in electric conductivity of pure metals demonstrated by Kamerlingh Onnes in the neighbourhood of the absolute zero.
One of the commonest objections to Debye's theory is the arbi- trary nature of the assumption of an abrupt limit of frequency v m . This assumption is made on account of its simplicity, but is highly improbable from a physical standpoint, though it might be expected to give results of the right order of magnitude. W. Sutherland (Phil. Mae.., 20, p. 657, 1910) had previously shown that the wave-length of the elastic vibrations of solids was of the same order of magni- tude as the distance between the atoms for frequencies correspond- ing to the optical frequencies in the infra-red, so far as these were known. If the forces holding the atoms in place in a crystal lattice are electromagnetic, as commonly assumed, we should expect that the energy would be shared between matter and aether, and that the natural frequencies of the optical and mechanical vibrations would be the same. The wave-length and velocity of the natural frequency as measured outside the crystal would be reduced inside the crystal in the same proportion as the ratio of the velocity of light to that of an elastic vibration, or of the wave-length outside the crystal to the lattice constant, i.e. in the case of rocksalt, NaCl, about in the ratio 2X10 6 to I. Since the energy in the cube of the wave-length remains constant, the energy-density of the external radiation of the natural frequency would be increased in the cube of this ratio, and would be of the right order of magnitude to explain the specific heat of the solid on the usual theory of resonance, as applied by Einstein. We have seen, however, that the assumption of Planck's radiation formula gives too low a value for the specific heat at low temperatures on Einstein's theory. If on the other hand we interpret Lord Rayleigh's formula, namely C'Tc'v"-dv, as representing the partial pressure pdv of radiation between the limits of frequency v and v-\-dv, the latent heat of emission or absorption of radiation per unit volume between the same limits, according to Carnot's principle, is represented by the expression,
T(<^AfT)=CTV(i+z) . . . (4)
and the total heat of a gram-atom of solid in equilibrium with radia- tion having this distribution of energy is given by,
H- 3 RT(i +)-, . . . (5).
The specific heat as ordinarily measured, when the external pres- sure is small as compared with internal pressure, will be simply,
-', . . . (6).
This expression, unlike that similarly deduced from Planck's for- mula, gives good agreement with the observed value of the specific heat in the case of rocksalt, when the optical frequency corre- sponding to 51 microns is assumed, at a temperature corresponding to the maximum of the frequency curve, where 3 = 2-732, T = loo , and 5 = 8-67 (doubled for a gram-molecule of NaCl). We should ex- pect to get good agreement at this point, in spite of the fact that the actual vibrations in a solid cannot be strictly monochromatic (as Einstein pointed out) but extend for a distance of an octave or more on either side of the maximum, as indicated by the absorption spectrum. The effect of this is to reduce the steepness of the mono- chromatic curve, bringing it into good agreement with observation at high and low temperatures, without materially affecting the agreement at the mean point corresponding to the maximum of the frequency curve. If we assume the value X m T =0-290 for the wave-length X m (corresponding to the maximum ordinate of the wave-length spectrum of full radiation at T), in deducing the appropriate value of Wien's constant /3A in formula (4), the maxi- mum ordinate comes out the same as in Planck's formula, provided that the same value of the Stefan-Boltzmann constant a is assumed in the fourth power law 4 for the total radiation. The two curves also agree so closely throughout their whole extent that it would be very difficult to decide between them by experiments on radiation. We should therefore be justified, according to Einstein's reasoning, in applying formula (4) in the deduction of the specific heat of a solid, especially when we find that the result gives such good qualitative agreement with the optical frequencies.
