Editing Equations of state for crystals of hard spheres
Jump to navigation
Jump to search
The edit can be undone. Please check the comparison below to verify that this is what you want to do, and then publish the changes below to finish undoing the edit.
Latest revision | Your text | ||
Line 6: | Line 6: | ||
:<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> | |||
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) | ||
Line 37: | Line 25: | ||
|- | |- | ||
| 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 \left(v-v_0\right) = 3 - 1.807846y + 11.56350 y^2 + 141.6000y^3 - 2609.260y^4 + 19328.09 y^5</math>, | |||
where <math> \left. v \right. </math> is the volume per particle, <math> v_0 \equiv \sigma^3/\sqrt{2} </math> is the volume per particle at close packing, | |||
and <math> y \equiv ( \beta p \sigma^3)^{-1} </math>. | |||
==References== | ==References== |