ON T H E T E M P E R A T U R E OF T H E S H A P E OF MAGNETIC DEPENDENCE RESONANCE LINES by MALCOLM B.Sc, University A THESIS THE SUBMITTED of B r i t i s h in Columbia, IN P A R T I A L F U L F I L M E N T REQUIREMENTS MASTER MCMILLAN OF FOR THE DEGREE OF SCIENCE the Department of Physics We accept to THE this thesis the required UNIVERSITY as standard OF B R I T I S H September, conforming 1959 COLUMBIA 1958 OF i i i ABSTRACT This thesis temperature lines dependence i n solids lattice which vibrations sufficiently resonance was i s devoted lines shape function lines i s defined (1950) which particular, that detail. the temperature helium also sample of nickel dependence resonance temperatures dependent and f r o m lines and t h a t on t h e shape of This procedure also A moments o f this formula i s discussed are applied fluosilicate special crystal. case these of the noticeable at characteristics sample. i n t o the case From i t follows of the c h a r a c t e r i s t i c s becomes line In the standard i s valid this by (1952). approximations. extent of of the resonance and s e c o n d moment resonance the shape a n d Kambe i n various that i s the case a t a n d was u s e d t h e shape of effect i s used. The g e n e r a l f o r m u l a e general discussions paramagnetic (19^+8) t h e q u e s t i o n t o what a spherical the method" and t h e f i r s t This To d i s c u s s and U s u i Van V l e c k f o r the second great when t h e d i r e c t describes are calculated study o f magnetic c a n be n e g l e c t e d . by Van V l e c k and S t e v e n s of remain t h e "moment Pryce of o f t h e shape low t e m p e r a t u r e s . introduced function to a theoretical of liquid a r e then In presenting t h i s thesis i n p a r t i a l fulfilment of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library s h a l l make i t f r e e l y available for reference and study. I further agree that permission for extensive copying of t h i s thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. It i s understood that copying or publication of t h i s thesis for f i n a n c i a l gain s h a l l not be allowed without my written permission. Department of The University of B r i t i s h Columbia, Vancouver Canada. Date g ^|, is, flyj. ACKNOWLEDGEMENTS I Professor problem valuable of this W. wish Opechowski help my advice gratitude to f o rsuggesting and f o rh i s c o n t i n u e d throughout interest the this and performance research. I wish Research t o express Council i n the form also t o thank o f Canada the National for financial of a Bursary. iv Table of Contents Page ACKNOWLEDGEMENTS ABSTRACT Table i of Contents . . . . . . . Chapter I - Introduction Chapter II Brief System Chapter III and Second 4 Shape of i t s F i r s t Function and Moments 7 3.1 General 3.2 Truncation of the Hamiltonian, and 3.3 of Chapter Discussion IV . . of the Interaction Definition Derivation Equations Moments 4 Moments from and Chapter V VI Sufficient . . 18 Equations • 29 C o n d i t i o n s f o rVan V l e c k ' s (1948) E x p r e s s i o n f o r t h e Second Moment t o be V a l i d An A p p l i c a t i o n and (4.19) c Second (3.3.13) a n d (3-3.15) • Chapter 9 "K.*\ and S t e v e n s ' (nso) f o r the F i r s t Directly ft * Calculation and Second of Pryce 7 Energy, o f f (v ) a n d i t s First i of the Physical of the Line Calculation i 1 Considered Definition i iv a n d Summary Discussion i of Equations 4l (4.18) 49 V Table of Contents (Cont'd) Page Chapter VII * ~~%r „ when 7.2 Application VIII Nickel Field IX . . . • and Absorption 78 f o r the Magnetic by a S p h e r i c a l l y Fluosilicate Crystal i s i n the D i r e c t i o n Writing when Shaped the Magnetic of the Optic and general of the form Kambe Showing 82 tf another field m = XX given by (195D that . { ? mr.+ f.mp ~\ Ishiguro, x 96 when there and when is parallel Usui, . ' = S field 93 form the i s no magnetic to the Z axis < Figures 97 F a c i n i l Page Theoretical dependence shaped II . «K = XX{ £«*p+fimp, j crystalline I Axis 91 of Rewriting in C VI Conclusions in B (7.1.4) t o t h e C a s e of Aggendices A 75 i n Chapter Calculation Resonance Chapter of 75 j t G e n e r a l Formulae Chapter a ^ _ a 7.1 Considered anj l^<^ ) f o r ^<t^> Expressions of nickel Theoretical dependence spherically crystal curves f o r the temperature #<&v) fluosilicate curve of for a f o r the £ <£^> a shaped nickel spherically crystal. . . 88 temperature for a fluosilicate 88 vi Table of,Contents (Cont'd) Page Bibliography 99 I Chapter Introduction This of magnetic concerned shape we thesis of give us magnetic ions and consider, i n the radiation width which both of the temperature effect of out very interactions low temperature of case the lattice to lattice thesis f o r which ions of the vibrations we restrict this effect interactions temperatures. which finite I t i s the i s the main para- field. The of mutual vibrations the of the topic of the neglected. paramagnetic temperature theory of line. with ourselves to be the from lattice width can In paramagnetic decreases between independent a (apart the the lines. vibrations, narrow", and of formulae magnetic between "infinitely practically effect The i s completely negligible). of mutual t o be effect i n general, range the the be In t h i s static are dependence lines. the branch We nuclear resonance of a paramagnetic effect temperature. at the contribute, However, turns would temperature or i n one resonance. for definiteness, of mutual line investigation absorption presence resonance ' magnetic the to electron disregarding interaction with resonance refer salt absence Summary theoretical in particular can and phenomena, namely, magnetic Let the is a I this this The ions except latter thesis. .3 In we order to discuss u s e t h e "moment (19^8) Kambe and used (1952). calculate classic in paper, in b u t when also introduce i s quite shall now After systems I I I , to define which then give describes we function. their independent by these under Our latter procedure, authors. o f the contents o f i n Chapter called the " l i n e o f magnetic these I I , we the proceed, shape resonance and second the s o - c a l l e d lines. moments o f expressions, interaction H a m i l t o n i a n and o f t h e o p e r a t o r w h i c h moment their consideration Before writing o f the magnetic general into have coupling extent expressions. factors. summary t h e shape this e q u a t i o n s f o r t h e moments used a function In with the To what h o w e v e r , we the performed lines.) and S t e v e n s discussing, have to find have expressions f o r the f i r s t shape than Pryce the temperature to a brief chapters. of magnetic line turn Is t o discussed illustrating to that Vleck one o f t h e t o p i c s their temperature similar rather lines and U s u i and method of these f o r t h e moments v i a B o l t z m a n n Chapter this factors by'Van (1950) on t e m p e r a t u r e . into a n d Kambe function" We the temperature remaining types lines the o t h e r hand, only We introduced t h e moment i s , i n fact, give fact, the On resonance not concern himself lines they equations in Van V l e c k does thesis. equations behind of resonance i s valid Boltzmann Usui idea o f resonance introduced via was by P r y c e and S t e v e n s (Thes attitude this also o f magnetic e x p r e s s i o n s f o r the shape dependence this method" which t h e moments analytical t h e shape "truncation" o f t h e sample of the represents to the 3 oscillating we rewrite function In with for magnetic field. the f i r s t i n a form Chapter their and t h e moments, second involving I V we a i d we are able and S t e v e n s . these formulae t o a case Chapter temperature the time the and give the t h e r e has been general than Chapter moments apply salt. the our exact and equations i n the form V I , we apply t h e one c o n s i d e r e d paper. conditions (involving of the i n t e r a c t i o n s ) no e x p l i c i t shape assumptions i n Chapter sufficient the magnitude line f o r t h e moments of their Kambe, moment under i s valid. discussion the which Up t o of this point literature. In we V we from Van V l e c k e q u a t i o n f o r the second this in i n 'section and these operators. simplifying Later, more Usui of this over to obtain, by Pryce In moments general formulae these authors following traces introduce given by Then, V I I we than that t h e s e more We first From these have been give exact and second included a more r e f i n e d given i n Chapter graphs graphs give formulae showing central some approximation f o r IV, and i n C h a p t e r to a paramagnetic the temperature moments general conclusions i n the f i n a l , very brief nickel variation of the l i n e shape of function. c a n be d r a w n ; Chapter VIII IX. these 4- Chapter I I B r i e f Discussion of the P h y s i c a l System Considered For the purpose of the discussions i n the l a t e r s e c t i o n s , we s h a l l consider the f o l l o w i n g t o be the p h y s i c a l s i t u a t i o n : We have a sample, which may be a c r y s t a l or a powder, containing i d e n t i c a l N " s p i n s " . For the general c a l c u l a t i o n s given i n the f o l l o w i n g sections i t i s not necessary t o s p e c i f y whether we are dealing w i t h e l e c t r o n i c systems possessing a magnetic moment or nuclear systems possessing a magnetic moment, that i s , the general formulae given can be applied e i t h e r to the case of paramagnetic resonance or nuclear magnetic resonance. Thus, we s h a l l speak