12.4 Anisotropic Pair Correlation Function and Static Structure Factor 231
12.4.7 Cubic Symmetry.
When a crystal melts, its highly anisotropic structure relaxes to an isotropic state
typical for a liquid in equilibrium. For an initial state with cubic symmetry, cubic
harmonics are appropriate to characterize the anisotropic pair correlation, as e.g.
studied in [25]. The pair correlation function is expanded with respect to the cubic
harmonics with full cubic symmetry, cf. Sect.9.5.2,viz.
g(r)=g(^0 )(r)+g 4 (r)K 4 (̂r)+g 6 (r)K 6 (̂r)+g 8 (r)K 8 (̂r)+.... (12.129)The functionsK 4 ,K 6 ,K 8 are defined in (9.33)– (9.35). In general, the radially sym-
metric partg(^0 )as well as the partial correlation functionsg 4 ,g 6 ,g 8 characterizing
the anisotropy ofg(r)are functions of the timet. Examples, from [25], are shown
in Figs.12.2,12.3and12.4. The ‘data’ stem from a molecular dynamics computer
simulation. Initially the particles are put on body centered cubic (bcc) lattice sites at
a temperature and density where the system is fluid. Thusg(^0 )approaches a pair cor-
relation function typical for a dense liquid, the functionsg 4 ,g 6 ,...associated with
the anisotropy decay to zero. The simulation [25] was performed for 1024 particles
interacting with ar−^12 potential cut off appropriately, which is also referred to as
‘soft sphere’ potential. Periodic boundary conditions and a ‘thermostat’ were used.
The decay of the cubic anisotropy, in the first coordination shell, is relatively
slow compared with the approach of the thermodynamic variables, like the pressure
and the energy, to their equilibrium values in the liquid state. This slowing down
of the relaxation becomes more pronounced when the initial state is closer to the
thermodynamic state where the solid phase is stable.
Fig. 12.2 Perspective view of the isotropic part of the pair correlation function. The variabler^2
runs right-upward, the timetruns left. The initial state shows the positions of the first, second,...
coordination shells in the bcc solid