Rigid Disks and Spheres at High Densities. III

A formal derivation of an asymptotic expansion for the Helmholtz free energy

F_{N}

of a system of

N

Λ‐dimensional rigid spheres of diameter

Σ

is given, which has the form,

F_{N} {/ Nk}_{B} T ∼ {lim}_{τ → 1} ν \ln (λ / σ) − ν \ln {(τ − 1) + C + D(τ − 1) + E(τ − 1)}^{2} + ···,

where

k_{B}

is Boltzmann's constant,

λ

is the mean thermal de Broglie wavelength, and

{θ = V / V}_{0}

is the reduced volume of a system of volume

V

, close‐packed volume

V_{0}

, at temperature

T

. Formal expressions for the constants

C, D

, and

E

are derived, and the cell‐cluster technique is applied to the calculation of

C

and

D

for the hexagonal‐close‐packed and face‐centered‐cubic lattices. The results are

C_{HCP} = 1.7786846···,

C_{FCC} = 1.7795003···,

D_{HCP} = 0.54187···,

D_{FCC} = 0.557994···.

A value for

D

for the two‐dimensional triangular lattice is found to be

D_{T} = 0.098993···.

Agreement with computer experiments is generally good.

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