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Diffraction by the ideal paracrystal

J. L. Eads & R. P. Millane, Acta Cryst. (2001). A57, 507–517

Paracrystal was the name given originally by Hosemann and co-workers (Hosemann & Bagchi, 1962) to a class of models used to describe structures with highly distorted lattices that exhibit only very broadened diffraction peaks. The models describe structures that are still crystalline in the sense that each lattice site can be indexed by pairs (in 2D) or triplets (in 3D) of integers but the shape and size of the unit cell is highly variable. There are many examples of such materials: for example, polymers, glasses and alloys (Hosemann & Hindeleh, 1995); colloidal suspensions and block copolymer films (Matsuoka, Tanaka, Hashimoto & Ise, 1987); polymer fibres (Granier, Thomas & Karasz, 1989); muscle proteins, biopolymers such as nucleic acids and some synthetic polymers (Millane & Stroud, 1991). The lattice models themselves deal only with the shapes and sizes of unit cells and not their contents.

[Figure 1] Figure 1. The cell closure constraint. The statistics of the intersite vectors must be specied such that d1 + d2 = d3 + d4.
In 1D the paracrystal model is well-defined and understood but extension to 2D and 3D is distinctly non-trivial, and has occupied numerous workers over the last ~40 years. The reason for this is easy to see by considering Fig. 1, which shows the general case in which the unit-cell edges can vary both in length and direction. Whereas for a 1D paracrystal the numbers of cell edges and lattice sites are equal, in 2D and 3D the number of edges far exceeds the number of sites, and this imposes severe constraints on the degree of variation that can occur.

This is the latest in a series of papers by Millane and co-workers that explore the properties of various types of such distorted lattice models. Their specific aim, not dealt with in any detail in the current paper, is to provide models suitable for describing more quantitatively the distribution of diffracted intensity from various real poorly crystalline materials. The aim is to be able to obtain better estimates of intensities of broadened Bragg peaks characteristic of these so-called fibre diffraction patterns, so that better structural information can be extracted from them. The work thus complements the authors’ work in analysing such materials as cellulose and other biopolymers (see, e.g., Finkenstadt & Millane, 1998; Stroud & Millane, 1995).

[Figure 2] Figure 2. The ideal paracrystal dened in terms of two 1D paracrystals.
In the introduction section of the paper the authors review the development of the paracrystal and other paracrystal-like models. The ideal paracrystal model originated with Hosemann many years ago. The unit cell in the ideal paracrystal has cell edges that can vary both in length and direction but each unit cell is constrained to be a simple parallelogram (see Fig. 2). This latter feature was always considered to be an unsatisfactory aspect of the model as it resulted in a number of unwanted properties of the diffraction pattern. In particular the model predicts low-angle scattering in excess of that observed in real systems and also shows a type of anisotropy in which the scattering behaviour in the direction of the lattice diagonal is different from that along the axial directions. There have been many and varied attempts to obtain models that do not have these shortcomings but despite this, the ideal paracrystal has continued to be of prime importance because of its amenability to analytical analysis, and consequent ease of computation.

The paper presents a description of ideal paracrystals in their most general form, develops the necessary equations by which their diffraction patterns can be calculated and explores the wide variety of peak-broadening characteristics they exhibit. The problem of incorrect low-angle diffraction is overcome by use of finite lattices, but the anisotropy between the axial and diagonal directions, however, remains. This is particularly problematical in attempts to construct an ideal paracrystal on a hexagonal lattice, an important case for crystals of molecules or assemblies that are approximately circular in cross-section. Despite this, the model described should provide a useful addition to the modelling possibilities in particular real example analyses. For example, it does seem likely that this anisotropy problem could be overcome for all practical purposes by taking an average of several different orientations of the ideal paracrystals described.

Finkenstadt, V. L. & Millane, R. P. (1998). Macromolecules 31(22), 7776–7783.
Granier, T., Thomas, E. & Karasz, F. (1989). J. Polym. Sci. Part B: Polym. Phys. 27, 469–487.
Hosemann, R. & Bagchi, S. (1962). Direct Analysis of Diffraction by Matter. Amsterdam: North-Holland.
Hosemann, R. & Hindeleh, A. (1995). J. Macromol. Sci. Phys. B34, 327–356.
Matsuoka, H., Tanaka, H., Hashimoto, T. & Ise, N. (1987). Phys. Rev. B, 36, 1754–1765.
Millane, R. P. & Stroud, W. J. (1991). Int. J. Biol. Macromol. 13, 202–208.
Stroud, W. J. & Millane, R. P. (1995). Acta Cryst. A51, 790–800.

Richard Welberry, Research School of Chemistry, Australian National U., Canberra, ACT, Australia