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 (face-centred cubic solid)

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\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.

Almarza equation of state

For the face-centred cubic solid phase (Eq. 19):

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle p\left(v-v_{0}\right)/k_{B}T=3-1.807846y+11.56350y^{2}+141.6000y^{3}-2609.260y^{4}+19328.09y^{5}} ,

where Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \left.v\right.} is the volume per particle, Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle v_{0}\equiv \sigma ^{3}/{\sqrt {2}}} is the volume per particle at close packing, and $y\equiv (p\sigma ^{3}/k_{B}T)^{-1}$; with $\left.\sigma \right.$ being the hard sphere diameter.

Hall equation of state (face-centred cubic)

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

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \xi =\pi {\sqrt {2}}/6-y}

Speedy equation of state

(, Eq. 2)

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

where

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle z=(N/V)\sigma ^{3}/{\sqrt {2}}}

and (Table 1)

Crystal structure	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c}
hexagonal close packed	0.5935	0.7080	0.601
face-centred cubic	0.5921	0.7072	0.601