The likelihood function in fiber diffraction

Abstract

The likelihood function is an appropriate target function for refinement of molecular structures using fiber diffraction data. However, its practical application to fiber diffraction faces two significant obstacles: (i) the intensities of layer lines in a fiber diffraction pattern usually arise from the superposition of several terms, each equivalent to a crystallographic structure factor, thereby making the calculation significantly more complex than for the crystallographic case; (ii) to describe a molecular structure at the atomic level based on fiber diffraction data, the radial and phase parts of the atomic coordinates must be treated separately owing to the uniaxial symmetry of the structure. These issues are addressed here in order to derive equations of likelihood functions for fiber diffraction. The special case of a single term on a layer line is treated first followed by extension of the method to the multiterm case. A practical difficulty in implementation of likelihood for the multiterm case is that each term has a different variance. An analytical technique is described that allows the conversion of the unequal-variance case to an equal-variance case. This makes it possible to express the likelihood by an explicit formula, allowing a direct implementation of the likelihood calculation. A cylindrically symmetric model is proposed for error distribution of the atomic coordinates in a helical structure. Variances and offset coefficients of the contributing terms in the likelihood functions are expressed in terms of the variance of the atomic coordinates in the cylindrical reference system.