly d i s t r i b u t e d paramagnetic c e n t e r s , then f o r low concentration c we have N,/Nx ^ c. This means that at concentrations of 0.1$, the p a i r r e l a x a t i o n could be-come n o t i c e a b l e . Not a l l p a i r l i n e s might have an e f f i c i e n t c r o s s r e l a x a t i o n to the s i n g l e ion l i n e . This would decrease the value of Ng. On the other hand, r a t h e r f a r neighbours might be e f f e c t i v e l y ooupled by i n d i r e c t Kramers-type exchange i n t e r a c t i o n . S t a t z et a l have I d e n t i f i e d p a i r l i n e s i n the Cr* + spectrum In ruby up to the 1 0 t h neighbour. This as w e l l as exohange ooupled c l u s t e r s of more than two ions can Increase N 2 considerably and If,N2..becomes comparable^to-N^-,: .the.:-model of separated s i n g l e ions and p a i r s would break down. The exohange I n t e r a c t i o n between rare e a r t h ions i s , how-ever, weaker than between the i r o n group ions and exohange i n -t e r a o t i o n , t h e r e f o r e , w i l l not couple neighbours which are too f a r away. The exchange i n t e r a c t i o n s between Eu -ions and CaO and SrO have been s t u d i e d by Calhoun and Overmeyer (1964). They f i n d 0.5°X<J„n < 1.5°K and J n n n <0.1°K, where J n n and Jno„ are the exchange c o u p l i n g constants between nearest neighbours and next nearest neighbours r e s p e c t i v e l y . The conclusion i s that exchange ooupled p a i r s are able to enhanoe the r e l a x a t i o n rates of the s i n g l e ions and might, t h e r e f o r e , cause the observed concentration dependence of T . I t i s d o u b t f u l , however, i f t h i s mechanism i s strong enough to 149 account f o r the l a r g e d i s c r e p a n c i e s between r e l a x a t i o n and ac o u s t i c absorption measurements. As has been mentioned e a r l i e r , some of our specimens have an app r e c i a b l e c o n c e n t r a t i o n of Eu x + - i o n s . These Ions have a around s t a t e s i n g l e t 7 F 0 . Sinoe the exchange I n t e r a c t i o n acts on the e l e c t r o n i c s p i n r a t h e r than on the t o t a l angular momentum Eu 4 + -ions can be coupled to neighbouring Eu*+ - p a i r s . I t i s pos-s i b l e that d u s t e r s i n v o l v i n g E u + -ions have a steeper *~ versus T dependence than those c o n t a i n i n g only Eu 2 + . This could e x p l a i n the steeper temperature dependence of T f o r the two specimens 1 and 2 which have much higher E u 3 + concentration than the other c r y s t a l s , as w e l l as the crossover at 1.9°K between the s p e c i -mens 2 and 3. Q u a n t i t a t i v e o r even semi-quantitative explana-t i o n s are impossible, due to the very complicated s t r u c t u r e of our system and to complete l a c k of knowledge outside the s i n g l e -ion system. An attempt to detect p a i r l i n e s In the already very oomplioated s i n g l e - i o n speotrum using a 34 Oc/s spectrometer was unsu c c e s s f u l . The c o n s i d e r a t i o n s above have therefore to remain r a t h e r s p e c u l a t i v e . 150 4.3 Erbium E t h y l s u l p h a t e 4.3.1 I n t r o d u c t i o n Er has a f r e e Ion ground s t a t e 1^ , whloh i s s p l i t 2. i n t o seven Kramers doublets by the e t h y l s u l p h a t e c r y s t a l f i e l d . At helium temperatures, only one doublet Is populated. I t has a h i g h l y a n i s o t r o p i c g-vslue. Bleaney and S c o v i l (1051) meas-ured f o r E r ^ d i l u t e d i n LaES g „ = I . 