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EINSTEIN AND SPECIFIC HEATS 395

for all n, where the value of the constant A is irrelevant. Mathematically, this is
the forerunner of the 5-function! Today we write a(E,v) = ^ 5(E — nhv).

From Eqs. 20.2 and 20.3 we recover Eq. 20.1. This new formulation is impor-
tant because for the first time the statistical and the dynamic aspects of the problem
are clearly separated. 'Degrees of freedom must be weighed and not counted,' as
Sommerfeld put it later [SI].
In commenting on his new derivation of Eq. 20.1, Einstein remarked, 'I believe
we should not content ourselves with this result' [El]. If we must modify the the-
ory of periodically vibrating structures in order to account for the properties of
radiation, are we then not obliged to do the same for other problems in the molec-
ular theory of heat, he asked. 'In my opinion, the answer cannot be in doubt. If
Planck's theory of radiation goes to the heart of the matter, then we must also
expect to find contradictions between the present [i.e., classical] kinetic theory and
experiment in other areas of the theory of heat—contradictions that can be
resolved by following this new path. In my opinion, this expectation is actually
realized.'
Then Einstein turned to the specific heat of solids, introducing the following
model of a three-dimensional crystal lattice. The atoms on the lattice points oscil-
late independently, isotropically, harmonically, and with a single frequency v

*I do not always use the notations of the original paper.

(20.1)

by introducing a prescription that modified Boltzmann's way of counting states.

Einstein's specific heat paper begins with a new prescription for arriving at the

same result. He wrote U in the form*

The exponential factor denotes the statistical probability for the energy E. The
weight factor us contains the dynamic information about the density of states
between E and E + dE. For the case in hand (linear oscillators), <o is trivial in
the classical theory: u(E,v) = 1. This yields the equipartition result U = kT.
Einstein proposed a new form for w. Let t = hv. Then o> shall be different from
zero only when ne < E < nt + a, n = 0, 1, 2, ... 'where a is infinitely small
compared with «,' and such that

(20.2)

(20.3)