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On the spin wave approximation in the theory of magnetically ordered crystals Pink, David Anthony 1964

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ON T H E SPIN WAVE APPROXIMATION IN T H E THEORY O F M A G N E T I C A L L Y ORDERED CRYSTALS by DAVID ANTHONY^PINK B . S c , Saiat Francis Xavier University, 1961 A thesis submitted in partial fulfilment of the requirements for the degree of Doctor of Philosophy in the Department of Physics We accept this thesis as conforming to the required standard T H E UNIVERSITY OF BRITISH COLUMBIA AUGUST, 1964 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of Bri t i sh Columbia, I agree that the Library shall make i t freely available for reference and study. I further agree that per-mission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. It is understood that;copying or publi-cation of this thesis.for financial gain shall not be allowed without my written permission. -Department of PHYSICS The University of Bri t i sh Columbia, Vancouver 8, Canada Date September 1,1964*  The U n i v e r s i t y of B r i t i s h Columbia FACULTY OF GRADUATE STUDIES PROGRAMME OF THE FINAL ORAL EXAMINATION FOR THE DEGREE OF DOCTOR OF PHILOSOPHY of DAVID ANTHONY HERBERT PINK B.Sc., St. Francis X. WEDNESDAY, SEPTEMBER 16, 1964, at 10^00 A.M. ROOM 10, HEBB BUILDING (PHYSICS) COMMITTEE IN CHARGE Chairman: I. McT. Cowan R. Barrie H. Schmidt C.W. Clark R.F. Snider M. McMillan L. de Sobrino External Examiner: Prof, J. Van Kranendonk Univ e r s i t y of Toronto Physics Department ON THE SPIN WAVE APPROXIMATION IN THE THEORY OF MAGNETICALLY ORDERED CRYSTALS ABSTRACT The question of the self-consistency of spin wave theory as applied to d i f f e r e n t spin arrangements i n magnetically ordered c r y s t a l s has been reinvestigated. A set of equations, involving the p r o b a b i l i t i e s of fi n d i n g a given number of spin deviations at a given s i t e f i r s t proposed by Van Kranendonk and Van. Vleck (1958) in connection with a simple cubic antiferromagnet at a temperature of 0°K, i s generalised and solved exactly for an a r b i t r a r y temperature. Two sets of equations are solved both for the case of a simple cubic antiferromagnet and for more general spin arrangementsj c o l l e c t i v e l y r e f e r r e d to as s p i r a l spin arrangements. In solving for the p r o b a b i l i t i e s a. method i s developed for easy c a l c u l a t i o n of the thermal average of c e r t a i n functions of number operators. F i n a l l y , numerical r e s u l t s are given for some p r o b a b i l i t i e s 1 connected withs ( i ) the simple cubic antiferromagnet and ( i i ) a model of the. rare earth metal, dysprosium. The l a t t e r i s of some in t e r e s t i n view of the investigations of s p i r a l spin arrangements i n recent years. GRADUATE STUDIES F i e l d of Study: Physics Elementary Quantum Mechanics Waves Electromagnetic Theory Nuclear Physics Advanced Magnetism Special R e l a t i v i t y Advanced Quantum Mechanics Related F i e l d s ; Mathematics -FoA. Kaempffer R.W. Stewart G.M Vol.koff JoB= Warren M. Bloom W. Opechowski FoA„ Kaempffer Theory and Applications of D i f f e r e n t i a l Equations C.W. Clark D i f f e r e n t i a l Equations Chemistry -S t a t i s t i c a l Mechanics C.A. Swanson R.F. Snider (ii) ABSTRACT The question of the self-consistency of spin wave theory as applied to different spin arrangements in magnetically ordered crystals has been reinvestigated. A set of equations, involving the probabilities of finding a given number of spin deviations at a given site first proposed by Van Kranendonk and Van Vleck (1958) in connection with a simple cubic anti-ferromagnet at a temperature of 0°K, is generalised and solved exactly for an arbitrary temperature. c a Two sets of equations are solved both for the case of a simple cubic antiferromagnet and for more general spin arrangements, collectively referred to as spiral spin arrangements. In solving for the probabilities a method is developed for easy calculation of the thermal average of certain functions of number operators. Finally, numerical results are given for some probabilities connected with: (i) the simple cubic antiferromagnet and (ii) a model of the rare earth metal, dysprosium. The latter is of some interest in view of the investigations of spiral spin arrangements in recent years. - i i i -T A B L E OF CONTENTS ABSTRACT ii T A B L E OF CONTENTS i i i LIST OF T A B L E S v ACKNOWLEDGEMENTS vi CHAPTER 1. Introduction and the Definition of the Spin Deviation Probabilities. i C H A P T E R 2. Canonical Transformations Used to Diagonalise the Hamiltonians 11 Section 1. Case (i): Simple Cubic Antiferromagnetic Hamiltonian 12 Section 2. Case (ii): A More General Hamiltonian 20 C H A P T E R 3. Mathematical Preliminaries and Formal Solutions of the Problem 36 Section 1. Normal Form Expansion of a Power of the Number Operator 37 Section 2. Formal Solutions for the Spin Deviation Probabilities 40 CHAPTER 4. Evaluation of Thermal Averages of Functions of Number Operators 48 Section 1. Thermal Averages for Case (i) 49 Section 2. Thermal Averages for Case (ii) 73 CHAPTER 5. Explicit Solution;!or JheuSpimfDevlfiaiiSniP^ 86 Section 1. Analytical Solutions for Case (i) ' and Some Numerical Results 87 Section 2. Analytical Solutions for Case (ii) 115 Section 3. Application of Case (ii) to Antiferromagnetic Dysprosium Metal 122 Section 4. Concluding R e marks 124B - iv -BIBLIOGRAPHY 126 APPENDIX 1. Outline of Walker's Approach in Order to Obtain Expressions for Probabilities 1Z7 APPENDIX 2. Outline of the Equivalence of the two Approaches to the Simple Cubic Antiferromagnet 130 APPENDIX 3. Derivation of the Relation nA„ + A n _, = A„ and Some Properties of the A'^' 136 APPENDIX 4. Equations Connecting the x^ and the A n f ) 139 APPENDIX 5. Equations Connecting the x f ' J and the A ^ 140 § APPENDIX 6. Alternative Derivation of an Expression for of Case (ii) (Equation (4. 2. 11) 142 APPENDIX 7. Proof that, for a Simple Cubic Antiferromagnet, G=0 145 APPENDIX 8. Proof that R as Given by (5„ 1. 5) + (5. 1. 6) is Equal to R as Given by (5. 2. 3) with h = O 147 APPENDIX 9. Alternative Expressions for |°„(T) of Case (ii) 148 - V -LIST OF T A B L E S Table 1. Values of A = W(T*O) as a Function of (J-t) « ar, Table 2. Values of tyato) and C/afy as Functions of a 105 107 Table 3 . Values of A = L(T»D) as a Function of (J-t) = at, Table 4. Values of |)oj(aS„q)and C0(a6,)as Functions of a 114 115 Table 5. Values of r, ft and pB(o)for Values of J 3 j and D j 124A - vi -ACKNOWLEDGEMENTS The author wishes to thank Professor W. Opechowski for suggesting the topic of this thesis, for the stimulating and enlightening discussions directly and indirectly concerned with the problems encountered, and finally for his continuous advice and help. Financial assistance in the form of a scholarship from the National Research Council of Canada is gratefully acknowledged. - '1 -C H A P T E R I Introduction and the Definition of the Spin Deviation Probabilities The object of this thesis is a study of certain aspects of the so-called spin-wave approximation in the theory of magnetically ordered crystals. The spin-wave approximation was first introduced by Bloch (1932) in his attempt to calculate the magnetisation of ferromagnetic crystals as de-scribed by the Heisenberg model (1928). The model is based on a spin Hamiltonian, whose form is very often taken to be (1.1.1) ft - ' £ J i j V 6 . where uandj refer to atomic sites in the crystal, and the sum runs over all such sites. S. denotes the operator of total angular momentum assoc-iated with the atom at site i and is usually referred to as the i-th spin operator. It is assumed that all atoms have the same magnitude of total angular momentum, S. cx The spin operators have components ( <*=-+•,—,% a n d operate on states defined, as usual (where we have put - I ), by: (1.1.2) 5 * l « i > - « i K > , etl*i>s •y/t*-O&+i*-s0 k+'> , 4>iK>- >/(*+«;Y>-*u+i)K-l> A more general form of the spin Hamiltonian will be considered presently. An approximation to (1.1.1) is often made by ignoring all interactions except those between nearest neighbours, and assuming that the coupling constants are independent of i and j . then becomes (1.1.3) H s - J I V S j <ij> J - 2 -where <<• j> denotis summation over all pairs of nearest neighbours. Equation (1.1.1) could also have been written as (1.1.4) where * b m Kronecker delta S&eJw&i interaction iscdlty isotropic. If, instead, fc/i^ K- a « p the interaction would be called anisotropic. Since we can put an anisotropic exchange interaction can always be written as (1.1.1) plus a suitable anisotropic exchange term. For example, auppoae that K r K for «. • z and K • 1 otherwise. Then the Hamiltonian would be written as Other interactions between the spins of the crystal can be taken into account by introducing appropriate additional terms in the Hamiltonian. For example, the lowest order term apart from a constant, suitable for describing an electrostatic crystal field of axial symmetry around the 6-axis would be of the form, (1.1.5) (1.1.6) where D is a constant. - 3 -The effect of an external magnetic field H can be taken care of by introducing, into the Hamiltonian, a term: (1.1.7) -yuj. T H - I , where /* ie the Bohr magneton and g is the electronic g-factor. Holstein and Primakoff (1940), in an attempt to circumvent the awkward commutation relations of the S* , simplified Bloch's approach by using a one to one correspondence between the operators S* and functions of harmonic oscillator operators. This correspondence is defined by (1.1.8) s: = Vai a?y>- « V * s , < = V a W / - "Was ^ , where a^  and a; operate on states defined by (1.1.9) ajr>p = V ^ K - i ) , a t ' O =V^7i K + O , • t a l O -The a\ ; a t operate tin an infinite-dimensional space spanned by the states |r>i) while the Sv operate in a (2S -*• l)-dimensional space, spanned by the states l^i) . The number n c is called the spin deviation number because it measures the deviation of the z-component of the i-th spin from its maximum value. The states . • •, i-2-15) correspond to the states |s>, . . .,l-s> while 12.&+0J • • • are "unphysical states". Due to the radical operator in (1.1.8) states do not contribute to observables. The radical operator is , however, inconvenient to work with despite the fact that it can be expanded into a finite series (Kubo, 1952): i ; x* ( l . l . 10) V i - *Ui/%c, * i 4. T. ^(ata) 1 ( O H ; ' is) - 4 -The inconvenience arises from the fact that the operators and S L are then expressed as linear combinations of products of one, three, five,. . . harmonic oscillator operators. This means that the exchange Hamiltonian is written as a linear combination of products of two, four, six,. o . operators, apart from a constant term. The diagonalisation of a spin Hamiltonian containing terms composed of products of four or more operators has not yet been performed. The spin-wave approximation as introduced by Bloch corresponds to replacing the radical in (1. L 10) by unity the argument being that at sufficiently low temperatures, the first term in the expansion of the radical in powers of 1/2 5 will be dominant. This approximation has the disadvantage that the "unphysical states" now contribute to observable quantities and further, that the amount of their contribution is unknown. However, Dyson (1956), has proved that for all practical purposes, the spin wave approximation is sufficiently accurate up to about ^-Tc in the case of ferromagnets with Hamiltonian (1.1. 3), that is, in the case of J ^ O. The situation is less satisfactory in the case of spin arrangements other than the ferromagnetic one. In the simplest case, Anderson (1952) used (1.1. 3) with J ^ O to describe a simple cubic antiferromagnet. The completely ordered state, which is known not to be the eigenstate of (1.1. 3) has all nearest neighbour spins oppositely oriented. This implies that the lattice is divisible into two interlocking sublattices, A and B. From now on we shall describe Anderson's approach in terms of operators which are linear combinations of the ones that he used. The operators - 5 -Sublattice-A spine are described by operators a t < sublattice-B spins by b-, i describing sites on sublattice-A, j on sublattice-B, and operators a t and bj commute. The spin wave approximation is used. After making the Fourier transformation, (1.1.11) where N is the total number of sites in the crystal, Anderson finds that the Hamiltonian is diagonalisable by a canonical transformation and can be expressed in terms of new operators, oc^and p^which are linear com-binations of operators a% and b% . The new ground state | &•) is defined by (1.1.1Z) <*xieO «= = O and Anderson is able to show that (G-IS* I &) = S a t , a , : l G ) = S - T where P= (cHa^O l^ &) ( f o r a n v Bfte i j n sublattice-A. He finds the quantity ^ to have a value of 0. 078. Also (G-| S* I &) = - S-v 0 .07? for any site j in sublattice-B. While this result shows that, at any given site i the z-component of the spin is nearly a maximum and, thus, that the assumption of " i being small is self consistent, it does not show to what extent the un-physical states contribute to any observables. In order to get a better idea about this point, Van Kranendonk and Van Vleck (1958) treated the same case as Anderson and have considered the probabilities, p n , of finding n v spin deviations on the site i when the system is in the ground state \ G-) . They have shown that the p„.. satisfy the set of equations 00 (1.1.13) Z « f t \ , . = (0rl(atAc/|GO U » o , i , x , . . . 0° = \) - 6 -and they have solved these equations for p„, p, and p* approximately on the assumption that (&lCo< Ct;)lCr) t which is equal to T introduced above* was small. {Because of translational symmetry within each sub-lattice, S^i. is independent of i). They found that p„ * I ~ ^  ~ 0 ^ 2-Since po 4- p( -t p^  « 1,their result shows that those probabilities p n v which correspond to "unphysical" situations, that is , for a, ZS, must indeed be very small. It is of some interest to have an expression for the probability of finding n; spin deviations on site i , when the system is at temperature T . Such a probability is written as \>n. ( T ) and the set of equations satisfied by the p„. (T) is derived as follows. Let I/O be an eigenstate of the Hamil -tonian with eigenvalue E ^ . It is possible to expand I/-0 as a linear com-bination of spin deviation states, each describing a state of the system, so that (1.1.14) I/O - £.&.<-'CtV I 'M "> where denotes the set of spin deviation numbers n u ^ and |l"\J) =TT . Then the probability of finding ti t spin deviations at site i , if the system is in state j /O is (1.1.15) ^ % J „ | C V » / The partition function is next defined in the usual way as (1.1.16) Z - Z C ^ W here (9= 'AT and jp„4 (T") is defined to be - 7 -(1.1.17) f,„f(T) = £ Moreover, if O is any operator, we define the quantity (0\. referred to as the thermal average of O at temperature T , by (1.1.18) <0>T = £ 2 6 " ^ (/*! Now (1.1.19) ( / x K a K / l i . ) = X 4 Z I C ^ J 1 Using (1.1.19) and definitions (1.1.17) and (1.1.18) we find (1.1.20) « « K r V = T 4p*.CT) (f*o,\t... o ° - 0 Introducing the number operator!* = equations (1. 1.20) can alBo be written as (1.1.21) 2 *ffc(T> - <Nf>T Putting T = O gives back equation (1.1.13). It would alaOiIbej useful to have an expression for the joint probability of finding n v spin deviations at site i and nj spin deviations at site j , when the system is at temperature T . This Joint probability will be writ-ten as p„.nj. (T) . The equations satisfied by the f^-CT) are derived in exactly the same manner as those for the P«,( T)- Using the same notation, we can write, OO <M> OO ( 1 . 1 . 