# Soft sphere potential

The soft sphere potential is defined as

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

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

## Equation of state

The soft-sphere equation of state[2] has recently been studied by Tan, Schultz and Kofke[3] [4] and expressed in terms of Padé approximants. For ${\displaystyle k_{B}T/\epsilon =1.0}$ and ${\displaystyle n=6}$ one has (Eq. 8):

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

and for ${\displaystyle n=9}$ one has (Eq. 9):

${\displaystyle Z_{n=9}={\frac {1+3.098829\rho +5.188915\rho ^{2}+5.019851\rho ^{3}+2.673385\rho ^{4}+0.601529\rho ^{5}}{1+0.262771\rho +0.168052\rho ^{2}-0.010554\rho ^{3}}}}$

## Virial coefficients

Tan, Schultz and Kofke[3] have calculated the virial coefficients at ${\displaystyle k_{B}T/\epsilon =1.0}$ (Table 1):

 n=12 n=9 n=6 ${\displaystyle B_{3}}$ 3.79106644 4.27563423 5.55199919 ${\displaystyle B_{4}}$ 3.52761(6) 3.43029(7) 1.44261(4) ${\displaystyle B_{5}}$ 2.1149(2) 1.08341(7) -1.68834(9) ${\displaystyle B_{6}}$ 0.7695(2) -0.21449(11) 1.8935(5) ${\displaystyle B_{7}}$ 0.0908(5) -0.0895(7) -1.700(3) ${\displaystyle B_{8}}$ -0.074(2) 0.071(4) 0.44(2)

## Melting point

For ${\displaystyle n=12}$

 pressure ${\displaystyle \rho _{\mathrm {melting} }}$ ${\displaystyle \rho _{\mathrm {freezing} }}$ Reference 22.66(1) 1.195(6) 1.152(6) Table 1 [5] 23.24(4) 1.2035(6) 1.1602(7) Table 2 [3]

For ${\displaystyle n=9}$

 pressure ${\displaystyle \rho _{\mathrm {melting} }}$ ${\displaystyle \rho _{\mathrm {freezing} }}$ Reference 36.36(10) 1.4406(12) 1.4053(14) Table 3 [3]

For ${\displaystyle n=6}$

 pressure ${\displaystyle \rho _{\mathrm {melting} }}$ ${\displaystyle \rho _{\mathrm {freezing} }}$ Reference 100.1(3) 2.320(2) 2.295(2) Table 4 [3]