STOICHIOMETRY, DEFECTS AND ORDERING · defects are important, involving some local modification of the crystal structure ; they include isolated defect clusters, planar faults such

JOURNAL DE PHYSIQUE Collogue C7, supplement au n° 12, Tome 38, decembre 1977, page C7-17

STOICHIOMETRY, DEFECTS AND ORDERING

J. S. ANDERSON

Edward Davies Chemical Laboratory, University College of Wales, Aberystwyth, Wales

Résumé. — La constitution des phases de composition variable — composés non stœchiomé-triques et solutions solides hétérotypiques — ne peut pas être comprise seulement en des termes de défauts ponctuels. Les défauts étendus sont importants puisqu'ils impliquent des modifications de la structure cristalline : ils comprennent des amas isolés de défauts, des défauts plans tels des plans de cisaillement cristallographiques (CC), des maclages chimiques, et des mises en ordre par surstructure qui transforment les défauts ponctuels en des éléments propres de la surstructure du nouveau cristal.

L'intercroissance absolument cohérente des structures topologiquement compatibles est un principe important. La mise en ordre des défauts étendus peut engendrer une suite de structures cristallographiquement déterminées, dont les compositions sont très proches et qui sont liées les unes aux autres du point de vue structural ainsi qu'à la structure d'hôte. Un déploiement désordonné de tels défauts étendus équivaut à une intercroissance des termes différents des séries homologues et simule les propriétés d'une phase de composition variable.

Il existe des indications de plus en plus fortes que quelques structures sont infiniment adaptives, vraiment non stœchiométriques mais monophasées et complètement ordonnées pour chaque composition. Deux types sont : les structures Vernier, contenant deux réseaux incommensurables partiels, et les empilements adaptables de couches démontrés dans des systèmes ayant une orientation variable de plans CC.

Le comportement des systèmes particuliers est déterminé par la balance entre l'enthalpie nette de la formation des défauts qui sont (le plus fréquemment étendus) structurels, et l'entropie confi-gurationnelle des défauts. Les systèmes sont ou contrôlés enthalpiquement, leur état d'équilibre étant conforme à une succession de composés intermédiaires stœchiométriques, ou contrôlés entro-piquement, ce qui conduit à une composition variable.

Abstract. — The constitution of phases of variable composition — nonstoichiometric compounds and heterotype solid solutions — cannot be understood solely in terms of point defects. Extended defects are important, involving some local modification of the crystal structure ; they include isolated defect clusters, planar faults such as crystallographic shear (CS) planes and chemical twinning, and superlattice ordering which transforms point defects into essential structure elements of a new crystalline order.

The completely coherent intergrowth between topologically compatible structures is an important principle. The ordering of extended defects can generate a succession of crystallographically defined structures, closely spaced in their compositions, and closely related in structure to each other and to the host structure. A disordered array of such extended defects is equivalent to an intergrowth between different members of the homologous series and simulates the properties of a phase of variable composition.

There is increasing evidence for infinitely adaptive structures : truly nonstoichiometric, yet mono-phasic and fully ordered for every composition. Two types are the 'Vernier structures', with two incommensurable partial lattices, and the adaptive layer stackings, exemplified in systems with changing orientation of CS planes.

The behaviour of particular systems is determined by the balance between the net enthalpy of formation of (usually extended) structural defects and the configurational entropy of the defects. Systems may be enthalpy-controlled, their equilibrium state corresponding to a succession of intermediate, stoichiometric compounds, or entropy-controlled, leading to phases of variable composition.

Inherent in the theory of the crystalline state is the principle of complete order : that translational symmetry completely defines the contents of the repeating unit of structure. This leaves no room for variability of composition. In practice, there is, however, a rather

restricted number of compounds which, by all operational criteria, can vary in composition, within a certain range, without any apparent change in crystal structure. They present a fundamental problem that is not yet fully resolved.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1977703

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C7- 18 J. S . ANDERSON

Genuinely nonstoichiometric compounds, with a thermodynamically bivariant range under equilibrium conditions (e.g. Fe,-,O, TiO) are fewer in number than was formerly thought. Ordering processes that would lead to defined intermediate phases are often extremely slow, or kinetically inhibited, so that the most careful experimental observations can actually relate to non-equilibrium conditions. At low temperatures, at least, apparently broad stoichiometric ranges have, in a number of instances, been found to comprise a succession of intermediate compounds, with closely spaced but definite compositions, structurally related to each other and to the parent nonstoichiometric compound, and this, as will be seen, is often the equilibrium situation. That stoichiometry can be genuinely variable (as shown, for example, by structural studies at high temperatures, under equilibrium conditions) is well attested in certain cases. Moreover, some structural interpretation is needed for the simulation of monophasic, bivariant behaviour under pseudo-equilibrium conditions. Hetero- type solid solutions (typified by calcium stabilised zirconia) present the same problem : a gross discre- pancy between the average number of atoms and the average number of sites per unit celi. The main differences between nonstoichiometric (mixed valence) compounds and heterotype solid solutions arise from the fact that the location of compensating charges, in the former, can be redistributed by electron transfer, whereas in solid solutions this can be achieved only by atomic (usually cation) diffusion. Steps essential for defect association and ordering are thereby impeded.

