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Low temperature paramagnetic resonance studies of the rare earth group ions using a new high sensitivity… Buckmaster, Harvey Allen 1955

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LOW TEMPERATURE PARAMAGNETIC RESONANCE STUDIES OF THE RARE EARTH GROUP IONS USING A NEW HIGH SENSITIVITY SPECTROMETER BY HARVEY AELEN BUCKMASTER A Thesis submitted i n p a r t i a l f u l f i l m e n t of the requirements f o r the degree of Doctor of Philosophy i n P h y s ics accept t h i s t h e s i s as conforming to the standard r e q u i r e d from candidates f o r the degree of Doctor of Philosophy Members of the Department of Physics The U n i v e r s i t y of B r i t i s h Columbia November 19^5 ABSTRACT A h i g h s e n s i t i v i t y , wide or narrow band, double f i e l d modulation paramagnetic resonance spectrometer operating a t a wavelength of 1.25 cm. f o r use at l i q u i d helium temperatures has been developed which i s described i n d e t a i l . This spectrometer employs a t r a n s m i s s i o n type c y l i n d r i c a l reson-ant c a v i t y operated i n the H ^ i mode. I n wide band oper-a t i o n , the magnetic f i e l d i s modulated at 60 cps. to an-amplitude i n excess of the resonance l i n e width and at kcs. to an amplitude l e s s than or equal to the l i n e width. For narrow band o p e r a t i o n o n l y , the h i g h frequency modulation i s employed and the s t a t i c magnetic f i e l d i s - 9 l i n e a r l y swept. A s i g n a l from 10 grams of dip h e n y l -t r i n i t r o phenyl h y d r a z y l [(G^H^)2'N-N» C^Hg. ( ^ 2 ) 3 ] has been observed at 290°K. u s i n g wide band o p e r a t i o n (8 kcs.) i n d i c a t i n g that s e n s i t i v i t i e s of the order of 1 0 " 1 2 grams can be achieved w i t h i t at ij..2°K. i n narrow band o p e r a t i o n (1 cps.). This s e n s i t i v i t y i s c l o s e to the t h e o r e t i c a l value p r e d i c t e d by Bleaney of l o " 1 ^ grams and i s s e v e r a l orders of magnitude greater than that r e p o r t e d f o r any other paramagnetic resonance spectrometer. Using t h i s improved s e n s i t i v i t y , higher order t r a n s i t i o n s A M = - 2, - 3 i n d i l u t e gadolinium e t h y l -sulphate [ i j . f 7 j 8 S ? / , 2 ] which were not p r e v i o u s l y obser-vable, have been stud i e d a t 90°K. as a f u n c t i o n of the 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 a x i s of symmetry of the c r y s t a l . These t r a n s i t i o n s show t h a t the e f f e c t of o f f - d i a g o n a l terms i n spin-Hamiltonian have a greater e f f e c t on the energy l e v e l s than had p r e v i o u s l y been appreciated from measurements of the A M = - 1 t r a n -s i t i o n s and help to e x p l a i n the d i s c r e p a n c i e s between the c a l c u l a t e d and observed zero f i e l d s p l i t t i n g s . = An e x c i t e d s t a t e i n d i l u t e dysprosium e t h y l sulphate j i | f 9 ; ^ 1 5 / 2 ] h a s D e © n observed at 1|.2°K. wi t h g H = 5 . 8 5 - 0.05 which had p r e v i o u s l y only been observed at 20°K. wit h g n = 5.80 - 0.02 and g J L = 8.1j.O - 0 . 2 . The l i n e width of d i l u t e praseodymium e t h y l sulphate | ^ . f 2 ; 3H|^j has been measured at Jf.2°K. and found t o be 3 5 - 5 gauss showing th a t the p r e v i o u s l y observed l i n e width of 200 gauss at 20°K. i s due t o s p i n - l a t t i c e broadening. Faculty of Graduate Studies P R O G R A M M E O F T H E Jffitrai ©rail ^ xamittatfan far i\\t J b j m af Jlmfar &f |Jfyifastfplj£ of H A R V E Y A L L E N BUCKMASTER B.Sc. (University of Alberta) 1950 M . A . (University of British Columbia) 1952 F R I D A Y , DECEMBER 16th, 1955, at 3:00 p.m. IN ROOM 300, PHYSICS BUILDING COMMITTEE I N C H A R G E D E A N H . F. A N G U S , Chairman A . D. M O O R E G. M . V O L K O F F T. E . H U L L K . C. M A N N B. N . M O Y L S J. B . B R O W N F. H . SOWARD A . M . C R O O K E R External Examiner—B. B L E A N E Y Clarendon Laboratory, Oxford, England LOW T E M P E R A T U R E P A R A M A G N E T I C RESONANCE STUDIES OF T H E R A R E E A R T H GROUP IONS USING A N E W HIGH SENSITIVITY SPECTROMETER ABSTRACT A high sensitivity, wide or narrow band, double field modulation para-magnetic resonance spectrometer operating at a wavelength of 1.25 cm. for use at liquid helium temperatures has been developed which is described in detail. This spectrometer employs a transmission type cylindrical resonant cavity operated in the H m mode. In wide band operation, the magnetic field is modulated at 60 cps. to an amplitude in excess of the resonance line width and at 462.5 kcs. to an amplitude less than or equal to the line width. For narrow band operation, only the high frequency modulation is employed and the static magnetic field is linearly swept. A signal from 10-u grams of diphenyl-trinitro phenyl hydrazyl [ ( C 0 H J 2 * N-N * C 6 H 2 * ( N 0 2 ) 3 ] has been observed at 290°K. using wide band operation (8 kcs.) indicating that sensi-tivities of the order of 10-1 2 grams can be achieved with it at 4.2 °K. in narrow band operation (1 cps.). This sensitivity is close to the theoretical value of 10- l a grams predicted by Bleaney and is several orders of magnitude greater than that reported for any other paramagnetic resonance spectrometer. Using this improved sensitivity, higher order transitions A M = ± 2 , ± 3 in dilute gadolinium ethyl sulphate [4f7; 8 S 7 / 2 ] which were not previously observable, have been studied at 90°K. as a function of the orientation of the magnetic field with respect to the symmetry axis of the crystal. These transi-tions show that the effect of off-diagonal terms in spin-Hamiltonian have a greater effect on the energy levels than had previously been appreciated from measurements of the A M = ± 1 transitions and help to explain the discrep-ancies between the calculated and observed zero field splittings. An excited state in dilute dysprosium ethyl sulphate [4fn; 8 H 1 5 / 2 ] has been observed at 4.2°K. with gx = 5.85 ± 0.05 which had previously only been observed at 20°K. with g, = 5.80 ± 0.02 and g 2 = 8.40 ± 0.2. The line width of dilute praseodymium ethyl sulphate [4f2; S H 4 ] has been measured at 4.2°K. and found to be 3 5 ± 5 gauss showing that the pre-viously observed line width of 200 gauss at 20°K. is due to spin-lattice broadening. PUBLICATIONS Radial Matrix Elements for the Quadrupole Transition with the Morse Potential. Canadian Journal of Physics. 30, 314 (1952). A New Paramagnetic Resonance Spectrometer. (Co-author H . E. D. Scovil l , Canadian Journal of Physics (submitted for publication). A M = ± 2 Transitions in Dilute Gadolinium Ethyl Sulphate. Canadian Journal of Physics (submitted for publication). A M = ± 3 Transitions in Dilute Gadolium Ethyl Sulphate. Canadian Journal of Physics (submitted for publication). G R A D U A T E STUDIES Field of Study: Physics Electromagnetic Theory W. Opechowski Quantum Mechanics G. M . Volkoff Magnetism W. Opechowski Electronics H . E. D. Scovil Special Relativity Theory W. Opechowski Quantum Theory of Radiation F. A . Kaempffer C W. Opechowski Advanced Quantum Mechanics «j G. M . Volkoff [F. A . Kaempffer Other Studies: Continuous Groups „ S. A . Jennings Network Theory A . D. Moore Theory of Functions of a Real Variable R. D. James Theory of Functions of a Complex Variable W. H . Simons Theory of Algebraic Numbers B. N . Moyls Modern Algebra D. Derry Integral Equations and Eigenvalue Equations - T . E. Hull ACKNOWLEDGMENTS The resea r c h described i n t h i s t h e s i s was sup-ported by the N a t i o n a l Research C o u n c i l of Canada through research grants to Dr. S c o v i l and the award of two Student-ships (1952-3, 1953-1+) and a Fellowship (195U~5) along w i t h Summer Scholarships (1952, 53, 5U» 55) "to the author. I am a l s o indebted'to N.R.C. f o r making i t p o s s i b l e f o r me to continue my s t u d i e s a f t e r the completion o f t h i s t h e s i s by the award of an Overseas Postdoctorate F e l l o w s h i p . I t i s not p o s s i b l e to adequately express the deep a p p r e c i a t i o n which I owe to Dr. H.E.D. S c o v i l . Without the mo t i v a t i o n through h i s a c t i v e i n t e r e s t , s t i m u l a t i n g d i s c u s -s i o n , and general h e l p , t h i s r e s e a r c h could not have been completed. I should l i k e to express my si n c e r e g r a t i t u d e to a l l my teachers, both at the U n i v e r s i t y of B r i t i s h Columbia and at the U n i v e r s i t y of A l b e r t a , f o r i n c r e a s i n g my know-ledge and enthusiasm f o r p h y s i c s . The members of the low temperature group, Dr. J.M. Da n i e l s , Dr. J.B. Brown and Mr. H. Zerbst, deserve s p e c i a l acknowledgment f o r t h e i r help i n the design of low temper-ature equipment and f o r supplying the l i q u i d helium r e q u i r e d f o r some of the experiments described i n t h i s t h e s i s . I am indebted to Dr. K.C. Mann f o r the lo a n of a current r e g u l a t o r which was used w i t h the electromagnet during the f i n a l experiments. The donation of a b i l l e t of s o f t s t e e l f o r the electromagnet by the S t e e l Company of Canada L t d . i s acknow-ledged. The equipment used could not have been designed and constructed so adequately without the i n t e r e s t , help and cooperation of the members of the Physics Department shop; Mr. A.J. F r a s e r , Mr. W. Morrison, Mr. W. Maier, Mr. S. P r i c e and Mr. J . Lees. G r a t e f u l acknowledgment i s al s o made to Miss A. R e d l i c k , Mr. .'D. Watts and Mr. J . E l l i o t t f o r d r a f t i n g most of the diagrams and to Miss L. S t a f and Miss P. Wood f o r t y p i n g the f i n a l d r a f t of t h i s t h e s i s . I wish t o express my a p p r e c i a t i o n t o a l l my f e l -low students who co n t r i b u t e d i n innumerable ways. TERMINOLOGY The f o l l o w i n g conventions, d e f i n i t i o n s , abbre-v i a t i o n s and symbols are adhered to i n t h i s t h e s i s . Conventions: a. 2.6i|. r e f e r s to subsection i|. of s e c t i o n 6 of Chapter I I . b. ( 111 ,7) r e f e r s to equation 7 which occurs i n Chapter I I I . A l l equations are numbered co n s e c u t i v e l y from the beginning of Chap-t e r I to the end of Chapter IV. e. Bleaney (B8) r e f e r s to a reference by Bleaney which occurs as the e i g h t h r e f e r -ence i n the "B" s e c t i o n o f the b i b l i o g r a p h y when these are arranged i n a l p h a b e t i c a l order. Minor references occur as ( E l ) . D e f i n i t i o n s : a. H a l f l i n e width r e f e r s to the f u l l width of a resonance l i n e at h a l f i n t e n s i t y , b. Room temperature means 290°K., oxygen temp-erature 90°K., n i t r o g e n temperature 78°K., hydrogen temperature 20°K. and helium temperature lj..2°K. A b b r e v i a t i o n s : Symbols: Constants: The meanings of a l l symbols employed i n t h i s t h e s i s are the same as those i n the l i s t i s s u e d by the P h y s i c a l S o c i e t y of London unless d e f i n e d otherwise and a l l values of constants are taken from the I n t e r n a t i o n a l C r i t i c a l Tables. TABLE OP CONTENTS Page INTRODUCTION A. Methods of Measuring the Paramagnetic P r o p e r t i e s of Matter i B. Magnetic P r o p e r t i e s of Matter i n the S o l i d State i i i CHAPTER I - QUANTUM THEORY OP PARAMAGNETISM 1 1.1 - I n t r o d u c t i o n 1 1.2 Paramagnetic Resonance Phenomena 3 1.21 The resonance c o n d i t i o n s 3 1.22 The f i n e s t r u c t u r e I4. 1.23 The hyperfine s t r u c t u r e 7 1.3 l i n e Width 8 1 . 3 1 Spin - l a t t i c e i n t e r a c t i o n 9 1.32 Spin - s p i n i n t e r a c t i o n 10 l . i i Energy L e v e l s of the Ion 12 l . i i l General Hamiltonian 13 1.14.2 Hamiltonian o f the f r e e i o n i n magnetic f i e l d l l j . 1.14.3 Ion i n the c r y s t a l l i n e e l e c t r i c f i e l d 16 l . i 4 4 M a t r i x elements o f the c r y s t a l l i n e f i e l d 20 1 . 5 Spin - Hamiltonian 23 l . ^ l A x i a l symmetry about z-axis 2I4. 1 . 5 2 Hyperfine s t r u c t u r e 27 1 .53 Non-axial symmetry 28 1 . 5 4 S-State ions 29 Page 1 . 5 5 Anomalous case 30 CHAPTER I I - PARAMAGNETIC RESONANCE SPECTROMETERS . . 33 2 . 1 I n t r o d u c t i o n 33 2 . 2 General Requirements 33 2 . 3 Noise 37 2.1; General C l a s s i f i c a t i o n 2 . 5 P o i n t - t o - P o i n t Spectrometer 43 2 . 6 Frequency Modulation Spectrometers . . . . k3 2 . 6 1 Iline Shape Spectrometer kh 2.62 Balanced Bridge Spectrometer .... . kk 2 . 7 F i e l d Modulation Spectrometers . . . . . . hS 2 . 7 1 Bolometer Spectrometer . . . . . . . lj.5 2.72 Superheterodyne Spectrometer . . . . 1|6 2 . 7 3 Video Spectrometer kl Z.Ik Double F i e l d Modulation Spectrometer, kl 2 . 8 Comparison of Techniques 5 l CHAPTER I I I - Experimental Apparatus 53 3 . 1 I n t r o d u c t i o n 53 3 . 2 Microwave Apparatus 53 3 . 2 1 Wave Guide Components Sh 3 . 2 2 Microwave Power Generator 55 3 . 2 3 Detectors 56 3 . 2 4 Cavity Resonators 56 3 . 2 5 Wavelength Measurement 57 3 . 2 6 K l y s t r o n S t a b i l i z a t i o n . . ' 58 3 . 3 Magnetic F i e l d Equipment . 58 3 . 3 1 Design and Construction of the Electromagnet 58 Page 3 . 3 2 Performance of the Electromagnet . . . 60 3.33 Electromagnet Power Sources 62 3 . 3 4 Magnetic F i e l d Measurements 63 3*35 low Frequency Modulation . . . . . . 65 3.36 High Frequency Modulation 66 3»4 D e t e c t i o n Apparatus 69 3 . 4 1 S i n g l e Modulation 69 3.I4.2 Double Modulation 70 3 . 4 3 Video D i s p l a y 71 Narrow Band Operation . . . . . . . . 72 3 . 5 A u x i l i a r y E l e c t r o n i c Apparatus 72 3 . 5 1 C a l i b r a t o r 72 3 . 5 2 Frequency Meter 73 3 . 5 3 Power Supplies 73 3 . 6 Low Temperature Apparatus 71+ 3.61 Single Modulation Head 74 3 . 6 2 Double Modulation Head 76 3 . 6 3 Helium Recovery System 81 3 . 7 C r y s t a l s 82 3 . 8 Performance 81+ CHAPTER IV - EXPERIMENTAL RESULTS 87 i f . l I n t r o d u c t i o n 87 lj..2 Gadolinium E t h y l Sulphate 88 1+.21 Theory 89 If. 22 Experimental R e s u l t s For D i l u t e Gadolinium E t h y l Sulphate . . . . . . 91 Page 4 . 3 Line Width o f Praseodymium E t h y l Sulphate at i+.2°K 98 E x c i t e d State i n Dy^>rosium E t h y l Sulphate at lj..2 0K 99 4 . 5 Future Experiments 100 E.IST OF LUSTRATIONS Fac i n g Page F i g . l a Energy l e v e l diagram of the ground s t a t e s p l i t t i n g s of C r 3 + i n a c r y s t a l l i n e e l e c t r i c f i e l d w i t h predominantly cubic symmetry but w i t h a sm a l l component of t r i g o n a l symmetry 5 F i g . l b Energy l e v e l diagram, S = 3/2, magnetic f i e l d p a r a l l e l to the a x i s of c r y s t a l l i n e f i e l d showing the allowed t r a n s i t i o n .... 55 F i g . 2 Energy l e v e l diagram, S = l / 2 , I = 5/2 i n a strong magnetic f i e l d showing the allowed t r a n s i t i o n s 7 F i g . 3 a P o i n t - t o - p o i n t paramagnetic resonance spectrometer 1+3 F i g . 3b Video paramagnetic resonance spectrometer . .43 F i g . 4 Balanced bridge paramagnetic resonance spectrometer 44 F i g . 5 Bolometer paramagnetic resonance spectro-meter 45 F i g . 6 Superheterodyne paramagnetic resonance spectrometer 4 ° F i g . 7 Double f i e l d modulation paramagnetic resonance spectrometer 47 F i g . 8 Block diagram o f 1.25 cm. microwave apparatus 54 F i g . 9 Block diagram o f a c a v i t y type k l y s t r o n frequency s t a b i l i ^ r 58 Facing Page F i g . 10 B l o c k diagram o f current r e g u l a t o r f o r electromagnet 62 F i g . 11 Block diagram of proton resonance u n i t . . . 64 F i g . 12 Block diagram of s l o t t e d resonant c a v i t y . 67 F i g . 13 Block diagram of double modulation d e t e c t i o n system 70 F i g i 14 C r o s s - s e c t i o n of K band resonator 75 F i g . 15a Diagram o f double Kovar s e a l 80 F i g . 15b Diagram of sandwich type i n s u l a t e d vacuum s e a l 80 F i g . 16 Helium recovery and vacuum system 81 F i g . 17 C r y s t a l s t r u c t u r e o f the rare e a r t h e t h y l sulphate s 83 F i g . 18 A M = - 2 t r a n s i t i o n s i n gadolinium e t h y l sulphate as f u n c t i o n o f the angle w i t h *• symmetry a x i s of the c r y s t a l 96 EIST OP TABLES Facing Page TABLE I The T r a n s i t i o n Groups Of The P e r i o d i c Table '. . v TABLE I I The Ground States Of The Free Ions Of The I r o n And Rare E a r t h Groups v i TABLE I I I The Values of P and H S a t i s f y i n g Kramer's Doublet R e l a t i o n When g = 2 35 TABLE IV The R e l a t i v e C h a r a c t e r i s t i c s And S e n s i t i v -i t i e s Of Paramagnetic Resonance Spectrometers . . . 51 TABLE V Experimental Data A M = - T r a n s i t i o n s For D i l u t e Gadolinium E t h y l Sulphate at 290 °K 95 TABLE VI Experimental Data AM = - 1 T r a n s i t i o n s of n D i l u t e Gadolinium E t h y l Sulphate at 90°K. . . . . 9 5 TABLE V I I J " g " Values And S p l i t t i n g Parameter For D i l u t e Gadolinium E t h y l Sulphate From AM = - 1 T r a n s i t i o n s 95 TABLE V3H Experimental Data A M = - 2 T r a n s i t i o n s For D i l u t e Gadolinium E t h y l Sulphate At 90°K. 96 TABLE IX AM = - 2 T r a n s i t i o n s For D i l u t e Gadolinium E t h y l Sulphate As Function Of Angle Be-tween D i r e c t i o n Of Magnetic F i e l d And C r y s t a l Symmetry A x i s At 5 ° I n t e r v a l s . . . 96 Facing Page TABLE X Values of g And g x And Splitting Parameters For Dilute Gadolinium Ethyl Sulphate From AM = - 2 Transitions At 90°K 96: TABLE XI AM = - 3 Transitions For Dilute Gado-linium Ethyl Sulphate As'Function Of Angle Between Direction Of Magnetic Field And Crystal Symmetry Axis At 5 ° Intervals 97 TABLE XII Values of g u And g u And Splitting Parameters For Dilute Gadolinium Ethyl Sulphate From AM = - 3 Transitions At 90°K 97 P l a t e P l a t e P l a t e P l a t e P l a t e P l a t e P l a t e LIST OF PEATES Facing Page I a. "g" marker s i g n a l from vide© spectrometer 49 b. "g" marker s i g n a l from vide© spectrometer w i t h r . f . f i e l d modulation on 49 c. "g" marker s i g n a l from double f i e l d modulation spectrometer w i t h l i n e a r detector 49 d. 10 - 9 grams "g" marker s i g n a l from double f i e l d modulation spectrometer w i t h l i n e a r detector 49 I I General view of experimental apparatus 53 I I I View of microwave components 55 IV View ©f electromagnet showing d . c . c o i l s , modulation c o i l s and spectrometer head w i t h the dewars i n the gap 60 V View showing s h i e l d i n g of r . f . a m p l i f i e r and detector w i t h s h i e l d s removed . . . 71 VI View ©f the double modulation head w i t h the dewars i n the magnetic f i e l d . . . 74 V I I View ©f helium temperature s i n g l e modulation head 76 Pacing Page P l a t e V I I I ' View ©f the double modulation head w i t h the dewars removed 77 P l a t e IX a. S t r u c t u r e on A M = - 1 t r a n s i t i o n s due to l a t t i c e d e f e c t s and p a i r s of Gd*3 ions 97 b. 4 of 5 AM = - 3 t r a n s i t i o n s near crossover p o i n t . . 97 c. AM = - 2 t r a n s i t i o n s near crossover p o i n t 97 d. 6 A M = - 2 t r a n s i t i o n s at magnetic f i e l d o r i e n t a t i o n where t o t a l separ-a t i o n i s $00 gauss 97 P l a t e X General view of experimental apparatus . 109 P l a t e XI General view of microwave bench 110 i INTRODUCTION A. Methods Of Measuring The Paramagnetic P r o p e r t i e s Of Matter I n t h i s t h e s i s , we s h a l l be concerned w i t h those magnetic p r o p e r t i e s of ions of the rare e a r t h group, which can be measured u s i n g paramagnetic resonance. Magnetic resonance i s a branch of r a d i o frequency spectroscopy which i s important f o r s o l i d s t a t e s t u d i e s . I t can be c l a s s i f i e d i n t o (a) n u c l e a r , (b) ferromagnetic, (c) antiferromagnetic and (d) paramagnetic resonance. The f i r s t i s concerned w i t h nuclear d i p o l e s while the remainder are concerned w i t h e l e c -t r o n i c d i p o l e s . (b) and (c) d e a l w i t h magnetic systems where the e l e c t r o n i c d i p o l e s are s t r o n g l y coupled togethe-r by ex-change f o r c e s while i n (d) the e l e c t r o n i c d i p o l e s are l o o s e l y coupled together; each paramagnetic i o n i s considered i n d i v -i d u a l l y . There i s one important d i f f e r e n c e between o p t i c a l and r a d i o frequency spectroscopy. The former deals w i t h spectra which are due t o the abso r p t i o n and spontaneous emission of r a d i a t i o n w i t h H i n s t e i n emission, which i s stim-u l a t e d by the presence of r a d i a t i o n of the same frequency, of at most secondary importance. The l a t t e r i s only concerned w i t h the absorption and st i m u l a t e d emission of r a d i a t i o n . As the energy hv> of the r a d i o frequency quantum i s always much smaller than the thermal energy kT, the number i i of o c c u r r i n g processes o f ab s o r p t i o n i s only s l i g h t l y l a r g e r than the number of o c c u r r i n g processes of s t i m u l a t e d emission. This net su r p l u s , which i s i n v e r s e l y p r o p o r t i o n a l t o the temperature T s, c h a r a c t e r i z i n g the s t a t i s t i c a l d i s t r i b u t i o n over the energy l e v e l s concerned, i s what i s detected i n an experiment. The magnetic moment of an i o n a r i s e s from c o n t r i -b u t ions by e l e c t r o n i c and nuclear d i p o l e s ; the e f f e c t of the former predominating. A study of magnetism from the micro-scopic viewpoint consequently deals w i t h the p r o p e r t i e s of e l e c t r o n s and n u c l e i . The paramagnetic p r o p e r t i e s of ions have been inves-t i g a t e d by measurements of the (a) s u s c e p t i b i l i t y , (b) spec-i f i c heat, (c) gyromagnetic r a t i o , (d) Faraday e f f e c t , (e) paramagnetic r e l a x a t i o n and ( f ) paramagnetic resonance. A l l but the l a s t of these methods, study the magnetic prop-e r t i e s of e l e c t r o n s i n atoms or ions' from the macroscopic viewpoint. I n i o n i c c r y s t a l s , these p r o p e r t i e s depend upon the behaviour of the occupied lower e l e c t r o n i c s t a t e s , i n p a r t i c u l a r , t h e i r s e p a r a t i o n , t h e i r r e l a t i v e p o s i t i o n and t h e i r a n i s o t r o p i c behaviour i n a magnetic f i e l d . Para-magnetic resonance, since i t i s e s s e n t i a l l y a spectroscopic method, s t u d i e s these same p r o p e r t i e s from the microscopic viewpoint which enables i t to d i f f e r e n t i a t e between impur-i t i e s . This i s of great value f o r s t u d i e s of the ions o f the rare e a r t h group where i t i s d i f f i c u l t t o o b t a i n pure H i compounds. Moreover, i t i s orders of magnitude more s e n s i -t i v e than any of the other methods. Consequently, i t can de a l w i t h very small q u a n t i t i e s o f paramagnetic substances which i s a l s o of importance f o r rare e a r t h group s t u d i e s . This s e n s i t i v i t y i s s u f f i c i e n t t o permit the p e r t u r b i n g e f f e c t s of the nuclear s p i n and nuclear e l e c t r i c quadrupole moment to be detected. This i s p a r t i c u l a r l y advantageous since t h i s method can be used t o determine n u c l e a r s p i n s , r a t i o of the magnetic moments of two isotopes of the same atom and nuc l e a r moments of those atoms which are not e a s i l y a c c e s s i b l e to other methods. U n f o r t u n a t e l y , i t s u f f e r s from s e v e r a l disadvantages. S i n g l e c r y s t a l s of s u i t a b l e s i z e , which are r e q u i r e d f o r accurate work, cannot always be grown. C e r t a i n substances cannot be i n v e s t i g a t e d because there are no allowed t r a n s -i t i o n s between the l e v e l s which give r i s e to the para-magnetism. At present, i n f o r m a t i o n can be obtained only about the ground s t a t e . Recent work i n d i c a t e s that reson-ances between the ground sta t e and the f i r s t e x c i t e d s t a t e may be detected. B. Magnetic P r o p e r t i e s Of Matter I n The S o l i d State I n the s o l i d s t a t e , atoms are very close together and consequently strong f o r c e s of i n t e r a c t i o n e x i s t between them. For pure elements, there i s an exchange i n t e r a c t i o n between the valence e l e c t r o n s . I n non-metals, the valence e l e c t r o n s are p a i r e d w i t h spins opposing and t h i s produces no magnetic i v e f f e c t . I n pure metals, the valence e l e c t r o n s form an e l e c t r o n gas since they are no longer bound to the atom. The remaining closed s h e l l s e x h i b i t no magnetic p r o p e r t i e s because of the P a u l i e x c l u s i o n p r i n c i p l e . I t can be shown by c o n s i d e r a t i o n of Fermi-Dirac s t a t i s t i c s or exchange de-magnetization that the e l e c t r o n gas has only a very f e e b l e magnetic moment. . Paramagnetism i s found i n s o l i d s only when some o f the atoms co n t a i n an incomplete e l e c t r o n s h e l l . I n i o n i c c r y s t a l s the tendency i s to form ions w i t h completed s h e l l s ( i n e r t gas c o n f i g u r a t i o n ) . Hence i n such compounds only those elements which have incomplete d or f s h e l l s are paramagnetic, since i n these elements incomplete s h e l l s remain a f t e r t h e i r normal valency requirements are s a t i s f i e d . I n covalent c r y s t a l s and covalent complexes the ten-dency i s to form bonds, each bond c o n t a i n i n g a p a i r of e l e c t r o n s w i t h opposed s p i n s . Hence, such c r y s t a l s are not normally paramagnetic. The e l e c t r o n s i n an incomplete d s h e l l f r e q u e n t l y take p a r t i n covalent bonding, the atoms i n v o l v e d e x h i b i t i n g an abnormal valency ( c o o r d i n a t i o n bond). Thus, many atoms which would be paramagnetic i n an i o n i c c r y s t a l , are diamagnetic i n a covalent c r y s t a l . I n p a r t i -c u l a r , covalent compounds of the 3d and l i f s e r i e s are gen-e r a l l y paromagnetic and there are reasons f o r b e l i e v i n g that many t y p i c a l i o n i c c r y s t a l s (e.g. CsTi(S0^) 2-12H 20) are not p u r e l y i o n i c , but that the paramagnetic i o n i s p a r t l y TABLE I T r a n s i t i o n groups of the P e r i o d i c Table Group Z Incomplete S h e l l n I r o n 21 to 29 3d 1 1 1 - 1 0 Palladium 38 to i+6 2|dn 0 - 1 0 Rare E a r t h 57 to 71 4 f n 0 - llj . P l atinum 72 to 78 5 d n 2 - 9 A c t i n i d e 88 to 96 (?) 