Helmholtz free energy of an anharmonic crystal to<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>O</mml:mi><mml:mo>(</mml:mo><mml:mrow><mml:msup><mml:mrow><mml:mi>λ</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:msup></mml:mrow><mml:mo>)</mml:mo></mml:math>. II

Abstract

Numerical calculations of the Helmholtz free energy $F$ to $O({\ensuremath{\lambda}}^{2})$ and $O({\ensuremath{\lambda}}^{4})$ from all the diagrams have been carried out in the high-temperature limit without making the leading-term approximation for a monatomic face-centered cubic crystal with nearest-neighbor central-force interactions. The numbers obtained for some diagrams and the total $F({\ensuremath{\lambda}}^{4})$ can differ by as much as 47 and 33%, respectively from those obtained in the leading term approximation, indicating that this approximation is not very good as far as absolute magnitudes are concerned. However, the ratio $\frac{F({\ensuremath{\lambda}}^{4})}{F({\ensuremath{\lambda}}^{2})}$ is nearly the same as in the leading-term approximation, indicating that the convergence of the perturbation expansion is satisfactory up to one third of the melting temperature. Finally, the improved-self-consistent (ISC) scheme of selecting the most important diagrams is probably as good as doing perturbation theory to order ${\ensuremath{\lambda}}^{4}$.