Novel aspects of the real paracrystal

Abstract

A revision of the mathematical concept of paracrystals is proposed. Hermann's IZ. Electrochem. (1940), 46, 425-4361 early theory of mesophases represents a special case of a paracrystal where the statistical parameters introduced were merely crystalline or gas-like. The theory of paracrystals removes the black-white picture of crystalline and amorphous phases by introducing 18 new statistical parameters into conventional crystallography. It is shown that a real paracrystal can be defined by a three-dimensional convolution polynomial in combination with an empirically derived +t* law. Hence, a missing corner-stone of colloid science is established.

The crystallite size–disorder relationship based on the spiral paracrystal

“…Since the variance of the¯uctuations depends on the degree of disorder between adjacent sites and the size of the domain, this requirement gives an implicit relationship between the size of a paracrystalline domain (crystallite size) and the degree of disorder within the lattice. Such a relationship has been observed experimentally from analysis of X-ray diffraction peak breadths, from which one can estimate average crystallite size and paracrystalline disorder (Hosemann et al, 1981), and has been called the`à rule' (Balta  -Calleja & Hosemann, 1980). Current explanations of the à rule are based on the one-dimensional paracrystal and are not particularly convincing (Hosemann, 1975;Balta  -Calleja & Hosemann, 1980;Hosemann & Hindeleh, 1995).…”

“…The resulting distorted two-dimensional lattice is made up of cells that are parallelograms and therefore represents a rather restricted, and unrealistic, form of disorder. A more general and potentially more realistic model is the spiral paracrystal (Janke & Hosemann, 1978;Hosemann et al, 1981). The cell shape of the spiral paracrystal is not restricted to a parallelogram; however, a disadvantage is that the statistics of the distortions are not well de®ned, and an analytic expression cannot be obtained for its diffraction.…”

“…The average crystallite size can be calculated by averaging over many realizations of such lattices. Janke & Hosemann (1978) and Hosemann et al (1981) described a termination condition as follows. With reference to Fig.…”

“…Janke & Hosemann (1978) and Hosemann et al (1981) generated a spiral paracrystal using the termination condition (2) described above and noted that its size satis®ed the à rule. However, this was a single simulation for a single value of ' and so does not provide a de®nitive result.…”

“…The spiral paracrystal is constructed as follows (Janke & Hosemann, 1978;Hosemann et al, 1981). We consider a square (undistorted) lattice and normalize the lattice spacing to unity.…”