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Studies in nuclear magnetic and nuclear quadrupole resonance spectra Cranna, Norman Greig 1954

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STUDIES IN NUCLEAR MAGNETIC AND NUCLEAR Q.UADRUP0LE RESONANCE SPECTRA by NORMAN GREIG CRANNA 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 »V-e accept t h i s thesis as conforming to the standard required from candidates for the degree of Doctor of Philosophy Members of the Department of Physics The University of B r i t i s h Columbia A p r i l 1 9 ^ 4 ABSTRACT Standard techniques of radio-frequency nuclear resonance spectroscopy have been applied to further studies of the in t e r a c t i o n between atomic nuclei i n c r y s t a l s and the c r y s t a l l i n e electric' f i e l d gradients at the nuclear s i t e s . Observations have been made on the nuclear magnetic resonance spectra of A l 2 ^ , L i ^ , L i ^ and S i 2 ? i n single crystals of L i A l ( S i 0 j ) 2 (spodumene) i n strong magnetic f i e l d s . Results from the A l 2 ^ spectrum provided improved values of the f i e l d gradient constants of spodumene and a check on the adequacy of second and t h i r d order perturbation theory i n describing the e l e c t r o s t a t i c per-turbation of the magnetic energy l e v e l s ; these r e s u l t s also provided an experimental check on a proposed new method of 6 7 nuclear spin determination. The L i and L i ' measurements provided a more accurate value of the quadrupole moment r a t i o f o r t h i s pair of isotopes. Observations on the 29 S i spectrum support; e x i s t i n g evidence that the spin 29 of S i i s 1/2. A super-regenerative spectrometer has been b u i l t for the detection of nuclear magnetic resonances and nuclear e l e c t r i c quadrupole resonances. Preliminary tests indicate that i t w i l l detect resonances i n so l i d s at low frequencies i i i . which could not be detected with the continuous-wave type of spectrometer. A pure quadrupole resonance i n NagB^ O^ .^ -HgO (kernite) has been detected at 1 .27 Mc./sec. using t h i s super-regenerative spectrometer. This represents a pure quadrupole resonance of the lowest frequency reported to date. THE UNIVERSITY OF BRITISH COLUMBIA Faculty of Graduate Studies PROGRAMME OF THE FINAL ORAL EXAMINATION FOR THE DEGREE OF DOCTOR OF PHILOSOPHY of NORMAN GREIG CRANNA B. Sc.- (Queen's) 1949 M. Sc. (Queen's) 1950. WEDNESDAY, MAY 5th, 1954 at 2:00 P.M. IN ROOM 301, PHYSICS BUILDING COMMITTEE IN CHARGE Dean W.H. Gage - Chairman FvA.'Kaempffer H„B.- Hawthorn J.M. Daniels T.-E. Hull H.E.D. Scovil V.J;.. Okulitch G.M. Shrum M.A>- Ormsby External Examiner - W.G,-. Proctor University of Washington PUBLISHED PAPERS Second-Order Effects i n Nuclear Electric Quadrupole Interaction of Al27 i n Spodumene (H-.-E. Petchj. G.M.- V'olkoff and N.G.. Cranna), Physical Review 88,: 1201,. 1952-. Second Order Nuclear Quadrupole Effects i n Single Crystals, Part II (H.E.. Petch, N.G. Cranna and G„M. V'olkoff) Canadian Journal of Physics, 31, 1185, 1953. ABSTRACT Standard techniques of radio-frequency nuclear resonance spectroscopy have been a p p l i e d to f u r t h e r , s t u d i e s of the i n t e r a c t i o n between atomic n u c l e i i n c r y s t a l s 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 g radients at the nuclear s i t e s v Observations have been made on the nuclear mag-n e t i c resonance s p e c t r a of Al27^ L i ^ , L i ? and Si 2-? i n s i n g l e c r y s t a l s of L i A . i ( S i O j ) 2 (spodumene) i n strong magnetic f i e l d s , ' Results from the A l 2 ? spectrum provided improved values of the f i e l d g r a -d i e n t constants of spodumene. and a check on the ade-quacy of second and t h i r d order p e r t u r b a t i o n theory' i n d e s c r i b i n g the e l e c t r o s t a t i c p e r t u r b a t i o n of. the magnetic energy l e v e l s j these r e s u l t s a l s o provided an experimental check on a proposed new method of nuclear s p i n determination. The L i 6 and L i ? measure-ments provided a more accurate value of the quadrupole moment r a t i o f o r t h i s p a i r of i s o t o p e s . Observations on the S i 2 9 spectrum support e x i s t i n g evidence that the s p i n of S i 2 9 i s l / 2 . A super-regenerative spectrometer has been b u i l t f o r the d e t e c t i o n of nuclear magnetic resonances and nuclear e l e c t r i c quadrupole resonances»• P r e l i m i n a r y t e s t s i n d i c a t e that i t w i l l detect resonances i n s o l i d s at low frequencies which could not be detected w i t h the continuous-wave type of spectrometer.- A pure quadrupole resonance i n Na2B4.O7.4H2O ( k e r n i t e ) has been detected at 1*27 M c / s e c using t h i s super-regenerative spectrometer. This represents a pure quadrupole resonance of the lowest frequency reported to date.-GRADUATE STUDIES Field of Study: Physics Quantum. Mechanics - H.A'.-Elliott Electromagnetic Theory - W,- Opechows.ki Theory of Measurements - A.M.- Crooker Nuclear Physics ;- K.C.-Mann Special Relativity . " - W.- Opechowski . Group-Theory i n Quantum Mechanics - H.'Koppe Other Studies: Differential Geometry - I. Halperin Integral Equations - T.E.-Hull Operational Methods i n Engineering - W.B.. Coulthard Physical Chemistry of High Polymers - B.A. Dunell Radio Chemistry - M. Kirsch and K.. Starke THE UNIVERSITY OF BRITISH COLUMBIA Faculty of Graduate Studies PROGRAMME OF THE FINAL ORAL EXAMINATION FOR THE DEGREE OF DOCTOR OF PHILOSOPHY of NORMAN GREIG CRANNA B. Sc.- (Queen's) 1949 M.;- Sc. (Queen's) 1950 WEDNESDAY,.- MA.Y 5th, 1954 at 2:00 P.M. IN ROOM 301, PHYSICS BUILDING COMMITTEE IN CHARGE Dean W.H. Gage - Chairman F. A.-Kaempffer H,B. Hawthorn J.M.- Daniels T.E. Hull H.E.D. Scovil V.J:.-' Okulitch G. M.- Shrum M.Av Ormsby External Examiner - W.G.. Proctor University of Washington PUBLISHED PAPERS Second-Order Effects i n Nuclear Electric Quadrupole Interaction of A l 2 ? i n Spodumene (H-.E. Petchj.- G.M.- Volkoff and N.G.. Cranna), Physical Review 88, 1201, 1952-. Second Order Nuclear Quadrupole Effects i n Single Crystals, Part II (H,E». Petch, N,G, Cranna and G.M. Volkoff) Canadian Journal of Physics, 31, 1185, 1953. ABSTRACT Standard techniques of radio-frequency nuclear resonance spectroscopy have been applied to further studies of the interaction between atomic nuclei i n crystals and the crystalline electric f i e l d gradients at the nuclear sites^ Observations have been made on the nuclear mag-netic resonance spectra of Al 27^ Li°, Li? and Si2-? in single crystals of LiAl(SiO^)2 (spodumene) i n strong magnetic f i e l d s . Results from the Al 27 spectrum provided improved values of the f i e l d gra-dient constants of spodumene and a check on the ade-quacy of second and third order perturbation theory i n describing the electrostatic perturbation of the magnetic energy levelsj these results also provided an experimental check on a proposed new method of nuclear spin determination.- The L i ^ and Li? measure-ments provided a more accurate value of the quadrupole moment ratio for this pair of isotopes. Observations on the Si 29 spectrum support existing evidence that the spin of S i 2 9 is l/2. A super-regenerative spectrometer has been built for the detection of nuclear magnetic resonances and nuclear electric quadrupole resonances»• Preliminary tests indicate that i t w i l l detect resonances i n solids at low frequencies which could not be detected with the continuous-wave type of spectrometer.- A pure quadrupole resonance in NapB^ Oy.4H2O (kernite) has been detected at 1.27 He./sec.- using this super-regenerative spectrometer- This represents a pure quadrupole resonance of the lowest frequency reported to date. GRADUATE STUDIES Field of Study: Physics Quantum Mechanics - H.A-. E l l i o t t Electromagnetic Theory - W„- Opechowski Theory of Measurements - A.M.- Crooker Nuclear Physics - K..C,--Mann Special Relativity . - W.- Opechowski Group - Theory i n Quantum Mechanics - H. Koppe Other Studies: Differential Geometry Integral Equations Operational Methods i n Engineering Physical Chemistry of High Polymers Radio Chemistry I. Halperin - T.E.- Hull - W.B.. Coulthard - B.-A. Dunell - M. Kirsch and K.- Starke i . ACKNOWLEDGMENTS 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 through the award of Summer Scholar-ships ( 1 9 5 2 , 1953) and a Fellowship ( 1 9 5 2 - 5 3 ) to the author. I also am grateful to the Research Council of Ontario for the award of Scholarships ( 1 9 5 0 - 5 1 , 1 9 5 1 - 5 2 ) . To Professor Volkoff, who supervised t h i s research, I wish to express my sincere appreciation for his active i n t e r e s t , for many il l u m i n a t i n g discussions;;:, and for his aid i n numerous other ways. *\ I would also l i k e to express my thanks to Mr. H, Waterman and Dr. R. Eades for h e l p f u l suggestions related to technical problems; to Mr. A.J. Fraser and Mr. W. Morrison of the Physics Department machine shop for t h e i r co-operation i n constructing parts of the apparatus; to Mr. J. Sample and Mr. J. E l l i o t t for taking the colored photograph shown i n Plate I ; and to Dr. R.M. Thompson of the Geology Department, Dr. B.A. Dunell of the Chemistry Department, and Dr. C.S. Samis of the Mining and Metallurgy Department, for kindly supplying c r y s t a l samples. F i n a l l y , I wish to acknowledge the help of my wife, Maria, who has not only typed t h i s thesis, but has contributed so much to the successful completion of t h i s work by :her constant encouragement and by shouldering so many of the parental r e s p o n s i b i l i t i e s which should have been mine. TABLE OF CONTENTS Page ACKNOWLEDGMENTS i ABSTRACT i i INTRODUCTION 1 PART I - THEORY Chapter 1'. Introduction 10 Chapter 2 . Summary of theory as applicable to the case.^J 14 Chapter 3. Theory for the pure quadrupole case (HQ=0). 2 5 Chapter 4. Theory applicable to the region where3*»£/\ 2 7 PART II - EXPERIMENTAL - EXTENSION OF RESULTS IN SPODUMENE AT HIGH MAGNETIC FIELDS Chapter 5 . Apparatus and experimental Procedure 3 0 Chapter 6. Experimental Results and Calculations 3 2 A. Inner s a t e l l i t e s of A l ' i n spodumene 32 B. Experimental check on new method of spin determination , 41 C. Ratio of the quadrupole moments of ZAP and L i 7 . 42 D. Observations on the S i 2 ? .'.resonance 44 Chapter 7 . Discussion 46 PART II I - EXPERIMENTAL - PURE QUADRUPOLE SPECTRA Chapter 8 . Apparatus and Experimental Procedure 5 2 k: Super-regenerative oscillators-General 5 2 B*. The super-regenerative o s c i l l a t o r used 6 0 C*. The quenching c i r c u i t 6 3 D'. Modulation methods 6 5 E: Frequency measurement 6 Chapter Experimental Results 6 A'. Preliminary t e s t i n g of the spectrometer 6 8 B; Pure quadrupole spectra 7 0 REFERENCES 74 LIST OF ILLUSTRATIONS Facing Page Fi g . 1 . Energy l e v e l s as a function of H for the case 1 = 5 / 2 , * [ = 0 1 1 Fig, 2 . Energy l e v e l s as a function of R for the case I = 5 / 2 , *|_ = 0 . 9 5 2 b Fi g . 3 . Transition frequencies as a function of R for the same case 2 6 Fig, 4. Relative t r a n s i t i o n p r o b a b i l i t i e s as a function of R for the same case 2 6 Fig . 5 . Projection of spodumene unit c e l l 3 1 Fig. 6 . Selected traces of the A l 2 ? spectrum i n spodumene 3 2 Fig. 7 . Y rot a t i o n frequency difference f o r A l 2 ^ "inner" s a t e l l i t e s 3 5 F i g . 8. T r o t a t i o n frequency s h i f t for A l 2 ^ "inner" s a t e l l i t e s 3 5 Fi g . 9. Y rot a t i o n frequency difference and s h i f t for A l ' "inner" s a t e l l i t e s 3 6 Fi g . 1 0 . L i ^ spectrum i n spodumene at maximum s p l i t t i n g 42 Fi g . 11. C i r c u i t diagram of the super-regenerative o s c i l l a t o r 6 0 Fi g . 12. C i r c u i t diagram of the quenching c i r c u i t . . . . 6 3 Fi g . 1 3 . Block diagram of the super-regenerative spectrometer 6 5 Fig . 14. Selected traces of signals obtained with the super-regenerative spectrometer 6 9 Fig . 1 5 . Recorded pure quadrupole absorption l i n e . . . . 7 2 LIST OF PLATES Plate I. Color Photograph of a 3 - dimensional model of the spodumene unit c e l l 3 1 LIST OF TABLES Page Table I. Magnetic f i e l d dependence of the frequency s h i f t 3 3 Table I I . Observed and calculated values of a, b, and R for the two spodumene samples 3 7 Table I I I . Summary of observations on two spodumene samples 3 8 1. INTRODUCTION The experimental work reported In t h i s thesis u t i l i z e s the technique of radio-frequency nuclear resonance spectroscopy. This technique provides a method of studying the interactions between atomic n u c l e i and t h e i r surround-ings. The work described i n t h i s thesis i s confined to the case of nuclei i n non-metallic c r y s t a l s , subject to the following two types of i n t e r a c t i o n , one or both of which may be present:-(a) a magnetic i n t e r a c t i o n between the n u c l e i and an externally applied s t a t i c uniform magnetic f i e l d H Q . (b) an e l e c t r o s t a t i c i n t e r a c t i o n between the nuclei and microscopic c r y s t a l l i n e e l e c t r i c f i e l d gradients. Fart X of t h i s thesis sets out the underlying t h e o r e t i c a l framework. Part IX reports experimental work at-high magnetic f i e l d s where interactions of both types (a) and (b) are present, but (a) greatly exceeds (b). Part I I I reports some work at lower magnetic f i e l d s , and some at zero f i e l d where only (b) need be considered. The experimental r e s u l t s so obtained y i e l d inform-ation on both c r y s t a l s and n u c l e i , as w e l l as broadening the knowledge of the technique of resonance spectroscopy. The information on cr y s t a l s i s related to the f i e l d gradient tensor at the s i t e s of n u c l e i : the or i e n t a t i o n 2. of i t s p r i n c i p a l axes with respect to the c r y s t a l axes, and i t s degree of asymmetry, i . e . the r e l a t i v e magnitudes of i t s components along the p r i n c i p a l axes. Such inform-a t i o n i s of p a r t i c u l a r i nterest to those studying the theory of bonding i n s o l i d s . The information which may be obtained about nuclei includes the determination of nuclear spins and, i n the case of cer t a i n isotope p a i r s , the determination of quadrupole moment r a t i o s . The study of nuclear magnetic resonance spectroscopy i s a comparatively new one, and that of nuclear quadrupole resonance spectroscopy (when H Q =0) i s even newer. The pa r t i c u l a r region of nuclear magnetic resonance spectroscopy where the interactions (a) and (b) are of the same order of magnitude has scarcely been touched, and any experimental r e s u l t s on frequencies and t r a n s i t i o n p r o b a b i l i t i e s as functions of c r y s t a l o r i e n t a t i o n and magnetic f i e l d are of value i n checking the e x i s t i n g theory, and i n the planning of future experiments. I f a sample containing n u c l e i with spin 1 ^ 0 and magnetic moment ^  i s placed i