Title:

Statistical thermodynamics of crystal lattices

This work is the development of a new approach to the theory of thermodynamical equilibrium in the ideal solid state, recently put forward by Professor Max Born (1951a, 1951b). His purpose was to overcome a fundamental expansion difficulty in the usual theory and to generalise it to include strongly anharmonic effects (such as exist in solid helium); it was also hoped to provide alternative solutions of the non linear vibrational equations which might I indicate intrinsic imperfections ( "block structure"  see Born (1947), Pìzrth (1949)) in the equilibrium state. Such solutions have not been found; even in the non linear case an ideal lattice configuration can be proposed as a solution (though perhaps not the only one), with some changes as discussed by Born (1951a), and the atoms can be taken to vibrate about this reference configuration with stable, fourth degree oscillations. By a method of adaptation independent harmonic modes of vibration can be chosen to be a close approximation to the atomic motion, whatever the reference configuration, and the corresponding thermodynamical formulae may be developed either for large anharmonic effects or by treating the third and fourth degree terms in the potential energy as a small perturbation. Instead of at the start taking the atoms to vibrate about the minimum energy configuration, which leads to the fundamental difficulty that thei mean displacements increasingly diverge from these positions as the size of the specimen increases, owing to the anharmonicity, we keep the reference configuration free and fix its coordinates as the quantum mechanical average positions at a later stage of the calculation; in this way both zero  energy and temperature effects can be properly accounted for and the vibrations can always be regarded as (relatively) small compared to the atomic spacing. At the same time, new effective harmonic lattice frequencies are established which reduce in a continuous fashion to those of the ordinary quadratic theory when the vibrations are very small. The general theory will first be developed, then the theoretical example of a monatomic linear viii. chain will be worked out in full, both to illustratF the three dimensional theory and to provide approximations for later work, and finally an application of the non linear results to the thermal behaviour of solid helium will be made.
