Invariants of spherical harmonics as order parameters in liquids

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Abstract
The application of the second- and third-order invariants of the even-l spherical harmonics for the geometrical characterization of clusters in disordered systems is discussed. Their use as sensitive order parameters is useful in the geometrical analysis of computer-simulated configurations. It is shown that the second- and third-order invariants give information on the three- and four-body angular correlations, respectively. The values of the invariants calculated for a given configuration set depend on the number of particles and their neighbours considered, i.e. on the size of the statistical sample. A renormalization is suggested to eliminate this size effect. By choosing a suitable method of averaging the spherical harmonics, they can be made characteristic either of the angular correlations within individual clusters only or also of cross-correlations between a set of clusters. Thus they can be used for the detection of traces of longer-range crystalline structures. The first five even-l invariants have been calculated for configurations of liquid argon, molten alkali-metal halides, and pure water simulated by Monte Carlo or molecular-dynamic methods. The results indicate that the structure of the first coordination spheres in liquid argon are slightly distorted hexagonal close-packed clusters which have practically no angular correlation with one another. Nearest-neighbour angular correlations in sodium, potassium, rubidium and caesium chlorides correspond to more or less distorted face-centred cubic lattices, while LiI and LiCl resemble the wurtzite and sphalerite structure, respectively. In comparison with the above cases, water is disordered to the extent that even the distorted tetrahedral clusters are barely recognizable. Peculiarly distorted tetrahedra which are in a characteristic angular correlation with one another are stabilized by a decrease in the density of water.

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