SPIN COMPLEXES IN FERROMAGNETISM BY ADEMOKUN IBIYINKA AGBOOLA B ,Sc, Physics CIbadan) M.Sc, Physics (Ibadan) (ton a A thesis in the Department of PHYSICS . Submitted to the Faculty of Science in partial fulfilment of the requirements for the degree of DOCTOR OF PHILOSOPHY UNIVERSITY OF IBADAN NOVEMBER, 1988 ii ABSTRACT The spin-wave theory in Heisenberg model of ferro­ magnetism is investigated with the Holstein - Primakoff transformation and with emphasis on the spin wave interactions. The temperature T below which the concept of magnons is valid is determined. By a special expansion formalism of operator (l-a+a/2S) 2 which yields 1+('1-(1-Jg) 2 )a+a it is shown that quantized spin waves which behave like spin 1 quasi­ particles (with dispersion relation oV~ k ) called magnons at temperatures T < T^, are Bosons with an effective (negative) electrochemical potesn„tial y that varies as T in the wave-wave interaction ap9pr>oximation. The various coeffi­ cients of Tv in the expression of the spontaneous magnetiza­ tion M (T )/M (o ) = l-(C1T3/2+C2T5/2+C3T7/2+C4T4 ) as well as the specific heat for some ferromagnets are calculated. The results are remarkabVQclose to the experimental values obtained by other investigators. The method used enables one to deal especially with regimes of small spin values S for which y differs substantially from zero. The influence of the chemical potential on some thermodynamic quantities are found for ferromagnets with Hexagonal-^close-packed structures, as well as for cubic crystals, The existence of the spin wave inter­ actions and hence of non-zero effective chemical potential is shown to give rise to a lowering of the thermodynamic internal Ill energy with the implication that spin waves, on the average, form bound states called spin complexes. The kinematical as well as the dynamical interactions on the thermodynamic quantities are also found for some ferromagnets, by sub-t- jecting the magnons to intermediate statistics. The influence of the spin-wave-spin-wave-spin-wave interactions on the coefficients of Tv in the expressiOon vof the spon­ taneous magnetization of some ferromagnets are found to be negligible in comparison with wave-wave interactions. An attempt is made to extend the above calculations to spin complexes in antiferromagn etism, a phenomenon which seems to be relevant to high temperature superconductivity. IV DEDICATION This work is dedicated to the Lord God Almighty in whom lies the mystery of wisdom and understanding. V ACKNOWLEDGEMENT I would like to express my profound gratitude and appreciation to my supervisor, DR. AKIN 0J0 for the great help he rendered and for his invaluable advice, guidance, understanding and encouragement all through tmhee pteriod of this work. I also wish to express my heartfelt gratitude towards my Parents MR, and MRS. D.I. ADEMOKUN, my sister and her husband DR,(MRS) and DR, R.A. ADENIRAN and my ' younger sisters, Ibidun and Yele, for their concern, love, care, moral and financial supOporvt. My sincere love and reciation goes to my fiance "DEOLA" (a better half a a friend to the letter) for 1 1 a his genuine love, care, concern, understanding, encourage­ ment and for showing so much patience all through the period of this work. VI CERTIFICATION I certify that this work was carried out by Miss Ibiyinka Agboola Ademokun in the Department of Physics, University of Ibadan, Ibadan. DR, AKIN 0J0 B.S.E, (Prin on), vii TABLE OF CONTENTS Page TITLE PAGE ABSTRACT DEDICATION xv ACKNOWLEDGEMENT : z = 8 . , , 79 3 .2- 4 Graph of chemical potential/exchange integral W against reduced temperature T for S = 5 , Z = 11 80 3 .2- 5 Graph of chemic:aal pPotential/exchange integral W against reduced temperature T for s = £, z = 12 ' ... 81 4. li A given atom with six nearest neighbours on the simple cubic lattice ,.. 92 4, lii A given atom with eight neighbours on the body centered cubic lattice , ,, 95 4,liii< 5A given atom with twelve nearest neighbours n the face centered cubic lattice 96 4, liv A given atom with twelve nearest neighbours on the Hexagonal close packed lattice 97 4, lv A given atom with twelve nearest neighbours on the Hexagonal close packed lattice 98 4 .3- 1 Graph of chemical potential/exchange integral W against reduced temperature T for 2 > z = 7 127 X Figure Page 4 .3- 2 Graph of Chemical Potential/Exchange integral W against reduced temperature T for s = 1 , z = 7.5 128 4 .3- 3 Graph of Chemical Potential/Exchange integral V against reduced temperature T for s = 5, z = 8 ... ■\ V129 4 .3- 4 Graph of Chemical Potential/Exchange integral W against reduced temperature T for s = 5 , z <= 12 ... \ 130 4 .3- 5 Graph of Chemical Potential/Exchange integral W against reduced temperature T for s = k, 12 V , 131 & sS^ xi LIST OF TABLES Table Page 2.1 Temperature T o of some ferrometals 54 4. li Nearest neighbour Distances on a simple cubic <&i i lattice in units a ... 91 4,lii Nearest neighbour Distances on body cen t r ea n cubic lattice in units a/2 ••• 93 4. liii Nearest neighbour Distances on a face-centred Vi cubic lattice in units a/2 ... 93 4, liv Nearest neighbour Distances on an Hexagonal close packed lattice in unit;s otf a. 94 4.1. Reduced curie temperature and the critical I t i point exponent for some metals 115 4.2 Coefficients of T3^2, T3/2 , t ^/2 an(j the spontaneous magnetization and the coeffi cients of T^/2 in the expression of the specific heat for some ferrometals. 116 4.3 Coefficients of T3 3̂ with chemical potential in wave-wave interactions and wave wave wave interactions .,, 126 5.2 Coefficients of T3^2, T5/2, T7/ 2 and T with kinematical interactions, 141 5.3 The coefficients of T,u3//2" (in units of 10 -6) 143 6.1 Values of C in units of (10~^/k3^2 ) 148 V 6.2 Ground State energies of antiferromagnets (by Kubo) 158 6.3 Ground state energies of antiferromagnets (by our technique) 159 CHAPTER I FERROMAGNETISM 1.1 Introduction Our aim is to elucidate new concepts and formalism that are germane to the subject of spin waves in low temperature ferromagnetism using the metals Iron, Cobalt, Nickel, Gadolinium and Dysprosium as concrete examples. Magnetism is a phenomenon displayed by, or a macroscopic property possessed by some charger-neutral material bodies whereby one b,±£ (1 .1- 2) and h is planck’s cons The conclusion that t his- suggests is that a moving electron can constitute an elementary magnet, , provided it iss< £inr a state with non-zero angular momentum.On the other hand, an electron moving round the nucleus may he in the state with zero angular momentum, (the S-state) in which case M *= 0, too. With the observations of the experiment carried out by Stern and Gerlach while sending a beam of silver atoms through a non-uniform magnetic field as well as the suspicion 5 borne from an analysis of atomic spectra, a quantum particle may possess an INTRINSIC ANGULAR momentum. This is inherent to the particle in addition to the orbital angular momentum caused by the motion of the particle in space. Therefore an electron in a state with zero orbital angular momentum possesses a non-zero value for the projection of its magnetic moment by virtue of its intrinsic angular momentum. This intrinsic momentum is also called SPIN, The projection of the intrinsic momentum of a particle can assume not only integral values but half integral values as well. The magnetic momen t of a free atom basically has four principal sources: (i) the spin with which electrons are endowed. Cii) their orbital angular momentum about the nucleus, (iii) the combination of both intrinsic and orbital angular momenta J, and (iv) the change in the orbital moment induced by an applied magnetic field. The first three effects give paramagnetic contributions to the magnetization, and the fourth gives a diamagnetic 6 contribution. The most natural way to classify the magnetic pro­ perties of a material is by its response to an applied magnetic field. The response is chi susceptibility X, in the relation M = XB.o where M is the magnetization, or magnetic moment per unit volume, and Bq is the applied field. Diamagnetic materials ha mall, negative tempera­ ture independent susceptibility X. The magnitude of X is of the order of 10~ 6 cm 3/mole. Since it is negative, the induced moment is directed oppositely to the magnetic field. This kind of magnetism is a direct consequence of Lenz's law applied to the motions of the elementary charges (generally electrons) of the system. All materials have diamagnetic contributions to their susceptibilities, but for most materials, the diamagnetic contribution to X is small compared to the total, and is usually neglected. For Paramagnets, the susceptibility is positive and temperature dependent, It is of the order of 10 -3 cm 3/mole at room temperature and varies approximately as 1/T, where 7 T is temperature. (see fig, 1.1). This kind of magnetic behaviour can be explained as a consequence of two opposing effects: one, the tendency of the applied field to orient the moments in the direction of the field, and the other, the tendency of thermal agitation to randomize the orientations of magnetic moments. The paramagnetic suscep­ tibility varies linearlv with B o for small fields and consequently vanishes for zero applied field. However, it is also well known that some crystals containing magnetic atoms develop a macroscopic magnetic moment in the absence of an applied field, if they are cooled to sufficiently low temperatures. These are ferro­ magnetic materials or simply ferromagnetics. Ferromagnetism does not exist at all temperatures. As temperature increases, the intrinsic spontaneous magnetic moment of a body decreases and vanishes at a certain tempe­ rature Tc called the Curie temperature (See fig, 1.2). This of course occurs if the external magnetic field is zero. Above the Curie temperature, ferromagnetic materials become paramagnetic. At high temperatures all ferromagnetic materials are paramagnetic but not all paramagnetic materials are ferromagnetic at low temperatures. Different materials 7 ' Fig. 1.2 Spontaneous m agne tiza tion as a fu n c tio n o f tem pe ra tu re . 8 have different values of the curie temperature Tc and spontaneous magnetic moment density M s at T 0. Neel predicted the existence of another kind of cooperative magnetic phenomenon, which he called Anti­ ferromagnetism, In the simplest form of an antiferro­ magnetic material, the lattice of magnetiic ato ms can be divided into two equivalent interpenetratbing sublattices, A and B, such that A atoms have only B atoms as nearest neighbours, and vice versa. The magnetic inter­ actions are such as to cause the sublattice magnetizations to be antiparallel. At absolute zero, each sublattice has its maximum saturation magnetization, and as the temperature increases, thermal agitation reduces the sublattice spon­ taneous magnetization in much the same way as for a ferro­ magnetic material. However, the net magnetic moment of the spontaneously magnetized antiferromagnet is zero at all temperaturesjS^ecause of the exact cancellation of the spontaneous magnetization of the two equivalent sublattices. The outstanding development in the phenomenological description of ferromagnetism is the theory of the molecular field by Weiss. . Shortly before this, Langevin had developed his theory of paramagnetism based on the funda­ mental idea that the orientation of molecular dipole of 9 moment y in a field B is governed by the Boltzmann's distribution law. Given N elementary dipoles per volume, each of magnetic dipole moment p, in a magnetic field B , the kinetic energy E is given by A E - -y ♦ B = Cos 0; y = |^|, B = \ B \ . (1.1-4) 6 is the angle between y and B. Writing m = = {yCos0> . . . (1 .1-5) B J yCose e^pB("os02irSin0 d10 m = . . . ( 1 .1- 6 ) | e3yBCos02TiS^^ de ° A M = Nm = Ny CotanhByB - _N_BB ... (1.1-7) For small B (high temperature T) ByB << for small x, tanhx = x-x /3, cotanhx = (x - -x~ 3- )x -11 - ----- 9 x(l-x^/3) - ¥l+l> m = yyt(1_3iyB_ + 3 _ ... (1 .1-8) MM = A 3- r" CN A3k Tb d.1-9) 10 The basic idea of the Weiss theory is that the effective field acting on an elementary magnet in a ferromagnetic medium is not to be identified with the applied field B, but is rather to be taken as B+qM where H is the iintensity of magnetization and q is a proportionality factor independent of temperature. The portion ql is called the molecular field and is clearly a manifestation of the cooperative phenomenon by virtue of which the atomic magnets tend to be parallel. The Weiss theory has the merit of simplicity, for any phenomena can be explained by taking B+qM instead of B in equation (1.1-7). With this modi­ fication with B replaced by B+qM, eqn. (1.1-7) becomes M = Ny [( Cotanh ^B+qM))_ (y(B+qM))-l] ..