# Voronoi tessellation to identify local structures¶

Voronoi tessellation can be used for identification of local structure by counting the number of faces of the Voronoi polyhedra of an atom [1][2]. For each atom a vector \(\langle n_3~n_4~n_5~n_6 \rangle\) can be calculated where \(n_3\) is the number of Voronoi faces of the associated Voronoi polyhedron with three vertices, \(n_4\) is with four vertices and so on. Each perfect crystal structure such as a signature vector, for example, bcc can be identified by \(\langle 0~6~0~8 \rangle\) and fcc can be identified using \(\langle 0~12~0~0 \rangle\). It is also a useful tool for identifying icosahedral structure which has the fingerprint \(\langle 0~0~12~0 \rangle\). In pyscal, the voronoi vector can be calculated using,

```
import pyscal.core as pc
sys = pc.System()
sys.read_inputfile('conf.dump')
sys.find_neighbors(method='voronoi')
sys.calculate_vorovector()
```

The vector for each atom can be accessed using Atom.vorovector. Furthermore, the associated Voronoi volume of the polyhedron, which may be indicative of the local structure, is also automatically calculated when finding neighbors using System.find_neighbors. This value for each atom can be accessed by Atom.volume. An averaged version of the volume, which is averaged over the neighbors of an atom can be accessed using Atom.avg_volume.

## References¶

- 1
J. L. Finney. Random Packings and the Structure of Simple Liquids. I. The Geometry of Random Close Packing.

*Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences*, 319(1539):479–493, 1970. doi:10.1098/rspa.1970.0189.- 2
Masaharu Tanemura, Yasuaki Hiwatari, Hirotsugu Matsuda, Tohru Ogawa, Naofumi Ogita, and Akira Ueda. Geometrical Analysis of Crystallization of the Soft-Core Model.

*Progress of Theoretical Physics*, 58(4):1079–1095, 1977. doi:10.1143/PTP.58.1079.