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A theoretical investigation of the nuclear resonance absorption spectrum of spodumene Lamarche, Joseph Louis Gilles 1953

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A THEORETICAL INVESTIGATION OF THE NUCLEAR RESONANCE ABSORPTION SPECTRUM OF SPODUMENE  GILLES LAKARCHE  A\ Thesis submitted i n p a r t i a l f u l f i l m e n t of the requirements f o r the degree o f Bfester o f A r t s i n the Department o f Physics  We accept t h i s t h e s i s as conforming t o the standard required f o r the degree of Master of A\rts  Members of the Department of P h y s i c s The U n i v e r s i t y of B r i t i s h Columbia  A p r i l , 1953  i ABSTRACT  The i n t e r a c t i o n of n u c l e i of spin P-g- and nonvanishing quadrupole moment with t h e i r surroundings c r y s t a l c o n s i s t s of two p a r t s !  in a  magnetic and e l e c t r o s t a t i c .  In  the absence of any external f i e l d , the i n t e r a c t i o n i s mainly the quadrupole i n t e r a c t i o n of the n u c l e i with the e l e c t r i c f i e l d gradient of the c r y s t a l , since the c r y s t a l l i n e magnetic f i e l d i s often small and contributes only t o the broadening o f the quadrupole l i n e s .  When an external magnetic f i e l d i s added, a  Zeeman e f f e c t i s introduced and both i n t e r a c t i o n s are present at the same time. In t h i s t h e s i s a study i s made of the resonance absorption spectrum of a nucleus subjected to both f i e l d s when the r a t i o of the two i n t e r a c t i o n energies assumes any given a r b i t r a r y value. A f t e r a b r i e f survey of the theory of both i n t e r a c t i o n s , and the various perturbation approximations, the problem f o r a nucleus of spin 1= 5/2 i s stated e x p l i c i t l y and a b r i e f a n a l y s i s shows that the s o l u t i o n i s p a r t i c u l a r l y simple i n the cases where the external magnetic f i e l d coincides with one of the p r i n c i p a l axes of the e l e c t r i c f i e l d gradient. For other d i r e c t i o n s of the magnetic f i e l d , the problem cannot be s i m p l i f i e d i n any obvious way and leads to much longer calculations.  numerical  The problem i s completely solved numerically i n a s p e c i a l case f o r A l 2 7 i n a spodumene c r y s t a l f o r one p a r t i c u l a r c r y s t a l o r i e n t a t i o n , but over the e n t i r e experimentally i n t e r e s t i n g range of the external magnetic f i e l d so as to f i t d i r e c t l y the conditions of an experiment proposed i n Dr. V o l k o f f ' s laboratory i n order to a i d i n the evaluation of i t s f e a s i b i l i t y .  The  expected v a r i a t i o n of the frequencies and the r e l a t i v e i n t e n s i t i e s of the resonance l i n e s as a f u n c t i o n of the a p p l i e d magnetic f i e l d i s e x h i b i t e d i n a s e r i e s of graphs.  iii M^KNOWLEDGffiNTS  The author wishes to express h i s g r a t i t u d e to Dr. G. M. V o l k o f f f o r suggesting the problem and guiding t h i s t h e s i s through numerous discussions and comments. He a l s o wants to thank him and the other members of the Department who through l e c t u r e s and discussions have increased h i s knowledge of and e n t h u s i a s m f o r Physics. L'auteur t i e n t a u s s i a remercier Le M i n i s t e r e du Bien Etre S o c i a l et de l a Jeunesse de l a Province de Quebec pour une genereuse bourse qui l u i a permis de poursuivre ses etudes pendant deux annees consecutives au Departement de Physique de l ' U n i v e r s i t e de l a Colombie Britannique,  TABLE OF CONTENTS ABSTRACT  i  &CKNOV/LEDGB/ENTS  i i i  INTRODUCTION SECTION  J.  I : PURE QUADRUPOLE SPECTRA &. Case of a x i a l symmetry B. Case of no.symmetry  8 9 12  SECTION I I : SPECTRA DUE TO COMBINED QUADRUPOLE AND ZEEMAN INTERACTIONS  16  A\. Pure Zeeman E f f e c t  16  B. Small Zeeman perturbation on the quadrupole spectra  16  C. Small Quadrupole perturbation on the Zeeman e f f e c t  20  SECTION I I I : EQUALLY STRONG QUADRUPOLE AND ZEEMSJil INTERACTIONS  22  A.. Secular determinant f o r the case 1-5/2 Bi Case o f 6 = 0°  22 26  C. Other o r i e n t a t i o n s of the magnetic field H  38  SUMMARY AND CONCLUSION  42  REFERENCES  43 LIST OF FIGURES  Fig. 1  Dependence on the asymmetry parameter of the energy l e v e l s of the nucleus i n the "pure quadrupole case" f a c i n g page  14  Fig.  2  Energy l e v e l s f o r the case I = 5/2, % = 0 and the magnetic f i e l d i n t h e ^ d i r e c t i o n of the z - a x i s , as a function of H f a c i n g page  18  Energy l e v e l s f o r the s p e c i a l case 1 = 5/2, % = 0.95 and 0-0° as a function of R, f a c i n g page  28  Square of the c o e f f i c i e n t s of the eigenfunctions f o r the eigenvalues of f i g . 3 Group L, f a c i n g page  29  The same as f i g . 4 f o r the group M, f a c i n g page  29  T r a n s i t i o n frequencies f o r the s p e c i a l case of f i g . 3 as a function o f R, f a c i n g page  32  Square of the matrix elements ( a r b i t r a r y scale) as a f u n c t i o n of R, f a c i n g page  34  Energy l e v e l s f o r the case I =5/2, % ~ 0.95 and 6=90° ( f i e l d along the x-axis) as a f u n c t i o n of R, f a c i n g page  39  t  Fig„ 3  Fig. 4  Fig. 5 Fig  e  6  Fig. 7 Fig. 8  LIST OF TABLES TABLE I :  T r a n s i t i o n f r e q u e n c i e s ( i n Mc/sec) and squares of the matrix elements ( a r b i t r a r y u n i t s ) f o r 0 R 4 , a t i n t e r v a l s of R=0.4 36 f e  TABLE I I :  t  Frequencies of the Zeeman l i n e s obtained by a quadrupole i n t e r a c t i o n perturbation theory compared with the frequencies c a l c u l a t e d d i r e c t l y f o r the s p e c i a l case of I -5/2, ^= 0.95, 6 - 0°, and quadrupole constant C = 2.96 Mc/sec, f o r R =2, 4, 8 and 20  37  TABLE I I I : Squares of the c o e f f i c i e n t s of the eigenfunctions f o r the case of the magnetic f i e l d along the x-axis and R-Q  40  z  INTRODUCTION The study of nuclear resonance absorption l i n e s i n c r y s t a l s gives information about both n u c l e i and c r y s t a l s . The properties of the n u c l e i t o be obtained are the spin I , the magnetic moment /T and the e l e c t r i c quadrupole moment eg. With regard to c r y s t a l s t h i s method gives information on the e l e c t r i c f i e l d gradient which i s c l o s e l y r e l a t e d to the symmetry p r o p e r t i e s o f c r y s t a l l i n e s t r u c t u r e (C2, T l C3), on r e l a x a t i o n f  mechanism (B2, B3, P3, P4, P5 W2) and a l s o on the p o s i t i o n of t  s p e c i a l n u c l e i (&2, I I , P I ) , A n a l y s i s of the various spectra obtained i s valuable to chemists  <&1, D l , L2, M l ) , Townes and D a i l e y ( T l )  discuss some o f the i m p l i c a t i o n s of the r e s u l t s obtained by t h i s method f o r the theory o f chemical binding. 54ich work has been done i n t h i s f i e l d during the l a s t few years.  Here we w i l l attempt only a b r i e f survey o f  that part o f the f i e l d i n which the present t h e s i s l i e s . In c r y s t a l s i n v e s t i g a t e d by nuclear resonance methods, we have t o deal with two general kinds of i n t e r a c t i o n s of n u c l e i w i t h t h e i r surroundings:  magnetic and e l e c t r o s t a t i c  interactions. Magnetic i n t e r a c t i o n s .  N u c l e i with spin  1*0  possess a magnetic d i p o l e f*. = gfi I , where g i s the gyromagnetic r a t i o o f the nucleus and 0 the nuclear magneton.  The nuclear  dipoles i n the presence of a magnetic f i e l d w i l l assume a  2, f i n i t e number of o r i e n t a t i o n s corresponding to d e f i n i t e energy levels.  The magnetic f i e l d s t o be considered are e x t e r n a l  magnetic f i e l d s t o which the c r y s t a l i s subjected i n experiments, and magnetic f i e l d s produced i n the c r y s t a l s  themselves  e  The energy l e v e l s of a magnetic d i p o l e i n the ->  presence o f a uniform magnetic f i e l d H only are the e q u i d i s t a n t Zeeman energy l e v e l s g i v i n g r i s e to a unique t r a n s i t i o n frequency Vo -  t  formed by the superposition of t r a n s i t i o n s between  adjacent l e v e l s according t o the s e l e c t i o n r u l e  Am=±l  .  U s u a l l y i n a c r y s t a l , a t the s i t e of the nucleus, other magnetic f i e l d s are a l s o present. presence of neighboring d i p o l e s .  These a r e due t o the  In some cases t h i s f i e l d turns  out to be q u i t e strong and w e l l o r i e n t e d , as when two ( P I ) , three (&2), f o u r ( I I ) prot^ons are present i n a c l o s e l y w e l l defined group and i n t e r a c t with each other.  spaced,  The net r e s u l t  i s a s p l i t t i n g of the s i n g l e Zeeman t r a n s i t i o n l i n e i n t o many components, since the energy l e v e l s are perturbed and no longer equidistant.  