THE QUANTUM THEORY
THE
QUANTUM THEORY
BY
FRITZ REICHE
PROFESSOR OF PHYSICS IN THE UNIVERSITY OF BRESLAU
TRANSLATED BY H. S. HATFIELD, B.Sc., Ph.D.. AND
HENRY L. BROSE. M.A.
WITH FIFTEEN DIAGRAMS
NEW YORK
E. P. BUTTON AND COMPANY
PUBLISHERS
66038
PRINTED IN GREAT BRITAIN BY
THE ABERDEEN UNIVERSITY PRESS
c
o
CONTENTS
^ CHAP. PAGE
^ INTRODUCTION 1
j, I. THE ORIGIN OF THE QUANTUM HYPOTHESIS .... 2
k\ II. THE FAILURE OF CLASSICAL STATISTICS .... 13
III. THE DEVELOPMENT AND THE RAMIFICATIONS OF THE QUANTUM
THEORY. * 16
IV. THE EXTENSION OF THE DOCTRINE OF QUANTA TO THE Mo-
^ LECULAR THEORY OF SOLID BODIES .... 29
V. THE INTRUSION OF QUANTA INTO THE THEORY OF GASES . 68
VI. THE QUANTUM THEORY OF THE OPTICAL SERIES. THE DE-
VELOPMENT OF THE QUANTUM THEORY FOR SEVERAL
DEGREES OF FREEDOM 84
VII. THE QUANTUM THEORY OF RONTGEN SPECTRA . . . 109
VIII. PHENOMENA OF MOLECULAR MODELS i ..... 117
) IX. THE FUTURE . . . ' 125
W MATHEMATICAL NOTES AND REFERENCES .... 127
A INDEX . . 181
THE QUANTUM THEORY
INTRODUCTION
r ~r" v HE old saying that small causes give rise to great effects
J[ has been confirmed more than once in the history of
physics. For, very frequently, inconspicuous differences be-
tween theory and experiment (which did not, however, escape
the vigilant eye of the investigator) have become starting-
points of new and important researches.
Out of the well-known Michelson-Morley experiment,
which, in spite of the application of the most powerful
methods of exact optical measurement, failed to show an
influence of the earth's movement on the propagation of
light as was predicted by classical theory, there arose the
great structure of Einstein's Theory of Kelativity. In the
same way the trifling difference between the measured and
calculated values of black-body radiation gave rise to the
Quantum Theory which, formulated by Max Planck, was
destined to revolutionise in the course of time almost all
departments of physics.
The quantum theory is yet comparatively young. It is
therefore not surprising that we are confronted with an
unfinished theory still in process of development which,
changing constantly in many directions, must often destroy
what it has built up a short time before. But under such
circumstances as these, in which the theory is continually
deriving new nourishment from a fresh stream of ideas and
suggestions, there is a peculiar fascination in attempting to
review the life-history of the quantum theory to the present
time and in disclosing the kernel which will certainly out-
last changes of form.
1
CHAPTEE I
The Origin of the Quantum Hypothesis
i. Black-Body Radiation and its Realisation in Practice
THE Quantum Theory first saw light in 1900. When, in
the years immediately preceding (1897-1899), Lummer
and Pringsheim made their fundamental measurements 1 of
black-body radiation at the ReichsanstaU, they could have
had no premonition that their careful experiments would
become the starting-point of a revolution such as has seldom
occurred in physics.
In the field of heat radiation chief interest at that time was
centred in the radiation of a black body (briefly called " black-
body " radiation), that is, of a body which absorbs completely
all radiation which falls on it and which thus reflects, trans-
mits, and scatters 2 none. We may shortly call to mind
the following facts. It is known that any body at a given
temperature sends out energy in the form of radiation into
the surrounding space. This radiation is not energy in a
single simple form but is made up of a number of single
radiations of different colours, i.e. of different wave-lengths A
or of different frequencies 3 v. In other words, it forms in
general a spectrum in which radiations of all frequencies
between v = and v = oo are represented. Further, these
radiations are present in varying " intensities." We define
this term thus. Consider the radiation emitted from unit
surface of the body per_second in a certain direction ; break
it up spectrally and cut out of the spectrum a small frequency
interval dv such that it contains all frequencies between v and
v + dv. The energy of radiation E v thus sliced out (namely,
the emissivity of the body for the frequency v) may be defined
in the following terms : *
E v = ^K v d v , . . (1)
2
BLACK-BODY RADIATION 3
provided that as we shall assume for the sake of simplicity
the surface. of the body emits uniform and unpolarised
radiation in" all directions.
The magnitude K,, thus defined is called the intensity of
radiation of the body for the frequency v. It is in general a
more or less complicated function of the frequency v, of the
absolute temperature of the body T, and of the inherent
properties of the body. The black body alone is unique in
this respect. For its radiation and therefore its K v is, as
G. Kirchhoff* was the first to point out, dependent only
on the frequency .v and the absolute temperature T, that is,
mathematically,
K v =f(v,T) .... (2)
This formula which gives the relation between the intensity
of radiation from a black body, the temperature, and the
" colour " is called the radiation formula or the law of radia-
tion of a black body.
To calculate this- relationship on the one hand and to
measure it on the other were unsolved problems at that
time. Unimpeachable measurements were of course possible
only if one could succeed in constructing a black body which
approached sufficiently near the theoretical ideal. This im-
portant step, the realisation of the black body, was taken by
0. Lummer and W. Wien* on the basis of KirchhofTs 1
Law of Cavity Eadiation, which states : In an enclosure
or a cavity which is enclosed on all sides by reflecting walls,
externally protected from, exchanging heat with its surroundings,
and evacuated, the condition of " black radiation " is auto-
matically set up if all the emitting and absorbing bodies at the
walls or in the enclosure are at the same temperature. In a
space, therefore, which is hermetically surrounded by bodies
at the same temperature T and which is prevented from ex-
changing heat with its surroundings, every beam of radiation
is identical in quality and intensity with that which would be
emitted by a black body at the temperature T.
Lummer and Wien, therefore, had only to construct a
uniformly heated enclosure wi.th blackened walls having a
small opening. The radiation emitted from this opening was
then " black " to an approximation which was the closer
4 THE QUANTUM THEORY
the smaller the opening, that is, the less the completeness of
the enclosure was disturbed. The manner in which the
intensity Kv of the black radiation thus realised depended
on the frequency v and the temperature T had next to be
determined. The above-mentioned investigation of Lummer
and Pringsheim was devoted to this purpose.
2. The Stefan-Boltzmann Law of Radiation and Wien's
Displacement Law
While experimental research was proceeding on its way,
theory was not idle, for valuable pioneer work was being
done inasmuch as two fundamental laws were set up. In
the first place, L. Boltzmann* proved, with the help of
thermodynamics, the law previously enunciated by Stefan, 9
that the sum-total of the radiation from a black body,
taking all the frequencies together, namely, the quantity
K = \ K. v dv, is proportional to the fourth power of its absolute
Jo
temperature : 10
K = y . T 4 (y = const.) . . . (3)
The laws proposed by Wien n entered more deeply into the
question. Wien imagines the black radiation enclosed in a
closed space with a perfectly reflecting piston as one wall,
and then supposes the radiation to be compressed adiabatically,
as in the case of gases (that is, no passage of heat to or from
the cavity is allowed during the process), by infinitely slow
movements of the piston. Now, if we express the change
which this process causes in the energy of a definite colour
interval dv in two ways, and if we take into consideration
that the waves reflected at the piston undergo a change of
colour according to Doppler's principle, we succeed in limiting
very considerably the unknown functional dependence of the
quantity K v on v and T. There is thus obtained a re-
lation of the form 12
in which c is the velocity of light in vaciio, the function F
being left undetermined. From this, Wien's Displacement
Law, the conclusion 13 may be drawn that the frequency
WIEN'S LAW OF RADIATION 5
I'max for which 1C (plotted as a function of v) is a maximum
is displaced towards higher values proportional to T as the
temperature increases :
Vmax = COnst. . T . . . (4ft)
If, as is usual in physical measurement, we use the wave-
length A = - instead of the frequency as the variable, Wieris
Law assumes a somewhat different form. For if we consider
the radiant energy of a narrow range of wave-length d\ cor-
responding to the frequency range dv, and write it in the
form E^dX, then EidK = K^dv, that is : E^ - K*> . ^. In
A
place of (4) and (4a) we then get the relations :
>-max . T . const. = S . . (5a)
3. Wien's Law of Radiation
To formulate the law of radiation it was therefore neces-
sary only to evaluate the unknown function F in (4) or (5).
But this was just the central point of the whole question,
and the most difficult part of the problem.
Here, too, Wien made the first successful attack. On the
basis of not entirely unobjectionable calculations, which were
founded on Maxwell's law of distribution of velocities among
gas molecules, he arrived at the following specialised form 14
of the function F :
p a e -pf ( a an d ft are two constants).
Thus the law of radiation (4) assumes the form
K,a.-4 . . . . (6)
which is called Wien's Law of Kadiation.
How far did experiment confirm these theoretical results ?
While the Stefan-Boltzmann Law and Wien's Displacement
Law were confirmed to a large extent by the observations of
Lunimer and Pringsheim, both experimenters found Wien's
6 THE QUANTUM THEORY
Law of Eadiation confirmed only for high frequencies, that is,
for short wave-lengths (more precisely, for large values of
m), and detected, on the other hand, systematic dis
for small frequencies, that is, for long wave-lengths. They
maintained with unswerving persistence that these discrepan-
cies were real in spite of objections from authoritative quarters.
For while F. Paschen 17 imagined that he had proved by his
work that Wien's Law of Radiation was universally valid,
Max Planck, in his detailed theory of irreversible processes
of radiation, 18 had arrived again at Wien's radiation formula
by a more rigorous method. Starting from Kirchhoff's Law
of Cavity Radiation, according to which the presence of any
emitting or radiating substance whatsoever in a uniformly
heated enclosure produces and ensures the maintenance of
the condition of black-body radiation, Planck chose as the
simplest schematic model of such a substance a system
of linear electromagnetic oscillators, and investigated the
equilibrium of the radiation set up between them and the
radiation of the enclosure. This is to be understood as fol-
lows : Each of the Planck oscillators as such we may, for
example, assume bound electrons capable of vibration pos-
sesses a fixed natural frequency v and responds, on account
of its weak damping, only to those waves of the radiation in
the enclosure whose frequencies lie in the immediate neigh-
bourhood of v, while all other waves pass over it without
effect. The oscillator thus acts selectively, as a resonator, in
just the same way as a tuning-fork of definite pitch com-
mences to sound only when its own " proper" tone, or one
very near it, is contained in the volume of sound which strikes
it. In this process of resonance, however, the oscillator ex-
changes energy with the radiation inasmuch as, on the one
hand, it acts as a resonator in abstracting energy from the
external radiation, and, on the other, it acts as an oscillator
and radiates energy by its own vibration. Hence a dynamic
equilibrium is set up between the oscillator and the radiation
of the enclosure, and, indeed, between just those waves of
the radiation which have the frequency v. In this state of
equilibrium the radiation of frequency v acquires an intensity
K t . which, according to Kirchhoff's Law, is equal to the intensity
THE QUANTUM HYPOTHESIS 7
of black-body radiation at this temperature. Secondly, the
energy U of the oscillator passes in the course of time through
all possible values, the mean value 19 U of which is found to
be proportional to the intensity K v , a result which seems im-
mediately plausible since the excitation of the oscillator will
be greater the more intense the radiation that falls on it.
The exact calculation of this relationship between S v and U
on the basis of classical electrodynamics this is the first
part of Planck's calculations leads to the fundamental
formula :
*,=*? - - ' (7)
In the second part Planck * determined U, although by a
method that is not free from ambiguity, as a function of v and
T on the basis of the second law of thermodynamics. He
obtained
ff-*-4 .... (8)
The combination of (7) and (8) gives us Wien's Law of Radia-
tion (6).
4. The Quantum Hypothesis. Planck's Law of Radiation
Lummer and Pringsheim, however, refused to surrender.
In a fresh investigation 21 in 1900 they showed that in the
region of long waves Wien's radiation formula undoubtedly did
not agree with the results of observation. As a result of this,
Planck, in an important paper 22 which must be regarded as
marking the creation of the quantum theory, decided to
modify his method of deducing the law of radiation, namely,
by altering the expression (8) which gives the mean energy
of the oscillator, but which is not unique. He proceeded as
follows. 23 In order to distribute the whole available energy
among the oscillators, he imagined this energy divided into
a discrete number of finite " elements of energy " (energy
quanta) of magnitude e, and supposed these energy quanta
to be distributed at random among the individual oscillators
exactly as a given number of balls, say 5, may be distributed
among a certain number of boxes, say 3. Each such distri-
bution (of 5 balls among 3 boxes) may obviously be carried
8 THE QUANTUM THEORY
out in a number of different ways, whereby, however, we are
not concerned with which particular balls lie in which par-
ticular boxes, but with the number contained in each. 2 * Now
since each such " distribution " corresponds to a definite state
of the system, it follows from what has just been said that
each condition may be realised in a number of different
ways, that is, each condition is characterised by a certain
number of possibilities of realisation. This number is called
by Planck the thermodynamic probability W of the condition
in question. For it is obvious that the probability of a con-
dition or state is the greater, i.e. it will occur the more fre-
quently, the greater the number of ways in which it may
be realised. By means of the usual formulae of permuta-
tions and combinations, of which the latter alone come into
consideration here, it was possible to calculate the probability
of any given distribution of the elements of energy among
the oscillators, and thus also the probability of a given
energetic condition of the system of oscillators as a func-
tion of the mean energy U of an oscillator and of the energy
quantum. Now, L. BoUzmann has given an extremely
fertile rule, which connects the probability of state W of a
system with its entropy S, a magnitude which, as is well
known, plays a similar role in the second law of thermodyna-
mics to that played by energy in the first. Thus S was ob-
tained as a function of U and c. If now, on the other hand,
one applied the second law itself, which expresses the en-
tropy S as a function of the mean energy U and the absolute
temperature T, the following result was obtained by this cir-
cuitous process : the entropy, as an auxiliary magnitude, was
eliminated, and a relation between U, T, and e was gained.
This fundamental result, first obtained by Planck, is as
follows :
U = _! - (k being a constant) . . (9)
But from (7) and TPtenVDisplacement Law (4) it follows
that for the mean energy U of an oscillator, a relationship of
the following form exists :
THE QUANTUM HYPOTHESIS 9
A comparison of (9) and (10) shows that U assumes the
form required by (10) only when e is set proportional to v, the
frequency. This is an essential point of Planck's Theory : if
we are to remain in agreement with Wien's Displacement Law,
the energy element c must be set equal to hv
t = hv . . . . (11)
The constant h, which, on account of its dimensions (energy
x time), is called Planck's Quantum of Action, has played,
as we shall see, a r61e of undreamed-of importance in the
further development of the quantum theory.
By combining the formulae (7), (9), and (11) the renowned
radiation law of Planck follows at once :
- (12)
- 1
which Planck first deduced in the year 1900 in the manner
above described, that is, by the hypothesis of energy quanta.
In the same year as well as in the following year this Law of
Kadiation was confirmed very satisfactorily by H. Rubens and
F. Kurlbaum 2 for long waves, and by F. Paschen Z1 for short
waves. The later measurements of radiation emitted by
black bodies,28 particularly the exact work carried out by E,
Warburg and his collaborators at the Eeichsanstalt, have also
demonstrated the validity of Planck's formula. In opposition
to this, W. Nernst and Th. Wulf& as the result of a critical
review of the whole experimental material available up to that
date, have recently shown the existence of deviations (up to
7 per cent) between the measured and the calculated values
according to Planck's formula, and hence feel themselves
constrained to decide against the exact validity of Planck's
formula. Whatever view is taken of this criticism, it is at any
rate a powerful incentive to take up anew the measurement
of the radiation emitted by black bodies with all the finesse
and precautions of modern experimental science, and thereby
to decide finally the important question whether Planck's
Law is exactly valid or not.
For short wave-lengths, i.e. high frequencies (more exactly,
10 THE QUANTUM THEORY
for high values of =-^\, Planck's formula assumes the form
and thus passes over into Wien's Law (cf. formula (6),
which, as we have seen, was confirmed by experiment for
these frequencies). In the other limiting case, i.e. for long
waves, low frequencies (more exactly for small values of ?-=,
Planck's formula assumes the form
*.- .... (H)
as is easily found by developing the exponential function
efk as a series. This limiting law, which has been confirmed
in the region of long wave-lengths, had been given pre-
viously by Lord Rayleigh. 30 Planck's formula thus contains
Wien's Law and Eayleigh's Law as limiting cases.
If we use the wave-length X instead of the frequency v,
Planck's Law takes the form
. . . (15)
To make this clear, the intensity of radiation E^ is plotted
in Pig. 1 as a function of A. for various values of T. The
curves which exhibit K. v as a function of v have a quite
similar appearance. The maximum of the ^-curves lies at
the point at which _A has the value 4-9651.
It follows that
Xinax ' T = TaSr-T = 6>042 x 1 9 ? = 8 (16)
4'yoOl . K K
a relation, which is identical in form with Wien's Displace-
ment Law (5a).
For the total radiation we get from (12) or (15)
CONSEQUENCES OF PLANCK'S THEORY 11
an equation which gives expression to the Stefan- Boltzmann
Law 31 (3).
From (16) and (17) we recognise that the measurement (a)
of the total radiation (K) and (6) of the
wave-length of the maximum (A. max ), at
a fixed known temperature, allows us to
calculate the two constants h and k of the
radiation formula. 32 From Kurlbaum's
measurements of the Stefan-Boltzmann
constant y, which were available at that
time, and from the constant 8 of Wien's
Displacement Law (measured by Lummer
and Pringsheim) Planck 33 found the fol-
lowing values :
h = 6-548 x 10- 27 [erg.sec.]
k = 1-346. 10 -"fl^l
Ldeg.J
(18)
Corresponding to the varying values
which have been found in the course of
time for the constants y and 8, the values
h and k have undergone changes which
are not worth while
recording here.
For particularly
_/l the measurement of
p IG i_ the total radiation
as we see from
the strongly varying values given in note 15 has not yet
reached a sufficient degree of certainty, to allow a very ac-
curate calculation of the two radiation constants h and k to be
based on the Stefan-Boltzmann constant. Methods which
allow h to be determined with undoubtedly much greater
accuracy will be described later.
5. Consequences of Planck's Theory
The deduction of the radiation formula and the determina-
tion of its constants did not, however, exhaust the successes
of Planck's new theory ; on the contrary, important relation-
ships of this theory to other departments of physics became
12 THE QUANTUM THEORY
immediately revealed. For it was found 3* that the constant
k of the radiation formula is nothing other than the quotient
of the absolute gas constant B (which appears in the equa-
tion of state of an ideal gas) and the so-called Avogadro
number N, i.e. the number of molecules in a grammolecule.
*-* '. . . . (19)
As the value of R is sufficiently accurately known from
thermodynamics
Planck,** by making use of the radiation measurements, was
able to calculate the value of N. By using (18) he found
N = 6-175 x 10 23 . \. . (20)
The agreement of this value with the values deduced by
quite different methods is very striking. 86 Avogadro s Law
forms the bridge to the electron theory. For it is known
that the electric charge which travels in electrolysis with
1 gramme-ion, that is, with N-ions, is a fundamental con-
stant of nature, which is called the Faraday. Its value was,
according to the position of measurements at that time,
9658 . 3 . 10 10 electrostatic units (the value nowadays ac-
cepted 37 is 9649-4 . 2-999 . 10 10 ). If now each monovalent-
ion carries the charge e, of the electron, the equation
Ne = 9658 . 3 . 10 10 . . . (21)
must hold. From this, by using (20), we get
e = 4-69 x 10- 10 electrostatic units .. (22)
The value of the electron charge thus calculated by Planck
from the theory of radiation differs only by about 2 per cent
from the latest and most exact measurements of R. A.
Millikan** who found the value
e = 4-774 . 10 - 10 electrostatic units. . (23)
A truly astonishing result.
CHAPTEE II
The Failure of Classical Statistics
I. The Equipartition Law and Rayleigh's Law of Radiation
IF these great successes had justified faith in Planck's
Theory, it was also soon recognised as had already been
emphasised by Planck in his first papers that the central
point of the theory lay in the Quantum Hypothesis, i.e. in the
novel and repulsive conception, that the energy of the oscilla-
tors of natural period v was not a continuously variable
magnitude, but always an integral multiple of the element of
energy, that is e = hv. The recognition of the necessity of
this hypothesis has forced itself upon us more and more in the
course of time, and has become established, more especially
through indirect evidence, inasmuch as every attempt to work
with the classical theory has led logically to a false law of
radiation. For when Planck turned the radiation problem
into a problem of probability for a definite amount of energy
was to be divided among the oscillators according to chance,
and the mean value U of the energy of an oscillator was to
be calculated it became possible to apply the methods of
the statistical mechanics founded by Clerk Maxwell, L.
Boltzmann, and Willard Gibbs. And the application of these
methods to the case in question appeared to be demanded
from the start, if the standpoint, self-evident in classical
physics, that the energy of the oscillator could assume in
continuous sequence all values between and CD were
adopted. What, then, did statistical mechanics require?
One of its chief laws is the law of the equipartition of kinetic
energy, * according to which in a state of statistical equilibrium
at absolute temperature T every degree of freedom of a mechan-
ical system, however complicated, possesses the mean kinetic
13
14 THE QUANTUM THEORY
energy ^TcT. In this expression the constant k is defined by
(19), and is thus the same constant as that which appears
in the Law of Eadiation. A system of/ degrees of freedom,
therefore, possesses at a temperature Ta mean kinetic energy
/ . $kT. For example, the atom of a monatomic gas is a
configuration which possesses three degrees of freedom, if we
regard it from the point of view of mechanics as a mass-
point. Its kinetic energy at the temperature T has therefore
a mean value* %kT, independent of its mass, a result which
has been known in the kinetic theory of gases since the time
of Maxwell, and which is deduced as a consequence of his
law of distribution of velocities.
Planck's linear oscillator, which is essentially identical
with an electron vibrating in a straight line, possesses one
degree of freedom ; its kinetic energy at the temperature T
has therefore the mean value -^kT. Now the mean potential
energy of the oscillator is equal to its mean kinetic energy.* 1
As a result, its mean total energy (kinetic plus potential) has
the value
C7= kT . . . . (24)
This result of classical statistics, when combined with the
relation (7) deduced from classical electrodynamics, gives
Rayleigh's Law of Radiation
K v = ^kT . . ... (25)
which, as we saw (cf. (14)), is contained in Planck's Law of
Radiation as a limiting case for small values of pL that is,
for long waves or high temperatures.
This Law of Radiation of Bayleigh which, deduced as it
is from the fundamental principles of classical statistics and
electrodynamics, should be able to claim general validity for
all frequencies and all temperatures, stands none the less in
glaring contradiction to observation. For while all observed
curves of distribution of energy of a black body (i.e. K v plotted
as a function of v, T being constant) always show a maximum,
the curve expressed by (25) rises without limit for rising
values of v, and therefore gives for the sum K 2 / Kvdv an
infinitely large value.
FRUITLESS ATTEMPTS AT IMPROVEMENT 15
2. Fruitless Attempts at Improvement
From very different quarters and in the most varied ways
attempts were made, as time went on, to escape from
Rayleigh's Law without discarding classical statistical
mechanics. All in vain. Thus 7. H. Jeans,* 2 without
making use of a "material" oscillator, considered only the
radiation as such in an enclosure, and distributed the whole
energy of radiation according to the Law of Equipartition
over the individual " degrees of freedom of radiation " (which
are here the individual vibrations that are possible in an en-
closure). Further, H. A. Lorentz** deduced in a penetrating
investigation the thermal radiation of the metals, starting from
the conception that the free " conduction electrons," which
carry the current, produce the radiation by their collisions
with the atoms, and applying the Law of Equipartition to
the motion of these electrons. The problem was attacked
in a somewhat different fashion by A. Einstein and L. Hop/.**
They imagined the Planck oscillator firmly attached to a
molecule, and then considered this complex exposed to the
radiation and the impacts of other molecules. The Law of
Radiation could then be deduced from the condition that the
impulse, which the impacts of the molecules give to the com-
plex, must not on the average be changed by the impulses,
which the radiation gives to the oscillator. We may also
mention a paper of A. D. Fokker** which was supplemented
by M. Planck.* 6 In this, by the aid of a general law due to
Einstein, the statistical equilibrium between the radiation
and a large number of oscillators was examined on the basis
of the classical theories. All these different ways ended,
however, at the same point ; they all led to Rayleigh's Law.
And finally, at the Solvay Congress in Brussels in 1911,
H. A. Lorentz showed, in the most general manner
imaginable, that we arrive of necessity at this wrong law,
if we assume the validity of Hamilton's Principle and of
the Principle of Equipartition for the totality of the pheno-
mena (of mechanical and electromagnetic nature) which
take place in an enclosure containing radiation, matter, and
electrons. Only in the limiting case of high temperatures or
small frequencies do the results of the classical theory agree
with the results of observation.* 8
CHAPTEE III
The Development and the Ramifications of the
Quantum Theory
i. The Absorption and Emission of Quanta
A S stated above, the conviction was bound to establish
Xl^tself that every attempt to deduce the laws of radiation
on the basis of classical statistics and electrodynamics was
doomed from the outset to failure, and it was necessary to
introduce a hitherto unknown discontinuity into the theory.
It was, of course, clear that this " atomising of energy " would
conflict sharply with existing and apparently well-founded
theories. For if the energy of the Planck oscillator was only
to amount to integral multiples of e = hv, and therefore was
only to be able to have the values 0, e, 2e, 3e . . . then, since
the oscillator only changes its energy by emission and ab-
sorption, the conclusion was inevitable that oscillators cannot
absorb and emit amounts of energy of any magnitude but only
whole multiples of e. (Quantum emission and quantum
absorption.} This conclusion is in absolute contradiction to
classical electrodynamics. For, according to the electron
theory, an electromagnetic oscillator, for instance a vibrat-
ing electron, emits and absorbs in a field of radiation perfectly
continuously, that is to say, in sufficiently short times it emits
or absorbs indefinitely small amounts of energy.
2. Einstein's Light-quanta ; Phenomena of Fluctuation in a Field
of Radiation
Thus at the very entrance into the new country there
yawned a gulf, which had either, in view of the previous
success of the classical theory, to be bridged over by a com-
promise ; or, failing this, tradition would have to be discarded
and the gap would be relentlessly enlarged. Einstein felt him-
16
EINSTEIN'S LIGHT-QUANTA 17
self compelled to take the latter radical course. On the basis
of very original considerations, 49 he set up the hypothesis that
the energy quanta not only played a part, as Planck held, in
the interaction between radiation and matter (resonators or
oscillators), but that radiation, when propagated through a
vacuum or any medium, possesses a quantum-like structure
(Light- quantum hypothesis). Accordingly, all radiation was
to consist of indivisible " radiation quanta " ; when the energy
is being propagated from the exciting centre, it is not divided
evenly in the form of spherical waves over ever-increasing
volumes of space, but remains concentrated in a finite numbe
of energy complexes, which move like material structures,
and can only be emitted and absorbed as whole individuals.
Einstein believed himself forced to this strange conception,
which breaks with all the observations that appear to
support the undulatory theory, by several investigations,
all of which led to the same conclusion. He was per-
suaded to this view by the result of calculations dealing
with certain phenomena of fluctuation, phenomena which
are familiar to us in statistics and particularly in the kinetic
theory of gases. It is well known that in a gas which
contains n molecules in a volume v , the spatial distribution
of these molecules is far from constant, being subject to vari-
ation on account of the motion of the molecules. Indeed, in
principle, extreme cases are possible as that, for example, in
which all n molecules are collected at a given moment in a
fractional part v(<v ) of the volume. The probability of
this rare constellation is known to be
. . . . (26)
an extraordinarily small number when n is great ; that is to
say, the event in question occurs extremely rarely.
Now, the spatial density of the radiation enclosed within a
volume v (} is subject to quite analogous variations. If E is
the total energy of the radiation (supposed to be monochro-
matic) and if its frequency v is so great, or its temperature
so low, that Wien's Law of Radiation holds for it, then the
probability that the whole radiation occupies the partial
volume -u V Q ) is, according to Einstein,*
18 THE QUANTUM THEORY
io-^\C . (27)
A comparison with (26) shows that the radiation, within
the limits of validity of Wien's Law, behaves as if it were made
wp of n ( = r- j independent complexes of energy, each of mag-
nitude hv.
Two other investigations si of Einstein led to the same
conclusion. In the first, a very large volume filled with
black-body radiation is considered, which communicates with
a small volume v. If E is the momentary energy of the
radiation of frequency v in the volume v, this energy varies,
as is known, irregularly with the time about a mean value
E ; the magnitude e = E - E is called the fluctuation of the
energy. Now, the general theory of statistics leads to the
following value fl2 for the mean square, that is, for e 2 ,
P-fcT'.g;. . . .- (28)
If we replace E by the value obtained from Planck's Law of
Eadiation, we obtain for the mean square of fluctuation an
expression with two terms, 93 in which only one term can be
calculated on the basis of the classical undulatory theory;
the second, which greatly exceeds the first in magnitude
when the density of radiant energy is low (that is, at high
frequencies or at low temperatures, in short, when Wien's
Law is valid), can only be understood when we again picture
the radiation as composed of indivisible energy-quanta.
The second of Einstein's two investigations, to which we
referred above, deals with the fluctuations of impulse which
a freely movable reflecting plate is subjected to in a field of
black-body radiation on account of the irregular fluctuations
of the pressure of radiation. If, in addition, the plate is sub-
jected to the irregular blows of gas-molecules, under the
influence of which it executes Brownian movements, there
must be equilibrium between the impulses which the mole-
cules on the one hand, and the radiation on the other, im-
part to the plate. If, now, we assume Planck's Law to hold
for the radiation, there again follows for the mean square of
the variations in impulse due to the radiation an expression
TRANSFORMATION OF LIGHT-QUANTA 19
in two terms, only one of which is explained by the un-
dulatory theory of light. The other term points to a
quantum-like structure of the radiation, and this suggests the
introduction of the light-quantum hypothesis.
3. Transformation of Light-quanta into other Light-quanta or
Electronic Energy
However strange this hypothesis appeared, it was not to be
denied that it was capable of explaining simply and naturally
a number of phenomena which completely baffled the un-
dulatory theory. A very striking example of this is afforded
by the laws of phosphorescence, investigated by P. Lenard
and his co-workers, and especially by Stokes' Law. For if
v p is the frequency of the phosphorescent light emitted,
and v e the frequency of the light exciting phosphorescence,
then, according to Einstein's conception, 34 one quantum hv e
of the exciting radiation is changed through absorption by
the atom of the phosphorescent substance into one quantum
hv p of the light of phosphorescence. According to the prin-
ciple of energy, we must have hv e !> hv p , i.e. v e > v p . And
this is Stokes' Law.
Further, another fact in the realm of phosphorescence
phenomena speaks against the undulation hypothesis and in
favour of that of light-quanta. According to the classical
undulatory theory, all molecules of a phosphorescent body
on which a light- wave impinges, should absorb energy from
the wave, and thus all simultaneously become able to emit
phosphorescent light. In reality, relatively only very few
molecules are excited to phosphorescence at the same time,
and only gradually, in the course of time, does the number of
molecules excited increase. It would thus appear as if the
light-wave falling on the phosphorescent body has not equal
intensity along its whole front as the classical theory
assumes but rather as if it consists of single energy-com-
plexes thrown out by the source of light, so that the wave-point
possesses, as it were, a "beady" structure, in which active
portions (light-quanta) alternate with inactive gaps.
This conception of the " beady " wave-front had played a
part before the advent of Einstein's hypothesis of light-quanta.
J". /. Thomson 8B had tried to make use of it to explain the
20 THE QUANTUM THEORY
fact that, when a gas is ionised by ultra-violet light or Kontgen
rays, only a relatively extremely small number of gas-mole-
cules are ionised. This is a phenomenon which is quite
analogous to the above-named phenomenon of phosphor-
escence ; for these, too, according to Lenard's view, the exci-
tation consists in the disjunction, through the agency of the
radiation, of electrons from the molecules of the phosphor-
escent body, and these electrons attach themselves to " storage
atoms." On the return of these electrons to the parent
molecules, energy is set free and sent out as phosphorescent
light. The ionisation of gases by ultra-violet light or Ront-
gen rays M is also capable of being explained naturally by the
light-quantum hypothesis. If we suppose with Einstein,
that one light-quantum hv is used up in ionising one mole-
cule, then hv > /, where / is the work required to ionise
one molecule, that is to say, to remove an electron from it.
We have under consideration here a phenomenon which be-
longs to the great branch of photo-electric phenomena,* 1 i.e.
the liberation of electrons from gases, metals, and other sub-
stances by the action of light. According to the hypothesis
of light-quanta, in all these processes light-quanta are changed
into kinetic energy of the electrons hurled off from the body.
If we again adopt Einstein 's standpoint, according to which
one light-quantum hv is transformed into the kinetic energy
of one projected electron, we must have the following re-
lation M for the energy of emission of the emitted electrons,
each having a mass ra :
fyn V 2 = hv - P . . . . (29)
This is called Einstein's Law of the Photo-electric Effect. In
this, P is the work that has to be done to tear the electron
away from the atom, and to project it from the point at
which it is torn from the atom up to the point at which it
leaves the surface of the body. For the energy of the emitted
electrons we thus obtain a linear increase with the periodicity
of the light which releases them. This law, which many in-
vestigators have attempted to prove, with varying success,
has recently been verified by B. A. Millikan* 9 for the normal
photo-electric effect w of the metals Na and Li with such a
degree of accuracy that we can actually use this method for
ELECTRONIC ENERGY 21
the exact determination of h. The value found by Millikan,
h = 6-57 x 10 ~ 27 , is in good agreement with the value
h = 6'548 x 10' 27 found by Planck from radiation measure-
ments.
In an entirely similar manner as was used for the
phenomena of phosphorescence, the phenomena of fluores-
cence in the regions of the Rontgen and visible radiations may
be explained by the hypothesis of light-quanta. The in-
vestigations of Ch. Barkla, Sadler, M. de Broglie, and
E. Wagner 61 have shown the following : if a body is inun-
dated with Eontgen rays, and if the absorption of these rays by
the body is measured whilst the hardness (i.e. the frequency
v e ) of the rays is varied, the absorption, as we pass from
lower to higher v e , suddenly increases to a high value for a
certain value of v e . At the same moment the body begins,
at the expense of the energy absorbed, to emit a secondary
Rontgen radiation characteristic of the body itself in the form
of a line spectrum. It further appears that all lines emitted
have a lower v than that of the exciting radiation. As a
matter of fact, the hypothesis of light-quanta requires that the
radiation-quantum hv of all rays emitted as secondary radia-
tion should be smaller than the quantum hv of the primary
exciting rays. For example, the region of frequencies which
serves to excite the " ^-series " stretches from a sharply
defined limit v k (the so called " edge of the absorption band ")
upwards towards higher frequencies; whereby v k is some-
what larger than the hardest known line (y) of the ^-series.
In other words, the excitation of secondary Rontgen radiation
by primary Rontgen rays also obeys Stokes' Law.
4. The Transformation of Electronic Energy into Light-quanta
It is very significant, that the transformation of light-
quanta into kinetic enei'gy of electrons is also, as it were,
"reversible," that is, the opposite process also occurs in
nature, by which light-quanta result from the kinetic energy
of charged particles. A good example of processes of this
kind is afforded by the generation of Rontgen rays by the
impact of quickly-moving electrons (cathode rays) on matter.
If, say, the characteristic X"-series of a certain element is to
22 THE QUANTUM THEORY
be generated by the impact of cathode rays upon an anti-
cathode formed of the said element, then the kinetic energy
E of an impinging electron must exceed a critical value E K .
For if we imagine E changed into a light-quantum hv e , then
v e must fall within the region of excitation of the JT-series, and
must thus be ^ V K (V K being the frequency of the edge of the
absorption band). It follows that E ^> hv K ( = E K ). From this
there follows an important relation between the frequency V K
of the edge of the absorption band and the critical value EK
of the electronic energy, i.e. the smallest value of the energy
at which the electron is just able to generate the required
secondary radiation. This quantum-relation E K = hv K has
proved quite correct according to measurements carried out
by D. L. Webster & and E. Wagner w and conversely presents,
when E K and V K are sufficiently accurately known, a method
for the determination of /i. 6 *
Now, it is known that the cathode rays, on striking the
anti-cathode, do not merely excite the characteristic Eontgen
radiation, that is a line spectrum, but excite a continuous
spectrum as well, the so-called " impulse radiation " (Brems-
strahlung). If we therefore select any frequency v of this
continuous spectrum, the ideas of the hypothesis of light-
quanta immediately suggest the conclusion that a definite
minimum energy E m of the impinging electrons is necessary
to excite this frequency v, and that we must have E m = hv.
The investigations of D. L. Webster & W. Duane and F. L.
Hunt**, A. W. Hull and M. Bice^E. Wagner, M F. Dessauer
and E. Back 66 have confirmed these formulae with the
greatest accuracy, and thus form the foundation of one of
the most trustworthy methods for the precise measurement
of the magnitude h. The following values were obtained :
h = 6-50 x 10-27 (Duane-Hunt) ; h = 6-53 x 10 - 2 ~ (Webster) ;
h = 6-49 x 10 ~ 27 (Wagner).
We also meet with similar phenomena in the visible and
neighbouring regions of the spectrum. Thus /. Franck
and Cr. Hertz m showed that the impact of electrons upon
mercury vapour molecules can be used to excite a definite
characteristic fluorescence line of mercury of wave-length
A = 25364 (i.e. v = 1-183 . 10 16 ), if the kinetic energy of the
HYPOTHESIS OF LIGHT-QUANTA 28
electron exceeds a certain critical value E Q . In this con-
nexion they found that the relation E = hv was again
fulfilled with great accuracy. 70 We shall return to these
experiments and others connected with them later, since they
play an important part in confirming the most recent model
of the atom.
5. Other Applications of the Hypothesis of Light-quanta
In a considerable number of other cases, which shall only
be noticed shortly at this point, the hypothesis of light-quanta
has proved of value, especially in the hands of /. Stark 71 and
Einstein. Thus Stark ra has made use of this hypothesis to
interpret the fact that the canal-ray particles emit their
" kinetic radiation " only when their speed exceeds a certain
value. He has also propounded general laws for the position
of band-spectra of chemical compounds by arguing on the basis
of the hypothesis of light-quanta. 78 Finally, Einstein 7 * and
Stark 78 have considered photo-chemical reactions from the
standpoint of the hypothesis of light-quanta and have enun-
ciated a fundamental law, which has been verified, at least
partially, by the detailed investigations of E. Warburg.
