THE EFFECT OF SURFACES ON THE PROPERTIES OF MAGNETIC MATERIALS

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THE EFFECT OF SURFACES ON THE PROPERTIES OF MAGNETIC MATERIALS

D. Mills

To cite this version:

D. Mills. THE EFFECT OF SURFACES ON THE PROPERTIES OF MAGNETIC MATERIALS.

Journal de Physique Colloques, 1970, 31 (C1), pp.C1-33-C1-48. �10.1051/jphyscol:1970106�. �jpa-

00213737�

JOURNAL DE PHYSIQUE Colloque C 1, supplément au no 4 , Tome 31, Avril 1970, page C 1 - 33

THE EFFECT OP SURFACES ON THE PROBERTIES OF MAGNETIC MATERIALS (")

D. L. MILLS (t)

Department of Physics, University of California, Irvine, California, 92664

Résumé. - Dans la première partie de l'article, on établit la forme de la matrice T qui décrit la diffusion d'un magnon par la surface d'un ferromagnétique type Heisenberg. Le modèle utilisé est tel que les constantes d'échange dans la couche de spins superficielle différents de leurs valeurs en volume. On obtient aussi la forme de la matrice sans qu'il soit nécessaire de spécifier l'arrangement géométrique détaillé des spins dans les couches parallèles à la surface où la portée de l'interaction d'échange dans les directions parallèles à la surface. La matrice T et la fonction de Green qui en résultent peuvent aussi être appliquées au calcul des propriétés du cristal semi-infini pour une grande diversité de géométries. Dans ce travail, on obtient la relation de dispersion des magnons de sur- face en examinant les pôles de la matrice T. Les résultats sont illustrés par des applications à un modèle spécifique considéré par Fillipov dans lequel des magnons de surface acoustiques endes sous de la bande des ondes de spin de volume ou des magnons de surface optiques au-dessus de la même bande peuvent apparaître. On utilise alors la matrice T pour calculer la durée de vie des ondes de spin après diffusion sur la surface. La durée de vie z(k) d'une onde de spin de vecteur d'onde k est donnée par l'expression simple z ⁼ LI1 G.vG@) 1 où L est l'épaisseur du cristal, 2 un vecteur unité normal à la surface et VG(k) la vitesse de groupe des magnons. Ce résultat est valide même pour les grandes valeurs de k et n'est pas affecté par les changements de constantes d'échange près de la surface.

Dans la seconde partie de l'article on fait une brève revue des travaux théoriques récents sur le ferromagnétique type Heisenberg semi-infini. On étudie également le comportement de la dévia- tion moyenne des spins près de la surface et l'effet sur la chaleur spécifique de surface des champs superficiels et des variations de constantes d'échange près de la surface. On décrit d'autre part quelques aspects de la théorie de la diffusion des électrons de basse énergie à partir des degrés de liberté magnétiques. Enfin, les propriétés des magnons de surface des antiferromagnétiques et la transition de renversements de spin en surface sont brièvement discutées.

Abstract. - In the first portion of the paper, we derive the form of the T-matrix that describes the scattering of a magnon from the surface of a Heisenberg ferromagnet. The model employed allows the exchange constants in the surface layer of spins to differ from the values appropriate to the bulk crystal. We also obtain the form of the T-matrix without the need to specify the detailed geometrical arrangement of spins in the layers parallel to the surface, or the range of the exchange interaction in directions parallel to the surface. The T-matrix, and the resulting Green's function may thus be applied to compute properties of the semi-infinite crystal for a wide variety of geome- tries. In this work, we obtain the surface magnon dispersion relation by examining the poles of the T-matrix. The results are illustrated with applications to a specific model considered by Fillipov, where acoustical surface magnons below the bulk spin wave band, or optical surface magnons above the bulk band may result. We then use the T-matrix to compute the lifetime of spin waves from scattering off the surface. We find the lifetime z(k) of a spin wave of wave vector k is given by the simple expression z ⁼ LI( /n: VG@) 1 where L is the crystal thickness, x a unit vector normal to the surface, and V G ( ~ ) the magnon group velocity. This result is valid even for large values of k, and is unaffected by changes in exchange constants near the surface.

In a second portion of the paper, we provide a brief review of recent theoretical studies of the semi-infinite Heisenberg ferromagnet. The behavior of the mean spin deviation near the surface, and the effect of surface pinning fields and changes in the exchange constants near the surface on the surface specific heat will be examined. Some features of the theory of low energy electron scattering from the magnetic degrees of freedom will be described. Finally, the properties of surface magnons in antiferromagnets and the surface spin flop transition will be discussed briefly.

(*) Supported in part by the Air Force Office of Scientific Research, Office of Aerospace Research, U. S. A. F. under AFOSR Grant No. 68-1448.

(t) Alfred P. Sloan Foundation Fellow.

Technical Report No. 69-27.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1970106

C l - 3 4 D. L. MILLS 1. Introduction. - There has been considerable theoretical interest in recent years on the effects of crystal surfaces on the excitation spectrum of crystals, and on the effect of the surface on the dynamical properties of the region near the surface. One finds that under a variety of conditions, surface modes exist which have the property that when excited, only the region of the crystal near the surface parti- cipate in the resulting motion. The first discussion of such modes is due to Lord Rayleigh [l], who studied surface waves (surface phonons) on the elastic continuum model of solids. As you have heard from Maradudin [2] earlier in this meeting, optical surface modes with complex properties are asso- ciated with free surfaces of polyatomic crystals.

In addition to giving rise to surface waves of various kinds, the presence of a surface leads to corrections to the thermodynamic properties of the material.

These corrections are proportional to the surface area.

The vibrational amplitudes of atoms in or near the surface also differ from the values appropriate to the bulk crystal. One thus has a wide variety of phe- nomena associated with the presence of the crystal surface. At the moment, the experimental study of these phenomena has been limited, because of the obvious difficulties associated with preparation of controlled surfaces, and the problem of probing the surface once it is prepared. Encouraging developments in the technique of low energy electron diffraction leads one to hope that the arnount of information available from experiment may greatly increase in the near future.

As remarked earlier, Maradudin has provided a discussion of recent work on the theory of surface phonons, and the effect of these modes on a number of properties of the crystal. In the present paper, we will examine the effect of surfaces on the spin wave spectrum of magnetic crystals, as well as other phe- nomena associated with the surface region.

We shall confine Our attention to the study of the effect of surfaces on the properties of the Heisen- berg ferromagnet. The effect of dipole interactions between the spins will be ignored. The dipolar inter- actions affect the nature of the excitation spectrum principally in the region of very long wavelengths, where a large fraction of the excitation energy of the spin wave comes from the Zeeman energy of the spin system in an external magnetic field, and from the macroscopic fields generated by the spin motion.

