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. 508516

Photographic singlecrystal techniques used for unitcell determination can be divided into three main groups:
In the past, only techniques belonging to groups (2) and (3) were used in absolute latticeparameter measurements. As recently shown by Carr, Cruickshank & Harding (1992), a single synchrotronradiation Laue photograph can provide all necessary information for the determination of unitcell dimensions on an absolute scale (though with low accuracy for the present).
The methods of the second group are popular movingcrystal methods or their modifications especially adapted for latticeparameter determination. Cameras and other equipment for performing these measurements – with the exception of special designs – are available in every typical Xray diffraction laboratory. At present, these methods of poor (1 part in 10^{2}) or moderate (up to 1 part in 10^{4}) accuracy are suitable only for preliminary measurements.
Less popular and more specific divergentbeam methods (third group) give satisfactory accuracy (1 part in 10^{4} or 1 part in 10^{5}), comparable with that obtained by counterdiffractometer methods, by means of very simple equipment.
In spite of the common use of counter diffractometers, and of the increasing use of imaging plates (and synchrotron radiation), traditional photographic methods of the second and the third groups are still popular and new designs are reported.
As based on polychromatic radiation, the Laue method is, in principle, useless for accurate latticeparameter determination. It is true that, from a single Laue diffraction pattern (in transmission), one can determine precisely the axial ratios and interaxial angles (a method based on the gnomonic projection is described by Amorós, Buerger & Amorós, 1975), but the unit cell determined will differ from the true cell by a simple scale factor.
The problem of absolute scaling of the cell is important nowadays, when synchrotronradiation Laue diffraction patterns are currently being used for collecting Xray data (from singlecrystal systems including proteins, for example). As shown by Cruickshank, Carr & Harding (1992), it is possible to estimate the scale factor using the minimum wavelength present in the incident Xray beam. A method proposed by the authors (Carr, Cruickshank & Harding, 1992) allows one to determine the unit cell and orientation of an unknown crystal (in a general orientation) from a single Laue pattern. The accuracy of the absolute latticeparameter determination depends on the accuracy with which the minimum wavelength is known for the experiment and is, at present, about 5% in favourable cases (while the error in axial ratio determination after refinement is typically 0.25%). To increase the accuracy, the authors propose either to record the Laue patterns with an attenuator in the incident beam that has a suitable absorption edge (λ_{min} can become a sharp and accurately known limit) or to locate the bromineabsorption edge, if the Xray detector contains bromine, as in photographic films and image plates.
Movingcrystal methods of latticeparameter determination apply basic photographic techniques, such as:

In the first of these methods, the film remains stationary, while in the others it is moved during the exposure. The principles and detailed descriptions of these techniques have been presented elsewhere (Buerger, 1942; Henry, Lipson & Wooster, 1960; Evans & Lonsdale, 1959; Stout & Jensen, 1968, Chapter 5; Sections 2.2.3 , 2.2.4 , and 2.2.5 of this volume) and only their use in latticeparameter measurements will be considered here.
The rotatingcrystal method – the simplest of the movingcrystal methods – determines the identity period I along the axis of rotation (or oscillation), , from the formula in which n is the number of the layer line and ν is the angle between the directions of the primary and diffracted beams.
The angle ν is determined from the measurement of the distance between two lines corresponding to the same layer number n from the equation where R is the camera radius.
All the lattice parameters may be determined from separate photographs made for rotations of the crystal along different rotation axes, i.e. the system axis, plane and spatial diagonals (Evans & Lonsdale, 1959), without indexing the photographs. In practice, however, this method is rarely used alone and is most often applied together with other photographic methods (for example, the Weissenberg method), but it is a useful preliminary stage for other methods. In particular, the length of a unitcell vector may be directly determined if the rotation axis coincides with this vector.
Advantages of this method are:

Drawbacks of the method are:

A twodimensional picture of a reciprocal cell from one photograph can be obtained by the methods in which rotation of the crystal is accompanied by movement of the film, as in the Weissenberg, the de Jong–Bouman, and the Buerger precession techniques. These methods give greater precision than the previous one (§5.3.2.3.1).
The advantages of the Weissenberg method in relation to the other two are:

On the other hand, the disadvantage, in contrast to the de Jong–Bouman and the Buerger precession methods, is that it gives deformed pictures of the reciprocal lattice. This is not a fundamental problem, especially now that computer programs that calculate lattice parameters and draw the lattice are available (Luger, 1980). In latticeparameter measurements, both the zerolayer Weissenberg photographs and the higherlayer ones are used. The latter can be made both by the normalbeam method and by the preferable equiinclination method. Photographs in the de Jong–Bouman and precession methods give undeformed pictures of the reciprocal lattice, but afford less information about it than do Weissenberg photographs.
The most effective photographic method of latticeparameter measurement is a combination of two techniques (Buerger, 1942; Luger, 1980), which makes it possible to obtain a threedimensional picture of the reciprocal lattice; for example: the rotation method with the Weissenberg (lower accuracy); or the precession (or the Weissenberg) method with the de Jong–Bouman (higher accuracy).
A suitable combination of the two methods will determine all the lattice parameters, even for monoclinic and triclinic systems, from one crystal mounting. This problem has been discussed and resolved by Buerger (1942, pp. 388–390), Hulme (1966), and Hebert (1978). Wölfel (1971) has constructed a special instrument for this task, being a combination of a de Jong–Bouman and a precession camera.
To measure with a precision and an accuracy better than is possible in routine photographic methods, additional work has to be performed. The first methods allowing precise measurement of lattice parameters were photographic powder methods (Parrish & Wilson, 1959). Special singlecrystal methods with photographic recording to realize this task (earlier papers are reviewed by Woolfson, 1970, Chap. 9) combine elements of basic singlecrystal methods (presented in §§5.3.2.3.1 and 5.3.2.3.2) with ideas more often met in powder methods (asymmetric film mounting). A similar treatment of some systematic errors (extrapolation) is met in both powder and singlecrystal methods.
Small changes of lattice parameters caused by thermal expansion or other factors can be investigated in multipleexposure cameras.
Bearden & Henins (1965) used the doublecrystal spectrometer with photographic detection to examine imperfections and stresses of large crystals. The technique allowed the detection of angle deviations as small as 0.5′′. A nearly perfect calcite crystal was used as the first crystal (monochromator), the sample was the second. The device distinguished itself with very good sensitivity. The use of the long distance (200 cm) between the focus and the second crystal made possible resolution of the doublet , and elimination of the radiation. An additional advantage was that the arrangement was less timeconsuming, so that it was suitable for controlling the perfection of growing crystals and useful for choosing adequate samples for the wavelength measurements.
Kobayashi, Yamada & Azumi (1968) have described a special `strainmeter' for measuring small strains of the lattice. The strain along an axis normal to the i plane results in a change of the interplanar distance : The use of a large camera radius R = 2639 mm makes it possible to obtain both high sensitivity and high precision (2 parts in 10^{6}) even in the range of lower Bragg angles . The device is suitable for the investigation of defects resulting from small strains and may be used in measurements of thermal expansion.
Glazer (1972) described an automatic arrangement, based on the Weissenberg goniometer, for the photographic recording of highangle Bragg reflections as a function of temperature, pressure, time, etc. A careful choice of the oscillation axis and oscillation range makes it possible to obtain a distorted but recognizable phase diagram (Fig. 5.3.2.1 ) within several hours. The method had been applied by Glazer & Megaw (1973) in studies of the phase transitions of NaNbO_{3}.

(a) Photographic recording of latticeparameter changes. (b) Corresponding diagram of the variation of lattice parameters in pseudocubic NaNbO_{3} (Glazer & Megaw, 1973). 
Popović, Šljukić & Hanic (1974) used a Weissenberg camera equipped with a thermocouple mounted on the goniometer head for precise measurement of lattice parameters and thermal expansion in the hightemperature range.
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 divergentbeam techniques initiated by Lonsdale (1947), and their numerous modifications belong to this group.
The excitation of the characteristic Xrays used in these methods can be performed by Xradiation (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 Xrays may be located either in the sample itself (the Kossel method), on the surface of the sample in a layer of target material (the pseudoKossel method), or outside the sample (the divergentbeam techniques). The divergent Xray 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 reciprocallattice 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 focustofilm 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 highprecision Kossel camera is described by Reichard (1969) and the generation of pseudoKossel patterns by the divergentbeam 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 backreflection region (Fig. 5.3.2.2). The second arrangement seems to be (Lutts, 1968) more suitable for latticeparameter 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 backreflection 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, pseudoKossel, or divergentbeam 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 backreflection region. The curves on transmission patterns can be considered to be conics; those recorded in the backreflection 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 latticeparameter determination, various features of the photographs may be used, i.e. intersections or nearintersections of Kossel lines, their neartangency, lensshaped figures, and the whole lines approximated with a function.
The basis of latticeparameter 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) Kα_{1} line. The other data needed (a sufficient number of equations, the solution of which leads to latticeparameter 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 10^{6}, 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 10^{5}] 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:

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 electronbeam probe may result from the thermal effects of the electron beam generating a divergent Xray 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 Xray 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 Xray 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 10^{3} to 1 part in 10^{5}. 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 10^{3}), 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 latticeparameter differences, a `point' source (i.e. as small as possible) is required and the highorder Kossel lines should be used to obtain both well resolved doublets and `thin' figures. The nearintersections of conic sections, applied in Lonsdale's (1947) method, the major axes of lensshaped figures, used in Heise's (1962) method, and the small spherical polygons formed by several Kossel cones are very sensitive to latticeparameter 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 latticeparameter changes or differences. It should be added that the Kossel method is especially suitable for small single crystals or finegrained polycrystals, whereas the other divergentbeam 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 divergentbeam 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 αiron and its alloys were examined by Lutts & Gielen (1971). Structure defects resulting from overpressure 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).
Divergentbeam 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 realstructure examination is the determination of angles between crystal blocks. A method has been worked out by Aristov, Shmytko & Shulakov (1974a,b).
Divergentbeam techniques can also be used in Xray topographic studies, realized either by means of Kosselline scanning (Rozhansky, Lider & Lyutzau, 1966) or by lineprofile analysis (Glass & Weissmann, 1969).
Schetelich & Geist (1993) used the Kossel method for latticeparameter 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 computeraided measurements of cell parameters of isotopically distinct samples of LiF over a wide temperature range of 15–375 K.
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