2 2 ) i /o = r i £ C ^ ^ I ^ H O where now [nk3 denotes the set of spin deviation numbers n ^ n i "j and (1.1.23) (CVO * J { , n k ) Then we have ( 1 - 1 - 2 4 1 i v v ' J . . | C W as the probability of finding n-k spin deviations at site i and nj spin deviations at site j in the state )/*) . Introducing the partition function, Z , as before, we define (1.1.25) (V , . CT) = Then d.1.26) c ^ i c - f c A ^ V ) = gjZstfK'ir Using equations (1.1.26) together with definitions (1.1.18) and (1.1.25) we obtain (1.1.27) < C » t « i M ^ > T - * 2 K T . . ( T ) Using the number operator, defined previously we can write (1.1.28) ^ . ^ . ^ " / ^ - j ^ * < M ' N J > T - 9 -la a theory which uses two kinds of operators, a and b, as in the two sublattice approach to antiferromagnetism, the p^^fT) given as the solutions of (1.1.28) will give the probability of finding two specified spin deviations on the same sublattice. To obtain jon.ny(r) where i is on sublattice-A but j is on sublattice-B, we merely replace a^ by b^  and solve for the p „ i r i j ( T ) . This thesis is devoted almost exclusively to a mathematical study of the probabilities l 3^^) and .fVHn J(T) » a s defined by equations (1.1.17) and (1.1.25), for a few typical case of magnetically ordered crystals. The program is as follows: Chapter 2 deals with a review of the diagonalisation of various Hamiltonians and includes a mathematical des-cription of the two cases for which probability expressions will be obtained. The two cases are: (i) The case of a simple cubic antiferromagnet (that is , the case consid-ered by Anderson, Kubo, and Van Kranendonk and Van Vleck). (ii) The case of a more general class of spin arrangements of which the so-called spiral spin arrangements (Kaplan, 1959, Villain, 1959, Yoshimori, 1959), ferromagnetic arrangement and case (i) are members. For brevity we shall refer to this case as "spiral spin arrangements". The reason why we investigate case (i) separately is because we need not con-sider some of the complicated expressions used in case (ii). In both cases the spin wave approximation, as defined earlier, will be used. Chapter 3 contains the statement and proof of a lemma which is used to - 10 -dispense with unnecessary work. Formal solutions of equations (1.1.21) and (1.1.28) for the probabilities, £ It) , and joint prob-abilities, fa.n.lr) , follow this lemma. In Chapter 4 we evaluate assorted thermal averages of operators. A large part of this chapter is taken up with simplifying the first problem encountered so that subsequent problems are relatively simple. Finally we combine the results of Chapters 3 and 4 in Chapter 5 to obtain explicit expressions for the two kinds of probabilities. In addition to giving expressions for these probabilities we calculate numerical values for the expressions in order to get some idea of the validity of the spin wave approximation in a simple cubic antiferromagnet. Chapter 5 ends with ah application of case (ii) to a spiral spin arrangement, and some con-cluding remarks. In order that the reading of this thesis be easier, each chapter is preceeded by an outline of its contents. It should be mentioned that the probabilities for case (i) have very recently been given by Walker (1963) who used a method of generating functions, which is different from that presented below. The results obtained here agree with his. In Appendix 1 an outline of Walker's method is given. 11 -CHAPTER 2 Canonical Transformations Used to Diagonaliae the Hamiltonians  Summary Chapter 2 is divided into sections 1 and 2 which correspond respectively to cases (i) and (ii) defined in Chapter 1. The purpose of this chapter is to review the methods used to diagonalise the two kinds of Hamiltonian which we shall consider. A precise mathematical definition of each case is given at the end of each section. In section 1 the Hamiltonian considered is that normally used to de-scribe a simple cubic antiferromagnet. Two kinds of operators, cor-responding to the two sublattices, are introduced. Section 2 treats a more general kind of Hamiltonian which makes use of only one kind of operator. This Hamiltonian is of interest primarily in connection with the so-called spiral spin arrangements, although, as is shown, it can be used to describe the simple cubic antiferromagnet of section 1 as well as a ferromagnet. Its usefulness is limited in that it is difficult to handle if an external magnetic field term is included, except in special cases. - 12 -Section 1. Simple Cubic Antiferromagnet Hamiltonian. First consider a simple cubic atomic lattice. At each atomic site we now place a localised spin, each spin having magnitude S, such that the magnetic lattice is face centered cubic. The magnetic lattice is divisible into two sublattice s, A and B, and each site on one aroMattice has, as nearest neighbours, only sites from the other sublattice. There are z such nearest neighbours. This is the conventional simple cubic antiferromagnet first considered by Anderson (1952). We assume a Hamiltonian of the form ( z . i . i ) * - ^ S v i ^ E ^ - p K ^ - h r ^ <t'j> J i " J ' * j i where J>0 and we have assumed all interactions to be zero except those between nearest neighbours, which are all equal. As before, the symbol <i j> means "sum over all i and j such that i and j are nearest neighbours". The terms in D are lowest order anisotropy terms caused by an electrostatic crystal field. 1A discussion on the significance of D has been given by Kubo (1952). The terms in h represent interactions between the spin operators and an external magnetic field, H = h/^ug, whereyu and g were previously defined. We choose O to ensure that the easy axis of magnetisation is along the c-axis which is chosen to be the quantisation axis. There are N sites in the crystal and each summation over i or j goes over N/2 sites. N is thus an even number. Sites on sublattice-A are described by spin operators S,;. while sites on sublattice-B are described by spin operators . We shall use the Holstein-Primakoff substitutions, already introduced in Chapter 1, - 1 3 -and make use of the spin wave approximation, as described earlier, putting If we substitute these equations into (2. 1. 1„) and drop all terms made up of products or more than two operators, (Kubo, 1952) the Hamiltonian becomes (2.1.2) - - J s N . z - D S M * ZJe>Z[^i>ui + • ata. t l ^ , , * ] i b where 6^  is a vector to a nearest neighbour site and the sum over 6 signifies a sum over nearest neighbour sites. We introduce the Fourier transformations 14 -P . 1.3) . J | £ t « t { . , , *x Inverting these equations and substituting them into (2. L2) we obtain (Zl 1.4) - * + 2 J $ z X [ a ^ t ^ t / x + (2.1.5) % - * . f I [ A , # K • + B x ( * t £ + a ^ > ) l (2.1.6) V = to + can be written as (2.1.7) % t = X A, 8 X Ax audi (2. 1.8) 7, . n - > 2A = /A, 4- Aj Jz - 15 -Now, X . 2 A is just a Miasmitian form which can be diagonalised according to the elementary theorem of Hermitian forms. The fact that a^ and at>, do not commute does not affect the theorem. The new operators, more-over, will be made up of a linear combination of a ^ and . In view of this we may write: (z.1.9) - r c c x « i « x + EX(L^P!>) where the canonical transformation, also called the Anderson transformation, is (2.1.10) ecx = < A X + 9~ir\ , p x « c f ^ + s j V * c°^ , s* , c^ , and s^ are real because A k , A^ and are real and we require that (2.1.11) = \ * > l $ y , & l = K * , t ^ & l ' O These commutation relations lead to the condition that (2.1.12) c f = i , - \ For physical reasons we require that E > , E > ^ O. These will be called "allowable" energies. We than have (Cooper et al. , 1962) (2.1.13) L ^ H J =• * > ( < £ < v + 6 / ^ ) - 16 -Using these two results we can calculate, in turn - [" 01^ . , OV-. ~^-x~^\ and . [ <V. ^ J ] and find that (2.1.14) (A,- E ^ ) c £ - S^sJ which implies that (2. 1. 15) whence (2. 1. 16) The allowable energy is thus 7* (2. 1. lfe) V Similarly, by calculating [ /2>-,u. , 'Ka.] allowable energy is we would find'that the -1'& -(2.1.18) = ftJSxVo*^*-^ - •» It should be noted for use in Chapter 5 that = and = In view of the fact that C* - S>\ =1, we will try to write and as where (2.1.20) *X * with similar equations for and We then find that (2.1.21) A, - E> _ A - V A * - B ; t A- - Ev A — yA^-Bt Kx - *-x  Bx ~ Bx (2.1.22) x^ " Thus - 18*-(2.1.23) «x - , rx= y f ^ ' V ^ - O It should be noted that c _^ = , s _ x = s„ . It can be easily »proved that Urn t x = O \ -* 0 To end this section we will give a mathematical statement of case (i). A Fourier transformation, (2.1.24) « x s J | ? e l W a i is used, We assume that This implies that The Hamiltonian, written in terms of the a^ andb-^ operators, is diagonalised by a canonical transformation (2.1.25) o<x = cxflx+ **"^ x , P x a C A + 5 A a I > , t x , * x T « a l where 19 -This implies that 4 - $t = . and we require that C x - , * x 3 *x • '• f^fefc''ground state |G) is defined by, and « | c i t e d states h^), l U j and are defined I v v ) - T T l v x ) | v x ) , with - 20 -The Hamiltonian is in the form and E Vv i s given by The partition function is VV where (2> - YkT and the thermal average <°X- of an operator O is < o > = JL 2 « - p E y V v | 0 i v v ) Section 2. A more General Hamiltonian. We now consider a more general Hamiltonian. We do not assume anything about the atomic arrangement except that it is invariant under the operations of translation and inversion, that it has a symmetry higher than monoclinic and that all the atoms are identical their sym-metry being higher than that of the atomic arrangement. Nor do we assume anything about the distribution of spins except that each one is located at an atomic site and that they are all of the same magnitude, S„ The equilibrium direction of the spins, which may vary from site to site, is described by two angles ^ and d. defined as follows: The angle between the equilibrium direction of the spin at site i and the c-axis - 21 -of the whole lattice is given by <Pi , while is the angle between the a-axis of the lattice and the intersection that the plane perpendicular to the c-axis makes with the plane perpendicular to the equilibrium direction of the i-th spin. It is assumed that and 9c vary from site to site in such a way that (2.2.1a) ©L « <? • "? (2.2. lb) " P-*1 ESr purposes of simplicity we shall assume either ^ to be constant and 0^  given by (2. 2. la) or 0 to be constant and ^ given by (2. 2. lb) The Hamiltonian is assumed to be (2.2.2) where we write J(j-i) instead of Jij- to stress the translational invariance of X . It should be noted that we have omitted the term describing the effect of an external magnetic field. This is because, with the approach that we are using, it is difficult to diagonalise the Hamiltonian if there is such a magnetic field present. The only case in which no difficulty arises is the case in which the angle between the field and the equilibrium dir-ection of the i-th spin is independent of i . It should be noticed that the exchange interaction of (2. 2. 2) is not multiplied by 2 as it was in equation (2. 1. 1). This is because, in summing over i and j in (2„ 2. 2) we include each interaction twice. The Hamiltonian (2. 2. 2) could be used to describe a spiral spin arrangement (Cooper et al, 1962; Yosida and Miwa, 1961). - 22 -Each spin will be quantises along the axis defined by its equilibrium direction. This necessitates a definition of a set of axes at each site. Such axes will be called "local axes" and their directions will be designated as and I with the J -axis being in the equilibrium direction of the spin vectors. The transformation from s. ,<S. and to s • , sc and 5. is given by (2.2.3) 6j - ^ W w ^ + * t if lj^tP*^ - ^. cw. ©t *i«npf t 1 * We define s ; = S- ± vs.. . Making the Holstein-Primakoff substitutions together with the spin wave approximations, we recall that we obtain (2.2.4) 6* = V a s ^ , c,\ = fea\ j 4 = « - « t « i Substituting (2. 2. 4) into <fe. 2. 3) we can write (2.2.5) * + at^ciii^ — i ^ £ ( « 4 - t t t ) c p » a & M « i — C^— ftt^O*^*6*****?; - 23 -If we now substitute (2. 2. 5) into (2. 2. 2) and again drop all terms made up of products of more than two operators, the Hamiltonian can be written as (2.2.6) VL = -fce + Ot, + ^ where (2.2.7) = 0 (2.2.8) tf, = 5 - 24 -(2.2.9) We shall now make the usual Fourier transformations defined by (2.2.10) ay = j= h e « . In order to be able to do this, we must assume that the atoms form a Bravais lattice. We shall consider as mentioned previously two cases: (a) 0i * t t , cp. = c/> . (b) 9i = 9 , q>i - P • i . We shall define (2.2.11) J(\> - £ j ( j - i ) e J Because of the inversion symmetry of the lattice, = J(V). Case (a) Substituting (2. 2. 10) into (2. 2. 7), (2. 2. 8) and (2. 2. 9) we get (2.2.12) i , B = - 6(£-M)N [j/G?>*twxcp f J"(O)£0*Np] + ^NDto^cp (2.2.13) ^ - | 2 {^ !>x + « x f l l ) [ « ( J ' Q > * i M V + J^to$V) Part of the linear term vanishes because 7{-">)=• T (A). The remainder of the linear term can also be made to vanish if we assume that the equations which describe a classical spin vector can be used in the linear term (Walker, 1963). These equations are (2.2.14) $J s - S s i r K P ^ e i , = 6 t i n ^ 6 i * , 6 * » 6 e o & ^ The procedure would be to substitute (2. 2. 14) into (2. 2. 5) and solve for ( - a-) and (a v + a.^ ) . The resulting expressions when substi-tuted into (2. 2. 8) would yield terms like 2 cos 0^ and <}L S | k i ®t - 26 -which are zero. We see, then, that <R is given by the sum of (2.2. 12) and (2.2. 13) only. Let us now consider some special cases of ~M-X • (i) <p= o , GU 0 . Then (2.2.15) <Xt = S £ - JU) + d](a{ox + which is the well known form of the ferromagnetic Hamiltonian with anils otropy added. (ii) <P« \ . We now get (2.2.16) ^ = | Z | (4*x + a A . X « • J ^ ) - * - • X) - X) + 2 D) which is the Hamiltonian of a simple spiral (Yosida and Miwa, 1961) having turn angle | Q I C where c is the interatomic spacing, in suitable units, along the c-axis. All spins lying in a plane perpendicular to the c-axis have the same equilibrium direction. Case (b) Substituting (2.2.10) into (2.2.7) and (2. 2. 9) we get (2.2.17) t0 = -6(&+0[NJ(P) - N D ( J . ^ t o 6 t ( P . i ) ) ] - 27 -(2.2.18) 3^ * | 2 | + <y*lX* J ( p > * - J ^ p +^ ~J(P-V> - | D j Again, some of the linear terms vanish because J (-*) = J(^) while the remainder can be eliminated by using the equations, (2. 2. 14) describing a classical spin vector. Consider now the lattice to be simple cubic with a spin localised at each site. Let the basis of the lattice be given by the three mutually orthogonal unit vectors, St . and hy where the symbol ^ means that ( t J = J . Finally, let (2.2, 19) P = 71 ( b, + K + %3) 2P is thus a vector of the reciprocal lattice, so that - A 1 Z~P =±7^ Further, we shall assume that (2.2.20) J C j - i ) = - U I if ( j - t ) - *i (t=i,*.,S) J ( j - i ) = O otherwise - 28 -and that D ^ O . Combining (2.2.20) and (2.2. 11) we see that where £ means "sum over all nearest neighbours". The expression h for "Jtj. now becomes (2.2.21) «XX » U I * * 2 [ c « t ? x + f l x t f t X ' + l>') + 7>(4 f l-x + aX f l-x)] where z is the number of nearest neighbours of any site, 7^ is defined by -» * (2.2.22) = L Z * l X ( x * b and D' is defined by »' * i k • v ' > ° Equation (2. 2. 21) is the Hamiltonian for a magnetic arrangement having face centered cubic symmetry. The lattice can be divided into two sub-lattices, A and B, and spins on both sublattices have their equilibrium directions along the c-axis. The equilibrium directions of spins on sublattice-B are opposite to those on sublattice-A. Thus (2.2.21) gives the Hamiltonian for the same system that was considered in section 1 except that the external magnetic field is put equal to zero. An outline - 29 -of the proof that (2. 2. 21) and U x of (2. 1„ 6) with h = o are identical is given in Appendix 2. We shall now diagonalise the Hamiltonian given in equations (2.2. 13) and (2. 2. 21) which are special forms of (2. 2. 9). Referring back to equations (2. 20 13) and (2.2.21) we see that, for case (a) and the ex-ample of case (b), tt^ can be written as (2.2.23) * t = + t ( V 4xX4«x * where, for case (a), (2.2.24) = |[*-(J((5Hiwa-(p + J(0)ct*V) ± ( J ( e + * ) - J ( S - 3 0 ) t o * « p (2.2.25) _$±^ = I [ j ^ s t i V * (j«5+X)-J «a-X)) to&«f - l (j(<3+\) + J«?-X)Xi -ew*<p) -D*»*V while, for the example of case (b), we have (2.2.26) - A - lJ |6r( i+D') If we define (2.2.27) K * Ky. * *-B\ We can write • # 4 as - 30 -(2.2.28) where r t (2.2.29) ^ X\ A. As in section l s this Hamiltonian can be diagonalised by a canonical transformation (2.2. 30) <a»d that w i H he in the form (2.2.31) 0^ = £ ( £ x « £ * x + The parameters Cc^ and i. S x are, of course, real because A ^  and B x are real. We require that (2.2. 32) K , * r J = which implies that (2.2. 33) Calculation of [<V • yields (2.2.34) * x ] = « £ ^ + ^ ) - 31 -Performing two more commutations we get , _ [ cy., E « ~ , and [ < V X*/~ , Kj] (2.2.35) ^ V ' E ^ V ~ B ^ V = = O which, for nonzero C C ^ and S implies that (2.2. 