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N.~ Bohr, Philos. Mag. {\bf 26,} 1 \hfill {\large \bf 1913}\\
\vspace{2cm}
\begin{center}
{\large \it On the Constitution of Atoms and Molecules}\\
\end{center}
\begin{center}
N.~Bohr,\\
{\it Dr. phil. Copenhagen}\\
(Received July 1913)\\
\end{center}
\section*{
{\it Introduction}}
\vspace{0.5cm}
In order to explain the results of experiments on scattering of $\alpha$ rays by
matter Prof. Rutherford\footnote{E. Rutherford, Phil. Mag. XXI. p. 669 (1911)}
has given a theory of the structure of atoms. According to this theory, the atom
consist of a positively charged nucleus surrounded by a system of electrons kept
together by attractive forces from the nucleus; the total negative charge of the
electrons is equal to the positive charge of the nucleus. Further, the nucleus
is assumed to be the seat of the essential part of the mass of the atom, and to
have linear dimensions exceedingly small compared with the linear dimensions of
the whole atom. The number of electrons in an atom is deduced to be
approximately equal to half the atomic weight. Great interest is to be
attributed to this \mbox{atom-model}; for, as Rutherford has shown, the
assumption of the existence of nuclei, as those in question, seems to be
necessary in order to account for the results of the experiments on large angle
scattering of the $\alpha$ rays.\footnote{See also Geiger and Marsden, Phil.
Mag. April 1913.}
In an attempt to explain some of the properties of matter on the basis of this
\mbox{atom-model} we meet, however, with difficulties of a serious nature
arising from the apparent instability of the system of electrons: difficulties
purposely avoid in \mbox{atom-models} previously considered, for instance, in
the one proposed by Sir. J.J. Thomson\footnote{J.J. Thomson, Phil. Mag. VII. p.
237 (1904).} According to the theory of the latter the atom consist of a sphere
of uniform positive electrification, inside which the electrons move in circular
orbits.
The principal difference between the \mbox{atom-models} proposed by Thomson and
Rutherford consist in the circumstance that the forces acting on the electrons
in the \mbox{atom-model} of Thomson allow of certain configurations and motion
of the electrons for which the system is in a stable equilibrium; such
configurations, however, apparently do not exist for the second \mbox{atom-
model}. The nature of the difference in question will perhaps be most clearly
seen by noticing that among the quantities characterizing the fist atom a
quantity appears -- the radius of the positive sphere -- of dimensions of a
length and of the same order of magnitude as the linear extension of the atom,
while such a length does not appear among the quantities characterizing the
second atom, viz. the charges and masses of the electrons and the positive
nucleus; nor can it do determined solely by help of the latter quantities.
The way of considering a problem of this kind has, however, undergone essential
alterations in recent years owing to the development of the theory of the energy
radiation, and the direct affirmation of the new assumptions introduced in this
theory, found by experiments on very different phenomena such as specific heats,
photoelectric effect, \mbox{R\"ontgen-rays}, \& c. The result of the
discussion of
these questions seems to be a general acknowledgment of the inadequacy of the
classical elecrtodynamics in describing the behaviour of system of
atomic size.\footnote{See f. inst., ``Theorie du ravonnement et
les quanta.'' Rapports de la
rennion a Bruxeless, Nov. 1911, Paris, 1912.} Whatever the alteration in the
laws of motion of the electrons may be, it seems necessary to introduce in the
laws in question a quantity foreign to the classical electrodynamics, i.e.,
Planck's constant, or as it often is called the elementary quantum of action. By
the introduction of this quantity the question of the stable configuration of
the electrons in the atoms is essentially changed, as this constant is of such
dimensions and magnitude that it, together with the mass and charge of the
particles, can determine a length of the order of magnitude required.
This paper is an attempt to show that the application of the above ideas to
Rutherford's \mbox{atom-model} affords a basis for a theory of the constitution
of atoms. It will further be shown that from this theory we are led to a theory
of the constitution of molecules.