of a " s p i n " as f o l l o w s : 1) t o r e f e r to an e l e c t r o n i c system with a r b i t r a r y but f i x e d spin quantum number S and " g - f a c t o r " , ^ , wherein jfc--^ g.S B where u,= magnetic moment of the e l e c t r o n i c system and p. = Bohr magneton = l ^ T - I f the e l e c t r o n i c system i s an atomic d i p o l e and i f no c r y s t a l l i n e f i e l d i s present, then the g-factor w i l l have the form tensor. g =. ^ EL where E Thus, i n t h i s case, we have i s the u n i t (JL--^^^^. • If the e l e c t r o n i c system i s an atomic d i p o l e i n the presence of a c r y s t a l l i n e f i e l d , the g-factor i n general become a non-unit tensor 3 , will 5 2) to refer fixed to a spin nuclear quantum <3 , w h e r e i n of the nuclear magneton We shall speak interacting We a "spin that the constant magnetic induces system from field, the In there rest be of to the physical field, and w h e r e we consider on the nature The a moment nuclear is a tensor). N N of which high high finds of N weakly the has been placed frequency oscillating frequency magnetic states that of energy in the field spin i s absorbed field. the spin spin latter the spin within the These lattice these spins a between the relationship procedure function g-factor, p - system energy between i n the frequency of the one between the included The the ^ a interactions interactions magnetic and as and The world. between fields. . interactions field, not where of N Q magnetic magnetic world H between oscillating the consist system a-n^t <3H, addition of and system" experimentally will nuclear magnetic (In general field, transitions and I and but spins. assume magnetic system with arbitrary where N . of a number ^ «^,«x ju.= N system spin system spin t o be between the oscillating system, and interactions system the system and or spin system and that system the part the of two absorption magnetic and the will the constant oscillating the "lattice", the physical magnetic of field energy will depend interactions. i n the depends following on chapter w i l l the above be interactions to and define which describes by the the spin magnetic relationship system field. and the between the frequency of absorption the of ene oscillating 7 Chapter III D e f i n i t i o n of the Line Shape Function and Calculation of i t s F i r s t and. Second Moments 3.1 General Discussion The contents of this "section w i l l be devoted to f i n d i n g the f i r s t and second moments of a function which we s h a l l c a l l the " l i n e shape function, f(-v)". Many functions may be used to describe the relationship between the frequency of the o s c i l l ating magnetic f i e l d and the power absorbed by the spin system. In section 3.2 we s h a l l define the l i n e shape function such that f(-f) i s proportional to the imaginary part of the high frequency s u s c e p t i b i l i t y , "X."(-») . Furthermore, we s h a l l define 5c© © To calculate the moments of f (ri) we s h a l l use a method, f i r s t developed by Van Vleck (1948), which demands a knowledge of the Hamiltonian, Hi , of the spin system. cussion i n Chapter II i t i s apparent that From the d i s - must i n general be written as where the four terms represent energies: the following interactions ft* = ipin-magnetic f i e l d crystalline field i n t e r a c t i o n plus spin- interaction when t h i s l a t t e r i s present. f£ CO - spin-spin interaction (which may include exchange interactions) S(* « spin- l a t t i c e i n t e r a c t i o n VL** " spin-high frequency magnetic f i e l d i n t e r a c t i o n . We have, furthermore, that fc *,, Z 1 . V* In the following, we s h a l l disregard Furthermore, we s h a l l assume that ti" » V^** (o) CO % » + 5 completely. i Ci"> V. and so that, to a f i r s t approximation,!^ can be written as %^ K We s h a l l consider as the perturbation (3.1.1) < 0 as the unperturbed energy and ti* energy. Van Vleck (1948) has pointed out, however, that use of (3<> 1.1) when finding the moments does not allow us e a s i l y bo compare our results to experimental findings. fact "truncate" the Hamiltonian. We must, i n The reasons behind t h i s are rather subtle and f o r this reason, we s h a l l devote the next section to this point. Usui and Kambe (for (1952) have pointed example, when a c r y s t a l l i n e f i e l d out that i n general i s present) a further truncation i s necessary, namely, truncation of the i n t e r a c t i o n energy section. ^ c i ) . This point w i l l also be discussed i n the next 3.2 Truncation and We to be of the Hamiltonian, of the i n t e r a c t i o n energy, are considering given be w r i t t e n could be of regarded levels highly Let $ us 0 > , that of the system label could being As induced proportional by . We known, ^ and 3 o ( E^, i s well with of a the a such energy In elements occur recall 9 o < that the various between the states square For we consider states (0) \^y ) >s that of the S*' WJ spin transition I^^O and of the matrix i s element ' (3.2.1) transition takes place the spin one u s u a l l y finds that system absorbs I E ^ - E ^ I . are non-zero but rather f o r certain values example, i f (3.2.1) that o f oi were not a l lsuch non-zero and non-zero ^ matrix i n addition 1F_E matrix elements . only i f L-^Ul and ft i s the degeneracy the p r o b a b i l i t y of a to the absolute practice only number of criK-i,,*-') When , the large and e i g e n v a l u e s , t r a n s i t i o n s between system. interactions levels. a n d w h e r e induces compared consist system of the system i s , i f a l lother the eigenstates the eigenvalue of the spin I f the Hamiltonian a s n e g l i g i b l e when degenerate °y of as , . the Hamiltonian (3.1.1). by could energy ft'" ft^V 1(3.2.2) Inconstant independent of * IO then the spin system S> frequency, would absorb , and f ( ^ ) would 0 energy have at a single the simple form -- , £ U - s > > (3.2.3) 0 S(>*-->>) where (3.2.3) (Condition o t o the case when When t h e H a m i l t o n i a n of the spin system , each o f t h e N s p i n s h a s t h e same c, a delta function. no crystalline i s present.) > the Dirac refers field y i s the familiar 0 case — a. L h.*\ h. . energy (3.2.2) The c o n d i t i o n i s written as values, say, corresponds to when = constant and, furthermore, only between f o rk = when adjacent 1,2, transitions levels ,R-1 can occur > (3.2.2) of the individual spins . If, _ Q Q V on t h e o t h e r hand, ' (3.2.4) constant A*i fc. \ (as then one u s u a l l y the spin frequencies, finds system say, -o when a c r y s t a l l i n e will 0 absorb -»>, -0 } energy ~>> p i } • field i s present) at several In this case different f(-o) would p have the form When values are S(v) = T I T .21 i s a d d e d t o Vtp> t h e h i g h l y split. g i v e n by f i r s t T h e new e n e r g i e s order degenerate energy a n d new e i g e n s t a t e s perturbation theory: II C.,W"U,0 E. . ^ + I X % M r f (3.2.5) 5a l O - W V l l l * , , • w h e r e we values use * v „ to label + . Wi) (3.2.6) t h e new e i g e n f u n c t i o n s and eigen- and where (3.2.7) UM^'U.gX^l^UO (3.2.8) U?\v"UO The zero order states are 1o( where A i s a unitary matrix and has been chosen so that: 1) 2) This a can always i s never be done. infinite. Now, the probability of ^ between and C4> inducing a t r a n s i t i o n Is proportional to the absolute square of the matrix element (3.2.9) Let us suppose f o r s i m p l i c i t y that ( 3 . 2 . 2 ) holds. It i s seen immediately that because of the l a s t two terms i n ( 3 . 2 . 6 ) , (3«2.9) w i l l be d i f f e r e n t from zero f o r many values of and £ %< . As a r e s u l t , energy w i l l be absorbed at frequencies many times larger or smaller than \> . The function f(s>) w i l l 0 now consist of several broad overlapping lines and w i l l have several maxima, the number of which depends on . As can e a s i l y be seen, these maxima w i l l occur at multiples of -» e when ( 3 . 2 . 2 ) holds. The following diagrams will illustrate the point: DiflCRnm x DlOCftflm A o %s"\ * TI 13 It should be noticed that this phenomenon i s not the result of the simple condition (3.2.2) but that i t i s the result of the fact that to f i r s t order the eigenstates of $ are linear combinations of a l l of the eigenstates of V^* . ; In general, then, when ft" i s added to ii^ two things happen to f ( ^ ) . F i r s t l y , the i n f i n i t e l y narrow lines represented by the delta functions broaden and secondly, secondary lines occur. (By secondary lines we mean those lines which occur because of the presence of the l a s t two terms i n ( 3 « 2 . 6 ) . We s h a l l refer to those lines which result from the broadening of the delta functions as primary l i n e s . See diagram II.) Experimentally, one observes one of the primary lines of f ( v ) . We seek to characterize f(i?) by Its moments, so that i f we are to compare the moments of the experimental curve with the moments of f ( v ) we must then find some way of eliminating the secondary lines from our function f ( v ) . If a • and \> . could be put equal to zero the d i f f i c u l t i e s would disappear. In this case, when con- d i t i o n (3.2.2) holds, transitions w i l l occur only between those states whose o r i g i n a l separation was o r i g i n a l separation we mean when W , where by i s taken to be zero. The function f ( v ) w i l l then consist of the broadened portion centred about v v , that i s , the primary l i n e about Q 0 In other words, no subsidiary secondary lines w i l l occur. In general, a „ ; . M and k . -; c M a n b e P u t to zero i n only one way, namely, by considering the equal Hamiltonian Vt" where o f the system i s that part Then, a s i s w e l l known, which are simultaneous These functions will the there »j;. - k . .