\(\renewcommand\AA{\unicode{x212B}}\)

DiffSphere¶

Description¶

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 modeled by fitting function ElasticDiffSphere, and an inelastic part modeled by InelasticDiffSphere.

Attributes (non-fitting parameters)¶

Name	Type	Default	Description
NumDeriv
Q
f0.Q
f0.WorkspaceIndex
f1.Q
f1.WorkspaceIndex

\(NumDeriv\) (boolean, default=true) carry out numerical derivative - \(Q\) (double, default=1.0) Momentum transfer

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
f1.Intensity	1.0	scaling factor
f1.Radius	2.0	Sphere radius, in Angstroms
f1.Diffusion	0.05	Diffusion coefficient, in units of A^2THz, if energy in meV, or A^2PHz if energy in ueV
f1.Shift	0.0	Shift in domain

Categories: FitFunctions | QuasiElastic

Source¶

C++ header: DiffSphere.h (last modified: 2021-03-31)

C++ source: DiffSphere.cpp (last modified: 2021-05-10)