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 THz.

When using InelasticDiffRotDiscreteCircle, he value of Q can be obained either though the Q attribute or can be calucated from the input workspace using the WorkspaceIndex property. The value calculated using the workspace is used whenever the Q attibute 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

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	1.0	Circle radius [Angstroms]
Intensity	1.0	scaling factor [arbitrary units]
Radius	1.0	Circle radius [Angstroms]
Decay	1.0	Inverse of transition rate, in nanoseconds if energy in micro-ev, or picoseconds if energy in mili-eV
Shift	0.0	Shift in domain

Categories: FitFunctions | QuasiElastic

Source¶

C++ source: DiffRotDiscreteCircle.cpp

C++ header: DiffRotDiscreteCircle.h