Ima j Appl Math 1968 Bell 375 98

J. List. Maths Applies (1968) 4, 375-398

Spectral Symmetry in Lattice Dynamical Models

R. J. BELL AND P. DEAN

Mathematics Division, National Physical Laboratory,Teddington, Middlesex

[Received 9 November, 1967]

Conditions are derived for the existence of symmetry in the squared frequency spectraof lattice dynamical models, and it is shown that the displacement eigenvectors associatedwith symmetrically related frequencies are simply related. It is also demonstrated that insome cases a delta function exists in the spectrum in addition to the symmetry relatedregions: the modes associated with the delta-function are vibrations in which certainatoms remain stationary. Various applications of the theory are discussed.

1. Introduction

ONE of the obvious features of the spectra of many lattice dynamical systems is thesymmetry or near-symmetry of the squared-frequency distribution function. Althoughthe literature abounds in references to this symmetry (or its absence) for particularsystems, there has been no investigation of the general conditions for the existence ofsymmetry, or of the possible significance of exact or near-symmetry in an observedor calculated spectrum. In this paper we present a theoretical study of the problem.Some of our results complement the work on sum rules and interatomic forces byBrout (1959), Rosenstock (1963, 1965) and Rosenstock & Blanken (1966).

One of the main results of our study is the derivation of conditions under whichspectral symmetry will occur. However, this is by no means the only significantresult which emerges. We find, for example, that displacement eigenvectors corres-ponding to conjugate (i.e. symmetrically placed) squared frequency values are simplyrelated. We can show when a band-gap will occur, and can predict the existence insome cases of a ^-function singularity in an otherwise symmetric spectrum. Ourresults hold not only for the full squared frequency spectrum, but also for the squaredfrequencies associated with any phonon wave vector in a regular lattice. The studyof systems with irregular boundaries leads to the remarkable result that, for certainclasses of disordered systems, the degree of localization of vibrational modes at highfrequencies is exactly reproduced in the low frequency region.

The plan of this paper is as follows. In Section 2 we present the basic mathematicaltheory: we establish that the eigenvalues and vectors of matrices of a particular typehave certain symmetry properties. In Section 3 we relate this analysis to the equationsof rigid-ion lattice dynamical systems, deriving the physical conditions which lead tospectral symmetry. In Section 4 we discuss the effects of boundary conditions. Section 5concerns the application of the theory of this paper to a range of simple regular lattices.In Section 6 we derive some interesting properties for a certain class of disorderedsystems. We consider extensions of the conditions for symmetry in Section-7.

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376 R. J. BELL AND P. DEAN

We emphasize one important point at this stage. The symmetric spectra we shalldiscuss in this paper are squared-frtquency spectra Gico2), where Giafydco2 representsthe proportion of squared frequencies of the system which lie in the infinitesimalinterval co2 to ofi+dco2. The function Gico2) is related to the frequency spectrum ^(cu)by the simple formula Gico2) = gico)l2co.

2. Mathematical Basis

In this section we shall investigate the solutions of the eigenvalue equation

(1)

where C is &p x q matrix of rank r and O is its hermitean conjugate, u and v are columnvectors of orders p and q respectively, and the I's are unit matrices of appropriateorders. The constants a and b are real, so that M is hermitean and its eigenvalues kare real. Lanczos (1958) has considered the more restricted eigenvalue problem inwhich a = b = 0.

Decomposing (1) we have immediately the relations

Cv = ik-a)u, (2a)Ctn = ik-b)v. (2b)

Premultiplying the first of these equations by O , and the second by C we obtainC+Cv = y2v, (3a)COu = y2a, . (3b)

where the eigenvalues y2 of O C and C O are given byy2 => ik-a){k-b). (4)

It can be shown that the matrices O C and C0 are each of rank r, and they possess rsimultaneous eigenvalues y2 > 0. Except in the case p = q = r there is one remainingeigenvalue, y2 = 0, which occurs with multiplicities p—r and q—r for C O and O Crespectively.

It is not possible, in general, to satisfy the eigenvalue equation (1) merely bycombining an arbitrary solution of (3a) with a solution of (3b). In order to analyse thestructure of the eigenvectors of M in detail, we shall find it convenient to separatethe cases (i) k # a or b; (ii) k = a # b; (iii) k = b # a and (iv) k = a = b.

(i) k j£ a or b

One may solve (3a) to obtain r solutions v (corresponding to the positive eigen-values y2) and then find the appropriate a's from (2a); alternatively, one may use (3b)to obtain the u's, and then construct the v's from (2b). Let us examine the lattersituation, having found a y2 and a corresponding u from (3b). It is evident from (4)that each y2 corresponds to a pair of eigenvalues,

k = \{a+b±[ia-bY+Ay2^}, (5)

of the original equation (1). For both these values of k we can use (2b) to obtain v.If one choice of sign in (5) gives an eigenvalue k and eigenvector (u, \) of (1), the otherchoice corresponds to the eigenvalue k' and eigenvector (u'.v7) where

u' = (A-fl)n,;.' = a+b-k; \

v' = ib-ky,.]

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LATTICE DYNAMICAL MODELS 377

This may readily be verified by substitution. The paired vectors (u,v) and (11',/)constitute 2r solutions of (1) with A ^ a or b. We shall refer to such pairs of eigen-vectors, related by (6), as conjugate eigenvectors.

In the case where A # a or b is a multiple eigenvalue associated with L linearlyindependent eigenvectors

(u,,vi), / = 1,2,...,L

then the application of (6) to these leads to a set of L linearly independent vectors

associated with multiple eigenvalue A' = a+b — X. However, if the former vectorsare chosen to be mutually orthogonal, the latter are not necessarily so unless a = b.

(ii) A = a # bFrom equations (2) we obtain

Cv = 0, (7)Oil = (a-b)v, (8)

so clearly, for non-trivial solutions, we require u # 0. Now the well-known com-patibility theorem for linear operators (cf. Lanczos, 1958) states that (8) is solvableif and only if the right-hand side is orthogonal to all solutions of the adjoint homo-geneous equation Cw = 0. Gearly, this condition can only hold in the presentcontext if v = 0, so the eigenvectors (u,v) of (1) must be given by

v = 0, Cu = 0, (9)

there being p—r such solutions. It is clear that we cannot obtain paired eigenvectorsof M from (6) in this case.

(iii) A = b ¥= aThe argument for A = b is analogous to that in case (ii). The eigenvectors of M

are now given byu = 0, Cv = 0; (10)

in this case there are q—r such solutions. The transformation (6) again gives n', v' = 0,showing that the eigenvectors cannot be paired.

