# Equations of state for crystals of hard spheres

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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)[edit]

^{[1]}

where where is the volume at close packing, is the pressure, is the temperature and is the Boltzmann constant.

## Almarza equation of state[edit]

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

- ,

where is the volume per particle, is the volume per particle at close packing, and ; with being the hard sphere diameter.

## Hall equation of state (face-centred cubic)[edit]

^{[3]} Eq. 13:

where

## Speedy equation of state[edit]

(^{[4]}, Eq. 2)

where

and (Table 1)

Crystal structure 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[edit]

- ↑ 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) - ↑ N. G. Almarza "A cluster algorithm for Monte Carlo simulation at constant pressure", Journal of Chemical Physics
**130**184106 (2009) - ↑ Kenneth R. Hall "Another Hard-Sphere Equation of State", Journal of Chemical Physics
**57**pp. 2252-2254 (1972) - ↑ Robin J. Speedy "Pressure and entropy of hard-sphere crystals", Journal of Physics: Condensed Matter
**10**pp. 4387-4391 (1998) - ↑ Marcus N. Bannerman, Leo Lue, and Leslie V. Woodcock "Thermodynamic pressures for hard spheres and closed-virial equation-of-state", Journal of Chemical Physics
**132**084507 (2010)