DOC.
21
MOLECULAR MOTION
IN SOLIDS
371
§ 3.
Dimensional
Argument Concerning
Lindemann's Formula
and
My
Formula
for
the Determination of
the
Proper Frequency
As
we
all
know,
dimensional
arguments
allow
us chiefly
to
find
general
functional
[13]
relations between
physical
quantities
if
all
physical
quantities occurring
in
the relation
in
question are
known.
For
example,
if
we
know
that the
oscillation
period
0
of
a
mathematical
pendulum can
only
depend on
the
length
of the
pendulum
l,
on
the
acceleration of free
fall
g,
on
the
mass
of the
pendulum m,
and
on no
other
quantity,
then
a simple
dimensional
argument
will
lead
us
to
the
conclusion
that the relation
must
be
given by
the
equation
0
=
C-
1
N
8
where
C
is
a
dimensionless number.
But
as
we
know,
there
is
still
something
more
that
can
be inferred
from
the dimensional
argument, even though
not in
a completely
rigorous
way.
Namely,
dimensional numerical
factors
(as
the factor
C in this
instance),
the
magnitude
of
which
can
only
be deduced
by means
of
a more or
less
detailed mathemati-
cal
theory,
are
generally
on
the order of
magnitude
one.
To be
sure,
this cannot
be
strictly
required,
because
why
should it
not be
possible
for
a
numerical factor
(12
it)3
to
appear
in
a mathematical-physical analysis?
But
such
cases are unquestionably rare.
Suppose,
therefore,
that
we
had measured the
oscillation
period
0
and the
pendulum
length
l of
an
individual
mathematical
pendulum
and that the
above
formula had
yielded us
1010
as
the
value
of the
constant
C;
in
that
case we
would
already
look
upon
our
formula
with
justified
suspicion. Conversely, our
trust would
grow
if
we
found
from
our experimental
data that
C
is,
say,
6.3; our
basic
assumption
that the
relation
sought
contains
only
the
quantities 0,
l,
and
g,
but
no
other
quantities,
would
gain
in
probability
in
our
eyes.
Let
us now
seek
to
determine the
proper frequency
v
of
an
atom
of
a
solid
by
means
of
a
dimensional
argument.
The
simplest possibility
would
evidently
be that the
oscillation
mechanism
is
determined
by
the
following
quantities:
1.
The
mass
m
of
an
atom
(dimension m);
2.
The
distance
d
between
two
neighboring
atoms
(dimension
l);
3.
The forces
with which
the
neighboring
atoms
oppose
a
change
in
their
distance
from each other. These
forces also
manifest themselves
in elastic
deformations;
their
magnitude is
measured
by
the
compressibility
coefficient
k
(dimension
lt2/m).
The
only expression
for
v
that
consists
of these three
quantities
and
has
the
right
dimension
is