Equations of state for crystals of hard spheres

The stable phase of the hard sphere model at high densities is thought to have a face-centered cubic structure. A number of equations of state have been proposed for this system. The usual procedure to obtain precise equations of state is to fit computer simulation results.

• Alder, Hoover and Young equation of state (Ref 1) is given by

$\frac{pV}{N{k}_{B}T}=\frac{3}{\alpha }+2.56+0.56\alpha +O({\alpha }^2).$

where $\alpha =(V-V_0)/V_0$ where $V_0$ is the volume at close packing, $p$ is the pressure, $T$ is the temperature and $k_B$ is the Boltzmann constant.

• Hall equation of state (Ref 2) Eq. 12:

$Z_{\mathrm{s}\mathrm{o}\mathrm{l}\mathrm{i}\mathrm{d}}=\frac{1+y+y^2-0.67825y^3-y^4-0.5y^5-6.028e^{\xi (7.9-3.9\xi )}{y}^6}{1-3y+3y^2-1.04305y^3}$

where

$\xi =\pi \sqrt{2}/6-y$

• Speedy equation of state (Ref 3)

References

1. B. J. Alder, W. G. Hoover, and D. A. Young "Studies in Molecular Dynamics. V. High-Density Equation of State and Entropy for Hard Disks and Spheres", Journal of Chemical Physics 49 pp 3688- (1968)
2. Kenneth R. Hall "Another Hard-Sphere Equation of State", Journal of Chemical Physics 57 pp. 2252- (1972)
3. Robin J. Speedy "Pressure and entropy of hard-sphere crystals", Journal of Physics: Condensed Matter 10 pp. 4387-4391 (1998)