(1.1-10) If we neglect saturation effects, and so make the approximation CLoottaannhn Un I5±k3T M2 . ((| i (Bk+Tq M)). -1 = 31 u (Bk+qTM) appro\xi>mate to small y(B+qM)/kT, then after solving for M, the relation (1 .1-10) reduces to X* = 3M/9B =• Ny /3k(T-T c ) . ?. (1 .1-11) 11 Here X denotes the susceptibility and Tc - Nii q/3k ,,, (1,1-12) Equation (1.1-11) gives an infinite solution, at T ■ = TC , and' so the Weiss theory immediately gives us a critical point or CURIE TEMPERATURE. Below Tc it is no longer allowed to make the approximation above because the moment ceases to be linear in the field strength, enormous magnetization can be obtained without the necessity of corresponding applied fields, and the behaviour becooimes ferromagnetic. This simple analysis furnishes a remarkably satisfactorjr description of the salient experimental facts. The two most important of the many successes of the Weiss theory are the following: (i) The linear relation which is predicted bjr equation (1 .1-111 between the reciprocal of the suscepti­ bility and the temperature above the curie point. V?The linearity is on the whole quite well confirmedexperimentally, As the temperature is lowered towards the curie point, the experimental curves begin to deviate more from linearity. The inter­ cept on the axis ^ i. e. ^ = 0, corresponds to infinite susceptibility, or in other words, to ferromagnetism. Usually it occurs at a lower temperature T' , than the value T which is obtained by extrapolation from the linear behaviour at higher temperatures. The quantities and Tc l are sometimes called the paramagnetic and ferromagnetic curie points, respectively. The difference between them is relatively small representing a second order effect. Cii) The other success of the Weiss theory is its prediction concerning the variation of saturation magnetization with temperature below the curie point. The great mystery of the Weiss theory was how to explain the large molecular fields. They were supposed to be a manifestation of powerful coulpplineg between elementary magnets, However, at the time? the only known interaction between them was the classical dipole - dipole coupling, whose potential is 13 where r .. is the distance between the two dipoles. However, this interaction is far too weak to yield the coupling required by the Weiss theory. It gives a maximum value 4 it for the constant q in the molecular field qM, whereas the successful application of the requires that q be of the order 10 5 . The development of quantum mechanics was a great help in the understanding of ferromagnetic phenomena. Tn the first place it was accompanied by the Uhlenbeck - Goudsmit concept of electron spin. The latter has a ratio of magnetic moment to angular momentum equal to e/m instead of the classical e/2m. This behaviour is to be expected if most of the orbital angular momentum is largely destroyed by interatomic forces in the solid state leaving only the spin. The quantum theory of ferromagnetism is usually developed on the basis that the orbital contributions to the magnetic moment are negligible. Actually they cannot be forgotten entirely as evidenced by the fact that the gyromagnetic ratios of ferromagnetics are usually nearer 1.9 than 2,0. One thing which quantum theory has obviously done is to introduce a discrete series of orientations rather than 14 a continuous distribution as in the classical Langevin theory. That is, the kinetic energy E of each magnetic dipole y in a magnetic field B, is given by H = —y • B = — eft • B — — y B * L O ...r(1.1-14) where L is the angular momentum operator with eigei ues l = -s, -s+1, 0, 1 , s The partition function given by Trace s( eixXp7s(S-B?H)) is Q. = ! e ^ B ... Cl.1-21) which can be separated into independent wave equations involving each electron. There are then solutions ip = ^(1)* (2), E° = E.+E . . . (1 .1-22) where \pi. and J ip .are solutions for a single electron moving in the potential V. If we apply first order per­ turbation theory to calculate the effect of the inter­ action, we find 2 E = E° + K C1H*C2) J- j(.2)4»i(l)]* , E° = E1+Ej ...(1.1-24) ... (1.1-26) This result suggests that the spin-dependent contributions to the energy arising from the Pauli Principle may for some purposes be regarded as caused by two body spin-spin inter­ actions of the form -2 E J. .S.-S . ...(1.1-27) i0 do happen, and it is they that explain the most spectacular magnetic property, namely, ferromagnetism. For our work, we takTe J.lj, >Q,'and our concern is not so much but on the consequence of the positive bondv^T in low temperature ferromagnetism. 1.2 The basic requirement of a fundamental theory in physics is that there must be a Hamiltonian. The question has always been' What Hamiltonian is appropriate to display spontaneous magnetisation? The Heisenberg model suggests that spontaneous magnetization arises from a coupling of 2Q the spin angular momenta, rather than the total angular momenta JL. This particular assumption is of course exact for atoms or ions with orbital angular momenta L , It is a reasonable approximation for most of the transi­ tion metal series but is inappropriate for the rare earths like cesium, ytterbium, lutecium and others; since L 0. The crystal contains atoms with magnetic moments =% associated v/ith their spin angular momenta. The magnetic atoms are assumed to interact in pairs according to (1,1-27) and to be subjected to an external applied field. In this case, neglecting the translational motion our starting model Hamiltonian for the crystal is H = -2 l.j . S..S. -gp'B £S. ... (1.2-1) i (1,3-5) k Then, In Q = ZlnO l & = Z exp(3Jn + 3B S, ) < c nk * ° Q = exp(3J+3Bo )+exp(3J-3Bo )+expC-3J)+exp(-3J) And 1 3 -| _ m = 6 JBnl Q (1.3-6)o or exp( 3J+3Bq ) exp( 3J-3Bq ) Nm = N- (1.3-7) exp(3J+3BQ)+expC3J-3BQ)+2e-0J As Bq -> 0,' m 0, and, there exists no spontaneous magnetization. 26 This result also holds for Heisenberg one dimensional model. Therefore the one-dimensional nearest neighbour Ising model does not exhibit ferromagnetism, although OJO,A. (1973) and others have shown that there exists ferromagnetism for a oner-dimensional Ising model, with long— range interaction. There is a considerable literature on the calculation of crystalnliine character­ istic values and curie points with the Ising model. Such treatments have the merit of being clear cut and rigorous for the assumed problem. However they have been confined primarily to one or two dimensional rather than three dimensional lattices. Evewn wthere a rigorous calculation with the Ising model is possible for the actual lattice pattern, the results should not be identified too closely with the actual magnetic behaviour of the material simply because of the inadequacy and arbitrariness of the model, This model of Ising and Lenz is too crude however to eluci­ date the low temperature thermodynamic properties of ordered magnetic systems, Fortunately another method, particularly adapted to the low temperature region was developed by Bloch (1930), 27 1.4 The Spin waves and Spin complexes The starting point of Bloch's attack is Slater's observations that the characteristic values of the Heisenberg exchange coupling can be rigorously determined if the spins of all but one atom are parallel. The solutions can be interpreted as representing waves of disturbance in which the reversed spin is propagated through the crystal with various possible wavelengths, The fundamental hypothesis of the Bloch calculation is that if there are k reversed spins, the solution can be obtained by additive superposition of k solutions in which a single spin is reversed. This will be an allowable approximation only Ai is small compared with N, so that the probability of two or more reversed spins being at the same point of the lattice is negligible. Hence the Bloch method of calculation with the Heisenberg model is only satisfactory in the immediate neighbourhood of complete saturation (M <=N y) and consequently only at temperatures near absolute zero. This theory of Bloch explicitly assumes that the density of reversed spins is so small that the effects of obstruction and interaction between two or more Spin waves can be neglected. This is an approximation that 28 will certainly be good at sufficiently low temperatures, less good at higher temperatures. At higher temperatures, BETHE (1931) made a thorough study of the effects of Spin wave interactions in a one­ dimensional chain of Spins, He showed that in addition to the elementary Bloch spin waves, there 1st excitations in which a block of two or more reversed spins travel together through the chain to form a BOUND STATE called a SPIN COMPLEX. On the average, the energy of such a state is less than the sum of the energies of free spin waves, 1.5 Stoner’s method The Heisenberg theory which have discussed is based on the Heitier-London model. This model represents a non-polar approximation. It supposes that the electrons responsible for ferromagnetism always remain on the same atom acndA do not participate in electrical conduction.This is an idealization never completely realized in fact, Anoth er limiting case is furnished by the model of ommerfeld. In this model of itinerant electrons, one supposes that the 3d electrons circulate independently and freely from one atom to another. The resulting momentary shortage or surplus 29 of charge on any particular atom makes the crystal instantaneously polar. Theoretical calculations from this point of view have been made by Bloch, by Slater and especially by Stoner, Evidently, the truth is some­ where between the Heiter-London model and that of itinerant electrons. The calculations of Stoner are based on a well defined, clear cut model, rt is assumed that the electron energy levels are attributes of the whole crystal rather than the individual atom, and can be handled by the Fermi-Dirac Statistics. A molecular fjiQelrd is used to represent the exchange interaction, The Stoner procedure can hence be characterized as the superposition of the Weiss molecular field on the Sommerfeld theory of electronic conduction, and depicts what is called collective electron ferro­ magnetism , < F One is apt to wonder whether from the agreement with experiment or other considerations, one can deduce whether the non polar Heisenberg model or the polar Stoner one comes closest to reality, ft is impossible to say anything very definite on this subject. 30 The actual intermediate case which is between the Heitler-London model and that of itinerant electrons is what we have attempted to explain in our work. Ins tead of considering the individual spins of electro ns, a 11 the spins should be taken as a collective because each spin interacts with several other spins simultaneously, thereby causing some fluctuations in the spin^vvaallues, 1.6 The motivation of our investigation A new attack on the problem of Spin wave interactions was opened by HOLSTEIN AND PRIMAKOFF (1940). They con­ sidered the behaviour & a * hree dimensional ferromagnetic array of spins in an external magnetic field. They succeeded in defining a set of coordinates which describe accurately the quantum state of the system. In terms of these coordinates, the Hamiltonian of the system splits into two parts, one quadratic in the amplitudes and one of higher order. The quadratic part alone would give a theory of non-interacting spin waves, Identical with the linear approximation of Bloch, DYSON (4956) invented a general theory of spin wave interactions. In his theory, he defines two kinds of interactions. One is the kinematical interaction which arises from the fact that more than 2S+1 31 units of reversed spin (S is the magnitude of atomic spin in units of h) cannot be attached to the same atom. The other is the dynamical interaction which represents the non-diagonal part of the Hamiltonian in his basic set of states. Dyson critizes Holstein and Primakoff spin wave theory saying that although the kinematical inter- action does not appear, the dynamical in.tteerac tion is so strong in their treatment that one cannot get rid of mathematical difficulties, Prior to Dyson's paper, sevCerral authors obtained correction terms in the expression of the spontaneous magnetization of ferromagnetism at low temperatures. Among these authors, SCHAFROTH (1954) and HEBERG (1954) followed the ideas of Holstein and Primakoff, they were neither in agreement with each other nor with Dyson. OGUCHI (1959) has shown that the origin of their incorrect results are not in HolsteinT-and-Primakoff's method itself, but in their poor approximations. Oguchi in his work, has shown that a careful treatment of Holstein and Primakoff's method gives the same results Cto first order in 1/S) as Dyson's, Oguchi's treatment which is an expansion in 1/S is suitable for large S, We in our work study the inter- 32 actions of spin waves in an Heisenberg model of a ferro- magnet using Holstein and Primakoff' s method. Our method as we shall show is very suitable for small 21 <.__ _S _<_ „2. In chapter two, we outline the theory of spin waves and spin complexes, with emphasis on wave~wave interactions#Spin waves being essentialolyO spin fluctuations, we examine the Temperature Limitation on the spin waves. The method of Holstein and Primakoff is outlined. In chapter three, we introduce the spin wave-spin wave interactions. We realize magnons as ideal Bosons, with an effective electrochemical potential , We calculate the modification by p of the Bloch coefficient of T2/2 in the expressions for the spontaneous magnetization and of specific heat, in the cases of cubic ferromagnetic metals. In chapter four, we apply our concept of effective chemical potential to the Hexagonal close packed ferro- magnets and calculate the influence thereof. In Chapter five, we calculate the additional effects due to the dynamical and kinematical interactions, And chapter six gives the discussion, summary and conclusion of our investigations, 33 As usual in many-body problems, our investigations use the language of second-quantization in quantum mechanics and of quantum statistical mechanics. Let us give an outline of the language. 1.7 The Secon.d... .....Q....u....a....n....t....i....z....a...t....ion Method In a quantum-mechanical investigation of the system consisting of a large number of ider .tictf particles which interact weakly in an arbitrary manner we often use the the second quantization mfeitethhoo a. - his method is particularly useful in a system where the number of particles is a variable quantity, If we consider a sy stem consisting of N identical non-interacting part icles for example, the free electrons in a metal, or the phonons in a crystal, the Schrodinger equation for a stationary state in this system may be writt:en il* l[l ̂=m& i + V(ri)U(r1 ,r2 , ,, . ,rN ) = E Yj[ > zi) E denotes the binding energy of the whole system in a given stationary state. The solution to the above equation is usually taken in the form ip — ipq^(r^ )<^Q2^r2^ ' * * ’ f ... (1.7-3) where labels a set of quantum numbers characterizing a given stationary state. Every q^ represents a full set of quantum numbers which describe the state as occupied by an individual particle. The functions î q̂ are the solutions to the Schrodinger equation for one particle, 1 a A i + VI ri)ltqi(.ri) = Eq^q^r., ) ... d.7-4) However,/the wave function (1.7-3) does not satisfy the symmetry requirement. In general, it is neither symmetrical nor antisymmetrical with respect to the exchange of the coordinates of any two particles. Consider for instance, a system of two identical particles. Clearly the possible wave functions are given by a combination which is either 35 symmetric or antisymmetric *s,a = 72* W W 4 ' M W V 1 (1.7-5) where s and a correspond to + and respectively. Each wave function is normalized to unity. The above result can be generalized system having an arbitrarily large number of identical particles, N. In this case we require for the total wave function to be either symmetrical or antisymmetrical with respect to the permutation of the coordinates belonging to any pair of particles. In the former case we say that* the particles cbey Bose-Einstein statistics, whereas in the latter case we say that the particles obey Fermi-Dirac statistics. The former particles are called Bosons which according to Pauli's investigations have an integer spin in units h Ce.g, photons, phonons, ir-mesons). The latter particles are called fermions which according to the same Pauli's investigation have a half-integer spin in units h (e.g. electrons, nucleons, deuterons), 'To expose the problem of second quantization in a complete form we start from the assumption that the system of N non-interacting particles, say bosons, is subjected to an external field, Then every boson occupies one state 36 which belongs to the set of states whose energies are Eq , E^, Eg... Let us denote the corresponding wave functions with Y q0 o 1 , ip(r } , q2 (r ) clearly the energy of such a general state is given by the matrix element of the single particle operator H^Cr), V E.1 = q ^ ^ ) H1 (r1 ) î qiCr1)dr1 ..,(1.7-6] whereas the interaction between the particles is given by the matrix element of the tjwCo-particle operator ^22^rl ,r2^ V.U . - & dq̂ (r) and ip $q^(r) by a set of annihilation and creation operators as follows: r) -v Nx2^qi(r)aj., Cl.7-9) + N 2^*qi(r)a^, where the operator a^ takes (annihilates) a boson from the state E^ to another state, and the operator a t brings (creates) to the state E^ a boson from another state. 