Equations of state for crystals of hard spheres: Difference between revisions

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:<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]].
==Almarza equation of state==
For the [[Building up a face centered cubic lattice |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>.


==Hall equation of state (face-centred cubic)==
==Hall equation of state (face-centred cubic)==
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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 \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==
<references/>
<references/>
{{Numeric}}
{{Numeric}}
[[category: equations of state]]
[[category: equations of state]]

Revision as of 16:04, 13 May 2009

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)

[1]

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

Almarza equation of state

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 .

Hall equation of state (face-centred cubic)

[3] Eq. 12:

where

Speedy equation of state

([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

References

This page contains numerical values and/or equations. If you intend to use ANY of the numbers or equations found in SklogWiki in any way, you MUST take them from the original published article or book, and cite the relevant source accordingly.