General remarks

Gałdecka, E.

doi:10.1107/97809553602060000597

International
Tables for
Crystallography
Volume C
Mathematical, physical and chemical tables
Edited by E. Prince

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International Tables for Crystallography (2006). Vol. C. ch. 5.3, pp. 505-506

Section 5.3.1.1. General remarks

E. Gałdecka^a

^a Institute of Low Temperature and Structure Research, Polish Academy of Sciences, PO Box 937, 50-950 Wrocław 2, Poland

5.3.1.1. General remarks

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The starting point for lattice-parameter measurements by X-ray diffraction methods and evaluation of their accuracy and precision is the Bragg law, combining diffraction conditions (the Bragg angle $[\theta]$ and the wavelength λ) with the parameters of the lattice to be determined: $[2 d\sin\theta=n\lambda, \eqno (5.3.1.1)]$ in which d is the interplanar spacing, being a function of direct-lattice 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 non-linear dependence of direct-lattice parameters on d is, in a general case, rather complicated (see, for example, Buerger, 1942, p. 103), it is convenient to introduce reciprocal-cell parameters ( [a^*] , [b^*] , [c^*] , $[\alpha^*]$ , $[\beta^*]$ , $[\gamma^*]$ ) and to write the Bragg law in the form: $[\eqalignno{ 4\sin^2\theta/\lambda^2 &=n^2/d^2 \cr &=h^2a^{*2}+k^2b^{*2}+l^2c^{*2}+2hka^*b^*\cos\gamma^* \cr &\quad +2hla^*c^*\cos\beta^*+2klb^*c^*\cos\alpha^*, &(5.3.1.2)}]$ 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 least-squares refinement to diminish the statistical error of the estimates. In some methods, extrapolation of the results is used to remove the $[\theta]$ -dependent systematic errors (Wilson, 1980, Section 5, and references therein) and requires several measurements for various $[\theta]$ .

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)]: $[\delta d/d=\delta \lambda/\lambda-\cot\theta\,\delta\theta, \eqno (5.3.1.3)]$ where δλ/λ is the relative accuracy of the wavelength determination in relation to the commonly accepted wavelength standard, and $[\delta\theta]$ 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 $[\sigma^2(d)]$ [or its square root – the standard deviation, $[\sigma(d)]$ ] of the measured interplanar spacing d as the measure of repeatability of experimental results.

The relative precision of lattice-spacing determination can be presented in the form: $[\sigma(d)/d=\cot\theta \sigma(\theta), \eqno (5.3.1.4)]$ where $[\sigma(\theta)]$ is the standard deviation of the measured Bragg angle $[\theta]$ .

Another mathematical criterion proposed especially for relative methods is the sensitivity, defined (Okazaki & Ohama, 1979) as the ratio $[\delta\theta/\delta d]$ , i.e. the change in the $[\theta]$ value owing to the unit change in d.

The main task in unit-cell determination is the measurement of the Bragg angle. For a given $[\theta]$ angle, the accuracy $[\delta\theta]$ and precision $[\sigma(\theta)]$ affect those of the lattice parameter [equations (5.3.1.3) and (5.3.1.4)]. To achieve the desired value of $[\delta d/d]$ , the accuracy $[\delta \theta]$ must be no worse than resulting from the Bragg law (Bond, 1960): $[|\delta\theta|=(|\delta d|/d)\tan\theta. \eqno (5.3.1.5)]$ An analogous equation can be obtained for $[\sigma(\theta)]$ as a function of $[\sigma(d)/d]$ . The values $[\delta\theta]$ and $[\sigma(\theta)]$ depend not only on the measurement technique (X-ray 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 $[h(\theta)]$ ], 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 $[h(\theta)]$ .

In the general case, $[h(\theta)]$ can be described as a convolution: $[h(\theta)=h_\lambda(\theta)*h_A(\theta)*h_C(\theta), \eqno (5.3.1.6)]$ where $[h_\lambda(\theta)]$ is an original profile due to wavelength distribution ; $[h_A(\theta)]$ is a distribution depending on various apparatus factors, such as tube-focus emissivity, collimator parameters, detector aperture; and $[h_C(\theta)]$ is a function (the crystal profile) depending on the crystal, its perfection, mosaic structure, shape (flatness), and absorption coefficient.

The functions $[h_A(\theta)]$ and $[h_C(\theta)]$ are again convolutions of appropriate factors.

Each of the functions $[h_\lambda(\theta)]$ , $[h_A(\theta)]$ , and $[h_C(\theta)]$ has its own shape and a finite width, which affect the shape and the width $[\omega_h]$ of the resulting profile $[h(\theta)]$ .

Since $[h_A(\theta)]$ and $[h_C(\theta)]$ are ordinarily asymmetric, the profile $[h(\theta)]$ is also asymmetric and may be considerably shifted in relation to the original one, $[h_\lambda(\theta)]$ , leading to systematic errors in lattice-parameter determination.

The finite precision $[\sigma(d)/d]$ , on the other hand, results from the fact that the two measured variables – the intensity h and the angle $[\theta]$ – are random variables.

The half-width $[\omega_\lambda]$ of $[h_\lambda(\theta)]$ defines the minimum half-width of $[h(\theta)]$ that it is possible to achieve with a given X-ray source: $[\omega_h\gtrsim \omega_\lambda. \eqno (5.3.1.7)]$ It can be assumed from the Bragg law that: $[\omega_\lambda=(w_\lambda/\lambda)\tan\theta, \eqno (5.3.1.8)]$ where $[w_\lambda]$ is the half-width of the wavelength distribution . In commonly used X-ray sources, $[w_\lambda/\lambda\approx 300\times10^{-6}]$ .

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 $[\delta\theta]$ to the half-width of the measured profile: $[{|\delta\theta|\over \omega_h}\le{|\delta d|/d\over w_\lambda/\lambda}. \eqno (5.3.1.9)]$ 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 X-ray 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 lower-symmetry systems only. In the last 35 years, many single-crystal 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 single-crystal sample, epitaxic layers).

Some elements are common to both powder and single-crystal 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 $[\theta]$ 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 single-crystal methods have developed separately, though some present-day high-resolution methods are not restricted to a particular crystalline form (Fewster & Andrew, 1995). In special cases, the combination of X-ray powder diffraction and single-crystal Laue photography , reported by Davis & Johnson (1984), can be useful for the determination of the unit-cell parameters.

Small but remarkable differences in lattice parameters determined by powder and single-crystal 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 refractive-index 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 $[\Delta d/d=2.05\times10^{-3}\pm10^{-4}]$ . Analysis of results of lattice-parameter 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 weighted-mean lattice parameter of silicon powder proved to be about 0.0002 Å smaller $[(\Delta d/d\simeq -4\times 10^{-5})]$ than that of the bulk silicon.

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International Tables for Crystallography (2006). Vol. C. ch. 5.3, pp. 505-506