6 d n 0 - 8 (?) f a c i n g page v c o v a l e n t l y bonded. Conversely, there are some p u r e l y co-valent compounds which e x h i b i t paramagnetism, e.g. NO which has 15 e l e c t r o n s and hence must have at l e a s t one of these e l e c t r o n s unpaired. In the pure metals of these groups, the exchange f o r c e s p l a y the predominant r o l e i n p e n e t r a t i n g the inner s h e l l s and u s u a l l y produce exchange demagnetization. I t i s p o s s i b l e f o r these f o r c e s t o a l i g n the spins of the e l e c t r o n s producing exchange magnetization which r e s u l t s i n the metal having a permanent magnetic moment, i . e . f e r r o -magnetism. This occurs w i t h c e r t a i n elements of the i r o n group such as i r o n , cobalt and n i c k e l . The f i v e t r a n s i t i o n groups of the p e r i o d i c t a b l e are l i s t e d i n Table I . The palladium, platinum and most a c t i n i d e group e l e -ments appear to £orm c o v a l e n t l y bound compounds which gen-e r a l l y e x h i b i t no paramagnetism except i n complexes. The i r o n and rare e a r t h groups form many i o n i c compounds which e x h i b i t paramagnetism and hence these groups have been s t u d i e d e x t e n s i v e l y . I n Table I I , the ions of these two groups are l i s t e d together w i t h t h e i r c o n f i g u r a t i o n of the incomplete s h e l l and the ground s t a t e as given by the Hundt r u l e s . These r u l e s , which determine the l o w e s t - l y i n g term f o r a  given c o n f i g u r a t i o n , are 1. a. Of a l l the terms allowed by the P a u l i p r i n -c i p l e , that one w i t h the maximum m u l t i p l i c i t y l i e s lowest (the m u l t i p l i c i t y i s equal to TABLE I I The Ground States of the Free Ions of the Ir o n and Ion I r o n Group Rare E a r t h Groups E l e c t r o n i c Ground State C o n f i g u r a t i o n Free Ion T i 3 + , 3 d 1 D 3 / 2 V3+ 3 d 2 3 f 2 C r 3 + , V 2* 3d3 S / 2 Mn3 +, C r 2 + 3d 1 * Fe3*, Mn 2 + 3d5 p e 2+ 3 d 6 Co 2* 3d7 S / 2 N i 2 * 3 d 8 \ C u 2 + 3d9 r Zn 2* 3 d l 0 \ E a 3 + kf° Ce3+ kf1 * F S / 2 Pr3+ i | f 2 \ Nd3 + 4 f 3 Pm3+ k& \ Sm3+ k** & H 5 / 2 E a 3 + kf6 7 P 8 Gd3 + kfl S 7 / 2 T b 3 + kfQ ? p 6 Dy3 + 4 f 9 % l 5 / 2 Ho3+ l|flO S I 8 Er3+ 4 f H Tm3 + 4 f 1 2 \ Yb3 + 2 p 7 / 2 Eu3 + kfii Rare . E a r t h f a c i n g page v i 2S + 1 where & i s the t o t a l s p i n quantum number). b. I f s e v e r a l terms have the same maximum m u l t i -p l i c i t y , that term w i t h the greatest E l i e s lowest (I* i s the t o t a l e l e c t r o n i c o r b i t a l quantum number). M u l t i p l e t s a r i s i n g from a c o n f i g u r a t i o n c o n s i s t i n g of l e s s than h a l f of the e l e c t r o n s i n a completed subgroup are u s u a l l y normal (smallest J lowest) and those from a c o n f i g u r a t i o n c o n s i s t i n g of more than h a l f are u s u a l l y i n v e r t e d ( l a r g e s t J l o w e s t ) . ( M u l t i p l e t s are terms formed through the v a r i o u s combinations of a given S and E and are charact-e r i z e d by the quantum number J = E- + S). CHAPTER* I QUANTUM THEORY OP PARAMAGNETISM An e x c e l l e n t summary of both the experimental and t h e o r e t i c a l aspects of paramagnetic resonance i s contained i n the review a r t i c l e by Bleaney and Stevens ( B l l ) . A com-plementary review by Bowers and Owens (B16), c o n t a i n i n g a summary of a l l the experimental r e s u l t s t o date, i s at present i n press. I n t h i s chapter, the t h e o r e t i c a l work des-c r i b e d i n these review a r t i c l e s together w i t h general theore-t i c a l papers by Bleaney (B7»8,10) and Abragam and Pryce ( Al) and s p e c i f i c t h e o r e t i c a l papers on the r a r e e a r t h group ions by E l l i o t t and Stevens (E1,2,3,S7) are summarized. Since the r e s e a r c h described i n t h i s t h e s i s i s experimental, emphasis i s placed on those aspects of the theory which are necessary to describe the r e s u l t s reported i n Chapter IV. These r e s u l t s are concerned w i t h only r a r e e a r t h group i o n s . No mention i s made o f those aspects of the theory which are a p p l i c a b l e to covalent complexes. This aspect i s considered i n papers by Stevens (S9)> and Abragam and Pryce ( A l ) amongst others. 1.1 I n t r o d u c t i o n The magnetic moment of a paramagnetic i o n i s due t o the o r b i t a l and s p i n angular momentum of the unpaired 2 e l e c t r o n s which determine the energy l e v e l s of the i o n . I t i s not unreasonable to expect t h a t these energy l e v e l s w i l l be a l t e r e d when the magnetic i o n i s surrounded by other ions and water d i p o l e s . These i n t e r a c t i o n s w i t h the other c o n s t i t u e n t s of the c r y s t a l are of three types. The i o n i s subjected to a strong inhomogeneous e l e c -t r i c f i e l d c a l l e d the c r y s t a l l i n e or Stark f i e l d which i s due to the other c o n s t i t u e n t s of the c r y s t a l . The main sym-metry of t h i s f i e l d i s determined by the c r y s t a l s t r u c t u r e ; however i t i s not unusual that the d i s t o r t i o n s from t h i s symmetry have a profound e f f e c t on the energy l e v e l s . The e f f e c t s of the c r y s t a l l i n e e l e c t r i c f i e l d i s to l i f t some of the degeneracy of the energy l e v e l s of the f r e e i o n . This e f f e c t w i l l be discussed i n 1.1}.. I t i s p o s s i b l e that the e f f e c t of an exchange be-tween the e l e c t r o n s of the paramagnetic i o n and the other e l e c t r o n s of the c r y s t a l w i l l a l t e r the energy l e v e l s . Such exchange i n t e r a c t i o n s , which can be c a l c u l a t e d only approx-imat e l y , w i l l be mentioned i n 1 . 3 1 . The t h i r d i n t e r a c t i o n couples the paramagnetic ions together i n t o a 'sp i n ' system. I t w i l l be considered i n 1 . 3 2 . I f the dis t a n c e between the paramagnetic ions i s increased by d i l u t i o n , then the l a s t two i n t e r a c t i o n s are decreased and i t i s only necessary t o consider the e f f e c t of the c r y s t a l l i n e e l e c t r i c f i e l d . We then speak of magnet-i c a l l y very ' d i l u t e ' s a l t s . 3 1.2 Paramagnetic Resonance Phenomenon i 1.21 The Resonance Condition When an i o n possessing a magnetic moment, and there-f o r e having a degenerate ground s t a t e , i s p l a c e d i n a steady magnetic f i e l d , the degeneracy i s l i f t e d and the l e v e l s undergo a Zeeman s p l i t t i n g . An o s c i l l a t i n g magnetic f i e l d of s u i t a b l e frequency w i l l induce t r a n s i t i o n s between the Zeeman l e v e l s i f these are allowed. The p r e c e s s i o n frequency of the magnetic moment about the a x i s of tne steady magnetic f i e l d H i s given by V=g-t—JL-\E (1) \i(.TT mc/ where g i s the spectroscopic s p l i t t i n g f a c t o r , g i s one i f only o r b i t a l motion of the e l e c t r o n s e x i s t s and p r e c e s s i o n frequency i s then the same as tha t given by the c l a s s i c a l theorem of Earmor which i s proved i n Van Vleck (V3). I f a c i r c u l a r l y p o l a r i z e d magnetic f i e l d i s a l s o a p p l i e d such that i t a l s o r o t a t e s about H i n synchronism w i t h the angular momentum ve c t o r , i t w i l l e x ert a constant couple on the l a t t e r , e v e n t u a l l y causing i t to t u r n over and reverse i t s p r o j e c t i o n on H. This " f l i p p i n g over" i s accompanied by an absorption of energy from the r a d i a t i o n f i e l d . Quantum mechanically, a f r e e i o n w i t h r e s u l t a n t angular momentum J i n a magnetic f i e l d H has energy l e v e l s corresponding t o the va r i o u s s p a t i a l o r i e n t a t i o n s of J w i t h energies MgPH, where M i s the e l e c t r o n i c quantum number and (3 i s the Bohr magnetbao. The s e l e c t i o n r u l e f o r magnetic d i p o l e t r a n s i t i o n s i s AM = - 1. These t r a n s i t i o n s can be induced by a f r e q -uency such that h » = gfSH (2) which i s i d e n t i c a l t o (1,1) since fl = e h (2a) 1} IT mc When a system o f ions i s i n thermal e q u i l i b r i u m w i t h t h e i r surroundings, the lowest energy s t a t e s have the greater p o p u l a t i o n . Since t r a n s i t i o n s up and down have equal a p r i o r i p r o b a b i l i t y , the net r e s u l t of the a p p l i c a t i o n of resonance r a d i a t i o n i s a ga i n i n energy from the r<padiation f i e l d , and a s h i f t towards a more equal p o p u l a t i o n o f the v a r i o u s l e v e l s . This corresponds to an increase i n the temperature of the system which i s sometimes c a l l e d the " s p i n " temperature as d i f f e r e n t i a t e d from the " l a t t i c e " temperature. The reson-ance a b s o r p t i o n can be detected by the l o s s of energy from the r a d i a t i o n , which causes a damping of the tuned c i r c u i t i n which the paramagnetic substance i s p l a c e d . 1.22 The Pine St r u c t u r e The degeneracy of the ground s t a t e of a paramagnetic i o n i n a c r y s t a l i s u s u a l l y l i f t e d by the c r y s t a l l i n e e l e c -t r i c f i e l d and other i n t e r a c t i o n s , eg. r; f o r C r ^ + i n FIGURE lb ENERGY LEVEL DIAGRAW^ -.S = 3/2 I N I T I A L S P L I T T I N G H I N C R E A S I N G M = + l/2 M=-3/c i ( 3 ) F ( 7 ) F R E E C U B I C FIGURE l a • ENERGY UZTZL DIAGRAM OF THE GROUND STATE SPLITTINGS OF Or3* IN A CRYSTAT.t.THE ELECTRIC FIELD S P I N O R B I T T R I G O N A L facing page 5 5 Cr ( M H 3 C H 3 )•(S0^) 2* 12H 20, the C r 3 + i o n i s surrounded by a d i s t o r t e d octahedron of water molecules which produce an e l e c t r i c f i e l d w i t h predominantly cubic symmetry but w i t h a small component of t r i g o n a l symmetry. The seven-fold degen-erate o r b i t a l ground s t a t e of the f r e e C r 3 + i o n i s s p l i t i n t o a s i n g l e t and two t r i p l e t s by the cubic component of the f i e l d , the s i n g l e t l i e s lowest. Taking i n t o account the s p i n of 3/2 t h i s l e v e l has f o u r - f o l d degeneracy and i s s p l i t i n t o two doublets by s p i n - o r b i t c o u pling. I t can be shown (Kramer's Theorem) that i f the number of e l e c t r o n s i n an i o n i s odd, a l l l e v e l s are doublets and hence magnetic. I f the number of e l e c t r o n s i s even, the l e v e l s may be s i n g l e t s which are non-magnetic. Although the energy d i f f e r e n c e be-tween the lowest and the next h i g h e r o r b i t a l l e v e l i s 0(10^) cm and corresponds to an o p t i c a l frequency, the s p l i t t i n g of the two lowest s p i n doublets i s 0(.2)cm'and corresponds to a microwave frequency. In Figure l a , an energy l e v e l diagram of these s p l i t t i n g s of the ground sta t e of Cr-3+ i s given. I n such a case, i t would be p o s s i b l e , i n p r i n c i p l e , to measure t h i s s p l i t t i n g d i r e c t l y by v a r y i n g the frequency of the a p p l i e d microwave r a d i a t i o n u n t i l an a b s o r p t i o n i s observed. However, i t i s e x p e r i m e n t a l l y u s u a l l y simpler t o apply a magnetic f i e l d , to vary the s p l i t t i n g s and examine the spectrum as a f u n c t i o n of f i e l d a t constant frequency (2.2). I n p r i n c i p l e , i t i s p o s s i b l e to pass from h i g h f i e l d 6 measurements to the zero f i e l d s p l i t t i n g s . As an example of t h i s , we consider an i o n of s p i n 3 / 2 w i t h an i n i t i a l s p l i t t i n g between the doublets M = * l / 2 and - 3 / 2 due t o the c r y s t a l l i n e e l e c t r i c f i e l d . The a x i s of q u a n t i z a t i o n i s taken to be the same as the a x i s of the f i e l d . When the magnetic f i e l d i s a p p l i e d p a r a l l e l to the a x i s , the energy l e v e l s diverge linea-tcly as i s shown i n Figure l b . This i s r e l a t e d t o the f a c t that the a x i s of pre-c e s s i o n does not change when the f i e l d i s a p p l i e d . The s e l e c t i o n r u l e i s A M = * 1 (magnetic d i p o l e t r a n s i t i o n ) . The c e n t r a l l i n e s are stronger since they are due to t r a n -s i t i o n s between sta t e s f o r which the p r o j e c t i o n of the mag-n e t i c moment on the d i r e c t i o n of the microwave magnetic f i e l d i s g r e a t e s t . When the e x t e r n a l magnetic f i e l d i s a p p l i e d at an angle to the a x i s of the c r y s t a l l i n e f i e l d there i s no unique a x i s of q u a n t i z a t i o n because there i s competition between the two f i e l d s . I n the l i m i t i n g case of very large magnetic f i e l d s the a c t u a l s t a t e s tend towards those d e f i n e d by u s i n g the d i r e c t i o n of t h i s f i e l d as the a x i s of q u a n t i z a t i o n . I t i s convenient to describe the a c t u a l s t a t e s u s i n g the same quantum numbers as f o r very l a r g e magnetic f i e l d s . Because the a c t u a l s t a t e s are not pure s t a t e s of a x i a l q u a n t i z a t i o n , the s e l e c t i o n r u l e A M = * 1 no longer holds s t r i c t l y and "forbidden" l i n e s ( A M = ± 2 , * 3 , etc.) appear. These are strongest when the s p l i t t i n g s due t o the e x t e r n a l magnetic f i e l d and the c r y s t a l l i n e e l e c t r i c f i e l d are about the same. > H FIGURE J2 ENERGY L E V E L DIAGRAM, S = 5 , I = 5 /2 I N A STRONG MAGNETIC F I E L D SHOWING THE ALLOWED TRANSITIONS facing page 7 7 Their i n t e n s i t y and p o s i t i o n vary i n a r a t h e r complicated manner w i t h the angle between these two axes. 1 .23 The Hyperfine Structure Hyperfine s t r u c t u r e i s observed i n paramagnetic resonance when the nucleus of the paramagnetic i o n possesses a r e s u l t a n t angular momentum and hence a nuclear magnetic moment which w i l l i n t e r a c t w i t h the magnetic f i e l d of the ele c t r o n s . . This hyperfine s t r u c t u r e i s perturbed when the nucleus a l s o possesses an e l e c t r i c quadrupole moment which w i l l i n t e r a c t w i t h the e l e c t r i c f i e l d gradient at the nucleus. The nuclear magnetic moment w i l l take up 21 + 1 o r i e n t a t i o n s i n the f i e l d of the surrounding e l e c t r o n s . The e x t e r n a l magnetic f i e l d i s neglected since i t i s sev-e r a l orders of magnitude smaller (0(1C-3)gauss). I f a microwave magnetic f i e l d i s a p p l i e d , t r a n s i t i o n s w i l l occur when the e l e c t r o n magnetic moment changes i t s o r i e n t a t i o n . The n u c l e a r magnetic moment w i l l not change i t s o r i e n t a t i o n because i t i s very s m a l l . These changes i n the o r i e n t -a t i o n o f the f i e l d a c t i n g on the nucleus w i l l change the energy of i n t e r a c t i o n and w i l l d i f f e r f o r each of the 2 1 + 1 nuclear o r i e n t a t i o n s thus s p l i t t i n g 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 2 1 + 1 components of equal i n t e n s i t y and spacing. I n Figure 2 , the s i t u a t i o n i s i l l u s t r a t e d when S = I and I = 5 / 2 . The allowed t r a n s i t i o n s correspond to 8 A M = - 1 and A m = 0. The e f f e c t of the nuclear e l e c t r i c quadrupole moment i s to make the hyperfine s t r u c t u r e more complex when the e x t e r n a l magnetic f i e l d i s at an angle to the symmetry a x i s of the c r y s t a l . I t t r i e s t o a l i g n the nucleus along the symmetry a x i s and the complexity a r i s e s from the f a c t t h a t the magnetic f i e l d of the e l e c t r o n s w i l l be simultan-eously t r y i n g to a l i g n i t i n the d i r e c t i o n of the e x t e r n a l magnetic f i e l d . When the two f i e l d s are p e r p e n d i c u l a r , i t can be shown tha t the t o t a l number of p o s s i b l e l i n e s i s 6 1 - 1 and t h a t they are of unequal i n t e n s i t y . 1.3 Eine Width The r e s o l u t i o n of the spectra i n paramagnetic reson-ance spectroscopy, i s not l i m i t e d by i n s t r u m e n t a l e f f e c t s , but r a t h e r by the c r y s t a l l i n e environment. I n t e r a c t i o n s between the paramagnetic ions and the l a t t i c e and between the Various ions themselves are the p r i n c i p a l causes of l i n e width. I t i s convenient to t r e a t these i n t e r a c t i o n s i n terms of r e l a x a t i o n e f f e c t s , c h a r a c t e r i z i n g them, by r e l a x a t i o n times: s p i n - l a t t i c e r e l a x a t i o n time and s p i n -s p i n r e l a x a t i o n . Paramagnetic r e l a x a t i o n * e f f e c t s may be s t u d i e d u s i n g frequencies of the order of the r e c i p r o c a l of the r e l a x a t i o n time. At such f r e q u e n c i e s , the l i n e s are not r e s o l v e d , and the l i n e width at h a l f i n t e n s i t y i s of the 9 order of l/Z sec" where "C i s the r e l a x a t i o n time. Such phenomena has been reviewed by Cooke (C3). I n paramagnetic resonance, where the l i n e s are r e s o l v e d because higher frequencies are employed, i t i s u s u a l l y necessary to reduce the l i n e w i d t h t o improve the r e s o l u t i o n . To achieve t h i s , i t i s necessary t o understand the u n d e r l y i n g processes which we w i l l now o u t l i n e . 1 . 3 1 S p i n - L a t t i c e I n t e r a c t i o n The inverse of the s p i n - l a t t i c e r e l a x a t i o n time i s a measure of the r a t e at which a s p i n reverses d i r e c t i o n and gives or r e c e i v e s a quantum of energy to the l a t t i c e . I f the l i f e t i m e of a s t a t e i s Z then by the u n c e r t a i n t y p r i n -c i p l e i t s energy i s u n c e r t a i n to the order of h/"C and hence the l i n e should have approximately t h i s breadth. Two t h e o r i e s have been developed. The f i r s t i s s e m i - c l a s s i c a l , based on a c o l l i s i o n model. Two processes are assumed to occur. One i s a d i r -ect process i n which quanta of energy are absorbed or emitted by the l a t t i c e . I t i s of the f i r s t order, but o n l y l a t t i c e waves of a c e r t a i n frequency are e f f e c t i v e since i t i s a resonance process. I n the second, the l a t t i c e waves are n o n - e l a s t i c a l l y s c a t t e r e d by the magnetic i o n s , g i v i n g a quasi-Raman e f f e c t of second order. The i n t e n s i t y of the l a t t i c e waves of low frequency which give r i s e t o f i r s t order processes i s proportioned to T, while the number of Raman processes i s proportioned to T^ f o r T ^ 0 and T f o r ,T <S» 9 where 9 i s the D ebeye temperature of the l a t t i c e . E x p e r i m e n t a l l y , i t has been found t h a t at helium temper-atures the f i r s t order processes may predominate while at higher temperatures the Raman processes predominate. Quantum mechanically, i t i s necessary t o consider the e f f e c t of an o s c i l l a t i n g c r y s t a l l i n e e l e c t r i c f i e l d caused by l a t t i c e v i b r a t i o n s which w i l l couple w i t h the o r b i t a l momentum of the magnetic i o n s . I t has been sug-gested that the coupling between the l a t t i c e waves and the spins i s produced by s p i n - o r b i t coupling. I t has al s o been found that the s p i n - l a t t i c e r e l a x a t i o n time i s s t r o n g l y dependent on the sepa r a t i o n between the ground s t a t e and the f i r s t e x c i t e d s t a t e ; "C i s longer as the sepa r a t i o n i n c r e a s e s . S p i n - l a t t i c e r e l a x a t i o n i s the predominant l i n e broadening mechanism at h i g h temperatures, but at tempera-tures below about 1 0 ° K, i t i s u s u a l l y n e g l i g i b l e compared w i t h the s p i n - i n t e r a c t i o n . 1 . 3 2 Spin-Spin I n t e r a c t i o n The other broadening process i s due to the i n t e r -a c t i o n of the magnetic d i p o l e s , which may be assumed to be s i t u a t e d i n a r i g i d l a t t i c e at low temperature. General t h e o r i e s have been developed by Pryce and Stevens (P3) and Van Vleck (V^) on the assumption that the s p i n - l a t t i c e r e l a x a t i o n time i s long and that the e f f e c t of the thermal motion i n broadening the resonance l i n e i s n e g l i g i b l e . Two 11 types of i n t e r a c t i o n between magnetic ions are i d e n t i f i e d ; the d i p o l e - d i p o l e and exchange. S e m i - c l a s s i c a l l y , each i o n may be regarded as a gyroscopic magnet; i . e . i t s angular and magnetic moments are p a r a l l e l , l o c a t e d at a f i x e d p o i n t i n space and i n an e x t e r n a l magnetic f i e l d H of f i x e d d i r e c t i o n . They w i l l process about t h i s d i r e c t i o n and can be regarded as equi-valent to a magnet f i x e d i n t h i s d i r e c t i o n together w i t h a magnet r o t a t i n g i n a plane perpendicular to i t . I f any magnet A i s considered, the steady f i e l d i t w i l l be the r e s u l t a n t of the e x t e r n a l f i e l d and the i n t e r n a l steady f i e l d of the f i x e d magnets as s o c i a t e d w i t h a l l the other magnets. A w i l l t h erefore precess about t h i s r e s u l t a n t f i e l d w i t h a frequency equal to K H where ¥ i s i t s gyromagnetic r a t i o and H i s the r e s u l t a n t steady f i e l d . . There w i l l be a spread i n the p r e c e s s i o n a l frequencies due t o the v a r i a t i o n i n the value and o r i e n t a t i o n of H from magnet t o magnet which w i l l r e s u l t i n a broadening of the l i n e c a l l e d "steady f i e l d broadening." I f two magnets have the same p r e c e s s i o n a l frequen-c i e s , the r o t a t i n g f i e l d from one w i l l be at the co r r e c t frequency to induce t r a n s i t i o n s i n the other, whereas i f the two magnets have d i f f e r e n t p r e c e s s i o n a l frequencies, the r o t a t i n g f i e l d s have l i t t l e e f f e c t . Magnets w i t h the same p r e c e s s i o n a l frequencies tend to induce t r a n s i t i o n s i n each other, reducing t h e i r l i f e - t i m e s i n gi v e n st a t e s and 12 t h i s broadening the resonance l i n e . Such broadening i s c a l l e d "resonance broadening." The quantum mechanical treatment based on these con-cepts enables the area and second and f o u r t h moments to be c a l c u l a t e d . I n the treatment due t o Van V l e c k (Vij.), i t i s found that exchange c o n t r i b u t e s to the f o u r t h moment. Exchange between s i m i l a r ions narrows the moment l i n e s while exchange between d i s s i m i l a r ions w i l l broaden i t . Exchange a r i s e s from the coulomb i n t e r a c t i o n between e l e c t r o n s . I t can be shown that t h i s can be replaced by an i n t e r a c t i o n between t h e i r spins of the form % J ( l + lj.s\«S*2). I n the theory of Pryce and Stevens ( P 3 ), exchange a l s o enters the second mom-ent, because i t i s probably more r e a l i s t i c . Since exchange has not been found to p l a y any r o l e i n ions of the r a r e e a r t h group we do not consider t h i s aspect any f u r t h e r . 1.4 Energy L e v e l s Of The Ion The p r i n c i p a l f e a t u r e s of paramagnetic resonance phenomena was o u t l i n e d i n 1 . 2 . Prom that d i s c u s s i o n ^ one might be tempted to f e e l that t h i s phenomena i s w e l l under-stood. F o r t u n a t e l y , the f a c t t h a t each i o n i n each c r y s t a l -l i n e e l e c t r i c f i e l d must be considered on i t s own m e r i t makes the i n v e s t i g a t i o n of paramagnetic phenomena a p r o l i f i c source of experimental and t h e o r e t i c a l problems. 13 l.lj . 1 General Hamiltonian The Hamiltonian f o r a paramagnetic i o n i n a c r y s t a l can be w r i t t e n as the sum of a number of terms whose e f f e c t on the energy of the i o n are i n decreasing order of magni-tude. An exact s o l u t i o n i s u s u a l l y obtained f o r the l a r g e s t term and the e f f e c t of the remaining terms taken i n t o ac-count by p e r t u r b a t i o n c a l c u l a t i o n s . The order of precedence of these terms v a r i e s w i t h the i o n under c o n s i d e r a t i o n . For ions of the r a r e e a r t h group i n a c r y s t a l l i n e s o l i d subjected to a magnetic f i e l d , the Hamiltonian i s H = H F + H L S + H V * H H + H M „ + H q + H w H F i s the Hamiltonian f o r the f r e e i o n H L S i s the e f f e c t of the s p i n - o r b i t coupling *Y i s the e f f e c t of the c r y s t a l l i n e e l e c t r i c f i e l d H H i s the e f f e c t of the e x t e r n a l magnetic f i e l d i s the e f f e c t i n t e r a c t i o n of the nuc l e a r magnetic moment w i t h the magnetic f i e l d H i s the i n t e r a c t i o n of the nuc l e a r e l e c t r i c q quadrupole moment H y i s the i n t e r a c t i o n of the nucleus w i t h the mag-n e t i c f i e l d of the e l e c t r o n s For the i r o n group Hy H^g whereas f o r the rare e a r t h group H £ S Hy. This i s due to the f a c t that 3d e l e c t r o n s are not as w e l l s h i e l d e d as 14 the 4 f e l e c t r o n s . This i s v e r i f i e d by the f a c t t h a t the o p t i c a l a b s o r p t i o n s p e c t r a l l i n e s of the rare e a r t h group atoms are very sharp. Another way of expressing t h i s i s to say t h a t the 4*" wave f u n c t i o n s do not p r o j e c t very f a r out-side of the atom and so only s l i g h t l y overlap the other atoms even i n the s o l i d s t a t e . To a good approximation, the r a r e e a r t h ions i n the s o l i d s t a t e may be considered as f r e e . 1.42 Hamiltonian Of The Free Ion In Magnetic F i e l d i o n i n a magnetic f i e l d may be enumerated i n order of de c r e a s i n g magnitude. They are as f o l l o w s : The terms which occur i n the Hamiltonian of the f r e e a. The coulomb i n t e r a c t i o n s of the e l e c t r o n s w i t h the nucleus (assumed f i x e d ) and w i t h each other. H + b. The magnetic i n t e r a c t i o n s between the e l e c t r o n spins and the o r b i t s may be w r i t t e n i n the form according t o Van Vleck (V3). Pryce (P4) has used a s p i n - s p i n i n t e r a c t i o n of the form H-to describe the i n t e r a c t i o n s between the e l e c -trons spins which i s based on the usual form 15 H = 2 of d i p o l e - d i p o l e c o upling. The orders of magni-tude are 1 0 2 - 103 cm - 1 f o r the s p i n - o r b i t coup-l i n g and 0(1)cm""1 f o r the s p i n - s p i n c oupling. The i n t e r a c t i o n w i t h the e x t e r n a l magnetic f i e l d i s H,- = ? , eh ( l + 2 s ) , .H = (I + 2$) -H ±J.£> 1 1|.T{ mc 1 where the diamagnetic term e2H 2 f ( x 2 + y 2 ) . 