6. Planck's Second Theory
In spite of all the successes which the quantum hypothesis
of light is able to show, we must not leave out of consideration
that this radical view, at least in its existing form, is very
difficult to bring into agreement with the classical undulatory
theory. Since on the one hand the phenomena of interference
and diffraction, in all their observed minutiae, are excellently
described by the wave-theory, but offer almost insuperable
difficulties to the quantum theory of light, it is easy to under-
stand that few scientists could make up their minds to ap-
prove of such a far-reaching change in the old and well-tested
conception of the propagation of light, a change that entailed
perhaps its complete abandonment. This more cautious and
conservative standpoint was taken up by M. Planck, who
retains it to this day, inasmuch as he preferred to locate the
quantum property in matter (the oscillators) or at least to
confine it to the process of interaction between matter and
24 THE QUANTUM THEORY
radiation while endeavouring to retain the classical wave-
theory for the propagation of radiation in space. None the
less, serious hindrances had already intruded themselves in
the development of his first quantum hypothesis (quantum
emission and quantum absorption). For H. A. Lorentz 1 ? 1
pointed out quite rightly that the conception, especially of
quantum absorption, leads to peculiar difficulties. He showed
that the time which an oscillator requires for the absorption
of a quantum of energy turns out to belong to an improbable
degree when the external field of radiation is sufficiently weak.
Moreover, it would be possible to interrupt the radiation at
will before the oscillator had absorbed a whole quantum. As
a result of these objections Planck determined to modify the
quantum hypothesis as follows. 78 Absorption proceeds con-
tinuously and according to the laws of classical electrodynamics :
the energy of the oscillators is therefore continuously variable, and
can assume any value between and oo . On the other hand,
emission occurs in quanta, and the oscillator can emit only
when its energy amounts to just a whole multiple of = hv.
Whether it then emits or not is determined by a law of prob-
ability. But if it does emit, then it loses its whole momentary
energy, and therefore emits quanta. Between two emissions its
energy -content grows by absorption continuously and in pro-
portion to the time.
According to this second theory of Planck, which is called
the theory of quantum emission, the mean energy U of a
linear oscillator is ~ greater than in the first theory. 79 While
in the former case the mean energy of the oscillator at abso-
lute zero was equal to zero (see equation (9) from which,
when T = 0, U 0), in the case of this second theory it is
equal to . The oscillators retain therefore at the zero-
point a zero-point energy of value - as a mean, inasmuch
2
as they assume, when T = 0, all possible energies between
and hv. Nevertheless, this theory also, when the relation
(7) is correspondingly modified, leads to Planck's Law of
Kadiation.
In the course of time Planck has made several further
ZERO-POINT ENERGY 25
attempts 80 to enlarge and modify this second theory too. For
example, he has temporarily assumed the emission also to be
continuous, and relegated the quantum element to the excita-
tion of the oscillators by molecular or electronic impacts. He
has, however, repeatedly returned in essentials to the second
form of his theory (continuous absorption, quantum emission).
7. Zero-point Energy
In more than one direction, this theory has had further
results. The appearance of the mean zero-point energy,
which is peculiar to this second theory of Planck, became
the starting-point of a series of researches, in which certain
physicists, going beyond Planck, postulated the existence of
a true (not mean] zero-point energy equal for all oscillators.
On this basis, Einstein and 0. Stern 81 have given a deduction
of Planck's Law which avoids all discontinuities other than
the existence of this zero-point energy.
In the year 1916, Nernst 82 took a still more radical step in
postulating the existence of a " zero-point radiation " which
was also to be present at the absolute zero of temperature
and was to exist independently of heat radiation, filling the
whole of space, and such that the oscillators, as well as all
molecular structures, set themselves in equilibrium with it by
taking up the zero-point energy. Even if we regard these
views more or less sceptically, one thing cannot be ignored :
many facts undoubtedly support the conception that at the
absolute zero by no means all motion has ceased. We need
only draw attention to the fact, that, according to the view of
F. Richarz*z P. Langevin& and according to the experiments
of Einstein, W. J. de Haas * and E. Beck** Para- and Dia-
magnetism are produced by rotating electrons and that this
magnetism remains in existence down to the lowest tempera-
tures.
8. Theory of the Quantum of Action
In yet another respect has Planck's theory proved stimu-
lating, in virtue of a special formulation which Planck gave
it 87 at the Solvay Congress in Brussels during 1911. For
here Planck gave expression for the first time to the idea
that the appearance of energy-quanta is only a secondary
26
THE QUANTUM THEORY
matter, being only the consequence of a deeper and more
general law. This law, which is to be regarded as the pre-
cursor of the latest development of the doctrine of quanta,
may be formulated as follows : Suppose the momentary state
of a Planck oscillator, say a linearly vibrating electron, to be
defined according to Gibb's method by its displacement q from
its position of rest and by its impulse or momentum p, and
suppose it to be represented in a q-p plane (the state- or
phase-plane). Every point of the q-p plane, that is, every
phase-point, corresponds to a definite momentary condition of
the oscillator. The postulate is then made that not all points
of this plane of states are equivalent. On the contrary, there
FIG. 2.
are certain states of the oscillator which are distinguished by
a peculiarity. The totality of the phase-points that cor-
respond to these peculiar states form a family of discrete
curves which surround one another. In the case of the
Planck oscillator these curves are concentric ellipses (see
Fig. 2) which divide the phase-plane into ring-like strips.
The postulate of the quantum theory now consists in this,
that these ring strips all possess the same area h. If we
calculate on this basis the energy possessed by an oscillator
in one of these unique states, we find 80 that it is a whole
multiple of hv. These special states (represented in the
phase-plane by the points of the discrete ellipses) are, there-
THEORY OF THE QUANTUM OF ACTION 27
fore, according to Planck's first theory, the only dynamically
possible and stable states of the oscillator. If an oscillator
emits or absorbs, its phase-point jumps from one ellipse to
another. The state of affairs is different if we accept Planck's
second theory. According to this, all conditions of the oscil-
lator, that is all points on the phase-plane, are dynamically
possible. On the other hand, emission takes place only in
the states specially distinguished by the ellipses. Seen from
this new point of view, the energy-quanta are, therefore, only
a result of the partitioning of the phase-plane. Mathe-
matically, we may express this " structure of the phase-plane "
thus : the ?ith unique curve encloses a surface of area nh, or,
in symbolic language,
I" [dqdp = \
nh . '. (30)
The double integral is taken over the surface ; the single
integral is taken around the boundary curve of the nth
ellipse.
On this basis for systems of one degree of freedom, which
is called Planck's theory of " the action-quantum " for h has
the dimensions of an action the modern extension of the
quantum theory for several degrees of freedom has, as we
shall see, been erected.
Further, a line of argument proposed and developed by
A. Sommerfeld takes its origin here. Starting from the fact
just mentioned, that Planck's constant h possesses the dimen-
sions of action (energy-time), Sommerfeld set up the hypo-
thesis w that for every purely molecular process, say the release
of an electron in the photo-electric effect, or the stopping of
an electron by the anticathode in the generation of Eontgen
rays, the quantity called action (L - V)dt, known to us
V J
from Hamilton's Principle, has the value - . Here L and V
Air
are the kinetic and potential energies of the electron respec-
tively, T is the duration of the molecular process 4 ! say, for
example, the time which is required for the release of the
electron from the atomic complex during the photo-electric
effect, or the stopping of the electron by the anti-cathode.
28 THE QUANTUM THEORY
This formulation of the quantum hypothesis is, as it were,
an expression of the well-known fact that large amounts of
energy are absorbed or given up in short times, whereas small
amounts are absorbed or emitted in longer times by the
molecules, so that on the whole the product of the energy
transferred and the duration of the time of exchange is a
constant. In fact, fast cathode rays, for example, are stopped
by matter in a shorter time and therefore generate harder
Kontgen rays than slow cathode rays. Sommerfeld has
applied his theory successfully to the mechanism of the
generation of Rontgen rays and y-rays. 90 Sommerfeld and
P. Debye w have worked out on the same basis a theory of
the photo-electric effect, which, like the hypothesis of light-
quanta, also leads to Einstein's Law (29).
CHAPTEK IV
The Extension of the Doctrine of Quanta to the
Molecular Theory of Solid Bodies 92
i. Dulong and Petit's Law
IT was a particularly fortunate circumstance for the con-
solidation of the doctrine of quanta that the failure of
classical statistics was not confined to the theory of radiation,
but, as appears later, extended to the molecular theory of solid
bodies. Thus there arose in quite another field a strong sup-
port for the quantum hypothesis, namely, in the field of Atomic
Heats. The Atomic Heat of a substance (in the case of poly-
atomic bodies we say the " Molecular Heat ") is defined as the
product of its specific heat and its atomic weight (or molec-
ular weight) ; or, otherwise expressed, it is that amount of
heat which must be communicated to a " gramme-atom " m
(or gramme-molecule) of the body, in order that its tempera-
ture may be raised by one degree. According to our present
conceptions, the thermal content of a monatomic solid, say
a crystal, is nothing more than the energy of the elastic
vibrations of its atoms, which are arranged in the form of
a space-lattice, about their positions of equilibrium. If we
apply classical statistics to these vibrations, and particularly
the law of equipartition of kinetic energy, we arrive at the
following conclusion : The mean kinetic energy of an atom
01L/TT
vibrating in space, i.e. with three degrees of freedom, is , and
its mean potential energy is equal to the same amount, 94 so
that its total energy is therefore 3kT. If we now consider
1 gramme-atom of the body, that is, a system of N atoms
(where N is the Avogadro number, approximately 6 x 10 23 ),
we get for the mean energy of the body, remembering (19),
E = 3kTN = 3RT . . . (31)
29
30 THE QUANTUM THEORY
where R is the absolute gas-constant. It follows that the
atomic heat of the body at constant volume becomes :
0,.^-3*-fr94[] . . (32)
This is the law of Dulong and Petit, 9 * according to which
the atomic heat (at constant volume) of monatomic solid bodies
has the value 5'94 jj , independently of the temperature. 96
This law is actually obeyed by many elements more or less
closely. 97 On the other hand, elements have long been known
which are far from following this rule, and which show
systematic differences, especially at low temperatures.
Thus, as early as the year 1875, F. H. Weber & found that
the atomic heat of diamond at - 50 C. is about 0'75 ?^. The
deg.
atomic heats of other elements as well (boron, beryllium,
silicon) have also been shown to be much too small at
ordinary temperatures. And altogether it appeared that the
defect from Dulong and Petit' 's normal value occurs quite
generally at low temperatures, and becomes the more pro-
nounced, the lower the temperature. The classical theory
offered no solution of these low values of the atomic heat. 99
2. Einstein's Theory of Atomic Heats
Einstein was the first to recognise 10 that in this case, too,
the quantum theory was destined to solve the difficulty.
Precisely as in the theory of radiation, the method of
classical statistics leads of necessity to a wrong law in the
field of atomic heats. Hence, here also, we must abandon
the laiv of the equipartilion of energy. In fact, we need only
imagine electric charges distributed among the atoms 101 and
then we see that, exactly like the Planck oscillators, they must
set themselves in equilibrium with the heat-radiation which
is always present in the body. This means, however, that
the relation (7), according to which U = ~a K,., must be set
up between the mean energy U of an atom vibrating linearly
with frequency v, and the intensity of radiation K,,. If we
now take Planck's radiation formula (12) as empirically
EINSTEIN'S THEORY OF ATOMIC HEATS 31
given, it follows immediately that the mean energy U of the
linearly vibrating atom must possess, not the value kT given
by classical statistics, but the value given by the quantum
theory, namely, U =
hv
For the atom which vibrates
in space we get, therefore by an obvious generalisation in
place of the classical value 3kT, the quantum value :
^j
^-
p--
..
X
/
/
I
/
1
/
.
o 4
1
j 0.3 a,
* <?* fl
s 0,7 0,8 0,9 io 1,5 i.tf zja
x THr
FIG. 3.
The heat-content of the gramme-atom will therefore be
3Nhv
. . . . (33)
w - 1
from which we get for the atomic heat at constant volume
Einstein's formula
According to this, the atomic heat of monatomic solid bodies
is not a constant which is independent of the temperature, as
Dulong and Petit' a Law requires, but is a function of JL and
32 THE QUANTUM THEORY
is therefore in the case of a definite body (i.e. with v fixed)
a funct'ion of the temperature. Its form is such (see Fig. 3),
that for T (i.e. x co ) the atomic heat itself is
zero, and then increases gradually with increasing tem-
perature, approaching asymptotically at high temperatures
(i.e. with small x) the classical value 3R. Dulony and
Petit's Law is therefore only true in the limit for small
values of-=-jL that is, for low frequencies of atomic vibration,
fCJ.
or high temperatures, exactly as is the case with Rayleigh's
Law of Radiation. The departures from Dulong and Petit's
Law, in passing from high to low temperatures, become marked
the sooner the greater the frequency of the atoms.
3. Methods of Determining the Frequency
This frequency v the only unknown magnitude in Einstein's
formula (34) may be determined by several independent and
very noteworthy methods. One way that is always possible
is of course the following : For a given substance we choose
an experimentally well-known value of the atomic heat C*
which corresponds to a definite temperature T*. From (34)
x 2 e x C*
it follows then that = ~, an equation from which
(e* 1) oH
x = ^ can be determined, and thence v. From the v thus
found the course of the whole G v curve can be calculated for
all temperatures, and compared with experiment.
Besides this "empirical" method of determining v, there
are a number of other more " theoretical " methods which
do not require the use of the values of the atomic heat.
Einstein, 102 as far back as 1911, discovered an important
connection between the frequency v and the elastic properties
of the body. That such a connexion must exist is easily
recognised from the following considerations : imagine the
atoms of the body arranged upon a space-lattice, as in a
crystal, and suppose a certain definite atom arbitrarily dis-
turbed from its position of rest, then this atom, when released,
will execute vibrations about its position of equilibrium. If
we suppose these vibrations to be simply periodic (" mono-
chromatic") we shall, however, soon recognise that this
DETERMINING THE FREQUENCY 33
supposition is an inadmissible approximation we see that
the frequency v is the greater the smaller the atomic mass,
and therefore also the atomic weight of the body, and the
greater on the other hand the force which restores the
atom to its position of equilibrium. This restoring force is,
however, for its part the stronger, the less extensible and
therefore compressible the body is. Hence v must turn out
the greater, the smaller the atomic weight and the compres-
sibility of the substance. The exact working out of this idea
led Einstein to the formula 103
2-8. 10 7 ,..
.... (35)
Where A is the atomic weight, p the density, and * the
compressibility of the body.
A further interesting relation, which connects v with ther-
mal data, namely, the melting-point, was found by F. A.
Lindemann 1M by working out the conception that the ampli-
tude of vibration of the atom at the melting-point is of the
order of magnitude of the distances between the atoms. If
T s is the absolute melting-point, then it follows that
. . . (36)
Another formula deduced by E. Gruneisen may also be
given here :
v = 2-91.10".^-^[cj.a-i.pi] . . (37)
Here C v is the atomic heat at constant volume, and a is the
coefficient of thermal expansion ; the index means that the
value of C\a.-*pk at absolute zero is to be used.
From formulae (35) and (36) we recognise at once the
abnormal behaviour of diamond, for example, in respect to its
atomic heat. For it is known that diamond has a high melt-
ing-point and very low compressibility accompanied by a low
atomic weight. Its v is therefore comparatively large, and
it follows therefore, according to the above considerations,
that its atomic heat falls below Dulong and Petit' s value of
3B = 5-94 at comparatively high temperatures. In fact
3
34 THE QUANTUM THEORY
the atomic heat of diamond at 284 abs. is only T35 -? '-, at
413 abs. it is 3'64 ~~ ' and even at 1169 abs. it reaches only
deg.'
the value 5'24
deg.
Finally, particular importance attaches to a relation, first
discovered by E. Madelung 106 and W. Sutherland, between
the frequency v of the atoms and the optical properties
of bodies. The two investigators started in this case from
the following conception : Crystals of diatomic compounds
(binary salts), such as rock-salt (NaCl), sylvin (KC1),
potassium bromide (KBr), and others, are known to be
cubical space-lattices, in which the single atoms carry electric
charges, and therefore appear as ions. In fact, the points of
the space-lattice are occupied alternately by the positively
charged Na + (or K+) atoms, and the negatively charged Cl~
(or Br~) atoms. If an electromagnetic light- wave of frequency
v falls upon this crystal, the two ions are thrown into forced
oscillations relatively to one another, and further, on account
of " resonance," the more strongly, the more exactly the fre-
quency v of the impinging wave agrees with the natural fre-
quency v r , which lies in the infra-red, of the ions themselves.
Since the ionic vibrations are set up at the cost of the energy
of the impinging wave, this energy will be weakened (ab-
sorbed) the more during its passage through the body, the
nearer v lies to v r . On the other hand, the vibrating ions
radiate back waves of frequency v since they are compelled
to execute these vibrations, when set into forced vibration,
doing so the more strongly, the more pronounced the reson-
ance is, again, therefore, the nearer v lies to v r . Hence a
region of maximum absorption and strongest (metallic) re-
flection will lie in the neighbourhood 108 of v = v r . These
regions of metallic reflection of a given substance may be
detected by the method of " Eeststrahlen " (residual rays)
worked out by H. Rubens and E. F. Nichols.' 109 For this
purpose we only require to reflect radiation of a considerable
range of frequency about v repeatedly from the substance.
In this way all waves will be gradually absorbed except those
most strongly reflected. These are, however, just those of
NERNST'S HEAT THEOREM 35
frequency v r . They are thus "residual." The ultra-red fre-
quency v f of the ions therefore agrees with the frequency of
the residual rays. 110 On the other hand, this vibration of the
charged atoms is dependent on the elastic properties of the
substance, as we recognised in considering the formula (35).
We thus conclude that the " elastic" frequency of the atoms
of binary salts agrees to a close approximation with the
"optical" frequency of their residual rays. But since the
" elastic " frequency of the atoms determines the behaviour
of their atomic heat, the ring is thereby closed, and W.
Nernst m was thus justified in propounding the fundamental
law, that in calculating the atomic heat of binary salts, we
may simply insert for the atomic frequencies v the frequencies
of the residual rays.
In this way a number of independent ways were opened
up for determining the atomic frequencies required for the
calculation of the atomic heat. A comparison of the various
values of v determined by these different methods shows in
general satisfactory agreement, at any rate in order of magni-
tude. 112 One could hardly expect more, as we shall soon see,
in view of the many idealised conditions that were used in
the theory.
4. Nernst's Heat Theorem
With a view to discovering experimentally the general law
for the decrease of the atomic heat when approaching low tem-
peratures W. Nernst 113 began in 1910, in co-operation with
his research students, a series of masterly and widely planned
researches. For, by an entirely different route from Einstein
namely, by way of thermodynamics he also had become
convinced that the atomic heat of solid bodies must become
vanishingly small on approaching absolute zero. In his
opinion this result was only one of several consequences of a
general principle, namely, a new law of heat. 11 * This Heat
Theorem of Nernst often called the Third Law of Thermo-
dynamics states, in its original form, the following fact : If
we regard a system of condensed (i.e. liquid or solid) bodies,
which passes at temperature T by means of an isothermal
reaction from one state to another, and if A is the maximum
work which can be gained from this reaction, then
36 THE QUANTUM THEORY
gy-0 for the limit T = . . (38)
tliat is to say, in the immediate neighbourhood of absolute zero,
the maximum work which can be gained is independent of the
temperature But it follows immediately from this, if we
apply the two laws of thermodynamics, 119 that for any
reaction which changes the system from the initial condition
with energy U^ to the final condition with energy U. 2 , the
relation holds that
^-^for the limit !T0 . . (39)
Now, since -5^, if we take a gramme-atom of the substance,
gives the atomic heat, we are led to enunciate the following
rule : in the immediate neighbourhood of absolute zero, the
atomic heat of condensed systems remains unchanged during
any transformation.
Planck m has given Nernst's Theorem a still more general
form : Not only the difference of the atomic heats (before and
after the reaction) is to assume the value at absolute zero, but
also each atomic heat itself is to do the same. Thus it follows
from the extended Nernst Theorem, in agreement with the
demands of the quantum theory, that the atomic heats of
solid bodies disappear at absolute zero.
5. The Improvement on Einstein's Theory of Atomic Heats
The experiments of Nernst and his collaborators proved
quite convincingly that the atomic heat of all solid bodies
tends towards a zero value as the temperature falls. In
the main, the courses of these decreasing values showed a
notable agreement with Einstein's formula (34). At low
temperatures, however, systematic discrepancies were found
in all cases, in the sense that the observed atomic heats fell
off much more slowly than Einstein's formula demanded. 1 "
W. Nernst and F. A. Lindemann "8 tried to take these dis-
crepancies into account by constructing an empirical formula,
and this actually expressed the observations much more
accurately than did the Einstein formula. This Nernst-
IMPROVEMENT ON EINSTEIN'S THEORY 37
Lindemann formula, which is now only of historical interest,
is as follows :
hv
where x . (40)
It receives a meaning if we suppose that one half of all the
atoms vibrate with the frequency v, the other half with the
frequency n While this supposition is untenable in this
raw form, it contains a kernel of truth, namely, recognition
of the fact that the " monochromatic " theory of atomic
heats, which assumes only a single fixed frequency v for all
atoms, goes too far, being an idealisation of the real state of
affairs. Einstein, who at first, for the sake of simplicity,
reckoned with only one frequency, had himself already
recognised how matters stood, and drawn attention to the
need for amending his theory. 119 Nowadays, in fact, we think
of a solid body, say a crystal, as built up of atoms regularly
arranged upon a space-lattice, according to Bravais' concep-
tion ; and this hypothesis has been verified as a certainty
through Lane's discovery of the interference of Rontgen rays.
In such a complicated mechanical system, however, the
single atoms do not vibrate independently of one another
with a single, frequency v. But the position of equilibrium
of each atom, and thereby the type of its oscillations about
that position, is determined rather by the forces which all the
other atoms of the body exert upon the atom in question.
We are confronted with a structure which is comparable to
the one-dimensional case of a vibrating string, and which
thus possesses a whole spectrum of natural frequencies,
corresponding to the overtones of the string. If the body
consists of N atoms, it possesses in general 3N natural
frequencies, 120 of which the slowest are sound waves, while
the quickest fall in the infra-red. The most general possible
movement of each atom then consists in a super-position
of all these natural frequencies. Now, since each natural
frequency represents a linear, i.e. simple periodic, motion,
exactly like the motion of a Planck oscillator, the idea
66038
88 THE QUANTUM THEORY
naturally suggested itself, in calculating the energy-content
of the body, to allot to each natural frequency of period v the
hv
theoretical quantum amount -^ as if the natural period
e Ff -l
were identical with a linear oscillator. The total mean
energy of the body then becomes
in which the summation is carried over all 3N natural
frequencies v v v 2 , v 3 , . . . v 3 jv, that is, over the whole elastic
spectrum of the substance. By differentiation with respect
to T we obtain the atomic heat
6. Debye's Theory of Atomic Heats
The kernel of the problem thus consists in calculating the
" elastic spectrum " of a given body, that is, in determining
for any body the position of its natural periods. In this sense,
the theory has been worked out from two different sides ; on
the one hand by P. Debye, nl who took an elastic continuum
as an approximation to the actual atomically constructed
body, and on the other by M. Born and v. Kdrmdn, who
replaced the crystal of limited size by one of infinite di-
mensions. The difference between these two methods of
approximation causes the main problem, namely, the working-
out of the elastic spectrum, to be solved quite differently in
the two cases. The Debye theory, which from the outset
leaves out of consideration the crystalline, and even the
atomic, structure of the body, rests upon the classical theory
of elasticity, which, of course, treats bodies as structureless
continua. From it follows the important law : the number
Z(v)dv of all those natural periods, the frequency of which
falls within the interval v, v + dv, amounts to 123
DEBYE'S THEORY OF ATOMIC HEATS 39
Here V is the volume of the body, GI and c t are the velocities
with which longitudinal and transverse waves, respectively,
are propagated within the body. In this case, however, the
following difficulty occurs in replacing the body, which in
reality consists of N atoms, by a continuum, namely, the
elastic spectrum extends to infinity, that is, the number of
natural frequencies becomes infinitely great. For example,
the number of natural frequencies (fundamental tone and
over-tones) of a linear string of length L are
Vi = c t ^f and vi = GI ^j. respectively (i = 1, 2, . . . x> )
2-Lv aJ
according as to whether we are considering transverse or
longitudinal frequencies. The series of overtones therefore
extends without limit to infinity. In reality, however, as the
body consists of N atoms (mass-points), it may not possess
more than 3N natural frequencies. In order to attain this,
Debye helps himself out by means of the following bold
supposition. Instead of calculating strictly the elastic spec-
trum of the real body consisting of N atoms, he replaces it by
that of the continuum as an approximation, but breaks it off
arbitrarily at the 3Nth natural period. Debye thus gets the
greatest frequency v m which occurs, that is, the upper limit
of the elastic spectrum, from the condition :
therefore
[9JV I
"(H)
. (44)
The atomic heat of the body, which follows from (42), is
40 THE QUANTUM THEORY
a result which can easily be brought into the following more
simple form : 1M
The atomic heat is therefore only a function of the magnitude
x m , that is, it depends only on the ratio ^ : here ^ = ^
This result may be expressed in Debye's terms thus : reckon-
ing the temperature T as a multiple of a temperature which
is characteristic of the particular body, then the atomic heat is
represented for all monatomic bodies by the same curve. Hence
we must be able to bring the C curves of all monatomic
bodies into coincidence, if only the scale of temperature be
suitably chosen for each substance. 128 For high tempera-
tures, the Debye formula passes over, as it must do, into the
classical value of Dulong and Petit, C v = 3.R, 128 just as do
the Einstein and Nernst-Lindemann formulae. On the other
hand, it differs from these latter in falling much more slowly
at low temperatures. For while the atomic heats, according
to both Einstein and Nernst-Lindemann, fall exponentially
/ 1 constN
(with 7- e ~~T~) at low temperatures, Debye's formula leads
to the fundamental law, 127 that the atomic heats of all bodies at
low temperatures are proportional to the third power of the
absolute temperature.
It is further remarkable, that we may write formula (44)
for the maximum natural frequency in a form such that only
measurable magnitudes occur in it. For if we express the two
velocities of sound Ct and c\ in terms of the elastic constants of
the body, and replace the volume Fof the gramme-atom by
,. . atomic weight (A) .
the quotient -- density g (p) ( >, * follows that
,5-28.10*.^)
where " '
DEBYE'S THEORY OF ATOMIC HEATS 41
In it K is again the compressibility of the body, <r the
Poisson ratio, that is, the ratio of the transverse contraction
to the extension. The similarity of this formula with the
Einstein relation (35) strikes one immediately. But in this
case the second elastic constant of the isotropic body, cr,
enters into the equation as well. Altogether, the upper limit
v m of the elastic spectrum, at which, as one can show, 129 the
natural frequencies always crowd together closely, plays in
the stricter theory an analogous role to that played by the
single natural frequency v in the " monochromatic " theory.
Comparison with experiment shows 130 that the Debye
formula, at any rate for the monatomic elements such as
aluminium, copper, silver, lead, mercury, zinc, diamond, de-
scribes the course of values of the measured atomic heats
very accurately. Particularly at low temperatures, the pro-
portionality between the atomic heat and the third power of
the absolute temperature receives fair confirmation. 131 In
view of the fact that the idealised view (replacement of the
actually atomic body by a continuum) is carried very far, we
must not regard the agreement between theory and experi-
ment as self-evident. At low temperatures, Debye's idealis-
ation will justify itself. For then ^ is large, and hence the
amount of energy v is small, excepting when v itself as-
ekf - 1
sumes small values. At low temperatures, there/ore, only long
waves will contribute sensibly to the energy of a body, and hence
to its atomic heat. For long waves, however, that is, for
waves, the length of which is great compared with the dis-
tance between the atoms, the specific atomistic construction of
the body plays no part ; for them the substance is almost a
continuum. The position is quite different at high tempera-
tures, at which the longer frequencies up to the maximum v m
(that is, the shorter waves down to the smallest) furnish con-
tributions of energy. For the waves which correspond to the
highest frequencies possess lengths, as can easily be shown, 132
which are comparable with the distances between the atoms,
and for these shorter waves the medium cannot fail to betray
its atomic structure. Here, therefore, its replacement by a
42 THE QUANTUM THEORY
continuum becomes questionable since the approximation is
only very rough.
7. The Lattice Theory of Atomic Heats according to Born and
Karman. The Elastic Spectrum of the most general Crystal
At this point the above-mentioned investigations of Born
and Kdrmdn intervene, which, going beyond Debye, take
account of the real crystalline structure of the body, that is
to say, the space-lattice arrangement of the atoms. In order
to overcome the great mathematical difficulties involved, they
imagined, as has already been said, the actual limited crystal
replaced by one extended indefinitely. Thus the disturbing
effect of the surface on the interior could be eliminated, so
that now all atoms were exposed to the same conditions.
Here also the main problem is again to determine the elastic
spectrum, or if we dispense with the exact calculation of
the proper frequencies at least to discover the law, accord-
ing to which the proper (or natural) frequencies are distributed
among the different regions of frequency. This problem was
first solved by Born and Kdrmdn for regular crystals. The
laws thus obtained were then extended to the case of simple
point-lattices of arbitrary symmetry, and finally, Born de-
duced them, in his "Dynamics of the Crystal Lattice," for
the most general form of space-lattice. 133
These most general space-lattices arise from the periodic
repetition in space of a definite group of atoms and electrons
(basic group) which on the whole is electrically neutral, and
is enclosed in a parallelepiped of space, the "elementary
parallelepiped." In Fig. 4 such a lattice, in this case, how-
ever, plane, is illustrated, in which the basic group consists
of three particles (-ox). All particles form together a
simple lattice, as do the o and x particles. We have in this
way three interlocked simple lattices.
Thus, for example, the halogen compounds of the alkalies
(NaCl, LiCl, KC1, KBr, KI, BbCl, EbBr, Ebl, and so forth)
form cubic space-lattices, in which the lattice points are
alternately occupied by the positive alkali ion and the negative
halogen ion (see Fig. 5). If we regard the whole cube here
pictured as the " elementary cube," then the basic group would
LATTICE THEORY OF ATOMIC HEATS 43
contain eight particles, namely, four ions of each sort (they are
numbered here). We have thus eight interpenetrating simple
lattices. Every four of them would, however, consist of the
same kind of particle. Hence it is advisable to select in this
case in place of the cube the rhombohedron (double-lined in
FIG. 4.
the figure) as the elementary parallelepiped. Then the basic
group consists only of the two different particles 1 and 8, of
which the one lies in a corner, the other in the middle of the
parallelepiped. In fact we can get the whole lattice by displac-
ing the basic group in the direction of the three rhombohedral
edges, a distance equal to a whole multiple of the length of
44 THE QUANTUM THEORY
the edge. The lattice consists therefore, according to this
view, of two interlaced simple cubical atomic lattices. Further-
more, they are " surface-centred " lattices, that is to say, such
that not only the corners of the cubes, but also the middle
points of the cube-surfaces, are occupied. If in the most
general case the basic group contains s different particles, the
lattice consists of s interlaced simple lattices.
FIG. 5.
In order now to get a general view of the laws which
govern the elastic spectrum of such a most general crystal,
we proceed according to Born and Kdrmdn as follows : We
imagine an elastic wave of definite wave-length and definite
direction (the normal to the wave front) passing through the
crystal. For each wave thus defined there are 3s natural
LATTICE THEORY OF ATOMIC HEATS 45
frequencies with periodicities v l v 2 v 3 . . . v 3 .. The first three
frequencies v lt v.,, v 3 correspond to those natural frequencies
of the crystal, by which the single interpenetrating simple
lattices are similarly distorted to a first approximation with-
out being compelled to move relatively to one another.
These are the three ordinary acoustic natural periods (one
longitudinal, two transverse). The remaining 3(s - 1)
frequencies, on the other hand, correspond to another type
of motion of the crystal, namely, to those natural frequencies
with which the single simple lattices oscillate with respect to
one another without distortion. If the basic group contains
only one particle (s = 1), i.e. if the crystal consists of only a
simple lattice, this second type of motion disappears alto-
gether, and we are left with only the three acoustic natural
frequencies v v v. 2 , v 3 . If, on the other hand, we are dealing
with a crystal, say of the halogen compounds of an alkali,
for example, rock-salt (NaCl), s = 2, there exist, as we have
seen, besides the three acoustic oscillations, three further
natural frequencies of the second type. In consequence
of the regular crystal character of the alkaline halides, these
three natural frequencies exactly coincide, at any rate for
long waves, and give rise to that motion in which the sodium
lattice vibrates approximately as a rigid structure against the
likewise rigid chlorine lattice. We see at once that it is
just the natural frequency last considered that will play the
chief part in the optics of these crystals. For when an
electromagnetic wave meets the crystal, the sodium ions
are driven by the electric force of the wave to the one side,
and the oppositely-charged chlorine atoms are drawn to the
opposite side. It is thus just the type of vibration described
above that is brought about. If the frequency of the
external wave approaches closely to that of the natural
period, resonance occurs. These infra-red vibrations, there-
fore, are what determine the course of the refractive index,
especially in the infra-red. They are the so-called " infra-
red dispersion frequencies." It is also in their neighbourhood
that the places of metallic reflection lie which are detected
by the method of residual-rays.
What has just been stated for the special case s = 2
(alkaline halides) may, of course, be immediately generalised.
46 THE QUANTUM THEORY
For if the basic groups consists of s different particles, it
is just the 3(s - 1) natural frequencies that determine the
dispersion of the crystal. Among them are those, in the
neighbourhood of which the regions of metallic reflection
(residual rays) lie. If the basic group contains p positive
atomic residues and s - p electrons, the frequencies v 4 . . . v 3 ,
fall correspondingly into two classes : the first class consists
of 3(p - 1) infra-red frequencies, which arise from the
atomic residues ; the second consists of 3(s - p) ultra-violet
frequencies, which are to be ascribed to the influence of the
electrons. The infra-red natural frequencies decide the
course of the refractive index in the infra-red, the position
of the residual rays, and, as we shall see, the atomic heats ;
the ultra-violet natural frequencies, on the other hand, deter-
mine chiefly the refractive indices in the visible and ultra-
violet. Incidentally, the general lattice-theory of Born 134
confirms the law previously enunciated by Haber 13S that the
frequencies of the first class (infra-red) bear the same ratio to
the second (ultra-violet) class, as regards order of magnitude,
as the square root of the mass of the electron bears to the
square root of the mass of the atom.
After this digression let us now return to our starting-point.
Up to the present we have always considered a wave of
definite length A. and with a definite normal direction n, and
we have seen that corresponding to it there are, in the most
general case, 3s natural frequencies i/j . . . v 3S . Let us now
allow the wave-length X to vary continuously, keeping the
wave-direction constant, by going from infinitely long waves
to the smallest. Then each of the 3s natural frequencies will
also vary continuously, and will pass through a continuous
range of values. In other words, the 3s natural frequencies
are certain functions of the wave-length X :
v< -/<(*).
From this, however, we learn the fundamental fact that all
these ranges of values of the single natural frequencies are
only finite in extent and that, therefore, each of the 3s continua
of frequencies automatically breaks off at a higJiest limiting
frequency. " Automatically," i.e. without our arbitrary assist-
ance (as in Debye's case), solely on account of the analytical
LATTICE THEORY OF ATOMIC HEATS 47
form of the function /,-. This is explained by the fact that
the wave-length X of possible waves in the crystal has a
lower limit set to it : waves of length below a certain lowest
value cannot exist. This is most simply recognised from the
following instructive example. If we consider a simple
cubical lattice having the atomic distance a, and examine,
for example, longitudinal waves, which are being propagated
along an edge of the cube so that all atoms on an edge at
light angles to this side oscillate in the same phase in the
direction of the edge then we see at once that the smallest
wave that is possible here has the length X m j n = 2a. For
this wave, namely, successive planes of the cube swing
in opposite phase, that is, " against " one another. The
functional relation between v and X assumes the special
form : 138
For infinitely long waves (X = co ), v = ; if we pass on
to shorter waves, v increases continuously, until, for X = 2a,
it reaches its maximum value v m . At this limiting frequency
v m the range of possible v's breaks off automatically.
Up to the present we have given the wave-direction (n, the
direction of the normal) a certain fixed value, and have
allowed the wave-length X to vary. We now give the wave-
direction by degrees other values, and at each step we allow
the wave-length to vary from the value GO to the least
possible value. Then the nature of the functional dependence
of the magnitude vt or X, and the position of the limiting
frequencies also change continuously with the wave-direction,
so that we may say : the 3s natural frequencies are, in
general, continuous functions of the wave-length X and of
the wave-direction n :
vi =/<(X, n), (i = 1, 2, 3, . . . 3s) . (48)
In it, each of the functions / t - breaks off automatically for a
minimum value of the wave-length at an upper limit
(vi)max, which itself still depends on the wave-direction.
These equations express the law of dispersion of waves in
crystals, for they determine for each wave the 3s frequencies
48 THE QUANTUM THEORY
vf and hence also tell us how the rates of propagation
q f = vi'X depend on the wave-length and the wave-direction.
The dispersion law becomes particularly simple in the region
of long waves : for the three acoustic vibrations the re-
lations 137
A A. A.
hold. In them the three magnitudes q^(n), q. 2 (n), and q s (n)
are three, in general different, functions of the wave-
direction. And further, these are the three velocities of
propagation of the three acoustic vibrations. In the region
of long waves, therefore, the three velocities of propagation
of the three slow acoustic vibrations are independent of the
wave-length to a first approximation.
The dispersion law (for long waves) assumes a very
different appearance for the 3(s - 1) rapid vibrations
v., V K . . v,.. It assumes the form
(i = 4, 5, . . . 3s) . (50)
A
here the v9's are constants, the p(w)'s are again certain
functions of the wave-direction. The velocities of propaga-
tion here assume the values
q t = Vi \ = v\ + p { (n) . . . (51)
and would thus be linear functions of the wave-length.
We may summarise thus : the elastic spectrum of the most
general crystal, the basic group of which contains s particles,
consists of 3s separate parts (" branches "). Each part consists
of a finitely extended continuum of frequencies. The three first
parts contain the totality of all sloiv, acoustic natural fre-
quencies (sometimes called "characteristic"). The remaining
3(s - 1) parts include the rapid (infra-red and ultra-violet)
natural frequencies, which play the chief part in determining
the optical dispersion and the positions of metallic reflection.
% 8. Continuation. The Law of Distribution of the Natural
Frequencies
While this knowledge of the general character of the elastic
spectrum is, as we shall soon see, of great value, it is none
LATTICE THEORY OF ATOMIC HEATS 49
the less insufficient for the question of the energy-content and
molecular heat of the crystal, inasmuch as, even for the
simplest crystal, a strict calculation of the elastic spectrum
is not possible at the present time. We know, however, on
the other hand, that we need not know the whole details of
the elastic spectrum to calculate the energy-content and the
molecular heat, but that it suffices to know the law according
to which the natural frequencies are distributed over the
elastic spectrum (or its individual "branches"). This is
the more true, the closer together the natural frequencies
lie. Now, in reality the finite crystal possesses, if it con-
sists of the basic group (of s particles) N times repeated,
SNs natural frequencies, which are distributed so that N fre-
quencies fall to each of the 3s branches of the spectrum. If N
becomes infinite, the N individual natural frequencies of each
branch merge into one another to form a continuum, and we
get exactly the elastic spectrum that we have just been con-
sidering. We see from this, that the more we are justified
in replacing the finite crystal by one of infinite extent the
better our results if we know only the distribution law of the
natural frequencies (without knowing their position exactly).