„ tf" which exists eigenfunctions be a 0, if" Thus iltonian when of sC" and Iv.O are zero d i s c u s s i n g f ( v ) we of the spin system ^ = spin . see i f it" 0> Because of immediately i s replaced shall consider t h e Ham- t o be (3.2.10) 0> refer to V as the "truncated Hamiltonian" o f the system. As a result secondary lines will have then of the t r u n c a t i o n of the Hamiltonian, disappear the from f(s>). the The f u n c t i o n f(-o) form «*1 if . w tf + shall the Hamiltonian condition (3.2.2) of the system holds, <0 set of functions e i g e n f u n c t i o n s we b ...„ K + t« of commutes w i t h the f u n c t i o n s and instead vF' by We of o r t h o g o n a l i t y of these that S*° + \F> t o be i s given by (3.2.10) and 15 or, if the form the H a m i l t o n i a n of the system condition We the have primary findings and f(v) (3.2.4) h o l d s lines. t h e moments that t h e moments of f ( ^ ) , we must any o f the primary lines For wish example, s> results then A -o. v with i f we we v the second (3.2.10) and r - 3 . e x p e r i m e n t a l l y one o b s e r v e s I f , t h e n , we by comparing f ( ) about ^» * mentioned and where i s g i v e n by must hope eliminate i n which to find we from the second interested. moment the primary our curve our function are not a r e t o compare moment experimental of the experimental eliminate i f we to interpret one o f of lines about theoretical o f f ( v ) about found experimentally. In must order to eliminate introduce another the unnecessary artifice. Instead primary lines of considering we -Je 111 as inducing transitions we s h a l l suppose that V induces l > t r a n s i t i o n s , where by the circumflex we mean that we are considering only that part of which i s relevant to the p a r t i c u l a r problem under consideration. For example, i f v we wish to consider only the broadened portion about , A then we only wish to consider transitions between the states U.O and Ip.k) ' of W 1E^-E \=-JU^ wherein p . In t h i s A case, then, we define the operator V u > such that i t has matrix elements For any other problem i t i s necessary to redefine A H'*' . In general, however, the circumflex indicates that the matrix elements can be non-zero only f o r p a r t i c u l a r values of <* and ^ The following diagram w i l l i l l u s t r a t e the point: AAA where and VP* induces the t r a n s i t i o n s , 17 but: A where and M - where + V ' and 7 induces t h e c i r c u m f l e x means, i n t h i s the case, transitions that the matrix element can be n o n - z e r o only 1 E„-€.l = £ v i f i It holds, should that necessary consists In with ing 1) 2) be n o t i c e d that i n the simple i s , no c r y s t a l l i n e to introduce of only then, t h e moments f o u n d present, the c i r c u m f l e x over one p r i m a r y general, field before {3*2.2) i t i s not since f ( v ) comparing t h e moments e x p e r i m e n t a l l y we have t o do of f ( v ) the f o l l o w - < the Hamiltonian secondary lines truncate S* from i n which f(^>) where line. two t h i n g s : truncate case (i> i n order to eliminate the from f ( v ) , i n order we to e l i m i n a t e any primary are not interested. lines I * We its c a n now first s>* and second the frequency are interested considering of proceed to defining moments. about the primary the following the primary That line discussion, to calculating For simplicity, which i s centered. f ( v ) and at we line v Thus, define Let of the ^ so u s now shall define label energy the high frequency and the eigenvalues of a function that the purpose (3.2.11) of i t s F i r s t the eigenvalues eigenstates probability from we a r e that ) consider corresponding we we and Moments .We If call i n which f o r the 1* D e f i n i t i o n o f f(s>) a n d C a l c u l a t i o n Second shall i s , i n what f o l l o w s , 1° 3.3 we i n one range by frequency v the spin that by E„ g(-o)^-v) system oscillating -\J+A->> to % and • g(-*) s u c h second eigenfunctions will magnetic i s the absorb field In then: (3.3.D where PCy") = p r o b a b i l i t y that from the state the spin In) system to the state undergoes U') i n one a transition second 19 2. -sum over a l l of the eigenstates of it n — Z. = sum over a l l of those eigenstates of ^ «=sum over a l l of those eigenstates of wherein ft wherein However, XZ « * PU>") * IT n' = IZ*PU>^ « «»' n o II* !?^ )-^^ 6 so that Now, c a „^ o I - (3.3.2) where - p r o b a b i l i t y that the system i s i n i t i a l l y i n the a state In} , that i s , i n the state high frequency f i e l d = In) before the i s applied. p r o b a b i l i t y that i n one second the system undergoes a t r a n s i t i o n to the state UO i s i n i t i a l l y i n the state \<v) From s t a t i s t i c a l given that the system mechanics we have * * r where I / k ^Boltamann's constant T" ^ i n i t i a l temperature of the spin system = l a t t i c e temperature Let us now consider y^o' • From standard quantum mechanical theory we have the p r o b a b i l i t y that the energy perturbing induces a t r a n s i t i o n between the states IrO )r,0 and is V. of the state i n the time l«> t - t Q where at t Q the system i s to where Now, i n our problem, - -(oscillating field)•(magnetic moment) — -H(t)-M -H(t)M of *jL Then, Now, M where M i s t h e component' i n the d i r e c t i o n ofH ( t ) . -H(t)M c we can write If H(t) i s resolved where energy crossing unit about into area frequency range classical electrodynamics, i t sFourier i n the d i r e c t i o n the frequency v components, the of H(t) per will unit be, a c c o r d i n g t Thus, f„ , 0 - 3TTC ' % (Co I mU')| Finally, so that In order which is then, that our depends a E^, on constant, results be independent experimental say U, Z g U w W i T V ' u of conditions, for a l l v . X ^ ( * " ^ - a ' ^ E we v , a quantity assume that E Then: T lc.ULOr ) (3.3.3) i Now, system range from \) E(^ * i f E(v)/w the i s the oscillating to power magnetic absorbed field by the i n the • (3.3» as is well known (see f o r example Andrew H(v)= 4TT^> H^-X'TA) where frequency then: 4v But, spin i s the susceptibility, and (nss )) (3.3.5) imaginary where SH, part i s the of the high amplitude frequency of the v oscillating magnetic Combining field. equations (3-3.3)» (3.3.^) and (3.3.5) we have The area under „.«^ t h e c u r v e xl^A I T (,-fe-.-^ i s then k.uuor t where ^ We means s u m m a t i o n shall w h i c h we shall W>»-J^ Our line shape T"(v) now define call over a l lstates a dimensionless the line shape wherein quantity ir(: -.^)I^W function, and then, n £(v) function: fe = E '>E^ i s proportional (3.3.6) t o the f u n c t i o n furthermore, O The first can be and t h e second can be moment of f ( v ) which i s defined as written written (3-3.7) moment o f f ( v )which i s defined as 33 We shall summations a knowledge would task (19*+3)» could that (3.3.8) (19*+8), Van V l e c k and others i n doing this i s apparent: obtained similarity evaluated from traces using the functions require I s , we =E ln") SEln) that equations i s , as c a n be , that this (3.3.7) over n (1932), by W a l l e r that the traces w the would problem form, a Evaluating stand of noticed by r e w r i t i n g i s i n v a r i a n t under c a n be as they the eigenvalue I t has been be a v o i d e d advantage hence to solve i n trace operator and two m o m e n t s . of a l l of the eigenvalues continuing. (3*3.8) The (3.3.7) i n be f o r c e d before Broer c a l c u l a t e these arduous and operators. The t r a c e of an transformation as a basis in) by a any and functions similarity transformation. The (3.3.8) Kambe i n trace (1952). formulae by procedure for these order form shall and be noticed <^y (3.3.7) and by U s u i and use t o rewrite i s identical It will ^v) we to that later given that our are identical general to those given authors. Consider, In which now, to avoid the denominator the rather (3.3.7) and summation 2_ of awkward (3.3.8). , i t i s tv convenient are defined to introduce the operators ir\ + and «\_ which, as (3.3.9) 34 Cn I rt\\</) ii (3.3.10) L CnliMnO = CnU \ ') "+CnU_\«0 ThUS + We c a n now IX n write KT II.'*"%U WX«'U,1*0 Cn\*"^\nO= But, so o = 2 / V t CnU m In') £ , that X," ^c«\«.w \ o KT XX Similarily, f - T Then, » * j XX («" - /^OlcoUo'^T^ (/^Lm where, + fe as i s customary, C > tnj] ~ *V.w\ + (3.3.11) . 35 Consider, immediately duce the be now, the noticed operators m numerator that here and + of (3»3»7). i t i s not It will necessary to intro- since >"» n' N O W , XX " * " ^ C E ^ E ^ k n l m U ' ^ ^ X l * {.E 'C«|AUXnUU') -E^nUU'XnlmU^ FET 0 r,' w h e r e we and have made u s e the r e l a t i o n s h i p n Since n £„ . = Col n fcloO X Z . * " & 1 n) C E ^ E ^ I C - U U ' ) ^ -XL/^CEw-E^lcnUUor have IZCEW-EX*"^)lcnUio9r Using in £ i t s consequences E ' (o \ m W ) = Cn I m W we of the form (3.3.11) and - aTr«*(/ (3.3.12) we can fe rewrite (3.3.12) (3.3.7) 5 * < V > ' It and T will Kambe -s f be » R . -A noticed (1952) for that (^y (3.3.13) - the expression given by Usui involves Tree However, T ? « c « ( / ^ L ^ . L as can e a s i l y be to the functions Consider, be necessary As before, 3 <\ we * ^ seen now, = a Tro.e (/^ by calculating the numerator to introduce of ft* £ P the traces with (3.3.8). the operators ro + It will and write n -t-CnlfclrTXnlwln")} CnU_U'Xn U j i O respect - w_< again : 27 = Z (nl e W - a « i ft <V I n X n | w> lr> = Tract m + V m_ ^ 1* I T + - -»• - a i f t . i i m 5 t n\_fl\ v tt + Similarity, ZZ = T~ f >r ce \ * Q In- X — * + ft ^ \ ft. j n , n\_ & Finally, in CnUjn'Xn'U L> CE^-O" ^ t h e n , we A . — - ^ ( T i ^ ft .