A semiempirical pair potential for molecular hydrogen and deuterium has been derived by fitting to solid state data. The potential is bounded to conform asymptotically to short- and long-range theoretical results. In the solid, many-body effects are accounted for by including a spherical form of the nonnegligible Axilrod–Muto–Teller three-body forces. The potential can be used to successfully describe isotropic properties in the solid and gas phases.
A self-consistent phonon approximation that includes odd-order anharmonic terms is found to give improvement over the usual lowest-order self-consistent phonon approximation when applied to computation of several properties of crystalline Ne and Ar. At low temperatures the improved theory joins smoothly onto the results of conventional perturbation theory.The failure of conventional lattice-dynamical perturbation theory when applied to ideal inert gas solids above -4/10 of their melting temperature (i.e., for rms amplitudes greater than -6%) has been pointed out recently. 1 At the same time the self-consistent phonon theory of Born and others has become well established. 2 The lowest-order form of this theory (hereafter called SC) has recently been applied to crystalline neon and argon by Gillis, Werthamer, and Koehler. 3 Since it omits odd derivatives of the interatomic potential (which are known to give substantial contributions at finite temperatures 1 ), the lowest-order theory is inadequate. It seems natural to investigate a self-consistent theory that includes odd-order anharmonic terms before investigating other problems such as hard-core effects and higher-order self-consistent theories.In the SC, frequencies oo^s 2 and wave vectors e a (qs) are obtained by iterating the following equations:^=e a (qs)D a^) e^s) 9 {
-i*% Gothic]Here, u+ =Hou+ (n+ +|), n+ ={exp(/3a>^ j-l}"" 1 , qs qs qs 2 ' qs L ^ qs J '(p(Rj) is the two-body potential, and /3 = #/&gT. These equations were applied to crystalline Ar and Ne by Gillis, Werthamer, and Koehler 3 with some disappointing results. For Ar they found C/,-26 J deg"" 1 mole" 1 at the triple point compared with the observed value 4 of 35 J deg"" 1 mole"" 1 , and wave velocities exhibited a temperature variation quite incompatible with the observed bulk modulus. 5 [Anticipating our results (Fig. 3) we will see the SC bulk modulus in reasonable agreement with experiment. Hence, Gillis, Werthamer, and Koehler's wave velocities are not those predicted by SC] An obvious omission from F$Q are terms involving odd derivatives of
We discuss the response of a crystal in the presence of a neutron distortion in the firstorder self-consistent approximation and we stress the importance of the self-consistency condition. The resultant integral equation for the phonon energies is solved by direct matrix inversion instead of truncating a series expansion. Numerical results are presented for solid Ne and fee He 4 at 10.0 and 11.5 cmVmole.
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