Order-Disorder phenomena in myelinated nerve sheaths: I. A physical model and its parametrization: Exact and approximate determination of the parameters

Abstract

An algorithm is developed for the analysis of the X-ray scattering spectra of lamellar systems, by reference to a precise physical model. The model consists of identical planar lamellae (the motif), all parallel and stacked in a one-dimensional crystal with four types of defect: stacking disorder, finite size of the crystallites, and presence of diffuse and blank scattering. In addition, the spectra are distorted by collimation aberrations. In order to evaluate the effects of these distortions, the following assumptions are made: (1) beyond some point Slimit the intensity curve can be expressed as a function of a (small) number of parameters; (2) the blank scattering, restricted to very small angles, can be identified and eliminated; and (3) the diffuse scattering is entirely defined by the values of idiff(h/D) at the lattice Sh = h/D (h is a positive integer less than or equal to DSlimit). These assumptions lead to an expression of the whole of the intensity curve as a function of a finite number of parameters: the average D and the variance sigma 2D of the repeat distance, the average number [N] of lamellae per crystallite, the set [idiff(h/D)] and the set [imotif(k/2D)] (where k is a positive integer), which defines the structure of the motif. An algorithm is proposed to determine the value of the various parameters. The derivation of the algorithm involves several operations: construction in real space of periodic functions whose motifs are step-sections of the autocorrelation function; expression in reciprocal space, and in terms of the experimental scattering curves, of the Fourier transform of those periodic functions; analysis of the properties of the two functions. The algorithm is tested using a variety of simulated scattering curves whose parameters [imotif(k/2D)], [idiff(2/D)], D, sigma D, [N] (and collimation distortions) are within the range commonly encountered in experimental conditions. The results show that the values of the parameters retrieved by the algorithms are very close to those used in the simulation. The calculations are fast and easy to implement on a computer. The main virtues of the algorithm are (1) to determine the values of all the parameters at once, eliminating most of the intermediate (and questionable) manipulations (separation of signal from noise, discrimination of overlapping reflections, integration of the intensities) and (2) to yield the continuous intensity curve of a single motif.