n a uniform magnetic f i e l d H Q , quantum mechanics predicts and experiment confirms that there w i l l be 2 I + 1 possible orientations which the nuc l e i may assume with respect to H Q , and that each o r i e n t a t i o n corresponds to a d e f i n i t e energy l e v e l d i f f e r e n t from the 3. others. These are referred to as the Zeeman l e v e l s . I f a l l other interactions between the n u c l e i and t h e i r surroundings are zero, or average to zero over a short time i n t e r v a l , the 2 1 + 1 l e v e l s w i l l be equally spaced by an amount ^ f - l o / l . Quantum mechanics also predicts that magnetic dipole t r a n s i t i o n s due to a perturbing radio-frequency magnetic f i e l d o s c i l l a t i n g i n the plane perpendicular to H© w i l l only take place between adjacent l e v e l s (hence a l l t r a n s i t i o n s w i l l represent the same energy difference ) and therefore only provided the frequency of i s equal to the c l a s s i c a l Larmor precession frequency V e = y * H o / l h . In the laboratory, magnetic f i e l d s as large as 10,000 gauss are comparatively easy to produce. For H Q of t h i s order and for a l l n u c l e i of known yu, , the frequencies given by the above expression f a l l i n the range from 1 MC/see. to 50 MC/sec. In 1946 i t was shown that i n a sample of protons, f o r example the hydrogen nuclei i n water, placed i n a magnetic f i e l d H Q, i t was possible t o induce t r a n s i t i o n s between the l e v e l s by means of a radio-frequency magnetic f i e l d perpendicular to H Q . Two independent groups, one at Stanford under Bloch ( l ) and one at Harvard under P u r c e l l (2) accomplished t h i s by d i f f e r e n t techniques and the f i e l d of nuclear radio-frequency spectroscopy was born. A great deal of inform-a t i o n on the spin'and magnetic moments of n u c l e i has been 4. compiled using these or related techniques. This simple picture of n u c l e i free from i n t e r -actions other than with H q and H^ i s never s t r i c t l y true, but to a good approximation i t describes the s i t u a t i o n for nuclei i n solutions or l i q u i d s . Here the l o c a l magnetic f i e l d due to neighboring dipoles and any e l e c t r o s t a t i c i n t e r -action i s averaged out by the thermal motion of the n u c l e i . Recent refined work has shown some l i n e structure even i n l i q u i d s , but we w i l l not be concerned with t h i s topic here. When the samples containing n u c l e i are i n the form of s o l i d s the picture changes somewhat. Now the n u c l e i , except f o r t h e i r v i b r a t i o n a l energy, are i n fixe d positions and each nucleus experiences i n addition to H Q a l o c a l magnetic f i e l d due t o the neighboring dipoles. Since the dipoles are oriented i n 2 I + 1 d i f f e r e n t ways the f i e l d produced at the s i t e s of s i m i l a r nuclei may vary from H Q by + H l o o a l where H i o c a l can be of the order of 10 - 20 gauss or more. For example, i n some cr y s t a l s containing water of c r y s t a l l i z a t i o n , t w o or more protons i n a c l o s e l y spaced group may be much nearer to each other than to other protons i n s i m i l a r groups. The mutual magnetic i n t e r a c t i o n of protons w i t h i n a group may be strong enough to s p l i t the proton Zeeman l i n e into components 1 3 ) . However, i n most cases the s h i f t of the l e v e l s due to variations i n H l o c a l i s diffuse i n nature and r e s u l t s only i n a broadening of the observed Zeeman l i n e (4). The quadrupole moment Q of a nucleus measures the deviation from spherical symmetry of the nuclear charge d i s t r i b u t i o n , being zero for the case of spherical symmetry.. I f a nucleus i s located i n a uniform e l e c t r i c f i e l d 15, i t w i l l experience no forces or torques even when Q^ 0. But i f Q^O, and at the nuclear s i t e V l ^ O , the nucleus experiences a torque and, analogous to the magnetic i n t e r a c t i o n described above, there w i l l be a f i n i t e number of possible orientations which the nucleus may take up with respect to the gradient of the f i e l d . The inhomogeneous e l e c t r i c f i e l d at nuclear s i t e s i n c r y s t a l s i s generated by the surrounding charge d i s t r i b u t i o n due to neighboring ions and the atomic electron s h e l l s and bonding electrons surrounding the nucleus i t s e l f . Quantum mechanical analysis shows the number of non-equivalent orientations, and hence the number of energy l e v e l s , to be I + 1/2 for the case of odd h a l f - i n t e g r a l spin. The spacing of the l e v e l s involves I and V l f . Tlie exact expression i s given i n the section on theory. More detailed theoretical treatment has been given by Kruger ($), Bersohn (6} and Cohen (7). The t r a n s i t i o n frequencies between these pure quadrupole l e v e l s i n the cases reported to date cover a much wider range than the Zeeman t r a n s i t i o n frequencies, 6 ranging from 1.5 MC/sec. to several hundred MC/sec. The pure quadrupole t r a n s i t i o n of lowest frequency up to the time of t h i s thesis was reported by Dehmelt ( 8 ) who ob-served a B 1 0 l i n e i n BtCH})^ at 1 . 5 2 MC/sec. at l i q u i d a i r temperature. Dehmelt has reported a great deal more work at high (9) and low frequencies (10, 11). Other re s u l t s have been reported by Watkins and Pound (12), Dehmelt and Kruger (13), Dean (14), Livingstone (15) and many others. Part I I I of t h i s thesis i s concerned p a r t l y with reporting some new work done i n t h i s pure quadrupole f i e l d . As can be seen from the frequency ranges of the magnetic and e l e c t r i c t r a n s i t i o n s mentioned above, the quadrupole i n t e r a c t i o n between a nucleus and i t s surround-ings may be much larger than, much smaller than, or of the same order of magnitude as the magnetic i n t e r a c t i o n between the same nucleus and a B^ of several thousand gauss. Both these interactions w i l l be present f o r n u c l e i of spin 1^0, quadrupole moment 0,^0, contained i n a c r y s t a l which has l o c a l l y non-cubic symmetry, at the nuclear s i t e s , and which i s placed i n a magnetic f i e l d . Since both involve the l i n i n g up of the spin axis of the nuclei they w i l l compete with each other. At the one extreme the quadrupole in t e r a c t i o n w i l l be a small perturbation on the Zeeman int e r a c t i o n while at the other extreme the Zeeman i n t e r a c t i o n w i l l be a small perturbation on the quadrupole i n t e r a c t i o n . Or the two interactions may be of the same order of magnitude. 7 . A great deal of work has been reported on the quadrupole perturbation of the Zeeman l e v e l s i n c r y s t a l s . Carr and Kikuchi ( 1 6 ) , Bersohn ( 6 ) , Pound ( 17) and Volkoff et a l . ( l 8 ) have reported t h e o r e t i c a l r e s u l t s i n the f i e l d , and the l a t t e r two among others have also reported ex-perimental r e s u l t s confirming the theory. Theoretical treatments of the Zeeman perturbation of the quadrupole l e v e l s have been given to f i r s t order by Kruger ( 5 ) , and Bersohn ( 6 ) calculated also the second order perturbation. Kruger and Meyer-Berkhout ( 19) report-ed observation of t h i s effect i n the quadrupole spectra of A s ^ and Dean ( 2 0 ) reported s i m i l a r observations on CI quadrupole spectra. From frequency measurements on observed spectra i n the above two cases, information i s obtained r e l a t i n g to Q of the nuclei concerned and to the magnitude and orientat i o n of V B with respect to the c r y s t a l axes. When the quadrupole and Zeeman interactions are of the same order of magnitude, perturbation theory i s no longer applicable and-direct but more tedious numerical calculations become necessary. Lamarche and Volkoff ( 2 1 ) have carried out t h i s c a l c u l a t i o n f o r the p a r t i c u l a r case of A l 2 ? i n spodumene at one p a r t i c u l a r o r i e n t a t i o n of the c r y s t a l with respect to H Q . For t h i s case they present complete information on the energy l e v e l s , t r a n s i t i o n 8 frequencies and t r a n s i t i o n p r o b a b i l i t i e s f o r a l l values of H Q from zero f i e l d to high f i e l d s where the quadrupole in t e r a c t i o n i s a small perturbation on the Zeeman i n t e r a c t i o n . Part I of t h i s thesis outlines the t h e o r e t i c a l framework into which the experimental work f i t s . Part I I describes work carried out on A l 2 ? , X i ^ , Li7 and S i 2 9 i n spodumene i n the high f i e l d region. This work was i n i t i a t e d with the following objects: to check the second-order and third-order perturbation theory-mentioned above, to obtain higher accuracy i n the values of the f i e l d gradient constants of spodumene, to check a proposed new method of spin determination which i s described i n the section on theory, and f i n a l l y to attempt to improve the accuracy of the Q^ .s / Q L. 7ratio previously reported by Schuster and Pake (22). These r e s u l t s have already been published (2?, 24). Part I I I of the thesis describes work undertaken with a view to checking the above theory at zero and low magnetic f i e l d s . I t was hoped to be able to present a series of observations extending from zero f i e l d to high f i e l d s so as to l i n k the three regions described above with experimental data confirming the theory. The t h e o r e t i c a l -l y available signal-to-noise r a t i o i s proportional to (25) so going to low f i e l d s presents serious d i f f i e u l t i e s . A super-regenerative spectrometer was b u i l t and with i t 9 the A l ' central l i n e i n spodumene was recorded i n a f i e l d of approximately 1100 gauss at a frequency of about 1 .2 MC/sec. This represents an extension by a factor of 2 i n frequency beyond the lower l i m i t attained here with other types of spectrometer, and i t i s possible that t h i s might be pushed another factor of 2 lower, but f o r a thorough study of the spectra of one type of nuclei i n a l l 3 regions i t i s l i k e l y that a more suitable combination of nucleus and c r y s t a l w i l l have to be found. A search f o r the predicted pure quadrupole l i n e of A l 2 7 i n spodumene has been made without success. How-ever, a predicted pure quadrupole l i n e of B 1 1 i n Na2B4.O7.4H2O has been recorded with t h i s super-regenerative spectrometer at 1 . 2 7 MC/sec._ This represents a pure quad-rupole l i n e of the lowest frequency reported to date. This work on B 1 1 l i n e s has not yet been submitted f o r publication. 10. PART I - THEORY Chapter 1 - Introduction. This part of the thesis consists of a summary of the theory of the nuclear e l e c t r i c quadrupole effects i n nuclear resonance spectroscopy. Although none of t h i s theory represents o r i g i n a l work by the author of the th e s i s , nevertheless i t s i n c l u s i o n i s required to enable the author to interpret his experimental r e s u l t s presented i n Parts I I and I I I of the t h e s i s . We consider nuclei i n single c r y s t a l s and i n p o l y c r y s t a l l i n e s o l i d s . The magnetic resonance spectra of these nuclei w i l l have a l i n e width determined mainly by c e l l - t o - c e l l variations of the l o c a l magnetic f i e l d due to neighboring nuclei, and perhaps by c e l l - t o - c e l l variations of the e l e c t r i c f i e l d gradient due to c r y s t a l l i n e imper-fections and impurities. We w i l l not consider these small perturbations further, nor w i l l we be interested i n l i n e structure of frequency s h i f t due to variations i n l o c a l magnetic f i e l d s , diamagnetic e f f e c t s , etc. We w i l l examine the magnetic and e l e c t r o s t a t i c interactions for the cases when one or both are present and are large compared to l i n e width. Suppose we have a system of nuclei with spin I Fig. 1. Energy l e v e l s , as a function of H, for the case 1 = 5/2, >i = 0 and H along the axis of symmetry of . to face page 11 11. and magnetic moment jj. placed i n a s t a t i c magnetic f i e l d H Q . I f there i s no V E , as i n s i t e s of l o c a l cubic symmetry, then I i s quantized with respect to the d i r e c t i o n of H Q . lach eigenstate of our magnetic i n t e r a c t i o n operator » w i l l be a pure spin-state represented by an eigenfunct ion, ^rfSay, where m w i l l take the 2 1 + 1 values + I , I - 1, I . The corresponding energy l e v e l s are equally spaced with spacing proportional to jjl H q as shown i n the dotted l i n e s i n Figure 1 for the case I = 5/2. Transitions induced by a l i n e a r l y or c i r c u l a r l y polarized r . f . magnetic f i e l d H]_ perpendicular to H Q w i l l occur between adjacent l e v e l s only, A m =±1. Since these energy differences are the same the resonance spectrum w i l l consist of a single l i n e . In the purely e l e c t r o s t a t i c case (H Q =0) f i r s t consider nuclei with spin 1^1/2 and quadrupole moment Q, located i n a c r y s t a l at s i t e s where the f i e l d gradient has a x i a l symmetry. This means that i f we choose the axis of symmetry as the z a x i s , the second derivative of the el e c t r o s t a t i c p o t e ntial with respect to z, j£jj7will be the maximum value of the f i e l d gradient; and i n the plane perpendicular to the .