But on top o f t h i s e f f e c t , and even when t h i s  strong i n t e r a c t i o n e x i s t s , the other nuclear dipoles d i s t r i b u t e d i n the c r y s t a l create a t the s i t e of the nucleus under consideration a weak but n o n - n e g l i g i b l e magnetic f i e l d .  Since the  macroscopic e f f e c t t o be observed depends on a great number of s i m i l a r n u c l e i i n analogous p o s i t i o n s i n each u n i t c e l l , and since the microscopic f i e l d s can vary from one c e l l t o another because of the varying c o n t r i b u t i o n s of the other d i p o l e s , the l i n e o r  3. l i n e s are broadened ( V I ) . The e f f e c t of these two d i f f e r e n t i n t e r n a l magnetic f i e l d s u s u a l l y appears as a perturbation on the Zeeman e f f e c t and can be c a l l e d r e s p e c t i v e l y " s t r u c t u r e " and " d i f f u s e " perturbation. Elar.t.rnstat.ie I n t e r a c t i o n s .  On the other hand  c r y s t a l l i n e s t r u c t u r e exposes n u c l e i to microscopic inhomogeneous e l e c t r i c f i e l d s , repeated i n each unit c e l l , that cannot be produced i n the laboratory.  I f these n u c l e i are not e l e c t r i c a l l y  of s p h e r i c a l symmetry, they possess an e l e c t r i c quadrupole moment eQ>, which i n t e r a c t with the gradient of the e l e c t r i c f i e l d . The energy of i n t e r a c t i o n w i l l depend on the o r i e n t a t i o n of the nucleus g i v i n g r i s e to a f i n i t e number of quadrupole l e v e l s that the nuclear resonance methods can detect, provided the spacing between these l e v e l s exdeeds the l i n e due to magnetic d i p o l e - d i p o l e i n t e r a c t i o n mentioned  tiidth  above.  T r a n s i t i o n s between energy l e v e l s r e s u l t i n g from t h i s i n t e r a c t i o n alone have been observed i n cases where the f i e l d gradient does or does not possess an a x i a l symmetry. given by Kruger (Kl),Bersohn ( B l ) .  The theory has been  Many r e s u l t s can be found  i n Kruger ( K l ) , Dehmelt and Kruger (B3-D4), Kruger and MeyerBerkhot (K2, K3), Dean (D2), Livingstone ( L I ) .  From these  r e s u l t s one can obtain the value of the quadrupole.coupling constant eQ.^  where cp^ i s the l a r g e s t eigenvalue of the axes system.  tensor when t h i s tensor i s taken i n an d i r e c t i o n i n which i t i s diagonal, and the asymmetry parameter 1% . When the s p l i t t i n g between the quadrupole l e v e l s  4 i s comparable to the d i p o l e - d i p o l e i n t e r a c t i o n , the quadrupole i n t e r a c t i o n w i l l merely c o n t r i b u t e to the l i n e broadening. e f f e c t has been considered by Bersohn.  This  In p o l y c r y s t a l l i n e samples  where a l l unit c e l l s are not o r i e n t e d i n the same d i r e c t i o n but randomly d i s t r i b u t e d the t r a n s i t i o n l i n e s a r e broadened.  Even  i n s i n g l e c r y s t a l s , the f i e l d experienced by s i m i l a r l y s i t u a t e d n u c l e i may vary from c e l l to c e l l due to c r y s t a l imperfections and i m p u r i t i e s and the l i n e s a r e a l s o broadened by t h i s d i f f u s e perturbation e f f e c t . Magnetic and E l e c t r i c  Interaction  I f we subject t o an external magnetic f i e l d a c r y s t a l i n which there i s an e l e c t r i c quadrupole i n t e r a c t i o n , the Zeeman l e v e l s w i l l be perturbed, and the s i n g l e Zeeman l i n e w i l l be s p l i t i n t o many components. According to the r a t i o of the Zeeman and quadrupole energy, e i t h e r e f f e c t can become a perturbation on the other. A t one extreme we have the case where the magnetic i n t e r a c t i o n of each d i p o l e w i t h an external magnetic f i e l d produces Zeeman l e v e l s separated by energies q u i t e large compared to the s p l i t t i n g due t o quadrupole i n t e r a c t i o n of the nucleus with i t s surroundings.  In these instances, the quadrupole i n t e r a c t i o n i s  taken as a perturbation on the Zeeman e f f e c t .  This perturbation  theory g i v i n g information on the perturbed energy l e v e l s , and the t r a n s i t i o n l i n e s from which the quadrupole coupling constant, the o r i e n t a t i o n , and asymmetry of the f i e l d gradient may be  obtained, has received much a t t e n t i o n . Carr and Kikuchi ( C l ) have obtained an expression f o r the frequencies to the f i r s t order i n the r a t i o  • Pound (P6) extended i t to the  t h i r d order i n case o f a x i a l l y symmetric f i e l d , and t h i s theory has been f u r t h e r extended t o case of asymmetry by Bersohn ( B l ) and V o l k o f f et a l (V2, V3). At the other extreme, Dehmelt and Kruger ( K l , D7) have i n v e s t i g a t e d t h e o r e t i c a l l y and experimentally the pure quadrupole case and have obtained information on the quadrupole coupling constant i n case o f a x i a l l y symmetric c r y s t a l s .  The  theory was then extended to non symmetric c r y s t a l l i n e f i e l d gradient by Kruger ( K l ) and observations published by Dehmelt A  and Kruger (D6, D3, D4, D5) g i v i n g the value of quadrupole coupling constant as w e l l as the degree of asymmetry of the f i e l d * Bersohn ( B l ) gives a general expression f o r the perturbation i n terms of an a x i a l asymmetry parameter up t o the fourth order. The i n t r o d u c t i o n of a weak external magnetic f i e l d can be treated as a perturbation on the pure quadrupole levels.  Kruger ( K l ) obtained a f i r s t order expansion i n terms  of the magnetic f i e l d strength.  Observation of t h i s e f f e c t i s  reported by Kruger and Meyer-Berkhout  (K3) and Dean (D2).  Bersohn ( B l ) has c a l c u l a t e d a l s o a second order perturbation. As the external magnetic f i e l d i s increased the perturbation theory from t h i s side breaks down while the perturbat i o n theory where the quadrupole i n t e r a c t i o n energy appears as  6  a perturbation on the magnetic l e v e l s i s not yet v a l i d .  »  The  object of our t h e s i s i s f i r s t to give a b r i e f d e s c r i p t i o n o f , and then to l i n k together the two extreme regions and to show that a complete knowledge of the energy l e v e l s , and the frequencies and i n t e n s i t i e s of t r a n s i t i o n s between them can be obtained by a d i r e c t numerical s o l u t i o n of the secular determinant. Unfortunately t h i s method of s o l u t i o n can not be c a r r i e d out with much g e n e r a l i t y . Our method i s somewhat s i m i l a r to the one used by Weiss(W3) f o r the  study of the paramagnetic resonance spectrum of Chromic Mums.  In our discussion we w i l l disregard completely the d i f f u s e perturbation, magnetic and e l e c t r o s t a t i c g i v i n g r i s e to broadening ;  of the l i n e s . The purpose of obtaining the information on the dependence of the t r a n s i t i o n frequencies and i n t e n s i t i e s on the external magnetic f i e l d strength i s to a i d i n the evaluation of the  f e a s i b i l i t y of a proposed experiment of observing the  resonance absorption spectrum of fSC i n spodumene over a complete 1  range of values of the magnetic f i e l d showing the gradual t r a n s i t i o n from a pure quadrupole spectrum to the other extreme of a s i n g l e Zeeman l i n e s p l i t i n t o a number of c l o s e l y spaced components.  Such an experiment w i l l y i e l d no f u r t h e r new informa-  t i o n on n u c l e i or on c r y s t a l s which cannot be obtained by experiments i n the ranges of H where one o r the other perturbation theory i s v a l i d , but i t w i l l serve as a c o n t r i b u t i o n to the f i e l d of resonance spectroscopy.  We a l s o compare the exact s o l u t i o n w i t h a perturbation expansion, showing the l i m i t of v a l i d i t y of the l a t t e r . In section I we review b r i e f l y the p r i n c i p a l aspects of the pure quadrupole i n t e r a c t i o n both i n the presence and absence of a x i a l symmetry of the f i e l d .  In section I I , the  i n t r o d u c t i o n of an external magnetic f i e l d i s discussed. In section I I I , we obtain an e x p l i c i t form f o r the secular determinant f o r the case of 1=5/2.  We solve t h i s determinant f o r the  p a r t i c u l a r case of the external magnetic f i e l d a p p l i e d i n the VP d i r e c t i o n of one of the p r i n c i p a l axes of the o r y s t a l .  Vie then  c a l c u l a t e the t r a n s i t i o n frequencies and t h e i r r e l a t i v e i n t e n s i t i e s , and present the information i n g r a p h i c a l form.  We a l s o discuss  b r i e f l y the case of the magnetic f i e l d along the two other p r i n c i p a l axes g i v i n g an e x p l i c i t example. We conclude that the problem leads to a simple s o l u t i o n only i f the external magnetic f i e l d i s along one of these axes.  