Copyright 1973. A II rights reserved

SOME ASPECTS OF STRUCTURAL

DISORDER IN SOLIDS 8544

Simon C. Moss

Department of Physics, University of Houston, Houston, Texas

INTRODUCTION

The subject of disorder in solids covers an enormous range of topics varying from purely structural considerations to the relation between the disorder and the fundamental excitations of the solid. In this review we confine our attention to selected examples of disorder ranging from the compositional disorder of alloys to the positionally disordered amorphous state encountered, for example, in metallic glasses. The unifying theme is the utilization of scattering data, such as X-ray, neutron, and electron diffraction, to characterize the disordered state. We try to emphasize similarities in phenomenology where they exist and to draw distinctions among truly disparate states that are often, or at least occasionally, mislabelled or confused with each other. The subject of spin disorder in otherwise regular solids is not treated; we do, however, devote a few words to the structural aspects of magnetic glasses. As to the influence on properties only occasKmal references are made. The literature there is vast and well beyond the modest scope of this article; it includes, for example, the electrical, optical, mechanical, thermal, magnetic, and dielectric properties of metallic alloys and glasses and amorphous, or glassy, semiconductors and insulators.

As a general guide there are a number of texts and symposia proceedings which both summarize the relevant formalism in this field and present diffraction results. These include the books by James (1), Zachariasen (2), Guinier (3), Hosemann & Bagchi (4), Wilson (5), Cohen (6), Warren (7), and Krivoglaz (8), and the proceedings edited by Cohen & Hilliard (9), Brumberger (10), and Mills, Ascher & Jaffee (11). There are articles in preparation specifically on the structure of amorphous semiconductors and insulators ( 12) and metallic glasses (13).

One must deal with two fundamentally different types of disorder. The first involves various deviations from an average lattice. Implicit in the treatment of

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such disorder is the existence of a crystalline lattice for which the Laue interference function (i-8) is more or less delta-like, depending only upon the physical extent of the lattice or the range over which regular crystalline order persists. For such solids there is a well defined reciprocal lattice and it is through the analysis of the distribution of intensity throughout the reciprocal lattice that we obtain information on the nature of the disorder. In this sense most of the classes of disorder in crystalline solids can be treated in terms of fluctuations in the various parameters of the real lattice such as the atomic coordinates, the average atomic coordinates or lattice parameters, the composition, the long range order parameter of an alloy, and so forth. The scattering experiments measure the squares of the amplitudes of the Fourier spectrum of these fluctuations. If the range of the fluctuations is large (or the fluctuations are periodic over a large distance) the scattering will have sharp structure; otherwise, it will be quite diffuse, becoming completely structureless in the case of pure randomness. In general, however, we refer to the disorder scattering as diffuse because, regardless of whatever sharp features it may reveal, it usually extends throughout the reciprocal lattice.

Examples of this first kind of disorder include single phase alloys of, say, two components. While the ground state may be either fully ordered or phase separated, above a critical temperature the disorder is characterized by fluctua­tions in composition whose Fourier coefficients are proportional to the ampli­tudes of scattering in the reciprocal space. In other words there is short range order, the scattering from which can be Fourier inverted to give the composition­al pair correlation functions in the alloy. The normal vibrations of a monatomic lattice belong to the same class of disorder except that they describe fluctuations in atomic position rather than in composition. Structural fluctuations are a natural extension of this form of disorder, in which, in some instances, certain of the normal modes soften or condense leading to a structural phase transition which can often be viewed as a linear disorder. Both crystalline damage (i.e. point defects, dislocation loops, vacancy clusters, etc) and atomic size disparity in solid solutions give rise to a static displacement field which can also be characterized by its Fourier spectrum. The diffuse scattering in this case is similar to that associated with thermal motion except that the character of the defect (its shape, symmetry, strength, etc) is crucial and, of course, the scattering is elastic rather than inelastic.

In the case of composition fluctuations the diffuse scattering off the Fourier waves obeys a sum rule where the integral of intensity over a Brillouin zone is constant and equal to that which would appear in the appropriate superlattice reflections were the lattice to be completely ordered. Generally (2, 1 4) the amount of intensity available in a diffuse scattering experiment is given by the difference <IAI&) - <IAli, where IAI is the amplitude of scattering for the complete lattice including all defects, fluctuations, and so forth. With thermal or static displacements the diffuse intensity can be thought of as coming from the ordinary Bragg peaks, which are accordingly attenuated by a Debye- Waller factor. For small displacements, or small wavevectors, this breakdown into

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STRUCTURAL DISORDER IN SOLIDS 295

attenuated Bragg peaks and diffuse scattering is appropriate. For large concen­trations of defects in severely distorted crystals this is no longer a sensible procedure and the disorder is then treated as strain (and particle size) broadening of the individual Bragg peaks. The reciprocal lattice is still, of course, preserved, and while the material is not "amorphous," so to speak, the analysis takes a different form. These distinctions are discussed by Krivoglaz (8) and the analysis of strain broadening is covered in great detail in the reviews by Warren (IS) and Wagner ( 16).

The general category of stacking disorders forms a subject too extensive for any justice to be rendered in this short article; we therefore refer the reader to Warren (7,15), Wilson (5), and Wagner ( 16), and to several articles by Kakinoki and co-workers (17). Essentially the stacking fault problem consists of imposing a sequence of regular, near regular, or random phase shifts in the scattering intensity depending upon the probability of encountering a coherent fault in the regular crystalline stacking sequence. The scattering thus remains coherent across the fault. The distribution of intensity in reciprocal space depends upon the sequence of faulting and the nature of the phase shifts. One is still dealing with modulations of the average lattice. The reciprocal lattice is maintained even though, for high fault probabilities just as, for example, with very severe damage, the Bragg peak may become quite diffuse. In the limit, however, as with some carbons, the reciprocal lattice can be partially lost due to complete randomness in phase shifts across a layer (7). At the other extreme, for regular faulting one introduces a new set of sharp reflections and thus a new structure. Because the fault is a planar defect its occurrence gives rise to diffuse streaking in reciprocal space; linear defects, in turn, produce sheets of intensity.