— (t\J« can write •+- m ft,_,-A + ^ f>? r0_ the numerator \ JJ . of (3.3.8) the form: CCm.^l.CSAll) Using ^ l = (3.3.11) and (3.3.1 *) 1 (3.3.1M we can write ^,(.ic[;., U^x) { as follows: <3S Our equations (3.3.13) and those given by Usui and Kambe £ S Dl"(v)Jv (3.3.15) (1952) are i d e n t i c a l with for and respectively. f x " ( v ) dv It should be noted that (3.3.13) and (3.3.15) have been derived without resorting to any assumptions as to the temperature T. These equations are quite general and are v a l i d at a l l temperatures. case of paramagnetic magnetic resonance. Furthermore, they can be applied to the resonance or to the case of nuclear an Chapter Derivation of and Moments D i r e c t l y Second Another line It shape will be b form found the first Equations and g i v e n by section equations g i v e n by the (1950) that Pryce and and First (3*3.15) and moments of Stevens (1950). introducing (3«3»13) these a one (3»3»15) can be authors. i t should be first second and second by f o r the (3.3.13) From E q u a t i o n s been i n this proceeding have of has assumption i n the E(v) Stevens' derivation shown Before Stevens and function simplifying written Pryce IV pointed out moments that of Pryce the and function £.V E(VW o where E ( V ) A ^ i s the the magnetic oscillating power a b s o r b e d field by the spin system from i n the frequency range S) t o V+A\> . i n the course their discussion B u t , b y (3.3.5) Pr£ce and have ca. that which Stevens the primary assumed where line under i s the consideration of frequency about i s centered. In 30 other and words, second which Pryce and S t e v e n s moments should explicitly instead, by P r y c e be n o t e d and (3.3.15) of ft shall define operator, operator of ^ t o > us 0 . Stevens Pryce to derive directly and of Stevens m and shall now from the our i n terms of do n o t ; they which introduce the truncations with an operator such consider have, perform projection f o r us so t h a t and 1£ these m (3.3*13) instead v , which we shall call a that we h a v e the matrix respect -1>2, P V o f (4.1) consider ( be a b l e d\ a consequence Let We perform C v K l p . O - \» As that c a n be w r i t t e n and projection and then, projection operators f o r them. which first (3-3.15). and introduced operators should, the truncation operations We We (3.3.13) equations It given c a l c u l a t e the of the f u n c t i o n i s our f(v>). expressions actually P.p vJ XA representation where (4.2) v * A t o the t o t a l i t y , _n- * P £ , of a general o f e i g e n s t a t e s \<s,k) ~n_ <= t h e number of I 31 «0 different is the eigenvalues degeneracy general matrix Now, the of of V the element matrix with respect written in -*£ of 0 elements of the respect of w The representation OP^ is then are using (4.1) representation t o t a l i t y of .) g^ of eigenstates any of 0 operator # can be form to the (*+.3)» we n That Let recall ^ u - we introduced tf'+vl" l<*k") easily , from and write section the operator commutes of and % in is J if with the i f and only i f P^f. M which V° with Vi writing # form given that - o 21 now which representations states find is us (4.3) 3.2 matrix CXl.f vf We of ^ matrix the this E where for section part Taking the in 1 , 2 , . . . . ^ _ In that to the 2L X 0= hy then, • eigenvalue C ^ l p ^ l p ^ V C/.ilol^,^) Clearly, ; k „ (^A) i n terms 3.2 ^ that of projection the circumflex operators. was introduced over V and hence over M i n order to eliminate from f(-a) any of the primary lines i n which we are not interested. we are interested only i n the primary l i n e at Here so that we s h a l l define the operator M such that i t has matrix elements US m 1^ p Consider the operator where ^ means sum over a l l values of ^ The matrix elements of MP 0 Thus we can write From (*+.5) we can immediately write «v = ^Z and are then: p. wherein 33 m » 22 P.^? 2-2- where (^.7) A means summation over a l l values of ^ o f and cJ<^ wherein single for - E L -.= E .(The symbol summation since (4.6) and large. speaking, (4.7) We have (4.7) i s an exact the assumption certainly that expression, S* i s not too 0J that f KV + tv> m =. equations recall that we (3.3.9) and t h e n we require V then a +-^r \ H o w e v e r , we and denotes use the notation -E^" (4.5) although involve 2 2 ( and We actually convenience.) Strictly and E„=E.-VRV* . X X U E,<Ep < o > have (3.3.10) defined respectively. and m_ by To w r i t e that i f U,0 - E ^ U,0 U,0 - E implies V < ; W,0 that E,.<E p t f o r a l l i and k. (4.6) 3f If i s not too large then this i n fact will be true. In shall and Appendix show (1952) Kambe be rather i s a (h.h) to transformed than Let shall i n Appendix Using can A we B into case of , and We Usui, (M-.5). and (3.3.15) It containing m form. of Sshiguro, (3.3.13) equations equations i n another (6) equation special * <T\_ ft^m that (*+.7), (^.5) rewrite and m + us c o n s i d e r t h e denominator of (3-3.13) and (3-3*15)• First, Now, L« L II P.TOP. ,H ^^t'l P r ^ e r o P - P„rr\P,0(\P \ o °> < £ Thus, * « * $ f c t 4 Lm_,m i« + UU " by Of» from (if.6) using (^.2) and p •> " 'P^mP,-, ^ K ^ P « . P p ] Of.7) where we Using (^.2) C Trace have again again Consider, (W.W), and ABC=Trace now, a n d 0+.5) we (W.2). used the f a c t CAB that f o r any c y c l i c a l l y , we the numerator of matrices A,B, have: (3.3.13). (W.2), Using can write and A Thus, Trace a ^ i t i l ^ ^ ] - = X T - T * c * [ u . ' ^ ( P , m P V P „ P - P,wf^mP -« ) t p r -ftm ( o4 • and \ N \ ? ( l + * 9 ) Consider, now, t h e n u m e r a t o r ( +.+), (*+.6), 1 l and ( 4 . 7 ) we (3.3.15). of Using (4.2), can write p" and E«,m "1 + so that Thus, - = X X1V ^ m P , L E «>-,«3,L« (h J Trace I I E C ^ + - P ^ P ^ P , ^ l l T,E R ^ T l ) = Trac-C P ^ P ^ V P^ ro - i> vp,mP^-WP m X4.10) J 37 We into have forms now transformed containing (3.3.13) equations and M r a t h e r ^ i< than m M . + I t should introduced By able i n writing introducing to produce (3) from YLP, We now so that e t This assume we - T JL Usui (1), we (4.10) be and respectively. (4.11) we to a first have o ^ approximation ^ A * ^ ^ a fcT — (4.12) their as shall (2), ^ a n d Kambe be w r i t t e n point, been (4.10). E ^ p ^ ~1 = o used general equations a t the beginning When a s s u m p t i o n can have and equations and > — j can write i s e s s e n t i a l l y the assumption when d e r i v i n g by - that at this ( 4 . 8 ) , (4.9), V°% ? -w'V (4.8), (4.9), and S t e v e n s ' -ttP« = ^ \ - » - Since no a s s u m p t i o n s one a s s u m p t i o n Pryce V - that equations our equations Using p be noted m > A and (3.3*15) and (4.12) follows: i s used by Pryce (1), (2), of section and and 4 Stevens (3)» of their and paper. ( 4 . 8 ) , (4.9) a n d ( 4 . 1 0 ) )ZXr*" Trace(^U.^l)a(l-*~^ Trace ( A mtfc.rni) Mi--»-* ^ T r a c e (* * T It ,L r r )Zl_A , l K P^m r Troc- . ( P.y fcmPjti \ (4.11) and = EJ-P rf Finally, the then, following ( 4 . 2 ) we +a^P r t . 14) ^ + Pd*P|V P ^ P ^ Using (^.13) J (4.15) can w r i t e , f o r example: ^\+P,^'^'^ when a p p r o x i m a t i o n relationships (4.12) holding: i s valid we have 31 *<*> - If* If ^ we and now d e f i n e (^^y - the second ^ - t n — e first moment moment o f f(s>) o f f(s>) about *— . si* C*.16) about then and so that -11* _ Z Z *~ • " ' T r a c * P ^ m «•»< ^ r ( l f - l 8 ) ifo =r ' r It should give t h e second that i s , since If this k^^y be n o t i c e d moment ' p that i s n o t t h e second *= s> , then by <^o^ <*.i9> L our o f f(->>) a b o u t quantity i s given and does not t h e mean v a l u e central moment - (^^V) t h e second o f s> , o f f,(v) • central moment o f f(->>) are equal. Equations and * Stevens' / ^ E ( ^ v (W.18) equations about and (W.19) are identical f o r the f i r s t the frequency and second v to the Pryce moments o f fl Chapter Sufficient V Conditions f o r Van Vleck's (1948) Expression for the Second Moment to be Valid Another derivation of an expression f o r the second moment of a l i n e shape function has been given by Van Vleck (19^8). Van Vleck has not defined e x p l i c i t l y the l i n e shape function he i s considering but we s h a l l show i n this chapter that by applying suitable assumptions to our expression which i s the second moment of f(%>)= ~C(^)f (3•3•15)« ^° » o the generalization of Van Vleck's equation (3) can be found. Van Vleck has considered the s p e c i a l case which we have called condition f i e l d i s present. m to be (3.2.2) that i s , the case when no c r y s t a l l i n e He has considered the operator portion of , and he has written the second moment of his l i n e shape function as (5.D where we use K. to denote the truncated Hamiltonian. As we have discussed e a r l i e r , i t i s necessary i n general to use M rather than M when discussing the problem. In the general case, then, t h e Van V l e c k expression becomes A It will be shown equal when field i s parallel Van before most the i s no c r y s t a l l i n e Vleck s to the used by these latter important point temperature, e x p r e s s i o n , Van V l e c k gave to the f a c t o r rise b e no d o u b t , the Van V l e c k results. general (that equations identical where with Usui, field a n d Kambe a crystalline i f given (195D field