8mc 2 i s omitted, because i t causes only a s h i f t i n these l e v e l s without a l t e r i n g t h e i r r e l a t i v e s e p a r a t i o n . The as s o c i a t e d energy i s 0(1) cm""1, d. The i n t e r a c t i o n between the magnetic moment of the nucleus and the magnetic f i e l d set up by the o r b i t a l and s p i n moments of the e l e c t r o n s i s L r i 3 ~J 3 -The l a s t term i s always zero unless there are S-e l e c t r o n s present, i n which case the other terms are zero. The energy i s 0 (10""2) cm--5-. The e l e c t r o s t a t i c i n t e r a c t i o n between the e l e c -• trons and the nuclear e l e c t r i c quadrupole mom-ent i s H q - e 2 Q . f ^ f ? ( I + D - 3 ( r * r ? ) 2 l 21(21-1) L r i 3 ' r ± 5 • J 16 which i s of magnitude 0(10"^)cm" 1. The i n t e r -a c t i o n of the nuclear quadrupole moment w i t h the c r y s t a l l i n e e l e c t r i c f i e l d i s u s u a l l y n e g l i g i b l e . f . The d i r e c t i n t e r a c t i o n of the nuc l e a r moment w i t h the e x t e r n a l f i e l d i s and has the magnitude 0(10"^-) cm"1. g. The i n t e r a c t i o n of c r y s t a l l i n e e l e c t r i c f i e l d i s H y = - f e V U±,7±>*l) I t s order of magnitude has already been discussed i n l . i | l . This term, w i l l be considered i n d e t a i l i n 1.43. 1.43 Ion I n The C r y s t a l l i n e E l e c t r i c F i e l d We s h a l l now consider the e f f e c t of the i n t e r a c t i o n of the f r e e i o n w i t h the c r y s t a l l i n e e l e c t r i c f i e l d which was mentioned i n l.l|.2g. I n a c r y s t a l , a paramagnetic i o n i s sub-j e c t e d t o a strong, inhomogeneous e l e c t r i c f i e l d which i s u s u a l l y c a l l e d the c r y s t a l l i n e e l e c t r i c f i e l d . This f i e l d a r i s e s from the e f f e c t of the other c o n s t i t u e n t s of the cr y s -t a l and c o n s i s t s of two componenets: a. s t a t i c and b. f l u c t -u a t i n g . The l a t t e r , which i s due to the thermal v i b r a t i o n of the l a t t i c e , has a n e g l i g i b l e e f f e c t on the energy l e v e l s and u s u a l l y c o n t r i b u t e s only to the l i n e width.• The s t a t i c component i s never e x a c t l y known because i t depends on the charge d i s t r i b u t i o n , the o r i e n t a t i o n of 17 the water d i p o l e s and the overlapping of the e l e c t r o n clouds. I f the c r y s t a l s t r u c t u r e i s known from X-ray data then, t o a f i r s t approximation, the f i e l d w i l l have the same symmetry as the c r y s t a l s t r u c t u r e . This f i e l d w i l l produce a Stark s p l i t t i n g of the energy l e v e l s of the f r e e i o n thereby remov-i n g some of the 2J + 1 degeneracy. The degree to which t h i s degeneracy i s l i f t e d has been shown by B e t h e (B^) to depend on the symmetry of the f i e l d . We now f o l l o w the d i s c u s s i o n of t h i s i n B I j l (Bjp). The a p p l i c a t i o n of group theory i s based on the f a c t t h a t S chrodinger 1s equation i s i n v a r i a n t under c e r t a i n transforma-t i o n s of the v a r i a b l e s of the system. Such tra n s f o r m a t i o n s always c o n s t i t u t e a group which may be c a l l e d the symmetry group of the system. The wave f u n c t i o n s o f an n - f o l d degen-erate energy l e v e l w i l l thus be l i n e a r l y transformed amongst themselves by a t r a n s f o r m a t i o n of the symmetry group,,i.e. the wave f u n c t i o n s transform according t o an n-dimensional r e p r e s e n t a t i o n of the group. The r e p r e s e n t a t i o n i s i r r e d u c -i b l e , i f the n-dimensional space spanned by these wave f u n c t -ions does not c o n t a i n an I n v a r i a n t subspace. An i r r e d u c i b l e r e p r e s e n t a t i o n of degree (2J + 1) , D j , of the space r o t a t i o n group i s induced by the (2J + 1) wave f u n c t i o n s o f an atom having an angular momentum J . When such an atom i s placed i n a c r y s t a l , i t s sym-metry group i s no longer the space r o t a t i o n group but one of lower symmetry which i s a sub-group of the o r i g i n a l group. The wave f u n c t i o n s of the atom now transform according to a r e p r e s e n t a t i o n of t h i s new group. I n ge n e r a l , t h i s represen-t a t i o n Dj w i l l be r e d u c i b l e and hence c o n t a i n i n v a r i a n t sub-spaces. An i r r e d u c i b l e r e p r e s e n t a t i o n of the new group i s r e a l i z e d i n each of i t s i n v a r i a n t subgroups. There i s no reason why the wave f u n c t i o n s belonging to d i f f e r e n t i r r e d u c -i b l e r e p r e s e n t a t i o n s should have the same energy and i n gen-e r a l t h i s i s t r u e . The o r i g i n a l l e v e l s of the f r e e i o n w i l l be s p l i t , i n a c r y s t a l , i n t o a number of other l e v e l s which can be c l a s s i f i e d according to the i r r e d u c i b l e represent-a t i o n s of the new group contained by the o r i g i n a l represent-a t i o n ; t h i s number w i l l be equal t o the number of i r r e d u c -i b l e r e p r e s e n t a t i o n s of the new group. I f J i s h a l f i n t e g e r , the re p r e s e n t a t i o n s are double valued and the l e v e l s remain degenerate even w i t h f i e l d s of lowest symmetry. Here "Kramers" degeneracy always remains, which can only be removed by a magnetic f i e l d . We thus see tha t group theory w i l l give the type of l e v e l that can occur the number of l e v e l s and w i t h a d d i t i o n a l i n f o r m a t i o n the r e l a t i v e spacings of the energy l e v e l s . P e r t u r b a t i o n c a l c u l a t i o n s are necessary to o b t a i n the magnitude of the s p l i t t i n g s . Only one s p l i t t i n g need be computed i f the r e l a t i v e separations have been obtained from group theory. For p e r t u r b a t i o n theory, the c r y s t a l l i n e e l e c t r i e f i e l d must be known. I f we assume th a t t h i s p o t e n t i a l s a t i s f i e s E,aplace's equation then i t can be expanded i n 19 s p h e r i c a l harmonics. where (©, 0) i s normalized to u n i t y and defined as (9, 0) = ("D n [ ^ 1 ? ( n - i m . ) i 1 l / 2 ^ . P ^ c o s G ) ^ » ' . L 2(n+|ml).» J f 2 l f n (4) w h e r e P* (x) = U 2 - l ) n • (W n 2 1 1 n j , n+m dx The number of terms i n (1,3) can be reduced considerably by the f o l l o w i n g arguments. For d - e l e c t r o n s , a l l terms f o r which n > i | w i l l have zero m a t r i x elements while f o r f -e l e c t r o n s , the same i s true when n >6. This a r i s e s from the f a c t that i n the e v a l u a t i o n of i n t e g r a l s of the form Jx*V(p dt where X and ip are e l e c t r o n wave f u n c t i o n s , the d e n s i t y X* <f can be expanded i n s p h e r i c a l harmonics and does not c o n t a i n terms where n > 21 (1 i s the quantum num-ber f o r the o r b i t a l angular momentum). By the o r t h o g o n a l i t y of s p h e r i c a l harmonics, the i n t e g r a l vanishes i f V i s a s p h e r i c a l harmonic w i t h n > 21. S i m i l a r l y , i f n i s odd, the i n t e g r a l i s zero because the product *XX i s unchanged by the s u b s t i t u t i o n x,y,z > -x, -y, -z whereas V reverses s i g n . The term f o r n = 0 i s dropped because i t i s an a d d i t i v e constant. To reduce the number of terms f u r t h e r , we must con-s i d e r the symmetry p r o p e r t i e s of the surroundings. For the r a r e e a r t h e t h y l sulphates, w i t h which we are concerned, 20 there i s t h r e e f o l d a x i s of r o t a t i o n symmetry and a r e f l e c t i o n symmetry plane which i s u s u a l l y denoted by 0^* E l l i o t t and Stevens ( E l ) have shown th a t f o r t h i s symmetry V can be w r i t -ten as .V = A 0 + A° ( 3 z 2 - r 2 ) .+ A£ ( 3 f ^ - 3 0 r 2 z 2 + 3 ^ ) + A£ (231z 6 -3l5r2z4 + 105r4z 2 - £ r 6) + A£ ( x 6 - l f r b 7 2 + I f r 2 ^ - y 6 ) (6) . o We now define the types of symmetry from the sub« s c r i p t m of the s p h e r i c a l harmonic which determine the va r i o u s p o t e n t i a l s . I f m = 0 the s p h e r i c a l harmonic has a x i a l sym-metry, while i f ra = - 2 i t has rhombic symmetry, m = * 3 t r i g o n a l symmetry and m = - i | t e t r a g o n a l symmetry. Any pot-e n t i a l c o n t a i n i n g terms of d i f f e r e n t symmetry i s s a i d to have an o v e r a l l symmetry, which i s the greatest symmetry common to a l l the terms. M a t r i x Elements Of The C r y s t a l l i n e F i e l d The method of c a l c u l a t i n g the m a t r i x elements of the c r y s t a l l i n e f i e l d p o t e n t i a l s f o r ions of the rare e a r t h group has been discussed i n d e t a i l by Stevens (S7) and E l l i o t t and Stevens ( E l , E2, E3). I n t h i s s e c t i o n we s h a l l o u t l i n e t h e i r general method of a t t a c k i n g t h i s problem. To evaluate the m a t r i x elements of the c r y s t a l l i n e f i e l d , i t i s necessary to f i n d r e p r e s e n t a t i o n s i n which the st a t e s are eigenstates o f the t o t a l angular momentum J and t o determine the m a t r i x elements j o i n i n g s t a t e s i n d i f f e r e n t J manifolds. These s t a t e s are formed by t a k i n g l i n e a r com-b i n a t i o n s of determinantal product s t a t e s of J+f one-electron s t a t e s . The rare e a r t h group ions show R u s s e l l - Saunders coup l i n g so the product s t a t e s are f i r s t combined to form s t a t e s f o r which I» and S are constants i . e . | 4 f n ; L,S,J> . These are then combined i n t o s t a t e s f o r which J i s constant i . e . l 4 f n ; L , , S,J,J Z^ . Since the b a s i c product s t a t e s are g e n e r a l l y not known, i t i s simpler to determine the matrix elements u s i n g methods which do not r e q u i r e t h i s i n f o r m a t i o n . These methods are based on the f a c t t h a t w i t h i n a manifold of s t a t e s f o r which J i s a constant, there are simple r e l a t i o n s between the m a t r i x elements of p o t e n t i a l operators and the appropriate angular momentum operators. Thus, f o r example, i n s i d e a manifold f o r which J i s constant J ( 3 Z 2 - r 2 ) e * r 2 [ 3 ^ - J ( J + l ) ] (7) These r e l a t i o n s can be v e r i f i e d u s i n g Wigner c o e f f i c i e n t s . Having e s t a b l i s h e d the above r e l a t i o n s , i t i s then necessary to determine the m u l t i p l y i n g f a c t o r . This i s done by u s i n g the f a c t that the p o t e n t i a l f u n c t i o n s do not show any dependence on the s p i n , so t h a t s i m i l a r operator eq u i v a l e n t s h o l d i n s i d e manifolds i n which E i s constant. A convenient s t a t e i n E,S,J,J q u a n t i z a t i o n i s chosen and expressed i n L,,S,J 2,S Z q u a n t i z a t i o n and an equation obtained by equating the two e x p e c t a t i o n values of the f u n c t i o n . Another s t a t e i n L.,S,L ,S„ q u a n t i z a t i o n i s then chosen and z z expressed i n one e l e c t r o n product s t a t e s and again expect-a t i o n values are equated. I n t h i s way, s u f f i c i e n t r e l a t i o n s are obtained t o determine the f a c t o r s i n terms o f r a d i a l i n t e g r a l s over l+f wave f u n c t i o n s . We now consider the c a l c u l a t i o n of m a t r i x elements coupling s t a t e s i n d i f f e r e n t J manifolds. I t i s no longer p o s s i b l e to use operator e q u i v a l e n t s and so the Wigner co-e f f i c i e n t s are used d i r e c t l y . The v a r i a t i o n of the m a t r i x elements w i t h J _ i s obtained i n the same way as above. To c a l c u l a t e the m u l t i p l y i n g constant, i t i s convenient to regard the p o t e n t i a l , say Vg, as a component of a vector i n a space w i t h m = 0. For the example ^J,J Z|VgIJ+2,J^> the c o e f f i c i e n t of / j , J ^ i n v£ | J+2, J^>is equal to a constant m u l t i p l y i n g the appropriate Wigner c o e f f i c i e n t . I t w i l l be independent of J and hence can be c a l c u l a t e d from values of J z . E l l i o t t and Stevens ( E l , E2, E3) have extended the above work of Stevens (S7) by showing general methods of f i n d i n g the ground sta t e of an i o n of the rare e a r t h group i n any c r y s t a l l i n e f i e l d . D e t a i l e d c a l c u l a t i o n s are done f o r the e t h y l sulphates. The e f f e c t s of the n u c l e a r s p i n and quadrupole moment are a l s o taken i n t o account. An important general theorem has been proved by E l l i o t t and Stevens (E2). I t s t a t e s that where A, B are d e f i n e d i n ( 1 , 1 3 ) and holds when the s t a t e s which describe a doublet are e i g e n s t a t e s of the t o t a l angular momentum. This r e l a t i o n i s a constant f o r an i o n independent of i t s environment and the degree w i t h which i t holds i s a good i n d i c a t i o n of whether or not i t i s a good approximation to assume that the c r y s t a l l i n e f i e l d s p l i t t i n g i s small compared w i t h the s e p a r a t i o n of the l e v e l s of the f r e e i o n . E x p e r i m e n t a l l y , i t i s more l i k e l y t o h o l d i n the second h a l f of the l\.f-shell where the f i r s t e x c i t e d l e v e l i s w e l l removed and i m p l i e s that the hyperfine s t r u c t u r e should be i s o t r o p i c when measured at constant frequency. 1.5 Spin-Hamiltonian I t i s d i f f i c u l t to c a l c u l a t e many of the parameters which are i n v o l v e d i n the t o t a l Hamiltonian as i t has been discussed i n l . i | . Furthermore, when they can be evaluated, the accuracy i s poorer than can be obtained e x p e r i m e n t a l l y . Consequently, a more convenient means of d i s c u s s i n g exper-imental r e s u l t s which a l s o permits t h e i r c o r r e l a t i o n w i t h other types of magnetic measurements has been introduced. Abragam and Pryce (Al) have shown that the behaviour o f the energy l e v e l s of a paramagnetic i o n can be represented i n the f o l l o w i n g phenomological manner. This semi-empirical r e p r e s e n t a t i o n i s c a l l e d the " s p i n " Hamiltonian.. I t i s based upon the concept of an " e f f e c t i v e " s p i n S which i s obtained by equating the m u l t i p l i c i t y of the l i n e s observed t o 2S. In the remainder of t h i s s e c t i o n , the var i o u s s p i n -Hamiltonians that have found s u c c e s s f u l a p p l i c a t i o n w i l l be considered. The eigenvalues i n terms of the adj u s t a b l e parameters w i l l a l s o be given. A x i a l Symmetry About z - A x i s . Bleaney (B7) M= g„- P- H zS z + g j / p • ( H X S X + HyS y) + D [ S z 2 - 1 / 3 S ( S + 1 ) " ] (9) where g l ( and g ^ are the values of the spectroscopic s p l i t -t i n g - f a c t o r p a r a l l e l and perpendicular to the z - a x i s . The term i n D represents the e f f e c t of the i n i t i a l s p l i t t i n g due to the c r y s t a l l i n e f i e l d . I n strong f i e l d s where g/3H^D, i t i s convenient to choose the axes so tha t elements i n S x and Sy do not occur i n the major term g/3H'S of the spin-Hamiltonian. I f g i s anisotropic, t h i s i s equivalent to choosing the a x i s about which the s p i n processes as the new z - a x i s . I f the magnetic f i e l d H makes an angle 9 to the symmetry a x i s of the c r y s t a l , then the allowed t r a n s i t i o n s are A M = * 1 . For the t r a n s i t i o n M * - * M - 1 we o b t a i n hV> = g^H + | (2M - 1 ) £ 3 S H 2 • c o s 2 9 - 1 j 25 /D- s i n 2 9\ 2 i P H + ) . X . f 2 S ( S + l ) - 6M(M-1)-?J (10) where g 2 = g„ 2 c o s 2 9 * g j 2 s i n 2 9 (10a) This shows that the f i n e s t r u c t u r e r e s u l t s i n a s p l i t t i n g i n t o 2S l i n e s which are e q u a l l y spaced i n the f i r s t approx-imation. The spacing v a r i e s w i t h angle, f a l l i n g t o zero JL by second order e f f e c t s which v a n i s h i n strong f i e l d s or where ^ cos 0 = 7=7- • This equal spacing i s d i s t u r b e d along the symmetry a x i s . I n the absence of the f i n e s t r u c -ture s p l i t t i n g , a l l of the l i n e s would co i n c i d e at a f i e l d H = Hg which i s used as the second order p e r t u r b a t i o n denom-i n a t o r . I t i s easy to see that the spectrum i s the same Independent of whether the measurements are made at constant frequency, as i s the case i n p r a c t i c e , or at constant mag-n e t i c f i e l d . The i n t e n s i t y of these l i n e s i s P = 7T V > 2 ( g ' £ H ' ) 2 . H . r S ( S + l ) - M ( M - l ) 1 - F ( V , ^ , Avo ) 4kT - - 2S+1 L J where P i s the power absorbed i n the c r y s t a l N i s the number o f paramagnetic ions t H i s the amplitude of the microwave f i e l d H ' COS 2iT P t g' i s the e f f e c t i v e g value along the normal to the a x i s along which the s p i n precesses. F( ^ » **o, A V >) i s a 'shape f a c t o r * depending on the l i n e width p( P' , V>0 , AV) « where i s the h a l f width at h a l f i n t e n s i t y . I t has been assumed that H' i s normal t o g'. I n p r a c t i c e , H* i s normal to H and hence i s at an angle t o g' r e s u l t i n g i n a s m a l l e r r o r i n the absolute i n t e n s i t i e s but none i n the r e l a t i v e i n t e n s i t i e s . T r a n s i t i o n s corresponding to changes i n M of greater than one have been observed which are weaker i n i n t e n s i t y by a f a c t o r of the order of (D/H) 2 and va n i s h only i f the e x t e r n a l f i e l d i s p a r a l l e l to the symmetry a x i s . The i n t e n -s i t y formulae are complicated, and show l i t t l e dependence on the angle between the e x t e r n a l and microwave magnetic f i e l d s . A + For A M = - 2 , the p o s i t i o n of a l i n e corresponding t o the t r a n s i t i o n M «—*• M-2 i s h V> = 2g fl H + 2D (M-l) I 3 g'« 2 • c o s 2 0 - l ( L 1 /Dg„ g. cos 0 s i n 0 \ 2 2 T 1 - X L • — • U s(S« . ) - 2 l|M(M - 2 ) - 3 3 l /Dg A 2 s i n 2 0 \ 2 •[ _ 1 • — . f 2 S ( S + l ) - 6 M ( M - 2 ) - 9 l \ g 2 / 2 s ^ H 0 J ( l 2 > where H Q has the same s i g n i f i c a n c e as f o r the A M = t l t r a n s i t i o n s . 27 1.52 Hyperflne S t r u c t u r e Bleaney (B8) M= A S z I z + B ( S x I x + S y I y ) + P [l z2-1/3.1(1+1)] - Y/^jjH.I (13) where A, B correspond to the i n t e r a c t i o n between the nuclear magnetic moment and the magnetic f i e l d of the incomplete e l e c t r o n s h e l l . P i s due t o the i n t e r a c t i o n o f the nuclear e l e c t r i c quadrupole moment w i t h the gradient of the e l e c t r i c f i e l d at the nucleus. The l a s t term takes i n t o account the d i r e c t e f f e c t of the e x t e r n a l magnetic f i e l d on the nuclear magnetic moment. I f t h i s Hamiltonian i s added to that of 2.51 (I»9) then the strong allowed t r a n s i t i o n s are those f o r which Am = 0 and the p o s i t i o n of the l i n e (M,m)«-*(M-l,m) i s found by adding to the r i g h t hand side of (1,10) the quan-t i t y . Km + _ B f L _ . A 2 + K 2 . + 1 } _ m2] 4 g H H o K2 B 2 A a . m (2M-1) 2g|4H Q K i2 ,„ .2 1 2g/jHQ P 2. (Ik) P c o s 2 9 s i n 2 9 2KM(M-1) ' ^ll\6xY • - [WX+U- 8m2 " l ] P 2 - P % i n 4 9 . [^±jk . m [21(1+1) - 2m 2 - l ] 8 KM(M-l) where K 2 g 2 = A 2g„ 2 c o s 2 0 + B 2 g ^ 2 s i n 2 0 ( l ^ a ) In the f i r s t approximation, the nuclear i n t e r -im-a c t i o n 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 f c ( 2 1 + 1) equally-spaced components w i t h s e p a r a t i o n K between successive l i n e s . There i s no f i r s t order e f f e c t from the quadrupole i n t e r a c t i o n i f P 4 B. The second order e f f e c t s of the quadrupole term are important since they produce a change i n the spacing of the hyperfine l i n e s , which means that P can be determined. I f H i s at angle w i t h the symmetry a x i s of the c r y s t a l , the quadrupole i n t e r a c t i o n w i l l break down the o r d i n a r y s e l e c t i o n r u l e A m = 0 when P « K and the new s e l e c t i o n r u l e s are A m = 0, * 1, * 2 . The i n t e n s i t y of the A m = - 1, * 2 t r a n s i t i o n s are of the order of (P / K ) 2 compared w i t h the A m = 0 t r a n s i t i o n s . Bleaney (B7) d i s -cusses these e f f e c t s i n great d e t a i l . 1.^3 Non-Axial Symmetry I f the symmetry of the c r y s t a l l i n e e l e c t r i c f i e l d i s not a x i a l , then a d d i t i o n a l terms i n the s p i n Hamiltonian must be introduced to account f o r t h e i r departure. I t i s u s u a l to use terms of the form E ( S X 2 - S y 2 ) . T h e i r e f f e c t i s taken i n t o account by second order p e r t u r b a t i o n c a l -c u l a t i o n s . F o r t u n a t e l y , f o r most of the resonance s p e c t r a observed to date, few departures from a x i a l symmetry have been encountered. 29 1.54 s ~ State Ions When the o r b i t a l s t a t e of the f r e e i o n has no degen-eracy the spin-Hamiltonian i s q u i t e d i f f e r e n t . According to the Hund r u l e , t h i s w i l l occur whenever as u n f i l l e d e l e c t r o n i c • s h e l l i s h a l f f i l l e d i . e . Mn 2 +, F e 3 + [^d^, 6S£/ 2] and G d 3 + , E u 2 + [ltf7, Q S 7 / 2 ] We f o l l o w the treatment of Stevens (S8). We already know from group theory (1.43) that the spin-Hamiltonian should only contain even powers of S x, Sy and S z and that i t should a l s o r e f l e c t the same symmetry as that of the c r y s t a l . We s t a r t w i t h a given value of S and a c e r t a i n f i e l d symmetry and consider the even powers of S^11, e t c . I t can then be shown th a t expressions where n ^ n-^  are e x p r e s s i b l e w i t h terms where h £ n-j_. I n t h i s manner i t has been shown by Bleaney and Stevens ( B l l ) that w i t h cubic symmetry and s p i n 5/2 and no nuclear s p i n jtf = gjiH-S+1/6 a Ji S^+Sy^+S^ - 1/5 S(S+1)(3S 2+3S-1)] (15) I n a s i m i l a r manner, E l l i o t t and Stevens (E l ) have found that the f o l l o w i n g spin-Hamiltonian # = g/iH-S + A ° 2 [ 3 S Z 2 - S(S + 1)] + A°^[35SZ^ - 30S(S+1)S Z 2 +25S Z 2 -6S(S+1)+35 2(S+1) 2 ] + A0 6 [231S z 6-315S(S+I)S z ^ + 735S 2 4+IO5S 2(S+I) 2S Z 2 - 525 S ( S + l ) S z 2 + 294 S 2 2 - 5 ( 3 + D 3 + 40 S 2 ( S + 1 ) 2 - 60 S(S+1)] + l>x + i V 6 + s y ) 6 l ( 1 6 ) w i l l f i t the experimental r e s u l t s f o r d i l u t e gadolinium e t h y l sulphate and magnesium n i t r a t e when the e x t e r n a l mag-n e t i c f i e l d i s p a r a l l e l to the symmetry-axis of the c r y s -t a l . 1 . 5 5 Anomalous Case I n a number of resonance s p e c t r a , i t has been ob-served t h a t , contrary to e x p e c t a t i o n , (B10,C4) the maximum i n t e n s i t y occurs when the microwave magnetic f i e l d was p a r a l l e l to the steady magnetic f i e l d producing the Zeeman s p l i t t i n g s . This phenomena has been explained i n the f o l -lowing way. The ground s t a t e of such ions i s a doublet whose l e v e l s c o n t a i n s t a t e s d i f f e r i n g i n J z by u n i t y but the normal type of resonance t r a n s i t i o n i s not allowed since the m a t r i x elements of J x and Jy between the two states are zero. A d i s t o r t i o n of the c r y s t a l l i n e l a t t i c e produced by the J a h n - T e l l e r e f f e c t ( J l ) (C4) w i l l remove the degeneracy of the l e v e l s , admixing the s t a t e s of the two l e v e l s , p e r m i t t i n g allowed t r a n s i t i o n s when the micro-wave and Zeeman magnets f i e l d s are p a r a l l e l . This s i t u a t i o n occurs f o r P r 3 + ^ f 2 , 3H^"j. I t has been s t u d i e d by Bleaney and S c o v i l (BIO) i n the d i l u t e e t h y l sulphate and Cooke and Duffus (€ f l ) i n the d i l u t e mag-nesium n i t r a t e . I t has a l s o been found by Bleaney, L l e w e l l y n , Pryce and H a l l (Blij.) to occur i n plutonium d i l -uted w i t h ( U 0 2 ) R b ( N 0 3 ) 2 i These workers have found t h a t the resonance r e s u l t s can be i n t e r p r e t e d u s i n g the f o l l o w i n g spin-Hamiltonian and f i c t i t i o u s s p i n S = l / 2 M = S„ |& H 2S Z + A S Z I 2 + P [ l z 2 - 1/3 1 ( 1 + 1 ) ] * AcSx + A y S y (17) where £ 2 = A J C 2 + * y 2 ( 1 ? a ) and the allowed t r a n s i t i o n s are given by n * = [(g„ p H cos 9 + Am) 2 + A 2 ] ^ 2 (18) where 9 i s the angle H makes wi t h the c r y s t a l a x i s and ef-f e c t of quadrupole i n t e r a c t i o n term has been neglected. The l a s t two terms represent the e f f e c t of departure of the c r y s t a l l i n e e l e c t r i c f i e l d from the f u l l summetry a s s o c i a -ted w i t h the c r y s t a l s t r u c t u r e . The l i n e shape of such t r a n s i t i o n s i s always as-symmetric; the i n t e n s i t y r i s e s slowly on the low f i e l d side and f a l l s s harply on the h i g h f i e l d s i d e . I t can be accounted f o r by assuming a random d i s t r i b u t i o n of s t r a i n s i n the c r y s t a l , g i v i n g a Gaussian d i s t r i b u t i o n centred on zero f o r the d i s t o r t i o n parameters A«k. and A T . i f the t r a n s i t i o n p r o b a b i l i t y i s assumed to be proportional, to A , the sharp cut-off In the Intensity on the high f i e l d side corresponds to the Gaussian d i s t r i b u t i o n of the d i s t o r t i o n energy. Consequently, the correct place to experimentally measure the p o s i t i o n of each resonance i s at the high f i e l d l i m i t . CHAPTER II PARAMAGNETIC RESONANCE SPECTROMETERS 2.1. Introduction To perform an experimental study of paramagnetic resonance phenomena, a suitable spectrometer i s required. The design w i l l be determined by the type of experiment to be performed, the s e n s i t i v i t y required, equipment avail a b l e , finances available and the taste of the designer. The fund-amental objective i n spectrometer design i s usually to achieve maximum, s e n s i t i v i t y . In t h i s chapter, the basic p r i n c i p l e s of spectrometer design and t h e i r a p p l i c a t i o n by various de-signers are•discussed i n d e t a i l . The object of t h i s survey was to gain an understanding of the l i m i t a t i o n s of the ex i s t i n g designs i n the hope that a new spectrometer of greater s e n s i t i v i t y could be designed. Such a spectrometer has been developed and constructed. Its design i s described i n t h i s chapter. (2 . 7 4 ) . F i n a l l y , a comparison i s made of ex i s t i n g types of spectrometers on the basis of s e n s i t i v i t y , s t a b i l i t y , type of operation, and ease of operation. 