^ a x i s , i n vir t u e of Laplace»s equation and a x i a l symmetry, <JV? = "~T <^£ regardless of the orientation chosen for the x and y axes 1 2 . i n t h i s plane, The existence or non-existence of a x i a l symmetry of V S i s usually described i n terms of a parameter \~ i^xyT' ^yyJ/fyfr where these three quantities are the values of the f i e l d gradient i n the d i r e c t i o n of the three p r i n c i p a l axes x, y, and z. The x d i r e c t i o n i s usually a r b i t r a r i l y chosen to be that of the smallest value, and the z d i r e c t i o n that of the largest value. For a x i a l symmetry = 0. Such conditions occur at p a r t i c u l a r s i t e s i n hexagonal and tetragonal c r y s t a l s . Assume Hg = 0. Then I i s quantized with respect to the axis of symmetry of V E. Each eigenstate of our quadrupole i n t e r a c t i o n operator spin-state with eigenfunction 0 ^ say, and i f we r e s t r i c t ourselves to the case of h a l f - i n t e g r a l I , the I + 1/2 states w i l l be doubly degenerate, and <t>-m corresponding to the same energy l e v e l . This i s shown i n Figure I at the l e f t where H Q i s zero. Again for a l i n e a r l y or c i r c u l a r l y polarized Hi perpendicular to H Q , t r a n s i t i o n s occur only for Am =±1,giving a resonance spectrum of I - 1/2 l i n e s . For I = 5/2 the frequencies of the two l i n e s are i n the r a t i o 2*1. These frequencies are independent of the ori e n t a t i o n of the c r y s t a l with respect to H]_. Proceeding to the case where both interactions are present,the theory i s simplest f o r the case when the axis of symmetry of y T i s l i n e d up with H Q . Then the two 13. axes of quantization coincide and the same eigenfunctions diagonalize both and . The t o t a l energy i s simply the sum of the i n d i v i d u a l energies as shown by the s o l i d l i n e s i n Figure I , but the differences are no longer equal. Since the eigenstates are s t i l l pure states, under the above mentioned conditions on the selection r u l e Am =±1 s t i l l permits t r a n s i t i o n s between adjacent l e v e l s only. But since they are of d i f f e r e n t frequencies i n the general case the resonance spectrum w i l l consist of 2 I l i n e s . The s i t u a t i o n i s not so straightforward i f either one (or both) of the following complications i s present. (a) V l i s a x i a l l y symmetric but t h i s axis i s not l i n e d up with H 0. (b) V S has no axis of symmetry. In either of these cases the eigenfunctions of the magnetic i n t e r a c t i o n £t are not eigenfunctions 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 3* and vice versa. We must seek the eigenfunctions of the sum : which w i l l not be eigenfunctions of either 3* or ^ taken separately. We can d i s t i n g u i s h three cases according to the experimental s i t u a t i o n . I f 3^^^/we st a r t with the eigen-functions of *H and treat the 3* operator as a pertur-bation. I f # « 7 w e s t a r t with the eigenfunctions of and treat the operator as a perturbation. I f 5^ and 14 'ft are of the same order of magnitude, perturbation theory i s not v a l i d and we have to e x p l i c i t l y f i n d the actual eigenfunctions o f ^ - f - ^ . In the laboratory a l -though we have no control over 5^ once a p a r t i c u l a r c r y s t a l has been chosen we have some control over -ft since we can vary H Q , and hence i n some cases we may be able to examine the spectra of a p a r t i c u l a r nuclear species i n a p a r t i c u l a r c r y s t a l i n a l l three regions. Chapter 2 - Summary of theory as applicable to the case We f i r s t introduce complication (a) of having a V E of a x i a l symmetry with the symmetry axis not l i n e d up along H Q . In t h i s case,each eigenfunction perturbed by the assumed small e l e c t r o s t a t i c i n t e r a c t i o n 5 ^ , i s la r g e l y the corresponding ^„ appropriate to fit (quantized with respect to H Q as a x i s ) , plus a small admixture of a l l the other states i . e . ^ ~ 2 E L where the c o e f f i c i e n t s cm are calculable functions of the angle between the axis of symmetry of V B and the quantization a x i s . In the more general case of 0, the eigenfunctions are s t i l l l a rgely the corresponding V^, appropriate to ^ plus a small admixture of the other states,but now the co e f f i c i e n t s c m are functions of >^  and of the angles between the p r i n c i p a l axes and H Q. i f we f i x H 0 and rotate our single c r y s t a l about an axis perpendicular to Ho,we then vary these angles with resultant variations i n 1 5 . the cm's and i n the energy l e v e l s . I t i s from an analysis of t h i s v a r i a t i o n that we get information regarding rj , the orientation of V E with respect to the c r y s t a l axes and the magnitude of the quadrupole i n t e r a c t i o n energy. A detailed t h e o r e t i c a l treatment of t h i s case has been given by Volkoff (26) among others. The electro-s t a t i c i n t e r a c t i o n ( i ) given by the scalar product of the nuclear e l e c t r i c quadrupole moment tensor with the e l e c t r i c f i e l d gradient tensor i s regarded as a perturbation on the Zeeman energy (2) and i n (26) the ca l c u l a t i o n Is carried to the t h i r d order using standard perturbation theory. The frequency ^JrT fea. " \.-±\/h o f t l l e m " ^ m " 1 t r a n s i t i o n i s given by: with = ± \(2m - 1 ) ^ + £ P,(m) ± ^(2m - 1) P3(m) + 0(X4) ( 3 ) 6C[ o o P2(m) = dim) P3(m) = ki(m) (V£)4:2 eq (V£)ii — C i O ) (YE)', ±1 eq (4) eq '(V-E)S eq — ki(m) ( V £ ) ± 2 eq (VE)'o eq h(m) (V£)V!2 (VE)i, + (VE)'J (V£)V2 ( 5 ) V6 (eq)3 In equations ( 3 ) - (5) the primes on the components of U E indicate that these are expressed i n terms of the laboratory 16 t t * , coordinate system x, y , z i n which z coincides with H . o I f eq= ^ represents one of the eigenvalues of the tensor then the other two eigenvalues may be written as 0VX = -eq(l -*|)/2 and ^ = ~eq(l + -*| )/2. I f the value of the asymmetry parameter \ i s r e s t r i c t e d to the i n t e r v a l 0 ^ ^ .^ 1, then 4>2>f, are arranged i n order of i n -creasing absolute values. However, the value of »^  need not be so r e s t r i c t e d , and i n that case eq i s not necessarily the largest eigenvalue, but merely the eigenvalue which corresponds to that p r i n c i p a l axis which has been a r b i t r a r i l y designated as the z axis. Three quadrupole coupling constants can be defined; with corresponding expressions for and C y. The un-perturbed Larmor frequency for the nucleus i s given by >^o= l/«|H./hl - • J Y | H . / C T T (7) Experimentally t h i s frequency may be defined as the resonance frequency for the nucleus i n question at a s i t e where the time average of the e l e c t r i c f i e l d gradient vanishes, as i s normally the case i n a l i q u i d . The dimensionless para-meter X , which i s a measure of the importance of the quadrupole coupling compared to the separation of the 1 7 . Zeeman l e v e l s , i s defined by: The c o e f f i c i e n t s c^m) and k-jjm) i n equations (4), (j?) are given by: ci(w) = 4[(/ + !-)(/ - i ) - 6(w - i ) 2 ] , c2(m) = 2 K / - + -|)(/ - i ) - 3(m - | ) 2 ] , = 12/(7 + 1) - 40m (m - 1) - 27, ( 9 } = 2[3/(7 + 1) - 5m (m - 1) - 6], h(m) = 3[8/(/ + 1) - 20m (w - 1) - .1.5]. The signs of the f i r s t and t h i r d order terms of equation ( 3 ) are both to be chosen ( i n the case of H Q coinciding with the + z f axis) opposite to the sign of the nuclear gyromagnetic r a t i o Y" . As may be seen from equations (9) 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 tran-s i t i o n , the c i(m) and k^m) remain unchanged. The sign of the second order term i n equation (3) i s therefore the same for both these t r a n s i t i o n s , while the signs of the f i r s t and the t h i r d order terms are reversed when m i s replaced by -(m - 1 ) . However, since experimentally there seems to be no way of distinguishing which one of these two t r a n s i t i o n s one i s dealing with (except, i n p r i n c i p l e at l e a s t , by the use of extremely low temperatures), the sign of \ , and therefore of the quadrupole coupling constant C g, w i l l remain undetermined. In the case of h a l f i n t e g r a l I the resonance l i n e 8 18. has a central component 1/2*-+ - 1/2, and I - 1/2 pairs of s a t e l l i t e s m-*—— m - 1 and -(m - 1 )•«—»-- m with I ^ - m> 1/2. Such a pair s h a l l be i d e n t i f i e d by st a t i n g the largest of the four m values involved. In the case of i n t e g r a l I there i s no central component, but there are I pairs of s a t e l l i t e s with I ^  m > 0. The frequency difference between the two o members of each pair of s a t e l l i t e s i s given by: tV- v ' l - z o » A ( W . + 4 * P , W + 0 ( \ 4 ) ] no) The displacement with respect to the unperturbed Larmor frequency v>0 of the center of gravity V m = ( v' + \>")m/2 of each such pair of s a t e l l i t e s , or of the central component, i s given by: v r o-v e = i\ 4v.p 2y+oa 4) • in) As i n Ref. (18) we introduce an a r b i t r a r y rec-tangular set of axes X, Y, Z fi x e d with respect to the c r y s t a l , and study the r e l a t i o n between the components of V E i n t h i s system of axes, and i n the laboratory system of axes x 1 , y 1, z 1 , as the c r y s t a l i s rotated about, say,its X axis which i s kept i n coincidence with the y' axis per-pendicular to H Q. The i n i t i a l p o s i t i o n of the c r y s t a l , i n which the angle of rot a t i o n i s zero, i s chosen so that Y, Z coincide with z 1 , x T . This transformation may, of 19. course, be obtained from the general theory of the rep-resentations of the r o t a t i o n group as indicated by Bersohn ( 6 ) . However, the r e s u l t s may also be re a d i l y obtained i n an elementary way by di r e c t substitution i n terms of Cartesian coordinates following the procedure of equations ( 7 ) , ( 8 ) of Ref. ( 1 8 ) . The transformation i s then given by: , (VE)'0 = U<fiyy+ <t>zz) + \(4>YY - 4>zz) cos 2dx - \4>yz sin 2dx, (v£)ii = — [ =F {<j>YZ cos 20x + i(<£y <j>zz) sin 26x} - i {<t>xr cos dx — 4>zx sin 0Y} L (12) -Y + <\>zz) — 7,{<I>YY — <t>zz) cos 29x + (j>Yz sin 26x ± 2f(</>zx cos Ox + 4>XY sin 0X) Combination of the f i r s t of equations (12) with the f i r s t order term of equation (10) leads to the following ex-pression: ("' — v")x = \{2m- 1) (flx + bx cos 28x + cx sin 20x) = \{2m - l)[ax + Rx cos 2(dx - 5X)] ( 1 3 ) with = |(2w - l ) [ i 0 + i i 5 x ] ax = i 0 + | i i = va\((j)YY + 4>zz)/eq, bx = Rx cos 28x = VO\(4>YY — <t>zz)/eq, Cx = sin 25 x = — 2voX<f>Yz/eq, Rx = iii> Sx = cos2 (0X - 5X) • (14) We may always choose i n the range -1/2 T?<.£x^-|r, so as to make R x ^ 0. A simultaneous reversal of sign of a x , b x, c x and a change of by ^/2 leaves R^- ^  0. Similar expressions for the Y and Z rotations may be ob-tained by c y c l i c permutation of subscripts. The quantities 2 0 a, b, c, R,& , L Q , L I , are expected to be independent of m and H Q , and can be obtained from a Fourier analysis of the experimental curve of v>'- V " against 0 . Substitution of the l a s t two equations ( 1 2 ) into equations ( 1 1 ) and (4) gives the extension of the above theory to the second order i n X • The second order contributes nothing new to the s p l i t t i n g of each pair of s a t e l l i t e s , but gives a s h i f t of the center of gravity sJtn of each pair and of the central component. Thus, for the X r o t a t i o n : (vm — v0)x = nx + px cos 26x + rx sin 29x + ux cos 4BX + vx sin 46x ( 1 5 ) = nx + Px cos 2(6X - yx) + Ux cos 4(0* - 5X) Similar expressions for Y and Z rotations may be obtained by c y c l i c permutation of the subscripts. The c o e f f i c i e n t s n, p, r , u, v, P, U are expected to be inversely proportion-a l to H Q , and to depend on m, and can be obtained from a Fourier analysis of the experimental curve of v>m~ v>a against 6 . For a r o t a t i o n axis p a r a l l e l to a p r i n c i p a l axis Y = £ holds, and specifies the orientation of a second p r i n c i p a l axis r e l a t i v e to 0 = 0 . For a r o t a t i o n axis perpendicular to a p r i n c i p a l axis V = S = 0 i f 6 i s measured from the p o s i t i o n i n which the known p r i n c i p a l axis i s either p a r a l l e l or perpendicular to H Q . In both these cases of Y= & equation ( 1 5 ) may be rewritten i n the form i v - v, = A'„ 4- KlS + K2s- ( 1 6 ) 21. with s s. cos 2(Q S ) and Kj_ related to n, P, U of equation (15) by: « = A"„ + i A \ + f K: P = H^i + K2), 2, (17) In a good c r y s t a l , i n which a l l the 21 components of a resonance l i n e are c l e a r l y v i s i b l e , the nuclear spin I can be obtained simply by counting the l i n e components. However, as w i l l be explained i n Chapter 7, c r y s t a l imper-fections and impurities may so broaden and weaken the s a t e l l i t e s that either none at a l l are v i s i b l e , or the outer ones are so blurred and weak, that i n counting the number of l i n e components one can not be ce r t a i n that none have been missed. Nevertheless, as long as the second order frequency s h i f t of the central component and the frequency difference of only one pair of s a t e l l i t e s (the strongest innermost pair) can be measured, the nuclear spin can be determined from a r o t a t i o n about any a r b i t r a r y axis by combining the experimental values of U from equation (15) for the central component (m = 1/2), and of R from equation (15) for the innermost pair of s a t e l l i t e s (m « 3/2) i n the expression (/ + - ! ) ( / - 4) = 32vaU/R2. (18) Here \>0 i s the unperturbed Larmor frequency corresponding 22 to the value of H Q at which U has been measured. The product \>„ U should be independent of H Q. I f the orientation of some one of the p r i n c i p a l axes of cfi^j i s known from c r y s t a l symmetry we can use i t as the X r o t a t i o n a x i s , and choose Y and Z r o t a t i o n axes i n the plane perpendicular to i t . We choose the X and Y axes to coincide respectively with the b and c axes of spodumene. I f we do not r e s t r i c t the value of the asymmetry parameter = -fiyy)/^ to the range 0 1 we may name the p r i n c i p a l axes at w i l l . We s h a l l choose the p r i n c i p a l axis coincident with the X rot a t i o n axis as the x p r i n c i p a l axis corresponding to the eigenvalue ^ x x = -1/2(1 - *^  ) S^JJ . The y and z p r i n c i p a l axes must then l i e i n the YZ plane. Let £ denote the angle between the Y r o t a t i o n axis and the y p r i n c i p a l axis corresponding to the eigenvalue jfi^ = -1/2(1 + \ ) • A positive S w i l l mean that a positive r o t a t i o n of the c r y s t a l about the X axis w i l l bring the yz axes into the position o r i g i n a l l y occupied by the YZ axes.. To analyze the ex-perimental data completely we must determine % , *^  , and the absolute value of the Quadrupole coupling constant referred to the z axis ( C z | s \e%<f>n /h| . For I = 5/2 the l a t t e r two quantities are related to a x and R x of equation (13) by: 1 (19) 2 3 . Since the T and Z axes are known to be perpendicular to a p r i n c i p a l axis, cy = = 0 andSy = ^ = °* I f s a t e l l i t e s are v i s i b l e , then ay, by = Ry, a^> ^z = are obtainable d i r e c t l y from an analysis of the experimental curves of the form of equation ( 1 3 ) , . a n d a check on a-^  and b-^  = R^ c o s 2 ^ may be obtained with the aid of the i d e n t i t i e s : aX = "2 ( V a y ) = T ^ V V • I, i ( 2 0 ) b x = - ^ a v + by) = l - ( 3 a z - b z ) I f IC z | , , and J are known from one ro t a t i o n , the values of V- \>" and "0 - for other rotations are easily computed. The following formulae for the case of I = 5 / 2 . are quoted since they w i l l be required i n Chapter 6 . The f i r s t order frequency s p l i t t i n g for the X, Y rotations are given by the l a s t of equations ( 1 3 ) with the c o e f f i c i e n t s L Q, determined by: X: L 0 - -^°z > L l =-TcPz^ + n > ( 2 1 ) Y: L 0 = - | ( l - \ ) C z , L x = ^ [ ( 3 * n > o s 2 S - 2 \ K The second order frequency s h i f t s of the inner s a t e l l i t e s for the Y ro t a t i o n i s given by equation (16) with the 24. c o e f f i c i e n t s as follows: K>= f 6 + «|)2 C| /8oo V0 K, =-|[2 t('5+r^) + i3 6 + v l ^ - v v ) c o s 2 ^ ] C|/8oo9 0 (22) The t h i r d order contribution to the frequency s p l i t t i n g for the r o t a t i o n i s given for the "inner" s a t e l l i t e s (m = 3/2) by: 9 C, 2000 11 2 , , ,288 + 402T? 4- 113r?2 ~~ 16 V + 5 x ( ~ + ^ - sx 2(3 + r,) 96 251 + 44r) 64 (23) + %(3 + v) ; 385 384. and for the "outer" s a t e l l i t e s (m = 5/2) by: 1000 K O I 16 rj - Sx(3 + i?) 432 4- 258T/ 4- 37r? 4- 5X2(3 4- ij) 96 2 279 4- 86?? 