Seventh Virial Coefficients for Hard Spheres and Hard Disks

The seventh virial coefficient B7 is expressed as a sum of modified star integrals instead of the usual Mayer star integrals. The new graphs contain both Mayer f functions and f̃(≡1+f) functions. This simplifies the calculation of B7. That is, instead of the 468 star integrals that appear in Mayer's formulation, only 171 modified star integrals now appear in the evaluation of B7. Furthermore, for D-dimensional particles with a hard core, these 171 integrals are strongly dependent on the number of dimensions D. For hard rods, hard disks, and hard spheres, respectively, at most one, 78, and 164 of these integrals give a nonvanishing contribution to B7. When Monte Carlo integration is used to evaluate these integrals using hard-sphere and hard-disk potentials we obtain the following values of B7: hard spheres, B7/(B2)6=0.0138±0.0004; hard disks, B7/(B2)6=0.1141±0.0005. For hard spheres, the truncated seven-term virial series for the pressure agrees within 10% with the results of the molecular dynamics data throughout the fluid phase. The Padé approximants to the pressure obtained from Bn(n8) data agree even better (≲5%) with the molecular-dynamics data on the fluid side. Estimated values of B8/(B2)7 obtained from the Padé approximants are 0.005 and 0.063 for hard spheres and hard disks, respectively. Furthermore, for hard disks, the Padé approximants obtained from the seven known values of Bn(n8) indicate that the virial series shows an unstable phase at ρ=ρ0. If this is in fact true, then the low-density study of the equation of state for hard disks would yield some information on the high-density side. We have also calculated the first seven coefficients Cn of the density expansion of the isothermal compressibility for hard spheres, hard disks, hard parallel cubes, and hard parallel squares. We find that these Cn alternate in sign. For two-dimensional continuum particles with a hard core, the corresponding compressibility series appear to converge faster than the usual virial series of the pressure. We also compare the present values of B7 with the various values of B7 calculated from approximate theories.