An obvious objection to Debye's theory in the case of transparent substances, such as quartz and rocksalt, is that, if the atoms have all possible frequencies below a certain limit, they ought to be com- pletely opaque in this region, and to become suddenly transparent when the limit v m is surpassed. Experiment shows, however, that quartz, for instance, which begins to be opaque about four microns, and has optical frequencies corresponding to 9 and 21 microns approximately, and possibly one lower, becomes almost perfectly transparent below 100 microns. The variation of its specific heat is of an entirely different type to that given by Debye s theory but corresponds closely, according to formula (4), with its optical fre- quencies. Ice and benzol, which are also hexagonal, show a varia- tion of specific heat similar to quartz, according to Sir J. Dewar. The corresponding optical frequencies have not yet been observed, but it appears that water must have some frequencies below 100 microns to account for its remarkable opacity to long wave-lengths, and the variation of its specific heat. We should naturally expect that the" torsional vibrations of an elastic solid, which are of the same kind as those of light, would be excited by radiation, and would be intimately connected with the optical frequencies. It is quite possible, however, that the compressional vibrations, which
are of a different type, and propagated with a different velocity (that of sound), would continue to exist at low temperatures with- out affecting the transparency. These acoustic vibrations, though not capable of being excited directly by radiation, would be neces- sarily excited by the impacts of the .molecules of the surrounding gas, with a distribution of energy corresponding to the Maxwellian law, and might be expected to provide a term in the specific heat of a somewhat similar character to the Debye term for compres- sional waves at low temperatures. It is noteworthy that Nernst and Lindemann in their latest reductions have found it necessary to retain the original Einstein term /(z) for transparent substances in their formula (3), but have replaced the hypothetical term/(z/2) by a term of the Debye type. The appropriate frequencies are cal- culated in most cases by Lindemann's semi-empirical formula from the molecular weight m, the atomic volume V, and the temperature of fusion T/, but with different values of the constants for the two terms, as follows:
- m =3-o8Xio> 2 (T//m)V-
of which the first gives the optical frequency of Einstein and the second that of Debye. The cube root of the atomic volume is pro- portional to the lattice constant, and the elastic constants of a solid must be closely related to the temperature of fusion. Nernst and Lindemann assign equal importance to the two terms, but we should naturally expect from elastic theory, as given by Debye and other previous writers, that the numerical coefficients should have differ- ent values, and should be proportional to i/' 3 , for the compres- sional waves, where u is the velocity of sound, and 2/u" 3 for the tor- sional waves, where u" is the velocity of light in the solid for the particular optical frequency considered. This may not fit so well with Planck s radiation formula for the Einstein term, but appears to give better agreement with experiment if formula (4) is substi- tuted for Planck's. The appropriate frequencies cannot be calcu- lated from the elastic constants for a discontinuous medium with- out introducing arbitrary hypotheses, which are unsatisfactory, because the effect of the hypothesis selected is most important at the point where the discontinuity commences, and it is difficult to avoid selecting an hypothesis to give the desired result. There is the further difficulty that the values of the elastic constants are somewhat uncertain, and liable to vary with temperature, and to depend on the particular specimen tested, especially with metals.
Sir J. Dewar (Proc. R. S., 1913, A, 89, pp. 158-169) has measured the mean specific heats of the elements between the boiling points of hydrogen and nitrogen by means of his liquid hydrogen calori- meter. The results for the specific heats, when plotted against the atomic weights, give a curve showing a most remarkable coinci- dence with the well-known curve of atomic volume as a periodic function of the atomic weight. In other words, the specific heat is nearly proportional to the atomic volume, or to the cube of the lattice constant, for similar substances, at this low temperature, corresponding to a mean about T = 5O. The relation does not pre- tend to be exact, though it is a fair approximation over the range 20" to 80, but it illustrates the point that the atomic volume is the most important factor in determining the frequencies.
In the case of the metals, which are opaque to all frequencies below a certain limit, we should expect the possible frequencies to extend over a considerable range, and to be grouped about a mean in a similar way to the velocities of gas molecules on the kinetic theory. But there are many possible alternatives to the somewhat arbitrary hypothesis of Debye. We might suppose, for instance, that of N molecules in a gram-molecule, the number possessing the frequencies between the limits v and v-^-dv was represented by an expression of the type,
(N/2) e-' xHx . . . (8)
in which x = vjv =j8v/T =ez, where z denotes 0v/T, and = T/T . Multiplying this by expression (6) divided by N for the specific heat of a single molecule of frequency v, at a temperature T, and inte- grating the product from o to , we obtain for the specific heat of a gram-molecule,
- =3R(9V_(i+0) 3 )(i+3/(i+0) + i2/(i+) 2 ) (9)-
This is much simpler than Debye's expression, but gives a very similar curve. The mean frequency, v m =yt>, is nearly the same as Debye's limiting frequency. More accurately, Debye's character- istic temperature corresponds to 2-giTo, in place of 3To, on account of the difference in the values of the constant ft, which are in the ratio 4-9651/4-8284 in Planck's and Rayleigh's formulae for radia- tion. If Debye's scale is multiplied by 2-91, his curve agrees very closely with (9) from 9 = 0-6 to = l-o. Below 9 = 0-6, (9) agrees better with the Nernst-Lindemann curve (3), except that (9) tends to vary as T 3 at very low temperatures, instead of vanishing expo- nentially. Above = 1, the curve (9) lies above Debye's by a quan- tity corresponding to the difference of the specific heats at constant pressure and volume. This is to be expected, because (9) represents the rate of change of total heat, which is the same as that of intrin- sic energy for all practical purposes under the condition of small external pressure and negligible expansion. Thus in the case of water under atmospheric pressure, the increase of total heat between o and 100 C. is 100 cals. C., and exceeds that of intrinsic energy