In the present first part of the paper the mechanism of the binding of electrons
by a positive nucleus is discussed in relation to Planck's theory. It will be
shown that it is possible from the point of view taken to account in a simple
way for the law of the line spectrum of hydrogen. Further, reason are given for
a principal hypothesis on which the considerations contained in the following
parts are based.
I wish here to express my thinks to Prof. Rutherford for his kind and
encouraging interest in this work.
\vspace{0.5cm}
\part*{
{\large Part I. -- Binding of Electrons by Positive Nuclei.}}
\vspace{0.4cm}
\section*{
$\S~1$. {\it General Considerations}}
\vspace{0.5cm}
The inadequacy of the classical electrodynamics in accounting for the properties
of atoms from an \mbox{atom-model} as Rutherford's, will appear very clearly if
we consider a simple system consisting of a positively charged nucleus of very
small dimensions and an electron describing closed orbits around it. For
simplicity, let us assume that the mass of the electron is negligibly small in
comparison with that of the nucleus, and further, that the velocity of the
electron is small compared with that of light.
Let us at first assume that there is no energy radiation.
In this case the electron will describe stationary elliptical orbits. The
frequency of revolution $\omega$ and the \mbox{major-axis} of the orbit $2a$
will depend on the amount of energy $W$ which must be transferred to the system
in order to remove the electron to an infinitely great distance apart from the
nucleus. Denoting the charge of the electron and of the nucleus by --
$e$ and $E$ respectively and the mass of the electron by $m$, we thus get
\begin{equation}
\omega = \frac{\sqrt{2}}{\pi} \cdot \frac{W^{3/2}}{eE \sqrt{m}}, ~~~~
2a = \frac{eE}{W}.
\end{equation}
Further, it can easily be shown that the mean value of the kinetic energy of
the electron taken for a whole revolution is equal to $W$. We see that if the
value of $W$ is not given, there will be no values of $\omega$ and $a$
characteristic for the system in question.
Let us now, however, take the effect of the energy radiation into account,
calculated in the ordinary way from the acceleration of the electron. In this
case the electron will no longer describe stationary orbits. $W$ will
continuously increase, and the electron will approach the nucleus describing
orbits of smaller and smaller dimensions, and with greater and greater
frequency; the electron on the average gaining in kinetic energy at the same
time as the whole system loses energy. This process will go on until the
dimensions of the orbit are the same order of magnitude as the dimensions of the
electron or those of the nucleus. A simple calculation shows that the energy
radiated out during the process considered will be enormously great compared
with that radiated out by ordinary molecular processes.
It is obvious that the behaviour of such a system will be very different from
that of an atomic system occurring in nature. In the first place, the actual
atoms in their permanent state to have absolutely fixed dimensions and
frequencies. Further, if we consider any process, the result seems always to be
that after a certain amount of energy characteristic for the systems in question
is radiated out, the system will again settle down in a stable state of
equilibrium, in which the distance apart of the particles are of the same
order of magnitude as before the process.
Now the essential point in Planck's theory of radiation is that the energy
radiation from an atomic system does not take place in the continuous way
assumed in the ordinary electrodynamics, but that it, on the contrary, takes
place in distinctly separated emissions, the amount of energy radiated out from
an atomic vibrator of frequency $\nu$ in a single emission being equal to $\tau
h \nu$, where $\tau$ is an entire number, and $h$ is a universal
constant.\footnote{See f. inst., M. Planck, Ann. d. Phys. XXXI. p. 758 (1910);
XXXVII. p. 612 (1912); Verh. Phys. Ges. 1911, p. 138.}
Returning to the simple case of an electron and a positive nucleus considered
above, let us assume that the electron at the beginning of the interaction with
the nucleus was at a great distance apart from the nucleus, and had no sensible
velocity relative to the latter. Let us further assume that the electron after
interaction has taken place has settled down in a stationary orbit around the
nucleus. We shall, for reasons referred to later, assume that the orbit in
question is circular: this assumption will, however, make no alteration in the
calculations for system containing only a single electron.