The law of distribution of the natural frequencies, which
was discovered by Born and Kdrmdn and extended by Born
in his " Dynamics of the Crystal Lattice " to the most general
type of crystal, may be formulated thus : Select from the
totality of all elastic loaves the small group, whose lengths lie
between A. and \ + d\, and ivhose normal direction lies in the
elementary solid angle 138 dO. Each of the 3s branches of the
Y
spectrum then contribute dXdto natural frequencies to this
group. Here V denotes the volume of the finite crystal.
9. Continuation. The Atomic Heats at Low, very Low, and
High Temperatures
The knowledge of this law of distribution allows us to
write down at once the thermal capacity of the crystal con-
sisting of Ns particles. From (42) it is :
50
THE QUANTUM THEORY
. (52)
This formula is to be interpreted as follows : the natural fre-
quencies vi are, by (48), to be expressed as functions of the
wave-length X and the wave-direction n : then the integra-
tion is to be performed with respect to X from the smallest
wave-length X m (w), which itself depends upon the wave-
direction n, up to the maximum X = oo . The result of this
integration still depends on the wave-direction and the index
i. Finally, integration is to be performed over all directions
(that is, over all elementary solid angles between and 4?r)
and summation over all 3s branches of the spectrum. But
we have seen that the 3s branches of the spectrum fall into
two groups. The first 3 branches (i = 1, 2, 3) contain the
totality of slow acoustic natural frequencies; for these
branches we have the dispersion law (49) which is valid for
long waves. The remaining 3(s - 1) branches contain the
totality of the quick (infra-red and ultra-violet) natural fre-
quencies, with the entirely different type of dispersion law
(50), which also holds for long waves. Hence the sug-
3t
gestion naturally occurs of dividing the sum J of (52) into
i = l
two parts, corresponding to the two different groups of fre-
quencies and of writing
r p = r? + r<*>
where
JL i (53)
r ( > = kv ; r<*> = kv
These still very complicated formulae may, according to
Born, be brought into a very simple and comprehensive form
LATTICE THEORY OF ATOMIC HEATS 51
by limiting our considerations to low temperatures and in-
troducing certain approximations. As we have already re-
cognised, at low temperatures only the long waves contribute
to the energy-content. Hence we shall apply in formula (53)
all those approximations which are introduced by confining
ourselves to long waves. Let us consider first r ( ;p. Here we
set in place of the v{S of (50) the constant values v., which
are independent of the wave-length A. and of the wave-
direction. If we do this, we can place the constant factors
in front of both integration signs, and write
!-*> V^,. FU>3 , where,,
The factor in square brackets has, however, a simple meaning.
From the law of distribution of the natural periods we see,
namely, that this factor gives the sum-total of all natural
frequencies that occur in one of the 3s branches of the
spectrum ; it therefore has the value N, which as has already
been said, is the number of basic groups which go to make
up the crystal. If we choose the piece of crystal under con-
sideration such that its size is so that N is equal to the
Avogadro number, then if we remember that Nk = R for
I* 2 ), the expression
3
follows. If we compare this result with (34) we see that rop
excepting for the missing factor 3 consists of 3(s - 1)
Einstein functions. We write the expression in the form
M where ^ =_^ . (55)
in which the abbreviation is obvious. The fact that, in using
52 THE QUANTUM THEORY
these approximations, we come across Einstein factors, i.e.
that we encounter the " monochromatic " theory, might have
been anticipated. For since we treated the v/s here as con-
stants that are quite independent of wave-length and wave-
direction, these vibrations represent processes which have
nothing to do with the propagation of elastic waves in the
crystal as a whole : and this means that the individual
particles, uncoupled as it were, perform 3(s -- 1) mono-
chromatic vibrations.
The approximate evaluation of the first part rW is quite
different. For here we have to use for the frequencies
v i V 2> V 3 * ne relations (49), which connect the three acoustic
natural frequencies with wave-length and wave-direction.
Here we have therefore to deal with three real elastic oscilla-
tions, which are propagated in the crystal with the three
different acoustic velocities q-^ri), q*(ii), q s (n), each of which
depends on the direction (n). The crystal acts here as a
dynamic whole, exactly as in Debye's point of view. Hence
we may conjecture that r ( J> allows itself to be brought
into the form of three Debye functions (45). The more
exact calculation confirms this supposition, and gives us 1M
(56)
which, taking Debye's formula (45) into consideration, we
may write in the following immediately intelligible form :
I^-IDft . . (57)
i=l
The three magnitudes Xi here play the part of three upper
limits of frequency. Their values are
where the three magnitudes qt represent certain mean direc-
tions of the acoustic velocities, which therefore no longer
LATTICE THEORY OF ATOMIC HEATS 53
depend on the wave-direction. From (55) and (57) we get for
the thermal capacity of the piece of crystal considered
- (59)
Now, since JV particles of each of the s different kinds of
particles are present, that is one gramme-atom of each kind
of particle exactly for N is the Avogadro number the piece
of crystal contains s gramme-atoms of different sorts of
particles. If, therefore, we cut the crystal into s equal
N
pieces in such a manner, that each piece comprises only -
basic groups, then each of these pieces contains a so-called
"mean" gramme-atom. Hence if we now consider only a
p
single one of these pieces, its thermal capacity is ; we call
S
it the " mean atomic heat " C v , and we may write
3
D + (* . . (60)
Here the #/s have the same meaning as in (58). For the
N
piece of crystal now under consideration consists of basic
s
groups, and has therefore the volume . Formula (58),
however, obviously remains unchanged when we replace in
it N and V by and . The quantity -, the volume of a
s s s
mean gramme-atom, is also called the mean atomic volume.
In the case of chemical compounds, in which several sorts
of atoms occur in the basic group, and also in the case of
polyatomic elements, in which the basic group contains several
particles of a like sort, we frequently speak of the molecular
heat. In doing so, we follow the usual chemical conception,
inasmuch as we imagine the s particles of the basic group
divided into one or several sub-groups, and regard each sub-
group, taken alone, as a molecule. If then the molecule
54 THE QUANTUM THEORY
contains q atoms, then qC,, is the mean molecular heat; for
example, the basic group of rock-salt (NaCl) contains one
sodium ion and one chlorine ion. The whole piece of crystal,
which, by definition, contains - = basic groups, comprises
S A
therefore sodium ions and the same number of chlorine
ions, that is to say - " NaCl-molecules." q is in this special
case equal to 2. Hence 2C,, represents the thermal capacity
of N " NaCl-molecules,'' that is, the mean molecular heat of
rock-salt.
If among the s particles of the basic group there are p
atomic residues and s p electrons, the number of Einstein
factors in (59) reduces to 3(p - 1), since the 3(s - p) ultra-
violet frequencies arising from the s - p electrons contribute
only in a vanishingly small degree to the atomic heat as com-
pared with the infra-red. We thus arrive at the law : the
mean molecular heat of a crystal wJiose basic group includes p
(similar or different) atomic residues, is made up, at a suf-
ficiently low temperature, of three Debye terms (with, in general,
three different upper limits of frequency) and 3{p - 1) Einstein
terms (in which the 3(p-I) infra-red natural frequencies for
long waves appear as frequency numbers).
When we descend to the lowest temperatures, the Einstein
terms disappear exponentially, and only the three Debye terms
remain, for these, as we know, decrease much more slowly.
In them we can further replace all the upper limits of the
three integrals (see (56)) by GO , so that the integrals thereby
become numerical constants. Eemembering (58) we get the
fundamental law, that the molecular heat of every crystal at
the loivest temperatures is proportional to the third power of
the absolute temperature. So the general lattice theory con-
firms Debye's result. The formula obtained has the following
simple form : 141
where VA is the " mean atomic volume "
mean atomic weights
- _ - . - i* - 1
TESTS OF THE BORN-KARMAN THEORY 55
and q represents a quantity which, if suitably defined, may be
called the mean acoustic velocity, introduced in place of the
three different acoustic velocities q lt q 2 , q s .
Also in the other extreme case, for high temperatures, a
very useful formula can be obtained, as H. Thirring iK
showed. He started from (52) and developed the exponential
functions in series. The following value is then obtained for
the mean atomic heat :
where the coefficients J v J 2 , J 3 , . . . depend in a complicated
manner on the elastic constants of the crystals, the atomic
masses, and the atomic distances.
10. Tests of the Born-Karman Theory
How do matters stand with regard to the testing of the
Born-Kdrmdn Theory? We see at once that it is incom-
parably more difficult than in the case of Debye's Theory : for
even in simple cases, the calculation of the mean atomic
heat of a crystal is very complicated, and requires above all
a more exact knowledge of its elastic behaviour than we at
present possess. Only by restricting our attention to low
and very low temperatures on the one hand, where the
formulae (60) and (61) may be applied, and, on the other,
to the region of high temperatures, within the limits of
applicability of Thirring's formula (62), are we enabled to
carry our calculations for a number of simple substances to
the point of comparison with experimental results. Born
and Kdrmdn themselves, in one of their first publications 14a
tested the formula (61), valid for the lowest temperatures
(Debye's TMaw), by comparing its results with those of
experiment. They limited themselves in this case to metals
(Al, Cu, Ag, Pb) which, however at any rate in the usual
form are not proper crystals, but irregular crystalline
aggregates. For this reason, they proceeded as if the metal
were an isotropic body, and obtained the mean acoustic
velocity the only quantity in (61) which in general requires
56 THE QUANTUM THEORY
extensive calculation from the following relation which
holds for isotropic bodies : 1H
Q 1 O
N+! < 68)
Here q l and q t are the velocities of propagation of the
longitudinal and transverse elastic waves, magnitudes, there-
fore, which may be simply calculated from the two elastic
constants of the isotropic body and its density. 1 * 9 The
agreement of the values of C v thus found with the experi-
mental data is, especially in the case of Al and Cu (and also Pb),
quite good. A. Eucken w has, however, pointed out rightly,
that no weight should be attached to this agreement. For
the values of the elastic constants which Born and Kdrmdn
used for calculating q t and q t are those which are correct at
the ordinary room temperature. If we take their dependence
on temperature into account, the good agreement between
theory and experiment disappears. Metals are, indeed, not
isotropic bodies, and hence it is not permissible to use the
observable elastic constants, which depend upon temperature,
in calculating q.
Matters are much more favourable in the case of real
crystals, in which, as experiments by E. Madelung i show,
the elastic constants vary very little with temperature. But
here the calculation of the mean acoustic velocity q gives rise
in general to notable difficulties, 1 * 8 which may, however,
be cleared away in simple cases by a very practical method
due to L. Hopf and G. Lechner. 9 Eopf and Leclmer were
thus enabled successfully to carry out the calculations for
sylvin (KC1), rock-salt (NaCl) fluor-spar (CaF 2 ) and pyrites
(FeS 2 ). They proceeded to calculate the quantity q from the
observed values of C v , assuming the correctness of formula
(61), and they then compared these with the value of q
calculated from elastic data. The result showed very satisfac-
tory agreement. 1 o
It is of particular interest to test the very clear formula
(60) which gives the mean atomic heat as a sum of three
Debi/e, functions and 3(s - 1) Einstein functions. Here the
three infra-red natural frequencies v, vj?, v coincide, and the
TESTS OF THE BORN-KlRMAN THEORY 57
three Einstein functions become equal to one another. If we
introduce the further approximation of replacing the three
different quantities Xi in the Debye formula by a mean value
x, it follows that
C, = \{D(x) + E(x)} . . . (64)
In this we use the value of x deduced from formula (58)
by merely replacing <?, in it by a mean value q, which can be
calculated by the method of Hopf and Lechner just mentioned.
j, o
x, on the other hand, according to (54), = ^ , where v is the
infra-red natural frequency of the crystal (for long waves),
which may be determined from the dispersion in the infra-
red or by the method of residual rays.
Formula (64) had already been given, previously to Born,
by W. Nernst, in who, however, based his argument on a
supposition which is no longer tenable. Nernst started
from the conception that, for example, in the case of rock-
salt, the NaCl-molecules are located upon the points of the
space -lattice, and that the most general state of oscillation
of the lattice arises from the superposition of two modes of
motion, firstly the oscillation of the whole molecules in the
lattice-structure, which give a Debye term, and secondly the
intra-molecular oscillations of the two atoms, which, being
almost monochromatic, lead to an Einstein term. The
agreement of the Born-Nernst formula (64) with the ex-
perimental data is not very satisfactory in the case of NaCl
and KC1, but much better in the case of AgCl, which belongs
to the same crystal type. 182 The reason for this is believed
by E. Schrodinger *** to lie in the excessively rough ap-
proximation inherent in formula (64).
Finally, Thirring's formula (62) has also been tested, by
Thirring himself, IM for NaCl, KC1, and, by neglecting certain
factors, for CaF 2 and FeS 2 . Taking into account the variation
of the elastic constants with temperature (which, however, is
to be regarded as uncertain and provisional since the values
are only obtained by interpolation) he found good agreement
between theory and experiment. In connection with the
Thirring formula, Born 1 ** has also calculated the atomic
heat of diamond and compared it with experiment. Since in
58 THE QUANTUM THEORY
this case, however, the elastic constants were unknown, Born
proceeded to evaluate the curves of atomic heat for various
possible values, and to select from them that curve which
conformed most closely to the results of observations. Thus,
for example, the value O63 x 10 ~ 12 , m was obtained for
LdyneJ
the compressibility ; this is in satisfactory agreement with the
value, probably too small, measured by W. Bichards, viz.
0-5x10-4^1.
LdyneJ
From all this we see that the possibilities of testing the
Born-Kdrmdn Theory of Atomic Heats, partly on account of
the great difficulties of calculation, partly on account of our
insufficient knowledge of the elastic behaviour of crystals, are
exceedingly sparse, so that for the present Debye's much more
tractable formula (if necessary, with the addition of Einstein
terms) appears more useful. If, in spite of this fact, so much
space has been devoted here to the Born-Kdrmdn Theory, the
reason is to be sought in the conviction that this theory has
gone much further than that of Debye into the kernel of the
matter. For, without a more exact treatment of the structure
of the space-lattice and its dynamics, our knowledge of the
nature of the solid state must without doubt remain faulty.
ii. The Equation of State of a Solid Body
Linking up with this new development of the theory
of atomic heats, a number of investigators, chiefly E.
Gruneisen, 1 * 6 S. Ratnoivski, 161 and P. Debye,*** have worked
out a theory of the solid state with the object of creating as
a counterpart to the Kinetic Theory of Gases a Kinetic
Theory of Solids. One of the main problems in this con-
nexion is to formulate an " Equation of State," that is, a
relation between pressure (p), volume (V), and temperature
(T), a problem, which, according to the doctrine of thermo-
dynamics, is to be regarded as solved as soon as the " free
energy " F of the body is known as a function of the tempera-
ture and the volume. 189 Then the pressure, for example, will
follow from the simple equation
EQUATION OF STATE OF A SOLID BODY 59
which, as a relation between p, V, and T, gives the equation
of state at once. If this is known, we have mastered quanti-
tatively the behaviour of the body for all changes of state.
For example, the coefficients of expansion a, and the com-
pressibility K, result from the well-known formulae
*
(F is the volume at the zero-point.)
P. Debye 16 was the first to draw attention to the fact that
the model of the solid body which forms the basis of the
atomic heat theories of Einstein, Debye, and JBorn-Kdrmdn,
is necessarily too highly idealised ; for this idealised solid
body has, as is easily seen, a zero coefficient of expansion. In
fact, if, as has always been assumed hitherto, the forces
which pull the atoms back into their position of equilibrium
are proportional to the first power of their relative displace-
ments (assumption of quasi-elasticity, Hooke's Law), then
the atoms will execute symmetrical oscillations about this
position of rest. If this supposition, viz. Hooke's Law, be
valid for all temperatures, then the mean volume of the
body that is, the volume that it possesses when all atoms are
exactly in their positions of rest must be just as often over-
shot as undershot, however great the amplitude of the heat-
vibrations may be. Hence, if we warm the body from zero
until it possesses the volume VQ, and if we assume that all
atoms are at rest at zero, then its mean volume at any tem-
perature will also be equal to F . The body, therefore, does
not change its mean, observable volume with rise of tempera-
ture ; its coefficient of expansion is therefore 0. If we desire to
represent the actual behaviour of the solid body, namely, its
expansion when heated, as known to us from thousandfold ex-
perience, we are necessarily obliged, according to Debye, to re-
place Hooke's Law of Force by an expression involving higher
powers of the variation of atomic distance. Then the oscilla-
tions of the atoms become unsymmetrical, and there occurs a
displacement of their position of rest as the energy of vibration
increases. If we arrange the generalisation of Hooke's Law
so that a greater force is necessary to bring the atoms nearer
together than to separate them, then the change in the
60 THE QUANTUM THEORY
position of rest occurs in such a manner that for increasing
energy of vibration, that is, for rise of temperature, the
relative distances of the atoms increase, and hence the body
increases in volume. Debye has extended the theory in this
sense. Among other things this gives us the law previously de-
duced by Griineisen m that at sufficiently low temperatures the
thermal coefficient of expansion a is proportional to the specific
heat. Moreover, the very small change in compressibility
with temperature is well accounted for on Debye 's Theory.
12. The Thermal Conductivity of Solid Bodies according to Debye
The importance of Debye' s Theory is by no means confined
to thermal expansion. On the contrary, it became manifest
that another important group of phenomena require this
generalisation of Hooke's Law. In the idealised solid body,
in which the elastic forces obey Hooke's Law, the elastic
waves will become superposed without disturbance, and will
penetrate the whole body without becoming weakened. If
we imagine the idealised body as a horizontal, infinitely
extended plate of finite thickness, and if we transmit a
powerful motion (high temperature) to the upper layer of
atoms, while we keep the lower layer at rest (i.e. at zero
temperature), then an elastic energy current (heat current)
will pass continually from above to below. An energy
gradient (temperature gradient) does not, however, exist
in the body, since, on account of the undamped character
of the wave, the mean density of energy is everywhere the
same. Since, in general, the conductivity for heat is equal
to the flux of heat divided by the gradient of temperature,
it follows that the idealised solid' body possesses an infinite
thermal conductivity. The case becomes different, however,
if we extend Hooke's Law in the manner described, and thus
pass over to the " real " solid body. The waves in the body
will then, on account of the departure of the equations of
motion from linearity, no longer pass over one another un-
disturbed. On the contrary, an oscillation already present
will, in consequence of the fluctuations in density caused by
it, disturb the oscillations superimposed upon it, with the
effect that a scattering, and therefore a weakening of the
waves in the body results, in precisely the same way as a
THE ELECTRON THEORY OF METALS 61
" cloudy " medium scatters and weakens light passing through
it. Hence, in the case taken, a temperature gradient is set
up in the plate from the top to the bottom. In the case of
the real body we thus arrive at a finite thermal conductivity.
The mathematical development of this conception led Debye to
the law 162 that the thermal conductivity of crystals is inversely
proportional to the, absolute temperature (if we confine ourselves
to temperatures which are so high that classical statistics are
applicable). This deduction seems to be in excellent agree-
ment with experimental results obtained by A. Eucken. 1 * 3
13. The Electron Theory of Metals and its Modification by the
Quantum Theory
If matters are already complicated in the intrinsically
clear case of crystals, the position becomes still more
difficult when we turn to metals which, in general, con-
sist of an irregular conglomerate of crystallites. In this
case the conductivities, namely, of heat and electricity, are
particularly deceptive. According to the classical theories of
P. Drudew E. Riecke, and H. A. Lorentz, these pheno-
mena are brought about by the free conductivity-electrons,
which, like gas-molecules, fly about in the space between the
fixed atomic residues, exchange energy with these upon
collision, and so take part in the establishment of thermal
equilibrium. Thus the conduction of electricity is explained
as follows : in a piece of metal of uniform temperature, an
equal number of electrons fly, on the average, in each
direction through an element of surface. Hence, on the
average, there is no transport of electrical charges through
this element of surface, that is, no electric current is flowing
in the piece of metal. If now we apply a potential difference
to the ends of the metal, an electric field exists in the metal,
and this field impresses upon the electrons during their " free
paths " (i.e. their paths between two encounters with atoms)
a certain one-sided additional velocity which is super-
imposed upon the irregular heat-motion. Now, therefore,
more electrons will pass per second through the element of
surface in one direction than in the other, and since the
electrons carry a negative charge, and so move against the
field, i.e. in a direction opposite to the field, we have now an
62 THE QUANTUM THEORY
electric current in the metal. The mathematical calculation
of this simple conception gives for the electrical conductivity
o- of the metal 167
N^
'-2S5 ' ' ' ' (67)
Here N is the number of electrons per unit of volume, e and
ra charge and mass of the electrons, q their average velocity,
and I their free path. If we write the expression (67) in the
form
we may, according to the assumptions of the classical theory,
replace the mean kinetic energy \m(f of the electrons by %kT.
For since, as we assumed, the electrons take part in
establishing heat-equilibruim, the law of equipartition of
kinetic energy applies to their motion, and there is thus
allocated to each of the three degrees of freedom of the
electrons the energy ^kT. In this way we arrive at the
formula
. - (676)
Analogously, we get from Drude's Theory the coefficient of
thermal conductivity 168
. . . . (68)
so that a combination of the two formulae leads to the
fundamental relation
I-f -T .... (69)
which is the law of Wiedemann- Franz and of Lorenz. m It
states that the ratio of the thermal to the electrical conductivity
has the same value for all pure metals and is proportional to
the absolute temperature.
Thus all appeared in the best of order. The classical
theory appeared here also to have worked successfully and
the law of equipartition celebrated a triumph. But upon
closer inspection, gaps appeared in the apparently solid
theoretical structure, and serious doubts arose. For if the
THE ELECTRON THEORY OF METALS 63
free electrons really took part in the thermal equilibrium,
and hence claimed their full share, fNfcT (per unit of
volume), in the equal division of kinetic energy, then this
share of energy should be plainly noticeable in the atomic
heat of the body, namely, to the extent of f N*&, where N*
denotes the number of electrons in a gramme-atom. Such
an increase in the atomic heat of the metals as compared
with the non-metals (which contain no, or vanishingly few,
free electrons) has never been observed. This difficulty could
be avoided by assuming that the number of electrons is small
compared with the number of atoms per unit volume, and
then their contribution to the atomic heat would be relatively
small. But then we should expect from (676) much smaller
conductivities than experiment has disclosed, unless we were
to assume high values, that are improbable, for the mean free
path, "a
Further, H. A. Lorentz^ 1 has shown, as we have seen,
that, if the law of equipartition for the motion of the electrons
is assumed, the metals would radiate in the region of long
waves according to Bayleigh's Law, whereas we have un-
questionably to expect, especially at low temperatures, the
radiation to take place according to Planck's Law.
The calculated dependence of the conductivity on tempera-
ture can only be made to agree with experience by making
particular assumptions at high temperatures, whereas no
assumptions seem to be able to make calculation and obser-
vation agree for low temperatures. At high temperatures
the resistance of the metals increases proportionally to
the temperature, that is, o- decreases with . This can
only be reconciled with (676), if the product Nlq is inde-
pendent of the temperature. If we assume with /. J.
Thomson 112 that N increases proportionately to *JT, then,
since q is likewise proportional to >JT, I must decrease with
=f a hypothesis which, as we shall see, has latterly been
upheld by several investigators.
Now, although the agreement between theory and experi-
ment could thus be compelled by special assumptions at high
temperatures, the region of low temperatures revealed itself
64 THE QUANTUM THEORY
as the vulnerable point of the theory. For experiments by
H. Kamerlingli-Onnes m in the laboratory for low tempera-
tures at Leyden had shown that the resistance of metals
at very low temperatures (the experiments extended as far
down as 1'6 abs.) falls away to a quite extraordinary degree,
and practically disappears before the zero-point is reached.
At any rate, the resistance cannot, as follows in view of what
has just been said from formula (67&), sink proportionately
to only the first power of the temperature ; on the contrary
the fall is without doubt proportional to a higher power.
That the Wiedemann- Franz Law also ceases to be valid in
this region, has been proved by experiments of C. H. Lees 174
and W. Meissner.
In order to escape from all these difficulties the quantum
theory was appealed to, and attempts were made, in the most
varied ways, to make it harmonise with the existing theory.
A first attack was ventured by W. Nernst 176 and Kamerlingh-
Onnes, 111 who gave for the resistance of the metals empirical
formulae which linked up directly with the form of Planck's
energy equation (9) and which gave the change in the resist-
ance with temperature satisfactorily. F. A. Lindemann m
and W. Wien 119 conceived more detailed theories. Linde-
mann accepts in his first paper J. J. Thomson's hypothesis,
according to which N is proportional to >JT, and retains the
equipartition law for the motion of the electrons, so that q
also becomes proportional to JT. Then, according to (676),
the variation of the resistance - with the temperature depend*
<r
entirely on the mean free path I. But this is, according to
well-known considerations of the theory of gases, the greater
the smaller the " radius of action " of the metallic atoms ; for
the electrons can pursue greater paths freely, i.e. without
collisions, the smaller the hindrances set in their path. The
novel part of Lindemann's Theory is the fact that he brings
the radius of action of the atom into relation with its ampli-
tude of swing in its heat-motion. For it is at once obvious
that the atoms in this heat-motion will cover a greater space
in a given time, and their sphere of action will be the greater,
the larger their amplitude of oscillation, i.e. the higher the
temperature. Thus the mean free path also becomes a
THE ELECTRON THEORY OF METALS 65
function of the temperature, inasmuch as it is brought into
relation with the energy of vibration of the atoms. But, for
the latter term, Lindemann inserts the value given by the
quantum theory, and finds for the resistance the formula 18
hv ' l~fc, ' ~ "" ' * ('")
F-I V^-i
where v denotes the frequency of the atoms (again the mono-
chromatic theory) ; A and B are constants. For high tem-
peratures W then becomes proportional to the temperature
T; for low temperatures W decreases exponentially with
hv
e 2kT to a constant value B 2 . With the help of this formula,
Lindemann succeeds in representing the observations quite
well (the formula contains two constants which can be mani-
pulated) ; but, since the law of equipartition has been retained
for the electrons, the difficulties of the excessive atomic heat
and of Rayleigh's radiation formula remain. Moreover, this
theory is unable to explain the departures from the Wiede-
mann-Franz Law at low temperatures; for the mean free
path I the only quantity dependent on T which occurs in a-
disappears entirely from the formula (69).
W. Wien attacked the question much more radically than
Lindemann. In order once and for all to get rid of the
contribution of the electrons to the atomic heat this is the
weak point of all theories which make use of the law of
equipartition he assumed that the electrons do not take
part in the thermal equilibrium, but possess a velocity q
which is independent of the temperature. Moreover, he
makes the number N of the electrons per unit volume equal
for all temperatures. Then, according to (67), the variation
of - with temperature is again determined only by the
dependence of the mean free path I on the temperature.
Wien, in a manner similar to that of Lindemann, connects I
with the energy of vibration of the metallic atoms, taking,
however, the complete elastic spectrum into account accord-
ing to Debye. He thus gets for the resistance the value
5
THE QUANTUM THEORY
v m
vdv
W = const. I A, ... (71)
( vdv
. I A*
3f--\
For high temperatures this formula gives W = const. T, i.e.
proportionality with the temperature. For low temperatures
it follows that W =- const. T 2 , i.e. a parabolic decrease.
The observations are very well represented by Wien's formula.
But, above all, the unsatisfactory fact remains that this
method does not lead us on to a theory of heat-conduction,
unless we make new assumptions, nor to the Wiedemann-
Franz Law. For, by the condition that the motion of the
electrons takes place quite independently of the temperature,
Wien has taken away the possibility of also ascribing the
transport of heat to the electrons.
This difficulty arises again in a more recent paper of
F. A. Lindemann in which, in continuation of the con-
ceptions of Born and Kdrmdn, the supposition is introduced
that just as the atoms in a crystal the electrons in a metal
form a lattice. F. Haber 181a has also adopted a similar hypo-
thesis. The conduction of electricity is then explained by
supposing this electron lattice to move practically as a rigid
structure relatively to the atomic lattice and so through the
metal. This model has many advantages. Since, in the
heat-motion, in which the electron lattice naturally takes
part, the electronic vibrations, on account of their mass, are
extremely rapid (high frequency), these vibrations of the
electron, according to Planck's formula for the energy, make
no appreciable contribution to the atomic heat. In addition
the abnormal conductivity (supra-conductivity) which has
been observed at very low temperatures may, if we use
earlier considerations by /. Stark, 1 ** be explained without
difficulty by the conception that at these very low tempera-
tures at which the atomic space-lattice is practically at rest,
the electronic lattice glides almost unimpeded through the
gaps of the atomic lattice.
G. Borelius, 1 ** in a sketch which was recently published,
uses ideas similar to those of Lindemann.
Finally, we may refer to a paper by K. Herzfeld 1M which,
THE ELECTRON THEORY OF METALS 67
in contrast to the preceding investigations, attacks the ques-
tion from a more phenomenological point of view without
making use of a particular model. For if, in the formulae
for o- and y, (67) and (68), we bring into evidence the energy
E = $mq 2 of the electrons by writing the equations thus : 183
. . . (72)
we get
as the expression which represents the Wiedemann- Franz
Law. Herzfeld then shows that, if we compare the result
with observation, the formula (73) can be made to agree well
with the actual measurements if we set Planck's expression
s -g~ for E. (The factor % has been introduced because
P -\
the energy of the electrons is solely kinetic.) The values for
v which have to be used stand in no recognisable relation to
the atomic frequencies. A paper by F. v. flawer 186 works
along similar lines.
If we survey the whole field of the conduction of heat and
electricity in metals we recognise that here the last word
has not been spoken, and that a great deal of hard work will
be necessary to clear up finally the extraordinarily com-
plicated relationshipsi But much would doubtless be gained
for the theory if in future the observations, as far as possible,
are no longer made on crystal aggregates, but on metal
crystals that are pure and homogeneous.
CHAPTEE V
The Intrusion of Quanta into the Theory of Gases
i. The Heat of Rotation of Diatomic Gases according to the
Quantum Theory
WHILE the molecular theory of the solid state thus
gained new nourishment from the doctrine of quanta,
the kinetic theory of gases could no longer be preserved from
the influx of the new views. W. Nernst 187 had pointed out
quite early that quantum-effects are to be expected in the
rotation of di- and polyatomic gas-molecules, and also in the
oscillation of atoms in the
molecule. Let us take as
an example the diatomic
gas hydrogen, the mole-
cule of which we may
picture provisionally as a
rigid " dumb-bell " (Fig. 6).
The knobs of the dumb-bell
are the hydrogen atoms,
the grip represents their
chemical bond. Such a
molecule is known to
possess, besides its transla-
tory motion (three degrees
of freedom), the possibility
of rotating about an axis at right angles to the line joining the
atoms (two degrees of freedom, corresponding to the two axes
dotted in the figure). Eotation about the line joining the
atoms does not if we accept Boltzmann's conception of the
absolutely rigid smooth atom come into play in the ex-
change of energy by collision and hence in the distribution
of energy among the separate degrees of freedom : for this
68
FIG. 6.
HEAT OF ROTATION OF DIATOMIC GASES
rotation cannot be changed by collision. Considered fror
the new point of view of quanta, which rejects " rigidity " and
" smoothness " as an unjustified idealisation, the position is as
follows : The moment of inertia of the molecule relatively to
the line joining the atoms is extremely small compared with
the moment of inertia about either of the axes at right angles
to this line. Buk it is known that rotations which take place
about axes with small moments of inertia occur with much
greater rapidity than those about axes with large moments of
inertia (the same energy having been imparted in each case).
If, therefore, we identify the revolutions per second with fre-
quencies, and use the Planck energy expression for the energy
of rotation (which is not strictly correct quantitatively), a line
of argument which has already been frequently applied shows
us that the rotation about the line joining the atoms possesses
only a vanishingly small share of the energy. For the same
reason (high frequencies) the degrees of freedom, which
correspond to the vibration of the atoms in the molecule,
become of importance only at high temperatures. As a result
JcT
of all this, classical statistics gives us the value 2 = TcT
for the mean energy of rotation of the hydrogen molecule ;
per gram me- molecule it therefore becomes NkT = RT.
Hence that part of the molecular heat which arises from
rotation is equal to R, that is, about T98 ^-l , and it is in-
deg.'
dependent of the temperature. In crass contradiction to this,
A . Eucken 188 found experimentally that the rotation part of
the molecular heat of hydrogen has the value R demanded by
the classical theory only at high temperatures. On the other
hand, it gradually decreases as we pass to lower temperatures,
and approaches asymptotically the value zero at the abso-
lute zero. In the immediate neighbourhood of absolute zero,
hydrogen behaves as a monatomic gas. Eucken's result was
confirmed by experiments conducted by K. Scheel and W.
Heuse,w who, however, measured the values of the molec-
ular heat only for three temperatures (92, 197, and 239
on the absolute scale). This falling off of the rotational heat
is without doubt a quantum-effect, similar to the decrease in
the atomic heat of solid bodies.
70 THE QUANTUM THEORY
The first attempt to calculate this phenomenon theoretically,
is due to A. Einstein and 0. Stern, 190 who proceeded as
follows : If / and v are the moment of inertia and the
number of revolutions per second of the molecule respectively,
then its rotational energy is
JSr-.(arv) . . . (74)
If, to simplify matters, we now suppose that all molecules
rotate with the same mean number of revolutions v per second,
then we can introduce for the corresponding mean energy of
rotation
E r = ?(2*W .... (75)
the theoretical quantum value 191
E r - -j^ (according to Planck's first theory) (76)
E r = A - V + - (Planck's second theory) . (77)
From (76) or (77), by combining with (75), we obtain v as a
dP
function of T. If finally we form -5, and multiply by the
Avogadro number N, then we get the share of the energy of
rotation in the molecular heat, and we see how it depends on
the temperature. It thus appeared that only by using the
expression (77) for E r would we be enabled to get a satis-
factory connexion agreeing with Eucken's measurements, a
fact which Einstein and Stern used at the time as an argu-
ment for the existence of a zero-point energy. It must, how-
ever, be emphasised that this theory can only be regarded as
a first attempt to find general bearings and that it does not
fulfil more rigorous requirements. For the Planck energy
formulae used, (76) or (77) are valid, as is shown by their
genesis, only for configurations whose frequency v is a con-
stant quantity independent of the temperature. Here, on
HEAT OF ROTATION OF DIATOMIC GASES 71
the contrary, we have made use of a mean speed of rotation
v, dependent on the temperature.
P. Ehrenfest i n 1913 built up a theory of the heat due to
rotation on a stricter basis. He had, however, to confine
himself to configurations with one degree of freedom, that is,
to rotations of the molecule around a fixed axis, as at that
time an extension of the quantum theory to several degrees
of freedom had not yet been worked out. The expression
thus obtained for the heat of rotation was then, in order to
take into account both degrees of freedom, simply multiplied
by 2, a method which readily suggests itself, but is not justi-
fiable. Ehrenfest started, in his calculation, from the original
form of the quantum hypothesis, according to which the
energy of linear oscillators may only be whole multiples of hv,
and accordingly made the condition that the rotational energy
of a configuration with one degree of freedom (fixed axis) may
only consist of whole multiples of 75-. The factor - appears,
because the energy of rotation in contrast with the vibra-
tional energy of the oscillator is solely kinetic by nature.
The Ehrenfest condition is, therefore, according to (74) :
(n-0,1,2,3 . . .) . (78)
hence
vn = ^j (n = 0, 1, 2, 3 . . .) . (79)
and by substitution in (78)
(n = 0, 1, 2, 3 . . .) . (80)
Hence tJie molecules can only rotate with quite definite, discrete
speeds v ll} and correspondingly acquire only a series of discrete
rotational energies E^\ quite in agreement with the sense of
Planck's quantum theory. It is noteworthy that these dis-
crete rotational energies are related to one another as the
squares of the whole numbers, whereas the energies of the
Planck oscillators are proportional to the whole numbers
themselves. With the discovery of the discrete values (80)
of the energy, the dynamical part of the problem was solved.
72 THE QUANTUM THEORY
But we require the mean energy E r of a totality of N similar
molecules. It is here, then, that the second, statistical part
of the calculation begins. If w n denotes the probability that
a molecule possesses the rotational energy E (n J at the tempera-
ture T (w n is therefore the "distribution-function " which has
been extended in accordance with the quantum theory), then
the mean rotational energy of a molecule is known to be equal
^ ' w n . Multiplication by N and differentiation with
n =
respect to T give us immediately the heat due to rotation. 193
Ehrenfest thus obtained for the relationship between the
rotational heat and the"Hemperature a curve which could, it is
true, be made to agree well with the measurements obtained
at low temperatures by choosing the arbitrary constant / (the
moment of inertia) suitably, but at high temperatures it showed,
before reaching the classical value B, a maximum and a sub-
sequent minimum, which did not correspond with the existing
observations.
We may note here an important consequence of equation
(79), since it has played a noteworthy r6le in the further
development of the quantum theory. If, namely, we write
down the angular momentum (the moment of momentum 194 )
of the molecule, that is, the quantity p = J x I 2irv, then it
follows from (79) that only the special quantum values
Pn=^ (n-0,1,2,3 . . .) . (81)
of the turning-moment exist. This relation may also be
deduced directly from the theory of the quantum of action as
formulated in (30). For, if we select as our general co-
ordinate, in this case q, the angle of rotation <f>, then the
corresponding momentum or impulse p is known to be
none other than the moment of momentum. 193 It follows
from this, since p is independent of </>, that
2irp* = rih . . . (82)
in agreement with (81).
HEAT OF ROTATION OF DIATOMIC GASES IS
In the same way, on the basis of Planck's first theory,
namely, the conception that the special quantum rates of
rotation v n are the only possible ones, and using the dumb-bell
model, the author 196 has recently carried out the strict calcula-
tion for structures with two degrees of freedom (free axes of
rotation), making use of the later ideas of the quantum theory.
This stricter method likewise gives us curves for the rotational
heat which are useless, for they also have a maximum and
a subsequent minimum, as in Ehrenfest's case. Only by
making special subsidiary assumptions, such as excluding
certain quantum states, can we get curves which rise steadily
with increasing temperature, and which agree, at least to a
certain extent, with observation. 197 . "* .
f-r E, E r E' r E r
FIG. 7.
Not much more satisfactory results were obtained in those
investigations which, again with the use of the dumb-bell
model, were based on Planck's second theory. According to
this theory, the discrete values v n of the rotational speeds are
not the only possible ones ; on the contrary, the molecule can
rotate with all rotational speeds between and oo , and hence
can assume all values of rotational energy between and oo,
exactly like the Planck oscillators in Planck's second theory.
The peculiarity of the special quantum values (80) for the
energy here consists in the following : imagine the energies
E r plotted as abscissae (Fig. 7) and the corresponding prob-
abilities w as ordinates ; then a step-ladder curve results,
74 THE QUANTUM THEORY
the steps of which lie exactly at the values Ety. The prob-
ability that a given value E r of the rotational energy will
appear is therefore constant within the range of energy between
E (r ? and JB,"* 1 * but changes suddenly at the ends of this range.
According to Planck's first theory, which allows only she
quantum values E%\ the encircled points alone have a
meaning. Only at those points is the probability other taan
zero, while all intermediate values of the energy possess the
probability zero, that is, do not occur.