(iv) A = a = bThe results given for cases (ii) and (iii) provide a full solution to (1) even when

a = b. Now, however, the eigenvectors of (1) need not be restricted to the form (u,0)or (0,v); we can obtain the p+q—2r required vectors in the more general form (u,v)by combining appropriate solutions of Du = 0 and Cv = 0.

Summarizing, for A # a or b, the eigenvalues of (1) occur in conjugate pairs A andA' = a + b—?.; the corresponding conjugate eigenvectors are related by the simpletransformation (6). This part of the eigenvalue spectrum of M is symmetrical aboutthe point i(a+b). If p> r there are also eigenvectors (u,0) corresponding to A = aand if q > r there are vectors (0,v) corresponding to A = b; these vectors do not occurin conjugate pairs related by (6). Finally, as an important additional point, we notefrom (5) that there is a region between a and b where no eigenvalues of M can occur.

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3. Application to Lattice Dynamical Systems

Consider a system in J-dimensions consisting of two types of atoms A and B, withrespective masses mA and mB. Suppose that each ^4-atom has nA nearest neighbours,!all of type B, and each i?-atom has nB nearest neighbours, all of type A. (We includealso systems of identical atoms, in which case the separation into types A and B ispurely formal.) We denote the equilibrium position vectors of .4-atoms and 5-atoms,respectively, by R, and R, (a,/? = 1,2,3,...) and define RJ/5 = R^-R,.

Let the forces between nearest neighbours be harmonic in character, with centraland non-central force constants (l-e)fc and ek respectively.J The time-independentequations of motion for a normal vibration about equilibrium, with frequency co,are then

a = kY' { ( l - e ^ R ^ + e a - R ^ ) } . ( r ^ - r j (A atoms), (lla)

= * £ ' {(l-e)ft,«ft/l,+8(l-ft,.ft,.)} . ( r . - r , ) (5 atoms), (lib)

where the primed summations are restricted to nearest neighbours. In equations (11),ia and TP denote the vibrational amplitudes of A and 5-atoms, ft^ is the unit vectorin the direction of RI/t and 1 is the d x d unit dyadic.

With the introduction of mass-dependent amplitudes

H (12)and the identification ofi — X, equation (11) may be written in the hermitean form

(13)

where the vectors °U and "V are themselves composed of {/-component vectors sa

and Sj, and the elements of M are dxd symmetric dyadics. <<f will often be a sparsematrix, whose dyadic elements are non-zero only if a and /? are nearest neighbours,when they take the form

«„= -^m^-HO-e^R^ + ea-R.A,)}. 04)si and Si are diagonal matrices with dyadic elements

-VU)}.(15)

-ft/.A«)},

where the primed summations are restricted as in equations (11).If the equilibrium configuration of the system is such that

.«*.« = el, (16a)St,, = M, (16b)

t For the sake of generality we take the nearest neighbours of A to mean simply those atomswhich interact directly with A. In most cases considered (but not all) these are the atoms which arespatially nearest to A.

% We denote the central and non-central force constants by (1 — e)k and dc, respectively, to ensurethat the entire range of central to non-central force constant ratios can be covered by varying thesingle parameter e between 0 and 1.

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it is clear that (13) assumes exactly the form of equation (1) of Section 2. It then followsthat the symmetry properties described in Section 2 apply to the atomic vibrationsof the dynamical system. For (16a) and (16b) to be valid, we require

kQ. -2e) £ ' ft,/,ft^ = (mja-kenjl, (17a)

*(1 -2e) £ ' » , . » , . = (mj-kenjl, (17b)

which, in turn, imply thata = {l + (d-2)e}knA/dmA, (18a)b = {l+(d-2)e}knBldmB, (18b)

sinceTr{fta/,fta/,} = l.

Conditions (17) are equivalent to specifying that each A-atom resides in a sphericallysymmetric potential VA(r) (in ^-dimensions), and each B-atom in a spherically sym-metric potential Vgfj), when the nearest neighbours are held fixed in their equilibriumpositions.

Equations (17) are trivially satisfied for any system with equal central and non-central force constants (e = %). In this case

]if a and ft are nearest neighbours | (19)= 0, otherwise J

and we can reformulate the problem as one of motion in one-dimension, by adoptingsolutions of the form sa = sjt, sf = spi where i is a convenient unit vector. Thistransformation enables us to reduce the order of the dynamical matrix by a factor of1/d, the reduced matrix being simply related to the adjacency matrix of the system(cf. Busacker & Saaty, 1965). The spectrum remains invariant under arbitrarydistortions of the system provided the topology, force constants and atomic massesare not changed. This property, for e = i, provides a method for investigating theeffect on the spectrum of the topology of the system, as distinct from the geometry.

If the atoms of a system lie in a hyperplane of dimension less than d, the out-of-plane components of H^ are zero; thus for (17) to hold (apart from the trivial casee = $) we must either discard the out-of-plane equations from (11) and consider onlymotion within the hyperplane, or vice versa. Thus, in the case of the one-dimensionallinear chain, equations (17) are satisfied if we consider longitudinal motion only,or transverse motion only, but (in general) not both together.

Equations (17) simplify considerably for two-dimensional structures, many of whichform useful illustrative systems (cf. Sections 5 and 6). Consider a typical A-atom ofa two-dimensional system, surrounded by nA 5-atoms. Let 9^ be the angular co-ordinate of Sa/J in a plane polar co-ordinate system in the plane of the atoms. Theunit vector fta/J then has Cartesian co-ordinates (cos 9^, sin 9^), and the conditions (17)are equivalent to

' exp 2i9xf = 0, (20a)

/0,a = O, (20b)

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implying

For any given v4-atoma, let us now renumber its nearest neighbours by /? = l,2,...nA.Then a simple choice of 0,^'s which satisfies (20a) is given by

0*i, = 0a+pnnJnA, 0 = l,2,...nA (nA^2). (22a)With a corresponding renumbering (a = l,2,...nB) for the neighbours of any given5-atom, f}, we find (20b) satisfied by

Op. = Qf+annplriB, a = l,2,...nB ( « f l >2) . (22b)Equations (22) are valid for any integers na and nf prime relative to nA and nB,respectively. The presence of the arbitrary constants 6t and 0f merely reflects therotational invariance of conditions (17). More generally, equations (17) are satisfiedif the nearest neighbours of each atom can be partitioned into several sets, each ofwhich obeys an equation of the type (22a) or (22b). Equations (22) and their generaliza-tion imply the circular symmetry of the potential at each atom when its neighboursare held fixed in their equilibrium positions.