The second category of disorder can also be both compositional and position­al. In this case, however, there is no average lattice, no translational symmetry, no measurable reciprocal lattice, and in fact, no crystalline order. The correlation range as defined by a spherically symmetric pair distribution function is usually of the order of 1 0-20 A. We emphasize that one state is not the natural extension of the other. The truly amorphous or glassy state, as this latter category is called, is not arrived at, except in extraordinary circumstances, by progressively disrupting a crystal. It is not, in any case, the limit of a high defect density, high stacking fault probability, or small crystallite size (however similar the spherical­ly averaged diffuse scattering patterns may initially seem to those of an actual glassy material); rather it is a state unto itself. For a real glass, of course, the solid state is merely the supercooled liquid state (a viscosity transition) and, given the lack of currency of crystallite models for the liquid state, should present no problem. There are, however, many amorphous materials which are not prepared by cooling liquids. It should be emphasized that where at least one- or two-, if not three-dimensional crystallite models fail, these materials are indeed noncrys­talline and possess none of the features of crystalline order save, in many cases, the near neighbor coordinations and distances which are arranged spatially in a way incompatible with translational symmetry. The preservation of this type of local order is crucial in determining many of the properties of the amorphous

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material and it may lead some to associate truly amorphous solids with defective crystals and vice versa, especially in the case of strained thin micropolycrystal­line films.

SELECTED EXAMPLES

Compositional Disorder In Binary Alloys

From the point of critical phenomena, equiatomic binary alloys should offer potentially the most ideal examples of Ising behavior and, indeed, pCuZn has been very carefully studied both with neutrons (I8) and X rays (I9) above and below its critical ordering temperature. The results of the various scattering experiments, vanishing of the long range order below I:: (I 9), decay of the short range order above I:: (I8), and so forth, agree well with Ising predictions for critical behavior in the vicinity of the order-disorder transition. There are, however, very few alloys that demonstrate true critical behavior both because most of the ordering alloys have first order transitions and often appreciable volume changes on ordering, and because all of the alloys that undergo critical unmixing have slight to large lattice parameter changes accompanying the phase separation and giving rise to an elastic strain contribution to the free energy which in turn suppresses critical fluctuations (8, 20, 21).

The majority of the work on compositional disorder in alloys has, therefore, tended to concentrate on measuring the diffuse scattering either well above the transition temperature or on a quenched sample, as quenching often preserves many of the features of the disordered state due to sluggish diffusion and the inability of the atomic arrangements to readjust during the quench. One physical motivation behind such studies is the recovery of the pair correlation functions needed to model via computer simulation the actual atomic arrangements in the alloy. These computer modeling studies have been pioneered by Cohen and co­workers (22-24) and have provided substantial insight into the ways in which the atoms are actually distributed in a disordered binary alloy. Another motivation (25-27) has been the recovery of the energies of interaction in the solid solution and the demonstration of long range oscillatory interactions in concentrated alloys denoting, in turn, the existence of a rather well defined Fermi surface (28, 29).

Among the many alloys that have been studied, Cu-Ni is an interesting example. The total scattering, as noted earlier, is separated always into diffuse scattering and Bragg peaks centered at the reciprocal lattice points. The former is expressed in electron units as a Fourier series (7, 25)

1.

where k is a general point in the reciprocal space, N is the number of atoms in the sample, rnA, mB are the concentrations of the species A and B of the binary

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STRUCTURAL DISORDER IN SOLIDS 297

alloy whose scattering factors areiA andia, rij is a vector connecting site i and site} in the lattice, and <G;G}) is the pair correlation function (26) often called (Xij' after Warren (7) and defined as <G;G}) == (Xij = I - [p/B/(mAm

B)]' p/B

is the probability of simultaneous occupation of an A atom on site i and a B atom on site}. Krivoglaz has shown (8) that an equivalent statement of the scattering intensity in Equation I, in the absence of any size disparity between the atoms, is per atom:

where Ck is understood as follows:

2(OjA - mA) = OJ where 0/ = I A on i = 0 B on i

2.

3.

The quantity (0/ - rnA) is the alloy equivalent of the spin deviation operator and represents the deviation of the composition on site i from the average composition. Ck is thereby the amplitude of the kth fluctuation mode describing the compositional disorder and characterizing the normal modes of the compo­sitional fluctuations. In this formalism the scattering intensity J(k) is just proportional to the square of the amplitude of the k th mode of the composition excursions in the lattice.

If instead of dealing with a single crystal one uses a polycrystalline sample the intensity J(k) must be averaged over all orientations with the familiar result (7)

4.

This expression has set the jth site to be an origin atom whereby (X; = I - (P//m A ) ' given a B atom at site). C; is the coordination number of the shell of atoms all at a distance li;1 from}, and It I = 4?r sin B/A, with A the radiation wavelength. Equation 4 is remarkably similar to the scattering equation for a liquid or glass. The Cu-Ni system emphasizes this similarity where in Figure I

we show the coherent neutron scattering from a polycrystalline foil of the alloy CU's2 Ni� (30). Igt(Ni)N;(62) was chosen because its scattering amplitude for neutrons b(Ni62) = -b(Cu). The Bragg peak intensities are thus all essentially zero (a so-called null matrix alloy), while the disorder scattering is proportional to J(k) ex N[b(Cu) + Ib(Ni62)1f and is quite intense. This permits a measure­ment solely of the fluctuations without the usual dominant contribution of the average lattice scattering. The notations in Figure I as to STOP 2, START 1, and FIT all have to do with details of the data reduction procedure in (30). The references to Bragg peaks (11 I), (200), etc, show how well the composition was balanced to nullify the average lattice scattering.

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'? o

(J') I­Z => o u

10

(111)(200) (220) I I I

L.-.--FI T------'

(I sin 8

(311) / (122) (410) (333) (331) (511)

/(120) (412) I

----- 01 ONLY -- ALL a's AND /3's

Figure 1 Neutron cluster diffuse scattering from a null-matrix CuNi alloy, furnace cooled from 1021 °C, which represents a quench from _550°C. The Bragg peak positions are

indicated by crosses. STOP 2, START 1, and FIT refer to data treatment. The dashed curve is a least squares fit to only one short range parameter, (Xl' The solid curve is the complete fit. From Mozer, Keating & Moss (30).