38 The above operators satisfy the following commutation relations U i ’ a-5+I = 6ij 10) [ ai? aj] *= [ a*, at] = 0 the second relation embodies the fact tha t any t wo bosons may be interchanged without changing the sign of the wave functions. To find a physical meaning for the first commutation relation in equation Cl.7-10), we introduce the state vectors which characterize the given state completely. The state vectors are designated as I "o' nlVn2’ * * *> . (1.7-11) to indicate that there are n.. identical bosons in a given state of energy E^, The matrix elements for the creation and annihilation operators are most conveniently defined by aiInD > ni ,...,n^,..,> — (n^+l)“ |nQ ,n^,...,n^+l ...> x a j | nQ , n^, .,,, n.., ,.. > — ( ^ 1 ’ * * ** f"l • • •> ... (1,7-12) We may observe that the occupation number operator 39 Ni = atai (1.7-13) has the following matrix element Ni I no ' nl ? ‘ ’ ni ’ ' * ̂ ~ nt I n0 A• • i (1*7—14) Now the first commutation relation has a set of e quivalent operational equations in the form atNi - 4 - atNiai - w 113a? - - N ^ N . - D C ^ - S ) .,,(1.7-15) aLti kaaji = a+i Ck-l)N 1 1 (k-1) = Nl. (Ni. -1).,.(Ni.-k+1) Therefore the matrrikx elements of the above operational identities become . . . |i +k k | | . • . > (1.7-16) Clearly the operator a+^ka_?k is a product of some operator and its hermitian conjugate fTherefore n^(n^-l)... (n^-k+l) cannot be negative. 40 Using equation [ Cl.7-6 - Cl.7-10)] , we may write the total Hamiltonian in the form (see NOVAKOVIC (1975)) H = ,EE-.ia.iai. + 2(N-1) £•(' V.l j. +W„I j.') a+l a+a.> J j j a.l C/r. (1.7-17) v/here E . , V. . and W. . denote the matrix elements of the single particle and1 J two particle oper*atvorys respectively. The summations are taken over all allowed states, so if N is sufficiently large then N/(N-1) may be replaced by 1. If we put = n^, Ei = hwi and a.+ +i a.J a.a3.i = ni. n1.,’ Vi.j .+Wi.J . = U. r., we have H = iE hw.1n .J- + i ̂ ij iJ n.1 n.J ... (1.7-18) The first term on the right gives the Hamiltonian of an ideal gas of quantum harmonic oscillators. The second term depicts the mutual interactions among the oscillators. Hamiltonians of this type are very important in many-body systems, And in our investigations, as we shall see, such a Hamiltonian is the starting point in spin-wave theory. CHAPTER II SPIN WAVES AND SPIN COMPLEXES In this chapter, we shall consider a collection of mutually interacting spin waves, and the existence of spin complexes, 2,1 Theory of Spin waves fj \ One approach to the low-temperature thermo­ dynamic behaviour of ferromagnets is provided by BLOCH (1930) theory of spin waves. In the ground state of the exchange Hamiltonian (1.2-1) which will be realized at zero degrees, each spin has the maximum allowable value of , namely S, A spin wave may be described as a sinusoidal distur­ bance of the spin system within simple cases, the amplitude at each magnetic ion site proportional to S-S^ . Let us consider a linear chain, S = ^ with periodic boundary conditions, The Hamiltonian will be written as H = - ^gPBBQEa^ - + + qfcTj)CSee Fluggae (1936)) (see Fluggae1 (1966) . (_2%1-J.) where H has been expressed by means of the raising and lowering operators + and of S^z, and for S = 1 use has been made of the Pauli matrices a x ’, c y and az where a + = a x + ia y a = aa - io y , —S i = a~a—.x . and 42 o - a X y = This Hamiltonian has the ground state eigenfunctions for N spins, *o 0ll0120l3 N ^ (2-1 where a is Spin-up state and 3 is spiinn?-down state. This is the state at zero decrees Kelvin, It represents the maximum alignment of spins or complete magnetization. As tempera­ ture increases the system will be excited out of the ground state. The next state may be thought of as one in which the th s—pin is. r.ev.ersed-. 4? , Jc' ^i aia2* ’ ’ ‘ ai-leiai+l* ' ’ ’ aN (2.1-3) However, this <|k is not an eigenfunction of (2.1-1). An eigenfunction may be formed from a linear combination of ̂, each member of the combination containing a reversal at a different magnetic ion site ♦ k s f t h (2.1-4) In a demonstration basic to Bloch’s theory, SLATER showed that each such combination is equivalent to a wave like disturbance of wave number k, and that the allowed values of k can be determined from periodic boundary conditions. 43 The wave-like properties of the solutions (2.1-4) for the case of a linear chain are demonstrated as follows. Periodic boundary conditions are equivalent to bending the chain around in a ring so that the first spin is also the N+l Spin. Each spin i has two nearest neighbours 1+1 and i-1 with which it interacts through the exchhange integral J. We then set out to evaluat = <(i>j iE I . .. (2.1-4) This yields [ E+|(N-2)gyBBo + KN-4)J]Cj + J(Cj+1+Cj^a ) = 0 ...(2.1-5) We may assume • solutions of the form c i < i ^ ^ elkja> . ••• (2.1-6) where a is the distance between spins. The periodic boundary conditions require Cjk to equal Ckj+^, anc* hence the allowed values of k are given by kaN/2w p 0,1,2,,.,,N-l . ( 2 , 1 - 7 ) The wave like properties of the solutions (2,1-3) is shown by (2.1-6), When the latter is inserted into (2.1-5) there results an equation relating the energy E and the wave number k. 44 E + |(N-2)gybD B o + |(N-4)J+2JCoska = 0 (2 .1-8 ) which may be written as the following dispersion relation E - E = e.k = hut,k gPrtj> B o + 2J(l-Coska. ) ... (2.1-9) where E = -!NgPBB0 - 4NJ . . . (2 .1-10) is the ground state energy. The energy is that required to excite the Spin wave In three dimensions, the disperson relation according to BLOCH (1930) becomes ek = gpBBo + . . . ( 2 .1- 1 1 ) where - n expC^ik.rn ) (2.1-12) Here -r n denotes the vectors to the z nearest neighbours. We can with the above say that the dispersion law has been derived for a single spin wave in an otherwise perfectly aligned system of spin vectors. The question arises: Can a disturbance in a non-perfect system be pictured as a spin wave? Bloch argued that this is reasonable at low temperatures and that the eigenstate of the ferromagnet should be very 45 nearly a linear superposition of non-interacting spin waves of the form {2.1-4), Bloch's point was that the presence of one spin wave cannot seriously modify the derivation of the Dispersion law for a second spin wave, and similarly on up to n spin waves, provided n is much smaller than the number N Bloch's conjecture, if correct spin wave theory, and the thermodynamic properties oi a ferromagnet can then ties of superposed sp The first study of the validity of BLOCH's super­ position conjecture was made by BETHE (1931) and limited to the linear chain with S = 02° + CTk2 . From this is inferred that spin waves (magnons) are well defined quasi-particles if ir/2a >> k >> kACT) ~ Cz-1)T^/1 equivalent if T << Tq and Tq ~ T^/(z-l}^. We proceed to outline this work. The Heisenberg ferromagnet with the following Hamiltonian is considered HI 45 ’-■JJJ.2L1 E S . ,S , Z.S.S. , ... (2,3-1) where {Sni are A J-l n J n the z nearest-neighbour spins to spin j and J is the binary bond. The dispersion relations of the spin waves are known to be IT = 'TT' l-exp(ik.a)! = ,,, (2.3-2) where |a| = a, the lattice constant. The same dispersion relations are obtained if one uses the 49 quantum dynamics ihdS . d—t " |sd'HI - [sj ' V ’ ,1, (2,3-3)' where H. - -JjS.,Sn , ... (2.3-4) and expand SjCt) = s^0) + T (t) . V ... (2.3-5) Under the approximation = S°3x? .= 0, S°z = S, for all m, and that T\ :< v + lTjy’ Tj ~ exPCtk.Ej) where R.J is, the posi.tti >oi nSpf spin S.M J, one obtains = i(ij^)P(k)T = iG(k)T (2.3-6) where the form factor is given by . O - r F(k) = t[ 1-expCik, a)I... (2.3-7) Inste^ad of consider’ ing one spin, all the spins were taken as a collective, and use was therefore made of the Bohm-Pines collective coordinate method in PINES and BOKM ([1952), AKIN 0J0 argued that the collective coordinate method is applicable because each spin interacts with 50 several other spins simultaneously adding that the inter­ actions are not of the binary collision type. At tempe­ rature T>0, the position R , of S . may differ from 3 J constant RJ°, The spin density at x is defined by N D ( x ) - j l a V C x - R j ) , R j = y t ) (2.3-8) and <$d ( x ) = jr r .Js C x - R . ) is the spin deviation density • Then pCk) = D(x)expC-ik.x)dx = £ S . exp(C--iikk.R ) f., (2.3-9) To first order pCk) f= p°(k) + xCk). with S J. = S?J + T.J, p o(yk)F £S exp(-ik.R ), .,. (2.3-10) £T^.exp(-ik,Rj), where Tj " Tjx + iTjy’ " TxCk* + iTy(1° and define the expectation value of -r(k) by 51 L(k) = = E exp(-ik,R,) (2.3-11) J J 3 using equation (2,3-6) twice in the form a r Y = iG(k) we have 2 = -G2(k)L + E (-ik*H , )2exp(-ik,E .) dt j J J A^ * + S-ik,[ 2iG(k)Rv W exp(-ik.Rj) (2.3-12)j The second summation is zei 2iG(k)R.+R. = 0, for all. 8? dR. 3 J ' ̂jJ, ' R. .1 = -Ad-t That is, A Rj(t) = Rj+rj exp(-iPt),fi = 0 or 2G = 2wQ ..,(2.3-13) This shows ^ fluctuation of the positions of the spins. We may use the Fermi-Dirac distribution at temperature T to average the velocities to obtain ( k R j of k2 = k26/M , .,(2.3-14) where 0 = kgT, kg is the Boltzmann constant, and M is the mass of the collection of electrons that give rise to each spin S,J in which case if |S.J | = S, then M = 2SM © , 52 M ® is the mass of an electron, and R.J is to be taken as the centre of charge or mass of such a collection. Consequently, equation (2.3-12) becomes d L^ = [-G2(k) - k20/M[L(k) , . (2.3-15) dt2 w hich g iv e s th e d is p e rs io n r e la t io n to2 = G2+k2(e/M) = to2 + (6/M)k2 , ( 2 . 3 - 1 6 ) It is well known that in the standard method, the tempe­ rature effects come in only when one determines the thermal average number of magnons , usvlk ng' the Bose-Einstein distri­ bution at temperature T. But now, equation (2.3-13) shows that at 9>o the positions of the spins vibrate and equation (2,3-16) demonstrates that w depends on 0, Equation (2,3-16) also enables one to estimate the value of k and of the temperature for which the quasi particles (to~k 2) called magnons are well defined, Obviously, > k >> h ( e / M ) a/2JSa‘ (2.3-18) and .. consequently, k1„3 T = e << (2JSa/h)2M - C2JSa/h )2/2S?vIt' A (2.3-19) are the conditions necessary to have magnons. These equivalent conditions are more concretely stated with the use of the RUSHBROOKE and WOOD (1958) expression k T . = ^(z-D(Hx-i), X = s(s+i; (2.3-20) for the cubic crystals, with kB = 1.38xl0-23, Mg = 9.1x10 tg, h = 1.0546 x 10"34JS_1, a = 2.5xl0~10 m and S((S+1)2) < | We obtain the inequalities 6xlOS > > k > > k A and T<>Tq , the second term in equation (2.3-16) dominates and then w ~ k, so that the spin waves are indeed waves. But then, the approximation S. = S^J +T.J is expected to be invalid, in which case the second order terms which feature wave-wave interactions must be included. We can say that at T > > T q , the wave-wave inter- actions becomes dominant and hence must not be neglected A way of accommodating them is by the method of Holstein and Primakoff, which we now ou1 S f 2 *4 The Method of Holstein and Primakoff HOLSTEIN and PRIMAKOFF (1940) worked out a spin wave theory which includes the important dipolar and pseudo dipolar interactions. The general program is to treat spin waves as quantized particles subject to creation and annihilation operators. To this end, spin deviation operators are introduced, N z = S-S„z ... (2.4-1) and the eigenstates of the Hamiltonian are expanded in eigenstate of these operators where 56 N£$n1 ' * ' - 'nN ~ V n ^ ' * ’nN , (2.4-2) Here N i}s the spin deviation of the atom, . and n? is the eigenvalue. The spin-raising and lowering operators have the following well known properties ■Sr ;+^ n = [ (s-s£zz )cs+s£zz +i)l,3 $n & ... (2.4-3) £ 9s V n = [ ( s+s« ) ( s - s * +1» 2W 5 ... (2.4-4) £ £ It is to be noted that the spin-raising operator lowers the number of spin deviations, and vice versa. Creation and Annihilation operators, working directly on the spin deviations are now introduced. V n t " \ + l ... (2.4-5) Vn, = (V * * n -1 (2.4-6) These operators have the property a a = hZ (2,4-7)Z Z and satisfy the commutation relations 57 (2,4-8) Prom the above equations the are easily deduced. (2,4-9) where 1 2 h - [ 1 - < < V 2S>1 (2.4-10) These equations, when inserted into a spin Hamiltonian, constitute the Holstein-Primakoff transformation. If S~ is applied repeatedly until the state n^ = 2S is reached, f becomes zero and further spin lowering has no effect. Therefore, although strictly speaking the Bose operators Ja. L arve defined by the above equations in a space of infinite dimensions (n allowed to run from 0 to «*), the presence of f^ in a spin Hamiltonian operator will ensure that n^ stops at 2S as it must in a real ferromagnet. Ano vay of deriving the Holstein-Primakoff transformation is to assume from the start the form (2.4-9). The require­ ment that the spin operators obey their usual commutation rules then yields the commutation relations (2.4-8) for 58 the a0 indicating that a correct transformation to creation and annihilation operators of Bosons has been achieved. The presence of the square root operator f^ A this H a m i l t o n i a n leads to many of the mathema- matical difficulties of spin wave theory. is claimed by Holstein and Primakoff that at low temperatures (i) [ 1-C/2S)I “ = 1 s T .,. (2,4-11) where is the average over a statistical distribution of the ferromagnetic eigenstates ipE> of the expectation value of n^ in these states. A further approximation made by Holstein and Primakoff is the neglect of terms of the form. Cii)y a„a„ am >am s N„Z Nm ... (2.4-12)Z Z .