In this case, too, the problem was first solved for one
degree of freedom (fixed axis of rotation). E. Holm lK and
J. v. Weyssenhoff 199 found, in agreement with one another, a
steadily rising curve for the rotational heat, which fitted the
observations well at low temperatures, but undoubtedly went
too high at higher temperatures (from about 140 abs. up-
wards).
But when the modern development of the quantum hypo-
thesis for several degrees of freedom, to which we shall be
introduced later, was available, a stricter calculation for
free axes of rotation, i.e. for two degrees of freedom, could be
carried out. This problem was attacked on the one hand by
M. Planck,* 10 on the other by Frau S. Rotszayn but was
treated differently in each case. Planck started with the
premise that this problem belongs to the category of so-called
" degenerate " problems. This term is to convey the follow-
ing: the molecule rotates, when no external forces act
on it, according to the doctrines of mechanics, with con-
stant speed in a spatially-fixed plane. The position of
this plane in space must, so Planck argues, be of no im-
portance for the statistical state of the molecule. Hence the
condition of rotation of the molecule in the sense of the
quantum theory is determined by a single quantity, namely,
the rotational energy. In spite of the fact, therefore, that
the problem is originally and naturally a problem of two
degrees of freedom for the position of the molecule in space
is determined by two angles we must, according to Planck,
treat it in the quantum theory as a problem of only one
degree of freedom. The two degrees of freedom coalesce, as
it were ; they are " coherent"
In contrast to this, Frau Rotszayn proceeds to turn the
HEAT OF ROTATION OF DIATOMIC GASES 75
problem into a non-degenerate one by the addition of an ex-
ternal field, and after solving this problem, reduces the field
of force till it vanishes. This method which was also used
by the author in the paper above cited, appears to be par-
ticularly advantageous, when the calculation is based on
Planck's first theory, for peculiar difficulties arise in " degen-
erate " cases. Success here decides in favour of the second
method. For while Planck finds a curve 2 02 which rises above
the classical value to a maximum, and then descends asymp-
totically towards the value R and is therefore of no use
the calculation of Frau Rotszayn gives a steadily rising curve,
which agrees well with the measurements for lower and higher
temperatures ; only the value observed at T = 197 abs. lies
about 10 per cent too low. 203
While all the above-mentioned investigations are based on
the dumb-bell model, which can only be regarded as a pro-
visional, schematic construction, P. S. Epstein* in 1916
carried out the corresponding calculations for another mole-
cular model proposed by N. Bohr.w* This model of the
hydrogen molecule, to which we shall return later, is built up
of two positive hydrogen atoms, each of which carry a single
positive charge, and around the connecting line of which two
electrons, diametrically opposite, rotate in a fixed circle at a
fixed rate (see Fig. 8). Since the equilibrium in this purely
electrical system is determined by the play of the Coulomb
attractions and the centrifugal forces, and since the radius of
the electron is determined by a quantum condition, this model
possesses the advantage that all its dimensions are completely
76 THE QUANTUM THEORY
fixed, so that there is no longer any question of the arbitrari-
ness of the moment of inertia. The " dumb-bell knobs " are
represented here by the two positively-charged hydrogen
atoms; the rotations of the molecule hitherto considered
would therefore correspond to those motions in which the
molecule rotates with a moment of inertia J" about an axis
at right angles to the line joining the atoms. But to this
there must very plainly be added the rotation of the system
about the axis of symmetry (i.e. the line joining the atoms),
which results from the extremely rapid rotation of the elec-
trons. The moment of inertia corresponding to this axis is,
in consequence of the extremely small mass of the electrons,
very small compared with J. The whole system obviously
possesses, if we regard it approximately as rigid, the properties
of a symmetrical top. Its motion is therefore, in consequence
of its own rotation about the axis of symmetry, not a rotation,
but instead the well-known motion, "regular precession." 206
Epstein treated the problem from this point of view but could
not obtain agreement at low temperatures with the moment
of inertia calculated from the model itself, 207 namely, / = 2 -82
x 10 ~ 41 . Presumably, this failure depends on the fact that
the model does not correspond with reality, and in fact we
shall see later, that well-founded doubts have arisen as to the
correctness of the Bohr model. We must therefore admit,
unfortunately as one of a number of instances in the quantum
theory, that the important problem of the rotational heat of
hydrogen still awaits solution.
2. The Bjerrum Infra-red Rotation-spectrum
N. BjerrumM* has applied the relation (79) in a very
interesting manner to the infra-red absorption of polyatomic
gaseous compounds. These gases (for example HC1, HBr,
CO, H 2 in the form of steam, but on the other hand not
the elementary gases H 2 , 2 , N 2 , C1 2 ) show, according to the
investigations of S. P. Langley, F, Paschen, H. Eubens, 211
H. Bubens and E. Aschkinass, H. Eubens and G. Hettner,
W. Burmeisterpu Eva v. Bahr,* u extensive absorption bands
in the short- and long- wave infra-red. While in the long-
wave infra-red we account for the absorption by the rotating
molecule, which, composed of positively and negatively
INFRA-RED ROTATION-SPECTRUM 77
charged atoms, act like electric double poles and hence in
turning emit and absorb radiation, Bjerrum was the first to
point out that the molecular rotation must also make itself
noticeable in the short-wave infra-red. For if there exists
in this region a linear vibration V Q of the ions in the molecule
relatively to one another and hence an absorption at this
point and if, in addition, the whole molecule rotates at the
speed v r , then it is known that there will be produced as a
result of the composition of the vibration with the rotation 218
two new vibrations (and, correspondingly, two new regions
of absorption) having the periods v e + v r and VQ - v r , sym-
metrically disposed on both sides of the ionic vibration VQ.
On the whole, then, we have three points of absorption :
v r , VQ ~ VT, vo + v r , to which we must add the non-rotational
state VQ as a fourth. But if now, according to Planck's first
theory, the molecule can only rotate with discrete speeds of
rotation v n [see (79)], we get symmetrically to the original
position of absorption v = VQ and, on both sides of it, a series
of further discrete equidistant positions of absorption :
v = i/o + v n = v + n~-}
* \(n - I, 2, 3 . . .) . (83)
v = v - v n = v - n
These discrete equidistant positions of absorption have
actually been found by Eva v. Bahr in the case of water
vapour and gaseous hydrochloric acid, and were measured
later with still greater accuracy by H. Rubens and G. Hettner
for water vapour. In an examination carried out on an
extensive scale E. S. Imes zl1 has once more thoroughly
investigated the hydrogen halides (HC1, HBr, HF) and con-
firmed the law (83) for the position of the absorption lines.
It was thereby found that the middle line VQ was always
missing. From the standpoint of the theory here described
this would mean that the non-rotational state does not exist,
that is, that the molecules always rotate (zero-point rotation).
A. Eucken? 1 * who discussed the results of E, v. Bahr,
which were at that time the only ones known, deduced from
the good agreement between observation and calculation that
Planck's second theory is not valid, for the experiments
78 THE QUANTUM THEORY
seemed so obviously to prove that the molecule can actually
only rotate with the discrete speeds v n . This conclusion,
however, is not inevitable, as Planck 219 showed in a pene-
trating investigation. On the contrary, the observations
may after all be explained, surprising as it may seem, on the
basis of his second theory (continuous " classical " absorp-
tion ; all speeds of rotation possible). This curious result is
explained as follows : Let w(E r )dE r be the probability that
a molecule possesses exactly the rotational energy E r ; hence
for N molecules Nw(E r )dE r will be the number that will
possess exactly the rotational energy E r . These molecules
rotate therefore according to (78) with the speed
_ 1 l2E r
"" \ </
The quantity w(E r ) is here, according to Planck's second
theory, the step-ladder curve pictured in Fig. 7. Planck's
calculation then leads to the following result : the absorption
of an external radiation of frequency v* is not as one should
expect -proportional to the number of molecules having a
rotational speed v r = v*, that is, to the quantity w(E r ) but to
its differential coefficient . This differential coefficient
dE r
is, however, as Fig. 7 shows, everywhere equal to zero,
excepting at the special quantum energy-values E^\ that is,
at the rotational speeds v n . It follows from this, that here
also, from the standpoint of Planck's second theory, absorption
takes place only at the special quantum rates of rotation v.
It thus comes about that, at present at any rate, the infra-
red absorption spectra of the polyatomic gases, contrary to
all expectation, do not decide one of the most fundamental
questions of the whole quantum theory, whether, namely,
Planck's first or second theory is correct. An important
remark must be added here. The deductions of the relation
which gives the position of the infra-red absorption bands is
half in accordance with the classical and half in accordance
with the quantum theory. For although the rotational
speeds v n are determined by the quantum theory, the resolu-
tion of the oscillation v into the two components VQ v n
are determined by the classical methods. How to attack this
THE DEGENERATION OF GASES 79
problem from a point of view entirely consistent with the
quantum theory will be seen later in Chapter VIII.
3. The Degeneration of Gases
The phenomena described above which were observed in
the case of polyatomic gases (falling-off of the molecular heat,
and infra-red absorption) justify fully the application of the
quantum theory to motions of rotation. On the other hand,
the attempts to go a step farther and to apply it to the
translational energy of gases rest upon a much more insecure
basis. If this step is taken, the hitherto exceptional position
occupied by the monatomic gases, whose molecules contain
only translational energy, becomes destroyed, for then they,
too, must succumb to the quantum law. This problem
has been attacked from various quarters [0. Sackur, H,
Tetrode m W. H. Keesom, W. Lenz and A. Sommerfeld,***
P. Scherrer , M. Planck.&S] Thus, for example, Tetrode,
Keesom, Lenz and Sommerfeld imagine the thermal motion
of the gas split up into a spectrum of natural frequencies, and
they then distribute the energy in quanta, that is, according
to formula (9), over the individual natural frequencies, quite
analogously to the manner of Debye and Born-Kdrmdn in
the case of solid bodies. Scherrer and Planck, on the other
hand, apply the quantum hypothesis directly to the motion
of the individual gas-atoms, basing their argument on the
modern formulation of the quantum conditions for several
degrees of freedom. How such a quantum resolution of the
translator^ motion is effected, is perhaps most easily seen
by the following simple example (Scherrer) : Let a gas-atom
of mass ra fly to and fro in a cube-shaped space of side a
with the speed v parallel to one of the edges. It then
executes a sort of oscillation with the period " = o~- If we
set its kinetic energy, E = %mv 2 , according to Planck's first
theory = n- (n = 0, 1, 2, 3 . . .) then it follows that
h v
80 THE QUANTUM THEORY
hence
Hence the velocity of the atom and its translatory energy
can acquire only discrete, quantum-determined values.
The calculations of the above-named investigators lead to
two important main results, at least in qualitative agreement ;
in the first place, there results an alteration in the gas laws
at very low temperatures. The necessity for this " degenera-
tion " of the monatomic gases had already been recognised by
Nernst, who deduced it on the basis of his new heat theorem. 226
For if the equation of state of ideal gases
= W
p = pressure
V = volume of a gramme-atom
IT- -prril ' .v^^-no vi u, Qiuimmv-aiw^ i , Q _,
J \R = absolute gas-constant
temperature
were exactly true for all temperatures down to the lowest,
then the maximum work A, which could be gained from
the isothermal expansion of the gas from the volume V l to
the volume F 2 , would have, as we know, for all temperatures
the value
v v
' '
= RT log, (pj).
A =
For all temperatures down to absolute zero, -Tm = R log ( -^
would differ from zero, in direct contradiction to the condition
(38) of Nernst' s Theorem. Hence it follows that in the region
of the lowest temperatures, the equation of state (85) must
undergo modification. In fact, experiments of 0. SacJcurW
on hydrogen and helium appear to speak in favour of the
existence of this " degeneration."
4. The Chemical Constants of Monatomic Gases
The second main result given by the application of the
quantum theory to monatomic gases, is an extremely in-
teresting relation of the Planck constant h to the so-called
CONSTANTS OF MONATOMIC GASES 81
"chemical constant" of the gas, a quantity which plays
an important part in changes of the state of aggregation
(vaporisation, sublimation) and in chemical states of equi-
librium. But it is here specially emphasised that the re-
lationship just mentioned is not bound to the undeniably
hypothetical resolution of the translatory energy into quanta.
On the contrary, 0. Stern 228 has succeeded in deducing it un-
objectionably, without applying the quantum tJieory to the gas.
The original method, which Stern adopts, may be shortly
sketched here. Consider the process of sublimation, i.e. the
passage from the solid into the vapour state. Let the
vapour obey the gas laws, and let its density be negligible
compared with that of the condensed solid. Then classical
thermodynamics gives for the pressure p of the saturated
vapour as a function of the temperature the following
equation :
~ + f log T ~ T + c (86)
Here X is the heat of vaporisation (per gramme-atom) at
absolute zero, E^ is the energy of the condensed solid (per
gramme-atom) at the temperature T; the constant C, which
is the chemical constant of the vapourising substance,
remains undetermined, according to thermodynamics. On
the other hand, the integral on the right-hand side of
equation (86), which contains the energy of the solid
material, may be completely calculated upon the basis of
our assured knowledge of the energy- content of the solids.
We only require to assume the solid to be a Born-Kdrmdn
crystal, and hence to use the quantum-theoretical value (41)
for E ( T\ If we now restrict ourselves to high temperatures,
to a region, therefore, in which the classical theory is valid,
(86) assumes the form
log p = ----- S r- - i log T + 3 log^|) + C (87)
6
82 THE QUANTUM THEORY
(The BN quantities v; here form the elastic spectrum of the
solid body ; v is their geometric mean.) The formulation of
this equation constitutes the first step of Stern's deduction.
It gives the result of thermodynamics, extended by the
application of the quantum theory to the condensed sub-
stance. The second step is the formulation, in accordance
with molecular theory, of a vapour-pressure formula for high
temperatures, in the region therefore of classical statistics.
Here, also, the Born-Kdrmdn solid model is used for the con-
densed substance, and, on the basis of known laws of
probability, the number of the atoms is calculated which are
in statistical equilibrium in the vapour phase. In this way
the density of the vapour, and hence, as a result of the gas
laws, its pressure, are given. So Stern finds
log,? - - ^ - i log T + log
Here m denotes the mass of an atom, and X' is the work
which is necessary to bring N atoms (N is the Avogadro
number) from complete rest to the gaseous state. An un-
determined constant naturally does not appear in this formula
deduced from pure molecular theory. For the molecular
model is completely determinate, and hence gives the absolute
value of the vapour pressure, not only its temperature co-
efficient, as in the case of thermodynamics. A comparison
of (87) with (88) shows, firstly, that
jp . (89)
i
and secondly, that the chemical constant C has the value
. . . (90)
Eelation (89) may be interpreted by making the supposition
that the solid body already possesses an energy amounting to
3tr
} -FT at the absolute zero, that is, a "zero-point energy," to
i
which the latent heat of vaporisation A must be added, i n
CONSTANTS OF MONATOMIC GASES 88
order to set the atoms completely free from their union in the
crystal. Equation (90) gives us the solution of the problem
before us. It gives the chemical constant of the monatomic
gases as a function of the atomic mass and the universal
constants h and k. Nowhere, however, in the whole
deduction this should be emphasised once more has the
quantum hypothesis been applied to the gas itself.
In order to make formula (90) available for comparison
with experiment 229 we may expediently introduce the molec-
7~>
ular weight M = mN, and set k >: then
C = C + a log M, where C = log ^7 = 10'17
If we finally use the base 10 instead of the natural base e for
our logarithms, and measure the vapour pressure not in
absolute measure but in atmospheres, we get the chemical
constant C' used by Nernst, which is related to Sterns vai-ie
for C thus :
C' = xJL* = 6-0057
For it we finally get the simple expression
C' = C' + 4 Iogi M, where C' = - 1-59 . (91)
This formula has been brilliantly verified by experiment. The
hitherto most trustworthy measurements of vapour pressure
and chemical states of equilibrium give in the case of hydro-
gen, argon, and mercury the values
- 1-69 0-15, - 1-65 0-06, - 1-62 0'03
We are therefore justified in saying with Stern that the
expression (90) for the chemical constant of the monatomic
gases is theoretically and experimentally one of the best
founded results of the Quantum Theory.
CHAPTEE VI
The Quantum Theory of the Optical Series. The
Development of the Quantum Theory for several
Degrees of Freedom 23
i. The Thomson and the Rutherford Atomic Models
THE greatest advance since M. v. Lane's discovery of the
method of Eontgen-spectroscopy for determining crystal
structure was made in the realm of atomic theory in 1913,
when the Danish physicist Niels Bohr placed the atomic
models in the service of the quantum theory. Bohr's labours
have in their turn reacted on the quantum theory and fertil-
ised it, and thus a marvellous abundance of notable successes
have been achieved in recent years through the interaction be-
tween the dynamics of the atom and "the quantum hypothesis.
Among serviceable atomic models, the one proposed by
J. J. Thomson long occupied a much favoured position ; accord-
ing to it, the electropositive part of an atom, which constitutes
the most important part of its mass, is supposed to be a
sphere of "atomic dimensions" (radius about 10 ~ 8 cms.)
filled with a positive space charge in the interior of which the
negative parts, the electrons, rest in a stable position of equi-
librium. This model has the great advantage of explaining
on purely electrical grounds the possibility of " quasi-elastic-
ally bound " electrons, i.e. such electrons as, being displaced
<from their position of rest, are drawn back into it by a force
ivhich is proportional to the displacement.^ 1 And it was just
with the help of such electrons that, as is well known, P.
Driide? W. Voigt** M. Planck,** and H. A. Lorentz
succeeded in building up large regions of theoretical optics,
namely, the theory of dispersion and absorption, and the
magneto-optical effects (magneto-rotation and Zeeman effect).
84
THOMSON AND RUTHERFORD MODELS 85
Moreover, the Thomson atomic model was able, by following
the classical doctrine of the theory of electrons, to do what
must be demanded of every serviceable atomic model, viz. to
explain the emission, as a result of the oscillation of its
electrons, of sharp, i.e. essentially monochromatic " spectral
lines," the position of which, on account of the quasi-elastic
restoring force, 236 was independent of the intensity of the
excitation, that is, of the energy of the oscillations.
In three important points, on the other hand, the model
failed completely. In the first place no success at all, unless
with complicated and artificial hypotheses invented ad hoc,
attended efforts to deduce from Tfwmson's model the formulae
for the optical series, for example, the simple formula for the
Balmer series of hydrogen. 237 Secondly, the model could not
account for the division of the spectral lines in an electric
field as observed and closely studied by J. Stark* (Stark
effect), in spite of the fact that it had been found most
valuable, in the hands of H, A. Lorentz, for explaining and
calculating the Zeeman effect. 239 Thirdly, it was not in a
position to explain the large individual deflections, sometimes
exceeding 90, which, according to H. Geiger and Marsden,
a-particles undergo in passing through thin metallic foils.
For on their way through the metallic foil, the a-particles,
which are known to be doubly charged helium atoms, come
into the neighbourhood of the metallic atoms and are more
or less deflected from their straight paths by the electric fields
of the atoms. If, now, the metallic atoms were Thomson
atoms, the electric field of these atoms would attain its
greatest value at the surface of the positive sphere, at a
distance therefore of about 10 8 cms: from the centre of the
atom. For from the surface outwards the field decreases,
according to Coulomb's Law, with - while it grows from the
centre to the surface proportionately to r. Those a-particles,
therefore, which pass close to the surface of the positive
sphere, must undergo the greatest deflection. An easy ap-
proximate calculation shows, however, that the field at this
distance from the centre is far from being strong enough to
explain the great deflections which Geiger and Marsden have
observed. This weighty reason led E. Rutherford 2* 1 to set up,
86 THE QUANTUM THEORY
instead of the Thomson model, a new one, which was able to
explain the large deflections of the a-rays. According to the
Rutherford atomic picture, the electropositive part of the
atom is compressed into an extremely small space M2 the so-
called nucleus. Its charge E consists in general of z positive
elementary charges e, so that E = ze. Here z is, according to
a hypothesis of van den Broek 3 the atomic number of the
element, i.e. the number which gives the position of the
element in the series of the periodic table. Thus, for example,
z = 1 for hydrogen, 2 for helium, 3 for lithium, and so on.
About this nucleus the electrons describe planetary paths,
that is, circles or Kepler ellipses with the nucleus as focus,
since the electrons are attracted by it in accordance with
Coulomb's Law (inversely proportional to the square of the
distance).
In the electrically neutral atom having the atomic number
z, e electrons circle round the nucleus. For example, the
neutral hydrogen atom consists of a singly charged nucleus
(E = e) around which one electron revolves in a circular or
elliptic path.
That this Rutherford model is actually able to explain the
cause of large deflections of the a-particles is seen at once ;
for the field-strength of the nucleus, in contrast to Thomson's
model, increases strongly up to the immediate neighbourhood
of the nucleus, in accordance with Coulomb's Law; hence,
if the positively charged a-particles come very close to the
nucleus that is, much nearer than 10 ~ 8 cms. then they are
exposed to the extremely powerful repulsion of the nucleus.
On closer examination, the Rutherford atomic model dis-
appoints us seriously : for the revolutions per second, v, of
the electrons depend on the energy of the system. 2 ** If,
therefore, we suppose, according to the classical electron
theory, that an electron revolving at v revolutions sends out
an electromagnetic radiation of frequency v, then, since the
system loses energy by this radiation, v must diminish cor-
respondingly. But this means that the atom is unable to emit
a sJiarp, homogeneous spectral line.
2. Bohr's Model of the Atom
It thus appears that we are obliged to reject this model at
the very outset. But the history of physics has decided
BOHR'S MODEL OF THE ATOM
87
otherwise. With deep-sighted intuition, Niels Bohr saw the
possibilities of Rutherford's model and brought it under the
quantum theory by making three bold hypotheses.^ I n the
first place, he assumed that the electron (or electrons) cannot
revolve around the nucleus in all paths possible according to
the view of mechanics, but only in certain discrete orbits
determined by the quantum theory. If we restrict ourselves,
as Bohr did initially, to circular paths, then only those paths
of an electron are allowable from the view of the quantum
theory for which the moment of momentum (angular mo-
mentum) of the revolving electron is a whole multiple of
- , in exact agreement with the quantum condition (81) or
(82) for the rotating molecule.
This gives, in the simplest
case for the quantum paths of
the electrons, a discrete family
of concentric circles around
the nucleus, with radii, which
are related to one another as
the squares of the whole
numbers (1 : 4 : 9 : 16 :
). See Fig. 9.
Secondly, these " allow-
able " orbits are stationary ; p I(Jj 9.
they are in a certain sense
stable states of motion. This stability is gained by making
the radical condition that the electron in striking contrast
with everything that the classical theory has taught us
shall not radiate when in the stationary paths. Since by
this " decree " the loss of energy is abolished, the electron
can continually revolve in such a "quantum path." That
there are such "non-radiating" paths in the atom, is be-
yond doubt. Among other things, the constancy, in time,
of the para- and ferro-magnetism of bodies, which is
generated by revolving electrons, speaks in favour of this
view. But how electrodynamics must be altered in order to
guarantee the non-radiation of the quantum paths, and only
of these, is a question which as yet remains unanswered. As
we have now abolished the "classical" radiation of the
88 THE QUANTUM THEORY
atom, the actually observed emission must be accounted for
by a new hypothesis. Here, again in direct connexion
with Planck's original quantum rule, Bohr's third condition
takes effect : when the electron passes from one allowable
quantum orbit, in which the energy is W 2 , into another
allowable quantum path of energy W lt energy amounting to
Wk - W\ is radiated in the form of an energy-quantum hv of
homogeneous, monochromatic radiation. The frequency of
the radiation emitted is determined by " Bohr's Frequency
Condition : "
We can follow Einstein 2 * 6 in imagining the passage from
the state of higher energy to the state of lower energy as a
sort of radio-active disintegration, the occurrence of which in
time is determined by chance. The details of this passage and
the release of energy accompanying it are, however, entirely
obscure up to the present.
3. The Hydrogen Type of Series according to Bohr's Atomic
Model
However bold and unorthodox Bohr's three hypotheses
may have appeared, their success was surprising. If we apply
them to a "hydrogen type" of Rut}ierford-a,tom in which
a single electron revolves around a positive nucleus with a
z-fold charge, we get 247 for the frequencies of the spectral
lines, which the electron emits in passing from the ?ith to the
sth quantum path, the following values :
fe, m charge and mass~|
J of electrons I (93)
~ 2 j \s, n whole numbers J
Nz*
If we here set z = 1 (hydrogen), s = 2, n = 3, 4, 5 . . . we
get in exactly the same form the empirical expression for the
Balmer series of glowing hydrogen M8
(n = 3, 4, 5 . . .) . (94)
THE HYDROGEN TYPE OF SERIES 89
For the constant N which appears in the empirical formula,
the so-called Eydberg number, Bohr's Theory therefore gives
the expression
N=*^ .... (95)
If we use here the well-known values
e = 4-774 x 10 - 10 (Millikan) h = 6'55 x 10 ~ 27 (Planck)
= 1-77 . 10 7
me
then it follows from (95) that
N = 3-27 . 10 1S
while the empirical Eydberg number has the value 3-29 . 10 15 .
This striking agreement and the resolution of the Bydberg
number into universal constants is one of the main achieve-
ments of Bohr's Theory, 249 and forms a strong argument for
its innate power. We may say that, according to Bohr's
original theory, the individual lines of the Balmer series (H a ,
Hp, H y , . . .) are emitted when the electron jumps from the
3rd, 4th, 5th ... orbit into the 2nd.
With this statement, however, the achievements of formula
(93) are not exhausted. For it includes, as we easily see,
further spectral series of hydrogen. Namely, if we set s = 1,
n = 2, 3, 4 . . . we get the ultra-violet series that was found
and measured by Lyman. 1 If on the other hand we set
s = 3, 71 = 4, 5, 6 . . . we get the infra-red Bergmann series,
the first two lines of which were measured by F. Paschen.** 1
The element which follows hydrogen in the Periodic System
is helium (atomic number z = 2). While, however, the con-
stitution of the neutral helium atom with its two electrons
is already more complicated according to the latest investi-
gations, the two electrons circle around the nucleus in two
different orbits the simply ionised helium atom, which has
therefore a single positive charge, is entirely " of the hydrogen
type ; " for it consists of a doubly-charged positive nucleus
around which an electron rotates. The sole difference, as
compared with the hydrogen atom, thus consists in the
doubling of the nuclear^charge, = 2. The series emitted
90 THE QUANTUM THEORY
from the positive helium atom may therefore, according to
(93), be comprised in the formula
>-*(?-*} < 96 >
where N is again the Rydberg number as defined in (98). If
we here set s = 3, n = 4, 5, 6 . . . we get the so-called
" principal series of hydrogen " which was observed by
Foivler 2fl2 and very recently measured with great care by F.
Paschen. 3 For s = 4, n = 5, 6, 7 . . . we get the so-called
" second subsidiary series of hydrogen," which was observed
by Pickering 28* and Evans. m Both series were, before the
advent of Bohr's Theory, falsely ascribed to hydrogen.
A new and extremely noteworthy result of Bohr's Theory is
revealed, if we allow for the movement of the nucleus in our
calculations. For, in reality, the nucleus is not stationary,
but nucleus and electron revolve about their common centre
of gravity. By taking this fact into account we are led to
a slightly altered expression for the Rydberg constant. In
place of (95) we get the formula
in which M denotes the mass of the nucleus. It follows
from this that for different elements, for instance, hydrogen
and helium, the Rydberg constant differs somewhat and is
smaller for hydrogen than for helium (since M H < Mff e ). In
general, the value of the Rydberg constant increases with
increase of atomic weight tending towards a limiting value.
All this is in perfect agreement with the results of many
years of spectroscopic research.
In the same way as emission, absorption has a quantum-
like character, according to Bohr's model. If light, say of
the first Balmer line (E a ), falls upon a hydrogen atom, a
quantum hv of this external H a radiation is used to "raise"
the electron into the third quantum orbit. An amount of
energy hvff a is taken from the external radiation, that is, light
from the line H* is absorbed.
STRUCTURE OF THE PERIODIC SYSTEM 91
4. The Structure of the Periodic System
Even in his earliest papers Bohr endeavoured to construct
for the higher elements as well (Li, Be, B, C, etc.), in con-
nexion with the Periodic System, suitable atomic models
with several rings of electrons, each occupied by several
electrons, in which, for example, the well-known octaves of
the system are reproduced by a regular arrangement of the
external electrons which recurs at every eighth element,
while the number of the electrons revolving in the outermost
ring is equal to the valency of the element in question.
W. Kossel 286 arrived at a similar structure of the atoms as
a result of a profound investigation of the formation of mole-
cules from atoms. Also, L. Vegard, 251 A. Sommerfeld* and
B. Ladenbnrg 2 s9 have constructed analogous atomic models,
particularly taking into account the well-known up-and-down
curve of atomic volumes, and using them to explain other
periodically varying properties (paramagnetism, ionic colour).
These considerations, although they are tending indisputably
along the right lines as far as the general principles are con-
cerned, are not yet firmly established in detail.
5. The Quantum Hypothesis for Several Degrees of Freedom
While the quantum hypothesis in its most primitive form
demonstrated in this way its innate power by entering the
field of atomic dynamics, it had, in doing so, gained little as
far as its own development was concerned. But the fruits of
Bohr's Theory ripened more rapidly than could have been
divined. Already the year 1915 brought a decisive develop-
ment : almost simultaneously, Planck and Sommerfeld inde-
pendently found the solution of a problem that had long been
a burning question, namely, the extension of tlie quantum
theory to several degrees of freedom. Sommerfeld^ retained a
close connexion with Bohr's Theory in attacking the problem.
The first main condition of this theory related to the choice
of " allowable " stationary orbits among all those mechanically
possible. According to this, as we saw, only those orbits
were allowed for which the moment of momentum (Impuls-
moment) p is a whole multiple of -. This may also be
92 THE QUANTUM THEORY
expressed according to (81) and (82) thus : among all mechan-
ically possible paths, only those are allowable and stationary
for which the pJuise-integral fulfils the condition :
nh . (98)
In this quantum condition we are to replace according to
(82) the general co-ordinate q by the angle of rotation (the
" azimuth ") <f>, the impulse ^> by the " impulse (or momentum)
corresponding to <f>," namely, p^ (the moment of momentum).
The integration is thereby to be extended over the whole range
of values of the variable q, that is, in the present case, from
to 2*-.
In the case of the original Bohr Theory, which considers
only circular orbits, there naturally exists only a single
quantum condition, namely, that for the case q = <, since
the angle of rotation <f> is the only variable of the path.
Matters are otherwise, when we reject the limitation to
circular orbits, and hence take .STe^/cr-ellipses into account.
Then each point of the path is determined by two variables,
namely, by the distance r of the electron from the nucleus,
which is at the focus of the ellipse, and by the angle </> (the
" azimuth ") which r makes with a fixed direction (say with the
straight line, which joins the nucleus to the perihelion). In
this case we are presented with a problem of two degrees of
freedom, with two generalised co-ordinates, r and $ (polar
co-ordinates). The simple extension of the quantum hypo-
thesis by Sommerfeld now consists in setting up in this case'
two quantum conditions of the form (98), one for the co-
ordinate <, which agrees with the single quantum condition of
Bohr's Theory, and a new one for the co-ordinate r, so that
the selection of the stationary orbits is here determined by
the two following equations :
nh. . . (99)
n'h . . . . (100)
n and n are here whole numbers, p$ and p,. are the impulses
(momenta) corresponding to the co-ordinates <f> and r. 261 The
THE QUANTUM HYPOTHESIS 93
integration in (100) is to be taken over the full range of
values of r, that is, from the smallest value r m i n (perihelion)
to the greatest value r max (aphelion) and back to the smallest
fmin. (99) is called the azimuthal quantum condition, n being
the azimuthal quantum number ; (100) is the radial quantum
condition, ri the radial quantum number.
In a corresponding manner the extension may be carried
out for more than two degrees of freedom. If the system has
/ degrees of freedom, and if it is therefore characterised by
the / generalised co-ordinates q v q.^, q s . . . and the corre-
sponding impulses p v p. 2 , p 3 . . ., then the " allowable "
movements of the system are limited by the / quantum
conditions :
\Pi d( li = n i h > p-A = nji, - - \Pfdqj- = njh . (101)
(n v n. 2 . . . HJ- are positive whole numbers).
In every one of the / phase-integrals the integration is to
be performed over the full range of values of the co-ordinate
in question.
A difficulty, which arose here from the outset, was the
question as to which co-ordinates ought to be chosen for the
application of the quantum rule (101), or whether the choice
is immaterial. In general, we may characterise a system of
several degrees of freedom by various types of co-ordinates ;
for instance, we may describe the Kepler movement of the
electron either by polar co-ordinates r and <f>, or by Cartesian
co-ordinates x and y. This question is the more urgent,
when one considers that the separate phase-integrals Ip^; do
not really become constants for every choice of co-ordinates,
as is required by the quantum rule (101) , 262 P. S. Epstein 263
and K. Schwarzschild 2 ^ have solved, independently of one
another, this problem of the " correct choice of co-ordinates "
to a certain extent. Incidentally, an interesting and sur-
prising relation of the quantum rules (101) to a long-known
theorem of classical dynamics was revealed, which had been
propounded by Jacobi and Hamilton, and had hitherto been
successfully applied in celestial mechanics. Finally, quite
lately, A. Einstein, 26 * by modifying the expression (101), has
94 THE QUANTUM THEORY
put forward a quantum hypothesis which has the advantage
of being independent of the choice of co-ordinates. But a
closer discussion of these abstract investigations would lead
us too far here.
The second formulation of the quantum hypothesis for
several degrees of freedom is due, as already mentioned, to
M. Planck.* It is, as it were, more cautious in its nature
than the more radical attack of Sommerfeld. Planck, con-
tinuing directly from the division of the phase-plane of linear
oscillators already discussed, starts from the so-called Gibbs
phase-space to deal with more complicated systems. For a
system of /degrees of freedom, which is characterised by the
co-ordinates q v q. 2 . . . qy and the impulses p lt p. 2 . . . pf,
the Gibbs phase-space is that 2/ dimensional space, the points
of which possess the 2/ 1 co-ordinates q 1 . . . p/. Each point
of the phase-space (phase-points) represents, therefore, a
definite momentary state of the system in question. Planck
now gives this phase-space, in exact analogy to the phase-
plane, a cellular structure, by bringing into prominence
certain specially distinguished boundary surfaces. At the
same time the size of the cells is proportional to h f . The
points of intersection of those boundary surfaces then repre-
sent the distinctive quantum states or phases of the system
(that is, according to Planck's first theory the only possible,
the "allowable" conditions). In contrast with Sommerfeld' s
Theory, in which the motion of a system of / degrees of
freedom is always determined by / quantum conditions, in
Planck's, under certain circumstances, the case may occur
that fewer quantum conditions than degrees of freedom exist,
so that several (" coherent ") degrees of freedom are limited
by a single quantum condition.
6. Sommerfeld's Theory of Relativistic Fine-structure
That these theories had found the kernel of the matter was
soon to be shown by applying them to Bohr's atomic model.
According to them from among all the mechanically possible
paths, which the electron can describe about the -fold
positively charged nucleus, the allowable, stationary paths
must be determined by the two quantum conditions (99) and
(100). This gives, in place of the discrete, quantised circles
RELATIVISTIC FINE-STRUCTURE 95
of Bohr, discretely quantised Kepler ellipses, among which
also the Bohr circles are included, as special cases. And
further, the ellipses are quantum-determined, both with re-
ference to their sizes (i.e. to their major axes), and to their
form (i.e. the relation of the axes to one another), so that here
every orbit, as compared with Bohr, is characterised by two
quantum numbers n and w'. 267 In place of formula (93) for
the hydrogen type of series, we get the general formula : a68
v = Nzf, _ J_ _ 1 "I (102)
L(s + s'Y (n + nj]
Here again N, the Rydberg constant, is given by (95), or
more exactly (the motion of the nucleus being taken into
account) by (97) ; s and s' are the two quantum numbers
(azimuthal and radial) of the final orbit of the electron ; n
and n' are the quantum numbers of its initial orbit. Since
also, as a result of this more complete view of Sommerfeld,
the number of allowable orbits is greatly increased, as com-
pared with those arising from Bohr's Theory (owing to the
addition of the ellipses), the electrons have a great many
more possibilities in passing from one orbit to another, that is,
the chances of generating spectral lines are multiplied. But
we easily recognise the following fact : if we choose as the
final orbit of the electron any one of those orbits, for which
the sum of the quantum numbers s + s' has a definite value,
say s + s' = 2, and as initial orbit, any one of those paths,
for which n + n' has a definite value, say n + n' = 3, then
all the different transitions of the electrons from any one of
these initial orbits to any one of these final orbits generate
always the same line (in the case of the figures above chosen
it will be the first Balmer line) ; for according to (102) the
frequency of the line emitted depends only upon the sum
s + s', and the sum n + n', and on the other hand not on the
separate values of s, s', n, n'. It would thus appear as if
nothing is gained physically by Sommerfeld' s elaboration of
the theory as compared with Bohr's original theory. How-
ever, as Bohr had already pointed out, the calculations are
incomplete in one important respect, which become of funda-
mental importance when consistently taken into account,
and which represents the main achievement of Sommerf eld's
96 THE QUANTUM THEORY
theory of spectral lines. Namely, the velocities of the
electrons, which appear in these problems, cannot be con-
sidered negligibly small compared with the velocity of light.
In this case, however, we cannot, as we know, calculate by
the methods of classical mechanics, which regards the mass
of the electron as constant, but must take our stand upon the
theory of relativity, and hence take into account the variations
of the mass of the electron with its speed. Sommerfeld com-
pleted the calculation in this respect. The paths of the
electron and the nucleus differ, in this refinement of the
theory, from the ordinary Kepler ellipse in that the perihelion
of the orbit advances in the course of time, and that the path
loses its closed character. This has the effect that the energy
of the electron in the stationary quantum-chosen orbits which
here also are determined by (99) and (100) are no longer
solely dependent on the sum of the quantum numbers as in
the case of the non-relativistic Kepler motion, but that the
quantum numbers n and n also enter, separately, into the
expression for the energy. Only as a first approximation,
therefore, i.e. when the relativity correction is neglected,
will the frequency v of the spectral line emitted depend on
the quantum sums s + s' and n + n' alone, as (102) shows.
If we take into account the relativistic change of mass of the
electron, on the other hand, v will also depend on the
individual values of s, s', n, w'. 269 It follows, therefore, that
the various possibilities, above considered, of the generation of a
definite spectral line, that is, the passage of an electron from
any one of the initial orbits s + s' = constant to any one of the
final orbits n + n' = constant, no longer produce exactly the
same line, but give rise to slightly different lines, which, how-
ever, on account of the smallness of the relativity effect, lie
very close together. This is Sommerfeld' s explanation of the
fine-structure of the spectral lines in the case of the hydrogen
type of spectra. For example, according to Sommerfeld, the
first line of the Balmer series (the red hydrogen line H a ) must
consist of five components, which are arranged in two chief
groups (of two and three each). The mean distance of these
two groups from one another should amount, according to the
theory, 270 to about 0'126A ; the best measurements of the
hydrogen doublet gave the value 0'124A (Paschen, Mciasner).
