The Partial Wave Sub-Amplitude Fourier Transform with Born Probability Interpretation in Crystallography

Abstract

We identify Maxwell-Schrödinger equation as the governing partial differential equation. We name the solution as unitcell function such that Born probability rule interprets electron density as modulus square of the unitcell function. The existing traditional full-amplitude Fourier transform relation with electron density as directly the inverse Fourier transform of structure factors therefore violates Born probability rule. We derive the partial wave Parse-val relation as the probability conservation law using Rayleigh expansion. We subdivide the experimental full-amplitudes into partial wave sub-amplitudes, which become amplitudes of spherical waves. The subscript on a sub-amplitude represents localization within the unit-cell and therefore spherical waves become localized wave packets. As a probability measure, the amplitude of a localized wave packet is exact and merits over the amplitude of a non-localized plane wave. We measure angle dependence as the precision angles data. We derive the partial wave sub-amplitude Fourier transform relation to determine orbital structures of individual atoms using the precision angles data. The partial wave sub-amplitude Fourier transform relation alone obeys Born probability rule and hence becomes the only acceptable Fourier transform relation in quantum crystallography.