64 3/0 , ,3 365 s* (3 4- n) g ^ j (24) Equations (21), (23), (24) give the correct r e l a t i v e sign of the f i r s t and t h i r d order contributions to s)" , but the absolute sign w i l l remain undetermined. 2 5 . Chapter 3 - Theory for the pure quadrupole case (Hp = 0 ) . The quadrupole i n t e r a c t i o n operator given i n equation ( 1 ) can be reduced to the following Hamiltonian ( l 3 ) where x, y, and z are the p r i n c i p a l axes of <&+j , I i s the angular momentum operator and Q i s the scalar nuclear quadrupole moment defined i n the conventional manner ( 2 7 ) « Q - ( i l l E e r i ' f r c o ^ f l . . - ( 2 6 ) In the case of h a l f - i n t e g r a l I the secular determinant can be factored into two determinants each of which leads to a c h a r a c t e r i s t i c equation of degree I + 1/2 ( 21 ' ) • The eigenstates defined by each of these sub-determinants are l i n e a r combinations of one of two subsets into which the functions which diagonalize 1^ may be subdivided. One set, L say, includes Hn with m = 1/2 + 2 n and the other, M say, includes Yin with m =-l/2 + 2 n , where n i s an integer. The eigenvalues are s t i l l doubly-degenerate as T 1 1 1 1 1 1 1 r I I I l I ' i ' i i 'I 0 0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2 3.6 4.0 F I G . Z . Energy values for / = 5/2, tj — 0.95 in terms of X= E/eQ<j>zz as a function of R^ iiiH<,/eQ<t>zz, for 0 < R < 4. Ho is along the z principal axis of <j>ij. The arrows represent the possible absorption lines, and are numbered from 1 to 9, the first five being the Zeeman transitions for R^> 1. to face page 2 6 F i g . 3. Transition frequencies V ( i n Mc./sec.) as a function of R (or of H 0 = ReQ03^/4/* ). Lines have been la b e l l e d intense or weak i n accordance with r e l a t i v e values of the squares of matrix elements being greater or smaller than unity on the a r b i t r a r y scale used i n Fig. 4. to face page 2 6 Fig. 4. Squares of the matrix elements of the perturbing operator ( i n ar b i t r a r y units) as a function of R = 4/*Ho/ eQ, 4>n to face page 26 2 6 . for = 0 . For I = 5/2 the secular equation r e s u l t i n g from (27) i s X a - T O ( 7 y \ 2 + z ' ) ^ - ^ o ( l - \ Z ) £ 0 ( 2 8 ) where A E/eQ <j>y^ . When and the quadrupole coupling constant are known from magnetic resonance spectra, the roots of t h i s equation w i l l give the pure quadrupole energy l e v e l s . Lamarche ( 2 8 ) gives approximate expressions for the roots of equation ( 2 8 ) for small q . For = 0 . 9 4 as at the A l ^ s i t e s i n spodumene, one root i s negative and very nearly zero and the other two, one posi t i v e and one negative, are nearly equal i n absolute value giving t r a n s i t i o n frequencies of .758 MC/sec. and . 7 8 9 MC/sec. For small Townes and Dailey ( 2 9 ) give 2 ( 1 - 1 . 3 0 m£ ) as the approximate expression for the r a t i o of the two frequencies. Another feature d i f f e r i n g from the = 0 case i s that due to the eigenstates being admixtures of states there i s a f i n i t e p r o b a b i l i t y of a t r a n s i t i o n occurring at the sum of these frequencies. Figures 2 and 3 taken from Ref. (21) represent the dependence of the energy l e v e l s and the t r a n s i t i o n frequencies on the magnetic f i e l d H Q . Figure 4 i s a revised version of the corresponding figure of Ref. ( 2 1 ) . The s i t u a t i o n des-cribed above f o r the sero f i e l d case corresponds to the 2 7 . l e f t hand side of each of these Figures. In Figure 2 , note that the v e r t i c a l scale represents energy i n units of the quadrupole coupling constant times h,while the horizontal scale i s the r a t i o of the magnetic i n t e r a c t i o n energy yuH0 to the quadrupole i n t e r a c t i o n energy e Q j ^ j / 4 . Recent unpublished calculations by Lamarche & Vblkoff have shown that,while the t r a n s i t i o n frequencies are independent of the o r i e n t a t i o n of the r . f . f i e l d H ^ with respect to the c r y s t a l , the t r a n s i t i o n p r o b a b i l i t i e s are strongly dependent on t h i s orientation. In the case of A l 2 ? i n spodumene the r e l a t i v e t r a n s i t i o n p r o b a b i l i t i e s for the . 758 MC/sec. l i n e for along the x, y and z p r i n c i p a l axes of the <fi^j tensor at the A l s i t e s are i n the r a t i o of 70 : 25 : 5» Chapter 4 - Theory applicable to the region where In Ref. (21) Lamarche & vblkoff report the r e s u l t of some calculations for the case of A l 2 ^ i n spodumene. These calculations include the case H Q = 0 and the p a r t i c -ular case of H Q coinciding with the ^ p r i n c i p a l axis of the 9^^- tensor f o r a l l values of H Q from zero up to the region where perturbation theory based on ^ < < ^ becomes 2 8 . accurate. This c a l c u l a t i o n involved the solving of the secular determinant ( f y + - t f ) w m . - E s , w | = o ( 2 9 ) for the energy l e v e l s and the computing of the frequencies and the r e l a t i v e t r a n s i t i o n p r o b a b i l i t i e s between them. The r e s u l t s are shown i n Figures 2 , 3 and 4 . The c l a s s -i f i c a t i o n of eigenfunctions described i n Chapter 3 p e r s i s t s when i s applied along the z p r i n c i p a l axis of VE,each L class eigenfunction now becoming a l i n e a r combination of , y£ ahc/ , and each M class eigenfunction a l i n e a r combination of % and 9^ . The c o e f f i c i e n t s i n each case are functions of HQ,and the character of some of the eigenstates changes markedly as the f i e l d i s varied. I Ml For example i n Figure 2 , while A", maintains mostly the character of {ff~% throughout, \\ which has mainly the character of at high f i e l d s , changes to mainly V^-5/2 i n the region of R = 0 . 8 to 1 .8 and then becomes c h i e f l y ^ + 3 / 2 as zero f i e l d i s approached. Note that the energy l e v e l s corresponding to eigenstates of d i f f e r e n t classes do cross but those corresponding to eigenstates of the same class do not cross. TheAm = ±l r u l e now indicates that t r a n s i t i o n s can occur between any pai r of l e v e l s belong-ing to dif f e r e n t classes. The resultant t r a n s i t i o n frequencies 29 are plotted ^ s-R i n Figure 3. At high f i e l d s the eigenstates are c h i e f l y those appropriate to (quantized with respect to H Q as axis) and only the f i v e t r a n s i t i o n s designated \ V to ~d& have s u f f i c i e n t p r o b a b i l i t y to make the resultant spectral l i n e s observable. But as the r a t i o R gets below 3 the percentage of the other spin functions i n the l i n e a r combination characterizing the states becomes appreciable. The r e l a t i v e t r a n s i t i o n p r o b a b i l i t i e s for the other four possible t r a n s i t i o n s , l a b e l l e d V 6 to y)Q , shown i n Figure 4 indicate that they should be observable i n c e r t a i n regions. The effect,on the observed nuclear resonance spectrum,of r o t a t i n g the c r y s t a l at zero magnetic f i e l d and at high magnetic f i e l d s has been described. In the region where , rot a t i o n of the c r y s t a l so that H Q i s not along a p r i n c i p a l axis of 7^ S w i l l i n general r e s u l t i n the eigenfunctions belonging to the various l e v e l s be-coming l i n e a r combinations of a l l the spin functions. Then i n p r i n c i p l e t r a n s i t i o n s can occur between any pair of level s and a t o t a l of 15 combinations are possible for the I = 5/2 case. Many of these w i l l not be observable because of low frequencies or low t r a n s i t i o n p r o b a b i l i t i e s . 3 0 . PART I I - EXPERIMENTAL  Extension of Results i n Spodumene at high Magnetic F i e l d s . Chapter 5 - Apparatus and Experimental Procedure. The recording nuclear magnetic resonance spectrometer, the electromagnet, and the c r y s t a l mount used i n the work described i n t h i s part of the thesis have been described f u l l y i n Refs. ( 1 8 , 3 0 ) . The two c r y s t a l samples of L i A l ( S i 0 ^ ) 2 (spodumene) used have also been described e l s e -where ( 2 3 ) . One, designated 2Y, has a volume of about 1 .3 c.c. and was rotated about the c r y s t a l " c n axis which co-incides with the a r b i t r a r i l y chosen Y axis mentioned i n Chapter 2 . This c r y s t a l was oriented with respect to the holder by means of the experimental curve obtained f o r the s p l i t t i n g of the inner s a t e l l i t e s . This was adjusted to make 9y = 0 correspond to the pos i t i o n of maximum s p l i t t i n g . The other, designated 3X, has a volume of about 1 . 5 c.c. and was rotated about the c r y s t a l b axis which coincided with the a r b i t r a r i l y chosen X axis mentioned i n Chapter 2 . This c r y s t a l was oriented with respect to the holder using two well-defined cleavage planes. Their i n t e r s e c t i o n defined the c axis and the bisector of the angle between these planes, l y i n g perpendicular to the c axis, defined the b axis. FIG. 5 . Projection of a unit cel of monoclinic spodumene on the (010) plane. The b axis, which is perpendicular to the page, was used as the X rotation axis, and coincides with the y and x" principal axes of the field gradient tensors fcj at the Li and Al sites respectively. The Y and Z rotation axes, and the x, z and y", z" principal axes of 4>ij at the Li and Al sites respectively are shown in the diagram. to face page c+ O to PIATS I. Color photograph of a 2 - dimensional model of the L i A K s i O ^ U (spodumene) J« unit c e l l . A l , L i , S i and 0 nuclei are represented respectively by the CD red, blue, s i l v e r and yellow b a l l s . On the central A l and L i nu c l e i the yellow l i n e s indicate the directions of the y and z p r i n c i p a l axes of the £J f i e l d gradient tensor at each of these s i t e s as determined"experimentally i n t h i s laboratory. A projection of the structure of spodumene on the (010) plane i s given i n F i g . 5. A detailed description of the structure i s given i n Ref. (18). A color photograph of a 3-dimensional model of the unit c e l l i s shown i n Plate I . The L i , A l , S i and 0 nuclei are represented respectively by the blue, red, s i l v e r and yellow b a l l s . Frequency measurements were made using^either a General Radio 6 2 0 A Frequency Meter or a BC - 2 2 1 A Frequency Meter, and a communications receiver. Regular frequency markers were made on the chart by zero beating the spec-trometer o s c i l l a t o r with the frequency meter. Frequencies corresponding to zero derivative were then measured d i r e c t l y from the charts using l i n e a r i n t e r p o l a t i o n between these markers. Where a knowledge of Vo was essential as i n the — N) 0 curves, -\)0 was checked p e r i o d i c a l l y by removing the c r y s t a l and i n s e r t i n g a test tube of aluminum chloride solution. In the experimental r e s u l t s i n section A and B below i t i s estimated that the i n d i v i d u a l frequencies could be measured to 1 1 KC/sec. and the angular positions of the c r y s t a l sample to ± 1 / 2 ° r e l a t i v e to an ar b i t r a r y zero on the c r y s t a l mount scale whose pos i t i o n with respect to the magnetic f i e l d could i n turn be determined to - 1°. FIG. 6. Selected traces of the recorded derivative of the Al27 absorption line in spodumene sample No. 3X at Ho = 6130 gauss corresponding to v0 = 6.806 Mc./sec. (a) $x = 86°. Sweep rate is approximately 54 kc./sec. per division. The lines and their approximate fre-quencies in Mc./sec. are from right to left: "outer" satelite of Al27 line 6.458, "inner" satelite 6.585, central component 6.769, Cu83 line 6.930, second "inner" satelite 7.009, second "outer" satelite 7.307, Cu65 line 7.420. The Cu lines are not symmetric because some dispersion mode is mixed in with the absorption mode owing to the phase shift of the Hi r-f. field penetrating the metal of the oscilator coil. The more abundant Cu63 isotope gives the stronger line. (b) 8X = 100.2°. Sweep rate is approximately 27 kc./sec. per division. The lines and their approximate frequencies in Mc./sec. are from right to left: central component of Al27-line 6.756, the two coalesced "inner" satelites 6.794, the two coalesced "outer" satelites 6.903, Cu63 line 6.930. (c) 0X = 100.5°. Same as b, except that both pairs of coalesced satelites have begun to split up. to face page 32 3 2 . Chapter 6 - Experimental Results and Calculations. 27 A. Inner s a t e l l i t e s of A l i n spodumene. The r e s u l t s on the inner s a t e l l i t e s i n conjunction with Fetch*s ( 3 1 ) r e s u l t s on the A l 2 ^ central l i n e were recently published ( 2 3 ) . They are presented below much as they appeared i n Ref. ( 2 3 ) . In making the f i n a l calculations for the constants of spodumene the best experimental values were chosen from both sets of r e s u l t s . In reproducing t h i s material those parts of Petch's r e s u l t s incorporated i n the calculations are necessarily included. The following t h e o r e t i c a l conclusions were v e r i f i e d . (a) I f , and only i f , second and higher order effects are negl i g i b l e a c r y s t a l p o s i t i o n can be found for which a l l the l i n e components simultaneously coalesce at the un-27 perturbed Larmor frequency \>6 • In the A l l i n e i n spodumene the components do not a l l coalesce for any c r y s t a l p o s i t i o n , i n d i c a t i n g an appreciable second order e f f e c t . (b) In the absence of measurable t h i r d order e f f e c t s , but i r r e s p e c t i v e l y of the presence or absence of second order e f f e c t s , the f i r s t order s p l i t t i n g V'— \>" between two members of a pair of s a t e l l i t e s should vanish i n a c r y s t a l p o s ition which i s the same f o r a l l s a t e l l i t e pairs for a given l i n e , and which i s independent of H Q . F i g . 6 shows representative traces obtained with sample No. 3 2 , described i n Chapter 5 , i n three positions i n a magnetic f i e l d of 3 3 . H Q - 6130 gauss, calculated from the Icnown gyromagnetic r a t i o for A l ^ and the measured value of \)0= 6 . 8 0 6 MC/sec. for A l 2 ? i n aluminum chloride i n solution. At Qy= 8 6 " the f i v e components of the Al^ l i n e are widely separated and are c l e a r l y v i s i b l e i n addition to the two l i n e s due to C u ^ and C u ^ i n the o s c i l l a t o r c o i l winding. As the c r y s t a l i s rotated about i t s b axis (chosen as the X rotati o n axis) the copper l i n e s remain at t h e i r f i x e d 27 frequencies, while the components of the A l l i n e gradually change t h e i r p o s i t i o n . By the t i m e 0 x = 100.2 has been reached the lower frequency member of each pai r or s a t e l l i t e s has crossed over to the high frequency side of the central component, and has coalesced with the other member of the same pair as shown i n F i g . 