It is now possible to recognize some guiding concepts for interpreting the structure of nonstoichiometric phases ; and until structure is understood, thermody-

POINT DEFECTS

SUBTRACTION of atoms from Unbalanced VA- parent structure CANCIES in one

sublattice

INTERPOLATION of atoms in Unbalanced parent structure INTERSTITIALS

in one sublattice

SUPERLATTICE ORDERING

Unbalanced SUB- STITUTION in one sublattice ;

A, or BA Superlattice based

on parent structure

narnic and other aspects of the problems cannot be explained. The first is that, rather than the averaged structure of a macroscopic crystal, with non-integral site occupancy, as found by the usual structural methods, emphasis must be placed upon the local structure from point to point within the crystal. Notable advances in this have been made by several methods, including the full exploitation of high resolution electron microscopy.

A second important concept is that of the extended defect. It is an unquestioned inference from statistical thermodynamics that all crystals are subject .to point defect disorder, and that this disorder results in some small variability of composition as the chemical potentials of the components are altered. Equilibrium constants for point defect disorder, however, are dominated by the endothermicity of defect creation. In general, defect creation energies are large, so that the equilibrium concentrations of point defects in stoichiometric crystals are correspondingly low, even at temperatures approaching the melting point. Point defects, and simple association between them, have undoubtedly provided a quantitative interpretation of the properties of the typical Daltonide compounds (e.g., alkali halides, the tetrahedral semiconductor compounds, etc.), but they cannot be satisfactorily invoked for nonstoichiometric phases in which of the order of 1-10 % of lattice sites may appear to be defective. The implication would be (for example, for stoichiometric Tio.8300.83) an unreasonably small value for the energy of creation of a Schottky defect pair. Interactions between defects are important, especially in ionic crystals where unscreened defect charges exert long-range Coulomb forces. For any chemically significant stoichiometric deviation, the

EXTENDED DEFECTS -

LOCALISED MULTI- PLANAR DEFECTS SITE CLUSTERS with Increased linkage bet- lowered atomjsite ratio ween coordination poly-

hedra. CS collapse ; chemical twinning.

LOCALISED MULTI- PLANAR DEFECTS SITE CLUSTERS with Decreased linkage bet- increased atomlsite ratio ween coordination poly-

hedra. Interpolation of atoms between layers.

Occupation of tunnel sites

Clustering Interpolated sheets of atoms.

Superlattice based on ordering of extended defects

STOICHIOMETRY, DEFECTS AND ORDERING C7-19

inference is inescapable that interaction effects must contribute some strongly exothermic terms to the free energy of the crystal. In now numerous instances, experiment has shown that these effects go far beyond simple association and clustering ; they involve sub- stantial modification of the local structure of the host crystal and replace point defects (which indubitably remain present at low concentration) by extended defects.

The third guiding concept is that of coherent intergrowth. Extended defects constitute strong per- turbations of the regular host structure; they have their own local structure, derived from that of the host and can - in virtue of that relationship - be inserted coherently, without significant strain, in the host structure. Replacement of host structure by the extended defect locally alters the composition of the crystal. On a point defect interpretation, the concentration of vacancies, interstitials or substituted atoms in a nonstoichiometric crystal is directly related to the departure from stoichiometry, but with extended defects there is not necessarily any simple relationship between the stoichiometric deficit and the concentration of wrongly occupied lattice sites. That there is an excess or a deficit of atoms, as compared with the parent structure, remains true; how this can be accommodated, as the stoichiometric deficit changes from the very low levels typical of semiconductor crystals to the high level found in nonstoichiometric compounds and heterotype solid solutions, is sum- marised in table I.

Extended defects interact with each other, through Coulomb forces and - particularly important in the case of planar defects - elastic forces [I]. If the neces- sary place-exchange and transport processes are sufficiently facile, these interactions can impose a superlattice ordering of extended defects. Each such mode of superlattice ordering represents a crystallographically defined compound of definite composition. It is clear from the experimental evidence that, in many systems, it is energetically advantageous to build up a homologous series of intermediate compounds, by the ordering of extended defects at certain discrete concentrations, rather than to form a nonstoichiometric phase, of broad range, with randomized defects. Much of the evidence for the nature of extended defects has come from the study of such ordered series.