2.2. General Requirements The e s s e n t i a l requirements of any paramagnetic resonance spectrometer are to place a paramagnetic substance 34 (usually a crystal) In a region where It can absorb electro-magnetic r a d i a t i o n , to apply a magnetic f i e l d , and to detect any absorption of power from the rad i a t i o n f i e l d as a func-t i o n of the magnetic f i e l d strength. A v e r s a t i l e spectrometer should operate over a wide range of temperatures. The temperature at which an i n v e s t i -gation i s carried out w i l l depend on the relaxa t i o n times of the substance under consideration. ( B l l ) . The best signals are obtained when the relaxation time Z i s long enough to permit the absorption of energy but not so long as to cause saturation. This i s usually of 0(10""^) seconds and can be obtained at room or oxygen temperatures f o r chemical com-pounds containing ions of the i r o n group whereas the rare earth group ions, with the exception of gadolinium, require hydrogen or helium temperatures. Paramagnetic resonance spectrometers are generally operated at microwave frequencies since i t i s t h e o r e t i c a l l y desirable (S2) to s p l i t the energy l e v e l s as f a r apart as possible. Moreover, the s e n s i t i v i t y i s proportional to the operating frequency ( 1 1 , 2 0 ) . Large magnetic f i e l d s are con-sequently required i f the ground state i s a KKrgmers doublet. Table I I I shows the corresponding values of V and V f o r d i f -ferent values of H when g = 2 , i . e . free electron. 35 Table III hV) = g (3 H V kilomegacycles = 2.80lj.0 H^iiogauss •^oersteds 256.63 1069.9 3 ,566.3 8 ,918.8 10,699.0 Vkmcs 1 3 10 25 30 A era 30 10 3.0 1.2 1.0 Experiments are performed i n small magnetic f i e l d s using radio frequency techniques when information about the various terms i n the Hamiltonian and the ground state energy leve l s i s desired. This region w i l l not be considered i n t h i s thesis although much work i s s t i l l required before a complete understanding of resonance phenomena i s achieved. Measurement techniques at microwave frequencies were highly refined during the war as a r e s u l t of the importance placed on the development of radar. These war-time investigations have been f u l l y documented and pub-l i s h e d as the Radiation Laboratory Series. This material has been found to be a valuable source of information on microwave and ele c t r o n i c techniques. The c i r c u i t r y i s of the d i s t r i b u t e d parameter type, instead of the lumped para-meter type used at radio frequencies. The a p p l i c a t i o n of these techniques to gas spectroscopy has been reviewed by Bleaney (B6), and Gordy (Gl) and to paramagnetic resonance by Bleaney and Stevens ( B l l ) . The existence at the end of the war of large quan-t i t i e s of radar equipment at 1 0 , 3 , and 1 . 2 5 cm. l e d to i t s e x p l o i t a t i o n f o r s c i e n t i f i c purposes. Consequently, most gas and paramagnetic resonance spectroscopy has been per-formed at these wavelengths. In paramagnetic resonance, i t i s necessary to place the sample i n a high Q tuned cavity because (a) the sample i usually a small, single c r y s t a l and the highest s e n s i t i v i t y i s obtained by concentrating the microwave magnetic f i e l d i n i t . Saturation e f f e c t s are increased by doing t h i s ; however they are not as serious as i n gas spectroscopy because here we are dealing with the t r a n s i t i o n p r o b a b i l i t i e s associated - 2 0 with magnetic dipoles (*»* 10 emu.) rather than with elec-t r i c dipoles (A»10~ 1 0emu.), and (b) the sample should be l o c -ated i n a strong, homogeneous s t a t i c magnetic f i e l d . Most paramagnetic resonance spectrometers are oper-ated at constant frequency due to the l i m i t e d tuning range of microwave power sources (Kij.) and because high Q tuned ca v i t i e s are required. It i s convenient to lock the freq-uency of the source e l e c t r i c a l l y to the frequency of the cavity (P-3), R2) . Resonances are thus investigated as a function of magnetic f i e l d at constant frequency. To gain information concerning the r e l a t i o n between the c r y s t a l symmetry axis and the d i r e c t i o n of the c r y s t a l -l i n e e l e c t r i c f i e l d , i t i s necessary to rotate either the c r y s t a l about one of i t s axes or the magnetic f i e l d with res-pect to the microwave magnetic f i e l d . Rotation of the cry-s t a l i n the cavity has the disadvantage that i t upsets the tuning of the cavity. This necessitates recheeking the tuning a f t e r each r o t a t i o n and remeasuring the frequency. I f the external magnetic f i e l d i s rotated then the inten-s i t y of the sig n a l i s a function of the r o t a t i o n since i t i s proportional to the square of the sine of the angle be-tween the magnetic f i e l d and the microwave magnetic f i e l d . It i s preferable but not e s s e n t i a l to be able to perform both of these operations. 2.3 Noise The s e n s i t i v i t y of a l l measurements i s l i m i t e d by spontaneous fluctuations c a l l e d noise which are a manifest-a t i o n of the s t a t i s t i c a l nature of physical phenomena. For el e c t r o n i c measurements, t h i s noise i s due to the random motion of the electrons. Noise o r i g i n a t i n g from mechanical v i b r a t i o n i s not considered although i t may be troublesome to eliminate. We assume that any sources of mechanical v i b r a t i o n can be eliminated. E l e c t r o n i c noise i n a paramagnetic resonance spectro-meter arises from three sources. a. Klystron or o s c i l l a t i o n noise b. Detector or converter noise c. Amplifier noise 38 In general, r e f l e x klystrons are used as the source of microwave power i n paramagnetic resonance spectroscopy because of t h e i r frequency s t a b i l i t y (3.26). We confine ourselves to a few .comments about the noise generated i n them. The theory of the noise i n r e f l e x klystrons i s d i s -cussed by Knipp and Kuper (Kif) and van der Z i e l (V2). I t i s s u f f i c i e n t to mention the following pertinent experi-mental f a c t s . 1. The noise-signal power r a t i o i s decreased only s l i g h t l y when operating at frequencies below the centre of a mode whereas i t increases by at l e a s t a factor of two at the high frequency half-power point. 2. Provided that the o s c i l l a t o r i s not overloaded and properly matched, the noise signal power r a t i o i s indep-endent of the mode of operation. Sligh t improvement may be obtained by the use of higher modes i . e . less negative r e f l e c t o r voltages. 3. The noise-signal power r a t i o decreases as the intermediate frequency i s increased. It i s usually found that k l y s t r o n noise does not appreciably affect the t o t a l noise of those systems i n which we are interested and, hence, we w i l l not consider i t further. In the microwave region, semi-conductor c r y s t a l diodes are generally used as power detectors or frequency converters. A bolometer can be used as a power detector. It does not suffer from the " f l i c k e r " or excess noise e f f e c t s that occur i n c r y s t a l s ; however, the upper l i m i t of i t s frequency response i s about 100 cycles. A spectro-meter using a bolometer i s considered i n 2.71. The theory of semi-conductor noise has been d i s -cussed by Torrey and Whitmer (T2) and van der Z i e l (V2), amongst others. Three noise components can be distinguished Johnson or thermal, shot and f l i c k e r . The f i r s t two are independent of frequency and f a i r l y well understood. No s a t i s f a c t o r y t h e o r e t i c a l explanation of " f l i c k e r 1 1 noise has been advanced. I t has been found experimentally, that the " f l i c k e r " noise power i s inversely proportional to the frequency and that i t i s of the same magnitude as the John-son and shot noise at a frequency 0(10 ) cycles. I t would, thus, be desirable to operate the c r y s t a l converter at f r e -quencies i n excess of 10^ cycles i f minimum c r y s t a l noise were the only experimental consideration. I t should be noted that t h i s upper frequency l i m i t has been set i n a wide range from 10^ to 2 x 10? cycles. It can be shown that the integrated " f l i c k e r " noise i n a given bandwidth i s reduced by a factor of 0(10^) when the band centre i s changed from 1 0 2 cycles to 10^ cycles. This c a l c u l a t i o n i s based on the assumption that the amplitude of the signal displayed on the oscilloscope i s proportional to the power absorbed i n the cavity by the t r a n s i t i o n . This means that the c r y s t a l converter operates as a square law detector when the ampli-tude of the modulation of the power l e v e l i s small compared with the power l e v e l . This assumption i s j u s t i f i e d by the f a c t that the r e l a t i v e i n t e n s i t i e s of the observed signals agree with the square 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 for the t r a n s i t i o n s . It has not been d i r e c t l y v e r i f i e d experi-mentally. L i t t l e improvement i s gained above this hbecause the " f l i c k e r " noise i s probably n e g l i g i b l e . The e l e c t r o n i c amplifiers used with c r y s t a l video detectors or converters can be e a s i l y designed to have noise figures of about 2 db. at frequencies up to 10 racs. Since the o v e r a l l noise figure of the c r y s t a l converter and ampli-f i e r i s about 15 db, very s l i g h t improvement i s achieved by decreasing the noise figure of the amplifier. This means that the matching network between the c r y s t a l and the input stage at low frequencies (-^.lOvmcs) i s n o n - c r i t i c a l and con-siderable mismatch can be tolerated without introducing ad-verse e f f e c t s on the o v e r a l l noise figure of the system (S9,V5,L1). Both Va l l e y & Wallman (VI) and van der Z i e l (V2) give a comprehensive analysis of the problems of noise and the design of matched low noise figure input amplifiers f o r c r y s t a l detectors. Prom the above considerations i t should be obvious the ultimate achievable s e n s i t i v i t y of a paramagnetic reson-ance spectrometer i s determined by the c r y s t a l noise. It follows that one of the primary design objectives i s to detect the resonance signal at a frequency where the noise from the c r y s t a l converter can be minimized. 2.4 • General C l a s s i f i c a t i o n Having discussed the problems which must be consid-ered the design of a paramagnetic resonance spectrometer*, we now c l a s s i f y the e x i s t i n g designs. I f video presentation of resonance phenomena i s desired then either the frequency or the magnetic f i e l d must be r e p e t i t i v e l y swept, i n time, over a small region of frequency or f i e l d . This leads, to the c l a s s i f i c a t i o n accord-ing to whether frequency or f i e l d modulation i s employed. The alternative to video i s recorder presentation. In theory, i t should be possible to sweep l i n e a r l y i n either frequency or magnetic f i e l d , as i s done i n nuclear magnetic resonance, and hence record detector output as a function of either frequency or magnetic f i e l d . I t has already been mentioned that the frequency cannot be swept over a s u f f i c -ient range with a r e f l e x klystron and i n any case the tuned cavity would have to be simultaneously kept i n tune with the klystron. At microwave frequencies, the magnetic f i e l d i s always swept when paramagnetic resonance phenomena i s d i s -played on an automatic pen recorder. Video and recorder display can be best c l a s s i f i e d by the term band width; I t has been shown (SlO).that the band-width required to reproduce accurately the l i n e shape of the resonance phenomena must be about 100 times the r e p e t i t i o n frequency of the modulation. Video and wide band operation are synonomous terms. The band width also depends on the amplitude of the modulation. As the amplitude or frequency-i s increased the bandwidth must correspondingly be increased, since i t i s proportional to dH-i . ^ ± = A - ^ c o s ^ t ( 1 9 ) where H x i s the amplitude of the f i e l d modulation at time T A-L i s the maximum amplitude of the. f i e l d modulation «1 i s the frequency of the f i e l d modulation Strandberg (S10) gives an analysis of t h i s e f f e c t . When recorder display i s u t i l i z e d , the bandwidth can be greatly reduced without a f f e c t i n g the lin e shape. Hence the term narrow band operation. The band width can then be of the order of l/lOOth the modulation frequency. Here, also, the signal amplitude depends on the amplitude of the modulation and should be less than or equal to the l i n e width of the signal (Sij.) i f undesirable e f f e c t s are to be eliminated. S t a b i l i t y i s a design c r i t e r i o n , which i s re l a t e d to band-width considerations. In general, the s t a b i l i t y of a system must be increased as the bandwidth i s decreased since, although the s e n s i t i v i t y i s greater i n narrow band operation, the time required to obtain the same amount of information as i n wide band operation i s increased. It i s convenient to speak of short and-or long term s t a b i l i t y thus r e l a t i n g s t a b i l i t y . a n d bandwidths. Another useful design c r i t e r i o n i s the concept of r e p r o d u c i b i l i t y . This i s always clo s e l y r e l a t e d to the MICROWAVE POWER SOURCE RESONANT CAVITY -^NC^-SAMPLE \ C R Y S T A L D E T E C T O R MICROAMMETER OR GALVANOMETER 1 .C.FIELD 1 FIGURE 3a POINT-TO-POINT PARAMAGNETIC RESONANCE SPECTROMETER MICROWAVE POWER SOURCE 7 R E S O N A N T CAVITY FIGURE 3b -^AND^-S A M P L E C R Y S T A L D E T E C T O R VIDEO AMPLIFIER -0 z H fi FIELD MODULATION | u_, • 1—1 D.C.FIELD P H A S E SHIFT NETWORK VIDEO PARAMAGNETIC RESONANCE SPECTROMETER 43 accuracy with which measurements can be made. In general, i t i s desirable to have the r e p r o d u c i b i l i t y greater than the accuracy although i t i s not infrequent that the accuracy i s li m i t e d by the r e p r o d u c i b i l i t y . 2.£. Point-to-Point Spectrometer The point-to-point method f u l f i l l s the basic require-ments of a spectrometer but i s considered obsolete because of the slowness with which r e s u l t s can be obtained. A block diagram of i t s components i s shown i n Figure 3 a . Two types of operation are possible; either the r e f l e c t i o n or trans-mission c o e f f i c i e n t of the resonant cavity can be measured as a function of magnetic f i e l d strength. Bleaney and Stevens (B l l ) give the necessary formulae f o r cal c u l a t i n g the mag-ne t i c absorption with t h i s method. Although i t i s very slow and tedious t h i s i s probably the best method of obtaining accurate data about the shape of resonance l i n e s . 2.6. Frequency Modulation Spectrometers Frequency modulation spectrometers employ eithe r e l e c t r i c a l or mechanical methods of modulating the output frequency of the klystron. The mechanical method introduces mechanical i n s t a b i l i t i e s which manifest themselves as elec-t r i c a l i n s t a b i l i t i e s i . e . noise. Frequency modulation of a klystron introduces an excess of low frequency noise which severely l i m i t s the s e n s i t i v i t y of the system. Moreover, the power varies considerably over a mode. Consequently, FIGURE k OSCILLATOR f 2 MICROWAVE POWER SOURCE Ifl FIELD MODULATION"!-RESONANT CAVITY P H A S E SHIFT NETWORK fl S A M P L E X 1 CRYSTAL DETECTOR Ifl FIELD MODULATION^-o E X T E R N A L RESONANT CAVITY CRYSTAL DETECTOR O AMPLIFIER fl AMPLIFIER 2 f 2 BALANCED BRIDGE SECOND DETECTOR BALANCED BRIDGE PARAMAGNETIC RESONANCE SPECTROMETER such systems are not very s a t i s f a c t o r y from the point of view of s e n s i t i v i t y . Also the s t a b i l i t y i s poor. 2.61. Line Shape Spectrometer Weidner and Whitmer (Wl) have constructed a frequency modulated spectrometer. Since the modulation r e p e t i t i o n rate i s very low (1 cps) i t i s e s s e n t i a l l y an automatic recording d.c. or point-to-point spectrometer and, hence, has the same s e n s i t i v i t y . S i m i l a r l y , i t also has the advan-tage of accurately reproducing line-shapes on a recorder. 2.62. Balanced Bridge Spectrometer Bagguley and G r i f f i t h s (Bl) have developed a f r e -quency modulation system employing a balance method which uses two resonant c a v i t i e s . A block diagram of t h i s spectro-meter i s shown i n Figure 4 . This method was developed with the aim of eliminating the " f l i c k e r " noise i n the.,crystal detector since the r e f l e c t o r was modulated at 1 mcs. The system has a number of inherent disadvantages. For optimum s e n s i t i v i t y , the Q of the two c a v i t i e s should be the same. This i s not possible since one cavity contains a sample which loads i t decreasing i t s Q. Moreover, since most experiments are performed at low temperatures, the Q»s of the c a v i t i e s would not be the same even i f they were both unloaded except at the expense of duplication of low temperature equipment. The use of two c r y s t a l converters also increases the noise figure of the system. This spectrometer appears to be no O CO MICROWAVE POWER SOURCE RESONANT CAVITY AND 7 S A M P L E HIGH Z BOLOMETER D.C. BRIDGE T fi FIELD MODULATION D.C. F I E L D T P H A S E SHIFT NETWORK f, AMPLIFIER P H A S E SENSITIVE D E T E C T O R GALVANOMETER FIGURE 5 BOLOMETER PARAMAGNETIC RESONANCE SPECTROMETER 45 more sensitive than a video spectrometer and suffers from the disadvantage that i t i s much more d i f f i c u l t to operate and i s l e s s stable. 2 .7. F i e l d Modulation Spectrometers In a f i e l d modulation spectrometer, the frequency of the klystron i s fi x e d and preferably looked to the resonant frequency of the cavity. The amplitude of the f i e l d modul-ation i s greater than the h a l f l i n e width In wide band oper-ation and about equal to or le s s than i t i n narrow band operation. F i e l d modulation eliminates the excessive noise . and i n s t a b i l i t i e s introduced by frequency modulation. How-ever, i t i s more d i f f i c u l t to eliminate the c r y s t a l " f l i c k e r " noise using t h i s technique. One successful method of achiev-ing t h i s Is described i n 3«74» This d i f f i c u l t y a r ises from the fact that i t Is impossible to modulate the f i e l d of an electromagnet to the amplitude required at frequencies much i n excess of one k i l o c y c l e because of the a.c. h y s t e r i s i s loss i n the i r o n of the magnet. The detection must occur at low frequencies where the c r y s t a l converter Is excessively noisy. 2.71. Bolometer Spectrometer A paramagnetic gas resonance spectrometer using a bolometer has been developed by Beringer and Castle (B2,B3). A block diagram of t h i s arrangement i s shown i n Figure 5 . This i s probably the most sensitive spectrometer that has been developed because there i s no problem of excess c r y s t a l MICROWAVE POWER S O U R C E f A MICROWAVE POWER S O U R C E f B | fi^ FIELD M O D U L A T I O N ! - ! MATCHING LOAD E MAGIC T E E H MAGIC T E E 7 R E S O N A N T CAVITY ( A N D SAMPLE fl FIELD MODULATION n  C R Y S T A L D E T E C T O R C R Y S T A L D E T E C T O R P H A S E S H I F T NETWORK f l AMPLIFIER BAND CENTRE f A - f B FIGURE 6 SUPERHETERODYNE PARAMAGNETIC RESONANCE SPECTROMETER AMPLIFIER fl S E C O N D D E T E C T O R k6 n o i s e . However, i t does s u f f e r from the disadvantage t h a t i t can only be used f o r narrow band ope r a t i o n due t o the l i m i t e d frequency response of a bolometer (T2). Moreover I t r e q u i r e s c r i t i c a l adjustment t o keep i n balance and i s not too s t a b l e since a d.c. bridge i s employed. I t i s r a t h e r strange t h a t , i n view of i t s reported s e n s i t i v i t y , i t has only been used to study paramagnetic resonance i n gases ( B l l ) . 2.72. Superheterodyne Spectrometer The a p p l i c a t i o n of the superheterodyne p r i n c i p l e to paramagnetic resonance spectrometer design has been discussed by s e v e r a l authors. T h e o r e t i c a l l y , i t should be a very s e n s i t i v e method but In p r a c t i c e i t f a l l s short of t h i s l i m i t by a wide margin,, s u f f e r i n g s e r i o u s l y from i n s t a b i l i t y s ince not only must one k l y s t r o n be locked to the resonant f r e q -uency of the c a v i t y but another k l y s t r o n must be locked at a constant frequency d i f f e r e n c e to the f i r s t one. Further-more, excessive noise seems to be introduced at the c r y s t a l converter. This noise i s probably due to low frequency beats between the two s i g n a l s which are caused by the f a c t that any o s c i l l a t o r does not generate a monochromatic spectrum ( f i n i t e Q of tank c i r c u i t ) . T h e o r e t i c a l l y , the use of a balanced mixer should e l i m i n a t e such noise (T2,M3) but t h i s does not seem to be achieved i n p r a c t i c e . A block diagram of a super-hetrodyne used by Schneider and England (SI) i s shown i n Figure 6. Recently, Hirshon and Fraenkel (H^) have reported an improved v e r s i o n . They have taken extreme care to FIGURE 7 MICROWAVE POWER S O U R C E f2F IELD MODULATION] I ' f i FIELD MODULATION \ > RESONANT CAVITY A N D S A M P L E fi FIELD MODULATION^ 1 Z± 1 f2 FIELD MODULATION 1 P H A S E S H I F T N E T W O R K fl C R Y S T A L D E T E C T O R P H A S E SH IFT NETWORK f 2 LINEAR D E T E C T O R f 2 t Af AMPLIFIER P H A S E SENSIT IVE D E T E C T O R f. AMPLIFIER DOUBLE FIELD MODULATION PARAMAGNETIC RESONANCE S P E C T R O M E T E R s t a b i l i z e the system; but u n f o r t u n a t e l y i t s u f f e r s from large spurious i n s t a b i l i t i e s which l i m i t i t s s e n s i t i v i t y . 2 . 7 3 . Video Spectrometer 'The s i n g l e modulation or video spectrometer i s the simp-l e s t and f a s t e s t instrument t o operate and i s probably the most widely used ( B l l , S2). A l a r g e b l o c k diagram of one i s shown i n Figure 3 b . I t can be used on wide or narrow band; however, i t s s e n s i t i v i t y i s poor since the modulation f r e -quency i s l i m i t e d to below 1 0 3 c y c l e s (2 . 7 ) and hence " f l i c k e r " c r y s t a l noise i s a s e r i o u s problem. I t s s t a b i -l i t y i s r e l a t i v e l y good. 2 . 7 4 . Double F i e l d Modulation Spectrometer The double modulation spectrometer shown i n Figure 7 and developed by the author i n t h i s l a b o r a t o r y (S3) has a l l the advantages of the video type ( s i m p l i c i t y , s t a b i l i t y , wide and narrow band operation) w i t h the added advantage of g r e a t l y increased s e n s i t i v i t y since i t e l i m i n a t e s most of the excess c r y s t a l n o i s e . The p r i n c i p l e of i t s design i s not new. I t was t r i e d i n microwave gas spectroscopy by Hartz and van der Z i e l (HI) without notable success. While we were i n the process of a p p l y i n g i t to paramagnetic resonance spectro-scopy at microwave f r e q u e n c i e s , Smaller and Y a s a i t i s (S5) reported the s u c c e s s f u l a p p l i c a t i o n o f i t t o paramagnetic resonance spectroscopy at r a d i o frequencies (10 mcs - 500 mcs). I n t h e i r a p p l i c a t i o n , i t was used to e l i m i n a t e f l i c k e r noise i n the a m p l i f i c a t i o n system. The g a i n i n s e n s i t i v i t y achieved at r a d i o frequencies "by t h i s method i s not l a r g e . Moreover, since the l i m i t i n g s e n s i t i v i t y i s p r o p o r t i o n a l to the d e t e c t i o n frequency, we w i l l not consider t h e i r method f u r t h e r . At r a d i o f r e q u e n c i e s , t h i s technique i s r e a l l y only a complicated approach to e l i m i n a t i n g the " f l i c k e r " noise I n the e l e c t r o n i c tubes. This e f f e c t can be e l i m i n a t e d much more simply by modulating at a s i n g l e frequency i n excess of 10^ c y c l e since there i s no i r o n necessary to produce the r e q u i r e d magnetic f i e l d s , which otherwise complicates the matter. I n a video type spectrometer, the magnetic f i e l d i s modulated at a frequency f]_ ( u s u a l l y l e s s than 100 cycles) w i t h amplitude (greater than the l i n e width at h a l f amplitude). I f the output of the c r y s t a l converter i s a p p l i e d , a f t e r a m p l i f i c a t i o n , on the Y p l a t e s of an o s c i l -loscope and the X p l a t e s are modulated at the frequency f, w i t h the proper phase then any resonances i n the r e g i o n (H - H^) where H i s the magnetic f i e l d , are d i s p l a y e d on the screen as s t a t i o n a r y s i g n a l s . I f we a l s o modulate the magnetic f i e l d a t a frequency f 2 w i t h amplitude H 2 ( l e s s than or equal t o the l i n e wid$h) :, feed the output of the c r y s t a l converter i n t o an a m p l i f i e r w i t h band centre at f 2 and band width o f about 100 £-±, r e d e t e c t at video frequencies and apply t h i s s i g n a l PLATE l a "g" marker s i g n a l from video spectrometer PLATE Ic "g" marker s i g n a l from double f i e l d modulation spectrometer w i t h l i n e a r detector -PLATE l b " g n marker s i g n a l from video spectrometer with r . f . f i e l d modulation on PLATE Id -9 10 grams "g" marker s i g n a l from double f i e l d modulation spectrometer w i t h l i n e a r d e t e c t o r f a c i n g page 49 to the Y p l a t e s , the d e r i v a t i v e or modulus of the d e r i v a t i v e of any resonance s i g n a l i n the re g i o n H - H i w i l l be d i s -played on the screen depending on whether a phase s e n s i t i v e or l i n e a r second detector i s used (H3,K1,C1). The choice of the frequency f 2 depends on two f a c t o r s a. I t should be above 1 mcs to enable the c r y s t a l converter t o operate i n a r e g i o n where the excess noise i s n e g l i g i b l e ; b. I t should be l e s s than the l i n e width of the resonance s i g n a l i n frequency u n i t s . For d i l u t e hydrated c r y s t a l s t h i s i s about 16 gauss or 25 mcs. In f r e e r a d i c a l . o r deuterated c r y s t a l s , i t may be as sm a l l as 1 or 2 gauss or about 3 mcs. 'Hence, the choice of a frequency of about 1 mcs f o r f 2 w i l l s a t i s f y both of the above'conditions. I t can be shown that the use of a frequency of one megacycle f o r should r e s u l t i n an improvement of s e n s i -r t i v i t y of 0(10-3) over that obtained by the use o f a s i n g l e f i e l d modulation, of low frequency. P l a t e l a . and c. shows o s c i l l o g r a m s of a s i g n a l obtained from 0 ( 1 0 " ^ ) grams of "g" marker' ( d i p h e n y l - t r i n i t r o phenyl h y d r a z y l ) [(C 6H£) 2' N-NC6H2. (N0 2 ) 3 ] o n b o t h s i n g 1 * 5 and double mod-u l a t i o n . The s i g n a l - t o - n o i s e r a t i o s are 5 0 : 1 and 5 * 1 0 $ : 1 r e s p e c t i v e l y . The t h e o r e t i c a l improvement i n s e n s i t i v i t y i s thus e x p e r i m e n t a l l y v e r i f i e d . I n P l a t e l b , the "g" marker resonance i s a l s o shown when detected w i t h the 50 s i n g l e modulation spectrometer but w i t h the h i g h frequency modulation a p p l i e d to an amplitude equal to the l i n e width. I t can be c l e a r l y seen that the amplitude of the s i g n a l has been decreased by a f a c t o r of two v e r i f y i n g that the ampli-tude of t h i s modulation i s a c t u a l l y equal to the l i n e width. P l a t e I d . shows an o s c i l l o g r a m o f 10" grams of "g" marker w i t h a s i g n a l - t o - n o i s e r a t i o o f 2 :1 . A l l of these s i g n a l s were obtained at room temperature u s i n g wide band o p e r a t i o n (8 k i l o c y c l e s ) . The use of a r a d i c a l as a c a l i b r a t i o n s i g -n a l i s convenient since the resonance i s due to an almost f r e e e l e c t r o n [ g r a d i c a l = 2.0038 (B£) and g e l e c t r o n = 2 . 