case (3.3.8) have \ , then applied i s present the and which There seems temperatures yields applied levels by V a n V l e c k . have account high considered and e n e r g y The not contain into (5.2), (1952) « The In deriving h i s for sufficiently i s , equation that does (3.3.15). n a n d Kambe those i n ± to the special and f o u n d a n d Kambe. (5.2) has not taken 5 that that Usui 1 appeared t o the problem. i s that introduced * i s , no c r y s t a l l i n e Ishiguro, case method, In f a c t , equidistant) are we however, the magnetic moment (3•3•15) d o e s . T, w h e r e a s which are and by us i n C h a p t e r I I I , approach to notice factors and when and U s u i authors, refined Boltzmann field f o r the second and Stevens, i n v o l v e a more S and 2-axis. derivation 1 S, that of Pryce general to C those methods merely there In Appendix correct their by Van Vleck of each their spin results Secondly, (5.2) and have to the stated that the results Pryce so obtained and S t e v e n s ' and a r e r e s t r i c t e d have checked From suitable possible to arrive give results such to sufficiently found when t h e are applied high (5.2), section, (3.3.15) which There point then, appear that i t should i s , i t should under moment. of this to that conditions i t would (5.2) be yields h a s , however, be d e v o t e d be possible valid been i n the l i t e r a t u r e . will by to no The finding t h e p r e s e n t , we shall find under which conditions expression which i s equivalent (3.3.8), be this temperatures. (5.3) I I tailor the to conditions. For the at discussion of this (3) and approximations f o r t h e second remainder been independently. sufficient explicit also the foregoing discussion applying to (1) equations case this have which following a valid to (5.2), i s equivalent to two a s s u m p t i o n s approximation of i s a valid (3*3.15). approximation of We are sufficient (3.3.8): shall see i f (5«3) that Is to We ^ *i *. 1) $ the exponentials i n W > » W so that (3.3.8) to a first approximation c a n be r e p l a c e d b y t h e i r values when i 2) for The temperature a l lvalues Let Chapter o f <* u s now shall where and perform I I I that eigenfunctions we i s high - E these calculations. We recall the f u n c t i o n s of E,. ^ E^ IE y U _ 0 ft t e > and (-/,; I \ . are the E. N . E„ I- •X using the f i r s t i=. assumption - Thus, L luuifcOP * above. (3.3.8), then, ; «;| X from simultaneous In equation written: ^=1 \« k~r p I n ) = U , 0 put so that enough (3.3.8) c a n be 15 using t h e d*s-«nVt.o of n ^ 6 » using the Thus, we (see Chapter I I I ) . SL second have M > assumption that when the above. above two assumptions are applied t o (3.3.8) we g e t n i l( _n_ n_ 1 V> which - is identical We have equation should most and *Jw shown (3*3.8), immediately unreasonable. the lowest except that f o r the by applying (5.3) equation be noticed The values of notation energy E . two can that be the equation assumptions found. second difference i s at with least to (5*3). our However, i t assumption between N£V* the so is highest that the second assumption Since i t has valid results would be by a of this of for Van to identical to It w i l l be with find for (W.17). to 1) given replace conditions compare equations convenient to rewrite (5.2) (k.k) Trace rather and 0+.5) & into Cro?*= T « - o « I Z and M, that (5-2). ^roP^,™^. 2) can done replaced the remainder much high validity for above. the assumed). is a (W.17) from This 2), valid and (3.3.15) is Therefore, assumption i s , we we approxi- (5.2), i t containing shall substitute using in can Thus, •= i t temperatures C+.12). form be in obtained (5*2) in a temperature, generally namely, to M that been order and be (W.l^) was that assumption mation will out has the what room sufficiently assumption, conditions equal l o experimentally assumption turn one under yields This (as recalled find be that are method simply In show 2» f>> (5.2) that condition. Vleck will that approximately temperature introducing order T desirable room the shown chapter. It by been much w e a k e r below implies (W.5) will 47 Thus, -— - — L - Using equations (4.11) and (4.2) we have, f i n a l l y : 4?\^ = S X r - ( P ^ % m f IX T^C< w m - P,m^ "4- • X X - r ^ c , P mp^ w We see that equation (5.*+) i s i d e n t i c a l to (4.17) hut with the Boltzmann-factors, last equation. A , replaced by unity i n this As before, i t seems that the procedure can only be j u s t i f i e d ifT » ^ . , We s h a l l see that this is not the case. The procedure now, w i l l be to consider (4.17) f o r the case when we can write: m= I (5.5) m- II Ml • (5.6) 4? where and, tC^ then, the i s symmetrical i n our results temperature to replace a l l exponentials containing by u n i t y . The r e s u l t s i f (5.W) identical to those then c o n d i t i o n s under give tials are by u n i t y i s valid. definitely that look upon then we must so obtained had been which the replacing I f we speak to write (5.5). of interest i n this field, forces o r exchange forces, c a n be w r i t t e n i s , i n the form It at this shall, a will valid that the this will be done approximation , a i spin , unperturbed a spin" f o r the second (where R and i f m respectively, ,a valid have Then, if? then, energy approximation of the system i fQ pairs spins", Secondly, most dipole-dipole forces, f o r t h e second VI. (5.2) which o f f(s>)i "energy each We We yields suppose values of levels t h e forms moment (5.5) values , equation of the of the N i s considered of the energy separation the c a l c u l a t i o n s and a r e l a b e l l e d values c a n be w r i t t e n 1 of physical i f the energy the phrase means t h e e n e r g y when t h e H a m i l t o n i a n expon(5.6) and I n Chapter moment a r e non-degenerate has and i n detail the c o n d i t i o n s under (W.12) h o l d s . unperturbed 4 state of these (5»6)X inconvenient to perform only be shall a s two-body very condition a, , a by f o r example, prove time; then, given We (5.5) the spins as separate be a b l e will of a "system forces that used. (The r e s t r i c t i o n s not severe. i s , i f we entities, found i n i and j (5.2) o f f (s>) i f spins t o be ^ and ), (5.6) a, , a ^ , & , 3 yields a Chapter VI An Application We s h a l l case ..,a now a p p l y which R labelling The that energy i tw i l l Q pairs where K-x<y< and f o r any r for ^ > } a then \ ^ \\lS ) such i s , i fA W p^..., Q> i f then The following a a Q Before method'of spin. spin I* . integers , a^ , that separation separation d a second a, , a i s such We c a n ( l , x , y , . — ) set of positive that , ^ ^ ' Ja^-cx I ^ r i si ntheset (l,x,y,....) and i f b i s n o t i n t h e s e t ( l , x , y , . . . . ) = convenience n to the suppose o f theunperturbed i n the s e t ( ^ + 0 ^ + 5 , That shall to establish energy a R values o f theunperturbed <£$R-i jfc . then have energy a set of Q positive - (^rt-a,*+^ have systems of levels construct labelling levels level always a values We (W.19) and has R energy be c o n v e n i e n t t h e energy energy integers spin a n d (W.19) (W.18) a r ea l l non-degenerate. o f these proceeding (W.18) equations when t h e u n p e r t u r b e d pairs that o f Equations t h e second example the energy we s h a l l call set i sthe should levels . ( I tshould noil illustrate be noticed set.) this of theunperturbed theset of Q positive method o f spin. (For integers 50 (l,x,y,..., level ) t h e s e t G.) L e t us suppose system f o r the unperturbed s p i n that the energy i s given by Diagram VII: Diagram VII Diagram VIII <x. R = 10 a n d Q » 4*. We have of the energy the set levels c^-o.,- and \ a _ < x l ^ -Rv* i n Diagram the second 6 r s t h e above method VIII. the labels We s e e t h a t s e t i s (8,9,10) a n d c x ^ a , . - a ^ a = a ^ a ^ X>+ us f i r s t 4 k>r- ^ ^ . . . . , 1 0 of a l levaluate (W.19) w i t h r e s p e c t ( p. are given <7 i s (1,2,W,6) that Let Using s r^-e^yo. the traces to the eigenfunctions 1,2,.... ,.n_ ; i = 1,2, i n (W.18) a n d of "S^ , \|A.>0' ,g , t h e d e g e n e r a c y o f e„ .) Trace = X P* MP^ M - Z U.LIP^P^U.O 3o< A - 3 u A. •»n ), - Wy CM-.O H IJ-M = 1 X C\>,l\ P ^ m ^ . O - u> p-.i I I ^ v ? \ v. I V S « *3a Cc/,ilv U k i k U l R {Xa;, U U . O - X X X c,, ' II?. »-< k = l i ; i * * «.