Soft sphere potential

The soft sphere potential is defined as

$\Phi _{12}\left(r\right)=\left\{{\begin{array}{lll}\epsilon \left({\frac {\sigma }{r}}\right)^{n}&;&r\leq \sigma \\0&;&r>\sigma \end{array}}\right.$

where $\Phi _{12}\left(r\right)$ is the intermolecular pair potential between two soft spheres separated by a distance $r:=|\mathbf {r} _{1}-\mathbf {r} _{2}|$, $\epsilon$ is the interaction strength and $\sigma$ is the diameter of the sphere. Frequently the value of $n$ is taken to be 12, thus the model effectively becomes the high temperature limit of the Lennard-Jones model . If $n\rightarrow \infty$ one has the hard sphere model. For $n\leq 3$ no thermodynamically stable phases are found.

Equation of state

The soft-sphere equation of state has recently been studied by Tan, Schultz and Kofke and expressed in terms of Padé approximants. For $k_{B}T/\epsilon =1.0$ and Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle n=6} one has (Eq. 8):

$Z_{n=6}={\frac {1+7.432255\rho +23.854807\rho ^{2}+40.330195\rho ^{3}+34.393896\rho ^{4}+10.723480\rho ^{5}}{1+3.720037\rho +4.493218\rho ^{2}+1.554135\rho ^{3}}}$

and for $n=9$ one has (Eq. 9):

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle Z_{n=9}={\frac {1+3.098829\rho +5.188915\rho ^{2}+5.019851\rho ^{3}+2.673385\rho ^{4}+0.601529\rho ^{5}}{1+0.262771\rho +0.168052\rho ^{2}-0.010554\rho ^{3}}}}

Virial coefficients

Tan, Schultz and Kofke have calculated the virial coefficients at $k_{B}T/\epsilon =1.0$ (Table 1):

	n=12	n=9	n=6
$B_{3}$	3.79106644	4.27563423	5.55199919
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B_4}	3.52761(6)	3.43029(7)	1.44261(4)
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B_5}	2.1149(2)	1.08341(7)	-1.68834(9)
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B_6}	0.7695(2)	-0.21449(11)	1.8935(5)
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B_7}	0.0908(5)	-0.0895(7)	-1.700(3)
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B_8}	-0.074(2)	0.071(4)	0.44(2)

Melting point

For Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n=12}

pressure	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \rho_{\mathrm {melting}}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \rho_{\mathrm {freezing}}}	Reference
22.66(1)	1.195(6)	1.152(6)	Table 1
23.24(4)	1.2035(6)	1.1602(7)	Table 2

For Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n=9}

pressure	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \rho_{\mathrm {melting}}}	Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \rho _{\mathrm {freezing} }}	Reference
36.36(10)	1.4406(12)	1.4053(14)	Table 3

For Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n=6}

pressure	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \rho_{\mathrm {melting}}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \rho_{\mathrm {freezing}}}	Reference
100.1(3)	2.320(2)	2.295(2)	Table 4