0 0 2 8 J . and the magnetic co n c e n t r a t i o n i s large (T3). The s e n s i t i v i t y achieved u s i n g t h i s method i n wide band oper a t i o n i s greater than that obtained u s i n g any other technique i n wide or narrow band op e r a t i o n except t h a t of Beringer and Castle (B2). U n f o r t u n a t e l y , we have not had the time to construct the apparatus necessary f o r narrow band operation. T h e o r e t i c a l l y , an improvement of 1 0 2 - 10^ should be p o s s i b l e u s i n g a narrow band phase s e n s i t i v e detec-t o r . I n p r a c t i c e a f a c t o r o f 5>0 i s considered good since i n s t a b i l i t i e s r a t h e r than noise now l i m i t the s e n s i t i v i t y . This would i n d i c a t e t h a t the u l t i m a t e s e n s i t i v i t y of a double modulation- spectrometer should be about lO-'^grams i n narrow band operation. The double modulation technique described here i s much e a s i e r to apply i n narrow band oper-a t i o n because of i t s i n h e r e n t l y greater e l e c t r i c a l s t a b i l i t y . S t a b i l i t y i n narrow band oper a t i o n of a video spectrometer Table IV Comparison of D i f f e r e n t Types of Paramagnetic Resonance Spectrometer Type Ref. Operation Mass S e n s i t i v i t y M min S t a b i l i t y Bandwidth Wide Band(lO^cps) Narrow Band (1 cps! Wide . Narrow Mmin(GRAMS) T(°K) Mmin(GRAMS) T(°K) T h e o r e t i c a l B l l Yes Yes 4 x 1 0 - 1 0 290° 4 x 1 0 " 1 2 290° 1 x 1 0 " 1 0 20° 1 x I O " 1 2 20° 5 x 1 0 " 1 1 4 ° \ 5 x l O - ^ . 4° P o i n t - t o - P o i n t B l l No Yes 1 x i o - 6 290° F a i r D.C. Recorder Wl No Yes 1 x 10~ b 290° • F a i r Balanced Bridge B l Yes Yes 1 x IO" 7 290° Poor Bolometer B2 No Yes 5 x I O " 1 0 290° Good Superheterodyne S1,H4 Yes Yes- 1 x.10-7 290° 5 x 10-9. 290° F a i r S i n g l e F i e l d Modulation (Video) B l l S2 Yes Yes 5 x 10-7 290° Cal c u l a t e d 5 x I O ' 1 0 4° Good 1 x 10-7 20° 5 x 1 0 ~ 8 4° Double F i e l d Modulation S3 Yes Yes 1 x 10-9 290° Ca l c u l a t e d 4° Very Good 1 x l o - W 4° . 1 x 10-** i 51 i s determined by the s t a b i l i t y of the microwave power l e v e l at the c r y s t a l converter. The double modulation spectro-meter I s l e s s s e n s i t i v e to f l u c t u a t i o n s i n the microwave power l e v e l because the s i g n a l i s f i r s t detected at h i g h frequencies and then redetected. S i g n a l s of good s t a b i l i t y have been observed w i t h the double modulation spectrometer which, when examined simultaneously w i t h the video spectro-meter, could h a r d l y be stud i e d because of microphonic d i f f i c u l t i e s . This a l s o e x p l a i n s why the c a v i t y can be allowed to f i l l w i t h l i q u i d helium when the double modulat-i o n spectrometer i s used. This cannot be pe r m i t t e d i f s a t i s -f a c t o r y s t a b i l i t y i s to be obtained w i t h the video spectro-meter. I t should be f u r t h e r noted that v i s u a l i n t e g r a t i o n of s i g n a l s i n c r e a s e s the apparent s e n s i t i v i t y of wide band operation. 2.8. Comparison Of Techniques s e n s i t i v i t i e s o f a l l the known paramagnetic resonance spectro-meters. The t h e o r e t i c a l s e n s i t i v i t y has been c a l c u l a t e d u s i n g the formula d e r i v e d by Bleaney and Stevens ( B l l ) Table IV i s arranged to show the c h a r a c t e r i s t i c s and 1/2 (20) where V i s the e f f e c t i v e volume of the c a v i t y d e f i n e d by (20a) 52 Q Q i s the unloaded Q of the c a v i t y NF i s the noise f i g u r e of the r e c e i v e r i s the a v a i l a b l e power from the microwave o s c i l l a t o r df i s the e f f e c t i v e bandwidth of the r e c e i v e r To convert from ^ min t o t l l e rainimura detectable mass, we r e q u i r e the equation X"_ TTW g1 2 ft2 N [s(S + 1) - M(M - 1)] f ( W ) 8 kT(2S + 1) (21) f o r the M«-» M - 1 t r a n s i t i o n where N i s the number of paramagnetic ions S i s the s p i n of the i o n W i s the frequency at which the resonance i s observed /3 i s the Bohr magneton ( /3 = -—2k_ ) Zj.Tr mc g' i s the spectroscopic s p l i t t i n g f a c t o r f ( P ) i s the normalized l i n e shape f u n c t i o n and the gram atomic weight of the paramagnetic i o n . The f o l l o w i n g values have been used i n the numerical c a l c u l a t i o n S = 1/2, P= 25,000 mcs; g = 2, QQ = 5000 , df = 1 cps, 10^ cps. V = 1 cc, T = k°>20°, 290° K., = lj.0 m i l l i w a t t s N.F.= 10 P/A^ « lO f o r a d i l u t e ..hydrated c r y s t a l , 100 f o r gram i o n i c weight. A l l p u b l i s h e d s e n s i t i v i t i e s have been normal-i z e d u s i n g these values. PLATE E • General view of experimental apparatus • CHAPTER I I I EXPERIMENTAL APPARATUS 3.1. I n t r o d u c t i o n I n t h i s chapter, the components of the wide band, double modulation spectrometer developed to perform the experiments described i n Chapter IV are described i n d e t a i l . Since i t can a l s o be operated (with a s l i g h t m o d i f i c a t i o n ) as a wide band, s i n g l e modulation spectrometer, we s h a l l des-c r i b e the components r e q u i r e d f o r both at the same time and i n d i c a t e , where i t i s necessary, those which are unique to one mode of o p e r a t i o n or the other.. Both f u n c t i o n from room to helium temperature. Block diagrams are employed to i l l u s t r a t e the design p r i n -c i p l e s used and the a c t u a l c i r c u i t diagrams are c o l l e c t e d together i n Appendix 1. Two general views of the apparatus are shown i n P l a t e s I I and IX. C h r o n o l o g i c a l l y , the s i n g l e modulation spectro-meter, which i s of standard design, was constructed f i r s t and the double modulation spectrometer was developed from i t . 3.2 Microwave Apparatus I d e n t i c a l microwave components are r e q u i r e d f o r the s i n g l e and double modulation spectrometers w i t h the 0-50 MICRO AMMETER I 60 CYCLE MODULATION CRYSTAL DETECTOR • ATTENUATOR POWER SUPPLY PRECISION WAVEMETER I CRYSTAL DETECTOR RUBICON GALVANOMETER • 2K33 REFLEX KLYSTRON ATTENUATOR I —-DIRECTIONAL COUPLER ' i CAVITY CRYSTAL DETECTOR BLOCK DIAGRAM OF I.25CM. MICROWAVE APPARATUS FIGURE 3 54 exception of the tuned c a v i t y i n which the sample under i n v e s t i g a t i o n i s placed. The choice of the frequency of opera t i o n was d i c t a t e d by two c o n s i d e r a t i o n s . a. the a v a i l a b i l i t y of apparatus i n t h i s l a b o r a t o r y from the microwave gas spectroscopy group, and b. the a b i l i t y t o operate over the widest range of "g" values. Consequently, the spectrometers operate at about 25 kmcs. or 1 .2 cm. The photograph i n P l a t e I I I shows the general arrange-ment of the microwave components. The block diagram i n F i g -ure 8 shows the same arrangement. 3 . 2 1 . Wave Guide Components The a t t e n t u a t o r s , "E" bends, "H" bends, c r y s t a l d e t e c t o r s , "magic" T's, tapered s e c t i o n , d i r e c t i o n a l coup-l e r s , and s t r a i g h t s e c t i o n s employed have been designed from the data i n Montgomery (M3) and constructed I n t h i s l a b o r a t o r y . E l e c t r o p l a t i n g techniques have been found use-f u l i n some in s t a n c e s . I t was considered economical to use commercial contact flanges RG - 1|25/U. 18 i n c h lengths of commercial f l e x i b l e waveguide are u t i l i z e d to connect the microwave equipment t o and from the spectrometer head to which the c a v i t y i s attached. This enables the c a v i t y t o •be moved i n and out of the magnetic f i e l d without breaking the microwave c i r c u i t and has considerable advantage when experiments are performed using, l i q u i d helium ( 3 . 6 2 ) . P L A T E IH • View of microwave components 55 3.22. Microwave Power Generator A paramagnetic resonance spectrometer r e q u i r e s a r a d i a t i o n source which a. generates an e s s e n t i a l l y monochromatic spectrum, b. possesses both long and short term frequency and power s t a b i l i t y (1 part 1C-5) c. generates s u f f i c i e n t power (100 m i l l i w a t t s ) d. can be tuned over a reasonable frequency range (10$) e. generates a minimum amount of n o i s e . The most s a t i s f a c t o r y source of microwave power i n the 1.2 cm r e g i o n which approximately s a t i s f i e s these c o n d i t i o n s i s the 2K 33A. r e f l e x k l y s t r o n s . This type o f tube i s f u l l y d iscussed i n (Klj.). I t i s e s s e n t i a l , i f s t a b l e , n o i s e l e s s o p e r a t i o n i s . to be obtained, to mount the k l y s t r o n at l e a s t s i x f e e t from the electromagnet and to s h i e l d i t i n a brass box since i t s o p e r a t i o n i s s e v e r e l y a f f e c t e d by s t r a y s t a t i c and a.c. magnetic f i e l d s . This box i s c l e a r l y shown i n P l a t e s I I I and IX. There are p a r t i c u l a r l y troublesome when the double modulation technique i s employed since the s t r a y r . f . f i e l d i s l a r g e . Improved temperature s t a b i l i t y i s another advantage of the brass box. A h i g h s t a b i l i t y , low r i p p l e power supply i s e s s e n t i a l i f c o n d i t i o n s a, b and c are to be s a t i s f i e d . A power supply of standard d e s i g n f o r the 2& 33^ w i t h a 56 voltage s t a b i l i z a t i o n f a c t o r of lC r and about 1 m i l l i v o l t r.m.s. of noise and hum has been constructed. The f i l a -ment i s d.c. heated by a 6 . 3 v o l t storage c e l l to prevent 60 n frequency modulation. Vacuum diodes, i n s e r t e d between the r e f l e c t o r and cathode and g r i d and cathode, prevent these elements of the k l y s t r o n from becoming p o s i t i v e w i t h respect t o the cathode and de s t r o y i n g the tube because of excess beam current and secondary emission from the r e f l e c t o r . A fuse i n the cathode a l s o l i m i t s the cathode c u r r e n t . To f a c i l i t a t e f i n d i n g the resonant frequency of the c a v i t y , prov-i s i o n i s made to modulate the k l y s t r o n r e f l e c t o r at 60 c y c l e s . For improved s t a b i l i t y , i t has been found u s e f u l t o l o c k the frequency of the k l y s t r o n to the frequency of the tuned c a v i t y . Such a system i s described i n 3 . 2 6 . 3 . 2 3 . Detectors Broad band c r y s t a l d etector mounts have been con-s t r u c t e d ( 3 . 2 1 ) . The 1JJ26 s i l i c o n c r y s t a l converter i s the only u n i t designed to operate at 25 kmcs. S p e c i a l l y s e l -ected, low noise temperature, low conversion l o s s u n i t s were obtained to ensure a minimum noise f i g u r e f o r the e n t i r e system. 3.2i j . . Cavity Resonators C y l i n d r i c a l 1 .2 cm. resonant c a v i t i e s o p e r a t i n g i n the mode have been designed from the data i n Montgomery (M3). They can be tuned by means of a threaded end plunger w i t h the proper choke t e r m i n a t i o n . This mode concentrates the microwave magnetic f i e l d a t the ends of the c y l i n d e r where the sample may be conveniently mounted on the face of the tun i n g plunger, ensuring maximum s e n s i t i v i t y because most of the magnetic f i e l d i s then concentrated i n the sample. The Q of the c a v i t y I s decreased from 0(10^) to 0(10-3) by heavy cou p l i n g to reduce i n s t a b i l i t i e s caused by microphonics and to permit the c r y s t a l converter t o op-erate with optimum s i g n a l - t o - n o i s e . To employ the double modulation technique, i t i s necessary to s l o t the w a l l s of the c a v i t y i n a plane through the a x i s of the c y l i n d e r . The microwave current i n the H m mode flows i n planes p a r a l l e l t o t h i s cut and, hence, the Q of the c a v i t y i s not a f f e c t e d by i t s presence, provided i t i s not too wide (<0.020 i n c h ) . The i n t r o d u c t i o n of the h i g h frequency modulation current i n s i d e the c a v i t y by means of t h i s s l o t w i l l be considered i n 3 . 5 . A diagram of a s l o t t e d c a v i t y i s shown i n Figure 12 and an a c t u a l one can be seen i n the photograph i n P l a t e "VtiT 3 . 2 5 . Wavelength Measurement A commercial, t r a n s m i s s i o n type wavemeter i s em-ployed to measure the wavelength of the microwave r a d i a -t i o n . I t w i l l measure frequencies i n the range from 22.20 - 27.00 kmcs. w i t h an accuracy of 1 p a r t i n 10^. FIGURE 9 POWER S U P P L Y R E F L E C T O R KLYSTRON C A V I T Y I N 2 6 C R Y S T A L D E T E C T O R M A T C H I N G N E T W O R K 4 7 5 K C S F I L T E R D Y N A M I C V O L T A G E C O N T R O L 10 7 M C S HIGH STABILITY O S C I L A T O R 10-7 M C S P H A S E SHIF^T N E T W OR K P H AS E S E N S I T I V E D E T E C T O R 107 M C S A M P L I F I E R B L O C K DIAGRAM OF A CAVITY T Y P E K L Y S T R O N F R E Q U E N C Y STABILIZER 58 The c a l i b r a t i o n has been c a r e f u l l y checked (M2) u s i n g the microwave frequency standard (accuracy 1 p a r t i n 10?) a v a i l a b l e i n t h i s l a b o r a t o r y ( M l ) . 3 . 2 6 . K l y s t r o n S t a b i l i z a t i o n I t has been found convenient to s t a b i l i z e the frequency of the 2K33A r e f l e x k l y s t r o n u s i n g a m o d i f i -c a t i o n of the Pound (P2) s t a b i l i z a t i o n c i r c u i t developed i n t h i s l a b o r a t o r y (R2). This s t a b i l i z e r , which i s shown i n Figure 9, l o c k s the frequency of the k l y s t r o n to t h a t of the tuned c a v i t y and w i l l f o l l o w changes i n the resonant frequency of i t . 3 . 3 . Magnetic F i e l d Equipment The design, c o n s t r u c t i o n and performance of the equipment r e q u i r e d to produce, c o n t r o l and measure the various magnetic f i e l d s r e q u i r e d f o r a double modulation spectrometer are described i n t h i s s e c t i o n . 3 . 3 1 . Design And C o n s t r u c t i o n Of The Electromagnet A small electromagnet, s u i t a b l e f o r a paramagnetic resonance spectrometer, has been designed and constructed. F l e x i b i l i t y and economy were the major design c o n s i d e r a t i o n s . At 25 kmcs. a f i e l d Of 9000 gauss i s r e q u i r e d to observe paramagnetic resonance absorptions when g = 2 and 18,000 gauss when g = 1. I t i s advantageous t o be able to observe resonances over as l a r g e a range of g values as 59 p o s s i b l e . The f i e l d homogeneity should be about 0 . 0 1 $ over the volume of the samples which are about 2 mm. on edge i f l i n e widths o f one gauss are to be observed. The a i r gap i s determined by the diameter of the c a v i t y and the temp-erature at which the experiment i s to be performed. At room temperature, where no dewars are r e q u i r e d , the gap can be 0 . 6 0 0 inches since the outside diameter of the c a v i t y i s 0 . 5 8 0 inches. Using standard s i z e s of g l a s s , the smallest outside diameter of the t a i l of a dewar i n t o which the c a v i t y w i l l f i t i s 0 . 9 0 0 inches. At helium temp-e r a t u r e s , two dewars, one i n s i d e the other, are r e q u i r e d and the outside diameter becomes 1 . 2 5 0 inches. Since e x p e r i -ments were planned i n v o l v i n g g = 1 .5 at helium temperatures, a f i e l d of 12 k i l o g a u s s i n a gap of 1 .270 inches i s r e q u i r e d . S a t i s f a c t o r y homogeneity can be obtained w i t h 2 i n c h d i a -meter p o l e faces although 4 i n c h faces would be b e t t e r . I t i s a l s o advantageous to be able t o r o t a t e the magnetic f i e l d ( 2 . 2 ) . On the b a s i s of the above c o n s i d e r a t i o n s , an e l e c t r o -magnet has been designed, which has the f o l l o w i n g f e a t u r e s ; a. A d j u s t a b l e gap up to 3 inches b. Interchangeable 4 i n c h c y l i n d r i c a l and 2 i n c h tapered pole faces c. Adjustable shims on the pole faces t o improve the homogeneity (Rl) d. Rotatable magnet yoke (360°) e. C a l i b r a t e d scale w i t h 0 . 5 ° d i v i s i o n s t o 60 measure r o t a t i o n . f. V e r t i c a l l e v e l adjustment g. T r o l l e y mounting f o r ease of moving h. Adjustable pads t o l e v e l and p o s i t i o n t r o l l e y i . Mater c o o l i n g A view of the magnet i s shown i n P l a t e IV w i t h the spectrometer head and dewars i n the gap. The yoke and pole pieces were made from a h i g h q u a l i t y s o f t s t e e l b i l l e t donated by The S t e e l Company of Canada L t d . w i t h the f o l l o w i n g chemical s p e c i f i c a t i o n s . 0.05#C, 0.010#P, 0 . 0 2 5$S, 0.08#Mn, 0 . 0 0 2 $ S i . The yoke was f a b r i c a t e d by Ross and Howard L t d . and the remainder of the u n i t was constructed i n the department machine shop. I t was estimated that 1|0,000 ampere-turns would be s u f f i c i e n t to o b t a i n 1 5 , 0 0 0 gauss i n a 1 i n c h gap assuming H i r o n - 80 ampere-turns / i n c h . Since a 125 v o l t 16 ampere D.C. generator was a v a i l a b l e , the magnet impedance c o i l s were matched to i t . Each c o i l was wound w i t h 3 , 3 0 0 turns of #13 double g l a s s i n s u l a t e d copper wire and has a r e s i s t a n c e of 16 ohms. When the c o i l s are connected i n p a r a l l e l , a magnetization f o r c e of 5 2 , 0 0 0 ampere-turns i s a v a i l a b l e . Two l a y e r s of 3 / l 6 " diameter copper t u b i n g are used to c o o l each c o i l . 3*32. Performance Of The Electromagnet Using the two i n c h tapered pole f a c e s , i t i s pos-s i b l e to o b t a i n a f i e l d of 1 3 , 0 0 0 gauss i n a gap of 1 . 2 7 6: inches, and 18,000 gauss i n 0.920 inches. This permits g values down t o 1 .5 to be observed at helium temperatures and to 1 at oxygen temperatures. L i n e widths of 3 .7 gauss have been observed f o r "g" marker i n agreement w i t h the reported value (H5) which con-f i r m s that the homogeneity i s adequate. Using samples of mi n e r a l o i l , 3/8" l o n g 3/16" diameter, very prominent wiggles ( 3 - 3 4 ) ( B l 4 f B l 5 ) on the proton resonance a b s o r p t i o n are observed w i t h the 4 i n c h diameter pole f a c e s . They are j u s t v i s i b l e w i t h the 2 Inch faces i n d i c a t i n g t h a t the homo-geneity i s considerably poorer. Prom the shape of the magnetisation curve, i t i s evident that the u l t i m a t e f i e l d obtainable i s l i m i t e d by s a t -u r a t i o n of the i r o n i n s i d e the current c o i l s r a t h e r than of the pole f a c e s . This e f f e c t i s due t o the l a r g e leakage f a c t o r which i s produced by shape of the magnet window neces-sary f o r a paramagnetic resonance spectrometer. Tapered pole pieces or 6 i n c h diameter p i e c e s w i t h tapered f a c e s would improve t h i s g r e a t l y . When a current of 9 amperes flows i n each of the c o i l s , the temperature of the c o o l i n g water r i s e s from 12°C. to 35°C. showing t h a t there I s not adequate thermal contact between the c o o l i n g s c o i l s and the wire. More c o o l i n g sur-face would be r e q u i r e d i f improved long term s t a b i l i t y were d e s i r e d . For maximum s t a b i l i t y , the magnet current i s l e f t at about 6 amperes f o r an hour i n advance of an experiment to a l l o w the magnet to reach thermal e q u i l i b r i u m . 240 450 V O L T S D.C. 0-10 AMR M A G N E T CURRENT M E T E R CURRENT REGULATOR T U B E S 3 8 — 6 A S 7 E L E C T R O M A G N E ' ( - ) MANGANIN RESISTOR CALIBRATION GALVANOMETER CALIBRATION B A T T E R Y POTENTIO — M E T E R A.C. [AMPLIFIER 4 0 0 ^ C H O P P E R STANDARD C E L L A . C AMPLIFIER PHASE SENSITIVE D E T E C T O R 4 0 0 'V/ OSCILLATOR 6 L 6 DRIV.ER FIGURE 10 BLOCK DIAGRAM OF CURRENT REGULATOR FOR ELECTROMAGNET 62 3.33. Electromagnet Power Sources Two power sources have been used w i t h the e l e c t r o -magnet. When a n o n - s t a b i l i z e d magnetic f i e l d r e p r o d u c i b l e to 1% i s s u f f i c i e n t , a one k i l o w a t t motor generator u n i t producing 16 amps at 125 v o l t s i s used. I t has the advan-tage that a l a r g e range of magnetic f i e l d s can be examined q u i c k l y . Current c o n t r o l i s obtained by v a r y i n g the r e s i s -tance i n the f i e l d c o i l . When accurate measurements are performed, i t i s necessary t o s t a b i l i z e the magnet c u r r e n t . Since the accur-acy of frequency measurements i s 1 p a r t 1 0 4 , i t i s necessary to s t a b i l i z e the current and to a l s o reproduce any r e q u i r e d current t o the same accuracy. A b l o c k diagram of the current s t a b i l i z e r used i s shown i n Figure 1 0 . The s t a b i l i z a t i o n f a c t o r i s 1 p a r t however, the r e p r o d u c i b i l i t y i s only 5 p a r t s l o 4 . The accuracy of resonance measurements i s con-sequently l i m i t e d by the r e p r o d u c i b i l i t y . When the current s t a b i l i z e r i s used, the d.c. power i s obtained from two 150 v o l t i+00 ampere D.C. motor-generator u n i t s i n the Physics B u i l d i n g . The output of one i s stacked on top of the other i n order to o b t a i n the necessary v o l t a g e . When s t i l l h igher voltages are r e q u i r e d , the output o f the 1 kva. motor-generator, mentioned above, i s stacked on top of the other two. 63 3 . 3 4 ' Magnetic F i e l d Measurements Three methods of measuring magnetic f i e l d s have been employed. a. F l i p c o i l b. Proton resonance c. Comparison w i t h known l i n e s . Where 1% accuracy i s s u f f i c i e n t , a f l i p c o i l and b a l l i s t i c galvanometer or fluxmeter i s used. With the proton resonance method, the p r e c i s i o n i s determined by the accuracy w i t h which frequencies 0(10) mcs. can be measured (1 p a r t 10^ i s e a s i l y o b t a i n a b l e ) . I f a combined paramagnetic resonance and proton head i s used, then • i t i s l i m i t e d by p r e c i s i o n w i t h which the proton resonance can be superimposed on the paramagnetic resonance (1 p a r t 10^"). This method has the advantage that the f i e l d i s measured sim-u l t a n e o u s l y w i t h the resonance. Inaccuracy i s introduced by the f a c t that the paramagnetic sample and the proton sample are not i n the same p a r t of the magnetic f i e l d . With a homo-geneous magnetic f i e l d t h i s o b j e c t i o n i s not s e r i o u s since the two samples can be arranged w i t h i n one centimeter of each other. U n f o r t u n a t e l y , t h i s technique cannot be employed at helium temperatures unless s o l i d proton samples are used. This n e c e s s i t a t e s a more complicated proton head, the l i n e width i s greater (10 gauss) and the accuracy i s correspondingly poorer (£ p a r t 10^-}. Consequently, the u s u a l procedure at helium temperatures i s to c a l i b r a t e the f i e l d i n advance, FIGURE 11 - MODULATION 60 C Y C L E MODULATION PHASE SHIFTER u C O I L S CATHODE FOLLOWER H A R M O N I C A M P L I Fl ER F R E Q U E N C Y STANDARD AUDIO AMPLI FIER C R Y S T A L D E T E C T O R DYNAM IC V O L T A G E C O N T R O L C A T H O D E FOLLOWER R F AMPLIFIER DC DIFFERENCE AMPLIFIER I N 34 CRYSTAL DETECT011 VACUUM T U B E VOLTMETER B L O C K D I A G R A M O F P R O T O N R E S O N A N C E UNIT t 6k u s i n g a proton head, or e l s e to place the proton head out-side the dewars. The theory and a p p l i c a t i o n of n u c l e a r magnetic resonance to the measurement of magnetic f i e l d s has been discussed i n the l i t e r a t u r e by numerous authors (Bl5,Bl6 , K 3,S2,C2,K5)• The gyromagnetic r a t i o of the proton has been very a c c u r a t e l y determined (Tl) ( Kp = 0 . 2 3 4 8 6 5 - O.OOOOOij. kilogauss/megacycle). Consequently, the proton can be used f o r accurate f i e l d mea-surements up to 15 k i l o g a u s s (63 mcs). Above t h i s frequency, L i t h i u m s i g n a l s are used (K5). The proton magnetic f i e l d spectrometer shown i n Figure 11 i s an improved v e r s i o n of one used at Clarendon Laboratory (S2). An important feature i s an automatic o s c i l l a t i o n l e v e l r e g u l a t o r which enables optimum s i g n a l - t o -noise r a t i o to be maintained over a wide range of frequen-c i e s . P r o v i s i o n to monitor the o s c i l l a t i o n l e v e l c o n t i n -uously i s a l s o made. Four p l u g - i n c o i l s , each covering about one octave of frequency, a l l o w the magnetic f i e l d to be mea-sured from 1 , 0 0 0 to 16 , 0 0 0 gauss. The o s c i l l a t i o n frequency i s measured w i t h a BC-221AH war surplus frequency meter ( 3 » 5 2 ) . The proton resonance head can be seen on the l e f t -hand side of the photograph i n P l a t e IX. Since i t was found to be d i f f i c u l t to reproduce mag-n e t i c f i e l d measurements t o the accuracy d e s i r e d (1 p a r t 10^-)-w i t h the proton spectrometer because of the d i f f i c u l t y of i n t e r c h a n g i n g i d e n t i c a l l y the p o s i t i o n s of the paramagnetic sample and the proton sample, i t was considered more accurate 65 to make use of the measurements of known absorptions obtained w i t h a combined pr o t o n and paramagnetic resonance head and t o c a l i b r a t e the magnetic f i e l d from these. Reproducible r e s u l t s , accurate to 5 p a r t s i n 10^, can be obtained u s i n g t h i s method i n c o n j unction w i t h proton resonance measurements. Unfort-u n a t e l y , the double modulation spectrometer does not permit the i n c l u s i o n of a proton head f o r mechanical reasons. More-over, s h i e l d i n g of the proton resonance head from the ij . 6 2 . 5 kcs h i g h frequency modulation would be a formidable task. 3 . 3 5 . L\ow Frequency Modulation Two Helmholtz c o i l s , each wound on b a k e l i t e bobbins w i t h 350 turns of #20, H.F. i n s u l a t e d , copper w i r e , supply the low frequency modulation f i e l d . The current I s obtained from the 60 c y c l e mains and i s c o n t r o l l e d by a v a r i a c . The voltage drop across a one-ohm r e s i s t a n c e i n s e r i e s w i t h the two c o i l s d r i v e s the X p l a t e s of the o s c i l l o s c o p e . A v a r i a b l e phase s h i f t network i s introduced to ensure that the two s i g n a l s obtained from each h a l f of the s i n u s o i d a l sweep can be brought i n t o coincidence. The amplitude of the f i e l d modulation i s obtained by c a l i b r a t i n g the current i n the c o i l s measured w i t h a substandard ammeter against the voltage induced i n a search c o i l p laced i n the centre o f the magnet gap. Since the a.c. h y s t e r e s i s depends on the magnitude of the d.c. f i e l d , c a l -i b r a t i o n s must be made at d i f f e r e n t magnetic f i e l d s . Magnetic f i e l d s i n the i n t e r v a l H * H i where H i s the magnetic f i e l d 66 and H i i s the amplitude of the f i e l d modulation are d i s -played on the h o r i z o n t a l t r a c e of the o s c i l l o s c o p e , however, since the sweep i s s i n u s o i d a l r a t h e r than l i n e a r , f i e l d s w i t h i n t h i s i n t e r v a l are s i n u s o i d a l l y r e l a t e d . 