<>-• Ciii) NJlam ,,, (2.4-13) These terms are smaller than the retained terms in the expression for the Hamiltonian operator. Spin waves are introduced by the following Fourier expansions in terms of wave vectors k within a Brillouin zone of the reciprocal lattice 59 a £ = N "2 E exp(jF i k . ) a ^ , afcak = nk ; a £ = a^ , (2 .4 -14) k ' where the permissible values of k are determined by periodic boundary conditions. With use of the commutation relations (2.4 -8), it is easy to verify that I ak • ak« wk,k’ (2.4-15) [ ak ^ 0 and hence the spin wave creation and annihilation operators ak*fc are also Bose operators and the transformation (2.4-14) is canonical . The eigenvalues of operators nk will be the numbers of spin waves nk . This Holstein-Primakoff formalism forms the basis of our further investigation reported in the next chapter. Gr CHAPTER III INTERACTIONS I By using a special expansion formalism we shall show here that spin waves, when quantized are ideal Bosons with an effective chemical potential effected by wave-wave interactions. 3.1 Spin wave - Spin wave Interactions an pansion Formalism The nearest neighbour exchange interaction model of a ferromagnet is described by the following Hamiltonian H - -2J V *j.,E£# S.j ,’ 8„Jt + B oe”pHBB j,-£1- S?j ., , (3.1-1) Here Sj is the spin operator at the j*'*1 atom, N is the total number of atoms, g the Lande-g factor, the Bohr magneton,the summation is taken over all nearest-neighbour pairs, and the external magnetic field Bq is directed along the z-axis. The spin variables in quantum mechanics are operators which obey the following commutation relations [S2 ,SZ] = 0, [SZ ,S+] = hs+, [SZ ,SJ= -hS_, [ S+ , S_1 = 2hSz where S± = Sx ± iSy ,S 2 = S2x + Sy2 + S2z » t • (3 • 1 -1 We take the simultaneous set of normalized eigenvectors of the operators S and S which mutually commute as the 61 basic representation. We denote the corresponding eigen­ values with S and M whereas their common eigenvector we denote with | SM> , Clearly, there are 2S+1 different values for M, -S < M < S and with AM = ±1, Also we have for h = S 2 ]SM> = SCS+l)jSM>, S\SM> = M\SM> ,,, (3.1^2) For a given value of S the eigenvectors |SM> span a (2S+1) dimensional space. These eigenvectors form a set of orthogonal unit vectors, - A(S-S’)ACM-M*) (3.1-3) It is the task of quantum mechanics to evaluate the non­ vanishing matrix elements of the operators S+ and S , Here we quote the final result: S+ | SM> = [ (S-MKS+M+1)] ̂ |SM+1> ... (3.1-4) SjSM> = [ (S+M)(S-M+1)1 * |SM-1> Instead of dealing with the quantum number M, we may introduce a quantum number n such that n = S-M, An = -Am ...(3.1^5) 62 The quantum number n gives a departure of the z component S„ from the fixed values S_ = S. Using (3.1-5) the matrix Z Z elements for the Spin components in units ft = 1 becomes S |S,n> p l C2S+l-n)nI 2 |S,n-l> S_|S,n> = [(2S-n)Cn+l)]21.|S,n+l> ... (3.1-6) Sz |S,n> - (S-n)|S,n> The quantum number n may further be replaced by a set of creation and annihilation operators as follows: n p a a a+ |s , n> - (n+1l)) 2|s ,a+l> (3.1-7) a |S , n> = /n S,n-1> where the operator a+ creates a departure of the z component Sz from the fixed value - S whereas the operator a annihilates a departure of this component from the fixed value -S. The minimum and maximum values for n determine the limiting eigenvectors |S ,n - 0> and |S ;n = 2S>, These eigenvectors are characterized by a Is , n 0> a+ |S ;n = 2S> - 0 (3.1-8) We shall call the state corresponding to the eigenvector |S?n = 0> the vacuum or ground state. 63 Using the introduced operators we may rewrite the matrix elements (3.1-6) in the form, S+ |S ,n> = (2S) A2 (l-e+)A2a |S ,n> S_ |s ,n> = (2S)2a+(l-e_)4 |S ,n> A (3.1-9) S IS ,n> = (S-a+a)|S,n> where _ n-1 n r e+ 2S e = 2S ... (3.1-10) We may present the matrix elements (3,1-6) in an alternative operational form. We have s + |s ,n> = (2S)2(l-a+a|2S)^a|s,n> SjS,n> = (2S)*a+(l-a+a/2S)^|s,n> (3.1-11) S |S,n> = (S-a+a)|s,n> Going back to our Hamiltonian (3.1-1), we express the Spin operators in the Jphns, sj = Sj + iSy = (2S)ilj(s)aj, j ^ r l s j ■ C 2s)S V s ) - ... (3,1-12) Sj ' S" V i ' where fj(s) - (1 2S 1 (3.1-13) and the operator a.3 a.3 = NJ. is called the number 64 operator and a. and a. are to be regarded as the creation and annihilation operator of the Spin deviation, and they satisfy the commutation rule a J. a + A„ - a+ j . a£ „ = ̂ (3,1.-16) Bq is the imposed magnetic field which we set equal to zero J>0 is the binary bond, and we assume that there are N spins in volume V , each with z nearest neighbours. Expansion Formalisms Several authors have used the above stated Hamiltonian and expanded f.3(s) up to some order, and thereby obtained spin waves as quasi particles which obey Bose-Einstein distribution with zero chemical potential. 65 NOVAKOVIC (1975) expanded 1 - a. fjCs) - 4SJ a .1. . ( 3 . 1 - 1 7 ) whilst OGUCHI (1959) went a step further. He expanded a +. a . a +. a ,a+.a . f r s) = l — —_1 _ _2_j— J— si ^ . . . ( 3 . 1 - 1 8 ) J 43 32S2 These expansions are good enough for large values of S. But for small values of S, we have discovered that in order to take into account the complete ^vave-wave ihteraction, Oguchi's expansion is not sufficient,Within the wave wave interaction approximation,neglecting wave-wave-wave and wave-wave^wave- wave interactions,etc,f .(s) must be expanded to all orders, and the two- product terms CaJt aJ. ) in f ,(s)f *(s) collected. Surprisingly the infinite series, in the two product terms, has a compact form. Noting that, Cl-xl® =^4^/2 - x 2 / 8 - X 3 / 1 6 ^ 5 x 4 / 1 2 8 - 7x5/256+ ... ^ ^ l - R ( x ) , C3,1-19) and NpuJtting a.J = a, a £ f b, for convenience, we obtain ̂ -\ _ -a + a a + aa + a a. +. aa + aa +a fj( ) “ -2T2S) ~ -8-(-2-S-)7‘5 ~ --1-6-(-2-S-)'~ .., (3,1-20) 66 We notice that every term in (3.1-20) contains a two product term a + a. Take for instance, a + aa + aa + aa +a, and use the commutation relation [ a,a I = 1, C3.1- 2 1 ) to see that a aa aa aa a = a (1+a a)(l+a a)(l+a a)a = a a+... . ,. C3.1-22) We take only these two product terms in equation (3.1t20), because in . equation (.3,1-16) aJ+ f.j f ~ a * makes such terms become 4-product terms, which are the only ones necessary in a wave-wave interaction approximation. With equation C3.1-22) and others like it, fj(s) becomes 1 + — -— „ + ----— ^ + ,.. )a+a (3.1-23) ^*b) ■ 8(2B) 16(2S)d = 1—R('2g )a a where “ (1- = -A < 0 ... (3,1-24) Thus, the operator fJ1 is approximated by 1+Aa a, and f J f ^ = (l+Aa+a)(l+Ab+b) - l+Aa+a + Ab+b ... (3.1-25) up to the two-product terms, Consequently, 67 H = -J j£e[2Sa^l+»a*aJ+Aa^a|t!alt - 2SaJaJ+a*aJa+a), I .. . (3,1-26) Using the Fourier transformations ak = N 2Eexp(-ik )a^; aT = N~2rexp(ik )â ,,, (3,1-27) where k denotes a reciprocal lattice vector and N is the total number of ferromagnetic spins, i Fourier- transformed Hamiltonian, we have expressions like v J(k) = EJ(j-£)exp[ ik(j-?,)l f= J(k)* = J(-k) ,,, (3,1-28) Such expressions are calculated in Appendix B for crystal lattices having a cubic symmetry and Hexagonal symmetry. As k -*■ o the expansion is given by J(k) - J{1—(ka)2/z + A ( Q K k a ) 4-BC«J.,0)(ka)6, ,, ,} ,.,(3,1-29) for the crystal lattice having a cubic symmetry; ,,, z denotes the number of nearest neighbours, a is the lattice constant, and J denotes an effective exchange integral. We can now transform every term appearing in the above Hamiltonian with the help of the fourier transform (3.1-27) EJCj-O *= zNJ ,.. (3,1-30) 2SEJ( j-£)a.a^ = 2SN 4EJ(j-Jl)a^aj_,expi(k.j-k' .&) - 2SN-1i:zJ(k)a^ak+pexp(-ip,«,) = 2sEzJ(k)akak ,,, (3,1,-31) 68 where we have used the substitution k* = k+p and N if p = 0 Eexp(-ip.£) = NA(p) = ( 3 . 1 - 3 2 ) z 0 if p f 0 This follows from the translational property for the reciprocal lattice vectors, Also the following relationship holds EJCj-Oa^aj = N'~1i:J(j-Ji)a^ak ,expl i(k-k') ,Rl = £zJa£ak ,,, (3,1-33) We also transform the remaining three terms in (.3.1,26) which depend on four operatorrj,s<.> The first of such terms is equal to Z J(j-£)a*a .a*a = J(j-^)a^ak,â Mak,(, Cexpi(k. j-k', j all j,l 3 J * all j, all k kM , j^k’*’' ,*)) Here we introduce the substitution k" = k ’+p’, k"’ = k+p, which leads the above exponential factor to exp[ i(k-k’) (j-fc)Ix exp[ i(p-p'). «,I . and the above term becomes = N 2 Z J(k-k’)al*rat. ',ka'* + p,,aa,k +p x exp[ i(p-p'). fclk,k',p,p' 69 The sum over £ may be performed giving the factor NA Cp-p'), therefore the above term becomes , O - T = N k j , _ pz'( W , | > A ' V tp ' V P There are two important contributions coming from the above term one with p = 0, the other with k = k r. The former contribution is equal to v = NT -1* E zj(k-k')a+k a+^,akak , 5i cN , E zJCk-k^a.a, k,k k,k» k k ... (3.1-34) Thhe latter contribution is equal to - N~^ E zJa a. a, , + N-1 E zJa,+a, , A(p) (3,1-35) k,P v y k k+p k,P k k+p By neglecting the second sum in equations (3,1-34) and (3,1-35) we finally obtain E J Cj-£ Ja^a^a^a^ = N_1 E z[ J+JCk-k*)Ja^a^,akakf ... (3.1-36) j , £ k , k ' Using a similar set of transformations we arrive at the result ■ J *i ~ j«-*)aK aA —_ *NT -1 E z[ J(k)+J(k')Jak+ ak+ ,akak , (3.1-37) k , k { 70 Now we can write the Hamiltonian in the transformed form as follows H = EQ+2szZ:(J-J(k))a^ak + N_1 E 4 J(k)+J(k' )-W(k-k')] +z(«-l)[ J(k)+J(k')] + + k,k< ' ^ V,k ' Y V •«• C3.1-38) Here Eq stands for the energy of the ground state EQ = -zNS J If we put k ! = q and we write the above Hamiltonian as H-E - EA. n, + N"1 E V, n, n + N^1 E W,._n,.n, C3.1.-39 0 k k k k,q k-o, the dispersion law for Magnons is A(k) = 2SJa2k2 « liw = Fk2 ,, (3,1-40) The second and third terms of expression (3,1-39), B and C, respectively describe an interaction wh. .i ch. is often called the magnon-magnon interaction or Spin wave Spin wave inter­ action, ,.2 2 With the expansion y(k) = 1 - k a to order k 2 a 2, in equation (3.1-39), the second term is zero on averaging over an angleCand this is why we shall need to expand Y(k) up to k 4 a 4 ),for the cubic lattice. Writing out the third term of expression (3.1-39) we have C E (ex — 1)[ 2z-k2a2-q2a2] n n (3.1-41) kq K q To this order of expansion of y(k) the effect of the wave- wave interaction is included in C as encapsuled by a(s). Rewriting our Hamiltonian, we obtain H = E{[ 2JSa2+(a—1)^| Ik2+ (gN~ - ^ <[ 2z-q2a2ln„>}n • • t (3.1-42) 72 Or H = Z(^k -y)n 1-43) k K ( a . where Fq - 2JSa 2 +(a-l)Ja 2 = 2JSa 2+FX (a 1-44) ~ 2JSa = F = - (aM- 1N ■ - ) JT< [f 0 ' 2z-q* 2 a2,- -1 n- q> In each case . averaging in a is done over the Bose Einstein distribution f(q) __1 8? (3. 1-45)exp(gFq -gy)-l & 3.2 Chemical Potential in Spin wave Interaction Theory Expression^ ('A. 1-43) which follows from (3,1^42) shows that there exists an effective chemical potential y . For a weakly interacting system, in the second quantization formalism, the Hamiltonian H = kEA ,k n,k + ,k , q k q .TkW, q n, n , . . (3 2-1) Note that the part of that is independent of k, we have called the quantity y, so that h = £ U k-y)nk k = A,k +F- k 2- 2 )k 1 ... (3, And the grand partition function is 73 -3(X -y)n Z = kji nz, e K K = kn i-exp(-3(Xk~y)) ... (3.2-3) which gives the Bose-Einstein distribution = expg(Xk-y)-l Thus the system behaves like an ideal Bose gas of chemical potential y. For this system we proceed to find out the nature of y (analytically and numerically). Rewriting y explicitly we hâ j O y = (1-a)^ <[2z-q a ] n^> ... (3.2-5) According to BOGOLIUBOV and SHIRKOV (1959), we make a transition from a discrete momentum representation to the continuous momentum representation by using the prescription ... (3.2-6) y = ( 1-a ) J[ 2ZV q 2 d, o a2V f q4 .dq ... (3.2-7) 2tt2N o exp3(Fq 2 -y)-l - 2tt 2 N ’I exp: 3(""F c 9- y')-l 74 V f — --- = I q2exp(-g(Pq2-y))l l-exp(-g(Fq2-y)).r^iq 2it“N qj exp$(Fq“-y )-l ' q 2ir N V f q2 ~ e~mB(Fq -P) 2ir N m=l dq « em$p mil 3/2 T Tv0 (3.2-8)Cm6F) ' 4tt NC4i>* < 7T > 1 F l4'4~7T2= 0,022, t b w P = 2JSa‘ we have, , 1 V , V r£ q2dq ^ = mSp N J = tt 3/' 2„r »r e (3.2t 9) q " 2iir2"N £ exp gCFq"-ji)-l m=l m 3/'2 Equation (3.2-7) becomes mey rx 5/2 - e1̂ . mil m 5/2I (3.2-10) y ** 2(l-a)rzJt3/2m|^expCSpm)/m3^2 , . ,(3t2-ll) Obviously y<0 , as expected of Bosons. With | y | = u, and for very large f5, we may approximate exp(-mgy) for some large L, by u = (a-l)2Zr J t3/2 < V > UL _ - - 1 m=^ l^ mL+3/2 75 u = [i2zlls£5]i 2jS t GCalkgT That is, M = -G(a)kBT ... (3,2-12) Equation (3,2-12) gives the approximate solution to equation (3,2-10), p as we can see is the EFFECTIVE ELECTROCHEMICAL POTENTIAL in the Bose Einstein distribution that governs the corresponding quasi-particles, with the dispersion relation hw = Fk 2 = 2JSa 2 k 2 . Thus far, we have been able to show that in a consistent expansion of fj(s) = (1 - *joWo 1 up to wave wave inter­ action only, for Heisenberg ferromagnet of N spins in a physical volume VQ , each of spin S, with z nearest neigh­ bours, bond J>0, the spin waves called magnons at temperature T are Bosons with effective chemical potential y , Instead of the above approximation, using a computer, we have solved numerically equation (3,2-10), Writing w ̂ 3 we have w = 2(a-l)rzx^/^ ^exp(-mw/2Sx)/m^^ ,,, (3.2-13) neglecting smaller terms, 76 Using iterative technique, w has been solved for the values of t between 0 and 0,50 and graphs of w against t is plotted and shown in Figs, ( 3 , 2 - 1 - 3 . 2 - 4 ) , By the method of least-square-error curve f: we obtain the following expressions of w, in terms of x for cubic lattices, for example, the simple cubic, body centred cubic and face centred cubicw, (i) Body-centred cubic (e.g. Fe) with. sv = §, z = 7,5 w = - 1 , 8 8 9 x 1 0 ~ 3 + 0 , 1 6 0 t + 0 , 2 3 t 2 * 0 . 1 6 t + 0 , 2 3 t 2 , , , ( 3 , 2 - 1 4 ) (ii) Body-centred cubic (e.g, Fe) with s = J, z = 8 w = - 1 , 9 7 4 x 1 0 ’_ 3 + 0 , 1 6 8 t + 0 , 2 4 t 2 = 0 . 1 7 t + 0 . 2 4 t 2 ( 3 . 2 - 1 5 ) (iii) Face-centred cubic (e.g, Ni) with s = f, z = 12 w = - 2 , 5 6 7 x 1 0 - 3 + 0 , 2 2 8 t + 0 , 3 0 t 2 = 0 . 2 3 t + 0 , 3 0 t 2 , . . ( 3 . 2 - 1 6 ) (iv) Simple cubicu with s - ,6, z = 6 w = -8.70 1x 10-4+6 . 80x 10'~2t+0 . 12t2 = 0 , 0 6 8 t + 0 .12t2 , , , ( 3 . 2 - 1 7 ) (v) Simple cubiicc, with s = .3, z = 6 w « - 5 , 6 7 4 x 1 0 - 4 + 4 , 2 8 x 1 0 " 2 t + Q . 0 8 6 t 2 = 0 , 0 4 3 t + 0 , 0 8 6 t 2 , , , ( 3 , 2 - 1 8 ) (vi) Simple cubic, with s = ,9, z = 6 w - - 4 , 8 9 4 x 1 0 _ 4 + 3 , 6 4 x 1 0 " 2 t + 7 , 6 4 x 1 0 ” 2 t 2 0 . 0 3 6 t + 0 , 0 7 6 t 2 (vii) Simple cubic, with s = 1,5, z = 6 ... (3.2-19) w = -2.708x10 4+l.95x10 2 t + 4 . 5 9 x 1 0 " 2 t 2 = 0 . 019t+0 . 046t2 t « f ( 3 . 