=' 3B SB 3 * 3 9 - X X X X ( i Iv'lp.rtf t|ft1»,0C^lm \ h n t t ^X^\m\p,0 We m u s t in t h e above 0+.19). as how f i n d equations T h e method that used.by some w a y t o p e r f o r m and t h e summations which Pryce t h e summations we u s e w i l l i n and be e s s e n t i a l l y (1950). and Stevens (W.18) This t h e same will now b e discussed. Let t " us denote spin so theunperturbed Ir^ by eigenfunctions and t h e corresponding of eigenvalue the by a r that ttjflr^ o. l(\) = r t = 1,2,...,N t r - 1,2, ,R (6.7) An eigenfunction can then be w r i t t e n For example, It true where should which fc> where we from AT c change la) x x (6.1) xlcO t o (6.6) since thefunctions I u'^) jl I K , a r e obtained to of the form . a t h e meaning symbol remain U ^ x U ^ l c ^ . - x U ^ + ,-.Eu now o n , t h i s 1 0 * 10 equations by of a l l functions o +Q + 0 v shall that i sreplaced diagonalize combinations i n the form + o,- E + o +Q+ » X_ be n o t i c e d i f lj.,0 a of o f t h e symbol will by taking linear I o D . ^ l t ) jcl^") * - K \<*\ Forthis ^fv^) reason 5 mean (6.8) , where This o + a o . . . . . -•• « o t+ should Now, + lead t o no 4 (6.9) , E^, confusion. we c a n w r i t e R E - X where o a r (6.10) r O*O IN W C R R and X nr where - (6.11) N r-i Furthermore, q Nl - If thedegeneracy - r i s y 1— o a when any eigenfunction characterized integers (6.12) ^ t function b y two s e t s a Xn 3 belongs, specifies 0 -s n \ t>$ R R that (a,b,c, r set ofR « N f o r r » 1,2,.... ,R t o which the us t h e ^--value. The second , d ) , where l*c«R We c a n , t h e n , The f i r s t us the eigenvalue i s , gives theparticular c a n be of integers. = N, t e l l s of N Integers , of (n, , n , n ,... . n ), where R where U a ^ R r 4 l r Thus, set of E u «i r-i and g 1-$<A-$R eigenfunction, that i s ,the i-value. replace (6.13) 54- r where L means summation over a l l combinations of the N integers wherein a -t- a, + a + + a, = E., . Further, we can replace L Si- r- L ^_ by 1 t « X where v means summation over a l l integers ,n ,°.H^,n n c ( a (6.14) R wherein 0 ^ n ^ r N for r = 1,2,...,R such that •r For example, we can write M R . Let us now consider the meaning of r e c a l l that II .We d< p means summation over a l l values of II •>< < «* and p wherein E^ - E^ = . Since E^ and E^ can be labelled by the sets of R integers (n, ,njO,....n ) and R (n' ( jn^ , ,n' ) respectively, the condition E -E^ = p and the condition X o - L n ^ N r imply that some r e s t r i c - tions must be placed on these 2R integers. We must, i n f a c t , have 1 n>< N f o r at least one g i n the set ^ 0 $ n << N for r ^ g 3 r (6.15) 55 and n,-l n n+1 = (6.16) Jfi 5»i n r\ for g,g 1. it Then E ~ 2 ( nl- rO a. E a Z Thus, 5« - X 3 a means integers (n, , ^ , . . . . , 0 where n -N I We form. rather this Trace c a n now w r i t e lo^.c, than summation 4 over a l l values of the ) and where- i ^ n . ^ N r For simplicity L,0= = (6.1) Wsome t o (6.6) i n w r i t i n g we g i n the set G of Q positive integers. i n a more convenient shall write a") t h e more cumbersome form given by (6.8). Using and (6.13) we c a n w r i t e - L* 71 CQ,^,... UU^'X^X..! ml „,(,, c,.l) (6.17) = Z" Trace H" (^U,.. l$rL',tY... * ( « 7 . c \ J mlo > ,cV...'Xa,i> c';...UIo>,c ... ) , ,, Z Trace P ^ ° f ^ = e ? 6 p ,, > / > Ca.t.c,... ? c>,--)Guv,oc<.*L"c';.-') 0 l > ; 1 t^la,* t V , . 0 57 Now, l e tus write (5.5) and (5.6) m and i n t h e forms respectively, given by that i s , M TH = (5.5) X 1$ TT where i s symmetrical i n i Before (5.5) and (5.6) following We proceeding shall Dashes t o evaluate i twill ) and i (6.17) be c o n v e n i e n t to (6.22) (5.6) using to establish the convention: use the l e t t e r will be used a to refer to the states o f t h e 1st spin b to refer to the states o f t h e 2nd spin e to refer to the states of the i t f to refer to the states of the J t g to refer to the states of thek h to refer to the states of the l * * to distinguish different states h h spin spin spin 1 spin o f t h e same spin. Substituting elements But o f the form m. U.b.c, SO t h a t (5.5) into, (QW f o r example, U (6.17) l o u "). IO\XIWXX.~.A C o X c ' , . . .Jm.U> c ...!)^ra1ayb lb )....(S U.lO--. * , i j -(s'ma S,A W N , yields matrix because of the orthonormality But, then i f a,' + a , +a.,+ cxod b a , b*= b c ' = c -for of the eigenfunctions. E - E. » +a- + -* Q a = a — (We shall i n future 5 - \ Ci use t o mean b c -a i£ <Xn«A Onlij l£ J is in t o m e a n " f i s i n t h e s e t G" " f i s n o t i n t h e s e t G".) Thus, i f Ca ^ c ^ . . . I mj then IQ.WCJ, .. o 5-or (6.23) S\c and Si ' • ' . > » * * aa »»b «_*«•! Si (6.2*0 so t h a t = a \ U ' X & . U - U ) S. S. > m ; - o Thus and + ; i; \ i Trace from (6.17), S. 5 „ = Trace ftnv = X lcelm;U+i^| 3 c€<i. But J_ means summation over a l l combinations of « the N integers a,b,... ,e,..... Thus, I 1 . where 0 . a + luU u^l - + A r q + 21 3 + > v V ' " and where 2 1 lr*U-l« 0l a ; 7-«va - "'" r E V.^.-Cn-.V. C means summation over a l l Q integers i n the set G. Then, Trace t w P « = X , IG»W U-MT ; Now, § X 21 « * where ; r p ot > (ni , „ , ^ has been explained before. Because of the term which i s co «n Pp , , (o-0l i f o=o ^values of the integers i n the denominator of (6.25) » we can take the summation over a l l k n ^ t i j , . . . ^ ^ where o^-n^R for a l l to sz. Thus, C o - 0 1 = 00 Since Trs.Ce Po,ir>:fftro- - ii- -_ n we 0 I I * R 1 I • n +i> -»"+n t-+(i -.H-l > 1 ( 0 $ t n $. N-I) r R 1 < I R JQ. have But so that 11 * 0 , fer r r a t e p p m n W . = ft lc«U-U,T/ L f c T fit NOW, Ce\«\;U+r) =• CilmU + i^ 5or i= i, . ..^ N since we are considering N i d e n t i c a l spins. Then Trace P ^ f l , H T ^ p a m ? p m . F i n a l l y , then, using (6.28) we get Let us now consider Trace (5.5) and (5.6) P^^&rrv^ffl we can write T r a c e P ^ ' Vm w h e r e we Using P m write (6.18), **C Thus, = Z. L I (t^VnA + ( l ^ m-^-Y } + £ Trace (6.23), a n d ( 6 . 2 4 ) we -S-=. 31H. /fe b («-*m n \ f l % have "•i", .»»(n-r>l....Cn..'>!... -l , ; i m > m Y- B L i 1 / Ti i Bll"t Z. i -°- Cof-n <N) R — ) "-^jT Ml. l i « « 4 4 '"RJ i-"»---(n-i). l«-i)'...o. n 1 0 ^n ^ N-i t (oi-n $H-j) r Thus o/< R * =ft*"*I I « Similarity, Z Z l(*UUT G.^L.v) fcT CscJ M ; N Y *" Z 2_ * " o i m U t . ^ m ^ ^ . ^ i ^ u , ^ ^ •A a k T (6.30) (6.319 ZZ «*< ^ X X a and, ' («;'• m * * J ( « feT ZZ -*^ 3 « C Z7 -4 * XX : i m . A ^ O *••*"(*«;•J m-m\« * ' .r *A'*' ^ < (<r * : i m ; \M > v CI ft «\= k fe. * * +Q U*.LUX*Jtn|$+v)C« (« ^ - IT M *• * p oi< ^ (6.38) = o. where we Using (6.30) ^ ill; use Trace f e to P^y °. P (6.38) we Rm P, m have (j--/-X>,sVpr" ii;^ A , ^ a (6.39) where (6.40) (ft(p In writing the above we have used »<0 since tt^ i s symmetrical i n i andj . Then, using (6.29) and (6.39), we can write (W.18) as follows: = «™ *«G where (6.W3) Evaluation of (6.20) to (6.22) involves the same methods as used above. I t should only be necessary then to give the results of our c a l c u l a t i o n s . After using equations we have found that: ZZ /^w ( p * V ? «tG $*C ? (5.6), (6.28), and (6.42) / VpWP,Ti\ - a ?„ ^tt '\^\ + ( V '\ y^) -- ^«-c 4 l l I , where: (5.5), vW'.O* (6>5) -5 I I tfeUJa^YBtllmU^C*^! V » | a , t y M I , k | « ? | * * , S ) 4- 1 J>5 J j<*0 ; i(*o where we use : L to mean summation over a l l positive integers Otbr«+4 a and b wherein t »^«*4 a „ + a , ^ a +a . mean summation over a l l positive integers o a and b wherein r a + a - a + a. a b and where furthermore a «G to mean summation over a l l positive integers a and b wherein »+<x. = a +<* . a t Now, using (6.29) and (6.W5) we can write (W.18) as follows: ILL l l i -*(-^ - ^ ( - ^ -Vw . I- ^ ( *€(» : 6 - ^ ) A „ ( . A f i ) n •70 (W.17) Equation c a n now (te)* + aU*L <KV>« where be w r i t t e n as f o l l o w s : (6.1+7) + Jt <*» > 3 i!<£*>> a and are given by (6.W3) and (6.W6). (6.W7) Equation moment 1) i s a valid o f f ( v ) whenever (W.12) Condition a the following holds, — * -a expression ir\ = X « W) There has 0. identical R energy values Q pairs of which We pointed method yields containing are by unity i n i n i and j of which energy results unperturbed i s non-degenerate separation chapter as does spin (W.17) the temperature and . that when i n this the Van Vleck a l l the latter expression unity. a l l exponentials (6.W7), conditions and e a c h out i n the l a s t t h e same Replacing spins each have exponentials replaced symmetric ; are N have hold: that i s , and 3) f o r the second hold, which containing is identical yields: with the temperature (W.17) when by the above 7/ NRT. I I I I NR* Let us justified. HflL for example, ice i«, 0 Aj^ , K M ^ ) I A U) 0 etc consider The A C) under which term feT A^G*) c a n be written conditions this step i s 7JL , 4- But, fi= 2_ a -6 -r- Thus, so that fcr rV Now, we define 6= _?«•-«« J L _ <: f? QLmax r r f -tvQT = L 2 * — Q.m»«. • 1 J so that A x _5 « a We can then write etc etc x.cr. J-.' N N X I f , now, *.T « -+- Z -e€G 6T~ -2a * S 3 so that a l l etc S«i «. A c,a 3il ~i JL GD 0 then Similarily, We found that equation of have shown t h a t s p i n s has R Rv* Condition 2) S4'= Z Z «V? 3) m= 1 C In (6.48) i s a investigated. valid the Van V l e c k method, that i s , approximation second when energy each f o r the of the N values, a.^a^ and Q p a i r s of which identical } Q. , r have a l l of energy holds ; and symmetrical i n i ahd j ^ H- Chapters hold, A. In (4.12) IT } c a n be ,i f : 1) to a valid a r e non-degenerate separation (6.47) (6.47). (5»2) y i e l d s unperturbed i n then of f ( i > ) f o r the case moment which terms i f approximation T h u s , we the other that — A V a n d V I we i s , we tL the next have •— c h a p t e r we have assumed considered condition (4.12) that £L shall find expressions f o r - R a n d $}<crtj when We shall these new identical show t h e n that the temperature expressions f o r with those found and by r e p l a c i n g independent ^*-<a-o*> terms i n are not a l l exponentials containing the temperature respectively. regarded Van Vleck It will as n e g l i g i b l e method d o e s be when by u n i t y shown that compared not yield i n (4.18) a n d i f with a valid (4.19) cannot > then be the approximation f o r Chapter Expressions f o r (7;1) General The A <^> l a n J M ^ U n Formulae e x p r e s s i o n s f o r ^<v> concerned VII and lv^> with w h i c h we a r e a r e g i v e n by (7.1.D (7.1.2) where It B = right hand side of equation (*+.8) C = right hand side of equation (^.9) D = right hand side of equation (^.10) should then we • be n o t i c e d first t h a t w h e n we put = can write: = (right hand side of everywhere (7.1.1) except when % i s replaced by i n the exponentials i n (7.1.1)), and, z (right hand side of everywhere (7.1.2) except when ^ i s replaced i n the exponentials i n (7.1.2)). (see Chap t e r Let IV f o r d e f i n i t i o n u s now take of by and fc< taT).