3 . 3 6 . High Frequency Modulation There are three p o s s i b l e methods of producing the h i g h frequency magnetic f i e l d at the c r y s t a l s i t e r a. Helmholtz type c o i l s outside the c a v i t y b. C o i l I n s i d e the c a v i t y c. S l o t t e d c a v i t y The f r a c t i o n of the a.c. magnetic f i e l d p e r p e n d i c u l a r to the a x i s of a conducting c y l i n d e r which penetrates t h i s c y l i n d e r can be deriv e d (S6). C a l c u l a t i o n s , based on t h i s formula, show that at 500 k c s , f o r a brass c a v i t y w i t h 0 . 0 1 0 i n c h w a l l thickness and 0 . 5 0 i n c h diameter, l e s s than one percent penetrates i n t o the i n t e r i o r . Since the. minimum l i n e width of a d i l u t e hydrated c r y s t a l i s about ten gauss ( 3 . 7 ) ( 1 . 3 )* at l e a s t one thousand gauss would be r e q u i r e d outside the c a v i t y which could only be produced by a l a r g e current f l o w i n g i n many tu r n s . The power consumed by t h i s and by the a.c. eddy current l o s s i n the i r o n of the magnet which, of n e c e s s i t y , must surround the c a v i t y to produce the r e q u i r e d s t a t i c magnetic f i e l d could o n l y be generated by a very h i g h power t r a n s m i t t e r . Moreover, the power consumed by the a.c. eddy current l o s s would heat the i r o n to very h i g h temperature, FIGURE 1 2 SLOTTED RESONANT CAVITY HHI MODE facing page 67 67 making f i e l d s t a b i l i z a t i o n d i f f i c u l t . Prom these consider-a t i o n s , i t can be seen that t h i s i s not a very f r u i t f u l approach. A small scale t e s t was performed to check these conclu s i o n s . An improvement of ten i n the s i g n a l - t o - n o i s e r a t i o was achieved when the resonance from a sample of f ,g" marker was examined which has a l i n e width of f o u r gauss (H5). The Q of the Helmholtz c o i l s was very small making i t d i f f i c u l t to match them to the t r a n s m i t t e r and the i r o n of the magnet became too hot to touch. The i n t r o d u c t i o n of a conductor i n s i d e a 1 . 2 cm microwave c a v i t y would reduce the Q of the c a v i t y to such a low value as to negate completely any of the advantages d e r i v e d from one ( B l l ) . I t was a l s o considered to be d i f -f i c u l t t o introduce the loop i n such a manner as to keep the r e f r i g e r a n t out of the cavity.' I t should be noted t h a t c o i l s have been s u c c e s s f u l l y Introduced i n t o l a r g e untuned c a v i t i e s which are e x c i t e d by many modes. This method was not considered s e r i o u s l y . Recently, Bleaney (B13) has reported a double modulation spectrometer u s i n g a h a l f t u r n i n s i d e a 3 cm c a v i t y . However, the s e n s i t i v i t y i s reported to be i n f e r i o r t o t h a t of the spectrometer described i n t h i s t h e s i s . I f a narrow s l o t (0 .020 inches) i s cut down the a x i s of the c a v i t y , as i n Figure 1 2 , then i t i s p o s s i b l e t o i n t r o -duce an a.c. magnetic f i e l d of any frequency i n s i d e the c a v i t y . The a.c. current flows down the outside o f one l e n g t h of waveguide connected to the c a v i t y , around both the i n s i d e and outside of the c a v i t y , and back up the other waveguide. The current f l o w i n g on the i n s i d e of the c a v i t y produces an o s c i l l a t i n g r . f . magnetic f i e l d i n the i n t e r i o r of i t . The current on the outside does not c o n t r i b u t e f o r the same reason as advanced above. The c a v i t y , from the p o i n t of view of the r ^ f . , acts l i k e a h a l f t u r n w i t h h a l f of the cur-rent being e f f e c t i v e . I n t h i s manner, i t i s p o s s i b l e t o produce r . f . magnetic f i e l d s i n s i d e a c y l i n d r i c a l conducting c a v i t y without a f f e c t i n g the microwave performance of the c a v i t y (3 . 2 4 ) . The s p l i t c a v i t y can be c l e a r l y seen i n the photograph i n P l a t e V I I I . The power r e q u i r e d to d e r i v e the h i g h frequency modulation current through the s p l i t c a v i t y i s obtained from a 400 watt 4 6 2 . 5 kcs. t r a n s m i t t e r which c o n s i s t s of a Clapp type, 6AC7 v a r i a b l e frequency o s c i l l a t o r , 807 Class C b u f f e r and four 8 l l A ' s i n p a r a l l e l . A low Q ( 10-15) tuned tank c i r c u i t i s r e q u i r e d t o permit c o r r e c t c l a s s C o p e r a t i o n of the 8 l l A ' s . The a.c. impedance of the s p l i t c a v i t y at 4 6 2 . 5 kcs i 1 .6 ohms and the d.c. r e s i s t a n c e i s 0 ( 1 0 " 2 ) ohms. The impedance match i s obtained by t u n i n g the s p l i t c a v i t y to resonance at 4 6 2 . 5 kcs and connecting i t to the appropriate tap p o i n t on the t r a n s m i t t e r tank o o i l . 100 amperes r.m.s. of modulation current can e a s i l y be obtained u s i n g t h i s pro-cedure. I t has been v e r i f i e d e x p e r i m e n t a l l y that the c a v i t y i s e q uivalent to a h a l f t u r n w i t h h a l f of the r . f . current being e f f e c t i v e i n producing the r . f . magnetic f i e l d . Thus, 25 gauss r.m.s. or 79 gauss peak-to-peak can be obtained. 3.1).. D e t e c t i o n Apparatus 3 . 4 1 . S i n g l e Modulation A two stage, low n o i s e , v a r i a b l e g a i n a m p l i f i e r o f standard design i s used to ampl i f y the s i g n a l s from the cry-s t a l converter to a s u f f i c i e n t l e v e l f o r p r e s e n t a t i o n on an o s c i l l o s c o p e . I t has been shown (S10) that the band width of a video a m p l i f i e r should be about 100 f1 where f-j^ i s the mod-u l a t i o n frequency i f u n d i s t o r t e d l i n e shapes are t o be reprod-uced. Since i s 60 c y c l e s i n our arrangement, the a m p l i f i e r was designed to be 3 db. down at 20 c y c l e s and 15 k i l o c y c l e s . The phase s h i f t i s then n e g l i g i b l e from 4 ° c y c l e s to 7 k i l o -c y c l e s . This was checked I n operation by i n c r e a s i n g the amplitude of the f i e l d modulation u n t i l the i n t e n s i t y of the s i g n a l began to decrease. This occurred at about twenty times the l i n e width demonstrating that the frequency r e s -ponse of the a m p l i f i e r was more than adequate. Since the o v e r a l l noise f i g u r e of the system i s p r i m a r i l y determined by the c r y s t a l converter, matching between i t and the input stage was found to be n o n - c r i t i c a l . The maximum gain of the ampli-f i e r i s 5 0 0 0 . P r o v i s i o n i s made t o mix e l e c t r o n i c a l l y proton resonance and c a l i b r a t i n g s i g n a l s w i t h the paramagnetic reson-ance s i g n a l . FIGURE 13 MICROWAVE SOU RCE MOOULATIOIV 475 KCS 60 C Y C L E MODULATION PHASE SHIFT NETWORK 4 7 5 K C S X TA L DETECTOR 6 O CYCLE PHASE SHIFT NETWORK X TA L MATCHING NETWORK PHASE SENSITIVE DETECTOR AMPLIFIER 475 KCS x © D C RECORDER INFINITE IMPEDANCE DETECTOR AUDIO AMPLIFIER B L O C K D I A G R A M O F D O U B L E MODULATION P A R A M A G N E T I C R E S O N A N C E S P E C T R O M E T E R 70 3-14-2. Double Modulation A three stage low n o i s e , stagger tuned, v a r i a b l e g a i n a m p l i f i e r of standard design i s used to a m p l i f y s i g n a l s when the double modulation technique i s employed. The band centre i s i|i>2.5 k c s . , the bandwidth i s 8 kcs. and the gain i s 10. The matching was found to be n o n - c r i t i c a l between c r y s t a l converter and input stage as i n 3-i+l (Ll» S10,T2). The s i g n a l from, the t h i r d stage i s fed i n t o e i t h e r an I n f i n i t e impedance l i n e a r d e t e c t o r or a phase s e n s i t i v e d e t e c t o r and then onto the Y p l a t e s of an o s c i l l o s c o p e . The bandwidth of the output of e i t h e r d e t e c t o r i s 100 f-]_ or about 8 k c s . (S10). The s i g n a l obtained from the i n f i n i t e impedance detector i s the modulus of the d e r i v a t i v e of the l i n e shape of the resonance (H3,C1) while the phase s e n s i t i v e d e t e c t o r y i e l d s the d e r i v -a t i v e . Figure 13 shows a b l o c k diagram of the d e t e c t i o n system. I t was found to be most e s s e n t i a l that the c r y s t a l converter and the tuned a m p l i f i e r be c a r e f u l l y s h i e l d e d . The a m p l i f i e r was constructed i n a sealed brass box and then enclosed i n another brass box w i t h the c r y s t a l converter. The only ground p o i n t i s the mounting of the c r y s t a l holder to t h i s outer box. The photograph i n P l a t e V shows the arrange-ment w i t h the l i d s removed. The closed box can be seen i n P l a t e I I I . The a m p l i f i e r box, a l l the c o n t r o l s and power and s i g n a l leads are i n s u l a t e d from the outer box. I t was found to be p o s s i b l e , u s i n g t h i s technique, to reduce the pickup from 71 the 400 watt 4 6 2 . 5 kcs. t r a n s m i t t e r , used to d r i v e the h i g h frequency s p l i t c a v i t y to below the input noise l e v e l of the a m p l i f i e r which i s 0 . 5 m i c r o v o l t s r.m.s. Pickup i n ex-cess of the noise w i l l produce phase s e n s i t i v e d e t e c t i o n of s i g n a l s of same magnitude i n the l i n e a r d e t e c t o r . When strong s i g n a l s are detected, the g a i n i s reduced so that the s i g n a l does not block the second d e t e c t o r . I n t h i s con-d i t i o n , the noise i n the absence of a s i g n a l i s square - law detected while that on the s i g n a l i s l i n e a r l y detected which gives an apparent improvement i n the s l g n a l - t o - n o i s e r a t i o . For weak s i g n a l d e t e c t i o n , the gain i s increased u n t i l l i n e a r d e t e c t i o n occurs on the n o i s e . P r o v i s i o n has been made to monitor the c r y s t a l current w i t h a microammeterj t h i s f a c i l -i t a t e s tuning the c a v i t y i n the absence of resonance s i g n a l s . 3 . 4 3 Video D i s p l a y The resonance s i g n a l s , from e i t h e r the s i n g l e or double modulation spectrometer are d i s p l a y e d on a 7 i n c h o s c i l l o -scope. The output of the phase s h i f t network ( 3 . 3 5 ) d r i v e s the X p l a t e s while the Y p l a t e s are d r i v e n by (a) video a m p l i f i e r ( 3 . 4 D ; (b) p h a s e - s e n s i t i v e detector ( 3 . 4 2 ) or (c) l i n e a r d e t e c t o r ( 3 . 4 2 ) . When photographs o f the reson-ances on the o s c i l l o s c o p e screen are taken, a 5 i n c h Cossor double beam o s c i l l o s c o p e w i t h 35 mm camera i s used. This u n i t has been modified to enable only one h a l f of the t r a c e to be recorded. The beam t r i g g e r i s c o n t r o l l e d by 60 c y c l e voltage of v a r i a b l e phase. By changing the phase, i t can be arranged t h a t one h a l f of the t r a c e I s blanked out. 3.l\k* Narrow Band Operation No narrow band ope r a t i o n has been attempted because i n s u f f i c i e n t time was a v a i l a b l e to construct the necessary apparatus r e q u i r e d to sweep and s t a b i l i z e the magnetic f i e l d . Moreover, the k l y s t r o n frequency s t a b i l i z e r (R2) has not yet been completed. The s e n s i t i v i t y of the double modulation spectrometer g r e a t l y reduces the n e c e s s i t y of employing nar-row band ope r a t i o n because there i s s t i l l considerable r e -search t o be performed fcin the l i m i t of i t s s e n s i t i v i t y . Moreover, narrow band i s a slower method of c o l l e c t i n g data. 3 . £ . A u x i l i a r y E l e c t r o n i c Apparatus 3 . 5 1 . C a l i b r a t o r I t i s u s e f u l to measure the r e l a t i v e and absolute i n t e n s i t i e s of resonance s i g n a l s to permit comparison w i t h t h e o r e t i c a l t r a n s i t i o n p r o b a b i l i t i e s . P r o v i s i o n has been made to i n j e c t " s p i k e s " of c a l i b r a t e d amplitude i n c o i n -cidence w i t h resonance s i g n a l s to perform these measurements. 60 c y c l e l i n e voltage of v a r i a b l e phase i s c l i p p e d by a blocked g r i d a m p l i f i e r . The 60 cy c l e square wave so produced i s d i f f e r e n t i a t e d and then e i t h e r the p o s i t i v e or negative spikes are a m p l i f i e d ; the spikes s e l e c t i o n i s accomplished by changing the d.c. l e v e l at which these spikes a r r i v e on 73 the g r i d of the a m p l i f i e r . One c o n t r o l sets the reference output amplitude of the s p i k e , a r e s i s t a n c e s w i t c h i n g net-work permits m u l t i p l i c a t i o n or d i v i s i o n by ten and another network d i v i d e s the r e s u l t i n g amplitude by one hundred i n u n i t s of one. These spikes are e l e c t r o n i c a l l y mixed w i t h the resonance s i g n a l s (3.I4.I) and then d i s p l a y e d on the o s c i l -loscope (3«43)« 3 . 5 2 . Frequency Meter A war surplus BC-221AH frequency meter i s used t o mea-sure the frequency o f the proton resonance s i g n a l s when mag-n e t i c f i e l d measurements are performed. I t i s accurate to 1 p a r t 10^ over the frequency range 2-80 mcs. The output above 20 mcs has been improved by the a d d i t i o n o f a harmonic a m p l i f i e r and a cathode f o l l o w e r has been added to d r i v e the cable that feeds the s i g n a l from t h i s u n i t to the proton head. 3 . 5 3 * Power Supplies A l l the power r e q u i r e d by the var i o u s e l e c t r o n i c com-ponents i n the two spectrometers i s obtained from standard e l e c t r o n i c a l l y r e g u l a t e d s u p p l i e s (EZj.). The s t a b i l i z a t i o n has been f u r t h e r improved i n some cases by the a d d i t i o n of an e l e c t r o n i c l i n e voltage r e g u l a t o r . I t has been found to be g e n e r a l l y a d v i s a b l e to employ d i r e c t current h e a t i n g of f i l a -ments to el i m i n a t e a.c. hum. This power i s obtained from t r i c k l e - c h a r g e d , l e a d storage c e l l s . P L A T E VI • View of the double modulation head with the dewars in the magnetic field • facing page 7 4 74 3.6. I»ow Temperature Apparatus Both spectrometers have been e x p r e s s l y designed f o r ope r a t i o n at helium temperatures. Consequently, considerable e f f o r t has been devoted t o the development of a f l e x i b l e system which would enable measurements to be made a c c u r a t e l y and q u i c k l y . The u n i t c o n s i s t i n g of the c a v i t y , connecting wave guide, pumping v a l v e s , dewars, e t c . i s h e r e a f t e r r e f e r r e d to as the spectrometer head. As has already been mentioned (3.21) the head i s mounted on bearings which r e s t on r a i l w a y t r a c k s p e r m i t t i n g i t to be smoothly r o l l e d i n and out of the magnetic f i e l d . The photograph i n P l a t e VI c l e a r l y shows t h i s aspect of the c o n s t r u c t i o n . 3.61. S i n g l e Modulation Head Since the c o n s t r u c t i o n of a low temperature s i n g l e modulation head presents fewer t e c h n i c a l d i f f i c u l t i e s , i t w i l l be discussed f i r s t . I f the head i s to be used at oxygen or hydrogen temperatures then the system i s g r e a t l y s i m p l i f i e d . Figure Ik shows a c r o s s - s e c t i o n a l drawing (S2) of such a head. A. c y l i n d r i c a l c a v i t y operated i n the H 1 : q mode and tuned w i t h a threaded end plunger (3.2!).) i s sealed w i t h a coned cap coated w i t h g l y c e r i n e or s i l i c o n e grease. E l l i p t i c a l holes couple the c a v i t y to the r e c t a n g u l a r waveguide (1/8" x 0 /4 " ) which i s constructed by drawing Ger-man s i l v e r tubing 6 mm diameter 0.1 mm w a l l t h i c k n e s s . German s i l v e r i s used t o reduce the heat leak i n t o the FIGURE 14 PUMPING TUBE MICA SEAL CROSS SECTION OF K BAND RESONATOR f a c i n g page 75 75 r e f r i g e r a n t . The cut-off frequency of t h i s waveguide i s decreased by f i l l i n g i t with a d i e l e c t r i c . Teflon ( E = 2.00) polystyrene, ( E = 2.50) or fluorethene ( E = 2.2) are suitable d i e l e c t r i c s , t e f l o n being preferable because of i t s f l e x i b i l i t y at low temperatures and chemical inertness. Electroformed tapered sections match the d i e l e c t r i c guide to the standard 1/1+" x l / 2 " O.D. K band guide. Thin mica sheets, beeswaxed to the ends of the tapered sections form a vacuum seal f o r the cavity without introducing serious d i s -continuity i n the microwave system. A 5 mm German s i l v e r tube mounted i n a hole i n the top of the cavity provides a sealed passage from the i n t e r i o r of the cavity to that portion of the head which i s at room temp-eratures. The top end i s terminated i n a female ground cone. A male cone forms a rotatable vacuum seal. A scale plate engraved i n f i v e degree i n t e r v a l s and a vernier permit the r o t a t i o n to be measured to one degree. A 3 mm German s i l v e r tube, connected to the male cone at the top, extends down the i n t e r i o r of the 5 mm tube to a c r y s t a l mount and choke which seals the cavity to microwaves. The sample i s mounted on the end of the plunger with polystyrene cement. In t h i s manner the c r y s t a l can be introduced f l u s h with one end of the cavity and can be rotated i n s i t u . Valves are provided to enable the cavity and wave guide to be evacuated and f i l l e d with a gas which does not freeze at the operating temperature. It i s also possible, using t h i s arrangement, to change crystal; P L A T E VII • VIEW OF HELIUM TEMPERATURE SINGLE MODULATION HEAD • f a c i n g p a g e 7 6 76 without warming the cavity. A metal cap, through which the waveguide and central tube pass, f i t s over the dewar and i s sealed to i t by a section of a rubber inner tube. Other tubes which pass through the cap permit the re f r i g e r a n t to be pumped when lower temperatures are required and the vapour pressure to be measured. When helium temperatures are required, two coaxial dewars are used. The outer one which i s f i l l e d with l i q u i d oxygen or nitrogen, i s suspended from the head i n a spring loaded cradle. A tube i s required to connect a syphon be-tween the dewar to be f i l l e d and the l i q u i d helium transport dewar. Additional pumping valves are required to f a c i l i t a t e the transfer by either pumping or blowing the l i q u i d over. Moreover, since helium gas i s expensive and, at times, d i f -f i c u l t to obtain, a recovery system i s required. A schematic diagram of the necessary pumping system i s shown i n Figure ljS and the pumping system and valves can be seen i n the photo-graphs i n Plates I I , VI, and IX. It i s advantageous to main-t a i n the top of the helium dewar at oxygen temperature to minimize the heat leak. The metal cap then involves a cup which can be f i l l e d with oxygen. A photograph of the helium temperature, single modulation head i s shown i n Plate VII. 3 . 6 2 . Double Modulation Head The application of the double modulation technique at temperatures which require l i q u i d r efrigerants introduces P L A T E 3ZTH • VIEW OF T H E D O U B L E MODULATION HEAD • 77 numerous technical d i f f i c u l t i e s , which w i l l be discussed i n d e t a i l i n t h i s section. I t i s unfortunately necessary to report that some of these d i f f i c u l t i e s have not yet been solved i n a s a t i s f a c t o r y manner.. Consequently, a consider-able amount of time and e f f o r t has been expended on these problems and some of the solutions reported are s t i l l of a makeshift nature. Plates VI and VIII contain photographs of the double modulation head. The pumping and f i l l i n g system has already been discussed i n 3 . 6 1 . I t should be noted that i t i s not possible, because of the nature of the construction of the double modulation cavity, to remove or rotate the c r y s t a l after the dewar has been positioned. Instead, the c r y s t a l i s mounted on the tuning plunger before the cavity i s tuned; no central tube, coned head, vernier scale, and mounting head are now required. It i s advantageous to prevent the re f r i g e r a n t from entering the i n t e r i o r of the cavity because any bubbling of i t due to b o i l i n g w i l l produce a modulation of the microwave power which w i l l r e s u l t i n very undesirable i n s t a b i l i t i e s . Moreover, some refrigerants are very lossy at microwave frequencies. Since the cavity i n the double modulation spectrometer has a slot i n i t , i t i s necessary f o r low temp-erature operation to seal the cavity to prevent the r e f r i g -erant from entering i t . Thermal setting A r a l d i t e , which has been used suc-c e s s f u l l y i n the low temperature laboratory at helium temp-eratures, was used to f i l l the s l o t . I n i t i a l l y , the s l o t i n a f i n i s h e d cavity was f i l l e d with A r a l d i t e , however, i t was found to be impossible to prevent some of i t from entering the i n t e r i o r of the cavity. Since Araldite proved lossy at microwave frequencies, the Q of the cavity was d r a s t i c a l l y reduced. This d i f f i c u l t y could be surmounted by soldering the German s i l v e r waveguide to a brass cylinder with a sl o t m i l l e d i n i t . The sl o t was then f i l l e d with the adhesive which was cured'by baking i t i n an oven at a suitable temp-erature f o r the recommended time. This assembly was then machined into a. microwave cavity of the correct dimensions. Nonetheless, because the thermal se t t i n g A r a l d i t e contains a considerable quantity of occluded a i r , i t was found, even after careful curing, to be porous, p a r t i c u l a r l y a f t e r being immersed i n l i q u i d helium because small cracks developed. Consequently, t h i s approach was abandoned; however, some sub-sequent work using cold setting A r a l d i t e which was heat cured i n an oven indicates that i t s t i l l might be successful... ' A more successful approach i s to enclose the cavity and associated waveguide below the oxygen cap i n a rubber condom. They survive immersion i n l i q u i d helium without cracking even when stretched, however, they unpredictably develop small holes due to s l i g h t flaws i n manufacture. For-tunately, i t has been found that l i q u i d helium can be allowed inside the cavity, although d i f f i c u l t y i s experienced i n 79 p r e d i c t i n g the amount by which the resonant frequency i s shi f t e d by i t s presence. Part of the reason f o r t h i s l i e s i n the f a c t that although the heat of vaporization i s small (4 calories/mole) the b o i l i n g takes place mostly at the sur-face p a r t i c u l a r l y below the X point. The same i s d e f i n i t e l y not true f o r either nitrogen or oxygen. Very anamolous effects occur which make operation almost impossible when they are present inside the cavity. Moreover, b o i l i n g occurs throughout the volume of these refrigerants. Another problem which has proven d i f f i c u l t to solve arises from the f a c t that the wave guide must be insulated from the bottom of the oxygen cap, yet pass through i t . This i n s u l a t i o n must also form a vacuum seal capable of hold-ing t i g h t at oxygen temperature since" i t i s undesirable to contaminate the helium gas with oxygen to f a c i l i t a t e easy cleaning of the recovered helium and also to reduce the helium b o i l i n g rate. I n i t i a l l y double Kovar seals, one end soldered to the oxygen cap and the other to the waveguide were t r i e d . A diagram of one i s shown i n Figure l£a. These seals are vacuum ti g h t and withstand l i q u i d oxygen i f they have been annealled i n a hydrogen oven. Unfortunately Kovar i s f e r r o -magnetic and a poor e l e c t r i c a l conductor and the induced eddy currents produce a large quantity of heat which i s most undesirable. Similar units employing housekeeper seals were then t r i e d , however they had a high breakage rate which occurred because of lack of mechanical strength and chemical FIGURE 15a • A S O F T S O L D E R KOVAR B R A S S S O F T S O L D E R <7 BRASS OXYGEN C A P G L A S S HARD SOLDER .GERMAN SILVER WAVE GUIDE DOUBLE KOVAR S E A L S O F T S O L D E R S O F T S O L D E R E D C O P P E R DISC \ < ffi B R A S S OXYGEN C A P N E O P R E N E GERMAN SILVER WAVE GUIDE M FIGURE 15b SANDWICH T Y P E INSULATED VACUUM S E A L facing page 80 80 reaction with water and l i q u i d oxygen. I t was considered to be possible to minimize the heating e f f e c t i n the Kovar by p l a t i n g to the skin depth of 4 6 2 . 5 k c s w l t h s i l v e r . This was t r i e d without success because i t was found to be impos-s i b l e to obtain s a t i s f a c t o r y bonding of the s i l v e r into the Kovar. Bakelite discs which were Araldited to the cap and also to the waveguide were t r i e d . Leaks developed i n the Ar a l d i t e for reasons similar to those mentioned i n the d i s -cussion of sealing the cavity. I n t e r f i t t i n g brass and bake-l i t e cone sockets, again A r a l d i t e d were unsuccessfully attempted f o r the same reasons. It was noted, during these e f f o r t s , that thermally cured cold s e t t i n g a r a l d i t e seals were much more r e l i a b l e than the thermal set t i n g v a r i e t y . I t i s now f e l t that these methods might have been successful i f the l a t t e r had been used. In the meantime, another approach to t h i s problem proved to be very successful. A t h i n c i r c u l a r disc of copper was soldered to each piece of waveguide. Discs of neoprene on each side of the copper of large diameter form a sandwich which i s compressed, after l u b r i c a t i o n with s i l i c o n e grease, against a metal seat soldered i n the bottom of the oxygen cap. The waveguide i s insulated from the cap by the neoprene discs yet a vacuum seal i s achieved. A drawing of t h i s arrangement i s shown i n Pig. 15b . HELIUM RECOVERY AND VACUUM SYSTEM T O GAS HOLDER IN CRYOSTAT ROOM FIGURE 16 'it Q = MICROWAVE MICA WINDOW AND V A C U U M S E A L •nu i — S > - A I R TO AIR T O INTAKE V T O E X H A U S T TO VACUUM PUMP 2 V ^ Q MERCURY MANOMETER HELIUM S Y P H O N LIQUID OXYGEN C A P -LIQUID HELIUM DEWAR J • WAVE GUIDE ^ C A V I T Y VACUUM PUMP No. I HELIUM TRANSPORT DEWAR V. V, V , V ! G L O B E V A L V E S V 5 , V 6 , V 7 • N E E D L E V A L V E S v * V 1 V i o ' V A C U U M V A L V E S V„ V / 2 V / 3 G L A S S STOP C O C K S V^. REGULATOR VALVE facing page 81 81 Since r . f . currents of up to 80 amperes rms, f l o w i n the German s i l v e r wave guide i n order to produce h i g h f r e q -uency f i e l d modulation of s u f f i c i e n t amplitude, a consider-able amount of heat i s generated which b o i l s o f f the l i q u i d helium. The heat l e a k , I n the absence of the modulation cu r r e n t , was found to be about 10 c.c. of l i q u i d helium per hour (0.015 w a t t s ) . When 25 amperes r.m.s. of r . f . current was passed through the system, the evaporation r a t e increased to 100 c.c. per minute (9 .0 w a t t s ) . C a l c u l a t i o n s based on these measurements i n d i c a t e d that t h i s evaporation r a t e could be d r a s t i c a l l y decreased by e l e c t r o p l a t i n g the guide w i t h s i l v e r to a thic k n e s s of about 0.0005 inches. At t h i s t h i c k -ness the conduction heat leak of the guide equals the heat generated by the r . f . c u r r e n t , i . e . , the t o t a l heat .input i s minimized. The r a t e was then found to be about 4 c.c. per minute (O .36 w a t t s ) . 