The above formula can only be correct if all the particles have the same proper frequency. As was mentioned above, this is not the case since a large number of lower frequencies can also occur. Strictly speaking therefore, one should substitute for the above represents the probability of a frequency between v and v+dv. The calculation of this probability is extremely complex and depends of course upon the assumptions made about the lattice and the intermolecular forces. It has been carried out in a num ber of the more straightforward cases by Born, but the expres sions are in general too complicated to be of practical use.
A simpler line of attack was developed by Debye. The slow frequencies really are nothing but the harmonics upon the fun damental frequency, i.e., the fundamental note of the solid. Each mode of vibration may be regarded as a stationary wave and the probability of any given frequency merely corresponds, on this view, to the number of ways in which one can arrange stationary waves of a given wave length in a solid. This num ber is obviously proportional to where p, q and r are whole numbers, hence the probability is proportional to the number of ways in which one can arrange a set of whole numbers so that the sum of their squares lies between given limits. It may readily be shown that this is proportional to the square of the frequency so that the probability function, according to Debye, may simply be written f(v) oc This derivation is of course accurate only for long waves and breaks down for the more numerous higher frequencies. Nevertheless, to facilitate computation and avoid complication Debye extended this for mula right up to the higher frequencies, determining his constant by the condition that the sum of all probabilities must be equal to one, i.e., the total of all the modes of vibration must amount to the total number of degrees of freedom, 3N. Thus he gave for These formulae of course only apply to monatomic substances. When one is dealing with substances in which the atoms are com bined in molecules, much more complicated conditions prevail. Strictly speaking all these problems could be attacked by the method developed by Born, but this requires a knowledge of the chemical forces as well as the cohesive force. This knowledge is at present almost entirely lacking but for practical purposes the method adopted by Nernst is adequate. He assumed that in a crystalline compound the molecules could be treated as forming a space-lattice and their contribution to the heat capa city expressed by Debye's formula. The oscillations of the atoms in the molecule he regarded as being little affected by the neigh bouring molecules and therefore applied to them Einstein's formula which corresponds to a line spectrum. By this means he was able to represent with great fidelity the molecular heats of various compounds. The large deviations found from Kopp's law are readily explained on this view; for the atomic frequencies under the comparatively large restoring forces inside the molecule, are often very high and their contribution to the specific heats, even at comparatively high temperatures, may be well below the classical value.
Like practically all substances, solids expand on heating. Their thermal expansion, however, is smaller than that of liquids and far less than that of gases. This is intelligible in view of the picture of their constitution which has been outlined. In a space lattice formed of molecules held in place by their mutual at tractions and repulsions, the primary effect of increasing the temperature is to increase the kinetic energy and consequently the amplitude of oscillation of the particles. If the restoring force is strictly proportional to the displacement, i.e., if the oscil lations are strictly harmonic, there will be no tendency to expand, for the molecules will remain equally long at either extremity of their swing and exert equal forces of attraction and repulsion upon their neighbours. This condition is of course not strictly possible in reality, for it leaves out of account the conditions at the surface. Here at any rate there must be asymmetrical oscil lations in which the molecules on their inward incursions are repelled by their neighbours, but on their outward swings are limited by the attraction of these same neighbouring molecules. As has been said, it is this very attraction which gives rise to the internal pressure in solids. It is clear that this same internal pressure limits the volume since this is determined by equating the total excess outward pressure of the oscillating particles to their mutual attractions. As has been shown, to a first approxi mation, the curve representing the force between the molecules near their positions of equilibrium, can be treated as a straight line, in other words a given deviation gives rise to just as much attraction from one side as repulsion from the other. This how ever is only a first approximation. Strictly speaking, the repul sion increases more rapidly than the attraction, otherwise the substance would not be stable and would collapse. Therefore any increase in the amplitude of the oscillations must give rise to an outward force equal to the aggregate of these differences which will lead to a general expansion of the solid. As has been stated, of course this is only a second order effect due to the nonlinearity of the function representing the restoring force, i.e., to the fact that the restoring force is not strictly elastic. If it were, that is to say if Hooke's law were strictly true and the extension of a solid were accurately proportional to the applied force, there would be no tendency to expand on heating. The extension which is observed, is due to and indeed a measure of this small devia tion from the perfect elastic properties which an idealized solid would possess and real solids at low temperatures approach. This being so it is plain that the expansion coefficient will vary with the temperature. At sufficiently low temperatures where the amplitude of oscillation is small the thermal expansion di minishes and approaches zero as the absolute zero is reached. To a first approximation it is proportional to the atomic heat.