amep.timecor.isf

amep.timecor.isf(start, frame, k)

Incoherent intermediate scattering function (ISF).

Notes

The incoherent intermediate scattering function is defined as

\[F_{\text{s}}\left(\vec{k},t\right)=\frac{1}{N}\left\langle\sum_{j=1}^{N}\exp\left[i\vec{k}\cdot\left(\vec{r}_j(t)-\vec{r}_j(0)\right)\right]\right\rangle\]

Here, we use the isotropic ISF

\[F_{\text{s}}(k,t)=\frac{1}{N}\left\langle\sum_{j=1}^{N}\frac{\sin\left(k\cdot\left|\vec{r}_j\left(t_0+t\right)-\vec{r}_j\left(t_0\right)\right|\right)}{k\cdot\left|\vec{r}_j\left(t_0+t\right)-\vec{r}_j\left(t_0\right)\right|}\right\rangle_{t_0}.\]

that can be obtained via taking the mean over all directions of \(\vec{k}\). A detailed derivation can be found in Ref. [1].

References

[1]

C. L. Farrow and S. J. L. Billinge, Relationship between the Atomic Pair Distribution Function and Small-Angle Scattering: Implications for Modeling of Nanoparticles, Acta Crystallogr. Sect. A Found. Crystallogr. 65, 232 (2009). https://doi.org/10.1107/S0108767309009714

Parameters:

• start (np.ndarray) – Unwrapped or nojump coordinates at the start time.

• frame (np.ndarray) – Unwrapped or nojump coordinates at a later time.

• k (float) – Wave number.

Returns:

Incoherent intermediate scattering function.

Return type:

float