6 b . The coalesced pairs l i e at frequencies recorded i n Table 1 . TABLE I . __. DE P E N D E N C E O N T H E M A G N E T I C F I E L D HO O F T H E F R E Q U E N C Y SHIFT V — vo O F T H E C E N T R A L j C O M P O N E N T A N D O F T H E C E N T E R S O F GRAVITY V — vo P $ ( » ' + v") — Vo O F T H E " I N N E R " A N D " O U T E ' R " S A T E L L I T E S O F T H E Al" L I N E IN S A M P L E No. 3X IN T H E POSITION 6X = 100" I N W H I C H T H E SPLITTING v' — v" O F T H E S A T E L L I T E PAIRS VANISHES* "0, Mc./ sec. Ho, gauss lAo, rela-tive Central component (m = « "Inner" satellites (m = 3/2) "Outer" satellites (m = 5/2) Mc../ sec. v — vo, Mc./ sec. V — Vo, rela-tive v' = v" = V, Mc./ sec. V — vo, Mc./ sec. V — Vo, rela-tive v' = v" = V, Mc./ sec. V — VO, Mc./ sec. v — vo, rela-tive 3.662 3300 1.000 3.570 -.092 ± . 0 0 1 1.000 3.643 -.019 ± . 0 0 1 1.000 3.843 .181 ± . 0 0 1 1.000 6.806 6130 0.539 6.756 -.050 ± . 0 0 1 0.54 ± . 0 1 6.794 -.012 ± . 0 0 1 0.63 ± . 0 6 6.903 .097 ± . 0 0 1 0.535 ± . 0 0 5 9.884 8903 0.371 9.849---.035 ± . 0 0 1 0.38 ± . 0 1 - -'9,876 -.008, ± . 0 0 1 0.42 ± . 0 5 9.952 .068 ± . 0 0 1 0.376 ± . 0 0 5 A further small r o t a t i o n of the c r y s t a l to 1 0 0 . 5 already produces a noticeable tendency of both coalesced 34. s a t e l l i t e pairs to s p l i t as shown i n F i g 6c. Thus the position of zero s p l i t t i n g can be found with considerable accuracy, and as expected turns out to be the same for both the "inner" (m = 3 / 2 ) and the "outer" (m = 3 / 2 ) s a t e l l i t e pairs. The magnetic f i e l d was then changed to H Q = 3300 gauss and H Q = 8903 gauss, corresponding to V 0 = 3 . 6 6 2 Mc./sec. and v>6 = 9.884 Mc./sec. respectively. The position of zero s p l i t t i n g was determined for each of the three values of H Q used, and was found to d i f f e r by not more than 1 / 2 between the two extreme values of H 0 used. This indicates that t h i r d order effects must be quite small at t h i s c r y s t a l p o s i t i o n . (c) In the absence of measurable fourth order effects the magnitude of the second order frequency s h i f t x> - \)a of the central component, or of the center of gravity of a pa i r of s a t e l l i t e s , should be inversely proportional to H q for a given c r y s t a l p o s i t i o n , and for a given l i n e component should have the same form of dependence on the angular p o s i t i o n of the c r y s t a l independently of H . Sample No. was used to obtain the data recorded i n Table I corresponding to the one po s i t i o n i n which the f i r s t order s p l i t t i n g vanished for both s a t e l l i t e p a i r s , so that >j'= "v/7 = i n both cases. In a l l cases - V a i s inversely proportional to \/0 within experimental error. 135 180 225 DEGREES 360 FIG. 7. Dependence on the angular position of spodumene sample 2Y of the frequency diference v — v" of the "inner" satelites of the Al27 line for the Y rotation about the crystal c axis. 8r = 0 corresponds to the position of the crystal in which the b axis is at 90° to the magnetic field H0. H0 = 5862 gauss, corresponding to vo = 6.508 Mc./sec. "T35 180 225" DEGREES FIG. Q. Dependence on the angular position of spodumene sample 2Y of the frequency shift v — va — i(V + v") — vo of the center of gravity of the "inner" satelites of the Al27 line for the Y rotation about the crystal c axis. 9Y = 0 corresponds to the position of the crystal in which the b axis is at 90° to the magnetic field Ho. Ho = 5862 gauss, corresponding to vo = 6.508 Mc./sec. to face page J>5 35 Figs. 7 and 8 represent respectively the ex-perimental values of \>- v>" and V - v) 0 = l/2( v>") -v>0 27 for the "inner" (m = 3/2) s a t e l l i t e s of the A l l i n e as a function of 9 y for the T ro t a t i o n of sample No. 2T. The magnetic f i e l d H 0 was held constant at approximately 5862 gauss corresponding to the observed Larmor frequency . for A l 2 ? i n aluminum chloride of 00 = 6.508 MC/sec. The zero of the Q y scale should be chosen at that p o s i t i o n of the c r y s t a l i n which the Z(or b) axis i s perpendicular to H Q . This p o s i t i o n was accurately located by making the observed >)" curve of F i g . 7 symmetrical about t h i s point. This same c r y s t a l p o sition was also chosen for 6Y = 0 for the "0 - >?o curve of F i g . 8, Since the accuracy of each experimental point i s i 1 k c . / s e c , and the t o t a l v a r i a t i o n of the curve i s le s s than 30 k c . / s e c , any seeming lack of symmetry of t h i s curve about 6 = 0* 90°, l80° etc., i s not s i g n i f i c a n t . The alignment of the ro t a t i o n axis at ri g h t angles to the magnetic f i e l d was achieved by adjusting the c r y s t a l holder u n t i l the two peaks of the ->)'' curve of F i g . 7 separated by 180° became of the same height. A very s l i g h t t i l t of the rot a t i o n axis with respect to the magnetic f i e l d produced a difference of as much as 15 kc/sec. i n the height of the two peaks. A Fourier analysis of the curve of F i g . 7 was made up to the s i x t h harmonic. The amplitudes of the terms T i f i i i i i i i r J i i i i i i i i i i 0 -I -2 •» » S -6 -7 -8 9 10 S-COS'9 F i g . 9 . The points of Figs. 7 , 8 replotted as a function of S Y ECos 2Gy. Each point of F i g . 9 i s the average of four symmetrically situated points of Figs. 7 , 8 . The s o l i d straight l i n e A i s ( V* - v " ) y = 2 ^ . 7 + 2 9 0 . 2 s Y which represents an empirical f i t to the points of F i g . 9 r e s u l t i n g from a Fourier analysis of the curve o f _ F i g . 7 . The s o l i d curve B i s the t h e o r e t i c a l curve ( V - Vo )y = 1 6 . 2 - 2 8 . 7 s y - f 1 . 9 s y 2 calculated from the data ootained from curve A,from the central Ald' l i n e X rotation i n Ref. ( 2 3 ) , and from spodumene sample 3X. The experimental points l y i n g on curve B are uncertain by ± 2 kc./sec. to face page 3 6 3 6 . of periods predicted by the f i r s t of equations ( 1 3 ) for the Y rotations were: Cy = 0 by the choice of the o r i g i n , and a Y - 1 7 0 . 8 ± 0.5 k c . / s e c , by = = 145.1 ±r 0.5 kc./sec The r.m.s. value of the amplitudes of a l l the other harmonics was found to be 0.43 k c / s e c , which d i f f e r s from zero by less than the experimental error. Expressing t h i s i n the l i n e a r form of the l a s t of equations (13) we obtain L Q = 2 5 . 7 £ 1 k c / s e c , and 1 ^ = 290.2 ± 1 k c / s e c . The s o l i d l i n e A of Fig . 9 shows this straight l i n e which has been empirically f i t t e d to the experimental points of F i g . 7 replotted against s = c o s * 6 y . Each point of F i g . 9 i s the average of the four points at 1 6, ± ( Tp + 0 ) of Fig . 7 . The points of Fig . 8 are s i m i l a r l y replotted i n Fig. 9 . The experimental accuracy i n t h i s case does not j u s t i f y an attempt to obtain an independent empirical set of constants i n equation ( 1 6 ) to f i t these points. Instead of t h i s , the t h e o r e t i c a l l y expected values of these constants were calculated from the data obtained from the preceding curves, as described below. The s o l i d curve B of Fig . 9 i s t h i s calculated curve which shows that the observed points are consistent with i t . Some information on the "inner" (m = 3 / 2 ) s a t e l -l i t e s was also obtained from the X rota t i o n of sample No. 3X i n a f i e l d of H Q = 6 1 2 9 gauss corresponding to \) 0 = 6.804 Mc./sec Zero s p l i t t i n g was observed at 0. = 100°± l°and 3 7 . O x = 1 9 2 6 1 l ! - The expected positions for maximum s p l i t -t i n g are then midway at $ x = 146* and 90 from the l a t t e r p o s i t i o n at Q^= 2 3 6 . The observed s p l i t t i n g s i n these two positions were V- \>" = - ( 8 8 5 - 2 ) and + ( 8 5 6 ± 3 ) kc./sec. respectively. Neglecting at t h i s stage any t h i r d order e f f e c t s , whose possible influence w i l l be discussed at the conclusion of t h i s section, su b s t i t u t i o n of these data i n equation ( 1 3 ) gives: & = 5 6 ° ± 1 ° a^ . = -14.5 - 4 k c . / s e c , R-£ = 8 7 0 * 4 k c . / s e c , b^ = R^ . cos 2 X = - 3 2 6 ± 3 0 kc./sec Table I I l i s t s the values of a, b, R obtained d i r e c t l y from the Y and X rotations of cry s t a l s 2Y and 3X respectively, and also computed from the above observations with the aid of i d e n t i t i e s (20). I t i s seen that the a and b values obtained from the two rotations agree within experimental error. Sample 3X provides a value of R j , while sample 2Y provides better values of Ry and R^ than sample 3X. TABLE I I  V A L U E S OF a, b, R (IN K C . / S E C . ) IN E Q U A T I O N [13]REQUIRED TO FIT T H E OBSERVATIONS O N S A M P L E S 2 Y , 3 X . F O R S A M P L E 2 Y , ay, by W E R E O B T A I N E D F R O M A F O U R I E R ANALYSIS O F I ( „ ' — v")Y G I V E N B Y T H E POINTS O F F I G . "7 FOR T H E " I N N E R " S A T E L L I T E S . T H E R E M A I N I N G I COEFFICIENTS W E R E C A L C U L A T E D F R O M T H E IDENTITIES[20]. F O R S A M P L E 3 X , ax, Rx, A N D 5 = 5 6 ° ± 1° ' W E R E O B T A I N E D B Y O B S E R V I N G T H E POSITIONS A N D A M O U N T S O F M A X I M U M POSITIVE A N D N E G A T I V E SPLITTINGS (v' — v")x. bx WAS C O M P U T E D F R O M bx = Rx COS 25, A N D T H E O T H E R QUANTITIES F R O M T H E IDENTITIES[20] Sample 2 Y Sample 3 X Axis X Y Z X Y Z . C a l c Obs. Calc. Obs. Calc. Calc. a - 1 3 ± 1 0 171 ± 0 . 5 - 1 5 8 ± 1 0 - 1 4 . 5 ± 4 170 ± 15 - 1 5 6 ± 15 b - 3 2 9 ± 1 0 145 ± 0 . 5 184 ± 1 0 - 3 2 6 ± 3 0 142 ± 15 184 ± 15 R — 145 ± 0 . 5 184 ± 1 0 8 7 0 ± 4 142 ± 15 184 ± 15 5 8 . We now apply the theory of Chapter 2 to determine the absolute value of the quadrupole coupling constant ( C2|, the asymmetry parameter , and the p r i n c i p a l axes orien-t a t i o n of the 4>.-j tensor at the A l 2 ? s i t e s i n spodumene. TABLE' I I I S U M M A R Y O F ax, Rx, S, | Cz I , i? F R O M O B S E R V A T I O N S ON-.TWO S P O D U M E N E S A M P L E S I Sample. Line"' , component aX, . kc./sec. Rx, kc./sec. degrees \cz\, kc. /sec. V 3X "Inner" sat. 870 ± 4 - 56 ± 1 2947 ± 20 0.93 ± .02 2Y "Inner" sat. -13 ± i:o (870 ± 4) ' 56.1 ± .2 2943 ± 15 0.941 ± .005 The re s u l t s are collected i n Table I I I . In the f i r s t case values of & , a x and R x are already given i n Table I I and |c z I and are calculated using equations (19). In the second case the res u l t s from Table I I combined with R x from the f i r s t case give better values of a x , and of S v - 1 / 2 cos" 1 ( b y / R x ) = 5 6 . 1 1 0.2°. Again | c z 1 and i^are calculated using equations ( 1 9 ) . The best values are seen to be those from the l a s t row. These values were substituted into the c o e f f i c i e n t s of equation ( 1 6 ) given i n equations (22) to give curve B of figure 9 . The following expression was obtained i n which the frequency i s i n kc./sec. (<J_Va) y= ( l 6 . 3 ± 2 ) - (28-7**)s + (\ 9 l >l)s 2 ( 3 0 ) As mentioned above, the observed points are 39-consistent with t h i s curve B but re c a l c u l a t i o n of si m i l a r c o e f f i c i e n t s for the central l i n e Y and Z rotations i n -dicated the p o s s i b i l i t y of a systematic error i n the form of a misalignment of the rota t i o n axes i n the c r y s t a l samples. In the analysis i t i s assumed that the three r o t a t i o n axes are mutually orthogonal, whereas ac t u a l l y they could be s l i g h t l y t i l t e d with respect to t h e i r inten-o ded positions. In view of t h i s , the estimated l i m i t s of error are increased and the values f i n a l l y adopted from Table I I I are: | Cz I = 2950 =fc 20 kc./sec., r, = 0.94 ± - 0 . 0 1 , 5 = 56° ± -§°. ( 31) We conclude t h i s section with some remarks on the size of the t h i r d order effects r e l a t i v e to the f i r s t order effects i n the X r o t a t i o n of c r y s t a l 3 X . Sub-s t i t u t i n g into equations ( 1 3 ) , ( 2 1 ) , ( 2 3 ) , (24) the constants of equation ( 3 1 ) , and V0=- 6.804 Mc./sec., corresponding to H Q = 6129 gauss used to obtain the data of Table I at zero s p l i t t i n g of both pairs of s a t e l l i t e s , and the data leading to part of Table I I at maximum s p l i t t i n g of the "inner" s a t e l l i t e s , we obtain the following expressions for v'- \)" ( i n kc./sec.) i n which the f i r s t and t h i r d order contributions are bracketed separately. For the 40 inner s a t e l l i t e s we have: - v")x = (885 - 1753s*) + (-1.53 + 79.1s* - 236s*2 + 155s*3) ; ( 3 2 ) — — . I and for the outer s a t e l l i t e s : j ( / - v")x = 2(885 - 17535*) + (-0.3 - 1465* + 440s*2 - 293s*3). (33) The t h i r d order contributions i n equations (32) and (33) have greatest deviations from zero near s x =? 0.2 and sx ^ 0.8 amounting respectively to about +6 kc./sec. and -10 kc./sec. i n the former case, and -14 kc./sec. and + 14 kc./sec. i n the l a t t e r case. However, at the po s i t i o n of zero s p l i t t i n g from the f i r s t order term ( ' s x * 0.5), the t h i r d order contributions are only -1.5 and +0.1 kc./sec respectively, and thus within experimental error do not influence the data of Table I. At the positions of maximum s p l i t t i n g of the s a t e l l i t e s the t h i r d order contribution to the "inner" s a t e l l i t e s i s -1.5 kc./sec. at s z - 0, and -3.4 kc./sec. at s z = 1. These values were used to ar r i v e at the quoted estimated errors of the measured maximum s p l i t t i n g s . 41. B. Experimental check on new method of spin determination. 27 The spin of the A l ' nucleus i s known from Ref. ( 3 2 ) to be I = 5 / 2 . The f i v e components of the resonance l i n e v i s i b l e i n F i g . 6 a would confirm th i s i f we were sure that no weaker s a t e l l i t e s had been missed. To convince ourselves that no weak outer components of the l i n e have been missed, and to i l l u s t r a t e the new method of spin determination proposed i n Chapter 2,we apply equation ( 1 8 ) to the values of TJ from Petchfs work on the central l i n e , and the values of R from Table I I . The values of U were as follows: U x = 2 5 . 6 ± 0 . 1 , Uy = 0 . 8 6 ± 0 . 1 , = 1 . 0 5 ± 0 . 1 . We set A)O= 7.4-53 Mc./sec. i n equation ( l 8 ) and compare the possible values of (I + 3 / 2 ) (I - 1 / 2 ) = 3 , 8 , 1 5 cor-responding to I = 3 / 2 , 5 / 2 , 7 / 2 respectively, with, the 2 experimental values of 3 2 sJeU/R , which for the X, Y, Z rotations are respectively 8 . 0 5 ± . 1 0 , 9 . 