36) whence The allowable energy value is thus We should note in contrast to the result of section 1, that E ^ , E ^ . - 32 -As we did in section 1, we shall try to write <> and Ss^ in the following form: (2.2.38) H * j = , 6 X = where (2.2.39) * \ = T% Substituting (2. 2. 37) into (2.2. 35) and using (2. 2. 39) we find that (2.2.40) t where A + 7*" Thus, T - ^ = 3^ a n a (2.2. 41) o_x - c x , e x « s x f i n a l l y we shall give a mathematical statement of case (ii). The Fourier transformation used is - 33 -(2.2.42) ^ = ^ \ € a; where The Hamiltonian, written in terms of and a^ , is diagonalised (2.2.43) oc^  » , s K* > ^ ^ r e a l which implies that 6 X - 6 * - I We require that c_x« c x and S_^ = -S^  The ground state I G") is defined, as usual, by <*xl&) = O and excited states l^") and li-O , by |v x ) = ^=(<*J)V*|&) , iv) - 77|vx) where *x>xl"0 - V A | V X ) The Hamiltonian is - 34 -where The partition function and the thermal average of an operator O are defined by Z - S * " ^ , <o>T = ^ I , e " p E v ( v l o | v 3 v Z v We shall now obtyim'.Ssome relations^ which will prove of some in-terest in Appendix; 7>. Consider the example given in case (b), the r Hamiltonian of which is (2. 2. 21) and A ^ and £ are given in (2.2.2 6). Bearing in mind the definition of T> in (2.2. 19) and ^ in (2.2. 22) we see that - 35 -Therefore ( 2 . 2 . 4 » y x , v = - T x where we have written "P as "P in the subscript attached to 7 x +-p Using the expression for T^. given in (2. 2. 40) together with the definition of B x given in (2.2.27) we obtain (2.2.45) 7J _ = - * M Together with equation (2.2. 38), this implies that (2.2.46) C x + p - <\ , $ x + p - - S x Furthermore, using (2.2. 37) we find that, (2.2.47) £ x + p - £ x - 36 -CHAPTER 3 Mathematical Preliminaries and Formal Solution of the Problem  Summary In the first section of this chapter we prove a lemma which gives i t \f M an expression for (ft; QJ expanded in terms of <2; <2t for n = 1,. . . , f. This expansion, referred to as the normal form expansion, is used to expand (a r;ai)*(ajap 5 and (a< a t) f (b* b,-) 9 In the second section we consider the formal solution of the equations satisfied by (T) and J?n;nj ( T ) • After some manipulations the solutions are put into a form containing the quantities ( a ; >T , \ a i 0.\ Qj / T or \Q; Pj Q,- bj ^ depending on which probabilities are being considered. - 37 -Section 1. Normal Form 0q>ansion of a Power of the Number  Operator The evaluation of the thermal averages ) T and (fy 4. which are necessary for the calculation of the probabilities defined previously, is facilitated by expressing N / as a sum of terms, each term being a product of annihilation and creation operators written in normal form. That is, each term has all creation operators standing to the left of all annihilation operators. Such an expansion for is given in the following lemma which is proven below. Lemma If a a - a a - l then (a + a)=N = 23 a a — — — — nil Proof: The lemma is true for f = 1 and 2, as may be seen easily by explicitly writing out the relevant functions. For f = 1, A^= 1 the lemma merely says that eta. - a a.. For f = 2, A? = I and i j - £ [ - ( * ) + (t)*] = ' The lemma then says that (eta)z * eta + a+1a*-This is also true because LotaY = <xf(e?a + \) A = + eta. Assume now that the lemma is true for any f. Then - 38 -It is a simple matter to prove that 1 -t »»—' fv A The proof of this is by induction (Direc 1958, }>\ 137). Thus, Hence Using the definition of given above, we prove the following relations in Appendix 3: Al*} = $ * / for all f Using these two results and equation (3. 1. 1) we see immediately Sb-athat (A1«.) s A, <ra + 2- *n a' a + A / + l a 7 a z ' / r v v . By induction the lemma follows Since the normal form expansion proved in the lemma depended only - 39 -upon the fact that a and a + satisfied commutation relations, that is this expansion is valid for any operators which obey.sueVi commutation relations. An example of another pair of operators is that of -~- and x, because It was discovered, after the normal form expansion for had been derived, that Schwatt (1924) had given such an expression for the case We can also obtain an expansion for (o-i a 0 ( Q j a j ) which is merely a product of two expansions like that given above: % £ A1** A & +" tm « m , . , v (3.1.2) N: r Nf - 2- I. An Am flj 4:: C^tt: , ( i / j ) . Similarly, the normal form expansion for (ai^i) (^j^j) is (3.1.3) ( ^ * t * j ) » - z £ A ^ A f f . r v j v v f . * 1 v J j «=l nr>=l J 1 J - 40 -Section 2. Formal Solutions for the Spin Deviation Probabilities. We shall first obtain the formal solution of (1. 1. 21). In order to do this, the following procedure is used: We shall solve for a given )pw (T) in the set of equations M (3.2.1-f) M K < T ) = <Nf>T ( * * 0 A . . . , M . 0 » « 0 , i and let M approach infinity. The equations of the set (3. 2. 1-f) are distinguished by the number f. First , eliminate from each equation (3, 2. 1-f) except from (3. 2. 1-0). From now on the subscript i will be dropped whereasver possible, but it will be understood. We get M I b„(T) = I n-o (3.2.2) M - n-o This set can be solved formally for f^ , (T) giving (3. 2. 3) bw(T) * . + JE x f [<W^>^ -where - 41 -(3o 2.4) = (-0 - W - w 1 ((w-i) - w ) ((wfi) - w) ( M - w ) The rows of the determinant in the numerator, except for the first one can be labelled by the exponents 1,. . . , M and the columns by the numbers O, (w) 1, o . . , M . The determinant in the numerator* of the expression for *»• has the row labelled by r and the column labelled by w missing. The determinant in the denominator can also have its rows and columns labelled by the exponents 1,. . . M and the numbers 0,1, . . . , M respectively, with the column labelled by w missing. The expression for X^ w ) is valid for w = O , . . o , M . (w) From now on the superscript (w) in * r will be dropped but will be understood. The * Y satisfy the following set of equations which do not - 42 -contain the bn ( r) # M (3.2.5-f) 1 + *-r = ° U=O yV..,(w-.yw+.;,.. vM). -r=o This set can be obtained by transposing both of the determinants in (3. 2. 4) and then working backwards to write out the set of equations (3. 2. 5-f) for the * r for which (3. 2. 4) with the determinants transposed, is a formal solution. The set (3. 2. 5-f) is not convenient to work with therefore we subtract (3. 2. 5-0) from (3. 2. 5-f), for f=l,. . . , (w-1, ) (w+1),..., M and obtain M (3. 2.6) 2 W < X t = I -r= i M (3.2.7) ^ 6 ' x f - 0 ( * = i , . . . , (w-0 , (w+i) , . •• , which can be combined as M (3.2.8-s) Z s \ = *>4W 0=>, Ssw= K-ro««lcer delta). -r=i Using (3. 2. 3) and (3. 2. 8-w) we see that M (3.2.9) }>W(T> = _2*T< Nr>T # The existence of the set (3. 2. 5-f) was suggested by a remark made by Muir (1911) in connection with a paper by Murphy (1833). A set of equations solved by Murphy is a set for the x/ 0 > which occur for jpotT) . The solution for the X r < 0 ) given by Murphy, however, would not be suitable for use in connection with f° Murphy's *v°' are svmmetrir fnnrtinno nf \A infon«» a C . - 43 -This equation can be further simplified by using the normal form expansion of <(r^  }T given in the Lemma of Section 1, together with a property of the which is proven in Appendix 3 and a set of equations involving the A„ and the * r which is derived A(r) in Appendix 4. The property of the Ar. is that, (3. 2.10) - o if -r < w. This property might be expected if we look back at the normal form expansion of N / in the Lemma. Evidently, the maximum power of a t^ or a ; in the normal form expansion must be f. This means that the coefficient of ct^d**, I M should be zero. This is whait equation (3.2.10) says. The Xir and the A„ are related through the following set of equations: (3.2. i i ) Z A t x; = — U ) where (w) = 0 if t w. The number M appearing as the upper limit in the sum over r is the same as that appearing in equation (3. 2. 1-f). Using the normal form expansion of M t we can then write M -r (3.2.12) hw(T) = ZA«} • Now, since A„ = U if r^n, we can replace the upper limit r in the sum over n by M . Then, if we interchange the two summations, we get m m . (3.2.13) bw(T) s £ ( Z *+An)<a\mai>T • Using equations (3.2. 11) we obtain - 44 -M rt *w (3.2.14) (.W(T) = (-,)" | t^" ( » ) < 4 X > T • Now we shall obtain the formal solution of (1. 1. 28). As before we shall consider a finite set of equations, M M (3.2.15-f.g) ^J.y^U'P * < N i N * > T > ^ ' r - . M y ^ l ) , and let M approach infinity. Each equation of the set (3. 2. 15-f, g) is distinguished by the pair of numbers f and g. The quantity pw, wy (T) is eliminated from all but one of the equations of (3. 2. 15-f, g) by multiplying i » (3. 2. 15-0,0) by wt and subtracting this from (3. 2. 15-f, g) for all f and g except f = g = O. The new equations are M M ? 2 fV.„ s<T> -(3.2.16) M M «• J J ^ ^ = o,. •:, M , except / = g = o-- 45 -The similarity between the set of equations (3. 2.16) and the set of equations (3. 2. 2) should be noted. We now have two w's, two exponents, f and g, and two labels, ni and nj, in the equations. The set of equations (3. 2.16) can be solved formally for pWiWj ( T ) giving; M M t 1 *=<> r (3-2.17) CT) . + j £ ( 0 ) X ^ [ < N f N J > T - ^ / ] where the notation means "sum as indicated but <K83J£ the term in which both f and g are zero". A determinantal form, similar to that of (3. 2. 4), may be written out for the . • Now, however, the rows, except the first in the numerator determinant, are labelled by two exponents, each one taking one values 1,. . , M and the columns are labelled by two numbers taking on values CK,. . . , and O j , . . . , Mj respectively. Because of the similarity of (3„ 2. 17) and (3. 2. 3), (Wt w,-j the determinants represetting the are similar to those representing the in equation (3. 2. 4) and so we do not write them out here. In the expression for X^ the determinant in the numerator has the row labelled by the pair of exponents f and g and the column labelled by the pair of numbers w^  and w^  missing. The determinant in the denominator has the column labelled by wi and Wj missing. The quotient of the two determinants is multiplied by a factor of (-1) raised to some power. The power is determined by the position of the row labelled by f and g in the numerator. Because we shall never be using the determinantal form of X ' 1 , it is irrelevant whether the quotient of h the determinant^;, is proceeded by a plus or a minus sign. - 46 -WO From now on the superscript (w^j) will be omitted from * ^ , but it will be understood. As before, we can transpose each determinant of the expression for Xj^ and work backwards to find a set of equations involving the.rX^ for which the expression for , with each of the determinants transposed, is a solution. We, thus, again find a set of equations involving the X ^ which does not contain the (3,;W. (T) and which is given by: M M (3.2.18-n,m) £ £ l O ( » f - ^ _ w f w » ) x = 0 „ t m = 0j , M ; if n = Wi, m ; ii m = wy ,11^^. From the set of equations (3. 2.18-n,m) we derive a set with which it is more convenient to work. Subtract (3. 2.18 - o, o) from all other equations (3. 2.18-n, obtaining: M M (3. 2.19>n,m) £ Z (°) >»'ro*X. = ^ f t W > w w . , w = 0, • . M. <j=o ' 1 j Using (3. 2.19-n,m) and (3.2.17) we see that M M (3.2.20) b IT) = £ Z ^ X , < N f N j > T . As we did for the case of (^CT) , we can simplify this expression. We first substitute the normal form expansion for < M< N j ) T and get - 47 -The upper limits f and g in the sums over n and mi^may be replaced by M (T) in both cases, because A s =0 if r s. If we do this, and interchange the sums over n and m and those over f and g, we get o.2.22) p (T) - z z (i z (o)x A ? f ^ y ^ c < a P T • w i w j n=i m=i i=o 3 j J T We now use the result of Appendix 5 iwhich gives a set of equations K-W)J .(t) (<}) involving and and A m . The set of equations is where M has the same significance as in equations (3.2. 15-f, g). We can now write pwi Wj• (T) as The next chapter will be concerned with finding expressions for < dt'Q^Q" a")T and ( 4 X ) r case (i) and case (ii). - 48 -C H A P T E R 4 Evaluation of Thermal Ave rage a of Functions of Number Operators  Summary Chapter 4 is concerned with evaluating \ a { Q.iy>T > \ a v a.j CLj / r and <,at toj ct^  Dj / T ^ and is made up of two sections. Section 1 deals with evaluating these three thermal averages for case (i) as defined in the introduction; while Section 2, which takes care of case (ii), is concerned with evaluating the first two thermal averages. The thjj-|d thermal average does not occur in case (ii) because only one kind of operator is involved. The first part of Section 1, which deals with ^Cl^ / is concerned with reducing the problem to its simplest form. After this, for the remainder of section 1 and both parts of sec-tion 2 the results can be written down quite easily. - 49 -Section 1. Thermal Averages for Case (1) The results of Chapter 3 showed that ^N^) and (N< Nj >T must be evaluated in order to get an expression for ("0 and p„ i K , . (T). We saw 'ht the end of that section that ( N l f } could be evaluated without difficulty once Q.^  ^ had been calculated. This possibility we shall find, makes the evaluation of <^N t y easier than if we tried to calculate it without using the normal form expansion, given in the lemma. Similar remarks hold for the evaluation of , ^ N t Mj /r for i =/ j , because it is a product of two normal form expansions. First we shall evaluate <(av ^ - I / T • We define a Fourier transformation (4.1.1) H = =^> £ e l X \ where N* is the number of sites in the lattice to which the site i belongs. i NJ , For case (i) N = , while for case (ii) N = N . Using this Fourier transformation we can write (4. i . 2) afr; = i I • • • S• • • £ I • • • £ Z t*\ • • V x e * V • • We now assume that the Hamiltonian of the system is diagonalised by a canonical transformation, (see equations (2. 1.25) and (2.2.43) ), (4.1.3) ATX - A T c > - B f t X , H ' *cx~*tx t + ^ t t For case (i) A c X = A and £ s „ = ^ . For case (ii) A c > = ^ = s > ^--^ . We now define A c i x a n d B s i X by the equations. (4.1.4) A f t i x - A f t X € i V t , 3 4 J X = B n « i V ' - 50 -Using (4. 1. 3) and (4. 1. 4) we can write (4. 1.2) as. If we multiply out the bracketed operators we shall get a sum of 2 terms, each term made up of a product of operators with one operator coming from each pair of bracketed operators. If we single out any one of these 2*" terms and include with it the 2n summations over the andyu's and the factor of ' / N ' m , we get what we will refer to as a product term. The general product term can be written as where k and k' are the numbers of " c i ; x s and A ^ ' s respectively in (4. 1.6). The expression (4. 1. 6) is, of course, made up of N ' 1 * terms because of the summations over the A s and •=> . We may note, first of all , that since we shall be taking the expectation value of (4. 1.6), we must have k' a k if that product term is going to make any non-zero contribution to the expectation value. This is ob-viously true if the A c ' s and the B s ' s refer to case (i) because, then, t , we must have the number of <x s equal to the number of s t t In case (ii) B 5 and B s represent <X and operators respectively. The total number of ex*'s is then (n-k1 + k) and the total number of tf's 1B (n-k + k' )• These two numbers must bq, equal which implies that k' = k. - 51 -Consider now the product term for which n = 2 and k' = k = O. Then (4. 1. 6) is In taking the expectation value of (4. 1.7) in the state J/*-) , (4. 1. 7) will give zero unless 'V/*; and , or and 1-^=/*, . Another way of pufjjting it is that (4. 1. 7) must be expressed as a product of pairs, B s B j , if it is to make a non-zero contribution to the expectation value. We can generalize from this example saying that, for each creation operator appearing in a| product term and labelled by a certain wavevector, there must be an annihilation operator, appearing in the same product term and labelled by the same wavevector, in order that the product term may make a non-zero contribution to the expectation value. We shall say then, that in order to pick out those product terms which make non-zero contributions, we must "pair" each annihilation operator of a given kind with a creation operator of the same kind. The process of doing so will be referred to as "pairing". Thus, in the example above, the two possible contributing cases were obtained if B s i X ( was paired with and B t i ^ was paired with B^y^ or if B^was paired with and Bsi.