4 7 and g x * 8.85. The paramagnetic resonance speotrum shows one strong c e n t e r l i n e due to the even Isotopes and eight h y p e r f i n e l i n e s a r i s i n g from the only odd isotope E r 3 + whloh has a n a t u r a l abundance of 23$. The t o t a l s e paration of the seven Kramers doublets i s 300 cm*1 and the doublet c l o s e s t t o the ground s t a t e i s at 44 cm"1. These values have been measured by Erath (lQ6l) l n the concentrated ErES. The s p i n - l a t t i c e r e l a x a t i o n of E r ^ d i l u t e d i n LaES and YES has been studied by Larson and J e f f r i e s (1Q66). In LaES and with H 11 z, they f i n d a r a t h e r weak d i r e c t process T^ = h,z T which i s dominant below 2°K, an Orbaoh process T ^ = 4 . 5 x l 0 -* °exp( -^ and a strong Raman process T^ R r 4.4 x 10*3 y9 which i s domi-nant above 2°K. Due t o the strong Raman process, the r e l a x a t i o n time changes by more than four orders of magnitude i n the l i q -u i d helium temperature range. Representative values are T^ s 150 mseo at 1.5 °K and T, = 14^seo at 4.2°K 151 No r e l a x a t i o n time measurements are as yet a v a i l a b l e f o r the concentrated ErES. As l n the case o f CeES and E u z + : CaF 2, the paramagnetio Faraday r o t a t i o n of ErES Is p r o p o r t i o n a l t o the magnetisation M. Since only one Kramers doublet i s populated <^ f and M are both p r o p o r t i o n a l to the population d i f f e r e n c e l n the ground st a t e doublet. Beoquerel, De Haas and Van den Handel ( 1 9 3 7 ) measured = A lonU f p -with /<- - 6 . 0 1 6 / 3 and A * -12.57 deg mm**1 I t should therefore be po s s i b l e to measure the spin-bath r e -l a x a t i o n by monitoring Cjp . In a n i s o t r o p i c c r y s t a l s , however, i s not e a s i l y measured. I n a d d i t i o n to the Faraday r o t a t i o n b i r e f r i n g e n c e occurs i f the l i g h t l a not propagated along the o p t i c a x i s . The observed angle of minimum l i g h t t r a n s m i s s i o n , i . e . the apparent Faraday r o t a t i o n , i s a f u n c t i o n of both b i r e -fringence and r o t a t i o n and i s not p r o p o r t i o n a l t o the magneti-z a t i o n . The problem, t h e r e f o r e , i s to a l i g n the o p t i c a x i s of the ethy l s u l p h a t e samples s u f f i c i e n t l y p a r a l l e l t o the l i g h t d i r e c -t i o n such that the b i r e f r i n g e n c e I s n e g l i g i b l e compared to the Faraday r o t a t i o n . No alignment problem a r i s e s i n the o p t l o a l l y i s o t r o p i c f l u o r i d e s . The apparent r o t a t i o n has been s t u d i e d as a f u n o t l o n of b i r e f r i n g e n c e and r o t a t i o n by Ramaseshan (1951). I t Is convenient t o v i s u a l i z e the e f f e c t of b i r e f r i n g e n c e 152 and Faraday r o t a t i o n on the s t a t e of p o l a r i z a t i o n by means of the Poincare sphere. For a d e t a i l e d treatment, we r e f e r to the a r t i c l e by Raraachsndran and Ramaseshan (1961). We give i n the f o l l o w i n g only a short review i n order to e x p l a i n the negative r e s u l t s i n ErES. 4.3.2 The Poincare Sphere Representation The e l e c t r i c v eotor