370
DOC.
21
MOLECULAR MOTION IN
SOLIDS
I believe
that Planck's relation between
u0
and
E
should be
retained, if
for
no
other
reason
than because
it has
led
to
an
approximately
correct
description
of
specific
heats
at low
temperatures.
On the other
hand,
we
have shown in
the last
section that
assumption
(3)
cannot be
supported.
Atomic oscillations
are
not
even
approximately
harmonic.
The
frequency
region
of
an
atom
is
so
great
that the
change
of the
oscillation
energy during
a
half–
oscillation
period
is
of the
same
order of
magnitude as
the
oscillation
energy.
Thus,
we
must ascribe to each atom not
a
specific frequency,
but rather
a
frequency
range
Av
that
is
of
the
same
order of
magnitude as
the
frequency
itself.
To
derive
rigorously a
formula
for the
specific
heats of
solids,
one
would have to
carry
out,
for
an
atom
of
a
solid,
an
analysis
that
is
based
on a
mechanical
model
and
is
completely analogous
to
the
analysis
carried
out
by
Planck
for the
infinitesimally
damped
oscillator.
One
would have to
calculate
the
mean
oscillation
energy
at which
an
atom,
when
provided
with
an
electric
charge,
emits
as
much
energy
in
a
thermal
radiation
field
as
it absorbs.
[8]
While
I
was
laboring
rather
fruitlessly on
this
project,
Nernst sent
me
the
proofs
of
a
paper5
that contains
a surprisingly
useful
tentative
solution of the
problem.
He
finds
that the
expression
\
(Pv~
(~2TJ
e
/
`2
I%V
~Iv
I
~3~Ie-~~T
R
___
+
___
is
an
excellent
representation
of the
temperature dependence
of atomic heat. The
fact
[10]
that
this
expression
shows
a
better
agreement
with
experience
than the
one
I chose
originally
is
easy
to
explain
in
the
light
of
what has
been
said above.
After
all, one
obtains
this
expression by assuming
that half of the time the
atom
performs
quasi–
undamped
sinusoidal oscillations with
the
frequency v,
and
the other half of the
time
with
the
frequency
v/2.
This
is
the
manner
in which
the considerable
deviation
of the
[11]
structure
from monochromatic behavior
finds its most
primitive expression.
It
is certainly
not
justified
to
consider
v
as
the
proper frequency
of the
structure;
instead,
a
value
between
v
and
v/2
is
to be
taken
as
the
mean
proper
frequency.
Further, it
should be noted that
an
exact
coincidence between thermal
and
optical
proper
frequencies
is out
of the
question, even
if the
proper frequencies
of the different
atoms
of the
compound
in
question
closely
coincide,
for
while
the
atom oscillates with
respect
to all
neighboring
atoms in
thermal
oscillations,
it does
so
only
with
respect
to
the
[12]
neighboring
atoms with
an
opposite
sign
in
optical
oscillations.
[9]
5
W.
Nernst
and
F. A. Lindemann,
Sitzungsber.
d.
preuss.
Akad.
d. Wiss.
22
(1911):
65-90.