*K was paired with B^%/i | SAisWe can thus write out the expectation value of (4. 1. 7) as: Generally only those product terms in whixih each annihilation operator is paired with a creation operator need be considered when the expectation value is being taken. - 52 -Expression (4, 1.8) is not in a convenient form. It can be rewritten as: Taking the thermal average of (4. 1.9). we get (4.1.10) ^ t i B ^ X 2 < B ^ B l i V i > T + ^ S < B ^ X > T . In all the cases in which we shall be interested the second term in (4. 1. 10) is of order 1/N'. Since N' is assumed to be very large and eventually to approach infinity, this term will be dropped. Furthermore, we will extend the summations in the first term to include the case of X 2 = A( . This will introduce an 0(1/N') term into (4. 1. 10). Finally, we get: " M * ? ^ 8 ' ^ 8 ^ ' ^ ? / 6 * ^ 8 * ^ + O(^)terms {omittea). This approximation is the large N approximation, customary in statistical mechanics of large systems, and will be referred to as the L N A * . If we now return to (4. 1.7) and u6e the pairing process and the L N A , that is, if we pair each B j X with some , which can be done in 21 ways, use the ^Subsequent investigations have shown that the results of this chapter, and equation (4. 1.11) in particular, are exact for all N*. Thus. - 53 -commutation rules and drop the 0(1/N) term, we can write the thermal average of (4. 1.7) as (4.1.13) . ! ^ I < B r t , Z<*,a/t« >T . ^ [ s - , I < 6 s i X , > T ] 1 Now consider the general term (4. 1. 6) and assume that <(AC B s ^ _ 5: 0. This assumption means that we are considering case (i). We have seen that, for (4. 1.6) to make a non-zero contribution to the expectation value we must have k' = k. The A \ ' S and the A c's may be paired in kf ways and the B s s and Bg'$ may be paired in (n-k)! ways. If we use the commutation relations as was done in the previous example, repeatedly, until every two operators which are paired are adjacent to each other, we may write (4. 1.6), after taking the thermal average, as (4.1.14) l / ^ ^ ^ ( JL £ /) where, as in the previous example, we do not commute the two operators which are paired. As long as we let N' approach infinity before any other quantities do, we may drop the 0(1/N') terms in (4. 1. 14) and write it as: (4.1.15, Hence, - 54 -(4. 1. 1 6) J. We may now draw the following conclusion: If, in a general term like (4. 1. 6) we pair operators, use the commutation relations repeatedly until every two operators which are paired are adjacent to each other and use the L N A , then the result of this process is as if we had paired operators and assumed that each operator commuted with all other;' operators except the one with which it was paired. We shall make use of this observation in all calculations which follow. We now define (4. 1.17) cik and introduce the following abbreviated notation: (4. 1. 18) and (4. 1.19) - 55 -VP H n n Each Ac^ hr is called an "operator" and each bracketed pair of operators, ( A c i -B S i )^ , is called a "bracket". We may thus write 0. • in (4. 1.5) as a product of n brackets ( A t - "B.^ and n brackets ( A c - T | ) . The subscript attached to each bracket is really only a label in order to show how many brackets there are and to keep them in order, (A c-~&sV preceeding ( A c - ~ B 5 ) t if k I . Thus, we may write (4. 1.5) as (4.1.20) &al = ( < - 5 5 ) , ^ i - » * \ • • • (At-B, ) , , where we have omitted the subscript i but it is understood and have written the product on the right hand side of (4. 1.20) in two rows for later convenience. The expression is read from left to right beginning in the upper row. We ; may recall that any product term is made up of a product of operators, each bracket contributing one and only one operator. Thus, we could obtain the general term, (4. 1. 6), from (4. 1.20) by choosing B s ' s from ( \ * and ( - \ \ , ltc's from ( K ~\)rt and (A* - ^ ) r „ :VmX s from (A c-"Bj) u and (A t-"Sj)v , A c 's f r o m ( A c - ^ ) h and (K'^)^ and suitable operators from other brackets depending upon the explicit - 56 -form of (4. 1.6). Of course, as pointed out previously, we need only consider those terms wherein k' = k. Note that "Bsu preceeds "B S v in the product term if u^v . Also, all operators from brackets like (A c-"B s) preceede all operators from brackets like (^c~^>s ) . In order to evaluate (a, a ;/Twe shall consider only those product terms in which A c '& are paired with Acs and B^>s  with ~%>s s » because we assumed that Ac's and TVs were different operators, i . e . , that (AI^JV 0 The expression resulting from the pairing of two operators and taking their thermal average will be referred to as a "contraction". Thus, if t we contract ^ c i k a n < i A c ^ we get (4. 1.21). -t where the identifying number, k, is omitted because (Ac.kAei.^is just a number and it is not necessary to identify it in this way. In our new parlance, then, the contraction of A c i . k and T3s;j. is zero. Thus, in evaluating <Q{ Cl^we shall have contractions only between the two rows in equation (4. 1. 20). These will be referred to as "cross contractions" and will be abbreviated to C C . Later on, we shall define three other sets of letters to denote Contractions different from C C . In those cases, as well as in the case of C C , the sets of letters will represent the noun (singular or plural), the verb (all tenses) and the adjective formed from the corresponding word. In order to see how all this is applied, let us first consider two simple cases: Those for which n=l and n=2 in equation (4. 1. 20). n=l: OfO.- is written as - 57 -(4. 1. 22) Each bracket contributes only one operator to a product term. The two contributing product terms are A C A C and Thus, we have (4.1.23) <at a i>T = <AtA t>T + < V l > T We will represent the process of contraction by a line connecting the t -t two contracted operators in (4. 1.22) and write (A t and ^ "^s)-,- as and respectively. Thus, (4. 1.24) < A i * i > T «AC-Bt), (Ac-»I\ n=2: We now have - 58 -(4. 1.25) The possible contributing product terms are A* A^AcA^ ) At,^^^C(Bt^  ^S3BSI^C2 , s^^ x^ a^ sa » ^* i^c i ^ i ^c i » and B s3s;A'^si • m the first and last terms we can contract A c t with A C i or with A c x , and £ s , withT3sa or with B S | . That is, in each of these terms there are 2! possible contractions. There is only one possible contraction in eac h of the second, third, fourth and fifth terms because ( A c B s > T = Q . The sum of all possible contractions gives (4.1.26) <*N?>T = *![<A f A> T • <& s Bj> T f . Using the notation that was defined above, we can represent each of the contributions from contractions by: ( A l - B ^ A t - B ^ (A ( ^ c - B l V A c - B f ^ - 59 -(AC-BI),(AC-BI> = <B«B^<B 5 t Bj 1 > T , (At-Bj),(A«rBl) ( A t - B s ) , ( ^ - B s \ ( A c -& I ) , ( V B l ) x M c - B t ) , ( A c - B t ) j L ( A f - B A M t - B A = <At,A„> T<BH lB/ 1> T, (Ac-Bt),(Ac- BI\ ( A . - B l ) f ( / C - B t ^ Thus, = *f<AtA«>T]* 4 *<4tA c > T <B w Bj> r * * [ < B , B J > T ] * - *![<AtA,> T + < B 4 B t > T ] 1 1" n n Now, referring to &• &v as written in (4. 1.20), we consider a particular product term formed as follows: From the upper row select t, A t 5 from each of the first k brackets and "Bs's from the remainder and from..the lower row select A' s from each of the first k brackets - 60 -t T and B s 's from the remainder. We now consider the case in which Aa T is contracted with A L( , I = 1 tZ, . . . . , k and 13 s r with T3 s r , r = k+1, , n. As was done previously, these contractions can be represented by Of. 1-27) (A£-B,) and the contribution is (4. 1. 28) <At,AC(>T • • • <AUAck>T<BsU)BSVL>T • • <B5nB^>T = Now consider another product term in which we select the operators as above, except that from the k-th bracket we choose and 3 S in the upper and lower rows respectively and in the |k + p)-th bracket we choose A c and A c in the upper and lower rows respectively. Now we contract Ac£ with (\ct for I = 1, . . . , (k-1), (k+p), and T i s r with 3 s r for r = k, . . . , (k+p-1), (kl+p+1), . . . , n. The contractions are represented by (4. 1.29) (Ac-*!),-• • (Ar»IVA' 8* V •' (V *t B t \ f f > ( A t - B $ \ ^ + i • • (VBD* > and the contribution is - 61 -• • • • < » . . B L > T - i«Uc>A<h*i>S'u• However, since after contraction, each operator commutes with all other operators except the one with which it is contracted, we may interchange (ATt - \ \ and (ATt-B^ , and ( A M \ and ( A ^ s \ t p ' Thus (4.1.29) is the same as (4. 1.2 7). The point is that in doing contractions within a product term characterized by the number k, which is the number of A*'5 (and Ajs ) in that product term, we need consider only one possible product of contractions for example (4. 1.27), because all possible contractions make the same con-tribution. We must, of course, multiply by a factor which will enable us to get the correct result. In the case considered here this conclusion is some-what trivial because the A and B operators commute. Later on, we shall consider cases in which the A and B operators do not commute and in which different kinds of contractions are possible. However, we can reduce those cases to this case without having to go through tedious algebra and so apply the results obtained here. Returning to the problem started above, the number of ways in which we may select the k %i and k At's is (£) in each case. That isjf^j is the number of product terms which contain k A^ c'$and k Ac's> . The number of ways of contracting k £'s and k Ac's , (n-k) Xs and (n-k) ' S is k! and (n-k)! respectively. In order to get the total contribution to < OL* AT/V > - 62 -we must sum over all possible values of k from O to n. Using the relation, k U n - k ) ! ( k ) -4n "\ we see that the contribution to<{Ql a^from CC is (4. 1.31, t.! E (^) [<AtA t > T ] k [^ 4 Bl> T ] n ' k = „! [<4At>T + <64Bt>T]n This result could have been obtained if we had assumed that we need not have gone to the trouble of contracting operators, but contracted, in-stead, brackets. A contraction of two brackets is defined by the equation (4. 1. 32) (At " B5) (AC- Bj) (A[-BS) (AC-BI) (AC-BJ) which gives (4.1.33) <A\Ac>r f < B 5 B ; > T The contribution from a product of contracted brackets is defined to be the product of the contributions from each contracted bracket. The contribution to<a"a")Twill be the sum of products formed from all possible contractions. For example, for n=2, (At-(4. 1. 34) 6 A ( & ' W t - h \ Mc-8 s +),(A t-B t\ CVB;),(AT-B+)JU - 63 -Alternatively, since each of these two products of contractions give the same contribution, we may consider only the contribution from any one product of contractions and multiply it by the number of possible products of contractions. The contribution from n CC brackets. (4. 1. 35) is given by 1<*[K\ + <B.Bt>J1 There are n! ways of forming products of CC and so the total contribution to <"Q- Q T >r from CC would be (4. 1. 36) t i ! [<AtA t> T + <B 5 Bt> T ] n A simple relation emerges from these calculations. According to (4. 1.23), t (o-, a , ) T is given by <4«i> * [<AjAt>T + <B,ft!>T] Using this in conjunction with equation (4. 1. 36), it is seen that, in the JLNA, - 64 -(4.1.37) < a t X > T = » ! [ < 4 « i > T r • If we refer back to the normal form expansion of ( o-Lav) in Chapter 3 we see that by combining it with (4. 1. 37) we have < ( 4 o ' > T - L ? » ! [ < « k > T ] " . T n = l T That is,(( a ; Q 0 / T has been expanded in powers of ( a v c O T • If we define P = <AU t>T , Q = <BsBfs>T , R = P*<2 = <a[aOr , then equation (4. 1. 37) can be written as (4.1.38) <afa?> T - • Equation (4. 1. 38) provides us with the last missing piece of information needed to calculate f>„ H") for case (i) and if we were interested only in this probability, we would stop here. However, we still have to calculate (al o.j a ; ^ r for case (i) in order to find the joint probabilities. Before continuing with this topic we shall introduce a graphical represen-tation for the contraction of brackets which will be used whenever convenient. Brackets involved in contractions will be denoted by a circle, © , a cross * , or some other suitable symbol which may or may not have attached a number signifying the position of the bracket in the expression. The contraction of two brackets will be denoted by a line joining the symbols representing - 65 -the two brackets. It should be emphasised that the diagrams resulting from this representation add nothing to the information that can be obtained by writing out the brackets explicitly. It is merely a device used in order to eliminate unnecessary labour and point up the fact that we need only consider contraction of brackets. As an example, we illustrate the diagrams for < a i C L - / t (compare equation (4. 1. 34) ). (4. 1. 39) We shall now consider another kind of contraction referred to as a "mixed cross contraction" or MCC which will presently be defined. We are still assuming that (A c k 1^=0, Instead of (4. 1.20), let us consider (4.1.40) A t n A ? V 0 j n - U f i - ^ V • ( A c V a ^ j - B ^ ) , - M f j - B 4 j ) m Now, if we contracted only Ac/s with A f/s, Acj s with S , "Bs,s with~B s,; s and £ > S j s withB5-'$f we could then use the results obtained above and write down the contribution: (4. 1. 41) * ! » ! [<AJAI> t • <B 5 iB, t i>T] ,'[<Al JA c. J>T + ^ . B ^ J " . However, we can also contract Ac/s with ACj s f AqS w i t h A c /s , "£>s; s with T f / t , •Ds, S and 5 with "Bs< s , Such contractions will5.be called MG<C. - 66 -Let us consider a product term in which k A c i ;s are MCC with k A£j'sbut we do not yet say in what way the other operators in the product term are contracted. Using the same notation for contraction of operators as before, we can write the information we are given as (4. 1. 42) (Ali - B,0, • • • < A l 4 - * * i U ( 4 - * i W • • (Atf l*tj-*l\ (Atj-B6 J)M Now, operators from the other (n-k) brackets,(^"B^J t must be contracted with operators from some (n-k) brackets, (Aci-BsJ, because, by definition, they are not contracted with operators from brackets like. (Acj-B^j) . Similar operators from the remaining (m-k) brackets, (Atj-U^) , must be contracted with operators from some (m-k) brackets, (A* -BSj). This leaves k brackets, (ATCj-BSj) and k brackets, (AC(.-B]t) , from each of which we must choose an operator for contraction. Evidently, these operators must partake in M C C . Thus, once we have chosen k brackets, from which to select operators to MCC with operators selected from k brackets, (Acj -B j^) we are forced to choose exactly k brackets, (Aci f from which to select operators to MCC with operators from k brackets / (ACj-\^) . Since all operators, except contracted ones commute, we can rewrite (4. 1. 4Z) as (4. 1. 43) (AIL-B^-^U-( A c i - A V - C V ' i U I I I I - 67 -This is divided into four parts and we may treat each part as (4. 1. 20) was treated because we are considering only contractions within each part. This means that we may forget about contracting operators and contract brackets within each part. Thus, we have shown that we may contract brackets rather than operators both in CC and in M C C . In order to write concisely the contributions of both CC and MCC to <av Qj a;a.j/T we first define (4.1.44) ?*<A^\'<*\tyry&*<B«*br -fc^A , U=<4iA tJ>T , and (4. 1. 45) P + $ - R U + V = W For the purpose of drawing diagrams, if we denote (Act' T^ ;) and - \ ) by ° and (Acj-'Bisj) and (ACj- by & , we can easily calculate the contribution to (4. 1. 40) from CC and M C C . The two simplest products of contractions are given below. The first is (4. 1. 46) and gives K K . There are n!m! such contributions (see (3. 1.41) )< The second is (4. 1. 47) i v« x, X, . . x n-i \ * / l * * - 68 -and gives K W . Each ° can be selected in (i) ways and each * in \ 1 ) ways. The (n-1) ° and the (m-1) * can be CC in (n-1)! and (m-1)! ways respectively. There are thus [(TXTjV-O'^-O1• = ">'™!(7)(T) such terms. The total contribution is easily shown to be I £ = n if -n <• m , (4.1.48) n!rn! L ( k X k ) R W 1 . Each "W" in equation (4. 1. 48) comes from a MCC between a circle, 0 , and a cross, x . Evidently then, the maximum power of W is 2n if n -^ m and 2m if m ^ n. This determines the upper limit of the sum over k. It should be noticed, using the definitions given in equation (4. 1.17) t of A C v and B s- that (4.1.49) R = <*\ai\ s <a]ai>r • (4.1.50) W - <at aj>T - <Ajtfi>T -Thus, (4. 