Applying the results of Section 2 to systems of the type considered above (i.e. whichsatisfy equations (17)), we find that (i) no frequencies occur in the range a<oJ1<bt

(ii) there may be special vibrations at a? = a and aP = b, in the former case only,4-atoms vibrate, while in the latter only .ff-atoms vibrate, (iii) the remaining squaredfrequencies are distributed symmetrically about ofi = $(a+b), i.e. they occur insymmetric (conjugate) pairs co2 and co'2 such that aP+co't = a+b, and (iv) the vibra-tional amplitudes associated with a symmetric pair of frequencies are related by thesimple formula (6).

It is appropriate, at this stage, to comment on a consequence of our choice of forceconstants. By denoting the central and non-central force constants by (1 — e)k and sk,respectively, we have ensured that the entire range of central to non-central forceconstant ratios can be covered by varying the single parameter e between 0 and 1.A result of this parametrization, for two-dimensional structures having spectralsymmetry, is that the lattice parameters a and b are independent of s; for two-dimensional systems, therefore, the position of the centre of symmetry, and thepositions of any predicted band gap or 5-function are independent of e. Thissimplification does not occur, however, for systems in three or more dimensions.

We conclude this section by describing another important property of systemswith spectral symmetry, namely the variation of the spectrum with atomic mass.Let X = aP- be a squared frequency of the system with masses mA and mB, and 1 = cS2

be the corresponding squared frequency for masses fhA and fnB. It follows from equa-tions (4), (14) and (15) that

m^siX-aXX-B) = mjjn^l-aW-b), (4a)where

mAa = mAa, mB5 = ntgb,

and a, b are given by (18). Thus, once the spectrum has been calculated for anyfinite mass-ratio, the spectrum for any other mass-ratio can immediately be deducedfrom (4a).

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Finally, if (r^r,,) are the vibrational amplitudes for a mode with squared frequency?. when the masses are mA and mB, the corresponding amplitudes (ra,r?) for masses fnA

and mB are given byf . = <™a> h = ™/J,

wherea I* = mA{_k-a)lmA(l-a) = mJI-h)lmB(k-b).

An important consequence of this relation is that, if localized modes occur for systemswith spectral symmetry, the degree of localization is independent of the atomic masses.

4. Boundary Conditions

The dynamical systems encountered in solid state physics are often so large on theatomic scale as to comprise, effectively, an infinite number of atoms. Clearly, for thesesystems the eigenvalue problem of equation (1) would be quite intractable withoutsome means of first reducing the equations of motion. One approach is to computethe vibrational properties, not of the whole system, but of a representative samplecontaining a reasonably large but finite number of atoms. A method of imposing aboundary condition is to consider atoms outside the sample as fixed in their equili-brium positions (without changing the values of any force constants); this constitutesthe well-known rigid-wall boundary condition. Another boundary condition—thefree-end condition—is obtained by severing the bonds (i.e. forces) connecting atomswithin the sample to those outside. In Section 4.1 we shall show that the impositionof the rigid-wall boundary condition on a sample of atoms in a lattice dynamicalsystem does not alter the features of spectral symmetry we have already discussed,provided the original system satisfies the two conditions of Section 3; on the otherhand, the imposition of the free-end condition does, in general, destroy the spectralsymmetry of a model. In Section 4.2, we shall consider the effects of the importantcyclic boundary condition.

4.1. The Rigid-wall and Free-end ConditionsConsider the situation where one of the atoms, say the /ith atom of a lattice

dynamical system, is held fixed in its equilibrium position. We take account of thisin the system of equations (11) by setting rM = 0 and discarding the /zth (dyadic)equation. The effect on (13) is simply to remove the /ith row and column of dyadicsfrom the dynamical matrix M (and the /zth vector amplitude sM from the amplitudeeigenvector). If the dynamical matrix is originally of the form in (1), the process ofremoving rows and the corresponding columns leaves this form unaltered. Thus, asystem which satisfies the conditions for spectral symmetry again satisfies thoseconditions if one atom—and, therefore, any number of atoms—are held fixed in theirequilibrium positions.

If the free-end boundary condition is imposed upon a sample of atoms in a system,it is clear (from equations (11) and (15)) that not all the diagonal elements of s4 and Siwill now take on the values a and b (cf. equation (16)), respectively. The free-endsample of atoms will therefore not, in general, satisfy the conditions for spectralsymmetry.

To summarize, if we have a large (or infinite) system of atoms which satisfies theconditions for spectral symmetry, and we restrain any number of these atoms to remain

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in their equilibrium positions, the resulting system will also satisfy the conditionsfor spectral symmetry. On the other hand, if we isolate a sample of atoms from theoriginal system by severing bonds between the atoms of the sample and those outside,the sample will not, in general, constitute a system which satisfies the conditions. Ofcourse, the effect of any particular boundary condition becomes less important as thesize of the sample to which it applies increases. Thus, very large samples may wellexhibit almost exact spectral symmetry even under the condition of free-end boundaries;this point is clearly of importance, for free-end boundaries probably simulate quiteclosely the surfaces of real solids.

4.2. The Cyclic Boundary Condition and Reduced Dynamical MatrixIf the dynamical system is an infinite regular lattice, the cyclic boundary condition

may be imposed in order to reduce the number of degrees of freedom. In applyingthis well-known condition one considers only a unit cell of the lattice, assuming thatthe vibrational amplitudes have a plane-wave-like spatial variation, differing onlyin phase from cell to cell. The full dynamical matrix of the original system thencondenses (cf. for example, Dean, 1967) to a reduced dynamical matrix Uifjj(k) whoseorder is the number of degrees of freedom in the unit cell, and whose elements arefunctions of a wave-vector k; it is usual to allow k to take values in a region ofreciprocal space known as a Brillouin zone.

Specifically, for a lattice vibration with wave-vector k, an equation such as (13)for the original system reduces to

where the subscript R indicates that we are now dealing with reduced matrices andvectors, ^ ( k ) and ̂ ( k ) contain k-dependent complex phase factors, arising from theinteraction between A and B atoms in neighbouring unit cells. If the system satisfiesthe conditions of Section 3, $iK and 3SR contain no phase factors, since neither A nor Batoms interact with atoms of the same kind: thus if, as indicated in Section 3, &/ and SSare of the form a\ and b\ respectively, sfR and 88R take exactly these same forms,although, of course, their orders are substantially smaller.

Provided, then, that the conditions for spectral symmetry apply to the lattice asa whole, (23) may be written as

^UwJ - Icwoi "bi'jL(k)J - W)J ' (24)

which is of the form (1). Consequently, our analysis and theorem of the precedingsections applies not only to the full dynamical matrix and the total squared frequencyspectrum, but also to the set of squared frequencies associated with each wave-vector k.This result is complementary to the sum rules discovered by Brout (1959), Rosenstock(1963,1965) and Rosenstock & Blanken (1966).