7 ± 1 , 7.4 1 1 . Thus the values of spin higher than 5 / 2 can be d e f i n i t e l y excluded, and the v i s i b l e f i v e components confirm I = 5 / 2 for A l 2 ? . The large uncertainties i n the experimental values of 3 2 %>0U/R2 for the Y and Z rotations are aaused by the large uncertainties i n Uy, U . These are poorly known because i n t h i s p a r t i c u l a r c r y s t a l the tensor i s such as to give only a very small cos 4 8 contribution to the frequency s h i f t for the Y and Z rotations. FIG. 10. Li6 absorption derivative curve in a single crystal of LiAl(Si03)2 showing the maxi-mum quadrupole splitting obtainable with the crystal b axis perpendicular to the external magnetic field. Frequency scale approximately 1 div. = 2.6 kc./sec. to face page 42 42. b n C. Ratio of the Quadrupole Moments of L i and L i ' . Schuster and Pake (22) reported a value for the rati o ' of the magnitudes of the quadrupole moments of L i ^ 7 6 and L i calculated from the quadrupole s p l i t t i n g of the L i n and L i ' magnetic resonances obtained from a single c r y s t a l of spodumene. Their c r y s t a l was mounted so as to rotate about i t s c axis. 7 The L i quadrupole i n t e r a c t i o n i n spodumene has been examined i n d e t a i l i n t h i s laboratory by Volkoff, Petch, and Smellie and has been reported elsewhere (18). This investigation yielded complete information on 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 f i e l d gradient tensor at the s i t e s of the L i n u c l e i . The result s show that maximum s p l i t t i n g of the L i l i n e s i s obtained when the c r y s t a l i s mounted to rotate about i t s b axis. This exceeds by 11.5% the largest s p l i t -t i n g obtained by a ro t a t i o n about the c axis. With a c r y s t a l so mounted, and oriented to give the maximum s p l i t t i n g of the L i ? l i n e s , the s p l i t t i n g of the L i ^ l i n e s was observed. The measurements were made at room temperature i n a f i e l d of about 8900 gauss. The res-o l u t i o n obtained i s shown i n F i g . 10, and represents an improvement over that reported by Schuster and Pake. I t i s f e l t that the improvement i s s u f f i c i e n t to warrant a 43. r e c a l c u l a t i o n of the quadrupole moment r a t i o . The im-provement i s due pa r t l y to the increase i n the A 0 being measured and par t l y to a better signal-to-noise r a t i o which allowed the use of a smaller modulation amplitude. The measured traces were recorded with a modulation amplitude equal to approximately one-sixth of the l i n e width at h a l f maximum. A reduction of t h i s modulation amplitude by a factor of 10 produced no appreciable further improvement i n resolution. From equation ( 10) since only f i r s t order effects are present here or Using ^, i n each case, A.^7 was measured to be 7 3 . 7 t 0 . 5 kc./sec. i n agreement with the quadrupole coupling constant eQd^/h for Li'' reported i n Ref. ( 1 8 ) . A ^6 was then obtained without moving the c r y s t a l by measuring the frequency difference between the positions of zero derivative of the two L i ^ l i n e s . The l i n e s i n Fi g . 10 and other s i m i l a r tracings seem to be we l l enough 4 4 . resolved so that the t a i l of one does not contribute to the peak of the other. The value of A-06 obtained from 30 such measurements i s 2 .18 * . 0 5 kc./sec. The quoted error i s the s t a t i s t i c a l probable error. This gives | CA 6/a 7| 5 (\-9t o i)x Id"2 ( 3 6 ) which i s s i g n i f i c a n t l y lower than the value ( 2 . 3 - 0.2) x 10~ 2 reported by Schuster and Pake. The quoted uncertainty i s s l i g h t l y greater than twice the s t a t i s t i c a l error. The values of neither ^ nor 0^ separately appear to be known at present ( 3 3 ) . 29 D. Observations on the S i Resonance. 27 7 Since the resonance spectra of A l ' and L i ' and L i ^ i n spodumene had been examined i n t h i s laboratory, i t 29 was decided to look for the S i resonance just for the sake of completeness. S i 2 ^ i s believed to have a spin of 1/2 ( 3 4 ) and hence a zero quadrupole moment but Sands and Pake (35) cast some doubt on t h i s when they reported a spin deter-mination for S i 2 ^ . Comparing the i n t e n s i t y of i t s magnetic resonance absorption with that of I^-2? seemed to indicate 3/2 as the correct value. They also referred to anomalous re s u l t s obtained by the Stanford group. 127 Watkins & Pound ( 3 6 ) however, suggest that the I 4 5 . i n t e n s i t y observed by Sands & Pake i n the EE crystals might have been reduced to.a f r a c t i o n of the expected value due to i n t e r n a l strains i n the c r y s t a l s . They have observed such e f f e c t s . Hatton, R o l l i n and Seymour ( 3 7 ) observed a single weak resonance l i n e i n beryl (Be^AlgSi^ 0-j_g) which they concluded was due to S i 2 ^ a-nd reported that there was no detectable s h i f t of the resonance on rot a t i o n of the c r y s t a l with respect to the magnetic f i e l d . The S i 2 ^ resonance consisting of a single l i n e was observed i n a f i e l d of about 8930 gauss at approximately 7 .575 Mc./sec. The signal-to-noise r a t i o of the l i n e was about 30:1 and there was no evidence of further l i n e s . A rot a t i o n was carried out and no s h i f t or s p l i t t i n g was detected. This would seem to add to the e x i s t i n g evidence 29 to confirm the spin of S i as 1/2. 46 Chapter 7 - Discussion, As outlined i n the Introduction, the objectives i n undertaking the work described In Chapter 6 were: to check the second order and t h i r d order perturbation theory results for the dependence of the resonance frequencies on c r y s t a l orientation, to obtain higher accuracy i n the values of the f i e l d gradient constants of spodumene, to check a proposed new method of spin determination, and f i n a l l y to attempt to improve the accuracy of the Q L , 6 A U 7 r a t i o . The check on perturbation theory was obtained for a complete ro t a t i o n of the c r y s t a l i n the case of the "inner" 27 s a t e l l i t e s of the A l l i n e i n spodumene, and for a few selected positions of the "outer" s a t e l l i t e s . I t indicated that second order terms were adequate to describe the s h i f t of the centre of gravity of the inner and outer s a t e l l i t e s and that t h i r d order effects i n the s p l i t t i n g of the inner s a t e l l i t e s were of the order of 1% of the f i r s t order effects at f i e l d s of the order of 6 0 0 0 gauss. In the case of the Y rota t i o n on the inner s a t e l l i t e s , the maximum f i r s t order s p l i t t i n g amounted to J l 6 kc./sec. and the t h i r d order effects were ignored since they were of the same order of magnitude as the experimental un-certainty. A complete ro t a t i o n for the outer s a t e l l i t e s 47. was not attempted. For some c r y s t a l positions these were weak, necessitating long time constants, with correspond-ingly long observation times and i t was f e l t that the agreement with theory was good enough at the several selected c r y s t a l positions that no further useful i n -formation would be obtained from extending the measurements. The values obtained for the f i e l d gradient constants i n spodumene quoted i n equation (31) above are s l i g h t l y more accurate than those obtained previously by Dr. Fetch from the central component analysis alone, but the l i m i t s of error are s t i l l greater than are inherent i n the method. However, the considerable e f f o r t involved i n determining the c r y s t a l axes more accurately and i n orienting them more accurately with respect to the axes of r o t a t i o n doesn Tt seem j u s t i f i e d at present i n view of the lack of any th e o r e t i c a l estimates of the same quantities to which they could be compared. From the quadrupole coupling constant, when Q, i s known from other sources, an empirical value can be obtained for M j y I . Such empirical information on I I , on the asymmetry parameter and on the orien t a t i o n of the f i e l d gradient tensor p r i n c i p a l axes, would be useful for comparison with t h e o r e t i c a l predictions, calculated on the basis of s p e c i f i c models assumed for 48. the c r y s t a l , i f such calculations were av a i l a b l e . Such comparisons would enable one to decide whether the bonding i n a p a r t i c u l a r c r y s t a l were la r g e l y of the i o n i c , or the covalent type, or intermediate between the two. No the o r e t i c a l calculations of the c r y s t a l l i n e f i e l d s i n spodumene have been attempted here but, i n a private communication, Prof. Bersohn of Cornell University has reported that some preliminary work for the spodumene and A I 2 O 3 cases was done at Cornell by one of his students. This indicated that the calculations would be of p r o h i b i t i v e length unless an electronic calculator were available, and they are try i n g to arrange for the use of one. The experimental check on the new method of measuring nuclear spin indicates that t h i s method w i l l give an unambiguous spin determination. The method depends on the quadrupole coupling constant for the nucleus i n question being large enough to make the second order* effect i n the angular dependence of the resonance frequency measurable with f a i r p recision. In p a r t i c u l a r , one must be able to compare the amplitude of the fourth harmonic term i n the second order effect with that of the second harmonic term i n the f i r s t order e f f e c t . Thus, that axis of r o t a t i o n of the c r y s t a l should be chosen which emphasizes 49. the fourth harmonic i n the second order e f f e c t , and then the f i r s t order effect should be observed for a r o t a t i o n about the same a x i s . When r e f e r r i n g , i n Chapter 2 , to spin determinations by counting l i n e components, the statement was made that s a t e l l i t e s might be broadened and weakened by c r y s t a l im-perfections and impurities. This effect was observed by Fetch et a l . ( 2 3 ) ; i n the f i r s t c r y s t a l samples used only the central Al27 l i n e was observable. This i s most l i k e l y caused by a spread i n the l o c a l values of V E at the various A l s i t e s throughout the c r y s t a l due to such im-perfections and impurities. In accordance with equations (13) and (13) the central component (m = 1/2) of the l i n e has no f i r s t order dependence of the resonance frequency on V E , while the s a t e l l i t e s (m = 3/2 and 5/2) do show f i r s t order e f f e c t s . Therefore l o c a l variations i n ^7E from one nuclear s i t e to another w i l l produce a much greater broad-ening of the s a t e l l i t e s as compared with the central component, and may smear them out so as to make them un-observable. A s i m i l a r explanation has been offered recent-l y by Watkins and Pound ( 3 6 ) to account for the anomalously weak i n t e n s i t y of resonance l i n e s of some nuclei i n cubic c r y s t a l s . The r a t i o of the quadrupole moments of the l i t h i u m isotopes i s calculated from the quadrupole coupling constants 50. by cancelling the <fijj term i n each. In cancelling out the <fiy.£ term we have assumed that the f i e l d gradient tensor at 6 7 ^ the L i and L i s i t e s i s the same. The two tyipes of nuclei w i l l be i n equivalent crystallographic positions, of course. But the f i e l d experienced by the nucleus i s a combined one due to the neighboring ions plus that due to the atomic electron s h e l l s and bonding electrons surrounding the nucleus i t s e l f . This l a t t e r contribution may perhaps be s l i g h t l y 6 7 dif f e r e n t i n the case of the L i and L i ' n u c l e i . In the f i r s t approximation i f the two nuclei are considered as point charges there should be no difference at a l l . How-ever, considering the f i n i t e size of the two n u c l e i , and the d i f f e r e n t charge d i s t r i b u t i o n s on each ( as evidenced by the large differences i n the experimentally observed quadrupole coupling constants), i t i s possible that the wave functions are distorted to some extent by the f i n i t e 6 V nucleus by amounts which d i f f e r i n L i and L i . This may, i n p r i n c i p l e , give r i s e to a small difference i n d-^y at the two s i t e s . I f the bonding of L i i n spodumene i s assumed to be of purely ionic type, then the only effect of the l o c a l s pherically symmetric electron d i s t r i b u t i o n of a L i + ion on fi-jy b e t o s c r e e n o u t p a r t i a l l y the e l e c t r i c f i e l d due to neighboring ions. Prof. Bersohn, i n the private communication referred to above, estimates t h i s shielding 51. effect i n the L i case to be of the order of 10%. I t seems reasonable to assume that the difference i n t h i s shielding between the two isotopes w i l l be an order of magnitude smaller. In view of t h i s , the l i m i t s of error placed on the quadrupole moment r a t i o , amounting to about 1 5?« i seem adequate. 5 2 . PART I I I - EXPERIMENTAL  Pure Quadrupole Spectra Chapter 8 - Apparatus and Experimental Procedure. A. Super-Regenerative O s c i l l a t o r s - General. I t was decided to attempt to observe the pure quadrupole spectrum of Al 2"? i n spodumene. Calculations of Lamarche and Volkoff (21) showed that the spectrum should consist of two l i n e s of medium strength at O.789 and O.758 Mc/sec. plus a much weaker l i n e at the sum of these frequencies. From t h e i r calculations on t r a n s i t i o n probabilities,and other calculations (25) on t h e o r e t i c a l l y available signal-to-noise an approximate c a l c u l a t i o n of the signal amplitudes to be expected was made. In the work at high magnetic f i e l d s reported i n Part II,which was carried out with the spectrometer previously con-structed by C o l l i n s and improved by Fetch,the size of the magnet gap had r e s t r i c t e d the size of the c r y s t a l samples to about 2 c.c. For such a sample, the c a l c u l a t i o n i n -dicated that the signals would be only of the order of the noise. Since there would be no such r e s t r i c t i o n on sample size i n the pure quadrupole case,it was hoped that the gain i n signals by increasing the samples 5 3 . to 50 - 100 c.c. or more would be s u f f i c i e n t to bring them above noise l e v e l . Larger samples would require a higher r . f . power l e v e l to give the same energy density i n the sample volume. Also i t was f e l t that the r . f . power l e v e l s being used i n the C o l l i n s spectrometer were many times below what the c r y s t a l samples would stand without onset of saturation. Since,in the absence of saturation,the absorbed power i s proportional to the r . f . power incident on the sample, a gain i n signal-to-noise should be obtained by increasing the power l e v e l , providing that there was not a proportionate increase i n the noise generated by the o s c i l l a t o r . Regenerative o s c i l l a t o r s of the C o l l i n s * type ( 1 8 ) , or of the Watkins and Pound type ( 3 8 ) , operate best at low power l e v e l s and the noise figure deteriorates as the l e v e l i s increased,so that nothing can be gained i n sign a l - t o -noise r a t i o i n t h i s way. Super-regenerative o s c i l l a t o r s have been shown ( 2 0 , 3 9 ) to operate w e l l at high power l e v e l s and can be swept i n frequency. They have some obvious disadvantages,including a complicated frequency spectrum which produces extra spectral l i n e s that may be d i f f i c u l t or impossible to resolve i n the case of wide l i n e s at low frequencies. But i n view of the fact that the quadrupole spectra of lowest frequency (as low as 1 . 