Crystallographic Shear and Non-stoicheiometry

1 Introduction

When the previous Report in this series was written, detailed experimental evidence about the microstructure of crystallographic shear structures (CS phases) was just becoming available. Two themes were therefore emphasized. The first was the relation between the concept of crystallographic shear and existing views of defects and non-stoicheiometry in inorganic compounds. Most CS phases do not appear to contain point defects in significant concentrations — i.e. in sufficient number to contribute materially to the apparent composition ranges of CS compounds. In a heuristic sense, at least, the collapse of the parent crystal structure which produces a CS plane eliminates point defects; it is not implied that the presence of point defects in high concentration is a necessary precursor stage in the formation of a CS plane. The mechanism of this transformation process is still not clear, and the role of point defects in certain structural types, notably the 'block' structure oxides, has continued to excite interest.

A second topic, which has also been actively pursued in the review period, was the importance of coherent intergrowth between structures that are topologically compatible, but of different composition, exhibited particularly by the CS phases. It was shown that in many macroscopically homogeneous CS phases there may be a considerable measure of internal disorder, associated with irregularities in the spacing between parallel CS planes. Even though this state may not represent a true equilibrium structure, it may be experimentally inescapable and it provides a basis for apparent non-stoicheiometric properties at the macroscopic level. The usual methods of characterization and structure analysis may fail to reveal and analyse such microscopic heterogeneity; to do so needs methods for determining the local microstructure, at the unit-cell level, as distinct from the averaged structure derived from diffraction methods and from postulated models for defect structures. Such methods were beginning to emerge from the application of electron microscopy.

In the intervening two years, the power of lattice-imaging methods in electron microscopy has developed markedly. Considerable attention has been devoted to coherent intergrowth (Wadsley defects) and other forms of faulting, as observed at or below the unit-cell level, and the topological constraints and relationships that determine the possibilities of intergrowth or structural relaxation in CS phases have been analysed. This advance in our knowledge of structure has not been matched by advances in knowledge of transport and reaction processes, and the physical properties associated with CS structures have received little study. In this review, we consider particularly the increasingly important role of electron microscopy, and the way in which structure adjusts itself to composition, both in materials that simulate non-stoicheiometric behaviour and those that are genuinely variable in composition. Crystallographic shear — the term has a very specific meaning, which should not be loosely used — is not the only transformation whereby inorganic structures can accommodate changes in the atomic ratio of metal : non-metal so as to maintain some high degree of order. Recent work has drawn attention to the formation of ordered structures, with large repeating units, where randomized solid solutions might be expected. Without venturing an answer to the difficult questions of how a solid compound should now be defined, or how complex ordering is established, we summarize also some recent developments in this wider field.

2 The Direct Observation of Structure in Crystals: Lattice Imaging

The newer experimental findings about CS phases have come largely from transmission electron microscopy and electron diffraction, rather than from X-ray diffraction, which has limitations imposed by the large unit cells, by the small differences in structure between one member and another in a homologous series, and, above all, by disorder in the crystals. The first results accruing from electron microscopy 1 indicated that complete order in crystals of CS phases was rarely attained; it may, indeed, be impossible even by the most careful preparative methods to obtain perfectly ordered crystals of many CS compounds for structure determination by single-crystal methods.

Transmission electron microscopy can present the essentials of the structure of CS phases and other suitable classes of compound very directly, although detailed metrical information — e.g. interatomic distances — cannot be extracted. For this, there is no alternative to precise structure determination by X-ray or, increasingly and for some purposes advantageously, neutron-diffraction methods. Lattice images can now be obtained, however, which show the projected structure at the level of the individual co-ordination polyhedron in oxide structures. Development of the technique has been largely pragmatic, based on the experimental finding that, provided that the crystals under examination were extremely thin, the contrast in micrographs made at 'optimum under-focus' approximated closely to a projection of the potential distribution in the crystal. Thus, for CS phases, in which the density of heavy cations, of high charge, is considerably higher in the CS planes than in the relatively open matrix of parent structure, the CS planes appear as fringes which collapse into dark lines of contrast when the crystal is so oriented as to bring the CS planes parallel to the incident beam. At the highest lattice resolution, individual corner-sharing (MO6) octahedral groups appear darker than the empty voids between them.

Interpretation of these lattice images is, however, not as straightforward as might appear at first sight. Ideally, an electron micrograph should be compared, point for point, with the intensity of the transmitted electron beam, as calculated from dynamical scattering theory. In practice, the theory of an electron-microscope image formed by the operation of many diffracted beams has been developed only in parallel with the experimental applications. Most results on CS and other structures have relied upon a less rigorous comparison with electron optic theory; much of the interpretation has, indeed, come from chemical intuition regarding the structural geometry that might be expected in the system concerned.