Other cases require l a r g e r amount of numerical work.  I.  PURE QUADRUPOLE SPECTRA  To obtain an expression f o r the energy of i n t e r a c t i o n of the nuclear e l e c t r i c quadrupole moment with the gradient of the e l e c t r i c f i e l d of the c r y s t a l , one must f i n d f i r s t an expression f o r the general e l e c t r o s t a t i c i n t e r a c t i o n energy o f the system.  Then i t i s p o s s i b l e t o i s o l a t e from i t  the quadrupole term F. This has been done previously (C2, P5) and here we w i l l quote only the r e s u l t s necessary f o r our calculations. F can be w r i t t e n as a complete s c a l a r product o f two tensors: F = Q>VE  (1)  Where Q i s the nuclear quadrupole tensor, and VE the e l e c t r i c f i e l d gradient tensor o f the c r y s t a l taken a t the s i t e o f the nucleus.  Using t h e methods of group theory, i t i s p o s s i b l e t o  w r i t e both tensors i n terms o f t h e i r i r r e d u c i b l e components. The i r r e d u c i b l e components o f the quadrupole moment tensor i n v o l v e a s i n g l e s c a l a r eQ and the angular momentum operators. The V£ tensor may be expressed i n terms o f the second d e r r a t i v e s of ^  e  the e l e c t r i c p o t e n t i a l .  are e a s i l y found.  Then the matrix elements o f F  The s c a l a r nuclear quadrupole moment eOj i s  defined i n the oonvential manner i n the I ( I I J X e i * f (3cc-3* 0«z)|ll)  representation by:  where the nuclear charge e- i s t  at a distance <*- from the o r i g i n and makes with the d i r e c t i o n o f t  quantization of 1^ anc' angle 0 sufficient to write  tI  • ^ot our purposes, i t i s  i n i t s simple diagonal form.  This i s  p o s s i b l e since the tensor i s symmetric  a  -^a^>  e t c  »  j  an<  * this  i n turn guarantees that a set of axes x, y, z, can be found to put t h i s tensor i n i t s diagonal form.  These axes x, y, z, are  c a l l e d the p r i n c i p a l axes o f the f i e l d gradient tensor, and i n t h i s system the i r r e d u c i b l e components of V£ a r e simply:  (2)  inhere ^  f o r instance i s the second d e r i v a t i v e of the e l e c t r o -  s t a t i c p o t e n t i a l y with respect t o z taken a t the s i t e of the nucleus. In t h i s p a r t i c u l a r case the energy operator can be w r i t t e n as:  Where I i s the angular momentum operator whose matrix elements i n the IiWj representation a r e w e l l known. Except when otherwise stated I w i l l always be understood  to be some odd multipe o f ^h.  When I i s an even  m u l t i p l e of -|-h, the d i s c u s s i o n i s s l i g h t l y d i f f e r e n t (Wl). A. Cqse of A x i a l Symmetry. The e l e c t r i c f i e l d a t the s i t e of a given nucleus i n a c r y s t a l depends on a l l the charges outside the nucleus: valence e l e c t r o n s , inner core of e l e c t r o n s , and ions at a distance of an atomic radius o r more from the nucleus i n question ( T l ) .  10. In many cases the main c o n t r i b u t i o n to the f i e l d gradient (which decreases as the inverse cube of the distance) i s from the atoms forming the u n i t c e l l around the nucleus under c o n s i d e r a t i o n , so that geometrical consideration of t h e i r charges help i n p r e d i c t i n g the nature o f the f i e l d gradient.  The simplest case of i n t e r e s t  i s when the f i e l d gradient i s a x i a l l y symmetrical  (the case of  s p h e r i c a l symmetry i s of no i n t e r e s t since then a l l components of V£ are z e r o ) . A good example of a p r a c t i c a l case i s BnCty  (Kl)  at the s i t e of Br, the symmetry a x i s passing through Br and C. Here F has a p a r t i c u l a r l y simple form since (j> x remember that  we  fty-j^if  r  X  S7'f- 0 a t the p o s i t i o n occupied by the nucleus.  Then the operator i s : =  . (311  -f )  <4)  z  In the I w j representation Fs i s diagonal and the energy eigenvalues are the diagonal elements:  £->*^ and each l e v e l i s doubly degenerate and belongs to two eigenfunctions: ^r»^ Q^d &*rty between the  adjacent l e v e l s .  transit*  0 1 1 8  c a n  D e  induced  I f we agree t o take w, >0 ,  then the allowed magnetic t r a n s i t i o n s ">v^-> Tn^-i have frequencies  which forw,-f,f^,... are i n the r a t i o 1:2:3:..; If the complete spectrum can be obtained i n a p a r t i c u l a r case, the number of l e v e l s w i l l g i v e I , while e Q<fs2 A. the quadrupole constant can be e a s i l y obtained from the knowledge  11. of t h e values o f the t r a n s i t i o n s frequencies.  The quadrupole  constant i n v o l v i n g the quadrupole moment and the biggest component i n absolute value o f the f i e l d gradient of the c r y s t a l i s c h a r a c t e r i s t i c of a p a r t i c u l a r nucleus a t a w e l l defined s i t e i n a given c r y s t a l . Let us note a t once that the v's are independent of the angle ^ that the r . f . magnetic f i e l d Hj used t o detect them, makes w i t h the z-axis o f the gradient. s p e c t r a l l i n e s on the contrary v a r i e s as  The i n t e n s i t y o f the • enabling us t o  determine experimentally the d i r e c t i o n of the z a x i s .  Cohen (C2)  has given an expression f o r the t r a n s i t i o n p r o b a b i l i t y .  We w i l l  reproduce i t showing how i t i s derived. The i n t e n s i t y of t r a n s i t i o n l i n e s between s t a t e m and s t a t e m' i s p r o p o r t i o n a l t o the square o f the absolute value of the matrix element mm* o f the time dependent perturbing operator 7^> • L e t the angular frequency o f the r o t a t i n g  field  Hj be to . The energy operator i s :  For an o s c i l l a t i n g f i e l d Hjco-aoot" l y i n g i n the x z plane and making an angle jr w i t h respect t o the z - a x i s , we have:  (8)  H^-  Hi  eeocot  12. % ^  can then be r e w r i t t e n as: (9) '/A ecou>~t~ c&> y ~J-y  when u> i s near the t r a n s i t i o n frequency between two adjacent energy l e v e l s we w i l l have induced t r a n s i t i o n s i f |w The square of the (jm^ > ^  -0-matrix  -Wjl = / •  5  elements a r e : (10)  and the i n t e n s i t y w i l l be proportional to t h i s term. i s then expected to decrease as /»y  The i n t e n s i t y  increases.  Since the i n t e n s i t y of s p e c t r a l l i n e s i s p r o p o r t i o n a l to  ^>t y t two r o t a t i o n s w i l l be necessary to l o c a t e the z a x i s . 2  The f i r s t a r b i t r a r y r o t a t i o n w i l l i n d i c a t e by i t s minimum a plane i n which the z a x i s l i e s . .  A second r o t a t i o n around an a x i s perpen-  d i c u l a r to t h i s plane w i l l g i v e the d i r e c t i o n o f the z a x i s at zero intensity. B.  Case of No  Symmetry  In cases where the s t r u c t u r e of the c r y s t a l i s such that the e l e c t r i c f i e l d gradient does not show any symmetry, i t i s u s e f u l t o describe i t s departure from a x i a l symmetry by means of an asymmetry parameter  ^ .  Taking the l a r g e s t component  i n absolute value of the diagonal tensor as f^,y J c  e  %  n o w  fittfyf  and the d i r e c t i o n s of the x and y p r i n c i p a l axes are no more arbitrary.  I t i s u s e f u l t o express them a l l i n terms of the s c a l a r  eq and t o define cA* - - fV-*) „ from which i t f o l l o w s that e  fyy = - *y C/+ °i) .  The parameter  •£ i s then defined by  13. From equation (3)  with the non-zero matrix elements i n the2 -w^ representation: ( - v n ^ l F . I ^ O = y<?  -l(L*ril  e  where p  (13)  ^(I-^ +i)atW))fr-^^fH>«^/) ?  The eigenvalues a r e obtained by the s o l u t i o n of the s e c u l a r determinant: Ifc*.^' - F  1=^  For any given l \ >  (14)  the determinant can be rearranged, leading to  the problem of s o l v i n g two equivalent equations of degree 3. * \  •  The eigenfunctions belonging t o each degenerate eigenvalue are no longer l i n e a r combinations of only two functions  a^^yu^as  in  the case of a x i a l symmetry, but of a l l the angular momentum eigenfunctions of the representation. The double Kramers degeneracy i s not removed by the i n t r o d u c t i o n o f the a x i a l asymmetry when I i s an odd m u l t i p l e of -J-h, and so the eigenfunction belonging to each eigenvalue appears as any l i n e a r combination of two sets 0-5  -  of f u n c t i o n s themselves g i v e n l i n e a r combination of 1+ •§• angular A  momentum eigenfunctions. The r i g h t l i n e a r combination of the two sets can be found to f i t continuously when the degeneracy i s removed by some magnetic f i e l d . P e r t u r b a t i o n c a l c u l a t i o n s can be c a r r i e d out when the departure from a x i a l symmetry i s small, i e . when  << I •  1  1  1  _  0.25  1  1  1  1  1  1  — _  b'A/ltT PERTVK BAT/OV A PPROxi MAT/OA/ 0.21 0.20  1  T  1  0.2  u  0.  1  1  1  1  1  0.1 3  1  1  i  0.6 1  H  i  i  0,8 1  1  I.O 1  Figure 1. Dependence on the asymmetry parameter of the energy l e v e l s of the nucleus i n the "pure quadrupole case". f a c i n g page 14.  