Unique or essentially unique results from one-wavelength anomalous dispersion data

Abstract

An experiment that is sensitive to anomalous dispersion effects will produce at one wavelength independent intensity information at a reciprocal-lattice point and its negative. These pairs of intensities are known as Bijvoet pairs. The usual analysis of the implications of Bijvoet pairs leads to the conclusion that they generate a twofold ambiguity in the evaluation of certain phase differences. In this paper, it is shown that additional information contained in the Bijvoet pairs, and not normally used in the analysis leading to the implication of twofold ambiguity, can be used to obtain unique or essentially unique values for the phase differences of interest with potentially useful accuracy. The accuracy, of course, depends upon the accuracy of the data, but a test example has shown considerable insensitivity to such errors. The analysis presented here is based on an exact algebraic analysis of the intensity equations associated with the anomalous dispersion technique. Although the theory is quite general, applying exactly to any number or type of anomalously scattering atoms at any number of wavelengths, the application here concerns the case of one type or one predominant type of anomalously scattering atoms in a one-wavelength experiment. It is noted that in the two equations associated with the Bijvoet pairs there are three unknown quantities. It is shown, however, that the two intensity data provide enough information to evaluate the three unknown quantities to good approximation in an essentially unique fashion, which, in addition, can be effected in a least-squares calculation. The phase information of interest that is obtained concerns the values of phase differences,φ 1,h n −φ 2,h n , between phases associated with the structure of nonanomalously scattering atoms and those associated with the structure of the anomalously scattering atoms, respectively, with all atoms scattering as if there were no anomalous dispersion.