2 - 2 0 ) 77 Fie. 3.2,1: Graph of Chemical ^otential/Exchange integral W against reduced temperature. T 0 0.4 REDUCED TEMPERATURE 78 Fig, 3,2.2' Graph of Chemical Dotential/Exchange integral W against reduced temperature REDUCED TEMPEJWTURE 79 Fig. 3.2.3: Graph of Chemical Potential/Exchange integral W against reduced temperature. REDUCED TEMPERATURE 80 Fig. 3.2.4: Graph of Chemical Potential/Exchange integral VT against reduced temperature. 5-1/2:2-11 0.2 0.4 81 Fig, 3,2,5; Graph of Chemical Potential/Exchange integral W against reduced temperature. 5=1/2:2=12 REDUCED TEMPERATURE 82 (viii) Simple cubic, with s = 1,2, z = 6 w = - 3 . 4 8 9 x 10~4+2. 54x 1 0" 2t +5. 72 x 10 2 t = 0 , 0 25t +0. 057T 2 ,..(3,2-21) We see that in each case, w may be written as W = W^T + WgT 2 so that G(a) from equation (3,2-12) becomes G(a) = 3 |p[ = w^/2S, 3,3 The Spontaneous Magnetization and Specific Heat Having found the nature of the effective electro­ chemical potential y, we ar&jinterested in finding out the effect of y on the coefficients of t in the expression for the spontaneous magnetization as well as on specific heat at low temperatures. The spontaneous magnetization is defined by a thermal average of the z component of the magnetic moment and summed over a unit volume of the crystal, M(T) p H(0)[1 r I ,,,(3,3-1) M(T) - M(o) = <5M(T) „ V 2 NS dk ... (3,g,-2)(2tt)3NS exp gCFk -y )-l 83 The third order approximation of F leads to F = 2SJ[ (ka)2-zA(<}>,0) (ka)4+zB(,0) and B((j>,0) are evaluated in the Appendix C putting equation (3,3-3) in (3,3-2), we obtain _ y dfi k2dk k NS " C 2 „)2NS o expf — -zA(,0 )-— ■-■- +zB(,e) ] -1 (3,3-r5) Expression (3,3-5) is a very difficult integration, so we apply the technique of NOVAKOVIC (1975) by introducing the following substitutions. S = C[l-Dk2 + Ek4] k2 C = — D = zA(4>,0)a* (3.3-6) E = z B C , 0 )a" According to NOVAKOVIC (1975), we obtain 84 Ck = S(l+D(-gS +. ^DS )-E(S^2 +. ~2D3S~2 +, —D2S2-.) +.„D 2 C,—Sr2r +, 2DS2 . D2S2c 0 c 0 ’c“r5 T — crT )1 (3.3-7) k2 = ±[S2+zA(<(>;e) S2+(2z2A20 M l - z B U , e ) 2S3+, , .] <4. , (3.3-8) 1 k2dk = --- g y 2 t 1+ |[zA(, 6 ) tS + (8 2A2(,6)- ẑB(<|>,e 2CT (3,3-9j Equation (3,3-5) becomes r 2 TT 3iv.q J dk d()) J Sine do k' (3,3-10) k NS (2tt)"NS exp(S)o o Substituting (3,3-9) in (3° tSn>d writing 2tt tt d A(4>,G)Sin0 de o 2 it 7 rd Jf A 0C ^ 0)Sin0 de (3.3-11) o 2 TT IT L3 - J d$ | B(<}>, 0 )Sin0 de o o we ob 5̂ " V V ^ - 3 / 2 ^ ^ -my NS 2tt r(w) + T-zL.rfe 5). e ( 2tt ) °NS m 3/2 4 ~ * V V2' m 5/'2 + (4z L2 - 47 zL3)T(72 T 2+.♦.] « t ♦ (3.3-12)m ' 85 where r is the gamrr.a function. The first three terms of expression (3,3-12) gives the. coefficients of T3/2 , T5^2 and T7^2 in the expression of <5M(t ) - p rp3/2 p J. p rp7/2 _ „4 “ ir^T ■ 1 C2T + C3T + C4T In evaluating (3,3t-12), we make use of y = ( = w^y2S which has been evaluated in section (3,2), The values of J are also needed for the various cubic structures, J is obtained from RUSHBROOKE and WOOD (1958) who arrived at the expression, e„ = = i96((zz--il)u(nHsS(S+l)-l) , , (3.3-13)/ where k„B is the Boltzman's constant,' T c is the transition temperature, z is the number of the nearest neighbours, s is the spin va lue5s and J is the exchange integral. Writing kgT 2JS 2 ST (3.3-14) With the substitution of (3,3-14) in (3,3-12), the various coefficients of T are obtained, and the effect of the electrochemical potential included. These have heen done, and the results are tabulated in table 4.2, Note here that the coefficients so computed are the lower bounds, because of the presence of spin fluctuations discussed in chapter two. 86 We know that, SjCt) = Sj + T Ctl where i(k .-wt) T . (t) = ia,e J S ■ Sj ’ a .3-15) where = N ■ k£E<(T^,k); E(T,k) = (Fk ■-y) and the interAnalr energy U(.y) is given by (F k 2- y ) k2 dk .3-15a) 2tt N > expg(Fk -y)-l a V \ 'yk ak expg(Fk2-y )-rl 2tt2M ) expg(Fk~-yi)-l mgy ^F|r(gF)"5/2j: - ^yr(gF) 3/2 E ̂ m m 3/2 87 _ 2JS 3r -5/2 5/2 T>r r„ -3/2, ,^3/2 „ " ^~ ~Q~ 2 ^ 2 j b ) E “m5 7 2 ~ ~ ~cTU T S ) ( k T ) E '̂ 5 / 2 m ' ... (3•3-i6) -hnBy CVCT) = “ f h j S r 5/*k5/2 T̂ 2 - g 3(2JS,3/V/V /2 I ^ -Hngy -Hngy = (2JS)'a/2k5/ V / 2| U ( y <0) * " ’ ~ “ " 1 The internal energy per spin is U (Fk 2 -y)k2* dk N C-l ... (3.3-17)2tt2N The number of magnons is 88 n V ( k ak 2n NJ C-l ,,, (3.3-18)2no where C = exp(3(Fk -y)) and 9 C on dC t-.2 97 = ~*C ‘> W = (Fk ^ )C 9n rV f ( F k 2 - y ) e k 2 dk 9 3 . . . ( 3 . 3 - la )2tt2N Jq (C-l)‘ 9(U/N) = _-V _ k2 dk _ V _ B 3y '2tt2N o C' 1 2,2 = -n-B 9n9 B t » t (3,3—20) Let 9 B + n = a ( B ) or 9n , 1 & a (B) , n 38 ‘ — ■ n("> = 0 ,,, (3.3-21) which has solution B n(B)-n(~) = a(B1)dB' (3.3.22 For large B, but finite, B n(B)-n(B-<$) = a(B')«B' ,,, (3.3-23) B-6 with o<6<<3, 89 By (3.3.19), we have 0 > n( 3 )-n( B-<5) -*■ a( 3)< 0 That is Therefore, U(v = 0) > U(y < 0). This implies that the existence of wave-wave interaction and hence of non-zero n, gives rise to a lowering of the internal energy. In other words, the spin waves mutually interact in such a way ( * T on the average they > -jWm bc>M.̂ c| states called spin complexes, C r CHAPTER XV THE HEXAGONAL CLOSE PACKED FERROMAGNETS In this chapter, we apply the formalism used in Chapter III to the hexagonal ferromagnets, 4.1 The Hexagonal close packed ferromagnets The ferromagnetic elements that have U ^ s c o v e r e d so far apart from the alloys includes Iron, Nickel, Cobalt, Dysprosium and Gadolinium, The first two have body centred and face centred cubic lattice structures respectively while the remaining three have Hexagonal close packed'S v lattice structures, In this section, we want to find the dependence of the coefficients of tv on the electrochemical potential (p) in the expression of spontaneous magnetization and specific heat for ferromagnets with Hexagonal close packed structures. This of course involves finding the correct expansion for the terms encapsuled in the dispersion relation. For the cubic crystals, the Hamiltonian is written as H = kE (Fk «-*p)n,k . (4,1-1) where F - 2zS[ JVJ(k)I/k" ,,, (4,1-2) 91 J(k) * I J l Jj»expUk-(Rj-Re>1 (4.1-3) - 1 i1iiJj*Coslk-tsr B »>I Let us put Rj = (0,0,0) and R^ <= (x^, , z£ The Reciprocal lattice vector is defined by k = tkx ’ ky ' kz> k: ■ = kCosSinek = kS.in<.i)S.ine x T k yz - kCose The nearest neighbour distances are given in Tables (4.1)i to (4,l)iii. Table (4.1)i - Nearest neighbour Distances on a Simple cubic < o attice in units a 2 3 4 5 6 -1 0 0 0 0 y o 0 1 r-1 0 0 See Pig,(4,l)i z 0 0 0 0 1 -1 92 -V Fig. 4.1i: - A given atom with* six nearest neighbours on the simple cubic lattice . o = given atom at (0,0,0) • = nearest neighbours. 93 Table (4,lii) - Nearest neighbour Distances on body centered cubic lattice in units a/2 Table (4,liii) - Nearest neighbour Distances on a Pace centered cubic lattice in units a/2 & 1 2 3 4 5 6 7 8 9 10 12 X 1 i -l -1 0 0 0 1 1 -1 -1°< y 1 -l l -1 1 -1 -1 0 0 0 03 z 0 0 0 0 1 -1 1 -1 1 -1 1 -1 (See Fig. (4,liii) > y Our esi is of the nearest-neighbour distances on a hexag close packed lattice in units of a, is shown in the following table 4,l(iv), 94 Table (4.1iv) _ Nearest neighbour Distances on an Hexagonal close packed lattice in units of a. 6 8 10 11 12A x 0 -0,5, 0,5, 0, -0.5, 0.5, 1, 1, -0.5, 0.5, -1, -1 y -0,577, 0.289, 0.289, -0.577, 0,289, 0.289, 0,00, 0 .722 , -1.154,-1.154 Q, 0.722 „ -1.632 -1.632,-1.632, 1.632, 1.632, 1.632, n A n A z --2 ’ --2 2 ~ 2~ 2 2 U’ U’ U> U’ U’ U< See our Pig. (4.1iv) 0^ For a simple cubic lattice, A T the Fourier transform J(k) = O^ 1 z X, A. J J. ~ Cost k. (R J,-R )] wr iting kx = k , ky = k and kz = k = ^ [Cos kxa + Cos(-lcxa $ Cos' Q (kya) + z Cos(-kya) + Cos(kza) Cos(-kza)J = gfCosCkxa) + CosCkya) + CosCkza)]. For a body centred cubic lattice, the Fourier transform J(k) = ^ [Cos(kxa-kya + kza)/2 + Cos(kxa + kya + kza)/2 + CosC-kxa+kya+kza)/2 + Cos(-kxa-kya+kza)/2 + Cos(kxa-kya-kza)/2 + Cos(kxa+kya-kza)/2 + Cos(-kxa+kya-kza}/2 + Cos(-kxa-kya-kza)/2l 95 Fig. 4.lii - A given atom with eight neighbours on the body centered cubic lattice. o = given atom at 00,0,0) 0 = nearest neighbours 96 Fig.4.1iii - At hegi vefna caet omc enwtitrhe d tcweulbviec neliagthtbiocures on 11 o = given atom at ('0,0,0) •= nearest neighbours B 98 , lv A given atom with twelve nearest neighbours on the Flg' 4 - Hexaeonal close packed lattice. 99 = g[ 2Cos(kxa+kza)/2Cos kya+2Cos(-kya+kza)/2Coskya +2Cos(kxa-kza)Cos kya+2Cos(-kxa-kza)/2Cos kya = J[ Cos Cos ^ Cos i p i For a face centred cubic, the Fourier transform J(k) = Cos(kxa+kya)+Cos(kxa-kya)+Cos(-kxa+kya) + Cos(-kxa-kva)+Cos(kya+kza)+Cos(kya-kza) / + Cos(-kya+kza)+Cos(-kya-kza)+Cos(kxa+kza) + Cos C kxa-kza) +Cos C^^a+kza) +Cos (-kxa-kza) I = Jr Cos_ kxa , k7S~ + Cos 7'■3 y'ZJ a *7," „T Co „s kza/2 We proceed to obtain the fourier transform J(k) for a hexagonal close packed lattice, For a Hexagonal close-packed lattice, we have Jy T 12 JT 12J(k) =• JT(k) = ££1 Co s k,R = z £„=r1. Cos(v k x x £ +k yyJ £ +k z„)Z £ y (4.1-4) Cos(kxX£+kyyjl+kzZ£) (v k x x„£ +k yJy.£ +k zz„)' (k x.+k y.+k z„)= 1 - X £ Y £ Z £ '2! 2 ! + . . . 100 (v k x x £ +k yJy.£ +k z z„£ 7) 6! ’ ' ' (4,1-5) To obtain J(k), we substitute the values on the table (4.1iv) into the expression obtained after putting (4,1-5) into (4.1-4). Taking each term separately, we have the following -1 ---1-2- --(-k-x-X-£-+- -k*vy--£-+ --k-z-Z-£-)2 1 212 £=1 2! -- (4.1-6) where 12 D"(,’ 0 ) (ka) - E1 (k■ 2 - £ = 1 X x 2. +. ,kL2£ y 1y 2 £ + kZ2“ z?“£ + 2k y k X x.£y‘ £+ 2k X k Z x„£ z„£ + 2k y kZvJ. £z .£) ' (4.1-7) 1i 12 (kyx x£D + ky 12 £=1 4!r! y»+ W . i1f2J^2L4,' A"(<}) ,6 )(ka) (4.1-8) where (C A (<)>, 6o)W (.k a), 4 12 = |I_ 1(,k.4x x4J + k,y4y4£ + k4zz4£ + S k2t ^2 x2 ^2 x+ C61k y2,k 2z y2£ Z2£ V& 6k 2,k 2 x 2 2 . 3, 3 x .,3, 3 ± ,.3, 3X Z £ z„£ + 4k X k y x £vJ £ + 4k X k z x„£ z £ + 4k y kX x„£yJ„£ + 4ky3 k Zy1' 3£ z £ + 4kZ3 k X x„£ z3£ + 4kZ3 k yJy £ z3£ 7) .’ .. (v 4.1-9)7 101 6 1 12 (kx V ky V kz V 1 V 1 12 £=1 6 ! = T^C7^o)[ B,,(4>, 6 )Cka)^l Writing B"( , 0 ) (ka)^ = a+b a = £0=E.1, (kx6 x?£ + k6yyJ ®£ + k6z z6£„ + 7k 4,k 2 z 4 v 2 +^ 7k 4,k 2 J z y £ x y x„ 4 £yJ „£ + 7k 4 , y lc 2 y4 2xJ £„ x£„ „ 2.4 2 4 ± ,,4,2 4 2 ^ „ 4,2 4 2 , - . 5 , 5 , . , 5 , 5 + 7kXkZX£Z£ + 7kXkZX£Z£ + 7kykZy£Z£ + 4 k Xk ZX£ Z£ v V + 4kykx V t + 4kykzy?zt+ Q1 3. 3 3 Sr+ 8k y k x x„£y „3 . Q, 3. 3 3 3 ^ 01 3. 3 3 3 ^ ...3. 2. 3 2J £ + 8k z k y z£j,y£n + 8k z k x x„£ z„£ + 16k yk xk y„z„x£„ £ + 1 6 k z3 k y kx2 Jy £ z 3£ x 2£ + 1 6 k 3k k 2x 3y- z2 + 1 6 k 2k 3k yx? xy3 zZ0 £ y x Z + 16ki 3ink, .2 3 2z x k y z£n x„£y^„£ + 1 16Ck. 3,y k ,2x k y 3 z 2 £ x ^+1,8,k2 ,k2 ,k2 £ y z x x„ 2 2 2 ZyJ „Zz „Z + 12 k 4x k y k z x 4z yJ „Zz .Z + 1 2 ky‘c k x k y4x . z £n + 12k x k y kz^ x . y . z 4£ and 12 b = £ - (2k3k x 3y + 2k3k x yf +2k°k x 3z 0 + 2k^k x . z 3 + 2 k 3k y 3z. £—1 x y Zy Z y x Z y Z x z £ £ z x £ £ + 20k1 5,k z 5v +^ 08k1 4.k2 x„4y „2 x+ o8kl 4,k2 x„2y „4 + 801k 4,k 2 x4z 2z y ZJZ X y ZJZ y X X Z £ „£ + 8qk, 4,k 2 x„2z „4 q | 4, 2 z x £ £ + 8k y k Z Jy „ 4 2 q , 4, 2 2 4 , q , 4, , 4£ £ z y J £ £ zZn + X 8yk £k J £y „z£„ + 18k k k x„y z, 102 + .1,80k,4 ,k ,k x„z.y.4 +. 118Qk1 4.k . 4 . 101 3, 3 3 3 . - 01 3. 3 3 3y x z £ £J£ x y k z x„£yJ.£z „£ + 12k x k y x„y„ + 12k y k Vyi„ zln + 1120k1 3,k3 x„3z „3 x+ 77021k 2,k2 .k 2 x„2 y„2z „2 , 3,2, 2 3x z £ £ x y z £ ̂ £ £ + 44k z k x k y x„£ z„Vyi„ + 4,4.,k3 ,k 2.k y„2 z 3 x ,,, 3, 2, z y x J£ £ £ + 44k ykz kx yJ„ 3 z2 ^,3, 3 2i£ £„x£ + 44k x k y x.£ z„£yJ.£ + 44ky2kx2kz yJ2£z£ x£2 + 44kx2 k^y k zx2y2z ) . (4.1-10) The fourier transform J(k) for an Hexagonal close packed lattice . is markedly different from that of the cubic lattice, ' „ Replacing F with F', for a Hexagonal close packed lattice, the transformed Hamiltonian is written as H = l(Fd'k2-y)nk . .. (4.1-11) where F ’ = 2Sz J[ D" (, e ) a^k2 - A"(, 0 ) (ka)4+B"(, 0 ) (ka)6+ . . .1 ... (4.1-12) The quantities D"(,e) and B" (4>,0) have been outlined and evaluated in the appendix D. We proceed to find the effect of the electrochemical potential y on the coefficients of in the expression of spontaneous magnetization and specific heat. As in eqn. (3.3-2) 103 ,0)a2k2 - A"(<)>, e) (ka)4 + B”( , 0 ̂ :a)6]| and ... (4.1-14) putting P = D"C, 0 )a 2 ,Ffk 9 = 2SzJS o> H = B"($,0)a4z. Equation (4.1-14) beeotnes = C[ P-Gk2 + Hk4] k2 (4.1-15) Ck = 1 1 + CP2 ♦ sCp& i i . » i - ^ (i .■ + 2 % + e V ) !o£j< p c2p2 CP2 C2p4 G2f S2 (1 2G& G2S? ,1 + 72 1 ~2~2 1 + 7C7P2 + 72—? ) (4.1 16) C P p3Hc 2)8?IC2P4 P2dk S* f 1 i 5G^ h 8g2^2 7 H s?i (4.1 k .17)2dk 2C3/2P3/ 2 ‘ ^k Vdk> A = s~ [ ! + 5 zA"(j> ,9) + 8z 2Aa"„ 2(,I.ejS.^2T 2 2cG/2p8/ 2 2 D"22(*,6) D H (,0) - 7o B''(43. ,0)ZT2a2 « «1 dSr • » * (4.1-18)D"2((b,0) • • • 104 Substituting (4,1-18) into (4.1-13) and by the techniques of numerical methods, we have been able to compute the first three coefficients of -Sm (t )/m(o). The first term of -6m(x)/m(o) is A 2tt tt dS d Sin 6 Sa2 e-mgy (2tt)2NS oJ 0 J oj 2C3/2P3/ 2 2 tt ^ T 3 / 2 r ( | ) m i l T / 2 - | J [ D ” ( * , e ) l - 3 / 2 s m e d e dSin0 )2£21x^+(Sin$Sin0 )2 ^ y ^ + CCose )2£|1zJl/24 (4,1-19) The second term of -e)Sin9 de dSin6) SinSin0 ^x^y^ + 4(Sin<|>Sin0)3Cos0 + + 6(Cos<))Sin0)2(SinSin0 )22(,C„ os0_ ),22 ^1g2^ y2^ z22 105 12 + 6(CosSin0)2(Cos0)2 Ê̂ x2z2 + 12(Cos0)2(Sin<|>Sin0)(CosSin0) " 12 2 A W i + 12(CosSin0) 2 (Sin ŜinO )Cos0 12 2 + 12(Sin<()Sin0)2 Cos(()Sin0Cos0 ^12 y 2 (4.1-20) The third term of -6(mCT))/m(o) is i4Z2f[ OX. , 0)Sin0d0d 7z f BH((|),0)Sin0 d0 d(f> ^ 2 *n h ^ Z i i ](2tt)3NS 2 xn/Z [DD"'(^m [D<'C4>,0O13 ... (4.1-21) where B"(,0) = (Cosij)Sin0). 6 12_6 + (Sin<|>Sin0) 6 ^12y ^6 + (Cos0)> 6 ^12z ^6 + 6Cos o (J>Sin c 0Sin<()J1l̂2x ^cy ^+ 6Sin6 ©Sin5 iCosl^l2^ x^y^2 + 6Gos 5 4>Sin 5 0Cos0̂ ~̂ĝ2 z^x^5 0£1=21 x z 2 + 6Sin 5 $Sin 5 0Cos0̂1£2^ z^y^2 + 6Cos 5 ©Sin̂ SinÔ g1-2j ŷ ẑ5 12 + 15Sin6 0Cos4 Cos 2.Sin4 0Cos 2 6 ^L x4 ? Xs JL X/ z JC + 15Cos 4 0Cos 2 <)>Sino 0 ^2 2X/ EJ-- x X zX + 15Sin4 <))Sin4 ©Cos 2 ^2 2 2 4 2 2 ^2 2 46z=lyzZi + 15Cos ©Sin 4>Sin 05g^y^z^ 106 4 12 12+ 30Cos 6Sin 2,eCos<}>Sin<}> E-x 1CTX/ y X/ z X» + 30Si . n 5„G_S,i. n4 ,<|)CosCos0 X/ g--L x X> v X> zX> 5 4 19 4 6 3 19+ 30Sin 0Cos ^Sin^Cose + 20Sin eCos<()Sin Q̂ JI-jX̂3ŷ3 12 12 2OSin%Sin36Cos30 ̂I-̂ ŷ z3 + 2CK3os3<}-Sin3eCos30££^x3z3 12 12 + 2QSin^0Cos2Sin2<}>Cos20 I^x^y^z2 + 6OCos30Sin30Ctos2Sin ]Lx2y z3 X,“ -L X/ X/ X/ ▲ ^ X»—"X X/ X/ X» 12 + 6OCos30Sin30Sin2(j)Cos4)̂ I:Ly^z^x^ + 6OSin^0Sin3(|>Cos<}>Cos20 ĵ|1y3z2x̂ 12 WV 12 + 6OSin^0Sin3Cos2Cos0 ̂I-̂ x3z2ŷ + 6OSin30Cos3$Sin2Cos0 ^£^y3x2ẑ 12 + 6OSinScos3SinCos20 Xj rL '-1x )jC XVj ]Xj /720 j T \ In evaluating our terms (4,1-19), (4,1-20) and (4.1-21), after integration, we require the value of J, the exchange integral for ̂ ferromagnets with hexagonal close packed lattice structure. 4.2 Evaluation of the Exchange Integral RUSHBROOKE and WOOD (1958) have in their work calculated the first six coefficients in the expansion of the susceptibility X, and its inverse X-1 in ascending powers 107 of the reciprocal temperature for the Heisenberg model of a ferromagnet with cubic symmetry. From these coefficients of X, estimates have been made of the reduced k„T curie temperature e C = — B J= —c for the simple, body centred and face centred cubic lattices. It was found that formula 5 0 96 (z-l)(llx-l), ... (4.2-1) where X = S(S+1) and z, the lattice coordination number reproduces the estimated curie temperatures fairly accurately. In evaluating J for the cubic lattices, we have made use of the above formula by writing T = V 02JS 2ST ... (4.2-2) Since (4.22-1)) does not apply to crystals with close packed Hexagonal lattice structure, following RUSHBR00KE and WOOD (1958) and DOMB and SYKES (1957) we have a technique of calculating the reduced curie temperature (6 c) from which J the exchange integral is obtained. 108 In what follows we write the reduced susceptibility X as JX Ng mB where N is the number of lattice sites, yB denotes the Bohr magneton and g the gyromagnetic ratio, been shown by Rushbrooke and Wood that xe = ±1