HIGHER ELEMENTS 97
If this agreement already speaks strongly in favour of Sommer-
f eld's Theory, the exact measurements, by F. Paschen, of the
fine-structure of the lines of positive helium (Fowler series) have
given a still more convincing proof of its correctness ; almost
without an exception, all the components required by the
theory of the fine-structure appeared on the photographic plate,
and thus proved strikingly the existence of the stationary paths
of the electron and its relativistic change of mass.
Two interesting consequences may yet be mentioned here ;
they are directly connected with Sommerfeld's Theory and
Paschen's observations. First of all they have rendered
possible the use of the fine-structure measurements for a
direct " spectroscopic " determination of the three funda-
mental constants e, m (mass of the electron at infinitely low
speeds), and h.^ 2 Secondly, K. Glitscher was able to
show that we only find the spectroscopic observations, for
example, the size of the hydrogen doublet, in agreement with
the theory, when we use for the variation in the mass of the
electron the formula given by the theory of relativity. On
the other hand, Abraham's Theory of the rigid electron leads
to formulae which do not agree with experiment.
7. Higher Elements
We thus see that Rutherford's atomic model as further
developed by Bohr and Sommerfield far exceeded the ex-
pectations which it could reasonably be expected to fulfil. At
any rate, it has revealed to us the optical series of hydrogen
and helium with undreamed-of precision as far as the finest
details. But beyond these primary gains, it has undertaken
a further series of successful attacks. Thus Landew* was
successful in calculating the two series-systems of neutral
helium (helium and parhelium) by taking, in contra-
distinction to Bohr, a model of the neutral helium atom in
which the two electrons circle around the double positive
nucleus in two different orbits, either co-planar or else
inclined at an angle to one another. In this case then, the
external electron, the leaps of which generate the radiation,
moves in a field in which the simple Coulomb Law no longer
holds, on account of the disturbing influence of the inner
electron. Examples of this type which differ from that of
7
98 THE QUANTUM THEORY
hydrogen have been generally investigated by Sommp.rfeld,
who has shown 273 that by giving up the Coulomb field we
arrive, to a first and second approximation, at the Bydberg
and Eitz forms of the series laws. A very promising
beginning in setting up a quantum theory of the spectral
lines was thus made.
8. The Stark Effect and the Zeeman Effect in Bohr's Theory
of the Atom
Under the circumstances the question forces itself upon us,
whether the atomic model in its present state of development
is able to account for the Stark effect, that is, the splitting up of
the spectral lines as a result of the action of an external electric
field on the electrons emitting the lines. For, as we may
remember, the original TJwmson model had completely failed
just at this point. And how do matters stand as regards the
Zeeman effect, the splitting up of spectral lines as a result of
an external magnetic field? Could the new model explain
these phenomena as well as the old ? Both questions have
fortunately been answered in the affirmative. As regards the
Stark effect, P. S. Epstein, in an important paper, succeeded
in demonstrating the following : if we calculate the motion of
the electron under the influence of the nucleus and the
external field, according to the methods usual in celestial
mechanics, and then choose from among all mechanically
possible motions the allowable stationary orbits by applying
the modern quantum rules for several degrees of freedom, and
if, thirdly, we allow the electron to leap from one of these
stationary paths into another (whereby we limit the infinite
number of possible passages by a "principle of selection"
presently to be discussed), then the Bohr frequency formula '(92)
gives with the most admirable accuracy and completeness, both
as regards position and number, all the components of the
resolved lines as observed by Stark in the cases of hydrogen
and positive helium. This astonishing result must be re-
garded as a further strong support of the correctness of
Bohr's model and its system of quanta. The theory of the
explanation of the Zeeman effect has up to the present not
been quite so successful, It is true that Debyc and
SELECTION OF RUBINOWICZ AND BOHR 99
Sommerfield 278 have been able to derive the normal Zeeman
effect (division of the original line into a triplet when the
line of observation is perpendicular to the lines of force) by
calculation from the model. The explanation, however, of
two important phenomena in this field has not yet been
accomplished : firstly, the anomalous Zeeman effect and its
laws (Runge-Preston rule), and secondly, the fact, discovered
by Paschen and Back, 1219 that even in the case of lines with a
complicated fine-structure, the normal triplet is formed as
the magnetic field grows. Further investigation will, it may
be hoped, unravel those difficulties.
9. The Principles of Selection of Rubinowicz and Bohr
Inasmuch as the foregoing considerations deal only with
the position of lines in the spectrum, i.e. with their frequency,
we are still confronted with the problem of their form of
vibration, i.e. their intensity and polarisation. Moreover, the
important question had yet to be answered, whether all leaps
of the electron from any one stationary path to any other
are possible, or whether the number of allowable transitions
must be limited by some " principle of selection." This
also is, fundamentally, a question of intensity, for the position
may be regarded as follows : the forbidden transitions corre-
spond to zero intensity. The solution of this whole complex
of problems has been greatly advanced quite recently. In
the first place, A. Rubinowicz, & by applying the law of the
conservation of the moment of momentum (impuls-moment)
to the system atom + radiated wave, arrived at a principle
of selection and a rule of polarisation of the following form :
in atoms of the hydrogen type, which are removed from the
influence of external fields of force, the azimuthal quantum
number n of the electron [see formula (99)] can only alter by
0, +1, or - 1, when emission takes place. In the first case,
the light radiated is linearly polarised, in the two other
cases circularly. The position of the plane of the orbit
remains unchanged during the process of emission. In
the case of atoms differing from the hydrogen type, and
of more complicated structure, the position is less simple;
if we set the total moment of momentum of all the
masses forming part of the system (we know that this
100 THE QUANTUM THEORY
impulse remains constant during the motion), equal to a
whole number, n*, times ^, it is just the changes in this
number n* during the emission which must be limited by the
principle of selection in the same manner, as, in the case
above, the alterations in the azimuthal quantum number of
the individual electron in its leaps were limited. Here also,
zero change in the azimuthal quantum number gives linear
polarisation, changes by + 1, on the other hand, lead to
circular polarisation. In place of the orbital plane we get
the " invariable plane " (at right angles to the total moments
of momentum or impulse-moments), the position of which in
space remains unaltered. If, finally, the atom is exposed to
an external field, say a homogeneous electric field (Stark
effect) or a homogeneous magnetic field (Zeeman effect), then,
as we know, only that component of the total turning
impulse remains constant during the motion of the masses
forming parts of the atom which is parallel to the external
field. If we set these components of impulse = n^, then
only the alteration of this number n will be limited by the
principle of selection (that is, the alterations must be Q l 1).
The principle of selection is thus clearly weakened in its
action by the external field, and can, if fields of irregular
strength and direction act on the atom, become completely
illusory, as, for example, in the case of electric discharges.
By means of entirely different considerations, N. Bohr 281
arrived at results which coincide, in essentials, with those of
Eubinowicz, but exceed them greatly in range. Bohr started
from the fact that in the limit for large quantum numbers,
when the successive stationary states of the atom differ very
little in the energy they involve, the frequency that the
electron emits in its passage between neighbouring states
becomes identical with the rate of revolution in the stationary
orbit. 282 The electron therefore emits, according to Bohr's
frequency condition, the same line that it sends out accord-
ing to the classical theory of electrons. In other words, for
very high quantum numbers, the quantum theory passes over
into the classical theory. (Bohr's "Principle of Correspon-
dence or Analogy.") Arguing from this principle, Bohr pro-
SELECTION OF RUBINOWICZ AND BOHR 101
ceeds as follows : according to classical mechanics, the motion
of the electron in Bohr's atom may be represented as the super-
position of component harmonic vibrations of the frequency :
"kl = Tl<l + T 2 0> 2 + . . . + TjWf . . (103)
Here, T I . . . T/ are whole numbers which in general may
have all values between oo and + oo ; the o^ . . . ay are
certain constants which depend on the character of the
motion : / is the number of degrees of freedom. Let the
amplitude of the partial vibration characterised by the
numbers TJ to T/ be A T i . . . A r f. Then, according to classical
electrodynamics, vki is the frequency of the radiated partial
wave (T! . . . T,) and A^ . . . A*f is a measure of its in-
tensity. On the other hand, the following result is derived
from the quantum theory (Bohr's frequency formula) for high
quantum numbers : in the transition from an initial state
characterised by the quantum numbers m v m 2 . . . w/ into a
final state corresponding to the quantum numbers n^ . . . Hf,
a line of frequency
VQU = (% - n^ + (ra 2 - w 2 )w 2 + . . . + (m f - n f )u>f . . . (104)
is emitted. Here the quantities <o 1 . . . <D/ are the same
constants as in (103). But, according to Bohrs Principle of
Analogy, for high quantum numbers vki = VQ U . Hence there
follows from a comparison of (103) with (104)
T i = m i ~ n i> T 2 = m 2 ~ n 2 > . . . T/ = ra/ - w/ . . . (105)
i.e. the " classical " partial vibration (r { . . . T/) corresponds to
that quantum transition, in which the quantum numbers alter
by exactly TJ . . . T/. The polarisation and intensity of the
wave emitted during this q^lantum transition may be calculated
from the form of vibration and amplitude of the " corresponding
classical" partial oscillation. This principle which has been
derived for high quantum numbers is extrapolated by Bohr
with great boldness over the region of all quantum numbers.
Thus the important " principle of correspondence " is obtained.
If in the development of the electronic motion in terms of
partial vibrations the term (r lt r 2 . . . ?/) is missing, then
the corresponding transition
102 THE QUANTUM THEORY
is not present. Hence there follows, for example, for atoms
of the hydrogen type in a field free from force, the law that
the azimuthal quantum number can in all emissions only change
by + 1 or 1, both of which lead to circularly polarised
radiation. This law is somewhat more limited in form than
that of Rubinowicz.
Both the principles of selection and the rules for the
polarisation and the intensity have stood the test of compari-
son with experiment. Bubinowicz himself showed that his
principle of selection and the rule of polarisation are in agree-
ment with Paschen's measurements of the fine-structure of
the helium lines, and further with the observations of the
Stark effect and the normal Zeeman effect. P. S. Epstein 283
and H. A. Kramers* went still further, and were able to
prove by profound investigations, based on Bohr's Theory, that
the calculations of intensity along the lines sketched above
were also in surprising agreement with observation. Finally,
Sommerfeld and Kossel 283 in an interesting study have applied
the Rubinowicz principle of selection to spectra differing from
the hydrogen type as well, and have shown that it is able to
explain why certain series appear more readily and are more
favoured than others, as it were, and that, by the selection of
the " possible " transitions, it sets a limit to the multiplicity
of possible combinations in a manner which, so it appears,
entirely agrees with experience.
10. Collision of Electrons on the Basis of the Bohr Atom
While in this way, through the interpretation and unravell-
ing of the universe and the almost bewildering abundance of
spectroscopic observations, the conviction of the correctness
of Bohr's atomic model deepened more and more, a series of
observations of quite another kind became known and contri-
buted considerably to the consolidation of Bohr's Theory.
These were the investigations already mentioned earlier in
connexion with the light-quantum hypothesis, which dealt
with the collision of free electrons with gas molecules and
atoms. These researches were conducted particularly by
J. Franck and G. Hertz 28fl and, in succession, by a considerable
number of American investigators in a systematic manner.
The manifold results of these interesting researches may be
COLLISION OF ELECTRONS
103
sketched here schematically by a simple example. What
have we to expect when electrons collide with a Bohr Atom ?
As a simple type of Bohr atom, let us choose a model in which
z electrons revolve around a 2-fold positively charged nucleus
in stationary quantum paths. The nature and spatial arrange-
ment of these paths, as well as the distribution of the electrons
among the individual paths will be left open, and we shall
FIG. 10.
make only the simplifying assumption that one electron the
so-called valency electron revolves alone in the outermost
orbit (1) (see Fig. 10). Let this be the " normal," unexcited
state of the atom. The hydrogen atom (z = 1) is, as we know,
constituted in this way, and, of the neutral complicated atoms,
the atoms of the vapours of the alkali metals (Li, Na, K, Kb,
Cs) very probably also fall under this scheme. If by any
addition of energy the electron is " raised " from its normal
104 THE QUANTUM THEORY
orbit (1) to a higher orbit (that is, one having more energy),
say into the orbit (2), (3), (4) and so forth, and if it "falls"
from these back into orbit (1), then the 1, 2, 3 . . . line of the
so-called " Absorption-series of the unexcited atom " (principal
series) is emitted. The frequencies of the lines emitted are
regulated by Bohr's frequency condition (92), i.e. that the loss
of energy W n W l incurred in passing from the nib. to the
first orbit is equal to a quantum hv n>l of the line emitted :
Wn-W^hv^. . . . (106)
The additional energy required to " raise " the electron to the
higher energy level can be obtained in two ways : firstly by
absorption of external radiation ; secondly (and that is the case
we are dealing with here) by electronic impact. If external
radiation of frequency v n<l falls upon the atom, a quantum
hv n ,i of this radiation is absorbed and is used to raise the
electron from the energy level W l to the higher level W n
= W l + hv n ,i- ^ n foiling from this to the original level, the
electron then emits the light corresponding to the line absorbed.
The circumstance is further noteworthy, that the electron,
when it is raised to the level (2), has no other choice than to
return to the initial level, whereas from orbit (4) it can make
one of three possible transitions to (3), (2), and (1). If,
therefore, the atom has absorbed light corresponding to the
line j/ 2a from the external radiation, it will re-emit this line
with its full complement of energy. The first line of the
absorption series is, therefore, in contrast with all other lines,
a so-called resonance line.
If the energy required -to raise the electron is furnished by
tfce impact of an outside electron, then as Franck and Hertz
were the first to prove the intruding foreign electron will be
reflected from the atom perfectly elastically (according to the
mechanical laws of elastic impact), as long as its energy
remains below a certain critical value ER. If this energy
value is reached, the impinging electron loses all its energy,
and gives it up to the electron of the atom which has been
struck (" inelastic impact "). What does this mean according
to Bohr's view of the atom ? Obviously ER is nothing other
than W 2 - W v that is, the energy which is necessary to raise
the electron from its normal state in the atom to the orbit (2).
COLLISION OF ELECTRONS 105
The result of this electronic impact, which adds energy of
amount ER to the atom must therefore be the emission of
the resonance line. If this view represents the kernel of the
matter, then the energy ER must be connected with the
frequency i/ 2n of the resonance line by the quantum relation
E R = hv^ .... (107)
This relationship has been excellently verified by experiment.
Thus Tate and Footed for example, find in the case of
sodium, that the first inelastic electronic impact takes place
when the impinging electron is accelerated by a potential
of VR = 2-2 volts, the so-called resonance potential. The
energy communicated by this potential to the impinging
electron is
On the other hand, the resonance line that is under con-
sideration here is the D-line, hence
We thus see that the relation (107) is fulfilled with great
accuracy. The same holds for potassium (Vp. = T55 volts,
.-. E R = 2-47 - 10- 12 , X 21 = 7-685 10~ 5 /. hv zl = 2-55 - 1Q- 12 ).
In the case of the inert gases (helium, neon, etc.) and the
vapours of mercury, zinc and cadmium, similar qualitative and
quantitative relations with some modifications occur. The
excitation, by electronic impact, of the mercury resonance line
A. = 2-536 10 - 5 , that is 2-536.4, discovered by Franck and
Hertz, and already referred to, presents a charactei-istic
example. The observed resonance potential is here 4-9 volts,
while from the relation
v 300 p 300, SQQhc
VR = - &R = - AT.* =
e e eAj,!
the value VR = 4-86 volts is deduced.
If the energy of the impinging electron is increased beyond
ER, then an "inelastic" impact, accompanied by complete
loss of the energy, is to be expected every time as soon as E
106 THE QUANTUM THEORY
has become equal to W n - W l (n = 3, 4, 5 . . .). By these
various additions of energy the electron attached to the atom
is raised successively to the 3rd, 4th, 5th . . . level of energy.
If, finally, E = E& = W& - W v then the energy of the
impinging electron is just sufficient to remove the electron
attached to the atom to infinity, i.e. to ionise the atom. E^
is thus the ionisation energy, and the voltage corresponding
SOOT' 1
to it, FOQ = 9P_ ( i s called the ionisation potential. From
the relation (106) we get immediately the important equation
E * = e m =/lv ' 1 (108)
That is to say, the ionisation energy is equal to the quantum
which corresponds to the last line of tJie absorption series, that is,
to the " series limit." This quantum relation has also been
excellently confirmed in all cases. For sodium, for example,
Tate and Foote found : FQQ = 5-13 volts, which gives an
ionisation energy of the value E^ = 8'17 10~ 12 . On the
other hand, the limit of the principal series has the wave-
length AQO x = 2-413 10- 5 , from which h v<x > 1 = 8-14 - 10' 12 ,
in striking agreement with the value of E^ .
For mercury vapour, the limit in question of the principal
series A.^ = 1-188 10 - 5 . From this follows, according to
(108), FOQ = 10-4 volts while the measurements of various
workers gave the value 10*2 to 10'3 volts (Tate, Bergen,
Davis and Goucher ; Hughes and Dixon ; Bishop 6 ). From
all these examples, which could be considerably multiplied,
the conclusion may be drawn with convincing clearness that
the Bohr conceptions have laid bare the nature of the con-
struction and the mode of action of the atom with un-
precedented lucidity.
11. Einstein's Deduction of Planck's Law of Radiation on the
Basis of the Bohr Atom
Under these circumstances the suggestion naturally arises to
refound the law of black-body radiation by taking as the ele-
mentary absorbing and emitting structure Bohr's model in
place of the linear oscillator used by Planck. Einstein 289 has
taken this step. In a highly important study he investigated
PLANCK'S LAW OF RADIATION 107
the equilibrium of energy and momentum between black-body
radiation and a generalised Bohr model, which, stripped of all
special properties, has only to fulfil the quantum condition of
being able to assume a discrete series of different states. For
the interaction between the radiation and the atom absorption
(Einstrahhing) and emission (Ausstrahlung) Einstein intro-
duces the following simple hypotheses : the frequency of the
emissions, i.e. the transitions, accompanied by loss of energy,
of the atom from a condition (2) of higher energy, E%, to
a condition (1) of lower energy, E v shall follow the same
statistical law as that which governs the disintegration of
radioactive bodies, i.e. the number of transitions 2 -> 1 in the
time dt, or, as we may say, the number of atoms (2) that " dis-
integrate " in this time is proportional to dt N v where N%
denotes the number of atoms momentarily in the state (2).
But, according to Einstein, a different law regulates the
processes called into existence by the effect of external radi-
ation. Under the influence of external radiation two things
may happen : either an atom may pass from state (1) to state
(2) by taking up energy, this is the " proper positive absorp-
tion." Or the case may also occur, that, as a result of the
phase-relation between the field of the external radiation and
the atom, the atom loses energy through the action of the im-
pinging radiation, and hence passes from state (2) to state (1)
("negative absorption"). The rate at which both kinds of
transition are repeated is then proportional to the intensity
K v of the external radiation : the number of transitions 1 -> 2
associated with positive absorption in the time dt is therefore
proportional to N-^dtK.,, ; the number of transitions 2 - 1 as-
sociated with negative absorption is proportional to N 2 dtK. v .
Here N^ is the number of atoms momentarily in the state (1).
Nj_ and N 2 are determined by the laws of distribution known
from the theory of gases and statistical mathematics and en-
larged in conformity with the quantum theory. There follows
from the energy equilibrium between in-coming and out-going
radiation at the temperature T
. . . (109)
108 THE QUANTUM THEORY
where k is Boltzmann's constant, and A is a constant inde-
pendent of the temperature. From Wieris Displacement Law
(4) it follows, firstly that A is proportional to v 3 and secondly
that E 2 - E! is proportional to v. If, therefore, we write
E 2 - E l = hv . . . . (110)
we recognise in this expression Bohr's frequency condition (92).
In this way K,, assumes the form of Planck's Law of Eadia-
tion, arising in a surprisingly simple and elegant manner from
a minimum of hypotheses of a general character. Einstein,
in pursuing and deepening these conceptions by writing down
the expression for the equilibrium of the momenta in addition
to the energies of the in-coming and out-going radiation, was led
to the remarkable conclusion that the radiation of Bohr atoms
cannot take place in spherical waves, as the classical theory
of electrons requires, but that the process of emission must
have a particular direction like the shot from a cannon. We
cannot fail to recognise that this brings the conception that
radiation has a quantum-like structure (light-quantum hypo-
thesis) within realisable bounds.
CHAPTEK VII
The Quantum Theory of Rontgen Spectra
i. The Analysis of Rontgen Spectra
T) AEALLEL with the development of the science of optical
[ spectra, a theory of Eontgen spectra has been developed of
late years upon the same basis. This theory has already shed
much light on the structure of atoms and thus forms a
desirable extension of the theory of optical spectra. The
investigations of Ch. Barkla, W. H. and W. L. Bragg, Moseley
and Darwin, Siegbahn and Friman,' 290 among others, have
shown that by the impact of cathode rays upon the anti-
cathode of a Eontgen tube two kinds of Eontgen rays arise :
first, the so-called " impact radiation " (Bremsstrahlung) con-
sisting of an extensive and continuous range of wave-lengths
(similar to the continuous background of visible spectra) ;
secondly, the " characteristic radiation," a typical line- spectrum,
the structure of which depends so essentially on the material
of the anti-cathode that a glance at this spectrum suffices us
to deduce immediately and unmistakably the nature of the
material of which the anti-cathode is composed. Thus along-
side the optical spectrum analysis of Bunsen and Kirchhoff a
Eontgen- or X-ray analysis presents itself. It has further
been shown that the characteristic X-ray spectrum is a
purely atomic property, and, indeed, an additive one. If we
examine, for example, the X-ray spectrum, which is emitted
by an anti-cathode of brass (copper + zinc), we find the
lines of both copper and zinc unaltered and occupying the
same positions as if only one metal were present in turn. No
new lines appear. Accordingly we are led to suppose that
the line-spectrum arises in the atoms of the anti-cathode, and
is generated there by the impinging electrons of the cathode
109
110 THE QUANTUM THEORY
rays. The further important fact appeared that the lines of
the characteristic spectrum may be arranged in series, just
like those of the optical spectrum. Thus we have discovered
up to the present a short-wave ^-series, a long- wave .L-series,
and a still longer-wave M-series.
The most curious feature of these spectra is their connexion,
by a definite law, with the atomic number of their element in
the periodic system. If we plot the position of a certain line
(say the first line K a of the ^-series) for the successive
elements of the periodic system, a perfectly regular progres-
sive shift is revealed : the line advances with increasing
atomic number steadily towards the shorter waves. The re-
gularity of this advance is such that we can recognise gaps or
false positions of elements in the periodic system immediately
by an excessive jump. Now, according to the hypothesis,
already mentioned, of Eiitherford, v. d. Broek, and Bohr, the
atomic number of an element is nothing other than the
number of its nuclear charge, that is, the number of elemen-
tary positive charges of its nucleus. If to this we add the
phenomenon just discussed, according to which the steady
advance of the nuclear charge in the series of the elements is
reflected in the steady displacement of the X-ray lines, then
we are forced to the view that the origin of tlie X-ray spectra
must be localised in the immediate neighbourhood of the nucleus,
that is, in the inmost part of the atom. For in this region the
nucleus clearly has the greatest power and is least disturbed
by external electrons, and hence it is here, too, that the growth
of the nuclear charge will make itself most felt.
The connexion between the position of the X-ray lines and
the atomic number z was first formulated by G. Moseley.
He found for the frequency of K a (first line of the JfT-series)
and L a (first line of the Z/-series) the empirical relation
(Ill)
where N is the Eydberg number.
The similarity of these relations, which are only approxj-
THE ANALYSIS OF RONTGEN SPECTRA 111
mately valid, with Bohr's formula (93) for the series of the
hydrogen type is so striking, that it was an obvious step to
seek to find the explanation of the Eontgen series by arguing
on the basis of Bohr's model.
This problem was attacked chiefly by W, Kossel, 2 A.
Sommerfeld, L. Vegard, P. Debye, J. Kroo,&* and A.
Smekal. 1 And thus, in addition to the theory of the optical
spectra which take their origin at the periphery of the atom,
a theory of the Kontgen spectra has arisen which leads us
FIG. 11.
into the inmost regions of the atom. According to this theory
we may picture to ourselves, in general terms, the emission
of the Rontgen spectra as follows : we consider a neutral
Bohr atom, consisting of a 2-fold nucleus, around which
z electrons revolve. These z electrons may be arranged in
different rings. The innermost, single-quantum ring, the so-
called -ST-ring, carries, let us say, p l electrons in its normal
state ; let the second ring, the ir-ring, be a two-quantum ring
occupied by p. 2 electrons, the third, three-quantum, the .M-ring
with^ 3 electrons, and so on (Fig. 11). The question whether
112 THE QUANTUM THEORY
we can reach our goal with this conception of the ring by
assuming the quantum numbers to increase as we go outwards,
and whether we are to take the rings as co-planar or inclined
to one another will be left open. The preparation for the
emission of the -BT-series consists in this, that by the addition
of energy whether by absorption of external radiation or by
electronic impact an electron of the K-ring is removed to in-
finity, that is, the atom is, so to speak, ionised " inside," i.e.
in the .BT-ring. If the energy of the atom before this inner
ionisation = W , and after the ionisation = W K , then the
amount W K - W of energy must be provided. Hence every
radiation, the energy quantum of which satisfies the condition
hv ^ W K - W , can on being absorbed effect the tearing of the
electron out of the K-r'mg. If we allow the v of the external
radiation to grow slowly from small values, then, at the point
"TT7" \JU
V K = *_ 9, a sudden increase of the absorption occurs,
because from this point onwards the external radiant energy
is used for the " ionisation of the K-ring." Thus an absorp-
tion-band extends from v = V K towards higher frequencies, the
edge of the band lying at V K . This phenomenon of the " edge
of the absorption-band " has already been interpreted above
in the sense of the hypothesis of light-quanta. If the addition
of energy is provided by the impact of a strange electron,
coming from without, then its energy must be E> W K - W ,
that is, E ^> ~hv K , a relation, which we have already deduced
earlier from the standpoint of the quantum hypothesis of light.
By ionisation of the .ST-ring the atom is now prepared for
^-emission. If now an electron falls from the 2-quantum
Zv-ring into the 1-quantum J5"-ring, filling up, so to speak, the
gap produced there, then the first line of the K- series, K a ,
will be emitted. If on the other hand the gap in the .ST-ring
is filled by an electron of the 3 -quantum If -ring, or the
4-quantum .W-ring, Kp or K y result respectively. The position
is quite analogous as regards the L- and .M-series. If, by the
addition of energy (absorption or electron-impact), an electron
of the Zi-ring is battered off, that is if the L-ring is ionised,
then the atom is prepared for the emission of the ZJ-series.
If, now, the gap in the 2-quantum L-ring is filled by an
electron of the 3-quantum M-ring, the first line of the Z/-series,
FINE-STRUCTURE OF RONTGEN LINES 118
L a , results; if it is filled by an electron of the N-riug, the
second line of the .L-series, Ly, results (the notation is not
quite consistent but will serve the present purpose), and so
forth.
The converse phenomenon to line emission, viz. line absorp-
tion, with which we are acquainted in visible spectra, appears
at first sight to be missing here. That is, however, as W.
Kossel 298 recently showed, an error. It is true that the ejected
electron of the .ST-ring, for example, cannot in general be
caught upon the L-, M-, or .N-ring, because all places on them
are already occupied. An absorption of the lines K a , Kp, K y ,
is therefore in this case impossible. But the electron of the
K-ring can certainly come to rest on an unoccupied quantum
orbit outside the occupied rings, that is, outside the surface
of the atom. In this process a " line " is actually absorbed,
namely, that line of which the hv is equal to the energy-
difference between the K-rmg and the final orbit of the ejected
electron. This refinement of our considerations shows, then,
that the electron from the .fiT-ring does not need to be raised
immediately to infinity, but that line absorptions may occur
before the edge of the band of absorption is reached.
2. The Fine-structure of Rontgen Lines
It is particularly noteworthy that Sommer/eld succeeded
also in the field of X-ray spectra in explaining the fine-
structure of the lines by calling in the aid of the theory of
relativity. Thus, for example, the 2-quantum .L-orbit is
"double"; it can occur as a circle (n = 0, n = 2) or as an
ellipse 2" (n = 1, n = 1). Hence the line which is emitted
by the electron of which the L-rmg is the initial orbit, namely,
K a , is a doublet (K a and K a -). In just the same way, those
lines for which the Iv-orbit is the final orbit of the electron
are doublets, namely, the line L a (more exactly L a >) to which
Lp is added to make a doublet ; further, L y which forms
a doublet with L&, and so forth. The distance between the
components of the doublets (expressed in frequencies) comes
out, according to Sommerfeld's Theory, as approximately pro-
portional to the fourth power of the atomic number z. Hence
here, in the X-ray region, where we are dealing for the most
part with elements having fairly high atomic numbers, the
8
114 THE QUANTUM THEORY
doublets appear microscopically enlarged as compared with
the microscopic hydrogen-doublet (z = 1). During the emis-
sion of X-rays the electron approaches very near to the
highly-charged nucleus, and hence the relativistic effects of
the resolution of the lines are much greater than in the case
of the optical spectra, in which the electron is moving at the
surface of the atom, where it is almost entirely screened from
the action of the strong nucleus by the remaining electrons.
With the help of the following relation deduced theoretically
and adapted to experimental evidence,
< - fr")' -.< 112 >
Sommerfeld was able to calculate the hydrogen-doublet from
the observed L-doublets, and compare it with the results of
experiment. The agreement is very satisfactory.
3. The Distribution of Electrons among the Rings. Objections
to the Ring-arrangement of Electrons
The quantitative calculation of the simplest case, namely,
the emission of K a , led Debye to the conclusion that the
.ST-ring in the normal state consists of three electrons. To
this Kroo, by elaborating the calculation, adds the con-
clusion that the L-r'mg contains in its normal state nine
electrons. With these two distribution numbers, p l = 3,
p 2 = 9, the position of K a could be represented as a function
of the atomic number z for all elements. The emission of K a
takes place according to the following obvious scheme :
I K-ring L-ring
Normal state | 3 9 . _ . . , , _ .
Initial state [ 2 | 9 > I ni8atlon of * Z-nng.
KEafBtate | 8 | 8 > EmiSS1On f *"
The two distribution numbers (Besetzungszahleri) thus found
for the two innermost rings excite our attention. For on
the basis of the Periodic System with its periods of eight
we ought to expect, according to Kossel, the numbers 2 and 8.
DISTRIBUTION OF ELECTRONS 115
The strange occurrence of the numbers 3 and 9 becomes
an objection, when we consider the case of sodium (z = 11).
Here, according to Kossel, we should expect the numbers 2,
8, 1, since in all probability an electron (the valency electron)
revolves alone, as in the case of all alkali metals, around the
outside quantum orbit (M-ring). In any case it is impossible
that the two innermost rings together should, in the normal
state, contain 12 (= 3 + 9) electrons. If we attempt to go
a step further still on the basis of Kroo's numbers 3 and 9,
and to set up a formula which represents for all 2*8 the
position of L a in conformity with observation, and thereby
to determine the number of electrons p 3 on the .M-ring, we
find, as A. Smekal 300 showed, that this mode of representation
is impossible with any combination 3, 9, p y Nor do we fare
better if we incline the various rings to one another, and take
their interaction into account. The suspicion is forced upon
us, that perhaps the whole conception of the arrangement
into plane rings does not correspond with fact, but that, rather,
the electrons in the atom form spatially symmetrical figures.
This suspicion is very much strengthened by a series of pro-
found investigations carried out by M.. Born and A. Landd. 901
Following on M. Bom's investigations of the dynamics of
the crystal-lattice, which we discussed in detail earlier hi
connection with the atomic heat of solids, the two in-
vestigators asked themselves the question, whether it is
possible to build up the cubic crystal-lattice of the alkaline
halides (NaCl, NaBr, Nal; KC1, KBr, KI, etc.) from ions of
Bohr atoms, by taking into account only the mutual electro-
static forces; and whether this method, if possible, would
enable them to prophesy the crystal properties (lattice-con-
stant, compressibility) from the atomic models of the two
constituent ions. The answer to this question has been, on
the whole, in the affirmative. But when the calculation of
the compressibility of these crystals was carried out, the
remarkable result manifested itself that crystals are found to
be too soft, that is, insufficiently rigid, if the conception of the
ring-arrangement of electrons in the atom is maintained. On
the other hand, we get good agreement with the observations
if, following Born, we introduce the hypothesis that the
electrons are arranged spatially. A complex of eight electrons,
116 THE QUANTUM THEORY
as occurs in sodium, potassium, etc., does not therefore occupy
a plane 8-ring ; the eight electrons describe paths of ciibical
symmetry. Into the still obscure region of these " spatial "
electron paths, A. Lande 302 has made some successful in-
cursions.
From all that has been said it would appear to be certain
that in dealing with Rontgen spectra, too, we can no longer be
content with the arrangement of the electron rings in planes,
and that the whole quantitative theory of the Eontgen series,
including Sommerfeld's fine-structure of the K- and the L-
doublets, must be built up on a fresh foundation.
CHAPTEB VIII
Phenomena of Molecular Models
i. Dispersion and Magneto-rotation of the H 2 Molecule
WHILE the X-ray spectra and the spectra of the optical
series arise from the atoms of the elements (and hence
their theory links up with the atomic models), there is a series
of phenomena which, in the case of polyatomic substances,
are peculiar to the molecules, and the theory of which,
therefore, is founded on the molecular models. Chief among
these are the normal dispersion, the rotation of the plane of
polarisation in the magnetic field (magneto-rotation), and,
further, the great and complicated subject of band-spectra.
Up till a few years ago, dispersion and magneto-rotation had
been exclusively treated from the standpoint of the Thomson
model, that is, with the help of quasi-elastically bound
electrons, and this explanation had served in turn as a
powerful support for this model. Nevertheless, discrepancies
in these theories had long been known. For example,
measurements calculated upon the basis of the dispersion
theories of Drude, Voigi, or Planck led to values for the ratio
of the charge to the mass of the electron ( ) which, in com-
parison with the direct measurements of this quantity (based
upon the deflection of the cathode- or /3-rays in the electric
and magnetic fields) which were much too small. When,
however, the Thomson model became displaced by the
Rutherford-Bohr model, and the successes of the Bohr atomic
model increased at an undreamed-of rate, the question arose
whether an unobjectionable theory of dispersion and magneto-
rotation could not be founded upon these new views. The
difficult position, into which we are brought by this problem,
117
118 THE QUANTUM THEORY
arises from the fact that we do not actually know a single
instance of the exact manner in which a polyatomic Bohr
molecule is built up from its nuclei and electrons. The
exact knowledge of this structure, and the motion of all the
electrons is absolutely necessary, if we desire to know how
the molecule reacts upon external waves (dispersion). It is
true that W. Kosseiw* has, in a detailed study already
referred to above, pointed out the general guiding lines along
which, from the chemical point of view, the building-up of
the atom from molecules must be carried out, but the details
of this construction remain open. Only in a few of the
simplest cases have detailed molecular pictures been con-
structed and closely tested. Thus Bohr, as we remarked in
discussing the atomic heat of gases, has already proposed a
model of the diatomic hydrogen molecule. It has the follow-
ing construction (see Fig. 8) : two singly-positive nuclei (that
is, each consisting of only a single positive charge) are
separated by the distance 26. In the vertical plane which
bisects the line joining the nuclei, two electrons rotate,
diametrally opposite one another, on a circle of diameter 2a.
The equilibrium of the Coulomb and the centrifugal forces
requires that a = b^/3. By means of this relation, and by the
quantum condition that each electron must have the moment
of momentum , the model is completely determined in all
2ir
its dimensions and speeds. It was this model which was
the first to be proposed : it was examined by P. Debye 3*
with reference to its dispersion. On account of its sym-
metrical structure the molecule possesses no electrical mo-
ment in its normal state. If, on the other hand, it is struck
by an external light wave, the motion of its electrons is
periodically disturbed ; they depart from the normal quantum
path, fall into forced vibration, and thus generate an electric
moment which changes periodically in step with the external
wave. Thus the original motion of the primary wave is
changed, and dispersion results. We may conceive this as
follows : Let c be the velocity of the primary wave in vacuo.
The oscillations of the electrons generate a secondary wave
which spreads out from the molecules. All these secondary
waves combine with the primary wave to a form new wave
OBJECTIONS TO BOHR'S MODEL 119
which moves with the altered velocity q, the value of which
depends on the frequency of the primary wave. But just
this is the phenomenon of dispersion. The electronic vibra-
tions which occur here are not oscillations about positions of
equilibrium, as in the case of the quasi-elastic model, but
oscillations about stationary paths. Moreover, here, the force
holding the electrons, as opposed to the usual classical
theories of dispersion, is anisotropic (that is, the electron is
held by different forces in different directions) ; above all, by
means of this anisotropy, it was possible to explain away the
disagreement in the value of , which had previously been
WIC
found to be too small ; and Debye succeeded, on the basis of
the normal value of , in deducing from the theory the
observed dispersion curve of hydrogen, that is, the curve
which shows how its coefficient of refraction depends on the
wave-length. It should be noted that in the formula for the
coefficient of refraction, no single constant is arbitrary, but
that the dispersion formula is made up entirely of universal
constants.
Using the same method (calculus of disturbances), P.
Scherrer 30 * has calculated the rotation of the plane of
polarisation which linearly polarised light undergoes in its
passage through hydrogen under the influence of a magnetic
field. His efforts were equally successful.
2. Objections to Bohr's Model of the Hydrogen Molecule
In spite of the successes which the Bohr model of the
hydrogen molecule has won, a list of weighty objections to
it has accumulated in the course of time. That the con-
tribution which the rotation (more accurately, the regular
precession) of this molecule makes to the molecular heat at
low temperatures, does not correspond with the observations
of Eucken, has been shown by P. S. Epstein, as we have
already mentioned. Also at high temperatures, when the
oscillations of the two nuclei relatively to one .another con-
tribute to the molecular heat, no agreement between theory
and observation has been found in the case of the Bohr model,
as G. Laski 306 recently showed,
120 THE QUANTUM THEORY
Further, the model must possess, in consequence of the
revolving electrons, an almost fixed magnetic moment parallel
to the axis of the nucleus, that is to say, it must be equivalent
to a molecular elementary magnet, which endeavours to set
itself, in an external magnetic field, parallel to the lines
of force. Hydrogen ought, therefore, to be paramagnetic,
whereas it is diamagnetic.
Another very important objection, to which Nernst in
particular drew attention, is the following : if we calculate
the work which is necessary to separate the molecule into
its two atoms, the so-called heat of dissociation, we get s 07 the
value, 61,000 calories. On the other hand, Langmuir 3 08 found
84,000 cals., Isnardi** 95,000 cals., /. FrancJc, P. Knipping
and Thea Kriiger 81,000 ( 5700) cals. In any case, the
calculated heat of dissociation comes out 25 per cent, too
small.sioa
Finally, W. Lenz 311 has recently increased the objections
to the hydrogen model by an important one based on a
theory of band-spectra, which we shall discuss below. He
proved that the band-lines of hydrogen and nitrogen can
exhibit the observed Zeeman effect, only if these molecules
possess no moment of momentum around the nuclear axis.
The fact that the two electrons in Bohr's molecular model
revolve in the same sense, however, endows it with just such
a moment of momentum. On the whole, the Bohr model does
not seem to correspond to reality ; the arrangement of the two
nuclei and electrons must plainly be quite different. No
satisfactory model, however, has yet been found.
