Validity of nearest-neighbor interaction and leading-term approximation in lattice-anharmonicity calculations of monatomic bcc structures

The validity of the nearest-neighbor central-force interaction (NNCF) and the leading-term approximation (LTA) in lattice anharmonicity calculations of monatomic bcc structures is investigated. Calculations are made of the lowest-order cubic (F₃) and quartic (F₄) anharmonic contributions to the Helmholtz free energy and the specific heat at constant volume (CVA) for Na from a number of central-force two-body potentials obtained by Cochran, Price et al., and the Morse potential of Lloyd. Excellent results for CVA are obtained with exact calculations for the nearest-neighbor anharmonic interaction. However, the good agreement is destroyed when more neighbors are included in the anharmonic interaction in the exact calculation. For example, the two-neighbor anharmonic interaction produces an almost exact cancellation of F₃ and F₄ for the Cochran and Ashcroft-Singwi potentials and CVA(calc)=-CVA(expt)4 for the Morse potential of Lloyd. The inclusion of the third neighbor in the anharmonic interaction in the exact calculation gives CVA(calc)=CVA(expt)8 for the Cochran potential and CVA(calc)=-CVA(expt)8 for the Ashcroft-Singwi potential. This behavior in CVA is due to a very low value of F₄ for the exact NNCF calculation and in this case it is even negative for the Ashcroft-Singwi potential. The second neighbors make a large positive contribution to F₄ but do not change the value of F₃ dramatically from its nearest-neighbor value. The results indicate that the NNCF and LTA approximations are totally unreliable in the calculations of F₃, F₄, and CVA.