5 Mc./sec), which have been reported i n the l i t e r a t u r e (8), have been recorded using a super-regenerative o s c i l l a t o r , the decision was made to bui l d a spectrometer of t h i s type i n the hope that i t could be made to operate at frequencies even below 1 Mc./sec. A super-regenerative o s c i l l a t o r i s a pulsed radio-frequency o s c i l l a t o r . I t consists of a resonant c i r c u i t and a regenerator tube, plus a means of turning the o s c i l l a t i o n on and o f f . This i s commonly accomplished by applying a periodic voltage wave from an external o s c i l l a t o r or pulse generator d i r e c t l y to the grid of the o s c i l l a t o r tube. The quenching wave, as i t i s c a l l e d , may be a sine wave, square wave, sawtooth, or of more complicated form. The effective conductance seen by the resonant c i r c u i t w i l l be a function of time, g ( t ) , alternately positive and negative, and equal to the sum of the positive conductance representing the inherent and coupled losses of the tuned c i r c u i t and the negative conductance supplied by the regenerator tube. Some control of the r e l a t i v e durations of the positive and negative periods i s desirable and t h i s may be accomplished by biasing the o s c i l l a t o r tube and varying the quenching wave shape. The quenching action, when g(t) i s p o s i t i v e , 55 erases the effect of the previous cycle, reducing the voltage across the tank c i r c u i t to the order of thermal noise voltages. The off-time i s usually made long enough compared to the decay time constant of the tank c i r c u i t (of the order of 10 to 15 times i t ) to allow the amplitude of the o s c i l l a t i o n s to f a l l to t h i s l e v e l . As the quenching i s withdrawn and g(t) goes through zero, o s c i l l a t i o n s w i l l be i n i t i a t e d by the voltages remaining i n the tuned c i r c u i t and the envelope of these o s c i l l a t i o n s w i l l r i s e exponentially as g(t) becomes negative. The tube i s l e f t on long enough to include many r . f . cycles and for the purpose of t h i s description we w i l l assume that i t i s l e f t on long enough for the o s c i l l a t i o n s to l e v e l o f f at the saturation amplitude U^g^. This i s c a l l e d the logarithmic mode. The time taken to reach U ^ B Y depends s e n s i t i v e l y on the minimum amplitude U mj_ n from which the o s c i l l a t i o n s b u i l d up. When signals from an external transmitter are present, the average Umj_n> consisting of signal plus noise, i s increased and the time average of the o s c i l l a t i o n amplitude increases. Any modulation of the signal w i l l appear as a change i n t h i s time average and may be de-tected, amplified, and presented using standard procedures. Analysis (40) shows that the output i s a greatly amplified reproduction of the signal-plus-noise amplitudes. I f a perfect square-wave quenching voltage i s assumed, the gain i s given i n Ref. (40) as: Sain = f ^ ^ L / R y l n ^ + V a)/V n where f = the quench frequency L = tank c o i l inductance Rip = net series tank resistance = R + R n R = series loss resistance of tank c i r c u i t R n = negative series resistance representing the regenerator tube V n = noise voltage at instant that o s c i l l a t i o n s s t a r t V s = signal » » » " " " This gain can be as much as 10 . For a sine-wave quenching voltage the gain i s higher. The same gain law holds but i s more d i f f i c u l t to apply since Rip can not be assumed constant. The e f f e c t i v e R T i s smaller but the maximum f ^ i s also smaller. In Refs.(41,42j a s l i g h t l y more general treatment of the super-regenerator i s given. Expressions are developed for s e l e c t i v i t y and s e n s i t i v i t y which can be applied to any quench wave form. In (41) Bradley concludes that the narrowest band and greatest s e n s i t i v i t y as well as the best signal-noise r a t i o appear to be obtainable with a conduct-ance function g(t) of the following form: g(t) has a large positive value for the quenching period, which i s made as short as i s consistent with thorough quenching. This i s 51. "followed by a period i n which g(t) = 0. This period i s made as long as possible. In the t h i r d period g(t) should become quite negative very quickly so as to give maximum amplif i c a t i o n of the voltages e x i s t i n g at the end of the second period. The t h i r d period should be as short as i s p r a c t i c a l . The spectrum of the r . f . output of a super-regenerative o s c i l l a t o r i s very complex. I f the quench-ing i s not complete (so that there i s coherence between the r . f . o s c i l l a t i o n s i n successive pulses), the spectrum consists e s s e n t i a l l y of a central c a r r i e r frequency with side-band frequencies, of decreasing i n t e n s i t y , displaced by i n t e g r a l multiples of the quench frequency f^. The int e n s i t y d i s t r i b u t i o n w i l l not, i n general, be symmetrical about the c a r r i e r frequency and the rate of f a l l - o f f i n in t e n s i t y w i l l depend on the Q, of the c i r c u i t . In addition to t h i s amplitude modulation there i s a frequency-modulation effect;;, of much smaller order, since the resonant frequency depends on tube voltages, damping, etc. I f quenching i s complete,so that there i s no coherence between pulses, i n the absence of signals the spectrum w i l l be continuous and extend over approximately 5 8 , the same band-width as i n the coherent case. Now i f signals are present,the phase relationship between successive pulses i s determined by the signal and the spectrum i s again a set of discrete frequencies but the apparent c a r r i e r frequency i s now a function of the signal frequency. The above description of a super-regenerative receiver has been given, purposely, i n terms of receiving radio signals from an external transmitter. The detection of nuclear resonance absorption by a super-regenerative spectrometer can take place by . two di f f e r e n t processes. When a sample i s placed i n the tank c o i l and the o s c i l l a t o r frequency i s swept through the resonant frequency, assuming the magnetic f i e l d or the frequency i s being modulated i n the usual way, the absorptive or *X component of the nuclear s u s c e p t i b i l i t y produces amplitude modulation of the c a r r i e r which may be detected by standard methods. This process i s not related to the reception of radio signals referred to above. Roberts ( 4 5 ) suggested, that i n addition to t h i s effect,there might be voltages induced i n the tank c o i l by precessing nuclei as i n the nuclear induction technique. These voltages would play the same role as signals from an external transmitter and the analysis for t h i s process would be the same as for radio reception. I t now seems to be 59. accepted that t h i s i s the important process,for the case of incoherent operation ( 3 9 ) at l e a s t . Dean (20) gives con-vincing arguments i n favor of t h i s picture and also fee l s that i t i s the important process for coherent operation. However, Williams (44), G-utowsky et a l . (45) describe t h e i r r e s u l t s i n terms of the straight absorption process. The voltage induced by the precessing nuclei i s given byAU--4T?Y Q'oc U m B s t where Q,' i s the quality factor of the tank c i r c u i t and o< i s a factor somewhat less than unity correcting for the intermittent nature of the r . f . f i e l d H-]_ and for variations i n H^ throughout the c o i l volume. An essential part of t h i s picture i s that the transverse relaxation time Tg be long enough so that a large degree of the spin coherence obtained during one on-period remains at the beginning of the succeeding on-period. The off-time for a t y p i c a l case (10 to 15 times the decay time constant) i s of the order.of 10 sees. The inverse l i n e width Z T T / V A H gives a measure of the transverse relaxation time. For the A l l i n e s i n spodumene using the observed l i n e width of the order of A H ~ 10 gauss t h i s i s of the order of 100 sees, or about 10 times the necessary off-period. The spectrum i n t h i s case i n the presence of signals, w i l l consist of discrete frequencies because of Q u e n c h ^ i n p u t ® o o a x> a <0 (0 0) O Outpu FIG 11 • T H E SUPER-REGENERATIVE SPECTROMETER 6o. the apparent coherence introduced by the signals. As the o s c i l l a t o r i s swept through the region of a nuclear resonance each of these frequencies, when coincident with the t r a n s i t i o n frequency, w i l l produce a resonance s i g n a l . A s i n g l e - l i n e nuclear resonance spectrum w i l l then appear as a m u l t i p l e - l i n e spectrum, the number of l i n e s depending on the band width of the tuned c i r c u i t and the quench frequency f^. The l i n e shapes depend on the phase r e l a t i o n between A U and U^n and i n general w i l l be mixtures between absorption and dispersion curves. B. The Super-Regenerative O s c i l l a t o r Used. The c i r c u i t of the super-regenerative o s c i l l a t o r , which was constructed, i s shown i n Figure 11. The basic c i r c u i t was designed by Dean (20) for use i n the region of 30 Mc./sec. Suitable modifications were made to allow i t s use i n the region of .73 Mc./sec. to 3 Mc./sec. In addition to these modifications, provision was made for operating the o s c i l l a t o r at low r . f . l e v e l s and a r . f . amplifying stage was added, preceding the detector, for low l e v e l operation. This i s by-passed for high l e v e l operation. The o s c i l l a t o r i s the grounded-plate version of the C o l p i t t s o s c i l l a t o r . The quenching action i s as follows: the two halves of a $J6 (71,2), i n p a r a l l e l , have t h e i r cathodes t i e d to the cathode of the o s c i l l a t o r tube, V3. 1 The 6J6 6 1 . plates are by-passed to ground for r . f . When a positive pulse from the external quenching c i r c u i t i s applied to the g r i d of 71,2 the additional cathode bias shuts o f f V3, since i t s grid-leak condenser does not discharge immediately. The time constant of the g r i d leak must be short enough for the bias to recover before the quench i s removed but, of course, long enough so as not to be able to follow the r . f . o s c i l l a t i o n s . At the same time as YJ> i s cut o f f , the cathode input impedance of VI, 2 i s shunted across part of the tuned c i r c u i t . This large positive conductance shortens the decay time constant of the tank c i r c u i t and aids i n quickly damping out the free o s c i l l a t i o n s . I t i s important that t h i s impedance should not have any damping effect during the o s c i l l a t i o n . , portion of the cycle and to insure t h i s , the gr i d of VI,2 must be made very negative - of the order of - lj>0 v o l t s . A fixed bias that may be set from 0 to -90 v i s provided by a battery and potentiometer. The o v e r a l l p o s i t i v e to negative swing of the quench voltage wave should be about 150 v o l t s . The v i b r a t i n g condenser i n p a r a l l e l with the tank c i r c u i t provides a means of frequency - modulating the o s c i l l a t o r . This i s an alternative to f i e l d modulation 62. that i s convenient for oscilloscope presentation of strong resonances. Dehmelt (J9) has also used t h i s type of modulation for chart recording of resonances but i t i s not as satisfactory for t h i s as i s the f i e l d modulation. A variable capacity-divider, following the tank c i r c u i t , allows some control of the f r a c t i o n of the tank voltage applied to the grid of the r . f . amplifier V4, or to the grid of the i n f i n i t e impedance detector V5. The time constant of the R and C combination, i n the cathode ov 75, i s selected to allow the cathode to follow the pulse envelope. Biasing considerations w i l l be the main factor i n determining R, but i t s value should be chosen so that the C giving the desired time constant i s not too small. There i s a greater tendency for V5 to act as an o s c i l l a t o r i f t h i s C i s small. The capacity-divider referred to above decouples 75 from the tuned c i r c u i t to a c e r t a i n extent and also helps to prevent such o s c i l l a t i o n s . The output from the cathode of 75 i s fed into a 4-section, RC f i l t e r network which integrates the pulse areas. The time constant of these i s chosen to suppress the quench frequency. Part of the output from the cathode of 15 i s taken to a pulseroutput terminal. This i s d i s -played on an oscilloscope and i§ an essential aid i n V 7 _ 6 C 5 V 6 I2AU7 V 5 o » FIG 12- THE QUENCHING CIRCUIT 0) 01 63 adjusting the o s c i l l a t o r . The output from the network w i l l contain, c h i e f l y , frequency components centred about the modulation frequency, f . This i s amplified i n a low gain audio stage, V6. Following 76 i s a second f i l t e r i n g network s i m i l a r to the f i r s t and another low gain audio stage, 77. C. The Quenching C i r c u i t . The quenching c i r c u i t shown i n Figure 12 i s also a modification of one used by Dean (20). The input to the c i r c u i t i s provided by a Hewlett-Packard audio o s c i l l a t o r . For the work described here, the frequency of t h i s input, fq, varied from 2 kc./sec. to 20 kc./sec. but with minor modifications the c i r c u i t was used for quench frequencies up to 100 kc./sec. The f i r s t three triode sections (71, 72, 73) square the sine-wave input. They are followed by a d i f -f e r e n t i a t i n g network. The r e s u l t i n g positive and negative pips appear at the g r i d of 74 which has a fixe d negative bias, w e l l below cut-off. Therefore only the positive pip a f f e c t s the tube current. The 1N34 diode was added across the 39K r e s i s t o r to suppress the effect of the negative pip further. The output pip from 74 triggers the "one-shot" multivibrator formed by 75 and 76. With no input, 75 i s 64. cut o f f and V6 i s conducting. 