1.48) is an expansion of QJ Q;Cij ) t in powers of (aiQ.v)T and (a* Q-j>T . That is, < We shall end section 1 by evaluating V \ ty a i . ^T . This expression appears in the normal form expansion of <((O-ia0(^^) /T as given in equation (3. 1.3) of Chapter 3, Section 1. We now give a few more definitions. It will be recalled that, for case (i), we defined 69 iV» and We now define (4.1.51) A 6 X = *->Ply , *l\ ~ cxPx , (4.1.52) B r t J , = *U*iX'J > A 5 j V = A i % e n V j and (4.1. 53) (Btjk - A *jk) = ~ A*}\ • Using these definitions we can write a,; b- a ; bj in the following way: (4.1.54) <S:ir]ma^rJ = ( A ^ B ^ • • • 8 , ^ ( 6 ^ • • • ( B ^ - A ^ ( A c i - B t i V - - ( A c i - B t i ) w ( B e j - A + , j ) | ••(B C j .-At j) t l - 70 -Now we can form contractions like (A^ Asv>r, C^BcjX-, A S j ) r a n d <(l»sv. ^cj)T - Such contractions will occur only within a row. Con-tractions occuring within a row between operators attached to different sites i and j will be called "mixed incontr act ions" and referred to as MIC. Now consider a product term of o^ . a t bj i n which k brackets ( A^ v -Bsi) contribute operators for MIC with operators from k brackets - Asj) . This leaves (n-k) brackets (A f C l - B s , ) t o Q Q W I T H some (n-k) brackets ( A c i - B ^ ) and (m-k) brackets ( B ^ - A s j ) to CC with some (m-k) brackets ( B c j - A*sj ) . No M C C are possible here. Thus, we are finally left with k brackets, - ^Bs-c) from which to select operators to MIC with operators selected from the remaining k brackets, ("&Cj - -Asj) . We can thus write (4. 1. 55) «Vir]\lir~ = (Af i -B4. . . (ATrBj k ^-^V--(Bl j -A, j ) k | ( ^ - B ^ ( A t r B ^ f B ^ A ^ ^ B J - A ^ 1 I 1 i I I ( A c i - 4 ) , - (V*^- A t , ) , - ( B ^ J ^ - B ^ V B ^ foj-tfcbf&iL The four regions defined by the dotted lines can be considered separately - 7 1 -because contractions occur only within each region. Consider the region in the upper left hand corner of the expression. It can be written as, (4.1.56) U ^ - ^ i ) , • • (Ati - B s i ) k t " ) k (A s j - fttj), • • • C A 4 j - B t j ) k This arrangement is similar to that obtained when considering CC , so that, using the results obtained for C C , we can immediately write down the contribution of (4. 1. 56): (4.1.57) l - i ) k k![<A^ s j > T + <6J iBtJ>T] In the same way we find that the contribution from the region in the lower left hand corner is (4.1.58) C- ' /k! [<AciAtj>T + <Bt iB t J>T] The contributions given in expressions (4. 1. 57) and (4. 1. 58) are identical, even though at first sight they might appear different when E ^ E ^ . Al l CC in the middle and right hand regions of the expression for a t bj a ; bj contribute (A. 1. 59) C"-k)! (*,-k)! [<A\.XA£i>T + < ^ \ i \ T [<8?j VT + <A*j^ J>T1 Now, if E ^ E ^ (4.1.60) [<AtiA t l>T + < & X > T ] * 0 ! j B £ J > T + <A 5 jAt j>T] - 72 -This may be seen easily by writing out these expressions explicitly. Let us define (4.1.61) [<*fciA*jX + <a*Xj>T] s [<Au<j>T+ <BtiBtJ>T] - L (4.1.62) [ < A t i O T + <B i iBj l> T] = R £ (4.1.63) [<*Jj*ej>T + <A s jA s+j>T] - * - ? where, if E x = Ex R = R_ E E = R (as defined previously) The number of ways in which we can select k brackets from a set of n brackets or m brackets is (k) and respectively. We must then sum over k from O to m if m<,n, or from O to n< if m> n. Using the relation k! (n-k)! (k) = we get finally, (4.1.64) <«V^V*">T - 73 -This expression will be used in section 1 of Chapter 5 to evaluate [>„.- (T) where i is a site in sublattice-A and j is a site in sublattice-B. The bar over the n, is put there to denote this fact. J We might note that (4.1.65) K E * <atfli>T ) (4.1.66) R f = <^ JVT J (4.1.67) L = < f l Hj> T = <*-|at>T , so that, as in the case found earlier in this section, (a- bj a ; bj is given as an expansion in powers of <[a[Q.^ >r , O*} kj7 r and (o.^ t>j>T • Section 2. The Thermal Average of Case (ii) In this section we no longer assume that (A c k "Bsf)r-0 though, of course, {Atlc"Bst>r=0 and 0 . This corresponds to case (ii) which deals with spiral spin arrangements. We shall first evaluate (a* a^)T by means of a procedure similar to that used in the last section. The possibility of non-vanishing contractions between operators A and B now arises, but since <^ Ack.^ >s(V>T - (A,.k~Bsi>T = 0 , this kind of contraction will occur only within a row. Contractions occuring within a row between operators attached to the same site i will be called "incontractions" and will be abbreviated as IC. Evidently, the only kind of contractions occuring in this case are CC and IC. Consider a product term of a" a" in which a number of A c ' s are IC with some ^ s'& , each lying to the right of the. A c with which it - 74 -is IC, and some T3 s's are IC with some A t ' s , each A c lying to the right of the B s with which it is IC. We do not say yet how the remaining operators are contracted. Each IC operator involves one bracket and thus each IC pair of operators involves two brackets. The total number of brackets involved in IC is thus a multiple of 2 and we can take this num-ber to be 2k where k is an integer. We may represent our information given thus far by writing out <x in the form previously used. (4.2. i.) (a[- a.), • • (<-&*)Xk i <*t- B * U + , • • • M + t - B 4 ) n I (AC-B1), (At— Bs)„ j where the brackets involved in IC lie to the left of the dotted line in the upper row. We have not drawn in the IC lines because we have not yet selected from the brackets any operators for contractions. We can say, however, that since by definition, the operators which can be selected from the remaining (n-2k) brackets are not involved in IC, they must be involved in CC with operators selected from (n-2k) brackets in the lower row. We can thus extend the dotted line, (4.2.2) (Al-B4), • • (Aj-B,)^  j (At-B5)LK+|- • (A£-B 5 ) n I I and say that all operators from the brackets lying to the right of the dotted line are involved in C C . This immediately implies that the 2k operators selected from the 2k brackets lying to the left of the dotted - 75 -line in the lower row must be involved in IC. Thus, an assumption of a certain number of IC in the upper row implies the existence of the same number of IC in the lower row. The contractions on either side of the dotted line may now be treated separately because there are no con-tractions "across" the line. The contribution from the right side has already been evaluated in section 1 when CC were investigated. Its con-tribution is, as before, (4.2.3) (»-*k)![<AtA t> T -r <B 6 Bt>J n " a k We now turn our attention to the IC. We may treat the remainder of each row separately because there are no contractions between the rows. Incase (ii), (fict Bsr)r=<Bse and {^c(\v\ = (\i . Thus, each row will make the same contribution from IC. We may then consider only the first row. In order to form a product term we must select an operator from each bracket and contract it with an operator from another bracket. The only contractions that will contribute are (&ct \-r}y and A c r ) T because (.^d Acr)= {&s»^>sr)T - 0 . Let us then select an operator from the first bracket. It does not matter whether it is A c or >^s There are (2k-1) ways of selecting such an operator. Let us select a second operator from a second bracket to contract with the first operator. Having made our selection we write down the two brackets from which we obtained the operators as (4.2.4) (At - B $), l-i)(B s - At), - 7 6 -where we have written the second bracket in the lower row and replaced its original subscript by the subscript of the bracket from which the first operator was selected. We now select a third operator from a third bracket and choose a fourth operator, from a fourth bracket, to contract with it. There are (2k-3) ways of selecting the fourth operator. We write down the two new brackets beside the first two brackets, change the subscript of the lower bracket as before, and obtain (4.2.5) ( A t - B ^ C A t - f t ^ C->) l (B*-At) 1 (B.-At!) 1 • This procedure is repeated until we have contracted an operator from each bracket and obtained (4.2.6) ( A t - & * ) , ( A t - & » ) x • - ( A t - B s ) k C - O k ( B , - A { \ ( B , - A t t ) J U - • • ( B 6 - A t ) k . The contribution from the contractions carried out must be multiplied by (2k-l)(2k-3). . . 3. 1 because this is the number of ways in which the contractions could be carried out. It should be noted that an operator from (Ac-S>s) can be contracted only with an operator from C&s - A c ) k because all other contractions, for example, an operator from (Ac -^>s\ with an operator from ^ , have been taken into account by the factor (2k-l)(2k-3). . . 3. 1. However, suppose that we did allow con-tractions between ( A t - B A and ("Bg-At).,, for V I . Then,since there - 77 -are k! such possible contractions, we would have to divide the result by k!. . But now, if we do allow all possible contractions between operators from (A c and operators from [\ - A c ) ^ , we are back to the problem of CC as discussed in the beginning of section 1. Using the results of that section, we know that the contribution from such contractions is (4.2.7) Ic! [<AlB 4>T + <B s A f t > T ] k . However, we must multiply by (-1) (2k-1)(2k-3). . . 3. 1. 1/k! . Using the result that (Ik)! ( tk-0( ik-3) . - 3-1 = we find the contribution from IC in a single row to be (4.2.8) M>h(*k)! [<4B,>T t < M t > T ] K ' s> k! The contribution from the two rows of IC and the C C , for fixed k, is then (4. 2. 9) C - i f V i k ) ! C<AtAt>T * <B,Bl> T r l l t [<AtB 5 > r + <t>A>Tf Now, there are (2k) possible ways of selecting 2k brackets from n brackets in either row. The total contribution is obtained by summing over k, either from O to n/2 if n is even, or from O to (n-l)/2 if n is odd. - 78 -We use the relationship that and obtain I I = | ever.), where K « <ACBI>T = <B SAJ> t , |_ = <AftBs>T = <BtAt>T , P = <A + TA T> T , * = <BSB£>T. If we further define (K+L) = G and (P + Q) = R then (4.2.11) < 4 * « ? > T = »• £ 0 & {tk)l*k) <*tk R""*k For those who did not like the derivation of (4. 2. 11) by reducing the IC problem to a CC problem, we have derived the same expression in a straight-forward fashion in Appendix 6. The above method was preferred because it brings out the essential unity of the four thermal averages treated so far. What might have been considered as an interesting coincidence obtained by a straight-forward derivation, is shown to be basically the same situation as previously encountered. That is, it is necessary to consider only the contraction of brackets instead of the contraction of operators. Equation (4. 2. 11) could be used as it stands to calculate fK^r) for case (ii). However, we shall write it in a different way so that f>^r) - 79 -will be calculated more easily in Chapter 5 and will appear inaa more convenient form. Equation (4. Z. 11) will, however, be used in Appendix 9 to derive a different expression for pn'T) . One may use whichever expression is more convenient. In order to alter the form of (4.2. 11), we rewrite ^ k ( ^  ) a s : and then use the following identities: (4.2.12) J- / to* 8de = — tf m is even o (4.2.13) J - / cos ddB = 0 if m ts odd. Using equation (4. 2. 12) we can write (4. 2.11) as (4.2.14, <#.;)- - * | J ! ( i , ) / ^ * k . « O - ' R — ' However, because of equation (4.2. 13) we can write (4.2.15) < . t " « " > r - ^ 2 ( S ) / W . ' » * » 6 H R - H A AM - 80 -Itt (4.2.16) <Qti',ar>T = "2i f (R.+ G,LO*9T<19 . We may thus write <<*-i °-i ) r as (4.2.17) <A T iX> T = £f(R+(*U*B)nd9. Equation (4. 2.17) will be used in the derivation of p* ("O in Chapter 5. Before moving on to the last part we might note that (4. 2. 18) R= <a fiai>T G = - < 4 * j > T = - < « C « J > T , so that (a?" a")T is given by (4.2.19) < « f X > T = (<«t«i> T - < 4 « 5 > T e p » 0 " r f B -We now move on to the last possibility to be considered in this section: that of evaluating (a; a ; ctj aj ) T for i^j, and (A t 3 j S ) T ^0 We now have all four possible contractions, C C , M C C , MIC, and IC, making non-zero contributions to the thermal average. As we might expect, the situation will turn out to be a combination of the four previously encountered situations. Before giving an expression for <at Q.^  ) r first we define some svmbols: - 81 -(4.2.20) < A ^ £ I > T + < B 4 I B ^ > T = < A C J A T J > T + <B^I J> T = R , <Aj iA t j>T + < B 5 I B ] j > t = < ^ A t l > T +<B^B j 1 > T = w , < 4 i ^ \ + <B 5 iA + £ i>T - <Bj.A c i> T + <A t i B $ t i > r <A!iB5j>T + <6 5 iAt j>T - <Bt.Ae j>T + <A t lBjJ>T - Y . In this chapter we already obtained expressions for contributions from the four possible kinds of contractions. No new kinds of contrac-tions appear in the calculation of (a^CLj*' CL? Q.j Xp s o that no new procedure is required in order to calculate an expression for the thermal averagei. Let us select 2k brackets, (A^ -"B>6i) , 2k1 brackets 2q brackets (Aci — B9<,) and 2qh brackets (Acj -B S j ) to take part in IC. These can be selected in (^ k)' (Tk') , [x<^ ) and (s-q.') ways respectively. The results of section 1 tell us that the contribution from IC is (4.2.21) £ L u k ) ( n X n ' A n : ; k i ^ k ' i ^ ! ( ' z / From the remaining brackets we select t brackets, and I brackets, (Aci "^sJ , I' brackets, (A^-T^j ) , and l' brackets (A cj -"Bsj) to take part in C C . These can be 82 /Ki-0.k\ (n-2q_\ /m-Z»<'» /<nn-Xq'\ selected in \ £ J ,{ l J, [ f J and ( ^ ) ways respectively. These contractions make a contribution of (4.2.22) ' £ E (" i* k ) ( - , t *- ) (" i? l 'X"l? < ) tl . 1 if From the remaining brackets we select u brackets, Met -^s<^ i t . and u brackets, ( ^ cj - "B&j ) f u ' brackets, ( A tj - Ssj J and u' brackets ( Aci - ) to take part in MCC. These can be selected ways respectively. According to the first part of this section, the contribution from these contractions is (4. 2. 23) We now have (n-2k- I -u) brackets, {^c-i -^,s^) which must be MIC with the remaining (m-2k'- t'-\x) brackets (AC J — 3sj ) and (n-2q-.f-u') brackets, which must be MIC with (m-2q1 - t-\x) brackets . This means that: ( n - l k - i - u ) = ( m - a t f - i ' - a ' ) (4.2.25) From these equations we can solve for u' and V, obtaining, u' - u + k - k' + £ (4. 2.26) V = m-n 4-k + i + ^ - k'-<£ . - 83 -The contribution from MIC, then, using ( £ . 2 . 2 6 ) is given by (4.2.27) C - l / ^ - k * 1 ( k / - l ' - k * l ) ! Y * - * - ^ . The total contribution to < Q ; Q.j Q . "j / T is given by the product of (4.2.21), (4.2.22), (4.2.23) and (4.2.27): (4.2. 28) < « M % > ? > T - S Z £ E Z t* &)U\)(&)&) • J J T k=o a = o k'=o o'=o l=o u=o ^ 1~ (aky.'(zi)!(alt)!fefl/)! / *-*k\/n-*o\/ "«-*k — \ I )\ \ Am-n + k + l + i-k'-o' M <tJ k ! ^ ! U + k - a - k ' + ^ V l u • k - i - V . ' + l ' J l u / (u + k-^-k' x X ) \ ( k ' - ^ - k n )! ( | ^ * . m-« *k + t» • V - k ' - l ' w i u + k - J - t f • 1' y k ' - l ' - k * 1 wbe-re. I = a even) > ? s ^ (n odd) , - 84 -Equation (4. 2. 28) can be made simpler by using the integral given in (4. 2. 12). Using the fact that (ak-i)(ak-3) • • • 3. I i we can write a k k! a H ? / too* 9 de. It is then possible to write (4. 2. 28) in the form (4.2.29) <ftt- f l t* d i r a «> T ZK rVt r2J! o Jo Jo *o £ L Z t E S <»!>Vtf k=0 <^=0 ^-0 8=0 Uro k!«jjk!<^l (k -k-^'+ <p! R. l [(.rt-zk-J-uV.r [(n-u_k-JU<j_+k'-^)1]1 u! ( 6 c o » ^ ) k C Otour)^ de dd> <*<r - 85 -It is difficult to reduce equation (4. 2. 29) to a simpler form due to the profusion of factorials appearing on the RHS. Because we have been unable to obtain a more satisfactory expression for \ ^ Q-j a i aj /-j-the problem of giving an explicit solution for the joint probabilities for spiral spin arrangements has not been further considered. - 86 -C H A P T E R 5  Explicit Solutions for the Spin Deviation Probabilities. Summary. In this chapter we combine the results of Chapters 3 and 4 to obtain explicit expressions for the various probabilities. We do not, however, calculate the joint probability of case (ii) (spiral spin arrangements) because of the complexity of the expression for ( Q j a, „ Section 1 deals with case (i) and is divided into three parts. In the first part pntO~) is obtained and some remarks are made concerning the temperature dependence of this function. In the second part we obtain an expression for the joint probabilities if both i and j are sites on sublattice-A, while in the third part we obtain a similar expression for the caset where i is on sublattice-A and j on sublattice-B. In the first section we comment upon the significance of numerical values obtained for the joint probabilities. Section 2 is concerned with case (ii) and an expression is obtained for bn(T). In Section 3 the results of Section 2 are applied to a physical model. The model is an approximation to dysprosium metal. Numerical results for (0 ) are obtained and discussed. Section 4 contains some concluding remarks. - 87 -Section 1. Analytical Solutions for Case (i) and Some Numerical  ResultSo Equation (3. 