The scattering in Figure 1 is really quite liquid-like, consisting of a series of decreasing intensity oscillations. The extraordinary thing about it is that the entire curve can very nearly be fit by a single correlation parameter a1' It is as if there were a small amount of nearest neighbor order with the alloy being compositionally random beyond the first coordination shell, i.e. /(k) ex sin krdkTj with no other contributions. This particular alloy of Cu-Ni was quenched from about 550°C, or well above its critical unmixing temperature. It should certainly not be terribly clustered into Cu rich and Ni rich regions, but the nearest neighbor clustering tendency (a) > 0) ought at least to propagate or fall off in some monotonic fashion. The answer to this question, as indicated by

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STRUCTURAL DISORDER IN SOLIDS 299

Mozer, Keating & Moss (30), lay in the application of the Clapp-Moss (26) equations to I(k). In that theory, which is similar to several others derived in different ways (11), the intensity I(k) is inversely proportional to the Fourier transform of the pair interaction energy, Vy = VyAA + VyBB - 2VyAB, for the alloy. By a suitable choice of the second to first neighbor energy ratio f2/J{ = -.60, Mozer et al were able to get an excellent fit to I(k), which means that in this quenched Cu-Ni sample an atom prefers a like nearest neighbor but an unlike second-nearest neighbor. The competition between nearest neighbor propagation and a direct second neighbor interaction randomizes the distant neighbor correlations. This, of course, implies a longer range oscillatory interac­tion.

A final point on Cu-Ni that deserves mention is one raised by Cohen and co­workers (24). Through computer modeling of the atomic arrangements commen­surate with the disorder measured by Mozer et ai, Cohen showed that the distribution function of Ni atoms with various numbers of (n) Ni neighbors is shifted. While the centroid is moved only slightly from the random value of n =

6, the number of Ni atoms with, say, ten (n = 10) Ni neighbors is enormously enhanced over the value in the random alloy. This enhancement plays an important role in the superparamagnetic behavior of these alloys. There is an exchange enhancement in Cu-Ni whose spatial extent is much greater than can be simply accounted for by, say, ten- or eleven-atom Ni clusters (31 ), but it may be greatly influenced by the small increase in local order.

The example of clustering in Cu-Ni, while interesting, is by no means isolated. The more frequent observations, actually, are on the phase-separation of more dilute alloys which, quenched from a temperature where the atoms are miscible, are then annealed in a two-phase field to develop a particular modulated structure often characterized as preprecipitates and called Guinier-Preston zones [although spinodal decomposition (20) is often the more appropriate description]. These are zones of enriched solute composition often with a unique internal structure as well as a shape dictated by the elastic properties of the matrix in which they form. (We presently discuss the scattering from solutions in which the composition fluctuations induce a strain field.) In these dilute alloys there are, in particular, important mechanical property improvements as a result of the development of the zones during thermal aging of the alloys. Aluminum alloys are hardened in this way and the systems AI-Ag, AI-Cu, AI-Mg, and AI-Zn have received a great deal of attention. Most of the scattering studies are confined to low angles where the fairly extensive scale of the composition fluctuations is revealed and where the influence of the static displacements is small. The work of Guinier and co-workers (3, 10) and Gerold (1 0) and co-workers is especially relevant.

As in some of the above cases when the atoms of the solid solution have different sizes the scattering is further complicated by the presence of static displacement scattering, including what is referred to as Huang scattering ( 32, 33) near the Bragg peaks as well as an asymmetric modulation of the normal disorder scattering discussed above. Sparks & Borie (34) have developed general

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procedures for separating these contributions and have successfully applied them to some single crystal Au-Ni data of the present author (35). It was shown, for a quenched alloy of Au60 Ni40, that composition fluctuations existed predominant­ly along the [100] cube axes of the crystal as predicted by Cahn (36), with a dominant wavelength of 8 A. The interpretation of this result by Cook & De Fontaine (37) in terms of a minimum elastic energy associated with that wavelength relied, again, upon extensions of the formalism of Krivoglaz (8) to which we can only allude briefly here. The reader is encouraged to consult the papers of Cook & De Fontaine, quoted in (37), on the microscopics of ordering and phase separation in the presence of elastic interactions.

What Krivoglaz has shown is that a good deal of the static displacement disorder scattering can be adequately treated by a linear coupling of the static displacement waves (the normal modes of the static displacement field) to the composition waves that produce them. If a composition wave of amplitude Ck exists in the lattice, a lattice parameter variation rides along with it if the atoms have different sizes. The displacement of the jth atom from its equilibrium site r; can be written 8rj = Y-I :p:kexp (-ik'�) where Y-I is introduced for convenience.

The coupling mentioned above is written in its simplest form as fk = Ak Ck. The coefficients ak are related to the elastic moduli of the material (8) and have a simple form in cubic crystals which is presently described. In terms of these elastic coupling coefficients the diffuse scattering in Equation 2 can be rewritten to lowest order as (8)

I (k) = N2 ICkl12 [J (k . Ak,) - (fB - fA)f J = mAfA + mBfa

5.

kl refers to a point in any Brillouin zone of the reciprocal space where in general k = k, + KN; KN = reciprocal lattice vector. Through a knowledge of the average elastic properties of the lattice, Equation 5 allows one to estimate the diffuse scattering away from Bragg peaks. Near Bragg peaks, it is a reasonable estimate of the true Huang scattering (33). Cook (38) has successfully applied the formalism to CU3 Au, CuZn, and to an artifically layered Ag-Au lattice.