Structured defect complexes can be regarded as closed elements of some new structural arrangement, embedded coherently within the host structure; in general, atomic positions within the defect complex are closely (but not conservatively) related to those of the host structure. As isolated units, defect complexes can be distributed randomly through the crystal and thus contribute significantly to the configurational entropy ; their concentration is directly proportional to the stoichiometric deficit. In a randomized confi- guration, the averaged translational symmetry remains

that of the host crystal, but diffracted intensities will be changed because, in terms of the original host structure, some proportion of sites is vacant, or interstitial sites occupied. Such defect complexes, through their interactions, may order into a superlattice.

That defect complexes replace point defects in highly nonstoichiometric phases was first recognized for the,cation-deficient wustite, Fe, -,O and the anion excess UO,,,. For wustite, Roth [2] showed by neutron diffraction that iron atoms were present on tetrahedral sites ; to produce these, octahedral sites must be vacated, over and above the vacancies that correspond directly to the deficit of cations. From structural studies on the partially ordered superlattice, formed by annealing in the low temperature metastable region, Koch and Cohen [3] diagnosed that the defect resembled an element of the zinc blende structure, with four tetrahedrally coordinated cations surrounded by thirteen empty octahedral cation sites, and with the anion sublattice substantially unaltered (figure lb). 'High temperature, neutron diffraction measurements carried out under equilibrium conditions, with controlled oxygen fugacity 141, established that the ratio of tetrahedral atoms to empty octahedral sites was consistent with the Koch-Cohen complex, and that the same complex appeared to account for the nonstoichiometry across the whole range of composition. Nearly all the cation vacancies are contained, as structure elements, in the defect complexes. The concentration of true point defect vacancies is small ; they play a minority role in structure, though an essential role in transport properties. It now seems likely that orientational degeneracy raises the apparent diffraction symmetry of the superlattice material, on which the Koch-Cohen model was based. Electron microscope lattice images of wustite (Iijima 1974), although not to be naively interpreted as projections of the atomic positions, show that the true symmetry of the superlattice is probably monoclinic, but that local ordering extends only over microdomains, of around 50 A in dimensions, on each of the three possible orientations, so simulating cubic symmetry.

In an analogous way, the nonstoichiometric fluorite structures are not based upon simple point defect anion vacancies on interstitials. Neutron diffraction in the UO,,, phase [5] revealed not only that oxygen atoms are displaced along [I101 from the high- symmetry interstitial site, but that each interstitial displaces oxygen atoms, along [I 111, from anion sites proper to the fluorite structure, thereby creating vacant anion sites and a second species of interstitial oxygen. The most probable basic defect complex is the 2 : 2 : 2 unit (figure la), with two interstitials of each kind and two vacancies ; associated with this is charge compensation on adjacent cation sites. Clusters of this kind can extend, by repeated processes of oxygen insertion and anion displacement, as the anion excess increases. Complexes or reconstructed clusters of this kind appear to be the generally favoured

J. S. ANDERSON

FIG. I . - Finite, structured defect complcxcs : (a) The Willis 2 : 2 : 2 clustcr in UO,,,, CaF,/YF,, etc. At right, extension to larger defect clusters, e.g. the 4 : 2 : 3 cluster shown, with increasing nonstoichiometry. (b) Fe,_,O. Above, the basic extended defect, with tetrahedrally coordinated Fe(iii) surrounded by empty octahedral cation sites. Below, the Koch-Cohen (Fe,),(VFe),, complex. (c) Sub-

stoichiometric fluorites. The basic complex (left) and the formation of single anion vacancies by a dissociation step.

means of accommodating excess anions in fluorite structures; they have been shown to underlie the heterotype solid solutions, typified by CaF,/YF,, up to compositions approaching MX2,, [6].

In anion-deficient fluorite structures, the usual defect complex is based upon the pairing of empty anion sites, across the diagonal of the cubic cation coordination polyhedron [7] (figure l c ) . Ordered phases (e.g. the M,O, , oxides) have these complexes aggregated in linear arrays ; the isolated defect can be coherently inserted into unperturbed fluorite structure, and is almost certainly dispersed randomly in the truly nonstoichiometric a-phase of Ce0,-,, Pro,-,, etc. ; there is some evidence that at least vestigial order is retained, even though the diffraction symmetry of the cr-phase is that of the fluorite parent structure.

Only a minority role is thus ascribed to point defects in these highly nonstoichiometric phases. It is therefore important to examine how this conclusion accords with point defect ideas ; there is undoubted evidence (e.g. from e.s.r.) that excess anions in fluorite structures, at low concentration, do occupy high symmetry (octahedral) interstitial sites. Some recent ab initio calculations of the interaction energies of defects in the NaC1-type structure of FeO, and in the fluorite structure of UO, [8] are therefore of great interest. They show that as the concentration of defects rises to significant values, structured complexes of the kind

inferred from experiment are precisely those pertur- bations that contribute large exothermic terms to the crystal energy.

