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Nuclear electric quadrupole interaction in single crystals Petch, Howard Earl 1952

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T H E U N I V E R S I T Y O F B R I T I S H C O L U M B I A F A C U L T Y OF G R A D U A T E STUDIES P R O G R A M M E OF T H E F I N A L O R A L E X A M I N A T I O N F O R T H E D E G R E E OF D O C T O R OF P H I L O S O P H Y H O W A R D E A R L P E T C H B.Sc. (McMaster) • 1949 M.Sc. (McMaster) 1950 FRIDAY, S E P T E M B E R 26th, 1952, at 2:30 P.M. IN R O O M 301, PHYSICS BUILDING of C O M M I T T E E IN C H A R G E Dean H . F. Angus, Chairman Professor G. M . Volkoff Professor K. C. Mann Professor J. B. Warren Professor G. M . Shrum Professor R. M. Thompson Professor T . M. C. Taylor Professor Basil Dunell Professor F. H . Soward E X T E R N A L E X A M I N E R : Professor W. D. Knight, University of California, Berkeley, Cal. G R A D U A T E STUDIES Field of Study: Physics. Selected Topics in Physics^-Professor M. W. Johns Atomic and Molecular Spectroscopy—Professor B. McLay Electromagnetic Theory—Professor W. Opechowski Quantum Mechanics—Professor G. M . Volkoff Ghemical Physics—Professor A. J. Dekker Nuclear Physics—Professor K. C. Mann Other Studies: Physical Chemistry of High Polymers—Professor B. A. Dunell Theory of the Chemical Bond—Professor C. Reid Radiochemistry—Professors M . Kirsch and K. Starke T H E S I S N U C L E A R E L E C T R I C Q U A D R U P O L E I N T E R A C T I O N IN SINGLE CRYSTALS The nuclear magnetic resonance absorption technique has been applied to the study of the interaction between atomic nuclei, possessing electric quadrupole moments, and the crystalline electric field gradient existing at the nuclear sites in single crystals. The theoretical section of this thesis outlines Dr. Volkoff's extension to non-uniaxial crystals of Pound's theory of the splitting of magnetic resonance absorption lines in a single crystal caused by the nuclear electric quadrupole interaction. Experiments have been performed on the splitting of the L i 7 and A l 2 7 absorption lines in a single crystal of L i A l (SiOs) 2 (spodumene) and the above theory has been used to analyse the results. The absolute value of the quadrupole coupling constant for the L i ' nuclei in spodumene is found to be |eQ<£zz/h|=:75.7 plus or minus 0.5kc. per sec. The axial asymmetry parameter of the field gradient tensor at the site of the L i nuclei is found to be 77 = (<£xx— <£yy) /<t>zz~0.79 plus or minus 0.01. One of the principal axes of this tensor (the y axis corresponding to the eigenvalue of intermediate magniture) is experimentally found to coincide with the b crys-tallographic axis of monoclinic spodumene as required by the known symmetry of the crystal. The other two principal' axes are in the ac plane, the z axis (corresponding to the eigenvalue </> zz of the greatest magnitude) lying between the a and c axes at an angle of 46.5° plus or minus 1° with the c axis. It is shown that the four L i atoms per unit cell all exist in equivalent crystal positions as required- by the crystal symmetry. • . The absolute value of the quadrupole coupling constant for the A l 2 7 nuclei in spodumene is found to be |eQ<£zz/h|—2960 plus or minus 10 kc. per sec. The axial asymmetry parameter of the field gradient tensor at the site of the Al nuclei is found to be 7) = (<£xx— <£yy)/<£zz=0.95 plus or minus 0.01. One of the principal axes of this tensor (the x axis corresponding to the eigenvalue of smallest magnitude) is experimentally found to coincide with the b crystallo-graphic axis of monoclinic spodumene as required by the known symmetry of the crystal. The other two principal axes are in the a c plane, the z axis lying between the a and c axes at an angle of 55.5° plus or minus 1° with the c axis. It is shown that the four A l atoms per unit cell exist in equivalent crystal positions. PUBLISHED PAPERS "Upper Limit for the Lifetime of the 411-Kev Excited State of Hg 1 M " (with R. E . Bell) . Physical Review 76, 1409 (1949) . Nuclear Spins of the 2.62 Mev and 3.20 Mev Excited States of Thorium D. (with M . W. Johns). Physical Review 80, 478 (1950). / Nuclear Electric Quadrupole Interaction in Crystals with Non-Axially Sym-metric Fields (with D. W. Smellie and G. M . Volkoff) . Physical Review 84, 602 (1951). i Design and Use of a Coincidence Circuit of Short Resolving Time (with R. E. Bell and R. L . Graham) . Canadian Journal of Physics, 30, 35 (1952) . Nuclear Electric Quadrupole Interaction in Single Crystals (with G. M . Volkoff and D. W. Smellie) . Canadian Journal of Physics 30, 270 (1952) . PAPERS P R E S E N T E D B E F O R E L E A R N E D SOCIETIES The Beta and Gamma Radiations from Rhenium 186 and 188 (with C. C. McMullen and M . W. Johns) . Presented before The Royal Society of Canada. Abstract in The Transactions of The Royal Society of Canada, Third Series, Volume XLIV, 194 (1950) . On the Angular Correlation Function for the Gamma Rays of C d 1 1 4 (with M . W. Johns and C. D. Cox) . Presented before The Royal Society of Canada. Abstract in The Transactions of the Royal Society of Canada, Third Series, Volume X L V , 174 (1951) . L i 7 Nuclear Electric Quadrupole Interaction in Crystalline L i A l (SiOa) 2 (with D. W. Smellie). Presented before the American Physical Society, June 1951. Abstract in Physical Review 83, 891 (1951). T H E U N I V E R S I T Y O F B R I T I S H C O L U M B I A F A C U L T Y OF G R A D U A T E STUDIES P R O G R A M M E OF T H E F I N A L O R A L E X A M I N A T I O N F O R T H E D E G R E E OF D O C T O R OF P H I L O S O P H Y of H O W A R D E A R L P E T C H B.Sc. (McMaster) 1949 M.Sc. (McMaster) 1950 FRIDAY, S E P T E M B E R 26th, 1952, at 2:30 P.M. IN R O O M 301, PHYSICS BUILDING C O MM Dean H . Professor G. M . Volkoff Professor K. C . Mann Professor J. B. Warren Professor G. M . Shrum I T E E IN C H A R G E '. Angus, Chairman Professor R. M. Thompson Professor T . M . C . Taylor Professor Basil Dunell Professor F. H . Soward E X T E R N A L E X A M I N E R : Professor W. D. Knight, University of California, Berkeley, Cal. G R A D U A T E STUDIES Field of Study: Physics. Selected Topics in Physics—Professor M . W. Johns Atomic and Molecular Spectroscopy—Professor B. McLay Electromagnetic Theory—Professor W. Opechowski Quantum Mechanics—Professor G. M . Volkoff Chemical Physics—Professor A. J. Dekker Nuclear Physics—Professor K. C. Mann Other Studies: Physical Chemistry of High Polymers—Professor B. A. Dunell Theory of the Chemical Bond—Professor C. Reid Radiochemistry—Professors M. Kirsch and K. Starke T H E S I S N U C L E A R E L E C T R I C Q U A D R U P O L E I N T E R A C T I O N IN SINGLE CRYSTALS The nuclear magnetic resonance absorption technique has been applied to the study of the interaction between atomic nuclei, possessing electric quadrupole moments, and the crystalline electric field gradient existing at the nuclear sites in single crystals. The theoretical section of this thesis outlines Dr. Volkoff's extension to noh-uniaxial crystals of Pound's theory of the splitting of magnetic resonance absorption lines in a single crystal caused by the nuclear electric quadrupole interaction. Experiments have been performed on the splitting of the L i 7 and A l 2 7 absorption lines in a single crystal of L i A l (SiOs) 2 (spodumene) and the above theory has been used to analyse the results. The absolute value of the quadrupole coupling constant for the L i 7 nuclei in spodumene is found to be | e Q ( £ z z / h | = 7 5 . 7 plus or minus 0.5kc. per sec. The axial asymmetry parameter of the field gradient tensor at the site of the L i nuclei is found to be TJ = (^ >xx— <f>yy) /<tzz=0.79 plus or minus 0.01. One of the principal axes of this tensor (the y axis corresponding to the eigenvalue of intermediate magniture) is experimentally found .to coincide with the b crys-tallographic axis of monoclinic spodumene as required by the known symmetry of the crystal. The other two principal axes are in the ac plane, the z axis (corresponding to the eigenvalue <t> zz of the greatest magnitude) lying between the a and c axes at an angle of 46.5" plus or minus 1° with the c axis. It is shown that the four L i atoms per unit cell all exist in equivalent crystal positions as required by the crystal symmetry.-The absolute value of the quadrupole coupling constant for the A l 2 7 nuclei in spodumene is found to be | e Q < £ z z / h | = 2 9 6 0 plus or minus 10 kc. per sec. The axial asymmetry parameter of the field gradient tensor at the site of the Al nuclei is found to be 7] = (<2>xx— </>yy)/<£zz =0.95 plus or minus 0.01. One of the principal axes of this tensor (the x axis corresponding to the eigenvalue o£ smallest magnitude) is experimentally found to coincide with the b crystallo-graphic axis of monoclinic spodumene as required by the known symmetry of the crystal. The other two principal axes are in the a c plane, the z axis lying between the a and c axes at an angle of 55.5° plus or minus 1° with the c axis. It is shown that the four A l atoms per unit cell exist in equivalent crystal positions. PUBLISHED PAPERS "Upper Limit for the Lifetime of the 411-Kev Excited State of Hg 1 0 8 " (with R. E. Bell) . Physical Review 76, 1409 (1949) . Nuclear Spins of the 2.62 Mev and 3.20 Mev Excited States of Thorium D. (with M. W. Johns) . Physical Review 80, 478 (1950) . Nuclear Electric Quadrupole Interaction in Crystals with Non-Axially Sym-metric Fields (with D. W. Smellie and G. M . Volkoff) . Physical Review 84, 602 (1951). Design and Use of a Coincidence Circuit of Short Resolving Time (with R. E. Bell and R. L . Graham) . Canadian Journal of Physics, 30, 35 (1952) . Nuclear Electric Quadrupole Interaction in Single Crystals (with G. M. Volkoff and D. W. Smellie) . Canadian Journal of Physics 30, 270 (1952) . PAPERS P R E S E N T E D B E F O R E L E A R N E D SOCIETIES The Beta and Gamma Radiations from Rhenium 186 and 188 (with C. C. McMullen and M. W. Johns) . Presented before The Royal Society of Canada. Abstract in The Transactions of The Royal Society of Canada, Third Series, Volume XLIV, 194 (1950) . On the Angular Correlation Function for the Gamma Rays of C d m (with M . W. Johns and C. D. Cox) . Presented before The Royal Society of Canada. Abstract in The Transactions of the Royal Society of Canada, Third Series, Volume X L V , 174 (1951) . . . . L i 7 Nuclear Electric Quadrupole Interaction in Crystalline L i A l (SiOa) 2 (with D. W. Smellie) . Presented before the American Physical Society, June 1951. Abstract in Physical Review 83, 891 (1951). NUCLEAR ELECTRIC QUADRUPOLE INTERACTION IN SINGLE CRYSTALS by HOWARD EARL PETCH 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 for the degree of Doctor of Philosophy i n Physics We accept t h i s thesis as conforming to the standard required from candidates f o r the degree of Doctor of Philosophy Members of the Department of Physics The University of B r i t i s h Columbia September 1952 ABSTRACT The nuclear magnetic resonance absorption 'technique has been applied to the study of the in t e r a c t i o n between atomic nucl e i , possessing e l e c t r i c quadrupole moments, 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 gradient existing at the nuclear s i t e s i n single c r y s t a l s . The theory of the dependence of e l e c t r i c quadrupole s p l i t t i n g of nuclear magnetic resonance absorption l i n e s i n a single c r y s t a l on the orientation of the c r y s t a l i n an external magnetic f i e l d was o r i g i n a l l y developed by Pound f o r crystals i n which the e l e c t r i c f i e l d gradient i s a x i a l l y symmetric. The theo r e t i c a l section of t h i s t h e s i s presents Dr. Volkoff*s extension of this theory to cover the case of a c r y s t a l with non-axially symmetric e l e c t r i c f i e l d gradient at the s i t e of the nuclei being investigated. It i s shown that i f the i n t e r a c t i o n i s weak, so that only f i r s t , order perturbation theory i s needed, an ex-perimental study of the angular dependence of t h i s s p l i t t i n g f o r three independent rotations of the cr y s t a l about any three mutually perpendicular axes w i l l y i e l d complete information about the orientation of the p r i n c i p a l axes and the degree of a x i a l asymmetry of the e l e c t r i c f i e l d gradient tensor. The absolute value of the quadrupole coupling constant f o r those nuclei w i l l also be given. Further, i f the quadrupole coupling constant i s so strong that second order perturbation e f f e c t s appear then a single rotation about any a r b i t r a r y axis w i l l y i e l d the complete informations i i i The author has performed experiments on the s p l i t t i n g 7 27 of the L i ' and A l ' absorption l i n e s i n a s i n g l e c r y s t a l of L i A l t S i O ^ ) ^ (spodumene) and has used the above theory to analyse the r e s u l t s . The absolute value of the quadrupole coupling constant fo r the L i ? n u c l e i i n spodumene i s found to be J £ tf^$M 1*75* 7 - 0. 5 kc. per sec. The a x i a l asymmetry parameter of the f i e l d g radient tensor at the s i t e of the L i n u c l e i i s found to be ^ - ^ ^ ^ " ^ ^ ^ S » -0.79iO.01. One o f l t h e p r i n c i p a l axes of t h i s tensor (the y a x i s corresponding to the eigenvalue of intermediate magnitude) i s experimentally found to c o i n c i d e with the b c r y s t a l l o g r a p h i c a x i s of monoclinic spodumene as r e q u i r e d by the- known symmetry of the c r y s t a l . The other two p r i n c i p a l axes are i n the ac plane, the z- a x i s (corresponding to the. eigenvalue of greatest magnitude) l y i n g between the a and c axes at an angle of . o 0 4o»5 - 1 w i t h the c a x i s . I t i s shown that the f o u r L i atoms per u n i t c e l l a l l e x i s t i n equivalent c r y s t a l p o s i t i o n s as r e q u i r e d by the c r y s t a l symmetry. The absolute value of the quadrupole co u p l i n g constant fo r the A l 2 ^ n u c l e i i n spodumene i s found to be j ^ ^ ^ / ^ j ~ * 2960 10 kc. per sec. The a x i a l asymmetry parameter of the f i e l d gradient tensor at the s i t e of the L i n u c l e i i s found to be ^ ( ^ ' ^ V ^ — 0.95*0.01. One of the p r i n c i p a l axes of t h i s tensor (the x a x i s corresponding to the eigenvalue of smallest magnitude) i s exp e r i m e n t a l l y found to c o i n c i d e w i t h the b c r y s t a l -l o g r a p h i c a x i s of monoclinic spodumene as r e q u i r e d by the known i v symmetry of the c r y s t a l . The other two p r i n c i p a l axes are i n the ac plane, the z axis l y i n g between the a and c axes at an angle p o of 55*5 i 1 with the c axis.. It i s shown that the four' Al atoms per unit c e l l e x i s t i n equivalent c r y s t a l positions. TABLE OF CONTENTS Pag;e ACKNOWLEDGMENTS * i ABSTRACT .......