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. 534536

Let us review the most important problems concerning accurate and precise latticeparameter determination.
The first, commonly known, requirement for obtaining the highest accuracy and precision is the use of highBraggangle reflections. The tendency to obtain, record, and use in calculation such reflections can be met in rotatingcrystal cameras in which Straumanis mounting is applied (Farquhar & Lipson, 1946; Popović, 1974), in the Weissenberg method (the use of zerolayer photographs), in multipleexposure cameras (Glazer, 1972; Popović, Šljukić & Hanic, 1974), in standard and special diffractometers (Kobayashi, Yamada & Nakamura, 1963), in doublecrystal arrangements with white Xradiation (Okazaki & Kawaminami, 1973a,b; Okazaki & Ohama, 1979; Ohama, Sakashita & Okazaki, 1979; Okazaki & Soejima, 2001), and in the triplereflection scheme realized by means of the doublebeam technique, proposed by Kovalchuk, Kovev & Pinsker (1975).
Increasing the value, in the case of photographic cameras or counter diffractometers, not only reduces the value of in the formulae describing accuracy and precision, but also decreases several systematic errors proportional to , , or (Kheiker & Zevin, 1963; Wilson, 1963, 1980). To find the minimum total systematic error, which would occur for ° (not attainable in practice), extrapolation of the results is used (Farquhar & Lipson, 1946; Weisz, Cochran & Cole, 1948; Smakula & Kalnajs, 1955; Kobayashi, Yamada & Azumi, 1968; Pierron & McNeely, 1969).
The problem of the choice of suitable reflections for the measurement, calculation, and reduction of systematic errors could be generalized. Not only are such reflections for which values are close to 90° desired but also those for which ν tends to 90° in rotatingcrystal cameras (Umansky, 1960), highorder Kossel lines in divergentbeam techniques, axial or nonaxial reflections in counterdiffractometer methods, etc. Zerolayer reflections in Weissenberg photographs or in twocircle (`Weissenberg') diffractometers are preferable to upperlayer ones, because they are less affected by crystal misalignment (Clegg & Sheldrick, 1984) and a larger range of reciprocallattice points can be recorded (Luger, 1980); they are not sufficient, however, for the determination of all the lattice parameters in the lesssymmetric crystal systems.
Use of the orientation matrix makes possible accurate crystal setting for an arbitrary reflection and identification of recorded reflections not only in the case of the automated fourcircle diffractometer or the twocircle diffractometer but also in photographic methods.
For obtaining high accuracy of latticeparameter determination, systematic errors depending on the radiation, the crystal, and the technique should be known, evaluated, and reduced or corrected. As a rule, those systematic errors whose part in the total systematic error is the most important should be removed first. Looking at the development of the Xray diffraction techniques, the following remarks can be made.
As far as the photographic methods are concerned, the errors due to the means of recording (film shrinkage, uncertainty of measurement of distances on the film) and camera construction (radius in the movingcrystal methods and the sourcetofilm distance in divergentbeam techniques) play a major role. They can be reduced to some extent by using the Straumanis mounting or the ratio method, or the resolved doublet . Various methods have been introduced for reducing the error due to the sourcetofilm distance in divergentbeam techniques.
In counterdiffractometer methods, which give more accurate determinations of the Bragg angles and intensities, several instrumental and physical factors should be taken into account (Kheiker & Zevin, 1963; Wilson, 1963, 1980; Berger, 1984, 1986a; Härtwig & Grosswig, 1989). The effects of some can be diminished by the use of Soller slits (Berger, 1984) and the effects of most can be reduced by the Bond (1960) geometry, in its basic form or in its various modifications (Kheiker, 1973; Mauer et al., 1975; Hubbard & Mauer, 1976; Wołcyrz, Pietraszko & Łukaszewicz, 1980; Kudo, 1982; Lisoivan, 1982; Grosswig et al., 1985), in particular in combination with double or triplecrystal spectrometers (Kurbatov, Zubenko & Umansky, 1972; Godwod, Kowalczyk & Szmid, 1974; Hart & Lloyd, 1975; Sasvári & Zsoldos, 1980; Fewster, 1982; Pick et al., 1977; Obaidur, 2002). Another arrangement giving a partial reduction of systematic errors is that proposed by Renninger (1937) and developed by Post (1975) and Kshevetsky et al. (1979, 1985), in which multiplediffraction phenomena are applied. In most one or doublecrystal asymmetric spectrometers, the uncertainty of the origin of the angular scale is the important problem, which can be resolved by introducing either the Bond (1960) symmetrical arrangement (combinations mentioned above) or an additional instrumental axis (or axes) for obtaining a multiplecrystal arrangement (Buschert, 1965; Skupov & Uspeckaya, 1975; Hart, 1981), or a second beam, used in multiplebeam methods (Hart, 1969; Kishino, 1973; Larson, 1974; Baker & Hart, 1975; Buschert et al., 1983) or in various combined methods (Ando, Bailey & Hart, 1978; Kovalchuk, Kovev & Pinsker, 1975; Buschert et al., 1980; Fewster, 1989; Shvyd'ko et al., 2000; Obaidur, 2002).
