Accurate structure analysis by the maximum-entropy method

Abstract

The electron-density map of an Si crystal is drawn by the maximum-entropy method (MEM) with 30 structure factors which were determined very accurately on an absolute scale by the Pendellosung method [Saka & Kato (1986). Acta Cryst. A42, 469-478]. It is shown that the following three beneficial points arise from the use of the MEM for accurate structure analysis. Firstly, a precise electron-density map can be obtained; the existence of bonding electrons is clearly visible in the maximum-entropy map, even though no forbidden reflections are included in the analysis. The structure factors calculated from the maximum-entropy map for the 222 and the 442 forbidden reflections show good agreement with those measured by different experiments. The final R factor was 0.05% as a consequence of the high accuracy of the measured structure factors. Secondly, there is no need to seek a better structural model which could be a complicated and time-consuming process in accurate structure analysis. Thirdly, the resolution of the maximum-entropy map is much higher than that of the map drawn by conventional Fourier transformation. It is also found in the case of silicon that phase information is not absolutely necessary because exactly the same density-distribution map can be obtained without any phase information as the map drawn with all the phase information.