3. Models of Higher Molecules
Matters are no better in the case of models of the more
complicated molecules. It is true that Sommerfeld 312 and
F. Pawer 313 have also worked out the theories of dispersion
and magneto-rotation in the case of the more general Bohr
models (N 2 and O 2 ) which are constructed on the lines of the
hydrogen model. According to Sommerfeld, four electrons
revolve about the line joining the two nuclei in the case of
oxygen, each of which acts with an effective charge + 2e ; in
the case of nitrogen, a ring of six electrons rotates about the
nuclear axis, while the nuclei carry triple effective charges.
QUANTUM THEORY OF BAND-SPECTRA 121
Sommerfeld was able to obtain agreement with observation only
by setting up for each electron of a valency ring of 2s-electrons
the unaccountably strange quantum condition : moment of
momentum = ^ ^ s > undoubtedly a most unsatisfactory
result. Gerda Laski 31 * obtained better results with some-
what different models, which she chose in such a way that
the specific heat of the two gases at high temperatures agreed
with the observations of Pier. 318 According to her ideas, the
nitrogen molecule must consist of two seven-fold positive nuclei,
each of which is closely surrounded by a 1-quantum ring of
two (or three) electrons. The " valency ring " in the central
vertical plane is 2-quantum and contains ten (or eight)
electrons. Analogously, the oxygen molecule consists of two
eight-fold positive nuclei, each encircled by a 1-quantum ring
of two (or three) electrons, whereas the 2-quantum valency
ring contains twelve (or ten) electrons. The same objections
apply to some extent to these models of Sommerfeld and Laski
as to the hydrogen model. For example, they give no account
of why oxygen should be paramagnetic, and nitrogen, on the
other hand, diamagnetic. Moreover, the above-mentioned
objection of Lenz applies in full force to these models ; for
they all possess moments of momentum around the nuclear
axis. In conclusion, we feel bound to admit that the exact
constitution of even the simplest models is at present unknown
to us.
4. The Quantum Theory of Band-spectra
To conclude this chapter, we shall turn our attention to the
band- spectra, and collect together shortly what the quantum
theory has been able to assert about them up to the present
time. That they belong to molecules and compounds may
nowadays be regarded as certain. The first attempt to con-
struct a logical quantum theory of band-spectra was under-
taken by K. Schwarzschild 316 who clearly recognised the
importance of the rotation of the molecule in the production
of these spectra. His conceptions may be defined as follows :
a system of electrons revolves at a definite quantum distance
around a molecule which itself rotates according to quantum
conditions, the assumption being made for the sake of
122 THE QUANTUM THEORY
simplicity that the motion of the electrons is not influenced
by the motion of the molecule. If E is the quantum energy
of the electrons, E r the quantised rotational energy of the
molecule, then E + E r = E is the total energy of the system.
If the three chief moments of inertia of the molecule / are
equal to one another, then it follows, just as in (80), that
where n denotes the rotational quantum number. Therefore
If, now, the system passes from one quantum state having
the electronic energy E Q and the rotational quantum number
n into another quantum state having the electronic energy E'
and the rotational quantum number n', then it follows from
Bohr's frequency formula (92) that the frequency of the line
radiated is given by
_ E -E' (n - n>
7 T o 2 T ' ' V A /
If we keep all the quantum numbers which occur here, except-
ing n, constant, and allow n to vary, then we get a series of
lines progressing towards the violet and having the frequencies
v = a + bn 2 (a and b are constants) . (115)
This is a formula which had already been given empirically
by Deslandres,* 11 and which is approximately true for the lines
of many bands.
Following Schwarzschild, T. Heurlinger 318 and W. Lenz* 19
in particular, have further developed and refined the quantum
theory of band-spectra. For example, Lenz has pictured the
molecule as a symmetrical top having two moments of inertia
and a rotational rigidity (moment of momentum) around the
axis of the figure, and hence deals from the outset with a regular
precession of the molecule in place of a rotation. Using
Bohr's frequency formula, and applying the principles of
selection, he obtained the following general foi-mula for the
lines of a band :
v a + bn + en 2 (a, b t c are constants) . (116)
QUANTUM THEORY OF BAND-SPECTRA 123
which is obeyed, according to Heurlinger, in the case of the
so-called " cyanogen " lines of nitrogen, for example. In
addition to the lines given by (116), Lenz's Theory requires the
occurrence of the series given by the formula
+ + 2 . . . (117)
for the case that the molecule really possesses a finite moment
of momentum about its axis of figure. A series which follows
this law does not, however, exist in the cyanogen bands, ac-
cording to Heurlinger. Lenz deduces from this the conclusion
already mentioned, that the nitrogen model does not possess
a rotational rigidity about its axis. By calculating the
Zeeman effects of the band lines, and comparing them with
observation, Lenz was able to confirm this, and to extend it to
the hydrogen molecule.
The infra-red Bjerrum absorption bands of the diatomic and
polyatomic gas compounds, which we had discussed at length
in Chapter Y, belong to the general type of band-spectra. If
we are to deduce them from a theory consistently founded on
quanta and not, as we did earlier, half according to the
quantum, half according to the classical theory we must
follow closely the course pursued above, with the difference
that, in place of the energy of the electronic system there will
appear the energy of the atoms, 920 with which the rotational
energy of the molecule is combined, as a first approximation,
additively. The logical carrying out of this calculation (in
which Bohr's frequency formula and the principle of corre-
spondence are applied), which was undertaken by Heurlinger & l
and the author, 322 gives for the structure of the " fluted " ab-
sorption bands an arrangement of lines which at first sight
does not appear to agree with the beautiful and exact measure-
ments of Imes. 323 The theory gives for the position of the
absorption lines a formula
l/ = Vo(n + i) JL, (n=l,2,3...) (118)
and therefore requires that all neighbouring lines be equi-
distant, including the two in the middle (n = 0). On the other
hand, Imes' observations show with indubitable clearness that
the interval between the two middle lines is twice as great as
124 THE QUANTUM THEORY
the interval between all neighbouring lines. This apparent
contradiction is explained, as A. Kratzer 32 * recently showed,
in a surprising fashion, if we take into account the intensity
of the absorption lines according to Bohr's Principle of Analogy.
For it then appears that the first absorption line to the right
of the middle v -line, namely, the line
h
" = "o + SW
(which is derived from formula (118) by setting n = and
using the positive sign for the second term) is of vanishingly
small intensity. This line is generated when the molecule
passes over from an initial rotationless and vibrationless state
into the final state in which the two ions oscillate relatively
to one another with one quantum, and in which, at the same
time, the molecule rotates as a whole with one quantum. The
rotationless and vibrationless state has, however, a vanishingly
small probability ; the number of transitions from this initial
state per second, and therefore the intensity of the correspond-
ing absorption line, is hence vanishingly small. By the dis-
appearance of the first line to the right of the middle position
v , the structure of the lines as observed by Imes is actually
reproduced, as one may easily recognise ; in the formula, the
" middle" of the line structure is displaced from the point v
to the right by the amount oZFr The absorption lines group
themselves equidistantly and symmetrically on both sides of
the missing " middle," v = v + Q-^J- This state of affairs
O7T J
may be expressed by writing, in formal agreement with (83),
" = "''4^7 (*-:i,8,8...)|
where . (119)
From the constant interval between neighbouring lines, namely
. (120)
the moment of inertia of the rotating molecule can be cal-
culated with great accuracy. 328
CHAPTEE IX
The Future
IN the preceding pages the author has attempted to give
in broad outline the most important features of the
doctrine of quanta, its origin, its development, and its
ramifications. If we now survey the whole structure, as
it stands before us, from its foundations to the highest story,
we cannot avoid a feeling of admiration ; admiration for the
few who clear-sightedly recognised the necessity for the pew
doctrine and fought against tradition, thus laying the founda-
tions for the astonishing successes which have sprung from
the quantum theory in so short a time.
None the less, no one who studies the quantum theory
will be spared bitter disappointment. For we must admit
that, in spite of a comprehensive formulation of quantum
rules, we have not come one step nearer to understanding
the heart of the matter. That there are discrete mechanical
and electrical systems, characterised by quantum conditions
and marked out from the infinite continuity of " classically "
possible states, appears certain. But where does the deeper
cause lie, which brings about this discontinuity in nature?
Will a knowledge of the nature of electricity and of the con-
stitution of the electromagnetic field serve to read the riddle ?
And even if we do not set ourselves so distant a goal, there
remains an abundance of unanswered questions. The
decision has not yet been made, as to whether, as Planck's
first theory requires, only quantum-allowed states exist (or
are stable), or whether, according to Planck's second formula-
tion, the intermediate states are also possible. We are still
completely in the dark about the details of the absorption
and emission process, and do not in the least understand
125
126 THE QUANTUM THEORY
why the energy quanta ejected explosively as radiation
should form themselves into the trains of waves which we
observe far away from the atom. Is radiation really pro-
pagated in the manner claimed by the classical theory, or
has it also a quantum character ?
Over all these problems there hovers at the present time
a mysterious obscurity. In spite of the enormous empirical
and theoretical material which lies before us, the flame of
thought which shall illumine the obscurity is still wanting.
Let us hope that the day is not far distant when the mighty
labours of our generation will be brought to a successful
conclusion.
Mathematical Notes and References
1 0. Lummer and E. Pringsheim, Wiedem. Ann. 63, 395 (1897) ;
Verhandl. d. deutsch. physikal. Ges. 1899, pp. 23, 215; ibid., 1900, p. 163.
Of. also O. Lummer and E. JahnJee, Drudes Ann. 3, 283 (1900), and O.
Lummer, E. Jahnke and E. Pringsheim, Drudes Ann. 4, 225 (1901).
2 Of. M. Planck, Vorlesungen iiber die Theorie der Warmestrahlung
(Leipzig 1906), 10.
3 Frequency () = velocity oflight in vacuo (c) ^
wave-length in vacuo (A.)
iCf., for example, M. Planck, Vorlesungen iiber Warmestrahlung
(1906), 17.
3 O. Kirchhoff, Gesammelte Abhandlungen (J. A. Earth, Leipzig 1882),
pp. 573 et seq. ; Berliner Akademieberichte, 1859, p. 216 ; Poggend. Ann.
109, 275 (1860).
6O. Lummer and W. Wien, Wiedem. Ann. 56, 451 (1895). Cf. also
O. Lummer and F. Kurlbaum, Verhandl. d. deutsch. physikal. Ges. 17,
106 (1898).
7 Of. Note 5.
8 L. Boltzmann, Wiedem. Ann. 22, 291 (1884).
9J. Stefan, Wiener Ber. 79, 391 (1879).
10 The Stefan-Boltemann Law is deduced as follows : Let the energy
of black-body radiation at the temperature T, which is enclosed in a
space of volume V having a movable piston, be U = Vu, where u is
the " spatial " density of the radiant energy. The pressure, equal in
all directions, which the radiation exerts upon the piston and walls is,
according to electrodynamics, p = $u. If we supply to this system at
the temperature T (that is, isothermally) an amount of heat d'Q, then
its energy increases by dU, and the radiation does work pdV in push-
ing back the piston. Therefore, according to the first law of thermo-
dynamics, and owing to the two relations above :
d'Q = dU + pdV = udV + Vdu + dV = | udV + Vdu.
According to the second law of thermodynamics, -Q must be a com-
plete differential. Hence the following relation holds :
\_ 3 m 4 d u\dT _1
,'ldt* u)dT_l
4 fl dw _ u]
3\TdT Z^J
127
128 THE QUANTUM THEORY
i.e. -^ = 4^, which, integrated, gives u = a!"
where a is a constant. Now, as we can easily see, the total radiation
00
K 2 / TH v dv is distinguished from the density of radiation u only by
a constant factor (see M. Planck, Lectures in Radiation (1906), 22),
hence the total radiation is proportional to the fourth power of the
absolute temperature and this is the Stefan-Boltzmann Law.
11 W. Wien, Sitzungsber. d. Akad. d. Wissensch. Berlin, 9 Feb. 1893,
p. 55 ; Wiedem. Ann. 52, 132 (1894). Of. also Max Abraham, Theorie
der Elektrizitat II, 43 (1914); M. Planck, Vorlesungen iiber die
Theorie der Warmestrahlung (Leipzig 1906), pp. 68 etseq. ; W. Westphal,
Verhandl. d. deutsch. physikal. Ges. 1914, p. 93; H. A. Lorentz, Akad.
d. Wissensch. Amsterdam, 18 May 1901, p. 607.
12 Formula (4) of the text (Wieris Law of Displacement) may be
obtained by means of a simple dimensional calculation, as L. Hopf
recently showed in the " Naturwissenschaften " (8, 109, 110 (1920)).
We assume that Kx depends only on v, T, and the velocity of light c.
The dimensions of Kx are obtained from the fact that, according to (1),
energy
surface x time
From this it follows that
[K,] - [-'].
If we set
Kx = const. v x T y c z
then, remembering that T has the dimensions of energy, we get
[m- 2 ] = const. [-* m y - V* r *" f t~ Z 1
= const. \m y ?V+* r*~ -*]
Hence x = 2; # = 1 ; 2 = -2
which gives us, Kx = const. . ?1 . T.
This relation is not, however, as we shall see, generally valid. In fact,
oo
it would give no finite value for K = 2 \"K. v dv. But, according to the
6
Stefan-Boltzmann Law (3), K = y T 4 . Hence the constant of Kx may
still depend on a dimensionless combination of the four variables
y, v, T, c. If, therefore, we set const. =f(y^T n c^y' a ) then the argument
of the function / must have the dimension 0. If, further, we remember
that
[ BfKf* surface x time
H = .
NOTES AND REFERENCES 129
it then follows that
= [t-t .mri.fr, t-*i .K.t -<T. m- 3 . l-**> . t ]
= [< -*--?+.
Hence const. = /[(?)". c- - 7 ,] = <
Therefore K , = Jr.* (j) = *. J.
or, finally, Kl> = 3 F (j}
13 If we plot K, - *(} function of v, keeping T constant, the
maximum of this curve if one is present lies at that point at which
= 0. This gives
where F 1 is the differential coefficient of F with respect to the argument.
This equation, in which only occurs as unknown, gives a definite value
for ^. In other words, for v = max, it follows that ^^ = const.
II W. Wien, Wied. Ann 58, 662 (1896).
13 0. Lummer and E. Pringslieim, Wied. Ann. 63, 395 (197) ; Drude's
Ann. 3, 159 (1900) : Veih. d. deutsch. phys. Ges. 1, 23 and *15 (1899).
The total radiation emitted per second from 1 cm. 2 in one direction is,
by formula (1) (
s =
According to the Stefan-Boltzmann Law, S is nroportional to T 4 , there-
fore S = ffT*. (The constant of proportionality a is related to the
constant y occurring in (3) by the equation v = iry.) The absolute
measurement of S gave the following values for a, in chronological order :
<r - 5-45 10' 12 r V ! a !' t ,"| according to F. Eurlbaum [Wiedem. Ann.
Lcm/ deg.*J
65, 746 (1898); Verhandl. d. deutsch.
physikal. Ges. 14, 576, 792 (1912)].
= 5-58. 10 ~ 12 ti according to S. Valentiner [Ann. d. Phys.
31, 255 (1910) ; 39, 489 (1912)].
= 5-90 . 10 " 12 .1 according to W. Gerlach [Ann. d. Phys.
38, 1 (1912)].
180 THE QUANTUM THEORY
a- - 5-30 . 10 ~ 12 T cm Tde 1 according to E. Bauer and M. Moulin
[Soc. Franc, de Phys. Nr. 301, 2-3
(1909)].
= 6*30 10~ 12 > according to Ch. Ffry [Bull. Soc. Franc.
Phys. 4 (1909)].
= 6-51. 10 ~ 12 > according to Ch. F&ry and M. Drecq
[Journ. de Phys. (5) 1, 551 (1911)].
= 5*67 . 10 ~ 12 i> according to G. A. Shakespear [Proc. Roy.
Soc. (A) 86, 180 (1911)].
==5-54. 10 ~ 12 M according to W. H. Westphal [Verhandl.
d. deutech. physikal. Ges. 14, 987 (1912)].
- 6-05. 10 " 12 >, according to L. Puccianti [Cim. (6) 4, 31
(1912)].
= 5-89. 10 ~ 12 ,, according to Keene [Proc. Roy. Soc. (A)
88, 49 (1913)].
= 5-57.10-12 according to W. H. Westphal [Verhandl,
d. deutsch. physikal. Ges. 15, 897 (1913)].
-= 5-85 . 10 ~ 12 , according to W. Gerlach [Phys. Zeitschr.
17, 150 (1916)].
As regards Wieris Law of Displacement, the relation (5a) was tested
and found to be confirmed. From Fig. 1, in which E*. is plotted as a
function of A. for different values of A., we see clearly how the maximum
of the curve becomes displaced towards shorter wave-lengths as the
temperature rises.
For the constant on the right-hand side of relation (5o) the measure-
ments gave the following values :
const. = 0-294 [cm. deg.] according to O, Lummer and E. Pringsfoim
[Verhandl. d. deutsch. physikal. Ges. 1, 23
and 215 (1899)].
= 0-292 according to F. Paschen [Drude's Ann. 6, G57
(1901)].
= 0-2911 according to Coblentz [Bull. Bur. of Stand. 10,
1 (1914)].
16 O. Lummer and E. Pringsheim, Verhandl. d. deutsch. physikal.
Ges. 1, 215 (1899).
17 F. Paschen, Berliner Ber. 1899, pp. 405, 959.
18 M. Planck, Absorption und Emission elektr. Wellen durch Resonant.
Sitzungsber. d. Berl. Akad. d. Wiss. 21 March 1895, pp. 289-301 ; Wiedem.
Ann. 57, 1-14 (1896). tJber elektr. Schwingungen, welche durch Re-
sonanz erregt und durch Strahlung gedampft werden. Sitzungsber. d.
Berl. Akad. d. Wiss. 20 Febr. 1896, pp. 151-170 ; Wiedem. Ann. 60, 577-599
(1897). tTber irreversible Strahlungsvorgange. (1. Mitteilung.) Sitzungs-
ber. d. Berl. Akad. d. Wiss., 4 Febr. 1897, pp. 57-68. (2. Mitteilung) ibid.,
8 July 1897, pp. 715-717. (3. Mitteilung) ibid., 16 Dec. 1897, pp. 1122-
1145. (4 Mitteilung) ibid., 1 July 1898, pp. 449-476. (5. Mitteilung) ibid.,
18 May 1899, pp. 440-480. (Supplement.) ibid., 9 May 1901, pp. 544-555 ;
NOTES AND REFERENCES 181
Drudes Ann. 1, 69-122 (1900). (Supplement.) Drudes Ann. 6, 818-831
(1901). Entropie und Temperatur strahlender Warme. Drudes Ann. 1,
719-737 (1900).
19 In place of the mean value, with respect to time, of the energy of a
single oscillator, we may use the spatial mean value of the momentary
energy of a whole system consisting of very many oscillators.
20 In this second, more difficult part of the calculation, Planck takes his
stand upon the second law of thermodynamics, and seeks, from this view,
to determine a phase-quantity S of the oscillator, which possesses the
well-known property of the entropy, that it increases in all irreversible
processes. He arrived at the solution :
This function possessed, as Planck showed, the required property of en-
tropy, but it was not the only function with this property. And in fact
it appeared later, that in the deduction of the above expression, a readily
suggested but unjustified supposition had been made. The expression
given in the text, formula (8), for the mean energy U follows from S by
applying the second law in the form :
MO. Lummer and E. Pringsheim, Verhandl. d. deutach. physikal.
Ges. 1900, p. 163.
22 M . Planck, Verb. d. deutsch. phys. Ges. 1900, p. 237. It is of
historic interest to note that Planck had already, in a somewhat earb'er
paper (Verh. d. deutsch. phys. Ges. 1900, p. 202), arrived at the true
law of radiation by a purely formal alteration of Wien's formula, which
was not further explained. Cf. also Ann. d. Phys. 4, 553 (1901) ; 4, 561
(1901) ; 6, 818 (1901) ; 9, 629 (1902).
23 Let N oscillators be present. Let the total energy to be divided
among them be UN = NU. The " state " or phase of the oscillator-
system, the probability of which is to be calculated, is then defined by the
fact that N oscillators possess the energy Ujt. We divide UN into P
energy elements , so that
UN = N . U = P*.
The number of possible ways of distributing P balls among N boxes is,
however,
(N+P- 1)1
(N - 1) 1 P ! '
This is therefore the probability of the state, which corresponds to the
distribution of P energy elements among N oscillators. P. Ehrenfest aud
H. Kamerlingh-Onnes give a very simple deduction of this formula in
Ann. d. Phys. 46, 1021 (1915).
The rule mentioned in the text, which is due to Boltzmann, states
182 THE QUANTUM THEORY
that the entropy Sjt of the oscillator system is connected with the prob-
ability TFby the fundamental relation
Sir = k log W
where k is a constant.
In this theorem of Boltzmann the following law of the growth of en-
tropy (second law of thermodynamics) is contained : if a system passes
from an improbable condition into a more probable one, then by this
transition W, and therefore the entropy S, increases. If we here insert the
value of W, and, since N and P are very large numbers, use Stirling's
approximation formula
\oge(Nl)=N(logeN- 1)
then, if we set for P, N , we get by an easy calculation
and hence the entropy S of one oscillator becomes :
But according to the Second Law (see note 20)
dS 1
If we carry out the differentiation on the left-hand side, and solve the re-
sulting relation between U, T, and e, with respect to U, we get the ex-
pression (9) of the text.
2< Of. the paper by Ehrenfest and Katnerlingh-Onnes cited in the
previous note.
2S This law is essentially identical with Boltzmann' 's H- Theorem. Of.
L. Boltztnann, Vorlesungen iiber Gastheorie Bd. I, p. 38 (1896) ; Sit-
zungsber. d. Wiener Akad. d. Wiss. (II) 76, 373 (1877). Of. also P.
Ehrenfest, Phys. Zeitschr. 15, 657 (1914).
36 H. Rubens and F. Kurlhaum, Sitzungsber. d. Berl. Akad. d. Wiss.
1900, p. 929 ; Ann. d. Phys. 4, 649 (1901).
27 F. Paschen, Ann. d. Phys. 4, 277 (1901).
28 L. Holborn and 8. Valentiner, Ann. d. Phys. 22, 1 (1907) ; Coblentz,
Physical Review, 31, 317 (1910) ; E. Baisch, Ann. d. Phys. 35, 543 (1911) ;
E. Warburg, O. Leithttuser, E. Hupka and C. Milller, Ann. d. Phys. 40,
609 (1913) ; E. Warburg and C. Milller, Ann. d. Phys. 48, 410 (1915).
W. Nernst and Th. Wulf, Ber. d. deutsch. phys. Ges. 21, 294 (1919).
80 Lord RayUigh, Phil. Mag. 49, 539 (1900).
31 The " Stefan-Boltzmann constant of total radiation" <r, introduced
in note 15, has therefore the value
NOTES AND REFERENCES 133
32 In order to determine the constants h and k which occur in the
radiation formula, we can, instead of using the equation : \max . T =
const., compare other relations with the measurement of the total
radiation. For example, we can proceed as follows : At a constant
temperature T we measure the ratio of the intensity of radiation for two
different wave-lengths \j and \^ (isothermal method). Now this ratio
is, according to (15)
&* \ A 1/ _C_ K
e* lT - 1
From this relation, since everything excepting C is known, C, that is,
- may be calculated. Another method is the following : we measure for
a fixed wave-length x the ratio of the intensity of radiation at two
different temperatures T l and T 2 (isochromatic method). Then it follows
that
This is a relation from which C, that is,- can again be calculated.
With the help of these methods, the researches, for example, of
Warburg and his co-workers cited in note 28 have yielded values for
C = ^ which lie in close proximity to C = 1-430. This value was taken
by Nernst and Wulf (see note 29) for their critical investigation.
For the constant of Wien's Law of Displacement in the form \inax . T
= b we would accordingly get from (16) :
4-9651
a value smaller, therefore, than that given by direct measurement (see
no'e 15). Whether Warburg's value, C = 1-430, or the measured values
of &(> 0-29) or both, are seriously affected by experimental error, or
whether after all as Nernst and Wulf maintain Planck's formula is
not right, must be left for the future to decide.
38 M. Planck, Ann. d. Phys. 4, 553 (1901).
84 If we apply Boltzmann's relation S = k log W (quoted in note 15),
which connects the entropy S with the probability of state W, to one
gramme-molecule of an ideal gas, then by calculating the probability of
a certain state, i.e. a certain distribution of velocities among the
molecules, we arrive at the following value for the entropy of the gas
S kN(l log* U + log V) + const.
184 THE QUANTUM THEORY
(Of., for example, M. Planck, Lectures on the Theory of Radiation
(1906), 143.) Here N is the number of molecules in a gramme-
molecule (Avogadro's number), U the energy, V the volume of the gas.
Now, according to the Second Law of Thermodynamics,
jo dU +
must be a complete differential, where p and T denote pressure and
temperature of the gas. Hence the relation
\dVju~ T
must hold. This gives
kV = P ie p _kNT
If we compare this with the equation of state of an ideal gas in thermo-
dynamics, p = ~, we get for the absolute gas constant R the value
B = kN
from which formula (19) of the text follows.
88 M. Planck, Ann. d. Phys. 4, 564-566 (1901).
38 Compare, for example, the table of the values of Avogadro's number
given in the report of J. Perrin at the Solvay Congress in Brussels
(1911). [A. Eucken, Die Theorie der Strahlung und der Quanten.
Abhandlungen der Bunsen-Gesellschaft Nr. 7, Wilh. Knapp, Halle 1914.]
37 R. A. Millikan, Phil. Mag. (6) 34, 13 (1917).
Mlbid., from the values given by Millikan for the electronic charge
e = 4-774 x 10- 10 (electrostatic units) and from the electrochemical
constant F = 969-4 . 2'999 . 1010 electrostatic units, there follows for
Avogadro's number the value N = 6-0617 . 1023.
39 Of., for example, W. Gibbs' Elements of Statistical Mechanics,
Chapter V.
WThe term "mean value" may be taken as referring to time or to
space. If we select a definite atom, and follow it a long time upon its
zig-zag path, and from the mean of the values which its kinetic energy
assumes in the course of time, we get the " time-mean." If, on the
other hand, we select a large number of identical atoms of the gas at a
particular instant and again form the mean of the values of the kinetic
energies which these atoms possess at the instant in question, we get
the " space-mean."
*1 If x is the elongation of the oscillator (electron) vibrating with the
natural frequency, then x = A sin (2mrf), where A is the amplitude and
t the time ; the mean kinetic energy becomes
NOTES AND REFERENCES 135
The mean potential energy is :
V
Hence, as stated, L = V: i.e. the mean kinetic energy = the mean
potential energy.
42 /. H. Jeans, Phil. Mag. 10, 91 (1905).
MH. A. Lorentz, Proc. Kon. Akad. v. Wet., Amsterdam 1903, p. 666.
The theory of electrons (Teubner, Leipzig 1909), Oh. II.
444. Einstein and L. Hopf, Ann. d. Phys. 33, 1105 (1910).
UA. D. Fokker, Ann. d. Phys. 43, 810 (1914).
46 M. Planck, Ber. d. Berl. Akad. d. Wiss., 8 July 1915, p. 512.
47 H. A. Lorentz. Die Theorie d. Strahlung u. d. Quanten ; Abhand-
lungen der Deutschen Bunsen-Gesellschaft. Nr. 7. v. A. Eitcken.
Halle, W. Knapp 1914 pp. 10 et seq.
48 By a suitable modification of classical statistics in the sense of the
quantum theory, we can obtain the expression (9) for the mean energy
of an oscillator in the following manner which is worthy of notice.
Let a number N of similar oscillators with the most varied values for
the energy be given. We require to find how great is the probability
w, that an oscillator possess a certain energy value U; or, otherwise
expressed, how many of the N oscillators possess the energy U. In
order to answer this question, we find it best to take first of all the
standpoint of Oibbs' statistical mechanics, that is, of " classical "
statistics. In place of the special case in question, namely, that of the
linear oscillator, let us consider at once quite generally a system of /
degrees of freedom, and characterise it by / generalised co-ordinates
2i2z ^ and by the corresponding impulses or momenta 2*1 .Pa . . Pf.
(Here, the impulse pi is thus defined : form the kinetic energy of
the system as a function of the generalised velocities qi J-, then
OT- . <**
pi = -~r. \ In particular, the linear oscillator (vibrating electron) will
qi'
be described by a co-ordinate q, namely, the elongation of the electron,
and the impulse p = m -%. In general, therefore, 2/ quantities are
necessary in order to define completely the momentary state of a
system. Hence we can represent this momentary state by a point
(" phase-point ") in the 2/- dimensional space in which 9i . . . Pf (of the
" phase-space ") are co-ordinates.
We now consider a number N of similar systems of this kind,
which are in thermodynamic equilibrium with a very large reservoir
at the temperature T. Then the probability that the co-ordinatea
and impulses lie in the small intervals q 1 . . . g t + dq v etc., and
Pi . . . Pi + rfpj, etc., that is, that the "phase-point" of the system lie
in the element dn = dq^dq^ . . . dqj, dp^p t . . . dpf of the phase-
space is, according to Oibbs,
136 THE QUANTUM THEORY
Here E is the energy of the system, and k is the constant defined in
(19). The integration in the denominator is to be taken over all
possible values of the 2/ quantities q 1 . . . pf, or, as we may say, over
all possible " phases," or over the whole region of the phase-space
concerned.
Among the N systems there are then Nw, whose phase-points lie in
the element dfi of the phase-space. This is therefore a " distribution "
of the N systems over the phase-space. This distribution is called
Canonical ; it represents a generalisation of Maxwell's familiar law of
distribution of velocities which may be deduced from it by special-
ising it for the case of the gas atom, that is, by setting/ =3.
The sum of all probabilities is naturally 1. Indeed, it is at once
clear that
L->
For the mean value of the energy E we get
J-M. -/*'**.
/."*>
If we apply this equation to the linear oscillator we get
/ f -
TT JjUe Wgqdp
ffe-^dp
N W ' <7 -!' + -0MV
If we introduce the auxiliary variables { and n, defined by
j = TW2m
j n = ,JL, and hence dqdp = ddn
* fj'2m TV
we get
U = f 2 + T?
and, therefore, it suggests itself to us to write
f|=
\rj=
where <f> is a parametric angle. If we interpret { and rj as Cartesian co-
ordinates of a point in the plane, then JU and <p are the polar co-ordinates
NOTES AND REFERENCES 137
of this point. The element of surface ddrj is written in polar co-ordinates,
as we know, thus
hence
Hence
dqdp =
r r -
J Jta M
tr=0*> =
in agreement with (24). This is the standpoint of classical statistics.
The quantum statistics of the oscillator may be immediately deduced
from this, if we elaborate the canonical law of distribution
i -
Je
kT dqdp
in a suitable manner
If we here again introduce dqdp = -dUdf, and integrate with respect
to $, we get
_
wu= e * r dU
Je'^dU
as the probability that the energy of the oscillators lies between U and
U H- dU.
Now the quantum theory demands that the energy U shall assume only
the discrete values U , U lt U , . . . Un- The transition may best be
effected by laying down the condition : E shall only be able to assume the
values contained in the narrow intervals between U and U + a, U^ and
I7j + o, and generally Un and U n + o. Then dU = a, and the integral in
the denominator changes into a sum. Thus it follows that
n n
thus a is eliminated ; if we now proceed to the limit a = 0, w remains
138 THE QUANTUM THEORY
unaltered. Hence w n is the canonical distribution function generalised
for quantum conditions, and hence, among N oscillators, Nw n have an
energy of the value U n .
We now get for the mean energy
V *
2ft* v
Now, according to the first form of the quantum theory,
U n = tie = nh v (n = 0, 1, 2, 8 . . . oo )".
Therefore
u =
2
V
o S,
o
If we set fc5>, for convenience, = x, then
Further,
from which we get
in agreement with (9)
The canonical distribution may be still further generalised by the intro-
duction of certain "weight factors," which are intended to express the
fact that the individual quantum states of the system considered have,
a priori, different probabilities. This happens, for example, if each quantum
state may be realised in different ways, and if the number of these possi-
bilities of realisation is different for the different quantum states. Then,
the different states will have different "weights," and a "weight factor"
v n
p n has to be included in the exper mental function e ~*r so that the can-
onical distribution function assumes the form
- = C . f^ff'
Here C depends on the temperature ; p n , on the other hand, doe not.
NOTES AND REFERENCES 189
194. Einstein, Ann. d. Phys. 17, 132 (1905) ; 20, 199 (1906) ; Verhandl.
d. deutsch. physikal. Ges. 11, 482 (1909) ; Bericht Einstein auf dem
Solvay-Kongress in Brussels 1911 ; cf. A. Eucken, Die Theorie der
Strahlung und der Quanten ; Abhandl. d. deutsch. Bunsen-Gesellschaft,
Nr. 7 (Halle, W. Knapp 1914), pp. 330 et seq. Cf. also W. Wien,
Vorlesungen iiber neuere Probleme der theoretischen Physik (Teubner,
Leipzig and Berlin 1913), 4. Vorlesung. H. A. Lorentz, Les theories
statistiques en thermodynamique (Teubner, Leipzig and Berlin 1916),
42 et seq.
SO A. Einstein, Ann. d. Phys. 17, 132 (1905).
31 A. Einstein, Phys. Zeitschr. 10, 185 (1909).
52 This formula may be deduced as follows : Firstly, from e = E - E
the frequently used relation
? = W - 2E . IS + (E)* = W - (IT) 2
follows. In order now to calculate the two quantities E* (mean of the
squares of the energy) and (E)* (square of the mean energy), which
are known to differ from each other in general, we do best to take the
standpoint of Gibbs' statistical mechanics (see note 48). According to
this, the probability that the co-ordinates and impulses lie in the small
intervals q l . . . q l + dq lt etc., p^ . . . p 1 + dp lt etc, that is, that the
"phase-point" lies in the element dq t dq 2 . . . dqfdp^dp^ . . . dp, = dn
of the " phase-space " :
Then the mean of the energy follows in the usual way :
Likewise,
We then form
dE
140 THE QUANTUM THEORY
Therefore,
dE
F-Wjy
We also arrive at the same formula, if instead of the classical
canonical distribution function, we start from the quantum distribution
function
68 The mean energy of radiation of frequency v in the volume v is
= vu v dv, where the monochromatic density of radiation is
if Planck's Law is taken as the basis. (Of., for example, M. Planck,
Lectures on the Theory of Radiation, Engl. Transl.)
According to formula (28) deduced in the previous note, it therefore
follows that
If we eliminate T on the right-hand side by substituting for e** its
value 1 + 8vhvS , it follows that
C'Uy
u v vdv ,
The second term on the right is required by the Undulatory Theory
for at each point of the volume v the most varied trains of waves of
radiation cross one another's paths with every possible amplitude and
phase. The interference of all these waves thus generates at the point
considered an intensity, which varies continually, and hence the energy
of the volume v also varies. If we calculate the mean of the square
of the energy, i.e. a , wo find precisely the second term of the above
formula. (Of., for example, H. A. Lorents, Les theories statistiques en
thermodynamique (Teubner, Leipzig and Berlin), 1916, pp. 114 et seq,)
The first term is not, however, explained by the classical undulatory
theory. On the other hand, it becomes endowed with meaning if we
suppose that the radiant energy consists of a certain whole number
NOTES AND REFERENCES 141
(n) of finite energy complexes of the value hv. For then E = n hy,
and therefore E = n hv, where n is the mean about which the number
n varies. If 5 = n -_ n be the variation of the number n, then it
follows that e = E - E = Shy^ where 2 = S 2 feV. But, according to
a well-known law of statistics, S 2 = n. (Cf., for example, H. A, Lorentz,
loc. cit., 26 and 27.) Hence ? = nfeV = ~E'hv. This is exactly the
first term in the above formula.
MA. Einstein, Ann. d. Phys. 17, 144 (1905).
95 J. J. Thomson, Conduction of Electricity through Gases.
96.4. Einstein, Ann. d. Phys. 17, 147 (1905).
97 Of. R. Pohl and P. Pringsheim, Die lichtelektrischen Erschein-
ungen. Sammlung Vieweg Heft 1 (Braunschweig 1914).
98 A. Einstein, Ann. d. Phys. 17, 145 (1905).
99 R. A. Millikan, Phys. Zeitschr. 17, 217 (1916).
60 According to Pohl and Pringsheim, we have to distinguish between
the normal and the selective photo-effect : in the case of the normal
effect the number of electrons torn off (per calorie of the Jight-energy
absorbed) is independent of the orientation of the electrical vector of the
light-wave, and increases, starting from an upper limit of the wave-
length, in general uniformly as the wave-length decreases. In the case
of the selective effect, on the other hand, which only appears when the
electrical vector of the light-wave possesses a component vertical to the
metallic surface, the number of electrons torn off (per calorie of light-
energy absorbed) shows a decided maximum at a definite wave-length.
61 Oh. Barkla, Phil. Mag. 7, 543, 812; 15, 218. Jahrb. d. Radioak-
tivitat u. Elektronik, 5, p. 239, 1908. Ch. Barkla and Sadler, Phil. Mag.
17, 739. Ch. Barkla, Jahrb. d. Radioaktivitat u. Elektronik, 1910, p.
12. if. de Broglie, G. R. 25 May and 15 June 1914, p. 1785. Ch. Barkla,
Phil. Mag. 16, 550. E. Wagner. Ann. d. Phys. 46, 868 (1915); Sit-
zungsber. d. bayer. Akad. 1916, p. 39.
62 D. L. Webster, Proc. Americ. Acad. 2, 90(1916); Physic. Review, 7,
587 (1916).
63 E. Wagner, Ann. d. Phys. 46, 868 (1915).
64 Of., for example, E. Wagner, Phys. Zeitschr. 18, 443 (1917). The
value that Wagner calculates for h is : h = 6-62 .10-27.
65 W. Duane and F. L. Hunt, Physic. Review, 6, 166 (1915).
86.4. W. Hull and If. Rice, Proc. Americ. Acad. 2, 265 (1916).
67 E. Wagner, Phys. Zeitschr. 18, 440 et seq. (1917) ; Ann. d. Phys. 57,
401 (1918).
68 F. Dessauer and E. Back, Ber. d. deutsch. physikal. Qes. 21, 168
(1919).
69 J. Franck and G. Hertz, Verhandl. d. deutsch. physikal. Ges. 16,
512 (1914).
70 The critical potential measured by Franck and Hertz amounted to
V= 4-9 volts = electrostatic units, and therefore the critical energy
300
of the electron is
v _ 4-774. 10 -10. 4-9
300
142 THE QUANTUM THEORY
The wave-length A of the mercury line emitted is
\ = 25364 = 2-536.10-5.
Hence we must get
17 j.e i. V* 4-774. 10 -10. 4-9. 2-536. 10 -5
,7= fc-, ,.e. h = = - 3 . 102-3 ^r
= 6-59.10-27
and this is in good agreement with the results of other measurements.
71 Cf., for example, J. Stark, Prinzipien der Atomdynamik II. (S.
Hirzel, Leipzig 1911), Chs. IV and V.
