DOC.
21
MOLECULAR MOTION
IN
SOLIDS
367
as
well
as
the
neighboring molecules,
proceeds
sinusoidally
during
the
half-oscillation
period, i.e., we
set
x
-
A sin
2n
vt,
£1
=
A/
sin
(2itvt
+
aj).
........
Multiplying
the
above
equation
by (dx/dt)dt
and
integrating
over
the time
indicated,
we
obtain the
expression
for the
change
of the
energy,
J
d
ji»^
+
£
(a
cos2p)-^j
=
Y,
a
008*
J
If
we
denote
by A
the total
energy
increase of the
atom,
and
by
n1, n2,
etc.,
the
amounts
of
energy
transferred
to
the
atom
from the
individual
neighboring
atoms
during
a
half-oscillation
period,
we can
write this
equation
in
the
form
A
=
£tln,
where
we
set
ri
=
a
cosp"Jf$"dt-dt.
n
With the
above
conventions for
x,
$1
...,
we
obtain
il"
=
|a
cos
p"
sin
an
A
An'.
From
this it follows
that the
individual
quantities
nn
are
as likely
to
be
positive as
negative,
considering
that the
angles
an
take
on
each
value with
equal
frequency and,
indeed, independently
of each other. For that
reason we
also have
A
=
0.
Now
we
form
the
mean
value
A2 as a measure
of the
energy change.
Due
to
the indicated
statistical
property
of
n1
etc., we
have
A*=E^.
Since, as can easily
be
seen,
sin2
a.A2A'nl
=
i^2,
we
have
I
-^cos2,.
and