While the effect of dipolar interactions on the nature of surface magnons has been the subject of a consi- derable amount of work [3]-[5], at most temperatures of interest, the contribution from the magnetic degrees of freedom to the thermodynamic properties of magne- tic materials comes from spin waves with wavelength sufficiently short that an adequate description of these modes may be obtained by confining one's attention only to the effect of the short range exchange interactions between the spins. In this paper, we shall

be interested primarily in the effect of the crystal surface on the thermodynamic properties of the crystal, such as the specific heat and the mean spin deviation near the surface.

We shall proceed by finding the T-matrix that describes the scattering of magnons from the surface of a semi-infinite Heisenberg ferromagnet. Since the T-matrix is related to the one-magnon Green's func- tion in a simple manner, these functions allow one to study the effect of the surface on a number of proper- ties of the semi-infinite crystal in the region of low temperatures, where spin wave theory is valid. We find the form of the T-matrix for the case in which the exchange constants in the surface differ from the values appropriate to the bulk crystal. To obtain the general form of the T-matrix, it is not necessary to specify the geometrical arrangement of spins within the atomic Iayers parallel to the surface, nor is it necessary to assume a particular model for the range of the exchange interaction within these planes.

We do assume the interaction has a short range in the directions normal to the surface, in the sense that a given spin is presumed to be coupled only to spins within the same atomic layer, and spins in the layer immediately above and below the given layer. Our model is sufficiently general enough to allow the T-matrix to be applied to a wide range of surface configurations and crystal geometries.

The dispersion relation for surface magnons is obtained by searching for poles of the T-matrix outside the frequency regime associated with bulk excitations. We apply the general form of the disper- sion relation to describe the nature of surface waves in a simple cubic ferromagnet, with nearest neighbor exchange coupling only, a (100) surface, and exchange constants in the surface layer that differ from the values appropriate to the bulk. This geometry has been considered previously by Fillipov 161.

We also calculate the lifetime of bulk spin waves that results from the scattering of a spin wave from the surface. This quantity enters the theory of the spin wave contributions to transport coefficients at low temperatures, where the spin wave mean free path remains finite only by virtue of the scattering of the magnons from the crystal surface. Our discussion is appropriate to the case where the surface is smooth, and the excitations scatter from the surface in a specular manner. We shall see that the lifetime z(k) of a spin wave of wave vector k is given by z(k) = L/(%.V,(~)) , where% is a unit vector normal to the surface, and V,(k) is the group velocity of the spin wave. This result is independent of changes in the exchange constants in the surface layer, and also applies to modes with large wave vector.

As we have discussed in the preceding paragraph,

surface magnon modes exist under a wide variety

of conditions. These modes are eigenstates of the

Hamiltonian of the semi-infinite crystal, in the spin

wave approximation. Thus, there will be a contribu-

THE EFFECT OF SURFACES ON THE PROPERTIES ( 2 1 - 3 5 tion to the specific heat from the surface waves.

At the same time, the mean spin deviation

will increase as one approaches the surface, where the enhancement of the mean spin deviation from the presence of the surface modes is encountered.

A number of theoretical discussions of the surface specific heat and dependence of A upon distance from the surface in ferromagnets 171, [9] and antiferro- magnets [IO] have appeared. In al1 of these calcula- tions, it is important to realize that in addition to giving rise to surface modes, the presence of the sur- face also modifies the distribution in frequency of the bulk modes and the form of the eigenfunctions associated with these excitations. Thus, the contri- bution to the change in A near the surface and the surface specific heat from the change in the nature of the bulk excitations must also be included in the calculation. In general, the contribution from the change in character of the bulk modes tends to partially cancel the surface magnon contribution.

We discuss this phenomena briefly in the present work by summarizing a recent study of the surface specific heat of the Heisenberg ferromagnet, in the presence of altered force constants in the surface layer [9].

I t should be remarked that very similar cancellations occur in the one electron theory of metals, in situations where the electrons move under the influence of a potential that produces very weakly bound states [Il].

In a final section of the paper, a very brief review of studies of other surface phenomena in magnetic crystals is presented, including the magnetic field induced surface spin flop transition in antiferro- magnets.

As mentioned earlier, the amount of experimental information available that bears on the phenomena discussed above is very limited. Recently Meyer and CO-workers [13] have reported studies of the specific heat of small particles of YIG in the liquid He tempe- rature range. These authors find a contribution to the specific heat of their sample that varies linearly with temperature. While the temperature dependence of the observed excess specific heat observed in this work agrees with that predicted for the leading surface contribution 171, 191 (in the absence of pinning fields in the surface [SI), the magnitude of the observed excess is much larger than that predicted by theoreticaI analyses of simple models of the kind described above. It is interesting to note that Meyer and CO-

workers have also measured the longitudinal nuclear relaxation Tl in their sample. They find Tl to be shorter than the bulk value by more than two orders of magnitude. 1 am not aware of any detailed theo- retical study of the expected decrease in Tl that would result from the enhanced values of A near the surface region. Nonetheless, since the theoretical analyses of the models discussed above show A in the surface layer to be twice the bulk value, it is difficult to see

how the very large decrease in Tl can be accounted for within the framework of a mode1 in which the crystalline arrangement of spins in the finite crystal remains perfect, with changes in the exchange cons- tants confined to a region near the surface only a few atomic layers in thickness.

II. The form of the T-Matrix for the semi-infinite Heisenberg ferromagnet. - (a) GENERAL CONSIDERA-

TIONS. - In this section, we shall consider the effect of a pair of free surfaces on the properties of the Heisenberg ferromagnet. The discussion will be carried out within the framework of the spin-wave approximation. In order to establish the notation that will be employed in the discussion, suppose we first consider the infinitely extended lattice of spins.

I t will be convenient to consider a crystal in the form of a large parallelepiped, with periodic boundary conditions applied.

In the spin wave approximation, one introduces the set of single spin deviation states 1 1 > defined by

where 1 O > is the ferromagnetic ground state of the crystal (in which al1 spins are directed in the + z direction), S is the spin angular momentum of each ion in the monatomic Bravais lattice of spins, and S(-,(l) is the lowering operator that creates a spin deviation localized at the lattice site 1. The eigenvalue equation from which the spin wave excitation energies are obtained has the form

The matrix Do(], Y) is a function only of (1 - Y), and assumes the form

In eq. (2.2), J(6) is the exchange interaction between a spin at site 1, and a spin at the site 1 + 6. The sum over the quantity 6 ranges over al1 values of this variable for which J(6) # 0.