2. 14) will now be solved for the probabilities (T) for case (i), the case of the simple antiferromagnet. For convenience we shall renumber the pertinent equations of Chapters 3 and 4 so that it will not be necessary to refer back to those chapters. It will be recalled that ^ w [T) was given by In Chapter 4; Section 1 we evaluated c^t^  a v We found that (5. 1.2) where (5. 1.3) R = P + £ and (5.1.4) If we put N'= N/2 , then, according to equation (4. 1. 17), ) where - 88 -We see then, that Evaluating the thermal averages we get (5 .1 .5 ) P - * £ C I 5* (5 .1 .6 ) $ = i £ — Using now (5. 1. 1) and (5. 1. 2) we can write M (5.1.7) b w (T) * H ) w £ H ^ l w ) ^ Letting M approach infinity we obtain (5.1.8) |>W(T) M - ^ ° ° » - 5 1 , if R < I (1+ R)"*' By substituting (5. 1. 8) into the set of equations (1. 1. 21) which are satisfied by the \>„(T)'$ we find that in order that - 89 -, w P w C O = w (i+R)"* 1 be a solution it is necessary only that R (5. 1.9) + R < I This requirement., places no condition on R. The condition R 1 is necessary only if we want a series expansion for pw(~f) in powers of R. The final result, then, is (5.1.10) P w ( T ) = £ ~ - ) W t l • We might mention at this point that for ferromagnetism we would obtain as we did for case (i) (see equation (4. 1. 37)). This means that equation (5. 1. 10) is valid for ferromagnetism. Now, for a ferromagnet at T=0, . ^ = 0.^=0 which implies that PA<L°) s ° <• n- > I ) , Pp.(o) = I • The explicit temperature dependence of R will depend upon the form of fc^and .5^ and thus cannot be discussed without making some assumptions about the Hamiltonian. As an example, thenrj* we shall consider the Mamiltonian introduced in Chapter 2 to describe the collinear antiferromagnet. We shall equate the external magnetic field to zero because retention of it does not change any of the results and only complicates the calculations. We have seen that in this case E ^ = E ^ , where (5.1.11) - ^ V O + D ' ^ - T X 1 - 90 -and D' = D / J , D being the anisotropy constant in the Hamiltonian. Using the expression for P and Q given earlier in this section we see that ( 6 . 1 . « ) R - • * h * * K 7 i t , ' Referring back to Chapter 2, equations (2. 1. 19) and (2.1.21), + is found to be given by (I + P ' ) (5.1.13) C* + = -V(l + D') x-Y> At low temperatures, which is the region in which we are interested, only long-wavelength excitations play a role and we can put, for any crystal, (Kubo, 1952): (5. 1. 14) y£ i= / - X* , so that we can write (3.1.15) £ x s= ZJSx-y/p'(* + !>') + X1 (5.1.16) c\ • s x si V n ^ B o T ^ Now we shall fix our attention upon the temperature dependent part of R which will be denoted by R 1 . We shall allow N to approach infinity and can thus make the customary replacement, - 91 -£ — ' CfrJ/X • where V is the volume to the system. V is allowed to approach infinity together with N in such a way that N/V = p remains finite. Furthermore, as is usual, we shall replace the integral over the first Brillouin zone by one over the whole of A -space. At this point we must distinguish between two cases: (a) D' « O . (b) D^Z + DO > >* . (a) D' ~ Q From (5. 1.15) and (5. 1.16) we have, (5.1.17) - ZJS z. X J l (5.1.18) c£ + s j ^ ^ Combining (5. 1. 17), (5. 1. 18) and the expression for R', using dJ>= tfdil s ih©d9df we get . oo (5-1.19) R = ptnr Jo > where kT (5.1.20) 9 = ZJ5x Defining X/0 = X t R' becomes - 92 -(5.1.21) * ' = \ J l • (b) D' (2 + D') ^ Now, (5.1.22) £ x ^ ZfSzJfi'Ct + V) [ I + gD'fr+p')^ (5. 1.23) * £ J S i [ L , + UjX] and (H-J>0 r I -vxl (5. 1.24) c> + S>x * VFaTDO" L " »9 J (5. 1. 25) * k, + . Defining © as above, we now find that - 93 -xK, (5.1.26, R = ^ ^ ^ — ^ If we define / L%/9 ^ = LJ then R 1 becomes, (5.1.27) R' * \ V J V * We now define (5. 1.28) 1 E «i = r . N X For the simple cubic lattice this constant was found by Anderson (1952) to be equal to 0. 078. Here we use the same notation as that used by Van Kranendonk and Van Vleck (1958). Using this definition, we can write (5.1.2 9, R = P + ^ T n • n = z if D ' a O j n * */» if D'(x+3>');> Combining equations (5. 1.2 9, and (5. 1. 10) we see that, if T is assumed to be sufficiently small, - 94 -M O [» + r * X „ T " ] n " |w + ' f 5- 1- 3 0* M T > « 7 T T F ? 4 r + P ( W + I + P ) ^ T . J At T = O we then obtain It should be noted that since T = 0. 078, we have ft(o)= 0. 92 8 (j,lo)= 0. 067 The approximation made by Van Kranendonk and Van Vleck (1958) in solving for p,(°), pj 0) and p^O) is thus seen to have been very good. It may be recalled that R = ^ a^  a i )r Now we have (5.1.32) <S?>T = S - R . Furthermore, it is easily shown, in this case, that (5.1.33) <6i?r - - — - 95 -where M(T) is the "sublattice magnetisation" as calculated from the free energy, and yu and g are the Bohr magneton and electronic g-factor resp-ectively. Combining equations (5. 1.2 9), (5. 1. 32) and (5. 1. 33) we see that if D'=0, the temperature-dependent part of M(T) has the customary T -dependence (Kubo, 1952 ; Oguchi, I960). If D' ^ O and is sufficiently large, then M(T) varies as T 3 / i , a dependence usually associated with ferromagnetis The expression for (a;"1 ay"1 ax Oj)T derived in Chapter 4 will now be used to evaluate f^ .w.(T)for case (i). As we did earlier in this section, we shall re-write the equations in which we are interested. We recall that p-(Wj. (T) was given in equation (3. 2.2 3) in terms of (cLia™ (X* Q^/V • Here both of the sites i and j are on sublattice-A. (5.1. 34) ^ t n - H ) - < ' " ' J J ^ f ^ ^ N j X a p , . The thermal average, (Ot^ Q-j Q, Oj ) r was evaluated in Chapter 4,Section 1 and found to be given by ^ - L i = m if n< m, (5.1.35) <4rtat%>7> - ^ZtyftW** k.. 3 = m if „ > m . where, as we may recall, (5.1.36) R - <A r tAi>T +<6 4 iBt>T , W = <At^cj>T +<*« ,** j> T • Using the definition given previously in equations (4. 1.17) and (4. 1. 4) that and we see that - 96 -(5.1.37) .ft - £ £ ( c * < * K > T + *x<Px*t>T) , and .-»_».. (5.1.38) W = * £ ( c f < 4 < > f 5*<8xp^x> \ e , V ( J " ° . We now use equation (5. 1. 34) together with (5. 1. 35) to obtain ( 5 . K 3 , , V . ( T ) . ^ U ^ ; ) " ! „ ! S ( K " ) ( " ) « ^ " ' " - L K . In equation (5. 1. 39) we may replace the upper limit 1 in the sum over k by M because (k) = 0 if k>n, while Ck)«0 if k> m. Interchanging summation signs we can write (5. 1. 39) as (s. i.40 a) = (-.f""'z n - v i<-O*(;.)(J)R"S Hr( :xr )R" Consider now the sum (5.1.41, £ H ) n ( ; ) ( k j R " - r n y t f X k K It can be written as If we assume that we may take the differential operator out from under the summation sign, we get 97 M M-W We now let (M - w) approach infinity and, using the result that, (Schwatt, p. 57) ,5.1.44, H < - . ) • ( - « ) „ . - i ^ j : , * We find that (5.1.45) 2<-.)"(;K)R" - , It seems likely that one can show in this case, as in the case of p^ f7") that the requirement R ^  1 is necessary only if we want p (T) written as a power series, and that the closed form of it can be "continued " to the region in which 1. Using (5. 1.40) and (5., 1.45) by letting M approach infinity in a suitable way, we arrive at the result, Further simplification has not been possible, so that this is the final result. The procedure would be then to select out the desired bW(fc/j (T) and calculate it by differentiating under the summation sign the required number of times. There should be no difficulty in evaluating the re-sulting sums. These sums can all be reduced to the form: (5.1.47) k7o - 98 -because, according to the definitions of R and W,we see that we always have |W| |R | . Equation (5. 1. 46) is also valid for a ferromagnet, where Now, at T = O both R and W are zero so that, for a ferromagnet, ( not V o t h Wj, W = 0 ) . Consider now the quantity (5.1.48) C T ( j - i ) = I ^ - I T ) - P B . ( T ) | ) 0 . ( T ; | . From what was said above and earlier in this section about the ferro-magnet we see that C 0 C j - i ) = o . This occurs because, at T = O, the completely ordered state is the ground state of the Hamiltonian. That is, spin wave theory applied to the ferro-magnet at T = O is exact. This is an illustration of the fact that if a Hamiltonian for a magnetically ordered system is written in terms of the - 99 -true excitations of the system, then at T = O, a suitably defined function like c r ( j - i ) should be equal to zero. We shall now evaluate Jp0ioj for a simple cubic antifer-romagnet in order to calculate c o ( j _ 0 and make some comments upon the result. Using equation (5. 1. 46) for w. = w^  = O, we have (5.1.49, ^ . ( T ) - £ « * - j i - Z [ ( ^ J . Using equation (5. 1. 47) we see that I (5-1.50) b 0 . 0 . (T) = O + R ^ w 1 ' Now, from equation (5. 1. 10) we see that (5. 1. 51) bp.(T) = 1+ R Using (5. 1. 50) and (5. 1. 51) then, (5.1.52) ^ ( T ) t Y T ) = ' [ ( , + # - W * J = That is (5.1.53) fJ^oCT) > pPiCr)bp.(T) J Now, the physical significance of fj (T) is the following: If at temperature T , we fix our attention on site i and ignore all other sites in the crystal, the probability of finding ni spin deviations at this site is given by That is, while observing site i we are in complete ignorance as to the number of spin deviations at any other site in the crystal. We might then say that the probability of finding - 100 -no spin deviations at both the i-th and the j-th site will be approximately equal to the product of Po;, a nd P o j C " ) -The result given in equation (5. 1. 53) shows that po^  "^r^ P°j is in fact a lower bound to the joint probability of finding no spin deviation at both the i-th and j-th site. Recalling that j 9 " * ^ = O. 92, w e n a v e We shall consider a simple cubic lattice made up of two interpenet-rating face centered cubic sublattices. Spins attached to sites on a given sublattice all have the same equilibrium direction. The equilibrium direction of spins in one sublattice is opposite to that of spins in the other sublattice. The results of Chapter 2, Section 1 where the antiferro-magnetic Hamiltonian was diagonalised can thus be used here. As in Chapter 2, Section 2, the basis of the simple cubic lattice is chosen to be three mutually perpendicular unit vectors e\ • °x "'5 • The basis of the face centered cubic sublattice from which we shall select sites t and j is given by: Since both \ and j refer to sites on the same sublattice, we shall denote the vector ( j ~ v ) by (5.1.54) C j _ ^ ) = d ^ + h-Yj. + C% } where a, b and c are integers. The general vector in "X-space is - 1 0 1 -where A A A Each site of a sublattice has, as nearest neighbours, six sites of the other sublattice. From equation (5. 1. 46) we see that W can be written as (5.1.55) w = g £ 6 J + g £ ( c f < 4 « > > T * ^<ptPx> T) e 1 * ^ . Now we define (5. 1. 56) A * «. L s A e J If we use (5. 1. 1 3) together with the relation 0^ -5^ =1 and put D=0, we find As we let N approach infinity we can write (5. 1. 56) as - 102 -In this case 3 b Using the expression for ^ we get It is more convenient to change to a different set of integration variables ^•i ' *^x and "X defining them by (5. 1. 57) zx' = ->,+ v,.*- x 3 , * j£ = x r x,.+ \ 3 j = + V - X i . We can then write (5.1.58) -y^ = ^l*0**', * tot*x + CosXjJ . Using (5. 1. 57) we find that In terms of the new variables,'> iand Aj,we have - 103 -(5.1.59) Cos (X- C j - t ) ) = ep&f^r + ^ A' + O 4 - (a +fc) X\] The new integral is now (5.1.60) . a - J L . / f € i M j L t > d > ; , (aw)' J-3T where, for s and e we must use (5. 1. 58) and (5. 1. 59) respectively. Using the formula 402.02 given by Dwight (1957) we can write (5. 1. 59) as (5. 1. 61) e©*(tr+t)}'- to&fA+c)^- tosfa+tOVj In order to evaluate (5. 1. 60) we shall expand the expression for in powers of and from now on drop the primes attached to the Xs We get 5* *lZ i ( \ k W x , + t n \ t + u»\tfk . (5. 1. 62) X *»««' fe 7 - 104 -We then combine (5. 1. 60), (5. 1. 61) and (5. 1. 62) . Before we can integrate we must eh©9§e (j - i j . Referring to (5. 1. 54) we shall perform the integration in (5. 1. 60) for two cases: a = 1, b = c = 0 and a = 2, b = c = 0. The integration is a tedious task, especially when k starts to get large, so that we shall use a procedure that renders unnecessary the calculation of the integral for large k. First we shall define and We then calculate these two functions for n = 1, 2 and 3, and plot _ along the ordinate against P We know that A = A> ^ > =1 and that • both A and P^ ^ approach their limiting values assymptotically. This means that, for very large values of n, 1 — =. i _ A which, in turn, implies that the slope of the curve becomes zero as the two quantities approach their limiting value. Finally we make use of the fact that - 1 0 5 -r < 0° ; = 0-078 and extrapolate the curve to this value of p^"1 ^ From the point at which the graph cuts the ordinate, we can easily calculate A. The fact that it is necessary only to calculate i and .A for n = 1, 2, 3 makes this approach very useful. The actual procedure employed was to extrapolate the curve after calculating n = 2 and then predict the value of / — as a check on the accuracy of the extrapolation. In both cases the agreement between the predicted and actual value was good. Two values of A are given in Table 1. a A = W (T = oj 1 0 . 0 3 S Z 0 . 0 1 4 -Table 1. Values of A = W ( T = O ) as a Function of - 106 -Having calculated A two values of \ j - v J the conditional probability. we can now evaluate fa^l0) for the considered above. Let us first define (5.1.63) M H ' T ) Referring to equations (5. 1. 50) and (5. 1. 51) we see that + R. Thus, i + r Using the values of P and A given above, we show the dependence of \>00 (j - i . ° ) and Col}-1), defined in equations (5.1.63) and (5. 1. 48) respectively, upon ( j — v ) in Table 2. - 107 -a 1 0. 929 0. 001 2 0. 928 0. 000 Table 2. Values of poo (<^ > 0 ) and c0 ( art ) as functions of a . Now, if spin waves were the true excitations in an antiferromagnet we would expect that C 0 far, ) would be zero for a = 1. For a = 2, J " L is large enough so that oscillatory effects of e u ^ gives rise to a low value of A . We see that, for a = 1, C0(o.r,) is very small. This result might be interpreted as further evidence that the ground state of the simple cubic antiferromagnet is approximately the same as the ordered state. It would be interesting to calculate the temperature dependence of f>00( i - 1 > T ) in order to compare the behaviour of a fer-romagnet with that of an antiferromagnet, in which only nearest neighbour coupling between spin operators in the Heisenberg spin Hamiltonian is considered. This would be the simplest case since we would choose the coupling constants, J , to be of equal magnitude, but of opposite sign in both cases. Because these calculations have not yet been carried out our comments on temperature-dependence are restricted to the case of - 108 -jpotOj t T ) £ o r a f e w v a i u e s of (~j - l) For a ferromagnet one can immediately see the; trend of j)OD (j 1- t , T ) as a function of T. We have seen before that |D (O)= pol(0) = | Thus, p0o ( f — i , 0) = | . This conditional probability is a true probability, and in particular, p 0 0 - I I . Thus, if jp^Cj-^  ,~0 does change with T , it must decrease with increasing T. Returning to (5. 1. 55) we see that as ( J* _ t* _) gets very large, the exponential oscillates very rapidly. This is interpreted in the usual way to mean that W will get smaller as ( j* - v. ) gets larger, and Ltro W = 0 We now consider the temperature-dependent part of W which we shall call W'. Thus W = A +• W We can see that W 1 is similar to R' except that, in the sum over A there is the exponential. In fact, if we write W.= W (j-i). then R « W (0) „ To calculate the temperature-dependence at low temperatures, we make the same approximation as in the case of R', in that we put 'Y = I - ^ . We also put H=0. We now assume that A • ( j - L ) is sufficiently small so that - 109 -where <f> is the angle between A and (f~^) • However, it can be seen that the first term determines the dominant temperature dependence. We can thus put: CP6 and we are back to the case of R' considered earlier in this section. We thus see that V/ = A + \ 7 n f n=2ifT>'*Oj ft - i/j. ii 3)'(t + D') > "X*; where is the same function given in (5. 