} •7(i> A in these (4.2) «, W ^ p iUw> expressions f o r and t h e f a c t (7.1.3) ^p; that Trace 45*-<e*>*> ABC = T r a c e . Then using CAB f o r a n y m a t r i c e s A , B , C we g e t i<rw>= -fr (7.I. *) ir (7.1.5) 1 where 6 =I X(/^- /^ ) ^« r r °<<ft + ~ : P ° X X ° - XX( A * { _ a + £ X X f. FCT ^ I X {-n^-c P.*' -«\P.«\ + ? v T^,« ^tt'f^W'^vnp^m + * P.^'V SMP, toP m f " T r ^ ( ? y ' V v'V.m P«\ - 3 P y V n ? . « V (0 B " ^ -aP^'Y^wpA^ Trac«( P ^ V ^ ^ W ' ^ W P , ^ 4-p v 'V^p^u 'V ^ p m') t c ? + c 0 o ( w P X P 0P «'P^ ' P^)} o w c ( ( ? > • ^7 Two points should be mentioned concerning equations (7.1.U) and ( 7 . 1 . 5 ) . F i r s t l y , the expansion indicated by i Pryce and Stevens (1950) on page h8 of t h e i r paper yields expressions which are s l i g h t l y cruder than (7.1.*+) and a. Secondly, taking ~ t- ^ K t then l e t t i n g (7.1.5). T-».OO i n ( 7 . I . * ) and ( 7 . 1 . 5 ) y i e l d s expressions which are d i f f e r e n t 1 i from those found when this procedure i s repeated with (*+.l8) and (W. 19) which are the expressions we found f o r £<t>*y .K*0^ > respectively when «. ~« ~ . lir and For example, E taking « kT then l e t t i n g -<--*oo - » - j£- ^<6v>- I I X X -w« whereas staking i n (M-.18) y i e l d s : I L (7.1.6) £mP m p then l e t t i n g ™ i- T**D i n (7.1.^) yields £ Jt* X - -us X T ^ C P ^ ' V ^ P , ^ -P,K i>in^(ri) lo L_! IS?.1.7) "Similarly, when this procedure i s carried out on equations (M-.19) and (7.1.5) the r e s u l t i n g expressions w i l l be temperature independent i n each case but they w i l l not be eqqal. In other words, i f we apply, f o r example, equations (^.18) ~7? and (7.1.^) to a special the results f o r high for this i s that imation ( a s we temperatures do will (7.1.6) i n writing ^ °" » c case we i n the next not agree. have chapter) The employed reason the approx- ft -> > <0 * P « we have 9. ~ ^ and the a taken < - >Y 0 — i assumption PU E P = a . and . should using find (7.1.6) Chapter 7.2 that form given spin shall (7.1.7) show Application We now of that rewrite when m (5«5) (5.6) and has R energy degenerate Since and VI only in case this fact, 0 > o-,,a a > then be written; we obtained i n i s true. considered i n a more i n Chapter VI convenient i n the forms and when each unperturbed ,0^ , a l l o f w h i c h a r e n o i j - 3 > energy are similar will used a i - » « c a n be w r i t t e n a «. not the r e s u l t s (7.1.^) o f which have c a n be & t o the case and « have In the a p p l i c a t i o n in fact equation we taken between this the r e s u l t s (7.1.^) have respectively, values Q pairs n (7.1.7) i s small. the calculations Chapter i P -jfcf ; we i f , (7.1.*0 f o r the case by V the d i f f e r e n c e and V I I I we ~i- In w r i t i n g ^L" ^ that E ** separation t o those given. We . given i n have found that where fcr L etc 4« c 3 ^•'Illa;*'.;*).-^^ +A Z Z Z {(;-& R -3 A 4 > J ^ V * ~ ^ A H > ^ ) ) 1 R +«IX I X( «*c n ^, I,-, 'Z Z C * ^ A J > l W + « * JLJ^S)):^ + -2 (7.2.3) 3=1 We have before, AJ*,5>> given A„(«\ JL,J*t$)t (see Chapter VI). The other and A's Jl C* i) 1>a 1 a r e as f o l l o w s : - I I Iccuu^ G,*IS«;?UO A G*>- I I 4a { laLU ,TL + Ma^M* + 5Xa|mla+iX*4'l»l*) + / A U,*) 4i = II {laUU*,")P Z la*.,*l«*!U,0| ir a-n, A G,s )= 45 A 4 U l3 I I ^ L ^auu.X4 iUs)C^l^ | * Q + lauu^LUM^U.^ -M <k$taJ** UA V- I I I . , A^ 3 > 3 )-II I A^G^^^H U . s V M * ^ ku^ta.*^ Kg)] H \X. • C . U U ^ L l d U . , i | ^ ! U J . y * l ^ l ^ ) > 1 A„(.,s,,,t).II I Equation l^wV^-s J ^ l * * " ^ ] L (7.1.5) + 3 l6i^.TtW«"W-(»#l<k*)]( ilt;ijA s> could also be written i n this, form but the labour involved would be quite considerable. In the next chapter we s h a l l r e s t r i c t ourselves to applying equation to a particular physical system. (7.2.1) Chapter Calculation Magnetic Nickel the Magnetic fluosilicate and resonance and expressions of absorption two (195D Kambe Van Vleck shall That to have lines has -ft*-<«^> case. 1 on . 1 a observed as as The relevance in the to I.U.K.) u s i n g as of question (1939) (see this was also this functions of recalled and nickel structure first Ishiguro, In the (7.2.1) the ** * for the by will not be temperature. (7.2.1) and equations that has P„« °P, ^ T crystal Becquerel we respectively 1 quantum and the -£<*v> -feW fluosilicate discussed I.U.K.) and * Usui, that assumption for theor- chapter £*<V-> of Kittel, frequencies by be that and found should Equation of (1950), It basis nickel Holden, (6.*+3)» ( 6 . ^ 6 ) , assumption structure by equations the is of absorption -SgS- derived crystal (5.2)). for been d e r i v e d by Field Axis. andStevens referred and apply Magnetic Optic been A<6\>> (6. +6) the mean s q u a r e equation and C J / ^ the (see special Spherically-shaped been Penrose for shall a the method i s , we (6.^3) by (hereafter give this by has 2 f o r the 6 absorption (NiSiF^-6H 0) etical £*< ^> C r y s t a l when D i r e c t i o n of (19^9) Yager and Absorption Fluosilicate in <£<&v) of Resonance VIII \ ). and mechanical and repeated been the problem Opechowski here. The main points 1) each of interest each R 3) N; i o n has the energy that single Thus, this case is The elements energy) ft"* I X levels spin S of each = 1 so of a that unperturbed of each unperturbed the s e t G defined i n Chapter ion * i o n are not VI consists 3, equidistant of the 1. equation (6.43), (6.46) and (7.2.1) t o take, f o r example, • itofi X reduces of > 4 expressions f o r interaction effective levels when a p p l y i n g tec ^ r , That of energy term, we i o n i s i n the presence + field, + + » number so N;'* paramagnetic crystalline 2) are t o one R<*v> , where energy. term and Z_Z_ If = i n this £ a = < W } contain K (exchange case. , the matrix spin-spin e n e r g y ) -+• ( d i p o l a r then + I I -4C( S;. O 3 a -3r- f, a S i .r 1 1 Y,S .r^](8.1) i In where | 5 If should then : (a the X I < tf \= not 5JS .+ S,.V , be .tu expressions 4 ; -SS^ M confused only matrix S 3 l | with for 6 the elements - of 3 M J» HS . B (o>o 4 used :this i n Chapter ft' VI) occurring 0 and are those in of the the operator Z I { f t i S..S. . 8 S 5 ; j + 4 ; 4 : } (8.2) where B ; i - and to -(a fjj) w h e r e we the small. d o e s -not a the d i r e c t i o n (I.U.K. have . At to high also cosine of 9j.= 3„= 3 taken in temperatures, i s dominant appear truncated so that ( the the + ^- relative their exchange resulting S^.) ; error this necessary.) (8.2) we expressions cs. * 5] as They have Using the use for contribution are r..* [ I w.*- i-axis. expression is J»J fcr have found for, become that and (6.*+3) End -K <&v > i l (6>6), we respectively: ft A fl found which when (8.2) Using expression for we h a v e -K<AV> found that we f o u n d (7.2.1), when * which i s the & * '^C- >p * ^) AT ' becomes: A<»>=-^|- «f^;(*-SO (8.5) kr where b=lC.U|^| ft L L lfl,U ^ ;^+/^K<VB;U + V r + a ^ - / ^ ) In t h e above -r-*oo As the other where fl hand, = 2_ A . , the r i g h t hand as , Q a constant T - * x > which goes side 6?.3") of %- - * o . f f -T ^ - * and to zero —•o as Q On (-£5*) goes to zero. o $hus, The the right difference, (7.1.7) In take Q hand side then, o f (8.5) i n the results ~~f^+ which goes applying (8.3) t o (8.5) i s the ?-axis to zero t o be t h e o p t i c shajhl a p p r o x i m a t e t h e l a t t i c e cubic lattice. cubic lattice.) We shall, i JC*i> furthermore, found —co as $ to nickel axis, using (7.1.6) goes and to zero. we and, f o r s i m p l i c i t y , a slightly approximation -r->co fluosilicate of the n i c k e l ( I ti s , i n f a c t , This as has a l s o ions by a distorted been shall we simple simple made b y I . U . K . take (8.6) S7 This We h o l d s , as shall also fi i f i s well-known, i and j are nearest has simple As a spherically shaped neighbours s i x nearest cubic (Each neighbours in a (8.7) lattice) otherwise. consequence of (8.6) and (8.7) we have and Finally, then, (6.43), (6.46) respectively: e crystal, take ion o for a + a*** 4. 5 " * and (7.2.1) b e c o m e FfKINC PftCE 88 FIGURE HE. T H E O R E T I C A L CURVE D E P E N D E N C E SPHER1COLI.X Fo*? THE ti* <as> ") FOR 3 S H R P E D N ; $ ; T E m P E R O T o R E F » CDH^O « CR><ST«L L E Q E N D ; 15 Mo SO ci* \»« o So ICO 83 .-R<6v> = -<*n 1 V— +A + a. 1 L V ; t T yg T + - a — A 5 g — A / B %1 - * j -t 4 * 0= KT — a. 0 w h e r e we ( d have i s the by are three x 3„M- l r'^C lattice and - That taken energy a ] * - o. tat N dl s c o n s t a n t ) , which values of each . O B (8.8) equations a virtual XX is a result obtained I.K.U. The I taken I I ) f o r the . (8.10) to case unperturbed we have As an plotted nickel ion illustration graphs (see of Figures when + 5,.|* H-& a i s , we H s a have taken constant = = + g„ 12 w + S kilogauss, . g„-=a.3(. .The function of experimental temperature data of and Penrose we have and taken Stevens We have , quantity £ is i t s values (1950). The a from the absolute value of the quantity IftU7.5*.o'%r s estimated transition. by I.K.U. t o be We have also used value. eigs We have, and <(*H) quantities Figure i n Figure I, plotted i n gauss. i s -iWfcv>= I I , we h a v e \&H > i n gauss 9, ^ <AH > b I I we have > 3N c moment As we results mentioned f o r -R(av)> ( 7 . 1 A ) , and agree of f ( v ) . that (seeFigure above example I t should A plotted (to> > = ((»- <v>y")> central as the ordinate . B Figure i n t h a these Similarily, also on i n e r g s and J r < ^ > The r e l a t i o n s h i p between 5 a . X i plotted . as the ordinate The r e l a t i o n s h i p between = <3^(u ( A H ) is the has been f o rthis 3 this fl these quantities be n o t i c e d a „ j 4*-<AV > a 1 i s , (*°*\ I t can easily i by t h i s t be shown i n Chapter VII, the high as given s h e second that application of by ( 6 . 