Lattice Dynamics of the Solid Inert Gases

The inert gases Ne, Ar, Kr, Xe which form one of the classic types of solid, crystallize in the fee structure. An exception is helium for which the atomic mass is so small that the quantum mechanical zero-point motion prevents it from solidifying at all, unless an external pressure is imposed. For this reason, solid helium is called a quantum crystal. Since in solid helium the displacements of the atoms are very large, anharmonic effects are of prime importance. 

This is the Lennard-Jones 6-12 -potential energy which is displayed in Fig. 

The second term represents the zero-point energy which we shall neglect in the following. This is a reasonable approximation for Ar, Kr, and Xe but the approximation breaks down for Ne and especially for He. We therefore obtain 

As we desire the volume under normal pressures which have negligible effects on solids, we can put P = 0 and at T = 0, we obtain from (4.8) and (4.6) 

Substituting o/Vq from (4.11) in (4.10), the static lattice energy per particle is given by 

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