The eigenvalue equation in eq. (2.1) is diagonalized by making a unitary transformation from the localized spin deviation states ] 1 > to Bloch wave states 1 k > :

The Bloch state 1 k > is an eigenstate of the operator D,(I, l'), with the eigenvalue Q(k) given by

Eq. (2.3) is the dispersion relation for spin waves

in the infinitely extended crystal. It is well known

that at low temperatures, where the number of spin

C l - 3 6 D. L. MILLS waves thermally excited in the crystal is small, the spin waves may be viewed to be a gas of bosons that interact weakly through the non-linear terms in the equations of motion.

Let us now include the effect of a free surface on the excitation spectrum. In this section of the paper, we employ a device that has been utilized frequently in studies of surface phenomena ; we suppose that two free surfaces are created by passing a ficticious mathematical plane between two adjacent layers of spins, and then reducing to zero the strength of al1 interactions between spins on opposite sides of the plane. For definiteness, we suppose that we consider a crystal constructed of atomic planes parallel to the x-y plane, and we imagine that the surfaces created by the above procedure are associated with the layers labeled by the indices 1, = O and 1, = f 1. After the

«bond breaking » process just described is carried out, one may treat the resulting configuration by the methods that have been developed to study crys- talline defects. 5y breaking the bonds, one has created an extended, two dimensional defect in an otherwise perfect crystal.

The excitation energies of the now imperfect crystal may be obtained by solving an eigenvalue equation of the same form as eq. (2.1) :

SZ 11 > ⁼ C D(I,It) Il'>

1'

where the matrix Do(l, 1') is the dynamical matrix appropriate to the pure crystal, and AD(1, 1') describes the changes in the equation of motion that results from the formation of the two surfaces.

An important property of AD(l,ll) is that it depends only on the differences (1, - 1:) and (1, - 1;) in the x and y coordinates of the sites I and 1'. (Recall that the x-y plane is parallel to the two surfaces.) This result follows simply upon noting that the process of forming two surfaces does not destroy the trans- lational invariance of the crystal, as far as transla- tions parallel to the x-y plane are concerned. In gene- ral D ( 1 , 1') will depend on both Z, and Zi, however.

For many purposes it will be convenient to intro- duce the Green's function

G(I,I';z)= < I ~ { z I - D ) - ~ ( ~ ' > , (2.5) where z is a (cornplex) frequency, 1 is the identity matrix on the N x N vector space spanned by the states I I >, and D is the operator constructed from the matrix D(1, 1').

The Green's function G has many useful properties since it may be related to a number of properties of the crystal. To see this, let 1 ^s > be an eigenstate of the operator D with frequency a,. Then since the exact set of eigenstates form a complete set on Our N x N dimensional space, we write

I l > = C l s > < S I ] > (2.6)

S

and

Now notice that we may write

The last step follows upon noting that the set ( 1 >

is complete. We then see that the density of states p(0) associated with spin waves in the crystal with two free surfaces is

Thus we can study the effect of the presence of the surfaces on the density of excited states of the material, once the Green's function is known. If

is the Bose-Einstein factor where /? = l l k , T, then the contribution to the interna1 energy from the magnetic excitations at temperature T is given by

while the spin wave contribution to the specific heat is

Consider the spin deviation

associated with site 1 when the mode 1 ^s > is excited.

Upon employing the well known relation between S,(l) and the boson annihilation and creation opera- tors a(l), a + ( ] ) and the properties of the states 1 ¹ >

introduced above, one has

When the system is in thermal equilibrium at tem- perature T, then

The results displayed in eq. (2. S), eq. (2.9) and eq. (2.10) combined with studies of the effect of the presence of the surfaces on the Green's function have formed the basis of investigations of the magnon contribution to the surface specific [7], [SI heat, and the variation of the mean spin deviation with distance from the surface [7].

We conclude this section with a direct physical

interpretation of the quantity G(1, 1' ; 0). Suppose

THE EFFECT OF SURFACES ON THE PROPERTIES C l - 3 7

for times t < O the crystal is in its ground state 1 O >.

Then at t = O, we create a single spin deviation at site 1.

We ask for the probability amplitude A(lr, 1 ; t) for finding a spin excitation on the site 1' at the time t.

Now by construction, for t < O, A(lr, 1 ; t ) r O since the crystal is in the ground state. Upon creating the spin deviation at t = O, we place the crystal in the state 1 1 >. At time t > 0, the state of the crystal is described by

l t > O Thus, if 8(t) =

O t < O , one has

i

- dOG(It 1; 0 + ig) e-"'.

271. -,

Thus, the Fourier transform with respect to time of G(lr, 1 ; z) with z = D + ig is just the probability amplitude A(1' 1 ; t), within a factor of i.

We next turn Our attention to the computation of the Green's function G(1, 1' ; 2).

(b) THE COMPUTATION OF G(11' ; Z) ; THE T-MATRIX We now turn to the computation of the Green's function. The Green's function G(l, 1' ; z) has the form

where G(z) = { zI - D )-'. We then employ the operator identity

to write

where Go(z) = { zI - ^{Do )} -' is the Green's function appropriate to the bulh crystal, in the absence of surfaces. Upon taking matrix elements of eq. (2.1 l), we have

In eq. (2.12), the propagator Go(ll' ; z) describes the pure crystal with no surfaces. As a consequence, this quantity depends only on the difference (1 - 1') between the quantities 1 and 1'. One rnay write Go in the form

where the sum is over the first Brillouin zone of the perfect crystal, and o(k) is the frequency of a spin wave cf wave vector k in the pure crystal.

We now introduce the T-matrix TQI' ; z ) that is

related to the Green's function G(11' ; z) in the following manner :

+ Go(llu; z)T(1"1"; z)Go(l"I'; z ) . (2.13)

l"1"'

The physical significance of the T-matrix rnay be appreciated by referring back to Our discussion of the relation between G(lll, D + ig) and the probability amplitude that a disturbance will propagate from site 1' in the time t. Suppose we create a disturbance at site 1 at time t = 0, and inquire how the disturbance will propagate to site 1'. There are two paths along which the excitation rnay propagate from 1 to 1'.