Figure 36 shows complexes, elements of the 2 Nb205 . 7 W 0 3 structure, coherently inserted in the tetragonal tungsten bronze structure of 4 Nb20,.9 W03, as a means of accommodating nonstoichiometry. In this case, the defects are aggregated into columns, so distributed as to suggest a tendency towards superstructure ordering.

Because isolated defect complexes can be disposed at random through the host crystal, they contribute substantially to the configurational entropy and can give rise to truly nonstoichiometric phases. Planar defects, which have been identified in an increasing number of systems, can contribute little to the configurational entropy ; their randomization is essentially a 1-dimensional disorder problem. Hence they tend to generate families of discrete intermediate compounds as a result of superstructure ordering. A distinction must be drawn between planar defects, in the present context, and antiphase boundaries, stacking faults, etc. These latter are conservative in lattice sites and involve no change in composition ; planar defects, in the sense used here, are nonconservative boundaries between slabs of parent structure, in which a structural rearran- gement is localised across a complete sheet of crystal structure, with some specific orientation. Within this sheet, the linkage of coordination polyhedra is


modified, so as to eliminate or to create lattice sites. In general, one sublattice (in most cases the anion sublattice) is thereby left unchanged and continuous across the boundary, which forms a 2-dimensionally localized change of composition. Regularly recurring planar boundaries, enclosing slabs of parent structure of constant width, produce a new crystallographically defined structure, of definite composition. Since the width of the slabs of parent structure can, in principle, vary by unit steps, the regular spacing of planar boundaries can generate a homologous series of compounds, with a subtle, progressive change in composition. The establishment of perfect order in such structures can be a kinetically very slow process, since it requires an extensive reshuffling of planar boundaries, sad even in well annealed materials, high resolution electron microscopy not infrequently reveals 1-dimensional disorder, with variable spacing between planar boundaries. The composition is then no longer defined ; they are phases of variable composition; but not equilibrium phases. The free energy difference between such a nonstoichiometric, imperfectly ordered state and the fully ordered state is likely to be small, but thedoiTynamic, as well as crystallographic, characterization is lost. Bivariant behaviour - e.g., continuous dependence of oxygen fugacity upon composition, in oxide systems - then replaces a discrete set of univariant equilibria between line phases and simulates a nonstoichiometric phase.

Three broad types of planar boundary, all serving to eliminate point defects, conform to these general considerations : crystallographic shear, chemical twinning and what might be termed layered chemical segregation.

Crystallographic shear, exemplified (but by no means confined to) the intermediate oxides derived from the ReO, and rutile structures (Magneli's Mo,O,,- ,, Tii02,-, series, etc.) can be formally regarded as effecting the elimination of a complete sheet of anion sites, lying on a particular crystallographic orientation, as a result of the simultaneous shear displacement and collapse of the structure on that plane [9]. ~har'acteristicall~, the anion sublattice remains, to a first approximation, unchanged ; the rearranged, closer linkage between cation. coordination s polyhedra at the crystallographic shear plane (CS plane) can be described in terms of shifts of the cations alone, across the boundary. Figure 2a shows a ball model of the (100) plane in the rutile structure, slightly idealised to bring the anions into hexagonal close packing. If cation rows are displaced by 112 [Ol 11 across a (011) plane, the process is conservative in sites ; it creates an,antiphase boundary with no change in composition. The same displacement, operating on any other plane in the [ l i l ] zone is non-conservative. On a (121) plane (the orientation of CS planes in the Magneli oxides Ti,O, to Ti,,O,,), each displacement of cation rows produces a group of face-sharing

1 i, I ? ; , , , ! ; I ! , ! ..,.:.a I . ,

FIG. 2. - (a) Ball model of (100) rutile plane (idealised). Upper left : notiona1 anion vacancies are eliminated by crystallographic shear collapse on the (121) plane (right), restoring the integrity of the anion lattice. Below, left, a (132) CS plane and, right, a (011) antiphase boundary, all formed by the same collapse vector. (b) The -Pb02 (recurrent APBs) and V,O, ((121) CS planes) structures stack to give a continuous succession of structures in the Fe203-Cr203-Ti0,-ZrO, system. (c) Adaptive structures : simple stacking sequences, the corresponding patterns of cation site occupations (figure 3b) and continuous change in superlattice in the Fe20,-Cr203-Ti0,-Zr0, system. (d) Adaptive structures. The ternier structure of Ba,+,Fe,S,. Left, Ba,Fe,S,,, centre Ba,Fe,S,,, right the end member, BaFe,S,.

