Liu hard disk equation of state

The Liu equation of state for hard disks (2-dimensional hard spheres) is given by Eq. 1 of .

For the stable fluid:

$Z_{v}={\frac {1+\eta ^{2}/8+\eta ^{4}/18-4\eta ^{4}/21}{(1-\eta )^{2}}}$

where the packing fraction is given by $\eta =\pi \rho \sigma ^{2}/4$ where $\sigma$ is the diameter of the disks.

The EoS for the stable fluid, liquid-hexatic transition region and hexatic:

$Z_{lh}=Z_{v}+{\frac {b_{1}\eta ^{m_{1}}+b_{2}\eta ^{m_{2}}}{(1-c\eta )}}$

The global EoS for all phases:

$Z=Z_{lh}$, $\eta <=0.72$

$Z=Z_{solid}$, $\eta >0.72$

where: $Z_{solid}={\frac {2}{\alpha }}+1.9+\alpha -5.2\alpha ^{2}+114.48\alpha ^{4}$

where Failed to parse (syntax error): {\displaystyle \alpha = \frac{2}{3^{1/3 \rho}}