366
DOC.
21
MOLECULAR MOTION IN
SOLIDS
1.
The
forces
binding
the
atoms to
their
positions
of
rest
are essentially
identical with
the
elastic
forces of
mechanics.
2.
The
elastic forces
operate
only
between
immediately
neighboring
atoms.
To be
sure,
the
theory
is
not
completely
determined
by
these
two
assumptions,
for
the
elementary
laws
of interaction between
immediately neighboring
atoms
can
still
be
chosen
freely
to
some
extent.
Also,
it
is
not
a
priori
clear
how
many
molecules
are
to
be
viewed
as
"immediately neighboring." However,
the
specific
choice
of
a pertinent
hypothesis changes
little in
the
results,
so
I will
again
stick with
the
simple assumptions
I
introduced
in
the abovementioned
paper.
I will also
use
the
same
notation
as
there.
In the
paper
cited I
imagined
that each
atom has 26
neighboring
atoms with which
it
interacts
elastically,
and that
all
these
atoms
may
be
viewed
as
mathematically
equivalent
with
respect
to
their
elastic effect
on
the
atom
under consideration. The
proper frequency was
calculated
in
the
following
way.
One thinks of the
26
neighboring
atoms
as being
at
rest,
while
only
the
atom
under consideration
oscillates;
the latter then
performs
an
undamped pendular
oscillation,
whose
frequency
one
calculates
(from
the
cubic
compressibility). Actually, however,
the
26
neighboring
molecules
are
not at rest,
but
oscillate
about their
equilibrium position
in
a
similar
way
as
to
the
atom
under
consideration.
Through
their
elastic
connections
with
the
atom considered,
they
influence
the
oscillations
of
the
latter,
so
that
its oscillation
amplitudes
in
the coordinate directions
are changing
all the
time,
orwhat
comes
to
the
same
thingthe
oscillation
deviates
from
a
monochromatic
oscillation.
Our first
task
is
to estimate
the
magnitude
of
this
deviation.
Let M be the molecule
considered,
whose oscillations in
the xdirection
we are
investigating;
let
x
be the
momentary
distance of the
molecule from its
rest
position;
if
M1'
is
a
neighbor
molecule of M that
is
in its rest
position,
but
is at the
moment at
the distance d
+
J1
from the
rest
position
of
M,
then
M1'
exerts
on
M
a
force of the
magnitude a(f1

x
cos
p1)
in
the direction
MM1'.
The
Xcomponent
of
this force
is
a(51

x
cos
p1)cos
p1.
If
m
is
the
mass
of
M,
then
one
obtains for M the
equation
of
motion
x

Achse
X
TM1'
m
d2x
dt2
=

x
£
a
cos2^
+ 52 «
cos
p,,
Fig.
1.
where
one
has to
sum over
all
of the
26
neighboring
atoms.
Now
we
calculate
the
energy
transferred
to
the
atom from
the
neighboring
atoms
during
half
an
oscillation.
We
calculate
as
if
the
oscillation
of the molecule considered