of a t o t a l l y p o l a r i z e d l i g h t wave de-sc r i b e s In general an e l l i p s e . The s t a t e of p o l a r i z a t i o n of the l i g h t wave i s s p e c i f i e d by two parameters, eg. the r a t i o ^ of the two axes of the e l l i p s e and the angle f between the major a x i s of the e l l i p s e and a f i x e d d i r e c t i o n , A one-to-one correspondence between the p o s s i b l e s t a t e s of p o l a r i z a t i o n (defined by ^ and •+ ) and the p o i n t s P on' a sphere (the s o - c a l l e d Poincare sphere) can be e s t a b l i s h e d i f P i s de-f i n e d as the point of l a t i t u d e 2 and l o n g i t u d e 2-f , where tan 4- = 3i ( See Figure 4,20). The poles of the sphere thus oor-a respond to c i r c u l a r l y p o l a r i z e d l i g h t and the points on the equator c h a r a c t e r i z e l i n e a r l y p o l a r i z e d l i g h t of d i f f e r e n t o r i -e n t a t i o n . The ooncept i s i l l u s t r a t e d i n Figure 4.20. Consider now a l i g h t wave of a r b i t r a r y p o l a r i z a t i o n P ( y / ) entering a c r y s t a l which possesses both b i r e f r i n g e n c e and Fara-day r o t a t i o n . The change In p o l a r i z a t i o n of the l i g h t wave i n passing through the c r y s t a l Is most e a s i l y discussed u s i n g the Poincare sphere. I t can be shown that f o r a given c r y s t a l and f o r a given d i r e c t i o n of propagation, there are two orthogonal 153 Figure 4.20 The Poincare sphere - General r e l a t i o n s 154 s t a t e s of p o l a r i z a t i o n whloh propagate unohanged through the c r y s t a l . These s t a t e s are c a l l e d the st a b l e modes C and C*. For an a r b i t r a r y p o l a r i z a t i o n P of the l i g h t e n t e r i n g the c r y -s t a l , the state P' of the emerging l i g h t a f t e r a path l e n g t h d l n the c r y s t a l can be found by r o t a t i n g the point P by an angle A about the a x i s J o i n i n g the two s t a b l e modes C and C . This a x i s passes through t h e center of the sphere due t o the f a c t that the s t a b l e modes are orthogonal. The angle of r o t a t i o n i s given by A =dAo = c/i/sl -fa.)*-where S0 i s the b i r e f r i n g e n c e per unit path l e n g t h i n absence of Faraday r o t a t i o n and <^ 0 Is the Faraday r o t a t i o n per u n i t path len g t h i n absence of b i r e f r i n g e n c e . The l a t i t u d e of the stable mode C i s given by tan zf = 4 ^ An analyser can be represented by the point A on the sphere which corresponds t o the l i g h t p o l a r i z a t i o n passing the analyser without a t t e n u a t i o n . I f l i g h t of p o l a r i z a t i o n P 1 Is i n c i d e n t on the a n a l y s e r A, only a f r a c t i o n p r o p o r t i o n a l to c o s Z A Is passed by the a n a l y s e r where 2.A*" i s the angle of the great c i r c l e between A and P'. Figur e 4.21 i l l u s t r a t e s the apeolal case which a p p l i e s i n our r e l a x a t i o n measurements. The l i g h t i n c i d e n t on the c r y s t a l Is l i n e a r l y p o l a r i z e d (point L ) . The c r y s t a l has a s t a b l e Figure 4.21 The Poincare sphere a p p l i e d to r e l a x a t i o n measurements 1 5 6 mode C w i t h , a 2<?o 2 <?. •/an T-<t = - r — = — — where oc Is the angle between the l i g h t d i r e c t i o n and