Let as now assume that, during the binding of the electron, a homogeneous
radiation is emitted of a frequency $\nu$, equal to half the frequency of
revolution of the electron in its final orbit; then from Planck's theory, we
might expect that the amount of energy emitted by the process considered is
equal to $\tau h \nu$, where $h$ is Planck's constant an entire number. If we
assume that the radiation emitted is homogeneous, the second assumption
concerning the frequency of the radiation suggests itself, since the frequency
of revolution of the electron at the beginning of the emission is 0. The
question, however, of the rigorous validity of both assumptions, and also of the
application made of Planck's theory, will be more closely discussed in $\S~3$.
Putting
\begin{equation}
W = \tau h \frac{\omega}{2},
\end{equation}
we get by help of the formula (1)
\begin{equation}
W = \frac{2 \pi^2 me^2E^2}{\tau^2 h^2}, ~~~ \omega = \frac{4 \pi^2 me^2 E^2}
{\tau^3 h^3}, ~~~ 2a = \frac{\tau^2 h^2}{2 \pi^2 me E}.
\end{equation}
If in these expressions we give $\tau$ different values, we get a series of
values for $W$, $\omega$, and $a$ corresponding to a series of configurations of
the system. According to the above considerations, we are led to assume that
these configurations will correspond to states of the system in which there is
no radiation of energy; states which consequently will be stationary as long as
the system is not disturbed from outside. We see that the value of $W$ is
greatest if $\tau$ has its smallest value 1. This case will therefore correspond
to the most stable of the system, i.e., will correspond to the binding of the
electron for the breaking up of which the greatest amount of energy is required.
Putting in the above expressions $\tau = 1$ and $E = e$, and introducing the
experimental values
$$
e = 4.7 \cdot 10^{-10}, ~~~ \frac{e}{m} = 5.31 \cdot 10^{17}, ~~~
h = 6.5 \cdot 10^{-27},
$$
we get
$$
2a = 1.1 \cdot 10^{-8}~ \mbox{cm}, ~~~
\omega = 6.2 \cdot 10^{15}~ \frac{1}{\mbox{sec}}, ~~~
\frac{W}{e} = 13~ \mbox{volt}.
$$
We see that these values are of the same order of magnitude as the linear
dimensions of the atoms, the optical frequencies, and the \mbox{ionization-
potentials}.
The general importance of Planck's theory for the discussion of the behaviour
of atomic system was originally pointed out by Einstein.\footnote{A. Einstein,
Ann. d.Phys. XVII. p. 132 (1905); XX. p. 199 (1906); XXII. p. 180 (1907).}
The considerations of Einstein have been developed and applied on a number of
different phenomena, especially by Stark, Nernst, and Sommerfield. The agreement
as to the order of magnitude between values observed for the frequencies and
dimensions of the atoms, and values for these quantities calculated by
considerations similar to those given above, has been the subject of much
discussion. It was first pointed out by Haas,\footnote{A.E. Haas, Jahrb. d. Rad.
u.El. VII. p. 261 (1910). See further, A.Schidlof, Ann. d. Phys. XXXV. p. 90 (
1911); E. Wertheimer, Phys. Zietschr. XII. p. 409 (1911),
Verh. deutsch. Phys. Ges.
1912, p. 431; F.A. Lindermann, Verh.deutsch.Phys.Ges. 1911, pp. 482, 1107; F.
Haber, Verh. deutsch. Phys. Ges. 1911, p. 1117.} in ann attempt to explain the
meaning and the value of Planck's constant on the basis of J.J. Thomson's
\mbox{atom-model}, by help of the linear dimensions and
frequency of an hydrogen atom.
Systems of the kind considered in this paper, in which the forces between the
particles vary inversely as the square of the distance, are discussed in
relation to Planck's theory by J.W. Nicholson.\footnote{J.W. Nicholson, Month.