> i i INTRODUCTION . . . . . . . . . . 1 CHAPTER I i THEORY Introduction 7 The Quadrupole Interaction . 9 Expressions f o r ^ E a n d Q 11 F i r s t Order Perturbation Theory • 18 Second Order Perturbation Theory ................. 24 CHAPTER II : THE SPODUMENE CRYSTAL.. 31 CHAPTER I I I : APPARATUS AND EXPERIMENTAL PROCEDURE . 36 CHAPTER IY - : EXPERIMENTAL RESULTS AND CALCULATIONS 39 Number of Line Components... 40 A p p l i c a b i l i t y of F i r s t Order Perturbation Theory to L i 42 Need f o r Second Order Perturbation Theory f o r A l 43 Detailed Analysis of the Measurements ,on L i ' . . . . . . 45 Detailed Analysis of the Measurements on A l 2 ' . . . . . 50 Line Broadening... 56 CHAPTER V : DISCUSSION * 57 REFERENCES .... 63 Table of Contents (Cont'd) Page LIST OF TABLES TABLE 1 : Summary of Experimental Results ••• 55 TABLE 2 : Line Width of A l 2 7 Central Component •••• 56 LIST OF ILLUSTRATIONS Facing Page F i g . 1 Spodumene unit c e l l 31 F i g . 2 Block diagram of the nuclear magnetic resonance spectrometer. 36 F i g . 3 C i r c u i t diagram of the o s c i l l a t -ing detector 36 F i g . 4 Lock-in detector..... 38 F i g . 5 Crystal mount..... 38 F i g . 6 L i 7 l i n e s i n spodumene 39 F i g . 7 (a) klfi l i n e s i n A 1 2 0 3 39 (b) A l ' l i n e s i n spodumene 39 F i g . 8 A l 2 ? signal i n AlClo solution and 7central component of the A l ' sign a l i n spodumene........ 39 F i g . 9 Dependence of the s h i f t of the A l 2 ' central component i n spodumene on H 0 44 F i g . 10 X rotation f o r L i 7 45 7 F i g . 11 Y rotation f o r L i 45 F i g . 12 Z r o t a t i o n for L i 7 46 F i g . 13 (a) X ro t a t i o n f o r A l 2 7 ; s h i f t plotted versus " ^ i •••^^ 50 (b) X rot a t i o n f o r A l ^ ' ; s h i f t plotted versus co**(^-th J 50 F i g . 14 (a) Y ro t a t i o n for A l 2 7 ; s h i f t plotted versus •«• • • 51 (b) Y rot a t i o n f o r A I 2 7 ; s h i f t plotted versus Cctffoyf 51 F i g . 15 (a) Z rot a t i o n f o r A I 2 7 ; s h i f t plotted versus 52 (b) Z rotation f o r A I 2 7 ; s h i f t plotted versus Ctx.*&2 . • 52 F i g . 16 V a r i a t i o n of the l i n e width of the c e n t r a l component with c r y s t a l orientation 56 i ACKNOWLEDGMENTS I am greatly indebted to Professor Volkoff f o r increasing my knowledge and appreciation of physics through numerous d i s -cussions. His enthusiasm and a i d , p a r t i c u l a r l y i n the t h e o r e t i c a l part, contributed much to the successful completion of t h i s work. Mr. D.W. Smellie deserves credit for f i r s t arousing our interest i n c r y s t a l s . Mr. N.G. Cranna very kindly helped i n 27 taking part of the experimental data on the Al '. I should also l i k e to express my thanks to Mr. A.J. Fraser, Mr. iJ.F. Brydle, and Mr. ¥. Maier of the Physics Department machine shop for t h e i r help i n constructing parts of the apparatus; to Mr. E. Price f o r h e l p f u l discussions on e l e c t r o n i c s ; to Mr. J. Lees f o r his glass-blowing; to Dr. K.C. McTaggart and Dr. R.M. Thompson of the Geology Department, U.B.C., f o r t h e i r advice and aid i n i d e n t i f y i n g the c r y s t a l axes and f o r supplying us with c r y s t a l s ; to Mr. J.A. Donnan of the same department fo r aid i n grinding the c r y s t a l s ; and to the Geology Department, U.C.L.A., for providing a large spodumene sample. The research described i n t h i s thesis was supported by the National Research Council of Canada through research grants to Dr. Volkoff and by the Research Council of Ontario through scholarships (1950-51, 1951-52) awarded to the author. I should also l i k e to express my thanks f o r a grant awarded by the B.C. Academy of Sciences which greatly broadened my outlook by making possible a v i s i t to several outstanding u n i v e r s i t i e s i n the U.S.A. 1 INTRODUCTION This t h e s i s deals with an application of the nuclear magnetic resonance absorption method to the study of the i n t e r -action between atomic n u c l e i and t h e i r surroundings i n a single c r y s t a l * The method consists i n placing the c r y s t a l i n an external uniform magnetic f i e l d , and observing the absorption of radio frequency energy due to t r a n s i t i o n s between the Zeeman l e v e l s of nuclear magnetic dipoles i n t h i s f i e l d . In the absence of further interactions the Zeeman l e v e l s would be evenly spaced, and would give r i s e to a single resonance absorption l i n e . The interactions between nuclei and t h e i r surroundings may i n general be of two types: (a), magnetic i n t e r a c t i o n with magnetic dipole moments of other n u c l e i , or with paramagnetic ions, (b} e l e c t r o s t a t i c i n t e r a c t i o n of a nucleus possessing an e l e c t r i c quadrupole moment eQ with an e l e c t r i c f i e l d gradient tensor VE set up by the surrounding charge d i s t r i b u t i o n . When weak, the above interactions r e s u l t only i n a broadening of the resonance absorption l i n e , but often they are strong enough to s p l i t the resonance l i n e into several components as was f i r s t shown by Pake (1) and Pound (2). This work i s primarily concerned with e l e c t r o s t a t i c interactions "in the spe c i a l 2 case when the int e r a c t i o n i s strong enough to separate the resonance l i n e into 21 components but the i n t e r a c t i o n energy-i s s t i l l much smaller than the Zeeman energy. Such studies y i e l d information about both n u c l e i and c r y s t a l s . As examples of the type of information about n u c l e i made available by such studies we may mention the determination of the spin of Be*? by Schuster and Pake (3), and the determination of the absolute value (but not the sign) of the r a t i o of the nuclear e l e c t r i c quadrupole moments of L i 6 and Li7 by Schuster and Pake (4)., and of Cu63 and Cu°5 by Becker and Kruger (5) and Becker (6) . Nuclear spin I i s determined by counting the 21 components into which the resonance l i n e i s s p l i t . From the magnitude of the s p l i t t i n g between the components one may determine the absolute value (but unfortunately not the sign) of the quadrupole coupling constant f o r the nuclei giving r i s e to the p a r t i c u l a r absorption l i n e i n the p a r t i c u l a r c r y s t a l . The quadrupole coupling constant i s the product of a nuclear property (eQ) and a c r y s t a l property (the largest i n absolute value of the. three eigenvalues of the tensorV E at the s i t e of the nucleus). I f t h i s largest eigenvalue ofVE were known, the absolute value (but not the sign) of eQ could be determined f o r the nucleus i n question, and would be of int e r e s t i n checking predictions of various nuclear models (7). In our present state of knowledge the eigenvalues of V£ are usually not known. However, f o r isotopes of the same element i n the same cr y s t a l the tensor 7 £ i s assumed to be i d e n t i c a l at the si t e s of the two isotopes., This leads to the determination of the 3 absolute value of the rati© of quadrupole moments of the isotopes. The p o s s i b i l i t y of determining such quadrupole moment rat i o s i s of inter e s t i n connection with some remarks of Kopfermann (8). He considered pairs of isotopes both members of which have the same spin. At the time the quadrupole moment r a t i o had been determined f o r f i v e such p a i r s . Kopfermann pointed out the empirical fact that i n a l l f i v e cases the r a t i o of the deviations, of the observed magnetic moments of the two members of each isotope p a i r from the value, common to both, predicted by the single p a r t i c l e model of Schmidt (9), was very nearly equal to the observed r a t i o of t h e i r quadrupole moments. This empirical i n d i c a t i o n , that the deviation of a nuclear magnetic moment from the corresponding Schmidt l i m i t i s proportional to eQ, might give some support to theories (10, 11) which ascribe the deviations from the Schmidt l i m i t s to the deviation of a nucleus from spherical shape. I t therefore seems of in t e r e s t from the point of view of nuclear physics to extend the measurement of quadrupole moment r a t i o s to other isotope pairs to see i f they further sub-stantiate Kopfermann's observations. The t h e o r e t i c a l analysis and the experimental techniques developed i n this thesis can be, and probably i n future w i l l be, used f o r such quadrupole moment r a t i o determinations. However, the actual experimental work reported i n t h i s thesis emphasizes the information obtained by nuclear magnetic resonance methods about c r y s t a l s rather than about nuclei. Although much has been known about the gross c h a r a c t e r i s t i c s of crystals f o r years, i t i s only r e l a t i v e l y recently that 4 powerful methods have been developed f o r studying the atomic structure of c r y s t a l s . X-ray analysis has played an extremely important part i n t h i s work and as a result the atomic structures are known fo r a great many c r y s t a l s . However, since the X^ray scattering cross-section of hydrogen atoms i s very low, X-ray techniques cannot be employed to locate the hydrogen of water of c r y s t a l l i z a t i o n . This l e f t a gap i n the knowledge of the atomic structure of a very important class of c r y s t a l s . Nuclear magnetic resonance techniques, through studies of the magnetic i n t e r -action between magnetic dipole moments of neighbouring protons, have added, and are continuing to add, to our knowledge of the po s i t i o n of the hydrogen atoms i n c r y s t a l s containing water of c r y s t a l l i z a t i o n (1). This technique has also been applied successfully to the study of int e r n a l motion i n c r y s t a l s containing protons (12). We s h a l l r e s t r i c t our discussion to the type of information made available about cr y s t a l s from a nuclear resonance study of 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 nuclei and t h e i r surroundings. The information obtained i s the orientation of the p r i n c i p a l axes and the degree of axial asymmetry of the f i e l d gradient tensor at the s i t e of the nuclei being investigated. Quantitative i n f o r -mation of t h i s type about the nature of the c r y s t a l l i n e e l e c t r o -s t a t i c f i e l d now being provided by the new magnetic resonance methods should serve as an impetus f o r further quantitative development of the theory of the chemical bond along the l i n e s described by Townes and Dailey (13, 14). 5 The various nuclear magnetic resonance techniques w i l l not.be discussed i n t h i s thesis as there are several excellent review a r t i c l e s i n the l i t e r a t u r e (15, 16, 17) • Pound (2) has given the theory of the dependence o f the s p l i t t i n g of a nuclear resonance absorption l i n e due to e l e c t r o s t a t i c i n t e r a c t i o n on the orientation o f the single c r y s t a l i n a magnetic f i e l d f o r the special case of u n i a x i a l c r y s t a l s , and f o r the s p e c i a l orientations of the un i a x i a l c r y s t a l i n which the axis of symmetry i s perpen-d i c u l a r to the axis about which the c r y s t a l i s rotated. This thesis gives an extension of Pound's theory suggested by Professor Volkoff to cover the dependence of the resonance frequencies oh the c r y s t a l orientation i n the general case of crystals of symmetry lower than u n i a x i a l , and f o r a r b i t r a r y orientations of the c r y s t a l . The theory i s then i l l u s t r a t e d by the author's experimental determination of the dependence of the frequencies of the L i ? and A l 2 ? absorption l i n e s i n a single c r y s t a l of spodumene L i A l f S i O ^ ) ^ o n the angles of r o t a t i o n of the c r y s t a l about three mutually perpendicular axes. The author's analysis of these rotations y i e l d s the p r i n c i p a l axes and the degree of asymmetry of V E at the L i and Al s i t e s , and the value of the quadrupole coupling constants f o r these two n u c l e i . This work on single crystals i s complementary to p a r a l l e l -work carried out by groups working with molecular beams, e.g. Kusch (18), Lew (19) and others, and those working with micro-wave techniques who study microwave spectra of gases, e.g. Gordy (20), Townes (21) and others, and pure quadrupole spectra in p o l y c r y s t a l l i n e materials, e.g. Livingston (22), Behmelt(23) and others. 