72 J. Stark, Ber. d. deutsch. phys. Ges. 10, 713 (1908) ; Phys. Zeitschr.
8, 913 (1907) ; 9, 767 (1908).
Canal-rays are positively charged particles of matter, which move in a
vacuum tube in the direction : anode to cathode ; the latter is pierced
with holes through which the canal-rays pass into the space behind the
cathode. If we generate such canal-rays in a vacuum tube rilled with
hydrogen, we find that the series lines of hydrogen are emitted. Now,
if we observe this emission spectroscopically " from the front," that is, so
that the canal-rays are moving towards the observer, we see, firstly, at
its usual place in the spectrum, the sharp series line (line of rest, "in-
tensity of rest ") ; secondly, we see displaced towards the violet, a
broadened strip (line of motion, "intensity of motion" or "dynamic
intensity"). These lines represent the series line emitted by the
moving canal-ray particles, which is displaced towards the region of
higher frequencies on account of the Doppler effect. Since the canal-rays
do not possess a single uniform velocity, and since particles with all
possible velocities occur, the displaced strip is not sharp, but softened
and broadened. The " intensity at rest " is therefore emitted when the
quickly moving canal particles strike " resting " molecules, i.e. gas-
molecules which are moving comparatively slowly and irregularly, and
excite these to emit the series lines. The " intensity of motion," on the
other hand, is excited by the unidirectionally moving canal particles
themselves, when they hit gas-molecules.
Now, it is very remarkable that the interval between the intensity of
rest and that of motion is not filled in, but that the emission of the in-
tensity of motion becomes observable only above a certain velocity.
Stark interpreted this fact in terms of the light-quantum hypothesis
thus : If %mv z is the kinetic energy of a canal-ray particle, and if the
fraction arau 2 (a > 1) is transformed into a light-quantum hv upon
collision with a gas-molecule, then we must have h < -me 2 ; that is, the
spectral line of frequency v can be generated only by canal-rays, the
Telocity of which >A/?5
\ can
The proportionality between the critical velocity and *Jv has been
fairly well borne out.
It should be remarked here that J. Stark has lately abandoned the
theory of light-quanta. (Cf. J. Stark, Verh. d. deutsch. physik. Ges. 16,
304 (1904) ; 18, 42 (1916).)
NOTES AND REFERENCES 148
73 J. Stark, Phys. Zeitschr. 9, 85, 356 (1908). J. Stark and W.
Steubing, Phys. Zeitschr. 9, 481 (1908). J. Stark, Phys. Zeitschr. 9,
889 (1908).
In these papers J. Stark defends the view that the band-spectra are
emitted when a "valency electron" belonging to the atom or molecule
is pushed out of its normal position and then returns again to its initial
position, counterbalancing the work done in displacement. If the
energy of deformation (valency energy) E is changed into a light-
quantum, then we must have hv = E, i.e. v ^- All lines of the
^ 7t
ET
band must therefore lie below the edge v = _ . If the valency energy
E is changed by chemical processes, the band-spectrum must be dis-
placed accordingly.
74,1 Einstein, Ann. d. Phys. 17, 148 (1905).
75 J. Stark, Phys. Zeitschr. 9, 889 (1908) ; Ann. d. Phys. 38, 467 (1912).
The fundamental law of photochemical decomposition enunciated by
Stark and Einstein states : If a molecule dissociates at all owing to j,he
absorption of radiation of frequency v, then it will absorb an amount of
energy hv when it dissociates. This energy, therefore, represents the
heat of reaction, which will be set free upon recombination of the
products of decomposition.
This law was later deduced by A. Einstein for the range of validity of
Wierfs Law of Radiation without the assistance of the light-quantum
hypothesis, by purely thermodynamical methods. (Of. Ann. d. Phys.
37, 832 (1912), and 38, 881 (1912).)
nE. Warburg, Ber. d. Berl. Akad. d. Wiss. 1911, p. 746; 1913, p.
644 ; 1914, p. 872 ; 1915, p. 230 ; 1916, p. 314 ; 1918, pp. 300, 1228. Cf.
also " Naturwissenschaften," 5, 489 (1917).
77 H. A. Lorentz, Phys. Zeitschr. 11, 1250 (1910).
78 M. Planck, Ber. d. deutsch. physikal. Ges. 13, 138 (1911); Ann. d.
Phys. 37, 6i2 (1912).
79 On account of the continuous (classical) absorption, all energy values
of the oscillator in an elementary region, say between n and (n + l)e,
are equally probable. The mean energy in the nth elementary region
is, therefore,
Prom the canonical law of distribution extended in the sense of the
quantum theory, it then follows that
144 THE QUANTUM THEORY
(cf. note 48). If we further set e = hy it follows that
In place of relation (7) of the text we get here
and this leads to Planck's Law of Radiation.
80 M. Planck, Sitzungsber. d. Kgl. Preuss. Akad. d. Wiss. 3 April,
1913, p. 350; ibid., 30 July 1914, p. 918; ibid., 8 July 1915, p. 512.
81 A. Einstein and O. Stern, Ann. d. Phys. 40, 551 (1913).
82 W. Nernst, Verhandl. d. deutsch. physikal. Ges. 18, 83 (1916).
83 F. Richarz, Wiedem. Ann. 52, 410 (1894J.
84 Report by P. Langevin at the Solvay Congress in Brussels, 1911.
Cf. A. Eucken, Die Theorie der Strahlung und der Quanten. Abhandl. d.
deutsch. Bunsen-Ges., Nr. 7 (W. Knapp, Halle 1914), pp. 318 et seq.
8M. Einstein and W. J. de Haas, Verhandl. d. deutsch. physikal. Ges.
17, 152, 203, 420 (1915). A. Einstein ibid., 18, 173 (1916). W. J. de
Haas, ibid., 18, 423 (1916).
86 E. Beck, Ann. d. Phys. 60, 109 (1919).
87 Report by Planck at the Solvay Congress in Brussels, 1911. See
A. Eucken, Die Theorie der Strahlung und der Quanten. Abhandl. d.
deutsoh. Bunsen-Ges., Nr. 7 (W. Knapp, Halle 1914), p. 77.
88 If q is the elongation of a linearly vibrating electron of mass ra (os-
cillator) and v its period of oscillation, then the energy of this configur-
ation is
The first term represents the kinetic and the second the potential energy.
Now the impulse (the momentum) is p = m ~Ti- Therefore, we may write
i.e.
NOTES AND REFERENCES 145
The curves U = const., that is, those curves in the phase-plane, which
correspond to the states of constant energy of the oscillator, are therefore
ellipses with the semi-axes
For a definite value of U we get a completely definite ellipse. The
" phase-point " of the oscillators would continually revolve in this ellipse,
if the electron, without emitting or absorbing, were to execute pure har-
monic oscillations : for then its energy would remain permanently constant.
If we allow U to vary continuously, i.e. if we give it other and again
other values in continuous succession, we get an unlimited manifold of
concentric ellipses.
The quantum theory, as formulated in (30) in the text, selects from this
infinite manifold a discrete set of ellipses, and distinguishes them as the
"quantised" ellipses which correspond to the " characteristic states " of
the oscillator. To these belong the "quantum energy- values " U , U lt
U a . . . Dn.
Now the nth ellipse encloses an area nh. The area of the nth ellipse
is, however,
hence we must have
= nh i.e. U n =
that is, in the nth quantum state the oscillator possesses an amount of
energy nt = nhv.
tOA. Sommerfeld, Phys. Zeitschr. 12, 1057 (1911). Report by A.
Sommerfeld at the Solvay Congress in Brussels, 1911. Cf. A. Eucken,
Die Theorie der Strahlung und der Quanten. Abhandl. d. deutsch.
Bunsen-Ges., Nr. 7 (W. Knapp, Halle 1914), p. 252.
80 Report by Sommerfeld at the Solvay Congress, 1911.
91 A. Sommerfeld and P. Debye, Ann. d. Phys. 41, 873 (1913).
92 Cf., for example, the recent summary by E. Schrddinger, Der
Energieinhalt der Festkorper im Lichte der neueren Forschung. Phys.
Zeitschr. 20, 420, 450, 474 (1919). A complete set of references accom-
panies this account.
93 One gramme-atom of a substance, the atomic weight of which is a,
is defined as the quantity a grammes of the substance. For example,
one gramme-atom of copper is equal to 63'57 grammes of copper, since
63-57 is the atomic weight of copper. Exactly analogous is the
definition of the gramme- molecule (also called " mol"). One gramme-
molecule of oxygen is 32 grammes of oxygen, for the molecular weight
of oxygen (diatomic) is 32.
If c is the specific heat of a substance of atomic weight a, it signifies
that one gramme of the substance requires an amount of heat c to raise
its temperature by 1 C. Hence we must communicate to a gramme-atom
10
146 THE QUANTUM THEORY
of the substance, i.e. to a grammeB of it, an amount of heat C = ea in
order to raise its temperature by 1 C. C is then called the atomic heat.
94 The equality of the mean potential and the mean kinetic energies is
true here as in the case of the linear Planck oscillator (vibrating electron),
of. note 41. This equality is, in general, always present when the forces
which act upon the atoms and restore them to their positions of rest
(zero positions) are Ivnea/r functions of the relative displacements of the
atoms, that is, when the force is "quasi-elastic," that is, proportional
to the displacement from the zero position. Cf. in this connexion
L. Boltzmann, Wiener Ber. 63 (11), 731 (1871), and F. Richarz, Wied.
Ann. 67, 702 (1899).
WDulong and Petit, Ann. de chim. et de phys. 10, 395 (1819).
96 The quantity usually obtained by measurement is not the atomic
heat at constant volume C,,, but the atomic heat at constant pressure C f .
For this we get values which in general fluctuate about the value 6'4
cal./deg. The calculation of C c from C p is based on the thenno-
dynamically deduced formula
C p - C, = a * VT
where a is the cubical coefficient of thermal expansion, K the (isothermal)
cubical compressibility, and V the atomic volume = atomic wei g ht .
density
97 E.g. we find
for silver at C C p = 6-00
aluminium 58 C p = 5-82
copper 17 C C p = 5-79
lead 17 C p = 6'33
iodine ,, 25 C C p = 6-64
zinc 17 C C p = 6-03
98 F. E. Weber, Poggend. Ann. 147, 311 (1872) ; 154, 367, 553 (1875).
99 As a possible way out, the "agglomeration hypothesis," supported by
F. Richarz [Marburger Ber. 1904, p. 1], C. Benedicks [Ann. d. Phys. 42,
183 (1913)] and others, has been put forward. According to this, as the
temperature falls the number of degrees of freedom of the system
diminishes by " freezing-in," as it were, in that certain linkages become
completely rigid. According to this, however, the compressibility should
decrease greatly as the temperature falls, which, according to E. Orii-
neisen's measurements is not the case [Verb. d. deutsch. phys. Ges. 13,
491 (1911)]. Compare also in this connexion the report of E. Schro-
dinger quoted in note 92.
100.4. Einstein, Ann. d. Phys. 22, 180, 800 (1907).
101 Cf. A. Einstein, Ann. d. Phys. 35, 683 ff. (1911), also the report by
Einstein at the Solvay Congress in Brussels, 1911 ; see A. Eucken, Die
Theorie der Strahlung und der Quanten. Abhandl. d. deutsch. Bunsen-Ges.,
Nr. 7 (W. Knapp, Halle 1914), pp. 330 et seq.
1024. Einstein, Ann. d. Phys. 34, 170, 590 (1911) ; 35, 679 (1911).
IWThe nature of the dependence of the frequency v on the three
NOTES AND REFERENCES 147
quantities A, p, K may, according to Einstein (loc. cit.), be obtained by a
simple dimensional calculation. If we assume that v depends only on the
mass in of the atoms, their distance apart d, and the compressibility K of
the body, then an equation of the following form must hold
v = C . m x . & . K z .
C is here a numerical constant ; x, y and z are numbers which remain to
be determined.
The dimensions of the frequency \y] are \t - 1] ; the dimensions of m and
d are \ni] and [i], and the dimensions of the compressibility K follow from
its definition :
_ _ increase in volume _
increase in pressure x original volume
K has therefore the dimensions
I! = [-surface "I
->"-" ' L force J L
We thus get the following dimensional equation
t~ l
Hence
x - 2 = 0; y -f z = ; "2z = - 1
from which we get
x- -J; y- +4; 2- -4
We have therefore,
Let N be Avogadro's number, i.e. the number of atoms in the gramme-
atom. Then the atomic weight of the body is numerically equal to the
mass of the gramme-atom, i.e.
A = mN.
If we imagine the atoms arranged upon a cubical space-lattice with
sides d, then the density must satisfy the equation
from this it follows that
and hence
i.
from which, it follows that
148 THE QUANTUM THEORY
Einstein determines the factor C by assuming simply that only the
twenty-six neighbouring atoms act upon the displaced atom.
IMF. A. Lindemann, Phys. Zeitschr. 11, 609 (1910). Lindemann's
formula may be shortly deduced thus : Let r = a sin (2-irvt) be the elonga-
tion of an atom which is vibrating with the amplitude a and the frequency
y. The mean energy of this atom is
E ' f (ar)" + f '
At the melting-point, according to Lindemann's conception, a is of the
same order as d (distance apart of atoms). On the other hand, the mean
energy of the atoms at high temperatures = 'SkT, or, at the melting-point
3kT s . (The melting-point, as a rule, is high.) From this it follows that
= SkT,
But we have (see note 103)
m =^j- d = m*,-i = AlN-*f-*>
Hence
v = const. I 7 ,* . 4~*JSM-*V = const. T,* . 4~* . />*.
105 E. Qrilneisen, Ann. d. Phys. 39, 291 et seq. (1912).
106 E. Madelung, Nachr. d. kgl. Ges. d. Wiss. zu Qottingen, mathem.-
physikal. Klasse 1909, p. 100, and 1910, p. 1.
107 W. Sutherland, Phil. Mag. (6), 20, 657 (1910).
108 If n and K are the coefficients of refraction and extinction of a
substance respectively, then, according to Maxwell's Theory, its reflect-
ing power is
n -
If we require the point of maximum reflection, we have to form the
equation ^^ = 0, which gives after reduction the following relation :
From this we see that the position of maximum reflection does not
coincide exactly with the position of maximum absorption (^ = Oj,
\ov /
but that it lies the nearer to it, the less the coefficient of refraction
varies with the frequency. On the other hand, the point of maximum
-absorption lies, according to the dispersion theory, in the immediate
^neighbourhood of the natural frequency v,..
109 H. Rvtens and E. F. Nichuh, Wiedem. Ann. 60, 418 (1897). Also
NOTES AND REFERENCES
149
77. Rubens and H. Hollnagel, Ber. d. kgl. preuss. Akad. d. Wiss. 1910,
p. 45; //. Hollnagel, Dissert. Berlin 1910; H. Rubens, Ber. d. kgl.
preuss. Akad. d. Wiss. 1913, p. 513 ; H. Rubens and H. v. Wartenberg,
ibid., 1914, p. 169.
As an example we give here the following small table in which \
denotes the wave-length of the " residual " rays, as given by the above
investigators.
A.
A.
NaCl
KC1
AgCl
HgCl
52 M
63-4 M
81 -5 M
98-8/i
T1C1
KBr
AgBr
TIBr
91-6 M
82-6 M
112-7 M
117 M
HO Of., however, note 108.
111 W. Nernst and F. A. Lindemann, Sitzungsber. d. kgl. preuss.
Akad. d. Wiss. 1911, p. 494 ; W. Nernst, Ann. d. Phys. 36, 426 (1911).
112 The following short table gives the values for v which are calculated
from Einstein's formula (35), Lindemann's formula (36), from the
"residual rays" (see note 109), and from the observed atomic heat
according to an empirical formula (40) proposed by Nernst and
Lindemann. For more detailed data with, in part, corrected numerical
factors see C. E. Blom, Ann. d. Phys. 42, 1397 (1913).
Substance
V E
"L
''residual rays
''atomic heat
(Nerntt-Lindemann)
Al
6-7 . 10 12
7-6 . 10"
8-3 . 10"
Cu
5-7 . 10 12
6-8 . 10 12
6-7 . 10 12
Zn
4-4 . 10 12
4-8 . 10 12
Ag
4-1 . 10 12
4-4 . 10 12
4-5.10 12
Pb
2-2 . 10 12
1-8. 10 12
1-5 . 10 12
Diamond
32-5 . 10 12
40 . 10 12
NaCl
7-2. 10 12
5-8 . 10 12
5-9 . 10 12
KC1
5-6 . 10 12
4-7 . 10 12
4-5 . 10 12
113 W. Nernst, F. Koref, F. A. Lindemann, Untersuchungen iiber die
spezifische Warme bei tiefen Temperaturen. I. u. II. Sitzungsber. d.
kgl. preuss. Akad. d. Wiss. 1910, 3 March. W. Nernst, idem III., ibid.,
1911, 9 March. F. A. Lindemann, idem IV., ibid., 1911, 9 March. W.
Nernst and F. A. Lindemann, idem V., ibid., 1911, 27 April. W. Nernst
and F. A. Lindemann, idem VI., ibid., 1912, 12 Dec. W. Nernst, idem
Vn., ibid., 1912, 12 Dec. W. Nernst and F. Schwers, idem VIII., loc.
tit., 1914. W. Nernst, Der Energieinhalt fester Stoffe. Ann. d. Phys.
36, 395 (1911).
HI W. Nernst, Die theoretischen und experimentellen Grundlagen des
neuen Warmesatzes. (W. Knapp, Halle 1918.)
115 The First Law states : If d'Q is the heat supplied to a system, d'A
150 THE QUANTUM THEORY
the work done on the system from outside, then the increase of energy
U of the system is given by
dU = d'Q + d'A.
The Second Law states : if d'Q is supplied reversibly at the temperature
T, then ^ is the complete differential of the entropy S, hence
Let us follow Helmholtz and introduce the " free energy" F denned by
F= U- T- S.
Then it follows that
dF = dU - T.dS - S.dT=d'Q + d'A - T.dS - S.dT
i.e. dF = d'A - S . dT
for every reversible process.
If the process is isothermal (dT = 0) then it follows that dF = d'A
or, for a finite change of state, F. A - F l = A. If we set A' = - A, so
that A' is the work gained, we get
Fj - F, = A'.
That is, the work gained in the isothermal reversible process which is,
as may be shown, the maximum obtainable is equal to the decrease of
free energy.
Further, it follows, since at constant volume F the work d'A = 0,
that
Therefore, formulating these expressions for two states, we get
/pjOffii - "a)~| _ ^ _ p^ _ ^J7 } _ {/ 2 j
or, finally, if we write for short t7j - U z = U'
an equation much used in physical chemistry.
Since, now, according to Nernst's heat theorem,
('),. =o
\OT Jhm T=Q
(A' - U') vanishes for T = 0, being above the first order.
Hence lim ?L4' ~ . ^') = Q
and hence also
NOTES AND REFERENCES 151
This is equation (89) of the text.
From -= - S, it follows further that ^ F ] ' " F ^ - S 2 - 8 lt or
Q _ O _ 9-4'
and hence Nernst's Theorem may be formulated thus
lim (S 2 - S,) =
r=o
that is, in /; neighbourhood of the absolute zero all processes proceed
without change of entropy.
116 Cf., for example, M. Planck, Lectures on Thermodynamics. Planck
goes further than Nernst inasmuch as he postulates that not only the
difference of the entropies S 2 - S x is zero at absolute zero (see previous
note) but also that the individual values themselves become zero. Hence,
according to Planck, at the absolute zero of temperature the entropy of
every chemically homogeneous body is equal to zero. From this the con-
clusion given in the text,
lim^^,j =
may be deduced immediately. It follows from the relation (occurring
in the last note)
F - U = - TS
and from Planck's version of Nernst's Theorem, that F - 0" vanishes for
T = 0, being of higher order than the first.
Hence
= or limf^-H S)
T=o\oT )
or, finally,
117 For low temperatures, that is, for high values of x = ^ Einstein's
formula (34) takes the following form : C v = 3Rx*e~ x . The falling-off at
low temperatures therefore follows an exponential law ; more exactly, it
varies as
i const
118 W. Nernst and F. A. Lindemann, Sitzungsber. d. kgl. preuss. Akad.
d. Wiss. 1911, p. 494 ; Zeitschr. f. Elektrochemie, 17, 817 (1911).
119 A. Einstein, Ann. d. Phys. 35, 679 (1911).
120 For if we regard the atoms as mass-points, then each atom hao three
degrees of freedom ; the whole body has therefore '3N degrees of freedom.
As is proved in mechanics, however (cf. R. H. Weber and R. Oans,
152 THE QUANTUM THEORY
Repertoriuin der Physik Bd. I. pp. 175 et seq.), a mechanical system of
3N degrees of freedom has 3N natural frequencies, and the moit general
small motion of each atom consists in a superposition of these 3N natural
frequencies.
121 P. Debye, Ann. d. Phys. 39, 789 (1912).
122 M. Born and Th. v. Kdrmdn, Phys. Zeitschr. 13, 297 (1912); 14,
15, 65 (1913). Of. also M. Born, Ann. d. Phys. 44, 605 (1914) ; M. Born,
Dynamik der Kristallgitter (Teubner, Leipzig and Berlin 1915).
123 Of., for example, R. Ortvay, tjber die Abzahlung der Eigenschwin-
gungen fester Korper. Ann. d. Phys. 42, 745 (1913).
Ortvay considers the natural frequencies of an elastic cube, each side
of which has the length L. There are found to be three groups of natural
frequencies. The first two groups are the transversal frequencies, the
third group is the group of the longitudinal frequencies. That the trans-
versal frequencies form two groups (moreover identical) is easily seen.
For in the case of a transversal vibration, which is propagated in, say, the
direction of the aj-axis, two equal alternatives are probable, namely, that
the particles vibrate parallel to the y- or to the 2-axis. In the case of the
longitudinal oscillations, however, there is naturally only one group ; for
in the case of propagation along the z-axis there is only one possibility,
namely, that the particles vibrate parallel to the x-axis. The frequencies
of the first two groups are characterised by the values
the third group by
Here c t and c t are the velocities of propagation of transversal and longitu-
dinal waves in the body, whereas a, b, c are arbitrary positive whole
numbers. If therefore we give a, b, c all possible values in all possible com-
binations, we get all the possible transversal and longitudinal natural fre-
quencies, which together form the elastic spectrum of the cube. If now
we inquire how many transversal natural frequencies of the first group
fall below v, this means nothing else than inquiring how many trios of
values (a, b, c) fulfil the condition
Imagine a, b, c as co-ordinates of a point in space. Then all possible trios
(a, b, c) of values are represented by the total "lattice-points" of the
positive space octant, and the above question is answered by counting how
many lattice-points are at a distance less than ''from the origin (0, 0,0).
c
NOTES AND REFERENCES 158
All these lattice-points lie within the positive octant of the sphere whose
radius is -. Since now one lattice-point is assigned to every volume of
c t
magnitude 1 namely, every elementary cube the required number of
lattice-points, provided that it is sufficiently large, is equal to the volume
of the positive spherical octant of radius ", i.e. is equal to
If V = L 3 is the volume of the given cubical body, then the number
of the transverse natural frequencies below v belonging to the first
group is
Zl = T F cJ
The number belonging to the second group is the same, that is
rr r? 4ir T r V
Finally, the number of the longitudinal frequencies corresponding to
these is
4ir v"'
We thus get for the total of all natural frequencies below v
The total of natural frequencies in the interval v . . . v + dv follows by
differentiation with respect to v
and this is just formula (43) of the text.
124 In formula (43) for Z(v)d v let us replace, according to formula (44)
of the text, the factor
Then it follows that
154 THE QUANTUM THEORY
If we now aet - and - m , we get
128 A table showing how the Debye function C depends on x m is given
by Nernst (Die theoretischen und experimentellen Grundlagen des neuen
Warmesatzes. W. Knapp, Halle 1918, p. 201). In it the simple
Einstein function [formula (34) of the text] is also tabulated.
126 If T is great, then x m is small compared with 1 ; then we may
replace in the integral of (45) e x by 1 in the numerator, and e x - 1 by x
in the denominator. It then follows that
127 If Tia small, then x m is large, and we may replace the upper
limit of the integral as a first approximation by oo . The integral will
thus become a numerical constant independent of x m , and it follows that
C. - j* . const. - 9 - . T3 . const.
128 From the theory of elasticity it follows that
and c =
J HE
\H-o
where K is the compressibility, p the density, and a the ratio
transverse contraction
longitudinal dilatation*
If we insert these values in (44) and note further that V = -, formula
(46) of the text follows.
129 As the number of frequencies below v is proportional to v*, we get,
for example, the following picture : if we divide the interval from to
v m into 10 parts, and if only one natural frequency lies in the first
division, then in the following divisions there will be 7, 19, 37, 61, 91,
127, 169, 217, 271 natural frequencies ; i.e. the natural frequencies crowd
continually closer together.
180 P. Debye, Ann. d. Phys. 39, 789 (1912) ; W. Nernst and F. A.
Lind&mann, Sitzungsber. d. Berl. Akad. d. Wiss. 1912, p. 1160.
131 A. Eucken and F. Schwers, Verhandl. d. deutsch. physikal. Ges.
15, 578 (1913) ; W. Nernst and F. Schwers, Sitzungsber. d. Berl. Akad. d.
Wiss. 1914, p. 355 ; P. Grttnther, Ann. d. Phys. 51, 828 (1916) ; W. H.
NOTES AND REFERENCES 155
Keesom and Kmnerlingh-Onnes, Amsterdam Proc. 17, 894 (1915). Cf.
also the graphic tables by E. SchrOdinger, Phys. Zeitschr. 20, 498 (1919).
182 If we introduce into equation (44) of the text,
a " mean acoustic velocity " c, by the obvious definition
then for the order of magnitude of the smallest wave-length \ m i n , there
follows
c
It now the atoms in the cubical space-lattice, for example, are arranged
so as to be a distance a apart, then Na* = V, and hence
VI-
Amin
\ 3
133 For references see note 122.
131 Of. Born, Dynamik der Kristallgitter, 19.
133 .F. Haber, Verb. d. deutsch. phys. Ges. 13, 1117 (1911).
For if the atomic residue (mass m) and the electron (mass /i) are held
to their zero positions by forces of the same order of magnitude, and if
they vibrate independently of one another (a simplifying supposition) the
equation of vibration of the atom is nix + a?x = 0, the solution of which is
x = A sin( ~~F^t ). The infra-red frequency of the atom is, therefore,
a
y r = 2a- / , and correspondingly, the ultra-violet frequency of the electron
Hence Habeas Law follows : v r : v v = <s//x : N/. The
general space-lattice theory of M. Born confirms this law and shows that
in the lattice, too, atomic residues and electrons appear upon an equal
footing, and are acted upon by forces of the same order of magnitude.
136 Cf. M. Born and Thos. v. Kdrmdn, Phys. Zeitschr. 13, 297 (1912).
We may treat this problem, which is of course one-dimensional, most
simply thus : If we imagine an endless chain of points of equal mass TO
disposed along the x-axis at a distance apart a, and if we suppose for
simplicity that each mass-point only acts upon its two neighbours, then
the equation of motion of the nth point is
mx n = a(x n+ i - X n ) - a(x lt - SC,,_i) = a(x n +i + X_i - 2 n ).
Here o is a constant, and n can assume all values between + o>and - <n .
As a solution let us set for trial
x n = A sin 2*yt - n\
156 THE QUANTUM THEORY
This represents a process which is periodic in space and time, that is, a
wave which is propagated along the chain in the direction of increasing
x. The frequency of this wave is v, its length is A. Then, if after p
points the same displacement is to recur, pa must = A, and hence it
actually follows that
f 2*-a"
x n+J> = A sin 2irvt - (n + p)
In order to find the relation between v and \ (that is, the " law of dis-
persion"), let us insert the above formula in the equation of motion.
Then it follows that
- (n + 1) 1 + sin [znt - (n - 1) 1
* J L A J
-n^l\
* Ji
- 2*1 sin
- *] . (l - cos *2\
\ J \ A /
That is,
= * J~sin ()- , m sin (Y if we set LJt = , m .
W \w \A/ \A/ ir Vro
187 Of. .Bom, Dynamik der Kristallgitter, p. 51.
From the special case treated in the previous note, we also recognise
the truth of law (49) ; for if A is much greater than a, the dispersion law
takes the form v = n , where q = v m ita, represents the velocity of
A A
propagation of the wave, and this is independent of the wave-length.
138 The statement that a given direction lies in the element of solid
angle dti is intended to convey the following sense : about an arbitrary
origin O describe a " unit sphere," i.e. a sphere of radius 1. Now let a cone
of infinitely small angle be constructed of rays passing through 0, the point
of the cone lying at O. Let this cone cut out of the surface of the unit
sphere a small element of surface dn. Now let the parallel ray to the
"given direction" be drawn through O (here, for example, the wave-
normals). If this ray lies in the cone just constructed, then we say that
the " given direction " lies in the elementary solid angle dfl.
139 The capacity for heat of a certain finite body is that amount of heat
which must be imparted to the whole body in order that its temperature
be raised by 1 C. If M is the mass of the body, and c its specific heat,
then its capacity for heat is
NOTES AND REFERENCES 157
From the mean energy content E of the whole body, r follows by
differentiation with respect to the temperature
r-* 1
140 This somewhat complicated calculation runs as follows : we start
from the formula
and first replace \ by g . Thus we get
and the integral with respect to A. is transformed into one with respect to
vi. The limits of this integral are
vi = g*'( n ) [corresponding to \ = A^n)]
and
vi = (corresponding to A. = oo ).
If we further set
we get
3 47T *<>
fe'T 3 ^ C dn ;'x 4 e*dx
In place of the quantities gi(i) and x(n) which still depend essenti-
ally on the direction, let certain mean values be introduced. Firstly, let
us set
4ir
In this way three mean acoustic velocities g lf 2 2 , g 3 , independent of the
direction, are defmed. We further introduce in place of Am(n) a mean
value independent of the direction, in the following manner. In deduc-
ing formula (55) we saw that
.
Am(n)
158 THE QUANTUM THEORY
If we carry out the integration with respect to x, we get
to
v r da x f
o
u,
47T
dn
Now, in a way analogous to that used for the acoustic velocities g, we
set
Hence
Into ^ n ) = \ we introduce in place of qi(n) and \ m (n) the mean
values qi and \ m , which are independent of direction ; thereby xi(n) also
becomes independent of direction, and is transformed into
It follows that
'- 1 * 1 ""'
141 At the lowest temperatures
.
32^2 Ji- - }
i=i Xl b
3 a
NOTES AND REFERENCES 159
Now the value of the integrals = ^w 4 . If we further set R = Nk, and
for z~J the value (59), we get
15** ' Zfe
l
If we introduce_in place of the three acoustic velocities^", 2j, 3s a mean
acoustic velocity q by means of the definition
it follows that
Finally for we can write V^ (mean atomic volume) and thus get the
formula
148 H. Thirring, Phys. Zeitsohr. U, 867 (1913) ; 15, 127, 180 (1914).
143 M. Born and Th. v. Kdrmdn, Phys. Zeitschr. 14, 15 (1913).
144 Cf. note 182.
148 Cf. note 128.
1484. Eucken, Verhandl. d. deutsch. physikal. Ges. 15, 571 (1913). Cf.
also A. E'ucken, Die Theorie der Strahlung und der Quanten (W. Knapp,
Halle 1914), pp. 386 et seq., Appendix.
147 Cf. A. Eucken, Die Theorie der Strahlung und der Quanten (W.
Knapp, Halle 1914), p. 387.
148 To calculate the mean acoustic velocity q, the relation given in note
141 is used
3 4ir
We have therefore to obtain from the "dispersion equation" of the
crystal in question (for long waves) the values of the three acoustic
velocities q^n), q 2 (n), q s (n) as functions of the wave-direction ; q is then
obtained from the above formula by integration over all directions and
finally summation.
149 L. Hopf and G. Lechner, Verhandl. d. deutsch. physikal. Gee. 16,
643 (1914).
180 The following short table is taken from the paper of Hopf and
Lechner cited in note 149 :
160
THE QUANTUM THEORY
Crystal
2 calc. from C y
5 calc. from elastic
data
Sylvin . . .
Rock salt .
Fluor-spar .
Pyrites
2-36 . 10 B
2-82 . 10 8
4-02 . 10 8
5-43 . 105
2-03 . 10 s
2-72 . 105
3-82 . 10 8
5-12 . 105
181 W. Nernst, Vortrage iiber die kinetische Theorie der Materia und der
Elektrizitat. Wolfskehl-Kongress 1913 in Gottingen (Teubner, Leipzig
and Berlin 1914), pp. 63 et seq.
182 W. Nernst, ibid., pp. 81 et seg_.
153 E. Schrodinger, Phys. Zeitschr. 20, 503 (1919). Schrodinger correctly
points out that apart from the substitution of one single mean x for
the three quantities x> in the Debye terms the approximation above all in
the second part of C v (i.e. the replacement of the 3(s - 1) frequencies
v t . . . v 3 by the constants ^ . . . i>,) may not be permissible in many
cases : namely, in those cases in which the masses of the various kinds
of atoms are not very different from one another. If we were to allow
so he argues the masses of the different kinds of atoms and the forces
acting upon them gradually to become equal to one another, a simple
atomic lattice would result, and during this process the 3(s - 1) branches
of the spectrum, which correspond to the second type of motion, would
merge into the three first branches. " They cannot therefore even be
approximately monochromatic if the masses differ only slightly."
184 H. Thirring, Phys. Zeitschr. 15, 127, 180 (1914).
188 M. Born, Ann. d. Phys. 44, 605 (1914).
186 E. Oriineisen, Ann. d. Phys. 39, 257 (1912).
187 S. Ratnowski, Verhandl. d. deutsch. physikal. Ges. 15, 75 (1913).
188 Vortrage iiber die kinetische Theorie der Materie und der Elek-
trizitat. Wolfskehl-Kongresz zu Gottingen, 1913. (Teubner, Leipzig
and Berlin 1914), Vortrag P. Debye.
189 If U is the energy, and S the entropy of the system, then the " free
energy " is defined according to Helmholtz by the relation
F = U - S . T.
It then follows from note 115 that
dF = d'A - S . dT
where d'A is the work done from without. If we set in the usual way
d'A = - pdV (p = pressure, V = volume)
then
dF=- pdV- SdT.
From this we get immediately the equation (66) in the text
NOTES AND REFERENCES 161
Similarly,
fdF\ _ _ s
and hence
160 P. Debye, loc. cit., note 158.
161 E. Oriineisen, Ann. d. Phys. 26, 211 (1908) ; 33, 65 (1910) ; 39, 285
(1912).
162 P. Debye, loc. cit., note 158.
1634. Eucken, Ann. d. Phys. 34, 185 (1911) ; Verhandl. d. deutsch.
physikal. Ges. 13, 829 (1911).
IMP. Drude, Ann. d. Phys. 1, 566 (1900).
163 #. Riecke, Wiedem. Ann. 66, 353, 545 (1898).
166 Of., for example, H. A. Lorentz, The Theory of Electrons (Teubner,
Leipzig, and Berlin 1909).
167 Let g be the average velocity of the electrons along the free path I.
Then the electron takes the time T = - to pass over this free path.
During this time it is exposed to the electrical force E of the external
field. Its increase in velocity due to this force is at the commencement
of the free path = 0, at the end of it = ?=, where e and m are the
fli
charge and mass of the electron respectively. In the mean, therefore,
the small additional velocity generated by the field is Ag = -^^- = |^--
The electrons stream unidirectionally with this velocity against the field.
If N is the number of electrons per unit volume, then through unit
area of the surface there streams per second a quantity of electricity
This is, however, the "current density" I which is
known to be connected with the field E by the relation I = ffE- The
expression (67) for the conductivity <r therefore follows.
A more thorough treatment is due to H. A. Lorentz (see note 166).
He does not give the electrons a single velocity q, but introduces Max-
well's supposition, known from the kinetic theory of gases, that all
possible velocities occur, which are distributed among the electrons
according to a fixed law, the so-called Maxwell Law of Distribution.
He thus obtained a formula of the following form :
-
\3ir mq
which therefore only differs by a numerical factor from Dnide's formula
(67) ; here q = v 2 a , the root mean square of the velocity.
168 Let a temperature gradient along the x axis be present in the piece
of metal. Let a section be taken (see Pig. 12) at right angles to the
11
162 THE QUANTUM THEORY
x axis; we shall calculate the energy transport across this section per
second. If we suppose that of all electrons wander in each of the three
directions in space, then move in the positive x direction ; and further,
^_ the number of electrons which
pass through the unit of sur-
face in one second, will be all
those which are contained in
the small shaded cylinder with
the base surface area 1 and the
height q (velocity), namely,
FIG. 12. iNg. We also make the sup-
position, usual in the theory of
gases (although not strictly true), that the energy, which each electron
transports through the cross-section, has the value corresponding to that
which it had at the point where it last collided.
Now the energy in the section itself at temperature T is equal to %kT,
and hence the energy = |fcr + ^j^ . I at the points which lie at a
distance I in front of and behind the section. Here, on the average, the
electrons coming from the right and the left meet with their last collisions.
The energy transport per second through unit of cross-section is
therefore
Hence 7 = ^JUlqk is the coefficient of thermal conductivity.
Here also U. A. Lorentz has deepened the theory by taking the distri-
bution of velocity into account, and finds that
where again q = v g 2
169 G. Wiedemann and R. Franz, Poggend. Ann. 89, 497 (1853) ; L.
Lorenz, Wiedem. Ann. 13, 422, 582 (1881). Of. also G. Kirchho/ and
G. Hansemann, Wiedem. Ann. 13, 417 (1881); W. Jaeger and H.
Diesselhorst, Abh. d. phys. techn. Beichsanstalt 3, 269 (1900).
The following short table is taken from the paper of the two investi-
gators last named ; it gives the ratio "L for various metals at a temperature
of 18 C.
NOTES AND REFERENCES
163
Metal
y . 10-10
a-
Al
6-36
Cu
6-65
Ag
6-86
Au
7-09
Zn
6-72
Pb
7-15
Pt
7-53
Bi
9-64
170 It follows from (67) by setting ^mq 2
, that is, a =
, that
ra
Now, let JY be the number of atoms per unit volume, .AT* the number
of atoms in a gramme-atom (Avogadro's number). If, further, A is the
atomic weight, M the mass of an atom, and p the density, then
(A = MN*
\p = MN
therefore
We next assume that N> the number of electrons per unit of volume, is
small compared with N, say
If we insert this value, then we get for the free path
200 <r
We shall make a rough calculation for copper at G. We have
ff <--> 5-4 . 10 17 (in electrostatic units)
A = 63-57
k = 1-4 . 10- 16
T= 273
m = 0-9 . 10- 27
N* = 6-1 . 10 23
P = 8-9
e = 4-77 . 10- 10 .
With these values we get
I is of the order 5-7 . 10-".
164 THE QUANTUM THEORY
Since the atomic distance is of the order of magnitude 2 . 10~ 8 , the
electrons would therefore only suffer collision after passing many thou-
sands of atoms. This is unacceptable, since the "radius of molecular
action " of the atoms itself has dimensions which fall within the order of
magnitude of about 10 ~ 8 .