the o p t i c axis and i s supposed to be s m a l l . =• ( n^~  n e ) i s the b i r e f r i n g e n c e per u n i t path l e n g t h , n ^ and n £ are the'two p r i n c i p a l r e f r a c t i v e i n d i c e s . A f t e r passing through the c r y s -t a l , the l i g h t can be represented, by the point L• which i s r o-tated by an angle A about CC' from L The a n a l y s e r at A gives minimum l i g h t t r a n s m i s s i o n (apparent Faraday r o t a t i o n ). For observation of the r e l a x a t i o n , the analyser i s set at 9'+ 45° (point A') and the transmitted l i g h t i n t e n s i t y i s p r o p o r t i o n a l to COS^A . A p p l i c a t i o n of a micro-wave s a t u r a t i n g pulse reduces the Faraday r o t a t i o n , thus moving both L' and C i n the d i r e c t ion Indicated by arrows In Figure 4.21. The f u n c t i o n a l dependence c o s 2 A * = f(<^ ) i s i n , general very complicated and expansions are p o s s i b l e only l n the s p e c i a l cases of 6 <-< 2<^  ( n e g l i g i b l e b i r e f r i n g e n c e ) o r S » 1 ^ ( n e g l i g i b l e Faraday r o t a t i o n ) . In the case of ErE3, the Faraday r o t a t i o n i s 1 0 deg mm"1 at resonance f i e l d and 1 . 5 ^ (Becquerel .et a l , 1 9 3 7 ) . The r e f r a c t i v e i n d i c e s are n^ * 1.490 and n £ = 1.480, g i v i n g a path r e t a r d a t i o n of o x = 115 mm"1. Hence = 0 . 0 0 3 . A few re p r e s e n t a t i v e values f o r the l a t i -tude 2 of the s t a b l e modes C f o r small misalignments << are 157 ~ <=•"-= 1 deg i_d- m Bh deg <^  s 2 deg ^ * 68 deg e 3 <3eg 2./ s 48 deg I t Is seen that f o r ErES, the alignment has to be very aoourate l n order t o keep the st a b l e modes on the pole of the Polncare sphere. I t has to be noted that the 2& v a l u e B given are the optimum values during the r e l a x a t i o n trace since microwave sa-t u r a t i o n d ecreases^ and hence 2 ^ . As has been described i n chapter 3.1, the c r y s t a l could be r o t a t e d at He-temperature about two f i x e d axes. I t i s gen-e r a l l y Impossible to obtain any d i r e c t i o n i n space by r o t a t i o n about two f i x e d axes. The p o s s i b i l i t y of proper alignment thus depends on the o r i e n t a t i o n of the two axes of r o t a t i o n ( i . e . on the p o s i t i o n of the t e f l o n plug i n the c a v i t y ) and on the p o s i t i o n of the c r y s t a l speoimen w i t h respect to the t e f l o n plug. I t Is impossible to adjust the alignment at room temperature, since c o o l i n g down to He temperatures can change the o r i e n t a t i o n s c o n siderably. The sources of misalignment are not reproducible from one experimental run to another. We have t r i e d t o achieve proper alignment i n seven runs with f o u r d i f f e r e n t c r y s t a l specimens. (The same c r y s t a l cannot be used f o r more than two runs, i n general, due to d e t e r i o r a t i o n of the p o l i s h e d faces.) The co n c l u s i o n i s that a more accurate mechanloal a l i g n i n g system has t o be found. This seems r a t h e r t r i o k y since the experiments are performed at He temperatures i n the t i g h t spaoe of a magnet gap. 158 BIBLIOGRAPHY Abel, V.R., Anderson, A.C., Blac k , W.C. and Wheatley, J.C. (1966) Phys. Rev. L e t t . 