6 The theory of the present method i s outlined i n the f i r s t chapter. The second chapter describes the spodumene crystals on which the'experiments were performed. The apparatus and experimental procedure are described i n chapter three. Chapter four deals with the experimental r e s u l t s and calculations. It includes observations of the dependence of the s p l i t t i n g of the L i 7 and A l 2 7 absorption l i n e s on c r y s t a l orientation. I t i s shown that the L i 7 nuclei have a small quadrupole coupling constant with the r e s u l t that f i r s t order perturbation theory gives an adequate treatment of the experimental r e s u l t s . In the case of the A l 2 7 nuclei the quadrupole coupling constant i s considerably larger, and second order perturbation theory is needed to obtain agreement with the experimental r e s u l t s . Some observations on the v a r i a t i o n of l i n e width with c r y s t a l orientation are given. The r e s u l t s are b r i e f l y discussed i n the concluding chapter. Chapter I THEORY Introduction -Inthis chapter we obtain the t h e o r e t i c a l expressions f o r the energy levels of a nucleus of spin I, magnetic moment^, and e l e c t r i c quadrupole moment eQ, placed i n a constant magnetic f i e l d H0, and an inhomogeneous e l e c t r i c f i e l d described by a f i e l d gradient tensor . The inhomogeneous e l e c t r i c f i e l d i s provided experimentally by the c r y s t a l l i n e f i e l d i n a single c r y s t a l . The e x p l i c i t dependence of the energy l e v e l s on the orientation of the c r y s t a l with respect to the magnetic f i e l d i s calculated, and leads to ac description of the dependence of the frequency of nuclear magnetic resonance absorption l i n e s i n single crystals on c r y s t a l orientation. A nucleus of spin I and magnetic moment^/placed i n a constant uniform magnetic f i e l d H Q has 2 1 + 1 Zeeman le v e l s with a constant energy difference h$Q =yU,H0/l. The 21 tra n s i t i o n s between adjacent l e v e l s — ± 1 ) under the influence of an external radio frequency magnetic f i e l d perpendicular to H C a l l correspond to a single resonance frequency VQ, giving r i s e to a single absorption l i n e . I f the nucleus i n question has I ^  i i t may also have an e l e c t r i c quadrupole moment which represents a deviation of the nuclear charge from a spherical d i s t r i b u t i o n . This moment can interact with an e l e c t r i c f i e l d gradient, caused by a non-spherical charge d i s t r i b u t i o n outside the nucleus, to produce a dependence of the e l e c t r o s t a t i c energy of the system on nuclear orientation. Two limiting:cases of t h i s type of i n t e r a c t i o n lend themselves to treatment by perturbation theory: (a) the e l e c t r o s t a t i c energy i s small compared to the magnetic (Zeeman) energy, (b) , the e l e c t r o s t a t i c energy i s much larger than the magnetic energy. The following discussion w i l l be r e s t r i c t e d to case (a) where the' e l e c t r o s t a t i c energy i s much smaller than the magnetic energy. The opposite case has been dealt with by Kruger (24), and i s not discussed here, since i t i s not required for the interpretation of the experimental r e s u l t s reported i n t h i s t h e s i s . The problem then i s f i r s t to f i n d the expression f o r 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 energy V and to i s o l a t e from i t the quadrupole i n t e r a c t i o n term F. This w i l l be just a r e c a p i t u l a t i o n of Pound's (2) work. Using standard perturbation theory, which may be found i n any text-book on introductory Quantum Mechanics, the energy of the perturbed Zeeman l e v e l s can.be written to the second order as S *n U/m ^ no , p o (1) i n which the prime on means, as usual,.that the summation i s carried out over a l l values of JL except x^/n. Thus to obtain ttie perturbed energy leve l s to the f i r s t order we require an e x p l i c i t determination of the diagonal matrix elements F ^ and f o r the second order we also require the off-diagonal matrix elements F ^ . 9 The Quadrupole Interaction The derivation outlined by Pound (2) w i l l be used to find F, the quadrupole i n t e r a c t i o n term. I t i s assumed that the charge d i s t r i b u t i o n giving r i s e to the e l e c t r i c f i e l d at the s i t e of the nuclei l i e s completely outside of the nuclear charges. Then the e l e c t r o s t a t i c energy of i n t e r a c t i o n between the n u c l e i and the surrounding charge d i s t r i b u t i o n can be written as V = Z ( 2 ) I d may be expanded i n the usual way i n terms of Legendre polynomials r e s u l t i n g i n whereJL^  refers to an element of the nuclear charge,^" to an element of external charge,/i^ and A,^ are the distances from the o r i g i n , taken at the center of the nucleus, to the charges J^i and JLQ respectively, and •Oy i s the angle between these radius vectors. By making use of the addition theorem of spherical harmonics, the i n t e r a c t i o n may be expressed i n terms of , (|)^  , "©"^  , and $j > t n e polar and azimuthal angles of and A^' respectively. Thus (2) becomes where (cos-©',<t) ) i s the normalized t e s s e r a l harmonic r 40-® A (u*+) x ( 5 ) 10 (6) and the functions 0^ (cos-6-), following Condon and Shortley's d e f i n i t i o n (25), are f o r | ) 0, with The i n t e r a c t i o n may then be written as where we have used Racah's (26) d e f i n i t i o n C ^ - J ^ " YlfcSL-frji) (9) Now we can write our i n t e r a c t i o n as a tensor product A * / — *4 ^ V = £ (Z ju AJ C%) '(Z C J ( 1 0 j defining I f successive k-terms of (10) are considered, i t becomes obvious that the f i r s t term, corresponding to k « 0, i s merely equal to the energy of the t o t a l nuclear charge concentrated at the o r i g i n i n the potential due to the surrounding charge d i s t r i b u t i o n . Although the k = 0 term may be large, i t i s not i n t e r e s t i n g to us, since i t i s independent of nuclear orientation. The k =• 1 term i s equivalent to the interaction, energy of an e l e c t r i c dipole i n an e l e c t r i c f i e l d . It i s generally assumed on the basis of some 11 suggestive t h e o r e t i c a l arguments (27) that nuclei and elementary-p a r t i c l e s have no e l e c t r i c dipole moments. Following t h i s general practice we s h a l l also assume that the k — 1 term i n our expansion i s zero. (However i t may be mentioned i n passing, that Pu r c e l l and Ramsey (.28 ) regard as inadequate the present experimental basis for assuming nuclear e l e c t r i c dipole moments to be zero, and propose a new experiment to test this point). Therefore the k - 2 term i n (10) i s the f i r s t one of intere s t to us. Presumably k — 3 and higher odd terms w i l l be zero f o r the same reasons of symmetry as the k — 1 term. The higher even terms W i l l o r d i n a r i l y be expected to be much smaller than the k =• 2 term. The only term of interest then i s the - k'= 2 term which represents the in t e r a c t i o n of an e l e c t r i c quadrupole with an e l e c t r i c f i e l d gradient. Following Pound t h i s can be written as the tensor product Qi - -03, (14) where and Expressions for and Q We require the matrix elements of F which may be written as ( a r^|r|^-')=(a-l^/-'/Q f ('v?) / ^l-') ( l 5 ) where Ot, refers to the degrees of freedom of the c r y s t a l other than nuclear spin. I t i s necessary to introduce <K because the 12 external charge d i s t r i b u t i o n i s not necessarily s t a t i c and i n f a c t at ro6m temperatures w i l l be carrying out rather complicated motions. Thus each component of ( V E ) becomes an average over the v i b r a t i o n a l states of the c r y s t a l . This averaging over<* i s always to be understood i n the following. It i s assumed that the above matrix element can be s p l i t up so that (16) In order to have a large number of i d e n t i c a l n u clei subject to the same H 0 and the same V £ so that each w i l l be subject to the same perturbation of i t s Zeeman l e v e l s , we consider a single c r y s t a l placed i n a constant uniform magnetic f i e l d H 0. I f a nucleus of the type under consideration occurs- only once i n each unit c e l l of the c r y s t a l , then i t i s guaranteed that the (which i s presumably due p a r t l y to the electrons i n the atom containing the nucleus i n question and the bonding electrons, and p a r t l y to the nuclei and the electrons of atoms at neighbouring l a t t i c e points) i s the same at the s i t e of a l l such n u c l e i . I f a nucleus of the type under investigation occurs more than once per unit c e l l , then i n some cases the symmetry properties of the c r y s t a l may .guarantee that V£ i s the same f o r a l l the possible positions of the given nucleus i n the unit c e l l . I f the various possible positions i n the unit c e l l are not equivalent i n t h i s sense, then the actual s p l i t t i n g observed w i l l be a superposition of several s p l i t t i n g patterns due to the-nuclei i n each of the several nonequivalent positions. The analysis below i s carried out e x p l i c i t l y f o r equivalent positions only. 13 The tensor V £ written i n rectangular coordinates i s symmetric (since 7 X B ' O ) , and traceless (since 7-E 5 8 0), and therefore has only f i v e independent components. We can write the V £ tensor i n Cartesian coordinates as D f x l>Ex\ (17) From the d e f i n i t i o n we obtain 4 A4 so that the V £ tensor may be written A4 1 3 ± -lil A? A >" (13) (19) V (20) When the v.^  are also xvritten i n terms of. the rectangular coordinates and substituted i n Equation (14), a comparison, with (20) indicates that the f i v e i r r e d u c i b l e components of the e l e c t r i c f i e l d gradient 14 tensor can be expressed i n terms of the Cartesian components by: (21) 2 W 6 * *3 x >3 *3 ) (22) \ I f we take an a r b i t r a r y set of right--handed rec-tangular axes XYZ; f i x e d with respect to the c r y s t a l , the symmetric tensor w i l l have the general form ^ x y ^yy ^ y * f z * Vyz. %2./ where ^ x } | f o r example, i s the second derivative of the elec-t r o s t a t i c potential at the s i t e of the nucleus i n question with respect to the X d i r e c t i o n . The tensor has only f i v e independent components since Vxx y + ^ z z ~ ^ • Because the tensor i s symmetric, i t i s guaranteed that a set of axes xyz, also fi x e d with respect to the c r y s t a l , may* be found to put t h i s tensor i n the diagonal form o \ (23) Here the axes are so l a b e l l e d that ^ n a s t n e largest and ^ZX~ "^Q^]) the smallest absolute value, r e s t r i c t i n g the value of the a x i a l asymmetry parameter .t In the notation of Pound's paper (2): 15 1 ^ The values of , ^  , and of any three quantities (e.g. the three Eulerian angles, or any three independent ones of the nine d i r e c t i o n cosines.) specifying the orientation of the xyz system with respect to the XYZ system constitute f i v e independent quantities defining the tensor.- In terms of the p r i n c i p a l axes the e l e c t r i c f i e l d gradient tensor components take on the simple form 0 4 = % = o We now require the matrix elements (V?)±/ - v (25) K (26) where, as given i n Uquation (13), the i r r e d u c i b l e components of the quadrupole moment tensor are: However, instead of computing these matrix elements i n which the 16 operator contains terms quadratic i n the coor d i n a t e s , we f o l l o w the procedure o u t l i n e d by Casimir ( 2 9 ) , and replace the operator by an equivalent one i n v o l v i n g p r o p e r l y symmetrized terms quadratic i n the components of angular momentum. The reason f o r r e p l a c i n g terms quadratic i n the coordinates by those quadratic i n the components of angular momentum i s that the m a t r i x elements of the l a t t e r are w e l l known. The-equivalent quadrupole moment operator i s then (27) where 2.Q ft-XIUI'I) (28) and the s c a l a r nuclear quadrupole moment Q i s defined i n the conventional manner ( 3 0 ) , +Q=(ll\ZA*A;(iu^6iI-i)\U) (29) In the above expression f o r the tensor operator Q ^ e x p l i c i t use has been made of the usual assumption t h a t the nuclear charge d i s t r i b u t i o n has a x i a l symmetry so that i n terms of i t s p r i n c i p a l axes the tensor may be c h a r a c t e r i z e d by a s i n g l e s c a l a r Q. The asymmetry parameter, analogous to ^  introduced i n connection with. V £ , i s here taken equal t o zero. 17 Since the e l e c t r i c quadrupole tensor i n the represen-tat i o n of Equation (27) involves only the well-known matrix elements of the angular momentum, the matrix elements of the quadrupole in t e r a c t i o n term can be obtained e a s i l y . Up u n t i l t h i s point the derived equations hold f o r any coordinate system. However, to use the matrix elements of the angular momentum we must specify.our laboratory coordinate system. Following the usual practice we s h a l l choose the laboratory system of axes x'y'z' i n such a way that the z' di r e c t i o n coincides with that of the constant magnetic f i e l d H 0. The x'y T axes may be chosen i n an a r b i t r a r y way to l i e i n the plane perpendicular to H 0. The matrix elements are then where A has the same d e f i n i t i o n as (28) and the prime on (y£". indicates that the tensor components of VE must be expressed i n terms of the laboratory coordinate system. Substitution of these matrix elements into Equation (1) leads to the e x p l i c i t dependence of the perturbed i energy l e v e l s on the orientation of the c r y s t a l with respect to the magnetic f i e l d . 7 (30) 18 F i r s t Order Perturbation Theory The f i r s t order perturbation i s considered separately because i t has been given the "s l i g h t l y more general treatments i n terms of a quite arb i t r a r y orientation of axes. It w i l l be adequate only i f the in t e r a c t i o n energy i s quite small compared to the Zeeman energy. Then the energy of the ni'th l e v e l i s given by F •= p 6 +• f ' - / • • V {^2) Substituting (31) and the fact that E£~styiHo/l into (32) gives • ^ - ^ * $ ^ « % - t t 4 * t (33) The frequency of the t r a n s i t i o n between the perturbed Zeeman lev e l s m and m-1 may then be written as (34) with K~ l^Ril^iL (351 Here the sign i s to be chosen opposite to the sign of the nuclear gyromagnetic r a t i o , and i s the second .derivative of the e l e c t r o s t a t i c potential at the s i t e of the nucleus i n question, the z' d i r e c t i o n coinciding with that of the constant magnetic f i e l d H 0. It i s e a s i l y seen from Equation (34) that i n t h i s f i r s t order - approximation the two resonance frequencies \^(m m-l) and\?