Size/strain analysis of line broadening necessarily begins with the removal of instrumental contributions, k, from the measured profile, g, in the presence of noise, n, to find the specimen function, f. This process is known as the ``inverse problem'', and requires the solution over range [a,b] of the equation
\begin{displaymath} g(x) = \int_{a}^{b} k(x,\xi)f(\xi)d\xi + n(x) \end{displaymath}
Traditional approaches to its solution, such as that of Stokes, often give rise to instability in the presence of noise [1]. We have addressed this difficulty by developing a maximum entropy (MaxEnt) method following the general algorithm of Skilling and Bryan [2].
The MaxEnt procedure is based on the probability formulations of Bayes and Laplace dating back to the 18th century. It is applicable to a large class of problems in which inferences are to be made about positive and additive distributions, such as the charge density of electrons in a crystal. The entropy, S, of a probability distribution is said to be a measure of its information content, defined by Shannon [3] for a discrete set of normalized probabilities pi as
\begin{displaymath} S = -\Sigma p_{i} logp_{i} \end{displaymath}
Jaynes [4] was the first to use S for assigning probability distributions subject to certain types of constraint; since then, the method has experienced a surge of popularity [5]. Its application to x-ray data is complicated by the fact that x-ray lines measure the Fourier intensities of the electron densities of the specimen but say nothing about phase, meaning that there is no certainty of uniqueness in the solution.
In this paper we discuss our implementation of the MaxEnt method, illustrating it in conjunction with the Warren-Averbach method applied to the Size/Strain 95 round robin specimens, and also to MgO powders prepared in a range of particle sizes.
1. Kalceff, W., Armstrong, N. and Cline, J.P.:
Adv. in X-ray Analysis (1995), 38 (in press).
2. Skilling, J. and Bryan, R.K.: Mon. Not. R.
Astr. Soc., 211,(1984), 111-124.
3. Shannon, C.E.: Bell Syst. Tech., 27, (1948)
379-423 and 623-656.
4. Jaynes, E.T.:IEEE Trans. Syst. Sci. Cybern SSS-4,(1968),
227.
5. Skilling, J. and Gull, S.F. :Maximum Entropy and Bayesian Methods in
Inverse Problems, eds. Smith and Grandy (Kluwer Academic
Publishers (1985).