Statistical direct methods revisited. Solving the constraints

Abstract

This paper focuses on statistical methods and formulas that solve the constraints Ȓ_q² (x₁, …, x_N) = R_q² for all q. To this end we invented several non trivial techniques. We introduce a functional measure that uses the Patterson function and show that this functional measure is equivalent to using a non uniform prior distribution of the x_i given by a product of delta functions δ(Ȓ_q²(x₁, …, x_N) – R_q². We then use a representation of the delta function for calculating probabilities. In this paper we don‘t aim to give a “final” formula (this we hope to do in a future paper) but we emphasize instead on the method. We also show in this paper that SAD gives almost the same conditional j.p.d. of the phases given the magnitudes of the structure factors. Finally we show that by using plausible prior densities of the x_i the principal part of the probabilities of phases given some neighborhood does not depend anymore on the number of atoms which is a completely new and unexpected result. The Bessel functions will play a dominant role in all our formulas.