3 . 6 3 . Helium Recovery System Due to the cost and the s c a r c i t y of helium gas, i t i s recovered by the low temperature laboratory.. The vacuum sys-tem incorporates a recovery l i n e so that when helium gas Is being pumped, i t can be returned to a r e s e r v o i r . Figure 16 i s a schematic diagram of the system employed. Some of the., pumping arrangement can be seen i n the photographs i n P l a t e s I I and IX. 82 3.7. C r y s t a l s The f i r s t step i n the i n v e s t i g a t i o n of a paramagnetic i o n i s to choose a s u i t a b l e c r y s t a l i n which to c a r r y out the study. This choice i s governed by a number of requirements; (a) The c r y s t a l s t r u c t u r e should be isomorphous throughout a group of s i m i l a r i ons to be s t u d i e d . This decreases the work i n v o l v e d i n c a l c u l a t i o n of the c r y s t a l l i n e e l e c t r i c p o t e n t i a l s and enables r e s u l t s on the i o n i c group to be cor-r e l a t e d . This group must i n c l u d e a diamagnetic i o n w i t h which to d i l u t e ' the c r y s t a l s . (b) The c r y s t a l s t r u c t u r e should be known from X-ray a n a l y s i s i n order t o s i m p l i f y the determination of the form of the p o t e n t i a l f u n c t i o n which must be c o n s i s t e n t w i t h the c r y s t a l symmetry'. (c) The s t r u c t u r e should have p r e f e r a b l y a x i a l sym-metry which f a c i l i t a t e s the i n t e r p r e t a t i o n o f the r e s u l t s . (d) I t i s advantageous f o r s u s c e p t i b i l i t y and s p e c i f i c heat measurements to have been made. The r e s u l t s of the resonance experiments can be compared wi t h the se. (e) There should be only one i o n i n u n i t c e l l . This makes i t e a s i e r to mount the c r y s t a l i n the proper plane and gives maximum i n t e n s i t y i n the spectrum. (f) S i n g l e c r y s t a l s must grow e a s i l y i n a d e s i r a b l e shape ( i n general one w i t h a low surface to F I G U R E 1 7 Elementary c e l l with the positions of the two metal ions. Projection of the lower half of the elemen-tary c e l l on a plane at r i g h t angles with the c-axis on 1/4 of i t s height. Probable positions of the nine watermolecules round the metal ion. CRYSTAL S T R U C T U R E OF T H E R A R E E A R T H E T H Y L S U L P H A T E S f a c i n g page 83 volume r a t i o ) . (g) The c r y s t a l must withstand low temperatures without c r a c k i n g and be chemically s t a b l e . I n the i r o n group, the alums, the f l u o s i l i c a t e s and the Tutton s a l t s have been found to be s u i t a b l e f o r paramagnetic resonance i n v e s t i g a t i o n s while i n the r a r e e a r t h group the e t h y l sulphates and the magnesium n i t -r a t e s are s a t i s f a c t o r y . I n Figure 1 7 , the e s s e n t i a l p a r t s of the c r y s t a l s t r u c t u r e , f o r the e t h y l sulphate as deter-mined by K e t e l a a r (K2), are i l l u s t r a t e d . I n concentrated c r y s t a l s , the width of the reson-ance l i n e i s determined by the s p i n - s p i n i n t e r a c t i o n (B11,P3). The r e s u l t i n g l i n e width (cor 200 gauss) i s u s u a l l y too great to resolve the hyperfine s t r u c t u r e and c e r t a i n l y f o r b i d s the making of accurate measurements. Penrose (P2) found that the l i n e width could be g r e a t l y reduced by d i l u t i n g the s a l t w i t h an isomorphous diamag-n e t i c compound. The width i s now l i m i t e d by the f l u c t u a t -i n g magnetic f i e l d of the protons i n the neighbouring water molecules. The c o n t r i b u t i o n of the protons t o the l i n e width can be c a l c u l a t e d . I t i s 16 gauss I n good agreement w i t h the e x p e r i m e n t a l l y observed value. This i s u s u a l l y s u f f i c i e n t l y s m a l l . I t can be f u r t h e r reduced by growing the c r y s t a l s from heavy water ( B l l ) . I n order t o reach the "proton" width, i t i s e x p e r i m e n t a l l y found t h a t the r a t i o of the paramagnetic to diamagnetic ions must be about 1 p a r t i n 200 f o r the e t h y l sulphates and 1 p a r t i n 20 f o r 81+ the magnesium n i t r a t e s . The mixed c r y s t a l s are grown by adding appropriate amounts of concentrated s o l u t i o n s and then evaporating. The s i z e of a s u i t a b l e c r y s t a l f o r use i n a 1.2 cm c a v i t y i s u s u a l l y about one or two m i l l i m e t e r s on edge I.e. 10" cm-3. I t was found t h a t a considerable amount of time was spent i n performing the necessary chemistry to prepare the v a r i o u s r a r e e a r t h compounds. (E2,V6,Y1) are valuable references on the subject. The techniques, although s t r a i g h t f o r w a r d , r e q u i r e considerable p r a c t i c e before they can be s u c c e s s f u l l y employed. 3.8. Performance The apparatus has been designed f o r ease, speed and accuracy of oper a t i o n because of the lar g e number of exper-imental runs which must be performed t o do a complete inves-t i g a t i o n of a given c r y s t a l . The resonance phenomena of any c r y s t a l must be i n v e s t i g a t e d at d i f f e r e n t temperatures, frequencies, and concentrations and a l s o f o r many angles between the c r y s t a l l i n e a x i s and the magnetic f i e l d . The apparatus discussed i n t h i s chapter performs most of these f u n c t i o n s very w e l l . The c h i e f l i m i t a t i o n i s the accuracy w i t h which the magnetic f i e l d can be measured and s t a b i l i z e d . This could be g r e a t l y improved by the c o n s t r u c t i o n o f a proton reson-ance magnetic f i e l d s t a b i l i z e r . To j u s t i f y the work i n -volved, i t would be necessary t o b u i l d a magnet which would give the r e q u i r e d magnetic f i e l d s i n the same gap but w i t h l a r g e r diameter pole faces so tha t both the paramagnetic resonance and the proton resonance heads could be s i m u l t -aneously p l a c e d i n close p r o x i m i t y i n a homogeneous f i e l d even when two dewars surround the paramagnetic resonance head. The f i e l d measurements could then be performed at the same time as the resonance measurements. With good f i e l d homogeneity, the c o r r e c t i o n f o r the d i f f e r e n t p o s i -t i o n s of the two heads should be small and e a s i l y measured. The other l i m i t a t i o n i s not s e r i o u s . I t i s d e s i r -able to be able to r o t a t e the c r y s t a l i n the c a v i t y and a l s o to be able to change c r y s t a l s d u r i n g a run. The mechanical nature of the double modulation spectrometer prevents t h i s . I n any case, a low temperature s i n g l e modu-l a t i o n spectrometer has been constructed to a l l o w t h i s to be performed. U s u a l l y , the h i g h s e n s i t i v i t y i s only used to i n v e s t i g a t e the f i n e r d e t a i l s of the resonances observed at lower s e n s i t i v i t y , i n which case s u f f i c i e n t i n f o r m a t i o n i s a v a i l a b l e to mount the c r y s t a l c o r r e c t l y . I t i s p o s s i b l e to t u r n the machine on and perform room temperature measurements w i t h i n f i f t e e n minutes. Even when helium temperature experiments are performed the time lapse i s only f o r t y - f i v e minutes. With the s e n s i t i v i t y a v a i l a b l e u s i n g the double modulation spectrometer (10"'' grams) almost a l l resonance phenomena can be presented on an o s c i l l o s c o p e which permits a spectrum t o be measured i n a matter of minutes. This s e n s i t i v i t y i s f a r i n excess of that obtained i n other magnetic measurements. The greater s e n s i t i v i t y , a v a i l a b l e when narrow band oper a t i o n i s employed, was not f e l t to j u s t i f y the com-p l i c a t i o n s which then a r i s e , p a r t i c u l a r l y f o r helium temp-erature operation. CHAPTER IV EXPERIMENTAL RESULTS 4 . 1 . I n t r o d u c t i o n The phenomenon of paramagnetic resonance was exper-i m e n t a l l y discovered by Zavoisky (Zl) i n 191+5- Concentrated c r y s t a l s of the s a l t s of the i r o n group were then r a p i d l y studied. The experimental technique was g r e a t l y advanced by Penrose (P2) i n 19^9, who found t h a t , by d i l u t i n g copper potassium sulphate w i t h the isomorphic diamagnetic mag-nesium s a l t , the s p i n - s p i n i n t e r a c t i o n was s u f f i c i e n t l y reduced to make i t p o s s i b l e to r e s o l v e the hyperfine s t r u c -ture of the copper i o n . A f t e r t h i s d i s c o v e r y , the exper-imental emphasis s h i f t e d to s t u d i e s of d i l u t e d s a l t s of the i r o n group. In 191+9, the Oxford group began a systematic study of the ions of the r a r e earth group i n the e t h y l sulphate c r y s t a l s t r u c t u r e . These r e s u l t s are reported i n Bleaney and Stevens ( B l l ) and i n Bowers and Owens ( B l 6 ) . I n 1 9 5 3 , i n t e r e s t s h i f t e d to the rare e a r t h double n i t r a t e s and coincided w i t h the p u b l i c a t i o n of the study of o p t i c a l a b s orption i n these double n i t r a t e s at helium, temperatures by Hellwege and Hellwege (H2). The paramagnetic resonance measurements of the rare e a r t h double n i t r a t e s have been 88 reported by Cooke and Duffus (C4»Cf?) and the c r y s t a l s t r u c -ture has been i n v e s t i g a t e d by Jantsch ( J 2 ) . One of the primary motivations f o r the development of the paramagnetic resonance spectrometer w i t h increased s e n s i t i v i t y described i n t h i s t h e s i s ( 2 . 7 4 ) (S3) was the de s i r e to attempt to measure the nuclear spin.and quadru-pole moment of r a d i o - a c t i v e isotopes of the rare e a r t h i o n s . I t was a l s o r e a l i z e d that higher order e f f e c t s i n the prev-i o u s l y observed spectra should be d e t e c t a b l e . The exper-imental r e s u l t s obtained u s i n g t h i s spectrometer are d i s -cussed i n the remainder of t h i s chapter (i+.2 - 4 ^ sum-marized i n l±.S 4 . 2 . Gadolinium E t h y l Sulphate The t r i v a l e n t gadolinium i o n , whose seven 4*" e l e c -o trons h a l f f i l l t h i s s h e l l , has S 7 / 2 ground s t a t e by the Hundt r u l e . I t s behaviour i s very d i f f e r e n t from that of the other r a r e e a r t h ions since i t has zero o r b i t a l mom-entum. The c r y s t a l l i n e e l e c t r i c f i e l d can a f f e c t i t s energy l e v e l s only through high-order i n t e r a c t i o n s . The 8 i n i t i a l s p l i t t i n g of the l e v e l * s very s m a l l (gener-a l l y l e s s than lcm~l) and the s p i n - l a t t i c e r e l a x a t i o n time i s so long that narrow resonance l i n e s can be observed even at room temperature. 89 4 . 2 1 . Theory . The theory of the Gd 3* i o n i n the e t h y l sulphate has been discussed by E l l i o t t and Stevens ( E l ) ( 1 . 5 4 ) • They show that i t should be p o s s i b l e t o f i t the r e s u l t s , when the a x i s of the e x t e r n a l magnetic f i e l d c o i n c i d e s w i t h t h a t of the c r y s t a l l i n e e l e c t r i c f i e l d , to a s p i n -Hamiltonian of the form S= g/3H.S+A§Pg(S)+A2p2( s) +A6P6 ( s ) + A6 P6 ( s ) ( 2 2 ) where each i s an operator f u n c t i o n which has the same tran s f o r m a t i o n p r o p e r t i e s as the corresponding s p h e r i c a l harmonic Y™.1 n The operator e q u i v a l e n t s f o r the P™s are PgtS) 5 3 S 2 2 - S (S + 1) (23a) PJ|(S) S 35S 2^ - [30S(S + 1) - 2 5 ] s z 2 - 6S(S+1) + 3S 2(S + l ) 2 (23b) P? ( s ) s 231S, 6 - 105S k f3S(S + 1) - 7 ] + s z 2 [ i o 5 s 2 ( s + 1 ) 2 - 525s ( s + D + 2 9 4 ] - 5 s 3 ( s + D 3 + 4 o s 2 ( s + D 2 - 6 o s ( s + i ) (23c) p|(S) 3 1/2 (S* + st) (23d) P 2 C S ) = 1/2 ( S 2 + S 2) (23e) When the magnetic f i e l d H i s p e r p e n d i c u l a r t o the symmetry a x i s of the c r y s t a l , i t i s convenient i n strong f i e l d s to use the d i r e c t i o n of H as the a x i s or quantiz-a t i o n and the spin-Hamiltonian (22) becomes k = gJlH • S 2 + 4 ( 3 / 2 P| •- 1/2 p|) + A° (35/8 pjj - 5/2 P^ + 3/8 P ° ) + A° (231/32 P^ - 63/16 pj* + 105/32 P^ - 5 / l 6 P£) + A^ (1/16 P°cos 60 + 15/32 p | + 3/16 pjf * l / 3 2 p£) (21+) where 0 i s the angle between the magnetic f i e l d and a side of the hexagon formed by the c r o s s - s e c t i o n o f the c r y s t a l normal t o the c r y s t a l a x i s , cos 60 - - 1 when the magnetic f i e l d i s p a r a l l e l to one of these s i d e s . The diagonal terms i n (21+) are Alt= g/JH-S2-l/2A°P°+3/8A°^P°+l/l6(A^cos6^-SA^)P^ (25) I t i s convenient, f o r computational purposes, to i n t r o d u c e new c o e f f i c i e n t s a m r e l a t e d to the A m as f o l l o w s . n n a 2 = 3 A 2 ' a4 = 6 0 A V a 6 = 1 2 6 0 A 6 » a 6 = 1 2 6 0 A 6 ( 2 6 ) The seven strong t r a n s i t i o n s corresponding to A M = - 1 have been s t u d i e d by Bleaney, et a l (B9, B12) p a r a l l e l and pe r p e n d i c u l a r t o the c r y s t a l symmetry a x i s and S c o v i l (S2) has v e r i f i e d that t h e i r angular dependence i s , to a f i r s t approximation, (3cos8 - 1) as expected f o r the l a r g e s t term P^, since i t has the same tr a n s f o r m a t i o n p r o p e r t i e s as the s p h e r i c a l harmonic Y^. 4.22. Experimental Results For D i l u t e Gadolinium E t h y l Sulphate T r a n s i t i o n s corresponding to A M = - 1, - 2 and - 3 have been observed i n c r y s t a l s c o n t a i n i n g gadolinium e t h y l sulphate d i l u t e d one p a r t i n about two hundred i n the i s o -morphic, diamagnetic lanthanum s a l t at l i q u i d oxygen temp-erature u s i n g the double f i e l d modulation spectrometer (2.74). Only the A M = - 1 t r a n s i t i o n s were v i s i b l e w i t h the same c r y s t a l i n the video spectrometer (2.73). Theor-e t i c a l l y , the i n t e n s i t y of the A M = - 2 t r a n s i t i o n s should be smaller than that of the A M = - 1 t r a n s i t i o n s by a f a c t o r of about (D/H) or about 0.006. The i n t e n s i t y of the A M = - 3 t r a n s i t i o n s should be smaller than t h a t o f the A M = - 2 t r a n s i t i o n s by a s i m i l a r f a c t o r . Exper-i m e n t a l l y , i t has been found that the r e l a t i v e i n t e n s i t i e s of the A M = - 1, - 2, and - 3 t r a n s i t i o n s are i n the k 2 r a t i o 10^" : 10 : 1 i n good agreement w i t h the above c a l -c u l a t i o n . The most intense A M = - 3 t r a n s i t i o n s occurred at s i g n a l - t o - n o i s e r a t i o s of 10:1 confirming the improve-ment i n the s e n s i t i v i t y of the double modulation spectro-meter of about 1000 over the video type. The p o s i t i o n s of the A M = - 1 t r a n s i t i o n s i n the p a r a l l e l and perpendicular d i r e c t i o n to the c r y s t a l symmetry a x i s have been measured at 290 K., 90 K., and 20°K. by S c d v i l (S2,B12). His r e s u l t s at 290°K. and 90°K. are compared w i t h ours i n Table V I I . The p o s i t i o n s of the energy l e v e l s f o r the p a r a l l e l d i r e c t i o n as p r e d i c t e d by the diagonal terms i n the s p i n -Hamiltonian ( I H , t t ) are E (-7/2) = 7/2G + 7 a ° + 7a£ + o l a 6 E (-5/2) = 5/2G + o - H - 5ag £ (-3/2) = 3/2G - 3 a ° - 3ag + E (-1/2) = 1/2G - • 9 a ° Q *a°6 E (+1/2) =-l/2G - * g + 9a° -E (+3/2) =-3/2G -* A + E (+5/2) =-5/2G + la?. 0 - 1 3 a ^ -E (+7/2) =-7/2G + 7 a ° + 7ag + * 2 where G = g ( (& H The t r a n s i t i o n s f o r the p a r a l l e l d i r e c t i o n cor' responding to A M = - 1 are, E - 7 / 2 « - - 5/2 = G + 6 a 2 + 2°a2 + 6 a 6 B - 5/2#*- 3/2 = G + U a 2 " 10&l " ^ a 6 E - 3 / 2 « — 1/2 = G + 2 a 2 • 1 2 \ + l U a 6 S-l/2«-* + 1/2 = G 93 E + l / 2 ^ + 3 / 2 = G - 2a| + 12a£ - Ufag -% 3 / 2 — + 5/2 = G ~ ^ 2 + 1 0 a £ * ^ a 6 E+5/fc + 7/2 = G - 6 a ° - 20a°. - 6a2 b A M = * 2 are, E - 7/2*+- 3/2 =2G + 0 10a 2 + - 8 a ° E - 5 / 2 - ~ - 1/2 =2G + / 0 6 a 2 - 22a° E - 3/2 •••+3/2 =2G + 2a° 2 - 12a° +I4a£ E - l / 2 * - * + 3/2 =2G -0 + 12a,° k - M E+l/2*-» + 5/2 =2G - 6a° + 22a° k =2G - 10a° - 10a° and f o r A M = - 3 are E-7/2<-»- 1/2 =3G + 12a° 2 - 2a° + 6a° E-5/2«e-+ 1/2 =3G + / 0 6 a 2 - 22a° k E - 3/2 3/2 =3G E-l/2« * - » + 5/2 =3G - 6a^ + 22a? k E + l / 2 * 7/2 =3G - 12a0 + < - 6 a ° S i m i l a r l y , the p o s i t i o n s of the energy l e v e l s f o r the pe r p e n d i c u l a r d i r e c t i o n as p r e d i c t e d by the dia g o n a l terms i n the spin-Hamiltonian (111,22) are, B(7/2) = 7/2G + 7/2a° - 2l/8a° - l/l6(a£cos60-5a2) E(5/2) = 5/2G + l/2a° + 39/8a°_ * 5/l6(a£cos60-5a£) E(3/2) = 3/2G - 3/2a° + 9/8a£ - 9/16 (a * co a60-5&l) E(.l/2) = 1/2G - 5/2a°2 - 2?/8a£ + £/l6(a|cos60-5a£) E(-l/2)=' -1/2G - 5/2*1 - 27/8a° + 5/l6(a|cos60-£ag) E(-3/2) = -3/2G - 3/2a° + 9/8a° - 9/l6(a|Cos60-£aj?) E(-5/2) = -5/2G + l/2a° + 39/8a° + 5/l6(a^cos6jZ(-5a°) E(-7/2) = -7/2G + 7/2a° - 2l/8a° - 6 0 l/l6ta 6cos6iZf-5a 6) where G : H The t r a n s i t i o n s f o r the perpendicular d i r e c t i o n cor' responding to AM = - l a r e E+7/2^*+5/2 = G + 3 * 2 - lS/2a° - 3/8(a^cos6jZ(-5a6) E+5/2*-»+3/2 = G + 2a° + l5Aa£ + 7/8(a^ Cos6^-5a^) E+3/2«-»-+l/2 = G + la° + 1/2*1 - 7/8(a^cos6j2f-5a2) E+l/2«-*-l/2 =  E - l / 2 < - - 3 / 2 = G G - 0 1&2 ~ « / o 9 / 2 % + 6 0 7/8(a6cos60-£a6) E - 3/24-»-5/2 = G - 4 " l5Aa° - 7/8(a^cos6^-5a^) E - 5 / 2 ^ - 7 / 2 = G - 3a°£ + l5/2a° + k 3/8(a£cos60-£a°) TABLE V Experimental Data AM = --1 T r a n s i t i o n s For D i l u t e Gadolinium E t h y l Sulphate at 290OK. H p a r a l l e l to c r y s t a l symmetry a x i s P= 2 i| .250 kmcs. T = 290°K. T r a n s i t i o n F i e l d R e l a t i v e Separation AM = * 1 (gauss) P o s i t i o n Between T r a n s i t i o n s - 7 / 2 • • - 5 / 2 - 5 / 2 * * - 3 / 2 9 5 6 7 . 2 +863.6 - 3 / 2 - * * - 1 / 2 9 1 7 4 - 5 + 4 7 0 . 9 - 1 / 2 * ^ + 1 / 2 8 7 0 3 . 6 0 + l / 2 * - + 3 / 2 8 2 3 0 . 6 - 4 7 3 . 0 + 3 / 2 « ~ + 5 / 2 7 8 3 5 . 3 - 8 6 8 . 3 + 5 / 2 * - » + 7 / 2 7 5 3 6 . 6 - 1 1 6 7 . 0 H perpendicular to c r y s t a l symmetry a x i s V>= 2 4 . 2 5 0 kmcs.- T = 290°K. T r a n s i t i o n F i e l d R e l a t i v e Separation C o r r e c t i o n Corrected AM = * 1 (gauss) P o s i t i o n Between / p 2 . F i e l d T r a n s i t i o n s 2' 392 . 7 4 7 0 . 9 4 7 3 . 0 3 9 5 . 3 2 9 8 . 7 + 7/2 +5/2 9 3 7 3 . 8 + 6 8 6 . 4 - 1 5 . 1 9 3 5 8 . 7 +5/2 « - + 3 / 2 280 . 8 9 0 9 3 . 0 + 4 0 5 . 6 + 2 . 5 9095-5 +3/2 «^+l/2 2 0 9 . 2 8 8 8 3 , 8 + 1 9 6 . 4 + 1 4 . 2 8898 .0 +1/2 ^ » - 1 / 2 1 9 6 . 4 8 6 8 7 . 4 0 +19.0 8 7 0 6 . 4 - 1 / 2 - ^ - 3 / 2 . 1 9 1 . 4 8 4 9 6 . 0 -191 .4 +16.3 8 5 1 2 . 3 - 3 / 2 ^ - 5 / 2 1 9 4 . 1 8 3 0 1 . 9 + 5 . 3 • 8 3 0 7 . 2 2 4 0 . 8 - 5 / 2 * - * - 7 / 2 8 0 6 1 . 1 - 6 2 6 . 3 - 1 5 . 4 8 0 4 5 . 7 f a c i n g page 95 TABLE VI Experimental Data A M = -1 T r a n s i t i o n s For D i l u t e Gadolinium E t h y l Sulphate at 90°K. H p a r a l l e l to c r y s t a l symmetry a x i s v> T r a n s i t i o n AM = * 1 - 7 / 2 • * - 5 / 2 - 5 / 2 ^ - 3 / 2 - 3 / 2 * - * - 1 / 2 - 1 / 2 * - * + l / 2 +1/2 +++3/2 +3/Z~+ +5/2 + 5/2*-*+7/2 H perpe n d i c u l a r to c r y s t a l symmetry a x i s p = 2^.375 kmcs. T = 90°K. T r a n s i t i o n F i e l d R e l a t i v e Separation C o r r e c t i o n Corrected AM = * 1 (gauss) P o s i t i o n Between , p 2 » F i e l d T r a n s i t i o n s { 2' +1/2 +->+$/2 9 4 6 0 . 9 + 7 2 8 . 2 - I 6 . 4 9 4 5 4 - 5 * 5 / 2 « - * + 3 / 2 299 . 8 9 1 6 1 . 1 + 4 2 8 . 4 + 2 . 7 9 1 6 3 . 8 +3/2*-*+l/2 2 2 0 . 5 8 9 4 0 . 6 + 2 0 7 . 9 +15 .4 8 9 5 6 . 0 + 1/2 <~>-l/2 2 0 7 . 9 8 7 3 2 . 7 0 + 2 0 . 7 8 7 5 3 . 4 -1/2++ - 3 / 2 2 0 3 . 2 8 5 2 9 . 5 - 2 0 3 . 2 +17 .7 8 5 4 7 . 2 - 3 / 2 * * - 5 / 2 2 0 4 . 8 8 3 2 4 . 7 - 4 0 8 . 0 + 5 . 8 8 3 3 0 . 5 - 5 / 2 ^ - 7 / 2 2 5 3 . 0 8 0 7 1 . 7 ' - 6 6 1 . 0 -16 . 8 8054.9 = 2 4 . 5 5 0 kmcs. T = 90°K. F i e l d R e l a t i v e Separation (gauss) P o s i t i o n Between . T r a n s i t i o n s 9 , 3 1 9 . 6 + 5 0 0 . 2 8 , 8 1 9 . 4 0 8 , 3 2 3 . 1 - 4 9 6 . 3 7 , 9 0 5 . 6 - 9 1 3 . 8 7 , 5 8 4 . 4 - 1 2 3 5 . 0 5 0 0 . 2 4 9 6 . 3 4 1 7 . 5 3 2 1 . 2 f a c i n g page 95 TABLE V I I "g" Values And S p l i t t i n g Parameters For D i l u t e Gadolinium E t h y l Sulphate. From A M = - 1 T r a n s i t i o n s 290°K. 90 °K. Buckmaster Sc o v i l ( S 2 ) Buckmaster S c o v i l g„ 1.9908-0.001 1.992-0.002 1.9880-0.001 1.992-0.002 g x 1.9919-0.001 1.992*0.002 1 .9915*0.001 1.992-0.002 (uncorrected) g± 1.9905-0.001 1.992-0.002 1.9901-0.001 1.992-0.002 (co r r e c t e d P 2) a° | ( +193-2*2.0 190 +206.2-2.0 204.7*2.0 a ° H - 3.84-0 . 3 -3.62 - 3.98-0 . 3 - 3-96*0 . 3 a £ H +0.44*0.3 +0.538 +0.41*0 .3 +0.63*0.1 a ^ j , + 1 9 3 . 3 -2 +204.5 (uncorrected) a ° ^ - 3.83-0 . 3 - 4 . 0 5 (uncorrected) (a£cos60-5ag)± -4.92*0.5 -5.58 (uncorrected) a° +192.8*2.0 +191 +204.7*2.0 2 (corrected' P 2) 0 k± 2 (corrected P2) (a^cos60-5a^) x -4.86*0.5 -6.22*0.5 ( c o r r e c t e d Pifj) a/ , +2.7*1.0 +2 .7 4-2-1.0 +3 .5-0 .5 o _i_ 2 ( c o r r e c t e d P2) A l l values o f a^J are i n u n i t s of 10 ^ cm a? . - 3 . 9 8 * 0 . 3 - 3 . 7 3 - 4 . 5 0 * 0 . 3 f a c i n g page 95 to AM = - 2 are E+7/2 «>_ + 3 /2 = 2G + 5a° - l5Aa° + l/2(a£cos60-5a°) E +5/2 —>+l/2 ~ 2G + 3a° + 3 3 A a J E+ 3/2*-* -1/2 " 2G + a° + 9/2a° - 7/8(a*cos60-5a£) E + l / 2 * - * - 3 / 2 = 2G - a° - 9/2a° + 7 / 8 ^ 0 0 3 6 ^ - ^ ) 3-1/2 — -5/2 = 2G - 3a° - 3 3 A a J •• E - 3 / 2 * - - 7 / 2 = 2G - 5a° + l5Aa° - l/2(a^cos6)Zf-5a^) and to A M = - 3 are E+7/2 «-+l/2 = 3G + 6a° + 3/8(a|cos60-$ag) E + 5 / 2 « ~ -1/2 = 3G + 3a^ + 3 3 A a ° S * 3 / 2 * ^ - 3 / 2 = 3G B * 1/2 *--5/ 2 = 3G "3a° - 3 3 A a J S - l / 2 « - 7 / 2 = 3G - 6 a 2 - 3 A a ^ + 3/8(a^cos60-5a°) Tables V and VI give the observed p o s i t i o n s of the A M = - 1 t r a n s i t i o n s at 290°K. and 90°K. f o r the p a r a l l e l and p e r p e n d i c u l a r d i r e c t i o n s of the magnetic f i e l d w i t h respect to the c r y s t a l symmetry a x i s . Table V I I compares the values of the c o e f f i c i e n t s (a™) and the "g" values as c a l c u l a t e d from the above data w i t h those of S c o v i l (S2,B12). The r e s u l t s are i n agreement w i t h i n the 6 experimental e r r o r . The value of a^ has been computed 5400h-5200 5 0 0 0 4 8 0 0 4 6 0 0 4 4 0 0 4 2 0 0 4 0 0 0 3 8 0 0 3 6 0 0 3400 x 2 < * '2 FIGURE IS J _ L J I L - 2 0 -10 0 10 2 0 3 0 4 0 5 0 6 0 70 8 0 9 0 100 9 , 1 9 DEGREES 9 _ L A M « ± 2 TRANSITION IN GADOLINIUM E T H Y L S U L P H A T E A S A FUNCTION OF T H E A N G L E WITH S Y M M E T R Y AXIS OF THE C R Y S T A L T « 9 0 ° K . = 2 4 - 4 5 5 kmcs . f a c i n g page 96 TABLE V I I I Experimental Data AM = - 2 T r a n s i t i o n s For D i l u t e Gadolinium E t h y l Sulphate At 90°K. H p a r a l l e l to c r y s t a l symmetry a x i s P= 24.455 kmcs. T r a n s i t i o n A M 2 - 7 / 2 « - * - 3 / 2 - 5 / 2 — - 1 / 2 - 3 / 2 — + 1 / 2 - 1 / 2 — + 3 / 2 +1/2 — + 5 / 2 +3/2 — + 7 / 2 F i e l d (gauss) 5473 5095 4636 4 1 3 2 3682 3308 T = 90°K. R e l a t i v e P o s i t i o n (Mean=4388) +1085 + 707 + 248 - 256 ' - 706 - 1 0 8 0 Separation Between T r a n s i t i o n s 378 459 504 450 374 T r a n s i t i o n AM = ±2 H perpe n d i c u l a r t o c r y s t a l symmetry a x i s P= 2 4 . 4 5 5 kmcs. T = 90°K. F i e l d R e l a t i v e Separation C o r r e c t i o n (gauss) P o s i t i o n Between /~2, /-p*\ . (Mean=4388) T r a n s i t i o n s 2] +7/2 + 5 / 2 ' +3/2 +1/2-- 1 / 2 . - 3 / 2 •+3/2 H - l / 2 - 1 / 2 - 3 / 2 - 5 / 2 - 7 / 2 4 9 5 4 . 5 4 6 7 9 . 2 4 4 6 0 . 8 4 2 5 8 . 0 4 0 6 0 . 0 3 8 4 6 . 0 + 5 6 6 . 5 + 2 9 1 . 2 + 72 . 8 - 1 3 0 . 0 - 3 2 8 . 0 - 5 4 2 . 0 2 7 5 . 3 218 .4 202 .8 198 .0 214.0 - 1 4 . 8 +15.3 + 3 4 . 8 +39.0 +26.0 - 7 . 6 Corrected F i e l d 4 9 3 9 . 7 4 6 9 4 - 5 4 4 9 5 . 6 4 2 9 7 . 0 4 0 8 6 . 0 3 8 3 4 . 4 f a c i n g page 96 TABEE IX AM = - 2 T r a n s i t i o n s For D i l u t e Gadolinium E t h y l Sulphate T = 90° 'K. • • W = 2 4 . 4 5 5 kmcs. E r r o r : - 0 . 5 gauss Angle 9 ° 2 2 2 2 2 2 2 2 2 2 -1^-1 2 2 - 1 0 3 , 3 4 2 . 9 3,721.3 4 , 1 6 7 . 4 4 , 6 3 4 . 4 5 , 0 6 0 . 1 5 , 4 0 5 . 5 - 5 3 . 3 0 9 . 7 3 , 6 9 3 . 1 4 , 1 4 6 . 0 4 , 6 3 8 . 2 5 , 0 8 9 . 4 5 , 4 5 6 . 5 S» 0 5 3 , 6 8 6 . 4 4 , 1 3 8 . 8 4 , 6 3 4 . 5 5 , 0 7 9 . 2 5 , 4 4 7 . 6 10 3 , 3 4 6 . 6 3 , 7 1 1 . 3 4,159.6 4 , 6 2 7 . 9 5,049.8 5 , 4 0 2 . 2 15 3,383.1 3 , 7 5 2 . 1 4,192.1 4 , 6 2 5 . 9 5 , 0 1 3 . 5 5 , 3 3 1 . 5 20 3 , 4 4 5 . 5 3 , 8 1 7 . 1 4 , 2 2 5 . 6 4 , 6 2 2 . 3 4 , 9 5 1 . 7 5,235.8 25 3,518.8 3,885.8 4,263.8 4 , 5 9 5 . 5 4 , 8 7 9 . 0 5 , H 9 . 5 30 3 , 6 1 1 . 1 3 , 9 6 7 . 6 4 , 3 0 4 . 2 4 , 5 7 7 . 0 4 , 7 9 6 . 4 4 , 9 8 5 . 1 35 3 , 7 1 6 . 3 4 , 0 5 4 - 2 4 , 3 4 1 . 5 4 , 5 5 0 . 1 4 , 7 0 7 . 3 4 , 8 4 5 . 1 ho 3 , 8 3 6 . 9 4 , 1 4 6 . 9 4 , 3 7 6 . 5 4 , 5 2 0 . 1 4 , 6 1 7 . 4 4,699.8 4 5 3 , 9 7 2 . 4 4 , 2 4 0 . 3 4 , 4 0 4 . 7 4,485.6 4 , 5 2 7 . 2 4 , 5 5 6 . 3 50 4,H5 .9 4,328.8 4 , 4 2 7 . 4 4 , 4 4 9 . 1 ' 4 , 4 4 9 . 1 4 , 4 2 7 . 4 5 5 • 4 , 2 7 1 . 2 4 , 4 1 5 . 4 4 , 4 4 6 . 0 4 , 4 1 5 . 4 4 , 3 5 7 . 0 4 , 2 8 6 . 9 60 4 , 4 2 0 . 4 • 4 , 4 8 9 . 6 4 , 4 5 4 . 3 4 , 3 7 7 . 3 4 , 2 8 5 . 0 4 , 1 7 5 . 9 65 4 , 5 5 9 . 9 4 , 5 4 9 . 9 4 , 4 6 1 . 2 4 , 3 4 4 . 8 4 , 2 1 7 . 8 4 , 0 8 0 . 9 70 4,692.0 4 , 6 0 0 . 7 4 , 4 6 2 . 9 4 , 3 1 5 . 1 4 , 1 6 1 . 2 3 , 9 9 7 . 7 75 4 , 8 0 3 . 5 4 , 6 3 6 . 8 4,462.7 . 4,290.2 "4,117.3 3,927.8 80 4,882.7 4 , 6 6 2 . 8 4 , 4 6 3 . 1 4 , 2 7 2 . 0 4 , 0 8 4 . 5 3 , 8 8 0 . 6 85 4,976.9 4 , 6 7 5 . 7 4 , 4 6 1 . 4 4 , 2 6 1 . 0 4 , 0 6 5 . 6 3 , 8 5 4 - 0 g x 9 0 4 , 9 5 4 - 5 4,679.2 4 , 4 6 0 . 8 4 , 2 5 8 . 