Bayesian inference of nanoparticle-broadened x-ray line profiles

Journal of Research of the National Institute of Standards and Technology, Jan-Feb, 2004 by Nicholas Armstrong, Walter Kalceff, James P. Cline, John E. Bonevich

A single-step, self-contained method for determining the crystallite-size distribution and shape from experimental x-ray line profile data is presented. It is shown that the crystallite-size distribution can be determined without invoking a functional form for the size distribution, determining instead the size distribution with the least assumptions by applying the Bayesian/MaxEnt method. The Bayesian/MaxEnt method is tested using both simulated and experimental Ce[O.sub.2] data, the results comparing favourably with experimental Ce[O.sub.2] data from TEM measurements.

Key words: Bayesian; fuzzy pixel; instrumental broadening; inverse problem; maximum entropy; morphology; nanoparticles; size broadening; size distribution; x-ray line profiles.

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1. Introduction

The analysis of x-ray line profile broadening can be considered as solving a series of inverse problems. There are usually two steps--removing the instrumental contribution (deconvolution), and determining the broadening contribution in terms of crystallite size and microstrain. Here we are concerned with quantifying only the size broadening, in terms of the shape and size distributions of the crystallites. We present a method that removes the instrumental broadening and determines the particle size distribution in a single step. The general theoretical framework developed makes it possible to determine the crystallite shape and average dimensions, and to fully quantify these results by also assigning uncertainties to them.

In general, there are two approaches that can be adopted. The first assumes functional forms for the size distribution and shape of the crystallites, and applies least squares fitting to determine the parameters defining the size distribution [1,2]. For pragmatic reasons, this approach is often used to ensure numerical stability; however, it is based on an explicit assumption for the crystallite size distribution and does not take into account the non-uniqueness of the solution.

The second approach takes into account the non-uniqueness of the problem of determining the size distribution P(D) from the experimental data, by assigning a probability to the solutions and enabling an average solution to be determined from the set of solutions; moreover, it also allows any a priori information and assumptions to be included and tested. This approach is embodied in the Bayesian and maximum entropy methods [3,4,5,6]. Essentially, Bayesian theory tells us how to express and manipulate probabilities. It might be said, therefore, that Bayesian theory helps us to ask the appropriate questions, while the maximum entropy method tells us how to assign values to quantities of interest.

2. X-Ray Line Profiles

2.1 Observed Profile

The observed line profile, g(2[theta]), can be expressed as

g(2[theta]) = [integral]k(2[theta] - 2[theta]')f(2[theta]')d(2[theta]') b(2[theta]) n(2[theta]) (1)

where k(2[theta]) defines the instrument profile and considers the imperfect optics of the diffractometer; f(2[theta]) is the specimen profile, which (apart from strain effects which are not covered here) characterizes the size broadening due to microstructural properties of the specimen (i.e., crystallite shape, distribution and dimensions); b(2[theta]) and n(2[theta]) are the background level and the noise distribution, respectively. The observed profile, Eq. (1), can also be expressed in terms of reciprocal-space units, s, centered about [s.sub.0] = [2sin[[theta].sub.0]]/[lambda], as

g(s) = g(2[theta])[[d(2[theta])]/[ds]] (2)

where d(2[theta]) = [[lambda]/[cos[theta]]]ds.

The problem we face is determining the size distribution and shape of the crystallites from Eq. (1), given our knowledge of the instrument kernel, k(2[theta]), and our understanding of the counting statistics, [[sigma].sup.2]. We also want to quantify the specimen profile and size distribution by assigning error bars to them. Before addressing these questions, we review line profile broadening from nanocrystallites.

2.2 Crystallite-Size Broadening

The line profile, [I.sub.p](s, D), from a specimen consisting of crystallites of the same size and shape can be expressed in terms of the common-volume function [7] as

[I.sub.p](s, D) = 2[[integral].sub.0.sup.[tau]]V(t, D)cos2[pi]std t, (3)

where [I.sub.p](s, D) is the intensity profile given by the dimensions of the crystallite, D = {[D.sub.i]; i = 1,2,3}. The common-volume function of the crystallite, V(t, D), quantifies the volume between the crystallite and its "ghost", shifted a distance t parallel to the diffraction vector. The dimension [tau] represents the maximum length of the crystallite in the direction of the diffraction vector, and can be expressed in terms of the dimensions of the crystallite, D, such that [tau][equivalent to][tau](D). The boundary conditions for the common-volume function are V(0, D) = [V.sub.0], where [V.sub.0] is the volume of the crystallite, and V([ or -][tau],D) = 0. Figure 1 shows a schematic diagram of a crystallite and its ghost shifted a distance t in the direction [hkl]; the shaded region represents the common volume between the crystallite and its ghost. V(t, D) is symmetrical about the origin over the range t [member of] [-[tau], [tau]]. This implies that V(t, D) is an even function over this range. A simple example is a set of spherical crystallites with diameter D, for which the common-volume is given by [7] as