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. 5.2, pp. 493494

The systematic errors are generally called aberrations; they differ from random errors in that in principle they can be calculated for any particular experimental arrangement and the observations corrected for them, leaving only the random error. In practice, of course, the calculation may be difficult. Aberrations can be divided broadly into two classes: (i) geometrical and (ii) physical. The geometrical aberrations are those that depend on the dimensions of the source, specimen, and detector (or of the slits that limit their effective dimensions). In angledispersive techniques, the physical aberrations depend on the intensity distribution in the range of wavelengths used, and in both angle and energydispersive techniques they depend on the response characteristics of the detector and associated circuits.
The aberrations shift and distort the diffraction maxima. The study of their effects can be divided into four stages, corresponding to four levels of mathematical difficulty, and the stage to which it is necessary to carry the calculation depends on the purpose in view and the identifiable feature (Subsection 5.2.2.1) of the wavelength distribution that it is intended to adopt as a measure of the position of the line profile. The three usual features are:
The first of these, the centroid, requires only the first stage of the calculation for the geometrical aberrations and the first and second for the physical; the second, the peak, logically requires all four stages, but approximations can be obtained at the second stage; and the third, the best overall fit, requires all four stages.
The first stage of the calculation is the determination of the effect of the aberration on the centroid of the diffraction maximum, and ordinarily this gives rise to no insurmountable difficulty (Spencer, 1931, 1935, 1937, 1939, 1941, 1949; Wilson, 1950; Ladell, Parrish & Taylor, 1959; Pike & Wilson, 1959). It is all that is required for the correction of centroid positions for geometrical aberrations, which should be strictly additive. There is some limitation for physical aberrations (Edwards & Toman, 1970; Wilson, 1970b).
The second stage is the calculation of the meansquare broadening (variance). This can be used to obtain a reasonable approximation to the correction of peak positions over a wide range of Bragg angle (Wilson, 1961; Gale, 1963, 1968). To this approximation, the position of the observed peak is given by where is the centroid and W the variance of the geometrical aberrations and I′′ and I′′′ are second and third derivatives of the observed line profile evaluated at its maximum. The physical aberrations of the centroid depend on the variance of the part of the wavelength distribution used in determining the centroid (Wilson, 1958, 1963; Wilson & Delf, 1961). Those of the peak depend on the ratio of the peak intensity I to its second derivative I′′ (Wilson, 1961, 1963, 1965c).
Often an aberration can be expressed in the form where the function F gives the angular variation of the aberration and K depends only on dimensions etc. that are fixed for a particular experiment but whose actual measurement is too difficult or tedious. The constant K can then be treated along with the lattice parameters as an adjustable parameter in leastsquares refinement (analytical extrapolation; see Subsection 5.2.3.2).
The third stage is the calculation of the line profile corresponding to each geometrical aberration. These aberration profiles can be combined by convolution (folding), either directly or by Fourier methods, and, in the fourth stage, the combined aberration profile can be convoluted with the emission profile of the Xray source (or the emission profile as trimmed by a monochromator, pulseheight analyser, filter etc.) and with the diffraction profile corresponding to the state of strain, crystallite size, etc. of the specimen. This calculation of the composite line profile would be a necessary preliminary to an exact use of peak positions or of overallprofile fitting in latticeparameter determination.
Such calculations were proposed many years ago (for example, by Alexander, 1948, 1950, 1953, 1954), and have been used by Beu and coworkers (see Section 5.2.9), and also by Boom and Smits (Boom & Smits, 1965; Boom, 1966). With the development of more powerful computer methods, such calculations can now be carried out routinely (e.g. Cheary & Coelho, 1992, 1994; Kogan & Kupriyanov, 1992; Timmers, Delhez, Tuinstra & Peerdeman, 1992). However, not all the relevant instrumental parameters can in general be determined with sufficient accuracy and overall instrumental line profiles are normally obtained by means of a suitable standard material, for which sample broadening is negligible (Section 5.2.12). There is, however, a related common approach, empirical rather than fundamental, based on the proposal of Rietveld (1967, 1969). Its use in structure determination is treated in detail in Chapter 8.6 , and its use in latticeparameter determination in Section 5.2.6. There seems to be no detailed published study of the accuracy attainable for lattice parameters, but the estimated standard deviations quoted (see, for example, Young, 1988) are comparable with those obtained for simpler structures giving resolved reflections.
Equation (5.2.1.4) indicates that for a given error in θ the fractional error in the spacing d approaches zero as θ approaches 90°. The errors in θ – expressed as in (5.2.3.2) – arising from any specified aberration may increase as θ increases, but ordinarily this increase is insufficient to outweigh the effect of the factor. In the simple cubic case, one can write where K is a proportionality factor and represents the angular variation of the systematic errors in the lattice parameter. The functions F in (5.2.3.2) and (5.2.3.3) are not exactly the same; they are transformed into one another by the use of (5.2.1.4). Functions suitable for different experimental arrangements are quoted in the following sections; see, for example, equation (5.2.8.1) for the Debye–Scherrer camera and Tables 5.2.4.1 and 5.2.7.1 for diffractometers. Simple graphical extrapolation is quick and easy for cubic substances, and by the use of successive approximations it can be applied to hexagonal (Wilson & Lipson, 1941), tetragonal, and even orthorhombic materials. It is, however, very cumbersome for noncubic substances, and impracticable if the symmetry is less than orthorhombic.
Analytic extrapolation seems to have been first used by Cohen (1936a,b). It is now usual even in the cubic case: programs are often included in the software accompanying powder diffractometers, and many others are available separately. Some programs that are frequently referred to are described by Appleman & Evans (1973), Mighell, Hubbard & Stalick (1981), and Ferguson, Rogerson, Wolstenholme, Hughes & Huyton (1987); for a comparison, see Kelly (1988). If the precision warrants it, the single function may be replaced by a sum of functions , one for each of the larger aberrations listed in Tables 5.2.4.1, 5.2.7.1, and 5.2.8.1. Two – the zero error and a function corresponding to specimensurface displacement and transparency – must be used routinely; one or two more may be added if the precision warrants it.
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