346 
DR. E. H. GRIFFITHS AND MR. EZER GRIFFITHS ON THE 
from which the following expression is obtained for the atomic heat in which for 
simplicity v m is inserted as v and /3 for k/k as before, 
o/dr 
^ m 
pv 
e T — 1 
The great point of difference between the theory of Einstein and that of Debye 
lies in the fact that the latter takes into account all frequencies from 0 to v m . 
The distribution of the different frequencies given by equation (14) appears to be 
capable of deduction from different standpoints, as shown by the researches* of Born 
and V. Karman. 
Callendar has drawn our attention to one important difficulty of the quantum 
theory which is mentioned by Einstein, namely, that for large values of e^ /T , but 
a small proportion of the atoms could have even one quantum. When /3i/T = 3 (as 
for iron at T = 130 C. abs., but one molecule in twenty could have a single 
quantum. Even at 100° C. it would be only one in three. Planck has 
endeavoured to avoid this by a more recent and artificial proof which has not 
received much support. 
Before proceeding to consider the other theories of specific heat, we will briefly 
compare the experimental results with the formulae based on Planck’s “ quantum ” 
theory. Debye’s equation (16) is acknowledged, for the representation of the facts 
at low temperature, to be superior to the formulae of Einstein and of Nernst and 
Lindemann, but we find that no single value of the limit frequency will make the 
formula fit the experimental results over the complete range. 
It is at very low temperatures that the assumption made by Debye, namely, that 
the vibrations whose frequency is greater than v m are negligible, can best be justified, 
since at such temperatures the slow heat vibrations would be the most important. 
Hence, in the following comparison, the value of v m is so chosen that Debye’s equation 
fits the results over the lowest portion of the range. 
It is of but little use to calculate the appropriate values of v m from the elastic 
constants of the metals, since these constants are considerably influenced by the 
nature of the previous heat treatment and by the temperature. 
The values calculated from the elastic constants are in accordance with those 
required by the atomic heat results, as wifi be shown subsequently. 
C„ = 3R 
4yVT 
5 \pvj 
* ‘ Phys. Zeitsclir.,’ vol. 14, pp. 15 and 65, 1913, 
