$$

DiffRotDiscreteCircle

Description

Summary

This fitting function models the dynamics structure factor of a particle undergoing discrete jumps on N-sites evenly distributed in a circle. The particle can only jump to neighboring sites. This is the most common type of discrete rotational diffusion in a circle.

Markov model for jumps between neighboring sites:

$$\frac{d}{dt}{p}_j(t)=\frac{1}{\tau }[p_{j-1}(t)-2p_j(t)+p_{j+1}(t)]$$

The Decay fitting parameter $\tau$ is the inverse of the transition rate. This, along with the circle radius $r$, conform the two fundamental fitting parameters of the structure factor $S(Q,E)$:

$$S(Q,E)\equiv =\int e^{-iEt/\hslash }I(Q,t)dt=A_0(Q,r)\delta (E)+\frac{1}{\pi }\sum _{l=1}^{N-1}{A}_l(Q,r)\frac{\hslash {\tau }_l^{-1}}{(\hslash {\tau }_l^{-1}{)}^2+E^2}$$

$$A_l(Q,r)=\frac{1}{N}\sum _{k=1}^N{j}_0(2Qr\sin (\frac{\pi k}{N}))\cos (\frac{2\pi lk}{N})$$

$${\tau }_l^{-1}=4{\tau }^{-1}{\sin }^2(\frac{\pi l}{N})$$

The transition rate, expressed in units of energy is $h{\tau }^{-1}$, with h = 4.135665616 meV ps.

This function is a composite of ElasticDiffRotDiscreteCircle and InelasticDiffRotDiscreteCircle.

When using DiffRotDiscreteCircle, the value of Q can be obtained either though the Q attribute or can be calculated from the input workspace using the WorkspaceIndex property. The value calculated using the workspace is used whenever the Q attribute is empty.

Example: Methyl Rotations

Methyl Rotations can be modelled setting N=3. In this case, the inelastic part reduces to a single Lorentzian:

$$S(Q,E)=A_0(Q,r)\delta (E)+\frac{2}{\pi }{A}_1(Q,r)\frac{3\hslash {\tau }^{-1}}{(3\hslash {\tau }^{-1}{)}^2+E^2}$$

If, alternatively, one models these dynamics using the Lorentzian function provided in Mantid:

$$S(Q,E)=A\delta (\omega )+\frac{B}{\pi }\left(\frac{\frac{\mathrm{\Gamma }}{2}}{(\frac{\mathrm{\Gamma }}{2}{)}^2+(\hslash \omega {)}^2}\right)$$

Then:

$$B=\frac{1}{\pi }h{A}_1$$

$$\mathrm{\Gamma }=\frac{3}{\pi }h{\tau }^{-1}=3.949269754meV\cdot THz\cdot {\tau }^{-1}$$

Attributes (non-fitting parameters)

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

$N$ (integer, default=3) number of sites - $NumDeriv$ (boolean, default=true) carry out numerical derivative - $Q$ (double, default=0.5) Momentum transfer .. properties:

.. categories::

Source

C++ header: DiffRotDiscreteCircle.h

C++ source: DiffRotDiscreteCircle.cpp