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<ref>[http://dx.doi.org/10.1063/1.1743957    B. J. Alder and T. E. Wainwright "Phase Transition for a Hard Sphere System", Journal of Chemical Physics '''27''' pp. 1208-1209 (1957)]</ref>, much of this work undertaken at the Los Alamos Scientific Laboratory on the world's first electronic digital computer  ENIAC <ref>[http://ftp.arl.army.mil/~mike/comphist/eniac-story.html The ENIAC Story]</ref>.
<ref>[http://dx.doi.org/10.1063/1.1743957    B. J. Alder and T. E. Wainwright "Phase Transition for a Hard Sphere System", Journal of Chemical Physics '''27''' pp. 1208-1209 (1957)]</ref>, much of this work undertaken at the Los Alamos Scientific Laboratory on the world's first electronic digital computer  ENIAC <ref>[http://ftp.arl.army.mil/~mike/comphist/eniac-story.html The ENIAC Story]</ref>.
==Liquid phase radial distribution function==
==Liquid phase radial distribution function==
The following are a series of plots of the hard sphere [[radial distribution function]] <ref>The [[total correlation function]] data was produced using the [https://old.vscht.cz/fch/software/hsmd/hspline-8-2004.zip computer code] written by [https://web.vscht.cz/~kolafaj/ Jiří Kolafa]</ref> shown for different values of the number density <math>\rho</math>.  The horizontal axis is in units of <math>\sigma</math> where <math>\sigma</math> is set to be 1. Click on image of interest to see a larger view.
The following are a series of plots of the hard sphere [[radial distribution function]] <ref>The [[total correlation function]] data was produced using the [http://www.vscht.cz/fch/software/hsmd/hspline-8-2004.zip computer code] written by [http://www.vscht.cz/fch/en/people/Jiri.Kolafa.html Jiří Kolafa]</ref> shown for different values of the number density <math>\rho</math>.  The horizontal axis is in units of <math>\sigma</math> where <math>\sigma</math> is set to be 1. Click on image of interest to see a larger view.
:{| border="1"
:{| border="1"
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| <math>\rho^*_{\mathrm {solid}}</math> || <math>\rho^*_{\mathrm {liquid}}</math> || Reference
| <math>\rho^*_{\mathrm {solid}}</math> || <math>\rho^*_{\mathrm {liquid}}</math> || Reference
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|-  
| 1.041(4)|| 0.943(4) || <ref name="HooverRee"></ref>
| 1.041|| 0.945 || <ref name="HooverRee"></ref>
|-  
|-  
| 1.0376|| 0.9391 || <ref name="FrenkelSmitBook">Daan Frenkel and Berend Smit "Understanding Molecular Simulation: From Algorithms to Applications", Second Edition (2002) (ISBN 0-12-267351-4) p. 261.</ref>
| 1.0376|| 0.9391 || <ref name="FrenkelSmitBook">Daan Frenkel and Berend Smit "Understanding Molecular Simulation: From Algorithms to Applications", Second Edition (2002) (ISBN 0-12-267351-4) p. 261.</ref>
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| 1.037 || 0.938 || <ref>[http://dx.doi.org/10.1063/1.476396 Ruslan L. Davidchack and Brian B. Laird "Simulation of the hard-sphere crystal–melt interface",  Journal of Chemical Physics '''108''' pp.  9452-9462 (1998)]</ref>
| 1.037 || 0.938 || <ref>[http://dx.doi.org/10.1063/1.476396 Ruslan L. Davidchack and Brian B. Laird "Simulation of the hard-sphere crystal–melt interface",  Journal of Chemical Physics '''108''' pp.  9452-9462 (1998)]</ref>
|-  
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| 1.033(3) || 0.935(2) || <ref name="Miguel"> [http://dx.doi.org/10.1063/1.3023062 Enrique de Miguel "Estimating errors in free energy calculations from thermodynamic integration using fitted data", Journal of Chemical Physics '''129''' 214112 (2008)]</ref>