5. Spectral Symmetry of Regular Lattices

In this section we consider the spectral symmetry properties of a number of regularstructures, drawing upon results in the literature where possible. In 5.1 we describe

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the longitudinal vibrations of the linear chain, and refer briefly to its transversevibrations and general vibrations in ^-dimensions. Several regular two-dimensionalstructures are discussed in 5.2: we consider the quadratic and hexagonal lattices, anda lattice of triangular molecules which illustrates particularly clearly some of theconcepts introduced in Section 3. In 5.3 we discuss the spectral symmetry of somecommon crystal structures.

5.1. The Linear Chain

Consider a linear chain of N atoms, with nearest-neighbour forces, and subject to(say) the rigid wall boundary condition. The time independent equations of motionfor longitudinal vibrations are

-mpP-X; = k(xj_i-2xj+xJ+1), j = 1,2,... N (23)xo = xN+i = 0,

where ntj is the mass and Xj is the amplitude of vibration from equilibrium of the^thatom, and k denotes the (central) force constant between nearest neighbours.

Suppose, now that all the even-numbered atoms are ^4-atoms with mass mA, andall the odd-numbered atoms are B-atoms with mass mB. To recast the equations inthe form of the matrix relation (13) we simply re-order the atoms so that the even-numbered equations are written first, and then we apply the transformation (12) withra s jca and r? = xf. It is clear that interactions occur only between unlike atoms,and that each atom resides in a symmetric (i.e. even) potential when its nearestneighbours are held in their equilibrium positions; therefore this system, the alternatingdiatomic chain, satisfies the conditions of Section 3 for spectral symmetry. We findthat the lattice parameters a and b (cf. equations (16), (17) and (18) of Section 3,with e = 0) are given by

a = 2klmA,b = 2k/mB.

From the theory of Section 2, there follows at once the well-known result that thesquared frequency spectrum is symmetric about the point co* = (k/mA+klmB), andthat a gap exists between ufi = 2k/mA and ofi = 2k/mB (cf. for example, Maradudinet a]., 1963, p. 62). If N is odd, the B atoms outnumber the A atoms by one, and wenote (again from Section 2) that there will be an additional frequency (that is, inaddition to the symmetric spectrum) at ufi- = b = 2k/mB.

Another interesting feature of the system follows from the conjugate eigenvectorproperty discussed in Section 2. If we consider the atomic displacements in a particularmode (with squared frequency aP, say), the displacements in the conjugate mode (withsquared frequency 2k/mA+2k/mB—a)Z) are given simply by multiplying the displace-ments of one species of atom (either even or odd) by a negative constant which dependsonly on co2- and the lattice parameters (cf. equation (6) of Section 2); if o& refers toan acoustic mode, the conjugate mode lies in the optical band, and vice versa. Thisproperty is illustrated in Fig. 1, where the horizontal scale refers to the positions ofatoms along the chain, and the vertical scale to the atomic displacements.

In the special case mA = mB = m, the system reduces to a monatomic chain andthe lattice parameters take the values a = b = 2k/m; we therefore find a spectrumwhich is symmetrical about co2 = Ikjm, but we do not predict a band gap. The

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well-known spectrum for the monatomic chain is indeed of this form (cf. for example,Maradudin et aJ., 1963). A point of interest for this system is that the conjugateeigenvector property (cf. Fig. 1) still applies between modes lying in the two halves ofthe spectrum (i.e. 0<afl<2kjm and 2klm<o>*<4k/m), despite the disappearanceof the band gap; thus one might, with some justification, refer to "acoustic" and"optical" regions of the single-band spectrum.

There is ample evidence that the introduction of longer-range forces into themonatomic and alternating diatomic linear chains destroys the symmetry of the0)2-spectrum (cf. Rosenstock, 1958; Czachor, 1963). Again, we note that a linear chainwith a more complex unit cell (such as —AAB— or —ABC—) does not, in general,have a symmetric squared frequency spectrum (Maradudin & Weiss, 1958; Pozubenkov,1966).

Acoustic mode

Fio. 1. Diagram illustrating the conjugate eigenvector property. Atomic amplitudes in the twocases are related by equation (6) in the text.

We now consider the general vibrations of linear chains in d-dimensions, and includenon-central forces between nearest neighbours. The equations of motion can bedecoupled into d independent systems by resolving the motion (i) along the directionof the chain and (ii) along (d— 1) mutually orthogonal directions perpendicular to thechain. The system of equations arising from (i) gives a squared frequency spectrumwhich is symmetrical about o2 = (l — e)(k/mA+k/mB), whereas each of those arisingfrom (ii) gives a spectrum which is symmetrical about co2 = e(k/mA+k/mB) (in thenotation of Section 3). Clearly, while the spectrum for longitudinal vibrations alone,or for transverse vibrations alone, is symmetric, the combined spectrum for generalvibrations will be symmetric in general if and only if 6 = •£. This a particular exampleof the general condition that, for our theorem to apply to a multi-dimensional system,the potential at each atom in the equilibrium configuration must have full sphericalsymmetry over all the dimensions of motion.

5.2. Two-dimensional Lattices

Here we consider the spectral symmetry of a number of regular two-dimensionalstructures. Five well-known planar lattices are shown in Fig. 2: they are (a) the simplequadratic, (b) the hexagonal, (c) the triangular, (d) the Kagom6 and (e) the four-eightlattice. Of these, the triangular and Kagome lattices cannot be divided into two typesof sites in the required way (cf. Section 3), and neither the Kagomd nor four-eightlattices satisfy the criteria (20) for circularly symmetric potentials. The remainingsystems both obey the conditions laid down in Section 3 for spectral symmetry:

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these structures, the simple quadratic and hexagonal lattices, are discussed below.We consider also the vibrational spectrum of a lattice of triangular molecules whichillustrates particularly clearly some of the properties predicted in Section 3.

