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

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==Hall equation of state (face-centred cubic)==
==Hall equation of state (face-centred cubic)==
<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:
<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. 13:
:<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}}) - \left[ (12-3\beta)/\beta \right]= 2.557696 + 0.1253077 \beta + 0.1762393 \beta^2 -  
1.053308 \beta^3 + 2.818621 \beta^4 - 2.921934 \beta^5 + 1.118413 \beta^6</math>
 
where
where
:<math>\xi = \pi \sqrt{2}/6-y</math>
 
:<math>\beta = 4(1-v_0/v)</math>
:<math>z(solid)=\frac{pV}{Nk_BT}</math>


==Speedy equation of state==
==Speedy equation of state==
Line 33: Line 37:
|-  
|-  
| face-centred cubic || 0.5921 || 0.7072 || 0.601
| face-centred cubic || 0.5921 || 0.7072 || 0.601
|-
| face-centred cubic <ref>[http://dx.doi.org/10.1063/1.3328823 Marcus N. Bannerman, Leo Lue, and Leslie V. Woodcock "Thermodynamic pressures for hard spheres and closed-virial equation-of-state", Journal of Chemical Physics '''132''' 084507 (2010)]</ref> || 0.620735 || 0.708194 || 0.591663
|}
|}
==References==
==References==
<references/>
<references/>
{{Numeric}}
{{Numeric}}
[[category: equations of state]]
[[category: equations of state]]

Latest revision as of 08:03, 23 February 2017

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)[edit]

[1]

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

Almarza equation of state[edit]

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 ; with being the hard sphere diameter.

Hall equation of state (face-centred cubic)[edit]

[3] Eq. 13:

where

Speedy equation of state[edit]

([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
face-centred cubic [5] 0.620735 0.708194 0.591663

References[edit]

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.