Line profiles for a collection of identical crystals: effect of a linear transformation of shape

Abstract

A linear transformation is defined as a distortion of the unit lengths along three Cartesian axes and of their angles, preserving the same covariant coordinates of the corresponding points. The mathematical relationship between the line profiles of two mutually transformed crystals is proposed. It makes it possible, for example, to get the line profiles for ellipsoids, parallelepipeds, distorted tetrahedra and octahedra from the exact results reported by Langford & Wilson [J. Appl. Cryst. (1978), 11, 102-113] for spheres, cubes, regular tetrahedra and octahedra.The line profile for a collection of identical crystallites of any shape is proportional towhere U is the volume of one unit cell, Vhkt(t ) is the volume common to a crystallite and its 'ghost' shifted a distance t in the direction perpendicular to the reflecting planes hkl, and s is the amount by which the radius vector in reciprocal space, S = (2 sin 0)/2, exceeds its value for the reciprocal-lattice point hkl (see, for example, Wilson, 1949, p. 41, equation 21). This line profile can be evaluated, in principle, for crystallites of arbitrary shape; the problem reduces to the purely geometrical one of determining V(t) and then finding its cosine Fourier transform. In practice most calculations so far have been confined to regular shapes with cubic or higher symmetry (see Langford & Wilson, 1978, for a survey), though some attention has been given to parallelepipeds (Allegra & Ronca, 1978, 1979 and hexagonal and circular cylinders (Langford, Lou~r & Wilson, 1980;Langford & Lou~r, 1982; * Present address: Crystallographic Data Centre, University Chemical Laboratory, Lensfield Road, Cambridge CB2 1EW.0567-7394/83/030280-03501.50Lou~r, Vargas & Langford, 1981). However, as was pointed out by Patterson (1939), results for a regular symmetrical shape may be altered to apply to any non-regular unsymmetrical shapes that can be derived from the regular one by linear transformations of coordinates. Thus results for a cube will lead to those for any parallelepiped, those for a sphere to those for any ellipsoid, and those for a regular tetrahedron or octahedron to those for a non-regular tetrahedron or octahedron. After a linear transformation the new values, Vl(t~), will be proportional to the old ones, V(t), the proportionality factor being independent of t, and if the crystallite is 'stretched' in the direction of t in the ratio I:R, then (1) shows that the line profile as a function of s is simply compressed in the ratio of R:I. This conclusion follows even more clearly from an alternative expression for the line profile (Wilson, 1949, p. 35, equation 5), in which the only effect of a linear transformation is to increase all the values of t in the ratio 1 :R. The alternative expression, however, is less convenient for the actual calculation of I(s). For the above regular shapes, except the circular cylinder, V(t) is a cubic in t, and remains so under any linear transformation of coordinates.We have, therefore, a cubic crystal with unit cell a=a|...

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“…Powder diffraction techniques are widely used in experimental condensed matter physics to characterize structural and microstructural properties of materials with statistical significance (Wessels, 1999;Bish et al, 2013;Solla-Gullon et al, 2015). Interpretation of diffraction data is often supported by theoretical models of the material's unit structure and an understanding of the contributions from features affecting its periodicity in space, such as the size/shape of crystalline domains (Scherrer, 1918;Langford & Wilson, 1978;Allegra & Wilson, 1983;Scardi & Leoni, 2001) and lattice distortions within these (e.g. strain field, defects and stacking disorder) (Williamson & Hall, 1953;Ungá r et al, 2005).…”