Phase Extension and Refinement Using Convolutional and Related Equation Systems

Abstract

The mathematical solution of a set of relations among the phases of the structure factors of a crystal may be attempted either ab initio, or by starting from some partial or approximate information concerning the desired solution. In the first case one speaks of a direct method of phase determination, in the second case of a direct method of phase extension or refinement. The preceding chapters have dealt principally with the first case, although phase extension and refinement have been mentioned at some length as .components of methods whose overall purpose is ab initio phasing. Starting with this chapter, attention will shift to the second case. Mathematically the shift will involve a change in the type of relation system solved. Chapter 8 will discuss methods of phase extension and refinement based on the solution of convolutional and related systems of equations, and Chapters 9 and 10 will deal, respectively, with methods involving the maximization of Karle-Hauptman determinants and the solution of relation systems arising out of structural redundancy. The shift will involve also a change in solution technique.