171 JET. A. Lorentz, loc. cit., note 43.
172 J. J. Thomson, The Corpuscular Theory of Matter.
173 H. Kammerlingh-Onnes, Leiden Communicat. 1913, 133.
171 C. H. Lees, Phil. Trans. (A) 208, 381-443 (1908).
175 W. Meissner, Ann. d. Phys. 47, 1001 (1915).
176 W. Nernst, Berl. Ber. 1911, p. 310.
177 H. Kammerlingh-Onnes, Leiden Communicat. 119, 22 (1911).
178 F. A. Lindemann, Berl. Ber. 1911, p. 316.
179 W. Wien, Berl. Ber. 1913, p. 184. Of. also Vorlesungen iiber
neuere Probleme der theoretischen Physik. (Teubner, Leipzig and Berlin
1913.) 3. Vorlesung.
180 If s is the radius of atomic action, N the number of stationary atoms
per unit of volume, then, according to a well-known result of the kinetic
theory of gases, the mean free paths of the electrons
Let us set
where s is the radius of atomic action for T = 0, that is, when the atoms
are at rest ; let a be the amplitude of atomic vibration. Now the mean
energy E of this vibration (frequency i>), on the one hand, = (2irv)*a z
(M is the atomic mass) ; on the other hand, it is, according to Planck-
Einstein,
From this it follows that
Now, according to formula (67) of the text, the resistance
w * 2m 2
r *r-I*
If we here set for q the value,*/ (cf. note 170), and for N, according
to J. J. Thomson's supposition, a^T, and for =- the value
*Ni* = *N(a' l + 2as + si)
NOTES AND REFERENCES 165
it follows that
an expression, which contains only a and s as unknown constants. If
we set
then W assumes the form given in the formula (70).
181 F. A. Lindemann, Phil. Mag. 29, 127 (1915).
181a F. Haber, Berl. Akad. Ber. 1919, pp. 506 and 990.
182 J. -Stark, Jahrb. d. Radioakt. u. Elektronik 9, 188 (1912).
183 G. Borelius, Ann. d. Phys. 57, 278 (1918).
184 K. Herzfeld, Ann. d. Phys. 41, 27 (1913).
185 If we set %mq* = E, therefore q = \K, the first of the two for-
mulae (72) follows from (67). If we further take into account that in Drude's
9 /7 77 1
Theory E = f kT, that is, that & - | ^ then from (68) the second for-
mula (72) follows.
186 F. v. Hauer, Ann. d. Phys. 51, 189 (1916).
187 W. Nernst, Berl. Ber. 1911, p. 65.
1884. Eucken, Berl. Ber. 1912, p. 141.
189 K. Scheel and W. Heuse, Ann. d. Phys. 40, 473 (1913). Of. also
L. Holborn, K. Scheel and F. Henning, Warmetabellen der physikal.-
techn. Reichsanstalt (Vieweg 1919).
1904. Einstein aud O. Stem, Ann, d. Phys. 40, 551 (1918).
191 The quantum formulae (76) and (77) properly correspond to the
Planck oscillator, that is, to a system of one degree of freedom, while
here, in. the case of rotation, we have to do with two degrees of freedom.
But the energy of the Planck oscillator is composed of two equal parts, a
kinetic and a potential part, while in the case of rotation only kinetic
energy comes into question. This is often expressed thus : the Planck
oscillator possesses one potential and one kinetic degree of freedom, while
the rotating molecule possesses two kinetic degrees of freedom.
192 P. Ehrenfest, Verhandl. d. deutsch. physikal. Ges. 15, 451 (1918).
193 According to note 48, the quantum canonical distribution function is
166 THE QUANTUM THEORY
and the mean energy is
~kT
If we here set all p n 's = 1, and if for E n we substitute the value E (n
from (80), there follows for the mean rotational energy of a molecule
o
and for the heat of rotation of hydrogen we get the expression
19* The turning impulse (moment of momentum) of a system, the
mass-points of which possess the mass mi, the velocities vi, and the dis-
tances n from a fixed point (say the origin of co-ordinates), is a vector of
the value
In the present case, the system consists only of the two atoms (mass M) ,
which rotate around a circle of radius r with the constant velocity
v = r 2irv.
Hence here
|U| =p = 2Mr a . 2 = J. 2w,
where J = 2Mr* is the moment of inertia.
198 The impulse (or momentum) pi corresponding to a generalised co-
(3L
ordinate qi is, according to note 48, defined by the relation pi = ^ . .'
where qi = -jr, and L is the kinetic energy of the system. Now here
the angle of rotation $ is chosen as a generalised co-ordinate. But the
kinetic energy of a body rotating about a fixed axis is known to be
= J ' (moment of inertia) x (angular velocity) 2 , hence
J/d<t>\*_J-
L= 2\-Tt) ~2*-
Hence
])Q = ov = J<p = 3 ' 2irv.
NOTES AND REFERENCES 167
196 F. Reiche, Ann. d. Phys. 58, 657 (1919).
197 The best curve was obtained by assigning the " weight " In to the
nth quantum state of rotation. The rotationless state (n = 0) thus
receives the weight zero, i.e. it does not exist. This amounts to the
same thing as the introduction of a zero-point rotation.
198 E. Holm, Ann. d. Phys. 42, 1311 (1913).
199 J. v. Weyssenhoff, Ann. d. Phys. 51, 285 (1916).
200 M. Planck, Ber. d. deutsch. physikal. Ges. 17, 407 (1915).
201 S. Rotszayn, Ann. d. Phys. 57, 81 (1918).
202 The curve is not drawn by Planck, but is discussed in the author's
paper cited in note 196.
203 See likewise the author's paper quoted in note 196.
204 P. S. Epstein, Ber. d. deutsch. physikal. Ges. 18, 398 (1916). Of.
also Phys. Zeitschr. 20, 289 (1919).
203 N. Bohr, Phil. Mag. 1913, p. 857.
206 During " regular precession " the top turns uniformly about its
axis of symmetry (axis of its figure), while at the same time this axis
describes a cone of circular section about an axis fixed in space.
207 A compilation of the moments of inercia of the hydrogen molecule
used by the various investigators is as follows :
.7.10.
Einstein-Stern 1-47
Ehrenfest 0'69
(2-214^
Reiche \ 2-293 \ different curves.
12-095J
Holm 1-36
Weyssenhoff 0-34
Rotszayn 2-12
Epstein (Bohr's model) . . . 2-82
MSN. Bjerrum, Nernst Festschrift 1912, p. 90. Bjerrum did not,
by the way, start from formula (79), but calculated with the values
y n = o~Tr> s ^ uce > following a proposal of H. A. Lorentz, he set the rota-
tional energy E^ equal to nhv n , in contrast to Ehrenfest's formulation
(78), which rests on a sounder basis.
209 S. P. Langley, Annals of the Astrophysical Observatory of the
Smithsonian Institution, Vol. I, p. 127, Plate XX (1900).
210 F. Paschen, Wiedem. Ann. 51, 1 5 52, 209 ; 53, 335 (1894).
211 H. Rubens, Berl. Ber. 1913, p. 513.
212 H Rubens and E. Aschkinass, Wiedem. Ann. 64, 584 (1898).
213 If. Rubens and G. Hettner, Berl. Ber. 1916, p. 167. See also
G. Hettner, Ann. d. Phys. 55, 476 (1918).
214 W. Burmeister, Ber. d. deutsch. physikal. Ges. 15, 589 (1913).
215 Eva v. Bahr, Ber. d. deutsch. physikal. Ges. 15, 710, 731, 1150
(1 M6 3 <3f . Lord RayUigh, Phil. Mag. 34, 410 (1892). Let an HC1 mole-
cule, for example, be considered, which consists of a positively charged
168
THE QUANTUM THEORY
hydrogen atom H+ and a negatively chlorine atom Cl~ (see Fig. 13). Let
its centre of gravity be S, and let a be the distance of the H+ atom from
S. Let the line joining the two atoms be the axis of x', and let this axis
turn in the positive direction about S at the rate of v r revolutions per
second with respect to the fixed x-7/-system. If, now, the two atoms
vibrate relatively to one another with the frequency r fl and the amplitude
A, then the x' co-ordinate of the H+ atoms may be represented thus
x' = a + A sin (2irv t).
If we project this vibration upon the fixed co-ordinate system, it follows
that
(x = x' cos (2*vJ) = o cos (2irv r t) + A sin (2trv t) cos (2*V)
\y = y' sin (2irv r t) = a sin (torvj) + A sin (2*V) sin (2W)
FIG. 13.
for which we may also write
fa = a cos (2y r O + 4 sin 2(r + , r )t + 4 sin 2*( v , - , r )t
|T/ = a sin (2w^) - ^cos 2 1 r(^ + , r )i + ^ cos 2(ir - Vf )t .
From the point of view of the system at rest we have thus three
oscillations :
(a) the left-circular oscillation
x = a cos (2^)|^ h ^ {requenc
T/ = oin(2x^)J
(6) the left-circular oscillation
with the frequency v u + v r
NOTES AND REFERENCES
(c) the right-circular oscillation
I with the frequency v n - 1
169
~ sin 2
217 E. S. Imes, Astroph. Journ. 50, 251 (1919).
218 A. Eucken, Ber. d. deutsch. phys. Ges. 15, 1159 (1913). Eucken has
here, on account of the asymmetrical form of the hydrogen molecule,
assumed two different moments of inertia
</! = 0-96 . 10- 40 , and J t =2-21 . lO" 40
and hence obtained two different series of numbers giving the revolutions
Vr per second, cf . the table given there. See also the table in Rubens and
Hettner, loc. cit., note 213.
219 M. Planck, Ann. d. Phys. 52, 491 ; 53, 241 (1917).
220 0. Sackur, Ann. d. Phys. 36, 958 (1911) ; 40, 67 (1913).
221 H. Tetrode, Phys. Zeitschr. 14, 212 (1913) ; Ann. d. Phys. 38, 434
(1912). ,
222 W. H. Keesom, Phys. Zeitschr. 15, 695 (1914).
2234. Sommerfeld, Vortrage liber die kinetische Theorie der Materie
und der Elektrizitat. Wolfskehl-Kongress in GQttingen 1913. (Teubner,
Leipzig and Berlin 1914), p. 125.
224 P. Scherrer, Gottinger Nachr. 8 July, 1916.
225 M. Planck, Berl. Ber. 1916, p. 653.
226 W. Nernst, Die theoretischen und experimentellen Grundlagen des
ueuen Warmesatzes. (W. Knapp, Halle 1918), pp. 154 et seq.
227 O. Sackur, Ber. d. deutsch. chem. Ges. 47, 1318 (1914).
228 0. Stern, Phys. Zeitschr. 14, 629 (1913) ; Zeitschr. f. Elektrochemie
25, 66 (1919).
229 For what follows cf. the paper by O. Stern quoted in the last note.
Further, W. Nernst, Die theoretischen und experimentellen Grundlagen
des neuen Warmesatzes. (W. Knapp, Halle 1918), Oh. XIII.
230 As regards this and the following chapter, the reader is referred for
more exact details to the article of P. S. Epstein in the Planck number
of " Naturwissenschaften " (1918, p. 230).
231 As the simplest Tiwmson atom, we are to imagine a sphere of radius
a, filled with the unit charge e of posi-
tive electrification, of space-density p, in
the middle of which an electron with
the charge - e rests. This structure is
externally neutral. If we draw the
electron out from the centre to a distance
r (see Fig. 14) the external (shaded)
hollow sphere exerts no force on the
electron, according to the well-known
laws of electrostatics. The inner solid
sphere of radius r, on the other hand,
acts on the electron just as if its total
charge were concentrated at the centre.
The force which draws the electron back into its position of rest is
170 THE QUANTUM THEORY
therefore,
that is, it is proportional to the distance of the electron from its position
of equilibrium.
232 Cf. also P. Drude, Lehrbuch der Optik. 2. Aufl., Chs. V and VII
(Hirzel 1906). There is an English edition of this work.
233 Cf . W. Voigt, Magneto- und Elektro-optik (Teubner 1908).
234 M. Planck, Ber. d. Berl. Akad. d. Wiss. 1902, p. 470; 1903, p. 480;
1904, p. 740 ; 1905, p. 382.
238 H. A. Lorentz, The Theory of Electrons, Chs. Ill, IV (Teubner 1909).
236 The electron oscillates, when bound quasi-elastically, according to
the equation of motion m ? = - fx, if we restrict ourselves to linear os-
cillations. Here m is the mass of the electron, x is its distance from the
position of rest, and / is a factqr of proportionality. The solution of this
differential equation is represented by the pure harmonic motion
x = A cos (nt + 8)
where the frequency is
n = V TO'
The frequency n is therefore, as we see, independent of the amplitude and
therefore of the energy of vibration.
237 The frequencies v of those spectral lines of luminous hydrogen,
which are included under the name " Balmer series," may be represented
with great accuracy by the following formula given by Balmer.
n = 3, 4, 5, 6 ... oo .
\&' n'/
N is here a constant, the so-called Rydberg number. If we set for the
current number n the values 3, 4, 5 ... we get in succession the fre-
quencies of the red line of hydrogen (H a ), the green line (H^), and the
blue line (H y ) and so forth.
238 J. Stark, Ann. d. Phys. 43, 965 (1914) ; J. Stark and G. Wendt, ibid.,
43, 983 (1914) ; J. Stark and H. Kirschbawn, ibid., 43, 991 ; 43, 1017
(1914) ; J. Stark, ibid., 48, 193, 210 (1915) ; J. Stark, O. Hardtke and G.
Liebert, ibid., 56, 569 (1918) ; J. Stark, ibid., 56, 577 (1918) ; G. Liebert,
ibid., 56, 589, 610 (1918) ; J. Stark and 0. Hardtke, ibid. 58, 712 (1919) ;
J. Stark, ibid., 58, 723 (1919).
239 Cf. H. A. Lorentz, The Theory of Electrons (Teubner, Leipzig and
Berlin 1909), Ch. III.
2*0 H. Geiger and Marsden, Phil. Mag. April, 1913.
241 E. Rutherford, Phil. Mag. 21, 669 (1911).
242 According to C. G. Darwin [Phil. Mag. 27, 506 (1914)], the radius
of the nucleus, taken as a sphere, is in the case of gold at the most
= 3 . 10 - 12 cms., in the case of hydrogen at the most = 2 . 10-13 cms.
NOTES AND REFERENCES 171
243,4. van den Broek, Phya. Zeitschr. 14, 32 (1913).
244 Cf. note 247.
243 N. Bohr, Phil. Mag. 26, 1, 476, 857 (1913).
2464. Einstein, Phys. Zeitschr. 18, 121 (1917).
247 The quite elementary calculation is as follows : let an electron of
charge e and mass m rotate around a nucleus of charge E = ez in a
circular orbit : then z is the atomic number (for hydrogen, in particular,
z = 1). If a is the radius of the circle, v the velocity, and a> the angular
velocity (frequency of rotation) of the electron in the circular orbit, then
the condition for equilibrium between the attraction of the nucleus and
the centrifugal force is
e _ = maw 2 or ma s w 2 = eE = e*z.
According to Bohr's second hypothesis the moment of momentum
p( = mva=ma?<u) is a multiple of , hence
ma*a = wA (n = 1, 2, 3 . . .).
From these two equations for a and u> we get for the discrete radii of the
permissible quantum orbits
and the corresponding frequencies of rotation
^BvWfm
The energy (kinetic + potential) is
(aT7 t \ 2tf
-T) -*"*-*-
therefore the discrete quantum values of the energy are
* Wn? '
If, in this expression, we set
we recognise, that W is a function of , and hence of v = -. The energy
of the electron in the Rutherford model therefore depends, as stated in
the text, on its frequency of rotation v.
If the electron passes from the ntb to the sth quantum path, then, ac-
cording to Bohr's third hypothesis, a homogeneous spectral line is emitted
of frequency
W n - W s 2irVw.2 2 /l 1
172 THE QUANTUM THEORY
where
N = Wm
2*8 Cf . note 237.
2*9 It is of historical interest to note that, before Bohr, A. E. Hems in
1910 (Sitzungsber. d. Wiener Akad. 10 March, 1910) succeeded in repre-
senting Rydberg's number in terms of the universal constants e, h, m ;
his result differed from that of Bohr only by a factor 8. He deduced his
result as follows. Starting from /. J. Thomson's atomic model, which
was generally accepted at that time, he calculated the maximum oscilla-
tion-frequency (no. of revolutions) /max of the electron in the simplest
atom (hydrogen atom) for the case when this atom, provided with one
energy-quantum, was circling just on the surface of the positive sphere.
He obtained
4Vw
"max = p
This maximum frequency was next identified by Haas with the series
limit (n = oo ) in Balmer's formula
Then it follows that
which is a value 8 times greater than JVjjohr- Haas used this relation to
calculate from the three quantities, the Bydberg number N, Planck's
constant h, and the ratio ~ , all of which he assumed known, the charge
m
e of the electron. In consequence of the factor 8 he obtained the value
e = 3'18 . 10~ 10 , a value that is too small according to our present know-
ledge, but which agreed well with the measurements of J. J. Thomson and
H. A. Wilson, which were available at that time.
230 Th. Lyman, Phil Mag. 29, 284 (1915).
251 F. Paschen, Ann. d. Phys. 27, 565 (1908).
252.4. Fowler, Month. Not. Roy. Astron. Soc. 73, Dec. 1912.
233 F. Paschen, Ann. d. Phys. 27, 565 (1908).
23* E. C. Pickering, Astroph. Journ. 4, 369 <1896) ; 5, 92 (1897).
238 E. J. Evans, Nature, 93, 241 (1914).
236 W. Kossel, Ann. d. Phys. 49, 229 (1916) ; Die Naturwissenschaften
7, 339, 360 (1919).
237 L. Vegard, Verhandl. d. deutsch. physikal. Ges. 19, 344 (1917).
238 A. Sommerfeld, Atombau und Spektrallinien. (An English edition
translated from the 3rd German edition (1922) is being prepared by
Messrs. Methuen & Co., Ltd.)
259 R. Ladenburg, Die Naturwissenschaften 8, 5 (1920).
280 .4. Sommerfeld, Ann. d. Phys. 51, 1 (1916).
NOTES AND REFERENCES 173
281 Expressed in terms of polar co-ordinates the kinetic energy L as-
sumes the well-known form :
L = (^ + rV).
In it, m denotes the mass of the electron, the dots represent differentia-
tion with respect to the time. The impulses p r and p^ are then defined as
follows (see note 48) :
p r =?>L = mr ; p. = !%L = mr^.
3r' ^
262 Only when each impulse _p f depends solely on the corresponding
2j (or when it is a constant), and when, in addition, the limits of the
phase-integral are independent of the g/s, does the phase-integral work
out to a constant. This is by no means the case for any arbitrary choice
of the co-ordinate-system.
263 P. S. Epstein, Ann. d. Phys. 50, 489 ; 51, 168 (1916).
264 K. Schwarzschdd, Sitzungsber. d. Berl. Akad. d. Wiss. 4. Map
1916. *
263 A. Einstein, Verhandl. d. deutsch. physikal. Ges. 19, 82 (1917).
266 M. Planck, Verhandl. d. deutsch. physikal. Ges. 17, 407, 438 (1915) ;
Ann. d. Phys. 50, 385 (1916).
267 The semi-major axis of the ellipse, which is characterised by the
values n and n', here has the value
The ratio of the axis is
b _ n
a~ n + n''
We see that n' = corresponds to the case of Bohr's circular orbits.
268 The energy of the electron moving in the Kepler ellipse (n, n') here
has the value
= _ 2TrVs a w _ _ Nhz*
h\n + n'Y ~ (n + n') a '
The series formula (102) of the text then follows from Bohr's Law of
Frequency
269 If account is taken of the influence of relativity, the series formula
for the spectra of the hydrogen type become to a first approximation
V = VQ + V l
where
174 THE QUANTUM THEORY
In these expressions the symbols N and a have the following meaning :
N= 2 f e * m o a = ^f; a 2 is of the order 5-3.10-5
m is the mass of the electron at vanishingly small velocities.
Hence whereas the first term i/ gives the old formula, which was
obtained by neglecting the influence of relativity, the small additional
term v l represents the influence of relativity. As we observe, v l does not
only depend on the quantum sums s + s' and n + n', but also on the
individual values s, s', n, n'. This member, v v is thus responsible for the
fine-structure.
270 If we apply the formula of the preceding note to H a , we have to set
z = 1, s + s' = 2, n + n' = 3. We then get
l s' 1 n'~\
+ r i + *
24 84 J
n
AVjj
j
1 [-
-i 1
r 1 1
t? *> *
i i
n
r c
I
1 i
1 "i
*
Pia. 15.
Corresponding to the possibilities of partition
and
2 + 01 circle
1 + I/ ellipse
2 ^^ orbits
n + ri = 3 = 3 + 0^ circle \
= 2 + 1 L ellipse L 3 initial orbits
= 1 + 2] ellipse )
(for dynamical reasons the azimuthal quantum number n cannot under
normal conditions assume a zero value), we should expect 2.3 = 6
possibilities of production and hence 6 components of the fine-structure
of Ha.. One of these components, however, namely, the one correspond-
ing to the transition of the electron from the circle (n = 3, n' = 0) to the
ellipse (s = 1, s' = 1) does not present itself under normal conditions, as
follows from the " Principle of Selection " enunciated by Rubinowicz
NOTES AND REFERENCES 175
and Sommerfeld (see Chapter VI, 9). Hence 5 components of the
fine-structure remain ; their position is exhibited in Fig. 15.
As we see, the 5 components arrange themselves into two main groups,
containing 3 and 2 members, respectively. The ' missing " line Ila is
dotted in. The distance AVH between I a and II a , Ib and lib, Ic and II C is
called the " theoretical hydrogen doublet."
According to the above formula the frequency-number of the line
la (3, 0-2, 0) is
The frequency-number of the line Ha (3, >1, 1) is
Thus
A "H = */J,, - "/a = ^=1-095.1010
corresponding to AA.H = 0'157A.
The hydrogen-doublet actually observed is measured from about the
middle of I a and It, to the middle of lib and II C , owing to the absence of
IIa. This leads to the value 0'8AA. H , that is, to 0-126A.
According to a principle of correspondence enunciated by Bohr (see
Chapter VI, 9), as a result of which the azimuthal quantum number can
only vary by + 1, the components Ib and II C are also absent.
271 F. Paschen, Ann. d. Phys. 50, 901 (1916).
272 From formula (97) of the text we get for the two Rydberg constants
for hydrogen and helium :
Moreover, according to note 269, we get the third formula giving the
value of the constant for the fine-structure :
From the first two relations, by using MHO = &Ma, we get
m a , __M^
~
176 THE QUANTUM THEORY
and hence
_L = JL N H-* N H*
m c MHC ' Nne - Na '
The two Rydberg numbers NH and NHC have been measured by Pastfien
with great accuracy :
Nil = (109677-691 + 0'06) . c
Nue = (109722-144 0'04) . c.
Moreover, = F is the electrochemical equivalent (Faraday'* num-
MH . c
ber), that is, the charge which, in electrolysis, accompanies one gramme-
atom (i.e. N = - atoms). This number has the value
F = 9649-4 electromagnetic units.
If we insert the three values of N a , NH* and * - in the relation above
MH.C
deduced, we get
JL = 1-7686 . 10 7 electromagnetic units,
m c
a value which agrees very well with those values of this quantity which
were obtained by direct methods (deflection of the cathode- and /5-rays in
the electric and magnetic field). Let us now write
or, using the value of - given above,
MH
^m^ = 3
h 4
The right-hand side of this equation is known. If we combine with it
m c
O /2
= = 7-290. 10- 3
which follows from Paschen's measurements of the fine-structure in the
case of helium, we have three equations in three unknowns e, m g , h.
From them we get
e = (4-766 0-088). 10 - 10
h = (6-526 + 0-200) . 10 - 27 .
According to Sommerfeld it is more advantageous to use Millikan's value
for e. We then get
fe =(4-774 0-004). 10 - 10
{ h = (6-545 0-009) . 1Q- 87
U = (7-295 0-005) . 10-.
NOTES AND REFERENCES 177
273 K. Glitscher, Ann. d. Phys. 52, 608 (1917).
274 A. Lande, Phys. Zeitschr. 20, 228 (1919) ; 21, 114 (1920).
278 Cf. A. Sommerfeld, Atombau und Spektrallinien. Ch. IV, 6.
276 P. S. Epstein, Ann. d. Phys. 50, 489 (1916).
277 P. Debye, Gottinger Nachr. 3 June, 1916.
278/1. Sommerfeld, Phys. Zeitschr. 17, 491 (1916). Cf. also Atombau
und Spektrallinien. Ch. VI, 5.
279 F. Paschen and E. Back, Ann. d. Phys. 39, 897 (1912) ; 40, 960
(1913).
2804. Rubinowicz, Phys. Zeitschr. 19, 441, 465 (1918).
281 N. Bohr, On the Quantum Theory of Line-spectra. Parts I and II.
D. Kgl. Danske Vidensk. Seisk. Skrifter, Naturvidensk. og Mathem. Afd.
8, Baekke IV, 1. Kopenhagen 1918.
282 The number of revolutions of the electron per second in the sth
quantum circle of Bohr is, in the case of hydrogen, according to note 247 :
On the other hand, it follows from formula (93) of the text, if we take s
considerably greater than 1 (high quantum numbers), and n = s + 1
(transition between neighbouring circles), that
283 P. S. Epstein, Ann. d. Phys. 58, 553 (1919).
284 H. A. Kramers, Intensities of Spectral Lines. D. Kgl. Danske
Vidensk. Selsk. Skrifter, Naturvidensk. og Mathem. Afd. 8, Raekke III,
3. Kopenhagen 1919.
283.4. Sommerfeld and W. Kossel, Ber. d. deutsch. physikal. Ges. 21,
240 (1919).
286 J. Franck and G. Hertz, Phys. Zeitschr. 20, 132 (1919) ; in which
references are also given. Cf . also J. Franck and P. Knipping, Phys.
Zeitschr. 20, 481 (1919) ; J. Franck, P. Knipping and Thea KrOger, Ber.
d. deutsch. physikal. Ges. 21, 728 (1919).
287 J. Tate and Foote, Phil. Mag. July, 1918.
288 References are given in the report by J. Franck and G. Hertz, men-
tioned in note 286.
289 A. Einstein, Phys. Zeitschr. 18, 121 (1917). Let us consider the
two quantum states (1) and (2) of the atom, with the energies E l and E z
(.E 2 > -EJ). The number of transitions 2 - 1 which take place in the
time dt owing to radiation is then, according to Einstein, NiA 2l dt, in
which N 9 is the number of atoms in the state 2, and, therefore, accord-
ing to note 48
~ &.
N,, = Nw,
N being the total number of atoms. 4 al is a factor of proportionality.
12
178 THE QUANTUM THEORY
The introduction of external monochromatic radiation of frequency /
and intensity K,, firstly brings about positive absorption, that is, transi-
tions l->2. The number of these in the time dt is, according to
Einstein, JVj.B 12 K,,, in which .B 12 is a factor of proportionality, N l is the
number of atoms in the state 1, and hence
5
jY, = NCpje M.
Secondly, the external radiation also effects transitions 2 - 1 (nega-
tive absorption). The number of these that occur in the time dt
= -ZV-j-B^Kj/, where jB 2] is a factor of proportionality. When the energy
exchange is in equilibrium the number of transitions 2 > 1 must be
equal to the number of transitions 1 > 2, hence
~ M'
i.e.
When the temperature increases indefinitely, K* must also increase to
infinitely great values ; from this it follows that
= 1.
Finally, if we set -^p- = A for shortness, we get the relation given in the
text:
^ A
290 Of. the resume* by E. Wagner, Phys. Zeitgchr. 18, 405, 432, 461,
488 (1917).
291 O. Moseley, Phil. Mag. 26, 1024 (1913) ; 27, 703 (1914).
292 W. Kossel, Verhandl. d. deutsch. physikal. Ges. 16, 898, 953 (1914) ;
18, 339 (1916).
293 A. Sommerfeld, Ann. d. Phys. 51, 125 (1916) ; Phys. Zeitschr. 19,
297 (1918). Of. also Atombau und Spektrallinien. Oh. Ill, Ch. IV
4, Ch. V 5.
29* L. Vegard, Verhandl. d. deutsch. physikal. Ges. 1917, pp. 328, 344 ;
Phys. Zeitschr. 20, 97, 121 (1919).
298 P. Debye, Phys. Zeitschr. 18, 276 (1917).
296 J. Kro6, Phys. Zeitschr. 19, 307 (1918).
297 A. Smekal, Wiener Ber. Ila 127, 1229 (1918) ; 128, 639 (1919) ;
Verhandl. d. deutsch. physikal. Ges. 21, 149 (1919). Of. also A. Smekal
and F. Reiche, Ann. d. Phys. 57, 124 (1918).
NOTES AND REFERENCES 179
298 W. Kossel, Zeitschr. f. Physik 1, 119 (1920).
299 Since the L-ring consists of several electrons, we take the expression
" elliptic motion " to mean the following type of motion : each electron
independently describes an elliptic path about the nucleus, whereby the
electrons are at each moment situated at the corners of a regular polygon
which shares in the motion of the electrons, alternately contracting and
expanding during this motion (" elliptical associates"), cf. Sommerfeld.
Atombau and Spektrallinien.
3004. Stnekal, Wiener Ber. Ha 128, 639 (1919).
301 M. Born and A. Lande, Berl. Akad. Ber. 1918, p. 1048 ; Verhandl,
d. deutsch. physikal. Ges. 20, 202, 210 (1918) ; M. Born, ibid., 20, 230
(1918) ; Ann. d. Phys. 61, 87 (1920).
302 A. Lande, Verhandl. d. deutsch. physikal. Ges. 21, 2, 644, 653 (1919) ;
Zeitschr. f. Phys. 2, 83 (1920). Cf. also A. Lande and E. Madelung,
Zeitschr. f. Phys. 2, 230 (1920).
303 W. Kossel, Ann. d. Phys. 49, 229 (1916).
3M P. Debye, Munch. Akad. Ber. 9 Jan. 1915.
305 P.. Scherrer, Die Botationsdispersion des Wasserstoffs. Dissertation,
Gottingen, 1916.
306 G. Laski, Phys. Zeitschr. 20, 269, 550 (1919).
307 Cf., for example, A. Sommerfeld, Atombau und Spektrallinien, Ch.
IV, 6.
308 Langmuir, Journ. Amer. Chem. Soc. 34, 860 (1912) ; Zeitsohr. f.
Electrochemie 23, 217 (1917).
309 Isnardi, Zeitschr. f. Elektrochemie 21, 405 (1915).
310 /. Franck, P. Knipping and Thea KriLger, Ber. d. deutsch. physikal.
Ges. 21, 728 (1919).
310a Planck has made an attempt to alter Bohr's model in such a way
that the right heat of dissociation results. See M. Planck, Berl. Akad.
Ber. 1919, p. 914. Cf. also H. Kallmann, Dissertation, Berlin 1920.
311 W. Lenz, Ber. d. deutsch. physikal. Ges. 21, 632 (1919).
3124. Sommerfeld, Ann. d. Phys. 53, 497 (1917).
313 F. Pauer, Ann. d. Phys. 56, 261 (1918).
314 G. Laski, see note 314.
315 M. Pier, Zeitschr. f. Elektrochemie 16, 897 (1910).
316 K. Schwarzschild, Berl. Akad. Ber. 1916, p. 548.
317 H. Deslandres, Compt. Rend. 138, 317 (1904).
318 T. Heurlinger, Phys. Zeitschr. 20, 188 (1919) ; Zeitschr. f. Physik
1, 82 (1920).
319 W. Lenz, see note 311.
320 A different view is upheld by /. Burgers (Versl. K. Ak. van Wet.
Amsterdam 26, 115, 1917), in which, also, jumping electrons produce the
middle line in the infra-red of band. In contrast with Schwartschild and
Lenz, Burger assumes that the motion of the electrons is influenced by
the rotation of the molecule. The energy of the system is then not com-
posed additively of the energy of the electrons and the rotational energy
of the molecule, but a third term has to be added, which is due to the
Coriolis force of the rotating system.
180 THE QUANTUM THEORY
321 T. Heurlinger, see note 318.
322 F. Reiche, Zeitschr. f. Physik 1, Heft 4, 283 (1920).
323 E. S. Imes, Astrophys. Journ. 50, 251 (1919).
324 A. Kratzer, Dissertation, Munchen 1920.
323 In the case of the gases investigated by Imes, namely HC1, HBr, and
HP, the following moments of inertia were found :
/HOI = 2 ' 6 * 10- 40 ; J H Br = 3-27 . 10 - 40 ; J HF = 1'37 . 10- 40 .
INDEX
Absorption band, edge of, 21, 112.
continuous, 78.
Acoustic vibrations, 157.
Action, quantum of, 9, 27.
Atomic heat, 29.
numbers, 86, 110.
Avogadro's constant, 12, 29, 163.
Law, 12.
Azimuthal quantum number, 92.
B
Back, 22.
Balmer, 90.
Barkla, 20, 109.
Beck, 25.
Benedicks, 146.
Bergen, 106.
Bergmann series, 89.
Bishop, 106.
Bjerrum, 75, 123.
Bohr, 75.
Bohr's model of the atom, 86, 119.
principle of analogy or corre-
spondence, 100.
selection, 98.
Boltzmann, 4, 13.
Born, 38, 42.
Bragg, W. H. and W. L., 109.
Bravais, 37.
Bremsstrahlung, 22, 109.
Broek, van den, 86.
Broglie, de, 21.
Bunsen, 109.
Burgers, 179.
Canal rays, 142.
Cavity radiation, law of, 3.
Characteristic Bontgen radiation,
109.
Chemical constant, 81.
Compressibility, 33.
Conductivity, thermal, 60.
Coulomb, 12.
Davis, 106.
Debye, 38, 39, 58.
Debye's formula, 40.
Degeneration of gases, 79.
Deslandres, 122.
Dessauer, 22.
Diamagnetism, 25.
Diamond, 33.
Dispersion, 47.
Displacement Law, Wien's, 4.
Distribution numbers, 114.
of velocities, Maxwell's Law of,
5.
Dix&n, 100.
Doppler, 4.
Drude, 61, 84, 17.
Duane, 22.
Dulong and Petit's Law, 29, 30.
Ehrenfest, 71.
Einstein, 1, 15, 16, 30, 107.
functions, 31, 32, 51.
Einstein's hypothesis of light-
quanta, 16, 17, 107.
Law of the photo-electric effect,
20.
Elastic collisions, 104.
spectrum, 42.
Electron theory of metals, 61.
Emissivity, 2.
Entropy, 8, 151.
Epstein, 75, 98.
Equipartition, 13.
Eucken, 56, 61, 69.
181
182
THE QUANTUM THEORY
P
Fine-structure, Sommerf eld's theory
of, 94.
of the Rontgen rays, 113.
Fokker, 15.
Foote, 105.
Franck, 22, 102.
Friman, 109.
Oeiger, 85.
Gibbs, 13, 26.
Glitscher, 97.
Ooiicher, 106.
Gramme-molecule, 145.
Grilneisen, 58.
Haas de, 25.
Haber, 46, 66, 155.
Hamilton, 15.
Hamilton's principle, 27.
Hauer, von, 67.
Heat theorem, Nernsfs, 35.
Helium, 97.
Helmholtz, 160.
.ZTerte, 22, 102.
Herzfeld, 66.
Heurlinger, 122.
Iftmse, 69.
floo&e's Law, 59.
flop/, 15, 56.
Hughes, 106.
fltiK, 22.
Hunt, 22.
Hydrogen type of series, 88.
Imes, 123.
Impulse or momentum, 26.
radiation, 22, 109.
Infra-red dispersion frequencies, 4i
Intensity, 2.
Isnardi, 120.
Jeans, 15.
K
Kammerlingh-Onnes, 64.
Karmdn, 38.
Kepler ellipses, 95.
Kinetic radiation, 23.
Kirchhoff, 3.
.KbsseZ, 91, 102, 105.
Kramers, 102.
Kratzer, 124.
K-rings, 111.
roo, 111, 115.
K-series, 21.
Kurlbaum, 9.
Ladenburg, 91.
Langmuir, 120.
Losfci, 119.
Lattice theory of atomic heats, 42.
Lowe, 35
Law, Avogadro's, 12.
equipartition, 13.
of cavity radiation, 3.
of radiation, Planck's, 10.
--- Bayleigh's, 10, 14.
--- Wieris, 5, 7.
Stefan and Boltzmann, 5.
L-doublets, 114.
Lechner, 56.
Lees, 64.
Lenard, 19.
Lews, 120, 122.
Light-quanta, Einstein's, 16.
Lindemann, 33, 64, 66.
Lorentz, 15, 24, 61, 161.
Lorenz, 62.
L-rings, 111.
Lummer, 2, 3, 4.
M
, 33.
Marsden, 85.
Maxwell, 5, 13.
Mean atomic volume, 53.
Meissner, 64.
Metals, electron theory of, 61.
Michelson-Morley, 1.
Millikan, 12, 20.
Molecular heat, 53.
Momentum, 26, 166.
moment of, 72, 166.
Moseley, 110.
INDEX
188
N
Nernst, 9.
and Lindemann's formula, 37.
Nernst's heat theorem, 35.
Nichols, 33.
Orbits, allowable, 87.
Ortvay, 152.
Paramagnetism, 25.
Parhelium, 97.
Paschen, 6, 9, 89.
Pauer, 120.
Phase-integral, 92.
Phase-space, 26, 92, 94, 136.
Photo -electrons, 20, 141.
Pier, 121.
Planck, 6, 15, 24, 27, 91, 117.
Poisson's ratio, 41.
Polarisation, 168.
Precession, 167.
Pringsheim, 2, 4.
Probability, thermodynamic, 8.
Q
Quanta, energy-, 7.
Quantum of action, 9.
Radial quantum number, 92.
Ratnowski, 58.
Bayleigh, 10.
Reflection, metallic, 45.
Relativity, 95, 96.
Residual rays, 35, 149.
Resonance lines, 104.
potentials, 105.
Resonators, Planck's, 6.
Bice, 22.
Richards, 58.
Richarz, 25.
Riecke, 61.
Rontgen radiation, 21.
Rotation spectra, 76.
Rotszayn, 75.
Rubens, 9, 33.
Rubinowicz' principle of selection,
98, 99, 174.
Rutherford, 84, 171.
Rydberg, 90, 110.
Sackur, 80.
Sadler, 21.
Scheel, 69.
Scherrer, 79, 119.
Schrodinger, 57.
Schwarzschild, 93, 121.
Selection, principle of, 98.
Sommerfeld, 27, 91, 94, 102, 120.
Stark, 23, 66.
effect, 98.
Stefan, 4.
Stefan-Boltzmann, Law, 5.
Stern, 25, 70, 81.
Stokes, 19.
Sutherland, 33.
Thirring, 55.
Thomson, J. J., 19, 63, 84.
atom, 169.
Vegard, 91.
117.
w
Wagner, 21, 22.
Warburg, 9, 23.
Weber, 30.
Webster, 22.
Weyssenhof, von, 74.
Wien, 3, 4.
Wien's displacement Law, 4.
Law of radiation, 5, 7.
WW/, 9.
Zeeman, 84.
effect, 98, 120.
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