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. 505508

The starting point for latticeparameter measurements by Xray diffraction methods and evaluation of their accuracy and precision is the Bragg law, combining diffraction conditions (the Bragg angle and the wavelength λ) with the parameters of the lattice to be determined: in which d is the interplanar spacing, being a function of directlattice parameters a, b, c, α, β, γ, and n is the order of interference. Before calculating the lattice parameters, corrections for refraction should be introduced to d values determined from (5.3.1.1) [James (1967); Isherwood & Wallace (1971); Lisoivan (1974); Hart (1981); Hart, Parrish, Bellotto & Lim (1988); cf. §5.3.3.4.3.2, paragraph (2) below].
Since only d values result directly from (5.3.1.1) and the nonlinear dependence of directlattice parameters on d is, in a general case, rather complicated (see, for example, Buerger, 1942, p. 103), it is convenient to introduce reciprocalcell parameters (, , , , , ) and to write the Bragg law in the form: where h, k and l are the indices of reflection, and then those of the direct cell are calculated from suitable equations given elsewhere (Buerger, 1942, p. 361, Table 2). The minimum number of equations, and therefore number of measurements, necessary to obtain all the lattice parameters is equal to the number of parameters, but many more measurements are usually made, to make possible leastsquares refinement to diminish the statistical error of the estimates. In some methods, extrapolation of the results is used to remove the dependent systematic errors (Wilson, 1980, Section 5, and references therein) and requires several measurements for various .
Measurements of lattice dimensions can be divided into absolute, in which lattice dimensions are determined under defined environmental conditions, and relative, in which, compared to a reference crystal, small changes of lattice parameters (resulting from changes of temperature, pressure, electric field, mechanical stress etc.) or differences in the cell dimensions of a given specimen (influenced by point defects, deviation from exact stoichiometry, irradiation damage or other factors) are examined.
In the particular case when the lattice parameter of the reference crystal has been very accurately determined, precise determination of the ratio of two lattice parameters enables one to obtain an accurate value of the specimen parameter (Baker & Hart, 1975; Windisch & Becker, 1990; Bowen & Tanner, 1995).
Absolute methods can be characterized by the accuracy δd, defined as the difference between measured and real (unknown) interplanar spacings or, more frequently, by using the relative accuracy δd/d, defined by the formula obtained as a result of differentiation of the Bragg law [equation (5.3.1.1)]: where δλ/λ is the relative accuracy of the wavelength determination in relation to the commonly accepted wavelength standard, and is the error in the Bragg angle determined.
The analogous criterion used for characterization of relative methods may be the precision, defined by the variance [or its square root – the standard deviation, ] of the measured interplanar spacing d as the measure of repeatability of experimental results.
The relative precision of latticespacing determination can be presented in the form: where is the standard deviation of the measured Bragg angle .
Another mathematical criterion proposed especially for relative methods is the sensitivity, defined (Okazaki & Ohama, 1979) as the ratio , i.e. the change in the value owing to the unit change in d.
The main task in unitcell determination is the measurement of the Bragg angle. For a given angle, the accuracy and precision affect those of the lattice parameter [equations (5.3.1.3) and (5.3.1.4)]. To achieve the desired value of , the accuracy must be no worse than resulting from the Bragg law (Bond, 1960): An analogous equation can be obtained for as a function of . The values and depend not only on the measurement technique (Xray source, device, geometry) and the crystal (its structure, perfection, shape, physical properties), but also on the processing of the experimental data.
The first two factors affect the measured profile [which will be denoted here – apart from the means of recording – by ], being a convolution of several distributions (Alexander, 1948, 1950, 1954; Alexander & Smith, 1962; Härtwig & Grosswig, 1989; Härtwig, Hölzer, Förster, Goetz, Wokulska & Wolf, 1994) and the third permits calculations of the Bragg angle and the lattice parameters with an accuracy and a precision as high as possible in given conditions, i.e. for a given profile .
In the general case, can be described as a convolution: where is an original profile due to wavelength distribution; is a distribution depending on various apparatus factors, such as tubefocus emissivity, collimator parameters, detector aperture; and is a function (the crystal profile) depending on the crystal, its perfection, mosaic structure, shape (flatness), and absorption coefficient.
The functions and are again convolutions of appropriate factors.
Each of the functions , , and has its own shape and a finite width, which affect the shape and the width of the resulting profile .
Since and are ordinarily asymmetric, the profile is also asymmetric and may be considerably shifted in relation to the original one, , leading to systematic errors in latticeparameter determination.
The finite precision , on the other hand, results from the fact that the two measured variables – the intensity h and the angle – are random variables.