16, 274. Abragam, A. (196l) The P r i n c i p l e s of Nuclear Magnetism (Oxford Univ. Press) P. 404. Anderson, A.C., Connolly, J . I . and Wheatley, J.C. ( 1 9 6 5 ) Phys. Rev. 1 3 5 , A910. A n d r o n l k a s h v l l i , E.L. (1956) S o v i e t Physics J e t p 2 , 4o6. Baker, J.M., Bleaney, B. and Hayes, W. (1958) Proc. Roy. Soc. A 2 4 7 , 141. Becquerel, J . , De Haas, W.S. , and Van den Handel, J . ( 1 9 3 7 ) Physica 4 , 3 4 5 . Becquerel, J . , De Haas, W.J., and Van den Handel, J . ( 1 0 3 8 ) Physica ^, 8 5 7 . Bleaney, B. and S c o v i l , H.E.D. ( 1 9 5 1 ) P r o c Phys. Soc. A64. 204. Bleaney, B. and Bowers, K.D. (1952) P r o c Roy. Soc. A214. 451. Bloembergen, N., Shapiro, S., Pershan, P.S., and Artman, J.O. (1959) Phys. Rev. 114, 445. Bloembergen, N. and Pershan, P.S. ( I 9 6 D Advances l n Quantum E l e c t r o n i c s (Columbia Univ. Press) J.R. Singer ed. P. 3 7 3 . Bogle, G.S., Cooke, A.H. and Whitley, S. ( 1 9 5 1 ) Proc. Roy. Soc. A64. 931. Brya, W.S. and Wagner, P.E. ( 1 9 6 5 ) Phys. Rev. L e t t . 14, 4 3 1 . B u r g l e l , J.C. and Meyer, H. ( I 9 6 6 ) B u l l . Am. Phys. Soc. I I , 11, 4 5 3 . Calhoun, B.A. and Overmeyer, J . (1964) J . a p p l . Phys. 21> $ 8 9 . Casimir, H.B.G. ( 1 9 3 9 ) Physica 6 , 1 5 6 . C h a l l i s , L . J . ( I 9 6 2 ) Proc. Phys. Soc. 80, 759. C h a l l l s , L . J . , Dr a n s f e l d , K. and Wilks , J . ( I 9 6 I ) Proc. Roy. S o c A 2 6 0 . 3 1 . Daniels, J.M., and Wesemeyer, H. ( 1 9 5 8 ) Can. J . Phys. J . 6 , 405. Devor, D.P. and Hoskins, R.H. (1961) B u l l . Am. Phys. Soo. I I i . 3 6 4 . Dobrow, W.I. and Broxroe, M.E. ( I 9 6 2 ) Magnetic and E l e c t r i c Reso-nance and R e l a x a t i o n , Prox. XI C o l l . Ampere, Eind-hoven ( I 9 6 2 ) J . Smidt, ed. (North-Holland P u b l i s h -ing Co. Amsterdam, 1963) p. 1 2 9 . Dobrow, W.I. (1966) Phys. Rev. 146, 2 6 8 . Dweck, J . and S e l d e l , G. (1966) Phys. Rev. 146, 3 5 9 . Dransfeld, K. (1958) B u l l . Am. Phys. Soc. I I J , 3 2 4 . E l l i o t t , R.J. and Stevens, K.W.H. (1952) Proc. Roy. Soc. A 2 1 5 . 4 3 7 . Erath, E.H. (I 9 6 l ) J . Chem.-Phys. J 4 , 1 9 8 5 . Eschenfelder, A.H. and Weidner, R.T. (1953) Phys. Rev. 92., 8 6 9 . Faughnan, B.W. and Strandberg, M.W.P. ( I 9 6 I ) J . Phys. Chem. So l i d s 1 2 , 155. F e o f i l o v , P.P. and K a p l y a n s k i i , A.A. ( I 9 6 3 ) Optics and Speotr. (USSR) 12, 1 2 9 . Finn, C.B.P., Orbach, R. and Wolf, W.P. ( 1 9 6 I ) Proc. Phys. Soo. 22, 2 6 1 . F i t z w a t e r , D.R. and Rundle, R.E. (1959) Z. K r i s t . 112, 3 6 2 . G-arwin, R.L., Hutchinson, D., Penman, S. and Shapiro, G. (1959) Rev. S c i . I n s t r . JO, 1 0 5 . Gesohwind, S., C o l l i n s , R.J. and Schawlow, A.L. (I96l) Advances i n Quantum E l e c t r o n i o s (Columbia Univ. Press, J.R. Singer ed.) p. 2 3 2 . Geschwind, S., D e v l i n , G.E., Cohen, R.L. and Chlnn, S.R. (1965) Phys. Rev. 137, A1087. G i l l , J.C. ( I 9 6 D Proc. Phys. Soc. 22., 58. G i l l , J.C. and E l l i o t t , R.J. (1961) Advances l n Quantum E l e o t r o -nics(Columbla Univ. Press) J.R. Sinerer ed. p. 399. Glordmalne, J.A. and Nash, F.R. (1965) Phys. Rev. 138. A 1 5 1 0 . Gorter, C.J., V-an der Marel, L.C. and Bolger, B. (1955) Physioa 2 ., 103. 160 G r i f f i t h s , D.J. (1965) Ph. D. T h e s i s , U n i v e r s i t y of B r i t i s h Columbia (unpublished). G r i f f i t h s , D.J. and G l a t t l i , H. ( 1 9 6 5 ) Can. J . Phys. 2 3 6 1 . Huang, C.Y. (1964) Ph. D. Thesis, Harvard U n i v e r s i t y (unpublished) (1965) Phys. Rev. A24l Hiifner, S. (1962) Z. f. Physik 162, -+17. Johnson, R.C. and L i t t l e , W.A. ( 1 9 6 3 ) Phys. Rev. 130. 