Defect Crystals Although defect crystals (ones with large equilibrium vacancy concentrations on selected sites) really fall into the general category of compositional disorder, they form an interesting class in themselves with diffuse scattering distributions which are often quite striking. Figure 2 is an electron diffraction pattern from a thin foil

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STRUCTURAL DISORDER IN SOLIDS 301

of TiOu9 (39) near a (11 1) orientation as quenched from -l300°e. At 990°C, TiO undergoes an order-disorder transition among the vacant lattice sites with 15% of both the Ti and ° sites missing in a NaCl lattice. The ordering of the vacancies occurs on 110 planes in a regular fashion. As one adds either ° or Ti the respective vacant sites are filled leaving a predominance of vacancies on the other. Above the critical temperature, which decreases to 750°C at Ti01.25, there is a short range ordering of the vacancies and the quenched TiOu9 sample demonstrates a local order mainly among the vacant Ti lattice sites. Because TiO is metallic, Castles et al (39) were prompted to interpret the diffuse streaks and their composition dependence along lines similar to the Kohn anomaly argu­ments applied to eu-Au alloys (28). Obtaining good X-ray single crystals is very difficult, however, and quantitative estimates of the correlation functions have yet to be made for comparison with structural models. An important aspect of the nonstoichiometric diffuse streaking is that it demonstrates maxima displaced somewhat from the reciprocal lattice spots of the ordered phase. This is a frequent observation in disorder scattering and has been attributed both to the energetics of ordering (26, 28, 29) and to the size effect (33, 34).

Figure 2 Electron diffraction pattern near a (Ill) orientation from a thinned Ti01.l9 foil quenched from l300°C. The white disks are overexposed Bragg spots. The rest is diffuse scattering from a disordering among vacant lattice sites, mainly at Ti positions. From

Castles, Cowley & Spargo (39).

A similar defect structure exists in transition metal carbides and the diffuse diffraction patterns have been analyzed by Sauvage & Parthe (40) based on the beautiful experiments of Billingham, Bell & Lewis (41). Their analysis of VC.75 uses short range order parameters similar to those described earlier with a vacant

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302 MOSS

carbon lattice site as an origin having excesses or deficiencies of vacancies in successive coordination shells. The results for YC.75 suggest a structure not unlike �C5'

Fel_xO is another example of a transition metal oxide in which vacancies play an important role and in which the ideal structure is NaCl, although the phase is stable only above 570°C. There have been a variety of reports on the structure and distribution of vacancies, the most detailed of which by Koch & Cohen (42) finds a consistent interpretation of the diffuse scattering in terms of a periodic distribution of vacancy clusters on octahedral cation sites. The surrounding cations are displaced inward toward the cluster while the anions (oxygen) are displaced outward. Such detailed analyses of diffuse scattering are essential in understanding the influence on properties of various categories of defect structures with their often complex distributions of vacant sites.

Thermal Scattering and Structural Instabilities

'Ihe literature on thermal scattering belongs largely to the field of neutron spectroscopy because through inelastic thermal neutron scattering one can directly measure both energy and momentum transfer and thus the phonon dispersion relations. X rays and electrons are also scattered by the lattice vibrations. Because the energy transfer (kT :::::: .03 eV) is small compared to the incident energies of 10 keY -100 keY, the scattering can be rigorously treated as elastic scattering off the normal modes of vibration of the (elemental primitive) lattice, as (I -8)

I (k) = Nj2exp (-2M)h2/m � ( k . eqn? nqn(w

) +! n=1 Wqn

6.

where j is the scattering factor, exp (- 2M) the Debye-Waller factor, m the atomic mass, q the phonon wavevector, eqn a unit vector describing the polarization of the wave, Wqn the frequency of the particular phonon wave, and nqn(w) the Bose-Einstein population number for phonons. If kB T » hw Equa­tion 6 reduces to

- kB T 3 - 2 I (k) = Nj2exp (-2M)m n�l(k . eqn) /w�n

7.

Equation 7 shows that for small q near any Bragg peak in the reciprocal lattice (k = q + KN, where KN = Nth reciprocal lattice vector) the diffuse intensity /(k) falls off approximately as Ijq2 because for small q, w/q = v (sound) = constant.

The intensity could also be viewed as the scattering off each sinusoidal normal mode giving rise to a side band intensity proportional to the square of the amplitude of the mode. Since the amplitude is proportional to n(w) and thus to 1/w, I(k) ex: 1/w2, or 1/q2. If all we had were one phonon we would have one side band. The envelope of side bands falling off as Ijq2 is called thermal diffuse scattering (TDS) and through a careful analysis of it one can determine w vs q for each of the 3(n = 1,-2, 3) branches of polarization in simple crystals. This is

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STRUCTURAL DISORDER IN SOLIDS 303

not done much anymore both because optical modes bring complications in more complex crystals and because neutron scattering is so much more powerful and straightforward a technique; however, the work of Walker on Al (43) is still a classic in lattice dynamics. The study of Collella & Batterman on V (44) was necessitated by the fact that the neutron cross section for vanadium is entirely incoherent. Their results were quite comparable with similar neutron studies on, say, Nb (45) or Ta (46).

TDS is also observable in electron diffraction where thermal streaking is often an important and sometimes anomalously strong component of the diffuse scattering. Its separation from other forms of scattering and its production of artifacts (the piercing of the Ewald sphere by thermal streaks to give sharp­looking spots) can be a complicated affair and some of the diffraction patterns are quite striking especially where dynamical (mUltiple scattering) effects are important. The reader is referred to the article by Komatsu & Teramoto (47) for a review of many of the observations in both X ray and electron diffraction patterns.

When the crystalline lattice develops dynamical instabilities prior to structural phase transitions, the distribution of thermal diffuse scattering can often be both dramatic and crucial in interpreting the instabilities. Again, much of the work has been done with neutrons, especially on the transitions in the perovskites (BaTi03, SrTi03, KTa03, KNb03, KMnF3, etc). The recent article by Shapiro et al (48) addresses the problem of mode softening in these crystals, in which the shifting and damping of the appropriate phonon frequencies in SrTi03 and KMnF3 (a zone boundary mode in both cases) were also accompanied by a strong central component (centered on zero energy transfer) of the scattering which increased with temperature in both the lightly damped SrTi03 and the heavily damped KMnF3. Their explanation of all of these features lay entirely in the dynamical response of the lattice. In fact, in all of the perovskite crystals the very intense X-ray diffuse scattering (49, 50), which had earlier been attributed to large static displacements (frozen thermal motion, etc) attendant upon the structural transition, now seems to have a dynamical origin. The [100] linear displacements of KTa03 proved fairly elusive and were finally resolved as dynamical only recently by Comes & Shirane (51). The existence of a tempera­ture-dependent central peak in the perovskite crystals, where a large component of intensity is centered at zero energy transfer for the soft-mode wavevector, could perhaps be thOUght of as an elastic contribution to the scattering, thereby legitimizing the static displacement interpretation. Its dynamical origin and its interpretation as critical scattering are thus especially satisfying.