The disturbance rnay travel directly from 1 to i', through the bulk crystal. The amplitude for this process is given by the first term Go(IIt, z) on the right hand side of eq. (2.13). On the other hand, the excitation rnay propagate from 1 to the surface region, then scatter from the surface, and the scattered wave will arrive at 1'. This second process is described by the second term of eq. (2.13). We shall see later that T(I" l", z) vanishes unless both 1" and 1" refer to sites disturbed by the « bond breaking » procedure that created the two surfaces. Thus the second term in eq. (2.13) describes propagation to the surface from the initial site 1 (the factor Go@", z)), the scattering of the excitation by the surface (the factor of T), and the propagation through the bulk back to the final site 1' (the second factor of Go). This process is illustrated schematically in figure 1. When 1 and 1' are on opposite

FIG. 1 . - The paths by which an excitation rnay propagate from a site 1 to a site 1'.

sides of the surfaces, the two terms in eq. (2.13) precisely cancel, so the amplitude for propagation of the distrubance across the gap between the surfaces is identically zero.

Thus, the information concerning the interaction of

spin waves with the surface region is stored in the T-matrix. There is one other property of T that we point out at this time. From eq. (2.7), we see that G(llt, z) has poles at the excitation frequencies of the crystal. Any surface modes present after the surfaces are formed must give rise to poles in G. Now suppose we examine the structure of eq. (2.13). The factors of Go have poles only at the excitation frequencies of the perfect crystal, as one can see from eq. (2.12a). (Of course, if the sum over k is replaced by an integration, the sequence of poles merge into a branch cut on the real axis.) Thus, any modes that lie outside the frequency region associated with spin waves in the bulk region must give rise to poles in the T-matrix.

Once we obtain the form of T, we can find the dispersion relation for surface spin waves by seeking the poles of T outside the bulk spin wave bands.

We may find the equation obeyed by T by substituting the form for the Green's function in eq.

(2.13) into eq. (2.12). After some straightforward algebra, one finds

We shall shortly solve eq. (2.14) for T, for a fairly general model of a Heisenberg ferromagnet. Before we proceed, one should note that TOl' ; z) is non-zero only if both 1 and 1' lie near the surfaces created by severing the exchange couplings between the layers 1, = O and 1, = 1. More precisely, the region of the crystal disturbed by creating the surfaces consists of the sites 1 for which the perturbation matrix AD(1,l') is non-zero. From the iterative solution of eq. (2.14), one may see that T(l1' ; z) is zero unless both 1 and 1' lie in this set of sites. Thus, so long as the bond breaking procedure directly affects only spins within a few atomic layers of the surface, TOI', z) will be zero unless both 1 and 1' are near the surface, inside the affected volume. Since forming the two surfaces destroys the periodicity of the structure in the z direction, ADO, 1') depends on both 1, and 1;' and not just on the difference of these quantities. It follows from the fact that G,(l", 1"' ; z) in eq. (2.14) depends only on (l" - 1") that the T-matrix is a function only of the four quantities (1, - Il), (1, - Ii), 1, and 1;.

The most convenient mathematical method for exploiting the partial translational invariance » discussed in the preceding paragraph is to make a partial Fourier transformation on al1 the quantities that appear in the equation. We introduce the quantities t B l l ; l, 1; ; z), Ad(kll ; l,l; ; z ) and g F l l ; l,l; ; z) by writing

with a similar relationship between AD and Ad, G

and Ag. In eq. (2.15), Ns is the number of atomic sites in the atomic planes parallel to the free surfaces, the sum of k l l is over the two dimensional Brillouin zone appropriate to the translational symmetry in the x and y directions, and k l l = Xk, + Pk,. If n is the number of atomic layers parallel to the surfaces in the large macrocrystal introduced before the free surfaces were formed, then one has the explicit form

In eq. (2.16), the sum over k, ruas from - (du) to

+ nia, where a is the spacing between the planes parallel to the surface.

Eq. (2.14) may now be transformed to an equation for t(kll ; IzlS ; z) :

Since the same wave vector kll appears in each quantity in eq. (2.17a), and the same complex frequency z appears both in g and t, we shall abbreviate the notation by suppressing explicit reference to these quantities in much of the discussion that follows.

We shall return to explicitly display these dependences when confusion may result. We thus rewrite eq. (2.17a) in the abbreviated form

The virtue of the form of eq. (2.17b) is that (except for the common dependence of the various quantities on the common variables k l l and z) only the plane indices 1, appear. If the formation of the free surfaces affects directly only a small number of atomic layers of spins because of the missing exchange bonds, then Ad(l,I;) is non-zero only for a small number of plane indices 1, and 1;. Just as we saw in Our earlier discussion, t(1, 1;) is non-zero only for the values of 1, and 1:

for which Ad(&, 1;) # O. Thus, if the formation of two surfaces layers by the bond breaking procedure described above affects the environment of N layers of spins, both Ad and t are N x N matrices, and we can find the form of t(kll ; 1,lS ; Z) for a given kll and z by inverting a small matrix, provided N is small.

In the next section, we shall obtain a simple analytic form for t(kll, lZ 1; ; Z) by applying eq. (2.17b) to a class of surface geometries that includes several cases treated in the literature.

(c) THE FORM OF THE T-MATRIX FOR A SPECIFIC CASE.

- We shall derive the explicit form of the T-matrix

for a model with the following propeities : We suppose

two free surfaces are created by the bond cutting

THE EFFECT OF SURFACES ON THE PROPERTIES C l - 3 9 procedure described above. The large crystal upon

which the periodic boundary conditions were imposed initially is thus severed into two semi-infinite crystals that consist of a series of atomic planes parallel to the two surfaces. We shall allow a spin in one such atomic plane to interact with its neighbors in the same plane with exchange interactions of arbitrary range.

However, a spin in a given atomic plane interacts only with spins in the same plane, the plane directly above it, and the plane directly below it. Thus, we presume the exchange interaction has a range of only one interplanar spacing in the direction normal to the surface. We shall also allow the exchange couplings between spins in the surface layers to assume values that differ from those appropriate to the bulk crystal.

Two modeis discussed elsewhere in the literature are special cases of the more general case considered here.

In the first work to discuss the dispersion relation of surface magnons in the exchange dominated regime, Maradudin, Wallis, Ipatova and Klochikhin [14]

examined the properties of the simple cubic ferromagnet with nearest and next nearest neighbor interactions, with the exchange constants in the surface taken equal to the values appropriate to the bulk.

Filipov [6] shortly there after considered the nature of the surface excitations in the simple cubic ferromagnet, with nearest neighbor coupling only, a (100) surface, and exchange constants in the surface that differ in value from those in the bulk.

We may determine t(l,, 1;) once Ad(l,, 1;) is known.

