DiffSphere.rst

DiffSphere

Description

Summary

This fitting function models the dynamics structure factor of a particle undergoing continuous diffusion but confined to a spherical volume. According to Volino and Dianoux 1,

S(Q,E\equiv \hbar \omega) = A_{0,0}(Q\cdot R) \delta (\omega) + \frac{1}{\pi} \sum_{l=1}^{N-1} (2l+1) A_{n,l} (Q\cdot R) \frac{x_{n,l}^2 D/R^2}{[x_{n,l}^2 D/R^2]^21+\omega^2},

A_{n,l} = \frac{6x_{n,l}^2}{x_{n,l}^2-l(l+1)} [\frac{QRj_{l+1}(QR) - lj_l(QR)}{(QR)^2 - x_{n,l}^2}]^2

Because of the spherical symmetry of the problem, the structure factor is expressed in terms of the j_l(z) spherical Bessel functions. Furthermore, the requirement that no particle flux can escape the sphere leads to the following boundary condition2:

\frac{d}{dr}j_l(rx_{n,l}/R)|_{r=R}=0 \,\,\,\, \forall l

The roots of this set of equations are the numerical coefficients x_{n,l}.

The fit function DiffSphere has an elastic part, modelled by fitting function ElasticDiffSphere and an inelastic part, modelled by InelasticDiffSphere.

Properties

Order	Name	Default	Description
1	Intensity	1.0	Intensity of the peak [arbitrary units]
2	Radius	2.0	Sphere radius [Å]
3	Diffusion	0.05	Diffusion constant [Å{}^2/ps \equiv 10 \cdot (10^{-5} cm^2/s)]

Category:Fit_functions