There are many metallic crystals that undergo displacive or structural transi­tions often called (often incorrectly) martensitic transformations. The literature on these is extensive; the review of Reed & Breedis (52) on their occurrence is useful, as is (23). One of the best known of the martensitic reactions occurs in nonstoichiometric {3CuZn where, even at the exact CuZn composition, there is a pronounced softening of the [110] shear modes (iill[l lO]) polarized in [TIO] «(;11 [T IO]). The elastic constant, small in this case, is C' = (ClI - C12) /2, a

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304 MOSS

frequently occurring soft vibrational branch in body centered cubic (bcc) crystals. The early (1952) measurements of Cole & Warren (7, 53) on TDS in pCuZn showed pronounced ridges of diffuse intensity along [110) directions that were due to the softness of the [110]: [110] shear modes.

Nb3 Sn is a technologically important alloy (its superconducting r;, � l 8°K) in which a displacive transition, or lattice instability, occurs at a temperature Tm = 45°K accompanied by a slight tetragonal distortion (54). Again, the instability occurs in the [ 110]: [110] acoustic shear mode, especially near the zone center. Shirane & Axe (55) also measured a three-peak spectrum in the frequency response of Nb3 Sn near 45°K whose shape seemed to be given by the strongly temperature-dependent anharmonic coupling of the unstable modes to other modes of the system. The appearance of a central peak is associated with critical scattering and is not to be taken in evidence of static displacements.

The general question of the lattice dynamical basis for structural instabilities in metals, including the martensitic reactions, is a fascinating one. The problems of identifying the soft mode(s) and associating it with the transition, unravelling the crystallography, and finally elucidating the dynamical response of the lattice have only been solved in a few cases, Nb3 Sn prominent among them. We choose as our final example one such instability which has received a great deal of attention and whose crystallography is reasonably well understood but whose dynamical aspects are only beginning to be clarified. The instability, called the omega (w) transformation, occurs in a large class of Zr and Ti alloys in which the bee phase is stabilized through the addition of neighboring transition metals such as Nb, V, and Mo. Over a limited composition range the stabilized bcc phase, when cooled, transforms spontaneously to the w phase which is nominally hexagonal. The transformation temperature is usually a very steep function of composition and for low solute contents the bcc phase transforms to hcp as with pure Ti and Zr. For more concentrated ailoys, CAl forms ; for still greater concentrations the bcc phase is preserved and the CAl is suppressed. De Fontaine (56) has shown how a single phonon, namely a [ 111] longitudinal mode with q = [2/3, 2/3, 2/3] 2",/ao(ao = lattice parameter), can transform the bee lattice to the hexagonal w lattice. It is a very simple structural transformation and has lately been documented by De Fontaine et al (57) in Ti-Mo and Dawson & Sass (58) in Zr-Nb. When it occurs, sharp Bragg spots appear at the 2/3[ 111] and all equivalent positions and the resultant reciprocal lattice is hexagonal. Above the transformation. temperature the reciprocal lattice is that of the bcc phase, but intense diffuse scattering occurs in the vicinity of 2/3[1 1 1] positions shown in the electron diffraction pattern of Figure 3 from Dawson & Sass (58). This is the (113) reciprocal lattice plane from Zr -22% Nb which has been overexposed to reveal the diffuse scattering. The large white circles are tQ.e overexposed bcc Bragg peaks. The rest is diffuse and is associated with structural fluctuations from the average lattice, which eventually lead to the transition, though not nominally in this alloy whose T", is below oaK. The marked shifting of the diffuse maxima from the exact 2/3[ 1 11] positions is an important aspect of the scattering.

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STRUCTURAL DISORDER IN SOLIDS 305

Figure 3 Electron diffraction pattern on a ( 1 13) orientation from a thinned foil of a Zr -22% Nb alloy quenched from 900°C. The large white disks are Bragg spots overexposed to reveal the diffuse scattering associated with the bcc phase above its omega transformation. The central spot is (000). To its right or left are { 1 1O} type Bragg peaks; directly above and below (000) are pairs of spots belonging to the {21 l} set. From Dawson & Sass (58).

The correlated displacements giving rise to the diffuse scattering of Figure 3 may be of a static or dynamic origin; a current neutron scattering (59) study is in the process of deciding the question. The preliminary results are that the scattering in the vicinity of 2/3[111] is elastic (to neutrons) and that a large fraction of all phonons are highly damped except at small wavevectors. The masses of Zr and Nb are very similar and it is unlikely that either spring constant or mass disorder is contributing significantly to the short phonon lifetimes. We are thus continuing this work to understand the dynamical response of the lattice, the nature of the structural fluctuations, and the basis of.the instability.

Static Displacements We have already discussed the inclusion of the static displacement field in compositional disorder scattering and shall in this section confine our attention mainly to other types of static displacements. A natural introduction, however, is the completely random disordered alloy in which the compositional diffuse scattering has only a perfectly smooth (dc) component called Laue monotonic scattering. Equation

' 5 may then be rewritten

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8.

Ak, takes a simple form for long wavelength static displacement fluctuations, although it may also be evaluated for all wavelengths (37, 38) in terms of interatomic force constants: Ak = ab ek Ilkll where, again kl refers to a point _ L- ':k.I I

inside any Brillouin zone, k = k} + KN(KN = reciprocal lattice vector) and Ck, is a unit polarization vector. For cubic crystals along high symmetry directions, the displacement waves are longitudinal, (ek, IIkl), although in a general direction they need not be and the values of ak, are simply given:

d Inv (Ie

(ak )[110] = 2 cl1 + 2cl2 d In Ii

, 3 Cll + C12 + 2C44 ---ac

(ak)[lll]= Cl1 + 2c12 dlnv , Cll + 2Cl2 + 4C44 �

v = atomic volume

C = composition

Knowing the elastic constants permits the calculations of the Huang diffuse scattering in a dilute random alloy. Conversely, measuring this static displace­ment scattering is a way of evaluating the elastic constants. These, of course, can be measured in other ways, but the equivalent formulae (37, 3 8) in terms of interatomic force constants permit their measurement as an alternative to neutron scattering. When the mass disparity is very large in a concentrated alloy this may perhaps become an important technique. For small wavevectors, kl' the intensity can easily be shown to fall off from a Bragg reflection as l/kl which is the same as the l/q 2 dependence for the TDS (kl = q). The analogies between static and thermal diffuse scattering are many, simply because they are both controlled by the force constants of the lattice.