1  14. These c a l c u l a t i o n s have been c a r r i e d through to the f o u r t h order i n i j by Bersohn ( B l ) .  For a l l p r a c t i c a l purposes an i t e r a t i o n  procedure gives a l s o an expansion i n power of ^ r a p i d l y f o r small value of  .  2  which  converges  This of course requires the  e x p l i c i t expression of the s e c u l a r equation of degree I-f equation (14). namely:  from  Bkny cases of i n t e r e s t have been i n v e s t i g a t e d  1-3/2, 5/2 ( K l ) , 1=7/2  (D7) and 1-9/2  (C2).  Cohen  obtains h i s approximations using a method of continued f r a c t i o n s . As an example we have p l o t t e d i n f i g . 1 the energy eigenvalue Ec =  ( i - 1, 2, 3) as a f u n c t i o n of-£  f o r the s p e c i a l case 1-5/2  and compared the values obtained by  exact s o l u t i o n of the s e c u l a r determinant with those obtained by i t e r a t i o n expansion up to the f o u r t h order i n «g  .  The s e c u l a r  equation i s : 3  (15)  v  w h i l e the expansions are:  (16) \  ~ - S ('  + O.HW  ^rrO  zdi)^  )  We can see that the expansions are reasonably accurate f o r ^<o.7, The use of higher terms i n ^ w i l l not improve the s i t u a t i o n f o r y  >0-7 (except f o r the highest eigenvalue) because a d d i t i o n of  terms w i l l r e s u l t i n an o s c i l l a t i o n around the true v a l u e .  15. No simple a l g e b r a i c formula f o r t r a n s i t i o n frequency has been obtained.  I n p r a c t i c e the t r a n s i t i o n frequencies are  more e a s i l y obtained by c a l c u l a t i n g the energy l e v e l s e i t h e r by exact s o l u t i o n of the determinant equation (14), o r by i t e r a t i o n method o r e l s e by pertabation theory ( B l ) (the f f t i w ^ ) have been tabulated f o r h a l f i n t e g r a l values of the spin by Bersohn i n h i s t h e s i s - o r can be e a s i l y computed) and then by taking t h e i r d i f f e r e n c e s divided by h. T r a n s i t i o n s take place between adjacent l e v e l s o n l y , when  i s small, though other t r a n s i t i o n s have a  small b'uf not n e g l i g i b l e t r a n s i t i o n p r o b a b i l i t y as soon as ^ ±0 When ^ a ^ i  t  „  information about t r a n s i t i o n p r o b a b i l i t i e s can be  obtained by exact s o l u t i o n of the secular determinant, equation (14), which gives compatible simultaneous equations f o r the eigenfunctions. The knowledge of the u n i t a r y matrix that diagonalizes  enables  one t o f i n d the matrix elements of the perturbing operator of which the squares are p r o p o r t i o n a l t o the t r a n s i t i o n s p r o b a b i l i t i e s . The procedure i s explained i n some d e t a i l s i n Cohen s t h e s i s , and w i l l not be repeated here since i t gives no simple way of f i n d i n g the d i r e c t i o n of the p r i n c i p a l axes contrary to the s i t u a t i o n when the f i e l d i s a x i a l l y  symmetric.  16. I I . SPECTRA DUE TO COMBINED OnADRnPOLE AND ZEEMAN INTERACTIONS When a nucleus with spin I i s placed i n a constant —•>  magnetic f i e l d H. i n the absence of a l l other i n t e r a c t i o n s the Hamiltonian has the simple form: y  hU  - - 3 f> So •?  Where 1 A.  (17)  i s the angular momentum operator i n the^-Wj representation.  Pure Zeeman E f f e c t  When n u c l e i w i t h spin I and zero  quadrupole  moment located i n a c r y s t a l are subjected to a constant e x t e r n a l magnetic f i e l d  H  e  , the net e f f e c t i s the appearance f o r each  nucleus of d e f i n i t e e q u i d i s t a n t energy l e v e l s , i f we neglect a l l kinds of d i p o l e - d i p o l e i n t e r a c t i o n s mentioned i n the I n t r o d u c t i o n * The s i n g l e frequency of t r a n s i t i o n observed i s n formed by the superposition of the 21 between the 2 l * i B.  and i t i s  allowed t r a n s i t i o n s  Zeeman l e v e l s of each nucleus.  Small Zeeman P e r t u r b a t i o n on the Quadrupole Spectra We want to consider here the e f f e c t of t h i s —>  e x t e r n a l magnetic f i e l d H. when i t i s imposed on a c r y s t a l i n which we have the quadrupole i n t e r a c t i o n discussed i n the previous section and to study what happens when Ho assumes l a r g e r and l a r g e r constant values.  When H  i s s m a l l , the energy due to  Zeeman e f f e c t i s small compared with the energy of the coupling.  quadrupole  As H« increases, the energy of the two i n t e r a c t i o n s —->  becomes comparable and f i n a l l y , f o r s t i l l l a r g e r value of H,  ,  17. the Zeeman energy i s greater than the quadrupole energy.  In t h i s  way the problem n a t u r a l l y s p l i t s i n t o three regions of i n v e s t i g a t i o n , &t both extremes we have the s i t u a t i o n where e i t h e r e f f e c t can be considered as a perturbation on the other, and i n between, the region where both e f f e c t s are e q u a l l y important. "We w i l l f i r s t t r e a t the case of a c r y s t a l placed i n a weak magnetic f i e l d perturbing the quadrupole spectra. To compare energies we use the f o l l o w i n g u s e f u l ratio:  The magnetic energy given by equation (17) has i t s  maximum absolute value when the spin i s almost a l i g n e d with the magnetic f i e l d , i e . when the magnetic quantum number >r?j=l . The quadrupole energy a l s o has i t s maximum energy when w =1 3  .  So f o r the r a t i o of magnetic t o e l e c t r i c quadrupole energy we s h a l l use: 4 % ""3 l s  Here again we introduce the u s e f u l d i v i s i o n between a x i a l l y symmetric and non symmetric f i e l d gradient cases. case of an a x i a l l y symmetric f i e l d gradient -  h i ; - T*7  In  «• o) the Hamiltonian i s (18)  —•>  I f the e x t e r n a l magnetic f i e l d W. coincides with the d i r e c t i o n of symmetry a x i s , the problem can be solved e x a c t l y f o r any value of H. , since }f i s diagonal and the energy l e v e l s are simply: 6», r 3 * i - i a + . ) 7 -9,4 (19) s  A l l other cases appear as more o r l e s s important deviations from t h i s very simple case. extent.  For t h i s reason we w i l l discuss i t to some  the magnetic f i e l d i n the d i r e c t i o n of the z - 9 x i s as a f u n c t i o n of H. f a c i n g page 18. t  The i n t r o d u c t i o n of the magnetic f i e l d has s p l i t the  previous doubly degenerate quadrupole energy l e v e l s i n t o two  non degenerate ones equally spaced from the o r i g i n a l by +-Wjl-^ft^ » fl  We have sketched roughly i n f i g . 2 the energy spectra*as a function of H f o r the case of I = 5/2. 0  For large values of 7Z we obtain Q  the Zeeman spectrum perturbed by comparatively small quadrupole interaction. There are 21 allowed t r a n s i t i o n s between the 21 + 1 l e v e l s , since each energy eigenvalue belongs to only one eigenfunction  of the J m ^ representation.  These frequencies are  given by:  'J  =•  »  (20)  and When Hq o r R i s l a r g e , and when we are w e l l i n the region i n which the Zeeman energy i s large compared to quadrupole energy, the spectra f o r h a l f i n t e r g r a l I can be described i n terms of the c e n t r a l component v.  which i s  )te>^ > P2) 3  surrounded by (21 - 1) s a t e l l i t e s appearing symmetrically i n p a i r s on each side of i t . —>  I f the r-A magnetic f i e l d M, i s perpendicular to N> as it u s u a l l y the case, the r e l a t i v e i n t e n s i t y of each l i n e i s proportional t o : JL ^ " j - O j  (21)  and so f o r the region of large magnetic f i e l d H the c e n t r a l 0  component "^»0$> ) i s the most intense with the f i r s t , second.  19. e t c . , s a t e l l i t e s on each side appearing with decreasing i n t e n s i t i e s . As soon as we depart from t h i s very s p e c i a l case, the problems become much more i n v o l v e d . I f +>=0 but the constant external magnetic f i e l d H  Q  i s no more p a r a l l e l to the z a x i s of the  c r y s t a l we already have complications. Assume that the magnetic f i e l d l i e s i n the x-z plane (where the x a x i s i s a r b i t r a r y ) and l e t the angle between the d i r e c t i o n o f H  Q  and the z a x i s be 0 .  Then from equation (18) we have f o r the Hamiltonian:  with the matrix elements given i n equation (5) supplemented by the new elements: C ^ / % / * * } ) = ~f/ * H  *»> *<*>^  (23)  The energy l e v e l s a r e obtained e x a c t l y by solving the s e c u l a r determinant of  }i  $  f o r E which i s of the form:  \Ji ^y»'^ - E &*<y>*')\ %  But i f R« ) a perturbation c a l c u l a t i o n (degenerate case) can be performed i n which the quadrupole part i s considered as the unperturbed Hamiltonian and the magnetic as the p e r t u r b a t i o n . The expression f o r the energy l e v e l s a r e given by Kruger ( K l ) who a l s o discusses the spectra t o be obtained.  