C7-22 J. S. ANDERSON

FIG. 3 . - (a) Faulted CS planes and variable qpacing'between them in an oxygen-deficient W03 phase. (b) Isolated columns of the structure of 2 Nb20, . 7 WO, (24 Ax24 A, marked in squares) coherently intergrown in the 4 Nb20, .9 WO, tetragonal tungsten bronze structure to accommodate a stoichiometric excess of WO,. (c) Domains of several Werent ordering patterns for the occupation of penta- gonal tunnel sites in the 4 Nb20,, 9 WO, tetragonal tungsten bronze structure, with excess Nb,O,. The unit cell of the TTB structure (principal structural feature) measures 12 A x 36 A, (d ) A nonstoichiometric phase in the Nb20,-MgF, system, showing a mosaic of coherently intergrown elements corresponding to different structures and stoichiometrics. The host structure is made up of columns

4 x 3 octahedra in cross section c14 A x 10 A).


cation coordination polyhedra, in place of the edge sharing of the rutile structure ; out of the plane of the diagram, these denser groups of cations extend along the [ I l l ] direction, and are structurally similar to elements of the corundum structure, with the composition M203. In the ordered titanium oxides between Ti,,O,, and rutile itself, the same displacement vector operates on the (132) plane. As can be seen, conservative antiphase steps and site-eliminating CS collapse steps alternate along the (132) boundary. In principle, not only this 1 : 1 alternation, but any regular sequence of the two kinds of step should generate a possible CS boundary, making every orientation in the [ l i l ] zone possible. This can, indeed, happen, and the point is taken up later.

Similarly, in the ReO, structure, CS planes involve segments of [I101 antiphase and (010) CS collapse displacements. In all the known, stable fully ordered phases with a single set of CS planes, the orientation is (100), (120), or (130), but in the niobium oxide block structures, with two intersecting sets of CS planes, these can swing around from { 100 ] to { 130 ).

Reference has already been made to imperfect ordering of CS planes, variations in their spacings, as one cause of variable composition. A second kind of disorder, resulting also in stoichiometric variability, is faulting in the CS interface itself (figure 3a). Whatever its orientation, the CS boundary is effectively planar' as long as the sequence of antiphase and CS collapse steps recurs regularly in i t ; errors in that repetition sequence distort the boundary and connote localised deviations from proper stoichiometry. Both irregula- rities in CS plane spacing (effectively an intergrowth between several different members of a homologous series) and kinks or changes in orientation of CS planes have been repeatedly observed in electron microscopy. When WO,, rutile, etc. are reduced, CS planes can be introduced on any of the equivalent orientations - e.g. on all 130 orientations, in the reduction of WO,. This can lead to complex tangles of wandering and intersecting CS planes [lo] ; such faults can be annealed out only with difficulty, and their presence implies deviations from the ideal composition of the reduction product.

By virtually eliminating anion sites, a single CS plane eliminates a very large number of point defects (anion vacancies or metal interstitials) from an oxygen- deficient oxide ; the residual point defect concentration is very small. It is of interest to enquire the situation for compositions very close to the ideal stoichiometry, apparently within the monophasic range of the parent compound - e.g., TiO,.,, +, W02.,, +. Examined at room temperature, such materials invariably contain isolated CS planes, or small groups, at such a density as to imply that the point defect concentration in the domains of parent structure is low-less than lo-? [I 11. At high temperatures, under equilibrium conditions, the matter is less clear. There is recent evidence from semiconducting properties [12] that these small depar-

tures from stoichiometry arise from point defects. If this is so, the point defects must aggregate and lead to the CS collapse of the structure as the temperature is lowered.

In twinning. as usually implied, a crystal structure is reflected across some composition plane common to both twins, with the condition that the operation involves no change of composition ; it is conservative in sites. Only a few orientations can provide possible twin planes that meet this condition. Reflection across any other composition plane would be nonconservative ; sites would be eliminated, or new sites created in the operation, which would thus impose a change of composition. Such chemical twinning [13] has recently been recognized as an important topolo- gical principle, throwing light on the relationships between simple and more complex structural types, and as a source of stoichiometric variability. A new crystal structure will be generated by regularly recurrent chemical twinning, with composition different from that of the type structure. Progressive change in the spacing between the twinning planes can again produce a homologous series of compounds, and fluctuations in that spacing confer stoichiometric variability.

The term layered chemical segregation can conve- niently be used for structures such as the K,NiF, structure and its derivatives (e.g. the Ruddleston- Popper phases, nSrO. SrTiO,) in which the repeating unit is itself built up from layers that differ in composition and structure, but are topologically compatible. Thus, in the Ruddleston-Popper phases, layers with the composition and structure of SrO, one or several sheets thick, are separated by 1-unit cell layers of perovskite structure. In the perovskite derivatives AnB,03n+2 (A = La, Ca, Na ; B = Nb, Ti) (Galy) and Bi20,An-,Bn03,+, (A = Bi, Ba; B = Ti, Nb) (Aurivillius). finite slabs of perovskite structure, n unit cells thick, are joined via a layer of cations or a sheet of Bi,02 respectively. In each case, one component of the composite layer can vary in thickness. If that thickness is constant, a crystallographically defined compound is again produced and homologous series of compounds become possible. If ordering is imperfect, with slabs of variable thickness, the resulting material is nonstoichiometric, but not in an equilibrium state.

