X-ray Diffraction by a One-Dimensional Paracrystal of Limited Size

Abstract

An explicit equation for X-ray diffraction by a ®nite one-dimensional paracrystal is derived. Based on this equation, the broadenings of re¯ections due to limited size and disorder are discussed. It depicts the paracrystalline diffraction over the whole reciprocal space, including the small-angle region where it degenerates into the Guinier equation for small-angle X-ray scattering. Positions of diffraction peaks by paracrystals are not periodic. Peaks shift to lower angles compared to those predicted by the average lattice constant. The shifts increase with increasing order of re¯ections and degree of disorder. If the heights and widths of the paracrystalline diffraction are treated as reduced quantities, they are functions of a single variable, N 1a2 g. The width of the ®rst diffraction depends mostly on size broadening for a natural paracrystal, where N 1a2 g 0.1±0.2. A method to measure the paracrystalline disorder and size using a single diffraction pro®le is proposed based on the equation of paracrystal diffraction. An initial value of size may be obtained using the Scherrer equation, that of the degree of disorder is then estimated by the à law. Final values of the parameters are determined through least-squares re®nement against observed pro®les. An equation of diffraction by a polydisperse one-dimensional paracrystal system is presented for`box' distribution of sizes. The width of the diffraction decreases with increasing breadth of the size distribution.

Probing surface and interface morphology with Grazing Incidence Small Angle X-Ray Scattering

“…Finite-size effects along the chain can also be included [155,239,240,234,241,242]. The contribution of correlations in SSCA [235,155] is illustrated in Fig.…”

“…This leads to an unphysical divergence of the scattering close to the origin and to scattering patterns that do not fulfill the symmetry of the mean unit cell. Those drawbacks can be cured by finite size effects [42,241] and symmetrization procedure [245]. Averaging over orientations allows to define a radial pair correlation function in dimension higher than one [27,239,240] (Section 6.3.4).…”

“…Finite size effects can be easily accounted for by restricting the sum to a given number of nodes [239,240,42,241,242]. For normally distributed distances,…”

Structure of cellulose and microcrystalline cellulose from various wood species, cotton and flax studied by X-ray scattering

“…For example, Busson and Doucet (1999) have calculated an analytical interference function for two-dimensional hexagonal paracrystals, which has been used in the analysis of keratin fibres (Briki et al 1998). According to Mu et al the size effect on the paracrystal diffraction has to be considered in the case of natural paracrystals (Mu 1998), but in our analysis the paracrystalline structure of the microfibrils was assumed to be infinite.…”

The crystallite size–disorder relationship based on the spiral paracrystal

“…The d j are taken to be independent. Properties of the one-dimensional paracrystal are described elsewhere (Hosemann & Bagchi, 1962;Mu, 1998;Millane & Eads, 2000). For our purposes, an important result is that the variance of the distance x j k À x j between kth nearest neighbors is equal to k' 2 , where ' 2 is the variance of the distance between ®rst nearest neighbors (Millane & Eads, 2000).…”