Let us begin by considering the dynamical matrix Do(ll') appropriate to the perfect, infinitely extended crystal. If the interplanar spacing is denoted by a, we begin with eq. (2.2), and write Do(l, 1') in the form

do(kll ; I,, 1:) = O for al1 other values of 1:. We have defined

We have presumed that the crystal structure is such that bl(kll) is real. From the fact that the three dimensional crystal is a Bravais lattice (and hence each site has inversion symmetry), it then follows that bl(kll) = b- ,(kIl). When the quantities do(kll, 1,I~) have the simple form given in eqs. (2.19), one may then easily show that the bulk spin wave dispersion relation is given by

Q(k) = A(kll) - B(kll) cos (k, a) , ^(2.20)

where

A(kll) = bo(0) - bo(kll) + ^{2 bl(o)}

and

B(kll) = 2 ~ I ( ~ I I ) .

A very convenient feature of the mode1 considered here, and the special cases that follow from is that the integrals that appear in the definition g(kII, lZ - zi ^{; Z)}

may be evaluated analytically, and also have a simple form [7], [a]. This enables one to obtain tractable, closed expressions for the Green's function G(ll', z) and the T-matrix.

From the form of eqs. (2.19), it is simple to see how do(kll, 2, 1:) changes upon breaking the bonds and creating two free surfaces associated with the planes 1, = O and 1, = 1. First consider the change Ad in the quantity do(kll, 1, IL) that results from the absence of exchange coupling between the layers 1, = O and l z = l :

Do(l, I')'= 6t,I; S J(6) ( S,,, ^y,l - 6,,, ,,,,, ^+&,, 60,d, ) - (i) The exchange field on the spins in the layer

6 1, = O and 1, = 1 is reduced by the absence of the

- 61.+1,~; S

~ ( a ; + ^6,,) ~ I , , , I , , , + S I , (2.18) inter-layer coupling. This effect may be accounted for by deleting one factor of bl(o) from do(kll, 00) and

- ~ ~ z - 1 , 1 ~ s ~ J ( - a ' + 6 ~ ~ ) ~ ~ ~ ~ ~ ~ ' l ~ + ~ ~ l ' do(kl,,ll).

S

(ii) The off diagonal terms that couple 1, = 0 In eq. (2. 18), the first term describes the contribution and 1, = are reduced to zero, i. e.

to Do from the exchange field seen by the spin 1,

and the term that allows the excitation to transfer Ad(kll, 01) ='Ad(kll, Io)= - do(kll ; l,lz + 1)= + ^bl(kll).

from 1 to spins in the same atomic plane, while the second and third terms describe the transfer of the excitation from 1 to spins in the layer just above and the layer just below that in which the spin lis located.

Now we introduce the quantity do(kll, Z, 1:) defined by the relation

Finally, we need to include the effect of altering the exchange constants in the surface layer. This effect may be included by replacing bo(kll) by bo(kll) - Abo(kll) in both do(kll, 00) and do(klI, 11). (We choose a sign convention so that Abo(o) > O when al1 the exchange constants are foflened.)

1 Upon combiiing the three changes in the dynami-

Do(IIf) = - _exp[ikll . ^{(1 -} If)] do(kll, 1, 1;) .

Ns

cal matrix just described, one has

A short calculation then shows that Ad(kll ; 10) = Ad@ll ; 01) = + S B ( k l l ) , (2.21a) and

~ o ( ~ I I ; lz,lz) = b0(0) - b0(kn) + 2 b ~ ( o ) (2.19a)

Ad(kll ; oo) = Ad(kll ; ^{= -} ' B(kb) (2.21b) do(kll ; lz, 1, +- 1) = - bl(kIl) = ; lz, ^jz - 1) , 2 IJ(kll) '

(2.19b) Ad(k II ; 1 , l ~ ) = O for al1 other 1, and 1; , (2.21~)

C l - 4 0 D. L. MILLS where we define

The properties of the quantity y(kll) will play a crucial role in the subsequent discussion. At this point, let us note that

lim y(kll) = 1 + terms of order (kll .

kli-+O

From eqs. (2.21), we see that Ad(l,, 1;) is non-zero only if 1, and 1: are either zero or unity. The discus- sion in the previous section demonstrates that the quantity t(l,, 1:) is non-zero only if 1, and 1: are both either zero or unity. Furthermore, from the fact that Ad(l,, 1:) is symmetric in 1, and IL, one sees that t(l,, 1;) is also symmetric in these indices. Thus, eqs. (2.17) degenerate into two equations in two unknowns. We shall solve explicitly for t(1, 1) and

~(0,1), noting that t(0,O) = t(1,l) and t(1,O) = t(0,l).

Upon noting that g(+ 1) = g(- l), and then inser- ting the form of Ad(&, 1,) into eq. (2.17), one obtains two simple equations for t(1,l) and t(0,l). We find the equations may be written in the form

Upon solving these two equations, we have.

t(1, 1) = t(0, O) =

C l - u21 g(0) + (g)

-

-

((Y + ^{1) (do) - g(l))} + 2) ^x

x ((Y - 1) (do) + ^{dl)) -} g)

- ^[y2 - 11 g o ) - g)

-

((Y + ^{1) (do)} - dl)) + 2) ^x

x ((Y - 1) (do) + g(1)) - g)

Eqs. (2.24) constitute a complete forma1 solution to problems of determining the effect of a surface on

the mode1 magnetic crystal in the spin wave regime, since knowledge of the form of T gives one the form of the Green's function. Before we proceed with an examination of the properties of the T-matrix, we shall place the result in a more convenient form. First of all, we employ a relation between g(1) and g(0) that one may prove from the definition of these quantities.

Bg(1) = 1 - (2 - A ) g(0) .

Then we introduce a quantity ~ ( k , k' ; z) related to the configuration space T-matrix in the following manner :

In the last line, we have introduced the quantity n = NINs, the number of planes of spins in the large crystal parallel to the surface.

The physical interpretation of z(kkf ; z) is that it is the amplitude that describes the scattering of an exci- tation of wave vector k to state k' by the surface. We shall employ this quantity to compute the contribu- tion to the lifetime of a bulk magnon from the pre- sence of the surfaces.

After some algebra, we find

In this expression, we have introduced the fre- quencies

and defined

The quantity 9, is the lowest frequency associated with bulk spin waves with wave vector cornponent k l l parallel to the surface (the mode has k, = O), while

QM is the maximum frequency associated with bulk

THE EFFECT OF SURFACES ON THE PROPERTIES C l - 4 1

waves with the wave vector component k l l parallel to the surface (this mode has k, ⁼ nla). The frequen- cies 9, and 9, thus bound the region of frequencies available to bulk waves with a fixed component of wave vector parallel to the crystal surface.

