Steinhardt’s parameters

Steinhardt’s bond orientational order parameters [1] are a set of parameters based on spherical harmonics to explore the local atomic environment. These parameters have been used extensively for various uses such as distinction of crystal structures, identification of solid and liquid atoms and identification of defects [2].

These parameters, which are rotationally and translationally invariant are defined by,

\[q_l (i) = \Big( \frac{4\pi}{2l+1} \sum_{m=-l}^l | q_{lm}(i) |^2 \Big )^{\frac{1}{2}}\]


\[q_{lm} (i) = \frac{1}{N(i)} \sum_{j=1}^{N(i)} Y_{lm}(\pmb{r}_{ij})\]

in which \(Y_{lm}\) are the spherical harmonics and \(N(i)\) is the number of neighbours of particle \(i\), \(\pmb{r}_{ij}\) is the vector connecting particles \(i\) and \(j\), and \(l\) and \(m\) are both intergers with \(m \in [-l,+l]\). Various parameters have found specific uses, such as \(q_2\) and \(q_6\) for identification of crystallinity, \(q_6\) for identification of solidity, and \(q_4\) and \(q_6\) for distinction of crystal structures [2]. Commonly this method uses a cutoff radius to identify the neighbors of an atom. The cutoff can be chosen based on different methods available. Once the cutoff is chosen and neighbors are calculated, the calculation of Steinhardt’s parameters is straightforward.

sys.calculate_q([4, 6])
q = sys.get_qvals([4, 6])
[1]Steinhardt, PJ, Nelson, DR, Ronchetti, M. Phys Rev B 28, 1983.
[2](1, 2) Mickel, W, Kapfer, SC, Schroder-Turk, GE, Mecke, K, J Chem Phys 138, 2013.