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)

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

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:

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

where

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

Speedy equation of state

(Ref 3) Eq. 2

${\frac {pV}{Nk_{B}T}}={\frac {3}{1-z}}-{\frac {a(z-b)}{(z-c)}}$

where

Crystal structure	$a$	$b$	$c$
hexagonal colse packed	0.5935	0.7080	0.601
face-centred cubic	0.5921	0.7072	0.601

and where

$z=(N/V)\sigma ^{3}/{\sqrt {2}}$

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 3688-3696 (1968)
2. Kenneth R. Hall "Another Hard-Sphere Equation of State", Journal of Chemical Physics 57 pp. 2252-2254 (1972)
3. Robin J. Speedy "Pressure and entropy of hard-sphere crystals", Journal of Physics: Condensed Matter 10 pp. 4387-4391 (1998)