He shows that we  have f o r c e r t a i n o r i e n t a t i o n of the c r y s t a l a non n e g l i g i b l e t r a n s i t i o n p r o b a b i l i t y between l e v e l s other than adjacent - which a t f i r s t glance seems to v i o l a t e the s e l e c t i o n rules f o r magnetic dipole t r a n s i t i o n s .  Expansions o f t h i s type hold as long as the  energy l e v e l s have not already s t a r t e d crossing each other.  20. The next and l a s t complication comes i n when we consider that the e l e c t r i c f i e l d gradient i s not a x i a l l y symmetric i  ' ) c>  • ^  Hamiltonian i n t h i s case i s given by equation (18)  Iie  to which we add the asymmetric part of the quadrupole i n t e r a c t i o n :  M  A  -&  +  (if - ip %  ( 2 4 )  The problem can be solved exactly with the r e s u l t i n g secular —>  determinant.  But again i f R, i s small the energy l e v e l s can be —•>  obtained by an expansion i n powers of K around the unperturbed energy l e v e l s which are i n t h i s case the quadrupole l e v e l s that can be found i n case of a x i a l asymmetry by the methods discussed i n the previous s e c t i o n , e i t h e r approximately procedure i s c a r r i e d out completely  or exactly.  This  to the t h i r d order by  Bersohn ( B l ) . H i s expansion requires that//I ^  and & (the  ->  angle between H„ and the z a x i s ) be small.  Perturbation theory  along t h i s l i n e i s cumbersome and does not i n general lead t o formulae of great a p p l i c a b i l i t y . C.  Small Quadrupole Perturbation on the Zeeman E f f e c t When the Zeeman energy i s large compared to the  quadrupole energy, i e . when/*^ » ^ f y i e  t  then the quadrupole e f f e c t  can be considered as a perturbation on the Zeeman pattern (section I I &).  This perturbation theory has been e x t e n s i v e l y  t r e a t e d by various authors.  The spectrum i s u s u a l l y described  i n terms of the c e n t r a l component V. , which appears when I i s h a l f i n t e g r a l , and which can be s h i f t e d , and p a i r s of s a t e l l i t e s , whose distances from the c e n t r a l components are functions of the  21. orientation  of the c r y s t a l with respect to the constant magnetic  field H  By r o t a t i o n of the c r y s t a l i n the f i e l d H  0  .  orientation  a  of the p r i n c i p a l axes can be recognized.  , the Explicit  expressionsfor t h i s perturbation can be found i n the l i t e r a t u r e f o r a x i a l l y symmetric (P6) as w e l l as f o r the asymmetric (V2, V3) cases.  I I I . EQUALLY STRONG QUADRUPOLE AND ZEEMN INTERACTIONS We a r e now l e f t with the problem of a double i n t e r a c t i o n where the e l e c t r i c quadrupole i n t e r a c t i o n i s comparable with the Zeeman energy.  This problem requires the complete  s o l u t i o n of the secular determinant, since as we have seen i n the previous section both perturbation theories break down i n that region.  I n t h i s section we w i l l proceed to set up t h i s secular  equation i n the case 1 5/2 and to give the s o l u t i o n f o r two 55  p a r t i c u l a r l y simple cases.  We have used i n the discussion that  f o l l o w s the data obtained i n Dr. V o l k o f f * s laboratory f o r the case of Al27 i  na  spodumene c r y s t a l and have stated our r e s u l t i n a  form that w i l l f i t d i r e c t l y a proposed experiment. A. Secular determinant f o r the case I = 5/2. We have set up i n eq. (24) the Hamiltonian f o r a nucleus with spin I , magnetic moment h , quadrupole moment eQj, placed i n a c r y s t a l with non a x i a l l y symmetric f i e l d gradient submitted to a constant external magnetic f i e l d H making with the z-axis an angle 6 i n the x-z plane.  I t should be emphasized at t h i s  point that the p a r t i c u l a r l y simple form of eq. (24) r e s u l t s from the f a c t that the xyz axes have been chosen to coincide with the p r i n c i p a l axes o f the f i e l d gradient tensor, which accounts f o r the absence of cross terms of the form I  x  I , etc. v  what the non-zero matrix elements are from equations  We a l s o know (19) and (23).  Examination of the problem reveals that we have to deal with 4 parameters: 1,^ , B , R.  Moreover the secular equation to be  solved i s o f degree 21 + 1 i n general, which means f o r our p a r t i c u l a r choice of I , the 6  t n  degree; hence there i s no hope o f  23. obtaining a general a n a l y t i c s o l u t i o n . We want t o f i n d the s o l u t i o n of the time-independent Schrodinger  equation; (i-1,2,3,4,5,6)  (25)  where "^i i s a l i n e a r combination of the eigenfunctions of the I,m , representation. Let the l a t t e r be l ^ - r ^ f / ^ - i , 4i z  ,tyi,  }  ~  then: /6  =  . Qi  ™  vi.-, J  (26)  m  Eq. (25) can be thought of as a matrix equation where J-J i s a A  6X6 matrix and J^i a column matrix of the form a  i,5/2  a  i,3/2  l  i.l/2  a  i.-l/2  a  i,-3/2 4,-5/2  the e q u a l i t y sign i n eq. (25) w i l l hold f o r c e r t a i n values of E j only, those values which make the system of simultaneous  equations  compatible; i e . , the E. which make the determinant of the c o e f f i c i e n t s vanish.  In matrix language t h i s a l s o means that  we f i n d the new representation i n which ^  A  i s diagonal; the  u n i t a r y transformation by which t h i s i s t o be e f f e c t e d , S, w i l l be given by the c o e f f i c i e n t s a j » m  24, For 1 = 5/2, the secular determinant i s : a  g  h  0  0  0  g  b  i  J  0  0  h  i  c  k  J  0  0  J  k  d  i  h  0  0  J  i  e  g  0  0  0  h  g  f  where a - -El  (27)  +A  •5"  i=. i- 3 6i«>e __A_  b •= - Ei - 3/3a<so& _ A  L  — F  T" h*  56^  " 7 ^ i ,_ y ?  *  and B  A t  =yuA/  The 73 terms o f the expansion reduce t o the f o l l o w i n g 6th degree equation:  where  c  =1  5 = G  c  4  c  3  200 - _ i L 200  200  28AB g 125 2  +  2  _ 56AB p 125 2  2  125  c  ZOO  '  C  _ 259^ - 6> B p + 625 125 2  2  2  123ft B q 2500  2  2  . / 17A2 2p2 _ 1 2 A M 0 * \ 625 2500 - _ 3B p g 625 5  3  88AB p 625  2  4  +  7&3 2p2 1000  f  c  Q  m  , /l2^B2p2 ( 5000 7&5 80000 . _  3B g i 5000 2  B  106AB p2 2 625  +  ^ ^ p q 625  q  _  5  _ MpjEq 2500  2  A\ B g 2500 3  2  2  2  ^ *  )* '  2  f  _  4  , 77A B p 5000 4  i  , / 9A B p I 5000 4  2  _ J27B6p qi ^ M B ^ a 625 2500 2  A2B4p2g2 _ A M q 1250 5000  4  &6 160000 4  1250 13& B p 80000 4  +  2  4  2  q  B  and where p H C O S 6  4  _  4000  T  _ 47A B g 80000  2  ^  2  _ 161A B2p2 _ 41&2B2q2 160000 160000  4  1250  . A\ B 2 3 __ / o&4 2 4000 ' \ 160000 4  +  '  4  7A\ I t 40000 ' '  q  2  4  2^1 + / i ^ A B l q 8000 V 625  2  2  e  _ 4 4 ^ ^ 625  _ 9B^c£ _ M ^ p i q 625 625  2  2  /  fiSbi 625 2  81A B g „ 1250  U%49& 160000  A\3 2 2 200 B  41A 160000 4  4  4  f  4  2  147& 80000  2  B  Ci  2  2  ' <  . 17A ^ 5000  2  f  2  4 q  _  A\ ) 80000/ 6  . 2  &6 \ ^ 160000 / c  and q =• s i n 6  (28)  The s i x eigenvalues of the problem are the roots o f t h i s equation. Once they are known, the a j > m  ;s  can be found with  any f i v e o f the s i x simultaneous equations (25) and t h e i r values  26. i n terms of any one of the a's with an a d d i t i o n a l condition of normalization: if  The determinant has a p a r t i c u l a r l y simple form i n the case of B- 0°, since the Zeeman energy i s then diagonal i n the representation chosen.  We w i l l see l a t e r that i t i s possible  to apply a transformation and t o obtain a determinant of the same form i n case where the f i e l d H i s i n the d i r e c t i o n of the two other p r i n c i p a l ax£s (section I I I C ) . Two parameters R a n d ^ are then l e f t that are to be f i x e d l a t e r . B. Case of 0^0° The determinant (27) has the form:  with:  a'  0  h*  0  G  0  0*  b'  0  j'  0  0  h'  0  c*  0  j'  0  0  J'  0  d'  0  h*  0  0  j'  0  e'  G  0  0  0  h'  0  _r +  a' b' c  f\TCo  -  >  1  -jr.  - - >; _  «•  c.  5"  j*  ~  (30)  0  f d'  =  e'  -  5-  <5  o  SLO  i 3"  = Vl_  2o\fz  A  1  27. By interchange of columns (the 3 d , one t o the l e f t r  and the 5th, two to the l e f t ) and of rows (the 3*d one up and the t  5 , t h  two up) det (30) now looks: a'  h*  0  0  0  0  h\  c*  j'  0  0  0  0  j*  e'  0  0  0  0  0  0  b'  j'  0  0  0  0  j'  d'  h'  0  0  0  0  h'  f*  L  0  0  M  (31)  or i n block determinant:  =  0  (32)  So the problem i s reduced i n t h i s case to the s o l u t i o n of two 3X3 determinants g i v i n g us the required s i x roots of (31). I f now we expand the two determinants we have two independent cubic equations to be solved: From  L > 4  and from  8  - ± r X \ ( j L r * - £ r . j ' \ k ' \  • - f+JL r>T\ (jl n'+JL n + f't £55-  L?0  J  *  :  f r y X  4'\n J  166  1  ' *  ) 1 J  ( 3 4 )  20  J  The s u p e r c r i p t s are being placed to i n d i c a t e i n the f o l l o w i n g discussion from which determinant the roots come from, since as we w i l l then see each set of three roots belongs t o a p a r t i c u l a r set of eigenfunctions.  