, •••>j b*g, are given as follows h * i zx b 2 - gZX[ 4 bg = 135z x I -4x -8x-6+10p^x + 5p^x] b4 = 4USZ X 1 (SOP^Pi-iG).3 + + (-54P1 + 96 - 45z)x + 45] Ill b5 = 42§25z x[ (2800P3^-3360P2-3360P^-1456P1+800)x 4 + (1400P3^6160P2^-3360P^4200zP1+12936P1+80+1120z )x : + C23 S8P1-4752- 3 7 8 0 z ) X t- 1 7 2 8 1 b6 = 127575 ̂ -4480q-26880P1P2+11200P4-13440P3-25984P2+5824P +24960P1-128) x5+(-4480ci-26880P1P2+5600P4 ‘ -' -27440P3 + 427842 - 224C0zP2 + 112896P2 - 11200ZP3 -62800P1 + 24640zP1 - 12960 :Z)X 1 < / +(-840r - 1260q-5640P1P2 - 12600P3 + 66612P2 - 15860zP2 +81648P2 - 6440zP| - 164760P1 + 61320zP]L - 204 -11844z + 58£022)x3 + (-210r + 26082P2 + 7623P2 + 35490zP1 + 27978 - 38052z + 9660)x" -19440P1 + 29116 - 29862z)x + 8694, ... (4.2-5) where X = S(S+1), 1 12 These coefficients have been computed with the aid of a computer for cobalt Co, Dysprosium Dy and Gd, Gadolinium, We have also extended it to cover, both Iron and Nickel, so as to make a comparison with the values of 6c calculated using Rushbrooke and Wood formula (3.4-1). The method originally used by OPECHOWSKI (1938) and ZEHLER (1950) for estimating the curie temperature from a knowledge of the first few a or b coefficients was to find the smallest positive real root of the equation l+b-.x + b0x2 + . . , = 0 , , , ( 4 . 2 - 6 ) 1 Denoting this by X , X~ = 0 when 0 = — r- so that — r is X X the estimate of the curoie' temperature. We have in this work, taken all the six coefficients into consideration and by the use of Pade approximants (see Appendixa expressed susceptibility as rational functions, v/ith a reduced number of terms X = 11 ____ = —:u 2—x* . . . ( 4 , 2 - 7 ) i In xp a=o a P Equation (4,2-7) after a cross multiplication gives for Nickel (Ni) ■ (l-6x+6x2+6x2+0,7 5 x 4 - 3 ,55x^ -28.389x^) ( d̂ +̂cL̂ x+d̂ x2) = 1+n^x ... ( 4.2-8) 113 Equating coefficients of like terms, we obtain with d, o = 1 = no d^ = n +̂6 -6d1+d2 = -6 6di-«2+d3 = -6 6d1+6d2-6d2+d^ = -0,75 0,75d^+6d2+6d2~6d^+d^ - 3,55 -3,55d^+0i75d2+6d2+6d^-6d^ = 28!. 389 « .2-9) Solving the equations above, we have d.1 = 1,6807,* dQ = 4,0842 2 d5 = 4,8105 and - - Therefore, 2 d ,x- X = = .1 b̂jj,x j=o (1-4,319X) ... 0 .2-1 0) Xc 4,3193 6 c = 4.3193 The critical point exponent . , 1,6807 , 4„0847 Y ̂ 4 3193 + ( . )^2 = 1.61.4 3193 114 0 has been obtained using this technique for, Iron, Nickel, Cobalt, Dysprosium and Gadolinium and the results compared with the results from Rushbrooke and Wood formula. This technique has enabled us to calculate the critical point exponents and all these have been tabulated and shown in the following table 4,1. These values of 0 c calculated now enable us to calculate the exchange integral J from equation (4.2-2). These values of J when substituted into equations (4.1-19), (4,1-20) and (4.1-21) give the coefficients T 3/' 2 , T 5/' 2 , and T 7/' 2 of spontaneous magnetization respectively and the effect of the electrochemical potential is examined. The coefficient of the specific heat is also computed by the method outlined in chapter III and the results tabulated, for the various ferromagnetic elements. All these coefficients are shown in the table 4.2. 4.3 Spin wave - Spin wave - Spin wave Interaction We have extended our formalism and technique to take into account the wave-wave-wave interactions and the effect thereof. As usual, by the method of second quantization and using Holstein and Primakoff transformation, the Hamiltonian 115 Table 4.1 - Reduced curie temperature (9C) and the critical point exponent for some ferrometals. Ferromagnetic Lattice Rushbrooke Pade* Critical Elements Structure and Approximinants Spin point Wood 0 0 £ 2 nent Body IRON (EE) Centred 7.656 8.057 1.45 Cubic (BGC) Hexagonal C0BALT ( a , ) S S S d 12-030 12.659 1.40 (HCP) Facer- NICKEL (Ni) Centred 4.153 .319 1 1.61 Cubic (PCC) 2 Hexagonal << DYSPROSIUM (Dy) P Calcoksee d 188.489 201.110 1.30 (KCP) y Hexs GADOLINIUM Close (Gd) Packed 98.680 105.180 7 2 1.31 (HCP) 116 Table 4.2 - Coefficients of T3/2,T5/2,T7/2 and T4 of spontaneous magnetization and the coefficient of Tr> 3 '/ 2 of specific heat for some f< i SPECIFIC ELEMENTS LATTICE SPIN (s)/ CHEMICAL TRANSITION REDUCED HEATSTRUCTURE NEAREST POTENTIAL TEMP. TEMP(0 ) c ^ > 2 < C3T7/2 c4t4 NEIGHBOUPS(z) (p) (TC)°K c t3/210-23f - . - SC s = i,z = 6 0.123 1043 1.88 5.203X10”6 1.345xl0“9 5.525xl0~13 3.590xl0-15 4.54 s = i,z = 7 0.152 2.34 2.292X10"6 0.828xl0“9 3.167xl0-13 2.669xl0-15 2.94 IRON (Fe) BCC s = i,z=7.5 0.161 1043 2.45 3.647X10"6 0.894xl0-9 3.420xl0-13 4.607xl0“15 2.73 s = £,z=8 0.169 2.64 4.000xl0“6 1.065xl0-9 4.OOSxlO-13 5.726xl0“15 2.70 s = i,z=ll 0.214 3.77 2.008x10“° 0.357xl0-9 1.355xl0~13 1.351xl0-15 1.27 COBALT (Co) HCP 1400 s = i,z=12 0.228 4.15 2.259x10^ 0.496xl0“9 2.271xl0-13 2.070xl0“15 1.25 NICKEL (Ni) FCC s = i, z=12 0.228 672 4.15 7.395xl0“6 0.360xl0“8 3.787xl0-12 5.146xl0“14 1.25 GADOLINIUM (Gd) HCP s = J, z=12 0.00215 292 105.19 2.279xl0“6 7.70xl0-7 9.275xl0-3 4.95xl0-10 1.728 DYSPROSIUM HCP s = 5, z=12 0.00103 85 201.11 1.671x10^ 2.497xl0-5 6.929X10-6 2.235xl0-7 1.731 (Dy) ( ■--------- 117 of a Heisenberg ferromagnet is written as, H =-2Jj E£ [ 2Sa^fj(s)f£(s)ap-2Sajaj+a^a^a*a£l - g y ^ E a ^ .,, (4.3-1) In equation (4,3-1), fj(s) = (l-a^+. aj/2S) * and the expansion of f.(s) takes into consideration all the two-product 3 and four product terms in order to account for the wave-wave wave interaction, Hence, fl a+ a ?. 1 _l, ,a+ + a^A -l, ^ra++ a,xz2O 1 ̂ ,a++ a^x30 5 .a+ av4A „7 ra+ a^5_ L ~ 2S j 2^ 2S } ~ 81 2S ; 1L6C(' 2S } " 128 Q 2S J ~ 256̂ 2S J * . (4.3-2) where (. a+ a), 2 = (, a + aa+ a)N = a+ .r(j|1 +a+ a. }a = a+ a+a+ a+ aa, (, a+ a)x 3 = (, a + aa+ aa+ a)x - a+ (,-1- +a+ a)x ,r( a+a+ a)x a = a+ ,(.-l +2 ̂a+ a+a4 - a+a+ a+ aa)Na = a+ a+, O3 a+ a+ aa+. a+ a+ a+ aaa = a+ a+3^ a+ a+ aa, - a+[ (l+a+a+a+a+aa)(l+a+a)la a+ a+. 7n a+ a + aa' +6a+ a+ a + aaa+a+ a+a +a + aaaa *= a + a+JT7J a+ a+ aa, 118 (, a4 - a)x 5 = (,a4 - aa+ aa+ aa4 - aa4 - a) f= a4 - (14-a4 - a)N (A-ll4 -a4 - a)w(-ll4 *a4 - a)N(l4-' a+ a)xa = a4 “ [ (l+7a4 * a+6a 4. a4 _aa+a a a aaa)(l+a a)]a = a+a+15a+a+aa+25a+a+a+aaa+9a+a+a+a+aaaa +1 o2 a+. a+ a+ a+ a+ aaaaa = a'a+lSa'a'aaa, ̂+ 6„ v VCa a) - (/.a4 - aa+ aa4 -a4a- aa+ aa4 a-)x - a+[r l+15a4 - a+25a4 ~ aaa+lOa4̂ a4 a4 aaa+a4 * a4 " a■ a+a4 - a+15a4 a+15a4 * a4 *aa 4-50a 4.-. a.. 4.-.a..a..4..“.2..5..a..4 ..-.a.4. .-.a.4. .- aaa4-30a+ a + a +aaa+10a 4- a +a 4- a4 -aaaa44a 4- a4 - a+ a4’*aaaa 4-a+a+afa+a+aaaaal a, = a (l+31a a+90a a a a aaa+15a a a a asaa+a a a a a aaaaaja - a + a+3„„1 a+ a+ aa, v X Equation (4t$/ •-2£) > becomes r & + ^\2 _ 1 ...................................\. ......l. .....5.....a ̂1 ̂ 4 1 j n 4-̂ (1 2S ) 1_ 2^2S 8^2S 16(2S) l28^2S) ,,,Ia a r H1^,21S, 2.+ 136,^21S,^ 3,+7 .125S,^12S,^4 .7+ .21556, ̂12,S 5.+ • ..Jaaaa ... (,,4, ,30- 30), un _ 4" i !2L3S. )} 2 1 + Aa 4 " a + Ya4 a-a4 ­a where 119 > = d - < 1 - 4 ^ ) A ir_ i A + A r JJ__^ 33 + l7i.5. 1 .4 7,15.1 v 58^2S; 16V2Sr; 12 8 ('2SS ; + 256 ^lV2S^ * *’ * * • (4 - Equation (4,3-1) can be wri tten with the above modification and HS o = 0 to give H = Z--2J[ 2Sa+(l+Aa+:a + Y a + a+a a ) (l+Ab+b+Yb+b+bb)b+(S- a~+a)(S-b+b)l ,, (4.3-5) utting a J. = a ', a ■* ’ b & E-2J[ 2Sa+(l+A(a+a+b+b)+Y(a+a+aa+b+b+bb)+A2a+ab+b)b _ A (S2-S(a+a+b+b)+a+ab+b)] -2J[2S(a+b+A(a+a+ab+a+b+bb)+Y(a+o+a+~ f ' fa aab+a+b+b+bb fc) V>VA2(a+a+ab+bb) + (S2-S(a+a+b+b)+a+ab+b] ,,, (4,3-6) S V + + Z-2J[2S(a b-a a)+2SA(a a ab+a b+bb)+a+ab+b + 2Sg(aVa+aab + a V b +bbb)+A2(a+a+ab+bb)I ,,, (4.3-7) Using the transformation 120 a,k - N _ i 2jEexppv( -ik j.') aj., a, + = N _ _ i ’ k 2£jexpl( vik J. ) + Ja . ... (4.3-8) where k denotes a reciprocal lattice vector and N is the total number of ferromagnetic spins. We can now transform every term which has not been transformed appearing in the above Hamiltonian with the help of the fourier transform (4,3-8) l Ja+a+ab+bb+ = EN”3Ja£ a+,a . .a+...a . a+ exp i(k,j+k1.j-k*1.j+k111.Jl all j K k1 k11 k111 k1V kv -kiv.«-kv ,n) where .k iii = k, i+ip. ii, ^ok iv = k, i+, pi, k, v = k. +, p on substitution, we have l Ja+ -a+ -a1b- +1b-*b- =- N 1 Ja.+ a+ a ..a + .. ..a . .a, expi, (~k ,j.+.k, i .j. -,ki i .j.+ki i.a all j k a1 k11 k11+p11 kVp1 k+p -p . t+p11 i t-k*. Jtr-p1-. £-k. 4 ) J * N-^J^yia^^piiy^^pexplCk+r-k1^ j, 1C1,j-t) expi(p11-p1-p), ,,, (4.3-9) There are three important contributions coming from the above term one with p^ = p = 0, k = k^, and k = k**. 121 The first contribution gives = N-3eJ(k+k^-k^)z a?"a+ .a+ ..a ..a .a. k k1 k11 k11 k1 k The second contribution gives Nt -2EJ(2k-k ii,) z ak+ a + .a 1±a ijL a ±ak K k1 k11 k11+p1+p k+p1 k+p The third contribution gives N, -2„E JT/(lk i,) z + + . . a . ^ a .x T X . a . . a .,k JLJ- ,k k-I 1 1 kT 1 1 k, 1+1p i+,p ■k,, + p. ix“ k,, xiii *+ p. Adding all contributions, we obtain, E Ja+ a+ a,b +,b b, = ,NT— 2zE[J(k+k4 3-ik,Xi.)+.J.(.2k, i -,k ixi).+, JT(/1k iA.),l ak+ a+ +iiaka ±_a i±±_. all .1 kx k " IV r k (4.3-10) Similarly, A c * E J(a+b+b+bbb) = N 3EJala+.a+..a .. .a . a exp i(k, j+ki.£+k'i'i<,Jl-k:i‘:i‘i'.Jl all j k1 k11 k111 klv kv -klv.*-kv.t) with k, lii f k, ii +p ii , k, iv = k,i ,+ p i , k,v = k, +. p. On substitution we have 122 EJ(a+b+b+bbb) = N^EJa~a+.a+. .a . . . ,a . .a, , expi(k, j+k'*', t+k^.t- k k1 k11 k11+p11 k V P k11,Jt-p"̂ . t-k .t-p.t-k. £-p. = N ' ^ J V ^ a ^ a ^ y a ^ ^ e x p i C k C j - O e x p - i C p ^ V p ) , ) ! ■Qj ... (4.3-11) = p = 0, k 11 £Ja+b+b+bbb + + + kakiaki.a ..a , ,,, (4.3-12)1 k k Now, we can write the Hamiltonian in the transformed form as follows a H = Eo+ lAtkJa^N-1 r V C k . k ^ V a -Hf2 J a V(k,k1,k11) K k k1 k k k,k ,k / ■ ak^a k+ii ak+ n akka f ci. ak n.. ... (4,3^13) where ACk) = 2zSJ(l-y (k)) V(k,k1) = 4S zJ[ YCkJ+YCk1)]-zJ[ l+YCk-k1)] VCkjk1,*11) = —4SYzJt y(k)+Y(k1)+Y(k'1’1)] —l2zJl Y(k+ki-kii)+Y(2Ici-kii)Y(ki)] and 123 E = -zNS2J, Y(k) = l exp(ik,a)/z; |a|3 = V /N a ° Writing V(k,kx,kxl) fully we have - i EV(k,kX,kXX) = N_2z4sYzJ(l . Itii + 1 _ ( J ! . (fcU a)' . .2 _ ,,±1i , ii,2 0 , 01 i . i i j / ^ /■ * i .2 + z v - (k+k z~ k ■ a2+l - (2k z- H O .n) a2+l - z 1 & ,,, (4.3-14) Averaging over k x i space we have —4sY!5-JE <(3z' -k2 a2 -. ki 2 a2 -. ki i2 a2 JE<3z-k2a2-6k1 a2-2ki;L a2n ..) N2 k1 ,,, (4.3-15) - ^ E(3z-k2a2-ki2a2)r x3/2 L ^ + ̂ W A W ^ r x 3^ 111=1 in /2 * m=z l e3 ... (4.3-16)m /'2 / = 4sYj[ (3z-k2a2)r2T3(n?1 -g^r)2 + A2J(3z-k2a2)r2T3(Ê ^ ) 2 ... (4.3-17) m Rewriting our transformed Hamiltonian, we obtain H = Z(A(k)-y+A1-B1k2a2)n1 = ZA(k)-(y-A1)-B1k2a2)n k k ... (4.3-18) where averaging over k ', we have 124 m By. A = 3zt 3 r 2J(4s y +A 2 ). (, °1° em=l m 3/'2- „B 1 = ,( 4A sv+^A, 2, . 2 3, j em6|)JrJv ,2( ^ - 575) . . . ( 4 . 3 - 1 9 ) m ' Equation (4,3-19) clearly shows the effect of the wave-wave- wave interaction on the dispersion relation of the magnons. The energy of a magnon is altered slightly by the additional term -B1 , while, the electrochemical potential of a magnon is altered by the additional term -A^. Our interest here is to find out tile effect of -A'*' on p , Writing w = |y|/J, We ha- ve V HTIW w = (a-1) 2rZx3/2 “ e-- -_!----+ (4sy+x2)rT32z[ | 2Sr_]2 ,(4.3-20) m3 / 2 lm=L m 3/'2 with the aid of the computer, we solve numerically the above equation and by the method of least square error fitting, we obtain the following expression for w, for Iron, Cobalt, Nickel, Dysprosium and Gadolinium, (i) ron Fe, (s = i, z = 7.5) w = -1,6S9x 10'"3+0,154t+0.28t2 = 0 . 1 5 t + 0 , 2 8 t 2 ... (4,3-21a) (ii) Iron Fe, (s = J, z = 8) w = -1,846x10-3+0,164t +0.27t 2 = 0,16t+0,27t2 ... (4,3-21b) 125 (iii) Cobalt Co, (s - §, z - 11) w = -2,304xl(T3+0,2C9x+0,32x2 = 0,21x+0.32x2 (4.3-21c) Civ) Cobalt Co. (s = f, z = 12) w - -2,441xl0~3+0,223t+0,33x2 = 0,22x+0.33x2 ... (4,3-21d) (v) Nickel Ni, (s « z = 12) w = -2.441x10 3+0.233x+0,33x2 = 0.22Jxt +O0 , 33x2 ... (4.3-21e) (vi) Gadolinium Gd. (s = §, z = 12) w = -1.919x10^+1,444x10 2x+4,172x10 2x2 = 1.44x10_2x +4.17xl0-2x 2 (4,3,21f) (vii) Dysprosium Dy (s = 5, z = 12) w = -1.319xl0”4+0,990xl0"2x+2,921xl0“2x2 = 0,S9xl0'‘2x+2.92xl0"2x2 ,,, C4.3-21g) G(a) from equation (4.3-7) becomes G(a ) = e |y | = v^/2S We proceed to find the effect of this new y on the coefficient of T3/ 2 in the expression of the spontaneous magnetization. This has been done and the table below compares this new ’ y ' with the y of chapter III, We see that the two p’s hardly differ in each case. That is, wave-wave-wave interactions is negligible in comparison with wave-wave interaction. The following figures shows the graphs of w against I. 126 Table 4.3 — Coefficients of T, 3/2 with chemical potential in wave—wave—wave interactions and wave wave wave interactions. CHEMICAL CHEMICAL POTENTIAL IN POTENTIAL IN c ^ 3/2 ELEMENTS WAVE-WAVE WAVE-WAVE-WAVE C,T3 / 2 C- experimen- INTERACTION INTERACTION C l 1 tal Cp -l) (p Cp2 ) BCC s = i, z=7.5 0.16069 0.15369 3,647xl0-6 e .g . Iron 3.701x10 -6 3.41x10 -6 (Iron) BCC S = y Z = 8 0,16880 0.16417 4.00x10 -6 4.039x10 -6 3.41x10 —6 (Iron)e.g. Iron HCP s = i, z = 11 0.21400 0.20940 2.008x10 -6 2.026x10 - 6 1.7x10, -6 (Cobalt)e.g. Cobalt HCP s = i, z = 12 0.2280 0.2233 2.259x10 -6 2.279x10 - 6 1.7x10v - 6 (Cobalt) e.g. Cobalt FCC S = 5 , z — 12 0,2280 0.2230 7.395x10 -6-6 7.459x10 -6 7.4x10v -w6 (Nickel) HCP H = 3,Z = 12 0.00215 0.00206 2.79x10 2.280x10 -6 (Gadolinium) HCP 5,z = 12 0103 0.00099 1.671x10 -4 1.6714x10 -4 (Dysprosium) — 127 Fig. 4.3-1: Graph of Chemical Potential/Exchange integral W against reduced temperature T. 128 Fig. 4.3-2 Graph of Chemical Potential/Exchange integral W against reduced temperature T 129 Fig, 4,3-3 Graph of Chemical Potential/Exchange integral W against reduced temperature T. REDUCED TEMPERATURE 130 Fig, 4,3-4: Graph of Chemical Potential/Exchange integral W against reduced temperature T. j i K U ill Th W U X*.T. ii li J' t: hL Xt-l. a0. LiiJ i} REDUCED TEM P ER ^U R E 131 Fig. 4,3-5; WG rapahg aiofn stC hermeidcuacle d Potteemnpteiraaltu/rEex cTh.ange integral H/S:Z=12 REDUCED TEMPERATURE CHAPTER V INTERACTIONS II It has been suggested that magnons, by virtue of their motion do mutually interact, dynamically and secondly they are strictly speaking not Bosons, In this chapter, we examine these two suggestions 5.1 The Dynamical Interaction To the best of our knowledge and in agreement with several authors, the spontaneous magnetization at low temperatures contains terms of the form T^^, T ^ ^ and T w h i c h correspond to the terms k^, k^ and k^ of the dispersion law respectively. In this section, we are interested in finding the next degree of T contri­ butes to the magnetization. We shall also find the effect of the electrochemical potential on the coefficient of this degree of T. Going back to our transformed Hamiltonian Expression (3.1-39) is H-E^ = £Aknk + N -k V, a ^kqn k^ q + N ~ kV,„q', kq k= q A+B+C where Vkk< = ZJI Y(k)+Y(q)-l-Y(k-q)] , Wkq f z(a-l)J[ y(k)+y(q)] Ak = 2sJz(l-y(k)] 133 If we expand y(k) to order k 4 a 4 , we have that . 