0 4 , 0 6 0 . 0 3 , 8 4 6 . 0 . 95 4 , 9 3 2 . 3 4,672.8 4 , 4 6 1 . 5 4 , 2 6 2 . 2 4 , 0 6 7 . 5 . 3 , 8 5 7 . 5 100 4,885.5 4 , 6 6 0 . 1 4 , 4 6 2 . 4 4 , 2 7 3 . 3 4,087.3 f a c i n g page 96 TABLE X Values of g„ And g A And S p l i t t i n g Parameters Fgv D i l u t e Gadolinium E t h y l Sulphate Prom AM = ± 2 T r a n s i t i o n s At 90OK. T r a n s i t i o n Mean F i e l d g n ± 7/2 * * ± 3/2 4 * 3 9 0 . 5 1 . 9 8 9 7 + 5/2 1/2 4 , 3 8 5 5 1 , 9 9 0 6 + 3 / 2 * * + 1/2 4 , 3 8 4 - 0 1 . 9 9 2 6 V = 2 4 . 4 5 5 kmcs. Mean Value g u = 1.9910 - 0 . 0 0 1 Mean Value g„ Excluding - 3/2«-» + 1/2 T r a n s i t i o n s 1 . 9 9 0 2 T r a n s i t i o n Mean g C o r r e c t i o n Corrected gj_ F i e l d Factor (p|) F i e l d Corrected * 7/2*»± 3/2 4 4 0 0 . 3 1 . 9 8 5 3 - 1 1 . 2 4 3 8 9 . 1 1 . 9 9 0 3 - 5 / 2 * » * 1/2 4 3 6 9 . 6 1.9992 + 2 0 . 7 4 3 9 0 . 3 1 . 9 8 9 8 ± 3 / 2 ^ 1/2 4 3 0 9 . 4 2 . 0 2 7 8 + 3 6 . 9 4 3 4 6 . 3 2 . 0 1 0 0 Mean Value g x Uncorrected 2 . 0 0 4 1 - 0 . 0 0 1 Corrected 1 .9967 - 0 . 0 0 1 Mean Value g . Corrected E x c l u d i n g - 3/2 "+ l / 2 T r a n s i t i o n s 1.9901 - 0.001 ^ 0 0 0 (~a6cos6jZf a 2 a 4 a 6 - 5 a g ) P a r a l l e l 2 0 5 . 4 - 2.0 - 3 . 6 1 - 0.2 0.98 - 0 . 3 + + + Perpendicular 204..7 - 2.0 -4.22 - 0.2 -4«5>9 - 0 . 3 Perpendicular 202.2 - 2.0 - 4 . 7 9 - 0 . 2 - 5 . 1 0 - 0 . 3 (Corrected) A l l values of a 1 s are i n u n i t s of 10 cm n 4 f a c i n g page 96 96 u s i n g 0 = 90° where cos60 = - 1 . However t h i s value i s sub-j e c t t o considerable e r r o r . Table V I I I summarizes the experimental data f o r the A M = - 2 t r a n s i t i o n s f o r d i l u t e gadolinium e t h y l sulphate when the magnetic- f i e l d i s p a r a l l e l and perpendicular to the symmetry a x i s of the c r y s t a l . The f i e l d values f o r the p a r a l l e l d i r e c t i o n are estimated from the r o t a t i o n a l data given i n Table IX and shown g r a p h i c a l l y i n Figure 1 8 , since the i n t e n s i t i e s of the A M = - 2 t r a n s i t i o n s f a l l s to zero as i s expected from the f a c t that the A M = - 1 t r a n s i t i o n s i n t h i s d i r e c t i o n are symmetrical w i t h i n experimental e r r o r as p r e d i c t e d by the diagonal terms i n the spin-Hamiltonian ( 1 1 1 , 2 2 ) . The behavior of the - 7 / 2 * * - 3/2 and - 5/2**- l / 2 t r a n s i t i o n s i s i n close agreement w i t h that expected theore-p t i c a l l y from the l a r g e s t o f f - d i a g o n a l term P^ i n the s p i n -Hamiltonian. The behavior of the - 3 / 2 * * + 1/2 t r a n s i t i o n s i s most c l e a r l y seen i n Figure 1 8 . Uo s a t i s f a c t o r y explan-a t i o n can be o f f e r e d . Approximate c a l c u l a t i o n s based on 2 the second l a r g e s t o f f - d i a g o n a l term P^ do not improve the s i t u a t i o n . The f a c t that these t r a n s i t i o n s occur at lower magnetic f i e l d s than expected i n d i c a t e s t h a t the zero f i e l d s p l i t t i n g i s greater than p r e d i c t e d by the present s p i n -Hamiltonian however t h i s i s not i n agreement w i t h the r e s u l t s of Bleaney, S c o v i l and Trenam ( B l l ) . The most s t r i k -i n g f e a t u r e s of the behavior o f these t r a n s i t i o n s are t h e i r complete l a c k of symmetry w i t h respect to each other. That each t r a n s i t i o n i s r e l a t i v e l y independent of the o r i e n t a t i o n PLATE IXa Str u c t u r e on bM«±l t r a n s i t i o n s due to l a t t i c e defects and p a i r s of Gd ions PLATE IXb 4 of 5 AM, ± 3 t r a n s i t i o n s near crossover point PLATE IXc A M » * * t r a n s i t i o n s near crossover point PLATE IXd . 6 &Hsi:3. t r a n s i t i o n s at magnetic f i e l d o r i e n t a t i o n where t o t a l separation i s 500 gauss f a c i n g page 9 7 TAEBE XI AM = - 3 T r a n s i t i o n s For D i l u t e Gadolinium E t h y l Sulphate As A Function Of Angle Between D i r e c t i o n Of Magnetic F i e l d And C r y s t a l Symmetry A x i s At 5 ° I n t e r v a l s V= 2k.$2 kmcs. T = 90©K. ©o - 7 / 2 * — 1 / 2 - 5 / 2 * * * l / 2 - 3 / 2 * * + 3 / 2 - l / 2 * * * 5 / 2 +l / 2 * * + 7 / 2 -10 - 6 . 5 ° - 5 10 11 15 16 20 21 25 26 30 31 35 36 40 4 1 4 5 46 50 51 55 56 60 61 65 66 70 71 7-5 76 80 81 85 86 *i.90 91 95 96 100 3 4 1 3 . 1 9 3 2 2 7 . 7 4 3 2 9 5 . 4 2 3269.14 3 1 7 6 . 1 0 3 1 4 9 . 2 8 3 0 5 7 . 7 8 3 0 4 5 . 8 7 2 9 2 4 . 6 4 2934.72 2 8 4 4 - 6 7 2 8 3 0 . 1 7 2 7 5 5 . 9 4 2 7 4 4 . 0 7 2 6 7 5 - 3 7 2 6 6 6 . 9 8 2 6 1 4 . 2 8 26OO.07 2 5 6 5 . 2 3 2 5 5 7 . 4 1 2 5 2 7 . 5 2 2514.91 2 5 0 5 . 1 8 2 5 0 0 . 0 3 2 5 0 2 . 4 7 2 5 2 2 . 0 3 3 3 7 2 . 7 3 3 2 9 . 1 5 3 3 1 9 . 6 6 3287.09 3277 .44 3 2 3 8 . 4 0 3 2 2 7 . 7 4 3 1 7 8 . 9 9 3 1 7 1 . 5 9 3 1 1 9 . 7 8 3 1 0 9 . 4 0 3 0 5 7 . 7 8 3 0 4 5 . 8 7 2992.81 2 9 8 3 . 2 3 2933.54 2925-47 2 8 7 2 . 7 5 2 8 6 5 . 0 1 2820.92 2 8 1 1 . 1 4 2778.29 2772.51 2 7 4 4 . 7 7 2 7 4 1 . 5 0 2 7 2 2 . 2 2 2 7 1 7 . 2 8 2 7 0 0 . 5 4 2 6 9 7 . 4 1 2692.38 2 6 9 7 . 0 2 2 7 0 9 . 4 4 2989.75 2975.97 2977.01 2986.43 2987.54 3001.65 3003.37 3006.13 3009.95 3010.33 3011.91 3 0 1 2 . 5 2 3008.25 3000.11 2997.49 2984.99 2985.75 2991.25 2968.80 2948.34 2947.69 2931.48 2927.63 2 9 1 4 . 0 8 2910.07 2901.58 2899.99 2893.26 2890.76 2889.59 2893-43 2901.03 2 6 3 4 . 4 7 2 6 9 4 . 6 8 2 7 4 2 . 0 2 2761.69 2 8 1 1 . 0 2 2820 . 23 2866.60 2 8 8 1 . 2 4 2 9 2 4 . 6 4 2934-72 2 9 8 4 . 9 9 2 9 8 5 . 7 5 3 0 1 1 . 1 1 3 0 2 3 . 0 8 3 0 4 9 . 0 9 3 0 5 5 . 2 8 3 0 7 5 . 6 8 3077.80 3 0 8 7 . 7 4 3 0 9 1 . 6 8 3 0 9 6 . 1 1 3 0 9 9 . 0 4 3 0 9 9 . 4 2 3 1 0 0 . 0 7 3 1 0 0 . 0 0 3 1 0 1 . 0 2 3 0 9 7 . 6 1 2 7 3 5 . 0 4 2 8 4 4 . 6 7 2 8 3 0 . 1 7 2 9 7 1 . 2 5 2 9 6 8 . 8 0 3049.09 3 0 7 9 . 9 8 3 1 5 5 . 3 7 3 1 7 2 . 7 6 3 2 3 9 . 1 4 3 2 5 3 . 1 3 3 3 0 3 . 1 2 3 3 1 4 . 8 4 3 3 4 6 . 6 0 3 3 5 4 . 0 8 3 3 6 3 . 6 6 3 3 5 6 . 6 0 3 3 1 9 . 4 7 f a c i n g page 97 TABLE X I I Values of g „ and g x And S p l i t t i n g Parameters For D i l u t e Gadolinium E t h y l Sulphate From AM=- 3 T r a n s i t i o n s At 90®K. N© .value f o r g ( | can he c a l c u l a t e d as n© t r a n s i t i o n s were observed i n t h i s d i r e c t i o n ©r s u f f i c i e n t l y near t o permit accurate e x t r a p o l a t i o n . Mean C o r r e c t i o n Corrected g x T r a n s i t i o n F i e l d g x Factor Pjj; F i e l d . Corrected +7/2*»*l/2 2932 1 .9915 * 4 2936 1 . 9 8 8 6 * 5 / 2**+l / 2 2897 2 .0156 +kl 2938 1 . 9 8 7 5 + 3 / 2 * > - 3 / 2 2893 2.0181). +54 2947 1 ,9814 Mean Value g A Uncorrected 2 . 0 0 8 5 * 0 . 0 0 1 Corrected 1 . 9 8 5 8 * 0 . 0 0 1 Mean Value g A Corrected E x c l u d i n g + 3 / 2 « * - 3 / 2 T r a n s i t i o n 1 . 9 8 8 1 * 0 . 0 0 1 I t i s not p o s s i b l e to c a l c u l a t e the values of the a^»s from the experimental data f o r the AM = - 3 t r a n s i t i o n s since there are not a s u f f i c i e n t number of l i n e a r l y independent equations. We, t h e r e f o r e , c a l c u l a t e the p o s i t i o n s from the data f o r the AM = - 1 and * 2 t r a n s i t i o n s and compare w i t h the observed p o s i t i o n s V>= 2 4 . 5 2 0 kmcs. T = 90°K. T r a n s i t i o n Measured C o r r e c t i o n Corrected C a l c u l a t e d C a l c u l a t e d AM = * 3 P o s i t i o n Factor P^ F i e l d AM=*1 Data AM=* 2 + 7/2+* +1/2 3364 + 1 .7 3366 3374 3369 + 5 / 2 * * - 1 / 2 3101 + 3 8 . 8 3140 3141 3138 +3/2 * - 3 / 2 2893 + 5 3 . 8 2947 2934 2934 +1/2 - * - 5 / 2 2692 + 4 4 . 2 2736 2727 2731 - 1 / 2 « - 7 / 2 2500 + 6 . 8 2507 2494 2500 f a c i n g page 97 97 of the magnetic f i e l d w i t h the symmetry a x i s over a r e g i o n o of about 20 i s the only s i m i l a r i t y . The values of the s p l i t t i n g parameters given i n Table X and c a l c u l a t e d from the AM = - 2 t r a n s i t i o n s are considered to be l e s s r e l i a b l e than those from the A M = - 1 t r a n s i t i o n s because of the above-noted behavior of the - 3/2+++ 1/2 t r a n s i t i o n s / This i s most c l e a r l y seen i n the "g" values where they a p p r e c i a b l y increase the value of the mean f o r the three p a i r s of t r a n s i t i o n s . Consequently, the g values have a l s o been c a l c u l a t e d e x c l u d i n g the data f o r the - 3/2«-* + l / 2 t r a n s i t i o n s and these values are shown i n Table X.- ) Table XI summarizes the experimental data f o r the AM = * 3 t r a n s i t i o n s f o r d i l u t e gadolinium e t h y l sulphate and Table X I I gives the values of the s p l i t t i n g parameters and spectroscopic s p l i t t i n g f a c t o r s f o r . t h e p a r a l l e l and' perpen d i c u l a r d i r e c t i o n s . This data shows that the e f f e c t s observed i n the AM = - 2 t r a n s i t i o n s are f u r t h e r magni-f i e d i n the A M = - 3 t r a n s i t i o n s . U n f o r t u n a t e l y , not a l l of these t r a n s i t i o n s could be observed at each o r i e n t a t i o n so the data i s incomplete and the c a l c u l a t i o n s correspond-i n g l y l e s s r e l i a b l e . The r e l a t i v e i n t e n s i t i e s o f the £ AM = * 3 are q u i t e asymmetric; the low f i e l d t r a n s i t i o n i s always the weakest and the two h i g h f i e l d t r a n s i t i o n s s t rongest. A s i m i l a r e f f e c t was a l s o observed i n the AM = - 2 t r a n s i t i o n s , however I t was not so marked. The behavior of those t r a n s i t i o n s between the + 1/2 and -1 / 2 l e v e l s and the + 3/2 - 3/2 t r a n s i t i o n as a 98 f u n c t i o n of the angle between the d i r e c t i o n of the s t a t i c mag-n e t i c f i e l d and the symmetry a x i s of the c r y s t a l are very asymmetric. Consequently, the data i n Table X I I i s not r e l i a b l e although i t has a l s o been correc t e d f o r the e f f e c t of the l a r g e s t o f f - d i a g o n a l term i n the spin-Hamiltonian P 2 ( S z ) . F o l l o w i n g a suggestion of Stevens (Bleaney, S c o v i l and Trenam B12), an attempt has been made to consider the 3 e f f e c t of a term o f the form P^(S 2) t h e o r e t i c a l l y ; however, t h i s was un s u c c e s s f u l . Moreover, the form of P 3 ( S 2 ) , when considered as a Legendre p o l y n m i a l , i s not c o r r e c t t o a i d i n symmetrizing the r o t a t i o n a l data and hence i s not considered as the c o r r e c t i n t e r p r e t a t i o n o f the asymmetry of these r e -s u l t s . k.3 L i n e Width Of Praseodymium E t h y l Sulphate At L.2°K. The paramagnetic resonance spectrum of P r 3 + | i | f 2 j ^Hj^] has been s t u d i e d at l i q u i d hydrogen temperatures i n the d i l u t e e t h y l sulphate by Bleaney and S c o v i l (BIO) and at l i q u i d helium temperatures i n the d i l u t e magnesium n i t r a t e by Cooke and Duffus (C^). The experimental r e s u l t s have been f i t t e d to the spin-Hamiltonian given i n 1 .55 ( 1 , 1 7 ) where a t h e o r e t i c a l d i s c u s s i o n i s in c l u d e d which considers the asymmetry o f the l i n e shape. D i l u t e praseodymium e t h y l sulphate has been examined at l i q u i d helium temperatures. S i x asymetric resonances were observed w i t h maximum i n t e n s i t y when the d i r e c t i o n of the s t a t i c and r . f . magnetic f i e l d s were p a r a l l e l i n confirm a t i o n 99 of the above r e s u l t s . I t was observed that the l i n e width i s considerably narrower at l i q u i d helium temperature than reported by Bleaney and S c o v i l (BIO). At i+.2°K. i t has been found to be 35 - 5 gauss whereas they r e p o r t 200 gauss at 20°K. i n d i c a t i n g t h a t s p i n - l a t t i c e broadening must occur at the l a t t e r temperature. U n f o r t u n a t e l y , the electromagnet broke down at t h i s stage of the measurements and the exper-iment was stopped. I t has not been repeated and thus no values f o r the s p l i t t i n g parameters are reported. k-k* E x c i t e d State I n Dy^brosium E t h y l Sulphate At 4«2°K. Recently, Bleaney (B13) has reported the d e t e c t i o n of an e x c i t e d state i n dysprosium e t h y l sulphate D/"* Jj+f^* ^ 1 5 / 2 ] a t i i ^ u i d hydrogen temperatures which d i s -appeared above 20°K. due to s p i n - l a t t i c e broadening and below 13°K due to suggested depopulation of the e x c i t e d l e v e l . T h e o r e t i c a l l y , the ground s t a t e i s probably J z = + 9/2 (g„ = 1 2 ; s u s c e p t i b i l i t y measurement g„ = 1 1 . 3 * 0 . 1 ) w i t h admixtures of J z = * 3/2 and * 15/2 g i v i n g no allowed t r a n s i t i o n s . The observed "g" values f o r the e x c i t e d l e v e l are g ( | = 5*80 - 0 . 0 2 and g j_ = O.ILO - 0 . 2 . They are i n good agreement w i t h the hypothesis that t h i s l e v e l i s an admixture of J z = * 7/2 and + 5 / 2 . D i l u t e d^prosium e t h y l sulphate has been examined at l i q u i d helium temperatures. A weak resonance w i t h a s i g n a l - t o - n o i s e r a t i o o f 2 0 : 1 has been observed w i t h the 100 double f i e l d modulation spectrometer ( 2 . 7 4 ) w i t h gU = 5 . 8 5 - 0 . 0 5 i n agreement w i t h the above measurements. The weak i n t e n s i t y observed supports the depopulation hypo-t h e s i s . U n f o r t u n a t e l y , t h i s experiment was a l s o h a l t e d due to electromagnet t r o u b l e s and consequently gj_ was not mea-sured. 4 . 5 . Future Experiments The apparatus has only r e c e n t l y been brought i n t o s u c c e s s f u l o p e r a t i o n at l i q u i d helium temperatures. Unfor-t u n a t e l y , whenever s u c c e s s f u l experiments have been under-way at t h i s temperature, the electromagnet gave t r o u b l e due to an i n t e r m i t t e n t short to one of the c o o l i n g c o i l s . Con-sequently, although resonances have been observed i n a num-ber of substances very few measurements have been made and those reported i n 4«3 and k.L have not been confirmed. I t would be worthwhile to study the i n t e n s i t y of the. e x c i t e d s t a t e t r a n s i t i o n i n dysprosium e t h y l sulphate as a f u n c t i o n of the temperature i n the l i q u i d helium temperature range to c a r e f u l l y check the depopulation hypothesis. An experiment has been planned to attempt to measure Pi 1+ 0 ^ „ w ~ * w « v w * * * — . * - . - w w x ^ w •*» 147 [ 4 ^ , ^ 4 ! * n t l l e magnesium n i t r a t e c r y s t a l environment. This i o n should e x h i b i t the same resonance p r o p e r t i e s as praseodymium which has been discussed i n 1 . 5 5 . The ground s t a t e i s not Kramer's degenerate however d i s t o r t i o n s from 101 t r i g o n a l symmetry due to the Ja h n - T e l l e r e f f e c t should produce t r a n s i t i o n s w i t h maximum i n t e n s i t y when the d i r -e c t i o n of the r . f . magnetic i s p a r a l l e l to t h a t of the s t a t i c magnetic f i e l d . No n a t u r a l l y o c c u r r i n g isotopes of promethium are known. Isotope li+7* which has a h a l f -l i f e of about f o u r years, can be produced a r t i f i c i a l l y by neutron bombardment of Nd U4.6 or as a f i s s i o n product. I t s s p i n i s not known; however, from the nuclear s h e l l model, i t i s probably 5/2 w i t h 7/2 another p o s s i b i l i t y . I n i t -i a l l y , 3 m i l l i c u r i e s of promethium 147 should weigh 0 ( 1 0 " 7 ) grams. I t should be p o s s i b l e to o b t a i n a lanthanum magnesium n i t r a t e c r y s t a l c o n t a i n i n g 0(10""^) grams of pro-methium. Our spectrometer ( 2 . 7 4 ) possesses s u f f i c i e n t s e n s i t i v i t y to observe a resonance from such a c r y s t a l . Some lanthanum magnesium n i t r a t e c r y s t a l s have been prep-ared from atomic weight p u r i t y lanthanum to reduce the pos-s i b i l i t y of rare e a r t h contamination. One of these c r y s t a l s w i l l be examined i n the spectrometer at l i q u i d helium temp-eratures to i d e n t i f y any such contamination and then r e -d i s s o l v e d i n a s o l u t i o n of promethium n i t r a t e and the mixed c r y s t a l grown. The promethium has been obtained from Atomic Energy of Canada L t d . as a f i s s i o n product t o reduce the p o s s i b i l i t y of other r a r e e a r t h ions being present. The spectroscopic a n a l y s i s shows a maximum of 2 . 4 mg./ml. i r o n , 0 . 8 mg./ml. aluminum, 0 . 5 mg./ml. l e a d , 0 . 5 mg./ml. n i c k e l and 1 . 5 mg./ml. c o b a l t . 102 I f t h i s experiment i s s u c c e s s f u l , there are a number of other r a d i o a c t i v e r a r e e a r t h Isotopes which have sho r t e r h a l f - l i v e s which could be examined. The determination o f t h e i r nuclear spins and magnetic moments would be most use-f u l . I t was f o r such experiments as these t h a t the h i g h s e n s i t i v i t y spectrometer was developed. Resonances from i m p u r i t i e s i n semi-conductor c r y s t a l s have been observed r e c e n t l y . I t i s hoped that resonances can be observed i n some s i l i c o n c r y s t a l s t h a t have been doped w i t h antimony, indium, a r s e n i c , g a l l i u m and boron. Some p r e l i m i n a r y t h e o r e t i c a l work by Dr. J . M. Da n i e l s supports t h i s hope. 103 REFERENCES A l Abragam, A. and Pryce, M.H.L., Proc. Roy. S o c , A, 205 135 ( 1 9 5 D B l Bagguley, D.M.S. and G r i f f i t h s , J.H.E., Proc. Phys. Soc. (London). A, 6£, 594 (1952) B2 B e r i n g e r , R. and C a s t l e , J r . J.G., Phys. Rev. Jd* 581 (1950) B3 B e r i n g e r , R. and C a s t l e , J r . J.G., Phys. Rev. 8 1 , 82 ( 1 9 5 D • Bk Be the, H.A. Ann. Phys. 3j . 133 (1929) B5 B i j l , D. Ph.D. Thesis, t e i d e n , (1950) B6 Bleaney, B. Rept. Prog. Phys. 1 1 , 178 ( 1 9 4 6 - 4 7 ) B7 Bleaney, B. P h i l . Mag. 1L2, 4 4 1 (1951) B8 Bleaney, B. Physica 3 J , 175 (1951) B9 Bleaney, B., E l l i o t t , R.J., S c o v i l , H.E.D. and Trenam, R.S. P h i l . Mag. ^ 2 , 1062. ( 1 9 5 D B10 Bleaney, B., and S c o v i l , H.E.D. P h i l . Mag. 4 3 , 999 (1952) B l l Bleaney, B. and Stevens, K.¥.H., Rept. Prog. Phys. 1 6 , 108 (1953) B12 Bleaney, B., S c o v i l , H.E.D., and Trenam, R.S., Proc. Roy. 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S o c , A, 218, 553 (1953) E3 E l l i o t t , R.J. and Stevens, K.W.H., Proc. Roy. S o c , A, 212, 387 (1953) E4 Elmore, W.C. and Sands, M., E l e c t r o n i c s , Div. V, V o l . 1. N a t i o n a l Nuclear .Energy S e r i e s , (McGraw-Hill, New York, 1949) , G l Gordy, W./Rev. Mod. Phys.; 20 , 668 (1948) HI H a r t z , T.R. and Van der Z i e l , A., Phys. Rev. J_8, 473, (1950) H2 Hellwege, A.M. and Hellwege, K.H., Z. Phys. 135, 92 (1953) H3 Hershberger, W.D., Journ. App. Phys. 19_, kll (1948) H4 Hirshon, J.M. and Praenkel, G.K., R.S.I. 26, 34 (1955) H5 Holden, A.N., K i t t e l , C , M e r r i t t , R.P., and Yager, W.A., Phys. Rev vJ_ 2 , 1^ 7 (1950) J l Jahn, H.A,, and T e l l e r , E., Proc. Roy. S o c A, 161, 220 (1937) J2 Jantsch, Y., Z. anorg. Chem. J6, 303 (1912) J3 Judd, B.R., Proc. Roy. Soc". A, 22J, 552 (1955) K l K a r p l u s , R., Phys. Rev. 13,1027 (1948) K2 K e t e l a a r , J.A.A., Physica i±, 619 (1937) K3 Knight, W.D., and Pound, R.V., R.S.I. 21, 219 (1950) KJ| Knipp, J.K., Kuper, J.B.H. and Hamilton, D.R. K l y s t r o n s and Microwave Tubes. V o l . 7. M.I.T. R a d i a t i o n Lab. S e r i e s (McGraw-Hill, New York, 1948) K5 Knoebel, H.W., and Hahn, E.E.-., R.S.I. 22, 904 (1951) E l Eawson, J.S. and Uhlenbeck, G.E., Threshold S i g n a l s V o l . 2k. M.I.T. R a d i a t i o n Lab. S e r i e s (McGraw-Hill, New York, 191+8) L2 Lavy, S.I., The Rare Earths - Their Occurrence, Chemistry Technology, (Edward Arnold and Co., London, 1924) Ml Mann, C.R., Ph.D. Thesis, U n i v e r s i t y of B r i t i s h Columbia, (1952) unpublished M2 McLay, D., Ph.D. Thesis, U n i v e r s i t y of B r i t i s h Columbia, (19$$) unpublished M3 Montgomery, C.A., Technique of Microwave Measurements V o l . 11, M.I.T. R a d i a t i o n Lab. S e r i e s (McGraw-Hill, New York, 1947) PI Penney, W.Y. and Schlapp, R., Phys. Rev. kl, 194 (1932) P2 Penrose, R.P., Nature, Lowd., 163, 992 (1949) P3 Pound, R.V/, R.S.I. J J , 490 (1946). Also Chapter 2 (M3) P4 Pryce, M.H.L. and Stevens, K.W.H., Proc. Phys. Soc. A, 62, 36 (.1950) P5 Pryce, M.H.L-., Phys. Rev. 80, 1107 (1950) ' R l Rose, M.E., Phys. Rev. £3_, 715 (1938) R2 Rundle, H.N., M.A. Thesis, U n i v e r s i t y of B r i t i s h Colum-b i a (1955) unpublished 51 Schneider, E.E. and England, T.S., Physica 17., 221 (1951) 52 S c o v i l , H.E.D., D . P h i l . Tnesis, Oxford ( 1 9 5 D . unpublished 53 S c o v i l , H.E.D. and Buckmaster, H.A. (to be published) 54 Smaller, B., Phys. Rev. 83, 812 (195D' 55 Smaller, B. and Y a s m i t i s , E.E., R.S.I. 24, 991 (1953) 56 Smythe, W.R., S t a t i c and Dynamic E l e c t r i c i t y , 2nd ed., p. 417, No. 15 (McGraw-Hill, New York, 1950) 106 57 Stevens, K.W.H., Proc. Phys. Soc. A, 6£, 209 (1952) 58 Stevens, K.W.H., Proc. Roy. Soc. A, 214, 237 (195 2) 59 Stevens, K.W.H., Proc. Roy. Soc. A, 212, 542 (1953) S10 Strandberg, M.W.P., Microwave Spectroscopy (Methuen, London, 1954) T l Thomas, H.A., D r i s c o l , R.L.., and H i p p i e , J.A.., Phys. Rev. 18, 787 (1950) T2 Torrey, H.C. and Whitmer, CA., C r y s t a l R e c t i f i e r s , V o l . 15, M.I.T. R a d i a t i o n Lab. S e r i e s (McGraw-H i l l , New York, 1948) T3 Tdwnes, C.H. and T u r k e v i t c h , J . , Phys. Rev. 77, 148 (1950) T4 Trenam, R.S., Proc. Phys. Soc. A, 66, 118 (1953) VI V a l l e y , J r . , G.'E. and Wallman, H., Vacuum Tube Ampli-f i e r s . Vol. , 1 8 , M.I.T. R a d i a t i o n Lab. S e r i e s (McGraw-Hill, New York, 1948) V2 Van d e r ' Z i e l , A.., Noise ( P r e n t i c e - H a l l , New York, 1954) V3 Van Vleck, J.H., The Theory of E l e c t r i c and Magnetic S u s c e p t i b i l i t i e s , (Oxford U n i v e r s i t y P r e s s , E,ond6n 1932) V4 Van Vleck, J.H., Phys. Rev. J4,-1168 (1948) • A V£ Van Voorki's, S.N., Microwave Receivers. V o l . 23, M.I.T. R a d i a t i o n Lab. S e r i e s (McGraw-Hill, New York, 1948) V6 V i c k e r y , R.C., Chemistry o f the Lanthanons, ( B u t t e r -worths / S c i e n t i f i c P u b l i c a t i o n s , London, 1953) Wl Weidner, R.S., and Whitmer, C.A., R.S.I. 23_, 75 (1952) Y l Yost, D.M., Rus s e l , J r . H., and Garner, C.S., The Rare-E a r t h Elements and Their Compounds (John WilleyaacU Sons., Inc., New York, 1947) Z l Zavoisky, E., J . Phys. U.S.S.R. 2 ; 211 (1945) 107 APPENDIX I The e l e c t r o n i c c i r c u i t diagrams of the components of the s i n g l e and double modulation paramagnetic resonance spectrometers described i n t h i s t h e s i s are c o l l e c t e d t o -gether i n t h i s appendix together w i t h s e v e r a l photographs of the equipment. Page P l a t e X A general view of the apparatus 109 P l a t e XI A general view of the microwave bench showing the 4 6 2 . 5 kcs. a m p l i f i e r w i t h the s h i e l d i n g cover p l a t e s removed . . 110 Figure 19 C i r c u i t diagram f o r proton resonance detector r . f . head I l l Figure 20 C i r c u i t diagram f o r proton resonance low noise audio a m p l i f i e r and cathode f o l l o w e r 112 Figure 21 C i r c u i t diagram f o r p r o t o n resonance r . f . l e v e l V.T.V.M. and o s c i l l a t o r l e v e l c o n t r o l and s t a b i l i z e r 113 Figure 2 2 500 kcs.. c l a s s C. power a m p l i f i e r and v a r i a b l e load matching s e c t i o n t o s p l i t c a v i t y 114 Figure 23 O s c i l l a t o r and b u f f e r f o r h i g h frequency modulation 115 Figure 24. C i r c u i t diagram f o r 500 k c s . t r a n s m i t t e r power supply 116 108 Pag© Figure 25 C i r c u i t diagram of video a m p l i f i e r . . . 117 Figure 26 C i r c u i t diagram o f c a l i b r a t o r 118 Figure 27 C i r c u i t diagram of t y p i c a l low r i p p l e , high, r e g u l a t i o n power supply 119 Figure 28 C i r c u i t diagram f o r h i g h voltage reg-u l a t e d k l y s t r o n power supply 120 Figure 29 Bl o c k diagram of magnet f i e l d c o n t r o l c i r c u i t 121 Figure 30 C i r c u i t diagram of current c o n t r o l s f o r electromagnet and 60 cps. f i e l d modulation 122 P L A T E X • G E N E R A L V IEW O F E X P E R I M E N T A L A P P A R A T U S FIGURE 19 R F 4 0 3 B f IK < 6 8 K 3 3 K W M © — ' v C I R C U I T D I A G R A M FOR P R O T O N R E S O N A N C E D E T E C T O R R- F- HEAD FIGURE 20 E F 37A 6 J 5 CIRCUIT D I A G R A M FOR P R O T O N R E S O N A N C E L O W NOISE AUDIO AMPLIFIER A N D C A T H O D E F O L L O W E R F I G U R E 21 6 S N 7 6 S L 7 6 S L 7 6 S H 7 2 2 5 V D C CIRCUIT D I A G R A M FOR P R O T O N R E S O N A N C E R F- L E V E L V T V M A N D O S C I L L A T I O N L E V E L C O N T R O L A N D S T A B I L I Z E R FIGURE 22 4 - 811 A 0 0 3 i W m H 5 0 0 M A (MA) r L o - 0 5 0 - 5 0 0 M A 4 V R I 5 0 _> T O 160 V D C 4 5 0 Pf •f-8 0 R F A M M E T E R looo o<-pf p ^ 0 - 5 A M P S -» T O SPL IT C A V I T Y I I 0 0 V D C 5 0 0 KCS C L A S S C POWER AMPLIF IER AND V A R I A B L E L O A D M A T C H I N G S E C T I O N TO SPLIT CAVITY FIGURE 23 6 A G 7 8 0 7 2 mHg 10 0 P F T -=K>^!50K 7 50-4-PF = 001 A ,\To. 6-3V AC 25 2-51 25 P F mH o o 01 150 I5K 6 3 V AC O T = 8 A F 6 K 5 0 W VR 105 VR105 i OK: 01 OSCILLATOR AND BUFFER FOR HIGH FREQUENCY MODULATION FREQUENCY RANGE 100 K C S . - I M C S ' . 4 0 0 P F TO 81 f " I 6 0 001^ H 7.5 mH 6 0 0 V D C F I G U R E H A M M O N D 2 6 7 6 0 0 V D C o 1300 S W 3 B R E D C I R C U I T D I A G R A M F O R 5 0 0 K C S T R A N S M I T T E R P O W E R S U P P L Y PROTON RESONANCE MACHINE FIGURE 25 5693 5693 50 H 30 MA 30 0 V.D.C. REGULATED 6.3VD.C. FILAMENT SUPPLY PIN 2 GROUNDED VIDEO AMPLIFIER FIGURE 26 H A M M O N D 167 E % F I G U R E 27 I IOVAC T Y P I C A L LOW R I P P L E , HIGH R E G U L A T I O N POWER S U P P L Y 2 5 0 - 3 7 5 V O L T S D C - 2 5 - 2 0 0 M A T -O 6 VOLT STORAGE BATTERY SW.IA O- • &-^o RED HAMMOND 30607 VARIAC W W 5 VOLTS 5 AMP *. W 2-HAMMOND 1120 X X' 6.3 V. I AMP HAMMOND 167 C Y /' 63 V 0.3 AMP HAMMOND I67B HAMMOND 156 50 H 30 MA 5 0 H 30MA 6.3 V GREEN fr. z . SW3B 6.3 V. I AMP RED HAMMOND I67C HIGH VOLTAGE KLYSTRON POWER SUPPLY FIGURE 0-600 VOLTS 10 OR 15 VR ISO'S 500 25 K FILAMENTS : o ANODE 0-25 MA 10 MA FUSE MA 450 K i 0-450 VOLTS CATHODE 150 K 0-150 VOLTS 5 0 K 0.01 0.05 0.25 4M •AVvW-IV GRID IV REFLECTOR -o SW.4 REFLECTOR MODULATION MICROWAVE POWER * SOURCE XTAL VIDEO AMPLIFIER Y X OSCILLOSCOPE A . C . MOTOR D.C. GENERATOR PHASE SHIFTER 0 ~ 2 0 AMPERE METER CONTROL PANEL 6 0 ~ MODULATION C O N T R O L •f ^ • FIGURE 29 FIGURE 3 0 6 W 110 VAC O G^ O-SW| RED VARIAC I 50 W T 0 MODULATION 0-5 AMP AC.COILS : Q H i © - ^ 3 y/^r-^ 0SCILLOSC0PE "TIME B A S E SW2 .01 , M HAMMOND 312 VOLTAGE o 2 0 0 2O0 12 1O0 W 1000 50 W vVWWVV-TO GENERATOR FIELD © : V W | «AA^*W»- •J^f&fiS SW 3 TO GENERATOR 0-I50VD.C. ( .V 0-20 AMPS D.C TO MAGNET CURRENT CONTROL CIRCUITS FOR DIRECT C U R R E N T MAGNETIZATION AND 6 0 C Y C L E MODULATION COILS ON PARAMAGNETIC RESONANCE ELECTROMAGNET 

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