.. index:: DiffRotDiscreteCircle
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 :ref:`ElasticDiffRotDiscreteCircle <func-ElasticDiffRotDiscreteCircle>` and :ref:`InelasticDiffRotDiscreteCircle <func-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.
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::
N (integer, default=3) number of sites - NumDeriv (boolean, default=true) carry out numerical derivative - Q (double, default=0.5) Momentum transfer .. properties:
.. categories::
.. sourcelink::