Not. Roy. Astr. Soc. LXXII. pp. 49, 139, 677, 693, 729 (1912).} In a series of
papers this author has shown that it seems to be possible to account for lines
of hitherto unknown origin in the spectra of the stellar nebulae and that of
the solar corona, by assuming the presence in these bodies of certain
hypothetical elements of exactly indicated constitution. The atoms of these
elements are supposed to consist simply of a ring of a few electrons
surrounding a positive nucleus of negligibly small dimensions. The ratios
between the frequencies corresponding to the lines in question are compared
with the ratios between the frequencies corresponding to different modes of
vibration of the ring of electrons. Nicholson has obtained a relation to
Planck's theory showing that the ratios between the \mbox{wave-lenth} of
different sets of lines of the coronal spectrum can be accounted for with
great accuracy by assuming that the ratio between the energy of the system and
the frequency of rotation of the ring is equal to an entire multiple of
Planck's constant. The quantity Nicholson refers to as the energy is equal to
twice the quantity which we have denoted above by W. In the latest paper cited
Nicholson has found it necessary to give the theory a more complicated form,
still, however, representing the ratio of energy to frequency by a simple
function of whole numbers.
The excellent agreement between the calculated and observed values of the ratios
between the \mbox{wave-length} in question seems a strong argument in favour of
the validity of the foundation of Nicholson's calculations. Serious objections,
however, may be raised against the theory. These objections are intimately
connected with the problem of the homogeneity of the radiation emitted. In
Nicholson's calculations the frequency of lines in a \mbox{line-spectrum} is
identified with the frequency of vibration of a mechanical system in a
distinctly indicated state of equilibrium. As a relation from Planck's theory is
used, we might expect that the radiation is sent out in quanta; but systems like
those considered, in which the frequency is a function of the energy, cannot
emit a finite amount of a homogeneous radiation; for, as soon as the emission of
radiation is started, the energy and also the frequency of the system are
altered. Further, according to the calculation of Nicholson, the systems are
unstable for some modes of vibration. Apart from such objections -- which may
be only formal (see p. 23)?????? --
it must be remarked, that the theory in the form
given dies not seem to be able to account for the \mbox{well-known} laws of
Balmer and Rydberg connecting the frequencies of the lines in the \mbox{line-
spectra} of the ordinary elements.
It will now be attempted to show that the difficulties in question disappear if
we consider the problems from the point of view taken in this paper. Before
proceeding it may be useful to restate briefly the ideas characterizing the
calculations on p. 5. The principal assumptions used are:
\begin{itemize}
\item[(1)]
That the dynamical equilibrium of the systems in the stationary states can be
discussed by help of the ordinary mechanics, while the passing of the systems
between different stationary states cannot be treated on that basis.
\item[(2)]
That the latter is followed by the emission of a {\it homogeneous} radiation,
for which the relation between the frequency and the amount of energy emitted is
the one given by Planck's theory.
\end{itemize}
The first assumption seems to present itself; for it is known that the ordinary
mechanism cannot have an absolute validity, but will only hold in calculations
of certain mean values of the motion of the electrons. On the other hand, in the
calculations of the dynamical equilibrium in a stationary state in which there
is no relative displacement of the particles, we need not distinguish between
the actual motions and their mean values. The second assumption is in obvious
constant to the ordinary ideas of electrodynamics, but appears to be necessary
in order to account for experimental facts.
In the calculations on page 5 we have further made use of the more special
assumptions, viz., that the different stationary states correspond to the
emission of a different number of Planck's \mbox{energy-quanta}, and that the
frequency of the radiation emitted during the passing of the system from a state
in which no energy is yet radiated out to one of the stationary states, is equal
to half the frequency of revolution of the electron in the latter state. We can,
however (see $\S~3$), also arrive at the expressions (3) for the stationary
states by using assumptions of somewhat different from. We shall, therefore,
postpone the discussion of the spacial assumptions, and first show how by the
help of the above principal assumptions, and of the expressions (3) for the
stationary states, we can account for the \mbox{line-spectrum}
of hydrogen.
