Polylogarithmic Structure of Bragg Diffraction in Finite-Coherence Lattices

Abstract

We develop a polylogarithmic structure for Bragg diffraction based on a weighted multi-plane interference model. Within this kind of construction, the scattering amplitude is expressed as a polylogarithmic generating function. By introducing extra contributions with power-law and the usual exponential decay, it takes the form $F(θ) = \mathrm{Li}_m\left(e^{iθ_{\mathrm{eff}} - ε}\right)$, where $ε$ is a finite coherence length. In the limit where $ε\rightarrow 0$, the argument of the polylogarithm approaches the unit circle and the classical Bragg condition corresponds to the approach of the polylogarithm argument toward its branch point $z=1$. This formulation provides a compact analytical framework for describing diffraction line shapes within a generalized correlation model in which peak positions, widths, and line shapes arise from a single analytic structure. Although we are able to recover the standard Bragg law for ideal crystals, the polylogarithm model captures deviations due to finite correlation length, disorder and non-uniform lattice coherence. We show that if Bragg peaks correspond to boundary singularities of the polylogarithm, a connection between diffraction theory and complex analysis arise. The proposed theoretical model may be particularly relevant for disordered or partially coherent materials, where conventional diffraction models often require additional phenomenological broadening assumptions.