396 THE QUANTUM THEORYaround their equilibrium positions (volume changes due to heating and contri-
butions to the specific heat due to the motions of electrons within the atoms are
neglected, Einstein notes). He emphasized that one should of course not expect
rigorous answers because of all these approximations.
The First Generalization. Einstein applied Eq. 20.2 to his three-dimensional
oscillators. In thermal equilibrium, the total energy of a gram-atom of oscillators
equals 3>NU(v,T), where U is given by Eq. 20.1 and N is Avogadro's number.
Hence,(20.4)which is Einstein's specific heat formula.
The Second Generalization. For reasons of no particular interest to us now,
Einstein initially believed that his oscillating lattice points were electrically
charged ions. A few months later, he published a correction to his paper, in which
he observed that this was an unnecessary assumption [E2] (In Planck's case, the
linear oscillators had of course to be charged!). Einstein's correction freed the
quantum rules (in passing, one might say) from any specific dependence on elec-
tromagnetism.
Einstein's specific heat formula yields, first of all, the Dulong-Petit rule in the
high-temperature limit. It is also the first recorded example of a specific heat for-
mula with the property
(20.5)
As we shall see in the next section, Eq. 20.5 played an important role in the
ultimate formulation of Nernst's heat theorem.
Einstein's specific heat formula has only one parameter. The only freedom is
the choice of the frequency* v, or, equivalently, the 'Einstein temperature' TE, the
value of T for which £ = 1. As was mentioned before, Einstein compared his
formula with Weber's points for diamond. Einstein's fit can be expressed in tem-
perature units by 7^E ^ 1300 K, for which 'the points lie indeed almost on the
curve.' This high value of TE makes clear why a light and hard substance like
diamond exhibits quantum effects at room temperature (by contrast, TE ~ 70 K
for lead).
By his own account, Einstein took Weber's data from the Landolt-Bornstein
tables. He must have used the 1905 edition [L2], which would have been readily
available in the patent office. These tables do not yet contain the earlier-mentioned
results by Dewar in 1905. Apparently Einstein was not aware of these data in
1906 (although they were noted in that year by German physicists [W3]). Perhaps
that was fortunate. In any case, Dewar's value of cv ~ 0.05 for diamond refers
*In a later paper, Einstein attempted to relate this frequency to the compressibility of the material
[E3].