28. Equations (33) and (34) can now be.solved e x a c t l y f o r ) as a f u n c t i o n of P function of ^  f o r a p a r t i c u l a r value of ^  f o r a p a r t i c u l a r value of T  ,  The A  or as a being  eigenfunctions of a p h y s i c a l problem are r e a l , and so the cubic equations have to be solved by trigonometric method, but the a n a l y t i c expressions obtained do not f u r n i s h a l g e b r a i c expressions that can be handled e a s i l y . As we have already stated we proceed to solve the problem with numerical values of immediate i n t e r e s t i n an a c t u a l experimental case.  We r e f e r to the case of A1^7  i n spodumene  c r y s t a l , f o r which the o r i e n t a t i o n of the p r i n c i p a l axes xyz of the f i e l d gradient tensor, the a x i a l asymmetry parameter ^=  0.95,  the magnetic moment j? and the quadrupole constant ^ ^ 3 ,  =  it  are known.  2.§tA**/  sec  h K  Using these data we i n v e s t i g a t e the behavior of the  eigenvalues, t r a n s i t i o n frequencies and p r o b a b i l i t i e s as a f u n c t i o n of the e x t e r n a l magnetic f i e l d H which can assume any constant value from 0 upwards, thus covering the complete range of energy —*  r a t i o R.  The r . f . f i e l d Hj, used to detect the t r a n s i t i o n l i n e s  i s assumed to be at 90° from H. Having i n s e r t e d  •= 0.95 i n equations (33) and  (34)  we have solved numerically these equations f o r X as a f u n c t i o n of r = J ?  .  The eigenvalues are p l o t t e d i n f i g u r e 3 f o r 0 * R t 4 . The three regions mentioned i n s e c t i o n I I  are  O - R ^ l , where the perturbation theory of a small Zeeman perturbation on a l a r g e quadrupole e f f e c t holds; R ^ l where no perturbation theory can be a p p l i e d s u c c e s s f u l l y , and R>>1  where the perturbation  theory of the Zeeman l e v e l s by small quadrupole i n t e r a c t i o n a p p l i e s successfully.  Figure 5.  Square of the c o e f f i c i e n t s of the eigenfunctions f o r the eigenvalues of f i g . 3, as a f u n c t i o n of R Group M. f a c i n g page 29»  Figure 4.  Square of the c o e f f i c i e n t s of the eigenfunctions f o r the eigenvalues of f i g . 3, as a f u n c t i o n of R Group L. f a c i n g page 29  29. I t can be seen a t once that the behavior of the eigenvalues i s d i f f e r e n t from that we have sketched i n f i g . 2 f o r the simplest possible case of double i n t e r a c t i o n s .  The eigenvalues  do not a l l cross each other, but some of them do, while others j u s t come near and repel each other as R increases. A c l e a r understanding of t h i s behavior w i l l be a v a i l a b l e when the eigenfunctions w i l l be found. By considering how equations (33) and (34) were a r r i v e d a t , i e . , how the det (31) was obtained i t should be c l e a r that the \  belong t o a l i n e a r combination of  while the \  belong t o a l i n e a r combination of  an<  * ^£  , t/J ^  and lps .  The c o e f f i c i e n t s of the eigenfunctions belonging t o the "X"; are found through equation (25):  4i>-\  »  V 2  „  (35)  A s i m i l a r expression can be derived f o r the eigenfunctions belonging t o the ^ i . The  covpespon  Aintj  coetficia-nts  Now, the squares of the a$ « m  s  are  denoted  by  6's  In  w i l l g i v e the  p r o b a b i l i t y f o r the nucleus t o be found i n the state  l|6 when  i t s eigenvalue i s ^« . We have p l o t t e d i n f i g . 4 and 5 these p r o b a b i l i t i e s f o r a l l A as a f u n c t i o n of R between 0 and 4. t  In the case of pure quadrupole i n t e r a c t i o n (R= 0) with  the degenerate s t a t e with the highest eigenvalue  consists almost e n t i r e l y of rV and^-£, while the p r o b a b i l i t y of f i n d i n g the nucleus i n the states 1/4  «  and  i s less  f^.S.  30. altogether than 5%.  The s t a t e of intermediate energy consists  mostly of tyz and the states ^  and  , and the chance of f i n d i n g the nucleus i n ty-±  z  and  in  and  i s about 18%.  Finally  the state of lowest energy consists mostly of l / i and  , but  again there i s a c o n t r i b u t i o n of 18% from ^ | andf'-i  , and 1%  from l ^ i and  • So we now can expect a non-zero t r a n s i t i o n  p r o b a b i l i t y from the highest to the lowest l e v e l s , which was completely forbidden  i n the case of a x i a l symmetry.  In t h i s l a t t e r  case the highest eigenvalue would belong e n t i r e l y to ^ and ty-i  the middle to  9  and the lowest to  and  andl/^-c , ; so that  t r a n s i t i o n s between the highest and the lowest l e v e l s would be s t r i c t l y forbidden by the s e l e c t i o n r u l e f o r magnetic dipole transitions.  In t h i s case also the i n t r o d u c t i o n of a magnetic  f i e l d H along the z-axis would s p l i t the degenerate l e v e l s as i n d i c a t e d on f i g u r e 2, r e s u l t i n g i n the crossing of the eigenvalues. In the present case, however, the s i t u a t i o n i s quite different.  The states are divided i n t o two classes (or "races"  to employ the term that H e i t l e r f o r instance uses i n a s i m i l a r situation)(H').The c l a s s L consist of a l i n e a r combination of ^-1 , l^i of  and  , while the c l a s s M c o n s i s t of a l i n e a r combination  , ^ - i and  .  Levels belonging to d i f f e r e n t classes may  cross each other, but l e v e l s o f the same c l a s s w i l l " r e p e l " each other. of R. L  1  Let us study c l o s e l y the states of c l a s s L, as a function I f we consider graph (a) i n f i g . 4, we can see that the state  up to R -0.4 c o n s i s t s mostly o f l^c . Then as R increases i t  becomes mostly characterized by increases.  , the more so as R f u r t h e r  So we would be j u s t i f i e d to l a b e l t h i s eigenvalue  31. for  s u f f i c i e n t l y large values of R as t h i s i s how we u s u a l l y l a b e l  i t i n the pure Zeeman e f f e c t ; i n other words, i f i t were not f o r the a x i a l l y asymmetric quadrupole i n t e r a c t i o n t h i s eigenvalue would belong uniquely to  ^-\«  A consideration of the s t a t e Lg now i v i l l e x p l a i n what exchange of character has taken place and how the phenomenal of "repulsion of l e v e l s of the same c l a s s " can be explained i n terms of the eigenfunctions. up to R~0.4. amount of ^  k> i s mostly characterized by  Then as R increases i t s eigenfunction shows a large .  L j and Lg have exchanged t h e i r respective r o l e s .  But eventually Lg would become a  Zeeman l e v e l , and i t i s  c l e a r that i t must again exchange r o l e t h i s time with L3. we can see again by looking at the graphs i n f i g .  This  4.  The states of c l a s s M c o n s i s t of l i n e a r combinations of the  . ^ ' i ; and ita. ; a s i m i l a r a n a l y s i s of t h e i r behavior  can be made with the help of the three graphs of f i g . 5. eigenvalues  The  of c l a s s M w i l l cross these of c l a s s L, since there i s  then no p o s s i b i l i t y of an exchange of characters, whereas among themselves they w i l l only come near each other and repel each other as R increases.  Mj, c o n s i s t i n g mostly when R =0,  of  is  already i n i t s proper p o s i t i o n f o r the Zeeman e f f e c t and i t s c h a r a c t e r i s t i c s simply become stronger and stronger as R increases. TRANSITION FREQUENCIES: magnetic d i p o l e t r a n s i t i o n s |m - m'|  The s e l e c t i o n r u l e f o r  - 1 i n d i c a t e s at once that  t r a n s i t i o n s are to be expected only between l e v e l s belonging  to  d i f f e r e n t c l a s s e s , but not between l e v e l s of the same c l a s s , i e . ,  32. only between the L and the M l e v e l s . as shown i n f i g . 3.  There are 9 such p o s s i b i l i t i e s  F i v e of them, numbered from 1 to 5 a r e the  expected Zeeman t r a n s i t i o n s , those l i n e s that represent t r a n s i t i o n s between adjacent l e v e l s .  The f o u r other t r a n s i t i o n s are from t h i s  point of view "forbidden".  Three of them, numbered from 6 to 8  are suras of three Zeeman l i n e s and the l a s t , 9, i s obtained by adding together the 5 Zeeman l i n e s . We have p l o t t e d i n f i g u r e 6 the frequencies of these 9 t r a n s i t i o n s as functions of R.  The frequencies i n Mc/sec have  been f i x e d with the help of the value of the quadrupole for & 1  2 7  i n spodumene 0?2)i 2.960 Mc/sec.  f i v e n by Vc; =H X -  constant  The frequencies are  I *  *  h  The c e n t r a l Zeeman c o m p o n e n t ^ (the one which i n equation (20) was c a l l e d only when R>1.6.  