The correspondence between lattice images and the structures postulated from a combination of X-ray structural information with chemical intuition has been remarkably good, and the validity of the interpretations has not been in serious doubt. This success has stemmed from careful selection of crystalline systems, with a known basic structure, that were appropriate for attack by microscopy. The electron-microscope image could then be so focused as to give optimum contrast and resolution which brought out structural features that harmonized with expectations based on the prior knowledge of the system. Aperiodic features of the image, faulting, disorder, etc. could then be interpreted in a manner consistent with the interpretation in perfect regions of crystal. The rather open block structures were particularly suitable for these tactics.

lf the lattice-imaging technique is to be extended to the study of crystals which have a less open and clearly projected structure, a rigorous theoretical basis for interpretation becomes indispensible. Lattice images at a sufficiently high degree of resolution to give some chance of displaying the positions of individual, highly scattering atoms (ca. 0.35 nm with current instruments, compared with the cation–cation distance 0.39 nm between apex-sharing octahedral groups) have hitherto been available only for the block structures. These have, accordingly, been the objects for calculations of image contrast as the electron optic theory has developed during the review period. The calculations have a much wider validity, however, for they define the conditions under which lattice images may be directly correlated with structure and thus now permit the technique to be employed for a wider range of chemically interesting systems. We therefore briefly discuss the relation between the recorded lattice image and the structure, seen in projection, of the observed crystal, before considering areas in which electron microscopy has shed new light on defect structures and chemical problems.

Theoretical Interpretation of Image Contrast. — In general terms, contrast in a micrograph reflects variations in the numbers of electrons falling upon the recording photographic emulsion; in an ideal optical system, the electrons arriving at a single point on the emulsion originate from a single point on the exit face of the object. The flux at each point on the emulsion depends upon the diffracting conditions within the crystal, upon instrumental factors, and upon operating factors. It is therefore possible to break down the theoretical problem of image formation from a large number of diffracted beams into three distinct parts: (a) calculation of the electron wavefunction at the exit face of the crystal; (b) modification of the wavefunction through the optical system, to take account of lens aberrations, moveable apertures, etc.; and (c) superimposed effects of operating adjustments and errors. Apart from the critical control of the orientation of the specimen, the major factor in (c) is the extent of under-focus; this is selected to compensate for the phase incoherence produced by spherical aberration and to give the best match between the observed image and that calculated theoretically.

Experimental experience has shown that useful lattice images are produced only from very thin crystals; even for crystals a few nanometres thick, realistic values for the intensity and phases of the various diffracted beams can be calculated only by using dynamical diffraction theory. When lattice fringe images are formed by the combination of more than one or two diffracted beams (for which the electron optic theory is well established), the dynamical formulation of Cowley and Moodie provides the most useful starting point for calculations.

The essence of this treatment is to consider the crystal to be divided into a number of thin slices perpendicular to the incident beam direction. The phase and amplitude of the incident wave is then modified by each slice. This is further simplified by considering each slice as a phase grating formed by projecting the potential distribution within each slice on to an internal plane. Fresnel diffraction then takes place between each grating. The treatment becomes less rigorous with increasing slice thickness, but provided this is kept small, no appreciable errors are found. Thus, each time the electron wave passes across a slice, it is multiplied by a transmission function which is dependent upon the potential distribution in the plane.

If the potential of the nth slice, of thickness Δz, is φn(x,y,z) then the projected potential, φn(x,y) is given by

[MATHEMATICAL EXPRESSION OMITTED] (1)

The transmission function for the slice, the phase grating, is then defined as

q(x,y) = exp(Iσφn) (2)

and gives the phase change imposed upon the wave as it passes the grating. The term in σ is a function of the relativistic electron wavelength λ of velocity υ and the accelerating potential W, given by

[MATHEMATICAL EXPRESSION OMITTED] (3)

where β is the relativistic correction term, v/c. The phase change between the planes must also be considered. This can be represented as exp [ik(x2 + y2)/2Δz] for the fast electrons that are involved. Adding these phase changes, by use of the principle of superposition, allows the wavefunction of the (n + l)th slice to be written in terms of the nth slice: thus

[MATHEMATICAL EXPRESSION OMITTED] (4)

where * represents convolution. The final wavefunction emerging from the crystal is obtained from equation (4) and an analytical solution can be derived. The form this takes, however, is rather unsuitable for numerical computation, and an iterative scheme based on the equation for Ψn + 1 has been used.