2 2 . y(k) = 1 - + A(, 0 )(ka) . Note here that the term B on averaging over an angle is not zero and Vkk< = ^ j r + A(,6)(ka)4+ 1 - +A(̂ *)(k1a)4-l-l+ (k~k-̂ a -A(a,g)(k-k'a)4a4X] . . . (5 ,1 -D zJ[-A(a,B) (k4+k' 4+4k2k'2) a4+A(*, e) (ka)4+A( <|> •, 6») (k' a)4J (5.1-1) Averaging in k' space and neglecting small terms, we have :Vkq nq> = -zJJ<̂<[ (Ai o « ; 6)4k2a4] nq q2> ,,, (5.1-2) Our fourier transformed Hamiltonian is written as II = z(9Fk f-u)n k (5.1-3)k where F = 2SJa2 - N 1zJ ... (5.1-4) F = 2SJa2[ 1-nJ ... (5.1-5) where n = N_1zJ/2SJa2 , 134 n = V zja4 A(g,e)4q dq 2tt2N exp (Fq 2 -v .) -l - 5 A where OS' 2tt T l i = da ACa,g)SinfrdB t f .C5.1-6) o We should note here that the presence of n has caused a modification of the Dispersion law. There is a shift in the energy of the Spinwave, because of the presence of other Spin waves. This kind of interaction between Spin waves is called the Dynamical interaction (Dyson (1959)), With the presence of the Dynamical interaction between Spin W&ves> we are interested in finding the effect of this interaction as well as the effect of the electrochemical potential on the coefficient of spontaneous magnetization, 1m ft 1 6nM. - £- - Q1S ^r8 ikrTJS ^3/2m =“l e3+/m23yri v (5.1-7)k m ' (1“n) where the number N of atoms per unit volume is Q/a , 135 Therefore for small n, = J_r m3yNS QS ̂ 8knTJ S ^ / 2m =vl em3 /2 u + 23n A _ 1 r kT 3/2 » 3 3ir T , kT ,4 t.e+mBl1 „em6p QS oirJS' n^lm3/2 + ̂ W ^5/2 ... (5,1-8) The second term of expression (5,1-8) clearly gives the coefficient of TA This coefficient has been calculated for some ferromagnetic elements, and shown in Table 4,2, This T 4 dependence, due to DynatmiOcal interaction, is experimentally negligible a^ compared with the other coefficients of T, 5.2 The Kinematical Interaction According to DYSON (1956), the kinematical inter­ action arises because the Spin wave states which contain more than one Spin wave are not members of an orthogonal set, Th^ non-orthogonality of these states produces an interaction between Spins which we call the kinematical interaction. The Physical cause of this interaction is the fact that more than (2S+1) units of reversed Spin cannot be attached to the same atom. There is therefore some form of 136 exclusion at work , to limit any dense packing of many spin waves within a given volume. The kinematical interaction is a purely statistical effect which reduces the statistical weight of states containing a high density of spin waves per unit volume, They appear in calculation of the statistical and thermo­ dynamical properties of the spin wave system, but not in the dynamics of individual spin waves. In this section, we show the effect of the kinematical interaction of Spin waves by recomputing the coefficients of T in the expression of the spontaneous magnetization, for Iron, Nickel, Cobalt, Dysprosium and Gadolinium, This time, all averaging in k space is done over inter­ mediate statistics. The intermediate statistics, is so-called because the largest numbe3r of particles p allowed in any state is intermediate between Fermi-Dirac Statistics with p - 1 and Bose-Ein\sVte*in statistics with p = For this statistics, it is thl«e case that the grand partition function is Z = P2 a,nkn ... (5.2-1)‘,k rfo~ akKk where 137 -3(Ek-y) a,k = e = bfcz, l V eOxp B(Ek1 -y)-l expB(E(k2S-+y2))( 2S+2)^1, (5,2-5)k Averaging under this distribution, the chemical potential y in (3.2-5) becomes y = (1-a) <̂1 2z-q 2 a 2ii nQ> (5.2-6) 138 = (l-a)Jl 2z-k2a2l — 5U k2 dk (2S+2)kdkP 2tt jN o expB(Fk -x I expgQFk -x)(2S+2)-l .,, (5.2-7) neglecting smaller terms, we have <^ - (nl -a)\Jt—2zVi Hr e^,g(Fk _P - 2irTtf - (5,2-8) mBx m$x(2S+2) = (l-a)2&j[ -( m~1 BF~ 3/2 ~ V (f ^ " C2S+2) E„ - £)3/2 4 BN 4 (m3F)(2S+2)l3/2 4vA d 4) Ji (5,2-9) Put Jr 5(r4iK) § , w - y|p.| T = W N2SJ , F = 2SJa ,4ir -mw < eT75>-Jl <^3^7729. C2̂ ) ' 5 (5,2.11) We proceed to solve y numerically from equation (5.2-11), for the values of t between 0 and 0.50. By the method of least square error fitting, we obtain the following expressions of w, for Iron, Cobalt, Nickel, Dysprosium and Gadolinium. 139 (i) Body-centred cubic (e,gt Fe) with s - z - 7,5 w - 1 . 4 7 6 x 1 ( T 3 + 0 . 1 0 8 t + 0 , 2 1 t 2 = 0 , 1 1 t + 0 . 2 1 t 2 ... (5.2-12a) (ii) Body-centred cubic (e,g. Fe) with s = z = 8 w = -1.564x10 3 + 0 . 1 1 5 t + 0 . 2 2 t = 0 . 1 2 t + 0 . 2 2 t 2 ^ ^(5.2-12b) (ill) Hexagonal close packed (e.g, Co,) with s ='•£, z ■ 11 w = -2.058x10 3+0,1 5 4 t + 0 , 2 8 t 2 - 0 , 1 5 t + 0 , 2 8 t'‘ ^ 7 (5-2-12C) (iv) Hexagonal-close padded (e.g, Co) with s = z = 12 w = - 2 . 6 5 9 x 1 0 _ 3 + 0 , 1 7 2 t + 0 . 2 9 t 2 = 0 . 1 7 t + 0 . 2 9 t 2 ... (5.2-12d) (v) Face centred cubic (e.g. Ni) with s - z = 12 = - 2 . 6 5 9 x 1 0 _ 3 + 0 , 1 7 2 t + 0 , 2 9 t 2 = 0 . 1 7 t + 0 , 2 9 t 2 .,. C5.2-12e) (vi) Hexagonal close packed (e.g. Gd) with s = z = 12 w = - 1 . 4 6 6 x 1 0 - 4 + 1 , 0 2 7 x 1 0 “ 2 t + 2 , 8 6 1 x 1 0 “ 2 t 2 = 1 , 0 3 x 1 0 _ 2 t + 2 . 8 6 x 1 0 ~ 2 t 2 , (5.2-12f) (vii) Hexagonal close packed (e.g, Dy) with s = 5, z = 12 w = -1,053x10^+7,4 2 2 x 1 0 ~ 3 t + 2 , 0 6 0 x 1 0 2 t 2 = 7 .4 2xl0~3t +2.0 6 x 1 0 ~ 2 t 2 ... 5.2-12g) As usualJ yT (the electrochemical potential) is computed for all tcheA ferromagnetic elements. We proceed to find the effect of this new y on the coefficients of TV in the expression for the spontaneous magnetization. 140 " \ [ k 2d k NS - 2S+2 k2dk l (2tt)3NS u expg(Fk2-P)~l u expg(Fk -y)(2S+2)-l (5,2-13) The third order approximation of F leads to = 2sJ[ (ka) 2 -zA. (, <. J), 0_ _) _(k a,) 4 +. zB.(.., .0. ). (k a,)v 6.+ ,,f, C5.2-14) Putting (5.2-14) into (5.2-13) and evaluating, vwe have S V . _ V _ c-3/2 e-n»y e-ny(2St2) . 3 NS (/02 tt)3N L mm»=ll t3l m732/2 _ m31 3//'22 ^ + ̂ 1(3) ̂“m5 7'2 e « 2 > (2s+2)-i3}x 2. 7 7 -my(2S+2) , „ ( 4 z"%2 - X4 )3r' (v^2')1 {7/^2 - 6 3/2 (2S+2) 2}t .. (5.2-15)m 7 m where r is the gamma function y = and all other terms retain their pr/ev•ioVus meanings and defkiTnitions. Expression (5,2-15) gives the effect of the kinematical interaction, as well as the effect of the electrochemical potential on the coefficients of spontaneous magnetization. Theitable 5,2 shows (the new effective chemical potential and the various coefficients of TV, The coefficient of T 4 show's the effect of both Dynamical and kinematical interactions, 141 Table 5.2 - Coefficients of T3/2,T3/2,T7/2 and T4 with Kin interactions. FERROMAGNETIC ELEMENTS CHEMICAL POTENTIAL C"CT 3/2, Cj.CT5/2) ( yM) SC s=5, z=6 0,113 4,513x10' 0,672xl0~7 2.762xl0~13 1.795xl0"15 BCC s=j, z=7.5 0,108 2.546x10 Q.543xl0~9 1.984xl0'~13 1.132xlQ~15 CFe) s=5, z=8 0,115 2,820x10,-6 0,655xl0"9 2 . 3 5 1 x K T 1 3 2.504xl0~15 HCP s=2, z=ll 0.154 0.235xl0"9 0.853xl0~13 0.659xl0”15 (Co) S=3, z=12 0,172 0.332X10-9 1.460xl0-13 1.036xl0-15 FCC s=j,z=12 (Ni) 0.172 5.529x10 -6 0.241xl0~8 2.434xl0~12 2.403xlQ-14 HCP s=5, z=12 0.00147 O 5 82x10-6 5.191xl0^7 6.231X10-8 2.298X10T"1° (Gd) HCP s=5, z=12 0.0C007442 , 1.217x10' l,787xl0“5 4.950xHf6 1.165xl0~7 CDy) — /P 142 The table 5,3 below gives the computed coefficients of T 3/2 for all the five ferromagnetic elements, with the inclusion of the effect of the presence of (i) the electrochemical potential y (ii) the kinematical interaction, (iii) both the electrochemical ootenti, al. a, s r well as the kinematical interaction, awnd (iv) no interaction. In table 5,3, we also give the available experimental values for some of the elements for comparative studies. t f G j - 9 S ' 143 Table 5.3 The Coefficient of IRON IRON COBALT COBALT GADOLINIUM DYSPROSIUM (s=5,z=7,5) (s=3,z=8) Cs=i,z=ll) 0=5,z= ,z=12) (s=7/2,z=12) (s=5,z=12) With the electro­ chemical potential u 3.647x10 6 4.000x1.0-^6 2.QQ8X1Q”6 2.259x10”̂ 7.393x1a”6 2 .2 7 9 x 1 0 6 1.671X 10-4 With the Kine- matica.l inter­ 2.818X10-6 3.239X10”6 1.495x1' 79X10-6 6.546x1a”6 2.635X10-6 1.940x10’ act ion With loth electro­ c0h0e miarc adl kpmoetmeanttiicaall 2.546xl0"6 2.820x10^ 1 1,689x10^ 5.529x1 .0-^6 1.582x10^ 1.217x10 interaction With ro inter­ 6,687x10„"-°6 6.479x10”° 4,102x10"° 4.726X10”6 15.49x10,”-°6 .-6 .-4action 4.27x10"° 2.95x10 Experimental value 3.41X10”6 a 3,41x1c”6 1.7X10”6 l^xlO”6 7.4X10”6 CHAPTER VI DISCUSSION In this chapter, we give a detailed discussion of our results and extensions thereof to antiferromagnetism. 6.1 Results The temperature Tq , below which magn Ions are well defined quasi-particles for cubic crysta shown in chapter two is given by To = 2x !0-3C ^ [)2 . . . ( 6 . 1- 1 ) Tq , for crystals with Hexag structure is & “ *•“ Tq = 1,23x10 3 S3( ^ ) 2 . . . ( 6 . 1- 2) where Tc is the transition temperature, 6c is the reduced curie temperature is the Boltzmann's constant, h is the plandk's constant, z is the number of nearest neighbours, a is the inter­ atomic distance and S is the Spin value, Thus for b^c c Iron T o = 44k,’ for fee Nickel T = 7k for fee Cobalt T o = 28k, for hi cp Gadolinium T o = 0.4k and for h up Dysprosium, T u = 0.02k, 145 For temperatures T>Tq , the interaction among the Spin waves is important. The spin operators S and S~ respectively contain the term fJ.(s), (see equation 3.1-12). Within the spin wave - spin wave interaction approximation, + f j ( S ) - [1 a . a j i where a.3 and a.3 are the creation and an hilation operators respectively and S, has been expanded to all orders. The infinite series in the two» product terms has the form V s ) - 1 + K a j ,,, (6.1-3) where . x . - A . .2S )2 - 1 The Fourier traann;sformed Hamiltonian of a ferromagnet as shown in chapter three is given by > -1 -1 ° k k k k , q kQ k Q k>q kq 'k q ... (6,1-4) where Eq = -zNS J is the energy of the ground state 146 Ak = 2SJz[ 1-y (k)l vkq ̂ZJlY^k) + Y(q)-i-r(k-q)] Wkq = z(a-l)JlyCk) + y(q )1 .1. (6.1-5) and a = a(s) = 4sA = 4s(l-(l^ 7̂1_2S) ' a2 ) < p Y(lc) = aEexp(ik.a)/z, n = a,k ak , 2 2 With the expansion y(k) = 1 - k a + ...Qj to o(rrd. z er 1k 2 a2 V.K q on averaging over an angle gives a zero, hence the effect of the wave wave interation is included in the third term of expression (6.1-4) as encapsuled by a(s). Rewriting the Hamiltonian, we obtain, H = E(Fk -y)nk k S where F = 2JSa 2 + («a -l)Ja2 — = 2JSaz9 . . . ( 6 ,1- 6 ) X T and v = - 2n z-q 2 a ] n q> 2 o - y is the effective electrochemical potential, OGUCHI (1959) in his attempts to find the correction to the spontaneous magnetization produced by spin-wave interactions has calculated the grand partition function of the system defined by Z = Trace exp(-gH), g = ^V . ,. (6.1-7) 147 With, BH = BE, A.k n.k + BEW,k q n.k n q = A+eB Instead of Oguchi’s Z of eqn. (6.1-7), we have used, Z = Tre"A (l-eB ) Z = Tre A - eTr Be A . . , (6. 1-8) = T„ r e -A -r- e[r -T-r- -B-e-~A-ArJ , Tr e-A Tr e Z r Tr e [ l-e] C6.1-9) Since A and B commute, [A, Bl = 0; then, In Z = In Z o - e where = I < « kqnkV “ E nk<£ZWklq^n q> pn. Essentially, Z = Tre ..., e )a2k2-A"( 4>, 6 ) (ka)^+B"(, 0 )(ka)6+ ., /k2 where D"(<}>,e), A"(;6) and B" (<{>, e ) have been outlined and evaluated in Appendix D. Following RUSHBROOKE and WOOD (1958) and DOMB and SYKES (1957) and by the use of Pade' Approximahts, we have calculated the Reduced Cur■bi^teihperature ec for some ferromagnets as well as their critical point exponent, Rushbrooke and Wood's for’mula for (eqn. 4.2-1) is for cubic crystals while our technique is adaptable for all types of crystals. From Table 4,1, we see that for cubic crystals, the Reduced temperature (0C) calculated by our technique differs slightly from the Reduced temperature calculated using Rushbrooke and Wood's formula. There is a remarkable difference for crystals with the hexagonal close packed lattice structures, This implies that Rushbrooke and Wood's formula cannot be used for crystals with Hexagonal close packed lattice structures. 150 The internal energy U with (p = 0) is given by UCy = 0) - ^Q 4£(2JS)~5/2(kT)5y/2i— im 5/2 , ^ 6.1-12) we have shown that 9u/3y>o and hence U(p = 0) is greater than U(p<0), U(l, X [1 - ijl* n (6,1-15) S xk| — xS yk, (2S)2[ 1 - 42SSjJ *b~,k ^zk ~ '"S+nic For a spin k on the other lattice, say the (-) lattice, the operators b and .b are naturally defined in the same way as and a and satisfy the equation 154 * * bk bk * nk- bkbk - bkbk ' 1 . . . (6.1 - 1 6 ) The simplest form of the Hamiltonian of an antiferro- magnet is usually assumed to be H ex - I1 J 1I . E,,k S j. ’, S.k s S < « >-17) Inserting equations (6,1-14) and (6.1-15) into equation (6,1-17) we obtain H-e x - - i ̂N z | j | S ^ + z|j|S(Ej n.J + iEc n, ) + |jis y k{fs(nj)ajfs(\ )bk + aj fsCnj ^ W •- (6-:.-18) where fs(nk ) " (1 ~ We have Hex = o +. „i A > ,,, (6,1-19) where H° = - ^Nz|J|S^+z|J|S( Eny-. + kEn ,k ) 4 |J|S and h1 ■ , !j is J y " i a] bk*aj * k bk + aj nj bk + aj V k > * * “|j|j^k(ajajbkbk) where X - (1 - ^ ) 2 1 • • ( 6 . 1- 20) 1 55 Note that X contains all the coefficients of the two product terms as shown in the ferromagnetic case. The classical treatment of spins in the limit of S ^ « is called the zeroth approximation, The first approxima­ tion is the approach by the Spin wave theory on the basis of the simplified Hamiltonian H°, By taking account of some of the higher terms omitted in the first approximation and applying the first order perturbation theory, we obtain the second approximation, and fortunately for us here, the results are convergent, By the canonical transformation defined by and using the following Fourier transforms, we have Q j We introduce also the fourier components of the creation and annihilation operators by 1 56 a, = 'N' = |CXU + X2X), * o i * -iAj i * * LX jj) 23je = 2^X1X+X2X .,, (6.1-22) bx - (l>iEV 1Ak - ! . * ,2.i . * iAk. 1 * v . bX = “bke 5 (X1X_X2X> where Xu = qlx + iPlx , Xlx - qlx - *plx. p * =

