Abstract:
A general derivation of the anharmonic coefficients for
a periodic lattice invoking the special case of the central
force interaction is presented. All of the contributions to
mean square displacement (MSD) to order 14 perturbation theory
are enumerated. A direct correspondance is found between the
high temperature limit MSD and high temperature limit free
energy contributions up to and including 0(14). This
correspondance follows from the detailed derivation of some of
the contributions to MSD. Numerical results are obtained for
all the MSD contributions to 0(14) using the Lennard-Jones
potential for the lattice constants and temperatures for which
the Monte Carlo results were calculated by Heiser, Shukla and
Cowley. The Peierls approximation is also employed in order
to simplify the numerical evaluation of the MSD contributions.
The numerical results indicate the convergence of the
perturbation expansion up to 75% of the melting temperature of
the solid (TM) for the exact calculation; however, a better
agreement with the Monte Carlo results is not obtained when
the total of all 14 contributions is added to the 12
perturbation theory results. Using Peierls approximation the
expansion converges up to 45% of TM• The MSD contributions
arising in the Green's function method of Shukla and Hubschle
are derived and enumerated up to and including 0(18). The
total MSD from these selected contributions is in excellent agreement with their results at all temperatures. Theoretical
values of the recoilless fraction for krypton are calculated
from the MSD contributions for both the Lennard-Jones and Aziz
potentials. The agreement with experimental values is quite
good.
