# 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)

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

where ${\displaystyle \alpha =(V-V_{0})/V_{0}}$ where ${\displaystyle V_{0}}$ is the volume at close packing, ${\displaystyle p}$ is the pressure, ${\displaystyle T}$ is the temperature and ${\displaystyle k_{B}}$ is the Boltzmann constant.

## Almarza equation of state

For the face-centred cubic solid phase [2] (Eq. 19):

${\displaystyle p\left(v-v_{0}\right)/k_{B}T=3-1.807846y+11.56350y^{2}+141.6000y^{3}-2609.260y^{4}+19328.09y^{5}}$,

where ${\displaystyle \left.v\right.}$ is the volume per particle, ${\displaystyle v_{0}\equiv \sigma ^{3}/{\sqrt {2}}}$ is the volume per particle at close packing, and ${\displaystyle y\equiv (p\sigma ^{3}/k_{B}T)^{-1}}$; with ${\displaystyle \left.\sigma \right.}$ being the hard sphere diameter.

## Hall equation of state (face-centred cubic)

[3] Eq. 13:

${\displaystyle z({\mathrm {solid} })-\left[(12-3\beta )/\beta \right]=2.557696+0.1253077\beta +0.1762393\beta ^{2}-1.053308\beta ^{3}+2.818621\beta ^{4}-2.921934\beta ^{5}+1.118413\beta ^{6}}$

where

${\displaystyle \beta =4(1-v_{0}/v)}$
${\displaystyle z(solid)={\frac {pV}{Nk_{B}T}}}$

## Speedy equation of state

([4], Eq. 2)

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

where

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

and (Table 1)

 Crystal structure ${\displaystyle a}$ ${\displaystyle b}$ ${\displaystyle c}$ hexagonal close packed 0.5935 0.7080 0.601 face-centred cubic 0.5921 0.7072 0.601 face-centred cubic [5] 0.620735 0.708194 0.591663

## References

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