We shall examine the properties of z(kkl ; z) in the subsequent sections. We conclude the present discussion by exhibiting the form of g(0). This func- tion may be obtained by a straightforward integra- tion of eq. (2.16). One has

In eqs. (2.26), the square roots are to be taken positive.

(d) THE SURFACE MAGNON DISPERSION RELATION. -

The dispersion relation for surface magnons may be obtained by finding the poles of the T-matrix that lie outside the frequency region allowed for bulk waves, as we have remarked above. Thus to find the frequency of a mode with a wave vector k l l parallel to the sur- face, one may study the singularities of z(k, k' ; z ) (eq. (2.25)) that lie outside the frequency range [Q,, a,]. The only possible factors in z(k, k' ; z) that may give rise to poles in ^.t outside the bulk spin wave frequency regime are the two factors in the denomi- nator. One easily sees upon employing eqs. (2.26) that for 9 < Q,, and for 9 > 52, each of the factors in the denominator vanish for precisely the same values of 9. We examine the two cases B < Q, and 9 > 9, separately.

(i) The case SZ < SZ, ; surface magnons below the bulk band. - Upon inserting the expression for g(0) in the region 9 < Qm (eq. (2.26a)) into the expres- sion for z, one finds the denominator has a zero when Q ⁼ QS(kll), and

Since the left hand side of eq. (2.27) is positive, one sees easily that if O < y(kll) < 1, eq. (2.27) admits a solution with Qs(kll) < O,(kll), i. e. we have a surface magnon mode below the bulk band. The criterion for the occurrence of a surface mode split off below the bulk band is thus simply

When y(kll) < 1, one can solve eq. (2.27) for a s , and write the result in the form

eq. (2.29) has also been obtained by a direct examina- tion of the equations of motion of the semi-infinite medium [9].

Notice that as k l l

0, Our earlier remarks on the behavior of y(kll) imply that SZs(kll) and Om(kll) differ only by terms of order (kll a)4 in the long wavelength limit. This appears to be a general feature of low frequency surface magnons, in the presence of exchange interactions only [6], [9], [14]. This behavior was first pointed out by Maradudin et al. [14].

We now apply the theory to the geometry studied by Filipov [6]. We consider the simple cubic ferro- magnet with nearest neighbor coupling only, a (100) surface, and exchange constants in the surface sof- tened by an amount AJ, i. e. the exchange interaction between spins in the surface is taken to be J - AJ rather than the bulk value J. Then

for al1 values of kx and ky. One then finds Qs(kll) = Qm(kll) -

2

16 S LV [sin2 ($ ^{k, a) + sin2} (; ^{k, a)]}

(ii) The case 9 > 9, ; surface magnons above the bulk band. - One finds for 9 > 9, that a pole in z occurs on the real axis if

The left hand side of eq. (2.31) is positive, and also greater than unity, since B > 9, > Q,. The right hand side of eq. (2.3 1) can only be positive and greater than unity only if y < - 1. Thus, for surface modes with frequency greater than 9, to exist, we require

y(kll) < - 1 .

When this condition is satisfied, one may show that the frequency Q,(kll) of the surface mode is

If we consider the case where the exchange in the sur-

face layer is stiffened by an amount A J (i. e. changed

from J to J + ^{AJ), then}

C l - 4 2 D. L. MILLS One interesting feature of this form of y is that for any specific value of AJ, the condition y < - 1 can be satisfied only if 1 k l l 1 falls inside a limitedrange of values. The condition that y < - 1 requires that

First of all, if A J < 3 J, the inequalities in eq. (2.34) cannot be satisfied. No surface mode above the band occurs at al1 in this case. This is in contrast to the modes below the band, which split off from the bulk spin wave region for arbitrarily small AJ.

If AJ > + J, then a surface mode splits off the bulk band only in a region of k space of limited extent In particular, as 1 k l l 1

O in any direction, the left hand inequality in eq. (2.34) will be violated. Thus, as 1 k l l 1 + O, in any fixed direction, the surface branch will merge with the continuum, and « cut-off » at some finite value of 1 k l l 1, that will depend on the direction of kl, . If AJ < 4 J , then there are some direc- tions in the two dimensional Brillouin zone where the right hand inequality of eq. (2.34) is violated. This means that as k l l is increased toward the zone boun- dary, there will be some directions in which the sur- face mode will merge with the continuum, and disap- pear for 1 k l l 1 greater than this value. Thus, the sur- face branch will exist only over a finite region of the zone, with a cut off at small k l l , and a second cut-off at large values of k l l in some directions.

This example, also discussed in detail by Filipov [6], indicates that surface magnon modes that split off above the bulk spin wave frequency bands may exhibit diverse behavior. We conclude this section by poin- ting out that Wolfram and de Wames [15] have recently completed a detailed study of the dispersion relation of surface magnons for a wide variety of surface geometries. We have sketched the behavior of the surface magnon dispersion relation in Fig. [2] for the two cases AJ positive, and A J negative.

SURFACE EXCHANGE STIFFEND

R

SURFACE EXCHANGE

FIG. 2. - Sketch of behavior of the surface magnon dispersion relation for the example discussed in the text. The kind of beha- vior that occurs when the surface exchange is softened is indi- cated, as well as a possible behavior when the surfaced exchange

is stiffened.

(e) THE MEAN FREE PATH OF BULK SPIN WAVES IN THE PRESENCE OF SURFACE SCATTERING, - In this section, we employ the T-matrix to compute the cross section for scattering of a bulk spin wave from the surface. Earlier, we remarked that the quantity z(k, kt ; z) in eq. (2.25) is the scattering amplitude that describes scattering of a bulk spin wave of wave vec- tor k from the surface to a state kt. More precisely, the contribution to the lifetime z(k) of a bulk spin wave from this scattering is given by

- 1 = 2 n C 1 ^{~ ( k ,} k'; O(&) - iy) l2 ^{8(o(k) - o(kl)).}

~ ( k ) k ' # k

(2.35) (We hope the reader will not become confused by the use of the single symbol z to denote both the lifetime of the spin wave and the appropriate trans- form of the T-matrix). From eq. (2.35), one sees that we require the « on shell» value of the quantity z(kk' ; z), i. e. we replace z by the frequency o(k) of the incident wave, and evaluate z(kkt ; o(k) - ir]).

There is a direct analogy between eq. (2.35), and the quantum mechanical theory of the scattering of a particle by a potential well.

We simplify eq. (2.35) by noting that z(k, k' ; z) is proportional to 6 k l l , k ~ l l . We exploit this by writting

Then eq. (2.35) becomes

- 1 = 2 n 1 3(kll; k,, k:) l2 ^{6(o(k) - o(kt))}

z(k) k:

k,

x 6(B(kll) [cos k, a - cos kS. a]) 2 n L ' dk:

= --- - 1 ^{3(kll ; k,, k@} l2 x B(kIi) a 5 ²

x G(cos k, a - cos k: a) .

