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# 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.

2.0

f1.Intensity

1.0

scaling factor

2.0

f1.Diffusion

0.05

Diffusion coefficient, in units of A^2*THz, if energy in meV, or A^2*PHz if energy in ueV

f1.Shift

0.0

Shift in domain

Categories: FitFunctions | QuasiElastic