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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:

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

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 \(S(Q,E)\):

\[S(Q,E) \equiv = \int e^{-iEt/\hbar} I(Q,t) dt = A_0(Q,r) \delta (E) + \frac{1}{\pi} \sum_{l=1}^{N-1} A_l (Q,r) \frac{\hbar \tau_l^{-1}}{(\hbar \tau_l^{-1})^2+E^2}\]

\[A_l(Q,r) = \frac{1}{N} \sum_{k=1}^{N} j_0( 2 Q r \sin(\frac{\pi k}{N}) ) \cos(\frac{2\pi lk}{N})\]

\[\tau_l^{-1} = 4 \tau^{-1} \sin^2(\frac{\pi l}{N})\]

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:

\[S(Q,E) = A_0(Q,r) \delta (E) + \frac{2}{\pi} A_1 (Q,r) \frac{3 \hbar \tau^{-1}}{(3 \hbar \tau^{-1})^2+E^2}\]

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

\[S(Q,E) = A \delta (\omega) + \frac{B}{\pi} \left( \frac{\frac{\Gamma}{2}}{(\frac{\Gamma}{2})^2 + (\hbar\omega)^2}\right)\]

Then:

\[B = \frac{1}{\pi}h A_1\]

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

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 (last modified: 2021-03-31)

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