# Structure of Argon Solid Phases Formed from the Liquid State at Different Isobaric Cooling Rates
## 2 Data and Methods
### The Method of Calculating the Concentration of FCC Cells  
To explain the method of quantitative calculation of structural characteristics, we present the basic concepts of solid state physics. To calculate, we need a certain unit of measurement, with the help of which it is possible to assess the degree of ordering of the arrangement of atoms. The definition of an elementary cell of a crystal lattice is suitable for this concept--the minimum cell corresponding to a single lattice point of a structure with  
Figure 1: Temperature dependence of argon density at a pressure of 4 MPa: blue line—results of modeling cooling at a rate of \(10^{8}\) K/s; red line—results of modeling cooling at a rate of \(10^{11}\) K/s; circles—reference data [19]; triangles—experimental studies [20; 21]. translational symmetry [22]. The main requirements for the construction of an elementary cell: the volume of the cell must be equal to the volume per one lattice node (the space inside the cell must be the geometric location of the points closest to the lattice node); the cells put together must occupy the entire space of the system. To construct an elementary cell, they resort to the Voronoi partitioning procedure [23; 24]: in the two-dimensional case, segments are drawn from a particle that is a lattice node to neighboring particles, which are split by perpendicular lines in the middle, the figure formed by these lines will be an elementary cell or a Voronoi polygon, in three-dimensional space, segments to the nearest neighbors are split by perpendicular planes and the geometric body formed by these planes will be an elementary cell or a Wigner-Seitz cell [22]. The particles of crystalline argon line up in the nodes of a face-centered cubic lattice [24; 25], each node of which, with a three-dimensional Voronoi partition, forms a Wigner-Seitz cell in the form of a rhombododecahedron (Figure 2a)--a 12-sided composed of identical rhombuses. The rhombodecahedron has three axes of symmetry passing through three mutually perpendicular long diagonals of the body, these axes of symmetry coincide with the translation vectors of the FCC lattice. Particle configurations (arrays of particle coordinates) obtained as a result of numerical experiments on argon cooling to a temperature of 40 K at different cooling rates and pressure values were used as input arguments for Delaunay triangulation procedures [26] and subsequent Voronoi partitioning. For this purpose, the standard Matlab R2013 software package functions Delaunay Triangulation and Voronoi [27] were used. The geometric bodies obtained at the output of the procedures were compared for compliance with the Wigner-Seitz reference cell of the FCC structure. The topological task of comparing two geometric bodies for similarity is mathematically difficult to implement, so, for example, a geometric body formed as a result of a Voronoi partition for one of the argon particles (Figure 2b) cooled at a speed of \(10^{8}\) K/s from the liquid state to a temperature of 40 K is visually similar to the reference cell, but due to fluctuation distortions in the dislocation of neighboring particles minor deviations from the standard appeared in this cell: additional edges, stretching of the faces. A programmatically implemented recognition algorithm would mark such a cell as non-conforming to the FCC, but such a cell is close to the reference both in shape and size. Figure 2: The Wigner-Seitz cell of the FCC crystal (**a**) and the geometric body obtained by spatial Voronoi partitioning for one of the particles of the argon system cooled at a rate of \(10^{8}\) K/s from the liquid state to a temperature of 40 K at a pressure of 4 MPa (**b**). Therefore, when implementing the comparative analysis procedure programmatically, the following criteria were selected: with a 5% tolerance, the equality of the volume of the cell under study and the reference cell and with a 6% tolerance, the equality of the longest diagonals.