(-(m-l)*-* - m) l i e equally spaced on opposite sides of V? , the frequency difference between them being given by twice t h e i r 19 separation from the unperturbed line "V& : n ' . (36) If the orientation of the c r y s t a l i s varied with respect to the constant magnetic f i e l d H Q the value of Vj'^/ along H 0 w i l l vary, and the observed s p l i t t i n g 2 w i l l be a function of c r y s t a l o r i e n t a t i o n . Let eq denote the value of fo r that p a r t i c u l a r c r y s t a l orientation which gives the greatest s p l i t t i n g . Then Equation (36) may be rewritten i n the form 41 ZI(ZT-I) X -If . (37) Here the f i r s t factor depends on the spin I of the nucleus and on the p a r t i c u l a r components of the l i n e being investigated, the second factor i s the so-called quadrupole coupling constant, and the t h i r d dimensionless factor of absolute value not exceeding unity depends on the orientation of the cry s t a l i n the magnetic f i e l d . In order to determine the dependence of 2 on cr y s t a l orien-t a t i o n we must show how depends on c r y s t a l orientation. Let us take an ar b i t r a r y externally e a s i l y iden-t i f i a b l e set of right-handed axes XYZ fix e d with respect to the c r y s t a l . This set may involve some of the crystallographic axes. Let X,Y,Z i n turn be made perpendicular to the uniform external magnetic f i e l d H Q. In each case l e t the c r y s t a l be rotated about thi s axis which remains f i x e d i n the (primed) laboratory system. Let the angle of rotation for the X rotation be measured from the position i n which the Y axis coincides with H Q, which i s chosen 20 to define z' i n the laboratory system. The other two directions x' and y' may be chosen i n the case of the X rotation to make x'y'z' coincide with the i n i t i a l p o s ition of Z,X,Y. The trans-formation between XYZ and x'y'z' i s then given by: From Equation (3#) i t follows that (33) (39) (40) (41) and consequently from Equation (36) with Similar relations for the Y and Z rotations may be obtained by c y c l i c permutation. Pound's r e s u l t s i n reference (2) refer to the X • rotation i n the spe c i a l case of a c r y s t a l with^-O, and oriented so that the Y axis coincides with the z p r i n c i p a l axis. In thi s case Yxx - V z z « - , tyy - £g , and the o f f -diagonal terms are zero, giving fl^s K*J/*f J Bx~ 3K*Q/t/ ) = °  and 21 By performing the three r o t a t i o n s i t i s p o s s i b l e to determine from the three experimental curves of 2^V^as a f u n c t i o n of •&• the nine constants* A,B,C. I t i s c l e a r from Equations (41) that the s i x A,B are not a l l independent, but s a t i s f y the f o u r independent i d e n t i t i e s : * 7 (43) plus two others obtained by c y c l i c permutation from the second one above. These i d e n t i t i e s are u s e f u l as a check on the experimental accuracy, and to f i x the r e l a t i v e signs of the three curves. ' From the nine experimental constants one may obtain w i t h the a i d of Equations (41): plus f o u r more r e l a t i o n s obtained by c y c l i c permutation. The r e s u l t i n g tensor  maY D e d i a g o n a l i z e d i n the usu a l way. The c h a r a c t e r i s t i c equation f o r the eigenvalues i s the cubic equation: (1*5) where (46) & The r e l a t i v e signs of these nine constants are f i x e d by the experimental curves, but the absolute s i g n i s not as long as there i s no way of i d e n t i f y i n g which of the two resonance frequencies belongs to the m*-»m-l, and which tb the -(m-l)«* -m t r a n s i t i o n . 'It might i n p r i n c i p l e be p o s s i b l e to make t h i s assignment i f the population o f the l e v e l s could be ap p r e c i a b l y a l t e r e d by going to s u f f i c i e n t l y low temperatures. 22 A r e v e r s a l of the undetermined common s i g n of A,B,C, leaves a unchanged, but a l t e r s the sig n of b, so that we may always choose b -|b|. The usual t r i g o n o m e t r i c s o l u t i o n o f the cubic w i t h three r e a l roots g i v e s : with ( n * l , 2 , 3 ) (47) 04,< ± If (43) Y$ i s p o s i t i v e and has the l a r g e s t absolute value, the other two y i a r e negative, and X" has the smallest absolute v a l u e , so t h a t i n accordance w i t h our convention we have, up t o a common s i g n : f33 (49) This leads to (50) (51) The absolute value of the quadrupole c o u p l i n g constant i s then given by: JL X 1 1 (52) The process of d i a g o n a l i z a t i o n also gives f o r the d i r e c t i o n cosines of the p r i n c i p a l axes x,y,z with respect to X,I,Z: ± / (n-1,2,3) (53) 23 where (54) The three values of X,lead to the d i r e c t i o n cosines of the cor-responding three p r i n c i p a l , axes. The r e l a t i v e signs of X^^U. fo r each value of n are fix e d by the r e l a t i v e signs of the , but the absolute- sign of each such set may be reversed by reversing the sign of the l a s t term of Equation (53), corresponding to the fact that f o r a second rank tensor only the orient a t i o n of the p r i n c i p a l axes i s s i g n i f i c a n t , but not the choice of the posit i v e d i r e c t i o n of each axis. I f desired, Equations (41) may be e x p l i c i t l y expressed i n terms of these nine d i r e c t i o n cosines, | Keq| and JJ , as was done in a preliminary communication (31)• However, the computational procedure outlined above i s considerably simpler. The largest s p l i t t i n g , that may be obtained for any p a r t i c u l a r c r y s t a l w i l l occur when the z p r i n c i p a l axis i s made to coincide with the magnetic f i e l d H 0. I t i s given by 2/»>?=|Keq(. For a c r y s t a l with ^ ^  0 and an unknown orientation of any of the p r i n c i p a l axes of the f i e l d gradient tensor at the s i t e of the nuclei being investigated the minimum number of readings required to completely determine t h i s tensor i s f i v e : f o r instance ©•x - &y - -0-z and any two of - "Oy = ^ ~ O w i l l s u f f i c e . An increase i n the number of c r y s t a l positions f o r which 24$ i s 24 determined w i l l increase the experimental accuracy with which the constants A,B,C, and therefore the tensor K ^ " i s determined. If from the symmetry of the c r y s t a l i t i s known that one of the X,Y,Z axes, say the X axis, coincides with one of the p r i n c i p a l axes of theVE tensor, then.we must have V Z x = Vxy - ° so that Cy = C z = o , and the diagonalization remains to be performed only i n the IZ subspace. In t h i s case only one, rotation about the X axis i s necessary to give: with t a n a 5 K = - C*/B% and the sign of S x opposite to that of C x. The three diagonal terms of the K tensor i n i t s p r i n c i p a l axes system are then given by: -s^ , fi*±(tf+c?>*. (56) The p r i n c i p a l axis corresponding to -2A X coincides with the X axis, while the other two p r i n c i p a l axes l i e i n the YZ plane, the axis corresponding to the plus sign i n the second expression (56) making an angle £ x with the Y axis. The minimum number of readings for a complete analysis i n t h i s case i s three, say •=• o", 45*, 90". However, more ~&x readings, and also Y and Z rotations , w i l l help to increase the accuracy attainable. Second Order Perturbation Theory The e l e c t r i c quadrupole i n t e r a c t i o n may be s u f f i c i e n t l y strong that f i r s t order perturbation theory^ i s inadequate, and the second order perturbation term of Equation (1) must also be used. In t h i s section we f i r s t give general expressions, i n terms of the 25 components of the f i e l d gradient tensor(v*E)in the (primed) laboratory-coordinate system, for the perturbed energy l e v e l s F^, and for the frequencies \? =(E^-, ~ En*. ) / f o r a r b i t r a r y I and m up to terms quadratic i n the quadrupole coupling constant. We then express the dependence of ( V E J on c r y s t a l orientation. It may be shown that i n t h i s case when second order e f f e c t s are appreciable, a complete rotation of the c o s t a l about a single arbitrary axis i s s u f f i c i e n t to y i e l d the complete information on the quadrupole coupling constant, , and the orientation of the p r i n c i p a l axes —» of VB . However, expressions f o r the angular v a r i a t i o n of the frequency s h i f t for an a r b i t r a r y axis of r o t a t i o n are a l g e b r a i c a l l y somewhat complicated, and the results presented below refer to the simpler case, applicable to our experimental s i t u a t i o n , i n which the d i r e c t i o n of one of the p r i n c i p a l axes i s known i n advance from c r y s t a l symmetry (a two-fold rotation axis passing through the n u c l e i i n question). Particular attention i s drawn to the central component (5 <—* - i ) of the resonance l i n e of a nucleus with h a l f - i n t e g r a l I, since t h i s component i s the strongest one, and shows no f i r s t order frequency s h i f t . It i s shown that i n the second order t h i s central component i s displaced from the unperturbed Larmor resonance frequency by an amount which i s inversely proportional to the constant magnetic field H Q, d i r e c t l y proportional to the square of the quadrupole coupling constant, and dependent on c r y s t a l o rientation. Using the notation of the preceding sections we introduce the dimensionless parameter 26 and f i n d on substituting the matrix elements (31) into (l) and simplifying; ir* The frequency corresponding to the t r a n s i t i o n m «—• m-1 i s then given by: r / \' 4 The sign of the f i r s t order term i s to be chosen opposite to the sign of the nuclear gyromagnetic r a t i o . I f m i s replaced by -(m-1), so that the -(m-1)*-* -m t r a n s i t i o n i s considered i n place of the m m-1, we see that the second order term remains unchanged, while the f i r s t order term changes sign. However, as was remarked e a r l i e r , there i s , without going to very low temperatures, no way of ex-perimentally deciding which of these two t r a n s i t i o n s one i s dealing with, so that the sign of the quadrupole coupling constant w i l l remain undetermined. We now introduce the e x p l i c i t e f f e c t s of crystal r otation. Let us assume that the X axis, f i x e d with respect to the c r y s t a l , coincides with one of the p r i n c i p a l axes, say the x axis of VE , 27 and that the c r y s t a l i s rotated about t h i s axis. The other two pr i n c i p a l axes y and z l i e i n the YZ plane but do not necessarily coincide with Y and Z; i n f a c t i n general they may be rotated through an angle £ xwith respect to the Y,Z axes. Since the angle 6K l i e s i n the same plane as the angle of r o t a t i o n we may d i r e c t l y write down the r e l a t i o n between x,y,z and the laboratory system x ^ y ' j z ' . As i n the section on f i r s t order perturbation we choose for the i n i t i a l p o s ition of the c r y s t a l the one i n which X,Y,Z coincide respectively with y', zO^x'. The transformation then i s (60) From equation (60) i t follows that i a (6D V3' = 0 2 a Substituting (61) into ( 21) gives A4 ( 6 2 ) and consequently from equation (59) the frequency of the m **m-l component for the X rotation w i l l be given by: where L J (63) A a X * ^5LZ^2>/) - W ~ " ' J - 3 j Formulae given by Pound ( 2 ) f o r the special case of Al in an a x i a l l y symmetric cry s t a l may be obtained up tb second order terms by setting I , ^ = 0 and S =/- 6/3*-^ , i . e . choosing - 90*, i n equations ( 6 3 ) . ~ • 29 The correct choice of A , ^  and SK inserted into Equation (63) should make the resultant t h e o r e t i c a l curve coincide with the experimental curve f o r the X rotation. The curve f o r the deviation \*-V% i s thus predicted to have a p e r i o d i c i t y of 180 , and to be symmetric about i t s maxima and minima. The value of fixes the position of these maxima and minima, while h and \ f i x the amplitude and shape of the curve. Thus a f i t t i n g of t h i s t h e o r e t i c a l curve f o r the X rota t i o n to the experimental r e s u l t s for any one component w i l l y i e l d Oj, 6^ , and the absolute value of the quadrupole coupling constant. Although the X rotation alone i s s u f f i c i e n t to give us the required information a second rotation, say about the Y axis, would give us an experimental check on the r e s u l t s . In both the Y and Z rotations the magnitude of the s h i f t depends on £„, as well as on % and X , so these rotations w i l l give a sensitive check on S. Let Y be made perpendicular to H 0 and l e t the c r y s t a l be rotated, about th i s axis. The angle of rotation -0y f o r the Y rotation w i l l be measured from the pos i t i o n i n which the X axis coincides with x* i n the laboratory system. The transformation between X,Y,Z and x f , y T z ' i s then given by In t h i s case &x i s not i n the plane of rot a t i o n and therefore must be introduced i n the transformation 30 % = X ^ - y o u S x + ( 6 5 ) from the x,y,z set of axes to the XYZ axes. Using Equation (65) the transformation from the xyz axes to the x'y'z' set of axes i given by (66) Following the same procedure as i n the development the X rotation we obtain f o r the Y ro t a t i o n : (AL fU^frc*1-*, fa +i 3^lsx)] S i m i l a r l y f o r the Z rotation we obtain: , W^$Ky^ . Wc^X) C ~ ^  (68) F i g . 1: P r o j e c t i o n of a u n i t c e l l of monoclinic spodumene on the (010)plane. The b a x i s which i s perpe n d i c u l a r to the page was used as the X r o t a t i o n a x i s , and c o i n -cides with the y p r i n c i p a l a x i s of the f i e l d g r a d i e n t tensor at the L i s i t e s . The Y and Z r o t a t i o n axes, and the x and z p r i n c i p a l axes of the f i e l d g radient tensor at the L i s i t e s are shown i n the diagram. The x p r i n c i p a l a x i s at the A l s i t e s c o i n c i d e s w i t h the c r y s t a l b a x i s . The y and z p r i n c i p a l axes (marked y t f and z ! t ) at the A l s i t e s are also shown i n the diagram. to face page 31 31 The values of Oj and 6X obtained from the X rotation can be substituted into the above Equations (67) or (68). The resulting expressions for the components of (Vif)' and the value of \a^Q/X I from the X rot a t i o n can be substituted into Equation (59) and the t h e o r e t i c a l curves so obtained can then be compared with the experimental r e s u l t s f o r the Y and Z rotations. Chapter II  THE SPODUMENE CRYSTAL . The preceding theory has been checked experimentally i n the case of the Li?'and A l 2 7 resonances i n a single c r y s t a l of LiAl(SiO-j) 2 (spodumene). Spodumene i s a monoclinic pyroxene with diopside structure. Its space group i s C^. It has four molecules per unit c e l l . F ig. 1, based on the data of Warren and Biscoe (32), gives a projection of the structure on a (0 10) plane. The dimensions of the unit c e l l are : o o o a = 9.50 A, b = a.30 A, c = 5.24 A, /S = 69*'40'. 