The Quadratic Lattice. We consider the in-plane vibrations of the quadratic lattice(monatomic or alternating diatomic) depicted in Fig. 2(a), a system which satisfiesthe conditions of Section 3 if interactions are restricted to nearest neighbours. Withforce constants defined as in equations (11), a and b take the same values as for thelongitudinal vibrations of the linear chain, so we again predict an co^spectrum

' f '1 «

(a) (b) (c)

/ / / \ /

X X XX: x x x >x x x x( x x x >

(d) (e)FIG. 2. Five two-dimensional lattices. The simple quadratic and hexagonal lattices satisfy the

conditions for spectral symmetry. The triangular, Kagomd and four-eight lattices do not.

symmetric about o2 = (klmA+k/mB), and a spectral gap between ofi = 2kjmA anda)2 = 2k/mB (in the notation of Section 3). In the monatomic case (mA = mB = m),the squared frequency spectrum will be symmetric about cu2 = 7k\m. If out-of-plane(rather than in-plane) vibrations are considered, the spectrum is again symmetric(and, in the alternating diatomic case, has a band-gap). It is clear that a general motionof the system (containing vibrational components both within and perpendicular tothe system) will, in general, lead to a symmetric spectrum only in the case wherecentral and non-central force constants are equal (i.e. e = i).

Montroll (1956) has studied the dynamics of the monatomic simple quadraticlattice, with nearest-neighbour interactions only, and his formulae show the G(a>2)function to be symmetric for all ratios of the central to non-central force constants.Simple quadratic lattices with second as well as first neighbour interactions have beenstudied by Montroll (1947) and Bowers & Rosenstock (1950); it is clear from theirresults that the introduction of second-neighbour forces destroys the symmetry ofthe G)2-spectrum.

The Hexagonal Lattice. The two-dimensional hexagonal lattice (depicted in Fig. 2(b))provides a dynamical system which satisfies the conditions of Section 3, if only

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nearest-neighbour interactions are allowed. According to equation (21) of Section 3,the lattice parameters for in-plane vibrations are

a = 3kl2mA\b = 3k/2mBj '

so we predict a spectrum which is symmetric about to2 = 3(k/mA+klmB)/4 andwhich has a gap between up- = 3k/2mA and co2- = 3k/2mB. A similar result holds forthe entirely out-of-plane vibrations.

Dean (1963) has considered the in-plane vibrations of the monatomic hexagonallattice. He finds a symmetric squared-frequency distribution, consisting of two distinctbands which just touch at the mid-point of the spectrum, as well as two symmetricallyplaced 5-functions (one at the bottom and one at the top of the continuousspectrum) each of which contains exactly one quarter of all the normal frequencies ofthe system. These ^-functions must be regarded as spectral bands of infinitesimalwidth, and are not to be confused with the 5-functions arising from the existenceof unequal numbers of A- and i?-type lattice sites. The out-of-plane motion of thissystem was described in an earlier paper by Hobson & Nierenberg (1953), who founda symmetric squared-frequency distribution. Their spectrum (suitably scaled) isidentical to that derived by Dean for the in-plane vibrations, but without the <5-functions. The work of Rosenstock (1953) indicates that the inclusion of longer-rangeinteractions destroys the symmetry.

The Molecular-triangular Lattice. We now describe the dynamics of a two-dimensional structure which has been of some interest to us in connection with work

FIG. 3. The "molecular-triangular" lattice.

on the atomic vibrations of glasses. This structure, which illustrates particularlyclearly some of the features predicted by the theory of Sections 2 and 3, is depicted inFig. 3: the unit cell, which is outlined by broken lines in the figure, contains two^4-atoms (denoted by O) and three 5-atoms (denoted by • ) . We regard nearest

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neighbours (which are not, of course, necessarily geometrically nearest neighbours)as pairs of atoms which are joined by full lines in the diagram; with this convention,interactions between like atoms do not occur. The three nearest-neighbour bonds ateach /4-atom are separated by angles of 2n/3, and the two at each 5-atom by an angleof 7i/2, so that each atom resides in a circularly symmetric potential when its nearestneighbours reside in the equilibrium position (cf. equations (22) of Section 3). Thesystem thus satisfies the criteria of Section 3 for spectral symmetry.

An interesting feature of the structure is that the B atoms outnumber the A atoms bythree to two: thus in addition to a symmetric squared frequency spectrum we expectto find at co2 = b a <5-function containing 20% of the total number of squaredfrequencies. We note from equations (21) that the a and b values for this lattice aregiven by

a = 3k/2mA,}b = k/mB,

so the squared frequency at which only B-atoms vibrate is given by co2 = k\mB.The rest of the spectrum will be symmetric about ofi = (3kl4mA+kl2mB), and willcontain no squared frequency between 3k/2mA and k/mB.

(26)

e = 0 or1

n

S) ri

/ N.

Fio. 4. Histograms of squared frequency spectra for the molecular-triangular lattice. The arrowsindicate the positions of 8-functions.

We have computed the squared frequency spectrum of this system for the casemA = 2, mB = 1, k = 1 for a range of values of e (where, in the notation of Section 3,(1 — e)k is the central and sk is the non-central force constant). The normal frequencieswere determined by direct diagonalization of the 10x10 reduced dynamical matrix,

26

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taking a total of 576 wave-vectors (k-values) in the Brillouin zone. The squaredfrequency distributions are shown as histograms in Fig. 4, in which the verticalarrows represent 5-functions. The <5-function at ofi = k/mB = 1 contains 20%of the frequencies, as anticipated, and the remainder of the spectrum is clearlysymmetrical about co2 = (3k/4mA+k/2mB) = J. As predicted, no squared frequencies

FIG. 5. Vibrational amplitudes of the molecular-triangular lattice for a "8-function" mode at2 >= 1. The A-atoms (O) remain at rest.

2 - - 2

Ikl

FIG. 6. a2-k relations for the molecular-triangular lattice for the direction [0,1] in the Brillouin zone.

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occur between 3k/2mA and k/mB, i.e. between J and 1. However, except for the casee = i , the actual band gap is wider than our predicted one; this is particularlynoticeable for e = 0 and 1, where the continuous regions of the spectrum havecontracted into <5-functions. Although our theory leads to the actual band gap forthe majority of common structures, it is worth stressing that for some special lattices,such as the present example, our prediction (which simply gives a region devoid offrequencies) may represent a gap smaller than that which actually occurs.

We computed the vibrational amplitudes for one of the modes with squaredfrequency to2 = 1 (in fact, the mode with wave vector k = 0): the result is shownfor the case e = 0 in Fig. 5, in which the directions and relative magnitudes of atomicdisplacements are indicated by arrows. As our theory predicts, for modes whosefrequency lies in the <5-function, only the 5-atoms are in motion. In Fig. 6, wegive the dispersion curves (a*2 versus k) for the direction [0,1] in the Brillouin zone.Apart from the horizontal line at a>2 = 1 (which contributes to the 5-function),the remaining branches lie symmetrically about afi- = %. This demonstrates ourconclusion of Section 4, that spectral symmetry occurs not only for the full squaredfrequency distribution, but also for those squared frequencies associated with anywave vector k.