In order to do this, it is convenient to handle the Fourier transform of equation (4), which is

[MATHEMATICAL EXPRESSION OMITTED] (5)

where Un(h,k) is the wave amplitude and phase from the nth slice, P is the propagation function

P(h,k) = exp[2πiζ(h,k)Δz] (6)

where ζ(h,k) is the excitation error, (u2 + v2)λ/2, for the (h,k) reflection in the reciprocal space co-ordinates (u,v), and Qn + 1 (h,k) is the Fourier transform of the phase grating function, qn + 1 (x,y) given by

[MATHEMATICAL EXPRESSION OMITTED] (7)

The equation in U, equation (5), is evaluated for a sufficient number of slices to give the correct crystal thickness, and a sufficient number of beams so that the sum of the intensities is close to the incident-wave intensity. In the calculations referred to, 435 beams were used in two-dimensional calculations. Normalization was 0.72 for a crystal thickness of 60 nm. This would be improved by including more beams, but at the expense of much greater computation time. The final result is to generate the diffraction pattern, which, because of the variable nature of Δz, is obtained as a function of thickness.

The contrast in the image is readily derived from the function Un by another Fourier transform. If ψH is the Fourier transform of UH so that

ψH (x,y) = F-1UH (h,k)(8)

the image intensity is given by

I (x,y) = ψ(x,y).ψ * (x,y) (9)

where F-1 is the inverse Fourier transform of UH (h,k), the amplitude and phase of the (h,k)th diffracted beam emerging from the crystal of thickness H. The representation of this image can be expressed in a variety of ways. The most immediately informative, however, is to plot out the expected contrast, using the half-tone print-out process developed by Head, to give a computed micrograph.

These results relate to a perfect, aberration-free electron microscope, and they now have to be modified to take into account the instrumental and operational defects mentioned earlier. These are principally the inclusion of terms to account for the objective aperture, the spherical aberration of the objective lens, and the defect of focus employed. The objective aperture is represented by merely excluding all beams intercepted by the objective aperture from the final Fourier transformation. The spherical aberration acts so as to retard the phase of a beam passing through the lens at an angle α to the optical axis by an amount πCsσ4/2λ with respect to the axial beam at the Gaussian image plane. Cs is the spherical aberration coefficient of the objective lens, which lies between 3 and 5 mm for most modern instruments. The defect of focus ε is taken into account by including a propagating function which is convoluted with the wavefunction at the exit surface of the crystal to allow the wavefunction at a distance a from the exit surface to be obtained. Thus equation (8) becomes modified to

[MATHEMATICAL EXPRESSION OMITTED] (10)

where P is a propagating function which accounts for defect of focus and spherical aberration, and is given by

[MATHEMATICAL EXPRESSION OMITTED] (11)

for a defect of focus a and spherical aberration coefficient Cs. ζ(h,k) is the excitation error for the (h,k)th beam. Hence we can write the image wavefunction for an apertured system

[MATHEMATICAL EXPRESSION OMITTED] (12)

where the summations are carried out over the n beams supposed to pass through the aperture.

Other electron-microscope defects — chromatic aberration, the divergence of the incident beam, and astigmatism — can all be taken into account. However, these are found to be rather less important, and astigmatism in particular can be ignored for a carefully corrected modern objective lens. Equation (12) can then be used to compute an aberrated apertured image which in turn can be compared with the ideal one and with experimental results.

Calculations were first carried out for the case where only one row of diffracted beams, typically (h00 or 00l), is excited. These give one-dimensional information, and a set of parallel fringes results, which can be compared with experimental lattice images obtained under conditions which are matched by the data used in the computations. Some of the results of these calculations are compared with the experimental results in Figure 1.

There are no theoretical problems associated with computing two-dimensional images using the same procedure. Once again, half-tone images are the easiest way of comparing the results of computations with experimental micrographs. Some examples are shown in Figure 2.

From these results it is possible to see that under certain circumstances the lattice images do represent accurately the structure of the crystal or, to be more precise, the projected charge density of the crystal. However, the calculations show that such an interpretation is not generally valid. In fact one must place severe limitations on the experimental technique used in order to avoid misinterpretation of lattice images. The most important limitation is that the crystal must be very thin, of the order of 10 nm. Of course, if the crystal thickness is accurately known, computations of contrast can still be made, but the image contrast does not reflect the crystal structure in simple terms and cannot be interpreted naively. Another very important result is that the most important microscope aberration appears to be spherical aberration, which causes severe perturbations of image contrast and necessitates rather careful selection of the objective aperture used. This is allowed for by computing images which are produced by beams having considerable phase retardation due to spherical aberration. This emphasizes, however, that lattice images can be interpreted intuitively only when care has been taken in obtaining the micrographs. The final criterion which must be carefully controlled is the defect of focus. For the lattice image to represent the crystal structure accurately, the microscope must typically be underfocused by about 80 nm. A degree of overfocus inverts the image contrast, but between these two positions, and outside them, the contrast varies in a rather complex way. Experimentally this is fairly easy to control, as the disappearance of Fresnel fringes at the edge of the crystal can be used to judge focus, and the correct defect of focus set by adjusting the objective lens by the appropriate amount. This is not necessary if the approximate crystal structure is known, for in that case the 'best' contrast can be judged by eye. Such a procedure has been successfully used in the block structures, for example.