596. K a l b f l e l s c h , H. ( 1 9 6 4 ) Z. f. Physik 181, 1 3 . K a l b f l e i s c h , H. ( 1 9 6 5 ) Z. f. Physik 188,- 186. Ka p i t z a , P. (1941) J . phys. USSR 4 , 181. K a s t l e r , A. (1951) C. R. Acad. S c i . , P a r i s , 232. 9 5 3 . K e t e l a a r , J.A.A. ( 1 9 3 7 ) Physica 4, 6 1 9 . Kramers, H.A. ( 1 9 3 0 ) Proc. Roy. Acad. Amst. Xl» 9 5 9 . Kronig, R. deL, ( 1 9 3 9 ) Physica 6 , 3 3 . Kuang, Wey-Yen (1962) Soviet Physics Jetp 1J5, 635. L a c r o l x , R. (1957) Helv. Phys. Acta. JO, 3 7 4 . Larson, G.H. and J e f f r i e s , C.D. ( 1 9 6 6 ) Phys. Rev. 141, 4 6 1 . Lee, K., Muir, H. and Catalano, E. (1965) J . Phys. Chem. S o l i d s 2 6 , 5 2 3 . Low, V/. Paramagnetio Resonance i n S o l i d s , Suppl. 2 of S o l i d State P h y s i c s , Selfe and Turnbull ed. Low, W. ( i 9 6 0 ) Nuovo Ciraento 1 2 , 6 0 7 . Meyer, H. and Smith, P.L. (1959) J . Phys. Chem. S o l i d s 9., 2 8 5 . M i l l s , D.L. (1964) Phys. Rev. I J l t A 8 7 6 . Mlms, W.B. (1965) Rev. S c i . I n s t r . J56, 1472. Opeohowskl, V7. ( 1 9 5 3 ) Rev. Mod. Phys. 2£, 2 6 4 . Orbaoh, R. ( I 9 6 D Proc. Roy. Soc. A264. 4 5 8 . Orbaoh, R. and Blume, M. ( I 9 6 2 ) Phys. Rev. L e t t . 8 , 4 7 8 . 1 6 1 Owen, J . (1961) J . a p p l . Phys. J 2 Suppl., 2133. Pe r s l c o , F., Stevens, K.W.H. and. Tucker, J.W. (1963) Phys. L e t t . It 1 6 . Peterson, R.L. (1965) Phys. Rev. 1 3 7 . A 1 4 4 4 . Ramachandran, O.N. and Raraaseshan, S. (1961) Encyclopedia of Physics (Flugge ed.) v o l . XXV/l, p. 1. Ramaseshan, 3. (1951) Proc. Indian Acad. S c i . A34. 32. Rannestad, A. and Wagner, P.E. (1963) Phys. Rev. 121, 1953. R l e c k o f f , K.E. ( 1 9 6 2 ) Ph.D. Th e s i s , U n i v e r s i t y of B r i t i s h C o l -umbia (unpublished). R l e c k o f f , K.E. and Weissbaoh, R. (lo62) Rev. S c i . I n s t r . 33. 1393. Ryter, C. (1957) Helv. Phys. Acta JO, 353. Schulz, M.B. and J e f f r i e s , C.D. (1966) Phys. Rev.,to be pub-l i s h e d . S c o t t , P.L. and J e f f r i e s , CD. (1962) Phys. Rev. 1 2 2 , 32. Shen, Y.R. (I963) Ph.D. Thesis, Harvard U n i v e r s i t y , unpublished . Shen, Y.R. (1964) Phys. Rev. 1J£, A511. Shlren, N.S. and Tucker, E.B.. ( 1 9 5 9 ) 1 Phys. Rev. L e t t . 2,, 206. S t a t z , H., Rimal, L., Weber, M.j., De Mars, O.A., and Koster, O.F. (1961) J . appl. Phys. 22 Suppl., 218 S. Stoneham, A.M. (I965) Proc. Phys. Soc. 86, 1163. Van den Broek, J . and Van der Marel, L.C. (1963) Physica 29,, 948. Van Kra.nendonk, J . end Van Vleck, J.H. (1958) Rev. Mod. Phys. 20, 1 . Van Vleck, J.H. (1939) J . Chem Phys. 2» 72. V an Vl e c k , J.H. (1940) Phys. Rev. J£, 426. Van V l e c k , J.H. (194l) Phys. Rev. ^9_, 724. Van Vleck, J.H. ( 1 9 5 9 ) Quantum E l e c t r o n i c s , C H . Townes ed. (Columbia Univ. Press) p. 391. 162 Van Vleck, J.H. ( 1 9 6 1 ) Advances In Quantum E l e c t r o n i c s (Columbia Univ. P r e s s , J.R. Singer ed.) p. 388. Van Vleck, J.H. and Hebb, M.H.(193*0 Phys* Rev. 4 6 , 17. Wyokoff, C r y s t a l S t r u c t u r e s , I n t e r s c i e n c e P u b l i s h e r s , New York. Ziman, J.M. ( 1 9 5 4 ) Proc. Roy. Soo. A 2 2 6 . 4 3 6 . 

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