##### Abstract

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.