32 The atomic coordinates i n decimal parts ^, ^ ) o f the a x i a l lengths are given by: LK.0Q-.31f .25) , A1(.00, .09, .25), Si(.21, .41, .25), Oi(.39, .41, .14), 0 2(.13, .25, .32), 0 3(.14, .49, .00). Twofold rotation axes p a r a l l e l to the b d i r e c t i o n pass through the L i and Al positions at ^ =• 0, ~£ . 5 , ^=^.25. Twofold screw axes p a r a l l e l to the b d i r e c t i o n pass through £ •=-± .25, J ^ — *25. Symmetry centers l i e at "^-0, — . 5, ^= 0, — . 5 , 5 = 0 > ~ ,*5 and also at ^ = ± . 25, 7j - ± . 25, ^ ~ °> • 5» The c r y s t a l consists of -0-Si02~0-Si02-0- chains indicated by broken l i n e s i n Fig. 1 running p a r a l l e l to the c d i r e c t i o n and held together by L i and A l atoms. Since both L i and Al nuclei l i e on twofold rotation axes of the c r y s t a l , i t follows that at the s i t e of each such nucleus some one of the p r i n c i p a l axes of the i^'tensor must coincide with the b d i r e c t i o n . The existence of symmetry centers then shows that the other two axes must have the same orientation at the four L i s i t e s (except f o r reversals of the positive axis directions which are immaterial f o r a second rank tensor), and also at the four Al s i t e s , the l a t t e r , however, not necessarily being the same as at the L i s i t e s . In Chapter IV below i t i s experimentally determined that the y p r i n c i p a l axis (corresponding to the eigenvalue of intermediate magnitude) of the f^i tensor at the L i s i t e s coincides with the crystaJ/axis, while the x and z p r i n c i p a l axes have the directions shown i n Fig. 1, whereas at the A l s i t e s the x p r i n c i p a l axis (corresponding to the eigenvalue of smallest 33 magnitude) coincides with the c r y s t a l b axis, while the y and-z p r i n c i p a l axes have the directions shown i n Fig. 1 (the axes i n the diagram marked y M and z T t)« The z p r i n c i p a l axis (corresponding to the eigenvalue of largest magnitude) of the tensor at the Al sites i s i n the same plane as the z;-principal axis of the tensor at the L i s i t e s , but is rotated about the b axis through an angle of 9°with respect to i t . The sample used f o r the study of the L i 7 resonance was taken from a cleavage mass of spodumene which was milky colored and nontransparent, presumably owing to impurities. (A smaller transparent single c r y s t a l was used for some preliminary work (33) and gave results i n agreement with the present ones). A piece i n the form of a rectangular prism based on a parallelogram was s p l i t o f f the large cleavage mass. Three of the four faces bounding the parallelogram were (110) cleavage planes, the edges formed by t h e i r intersection defining the d i r e c t i o n of the c axis. The fourth face was ground down to give a c r y s t a l of convenient size. The bases were ground to planes perpendicular to the c axis (the ri g h t angle between the c axis and the ground base planes i s believed to be accurate to about 1°). The b axis was then taken to l i e i n the plane of the base bise c t i n g the parallelogram's obtuse angle (2 t a n - 1 ( a s i n 0/b) =-94°) between two cleavage planes. I t i s believed that t h i s d i r e c t i o n could be i d e n t i f i e d to within perhaps two degrees. The spodumene c r y s t a l i s very hard, and could not be marked with a sharp s t e e l point, so a p e n c i l mark was used. As the cleavage planes are not p e r f e c t l y smooth there i s some error introduced i n bisecting the angle between them. Additional error comes from making settings on the rather thick pencil mark 34 d e f i n i n g the b a x i s . The b d i r e c t i o n was chosen as the e x t e r n a l l y e a s i l y recognizable X a x i s , the edge between the cleavage planes (c a x i s ) as the Y a x i s , and the d i r e c t i o n mutually p e r p e n d i c u l a r to the other two as the Z a x i s . O p t i c a l examination of our c r y s t a l showed that the c r y s t a l l o g r a p h i c a a x i s l i e s between our Y and Z axes. The Y and Z axes are marked i n F i g . 1. The dimensions of our c r y s t a l were approximately 8 by 10 mrn^  along the p a r a l l e l o g r a m s i d e s , by 10 mm. along the c (or Y) a x i s . This sample was destroyed during the o p t i c a l examination to determine the d i r e c t i o n of the a a x i s . For work on A l three d i f f e r e n t sources of spodumene were used. 1. Samples of spodumene Nos. IX, 1Y, 1Z were -taken from the same milky cleavage mass f o r the study o f the A1^7 l i n e s as was used f o r the work on L i 7 . To improve the i n t e n s i t y of the l i n e s separate pieces were taken f o r the X,Y, and Z r o t a t i o n s . These pieces were roughly c y l i n d r i c a l w i t h the axes of r o t a t i o n along the c r y s t a l l o g r a p h i c b,c, and bxc d i r e c t i o n s f o r the X,Y, and Z r o t a t i o n s r e s p e c t i v e l y . The pieces were approximately 30 mm. long and 15 mm. i n diameter. The X,Y, and Z r o t a t i o n s were c a r r i e d out with these c r y s t a l s . The a x i s of r o t a t i o n was the only d i r e c t i o n a c c u r a t e l y known i n each of these c r y s t a l s so the L i 7 l i n e s were used to determine the p o s i t i o n of the other two c r y s t a l axes. Therefore the d i s c u s s i o n on c r y s t a l alignment e r r o r s given above i n connection with L i measurements a p p l i e s e q u a l l y w e l l here. The * c e n t r a l component of the A l 2 7 resonance was quite intense i n these c r y s t a l s but the other components d i d not appear. This was 35 believed due to crystal imperfection and impurities i n the c r y s t a l as explained i n Chapter IV. 2. A nearly perfect c r y s t a l was obtained from V.D.-H i l l which was colourless and almost free of cracks. However, only the di r e c t i o n of i t s c axis' could be determined from external evidence. This No. 2 c r y s t a l was cut i n the shape of a cylinder with i t s long axis coinciding with the cr y s t a l c axis. Although t h i s crystal was much smaller (14 mm. long with 11 mm. diameter) than the milky ones, the f i v e components of the Al l i n e due to the 21 tran s i t i o n s were c l e a r l y evident as shown i n Fig. 7b. A few measurements on the inner s a t e l l i t e s (- g -\) were made f o r the Y rotation of t h i s c r y s t a l . • 3. The results from the Y and Z rotations of the milky No. 1 samples which are reported i n greater d e t a i l i n the next chapter indicated a possible error of 2 ° i n the determination of the orientation of the Y and Z axes with respect to the magnetic f i e l d . To check on the p o s s i b i l i t y of t h i s error spodumene c r y s t a l No. 3 with two nearly perfect (110) cleavageas planes was obtained from Schortmann's Minerals. The d i r e c t i o n of the c axis was defined by the edge formed by the int e r s e c t i o n o f the two cleavage planes and the b axis was taken to l i e i n the plane of the bisector of the angle between the two cleavage planes. This angle was 0 almost exactly 94 • The other cr y s t a l faces were ground down to make the c r y s t a l a convenient size (approximately 11 mm. by 13 mm. by 12 mm.) This crystal was rotated about its b axis and the pos i t i o n of i t s J FREQUENCY METER COMMUNICATIONS RECEIVER 3= AUDIO AMPLIFIER A.F.^  LOCK-IN DETECTOR D.C., V. T. V. M. OSCILLOSCOPE OSCILLATING DETECTOR FIELD SWEEP ... /? r PHASE SHIFTER RECORDING MILLI-AM METER AUDIO OSCILLATOR F I G . 2. Block diagram of the nuclear magnetic resonance spectrometer. F I G . 3. Circuit diagram of the oscillating detector: 7?, = 68 K, R2 = 10 K, Rs = 1.5 K, -i?4 = 220 n, 7?5 = 12 K, R6 = 220 K, Ri = 10 K, Rs = 1 meg., R, = 1.2 K , Ct = 2-7MMF, C 2 = 0.001 MF, C3.'= 300 w F , Ct = 0.01 MF, C 6 / = 0.01 m F , C 6 = 0.005 MF, C 7 = 0.002 m F , C 8 '= 0.001 juF, G = 0.001/xF, Cio = 0,2.5 fiF, Cn = 140 W*F, Ci 2 = 20j"AtF. L 1 tune's 9 - 17.5 Mc. per sec. L 2 resonates with 10 MMF at 17 Mc. per sec. to face page 36 36 c axis with respect to the magnetic f i e l d could be determined to o 7 within 1 . The positions of zero s p l i t t i n g f or the L i ' l i n e s were observed with t h i s c r y s t a l (no Al observations were made)} and they indicated that an error of 1.5 had been made i n the c early work on L i i n determining the zero position on the scale. The value of Sx for Al and the new, better value of 6^  for L i are shown i n Fig. 1. Chapter I I I APPARATUS AND EXPERIMENTAL PROCEDURE A recording nuclear magnetic resonance spectrometer designed and constructed by Dr. T.L. C o l l i n s (34) during his doctorate research, was made available f o r t h i s work by the Physics Department of the University of B r i t i s h Columbia. It i s sim i l a r to the instrument described by Pound and Knight (35) but uses a d i f -ferent type of o s c i l l a t i n g , detector and l o c k - i n detector. F i g . 2 shows a block diagram of the apparatus. The electromagnet, with pole faces 6 i n . i n diameter and a gap of 1^ i n . , i s capable of producing f l u x densities up to 12,000 gauss. The pole faces were c a r e f u l l y machined with a rectangular ridge turned at the outside edge i n order to make the f i e l d more uniform at the center of the magnet. A region near the center of the pole faces has an inhomogeneity of less than 37 0.2 gauss i n 8000 gauss over the area of samples normally used (approx. 1 cm. ). The electromagnet i s s u p p l i e d w i t h e l e c t r o n -i c a l l y regulated current and i n a d d i t i o n the magnetic f i e l d i s i t s e l f f u r t h e r s t a b i l i z e d by a c o n t r o l voltage derived from a proton resonance s i g n a l as f i r s t suggested by Packard (36). The o s c i l l a t i n g d e t e c t o r , F i g . 3 , c o n s i s t s of a p a i r of 6AG5's operating i n " p u s h - p u l l " i n a C o l p i t t s c i r c u i t . The tank c o i l containing the sample under i n v e s t i g a t i o n i s placed between the poles of the electromagnet while the r e s t of the o s c i l l a t o r i s assembled i n a brass box outside the magnet. The o s c i l l a t o r ' s t uning condensers are d r i v e n by a synchronous motor and chain of r e d u c t i o n gears so t h a t the frequency can be v a r i e d at the d e s i r e d r a t e . (Actual :ra;te used was i n the neigborhood of 160 kc. per sec. per hr.) One stage of r - f . a m p l i f i c a t i o n i s used before the c r y s t a l r e c t i f i e r . An A.V.C. network feeds part of the d-c. r e c t i f i e d voltage back through an R.C. f i l t e r i n such a way as to keep the l e v e l of o s c i l l a t i o n very low. Thus the c i r c u i t w i l l be extremely s e n s i t i v e to small changes i n the shunt r e s i s t a n c e o f the tank c o i l such as occur on passing through a nuclear resonance i n the sample placed i n s i d e the c o i l . The r - f . voltage.across the c o i l may be adjusted by v a r y i n g the c a p a c i t i e s C^. The l a r g e s t a t i c magnetic f i e l d i s modulated (by means of a p a i r of pancake f i e l d c o i l s i n the gap of the magnet) at 200 c.p.s. with a constant amplitude much smaller than the resonance l i n e width. As the frequency of the o s c i l l a t o r s l o w l y passes through a nuclear resonance, the l e v e l of o s c i l l a t i o n i s A.C. INPUT .5 meg • < —<SLSLSU-^f- TO AUDIO -S-\ OSCILLATOR ? X I meg -WW— D.C. OUTPUT 9^f 20^f F i G . f . Circuit diagram of the lock-in detector. Crystal rectifiers Sylvania Type 1N40. R. F COIL-DIAL CALIBRATED IN 1° INTERVALS I LEVELj ADJUSTMENT F I G . 5 . Diagram of the crystal mount (not to scale). to face page 3a modulated with an amplitude which v a r i e s as the slope of the absorption curve, f i r s t i n phase w i t h the 200 c.p.s. modulation s i g n a l then r e v e r s i n g i t s phase at the center of the resonance l i n e . The l o c k - i n d e t e c t o r , F i g . 4, which i s al s o f e d the 200 c.p.s. modulation s i g n a l through an a d j u s t a b l e phase network, converts the a m p l i f i e d audio s i g n a l i n t o a d-c. voltage which i s approximately-p r o p o r t i o n a l t o the f i r s t d e r i v a t i v e of the resonance absorption l i n e w i t h respect to frequency. This d e r i v a t i v e s i g n a l i s recorded by a vacuum-tube voltmeter and recording milliammeter. Examples of such recorded absorption d e r i v a t i v e curves are shown i n Fi g s . 6, 7 and S. An R.C. f i l t e r c i r c u i t w i t h a d j u s t a b l e time constant ( o r d i n a r i l y 9 sec.) precedes the V.T.V.M. In the case of L i 7 t h e frequency d i f f e r e n c e between the two outer components was measured. In the case of Al r 7 t h e s h i f t of the c e n t r a l component was measured as the frequency d i f f e r e n c e between the c e n t r a l component of the A1^7 resonance i n spodumene 27 and the unperturbed A l ' s i g n a l from a s o l u t i o n of A I C I 3 . The spodumene sample was immersed i n the s o l u t i o n and the spectrometer set to run through both the s i g n a l from the c r y s t a l and the s i g n a l from the s o l u t i o n f o r each c r y s t a l s e t t i n g . F i g . a shows the two s i g n a l s f o r v a r i o u s o r i e n t a t i o n s of the c r y s t a l i n the X r o t a t i o n . Frequency measurements were made by means of a General Radio Type 620A Heterodyne Frequency Meter with the a i d of a standard communications r e c e i v e r . As the o s c i l l a t o r passed through zero beat with a s e r i e s of s e t t i n g s of the frequency meter, markers were made on the recorder chart. Over the small frequency F I G . 6. Exaniplcs of recorded absorption derivative curves. The first three show selected .'splittings for the X rotation. The last three demonstrate the accuracy with which the position of zero splitting at BY'— 147j was determined for the Y rotation. A change of Or by ± 1° produces a marked decrease in the maximum of,the der-ivati'vejcurve. The sweep rate in all cases is 10 kc. per sec. per division. to face page figm 7i .