Gaussian growth-disorder models and optical transform methods

Abstract

M. LABEAU et al. 761Many details relating to the formation of the microdomains and the structural modifications at the domain boundaries remain unknown and further studies on this interesting material are warranted, using high-resolution electron microscopy as well as inelastic neutron diffraction. ReferencesALARIO-FRANCO, M. A., GREY, I. E., JOUBERT, J. C., VINCENT, H. & LABEAU, M. (1982). Acta Cryst. A38, 177-186. AbstractThe properties of Gaussian growth-disorder models are explored and their use for producing realizations of disordered lattices for optical transform analogue experiments is described. Use of Gaussian variables provides greater flexibility than previously described binary ones and in particular enables realizations to be produced in dimensions greater than two without restriction on the values of nearest-neighbour correlation coefficients. A method of converting Gaussian realizations to binary ones is also described.

The rôle of phase in diffuse diffraction patterns and its effect on real-space structure

“…2(b) for which random phases were used. In addition we showed (Welberry & Carroll, 1982) that when there is correlation, CGaussian , between two such normally distributed variables, the application of (1) results in a different value, Cbinary , for the correlation between the resulting binary variables, such that Cbinary = (2/rr)arcsin(fGaussian ).…”

Diffraction by Polycrystalline Fibers with Correlated Disorder

“…For distortion vectors with stationary Gaussian statistics, the average of this function over all disorder states is given by (Stroud & Millane, 1996) Following de Graaf (1989), we assume that correlations between the displacements of two sites depend only on their average separation Irjk I. Guided by the form of the correlation fields of Gaussian growth disordered lattices (Welberry, Miller & Carroll, 1980;Welberry & Carroll, 1982), we use circularly symmetric exponential correlation functions (Stroud & Millane, 1996) …”

“…Nonetheless, statistics for two classes of perturbed lattices have been determined. The statistics of lattices belonging to the class of perturbed lattices referred to as 'Gaussian growth disordered lattices' (Welberry, Miller & Carroll, 1980;Welberry & Carroll, 1982) are anisotropic, with correlations along the principal lattice axes decaying more slowly than those along the diagonals. Members of the second, more general, class of perturbed lattices, referred to as 'general symmetric Gaussian disordered lattices" (Welberry & Carroll, 1983), have more complex correlation structures and are capable of producing a greater variety of diffraction effects.…”

“…Incorporation of correlated lattice distortions into a model of polycrystalline fibers, and calculation of associated diffraction patterns, is the subject of this paper. Two models of lattices with correlated distortions are the paracrystal (Hosemann & Bagchi, 1962) and the perturbed lattice (Welberry, Miller & Carroll, 1980;Welberry & Carroll, 1982, 1983. In the paracrystal model, a distorted lattice is generated by replacing vectors that form the edges of the unit cell of a periodic lattice with vectors that vary in both length and direction.…”

Hexagonal paracrystals

“…Following the line advocated by Hammersley [3] Welberry and co-workers [4][5][6] developed a model for distorted lattices in which sets of Gaussian distributed random variables were associated with each site of a regular lattice and then correlations introduced between these variables. In the remainder of this paper the term paracrystal is used (rather loosely) to refer to highly distorted examples of these perturbed regular lattices.…”