476
DOC.
21 MOLECULAR MOTION
IN
SOLIDS
[6]See
Planck
1900a, p.
93. Einstein had earlier
presented
the
following
considerations
in
Einstein
1905i
(Vol.
2,
Doc.
14), §1.
[7]The
following argument
summarizes Einstein's
line
of
thought
in Einstein
1907a
(Vol. 2,
Doc.
38),
pp.
184-186.
[8]Einstein
continued to
work
on
this
generalization
of Planck's
analysis
of
an
oscillator
in
a
radiation
field;
see
Einstein to H. A.
Lorentz, 23
November
1911,
where
he writes:
"I
am
occupied
with the
case
of
damped resonators;
it
involves
quite some
calculations"
("Mit
dem
Fall der
gedämpften
Resonatoren bin
ich beschäftigt;
es
ist eine
ziemliche
Rechnerei").
For
further
comments by
Einstein
on
this
problem,
see
Einstein
1914 (Doc.
26), pp.
338-339.
[9]Nernst
and Lindemann
1911a,
which
was
delivered
to
the Prussian
Academy on 6 April
1911.
[10]For
a
comparison
of
the formula
by
Nernst and Lindemann with
empirical
data,
see
Nernst
and
Lindemann
1911a,
§4.
[11]Nernst
and Lindemann
gave
a
different theoretical
justification
for their
formula,
see
Nernst and
Lindemann
1911a,
§6.
They
proposed
a
modification of
the
quantum theory
accord-
ing to
which the
heating
of
a
solid
body
results
in increasing
the
potential
energy
by
quanta
that
are
halves of the usual
quanta. Before
the
publication
of the
present paper,
Einstein had
communicated
his
different
explanation
of the Nernst-Lindemann formula
to Nernst;
see
Einstein to Walther
Nernst, 20
June
1911.
Einstein's
argument
in
the remainder of
§2
is
elabo-
rated
upon in
the
note
added
in proof.
[12]Einstein
made
a
similar
point
in criticizing
Sutherland's
approach,
in
Einstein
1911b
(Doc.
13),
p.
170.
[13]Following
the
example
of
Jeans,
Einstein had earlier used dimensional considerations
in
his
exploration
of
the
consequences
of
the quantum hypothesis,
see
Einstein
1909b
(Vol. 2,
Doc.
56), p.
192.
[14]Einstein's numerical
constant
should
actually
be
larger
if
a
standard value of N
is used,
such
as
the
one
used
in
the second
equation
on p.
692.
[15]See Einstein
1911b
(Doc.
13), p.
173.
[16]v
is
derived from
A
for
copper
cited
in
Einstein 191lb
(Doc.
13),
p.
173.
[17]Nernst
and Lindemann
1911a,
p.
496,
cite
ßv
=
320
for
copper,
and Nernst and Lindemann
1911b, p. 820, give ßv
=
321, as
derived from their formula. This
yields
the value cited
by
Einstein.
[18]Lindemann
1910.
[19]See Lindemann
1910,
p.
612,
where the numerical factor
is
given as
2.06.
The factor
2.12
given
by
Einstein
is
found
in
the
paper by
Nernst cited
in note
2; see
Nernst
1911b,
p.
311.
A
similar formula
was
first found
empirically by Magnus
and
Lindemann;
see
Magnus
and
Lindemann
1910, p.
271.
[20]Nernst
1911b, p. 311,
table
I,
contains
a
favorable
comparison
of
"proper frequencies"
calculated
using
Lindemann's formula and those derived from the observed
specific
heat
curves
of
seven
substances.
[21]Lindemann
derived
his
formula from the
assumption
that
at
the
melting point
of
a
sub-
stance
the
amplitudes
of the atomic oscillations
are so
large
that
the
atoms,
more
precisely
their
spheres
of action
("Wirkungssphären"),
touch each
each;
see
Lindemann
1910,
pp.
609-
610.
[22]Grüneisen
1908,
table
17
on
p.
848.
[23]For
a
review
of later work
by
Grüneisen and others
on
the
relationship
between Einstein's
and Lindemann's dimensional formulas
as
well
as on
the
equation
of
state
for solid bodies
in
general,
see
Born
1923,
in particular
pp.
659-661.
[24]For
evidence of Einstein's intention
to
undertake
compressibility measurements,
see
Einstein
to
Michele
Besso, 21
October
1911.
[25]See Nernst
1911b, pp.
309-310,
where Nernst
writes:
"The fact
that
potassium
chlorine
behaves with
respect
to
the
dependence
of
its
atomic heat
exactly
like
an
element whose atoms
are
bound in the
same way
would
indicate,
according
to
Einstein's
view,
that
both
atoms
possess only slightly
different
proper frequencies,
which
seems entirely plausible
in
this
case,
because
according to
Lindemann's
formula, elementary
chlorine and
potassium
do
not
have