\vspace{0.5cm}
\section*{
$\S~2$.{\it Emission of Line-spectra}}
\vspace{0.5cm}
{\it Spectrum of Hydrogen.} -- General evidence indicates that an atom of
hydrogen consist simply of a single electron rotating round a positive nucleus
of charge $e$.\footnote{See f. inst. N. Bohr, Phil. Mag. XXV. p. 24 (1913). The
conclusion drawn in the paper cited in strongly supported by the fact that
hydrogen, in the experiments on positive rays of Sir. J.J. Thomson, is the only
element which never occurs with a positive charge corresponding to the lose of
more than one electron (comp. Phil. Mag. XXIV. p. 672 (1912).}
The reformation of
a hydrogen atom, when the electron has been removed to great distances away
from the nucleus -- e.g. by the effect of electrical discharge in a vacuum tube
-- will accordingly correspond to the binding of an electron by a positive
nucleus considered on p. 5. If in (3) we put $E = e$, we get for the total
amount of energy radiated out by the formation of one of the stationary states,
$$
W_r = \frac{2 \pi^2 me^4}{\tau^2 h^2}.
$$
The amount of energy emitted by the passing of the system from a state
corresponding to $\tau = \tau_1$ to one corresponding to $\tau = \tau_2$, is
consequently
$$
W_{r_2} - W_{r_1} = \frac{2 \pi^2 me^4}{h^2} \cdot \left( \frac{1}{\tau^2_2} -
\frac{1}{\tau^2_1} \right).
$$
If now we suppose that the radiation is question is homogeneous, and that the
amount of energy emitted is equal to $h \nu$, where $\nu$ is the frequency of
the radiation, we get
$$
W_{r_2} - W_{r_1} = h \nu
$$
and from this
\begin{equation}
\nu = \frac{2 \pi^2 me^4}{h^3} \cdot \left( \frac{1}{\tau^2_2} -
\frac{1}{\tau^2_1} \right).
\end{equation}
We see that this expression accounts for the law connecting the lines in the
spectrum of hydrogen. If we put $\tau_2 = 2$ and let $\tau_1$ vary, we get the
ordinary Balmer series. If we put $\tau_3 = 3$, we get the series in the
\mbox{ultra-red} observed by
Paschen\footnote{F. Paschen, Ann. d. Phys. XXVII. p.565 (1908).}
and previously suspected by Ritz. If we put $\tau_2 = 1$ and $\tau = 4, 5,
\ldots,$ we get series respectively in the extreme ultraviolet and the extreme
\mbox{ultra-red}, which are not observed, but the existence of which may be
expected.
The agreement in question is quantitative as well as qualitative. Putting
$$
e = 4.7 \cdot 10^{-10}, ~~~ \frac{e}{m} = 5.31 \cdot 10^{17} ~~~
\mbox{and} ~~~ h = 6.5 \cdot 10^{-27},
$$
we get
$$
\frac{2 \pi^2 me^4}{h^3} = 3.1 \cdot 10^{15}.
$$
The observed value for the factor outside the bracket in the formula (4) is
$$
3.290 \cdot 10^{15}.
$$
We agreement between the theoretical and observed values is inside the
uncertainty due to experimental errors in the constants entering in the
expression for the theoretical value. We shall in $\S~3$ return to consider the
possible importance of the agreement in question.
It may be remarked that the fact, that it has not been possibly to observe more
than 12 lines of the Balmer series in experiments with vacuum tubes, while 33
lines are observed in the spectra of some celestial bodies, is just what we
should expect from the above theory. According to the equation (3) the diameter
of the orbit of the electron in the different stationary states is proportional
to $\tau^2$. For $\tau = 12$ the diameter is equal to $1.6 \cdot 10{-6}$ cm, or
equal to mean distance between the molecules in a gas at a pressure of about 7
mm mercury; for $\tau = 33$ the diameter is equal to $1.2 \cdot 10{-5}$ cm,
corresponding to the mean distance of the molecules at a pressure of about 0.02
mm mercury. According to the theory the necessary condition for the appearance
of a great number of lines is therefore a very small density of the gas; for
simultaneously to obtain an intensity sufficient for observation the space
filled with the gas must be very great. If the theory is right, we may
therefore never expect to be able in experiments with vacuum tubes to observe
the lines corresponding to high numbers of the Balmer series of the emission
spectrum of hydrogen; it might, however, be possible to observe the lines by
investigation of the absorption spectrum of this gas. (see $\S~4$).