(a) &&&m tfee five' Al s 7 compoaeate .froa AlgQa (arti-£|efeiX% groun). wtese fre$g&n6l$& are af^ rd&lEmteisr 7*2ff '7*350' $c«/8$$?» 7-«4$6 «u»./$ee»*"7.«507 ias*/s9f:«, and 7*600 m©»/secu. frees r i ^ t to left* • ffeg*, assail signal interfering taifch tfe* 7.600 aeu/sea. Al**' * eoap©a.#3a.t .is the Sia*r pipi&i g!Ta& to© copper iii t&e $oii* J'fe) shafts tjfce five A l ^ e©m|®iiea«i from t&e Ho* 2 -erirital of gpoduaene #*o$© frequencies are spg^^aafcely 7*245 kQ*/86Q*i 7#360 -ias#/s#e40 7*415 roe^/see** 7.601 to face page 39 Fig.- $t the above examples e£ r@eorde4 absorption derivative .curves show the A l 2 ' signals from Aluljj i n solution and. from spoduraene. |he eiaall. narrow signal i s in. each case from the AX*< in AICI3. I t i s unperturbed and appears at the Larmor frequency. fhc?so examples show the shi f t of the central component for various . crystal orientations for the X rotat ion. to face page 39 39 ranges used the sweep rate was n e a r l y l i n e a r and the frequency d i f f e r e n c e s f o r w e l l r e s o l v e d resonances were measured d i r e c t l y from the recorder charts. For p o o r l y resolved and unresolved resonances i n the case of L i the composite d e r i v a t i v e curve was g r a p h i c a l l y broken up i n t o i t s three component d e r i v a t i v e curves and the s p l i t t i n g estimated from these. The c r y s t a l mount, F i g . 5, l o c k s t i g h t l y between the ridges of the pole faces so that the a x i s of r o t a t i o n i s perpendicular to the magnetic f i e l d . The d i a l i s c a l i b r a t e d i n one-degree i n t e r v a l s and r e l a t i v e measurements can be repeated c o n s i s t e n t l y t o b e t t e r than h a l f a degree. The two c h i e f c r y s t a l p o s i t i o n e r r o r s a r i s e f i r s t i n determining the c r y s t a l axes, and secondly i n a l i g n i n g the c r y s t a l axes with respect to the a x i s of r o t a t i o n , and to the magnetic f i e l d H 0. Chapter IV EXPERIMENTAL RESULTS AND CALCULATIONS The s p l i t t i n g s of the L i ? and Al27 resonance l i n e s i n spodumene were st u d i e d as f u n c t i o n s of c r y s t a l o r i e n t a t i o n . The number of l i n e components i n each case agreed w i t h t h e o r e t i c a l p r e d i c t i o n s . The s p l i t t i n g between components i n the L i case was • found to be s u f f i c i e n t l y small to be described by f i r s t order p e r t u r b a t i o n theory. However, second order p e r t u r b a t i o n theory 40 was d e f i n i t e l y r e q u i r e d to account f o r the frequency s h i f t s i n A l . Complete informa t i o n on the VJ? tensor at the L i and the A l s i t e s i s obtained from the a n a l y s i s of the frequency s h i f t s . Some observations on.the v a r i a t i o n o f the l i n e broadening as a f u n c t i o n of c r y s t a l o r i e n t a t i o n i n the case of the A1^7 resonance are presented i n the l a s t s e c t i o n . Number of l i n e components 7 3 The s p i n of L i ' i s 1= 2 s 0 t h a t i t s resonance absorption l i n e should break up i n t o 21 ^ 3 components. The s p l i t t i n g of the L i ? resonance l i n e as a f u n c t i o n of c r y s t a l o r i e n t a t i o n has been stud i e d f o r three r o t a t i o n s about the X,Y and Z axes. Only three components of the absorption l i n e appear i n each of the three r o t a t i o n s (see F i g . 6), with the dependence of the frequency s p l i t t i n g 2 A>P between' the two outer components being given i n each case by curves of the expected form (40). This corroborates the deduction made i n Chapter I I on the ba s i s of the known symmetry p r o p e r t i e s of spodumene that the f i e l d gradient t e n s o r has the same eigenvalues and p r i n c i p a l axes at a l l f o u r L i s i t e s i n the u n i t c e l l . The s p i n of A l ^ ? i s I s= and one would th e r e f o r e expect the Al^ ? resonance absorption l i n e to s p l i t up i n t o 21 = 5 components. However, i n the milky-coloured No. 1 spodumene samples the components corresponding to the t r a n s i t i o n s ( ± 1 were b a r e l y d i s c e r n i b l e above the noise and f o r many 2 2 c r y s t a l p o s i t i o n s d i d not appear at a l l . The components c o r r e s -p o n d i n g to the t r a n s i t i o n s ( 4 2. «-* "+ £.) were never found i n " 2 " 2 41 the No. 1 samples. This i s i n spite of the f a c t that the central component was very intense. This led us to suspect that eith e r 1) something was wrong with our technique, or 2) something was wrong with our c r y s t a l s . We f e l t that the trouble was not instrumental, and suspected that impurities i n our c r y s t a l were making the components so broad that t h e i r derivatives were too small f o r us to record. To check the above suspicion we obtained an a r t i f i c i a l l y grown crystal of A^O^ which should be r e l a t i v e l y pure. Five components corresponding to'the 21+ 1 27 tr a n s i t i o n s i n the Al resonance l i n e were e a s i l y found. F i g . 7a i s a reproduction o f t h i s spectrum. This proved that our d i f f i c u l t y was not instrumental. We then obtained a much more perfect spodumene c r y s t a l (No. 2 c r y s t a l of Chapter II) i n which we found the f i v e components of the A l 2 ? resonance l i n e (see Fig. 7b). The f a c t that we could f i n d the four A l 2 ? s a t e l l i t e s i n a small but more perfect c r y s t a l and not i n the large milky crystals which gave a stronger c e n t r a l component led us to believe that the source of the trouble was a v a r i a t i o n of the e l e c t r i c f i e l d gradient at the Al s i t e s i n the milky c r y s t a l s . The milky c r y s t a l s probably contain a great deal of impurities, and c r y s t a l defects would tend to concentrate along the cracks. Assuming that these impurities and c r y s t a l defects are not dis t r i b u t e d uniformly throughout the c r y s t a l , one would expect the e l e c t r i c f i e l d gradient to vary to some extent between the s i t e s of the various Al n u c l e i . This would have l i t t l e e f f e c t on the central component of the Al li n e f o r which the f i r s t order perturbation i s zero, but would appreciably broaden the 42 other components. Presumably then the outer components of the A l 2 ? resonance i n the milky spodumene are broadened to such an extent that t h e i r d e r i v a t i v e s are so small t h a t we cannot even detect them, but the c e n t r a l component which w i l l be almost unaffected i s quite strong* The f a c t t h a t never more than f i v e components of the A l 2 ? absorption l i n e appear corroborates the deduction made i n Chapter I I on the basis of the known symmetry p r o p e r t i e s of spodumene that the f i e l d g r a d i e n t tensor has the same eigenvalues and p r i n c i p a l axes a t a l l f o u r A l s i t e s i n the u n i t c e l l . A p p l i c a b i l i t y of f i r s t order p e r t u r b a t i o n theory t o L i The L i measurements were made i n a magnetic f i e l d of approximately H 0 = 7190 gauss c a l c u l a t e d from the known value of the L i ^ gyromagnetic r a t i o and the measured value of rf>-11.90 Mc. per sec. The frequency d i f f e r e n c e 2 between the two outer components (m-3/2 l/2} - 1/2*-* - 3/2) of the absorption l i n e was measured as a f u n c t i o n of the angle of r o t a t i o n of the c r y s t a l about three orthogonal XYZ axes each i n turn kept perpendicular to the magnetic f i e l d H Q. I t turns out that the 'quadrupole coupling constant f o r L i ? i n spodumene i s small enough f o r f i r s t order p e r t u r b a t i o n theory to be adequate. F i r s t order p e r t u r b a t i o n theory p r e d i c t s t h a t : , 1) the c e n t r a l l i n e does not s h i f t from the Zeeman frequency. •...43 2) the outer s a t e l l i t e s are symmetrically situated with respect to the central l i n e . 3) AV^between a s a t e l l i t e and the central component does not depend on the external magnetic f i e l d H Q. 4) a l l of the components converge f o r some orientation of the c r y s t a l . It was observed experimentally that the central component of the Li7 absorption l i n e did not s h i f t from the Larmor frequency f o r any orientation of the c r y s t a l and that the outer s a t e l l i t e s are symmetrically located with respect to the central l i n e as shown i n Fig. 6. To v e r i f y that the observed s p l i t t i n g oft the Li? resonance i s actually independent of the value of H Q, a series of measurements was taken f o r one p a r t i c u l a r c r y s t a l p o sition (-^ = 40°) which gave a large value of 2 ^ i n magnetic f i e l d s varying over the range of 6500 to 8000 gauss. The observed frequency difference 2A\? remained constant well within our experimental error of about 1 kc. per sec. It was also found that the three l i n e s do converge to a single l i n e f o r certain angles. Need for second order perturbation theory f o r Al iThe A l ^ 7 measurements were made i n a f i e l d of approximately H Q = 6690 gauss corresponding to the Larmor frequency f o r Al27 of V5 = 7.450 mc/sec. In the experimental 27 study of the Al absorption l i n e s i n spodumene, s t r i k i n g d i s -crepancies from the f i r s t order perturbation theory show up + 1 0 KC./, -4 0 -— 5 0 -140 160 180 200 220 240 DEGREES F i g . 9: Frequency s h i f t of the c e n t r a l component of the A l line" i n spodumene at various magnetic f i e l d s t r engths. The dots represent experimental p o i n t s taken at H Q = 5640 gauss, the crosses at H 0 = 6690 gauss and the t r i a n g l e s at H 0 == 8320 gauss. to face page 44 44 which can on l y be explained i n terms of the p e r t u r b a t i o n c a r r i e d to the second order. Second order p e r t u r b a t i o n theory p r e d i c t s : 1) the c e n t r a l l i n e does s h i f t from the Larmor frequency, 2) the outer s a t e l l i t e s need not be symmetrically s i t u a t e d , w i t h respect to the c e n t r a l l i n e , 3) the second order s h i f t v a r i e s i n v e r s e l y as the ex t e r n a l magnetic f i e l d , 4) a l l of the components w i l l not converge f o r any o r i e n t a t i o n of the c r y s t a l . I t was observed experimentally that the c e n t r a l component of the A l ^ 7 l i n e does s h i f t from the Larmor frequency. The dependence of the c e n t r a l l i n e s h i f t on c r y s t a l o r i e n t a t i o n was measured f o r r o t a t i o n s about the X,Y and Z axes. The s a t e l l i t e s were not symmetrically l o c a t e d w i t h respect to the c e n t r a l component as shown i n F i g . 7b. The s h i f t of the c e n t r a l component from the Larrnor frequency was measured f o r various values of & x at three Larmor frequencies \£ = 9*260 mc/sec, \£s= 7*453 mc/sec, and \^  = 6.2S0 mc/sec. corresponding to f i e l d s of approximately H 0 « 8315 gauss, Ho-=*6690 gauss and ^o = 5640 gauss r e s p e c t i v e l y . F i g . 9 shows the r e s u l t s obtained which v e r i f y w i t h i n our experimental e r r o r t h a t the second order term i s i n v e r s e l y p r o p o r t i o n a l to H 0. For "vr*- 192 the r a t i o s of the s h i f t are O.664 : 0.#36 : 1.00 and the inverse r a t i o s of the Larmor frequencies are 0*67$ : 0.$43 : 1.00 r e s p e c t i v e l y . I t was al s o found that the f i v e components never converged f o r any o r i e n t a t i o n of the c r y s t a l . 0 45' '90 135 180 225 / 270 315 360 , DEGREES « ' , FiG.iQ Frequency difference between the two outer components of the L i 7 line for the X i rotation about the b axis of spodumene. The circles are the experimental points. The solid curve represents (2AC)A'~ = 34.0 + 41.8 cos 2 (Ox —AS"). 9x = 0 when thecaxis of spodumene coincides with the magnetic field lit,. The 48° phase shift indicates that the s principal axis, which corre-sponds to the maximum possible splitting, makes that angle with the c axis. ' , • DEGREES . . . . . . FlG.fl-.- Freqiiency-'di(Terence between the two outer components of the L i 7 line for the Y rotation about the c axis of spodumene. The circles are the experimental- points.. The solid curve represents (2Av)x = ^14.S+53.2 cos 2(dr —'3°).' OY=Q was meant to mark the position of the crystal when the b axis is at 90° to the magnetic field Ho, but the 3° phase shift shows a small I alignment error. l_ - - -- --- -II - - - - • t o face page 45 45 D e t a i l e d A n a l y s i s of the Measurements on L i7 7 The r e s u l t s on L i ' are presented below i n the same form i n which they were r e c e n t l y published (37)• A b e t t e r value of $ A due to subsequent work i s given a t the end of t h i s s e c t i o n . F i g s . 10, 11, and 12 represent (up to a p o s s i b l e common minus sign) the frequency d i f f e r e n c e 2 between the two outer components of the L i 7 l i n e i n spodumene as a f u n c t i o n of •0- f o r the X,Y,Z r o t a t i o n s about the b and c axes and the t h i r d perpendicular d i r e c t i o n r e s p e c t i v e l y . Each dot i n these f i g u r e s corresponds to a recorded absorption d e r i v a t i v e curve of the type shown i n F i g . 6. The c r y s t a l p o s i t i o n could i n each case be determined to b e t t e r than h a l f a degree w i t h respect to the zero on the c r y s t a l mount s c a l e . However, a somewhat l a r g e r e r r o r could have been made i n a l i g n i n g the zero of the s c a l e with respect to the magnetic f i e l d H 0 and al s o with respect to the c r y s t a l l o g r a p h i c axes, i n view of the d i f f i c u l t y mentioned i n Chapter I I of p r e c i s e l y l o c a t i n g and marking the d i r e c t i o n of the axes, p a r t i c u l a r l y of the b. a x i s . The frequency d i f f e r e n c e could i n each case be measured to somewhat b e t t e r than 1 kc. per sec. Curves of the t h e o r e t i c a l l y expected form A+D cos 2(0-+$ ) given by Equation (40) were f i t t e d to the experimental p o i n t s g r a p h i c a l l y , as f o r the purposes of t h i s t h e s i s i t was not considered worth-while to expend the e f f o r t f o r a l e a s t squares f i t . The s o l i d curves i n F i g s . 