5.3. Three-dimensional Structures

We find that many of the common crystal structures satisfy the conditions of Section3. There are, however, some surprising exceptions: thus, whereas the simple cubic andbody-centred cubic systems have spectral symmetry, the face-centred cubic does not.In the face-centred cubic structure, atoms at the centres of faces are connected not

TABLE 1

Some common crystal structures which (a) conform and (b) do not conformto the conditions for spectral symmetry if only nearest-neighbour interactions occur

(a) (b)

Simple cubic Face-centred cubicBody-centred cubic Hexagonal close packedDiamondSodium chlorideCaesium chlorideZinc blendeFluorite (CaF2)Wurtzite

only to nearest neighbours at the cube vertices, but also to atoms on adjacent facecentres, and therefore the system does not satisfy the first criterion of Section 3,i.e. that interactions occur only between unlike atoms. In Table 1, we list a numberof the common crystal structures which satisfy, and others which do not satisfy, theconditions for spectral symmetry if only nearest neighbour forces are taken into

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consideration. We shall specifically consider the spectral symmetry properties of thesimple cubic, body-centred cubic, and CaF2 (fluorite) structures.

The Simple Cubic Lattice. The cubic lattice (monatomic or alternating diatomic),with nearest neighbour forces, is the simplest three-dimensional structure whichsatisfies our conditions for a symmetric spectrum. The lattice parameters for thediatomic structure are (in the notation of Section 3):

a = 2(l + e)k/mA,\b = 2(1+ e)k/mB;f

it follows that the spectrum is symmetrical about co2 = (1 + e)(k/mA+k/mB), andshows a band gap between up- = 2(1 + e)k/mA and aP- = 2(1 + e)k/mB. All this isconsistent with the exact formulae for the spectrum, as derived by Maradudin et al.(1958). The monatomic case mA = mB has been treated in detail by Montroll (1956)who finds that the spectrum is symmetric for all ratios of the central to non-centralforce constant, a result which is again to be expected from the theory of Section 3above. Work by Rosenstock & Newell (1953) on the monatomic lattice, and byMazur (1958) on the diatomic lattice, confirms that the spectral symmetry is lost ifsecond-neighbour forces are included.

For the generalization of the diatomic cubic lattice to ^-dimensions, the latticeparameters are

a = 2[l+(d-2)e]k/mA,\b = 2[l+(d-2)s]k/mBJ

so we predict a squared frequency spectrum which is symmetric about[l+(d-2)e](k/mA+k/mB), and has a gap between 2[l+(d-2)e]k/mA and2[l + (d-2)e]k/mB. Montroll (1956) has calculated the squared frequencies for aJ-dimensional NxNx...xN cubic array of identical atoms, with rigid wallboundaries and nearest neighbour interactions. Our prediction is clearly consistentwith his result,

where m is the atomic mass and, in the notation of Section 3, ki = Jfc(l — e) andk2 = k$ = ... = kx = ke are the central and non-central force constants, respectively.It is worth noting that, when there are more A atoms than there are 5-atoms (orvice versa), i.e. when N is odd, Montroll's formula gives modes at

in agreement with our predictions (for the <5-function modes).

The Body Centred Cubic Lattice. Another crystal structure which satisfies theconditions for spectral symmetry, if only nearest neighbour interactions occur, is thebody centred cubic lattice in its monatomic or diatomic (CsCl) form. The latticeparameters for the diatomic case are

a = S(l + e)k/3mA,\b = 8(l+e)k/3mBJ

so here we expect a squared frequency spectrum which is symmetric about the pointCD2 = 4(l + e)(klmA+k/mB)l3, with a band gap between ofi- = 8(1 + e)k/3mA and

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0)2 = 8(1 + e)k/3mB. We have not found, in the literature, any exactly calculatedspectrum for the monatomic or diatomic cases. However, Rosenstock (1955) hasinvestigated the spectrum for the monatomic structure, via the critical points, andfound that the critical points lie symmetrically about the mid-point of the spectrum.Although Rosenstock's sketch of the a)2-distribution (top diagram of Fig. 11 of his1955 paper) appears to have an overall asymmetry, Rosenstock has stated (pers.comm.) that errors occurred in the actual preparation of the figure, and that thespectrum should be "entirely symmetric about the center point".

The Fluorite Structure. The fluorite lattice is interesting in that it contains unequalnumbers of two atomic species, and also satisfies the conditions for spectral symmetrywhen only nearest neighbour forces are considered. The lattice may be regarded asa body-centred cubic structure, with the cube vertices (A-type sites) occupied byfluorine atoms and alternate body centres (5-type sites) by calcium atoms; accordingto equations (18), therefore, the relevant parameters are given by

b = 8(l + e)A:/3mJ y '

Lattice dynamical models of CaF2 studies in the literature, to date, have includedlonger range forces than those between nearest neighbours, so the exact spectralsymmetry of the simple force model is not apparent. However, we note here that, ifonly nearest neighbour forces are considered, the squared frequency spectrum for thisstructure will be symmetric about co2 = (l + eX2fc/3m^+4fc/3mB) and there will beno squared frequencies between 4(1 + e)k/3mA and 8(l + e)fc/3mB; in addition therewill be a 5-function at afi- = 4(1 + e)k/3niA corresponding to normal modes inwhich only the fluorine atoms vibrate.

6. Systems Lacking Spatial Periodicity

In the previous section we considered examples of regular lattices with spectralsymmetry. Here we shall be concerned with the application of the theory of Sections 2and 3 to systems which lack overall spatial periodicity.

It is clear from the discussion of boundary conditions in Section 4.1 that thesymmetry properties described in Section 3 apply not only to a range of regularlattices, but also to structures which include isolated fixed atoms or clusters of fixedatoms and, further, to finite or infinite clusters of vibrating atoms bounded by rigidwalls of fixed atoms. Three two-dimensional structures containing these features areillustrated in Fig. 7, in which the fixed atoms are represented by unoccupied latticesites, the A atoms by open circles (O) and the B atoms by closed circles ( • ) .

As a particular example, let us consider the structure shown in Fig. 7(c). This systemobeys the equations for a circularly symmetric potential, and there are no forcesbetween atoms of the same kind. The lattice parameters are the same as those for theregular diatomic quadratic lattice discussed in Section 5, viz.

a = 2k/mA

b = 2k/mB.We therefore expect a squared frequency spectrum which is symmetric about thepoint co2 = k(jn2l+mB

l) and which has no frequencies between co2 = 2k\mA anda>2 = 2kjmB. Since there are 13 4-atoms as compared with 7 5-atoms, the matrix C

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of equation (1) has dimensions p = 26 by q = 14, and at least 12 of the 40 squaredfrequencies will occur at 2k/mA. (More precisely, if there are n squared frequenciesat b = 2k/mB, there will be 12+n at a = 2Jc/mA.)

