International
Tables for Crystallography Volume B Reciprocal space Edited by U. Shmueli © International Union of Crystallography 2006 |
International Tables for Crystallography (2006). Vol. B. ch. 4.2, pp. 428-429
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In this context the structure factor of a chain molecule is neglected. Irregular distances between the molecules within a chain occur owing to the shape of the molecules, intrachain interactions and/or interaction forces via a surrounding matrix. A general discussion is given by Guinier (1963). It is convenient to reformulate the discrete Patterson function, i.e. the correlation function (4.2.4.39). in terms of continuous functions which describe the probability of finding the µth neighbour within an arbitrary distance
There are two principal ways to define . The first is the case of a well defined one-dimensional lattice with positional fluctuations of the molecules around the lattice points, i.e. long-range order is retained: , where denotes the displacement of the µth molecule in the chain. Frequently used are Gaussian distributions: ( = normalizing constant; = standard deviation). Fourier transformation [equation (4.2.4.45)] gives the well known result i.e. a monotonically increasing intensity with L (modulation due to a molecular structure factor neglected). This result is quite analogous to the treatment of the scattering of independently vibrating atoms. If (short-range) correlations exist between the molecules the Gaussian distribution is replaced by a multivariate normal distribution where correlation coefficients between a molecule and its µth neighbour are incorporated. is defined by the second moment: . Obviously the variance increases if the correlation diminishes and reaches an upper bound of twice the single site variance. Fourier transformation gives an expression for diffuse intensity (Welberry, 1985): For small Δ, terms with are mostly neglected. The terms become increasingly important with higher values of L. On the other hand, becomes smaller with increasing j, each additional term in equation (4.2.4.46) becomes broader and, as a consequence, the diffuse planes in reciprocal space become broader with higher L.
In a different way – in the paracrystal method – the position of the second and subsequent molecules with respect to some reference zero point depends on the actual position of the predecessor. The variance of the position of the µth molecule relative to the first becomes unlimited. There is a continuous transition to a fluid-like behaviour of the chain molecules. This 1D paracrystal (sometimes called distortions of second kind) is only a special case of the 3D paracrystal concept (see Hosemann & Bagchi, 1962; Wilke, 1983). Despite some difficulties with this concept (Brämer, 1975; Brämer & Ruland, 1976) it is widely used as a theoretical model for describing diffraction of highly distorted lattices. One essential development is to limit the size of a paracrystalline grain so that fluctuations never become too large (Hosemann, 1975).
If this concept is used for the 1D case, is defined by convolution products of . For example, the probability of finding the next-nearest molecule is given by and, generally: (µ-fold convolution).
The mean distance between next-nearest neighbours is and between neighbours of the µth order: . The average value of , which is also the value of w(z) for , where the distribution function is completely smeared out. The general expression for the interference function G(L) is with .
With , equation (4.2.4.47) is written: [Note the close similarity to the diffuse part of equation (4.2.4.5), which is valid for 1D disorder problems.]
This function has maxima of height and minima of height at positions and , respectively. With decreasing the oscillations vanish; a critical L value (corresponding to ) may be defined by Gmax/Gmin ≲ 1.2. Actual values depend strongly on .
The paracrystal method is substantiated by the choice , i.e. the disorder model. Again, frequently used is a Gaussian distribution: with the two parameters .
There are peaks of height which obviously decrease with and . The oscillations vanish for , i.e. . The width of the mth peak is . The integral reflectivity is approximately and the integral width (defined by integral reflectivity divided by peak reflectivity) (background subtracted!) which, therefore, increases with . In principle the same results are given by Zernike & Prins (1927). In practice a single Gaussian distribution is not fully adequate and modified functions must be used (Rosshirt et al., 1985).
A final remark concerns the normalization [equation (4.2.4.39)]. Going from (4.2.4.39) to (4.2.4.45) it is assumed that N is a large number so that the correct normalization factors for each may be approximated by a uniform N. If this is not true then The correction term may be important in the case of relatively small (1D) domains. As mentioned above, the structure factor of a chain molecule was neglected. The H dependence of , of course, obscures the intensity variation of the diffuse layers as described by (4.2.4.47a).
The matrix method developed for the case of planar disorder was adapted to 2D disorder by Scaringe & Ibers (1979). Other models and corresponding expressions for diffuse scattering are developed from specific microscopic models (potentials), e.g. in the case of Hg3−δAsF6 (Emery & Axe, 1978; Radons et al., 1983), hollandites (Beyeler et al., 1980; Ishii, 1983), iodine chain compounds (Endres et al., 1982) or urea inclusion compounds (Forst et al., 1987).
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