Attainment of order, from a disordered state, in such materials involves complex diffusional readjustments ; the experimental. evidence suggests that some strong control of ordering is exerted in the processes of reaction and crystal growth. The layer ferrites, an extensive series built up from the ordered stacking of two quite complex units - BaFel2OIg and Ba,Me2Fe,,02, - strikingly illustrate this. Electron microscopy has confirmed the high per- fection of ordering in stacking sequences that result in repeating units ranging from thirty to several hundred Angstroms in length [14].

C7- 24 J. S . ANDERSON

Perhaps the most interesting development in recent years has been the recognition of what have been termed adaptive structures [15]. The preceding section of the paper has dealt mainly with how small changes in composition may lead to a set of discontinuous structural adjustments and the formation of homologous series of discrete compounds as the equilibrium state. In an equilibrium diagram, these would constitute a progression of line phases, separated by biphasic ranges of composition. Hence any material with a composition deviating from one or other of the intermediate compounds would show, in its diffraction pattern, the superlattice lines characteristic of each constituent of a 2-phase mixture. By contrast, certain systems have been found to display the remarkable property that, no matter how closely spaced in composition a series of preparations may be, each specimen has its own, single, sharply defined superlattice ; it is always uniquely monophasic. This is the operational definition of an injinitely adaptive structure. Its implication is that, within some general structural pattern, holding over a range of composition and usually based upon some small, notional, parent subcell, periodic fluctuations in composition (and therefore in structure) can establish a close approximation to complete order, for every atomic ratio of the components. The periodic fluctuations can be of two kinds, which may be termed adaptive layer stacking and vernier structures respebtively.

In adaptive layer stacking, the concept is that of a structure with a long over-all periodicity, built up by the regular stacking of two kinds of laminar subunits, which differ in composition but could be interconvertible through small structural readjustments, in response to. the incorporation or removal of atoms of the variable component. Such subunits may, for example, have the structures of successive members of a homologous series of phases ; by definition, these have some common welding plane that, permits of perfect intergrowth. Subunits of type A (with composition XA) and type B (composition XB) stack to form a superlattice with the repeat unit A, B,. Provided that order could be attained over a sufficiently long repeating unit (i.e. p, q large), a regular structure could be built up for every composition between XA (p = 1, q = 0) and XB (p = 0, q = 1) by suitable adjustment of p and q. Rather surprisingly, this flexibility seems to be shown, to a first approximation, by a number of known systems, through a small set of shuffling readjustments as the composition. is changed. In practice, the kinetic and energetic problems of attaining perfect stacking order for very high superlattice multiplicities must limit the degree to which perfect long range order can be retained in very small changes of composition.

An approa'ch to the simultaneous attainment of variable composition and .perfect order can be illustrated by several CS systems derived from the rutile structure, where the operative mechanism is

a progressive reorientation of CS planes, already referred to. The titanium oxides between Til1O2, (TiOl.,091) and Ti16031 (Ti01.9375) and the Cr203- TiO, system are of this type. In the latter 1161, the CS planes change their orientation progressively from close to (121) to close to (132) over the composition range 0.22 > x 3 0.087 in Cr,Til-,02 -,.,, (i.e. from Cr2Ti7017 to Cr2Ti210,5, approx.). In this transition, lattice imaging electron microscopy shows that the slabs of rutile structure between CS planes remain very uniform in breadth, and the CS boundaries remain free from kinks. The implication is that although the irrational orientation of CS planes must impose a complex sequence of CS collapse and antiphase steps, practically perfect regularity is reached at every composition. Similar considerations apply to the Fe203-Cr203-Ti02 system [17], which can be regarded as based either upon the rotation of CS planes or upon the regular stacking of V305-like and a-Pb02-like layers (figure 2b). Each composition yields its own sharply defined superlattice, indicative of good ordering. That the superlattice is sensitive to minute differences in composition appears from subtle differences in the superlattice reflections obtained from different crystalline particles picked out of the same preparation. In such a system, there can well be vestigial residual variations in composition in the product of a solid state reaction, even after annealing. The ratio p/q of the two layer types will be sensibly constant in a very nearly uniform material, but minute differences in composition would require differences in the absolute values ofp and q as between different crystals ; although such variations would undoubtedly require the total superlattice multiplicity, p + q, to become very large, the occurrence of such adjustments is evidenced by the small shifts observed in the superlattice reflections.

The so-called vernier structures present a second way in which a long-period modulation can be impress- ed on a basic structure, so as to combine variable composition with retention of order. The essential characteristic is that the crystallographic unit can be resolved into two independent sublattices which are not fully commensurable ; along one axial direction, they are defined by different 'translation vectors. One sublattice has a constant repeat length, independent of wmposition, and ba,sed upon the corresponding sublattice of some parent, structure with awmall cell. The other sublattice has a variable population and its periodicity depends upon the composition. The regular periodic structure, as a whole, is defined by that translation which is a common multiple of the repeat length for each of the two sublattices.

