International
Tables for Crystallography Volume C Mathematical, physical and chemical tables Edited by E. Prince © International Union of Crystallography 2006 |
International Tables for Crystallography (2006). Vol. C. ch. 5.3, pp. 510-516
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Another group of methods with photographic recording has been developed in parallel with those discussed in Subsection 5.3.2.3. These are the methods in which the crystal remains stationary and the diffraction conditions are fulfilled, simultaneously for more than one set of crystallographic planes, by the use of a highly divergent beam, dispersed from a point source (Fig. 5.3.2.2 ). The Kossel method (Kossel, 1936, and references therein), the divergent-beam techniques initiated by Lonsdale (1947), and their numerous modifications belong to this group.
The excitation of the characteristic X-rays used in these methods can be performed by X-radiation (Lonsdale, 1947), by electron bombardment (Kossel, 1936; Gielen et al., 1965; Ullrich & Schulze, 1972) or by proton irradiation (Geist & Ascheron, 1984) of a single crystal. The source of emitted X-rays may be located either in the sample itself (the Kossel method), on the surface of the sample in a layer of target material (the pseudo-Kossel method), or outside the sample (the divergent-beam techniques). The divergent X-ray beam diffracts from sets of crystallographic planes. The diffracted rays for each Bragg reflection form a conical surface whose semivertical angle is equal to 90° − and whose axis is normal to the Bragg plane (i.e. coincides with the reciprocal-lattice vector).
The conical surface of an hkl reflection can be described in the form (Morris, 1968; Chang, 1984): where is an orthogonal coordinate system with its origin at the vertex of the cone and with z′ along the axis of the cone and normal to the plane of interest, and α is the semivertical angle. Since α depends on the Bragg angle, it is possible to combine (5.3.2.6) with the Bragg law [equations (5.3.1.1) or (5.3.1.2)], and so with the lattice parameters. In particular, the dependence can be presented as: where .
In another convenient coordinate system (x, y, z) common for all the cones, say with z along the direction of the incident beam, (5.3.2.6a) will take the form: where are direction cosines of the angles between the axis and the axes x, y and z, respectively. Since the origin of the coordinate system has not been changed, The Kossel lines (Fig. 5.3.2.3 ) are formed at the intersections of the cones with a flat film placed parallel to the specimen surface (Fig. 5.3.2.2). When the film plane is normal to the z axis, and the focus-to-film distance is equal to Z, putting z = Z in (5.3.2.6b,c) gives the formulae describing the conic section on the film.
(a) The Kossel pattern from iron and (b) the corresponding stereographic projection (Tixier & Waché, 1970). |
A high-precision Kossel camera is described by Reichard (1969) and the generation of pseudo-Kossel patterns by the divergent-beam method has been described by Imura, Weissmann & Slade (1962), Ellis, Nanni, Shrier, Weissmann, Padawer & Hosokawa (1964), and Berg & Hall (1975).
The photographs may be in either the transmission or the back-reflection region (Fig. 5.3.2.2). The second arrangement seems to be (Lutts, 1968) more suitable for lattice-parameter determination, since the background is less intensive and the lines on the photographs have greater contrast. Both possibilities are used in practice. Photographs in the back-reflection region have been reported by Imura, Weissmann & Slade (1962), Ullrich (1967), Newman & Weissmann (1968), Newman & Shrier (1970), and Berg & Hall (1975). Examples of the use of the transmission region are given by Yakowitz (1966a), Reichard (1969), and Glass & Weissmann (1969).
The recommended crystal thickness t for work in the transmission region, according to Hanneman, Ogilvie & Modrzejewski (1962), is given by: where is the linear absorption coefficient for Kα radiation generated in the crystal. A more detailed study of the effect of sample thickness, as well as operating voltage, on the contrast of Kossel transmission photographs is given by Yakowitz (1966a).
The picture geometry does not depend, in principle, on whether the Kossel, pseudo-Kossel, or divergent-beam technique is applied. Imura (1954) has studied in detail the form of the curves of the light or deficiency type, and recorded both in the transmission and in the back-reflection region. The curves on transmission patterns can be considered to be conics; those recorded in the back-reflection region are related to ellipses, but of higher order. In general, the photograph has to be indexed before performing measurement on the film. For this purpose, the pattern may be compared with a calculated pattern (gnomonic, orthogonal, cylindrical, or stereographic projection). For lattice-parameter determination, various features of the photographs may be used, i.e. intersections or near-intersections of Kossel lines, their near-tangency, lens-shaped figures, and the whole lines approximated with a function.
The basis of lattice-parameter determination involves measurements performed on the film. There are various methods covering most of the different geometrical features of the cones and recorded pictures. These were reviewed by Lutts (1968), Yakowitz (1966b, 1969) and Tixier & Waché (1970). In each case, the wavelength of the excited radiation has to be known. Often, the resolved doublet and/or Kβ radiation is applied rather than a single (but most pronounced) Kα1 line. The other data needed (a sufficient number of equations, the solution of which leads to lattice-parameter determination; camera geometry; crystal system; and indices) depend on the method.
Biggin & Dingley (1977) propose a classification of all the methods using a divergent beam based on the information required.
Although the precision theoretically obtainable by means of the Lonsdale (1947) method is of the order of 1 part in 106, this limit is unattainable in practice. The reported values are in the range of about 10−4–10−5 Å, depending not only on the method but also on the crystal – its symmetry and perfection. The highest accuracy known by the author was achieved by Lonsdale [(1947), ±5 × 10−5 Å, for diamond], Morris [(1968), 2 parts in 105] and Aristov & Shmytko [(1978), |δd|/d ∼ 3 × 10−5, 1–5 × 10−5 rad for angles between crystallographic directions].