4 3 ) and ( 7 . 2 . 1 ) , I ) . chosen t o be q u i t e have and, as expected, H (*»} '-i c temperature i s ,as given We on where c t that (4.18) do n o t large i n the difference i nthe results f o r JUv> = W is small For n + A<V> a t high (8.11) temperatures. T = 100°K, (8.11) yield ( i f , f o r example, i s negative): -«•» _o whereas, example, = 100°K, f o rT H = -n (8.11) and ( 7 . 2 . 1 ) i s negative): _n & <V) and ( 6 . 4 3 ) ^.ogXIO ir-g -o +O.»aX.l0 -n sr-^ - 3 0 . I 4 / . I O -»rtj yield ( i f , f o r The difference In the the next conclusions particular even at 100°K chapter that physical can i s indeed small. we shall very briefly be drawn from this system. mention application some to of a HI Chapter In this conclusions. the 1) 2) c h a p t e r we The f o l l o w i n g application i n Chapter The dependence magnetic to less The dependence helium observe of sample of our are apparent from of the c h a r a c t e r i s t i c s lines i s negligible of even down 1°K. resonance temperatures, paramagnetic some VIII of our general formulae: on t e m p e r a t u r e a change outline g e n e r a l remarks resonance than paramagnetic very briefly on temperature nuclear much shall IX lines that of the characteristics should i s , i t should i n the resonance resonance i s lowered from be O b s e r v a b l e lines room of at be p o s s i b l e liquid to f r e q u e n c y and t h e as the temperature temperature to liquid shape of the helium temperatures. 3) Both the resonance line depend We pertain can also when slightly make i n particular fluosilicate which the energy temperature, f r e q u e n c y and t h e shape we on t h e shape a few remarks from to the spherically £ o n e o f s> sample. Figures shaped considered i n Chapter difference, keeping of the of the resonance I and sample VIII: I I which of Below nickel 20°K, , i s approximately constant with or the H c o n s t a n t and v a r y i n g other, one s h o u l d mean v a l u e value Coupled Our extreme be this will - physical low temperatures, take This by f i n d i n g * That . P i s ppesitive. of the resonance ^<^v> i n both find line. attains I ) . an I t should of our an extreme c= *. ~* ~ value and when F we N ) f o r this however, i s not apparent; i t i s highly i s valid. This ( ^ - ^ a t these probable could the expression i n t h i s ^ fi of the v a r i a b l e that i s , we the coefficient, i s then occurs A P « ' - > approximations * ^ . take reason shown value ( i- The course, also i s lowered, I f t h e mean 1°K a n d 2 ° K ( s e e F i g u r e f * ft decreases, i n the width have f o r ^»<o.^> w h e n we these , the exchange i n t h e mean v a l u e the extreme -5*5. we shift between that for of fi calculations expressions take q u a n t i t y changes. be a d e c r e a s e value noted then as the temperature i f t h e mean v a l u e with quantity that of the v a r i a b l e increases, negative; find case that very neither be c h e c k e d , f o r + would involve c o n s i d e r a b l e l a b o u r and h a s n o t b e e n undertaken i n this thesis. of when Appendix Writing m ^ ~LT. of { P^mP. + Pstnp ,) Q -t < ^ in Let us Chapter energy which in have this where we values, p " J another general suppose VI, each A of that equation the N (5.5) identical a l l of which energy form. holds . that, unperturbed spins are non-degenerate separation and We shall and show Q as has R pairs here i n of that case XX write, - I "and as is operator 1) P. been discussed previously Xlx P P«. i m P w l ) ( ix •-- i s such * i. that: ?• = P« I, 2) X 3) p.UO. P. 4 ^ I : --lo. ^ and customary, 1*1x1* The have (A.2) , the identity W^u^'-a uy operator (A.3) ( A » where It and should Now, in i t should terms that f o r some *e<5 n^x where E .=I n a . be r e c a l l e d be p o s s i b l e R operators of the , and t h a t to write Z.r> -N where r - o,-+ E the operator P (ar.,^ . R N -» — tt n '. r=i of the d i r e c t product of In fact, 1 we can write as a sum o f terms 5 r each of which i i' , ,n «5 • n P s R The d i f f e r e n t terms p correspond consists i n the expression to the d i f f e r e n t combinations of the P Similarily c a n be w r i t t e n R P'S n as a sum o f n P's for P^ n. P \ , . . . . , n P ' R S . 2L-- terms. Let we P^m-P^ us c o n s i d e r P «. Pp see that rf . consists ; From the above discussion of \/L- NI N' T — *- n' l...f..,V.( ^!..^[ n terms. n Because non-zero. of There (A.2), however, can, i n fact, n o t a l l o f these terms a r e be a t most CN-Ol I- «..•-,• etc non-zero ...(o-,V.(, ,v... ! + terms Thus, Using (A.3) ponding form H OR i n P P*™^ , P A i t i s possible to each will t o group of the Q values of consist the e R into of Q R terms terms corres- one t e r m o f t h e Finally, then P, II < f Ilf.^P^ Similarily d< p ' Thus we N can write N N P« = I P„ . p . ••I • «€<. m F Z ltd j ? ^.9 *t * > j Appendix Rewriting form given Using A of tfWS m. = II* X p by I s h i g u r o , we x have B + P„mpJ Usui, from i nthe (195D a n d Kambe (A.l) N S * I x I { P.S..P ,. P .s,.P,.] + If, now, only one p a i r x V I* Equation Ishiguro, energy the ' of only the integer unity, , we from a i + have P (B.2) Usui, ^ a ; p i ; separation 1 and levels i s 2. that spin i s , wifla (B.l): (B.2) } i s identical a n d Kambe the energy integers ' of the unperturbed x labelled ' of levels { P,S .P I (B.l) fi " Q consists separation S + * esq with (195D. equation They have, of the unperturbed by •+ o (6) of however, spin whose whereas we u s e Appendix C A Showing that field a n d when the axis. Let Z us suppose (1,2,3,....,R-1). (3.2.2) h o l d s , unperturbed only - 5 spin between This i s no field crystalline i s parallel the s e t G i s the s e t of corresponds to the case i s , when the energy are equidistant adjacent there levels. levels and when Then, from to integers when of condition each transitions ( B . l ) we occur have: R-l A »• a.=i (CD > If we when the magnetic that that X » a A U \ then, using (k.h) and ( C . l ) have .C-'ls .U").= G!s,.U,y X, L x for It «:= l , a / i i s well eigenfunction .R-| known, h o w e v e r , of 5^ then: that i f UV denotes an S3 Thus, C i f V t , , then c."ii..i.-y..c.-is..i.o ; Since equation 1) Q we ( 5 . 2 ) , we consists (3.2.2) [ V- 2) These by Van to use are concerned can replace of the the 5* traces of operators with s e t (1,2,3,...R-l) 5* in i f le condition holds S*>o two conditions are satisfied Vleck S with x (19^8) instead and of hence S x f o r the case i t was considered not necessary f o r him 99 Bibliography Andrew, E.R., N u c l e a r Press, Becquerel, Broer, J . , and Opechowski, L.J.F., Holden, Ishiguro, Cambridge University (19^9). E., Usui, W., Physica 6 1039(1939). 10, 801 (1943). Physica A.N., K l t t e l , 1443 Magnetic Resonance, 1955* C , and Yager, T . , a n d Kambe, W.A., P h y s i c a l K., P h y s i c a Review 75, 1£ 310 (1951). P e n r o s e , R . P . , a n d S t e v e n s , K.W.H., P r o c e e d i n g s o f t h e P h y s i c a l S o c i e t y A j £ , 29 (1950). P r y c e , M.H.L., a n d S t e v e n s , K.W.H., P r o c e e d i n g s o f t h e P h y s i c a l S o c i e t y A 6 J , 36 (1950). Usui, Van T . , a n d Kambe, 'K., P r o g r e s s 302 (1952). Vleck, Waller, Physics 2it» H68 (1948). f u r P h y s i k 22, 380 (1932). J.H., P h y s i c a l I., Zeitschrift of Theoretical Review 8,
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On the temperature dependence of the shape of magnetica resonance lines McMillan, Malcolm 1959
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Title | On the temperature dependence of the shape of magnetica resonance lines |
Creator |
McMillan, Malcolm |
Publisher | University of British Columbia |
Date Issued | 1959 |
Description | This thesis is devoted to a theoretical study of that temperature dependence of the shape of magnetic resonance lines in solids which remain when the direct effect of lattice vibrations can be neglected. This is the case at sufficiently low temperatures. To discuss the shape of resonance lines the "moment method" is used. This procedure was introduced by Van Vleck (1948) and was used also by Pryce and Stevens (1950) and Usui and Kambe (1952). A line shape function which describes the shape of the resonance lines is defined and the first and second moments of this function are calculated in various approximations. In particular, the question to what extent the standard formula of Van Vleck for the second moment is valid is discussed in great detail. The general formulae are applied to the case of a spherical sample of nickel fluosilicate crystal. From the general discussions and from this special case it follows that the temperature dependence of the characteristics of paramagnetic resonance lines becomes noticeable at liquid helium temperatures and that these characteristics are then also dependent on the shape of the sample. |
Subject |
Magnetism Thermomagnetism |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2012-01-12 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0085389 |
URI | http://hdl.handle.net/2429/40048 |
Degree |
Master of Science - MSc |
Program |
Physics |
Affiliation |
Science, Faculty of Physics and Astronomy, Department of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
Aggregated Source Repository | DSpace |
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