Editing Equations of state for crystals of hard spheres
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A number of [[equations of state]] have been proposed for this system. The usual procedure to obtain precise equations of | 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 techniques | computer simulation]] results. | state is to fit [[Computer simulation techniques | computer simulation]] results. | ||
==Alder, Hoover and Young equation of state | ==Alder, Hoover and Young equation of state== | ||
<ref>[http://dx.doi.org/10.1063/1.1670653 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)]</ref> | <ref>[http://dx.doi.org/10.1063/1.1670653 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)]</ref> | ||
:<math>\frac{pV}{Nk_BT} = \frac{3}{\alpha} + 2.56 + 0.56 \alpha + O(\alpha^2).</math> | :<math>\frac{pV}{Nk_BT} = \frac{3}{\alpha} + 2.56 + 0.56 \alpha + O(\alpha^2).</math> | ||
where <math>\alpha = (V-V_0)/V_0</math> where <math>V_0</math> is the volume at close packing, <math>p</math> is the [[pressure]], <math>T</math> is the [[temperature]] and <math>k_B</math> is the [[Boltzmann constant]]. | where <math>\alpha = (V-V_0)/V_0</math> where <math>V_0</math> is the volume at close packing, <math>p</math> is the [[pressure]], <math>T</math> is the [[temperature]] and <math>k_B</math> is the [[Boltzmann constant]]. | ||
==Hall equation of state== | |||
<ref>[http://dx.doi.org/10.1063/1.1678576 Kenneth R. Hall "Another Hard-Sphere Equation of State", Journal of Chemical Physics '''57''' pp. 2252-2254 (1972)]</ref> Eq. 12: | |||
:<math>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}</math> | |||
==Hall equation of state | |||
<ref>[http://dx.doi.org/10.1063/1.1678576 Kenneth R. Hall "Another Hard-Sphere Equation of State", Journal of Chemical Physics '''57''' pp. 2252-2254 (1972)]</ref> Eq. | |||
:<math> | |||
where | where | ||
:<math>\xi = \pi \sqrt{2}/6-y</math> | |||
:<math>\ | |||
==Speedy equation of state== | ==Speedy equation of state== | ||
(<ref>[http://dx.doi.org/10.1088/0953-8984/10/20/006 Robin J. Speedy "Pressure and entropy of hard-sphere crystals", Journal of Physics: Condensed Matter '''10''' pp. 4387-4391 (1998)]</ref>, Eq. 2) | (<ref>[http://dx.doi.org/10.1088/0953-8984/10/20/006 Robin J. Speedy "Pressure and entropy of hard-sphere crystals", Journal of Physics: Condensed Matter '''10''' pp. 4387-4391 (1998)]</ref>, Eq. 2) | ||
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| face-centred cubic || 0.5921 || 0.7072 || 0.601 | | face-centred cubic || 0.5921 || 0.7072 || 0.601 | ||
|} | |} | ||
==Almarza equation of state== | |||
For the face-centred cubic solid phase <ref>[http://dx.doi.org/10.1063/1.3133328 N. G. Almarza "A cluster algorithm for Monte Carlo simulation at constant pressure", Journal of Chemical Physics '''130''' 184106 (2009)]</ref> Eq. 19: | |||
:<math>\beta p (v-v_0) = 3 - 1.807846y + 11.56350 y^2 + 141.6000y^3 - 2609.260y^4 + 19328.09 y^5</math> | |||
==References== | ==References== |