We computed the frequencies and eigenvectors for this system when mA = 2,mB = 1,1c = 3, e = $ (where, as stated in Section 3, (l-e)k is the central, and ek

*

, J

1

, J

, I

(a) (b) (c)

FIG. 7. Structures with irregular boundaries. Provided the missing atoms in this diagram are regardedas fixed and the forces between the vibrating atoms and the fixed atoms are not altered in any way(rigid-wall boundary condition) these structures satisfy the conditions for spectral symmetry.

FIG. 8. The squared-frequency spectrum of the system of Fig. 7(c) for the case mA = 2, ms = 1,k = 3, e = i (n\A denotes the mass of an atom represented by O (A atoms), ms the mass of an atomrepresented by • (S atoms)).

(a) (b) (0

FIG. 9. Vibrational amplitudes for three modes of the system of Fig. 7(c). (a) and (b) representconjugately related modes at u2 — 2-07 and uA = 6-93, (c) represents a mode of the special S-functionat ca2 = 3; note that in this mode the S-atoms remain at rest.

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the non-central force constant). The squared frequency spectrum is shown in Fig. 8as a "stick" diagram. The symmetry about co2 = kim^+mg1) = f is evident, as isthe "band gap" between 2k/mA(= 3) and 2k/m^= 6); also, 12 squared frequenciesoccur at co2 = 3. The vibrational amplitudes are displayed in Fig. 9 (the arrowsindicate the directions and relative magnitudes of atomic displacements) for theconjugate modes at co2 = 207 and aP- = 6-93, and also for one of the "5-function"modes at o£ = 3. Figure 9(a),(b) demonstrates clearly the conjugate eigenvectorproperty described in Section 2, while from Fig. 9(c) we note once again that onlythe more plentiful species of atom vibrates at the special frequency co2 = 3.

Consider, now, the case where atoms located at A and B sites in a system containingalso fixed atoms are, in fact, of the same kind. The two-component isotopicallydisordered structure in which one component has infinite mass is a system of just thistype. We therefore arrive at the following result:

THEOREM. If a monatomic dynamical system obeys the conditions of Section 3 forspectral symmetry then, neglecting the zero frequency modes associated with infinitelyheavy atoms, the corresponding isotopically disordered system satisfies the conditionsfor spectral symmetry in the limit of infinite mass ratio. If the infinitely heavy massesare unevenly distributed between A- and B-type sites, an additional 5-function, corres-ponding to modes in which atoms at either A- or B-sites remain stationary, will occur inspectrum.

This result promises to be of some interest in the theory of isotopically disorderedsystems, in view of the remarkable connections which have already been establishedbetween the spectra of lattices with finite and infinite mass ratio (Borland, 1964;Hori, 1964; Matsuda, 1966). It is already known (cf. Domb et al, 1959) that thesquared frequency spectra of one-dimensional systems of this kind (i.e. disordered

Final level(fixed)

1 1 1 2 1 3 Second level

First level

0 Root point(fixed)

FIG. 10. Illustration of the second-order Cayley tree of branching-ratio 3.

two-component linear chains with infinite mass ratio) are symmetric, and there isevidence in the literature (Fig. 7(e) of Pay ton & Visscher, 1967) that the same istrue of similar two-dimensional systems.

An interesting one-dimensional aperiodic system, to which our theory of Section 3applies, is the Cayley tree considered by Rubin & Zwanzig (1961). In this system

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a trunk runs from the root-point (0) to the first-level branch-point (1) where it dividesinto m parallel branches; each of these m branches then itself splits into m sub-branchesat the second level branch-points (l,i), and so on. The number of interior levels iscalled the order of the tree, while m is called the branching ratio. A second-orderCayley tree, with branching-ratio 3 is illustrated diagrammatically in Fig. 10.

The root-point and final level branch-points of the Cayley tree are considered fixed,and the interior branch-points are taken to be particles; the branches are regarded assprings constraining the particles to their equilibrium positions. Suppose that theparticles are permitted to move in a direction parallel to the springs; then the treerepresents a system vibrating in one dimension. If we take the branch points on oddlevels to be y4-atoms and those on even levels to be 5-atoms, it is evident that onlyA—B interactions occur. In addition the potential is symmetric (because of the one-dimensional nature of the system), so the system satisfies the conditions of Section 3for a symmetric spectrum. Considering, for simplicity, the case mA = mB = 1, thelattice parameters (cf. Section 3) are

a = b = k(m+l).We therefore find that the squared frequency spectrum i s symmetric about co2 = k(m +1),a result which may be verified by referring to Rubin & Zwanzig's calculated spectrum.

For an Mh order tree, the total number of atoms is given by

N(A)+N(B) = (m*-l)/(m-l) ,

while the difference between the numbers of the two types of atoms is

\N(A)-N(B)\ = (m*- l)/(/n +1), N even

= (mN+l)/(m + l), iVodd;

the proportion of modes having the special squared frequency co2 = k(m+1) is there-fore

/ = , N evenm + 1

This is in agreement with the result

( I

j/->( I as N-* co

given by Rubin & Zwanzig.Our results for the Cayley tree can easily be generalized to the case where mA # mB

and where the branching ratio is different for odd and even levels. Moreover, theoccurrence of spectral symmetry and special frequencies persists when arbitrarybranch-points are taken to be fixed, and even when the tree is "pruned" at some ofthese fixed branch points.

Finally, it is worth noting that for m = 1 the Mh order Cayley tree becomes alinear chain of N atoms with fixed boundaries, and the even and odd levels are justeven and odd points along the chain. The occurrence of spectral symmetry for thelinear chain has already been discussed in Section 6.

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7. Extension of the Conditions for Spectral SymmetryIn Section 2 and 3, we proved that spectral symmetry occurs for a dynamical system

if (i) the atoms can be classified into two types such that interactions only occurbetween unlike atoms and, (ii) each atom resides in a spherically symmetric harmonicpotential when the remaining atoms are in their equilibrium positions. Our theory,of course, applies directly (like most physical theories) to idealized models rather thanreal systems. It applies, for example, to many common crystal structures (as wenoted in Section 6) provided interactions are limited to nearest neighbours. In realsolids, on the other hand, second- and higher-neighbour interactions are alwayspresent to some extent, leading inevitably to the violation of condition (i). In spite ofthis the theory has a relevance to real solids. If second- and higher-neighbour con-tributions represent only a small perturbation of the dynamical matrix, and theunperturbed system (with nearest-neighbour forces only) is one which satisfies theconditions for spectral symmetry, it is a reasonable hypothesis that the perturbedsystem will retain some features of this symmetry. Indeed, the departure from idealsymmetry may be regarded as a measure of the relative importance of first- and higher-neighbour forces. The persistence of near-symmetry, when a small proportion ofsecond-neighbour force is included, is clearly borne out, for example, by the theoreticalwork of Bowers & Rosenstock (1950) and Rosenstock (1953, 1955).