1.29). Since, using equation (5. 1.29) we see that the temperature dependence of p 0 i 0 ) LV is like T 2 i f D ^ O and like if T>'(2+D'• To end this section we shall evaluate pw{wj ( 1 ) where, it will be recalled, the bar over wy means that site j is on sublattice-B whereas site i is on sublattice-A. It may also be recalled that we can write pw.^r ( T ) as 15. 1. 64) 110 -Now we shall use the expression given for (a.; b j &i Oj /T in equation (4. 1. 64) together with equation (5. 1. 64) in order to evaluate p __ ( T ) . Combining these two equations, we see that ( 5 . . . 6 5 , fci,.tT). ( - o ^ " ' i | / - o " * " ( s , ) ( ; J ) i u ) ( T ) C R r L ' 1 . Let us write out R £ , R^ and L explicitly. Using the definitions of equations (4. 1. 65) to (4. 1. 67) we see that * E = [<AtiAel>T + <BX> T ] (5.1.66) R E = J 2 ( ' x ( - K > T * *l<Mi\) (5.1.67) R£- = £ ( <plfc>T + *x<«x*>> T) L = [<AtiAtj>T + <6l l 8 £ j> TJ (5. 1. 68) L = j £ (<* X*]> T + < ^ > T ) € , > ^ - ° - I l l -(5.1.69) L = JL S c A « ^ > T + <(3^ t x>T)e l X' ( l" j ) and that while (5. 1. 68) and (5. 1. 69) are equal, (5. 1. 66) and (5.1. 67) are not equal, unless which occurs only if E ^  = . If E ^  = E ^  then (5. 1. 65) becomes identical in form to (5. 1. 39) whereupon we can immediately say that v7 * s g l v e n by equation (5. 1. 46) with W replaced by L . Let us then consider the case in which E ^  £ E^. As we did before, we can replace the upper limit £ in the sum over k in equation (5. 1. 65) by M because ( £ ] = 0 if k > n and ^ )=0 if k i m. If we interchange the summation over k with those over n and m, we obtain (5.1.70) M , M M p^IT) , c - ^ - i £ R ; k ^ - ^ ^ S H r ( . ^ ( ^ R e Z ( - . r ( ; ) ( T ) R ; This equation can be treated in a way identical to that which was used to simplify ^ w (T) earlier in this section. All we need do, apart from remembering to replace W by L , is to distinguish between R g and Rg- . To do this, we need only note that R e is associated with wc , R-g-with Wj and that instead of R we have K E Kg- , Using the result, then, obtained for pw.w. (T) given in equation (5. 1. 46) we see that (5.1.7D b W i _ C T) - {^(^rrA^uT^r') - 1 1 2 -As before, further simplification has not been possible but the method of evaluating any given f^w- ^ _ r -' is the same as that for evaluating kWl-Wj (~r) • After working out the differentials in equation (5. 1. 71), evaluation of P^ wy ( _ r) would entail only repeated use of the sum: (5.1.72) oo r L*. -.k (i + R^i+Rg) l V < [. t We shall now evaluate fVSj i*1 order to calculate and U sing equation (5. 1. 71) for w c = w^ . = 0 we have f> (T) = g p - k P."' R e R r - 113 -X 2 If we use the same expressions for E ^ , ^ and as were used when previously discussing W and R, it is easily seen that for j ~^ ' sufficiently small, the temperature dependence of R E , R -g and L is like T2" if D ' « O and like if D'(2 + D') >>l . Because of the similarity between (5. 1. 72) and (5. 1. 50) the temperature dependence of would also be either like T or like T It remains only to evaluate the temperature-independent part of L which will be referred to as A . That is, from (5. 1. 69) (5.1.7fr) A - £ £ C * 6 > e i where i is on sublattice-A and j is on sublattice-B. This means that ( j - i ) is now written as j The evaluation of A may be carried out in a way similar to that - 114 -used to evaluate A . We can put (5. 1.75) In this case C-^ S- is given by (5. 1. 76) * * *~ 2 4 > - Y We shall make use again of the transformation from to ^ y as integration variables. .As before, we shall evaluate (5. 1. 75) for two choices of ( j - 0 : a = b = c = 0 , that is ( j - ) = S{ and a = -1, b = c = 1 which means that ( j - i ) = 2> o, • Now we define ixtf ( !\ I . ^ ^ ( T T * I I + * * * * * ^ ^ ^ ^ (5. 1. 77) X where a = 1 or 3. Using the previous definition of T , T - A was plotted along the ordinate, against I and the curve extrapolated to ' = 0. 078. As before, the curve is perpendicular to the line ) = 0. 078. The values of JV obtained are given in Table 3, while in Table 4 we give values of c 0 ( j - i ) a n ^ J^o (T~• ^ 1 a A = L(T-O) 1 0. 116 3 0.010 Table 3. Values of A = L(TrO) as a function nf ( i - i l - « - 115 -a Poo < °) c0(o.<5,) 1 0. 939 0. 010 3 0. 928 0. 000 Table 4. Values of poS (aS,, 0) and C Q ( °- ) as Functions of a. In this case, if i and j are nearest neighbours, C 0 (<£, ) might be considered to give the best estimate as to what extent the ground state ( G) differs from the true ground state of the system. It must be emphasised, though, that this interpretation is open to question and is at present, being examined. Likewise, the temperature dependence of jp0^  (CL is being studied. Section 2. Analytical solutions for Case (ii) In this section we shall solve explicitly for the jf^(T) f ° r case (ii), the case of spiral spin arrangements. It will be recalled that the equation satisfied by pw. (T) was (5 .2 . . , |, (T) - C - 0 " ' £ fcf(4)<*.J>T • We now use the result of Chapter 4;Section 2 where an expression was obtained for < \ a v a ^ T • This expression was (5.2.2) < « f O T ' -- 116 -Before continuing, it is worthwhile to write down the expressions for R and G and investigate their relative magnitudes. Referring back to equation (4. 2. 18) we see that R = <AtxAci\ + <B 5 iBli>T t (5.2.3) R. = j L £ s f + jjS(ct*5j)<^t*>> T , where we have used the expressions given in (4. 1.17) for A c a and B 6 i . Similarly, G is given by & - <'ci&!i>T + <Ati^>r , = FJ £ V x ( < * x « I > T + <<*K>T) > (5.2.4) & = J T ^ ^ X + ^ £ cxSx<*x>**>T • Now, let us consider the quantity (1+R)-G. According to (5.2. 3) and (5. 2.4) it is given by (5.2.5) l S ( * x + l - C x * x ) + £ 2 ( C x - i ^ + ^ X ^ • We now use the relation - 1 1 7 -and rewrite (5. 2.5) as (5.2.6) i S c x ( c x - 5 0 + JiL(cx-sxj f < * K > T • Now, and s^  are just cosh and sinh functions respectively. This implies that, for all"X , except "X = O when both c^ and s^  become infinitely large, c%> s^  . This, in turn, implies that the expression given in (5.2. 6) is greater than zero, which leads to the conclusion that If we interchange the order of summation and integration, we get 1 + Rb-G. Since, for any real angle 0, cos 0- 1, we then have that 1 + R 3 G cos 0 . We now combine (5. 2.1) and (5. 2.2) to obtain (5.2.7) pw(T) If we now let M approach infinity and use the result that - 1 1 8 -oo (5.2.9, S w f C K 1 « ^ we obtain that (5. Z. 10, fc,(T) « ± ( » * C » > 8 r t if ft (n-R + c»e.o»e) We see that pw(T1 for spirals appears as an integral of a term of a geometric progression. This may be compared to p w (T) obtained in case (i) which was given by a term of a geometric progression. It is easily verified that oo (5.2.11) 2 P « ( T ) = '• Substituting (5.2. 10) into (5.2. 11) we obtain ( 5 . 2 . 1 2 ) j_ Z f ^ * f t t t " > w | < e . If we interchange the order of summation and integration we obtain, 2 * R.+ Oco»e ^ I - 119 -Because I + R. + G Co* © is always satisfied without any conditions being put upon (R + Gcose) we see that the expression, (5.2.10) for pw(T) can be "continued" into the region in which .tit Finally, then, we have j_ f (R » Gcose)* A Q (5.2.14) bwCT) = 2 n < / o ( 1 + R f & C o f ' We shall now use equation (5.2. 14) to evaluate p0(T) and f3,(T). For w-s O, in p (T) = S f ! K o ' mj (I + R + - / j ^ > ton" ,'+*-<* t a n g ] lit because G * l + R. This gives (5.2.15) M T ) = V0 + For use in Section 3 we define r = R(T=O) and SI = & ( T = O) . Thus I k ( 0 ) = V(i + r ) 1 -^ - 120 -For w = 1 , JL / —?e— t + G / C o » 9 d e , - a* ^  0 + R.+GCO*B) 2n J (i +R+ crc©&e)a [Ci+R)l_&r] «/„ (' + Creole) zn [o+fo'-tf] J o + 6C we) R.Q + R) - fr* (5.2.16) p.tT) = [ ( I + R ) * _ 6 * ] * " These two results will also be obtained from the second representation of (^ (T") given in Appendix 9. It should be noted that (5.2. 14) is valid for any spin arrangement provided only that the definitions and requirements, given in the mathe-matical statement of case (ii), are adhered to. If some spin arrangement is treated in the spin wave approximation and only quadratic terms appear in the undiagonalised Hamiltonian, then the canonical transformation parameters c^ and S> , will be independent of the magnitude of the spin, S. If the treatment is such that either case (i) or case (ii) may be used to investigate the theory, pw U n either case (i) or case (ii)) will be independent of the spin magnitude. If in equation (5. 2. 14) we put G = O and carry out the integration - 121 -we get R w (5.2.17) bw(T) = ( | + R ) « i This is identical in form to equation (5. 1. 10). The result is not a coincidence but arises in the following way. As was mentioned in Chapter 2, the simple cubic antiferromagnet may be treated either by introducing two kinds of operators from the start (the conventional approach) or by using the local axis approach defined in Section 2 of Chapter 2. The proof of the equivalence of the two approaches as N approaches infinity is outlined in Appendix 2 for the case in which the external magnetic field H is zero. The proof of equivalence of the two approaches as N approaches infinity if H is not zero is a little longer and is not given here, but is quite straightforward. The hardest part is to prove the existence of two kinds of excitations instead of assuming them from the start. Because of this equivalence the expression for fa(.T) for a simple cubic antiferromagnet in the absence of an external magnetic field, is given by (5. 2. 14). In Appendix 7 it is proved that, in this case, G = O whence (5. 2. 14) becomes ^ R which is what was obtained before. The quantity R appearing in (5. 2. 17) is given by (5. 2. 3) while the quantity R in (5. 1.10) is given by the sum of (5. 1. 5) and (5. 1.6). In (5. 2. 3) the sum over A runs over N wavevectors while those in (5. 1. 5) and (5. 1. 6) run over N/2 wavevectors. However, as is shown in Appendix 8 - 122 -for the case in which we are interested, the two R's become identical in the limit as N approaches infinity. Section 3. Application of Case (ii) to Antiferromagnetic Dysprosium Metal The results of the previous section for b0(0) will now be applied to the particular case of a simple spiral arrangement, ;that is, one having the equili-brium direction of its spins perpendicular to the spiral axis, which is taken to be the c-axis. The lattice is assumed to be simple hexagonal. Spins attached to sites in the same hexagonal plane, these planes being perpendicular to the c-axis, are assumed to have the same equilibrium direction. The angle be-tween equilibrium directions in adjacent planes is called the turn angle and is assumed to be constant. We shall be particularly interested in the case of dysprosium, one of the heavy rare earth metals, which has this spin arrangement between about 9 0 ° K and 1 7 9 ° K (Elliott, 1 9 6 1 ) . Objection could be raised as to the significance of bo(o) based on data measured at about 9 0 ° K , especially since dysprosium is apparently ferro-magnetic at TbO ° . The significance of b0(o) depends upon the probability of realising the state |G) at about 9 0 ° K . One might point out also that the lattice appropriate to dysprosium metal is hexagonal close packed and not simple hexagonal. However, the results of the calculations done here depend more upon the parameters describing the spin arrangement than on the lattice structure. Furthermore, nearly all treatments (Yosida & Miwa, 1 9 6 1 ; Cooper et al . , 1962 ; Miwa & Yosida, 1961) of dysprosium metal use the assumption that the lattice is simple hexagonal, so that these results would be applicable to those treatments. - 123 -Elliott (1961) has shown that much of the experimental data obtained on the heavy rare earth metals can be explained by assuming a Hamil-tonian made up of equation (1. 1. 1) together with anisotropy terms which represent additional interactions between spins. These anistropy terms are made up of a large axial term and smaller hexagonal anisotropy terms. Cooper et al. (1962) used such a Hamiltonian in investigating magnetic resonance in the heavy rare earths. We shall use a Hamiltonian such as was given by Cooper et al. but with all anisotropy terms Omitted except the axial anisotropy term. The Hamiltonian used, then, is the one con-sidered in Chapter 2 Section 2, *X = S j i j V ^ j + D Z & J 1 D > o . The first term is the isotropic exchange interaction and the second is the lowest order term, in the 5?^  operator, of an interaction between the spins and the crystalline electric field of hexagonal symmetry about the c-axis (Cooper et al. , 1962; Miwa and Yosida, 1961). Any anisotropic exchange interaction is ignored since this does not seem to be an essential feature of the theory. D is taken to be greater than zero so that the easy axis of magnetisation will be perpendicular to the c-axis (Wilkinson et. al. , 1961). The term containing the operators (s\h + svfe] has been omitted because the interaction constant associated with this term has been shown by Miwa and Yosida (1961) to be of order 10 or 10 compared to other interactions which will be considered here. This term however, is significant when investi-gating the change of the turn angle with temperature. - 124A -All coupling constants are assumed to be zero except: (a) J}, between nearest neighbours along the c-axis. (b) J 2 , between second nearest neighbours along the c-axis. (c) J 3 , between nearest neighbours in the hexagonal plane perpendicular to the c-axis. This choice of coupling constants appears to be conventional and has been used by Miwa and Yosida (1961). o The turn angle is taken to be 2 6. 5 . This model, then, corresponds o approximately to the case of dysprosium at about 90 K (Elliott, 1961). The value of Jz 1 = 3^13\ w a s determined by minimizing the zero-point energy with respect to the turn angle. P and £1 were calculated approximately by a numerical method for various values of J3l=J3/Ji and Dj = D/J}. The singularities in -5^  and c xsA at the origin necessitated expanding the functions in a power series around the origin. The results are given below in Table 5. 1 1 <* i 1 k 1 - i 1 1 1 2 \ i 2 1 2 X X r 0.009 0.004. 0.003 0.02/ 0.008 0.00 7 0.0^ +7 0.019 O.o\k n -0.070 -0.035 -O.Oii" - 0 . Ufa -0.06O -0-043 -0. i9o -0.1° 1 -0.O74-0.994 0.997 0.997 o.9$b 0.9 94- 0.994- 0.972. 