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):

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 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 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} one has (Eq. 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 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 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 k_BT/\epsilon=1.0} (Table 1):

	n=12	n=9	n=6
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_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)
$B_{5}$	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	$\rho _{\mathrm {melting} }$	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 (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
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