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