Angular and bond length distributions

The angular distribution function (ADF) and the bond length distribution function (BLDF) are the two simplest two- and three-body distributions one can build from a neighbor list. They are diagnostic fingerprints of the local environment and complement the radial distribution function \(g(r)\).

Angular distribution function

For each central atom, every pair of neighbors \((j, k)\) defines a bond angle

\[ \theta_{ijk} = \arccos \bigg( \frac{\pmb{r}_{ij} \cdot \pmb{r}_{ik}}{|\pmb{r}_{ij}|\, |\pmb{r}_{ik}|} \bigg) \]

The ADF is the histogram of \(\theta_{ijk}\) over all atoms and all neighbor pairs.

import pyscal
from ase.io import read

atoms = read('conf.dump', format='lammps-dump-text')
pyscal.find_neighbors(atoms, method='cutoff', cutoff=3.5)
hist, angles = pyscal.angular_distribution_function(atoms, bins=180)

The histogram and bin centers are also stored as atoms.info['pyscal_adf'] and atoms.info['pyscal_adf_angles'].

Bond length distribution function

The BLDF is the distribution of pair distances \(r_{ij}\) restricted to the bonds in the current neighbor list. It differs from \(g(r)\) in that it carries no shell-volume normalisation, so it more directly reflects the geometry of the chosen coordination shell.

hist, r = pyscal.bond_length_distribution(atoms, bins=100)

The histogram and bin centers are stored as atoms.info['pyscal_bldf'] and atoms.info['pyscal_bldf_r'].