The accuracy of atomic co-ordinates derived from Fourier series in X-ray structure analysis. III

Abstract

Further work on the problems considered in the previous papers of this series has resulted in a more satisfactory treatment of finite summation errors in the three-dimensional diatomic case. The results are extended to the two- and one-dimensional series, and the interesting result emerges that finite summation errors are of the same order of magnitude whatever the dimensions of summation. Using the new results a more quantitative examination of the effects of real thermal motion becomes possible. It is shown that the relative accuracies of parameters in structures, the higher order reflexions from which are suppressed by thermal motion, follows a simple power law in the corresponding reciprocal spacings. These considerations lead to an examination of the artificial temperature factor method of securing convergence, and it is shown that this produces greater errors due to overlapping than those it is designed to eliminate. A method of correcting these distortions is suggested. Finally, the treatment of the effect of experimental errors is extended to two and one dimensions, and it is shown that the three-dimensional summation is least affected by experimental inaccuracy. The errors for three-, two- and one-dimensional summation, in a particular case, are calculated to be in the ratio 1: 3: 10.