The halfwidth of defines the minimum halfwidth of that it is possible to achieve with a given Xray source: It can be assumed from the Bragg law that: where is the halfwidth of the wavelength distribution. In commonly used Xray sources, .
Combination of (5.3.1.5) and (5.3.1.8), and with (5.3.1.7) taken into consideration, gives an estimate of the ratio of the admissible error to the halfwidth of the measured profile: To obtain the highest possible accuracy and precision for a given experiment (given diffraction profile), mathematical methods of data analysis and processing and programming of the experiment are used (Bačkovský, 1965; Wilson, 1965, 1967, 1968, 1969; Barns, 1972; Thomsen & Yap, 1968; Segmüller, 1970; Thomsen, 1974; Urbanowicz, 1981a; Grosswig, Jäckel & Kittner, 1986; Gałdecka, 1993a, b; Mendelssohn & Milledge, 1999).
Measurements of lattice parameters can be realized both with powder samples and with single crystals. At the first stage of the development of Xray diffraction methods, the highest precision was obtained with powder samples, which were easier to obtain and set, rather than with single crystals. The latter were considered to be more suitable in the case of lowersymmetry systems only. In the last 35 years, many singlecrystal methods have been developed that allow the achievement of very high precision and accuracy and, at the same time, allow the investigation of different specific features characterizing single crystals only (defects and strains of a singlecrystal sample, epitaxic layers).
Some elements are common to both powder and singlecrystal methods: the application of the basic equations (5.3.1.1) and (5.3.1.2); the use of the same formulae defining the precision and the accuracy [equations (5.3.1.4) and (5.3.1.3)] and – as a consequence – the tendency to use values as large as possible; the means of evaluation of some systematic errors due to photographic cameras or to counter diffractometers (Parrish & Wilson, 1959; Beu, 1967; Wilson, 1980); the methods of estimating statistical errors based on the analysis of the diffraction profile and some methods of increasing the accuracy (Straumanis & Ieviņš, 1940). In other aspects, powder and singlecrystal methods have developed separately, though some presentday highresolution methods are not restricted to a particular crystalline form (Fewster & Andrew, 1995). In special cases, the combination of Xray powder diffraction and singlecrystal Laue photography, reported by Davis & Johnson (1984), can be useful for the determination of the unitcell parameters.
Small but remarkable differences in lattice parameters determined by powder and singlecrystal methods have been observed (Straumanis, Borgeaud & James, 1961; Hubbard, Swanson & Mauer, 1975; Wilson, 1980, Sections 6 and 7), which may result from imperfections introduced in the process of powdering or from uncorrected systematic errors (due to refractiveindex correction, for example; cf. Hart, Parrish, Bellotto & Lim, 1988). The first case was studied by Gamarnik (1990) – both theoretically and experimentally. As shown by the author, the relative increase of lattice parameters in ultradispersed crystals of diamond in comparison with massive crystals was as high as . Analysis of results of latticeparameter measurements of silicon single crystals and powders, performed by different authors (Fewster & Andrew, 1995, p. 455, Table 1), may lead to the opposite conclusion: The weightedmean lattice parameter of silicon powder proved to be about 0.0002 Å smaller than that of the bulk silicon.
The essential feature of singlecrystal methods is the necessity for the very accurate setting of the diffracting planes of the crystal in relation to the axes or planes of the instrument; this may be achieved manually or automatically. The fully automated fourcircle diffractometer permits measurements to be made with an arbitrarily oriented single crystal, since its original position in relation to the axes of the device can be determined by means of a computer (calculation of the orientation matrix); the crystal can then be automatically displaced into each required diffracting position. Misalignment of the crystal and/or of element(s) of the device (the collimator, for example) may be a source of serious error (Burke & Tomkeieff, 1968, 1969; Halliwell, 1970; Walder & Burke, 1971; Filscher & Unangst, 1980; Larson, 1974). On the other hand, a well defined singlecrystal setting allows the unequivocal indexing of recorded reflections (movingfilm methods, counter diffractometer) and the accurate determination of Bragg angles. In the particular case of large, specially cut and set, single crystals and a suitable measurement geometry, it is possible to avoid some sources of systematic error (Bond, 1960).
Singlecrystal methods are realized by a great variety of techniques.
When making a choice of the method, the aim of the measurement, the required accuracy and/or precision as well as the laboratory equipment available should be taken into account.
In the present review, all the methods are classified with respect to the measurement technique, in particular into photographic and counterdiffractometer techniques. Moreover, the methods will be described in approximately chronological sequence,^{1} i.e. from the earliest and simplest rotatingcrystal method to the latest morecomplex nondispersive techniques, and at the same time from those of poor accuracy and precision to those attaining the highest precision and/or accuracy. In each of the methods realizing a given technique, first the absolute and then the relative methods will be described.
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