Radiation damage has provided a strong impetus for the study of defects in solids. Perhaps the simplest of these occur in X-ray irradiated alkali halides. The report of Lohstoeter et al (60) characterized the anion vacancy-anion interstitial (Frenkel) pairs formed during low temperature irradiation of KBr, using the anisotropy and strength of the llq 2 Huang scattering from these defects to identify them. The results of their X-ray scattering study showed, via the theory of Trinkaus (61), [Dederichs has developed equations governing similar situa­tions (62)] that the defects were cube-edge oriented and probably consisted of Br-Br pairs centered on an anion site with an internuclear axis along [ 100] . In general, for radiation damage Goland pointed out (63) that there are many tools which complement each other, including lattice parameter data, electron micros­copy, and X-ray'diffuse scattering. This complementarity is discussed by Larson & Young (64).

Much more complicated defects can be studied if one has a model of the defect configuration, which can then parameterize the attendant displacement field calculation to fit the measured diffuse scattering away from Bragg peaks where it may assume rather odd shapes. Such is the case in neutron-irradiated

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STRUCTURAL DISORDER IN SOLIDS 307

BeO where collapsed interstitials form dislocation loops on the basal planes of the hexagonal lattice. Keating & Goland (65) analyze both the Bragg peak attenuations, which are strongly anisotropic along [00.1), and the diffuse scatter­ing. Their computer maps of diffuse scattering contain all of the characteristic features of the measured diffuse scattering in irradiated BeO; they are thus able to evaluate the size and character of the defects as a function of damage.

Another example of interest is the progressive damage in covalently bonded neutron-irradiated diamond. In this case, with increasing irradiation the dia­mond cubic Bragg reflections are attenuated and strong diffuse scattering builds up in their stead until one seems left with only diffuse scattering and a rather liquid-like pattern (66). Unfortunately, the diamond begins to graphitize at the same time and the limit of heavy irradiation is hard to explore. Perret & Keating (67) have also studied the low angle scattering from moderately irradiated diamond crystals where the defects are probably simple vacancies and intersti­tials. In the course of this work they revealed a new source of diffuse low angle scattering which arises from a multiple scattering between the Bragg and Huang diffuse scattering within the crystal. Because the Huang scattering off the displacement field is so intense and spread out, the sharp diffraction conditions

( I ii I

Figure 4 Low angle X-ray scattering photograph from neutron-irradiated diamond. The intense scattering about the forward beam is not directly off density fluctuations. Rather, the (111) reflection has been satisfied which has then been Huang scattered off the defect displacement field into the incident beam direction. From Perret & Keating (67).

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308 MOSS

are somewhat relaxed and diffuse and Bragg scattering can interact. Figure 4 shows an apparent intense low angIe scattering arising from satisfying the (Il l )

reflection, which then is Huang diffuse scattered back into the main beam direction. Such artifacts may always play a role in low angle scattering from defective crystals.

Ion implantation of Si or Ge is the last example of damage in covalently bonded crystals that we shall mention. It leads us, for rather unusual reasons, into our final topic of amorphous solids. After all that has been said so far on defects, damage, average lattices and their displacement fields, and the charac­terization of diffuse intensity as the scattering off fluctuations in various parameters of the lattice, ion-implantation damage comes as a bit of a shock. If one implants a crystal of Si with 5 x 1015 ions of Si per cm2 at an energy of 100 keY, the implanted layer of 1000-2000 A seems to be rendered amorphous in the same sense that evaporated or sputtered Si is amorphous. The situation has been discussed by Moss & Adler (68) and Moss, Flynn & Bauer (69); however, much of the evidence comes from the earlier work of Large & Bicknell (70) supported by a recent study of Graczyk & Chaudhari (71) showing that both the scattering intensity from the implanted material and its Fourier transform (the radial density function) are incompatible with a defective crystal model, being identical to these functions in- vapor-deposited amorphous Si (72) or Ge (73). This can

Figure 5 Transmission electron micrograph from a single crystal of silicon implanted with 100 keY silicon ions to 8 x 1015/cm2 and thinned from the back to reveal both the implanted layer above and the thicker crystal (plus implanted overlayer) below. The boundary is clear. The sample has been a given a short anneal below 200°C and shows void formation in the implanted layer (average void size -100 A). The wormy features in the lightly damaged crystaijust below the boundary are recrystallized damage tracks. From Moss, Flynn & Bauer (unpublished); see also (69).

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STRUCTURAL DISORDER IN SOLIDS 309

mean only that implantation damage creates pockets of locally vapor-quenched material which, with increasing damage, continually consume the crystalline layer. There doesn't seem to be any other type of damage which does that. Annealing of this damage can produce an epitaxially crystallized layer over the old crystal (70); however, the early stages « 200°C in Si) show pronounced void formation in the implanted layer and a recrystallization of the lightly damaged underlying crystal in a pattern which is probably replicating the original particle damage tracks rather like a crystalline "cloud chamber." Figure 5 shows these effects.

The Amorphous Solid The ion-implantation results of the preceding section really suggest a method of preparing an amorphous solid which is more in analogy with vapor quenching than with the usual forms of radiation damage which, via displacement spikes, produce vacancy and interstitial clusters and a defective crystal. For low implantation levels with light ions the defective crystal model is probably still appropriate. But when we consider heavy levels of implantation which are confined, regardless of dosage, to a few thousand angstroms of the surface and have incident energies of, say, 1 00 keY compared with crystal binding energies of 5-10 eV, the concomitant collision cascade processes are both very short-lived and extremely disruptive, producing as noted a state locally much like a dense vapor. To be sure, this is not the most conventional method of making a glassy or noncrystalline solid and it should not, therefore, be thought of as a link between our two classes of disorder. The articles referred to earlier (68-73), as well as the review of Turnbull & Polk (74), discuss at length the structure of amorphous silicon and germanium and we shall not dwell further on these materials.