This principle was first recognised in the so-called chimney ladder transition metal silicides and ger- manides studied by Nowotny [18]. In these, the metal atoms form a sublattice of constant pitch, as. in the


parent structure, TiSi,. Only in the end member MX, of the series, however, do the metalloid atoms follow the same periodicity. By changing the metallic component, or by alloying, phases with a wide range of compositions MX,-, can be obtained, in which the pitch of the metalloid sublattice increases with increase in x. Thus, in MnllSi19 and V,,Ge,, the two sublattices come into coincidence every 19 and every 31 subcells, respectively. Alloying experiments strongly indicate that the periodicity of the metalloid sublattice is a continuous function of the electron- atom ratio, so that, in general, perfect periodic order could be attained only in a structure having, for its long axis, an extremely large multiple of the subcell dimensions.

The nonstoichiometric barium iron sulphides Bal+,Fe2S4 provide another example 1191. In these, a rigid 3-dimensional framework, with the composition Fe2S4 is provided by columns of linked, tetrahedral [FeS,] groups. The net anionic charge on that framework is determined by the Fe(iii)/Fe(ii) ratio, and must be balanced by a corresponding number of

barium cations occupying tunnels in the framework. As the oxidation state of the iron is changed, the population of barium cations changes with x, and the linear periodicity of the barium sublattice alters ; there are no point defect interstitial barium. The fixed Fe2S, and the continuously variable Ba sublattices come into coincidence every Ilx subcells; every composition has its own, single set of superlattice diffraction lines (figure 2c). This structural principle is well established by detailed structural work on two (rational composition) members of the family of phases, and by the crystallography of other vernier structures in the yttrium oxyfluoride series YO, -,F, +,, (0.13 < x < 0.22) and the related Zr0,-Nb20, compounds.

At the present stage of knowledge we can draw certain summarising, general conclusions that apply to at least the majority of the experimental evidence.

1) The effect of defect interactions is to replace point defects by extended defects as the deviation from ideal stoichiometry becomes significant, so

INCREASING CONFIGURATIONAL ENTROPY +

ENTROPY CONTROLLED SYSTEMS

Point defect unbalance

Randomised defect complexes

Point defects assimilated as structure elements

of ordered microdomains

Superlattice ordering

Adaptive structures

Planar extended defects : CS planes, chemical twinning

Discrete intermediate phases

1 ENTHALPY CONTROLLED SYSTEMS

C7- 26 J . S. ANDERSON

that the concentration of point defects always remains small. This transition in defect structure is the direct consequence of minimizing the total crystal energy.

2) This virtual elimination of point defects may be achieved either by local reconstruction of the crystal structure or, directly, by superlattice ordering of empty sites or interstitials.

3) Extended defects are themselves subject to ordering effects. As a consequence, and as the composition (and therefore the concentration of extended defects) is altered, the equilibrium state in many systems is a succession of defined ,line phases, not a phase of variable composition.

4) The requisite condition for forming genuinely nonstoichiometric compounds is a delicate balance between the net energetic cost of creating extended defects and the configurational entropy. We can broadly classify behaviour as a progression from entropy controlled systems, forming nonstoichiometric compounds, to enthalpy controll~d systems, forming only families of discrete intermediate compounds (Table 11).

5) There are some structures with such flexibility of ordering that many free energy minima can be attained, at different compositions, without significant contributions from configurational entropy. They constitute a special case, a second class of genuinely nonstoichiometric systems, the adaptive structures.

6 ) Although a discrete set of ordered phases should be formed in many systems as the equilibrium state, this may be kinetically unattainable (e.g. in heterotype solid solutions). The resulting pseudo-bivariant phases are heterogeneous in their local microstructure, on a very fine scale ; elements of different structures - in general derived from the notional set of ordered intermediates - are coherently intergrown.

It may be noted, finally, that whilst chemically specific factors - the energetics of some change in valence state of metallic or nonmetallic atoms, crystal field stabilisation effects, cation-anion or cation- cation orbital overlap, etc. - unquestionably deter- mine the chemistry of any system, the structure of the extended defects formed, depends more upon the parent crystal structure type than on the particular chemical compound involved. Certain crystal structures that comprise a wide range of compounds, of differing character, have been found repeatedly to give rise to extended defects of characteristic type. This is exemplified, for some common structures, in table 111.