V. ) takes on i t s usual s i g n i f i c a n c e  For large magnetic f i e l d s i t i s the c e n t r a l  l i n e discussed i n the l i t e r a t u r e on quadrupole perturbation on the Zeeman e f f e c t . Of these 9 l i n e s , some w i l l be too weak to be observed, while others w i l l escape detection because of t h e i r very low frequencies. TRANSITION PROBABILITIES: t r a n s i t i o n s between two energy l e v e l s ,  The net number o f and Mj i s p r o p o r t i o n a l  to the d i f f e r e n c e i n population o f the two l e v e l s and to the square of the absolute value of the i j - m a t r i x element of the time dependent perturbation operator causing the t r a n s i t i o n s .  The  energy d i f f e r e n c e involved i n the t r a n s i t i o n s between the various  o r i e n t a t i o n s of the n u c l e i i s very small compared t o the energy o f thermal a g i t a t i o n . where N  Q  The population of the l e v e l s i s given by f v i ^ l U ^ *  T  i s the population of the ground l e v e l and E j i s the energy  d i f f e r e n c e between the given l e v e l and the ground l e v e l . E^^<fel f o r a l l i  f  Since  a l l l e v e l s are almost equally populated,  their  population being given i n f i r s t approximation by Nj = N (|- ^ 0  )  Hence the f r a c t i o n a l d i f f e r e n c e i n population between the j ^ a.t room temperature 4  and the ground s t a t e i s given >i =-30 Mc/sec.  byj^J  (of the order of 10"^ i f  Thus the net number of t r a n s i t i o n s between two  states w i l l depend on the frequency of t r a n s i t i o n and on the square of the absolute value of the matrix element. Since the r e l a t i v e i n t e n s i t y of the expected  lines  i s thus seen t o be determined t o a l a r g e extent by the square of the absolute value of the matrix element we now proceed t o c a l c u l a t e i t and t o p l o t i t as a f u n c t i o n of R f o r each o f the I l i n e s f o r which i t does not vanish i d e n t i c a l l y . I f the r - f magnetic f i e l d H,is perpendicular to H  t  the perturbing operator i s simply  ^  - Ml [ (Z « T / ; e- t; C I / - f y ' t e ^ J  (36)  UZ  and 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 given by: W-^j  \Ci  "  Ify'l  (37)  where PT£ i s taken i n the system o f representation which diagonalizes p-/  A  . But we already know the u n i t a r y transformation S which  diagonalizes j V «= S-'# ?. A  4  now that we know e x a c t l y the a j i . m  s  Hence: (38)  34. And so:  W^*  K'lS-'Of. -HTy)S/pjV  (39)  We have p l o t t e d with an a r b i t r a r y ordinate the Wj_>j i n f i g . 7 as a f u n c t i o n of R between 0 and 4, f o r the 9 l i n e s .  They appear  i n two groups.  In a) we have the f i v e Zeeman l i n e s , and i n b) the  f o u r "forbidden  lines". Of the Zeeman l i n e s , v, >y and V z  3  a r e very weak i n  the region of l a r g e quadrupole e f f e c t and t h e i r t r a n s i t i o n p r o b a b i l i t y increases r a p i d l y around R ~ 0 , 6 v., and V - a r e expected t  to be observed f o r a l l values of R through  s  might be too weak i n  region 0.8 < R < 1.6 and V - of too low frequency f o r a l l R  <4.  s  When R>2.8, we have the same kind of spectrum as observed f o r the case of a large Zeeman energy perturbed by a small quadrupole e f f e c t , (P.6).  The c e n t r a l component  v  3  u s u a l l y denoted by >>*  has the greatest i n t e n s i t y , while the inner s a t e l l i t e s are weaker but o f equal i n t e n s i t y , and the outer s a t e l l i t e s weaker s t i l l . I t i s to be remarked that the s o - c a l l e d "forbidden l i n e s " received t h e i r name from the f a c t they correspond to t r a n s i t i o n s between energy l e v e l s that are not adjacent.  But the  d e t a i l e d a n a l y s i s we have given above of the eigenfunctions  should  make i t c l e a r that they are as allowed as any other inasmuch as they contain a c e r t a i n amount o f where jm - m*\ =1.  f o r one of them and  ty**'  f o r the other,  For instance, inspection of graph c ) f i g . 4,  and of graph b) f i g . 5, w i l l show that a t r a n s i t i o n i s to be expected between l e v e l s / and A$ , since a t r a n s i t i o n between the 2  eigenfunctions  i n Mg and  i n Lg, as w e l l as a t r a n s i t i o n  35. between  i n Lg and  i n Mg a r e p o s s i b l e . I n f a c t these t r a n s i t i o n s  are responsible f o r the large i n t e n s i t y of the l i n e V graph b) of f i g u r e 7 when 0tR<r2.8.  7  as shown by  Of the forbidden l i n e s v  might a l s o be expected i n the region of small R.  6  I t i s one of the  components of the allowed t r a n s i t i o n s f o r the pure quadrupole, the other being V  a  . The forbidden t r a n s i t i o n of the pure  quadrupole which i s i n f a c t formed by the superposition of and ^  i s very weak.  I t s components w i l l not be detected when H  increases, except perhaps V  s  around R~0.8.  More p r e c i s e information oa the frequencies and the squares of the matrix elements w i l l be obtained from Table I f o r R from 0 t o 4, a t i n t e r v a l s of 0.4. On the other hand we have Out t r i e d to b r i n g ^ t h i s information on the graph of f i g . 6 by using p l a i n l i n e s f o r r e l a t i v e l y intense t r a n s i t i o n s ( > 1 i n aiir u n i t ) and dashed ones otherwise. COMPARISON WITH PERTURBATION THEORY: .It i s i n t e r e s t i n g to compare the approximation given by a t h i r d order perturbation theory expansion of the type used f o r large R.  Using  the formula derived by Drj V o l k o f f , (V#) f o r the frequencies we have tabulated i n Table I I the values so obtained as compared to the exact s o l u t i o n obtained through a s o l u t i o n of the s e c u l a r equation f o r R=2.0, 4, 8, 20. In our case the perturbation formula used f o r the 5 Zeeman t r a n s i t i o n s reduces t o : . V ^ » . , - V j | -4 22HiL A t ff  with,  )-  <  1  7 2  c  ! ( w  ) T^(i-m-Oktl^)  /vy  J  (40)  36. We have again made use of the value 2.96 Mc/sec f o r the quadrupole coupling constant.  I t i s seen from Table I I that the expansion i s  c e r t a i n l y v a l i d f o r R>4. For a l l values of R>4, there aV€ no d i f f i c u l t i e s and the problem i s more e a s i l y solved by a perturbation expansion of t h i s type.  Then only the transitions between adjacent l e v e l s  are allowed and t h e i r frequencies can be c a l c u l a t e d w i t h s u f f i c i e n t accuracy by t h i s shorter method.  TABLE I  V  R  V  0.0 0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2 3.6 4.0  0.000 0.572 1.035 1.190 1.320 1.446 1.569 1.691 1.813 1.935 2.054  0.107 0.230 4.064 4.959 5.000 5.011 5.014 5.013 5.013 5.012 5.009  0.789 0.623 0.531 0.702 0.861 1.003 1.135 1.261 1.385 1.507 1.627  0.243 0.560 5.483 6.983  0.0 0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2 3.6 4.0  0.758 0.888 0.963 0.789 0.643 0.542 0.504 0.533 0.605 0.698 0.801  3.147 2.867 1.112 0.590 1.176 2.304 4.340 6.126 7.030 7.442 7.645  0.000 0.035 0.046 0.024 0.021 0.061 0.063 0.009 0.080 0.185 0.297  V  1  7.716 7.830 7.888 7.922 7.940 7.949  0.000 0.252 0.371 0.148 0.122 0.392 0.624 0.804 0.948 1.078 1.201  0.000 0.014 2.679 3.354 3.562 4.593 6.413 7.809 8.426 8.671 8.784  7.014 6.656 5.553 4.364 3.777 3.926 4.590 5.012 5.111 5.109 5.090  0.789 0.943 1.195 1.744 2.303 2.840 3.328 3.757 4.146 4.518 4.882  4.757 4.727 0.935 0.053 0.030 0.011 0.009 0.007 0.005 0.004 0.003  I.Alt  -vr "V  V  R  V  0.0 0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2 3.6 4.0  0.758 0.671 0.638 0.664 0.744 0.873 1.066 1.328 1.634 1.961 2.299  0.933 1.707 2.921 4.086 4.505 3.800 2.180 0.890 0.354 0.157 0.072  1.547 1.259 1.123 1.342 1.626 1.937 2.263 2.598 2.939 3.282 3.629  0.005 0.000 1.015 0.857 0.431 0.216 0.115 0.065 0.040 0.025 0.017  1.547 0.251 1.866 0.124 2.204 0.064 2.556 0.034 2.925 0.018 3.321 0.008 3.770 0.003 4.281 0.001 4.832 0.000 5.402 0.000 5.980 0.000  T r a n s i t i o n frequencies ( i n Mc/sec) and square of the matrix elements (arbitrary units) f o r 0 R ^ 4 , at i n t e r v a l s of R-0.4. t  TABLE I I R- 8  R= 4  R •- 2  R -20  PVC  Calc.  PVC  Calc.  PVC  Calc.  PVC  Calc.  -5/2  1.43  1.44  2.05  2.05  3.25  3.25  6.80  6.80  -3/2  0.891  1.00  1.67  1.63  2.81  2.81  6.36  6.36  -1/2  0.659  0.392  1.21  1.20  2.38  2.38  5.93  5.93  +1/2  0.293  0.542  0.786  0.800  1.94  1.94  5.48  5.48  *3/2  —  0.061  0.277  0.297  1.47  1.47  5.03  5.03  m  Frequencies of the Zeeman l i n e s obtained by a quadrupole i n t e r a c t i o n perturbation theorjTcompared with the frequencies c a l c u l a t e d d i r e c t l y f o r the s p e c i a l case of I * 5/2, \ - 0.95, e - 0°, and quadrupole constant C -2.960 Mc/sec, f o r R ^2, 4, 8 and 20. z  38. C.  