| 1.035(3) || 0.936(2) || <ref name="Miguel"> [http://dx.doi.org/10.1063/1.3023062 Enrique de Miguel "Estimating errors in free energy calculations from thermodynamic integration using fitted data", Journal of Chemical Physics '''129''' 214112 (2008)]</ref>
|-
| 1.03715(9) || 0.93890(7) || <ref name="MoirEtAl2021"> [https://doi.org/10.1063/5.0058892 Craig Moir, Leo Lue, and Marcus N. Bannerman "Tethered-particle model: The calculation of free energies for hard-sphere systems", Journal of Chemical Physics '''155''' 064504 (2021)]</ref>
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|}
The coexistence [[pressure]] has been calculated to be
The coexistence [[pressure]] has been calculated to be
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| 11.43(17) || <ref>[http://dx.doi.org/10.1063/1.3244562 G. Odriozola "Replica exchange Monte Carlo applied to hard spheres", Journal of Chemical Physics '''131''' 144107 (2009)]</ref>
| 11.43(17) || <ref>[http://dx.doi.org/10.1063/1.3244562 G. Odriozola "Replica exchange Monte Carlo applied to hard spheres", Journal of Chemical Physics '''131''' 144107 (2009)]</ref>
|-
| 11.550(4) || <ref name="MoirEtAl2021"></ref>
|}
|}
The coexistence [[chemical potential]] has been calculated to be
The coexistence [[chemical potential]] has been calculated to be
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|-  
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| 15.980(11) || <ref name="Miguel"></ref>
| 15.980(11) || <ref name="Miguel"></ref>
|-
| 16.053(4) || <ref name="MoirEtAl2021"></ref>
|}
|}
The [[Helmholtz energy function]] (in units of <math>Nk_BT</math>) is given by  
The [[Helmholtz energy function]] (in units of <math>Nk_BT</math>) is given by  
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==Solid structure==
==Solid structure==
The [http://mathworld.wolfram.com/KeplerConjecture.html Kepler conjecture] states that the optimal packing for three dimensional spheres is either cubic or hexagonal close [[Lattice Structures | packing]], both of which have maximum densities of <math>\pi/(3 \sqrt{2}) \approx 74.048%</math><ref>[http://dx.doi.org/10.1038/26609 Neil J. A. Sloane "Kepler's conjecture confirmed", Nature '''395''' pp. 435-436 (1998)]</ref>
The [http://mathworld.wolfram.com/KeplerConjecture.html Kepler conjecture] states that the optimal packing for three dimensional spheres is either cubic or hexagonal close [[Lattice Structures | packing]], both of which have maximum densities of <math>\pi/(3 \sqrt{2}) \approx 0.74048%</math><ref>[http://dx.doi.org/10.1038/26609 Neil J. A. Sloane "Kepler's conjecture confirmed", Nature '''395''' pp. 435-436 (1998)]</ref>
<ref>[https://www.newscientist.com/article/dn26041-proof-confirmed-of-400-year-old-fruit-stacking-problem/ Jacob Aron "Proof confirmed of 400-year-old fruit-stacking problem", New Scientist daily news 12 August (2014)]</ref>
<ref>[https://www.newscientist.com/article/dn26041-proof-confirmed-of-400-year-old-fruit-stacking-problem/ Jacob Aron "Proof confirmed of 400-year-old fruit-stacking problem", New Scientist daily news 12 August (2014)]</ref>
<ref>[http://dx.doi.org/10.1103/PhysRevE.52.3632 C. F. Tejero, M. S. Ripoll, and A. Pérez "Pressure of the hard-sphere solid", Physical Review E  '''52''' pp. 3632-3636 (1995)]</ref>. However, for hard spheres at close packing the [[Building up a face centered cubic lattice |face centred cubic]] phase is the more stable
<ref>[http://dx.doi.org/10.1103/PhysRevE.52.3632 C. F. Tejero, M. S. Ripoll, and A. Pérez "Pressure of the hard-sphere solid", Physical Review E  '''52''' pp. 3632-3636 (1995)]</ref>. However, for hard spheres at close packing the [[Building up a face centered cubic lattice |face centred cubic]] phase is the more stable
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