As will appear below several important characteristics of solids depend entirely upon the deviations from the simple linear ex pression. If one confines oneself to the first order and considers only a quasi-elastic restoring force a and it follows that the molecules in a solid will oscillate with a definite proper frequency m being their mass. The reality is not quite so simple since the motion of one molecule will be communicated to its neighbours and the whole system will quiver with a number of different proper frequencies which depend upon the lattice structure, the force between the molecules and the molecular weight. The highest frequency, however, will be that given by the above formula for a body of mass m vibrating under the elastic restoring force a. This can be observed directly in solids which contain ions. In the case for instance of rock salt or sylvin, the crystal is built up of a cubical array of alternating alkali ions and halogen ions. The restoring forces acting on these may be derived from the compressibility, and their mass, the atomic weight divided by Avogadros number, is of course known. Hence their frequency of oscillation can be calculated; for NaC1 it is about —6 X and for KC1 about 5 x ro". If an alternating electromagnetic field, in other words, radiation of this frequency, is allowed to fall upon these crystals the ions, being in resonance, are set in motion and the radiation is reflected.
Another method of estimating molecular frequencies makes use of the definition of the melting given above, namely that the melting point is the temperature at which the amplitude of oscil lation of the molecules enables them to collide with one another. As in the case of a musical string or pendulum, the amplitude of oscillation is completely determined if the mass, frequency and energy are known. Since the energy at the melting point and the atomic mass are ascertainable, the frequency can be derived if the free space between the molecules is known. Conversely if the frequency is known the amplitude of oscillation at the melting point, in other words the distance between the surfaces of the molecules, can be calculated. It appears to be much the same fraction in all monatomic substances, o% to 12% of the distance between the centres of neighbouring atoms.
The frequencies are of importance, since it has been shown by Einstein following Planck, that they determine the course of the specific heats of substances at low temperatures. The classical statistical mechanics of Boltzmann, showed that the mean kinetic energy resolved along any one axis of particles capable of motion and capable of interchanging momentum, averaged over the time, should be equal provided the particles are in thermal equi librium, i.e., at the same temperature. Hence the molecules in a solid, since they are capable of oscillation along each of three axes, just as the molecules in a gas are free in three dimensions, should contain the same amount of kinetic energy as a monatomic gas, i.e., RT. In a solid, as we have seen the molecules sions are restrained by the repulsion of their neighbours. During part of their oscillations therefore, the kinetic energy is converted into potential energy. If the restoring force is proportional to the displacement the oscillations are harmonic and it may be shown that the potential energy averaged over a long time is equal to the kinetic energy. Hence the atomic heat of a solid at con
stant volume, should be exactly twice that of a monatomic gas at constant volume, which latter is 3 R. This result is of course in good agreement with Dulong Petit's and Kopp's law and was at one time held to be an important confirmation of these empyri cal approximations. Ordinary measurements of atomic heats of solids are of course carried out at constant pressure and not at constant volume, but a simple thermal dynamical consideration enables one to show that the difference cp—cv= T where 13 is the thermal expansion coefficient, K the compressibility and T the temperature.
Without making too extended an incursion into that large body of hypotheses and doctrine usually grouped under the heading Quantum Theory (q.v.), it is impossible to give an accurate der ivation of the formula which represents the atomic heat as a function of the temperature. Roughly, one may summarize the quantum condition by saying that the oscillating system at any given time can only be in one of a number of discrete states, these states corresponding to energy levels, differing from one another by one quantum of energy. If the classical statistical mechanics were true, every energy level would be possible and the prob ability of any one corresponding to an energy e would be given by being Boltzmann's constant and T the temperature. The sum of all the energies of N molecules each having three degrees of freedom would thus be U=3NkT=3RT for one gramme molecule. With the quantum restriction a system of oscillating particles will contain less energy, for the energy which on the classical view would be contained by molecules having less than one quantum E, would not be present, similarly all the molecules which on the classical view would contain energies between one and two quanta, would only have one quantum, and again those molecules which should contain between two and three quanta, would have exactly two quanta. Hence the total energy instead of being given by the above expression is equal to reaches a value 3R asymptotically at temperatures high corn pared to €/k but that it falls far below this value and indeed to zero at low temperatures. It is clear that if the quantum e be comes zero one arrives at the classical result; it is its finite value which leads to the discrepancy. One of the most fundamental requirements of the quantum theory is that the quantum is pro portional to the frequency v i.e., E = hp. Hence elements with high proper frequencies have large quanta, and elements with low frequencies small quanta. But as we have seen high proper frequencies are characteristic of elements which combine a large restoring force with a small atomic weight. Hence deviations from the classical value for the atomic heat, i.e., Dulong Petit's law, are to be expected at comparatively high temperatures in elements such as carbon or boron, which combine small com pressibility or high melting point with low atomic weight. On the other hand elements of high atomic weight, which are soft and have low melting points, such as lead or mercury, have a low frequency, i.e., they obey Dulong Petit's law down to very low temperatures. That carbon and boron deviated from the normal was known to Dulong Petit, but it is only within the last 20 years that the explanation has been found.