0.987 0.989 Table 5. Values of P , _fl and b0(o) for values of J - J J and -124B--While one should consider these results to show the trend of p>0 (o) , due to the approximations they should be considered numerically reliable only to the second figure. It should be noticed that p 0 ( ° ) is relatively insensitive to changes in T 3 l and and that for 7 1 > p„(°) is effectively constant. The decrease in p0(o) as J^j decreases would be expected since as the ferromagnetic coupling in the hexagonal plane decreases, the spins can depart more and more from their equilibrium direction. The values of f and .Q given above should not be used in calculations in which small changes in T and Q. would lead to significant changes in the results. In the case of p0(o) even large changes in f (of order 10% or 20%) will have very little effect on the value of (I + ^ ) which, in turn, completely dominates -Q. . Therefore -O- could undergo large changes (10% to 20%) without affecting the value of f=>o (0) . In the above calculation D^o(0) appears to have converged to the second figure. Section 4. Concluding Remarks The calculations carried out here serve to support the view that the spin wave approximation is applicable to a variety of spin arrangements at low temperatures. In particular, the results obtained fromthe joint probabilities for nearest neighbours in the case of the simple cubic anti-ferromagnet might be interpreted as further evidence that the true ground state is near to the state \G) , the ground state of the excitations described by the operators « and (3 At the present time some further calculations are being done with a view of saying something about the joint probabilities associated with - 125 -case (ii), the interest being primarily connected with spiral spin arrangements. As was mentioned, the case in which an external magnetic field term is introduced into the more general Hamiltonian of case (ii) is as yet unsolved. This interesting problem is also being given attention at the present time. - 126 -BIBLIOGRAPHY Anderson, P. W. , 1952 Phys. Rev. 86, 694 Bloch, F. , 1932 Z. Physik 74, 295 Cooper, B . R . , Elliott, R . J . , Nettel, S .J . & Suhl, H. , 1962 Phys. Rev. 127, 57 Dirac, P. A. M. , 1958 The Principles of Quantum Mechanics (Oxford University Press, 4th edition) Dwight, H. B. , 1957 Tables of Integrals and Other Mathematical Data (The MacMillan Co. , New York, 3rd edition) Dyson, F . J . , 1956 Phys. Rev. 102_, 1217 Elliott, R . J . , 1961 Phys. Rev. 124, 346 Heisenberg, W. , 1928 Z. Physik 49, 619 Holstein, T. & Primakoff, H. , 1940 Phys. Rev. 58_, 1089 Kaplan, T. A . , 1959 Phys. Rev. _U6, 888 Kubo, R. , 1952 Phys. Rev. 87, 568 Miwa, H. & Yosida, K. , 1961 Progr. Theoret. Physics (Kyoto) 26, 693 Muir, T. , 1911 Theory of Determinants, Z_ (MacMillan and Co. , London) Murphy, R. , 1833 Trans. Camb. Philos. Soc. 5, 65 Oguchi, T. , I960 Phys. Rev. 117, 117 Schwatt, I .J . , 1924 An Introduction tothe Operations With Series. (The Press of the University of Pennysylvania, Philadelphia) Van Kranendonk, J . & Van Vleck, J . H. 1958 Rev. Mod. Phys. 3_0, 1 Villain, J . , 1959 Phys. and Chem. of Solids 11, 303 Walker, L . R. , 1963 Magnetism, 1, 299 (Academic Press, New York) Wilkinson, M. K. , Koehler, W . C . , Wollan, E . O. & Cable, J . W. , 1961 J . Appl. Phys. 32_, 48S Yoshimori, A . , 1959 J . Phys. Soc. Japan J_4, 807 Yosida, K. & Miwa, H. , 1961 J . Appl. Phys. 32, 8S - 127 -APPENDIX 1 Outline of Walker's Approach in Order to Obtain Expressions for Probabilities. We shall outline Walker's approach for the case of ferromagnetism. For the case of antiferromagnetism we shall just give the results because the calculations are somewhat longer than in the ferromagnetic case. His notation will be altered to conform to that used in this thesis. Consider a general state of a ferromagnet, (Al . l ) 1 Vn . i |n) = IT 10) = 17 K ) • i The probability that this state is occupied at temperature T is (Al. 2) pA = ± (o| € |n) . Consider a generating function, F(x, ; x X i . . . ; T), such that oo where [n;] denotes the set of all spin deviation numbers attached to sites in the crystal. Walker finds F(x, ( x x , . . . ; T) to be given by (Ai.4) F ( * , , * i , - - - i T ) = Mole* e e< lo) ? where b^  and b^  are dummy Bose operators with 1°) being also the ground state of the operators b< and b^ . Equation (Al. 4) is then written in the form - 128 -( A 1 . 5 ) F ( x , , 3 T ) = ^ ( o | € « €« e e * l o ) Walker finds that ( A I . 6) i . « 1 € r e 1 l o ) = e 4 J l o ) 21 where ( A I . 7 ) R( t - j ) = — i £ a ( T - t ) < ^ x > T N * iV l - i ) R(i-j) is equivalent to what we called W in Chapter 5, Section 1, when discussing the case of pw^ Wj (.T~), if we put 6 ^ = 0 and ' cA = 1 so that the expression for pWiW.(r) would then be applicable to a ferromagnet. Now let i„ be a set of subscripts, i , for spins in which we are interested. To obtain the pertinent generating function, F ' , we set all x- = 1 for those i which are not contained in i 0 - Walker finds that (A1.8) F ' l * i . i T ) « ^ 7 T Y ) ' where (A1.9) Y 4 t / = ( ^ f ' ' ) R ( i „ - O , are the matrix elements of the matrix Y. For a single spin, i c he finds that - 129 -(ALIO) F'lijf) s If we expand this we see immediately, by using (Al. 3) that ( A l . l l ) h = — r w . + , r w * [l + 9.(0)] 1 For spins i and j , (A1.12) F ' h ' , j ; T ) = j[,-(x 4-.)Rto)J[i-(x r.)Rto)J-(xHXXjH)R(i-j)RCj-o} For a simple two-sublattice antiferromagnet, he defines a generating function, F{(X(.),(yj) , T j where i refers to sublattice-A and j to sublattice-B. The symbols ( TO and t^ j,) represent sets of parameters X ; and y^  . He finds, for a spin on sublattice-A, (A1.13) F ( x t ; T ) = , „-p&x»-P^ where he assumes that there are N sites on a sublattice. It is believed that equation (Al . 13) contains a misprint and that E^should be intern changed with E x . Otherwise, the expression for pwt(T) obtained in Chapter 5, Section 1, agrees with that obtained from (Al . 13). The generating function for the joint probabilities is quite long so that we do not give it. - 130 -APPENDIX 2 Outline of the Equivalence of the two Approaches to the Simple Cubic Antiferromagnet From equations (2.1.5) and (2. 1. 6) we have the Hamiltonian of the first approach which employs two kinds of operators for h • 0 given by (A2.1) = *UlSx E [ d + l M v + yxUxVx + , where 52 signifies a summation over N/2 wavevectors, ^ , lying within the first Brillouin zone from now on referred to as Bj , of a face centered cubic lattice. The superscript in signifies that (A2. 1) is the Hamiltonian of the two operator approach. We have written tJI rather than J because J O . In this appendix we shall reserve the symbol A for wavevectors lying within From equation (2.2.21) the Hamiltonian for the same physical system, but using instead, one kind of operator, is (A2.2) < > = U I S J L Z + + + y j ^ a ! * , where j-^j signifies a summation over N Wavevectors, , lying within the first Brillouin zone, denoted by B s , of a simple cubic lattice. The superscript in otj, signifies that only one kind of operator is used, and we have written a^ as a^  to distinguish it from the operators in ^j. . The symbol yu. is used to denote wavevectors of B s • Let us take B s and B^ to have a common origin. Then, if V s is the volume of B& and Vj is the volume of (A2.3) V s = . - 131 -As N approaches infinity, then, N/2 wavevectors of B s will lie within Bf. Let us denote this set of N/2 vectors by denotes the set of wavevectors of B s ; we have that where the sum denotes the union of two sets, and P is, from (2.2. 19) The vectors . &x and ^ 3 are the basis vectors of the simple cubic lattice. The proof of (A2. 4) consists in selecting any > from B s and then forming A 4 P. Since P is not a reciprocal lattice vector, ^ and A + P both lie in B & and are neither equivalent to one another nor equivalent to any other wavevector in B s . We now make the key assumption which is, that as N approaches infinity, every wavevector A £ B^  and contained in I s equal to some wavevector yU-^  £. B s which is contained in £-C^  /<• j We have written yu.^  to indicate that = "X . This means that ( 2 H • 1 5 A } , • Now, P is a vector of the reciprocal lattice of a face centered cubic direct lattice. This means that, for every A £ , (A2.6) X + P = A But - 132 -(A2. 7) X + P = fix + f • Therefore, (A2. 8) Ju^ + P = \ . Having obtained (A2. 8) we consider the Fourier transformations (2. 1. 3) connecting a^ and b^  with a^ and b ^  , and (2.2.10) connecting with . From (2. 2. 10), , A 2 - 9 ) a ^ i . V "* ' where JNt^is the set of N sites in the simple cubic lattice. If we let "tj denote the set of N/2 sites in the face centered cubic sublattice-A it is obvious that for every site i £ \jx ^ \ \ there exists a site t x ^ ( N ^ j c and for every site i + where i £ L K there exists an ^ v + 6 , We can then say that (A2.10) a ; . = a{ (A2.i i ) a, = We can then write (A2. 9) as (A2.12) v T ^ N f N ^ ^Nmt} Now, if p. "/^^ where, as above ^ ^ ^j . we can write (A2. 13) £ * x ) - 133 -However, if ^ T u a + " p then, since e = —I and e = 1 (A2. 14) - ( a x - ^ ) It is now easy to show that (A2.16) = ^= K - f>x) • Substituting (A2. 13) and (A2. 14) into (A2.2) we see that In the case that the external magnetic field, H , is not zero, will be given by (2.1.5) but will contain extra terms like a/* + f It is not difficult to prove that it is necessary to have two kinds of operators in terms of which ^x is diagonal. ^ will be written as where - V 7 The proof is based on the following assumptions: - 134 -(i) There exists a matrix, T .^ i O 5 , where Pyu = ^yu^i a n d ^lyi = ^ ^ y ^ ^ i ^ / i ] i s diagonal, (ii) Let t,* be any matrix element of T^ . Then (a) t^  is real (b) no t^  is zero (c) t.^ = y If and yL. are the operators in terms of which "J^is diagonal, we find that ( A 2 . 1 9 ) Uyu * 0 > , V * - 135 -where was defined in Section 2 of Chapter 2. The splitting of the spectrum into \] A 1 - B% - h is obtained. - 136 -APPENDIX 3 Derivation of the Relation n ^ n + -A n _i = A and Some Properties of the A ^ ; . Derivation of the Relation nA +A , =A n n-1 n According to the lemma of Chapter 3, Section 1, A n is defined by .('> 1-0" " (A3.1) A? - ^ S H / ( 5 ) * ' Using this definition we are able to write Collect together the second sum and the first (n-1) terms of the first sum. This gives (A3.2) + An_{ * [*!(«-6>i. 6 ! (^,_ 6 )>J e * ^ ( , ) We now use the identity whence (A3. 2) becomes - 137 -Some Properties of the A n Using the definition of , a function <p^ can be defined: (A3. 3) An - -77 • The proof that ^ =0 for f^n is given by Schwatt (1924). W We shall repeat the proof here since we must also show that -Consider the function $-0 p Expanding the exponential, f»9 r 7-0 T • - 138 -Thus, using (A3. 1) and (A3. 3) (A3. 4) • < - r S i > ? + <-->« RHS : coefficient of x* is t! r or-LHS where the lowest power of x is x . Therefore the coefficient of X (Y> is zero, where ~) - 1, . . . , <*-!; whence = 0 "Y^ ^ • Further, the coefficient of x in (<2 -i) is 1. Using (A3. 4) with .0) . A(f) , we see that = I . That A, = 1 can be seen by directly writing out the expression for A{ . - 139 -APPENDIX 4 Equations Connecting the * r and the Hn (w) In equation (3. 2. 8-s) of Chapter 3 we saw that the x r satisfied a set of equations, M S S ^ x J 0 = (*= S $ w = k-ronecketr delta) . •r=i The equations of this set can be characterised by the integer s. Let us multiply the I -th equation, that is, the equation for which s=l, by ^ ID and sum over t from I to t, where t^ M. We then get that We may interchange the summations on the LHS of (A4. 1). If we do this, at the same time carrying out the summation over I on the RHS we obtain (A4.2) H The part of (A4. 2) in brackets on the LHS is just, by definition, A^ ( w) We then obtain the set of equations connecting the x^  and the A^ : - 140 -APPENDIX 5  Equations Connecting the x ^ and the A n This Appendix is similar to the one in which the set of equations connecting the x and the A (f) was derived. According to equation (3. 2.19-n,m) of Chapter 3, Section 2, the satisfy the set of equations M M (n, r* * o , l , . toy Each equation of this set can be labelled by the pair of integers n. m. Consider the equation for which n = ^ and m=k. Multiply it by b! U and sum over t from 1 to p and over k from 1 to q where both p and q M . We obtain W« may interchange the summations over £ and k with those over f and g on the LHS of this equation and carry out the summations over I and k on the RHS. We thus get £ £ »«r'f^  <^ !^(f j'A^l - t - r ^ - y v i - 141-The two brackets on the LHS of this equation are and A ^ respectively. Substituting these in, we obtain the set of equations relating the J to the A„ : - 142 -APPENDIX 6 Alternative Derivation of an Expression for <^X^  ixx ) r of Case (ii) (Equation (4. 2. 11) As was done in Chapter 4 we express ai as It will be recalled that operators may take part in both CC and IC in <aT" a^ T . That is, in this case, (A^ B S ) T 0 . We shall now consider a typical product term. From the upper row k A c ' s are IC with k Bs's each B S lying to the right of the A*c with which it is IC and £ B S 's are IC with I A*C'S each B S lying to the left of the A c^ with which it is IC. In the lower row k' A c 's are IC with k' B ^ ' s each A c lying to the right of the B * S with which it is IC and I' A^s are IC with t' B\'S each A c lying to the left of the B ^ with which it is IC. There thus remains n-2k- 2.1 operators in the upper row to CC with n-2k'- 2.1' operators in the lower row. These two numbers must evidently be equal so that we are forced to choose V =k + t - k 1. The k A c ' s and K C B s ' s , £ B s ' s and the t A T/s, k> A ' s - 143 -and k' B\JS and V A t J s and (' BV-S can be IC in (2k-l)(2k-3). . . 3 . 1 , (Zl-\){zl-l) ..-3.1 , (2k'-l)(2k ,-3)...3. 1 and (2k+2* -2k1-1) (2k+2£ - 2k'-3). . . 3. 1 ways respectively. Furthermore, we can choose the k A*'s and k B s ; s , the I A-Vs and I Bs's, the k' Ae's and k« B\'S and? Ac's and i' B\JS in ,(n^k], (^) , and (j^ +i^ -z|<,) ways respectively. Finally, the contribution from the CC is, as we know, Noting that <A T T B I > T - < B ^ T > T , <A T B, F > R = <B ,A F T > T we can write the contribution to ^ a t V ^ from a single product term, as selected above, as where R. - <AtAt>T + <B,BI>T - 144 -We now sum the quantities contained within the square brackets over k', the sum running from O to k + I . We get 1 ^ <*t*Jr<K*\>l ^ K * - U k ¥ i ) * x ( k + ° ( r > - Z ( k * l > ) ! ( k * i M M l ! where G = We now define * * It + i and sum over k from O to f: ( V ) < < I « » : > V K M * £ ( { ) < 4 8 , > I < A c B t > ; t t which gives » ! (^) £xf (*/) G f ^ R " " * ' . Finally summing over f from O to ? where J = n/2 if n is even, and 7, = (n-l)/2 if n is odd, we get which is what was obtained before. - 145 -APPENDIX 7 Proof that, for a Simple Cubic Antiferromagnet, G=Q From equation (5. 2. 4), G is given by, Gr = £ S ^ + ^ £ c x s x <4*)> T which is (A7.U o - i f ' A * ^ ' A p ^ ' The sum over ^ runs over N wavevectors in the first Brillouin zone of aample cubic lattice and N is assumed to be an even number. If we denote the set of wavevectors in this Brillouin zone by and denote the set of any N/2 wavevectors, also taken from this Brillouin zone, by , j we have proven (see Appendix 2) that (A7.2) JN.X} * + | * . W P} In view of (A7. 2) we may write (A7. 1) as where 21 means "sum over any N/2 wavevectors". Now, from equations (2. 2. 46) and (2. 2. 47) we have that - 146 Combining (A7. 3) and A7. 4) we obt - 147 -APPENDIX 8  Proof that R as Given by (5. 1.5)+ (5. 1. 6) is Equal to  R as Given by (5. 2. 3) with h = O From (5. 2. 3) we have where we are using the notation of Appendix 2„ Bearing in mind (A2. 5), (A2. 15) (A2. 16) and (A7. 4) we can write (A8.2) R - J- £ U j + *x>i<«K+£Px>T + * 2 *x But, when h = O, Therefore, ( A S . 3) R. = 4 z *t + a 2, <<x * - * ^ - p r ~ M « » l , which is just the sum of (5. 1= 5) and (5. 1. 6) - 148 -APPENDIX 9 Alternative Expression for j=>„ ("*") of Case (ii) We shall now give a second derivation of an expression for This expression will appear in a quite different form from that given in equation (5.2. 14). We shall start with equation (4. 2. 11) which gives an expression for ^oJ" ) and equation (3. 2. 14) for pw (T) . Combining these two we obtain.-We may replace the upper limit, ? in the sum over k by M because =0 if 2 k n. We may then interchange the summations over k and n in (A9. 1) and write p as (A9.2) bwCT) - ^ ^ ( f t ^ ^ ^ H H X j R " . Consider the sum over n: We can write it as M M W n = i OK. - 149 -If we assume that we may take the differential operator out from under the summation sign, this becomes (A 9 . 3) £ l-o"tt)U° K)R" - £ £ £ ( - ' ) " & ) R " • We now let M approach infinity in a suitable way and make use of the formula (A9.4) Z(-V[!k)*n = TTTRF^' ' R ^ The condition that R be less than unity is similar to the condition R <1 1 of (5. 1. 8) which was subsequently shown to be too restrictive. It is believed that the condition here is also too restrictive because, as is seen from the results of Appendices 2, 7 and 8, the expression for ("0 of case (i) must be obtained from the expression given in this part. This conjecture, on the freedom of the magnitude of R, has not yet, however, been proved. It is also possible that the restriction on G, found in (A9. 6) and (A9. 7) is too strong but this has not been proved. Using equation (A9. 2) together with equations (A9. 3) and (A9. 4) we obtain (A9. 5) bw(T) - S_ R Lo zXkU)lv) J R - ( T ^ T + . Further simplification of (A9. 5) has not been possible. For applications,, the procedure would be to evaluate the Pw^ "0 °f interest. This evaluation would pose no computational problem as far as summations are - 150 -concerned, because, after the necessary differentiations, we would be left with various sums like , O £ * * j -o v The procedure will be illustrated for f> ( »") and p, ( ' ) „ This equation will be used as many times as is necessary. First , define G 1 x ^ o Then, for w = O, For w = 1, X < - 2. (I + R ) J Thus oo - 2* G3-- 151 -which may be rewritten as g(l + R) - (y-These two results are indentical to those obtained by way of the first method. 

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