10 - 12 are given r e s p e c t i v e l y by: w, 34.0 + 41.8 cos 2 ( -e x - 48°) (p.&0)y ~ - 14.8-+53.2 cos 2 ( 6 7 - 3 ° ) (63) (2*V)Z = - 19.2 - 47.8 cos 2 (<ez+l°), w .^^ -. -ow 225 2 7 0 315 3 6 0 .• , , DEGREES A • • FIG. 12. Frequency difference between the two outer components of the L i 7 line for the Z rotation about the direction orthogonal to the b and c axes of spodumene. The circles are the experimental points.. The solidcurverepresents(2Ai>)z = —19.2.— 47.8cos2((?z + l ° ) . 0z=Owas meant to mark the position of the crystal when the 6axis coincides with the magnetic field Ha but the 1° phase shift shows a small alignment error. : t o face page 46 46 where the constants A and D are i n k i l o c y c l e s per second. Our rough estimate of the e r r o r s , based, on g r a p h i c a l f i t t i n g of the curves, i s that the A and D constants are accurate to about h a l f a k i l o c y c l e per second and the phase angles $ (with respect to the zero of the scale) to about h a l f a degree. The d i s c u s s i o n o f symmetry p r o p e r t i e s of spodumene i n Chapter I I showed t h a t the twofold symmetry a x i s i n the b d i r e c t i o n must be one of the princ i p a l axes of the V!t'^  tensor at the s i t e s of the L i n u c l e i . Consequently i f we had suc-ceeded both i n p r e c i s e l y i d e n t i f y i n g and marking the b a x i s , _ O and also i n making i t c o i n c i d e w i t h H Q f o r x?y •= - 90 , and f o r •©-£•= 0 ° , we should have obtained zero phase a n g l e s ^ f o r the Y and Z r o t a t i o n s i n Equations ( 6 9 ) . . The small d e v i a t i o n s of 6^  and 5^ from zero i n d i c a t e the presence of one or both of these e r r o r s : (1) the c r y s t a l l o g r a p h i c axes were p r o p e r l y i d e n t i f i e d , but an e r r o r was made i n l i n i n g them up w i t h respect to the a x i s of r o t a t i o n o f the c r y s t a l mount and the magnetic f i e l d H Q, or (2) the X,Y,Z axes as a c t u a l l y picked out were l i n e d up p r o p e r l y with respect t o the c r y s t a l mount and the magnetic f i e l d , but did not quite c o i n c i d e with the b and c c r y s t a l l o g r a p h i c axes and the t h i r d perpendicular d i r e c t i o n . The a n a l y s i s of the r e s u l t s w i l l d i f f e r s l i g h t l y depending on which of the above two sources of e r r o r i s assumed to be the dominant one. In Case (1) we assume that the b a x i s was properly i d e n t i f i e d . Then the -3° and l°phase angles i n d i -cate that e r r o r s of that magnitude were made i n determining the reference p o s i t i o n s of the c r y s t a l i n which the b a x i s i s 47 supposed to be perpendicular to H Q f o r TTy-O, and along H Q f o r X7Z - 0 . To compensate f o r these e r r o r s we merely s h i f t the zeros of the ~&y and -Q-z scales to make - 0° i n Equation (69). The phase angle 8 X - - 4$° can presumably also be i n e r r o r by a comparable amount but we have no d i r e c t i n d i c a t i o n as t o the p o s s i b l e magnitude and s i g n of t h i s e r r o r . We then o b t a i n the f o l l o w i n g set of constants ( i n kc. per sec.) f o r Equations (40): A x - 34.0 B x = - 4 .7 C x = 41.6 A Y =r - 14.8 B Y = 53.2 C y = 0 A z ^ - 19.2 B z = - 47.8 C z - 0 (70) The consistency of the set of s i x constants A,B above may be checked with the a i d of I d e n t i t i e s (43)• Since from these i d e n t i t i e s the three q u a n t i t i e s : A x = 34.0, (By - A Y.)/2« 3 4 . 0 ,-(B z + A z ) / 2 •= 33.5 and a l s o the three q u a n t i t i e s : B x = - 4-7, - (3A y+- B y ) / 2 = - 4-4, (3AZ - B z ) / ? = - 4-9 should be equal, we choose the average values. A x •= 33»8, B x = - 4»7« The spread of the values obtained from the three r o t a t i o n s i s w i t h i n our estimate of h a l f a k i l o c y c l e per second e r r o r f o r the i n d i v i d u a l constants. Using these'averages and C x = 41.6 i n Equations (56) we o b t a i n the f o l l o w i n g r e s u l t s . The three eigenvalues of the K 1^'tensor (up to a common sign) a r e: K ^ * r - 7 . 9 , / f 1 ^ - - 67.6, 75.7 The y a x i s l i e s along the b (or X) a x i s and the x and z axes l i e i n the ac (or YZ) plane with the z a x i s making an angle S x — - 48 48 w i t h the c (or Y) a x i s . The values of the a x i a l asymmetry parameter \ and of the magnitude of the quadrupole c o u p l i n g const-ant a r e : Y 3 3 A I ' d < " ( 7 i ) In Case (2) we assume that the e r r o r l i e s p r i n c i p a l l y i n the i d e n t i f i c a t i o n of the p o s i t i o n o f the c r y s t a l l o g r a p h i c axes, and that there i s no e r r o r i n a l i g n i n g the chosen XYZ d i r e c t i o n s with respect t o the c r y s t a l mount and the magnetic f i e l d . In t h i s case Equations (69) should be used as they stand. The - 3° and 1° phase angles i n Equations (69) i n d i c a t e that an e r r o r of approximately 3° has been made i n l o c a t i n g the b a x i s i n the plane p e r p e n d i c u l a r to the c a x i s , and an e r r o r of approximately 1° has been made i n l o c a t i n g the b a x i s i n the be plane. The f o l l o w i n g set of A,B,C ( i n kc. per sec.) i s obtained from Equations (69): A2, 34.0 Bj. =s - 4.7 C x •= 41.6 Ay; — - 14.8 By,. •=. 53.0. c Y =• 5.4 Az. t= - 19.2 B Z = - 47.8 C z = 1.9 (72) The r e l a t i v e signs of these nine q u a n t i t i e s are determined by the experiment, but not the absolute signs. The above values of A and B l e a d , i n accordance w i t h Equations (44) to the f o l l o w i n g a l t e r n a t i v e expressions f o r the diagonal terms of the tensor K : - 2 A X = - 68.0 -2Ay = 29.6 - 2 A Z =38.4 A y - By = - 67.8 A z - B z = 2 8 . 6 A x - B x =38 .7 Az + B z -= - 67.0 A x + B x = 2 9 . 3 Ay + By-38.2 The spread of values i n each column i s c o n s i s t e n t w i t h our estimated e r r o r o f about h a l f a k i l o c y c l e per second for.the i n d i v i d u a l constants. Averaging the three columns we o b t a i n : K ftx - - 67.6., K f y y « 29 . 2 K%,z-= 3 8 . 4 which add up to zero as re q u i r e d . From the C's and Equation (44) we have f o r the o f f - d i a g o n a l terms: K ^ y 2 = - 41 .6 , K S J X - - 5-4, K / x y = -1*9. The process of d i a g o n a l i z a t i o n o u t l i n e d i n Equations (46) to (52) of Chapter I g i v e s : a e - 5.21 x 1 0 3 , b - 3 .93 x 1 0 4 , 01=24*25' •^= 0.798 |e2qQ/h|« 75.7 kc./sec. (73) S u b s t i t u t i n g the three values o f (h = 1,2,3) from Equations (47) and (4$) i n t o Equations (54) we ob t a i n from Equations (53) the three sets of d i r e c t i o n cosines X,*. V£v each set d e f i n i n g the d i r e c t i o n o f one of the p r i n c i p a l axes x,y,z, w i t h respect to the r o t a t i o n axes X,Y,Z. These d i r e c t i o n cosines a r e : )\, = - .083 X x - .996 A 3 = .019 ytl, a .744 ^ = . 0 5 0 /i3 - .666 = .663 = .070 - - .746 (74) Here only the r e l a t i v e signs w i t h i n each set X,*.,^^, l ^ , are s i g n i f i c a n t , as the absolute signs d e f i n i n g the p o s i t i v e d i r e c t i o n s of the p r i n c i p a l axes are immaterial f o r a second rank tensor. The d e v i a t i o n s of A^from u n i t y and of'A i , X 3 y^ x , Vx, from zero correspond to the d e v i a t i o n of the a c t u a l d i r e c t i o n of the y p r i n c i p a l a x i s from that, of the X r o t a t i o n a x i s , which was meant to coincide with the c r y s t a l l o g r a p h i c b a x i s . This small d e v i a t i o n i s w i t h i n our experimental e r r o r of i d e n t i f y i n g (a) + 10 ! !KC./, 0 . 4 5 1 3 5 1 8 0 2 2 5 2 7 0 3 1 5 3 6 0 D E G R E E S (<*> 'Key, - 4 0 -4 .5 .6 .7 cosT-G +^fiw) 97 F i g . 13: Frequency s h i f t of the A l * - l i n e f o r the X r o t a t i o n about the b a x i s of spodumene p l o t t e d versus "^"x i n (a) and versusSsc^w^+^Ln (b). The dots are the experimental p o i n t s . The' s o l i d curve of (b) rep-resents V3r -V^ » 1.46 kc./sec. f 3.610 - 139-8-5 + 140.4 5*-} ."£"x= o when the c a x i s o f spodumene coincides w i t h H Q. to face page 50 50 the b a x i s , and i s c o n s i s t e n t with the symmetry requirement that the b a x i s should c o i n c i d e with one of the p r i n c i p a l axes. The other four cosinesy"-, , ^yU2j i n d i c a t e t h a t the z,x axes are r o t a t e d away from the YZ axes through an angle S x = - 48° . Combining the r e s u l t s of our two a l t e r n a t i v e analyses given by Equations (71)> (73), and (74) we conclude f i n a l l y that at the s i t e of the L i n u c l e i the y p r i n c i p a l a x i s c o i n c i d e with the b a x i s , and t h a t : )e 2qQ/h| - 75.7 ± 0 . 5 k c . / s e c , 0.79 ± 0 * 0 1 , - - 43° ± 2° (75) The u n c e r t a i n t i e s shown are our estimates of our l i m i t s of e r r o r . A redetermination of the zero p o s i t i o n on the •-^c s e a l has been made, since the above r e s u l t s were published (31, 37) , using & c r y s t a l (No. 3) i n which the c and b axes were b e t t e r defined than i n the o r i g i n a l c r y s t a l used f o r the above L i work This leads to the new value of E>x = •- 4 6 . 5 ° + 1 ° (76) • which i s w i t h i n the l i m i t of u n c e r t a i n t y on the e a r l i e r r e s u l t s D e t a i l e d Analysis of the Measurements on A l 2 ? The work reported i n t h i s s e c t i o n has not yet been published and i s presented here f o r the f i r s t time. F i g s . 13 (^a), 14 ( a ) , and 15 (a) represent the frequency s h i f t of the . c e n t r a l component of the Al27 i n the No. 1 c r y s t a l s of spodumene as a f u n c t i o n of "0" f o r the X,Y, and Z r o t a t i o n s (a) I i + 3 0 + 2 0 + 10 0 10 - 2 0 - 3 0 - 4 0 "i 1 1 r "i r i i i i L 0 4 5 9 0 135 180 2 2 5 2 7 0 315 3 6 0 D E G R E E S + 2 0 -+ 10 -KC/< S E C - 2 0 - 4 0 -F i g . 14: Frequency s h i f t of the A l 2 ? l i n e f o r the Y r o t a t i o n about the c a x i s of spodumene p l o t t e d versus i n (a) and versus s s ^ 1 ^ i n (b). The dots are experimental p o i n t s . The s o l i d curve of (b) represents V»v-V0 = 6.59 kc./sec. f 3.467 - 10.176* + 0.8025S* I t o f a c e page 51 51 r e s p e c t i v e l y . The Larmor frequency was 7«453 mc./sec, cor-responding t o a f i e l d of approximately H Q •= 6690 gauss f o r these r e s u l t s . Each dot i n these f i g u r e s corresponds to a recorded absorption d e r i v a t i v e curve of the type shown i n F i g . 8. The L i ? r e s u l t s , d e s c r i b e d above, were used to a l i g n the c r y s t a l s i n the magnetic f i e l d . Therefore the p o s i t i o n s f o r maximum s h i f t i n the Y and Z r o t a t i o n s are d i s p l a c e d from S - 0 by - 3 ° and l e r e s p e c t i v e l y . For the work on the aluminum i t w i l l be assumed that the b a x i s was p r o p e r l y i d e n t i f i e d but o o the - 3 and 1 displacements i n d i c a t e t h a t e r r o r s of tha t magnitude were made during the work with L i i n determining the reference p o s i t i o n s of the c r y s t a l i n which the b a x i s i s sup-posed to be perpendicular to H Q f o r •©•y = 0° and along H 0 f o r &z - 0 . To compensate f o r the se e r r o r s the zeros of the "S"^ and &z s c a l e s have been s h i f t e d to make the p o s i t i o n s of greatest s h i f t occur at &y -•Qmz-0 as shown i n F i g s . 14 (b) and 15 (b) where the s h i f t of the c e n t r a l component.is p l o t t e d versus c o s ^ ^ . The p o s i t i o n f o r maximum p o s i t i v e s h i f t of the c e n t r a l component i n the X r o t a t i o n g ives d i r e c t l y e x = - 55»5 • The r e l a t i v e p o s i t i o n s f o r maximum s h i f t of the c e n t r a l component 27 i n the case of A l ' and f o r maximum s p l i t t i n g i n the case of L i ? can be obtained w i t h i n h a l f a degree. Their d i f f e r e n c e (9° ) gives the angle i n the YZ plane between at the s i t e of the A l n u c l e i and at the s i t e of the L i n u c l e i . This 9* added to the 4 6 . 5 ° of Equation (76) gives the above value of S x » - 55.5 * f o r A i . . <<0 135 180 2 2 5 2 7 0 D E G R E E S (A) + 2 0 + 10 -I K C / - 3 0 F i g . 15: 27 Frequency s h i f t of the A l ' l i n e f o r the Z r o t a t i o n about the d i r e c t i o n orthogonal to the b and c axes of spodumene p l o t t e d versus i n (a) and versus s s c*/-*1-©* i n (b). The dots are experimental o o i n t s . The s o l i d curve i n (b) represents ^Z-^O ' 6 . 5 9 kc./sec. ^ - 5 . 9 5 4 + 8.196S+ 1.2275*-] to face page 52 52 F i g . 13 (b) represents the frequency s h i f t of the c e n t r a l component of the A l 2 ? l i n e i n spodumene as a f u n c t i o n of S = cos^+£x)for the X r o t a t i o n about the b a x i s . The general Equation (63) of Chapter I may be s p e c i a l i z e d to the case of the c e n t r a l component i n A l by s e t t i n g m» J and \ _ 1 le 2qQ j 9 I t then becomes " 20 U v ? I (77) whe re (78) A curve of t h i s t h e o r e t i c a l l y expected form was f i t t e d to the experimental p o i n t s g r a p h i c a l l y . This was accomplished by choosing a value of ^  i n above to make the r a t i o s ^(s-o) : ^ 5 - 0 . 5 ) and^S-t):^{S'0.5) correspond as c l o s e l y as p o s s i b l e to the experimental values. Then the e l e c t r i c quadrupole coupling constant was chosen t o make the t h e o r e t i c a l curve of Equation (77) f i t the - experimental r e s u l t s . This method of f i t t i n g the t h e o r e t i c a l curve to the experimental r e s u l t s gives .2. 1 ^ 0.95 e ~ 2956 kc./sec. h (79) The s o l i d l i n e i n F i g . 13 (b) represents the t h e o r e t i c a l curve (using the above values of ^  and j6^0 1^ j and \j?r 7.453 mc.) 53 The small d e v i a t i o n of ^ from u n i t y shows up i n the very s l i g h t but r e a l d i f f e r e n c e between the two sets of p o s i t i v e maxima i n F i g . 