//2 < J r (6 1-23) ' S X “ ( P 1X - P 2 A ^ /2 RX = (0- l x - °-2x ) / ' ' 2^ < is written as, H1 = X|J|SN j ^ p K \ r t * * l 1\b1,bv , V A lk •24) Using the following averages, . " V x * = ‘V l * = 3 • . . . C6. 1-25) = = = = 0 = 157 To the first order of H , the Partition Function can be written as F e = Trace! exp(-H /kT)(l-H /kT)l =[ Trace(expC-H°/kT) )1 (l-

/kT) ... C6.1-26) and the free energy as = log [Trace(exp(-K°/kT))]-

/kT (6,1-27) The derivation of (6.1-27) from (6.1-26) is known to be unsatisfactory, from the mathematical point of view, but (6.1-27) is rigorous in the first order approximation of H1 . Noticing that all products of operators such as a^a^, a^b^, b^b^, a*} a,, ^ and so on have averages equal to zero, if the wave number A and y are different, we find that = - 1 (A+C) + N J (1+4AS)AC (6.1-28) where AA = rrE \ > * = r2NF XE x x and C = Cj^)S \ 2 * * = N ^ aihx> 158 If we insert, the expressions of Eq and Ê , given in table E-l in appendix E, we obtain the energy in the second approximation E = - i ̂Nz|j|1 S(S+C o -ACCo2S'-1) + Nz|J|S(1+2XDCqS — 1 )C104̂-1 where C o ’, and CL1 are easily found in Apjj endlx E, and D is the Dimentionality of the lattice. >P6na: The following tables shows the ground state energies of antiferromagnets calculated respectively by Kubo and by us using our method for s = ^ TABLE 6.2 - Ground state energies of antiferromagnets (by Kubo) ( r Lattice * -Eo/(Nz|j|S/2) Linear chain - - - - - 1 S+0,363 + 0,033s-1 = S+0,363 + 0,066 Quadratic layers 4 S+0,158 + 0.0062S-1 = S+0.158 + 0 .0 1 2 NaCl-type V 6 S+O.OS7 + 0.0024S-1 = S+0,097 + 0 ,0 0 5 CsCl-type 8 S+0.073 + 0.0013S"1 = S+ 0.073 + 0 .0 0 3 159 TABLE 6 . 3 - Ground state energies of antiferromagnets (by our technique) Lattice z -Eo/(Nz|j|S/2) Linear chain 2 S + 0,363 + 0,130 Quadratic layer 4 S + 0.158 + 0,024 NaCl-type 6 S + 0.097 + 0.009 CsCl-type 8 S -^9.073 + 0,005 Indeed our expansion formalism can be extended to antiferromagnetism, This i encouraging in view of the recent researches (Mackintosh 1988, Sogo, 1987) which indicate that Spin waves in antiferromagnetism are relevant to high temperature superconductivity. In other words, unlike low-temperature superconductors which are diamagnetic, the age-old spin-wave theory may be the starting point for the exciting new phenomenon of super­ conductivity in antiferromagnetics. YIe envisage that, since an antiferromagnet is essentially an interlace of two ferromagnets, our investigations reported here will be highly germane to the current effort to understand high temperature superconductivity and ferromagnetism itself. 160 6.3 Conclusion The Developed Formalism of Holstein-Primakoff-Oguchi actually deals with a system consisting of some abstract quasi-particles whose statistical behaviour is to be determined, A point of view suggests that such a gas consists of quasi-particles whose field amplitudes obey a set of commutation relations. Therefore it looks as if the system is a system of Bosons, However such a boson system is not real owing to the fact that the occupation number operator a+a . is restricted to the eigenvalues n^ = 0,1,,..,2S whereas the real boson system would require also the eigenvalues for nj>2S, As we know, these latter values are unphysical. However, there are tso limiting situations, where the boson picture represents a fair approximation to the real system, The first is the case where S is large enough to justify the use of commutation relations for bosons, The other case is that of the Dynamical behaviour of the system in the ground state or state very close to it, In both cases, NOVAKOVIC (1975) and OGUCHI (1959) considered the op.e rator aJ+ aJ./2S when applied to the eigenvectors as a small quantity compared to unity, Therefore expansions were made in powers of a.J a.J/2S to some order to investigate the thermodynamic properties of an exchange interaction acting between the ferromagnetic spins at low temperatures, In our work, we have expanded a*a./2S to all orders in S U J and we have shown that in a consistent expansion of a -ja -i a y s ) - (i - -is1 ) 2 up to wave-wave interaction for Heilse,nberg Ferromagnet of N spins in a physical volume VQ each of spins S, with z nearest neighbour nd J>0, the quantized spin waves called magnons at temperature T are Bosons with effective chemical potential p. The existence of wave-wave interaction and hence of non­ zero p gives rise to a lowering of the thermodynamic internal energy and entropy. The spin waves, on the average thus form spin complexes, The wave-wave-wave interactions is negligible in comparison with the wave- wave interactions, The dynamical term which is depicted by the coefficient of T 4 is both numerically and experimentally negligible, as compared with the other coefficients of Tv. 162 The kinematieal term accounted for with intermediate statistics is negligible for ferromagnets with large exchange integral J values. The phenomenon of magnetism still remains incompletely understood, even though it is one of the oldest observed in the annals of physics. In the quest to elucidate it, spin-wave theory remains an amenable theory at low temperatures, Within the theory, for both Ferro- and antiferro-magnetism, our expansion formalism simplifies, the logic of spin wave- spin wave interaction by enabling one to treat the waves as ideal magnons with effective chemical potential and obeying Bose-Einstein or intermediate statistics, 6.4 Suggestions for Further Work Our work is basically on the Ferromagnets, The following are some suggestions for further research work (i) A consistent expansion of (1-a a/2S) (i) 2 up to wave-wave interactions can be done for the antiferromagnets as we have attempted and outlined in the discussion (section 6,1), the perpendicular susceptibility can be examined and its dependence on T obtained, 103 (ii) The fundamental question "Are the magnetic electrons localized or itinerant needs to be fully answered, (iii) The intimate link between magnetism and super­ conductivity in the high temperature supercon­ ductors can be found since the progenitor of the high temperature superconductors, LaCuO^ exhibits antiferromagnetism, even though band calculations indicate that the exchange interaction is far two small to induce magnetic ordering in a pure, perfect crystal, 164 REFERENCES Anderson, P, (1952). 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Electrons and Phonons - The theory of transport phenomenon in solids, Oxford University Press, 170 APPENDIX A COMMUTATION RULES To show that the spin operators satisfy the commuta­ tion rule aj % - aA = sJlt The spin operators are written in the following forms J r S+ = (2S)zfa, S_ = (2S) 2 a+f where xf L-1 - &2_ + S& \; 2 1 o2 _ - _+’ 1 ~ R2SRl writing e = [fa,a fl = f[a,a fl + [f,a fla = f[a,a+lf + fa+[a,fl + [f,a+]fa + a+[f,fla = faf + fa+%j/ y+fa Cl) [a,fl+ = (af-fa)+ = fa+ -a+f = [f,a+] where + A T a = [a,a+], y = [a,f] or af-fa = y (f2,a+£ V f [ f , a +l + [ f , a+] f - - ,a+] = ^ [f2,al' = f[ f ,a]+[ f,ajf = - , a] = +§| 1• • ®_ * Y + Y f -a2 aC •. < * C4) 171 fy+yf = -aa2S , C4») [y,fl = [[a,fj,fj i= [af-fa,f] = aff-faf = -faf+ffa p aff+ffa-2faf = aff+ffa - 2fCfa+y) = aff-ffa-2fY - [a,ffl -2fY = ^| | --2fiyY 4 . yf« -fo y = —oa2q a ' 20fn S y or yf n = - arjag - f„y ...C5) + U -Y+f - / f - - g -fr+ \ » • t (51 ) e = aff+fa+y+Y+fa = aff+fa+Y-fY+a - a a " - 2Sa+a + f (a+y-Y+a ^ ^ ^ ' ( 6 ) a + y = a +rl a,f_ j| = a+ a„f -a+ fa C7) y + a = fn a + a-a + fn a ( 8) a + Y-y + a - a + af-fa + a = 0 (9) 2Se = 2S(aff-a|^) p 2SaCl - ^ ) - a a +a = 2«CS-a+a) = 2(S-a+a) N J i 172 APPENDIX B PADE APPROXIMATES The Pade approximants are a particular type of rational fraction approximation to the value of function. The idea is to match the Taylor series expansion as far as possible. For example, we would like to pick an appro­ ximation of the form (a+bx)/(c+dx) & CB1) so that it would tend to a finite limit as x tends to infinity. We denote the L,l! Pade approximant to A(x) b y < ^ r [L/M] = Pl OO/Q jjCx ) ,,, (B2) where PL(x) is a polynomial of degree at most L and Q m(x ) is a polynomial of degree at most M, The formal power series A(:sx>) = J.=r o a j.x‘ / * • ( B3) d• • etermines the coefficients of P^(x) and Q^x) by *t he • \ equation A(x )-Pl (x )/Qm Cx ) - 0(xL+M+1) .,, (B4) Since we can obviously multiply the numerator and denominator by any constant and leave [ L/Ml unchanged, we impose the normalization condition 173 Q,jjC° ) ~ l . Q CB51 Finally we require that and Q̂ j have no common factors, If we write the coefficients of P^Cx) and as p Cx) = Pq+P-jX* ,,, + PxxL, y y S T ” • CB6) QmCx ) - l+q-̂ x + ttt + qyc1 then by eqn, (B5) we may multiply eqn,CB4) by which linearizes the coefficient equations, We can write out eqn,(B4) in more detail as o " Po al+aoql a2+alql+aoq2 = P, . . . (B7) V aL-lql*’' '+aoa-L - PL aL+l+aLql+ ’',+aL-MtlQM ” 0 aL +M +’ attL +M, -1qH11+,,,*aLqM - 0 where wm ,, CBS) 174 APPENDIX C CUBIC LATTICES Basic features of the simple, body centred, and face centred cubic lattices are given in table below. Lattice SlmPle J10?? FaceCentred Centred Unit cell volume a a Nearest-neighbour distance a/3/2 a//2 The number of nearest neighbours z A 6 8 12 The number of sites per unit cell 2 4 Using these data the following sum is read1ily calculated J(k) - Z• - eXp[iikk..((RRr,-BRt„)J ... C(l) 1 Z - Z Cos ... C(2) with R . = (0,0,0), R£ - (X^, , Z^} ... C(3) The Reciprocal lattice vector is defined by k tf*{k x ’, k y ’, k z } kCos(j)Sin0 k^ = kSinc()Sine k„ = kCose Nearest-neighbour distances are given below 175 Table CC1) ~ Nearest-neighbour distances on a simple cubic lattice in units a 1 1 . 2 3 4 5 X 1 -1 0 0 0 Y 0 0 1 -1 0 Z 0 0 0 0 1 Table (C2) - Nearest -neighbour Distances centred cubic lattice in un: £ 1 . 2 3 4 5 7 8 X 1 1 - 1 - 1 1 -1 -1 Y -1 1 1 - 1 - 1 1 1 - 1 Z 1 1 1 r 1 y -1 -1 -1 -1 Table B3 - Nearest-neighbour Distances on a face centred cubic lattice in units a/2 « 1 £ / 3 4 5 6 7 S 9 10 11 12 — X -1 -1 0 0 0 0 1 1 -1 -1 V 1 -1 1 -1 1 1 -1 -1 0 0 0 0 z 0 0 0 0 1 -1 1 -1 1 -1 1 -1 176 The Fourier transform is expressed as follows, for a simple cubic lattice J(k) = ^o lCos(k x a'J+CosCk y al+Cos(k za)l A . ™ For a Body centred cubic lattice, k a _k a. k a J(k) = JCos x C—os z_ cos -§- (C8) For a face centred cubic lattice, k a k a JCk) *= . Cos x. + Cos * ■ + Cos -J-] . , (C6) Introducing the abbreviations, Cos = a Sin<}> = 3 Cose = y / Sine = ,e)(ka)6+, . .} ... (C7) where the expansion coefficients are given by, for a simple cubic lattice, . 177 A = y7>{(aS) + (g ,,, (CIO) 178 APPENDIX D ON SOME INTEGRATIONS In the present analysis we use integrals of the form, 2x1+2, InCx] = q dq - (Dl) ° exp[ Cl-n 1 -1 where n = 0,l,2,.,,,n is some parameter such that t\ > 0, t is a dimensionless temperature defined by T ri kT2SJ (D2) By the substitution Q-^n] we obtain, ,,, CD3) CD4) CD5) TCV) 1 /tp/2 1 3/tt/4 2 15/tt/8 The Riemann Zeta Function 179 3 5 7 9v —2 2 2 2 2 e(v) 2.612 1,645 1.341 1,202 1,127 1.082 1,055 also use integrals of the form 00 2tT 7T 2n+2 Jn(x) = da d Sin0d0 — ) exp x-1 o where x = .1 (l-n)(qa).‘i -zAC<}>,e)(qa)‘:t+zBC,e)Cqa&)6+ t ,,] (D7) We introduce the substitution X = C(l-Dq2+Eq4)q2 , C = 4(l-n)a2 , D = zAU1>-9n) a' (D8) E = 6 Cq“2 = ----------\f-—--4- * x[ l+Dq2-Eq4+(Dq2-Eq4 )2]l-(Dq -Eq’ ) ,..(D9) C„ q 2 ^ X D ^ 2D“-E 2 .= x ( l + p X + ------ p— x + , , . ) = x{l+zA(4),e)Cl-n)“2Tx + [ 2z2A2(,e)Cl-n)"4 - zB( ij), 0 ) (1-n )~31 Ct x )2+, , . } ,,.(DIO) 180 There follows the result of Integration for the integral ■ v 2 TT V O - K 3 / 2 d(j> Sinede x 3 f(x ) dx exp x-1 o o o where f(x) = 1 + |zA(, 0 )(l-n)"2tx+[ 8z2A2(>,6)(1-n)”4 * (tx)2+., , 2tt 4. d<}> A(,0)Sin0d0 O O 2tt IT L2 = d A 2 ($,0)Sin0d0 ^ o o r 2tt ^ dcj) B(y lz ,l = 0 12 (v k x x XL + k y-̂y t + k z z £ )4 _ -1 Alt/, „w ,__x4 ^ 1 4! = ^ A" C (}>, 0 ) ( ka) AA"M<(. 12 *,e„)x = tir l ,k4x x,4 +, k,4y y4t +, ,k4z z4t +, 6..k2x.k y2X2ly2t + 6-.k2 .k 2y 2 z,2 + 6_ k 2 k 2 x„2z .2 ... 3, 3yz-'tt + 4k k x„y„ x z XL XL x y XLy XL + 4kx2 k z x2t z t + 4ky2 k x xXnL y2l + 4ky2kz y‘2tzt„ 4k^zkx xt zt2 + 4kZ2 k V Jy XL z2XL ') 182 12 ( k ^ t kyyt+. kazt)6 j Z =1 6! yfo E”Oi>,e)(kar B"(*,e) = + k®y® + k®zt + * 7$ y V * +7k4",k 2 4y x yJ„Zx 2 .Z + 7kx 2,k4 zx „2 4 4, 2 4 2 4,2 4 2rzz „Z + 7kx kz x .Zz „Z + 7k k y.z. +4kx^ ky x̂ y„ + 4k^k x^zn + 4k^k x y; + 4k^k ŷ z. zyz xrz z z y x z*z +4k5 , 5 , , 5 , 5 Q, 3 , 3 z k x x z + 4k k z v + 8k k x„ 3y z y. 3 + 8OkI 3 k, 3 z.3y .3 Z Z ZJ Z z y Z* Z x+08,k3 ,k3 x 3 z 3 x+ 11C6, 3, 2, 3 2 ^ 3. 3 2z x z Z ky k x k z yJ„z zZA x„Z + 16k k kx *yzA z„Z xZA 3, , 2 3 2 3, ,2 3 2 +16k? y k"zX?t z : + 16k K k z y M 1 6kz V y W t +x11c6,k3 .k k,2 3 2 x 10,2,2.2 2 2 2 , 1014. , 4 y x z 7v„i z.x„ z+ Z1 8ky k zk x x „Zy*.Zz .Z + 12kx ky k z x AyAzZA +12k4yk x k yfx.z + Z1 2k k k4xnynzf + 2k\ x^y z J Z Z yx z x y r Jl -u2ky5k x x v ZWJ Zxk z x^z +2k\ xnzfZ +Z2 zkx\ V„zn y z'vZ Z +.25k ,k z 5 v ^+8.k4 ,k2 x.4y .2 X+ 80k, 4k,2 2z y ZJZ x y ZJZ x.Jy .4 , ot4,2 4 2r £ + 8kx k zxAzA +8k̂zk̂ x xfz ẑZ +8ky^ kz^ yfz0+8k̂ k*vfzf +1J8z k^Z k zxk xy. yJr zf 183 +18k4y k x k z xfzj„ 4 it Z* Z + 18k 4 k k x4" yz„ + 1■ 2k* 3 q x *3 y*3k x y x y JT£ +^102,k3 ,k3 y3 3 ^ 10y zrz z Z + 12 1k.3tx k 3 x3 z„3 ^+ 7_2otz k 2 kT 2 ,k 2 x „2y .2 2 zz „Z +44kz3k x2k y x 2Z z3ZyJ Z + 44kz3ykx2^ki yt2izt3ixt + 44ky3 kz2 kx 'y f3cz2 +^4,4,k3 ,k2 ,k x„3z „2y „ . + 44k k k y„x„z„ + .4,43 3kt k 3k 2x .y.z, ... 3T 2, > 3 2x z y Z Z*Z y x zJz Z Z x y z » Z 184 APPENDIX E TABLE El: Thermodynamical quantities of antiferro- magnets calculated by the spin wave theory, Linear Quadratic chain Layer NaCl-type CsCl-type Eo S+0,363 S+0,158 S+0.097 S+0.073 (§)z|j|S ifi2 4.808 Oa3 3 = ^ 2 4 4*2e4Nz | J | S 3 6 TT 15 sT 3*8 7.212 Q 2 5?- 33/2,283 1i«6 TT 2 63TOE 15 '45 Mso S-0.127 S-0,078 S-0.075 x log2a M^r C2ir)“ 0̂ (7 log 34 02 2 9 2q23* N? a kT (2a" 4 l0g ̂ )92 3* e3 1 03 3 9 8Ao n7 2 y 2 7 l0g 2a finite 0.396S O „X2 l , 2 £/*_1 2 4 i .2 -\ u 2 - a & W ira £ O ’ (20)* * (2*)*’ 2 Q Xk 14,424 q2r e 33/2 2.3• TT 16 2.3 15 ’ 9 l T 7