For our present purposes the amorphous or glassy state is well exemplified by the diffuse scattering (and its transform) from metallic glasses. These have been prepared by several methods, including rapid quenching of the melt, pioneered by Duwez (75), flash evaporation or simultaneous coevaporation of the constit­uents, and chemical and/or electrodeposition and sputtering. Among the most common of these materials are glasses based upon low lying eutectics, as in the Pd-Si and Ni-P systems, and there are numerous examples of ternary glasses including, Fe-Pd-P, Co-Ni-P, Au-Ge-Si, etc. It is generally conceded (76) that the smaller (P or Si) atoms impede the motion of the larger atoms in the liquid state by filling selected interstices, thereby drastically reducing the freezing point and permitting the preservation of the liquid-like structure upon supercooling or quenching. A crucial test of the glassy nature of these metals was provided by the observation of Chen & Turnbull (77) of a glass transition temperature in a Au-Ge-Si alloy.

The conditions of glass formation are discussed by Turnbull and co-workers (76-79). We choose from among the many metallic glasses the Ni-P alloys as prototypical, mainly because Cargill (80) demonstrated so conclusively the inapplicability of crystallite models for glasses in this system. The scattering

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intensity from a (monatomic) liquid or amorphous solid is given by a spherically averaged intensity function (1, 3, 7, 75) where k = 4'17 sin 8/>.. (the magnitude of the diffraction vector) and r is a radial distance from an origin atom:

co

2 f 2 sin kr /(k) = Nf [I + 4'ITr [p(r) - Pol----,a:- dr

o

k[/(kl - IJ == F(k) = j G(r) sin kr dr Nf 0

co

where: G(r) == 4'ITr[p(r) - Pol = � f F(k)sin rk dk o

9.

p(r) is the radial atomic density or the number of atoms per unit volume at a distance r from the origin and Po is the average atomic density. There are many formulations of the radial atomic density including 4'lTr2 p(r), which is the radial density function whose integral between appropriate limits gives the coordina­tion number of the "shell" of atoms defined by those limits. A useful quantity is

1.5

-:: 1.0

"

� 0.5

I I

I I I I I I , --,-I-I I I I I I I , I I I I I •

O'O L-----�2�----�4------�6------�8------� 1�0------IL2�

r in A

Figure 6 Pair distribution W (r) = p(r)/po for a dense random packing of hard spheres [see Finney (82)] scaled to compare with the experimental W(r) for a N,76 P,24 metallic glass determined by X-ray diffraction. The histogram is the Finney model, the broken curve is the data. From Cargill (81).

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STRUCTURAL DISORDER IN SOLIDS 311

the pair distribution function, W(r) = p(r)/po' which tends to unity at larger r and describes the deviations from the average atomic density in a useful way. This elemental glass analysis is complicated somewhat by the addition of other constituents as discussed, for example, by Giessen & Wagner (75).

Figure 6 shows a comparison from Cargill (81) of the X-ray determined pair distribution function of a Ni.76 �24 alloy, prepared by electrodeposition, with the histogram of distances measured by Finney (82) for the dense random packing of hard spheres. This is a classical noncrystalline solid whsse properties have long been the interest of the group of the late J. D. Bernal. The coincidence of the oscillations in Figure 6 is impressive and the reproduction of the split second peak in W(r) is strong evidence that the metallic glasses are well modeled by a noncrystalline dense packing of spheres. The principal difficulty encountered in applying crystallite models was that, for J(k), crystallites perfect enough to reproduce the sharpness of the first diffuse oscillation were too well defined to give the rapid fall off and diffuseness in the rest of J(k). This, together with other systematic inconsistencies that could not be removed with various kinds of distortions or faults, ruled out a microcrystalline interpretation (80).

As a counter-example of positional disorder, which can be accounted for by a small crystal calculation, we mention the work of Wagner et al (83) on a vapor­quenched Cu-Ag alloy. This study shows that even for fcc crystallites of a diameter less than 16 A the preservation of crystalline order in the diffraction pattern is quantitatively given by the Debye interference function for small crystals. The lesson is clear: in those cases where the films [the question is raised most often in the case of vapor"quenched films] are composed of tiny distorted crystallites, the spherically averaged diffraction pattern when it is analyzed carefully reveals that fact fairly unambiguously. The reciprocal lattice is thus preserved in at least one or two, if not three, dimensions and some translational symmetry exists. A careful visual estimate of an electron diffraction photograph is usually insufficient to make the distinction between this state and the truly amorphous state. For example, the observation of diffuse haloes on films either splat-cooled (84) or vapor-quenched (85) requires a quantitative analysis of the intensity and its transform to decide whether one is dealing with a true loss of translational symmetry or a messed up crystal. When the former prevails, no distorted crystal calculation suffices (80); when the latter condition exists, as it very well may in the films of Collver & Hammond (85), (alloys among neighboring transition metals of similar atomic masses and sizes are not notorious glass formers), proving it should be a relatively straightforward matter.

A final comment on metallic glasses has to do with the interesting observations of Rhyne et al (86) on the spin distribution in amorphous TbFe2 well below the Curie point. The critical behavior near the Curie point should also be measurable through an analysis of the temperature dependence of the low angle neutron scattering with and without a field parallel to the diffraction vector. In such a material, when the nearest neighbor exchange interaction is dominant, the critical point could be essentially as sharp as in a crystal if, as with many glasses, the nearest neighbors have no more variation in interatomic separation than their

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312 MOSS

crystalline counterparts. In the binary or ternary glasses, however, there may be an appreciable variation in nearest neighbor distance due to a large atom size disparity and some randomness in the identity of the nearest neighbors (compo­sitional disorder). When considering rare earth, transition metal glasses (assum­ing that they are, indeed, glasses) such as Tb:Fe, Gd:Co, or Gd:Fe (88), the combination of atom size disparity and variation in nearest neighbor identity should have a large effect on their magnetic properties, especially when the transition metal atoms have parallel spins while the transition-rare earth pair can be aligned antiparallel, at least in the crystal. Each of these interactions is highly distance-dependent and one might imagine a strong coupling between short range compositional order and short range magnetic order, which among other things might smear out the critical point. If longer range exchange interactions are important, the problem is even worse, except, perhaps, if the exchange is infinite-ranged as in a mean-field crystal as discussed, for example, by Handrich and co-workers (87).

