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.

When using InelasticDiffSphere, he value of Q can be obained either though the Q attribute or can be calucated from the input workspace using the WorkspaceIndex property. The value calculated using the workspace is used whenever the Q attibute is empty.

Attributes (non-fitting parameters)

Name Type Default Description
NumDeriv      
Q      

Properties (fitting parameters)

Name Default Description
f0.Height 1.0 Scaling factor to be applied to the resolution.
f0.Centre 0.0 Shift along the x-axis to be applied to the resolution.
f0.Radius 2.0 Sphere radius
Intensity 1.0 scaling factor
Radius 2.0 Sphere radius, in Angstroms
Diffusion 0.05 Diffusion coefficient, in units of A^2*THz, if energy in meV, or A^2*PHz if energy in ueV
Shift 0.0 Shift in domain

Categories: FitFunctions | QuasiElastic

Source

C++ source: DiffSphere.cpp

C++ header: DiffSphere.h