Systematic errors due to the methods in which a divergent beam is applied have been discussed by Hanneman, Ogilvie & Modrzejewski (1962), Gielen, Yakowitz, Ganow & Ogilvie (1965), Beu (1967), Lutts (1968), and Aristov & Shmytko (1978). The main sources of systematic error are:
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The errors of the second group may be to some extent removed if small differences of the length resulting from the resolved doublet are measured on the film rather than distances due to only one wavelength, and/or if the camera dimensions can be eliminated from the equations used in the calculations of lattice parameters (see §5.3.2.4.2). A relative misorientation between the specimen and the flat film has been analysed by Lutts (1973).
An error typical for methods realized by means of an electron microscope or an electron-beam probe may result from the thermal effects of the electron beam generating a divergent X-ray beam at the crystal surface. Uncontrolled thermal effects may also occur in the case of the Kossel method, since the sample is situated inside the X-ray tube. In the latter method, the wavelength of the radiation emitted depends on the chemical composition of the sample, since the sample plays the role of the anode of the X-ray tube.
The reported precision of the methods, limited by the finite width of the lines on the photograph, and depending also on the geometrical features taken into account, is 1 part in 103 to 1 part in 105. The highest is reported by Hanneman, Ogilvie & Modrzejewski (1962), Gielen, Yakowitz, Ganow & Ogilvie (1965), and Lider & Rozhansky (1967). On the other hand, the lowest (1 part in 103), obtained by Harris & Kirkham (1971), is attributed to the method in which neither the indexing of the lines nor a knowledge of the crystallographic system or camera geometry is required.
For precision determination of lattice-parameter differences, a `point' source (i.e. as small as possible) is required and the high-order Kossel lines should be used to obtain both well resolved doublets and `thin' figures. The near-intersections of conic sections, applied in Lonsdale's (1947) method, the major axes of lens-shaped figures, used in Heise's (1962) method, and the small spherical polygons formed by several Kossel cones are very sensitive to lattice-parameter changes, so that these figures can be taken into account in the precise measurements reported in §5.3.2.4.4.
As was mentioned in §5.3.2.4.3, the methods in which a highly divergent beam is used are applied both to the accurate determination of the unit cell and to the precision detection of lattice-parameter changes or differences. It should be added that the Kossel method is especially suitable for small single crystals or fine-grained polycrystals, whereas the other divergent-beam techniques need larger specimens (Lutts, 1968).
Since all the methods are relatively simple (stationary specimen, stationary film, simple construction of the camera) and, on the other hand, are applicable mainly for highly symmetric systems, they proved to be particularly useful in studies of metals and semiconductors. Various applications of the Kossel method and other divergent-beam techniques for this task have been discussed by Ullrich (1967), Ullrich & Schulze (1972), and Geist & Ascheron (1984). The latter paper relates especially to semiconductors.
A task that arises both in metallurgy and in the semiconductor industry is the examination of the real structure – in particular, measurements of strains introduced by variation in temperature, pressure, mechanical stress (elastic strains) or by point defects, deviation from exact stoichiometry, irradiation damage, and phase changes (permanent strains).
Measurements of small changes in interplanar spacings of independent sets of crystal planes enable a stress–strain analysis to be made (Imura, Weissmann & Slade, 1962; Ellis et al., 1964; Slade et al., 1964; Newman & Weissmann, 1968; Berg & Hall, 1975). A special case of strains is an extensional deformation of the lattice in the direction of crystal growth (Isherwood, 1968).
A typical metallurgical problem is the effect of heat treatment on the microstructure of alloys. An example of the application of the Kossel method to the task is given by Shinoda, Isokawa & Umeno (1969), who reported a study of precipitation of α from β in copper–zinc alloys. The lattice parameters and thermal expansion of α-iron and its alloys were examined by Lutts & Gielen (1971). Structure defects resulting from over-pressure experiments and annealing were investigated by Potts & Pearson (1966). Irradiation effects caused by neutrons were the subject of papers of Hanneman, Ogilvie & Modrzejewski (1962), Yakowitz (1972), and Spooner & Wilson (1973); those caused by electron bombardment were reported by Ullrich (1967).
Divergent-beam techniques are considered to be a suitable tool for studying strains in epitaxic layers (Hart, 1981), since corresponding lines of the layer and substrate, observed on one photograph, can be readily identified. Relevant examples are given by Brühl (1978), Chang, Patel, Nannichi & de Prince (1979), and Chang (1979), who examined lattice mismatch in LPE heterojunction systems, and by Brown, Halliwell & Isherwood (1980), and Isherwood, Brown & Halliwell (1981, 1982), who reported characterization of distortions in heteroepitaxic structures together with a theoretical basis (multiple diffraction) for the method.
Another task of real-structure examination is the determination of angles between crystal blocks. A method has been worked out by Aristov, Shmytko & Shulakov (1974a,b).
Divergent-beam techniques can also be used in X-ray topographic studies, realized either by means of Kossel-line scanning (Rozhansky, Lider & Lyutzau, 1966) or by line-profile analysis (Glass & Weissmann, 1969).
Schetelich & Geist (1993) used the Kossel method for lattice-parameter determination and a qualitative estimation of the crystal perfection of quasicrystals and showed that the fine structure of Kossel lines of quasicrystals is the same as observed for conventional crystals.
Mendelssohn & Milledge (1999) used a Dingley–Kossel camera for quick and simple computer-aided measurements of cell parameters of isotopically distinct samples of LiF over a wide temperature range of 15–375 K.
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