These are cases of spin disorder on a compositionally and positionally disordered matrix. While the pair correlation function for the spins can thus have a positional correlation range no greater than that for the atoms, certainly all the spins can in principle be ferro- or ferrimagnetically aligned at OaK. This would yield a delta function in the neutron spin scattering at k = 0 in the same sense that a liquid has a delta function in its scattering at k = 0 of height N2j2, where j is the scattering factor and N is the number of atoms in the sample. Over the rest of the range of k the spin scattering would look much like the nuclear scattering which is liquid-like as in our Figure 1. Therefore, as T approaches OaK, the spin scattering near k = 0 might be very sharp. In fact, on approaching 7;; from above it should sharpen as in critical scattering, while below 7;; its breadth should be given by fluctuations in the long range order. The fact that Figure 3 of Rhyne et al (86) looks like our Figure 1 at low k is initially a bit surprising. It is definitely suggestive of spin clustering (ferro- or ferrimagnetism) but with fluctuations in long range spin order on a range comparable with the atomic pair correlation function for the glass; that is not a great deal of long range spin order for a system with a well defined 7;; . We emphasize that because the spins are merely decorating the atoms the scattering outside the low angle regime must always be liquid-like even if all the spins are parallel.

Rhyne et al (86), of course, recognized this fact and noted that the spin scattering at 4°K is quite broad at low angles, concluding that there must be regions analogous to domains which define the range of spin alignment well below 7;; (7;, R:: 4000K). From an extrapolation of the low angle data in their Figure 3, one might estimate the size of these magnetically clustered regions of long range order to be roughly 20--30 A.

This result perhaps has a qualitative explanation in the nonuniqueness of any direction or axis along which to order the spins. While they wish to align, say, in a ferrimagnetic fashion, there is no lattice to define a preferred direction. They, therefore, take their clue from their neighbors where, by chance, two or more atoms are closer together and thus strongly coupled. The direction for alignment

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STRUCTURAL DISORDER IN SOLIDS 3 13

i s thereby self-induced and has meaning only over a distance for which positional or compositional correlation has meaning. In other words, the spins all align below 'Fc but the directional information present in a crystal is absent in the glass unless induced during the deposition of the film through an applied field or other deposition parameter (88). All directions are thus equally good and the range over which the spins in any part of the glass align parallel to some external direction is no greater than the range over which we can define positional correlation. The model based on the results of Rhyne et al is thus one in which the magnetic ground state, in the absence of any easy or imposed axis, consists of fluctuating spin clusters on a scale comparable with the actual positional correlation range of the glass.

SUMMARY AND ACKNOWLEDGMENTS

We have discussed several types of structural disorder in solids, demonstrating analyses where they seemed appropriate, and stressing those aspects of fluctua­tion theory which serve to unify the subject. The treatment has largely been restricted to categories of disorder which have been of personal interest; a bias remedied in part, hopefully by a reasonable distribution of general references. We have not mentioned molecular solids (91), the vibrational properties of which are fascinating; nor polymeric solids which form a large class of more or less disordered materials (89) of great technological importance; nor layer disorder, from the semiconducting supperlattice structures discussed by Segmiiller & Blakeslee (90) at the ordered end of the spectrum, to glassy carbon at the other end. These omissions serve additionally to emphasize how much of what we treat in solid state physics belongs within the domain of disorder.

Drs. G. S. Cargill, J. M. Cowley, D. T. Keating, and S. L. Sass kindly supplied both the photographs reproduced here and the permission to use them. For many of the diffractionists referenced herein, enthusiasm for the variety of problems discussed in this review has derived in some measure from an association either directly or indirectly (through each other) with B. E. Warren. This seems a suitable place to acknowledge that association. The contribution of M. A. Krivoglaz is also noteworthy. He has provided the union between various disciplines, such as fluctuation theory and microscopic elasticity theory, which through a Fourier space representation is naturally written in diffraction language.

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314 MOSS

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CONTENTS

EXPERIMENTAL AND THEORETICAL METHODS

Theoretical Approaches to the Determination of Phase Diagrams, Larry

Kaufman and Harvey Nesor

PROPERTIES, PHENOMENA

The Fracture Crack as an Imperfection in a Nearly Perfect Solid, Robb M.

Thomson 31

Critical Phenomena in Solids, M. E. Lines 53

Mass Transport in Solids, N. L. Peterson and W. K. Chen 75

Electrical Properties: Charge Injection Phenomena, Peter Mark and Myron

Allen 111

SPECIAL MATERIALS

Layer Compounds, A. D. Yoffe 147

High Temperature Compounds, Hans Nowotny and Stephan Windisch 171

Carbon and Graphite Science, Douglas W. McKee 195

Cement Paste and Concrete, Torben C. Hansen, Fariborz Radjy and Erik J.

Sellevold 233

Recent Advances in Liquid Crystals, Georges Durand and J. D. Litster 269

STRUCTURE

Some Aspects of Structural Disorder in Solids, Simon C. Moss 293

PREPARATION, PROCESSING, AND STRUCTURAL CHANGES

Chemical Vapor Deposition of Electronic Materials, James J. Tietjen 317

Phase Transformations in Metals and Alloys, Glyn Meyrick and Gordon W.

Powell 327

Synthesis of Materials From Powders by Sintering, A. L. Stuijts 363

Crystallization, A. A. Chernov 397

REPRINT INFORMATION 455

INDEXES

Author Index

Subject Index

Cumulative Index of Contributing Authors, Volumes 1-3

Cumulative Index of Chapter Titles, Volumes 1-3

457

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481

482

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