O t h e r o r i e n t a t i o n s o f t h e magnetic f i e l d  H  When the quadrupole energy matrix i s w r i t t e n down i n a representation i n which the xyz axes t o which I x , I and >  I  z  y  are r e f W e d , c o i n c i d e with the p r i n c i p a l axes of the c r y s t a l ,  i t has" diagonal and | m - m*/ = 2 non vanishing matrix elements. In a coordinate system that does not coincide w i t h the p r i n c i p a l axes, the | m - m'| = 1 matrix elements a l s o do not vanish. The magnetic matrix has only diagonal elements i f the a x i s o f quantization of spin coincides with the d i r e c t i o n of ->  the magnetic f i e l d H. Otherwise i t has non-vanishing | m - m*| - 1 matrix elements as w e l l . So i f , and only i f , H coincides with any one of the p r i n c i p a l axes of the c r y s t a l can the fm - m*l = 1 matrix elements of both i n t e r a c t i o n s be made t o vanish simultaneously.  The  above a n a l y s i s ( s e c t i o n I I I , B) i s then a p p l i c a b l e . For then a "checkerboard" determinant  can be obtained from eq. (30) by  a simple r e l a b e l l i n g of the axes and i t s p l i t s up i n t o two subdeterminants.  For h a l f i n t e g r a l I we have two "races" of  eigenvectors of I f 1/2 members each.  The t r a n s i t i o n s are p o s s i b l e  only between members of d i f f e r e n t races and we have ( I + l / 2 )  2  l i n e s instead of I (21+ 1). For example, i f the f i e l d i s along the x - a x i s , we have to change the d e f i n i t i o n of A a n d ^ i n eq. (30) by the f o l l o w i n g transformation: A-*  ~4  O - - , )  (41)  39 The secular determinant i s then: a"  0  h-  0  0  0  o  r  0  0  0  c"  0  r  0  0  J"  0  d"  0  h  0  0  r  0  e"  0  0  0  0  h"  0  f  0 h"  where a  M*-%)  M  ft  h"  4  -  *  d" = -LA  G  (42)  0-*,) f & - A  s  -&- *'*>  r  (  A f t e r rearranging the terms i n eq. (42) the s e c u l a r equations were obtained and solved f o r A f u n c t i o n o f R.  as a  The r e s u l t i s given i n g r a p h i c a l form i n f i g . 8.  The pattern obtained d i f f e r s from the one obtained f o r the f i e l d along the z-axis ( c f . F i g . 3 ) . The d i f f e r e n c e between the two patterns i s explained i f we r e a l i z e that the eigenfunctions belonging to the puce quadrupole l e v e l s (R»0) i n t h i s new representation are not nearly pure eigenstates of the spin operator as they were i n the previous case ( c f . page 30 f o r the d i s c u s s i o n ) . Using the transformation matrix to pass from a representation w i t h the a x i s of quantization along the z-axis to that along the x - a x i s and the known values of the c o e f f i c i e n t s of the eigenfunctions f o r the pure quadrupole l e v e l s i n the Im^ representation the squares o f the c o e f f i c i e n t s of the angular  momentum eigenfunctions denoted by 1 ^  i n the new representation  Imx were found and tabulated i n Table I I I .  They can be compared  with the squares of the c o e f f i c i e n t s o f the v j ^ j ( f o r R ^0) i n f i g s . 4 and 5. TABLE I I I Squares o f C o e f f i c i e n t s o f andf 1  ^i nd^i a  (fkandf-i  HIGHEST LEVEL  0.17  0.51  0.32  MIDDLE LEVEL  0.64  0.00  0.36  LOWEST LEVEL  0.19  0.49  0.32  Squares o f the c o e f f i c i e n t s o f the eigenfunctions f o r the case o f the magnetic f i e l d along the x-axis, R = 0. When the magnetic f i e l d i s applied along the x - a x i s , the double degeneracy i s removed and we have two classes of eigenfunctions.  Class L' c o n s i s t s o f ^ - | , < ^ and(ff  c l a s s M*, of ij>,£ tjftt  and  and ^5 . An a n a l y s i s very s i m i l a r t o the  one given i n s e c t i o n I I I , B, can be made.  For instance: the  s t a t e L ] c o n s i s t s , when R« 0, o f 51% of tfi , 31% of fx^ , and 17% o f  . As R increases ^ - | w i l l become predominant. When  R * 0 but very small, the corresponding eigenvalue does not increase with R a t f i r s t .  We can v i s u a l i z e the system i n t h i s s t a t e as  being composed o f 51% o f spins i n they)s s t a t e , 31% i n  and  17% i n C^js . The Zeeman e f f e c t on the spin i n s t a t e ^ - { i s an increase i n energy when R increases and a decrease i n energy f o r the spins i n the states  Since we have i n a l l 51% o f  the f i r s t against 49% of the others, the net r e s u l t i s that t h i s  41. l e v e l i s almost constant at the s t a r t as we can see i n f i g . 8. But as R increases f u r t h e r , ^ - i becomes predominant.  In t h i s  manner the dependence of the eigenvalue on R can be explained readily. In t h i s case the gradual change of character of the eigenstates i s not so obvious from the energy l e v e l diagram of f i g . 8, but could be traced e a s i l y by drawing graphs s i m i l a r to those of f i g s . 4 and 5. I f the r - f f i e l d used to i n v e s t i g a t e the frequency spectrum i s placed at r i g h t angles to H, then, as before, we s h a l l expect only 9 t r a n s i t i o n s between d i f f e r e n t c l a s s e s . In the most general case when i t does not coincide w i t h any of the p r i n c i p a l axes of the c r y s t a l there w i l l always be the /m - m'/ = 1 non-vanishing matrix elements present no matter what representation i s used.  The "checkerboard e f f e c t " i s l o s t ,  and a l l s i x l e v e l s i n v o l v e a l l s i x angular momentum eigenfunctions. Then a (21 +1) degree equation ( c f . eq. (28) ) has to be solved, and (21-+1)1 l i n e s are expected with varying i n t e n s i t i e s .  No  d e t a i l e d c a l c u l a t i o n s of t h i s case have been c a r r i e d out, but the procedure to be followed has been o u t l i n e d  above.  SUMMARY AND CONCLUSION A complete numerical s o l u t i o n has been obtained f o r the dependence on the uniform e x t e r n a l magnetic f i e l d H of the expected l i n e frequencies and l i n e i n t e n s i t i e s f o r the nuclear of Al* 1  resonance t r a n s i t i o n spectrum4n spodumene f o r one p a r t i c u l a r c r y s t a l o r i e n t a t i o n (H along one o f the p r i n c i p a l axes^). At H= 0, the pure quadrupole spectrum w i l l consist of two strong l i n e s of almost the same frequency (0.789 and 0.758 Mc/sec) plus a weaker l i n e a t the sum of these two frequencies as shown i n f i g . 6* -»  As H increases, the spectrum g r a d u a l l y changes as shown i n f i g . 6 o r i n Table I I , u n t i l eventually the Zeeman l i n e s p l i t i n t o f i v e components by the quadrupole i n t e r a c t i o n i s established.  A t intermediate f i e l d s (around 260 gauss) there i s  apparently a chance of f i n d i n g "extra, l i n e s " , as shown i n f i g . 6 ( V f o r 200 < H< 400 gauss and V e  —>  7  f o r 100< H<750 gauss).  As long as H i s along one of the p r i n c i p a l axes a maximum of 9 components i s p o s s i b l e , although some may be of too low i n t e n s i t y o r of too low a frequency to be observed. I f H i s a t an angle to a l l the p r i n c i p a l axes, 15 components become p o s s i b l e i n p r i n c i p l e , although again some may not be observable f o r reasons of low i n t e n s i t y o r frequency. The complete spectrum can be c a l c u l a t e d f o r a n y o r i e n t a t i o n by ft  the method used i n t h i s t h e s i s .  REFERENCES Al A2  A l l e n , J.of Am. Chem. See. Z4, 6074, 1952. Andrew and Bersohn, J . Chera. Phys. l f i , 159, 1950.  Bl B2 B3  Bersohn, J . Chem. Phys. 2Q, 1505, 1952. Bloembergen, Physica 386, 1949. Bloembergen, P u r c e l l and Pound, Phys. Rev. 13., 679, 1948.  CI C2 C3  Carr and K i k u c h i , Phys. Rev. Zfi, 1470, 1950. Cohen, Thesis, U. o f C a l i f o r n i a , 1952. Cohen, B u l l . Am. Phys. S o c , F.9, Jan. 1953.  Dl D2 D3 D4 D5 D6 D7  Dean and Pound, J . Chem. Phys., 2Q, 195, 1952. Dean, Phys. Rev., 607, 1952. Dehmelt and Kruger, Naturwiss, 31, 111, 1950. Dehmelt and Kruger, Naturwiss, 22, 921, 1951. Dehmelt and Kruger, Z.f. Phys., 122, 401, 1950. Dehmelt, Z.f. Phys., 12Q, 356, 1951, and 12Q, 480, 1951 Dehmelt and Kruger, Z.f. Phys., JL2Q. 385, 1951.  Fl  F e l d and Lamb, Phys. Rev., £2, 15, 1945.  HI  H e i t l e r , "Elementary Wave Mechanics", Oxford, 1945, Pages 118 f f .  II  I t o h , Kusaka, Yamagata, Kiriyama and Ibamoto, J . Chem. Phys., 20, 1503, 1952.  Kl K2 K3  Kruger Z.f. Phys., 120, 371, 1951. Kruger and Meyer-Berkhout, Z.f. Phys., 122, 171, 1952. Kruger and Meyer-Berkhout, Z.f. Phys., 122, 221, 1952.  LI L2  L i v i n g s t o n , J . Chem. Phys., 20, 1170, 1952. L i v i n g s t o n , t o be published i n J . Phys. Chem.  Ml  Meal, J . of Am. Chem. Soc. 21,6121 , 1952.  PI P2 P3 P4 P5 P6  Pak, J . Chem. Phys., 1&, 327, 1948. Petch, Votkoff and Cronna, Phys. Rev. £&, 1201, 1952. Pound, Phys. Rev., 12, 1273, 1948. Pound, Phys. Rev., 22.. 523, 1948. Pound, Phys. Rev., 22, 1112, 1948. Pound, Phys. Rev., 22, 685, 1950.  Tl  Townes and D a i l e y , J . Chem. Phys. 12, 782, 1949, and 2Q, 35, 1952.  VI V2 V3  Van Vleck, Phys. Rev., 24., 1168, 1948. V o l k o f f , Eetch and Smellie, Can. J . Phys., 2Q, 270, 1952. Volkoff and Petch, Cranna and V o l k o f f , to be published i n Can. J . Phys.  Wl W2 W3  Watkins and Pound, Phys. Rev., £5, 1062, 1953. Waller, Z.f. Phys., 22, 370, 1932. Weiss, Phys. Rev., 23., 470, 1948.  

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