DOC.
26
THE PROBLEM OF SPECIFIC HEATS
407
frequencies or, more
generally,
the
oscillations
of the
atoms
about their
equilibrium
positions, are essentially
identical
to
the forces
opposing
the deformation of
solids.
Motivated
by
this
idea,
Madelung6
and
I7
have
tried
to make
an approximate
calculation
of these
proper frequencies
from
the
elastic
constants,
with
Madelung turning
his
attention
to
optical
proper frequencies
of
simple
compounds,
while I
turned
my
attention
to
those
proper frequencies
that determine the
specific
heat. The
following
is probably
[12]
the
most
primitive
model
on
which
one can
base the
calculation.
Starting
out
from
a
representation
in which
the
atoms
are
arranged
in
a
cubic
lattice, one comes
up
with
a
picture
in which each atom has 26
neighboring
atoms,
all
of
which
are
located
at
approximately
the
same
distance
d from
it.
Let
each
change A
of
this distance
d
be
opposed
by
a
force
a
A,
where the
constant
a
determines the
degree
of
rigidity
of
this
model
body.
The
compressibility
k of
this
model
body,
as
well
as
the
proper frequency
of the
atom
v, can
then be
expressed
as a
function of
a.
We then obtain the latter
frequency
by
keeping
the
26
neighboring
atoms
in
their
rest
position,
while
the
atom
under consideration
is supposed
to oscillate.
Eliminating
the
auxiliary
variable
a
from
these two
relations,
we
obtain the
following
relation
between
v
and
k:
(5)
=
A
=
1.08
1O~M~p~k~
where
c
denotes the
velocity
of
light
in
vacuum,
A,
the
wavelength
in
vacuum
that
corresponds
to
v,
M the
gram-atomic weight,
and
p
the
density.
Using
this
formula,
I
obtained
for silver
A
•
104
=
73,
whereas Nernst obtained
A
•
104
=
90
from the
specific
heat.
Since this
good
agreement
in
the order of
magnitude is hardly
a
matter
of
chance,
the essential
identity
of the
forces
determining
the
degree
of
rigidity
and those
determining
the thermal
proper frequency can
be
considered
firmly
established.
Naturally,
such
a
formula
can
give only a
rough
approximation,
because it
does not
take into account
the
individual
properties
of the
substance
(e.g.,
the
crystal
structure),
which
do
not
occur
in
the
formula.
The
degree
of
approximation
with which
formula
(5)
is
able
to represent
the
actual
situation
depends,
ultimately, on
the
extent to which
a
particular
body
is
characterized,
if
at
all, by
the
distance
d between
neighboring atoms,
the
mass
of
the individual
atoms,
and the
compressibility.
Insofar
as
this
is
the
case,
then
in
place
of the
compressibility,
for
example,
one can posit
some
other fundamental characteristic
as
the
defining
quantity
of the
substance,
and
derive
an expression
for the
proper frequency
by
dimensional
[13]
6 E.
Madelung, Physik.
Zeitschr.
11
(1910):
898.
7 A.
Einstein,
Ann.
d.
Phys.
(4)
34
(1911):
120.
[10]
[11]