13 (a). As an experimental check o n . t h e i r accuracy, the above values of Sx , Of and {e^qQJ together w i t h I« m = l', 1 2 1 ' h ' 2 1 L 1 e oQ/hj were s u b s t i t u t e d i n t o the t h e o r e t i c a l equations 20 I ' (59), (67) and (6&) f o r the Y and Z r o t a t i o n s . The t h e o r e t i c a l l y p r e d i c t e d curves are then (81) and I J (82) f o r the Y and Z r o t a t i o n s r e s p e c t i v e l y . The s o l i d l i n e s i n F i g s . 14 (b) and 15 (b) represent these curves p l o t t e d versus cos *0~and the dots represent the experimental p o i n t s i n each case. The t h e o r e t i c a l l y p r e d i c t e d and experimental r e s u l t s agree i n both cases w i t h i n the l i m i t s of accuracy of the experiment. A r a t h e r h u r r i e d check on the s h i f t o f the components corresponding to the ( ± h *-* ± \ ) t r a n s i t i o n s f o r the Y 2' 2 r o t a t i o n i n c r y s t a l No. 2 shows s a t i s f a c t o r y agreement with theory, but i n d i c a t e s t h a t there i s probably an e r r o r of about h a l f a degree i n our determination of SK . These very incomplete r e s u l t s would i n d i c a t e b x * 56 . A thorough study of the components belonging to the {- ^ * 2) a n d •x 3 5 (- 2 *~~*~2 ) t r a n s i t i o n s would probably extend the accuracy - 54 je 4qQ i of the r e s u l t s f o r ^ , £ x , and I h J one more f i g u r e since the s p l i t t i n g i s much l a r g e r . However, at the present state of our t h e o r e t i c a l knowledge regarding these e l e c t r i c f i e l d gradients i t i s d o u b t f u l t h a t extreme accuracy i s worth the a d d i t i o n a l work i n v o l v e d . F i n a l l y i t i s concluded that at the s i t e of the aluminum n u c l e i the x p r i n c i p a l a x i s c o i n c i d e s w i t h the b a x i s , and th a t : \ e 2qQJ = 29 60 kc./sec. ± 10 k c . / s e c , Oj = 0.95 ± 0.01 '  h S* = - 5 5 . 5 ° ± 1° The u n c e r t a i n t i e s shown are our estimates of our l i m i t s of e r r o r . A l l of the above r e s u l t s on L i and A l are summarized i n Table I . 55 TABLE I Summary of Experimental Results i n the Spodumene C r y s t a l ( L i A l ( S i O ^ L i ? s i t e s ' A l 2 ? s i t e s Quadrupole Coupling, Constant 75.7 t 0.5 kc./sec. 2960S10 kc./sec. P r o p e r t i e s of the F i e l d Gradient Tensor x p r i n c i p a l a x i s between c and -a axes along b a x i s at an angle of 43*5 with the c a x i s y p r i n c i p a l a x i s along b a x i s between a and -c axes at an angle of 145.5 w i t h the c a x i s z p r i n c i p a l a x i s between a and c axes between a and c at 46.5 w i t h the a x i s at 55.5 w i t h c a x i s the c a x i s \ 0.79 0.01 0.95 0.01 _ 0.10 - 0.02 r^ cf/ '11 - 0.90 - 0.93 o •P F i g . 16: V a r i a t i o n of the width .of the c e n t r a l component of the A l 2 ' l i n e i n spodumene f o r s e l e c t e d o r i e n t a t i o n s of the X r o t a t i o n 56 Line Broadening I t was observed that the width of the L i ? and A l 2 ? l i n e s v a r i e d as the c r y s t a l o r i e n t a t i o n was changed. Some quan-t i t a t i v e measurements were taken on the v a r i a t i o n o f the l i n e broadening as a f u n c t i o n of c r y s t a l o r i e n t a t i o n f o r the X r o t a t i o n i n the case of the A l 2 ? c e n t r a l component. No t h e o r e t i c a l attempt has, as y e t , been made to i n t e r p r e t these r e s u l t s . I t i s f e l t that d i p o l e - d i p o l e broadening as des c r i b e d by Van Vleck (38) w i l l at l e a s t p a r t i a l l y account f o r t h i s broadening. The abs o r p t i o n d e r i v a t i v e curves of the c e n t r a l 27 component of the A l l i n e are shown i n F i g . 16 f o r various o r i e n t a t i o n s of the c r y s t a l taken du r i n g the X r o t a t i o n . Table I I gives the width (distance between i n f l e c t i o n p o i n t s on the absorption l i n e ) of the c e n t r a l component f o r a number of R v a l u e s . TABLE I I Line width«AH ( i n gauss) f o r A l 2 ? as a f u n c t i o n o f c r y s t a l p o s i t i o n AH AH 136.5 3.6 154.5 4.0 189.5 5.6 224.5 3.7 140.5 159.5 4.2 199.5 4.2 229.5 3.0 145.5 3-1 169-5 4.9 209.5 4.5 234.5 3.0 150.5 3-1 179.5 5.2 215-5 4.0 239.5 3.1 57 Chapter V DISCUSSION The experimental measurements confirm w i t h i n experimental e r r o r the only t h e o r e t i c a l r e s u l t s a v a i l a b l e at present about the f i e l d . g r a d i e n t tensor at the L i and A l s i t e s i n spodumene: the f i e l d gradient tensor i s the same at the f o u r L i s i t e s i n the u n i t c e l l , i t i s again the same at the four A l s i t e s (but d i f f e r s from that at the L i s i t e s ) and i n each case one of the p r i n c i p a l axes of the two tensors involved does coi n c i d e w i t h the c r y s t a l l o g r a p h i c b twofold a x i s of symmetry. The other experimental r e s u l t s , i . e . the determination of which p a r t i c u l a r p r i n c i p a l a x i s c o i n c i d e s with the b a x i s , the o r i e n -t a t i o n of the other two p r i n c i p a l axes, the degree of a x i a l asymmetry of these t e n s o r s , and the absolute value of the quadrupole coupling constant f o r L i and A l i n spodumene, have at present no t h e o r e t i c a l counterparts with which they might be compared. To the author's knowledge t h i s represents the f i r s t complete experimental a n a l y s i s of the f i e l d gradient tensor at a nuclear s i t e i n a s i n g l e c r y s t a l i n the general case of a x i a l asymmetry. No attempt has been made as yet t o i n t e r p r e t these a d d i t i o n a l data i n d e t a i l i n terms of c r y s t a l s t r u c t u r e , but there are some i n d i c a t i o n s t h a t t h i s may be p o s s i b l e : on the one hand there i s an i n t e r e s t i n g c o r r e l a t i o n between the d i r e c t i o n s of those p r i n c i p a l axes of the f i e l d gradient tensor at the L i and A l s i t e s which correspond to the l a r g e s t eigenvalues, and the p o s i t i o n s of neighbouring oxygen atoms; on the other hand the magnitudes of the eigenvalues suggest i o n i c b i n d i n g . 5$ In Fig. 1 the direction marked z at an angle of'4 6 . 5 0 to the c axis denotes the principal axis corresponding to the largest eigenvalue ^ 3 at the site of the Li nuclei, and the direction marked z , f at an angle of 55«5° to the c axis denotes the direction of the principal axis corresponding to the largest eigenvalue at the sites of the Al nuclei. The angle between these directions, according to our measurements is 9o0° * 0.5°« The experimental error is less here than is quoted in Table I for the two angles separately because relative angle measurements can be made to within half a degree. The interesting point is that the line joining the projections of the two oxygen atoms marked .09, located on either side of the Li atoms, makes an angle of 4 6 0 1 2 t with the c axis and a similar line joining the projections of the oxygen atoms marked .25 makes an angle of 5 5 0 2 f with the c axis, giving a difference of 8 ° 5 0 t. The oxygens- marked .09 are the nearest oxygen neighbours of the Li nuclei and those labelled ,25 are the nearest oxygen neighbours of the Al nuclei. It appears that in each case the direction of the principal axis corresponding to the largest eigenvalue of the electric field gradient tensor coincides, within experimental error, with the line joining the nucleus in question and the projection of the nearest oxygen neighbour on the ac plane containing the nucleus. If the values of the quadrupole moment for Li? and Al '-which are given in Mack*s tables (39) as 59 Qu? = G.G2 + .02 x 10~ 2 4 cm2 QF,/1 - 0.156 ± .003 x 10"2/f cm2 are inserted into our values for the quadrupole coupling constants in spodumene, i t appears that ftll)ur 5 x 10 1 3 esu / cm2 (^"J'^= 3 x 10 1 4 esu / cm2 The order of magnitude of the above eigenvalues is what one would expect for ionic bonds i f one allows a factor of 10 decrease due to' shielding as Townes and Dailey (14) suggest. The fact that in this special case of spodumene the field gradient tensors at the Li and Al sites differ in a l l their elements — the orientation of the principal axes, the degree of asymmetry, and the magnitude of their eigenvalues — emphasizes that one should be very cautious in attempting to obtain (as Hatton, Rollin, and Seymour (40) have done for Be^ and Al 2? in beryl) the approximate ratio of quadrupole moments for two different elements from the ratio of their quadrupole coupling constants by assuming the approximate equality of ^ at the sites of two different but somewhat similarly situated nuclei. One would have greater confidence in this assumption i f in the case of a non-axially symmetric ^* i t could be shown that the value of \ and the principal axes orientation are roughly the same at the sites of the two kinds of nuclei. The following further experiments employing the theoretical and experimental methods described in this thesis suggest themselves. From the point of view of nuclear physics three types of investigation are of interest. 60 1) One might try to fix definitely the spin of Mg ' which is reported as probably 5/2 by Mack (39)• This could be attempted by looking for the splitting of the Mg25 line in a single crystal of diopside (CaMg(SiO-j) 2) or some other crystal containing Mg. 2) To test further the observation of Kopfermann outlined in the Introduction one might look for quadrupole moment ratios of further isotope pairs, e.g. of Mo^ S and Mo^?. The relative abundances.and gyromagnetic ratios of these isotopes are such as to make the outcome of such experiments promising. When com-paring two isotopes of the same element in the same crystal one presumes that is the same for both isotopes, and compares the observed splittings for the two isotopes. It is advantageous to have these splittings as large as possible, and the method of this paper enables one to determine uniquely the position of the crystal which will give the maximum splitting. 3) To obtain a check on the validity of Hatton, Rollin and Seymour's attempt (40) to ;determine the quadrupole moment ratio of Be^ and Al 2? (or for any other non-isotopic pair of nuclei) one might investigate the ratio of the quadrupole coupling constants of these two nuclei in a variety of uniaxial and non-uniaxial crystals containing the same two elements. From the point of view of crystal physics i t is of interest to learn as much as possible about a l l the properties of the field gradient tensor at the nuclear sites. From the point of view of 61 ? nuclear physics i t is of particular interest to obtain by calculation, or otherwise, the largest eigenvalue!^/ of this tensor for with its aid the magnitude of the nuclear quadrupole moment could be determined from the experimental absolute value of the quadrupole coupling constant j-* Q j j j ^ I • Presumably the tensor V*^' at a nuclear site can be broken up into two contributions: a "local" one arising from the atomic and bonding electrons in the immediate vicinity of a given nucleus, and a "distant" one arising from charge distributions at other lattice points. The calculation of the former would require a detailed knowledge of the electronic wave functions. The latter might perhaps be easier to estimate, for i t might turn out that sufficient accuracy is attained by replacing the atoms at more distant lattice points by equivalent point charges. At present i t does not seem to be definitely known as to which is the predominant contribution. If one succeeded in obtaining a theoretical estimate of only the "distant" contribution to YA^ in some crystal i t would be interesting to compare the theoretical value of the asymmetry parameter ^ and the principal axes orientation of this "distant" contribution with the experimentally observed ones. A marked discrepancy between the two would definitely indicate that the "local" contribution predominates. Alternatively, two isomorphic crystals in which, say, only Li is replaced by Na, everything else remaining the same, could be experimentally compared to each other. A marked difference in the two values of J and in the orientations of the principal axes at the Li and Na sites would again indicate the importance of the "local" as against the 62 "distant" contributions to . A systematic search for such isomorphic crystals should be made. If one succeeded in theoretically estimating both the "local" and the "distant" contributions to $ji n some crystal, greater confidence could be placed in the theoreti-cal estimate of the largest eigenvalue Jif the calculated value of ^  and the principal axes orientation could be checked directly experimentally. Such a calculated value of / I could then be used to obtain I-LQ\ from the quadrupole coupling constant I Q \ % / ^ - I • Finally, from the point of view of nuclear resonance absorption spectroscopy i t might be of interest to have an example of a pure quadrupole spectrum (no magnetic splitting of levels)' in the same substance In which one also observes the case discussed in this thesis1 viz. when the electrostatic splitting is small compared to the magnetic splitting. The 27 case of Al in spodumene seems favorable, as i t s pure quadrupole lines should lie in an experimentally easily accessible region near 1 mc./sec. 63 REFERENCES., (1) Pake, G.E., J . Chem. Phys., 16, 327 (1948) (2) Pound, R.V., Phys. Rev., 22. 6 g5 (1950) (3) Schuster, N.A., and Pake, G.E., Phys. Rev. dl, dd6 (1951) (4) Schuster, N.A., and Pake, G.E., Phys. Rev. d l , 157 (1951) (5) Becker, G. and Kruger, H., Naturwiss. ^d, 121 (1951) (6) Becker, G., Z. Physik, 110, 457 (1951) (7) Townes, C.H., Foley, H.M., and Low, W., Phys. Rev. Jki W 5 (1948) (8) Kopfermann, H., Naturwiss. ^d, 29 (1951) (9) Schmidt, T., Z. Physik 106, 358 (1937) (10) Bohr, A., Phys. Rev. d l , 134 and 331 (195U (11) Rainwater, J . , Phys. Rev. 22> 432 (1950) See a l s o Feenberg, E. and Hammock, K.C., Phys. Rev. d l , 2d5 (1951) (12) Gutowsky, H.S. and Pake, G.E., J . Chem. Phys. I d , 162 (1950) (13) Townes, C.H. and D a i l e y , B.P., J . Chem. Phys. 12, 7d2 (1949) (14) Townes, C.H. and D a i l e y , B.P., J . Chem. Phys. 20, 35 (1950) (15) Bloch, F., Phys. Rev. 22, 460 (1946) (16) Pake, G.E., Am. J . Phys. 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