In the last expression, we have converted the sum over k, to an integral. The thickness of the crystal is L, and a is the distance between the planes parallel to the surface. The integral contains a contribution from the case k: = - k,. Thus,

1 L

- = - - 1

z(k) a2 B(kll) 1 sin k, a 1 13(kll;k,,-k3I2. (2.36)

The expression for J(kll ; k,, - kz) may be obtained

by finding the form of z(kkt ; D - iy) for Dm < D < DM,

then replacing D by w(k), and noting the definition

THE EFFECT OF SURFACES ON THE PROPERTIES C l - 4 3 of 3 that follows eq. (2.35). After some algebra, we

find

aB(k11)

~ ( k k ' ; - i l ) = 6klIir. -T, k a

(1 - y) cos "- ₂ - i(1 + ^{y) sin}

x ( 2 y sin k, a + ^{i(1 - ) { 1 +} eia(kL-kz)

[ ¹

( (1 + Y') sin k, a + i(l - y2) ^COS k, a ) x

and

aB(k11) J(kll ; k,, - k,) ⁼ + i- L ^X

[y exp ( - ik, a) - 11

x sin (k, a)

[y exp( + ik, a) - 11 ^'

It will prove useful to introduce an angle q(kll k,) defined by

sin (kZ a) [1 -

q(k,, kJ = tan-' . (2.37)

2 y - (1 + ^{y') COS (k, a)}

We then find that

aB(k,,) J(kll ; k,, - k,) = + ^{- i} L ^X

x sin (k, a) exp[- iq - ik, a] . (2.38) Upon inserting eq. (2.38) into eq. (2.36), one finds the simple result

1 aB(kll) 1 sin (k, a) 1 .

z(k) - L (2.39)

Eq. (2.39) may be written in a simple illuminating form by noting that since

1 ^G.~,o(k) 1 ⁼ 1 ; . ~ , ( k ) 1 ^{= aB(kll)} 1 sin (k, a) 1 .

Thus, for the lifetime z(k), one obtains the very simple result

where n is a unit vector normal to the surface.

Eq. (2.40) is a striking result that may be inter- preted very simply in kinematical terms. We should note that since we have not had to resort to a study of the detailed form of y(kll) to obtain this result, the expression for z(k) is independent of changes in the exchange constants in the surface layer, and the detailed geometrical arrangement of the spins in the

layers parallel to the surface. The result of eq. (2.40) applies also to magnons with arbitrary values of k, including those near the zone boundary.

III. Other phenomena associated with the crystal surface-A summary. - In the last section, the dis- persion relation for surface magnons was discussed, and the scattering of bulk magnons from the crystal- line surface was examined. In this section, we provide a brief review of studies of some other phenomena associated with the presence of surfaces in magnetic crystals. Since the material discussed in this section either has appeared in the iiterature at this time, or will appear shortly, we shall confine ourselves to a brief examination of the results and some features of the calculations. The reader will be directed to the literature for the details.

(a) THE SURFACE SPECIFIC HEAT, - In most text- book discussions of the thermodynamics properties of crystals, one assumes one is dealing with a very large crystal, with the appropriate periodic boundary conditions applied to the wave functions that des- cribe excited states of the crystal (Born-von Karman boundary conditions). The various intensive thermo- dynamic functions such as the specific heat, the inter- na1 energy, etc. are then found to be proportional to the volume of the crystal. In fact, if the crystal has a finite volume, and if realistic boundary conditions are imposed, the specific heat is not simply proportional to the volume of the crystal. The leading correction to the infinite volume form of the specific heat is pro- portional to the surface area of the crystal, and is presumably shape independent so long as the interac- tion between the ions has a short range.

A study of the surface correction to the specific heat of the Heisenberg ferromagnet was first discussed by Maradudin and Mills 171. In this work, a Green's function approach similar to that described in sec- tion II of the present paper was employed. The ana- lysis was carried out for a particular model crystal- the simple cubic Heisenberg ferromagnet with nearest neighbor and next nearest neighbor interactions between the spins, and a (100) surface with no changes in the exchange constants near the surface. It was previously pointed out that in this model, one has a surface spin wave branch below the bulk spin wave frequency region, with properties very similar to the surface mode described in the previous section of the present paper.

The surface correction to the specific heat C,(T) was found to have the form

where S is the total surface area of the crystal, ('(2) is

the Riemann zeta function of argument two, and D

is the curvature of the bulk spin branch near k = 0,

i. e. the bulk spin dispersion relation is o(k) ⁼ Dk2

in the long wavelength limit.

C l - 4 4 D. L. MILLS There are several comments that one should make concerning the result exhibited in eq. (3.1).

As mentioned in the introduction, recent experi- mental studies of a sample of small YIG particles have shown that a term linear in the temperature is present in the specific heat a t temperatures in the liquid He range [13]. When the result of eq. (3.1) is applied to the data, and the surface to volume ration appro- priate to this sample is employed, one finds that the observed linear term is larger than the prediction of eq. (3.1) by more than an order of magnitude. One might suppose that if the exchange constants near the surface (say, in the surface layer) are softened, then the mean spin deviation near the surface would be enhanced, and the low temperature specific heat then increased over the value exhibited in eq. (3.1). In fact, the model discussed in section II is appropriate for such an investigation, since we have found the explicit form of the Green's function. The changes in the exchange constants in the surface layer are incor- porated into the factor y&,,). Such a study has been carried out, although the change in density of states from the presence of the surfaces was constructed directly from the equations of motion of the appro- priate spin operators, rather than from the Green's function [9]. It is found that the result of eq. (3.1) is valid even in the presence of changes in the exchange constants in the surface layer. This result is a remar- kable one, since as we have seen, the nature of the excitation spectrum of the material is sensitive to these changes. It should be remarked that it is only the leading term in the low temperature speficic heat (the term linear in T ) that is insensitive to the character of the surface region. The corrections higher order in (TIT,) are, in fact, sensitive to the details of the surface geometry.

The lack of sensitivity of the result to the softening of the exchange interactions near the surface of eq. ( 3 . 1 ) means that it is difficult to account for the specific heat data [13] ou YIG particles by a model with spins arranged in a perfect crystalline array, with changes in the environment only in the surface layer. While the model employed in this last calcula- tion allowed the exchange constants to be altered only within the surface layer, we feel the result is very likely valid so long as the perturbed region near the surface is localized well within a thermal magnon wavelength A,, N 2 n(DlkB T)% of the surface.