Some guiding principles have been developed in the foregoing sections, chiefly in relation to mixed valence compounds. In these, the possibility of forming nonstoichiometric phases under equilibrium condi-

Types of extended defect recurring in some common structural types

Structure type Extended defects - -

MX NiAs type Lamellar ordering of empty sites in cation sublattice

Omission of cation layers MX, CdCI, or CdI, types Interpolation of complete or in-

complete (but ordered ?) cation layers

Fluorite type Defects confined to anion sublattice

MX, -, Bevan cluster MX,, , Willis cluster

Rutile type Crystallographic shear or other planar defects eliminating sites in anion sublattice

MX3 ReO, type

tions depends upon favourable entropic factors (discrete extended defect complexes) or unusual structural and kinetic flexibility (adaptive structures) ; lacking these, the equilibrium state must be some succession of ordered structures. If diffusion and place exchange processes are too slow, extended defects cannot be ordered and a nonstoichiometric phase results, in a frozen-in, nonequilibriurn state. Because, in polar structures. there is a reauirement for local charge - compensation, this false equilibrium condition is particularly likely in heteiotype solid solutions. The general model that emerges for the structure of such materials was foreshadowed by Ariya's idea of submicroheterogeneity : a mosaic of coherently intergrown microdomains, having structures of, or related to, those of possible ordered intermediate phases. There is now ample evidence that this is the case for systems that have proved amenable to study by lattice imaging electron microscopy-block structures, tunnel structures, layer structures with l-dimensional disorder, etc. Local order is not lost, but structure and composition fluctuate from point to point within the crystal, on the unit cell scale, to form a mosaic of the elements of different real or possible ordered phases (figures 3b, d ) . The evidence that impurities in Nb,O, can be so segregated in domain walls of appropriate structure, surrounding domains of perfect crystal, exemplify this [20]. It is a reasonable inference that the same model should be applicable to the derivatives of other structures - e.g., to solid solutions derived from the fluorite and other structural types. The recent characterization of a family of ordered vacancy structures in the CaO-ZrO, and CaO-HfO, systems [21] is suggestive evidence for the microdomain constitution of such phases as the stabilized zirconias.

STOICHIOMETRY. DEFECTS AND ORDERING

References

[I] STONEHAM, A. M. and DURHAM, P. V., J. Phys. & Chem. Solids 34 (1973) 2127.

[2] ROTH, W. L., ~ c t a Crystallogr. 13 (1960) 140. [3] KOCH, F. and COHEN, J. B., Acta Crystallogr. B 25 (1969) 275. fq ,CHEET~IA&Z, A. K., FENDER, B. E. E. and TAYLOR, R. I., J.

Phys. ' C 4 (1971)- 2160. [5] WILLIS, B. T. M., Proc. R. SOC. A 274 (1963) 134 ; J. Physique

25 (1964) 431. [6] CHEBTHAM, A. K. and FENDER, B. E. F., J. Phys. C 4 (1971)

3107. [7] THORNBER, M. R. and BEVAN, D. J. M., J. Solid State Chem.

1 (1970) 526, 545. [8] CATLOW, C. R. A., P ~ o c . R. SOC. A 353 (1977) 533. [9] ANDERSON, 3. S. and HYDE, B. G., J. Phys. & Chem. Solid!

28 (1967) 1393. [lo] ALLPRESS, J. G. and GADO, P., Crystal Lattice Deyects 1

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ANDWSSON, S., LEYGRAF, C. and JOHNSSON, T., J. Solid State Chem. 14 (1975) 78.

[14] MCCONNELL, J. D. M., HUTCHISON, J. L. and ANDERSON, J. S., Proc. R. Soc. A 339 (1974) 1 ;

VAN LANDUYT, S., AMELINCKX, S., KOHN, J. A. and ECKART, D. W., Mat. Res. Bull. 8 (1973) 389, 1173.

[15] ANDERSON, J. S., J. Chem. Sot.\ Dalton Trans. (1973) 1107. 1161 PHILP, D. K. and BURSILL, L. A., J. Solid State Chem. 10

(1974) 357. [17] GREY, I. E., REID, A. F. and ALLPRESS, J. G., J. Solid State

Chem. 8 (1973) 86. [la] VOLLENKLE, H., WITTMANN, A. and NOWOTNY, H., Monatsh.

Chem. 97 (1966) 506. [19] GREY, I. E., J. Solid State Chem. 11 (1974) 128.

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[ll] BURSILL,. L. A. and HYDE, B. G., Phil. Mag. 23 (1971) 3 ; [20] ANDERSON, J. S., BROWNE, J. M. and HUTCHISON, J. L., TILLN, R. J. D., Mat. Res. Bull. 5 (1970) 813. Nature 237 (1 972) 153.

[12] BAUMARD, J. F., PARRIS, D. and ANTHONY, A. M., J. Solid [21] ROSSELL, H. J. and SCOTT, H. G., J. Solid State Chem. 13 State Chem. 20 (1977) 43. (1975) 345.

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STOICHIOMETRY, DEFECTS AND ORDERING · defects are important, involving some local modification of the crystal structure ; they include isolated defect clusters, planar faults such