Averaged Steinhardt’s parameters

At high temperatures, thermal vibrations affect the atomic positions. This in turn leads to overlapping distributions of \(q_l\) parameters, which makes the identification of crystal structures difficult. To address this problem, the averaged version \(\bar{q}_l\) of Steinhardt’s parameters was introduced by Lechner and Dellago [1]. \(\bar{q}_l\) is given by,

\[\bar{q}_l (i) = \Big( \frac{4\pi}{2l+1} \sum_{m=-l}^l \Big| \frac{1}{\tilde{N}(i)} \sum_{k=0}^{\tilde{N}(i)} q_{lm}(k) \Big|^2 \Big )^{\frac{1}{2}}\]

where the sum from \(k=0\) to \(\tilde{N}(i)\) is over all the neighbors and the particle itself. The averaged parameters takes into account the first neighbor shell and also information from the neighboring atoms and thus reduces the overlap between the distributions. Commonly \(\bar{q}_4\) and \(\bar{q}_6\) are used in identification of crystal structures. Averaged versions can be calculated by setting the keyword averaged=True as follows.

sys.calculate_q([4, 6], averaged=True)
q = sys.get_qvals([4, 6], averaged=True)
[1]Lechner, W, Dellago, C, J Chem Phys, 2013.