Spins in the surface of a magnetic crystal are often subject to pinning fields not present in the bulk mate- rial [17]. The presence of surface pinning can greatly inhibit the motion of spins in the surface ; through the exchange coupling of surface to interior spins, these fields can in fact alter the mean spin deviation in the region near the surface. Thus, one suspects that the surface specific heat should be quite sensi- tive to the presence of surface anisotropy fields. A study of the effect of surface pinning fields on the magnon contribution to the surface specific heat has

recently been completed [8]. In the presence of pinning fields, we write the surface specific heat in the form

C U ) = R ( T ) ,

where C,(T) is the « no pinning » result of eq. ( 3 . l), and R ( T ) is a reduction factor that measures the abi- lity of the surface pinning field in reducing the sur- face specific heat. The factor R ( T ) is plotted as a function of T in figure 3. If 2 J is the strength of the nearest neighbor exchange interaction, the tempera- ture is measured in units of ( 2 J S / k B ) , and the para- meter ^E = gpg Hs/2 J S provides a dimensionless mea- sure of the strength of the surface pinning field, mea- sured in units of the exchange field felt by a spin from one of its neighbors.

FIG. 3. - The reduction of the surface specific heat by a surface pinning field. If the exchange interaction between nearest neigh- bor spins is 2 J, then

= ( g p ~ Hs/2 JS), and k~ Tz = 2 JS, where Hs is the magnitude of the pinning field in the surface

layer.

When E is small, and T sufficiently large, one has

R ( T ) x 1, and the pinning field is ineffective in reducing

the surface specific heat. For fixed E, T is lowered,

R ( T ) decreases and eventually approaches the limit

of - 1 as T + O. Thus for sufficiently low T, the sur-

face correction to the specific heat is negative. Phy-

sically, the reason for this is that at low temperatures

the pinning field inhibits the spin motion near the

surface, and reduces the contribution to the specific

heat from the surface region. Of course, the total

specific heat is always positive. The notion of decom-

posing the total specific heat into the sum of a term

proportional to the volume, and a correction propor-

tional to the surface area breaks down as soon as the

surface and volume contributions become compa-

rable [8].

THE EFFECT OF SURFAC :ES ON THE PROPERTIES C l - 4 5 We conclude the discussion of the surface specific

heat with some comments on the details of calcula- tion. There is one feature of al1 the studies cited above that involve geometries in which (low frequency) surface modes are present. When on writes down the magnon contribution to the specific heat for a geometry in which surface modes are present, quite clearly there will be a contribution to the specific heat from thermally excited surface magnons. From our discussion the nature of the dispersion relation in section II, one may easily write down the surface magnon contribution to the specific heat, and evaluate this quantity in the limit of low temperatures. This portion of the specific heat is proportional to the temperature T, and the surface area S of the crystal).

However, the surface magnon contribution to Cs(T) is found to be four times larger than the result exhi- bited in eq. (3.1). When the theory is examined in detail, one finds the presence of the surfaces alters the distribution in frequency of the bulk modes. In fact, the number of bulk modes of low frequencies is decrea- sed in such a way that 75 % of the surface magnon contribution is cancelled, leaving the remainder in eq. (3.1).

Let us make the above remark a bit more precise.

For the models examined so far (we ignore the effect of surface pinning fields in the present discussion), the change in the number of modes with frequency between SZ and SZ + d 9 , and wave vector k l l parallel to the surface may be written in the form

In the theory of the low temperature specific heat, one is concerned with the behavior of Ap(kll, SZ) for small k since the long wavelength modes make the domi- nant contribution to the low temperature thermodyna- mic properties.

In figure 4, we sketch the behavior of cp(kll, SZ) as a function of SZ, in the region of small k l l . The jump of 2 n at the surface mode frequency SZ, gives a change in Ap(kll, IR) of just 2 6(Q - 9,). This describes the

FIG. 4. - The angle y> as a function of frequency, for small values of kll .

contribution of the surface magnon branch to the change in density of states. (In the geometry described in section II, there are two surfaces, and one surface mode for each surface. This is the origin of the factor of two.) Note that there is then a jump of - n/2 in cp at 51 = a,, the bottom of the bulk spin wave band associated with the wave vector component k l l parallel to the surface. In the long wavelength region, where 9, and 9, differ by only terms of order (kll a)4, the dif- ference in SZ, and 0, may be ignored. This jump thus cancels 25 % of the surface magnon contribution to Ap(kll, 9 ) . As one moves past 52, in frequency into the bulk spin wave band, the angle cp plunges rapidly from 3 4 2 to a value roughly equal to 7~12. This rapid variation in cp comes about through the contribution of bulk modes that strike the surface at glancing inci- dence, their wave vector making an angle of roughly (kll a) with the surface. This strong variation of cp very close to 9, produces an antiresonance in the density of states. This anti-resonance has the effect of cancel- ling 50 % of the surface magnon contribution to the specific heat. The end result of the two negative contri- butions just described is to cancel75 % of the contri- bution of the surface manons to Cs, thus reducing the total surface contribution to the specific heat to 25 %

of the surface mode contribution.

In the frequency region 52, and SZ,, the angle cp that appears in the expression for Ap(kll, SZ) is precisely the same cp defined in eq. (2.36). The quantity k, and the frequency SZ are related by SZ = A - B cos (k, a).

Thus, the anti-resonance in the density of states and a rapid variation of the phase of the T-matrix both undergo rapid variation with frequency (or k,) for 52 near 9, (or k, near zero). I t is interesting that the modulus of T-matrix varies smoothly through this region.

(b) THE MEAN SPIN DEVIATION NEAR ^THE SURFACE.

- In the study of the surface specific heat described earlier [7], the mean spin deviation A , = S - < ^Sf >

was studied as a function of distance from the surface.

By employing the Green's function method, an analy- tical expression for this quantity in the form of a rapidly converging infinite series is obtained. There are two features of the result worth noting :

(i) The spin deviation in the surface layer is found to be precisely twice that in the bulk.

(ii) The excess spin deviation near the surface falls off to zero in a distance the order of a thermal magnon wavelength A,, ^N 2 n(D/k, T)% .

The effect of changes in exchange constants near the

surface on A , have not been explored in detail. Howe-

ver, the insensitivity of C,(T) to these changes strongly

suggests that A , is not sensitive to alteration of the

exchange constants near the surface at temperatures

low compared to the Curie temperature. It has also

been pointed out [8] that in the limit of infinite sur-

face pinning field, the change in A , near the surface is