Statistical inference and crystallite size distributions

Abstract

An information theory approach is devised in order to obtain crystallite size distributions from X-ray line broadening. The method is shown to be superior to those based on Fourier expansions, as illustrated by numerical examples and a realistic situation. The powder model of Warren and Averbach is considered, in which the sample is thought of as a 'column-like' structure of u.sfit cells perpendicular to the diffraction plane. Errors in excess of 100% arise as a result of truncating the diffraction peak. It is shown that, with the present approach, the corresponding figure is reduced to 5%, which confirms the power of information theory, and makes this method especially convenient in those cases in which there are large overlaps between the tails of two diffraction peaks.

Disjoint states and quantum games

“…using orthogonal states permits one to dispense with the denominator. However, if the pertinent Hamiltonian is unknown and one has to infer it from some empiric moments using, for instance, the MaxEnt approach (see, for instance, [22][23][24][25] and references therein), one gets immersed in a situation that can be assimilated to a game, and our just found multiplicity of Nash points makes the theorist task much simpler.…”

Expansion of Natural Na+– and Ca2+–Montmorillonites in the Presence of NaCl and Surfactant Solutions

“…The particle size distribution can be calculated on the basis of the hypothesis that every diffraction peak can be described by the mean of the following equation: I false( normalψ i false) = K ∑ N = 1 ∞ normalγ ( N ) G ( N , ψ i ) where I is the intensity for a determined set of diffraction angles ψ i calculated as follows ψ i = 2π d sin θ/λ (where θ is the diffraction angle, d is the perpendicular distance between diffraction planes, and λ is the Cu Kα radiation length), γ­( N ) is the size distribution of columns in the crystal pattern, N is the number of scattering centers, K is an adjustment constant, and G ( N , ψ i ) is the interference function of a column of N scattering centers. Approximation of the unknown γ­( N ) is obtained by means of a statistical method based on the information theory, and the measurements are obtained for a determined set of angles ψ i and expressed by the statistical distribution function ρ­( N ) as given in ref .…”

Statistical analysis of a mixed-layer x-ray diffraction peak

“…As a first example, we believe it to be instructive to analyse an illite sample which is known to be in a single phase (that is, a sample with a negligible degree of interstratification). In previous works [20,21], it was shown that the particle size distribution of sodic montmorillonite samples was such that most stacks had fewer than six layers. For this reason (and according to the results arising from this first example) we shall consider N m = 10 (which entails dealing with 2046 configurations) and we shall assume that configurations with more than ten layers are of the single-phase kind only (so that in the concomitant states the maximum number of layers can be safely set to be equal to 50).…”