ON THE SINGLE-PIXEL APPROXIMATION IN MAXIMUM-ENTROPY ANALYSIS

By a recent development of the maximum-entropy method (MEM) following Sakata & Sate [Acta Cryst. (1990), A46, 263-270], electron- (or nuclea r-) density distributions have been obtained for crystalline materials of simple structures from single-crystal or powder diffraction data. In order to obtain a ME density map, the ME equation is solved iterati vely under the zeroth-order single-pixel approximation (ZSPA) starting from the uniform density. The purpose of this paper is to examine the validity of the ZSPA by using a one-dimensional two-pixel model for w hich the exact solution can be analytically obtained. For this model, it is also possible to solve the ME equation numerically without ZSPA by the same iterative procedure as in the case of ZSPA. By comparison of these three solutions for a one-dimensional two-pixel model, it is found that the solutions obtained iteratively both with and without ZS PA always converge to the exact solution so long as the value of the L agrange undetermined multiplier, lambda, is chosen to be sufficiently small. This means the ZSPA solution does not depend on lambda when the convergence is attained. When lambda exceeds a critical value, iterat ion with ZSPA gives oscillatory divergence but iteration without ZSPA converges to a different value from the exact solution. It is conclude d that the introduction of ZSPA does not cause any serious problem in the solution of the ME equation, when a sufficiently small lambda valu e is used in the ME analysis.