75 and 74 have a common plate load. Thus the positive pip at the g r i d of 74 appears as a negative pip at the plate of 75 and the grid of 76, i n i t i a t i n g the change to the metastable state. The duration of t h i s state i s determined by the R and C values i n the grid of 76 and the s e t t i n g of the potentio-meter i n the g r i d of 75. The output, taken from either plate, i s then e s s e n t i a l l y a square wave i n which the r e l a t i v e duration of the positive and negative portions may be varied. In Dean*s o r i g i n a l c i r c u i t t h i s square wave formed the quenching voltage. After some amp l i f i c a t i o n and smoothing i n 77, i t was rounded somewhat by a 10K r e s i s t o r and the capacitance of a short length of cable, before being applied to the grids of the 6J6. This ar-rangement was also t r i e d here, before the modifications shown i n Figure 12 were made. The combination of d i f f e r e n t i a t i n g and integrating networks, shown following the multivibrator, was inserted i n an attempt to reproduce an approximation to the optimum conductance wave form described by Bradley (41). The values of R and C were chosen so that the " d i f f e r e n t i a t i o n " and "integration" would be very poorly done. The r e s u l t i n g wave shape was then amplified i n 77 and applied to the 6J6 grids. A comparison of the signals obtained with t h i s quenching wave shape and with the "rounded square-wave" Frequency Meter Recording Mill i ammeter Commuriicat ions Receiver I Audio Osc i l la tor Phase Mixer Time Const-I - IQOsec Quenching C i rcu i t 1 4 4 cps Tuned Ampl i f ier ' Alternative _ Phase-Sh i f t ing Network Audio Oscillator 2-15 Kcps-FIG 13- BLOCK DIAGRAM OF THE SUPER-REGENERATIVE SPECTROMETER 6 5 was made i n the case of D signals from D20 displayed on the oscilloscope. No marked difference was noted. However, the author feels that no f a i r comparison can be made on the basis of t h i s b r i e f check. The c i r c u i t as shown was used for the work described i n Chapter 9. D. Modulation Methods. Figure 1 3 shows a block diagram of the spectrometer. With the exception of the method of modulating the signals, the recording systems used for the pure quadrupole work were simi l a r to the one used i n the magnetic resonance work and described i n Ref. ( 1 8 ) . No further description of them i s needed here. The method of f i e l d modulation used for the magnetic resonance work i s also described i n Ref. ( 1 8 ) . To record the pure quadrupole resonances, an on-off square-wave magnetic f i e l d at 44 c.p.s. was applied to the sample. During the on-period,the resonance i s smeared out by the Zeeman s p l i t t i n g of the leve l s and 44 c.p.s modulation of the signal r e s u l t s . There i s no pa r t i c u l a r significance i n the choiee of 44 c.p.s. as the modulation frequency. Because of the time constant of the modulation c o i l s , a square-wave f i e l d i s more e a s i l y 66. produced at low frequencies. An available audio am p l i f i e r , o r i g i n a l l y tuned at a higher frequency, tuned at 44 c.p.s. af t e r minor modifications, so t h i s frequency was used. In some cases the f i e l d was applied by means of a solenoid; i n other cases Helmholtz c o i l s were used. The inte n s i t y of the smearing f i e l d s was about 25 gauss as t h i s was the maximum obtainable with the equipment as i t was f i r s t set up. This f i e l d seemed adequate for smearing the 3 5 CI" resonances, as observed on the oscilloscope, but i t i s not so cer t a i n that i t i s adequate i n the case of wide l i n e s . Dehmelt (39) uses smearing f i e l d s of about 100 gauss. Watkins ( 3 8 ) reported using f i e l d i n t e n s i t i e s of about 25 gauss on N-^-4 resonances but these were very narrow l i n e s and the "negative wings" were quite pronounced in d i c a t i n g incomplete smearing. The current for the modulation f i e l d was supplied by a 110 v o l t battery or d.c. generator. The d.c. voltage was applied to the plates of a bank of 19 - 6AS7's i n p a r a l l e l . The modulating c o i l s plus a variable' resistance formed the cathode load of the 6AS7 Ts. A square-wave voltage with a peak-to-peak amplitude of about 200 v o l t s was then applied to the grids, cutting the tubes o f f f o r half of each period. To supply the square-wave,a Hewlett-Packard audio o s c i l l a t o r was used to supply the input to a simple square-wave generator c i r c u i t (46). The output 6 7 . of t h i s was fed into a Williamson amplifier ( 4 7 ) , which was provided with a dummy load matched to the output impedance of i t s step-down output transformer. This assured best reproduction of the square wave input. The square-wave to drive the 6AS7 grids was then itaken, v i a a coupling capacitor, from one side of the output transformer primary, where the voltage amplitude i s about 25 times the secondary output voltage. E. Frequency Measurement. Accurate measurement of nuclear resonance frequencies i n the case of chart-recorded signals i s d i f f i c u l t to a t t a i n . The method,described i n Chapter 5 for the c.w. o s c i l l a t o r , i s not suitable because of the p o s s i b i l i t y of confusing the various harmonics with the central frequency and because the frequency i n each case i s not sharply de-fined, but "fuzzy", due presumably to frequency-modulation e f f e c t s . This makes i t impossible to get a sharply defined zero-beat with a frequency meter. I f the quench i s turned o f f , the c.w. o s c i l l a t i o n frequency can be measured, but t h i s again does not coincide exactly with the central frequency when the quench i s on. The method used was as follows: The o s c i l l a t o r drive and chart drive were stopped simultaneously. The quench was then turned o f f and the c'.w. frequency located. 6 8 . Then the quench was turned on again and the central frequency was measured as accurately as possible and a marker was made. This process was repeated i n the neighborhood of the strongest resonance and l i n e a r i n t e r p o l a t i o n used as before. Chapter 9 - Experimental Results. A. Preliminary t e s t i n g of the spectrometer. The super-regenerative spectrometer described i n the previous chapter was f i r s t tested for low-frequency operation as a magnetic resonance spectrometer using the magnet and modulation equipment described i n Ref. ( 1 8 ) . The c r y s t a l used, i n most cases, was the milky spodumene c r y s t a l referred to i n Chapter 7, with which only the 27 central A l l i n e was v i s i b l e . This was chosen because the m u l t i p l e - l i n e spectrum produced by the super-regenerator i s easier to interpret i f i t i s due to a single nuclear resonance, and at t h i s time we were c h i e f l y interested i n studying the operation of the spectrometer. As was expected, i t was found that much higher values of Hj_ could be used without saturating the samples. With the c.w. spectrometers ( 3 0 , 3 8 ) the H-^  applied to the sample i s usually a few milligauss. With the super-regenerative spectrometer r . f . f i e l d i n t e n s i t i e s of the 1 Pig. 14. Selected derivative curves for A l 2 7 and L i i n spodumene, recorded with the super-regenerative spectrometer. Figs.14a and 14b were recorded using a magnetic f i e l d of 1500 gauss corresponding to an unperturbed Larmor frequency for Al2,} of Vp= 1 . 6 8 6 Mc./sec. In Fig. 14a the central A l ' l i n e i n the milky spodumene i s recorded at about 1 . 6 Mc./sec.In Fig.14b the central and inner s a t e l l i t e s are shown centered at about 1 . 6 Mc./sec.The quench frequency i n each case i s 10 kc./sec. The frequency scale i n Fig.14a i s 25 kc./sec. per d i v i s i o n and i n Fig.14b i s 50 kc./sec. per d i v i s i o n . Figs.14c and 14d were recorded-using a magnetic f i e l d of 1100 gauss cor-responding to an unperturbed Larmor frequency for A l ^ 7 of » 0= 1 . 1 9 5 Mc./sec. and for Li7 of v n= 1 . 7 8 0 Mc./sec. Fig.14c shows the central A l 2 ' l i n e i n the milky spodumene at about 1.2 Mc./sec. The quench frequency was 7 kc./sec. and the frequency scale i s about 12 .5 kc./sec. per d i v i s i o n . Fig.l4d shows the A l ' resonance centered at about 1.2 Mc./sec. and the L i ' resonance centered at about I . 7 8 Mc./sec. The frequency scale averages about 1 5 0 kc./sec. per d i v i s i o n and the quench frequency i s 15 kc./sec. to face page 69 69. order of 0.5 gauss or more have been used without apparent saturation of the A l 2 ? resonance i n spodumene. The central A l l i n e from the milky spodumene c r y s t a l was observed at several values of the magnetic f i e l d H Q , the lowest value used being about 1100 gauss. For comparison, a few traces were recorded using a clear c r y s t a l of spodumene, i n which s a t e l l i t e s were v i s i b l e . The volume of t h i s c r y s t a l was s l i g h t l y less than half that of the milky c r y s t a l . Representative traces are shown i n Figure 14. The traces of Figures 14a and 14b were record-ed i n a f i e l d of about 1500 gauss. Figure 14a shows the 27 central A l ' l i n e obtained with the milky c r y s t a l at about 1 . 6 Mc./sec. In Figure 14b, obtained with the clear c r y s t a l , the central l i n e frequency i s approximately the same. The signal amplitude i s down mainly because of reduction i n sample volume. One "inner" s a t e l l i t e i s v i s i b l e but the other i s so close to the central l i n e that the super-regenerative s a t e l l i t e s of each overlap. Figure 14c was recorded with the milky c r y s t a l using a f i e l d of about 1100 gauss. The approximate frequency of the A l 2 ? resonance i n t h i s case was 1.2'Mc./sec. Figure 14d,. also recorded at 1100 gauss but with the clear c r y s t a l , i s reproduced here as an i n d i c a t i o n of the s t a b i l i t y of the spectrometer. The frequency range covered i n the trace i s over 6 0 0 kc./sec1. At the r i g h t are the A l 2 7 resonances centered at about 7 0 1.2 Mc./sec. and at the l e f t are the L i ' resonances centered at about 1 . 7 8 Mc./sec. The signal amplitudes are down because of the high sweep rate. I t i s of interest to compare the signal-to-noise obtained with t h i s spectrometer, for example i n Figure 14b, with that obtained with the C o l l i n s spectrometer using a sample of comparable s i z e . At 2.3 Mc./sec, the s i g n a l -27 to-noise of the Al.. central l i n e with the l a t t e r spectrometer was about 2 : 1 . - At 1.6 Mc./sec. the t h e o r e t i c a l l y - a v a i l a b l e signal-to-noise would be a factor of 2 lower than at 2.3 M c / s e c so the central l i n e would be down to the order of the noise. Having established that the super-regenerative spectrometer would detect comparatively weak nuclear resonances at frequencies as low as 1 Mc./sec., the t e s t -ing was discontinued and searches for pure quadrupole resonances were begun. B. Pure Quadrupole Spectra. The search for the i n spodumene, the calculated Mc./sec and O .789 Mc./sec, A search for one of the pure AlpO,, predicted to occur at pure quadrupole A l ' resonances frequencies of which were 0 . 7 5 8 has so far been without success. 27 quadrupole A l resonances i n O .718 Mc/sec. and unsuccessfully 71. searched for by Pound ( 1 7 )» has also been made without success. P o l y c r y s t a l l i n e samples, with volumes ranging up to about 5 0 0 c . c , have been used. The author feels that, although these resonances are probably quite weak, they might be detectable with the super-regenerative spectrometer when more experience i s gained i n the operation of the spectrometer at low frequencies. The spectrometer has detected a pure quadrupole resonance at 1 . 2 7 1 . 0 1 M c / s e c i n Na 2B^0 7.4H 2 0 (kernite). Mr. H. Waterman, i n th i s laboratory, i s investigating the magnetic resonance spectrum of B*^ i n kernite, i n a magnetic f i e l d of about 7 0 0 0 gauss. Preliminary r e s u l t s indicate that there are 4 non-equivalent boron positions i n a unit c e l l . An analysis of the dependence of the frequencies of the B ^ l i n e s , on c r y s t a l orientation^ f o r one of these four positions, gave an estimate of the quadrupole coupling constant and of . From these values, a pure quadrupole t r a n s i t i o n was predicted at a frequency of 1 3 3 0 t 8 0 kc./sec A private communication,from Prof. W. Proctor of the University of Washington,states that Mr. Blood of that Department has calculated a pure quadrupole l i n e at 1 2 5 0 kc./sec. on the basis of his high f i e l d measurements. The t r a n s i t i o n observed i s consistent with t h i s predicted l i n e but d e f i n i t e assignment of t h i s observed l i n e to B 1 1 can not be made without further Fig. 1 5 . A recorded, pure quadrupole resonance i n EapB^O rj .AE^O. In Fig. l ^ a the quench frequency i s 1 5 kc./sec. and the frequency scale about 29 kc./sec. per d i v i s i o n . In Fig. 1 5 b the quench frequency i s 2 kc./sec. and the frequency scale the same as i n Fig . 1 5 a . The measured frequency of the peak i n F i g . 1 5 b i s 1 . 2 7 Mc./sec. and the pattern i n Fig. 1 5 a i s centered at t h i s same frequency. The pattern of the resonance i n Fig. 1 5 a i s complicated due to the fact that the resonances are a mixture of absorption and dispersion curves which are not completely resolved and not completely smeared out during the "o n - f i e l d " part of the modulation cycle„ tp face page 72 7 2 . investigation. The magnetic resonance spectrum of Na i n kernite has not yet been examined. I t i s possible that Na 2^ might give r i s e to a pure quadrupole resonance i n . t h i s same region. Prof. Itoh, of Osaka University, i n a private communication,has predicted on the basis of high f i e l d work 22 a Na pure quadrupole t r a n s i t i o n of 1 . 2 3 Mc./sec. i n NagSgO^^HgO. Since the values of I remain of the same general order of magnitude i n various compounds of the same general type, the p o s s i b i l i t y that the observed t r a n s i t i o n i s due to Na 2^ cannot at t h i s stage be excluded. Recorded traces of the observed resonance are shown i n Figure 1 5 . In Figure 1 5 a , with a quench frequency of 1 5 kc./sec. the super-regenerator s a t e l l i t e s are not re-- solved. In Figure 1 5 b the quench frequency has been reduced to 2 kc./sec. This has caused the super-regenerator s a t e l l i to f a l l -within the resonance l i n e but has also reduced the gain by a factor of 4 or 5 . The pronounced "negative wings" are believed to be due to an inadequate smearing f i e l d . The quoted uncertainty i n the frequency of t h i s l i n e has been made quite high because of the d i f f i c u l t i e s i n frequency measurement discussed i n Chapter 8. The observation of a pure quadrupole l i n e at 1 . 2 7 Mc./sec., which i s the lowest pure quadrupole frequency 73. reported i n the l i t e r a t u r e to date,concludes the preliminary stage of the inve s t i g a t i o n of the s u i t a b i l i t y of super-regenerative detectors for the observation of pure quadrupole spectra i n this low frequency region. Further extensions and applications of t h i s technique w i l l be l e f t f or future investigations. 74. REFERENCES Bloch, F., Hansen, W.W. and Packard, M., Phys. Rev. 70., 4 7 4 ( 1 9 4 6 ) P u r c e l l , E.M., Torrey, H.C. and Pound, R.V. , Phys. Rev. 6 9 , 3 7 ( 1 9 4 6 ) Pake, G.E., J . Chem. Phys. 16., 3 2 7 (1948) Van Vleck, J.H., Phys. Rev. 7_4, 1 1 6 8 ( 1 9 4 8 ) Kruger, H,, Z. Phys. 1 ^ 0 , 3 7 1 ( 1 9 5 1 ) Bersohn, R., J. Chem. 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