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

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(Added Hall equation of state)
(Added Speedy equation of state)
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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 (Ref 1) is given by
==Alder, Hoover and Young equation of state==
(Ref 1)
:<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 2) Eq. 12:
==Hall equation of state==
(Ref 2) 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>
:<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>\xi = \pi \sqrt{2}/6-y</math>
* Speedy equation of state (Ref 3)
==Speedy equation of state==
(Ref 3) Eq. 2
:<math>\frac{pV}{Nk_BT} = \frac{3}{1-z} -\frac{a(z-b)}{(z-c)}</math>
where
:{| border="1"
|-
| Crystal structure || <math>a</math> || <math>b</math> || <math>c</math>
|-
| hcp || 0.5935 || 0.7080 || 0.601
|-
| fcc || 0.5921 || 0.7072 || 0.601
|}
==References==
==References==
#[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- (1968)]
#[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)]
#[http://dx.doi.org/10.1063/1.1678576  Kenneth R. Hall "Another Hard-Sphere Equation of State", Journal of Chemical Physics  '''57''' pp. 2252- (1972)]
#[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)]
#[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)]
#[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)]
[[category: equations of state]]
[[category: equations of state]]

Revision as of 14:19, 12 December 2008

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

(Ref 1)

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

Hall equation of state

(Ref 2) Eq. 12:

where

Speedy equation of state

(Ref 3) Eq. 2

where

Crystal structure
hcp 0.5935 0.7080 0.601
fcc 0.5921 0.7072 0.601

References

  1. 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)
  2. Kenneth R. Hall "Another Hard-Sphere Equation of State", Journal of Chemical Physics 57 pp. 2252-2254 (1972)
  3. Robin J. Speedy "Pressure and entropy of hard-sphere crystals", Journal of Physics: Condensed Matter 10 pp. 4387-4391 (1998)