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*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. The cell closure constraint. The statistics of the intersite vectors must be specied such that **d**_{1} + **d**_{2} = **d**_{3} + **d**_{4}.

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. The ideal paracrystal dened 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 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.

**References**

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.* B**34**, 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.* A**51**, 790–800.

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