Many lattice dynamical models, while not directly satisfying the conditions ofSection 3, can easily be decoupled into systems which do possess spectral symmetry.One simple example, already discussed in Section 5, is the linear chain vibrating in^-dimensions; here the longitudinal and transverse vibrations separately providesymmetric spectra. Another, less obvious, example is the diatomic lattice (withnearest neighbour central forces) depicted in Fig. ll(a). By a simple transformationof co-ordinates, the dynamical matrix for this system can at once be partly diagonalizedto yield 30% of the squared frequencies; these frequencies, at o2 = 0, correspond tomodes in which the iJ-atoms move perpendicular to their nearest neighbour bonds. Theremainder of the dynamical matrix is of the form (1), with a = 3k/2mA, b = 2k/mB

andp = 4q/3. The lattice will therefore have 30 % of its squared frequencies at co2 = 0,10% (corresponding to modes in which only the ^4-atoms vibrate) at co2 = a and theremainder distributed symmetrically about (a+b)/2; further, there will be a spectralgap between ofi = 3k/2mA and ofi = 2k/mB. In Fig. ll(b) we give a histogram of thespectrum, computed for the case mA = 2, mB = 1, k = 1, E = 0; it is clear that itbears out our predictions. The arrows in the figure denote <5-functions; the <5-functions at aP- = \ and aP = ^- each contain 10 % of the squared frequencies of thesystem, while that at ©2 = 0 contains 40 % (30 % of these arising from the originaldecoupled modes, with 10% corresponding to those in the ^-function at co2 = ^).

A decoupling can also be performed for the model of Fig. 11 (a) with completelynon-central forces between nearest neighbours. The spectrum is identical to that forcentral forces, but the decoupled modes at to2 = 0 now correspond to vibrations ofthe 5-atoms along the nearest-neighbour bonds.

The cristobalite lattice, shown in Fig. 12(a), is a three-dimensional structure whichcan easily be decoupled to provide a symmetric spectrum, if only central forcesbetween nearest neighbours are allowed. This lattice is based on the diamond structure,with y4-atoms on the diamond sites (denoted by open circles O) and 5-atoms on the

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mid-points of A-A lines (denoted by closed circles • ) . In this case the decoupledmodes, at co2 = 0 correspond to vibrations of the .ff-atoms perpendicular to theirnearest neighbour bonds; there are two such modes for every 5-atom. The dynamicalmatrix for the remaining modes is of the form (1), with a = 4k/3mA, b = 2k/mB,

•vvw-

(b)

FIG. 11. Illustration of a planar lattice the equations of motion of which can be decoupled asdescribed in the text. The histogram represents the squared-frequency spectrum of the lattice for the

2, ma = 1, k = 1, e = 0, the arrows representing 8-functions.

(a) (b)

-k-

(0

-s-Fio. 12. The Cristobalitc lattice, with histograms of the squared frequency spectra for the cases

(b) r»A = | , n>B = I, k = 1, e = 0 and (c) mx = f, ma = 1, k = 1, e = 1. The arrows indicate8-function positions.

p = 3N(A) and q = Af(5) where N(A) and A (̂5) are the numbers of A- and 5-atoms,respectively. The spectrum will therefore have 2N(B) squared frequencies at co2- = 0,3N(A)—N(B) at OJ2 = 4k/3mA and the remainder symmetrically distributed aboutco1 = 2kl3mA+k/mB with a gap between 4k/3mA and 2kjmB. We have computed the

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spectrum for a sample of this lattice, subject to rigid wall boundary conditions, withN(A) = 192, N(B) = 308, mA = f, mB = 1, k = 1 and e = 0. The squared frequencydistribution, shown in Fig. 12(b), agrees with our predictions. The 5-function ata)2 = 2-8 contains 117 frequencies, the one at co2 = 0-8 contains 268, while that atop = 0 contains 733 (of which 616 arise from the decoupled modes and 117 correspondto those at ofi = 2-8).

A similar result holds for the cristobalite lattice with completely non-central forcesbetween nearest neighbours; one can decouple N(B) modes at co2 = 0 correspondingto vibrations of the 5-atoms along the bonds, and the relevant parameters for theremaining modes are a = $k/3mA, b = 2k/mB, p = 3N(A) and q = 2N(B). In thiscase, therefore, there will be N(B) squared frequencies at a? = 0, 2N(B)-3N(A) ata)2 = 2kJmB and the rest of the spectrum will be symmetric about co2 = 4k/3mA+k/mB

with a gap between Sk/3mA and 2k/mB. This is borne out by the squared frequencydistribution in Fig. 12(c), calculated for e = 1 (with N(A), N(B), mA, mB and k thesame as before).

8. Summary

We have given, in Section 3, conditions under which the squared frequency spectrumof a lattice dynamical rigid ion model will be symmetric; we have found, also, thatin some cases an additional "special" ^-function occurs in the spectrum. For certainsystems a prescription has been given for rinding a region of the spectrum devoid offrequencies (i.e. we have indicated a minimum band gap). We have shown thatdisplacement eigenvectors corresponding to symmetrically related (conjugate) squaredfrequencies are simply related, as indicated in Sections 2 and 3; eigenvectors corres-ponding to frequencies in the special (5-function which may occur in the spectrumhave been shown to correspond to modes in which atoms located at certain sitesremain at rest. For periodic systems the theory applies to specific k-values, and thuscomplements previous work on sum rules.

We have applied the general theory to various types of lattice dynamical model,verifying results which have been obtained by other authors and predicting featuresof the spectra and dynamical properties of systems which have not yet been fullytreated. The general theory has applications to simple lattice dynamical models,to models of disordered structures (as indicated in Section 6) and to realistic crystalstructures; in this last case it suggests a simple explanation for the near-symmetrypresent in some experimental spectra.

This work has been carried out at the National Physical Laboratory.

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McGraw-Hill.CZACHOR, A. 1963 Acta phys. pol. 24, 351.DEAN, P. 1963 Proc. Camb. phil. Soc. math. phys. Sci. 59, 383.

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