The other important problem in singlecrystal methods, in the case when high accuracy is desired, is the misalignment of the crystal(s) and some elements of the device. The errors due to the misalignment can be reduced to some extent experimentally, if a respective dependence is known (Baker, George, Bellamy & Causer, 1968; Burke & Tomkeieff, 1968, 1969; Bowen & Tanner, 1995), or by calculation of the exact correction (Burke & Tomkeieff, 1968, 1969; Gruber & Black, 1970; Filscher & Unangst, 1980; Larson, 1974) if the inclinations of the beam(s) and the crystal(s) can be determined (Halliwell, 1970; Nemiroff, 1982).
Along with an increase in the accuracy of the angle determination, the error due to the wavelength determination becomes more important and a correction for the refraction should be introduced [the problem has been more intensively argued by Hart (1981, Section 2.3)]. Among others, measurement performed in the case of the Kossel method and divergentbeam techniques can be strongly affected by uncertainty of the wavelength, especially when the exact value of the wavelength is to be determined from the experiment. The combination of Xray and optical interferometry (Deslattes, 1969; Deslattes & Henins, 1973; Becker et al., 1981), the use of new latticeparameter standards (Hölzer et al., 1997), and new Xray or γray sources and wavelength standards (Shvyd'ko et al., 2000; Obaidur, 2002) gives new possibilities for very accurate latticespacing measurements, up to the subparts in range.
The accuracy of the method can be evaluated theoretically taking into account and estimating the errors that have not yet been eliminated. In the case of some dependent errors, the maximumlikelihood method proves useful for this task (Beu, Musil & Whitney, 1962). Sometimes computer simulation is used to estimate some errors difficult to express in an analytical form (Hubbard & Mauer, 1976; Urbanowicz, 1981b; Berger, 1986a; Härtwig & Grosswig, 1989). A more objective test of the accuracy would be an interlaboratory comparison, like that reported by Parrish (1960) many years ago. Usually, the authors introducing new methods, techniques or devices try to compare the results of their latticeparameter determination with those obtained by other experimenters (Batchelder & Simmons, 1965, p. 2868; Hubbard, Swanson & Mauer, 1975, p. 46; Hubbard & Mauer, 1976, p. 2; Łukaszewicz et al., 1976, p. 70; Łukaszewicz et al., 1978, p. 566; Lisoivan, 1982, p. 94; Grosswig, Härtwig, Alter & Christoph, 1983, p. 506; Chang, 1984, p. 254; Härtwig, BąkMisiuk, Berger, Brühl, Okada, Grosswig, Wokulska & Wolf, 1994; Fewster & Andrew, 1995, p. 455; Herbstein, 2000).
In view of the new achievements in highaccuracy absolute latticeparameter measurements, such as the results reviewed in §§5.3.3.8 and 5.3.3.9, and the accurate error analysis [Härtwig & Grosswig (1989); §5.3.3.4.3.2, point (7)], Härtwig, with his coworkers, initiated and organized an interlaboratory comparison of various `traditional' methods and techniques, including the Bond method (cf. §5.3.3.4.3), the `Sollerslit' method (§5.3.3.4.2) and the photographic multiplediffraction (Umweganregung) method (§5.3.2.4.2). In all measurements performed, a common silicon standard (WASO 9, mentioned above) of accurately known lattice parameter was used as the specimen, so it was possible to check independently the validity of all corrections introduced for systematic errors. As shown by Härtwig, BąkMisiuk, Berger, Brühl, Okada, Grosswig, Wokulska & Wolf (1994), the agreement of their independent results was not worse than 3 parts in 10^{6}. The quantity defines the presentday possibilities of accurate `dispersive' methods.
Herbstein (2000) concluded, `there has been little improvement in claimed precision over the past 40–60 years', but his analysis relates to widely used `dispersive' methods [including the Bond (1960) method] rather than to presentday highprecision comparison methods, which are also used for absolute measurements.
For obtaining high precision, a monochromatic and well collimated Xray beam is desired. The first requirement can be fulfilled by using a filter or monochromator, the second can be satisfied by introducing a point source and a choice of collimation parameters, and both can be accomplished by introducing one or more additional crystals. It is not necessary for a well collimated beam to be very narrow if local measurements in selected small areas of the specimen are not the object of the experiment. The use of a large, parallel Xray beam may be advantageous in some comparative measurements (Buschert, Pace, Inzaghi & Merlini, 1980).
Very high accuracy and highprecision measurements are, however, expensive; they need additional instrumental axes and detectors, and sometimes additional sources. Relatively simple divergentbeam techniques (photographic recording, stationary crystal) require fine Xray sources (an electron microscope or an electronbeam probe) if high precision is to be attained.
Some authors have managed to reduce the cost of the experiment using simpler equipment. An example may be the triplereflection scheme realized by the use of a doubleaxis spectrometer proposed by Ando, Bailey & Hart (1978) and Häusermann & Hart (1990). An additional advantage of the device is excellent stability.
Apart from the magnificent development of measuring techniques, an increasing role of mathematics is noticeable in contemporary papers. Matrix algebra is used for description of the crystal orientation in relation to the instrumental axes. Mathematical interpretation of the results is a very important part of the measurement, in particular in multiplediffraction methods, both divergentbeam photographic and collimatedbeam counter diffractometric. Syntheses of diffraction profiles are easily done with the aid of a computer. Algebraic and geometrical considerations make it possible to calculate proper corrections for systematic errors. Statistical methods give a tool for estimating and increasing the precision, in particular by means of leastsquares refinement, and also for testing the hypothesis of `no remaining systematic errors'. Commonly used computers are useful in numerous calculations and in controlling automatic devices.
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