It will be observed that we in the above way do not obtain other series of
lines, generally ascribed to hydrogen; for instance, the series first observed
by Pickering\footnote{E.C. Pickering, Astrophys. J. IV. p. 369 (1896);
v. p. 92 (
1897).} in the spectrum of the star $\zeta$ Puppis, and the set of series
recently found by Fowler\footnote{A. Fowler, Mouth. Not. Roy. Astr. Soc. LXXIII.
Dec. 1912.} by experiments with vacuum tubes containing a mixture of hydrogen
and helium. We shall, however, see that, by help of the above theory, we can
account naturally for these series of lines if we ascribe them to helium.
A neutral atom of the latter element consists, according to Rutherford's theory,
of a positive nucleus of charge $2e$ and two electrons. Now considering the
binding of a single electron by a helium nucleus, we get putting $E = 2e$ in the
expressions (3) on page 5, and proceeding in in exactly the same way as above,
$$
\nu = \frac{8 \pi^2 me^4}{h^3} \cdot \left( \frac{1}{\tau^2_2} -
\frac{1}{\tau^2_1}
\right) = \frac{2 \pi^2 me^4}{h^3} \cdot \left( \frac{1}
{\left(\frac{\tau_2}{2}\right)^2} -
\frac{1}{\left(\frac{\tau_1}{2}\right)^2} \right).
$$
If we in this formula put $\tau_1 = 1$ or $\tau_2 = 2$, we get series of lines
in the extreme \mbox{ultra-violet}. If we put $\tau_2 = 3,$ and let $\tau_1$
vary, we get a series which includes 2 of the series observed by Folwer, and
denoted by him as the first and second principal series of the hydrogen
spectrum. If we put $\tau_2 = 4$, we get the series observed by Pickering in the
spectrum of $\zeta$ Puppis. Every second of the lines in this series is
identical with a line in the Balmer series of the hydrogen spectrum; the
presence of hydrogen in the star in question may therefore account for the fact
that these lines are of a greater intensity than the rest of the lines in the
series. The series is also observed in the experiments of Fowler, and denoted in
his paper as the Sharp series of the hydrogen spectrum. If we finally in the
above formula put $\tau_2 = 5,~6, \ldots$, we get series, the strong lines of
which are to be expected in the \mbox{ultra-red}.
The reason why the spectrum considered is not observed in ordinary helium tubes
may be that in such tubes the ionization of helium is not so complete in the
star considered or in the experiments of Fowler, where a strong discharge was
sent through a mixture of hydrogen and helium. The condition for the appearance
of the spectrum is, according to the above theory, that helium atoms are present
in a state in which they have lost both their electrons. Now we must assume that
the amount of energy to be used in removing the second electron from a helium
atom is much greater than that to be used in removing the first. Further, it is
known from experiments on positive rays, that hydrogen atoms can acquire a
negative charge; therefore the presence of hydrogen in the experiments of Fowler
may effect that more electrons are removed from some of the helium atoms than
would be the case if only helium were present.
{\it Spectra of other substances.} --- in case of systems containing more
electrons we must -- in conformity with the result of experiments -- expect more
complicated laws for the line-spectra than those considered. I shall try to show
that the point of view taken above allows, at any rate, a certain understanding
of the laws observed.
According to Rydberg's theory --- with the generalization given by
Ritz\footnote{W. Ritz, Phys. Zeitschr. IX. p. 521 (1908).} -- the frequency
corresponding to the lines of the spectrum of an element can be expressed by
$$
\nu = F_{\tau}(\tau_1) - F_s(\tau_2),
$$
where $\tau_1$ and $\tau_2$ are entire numbers, and $F_1,~F_2,~F_3, \ldots$ are
functions of $\tau$ which approximately are equal to $\frac{K}{(\tau + a_1)^2}$,
$\frac{K}{(\tau + a_2)^2}, \ldots$ $K$ is a universal constant, equal to the
factor outside the bracket in the formula (4) for the spectrum of hydrogen. The
different series appear if we put $\tau_1$ or $\tau_2$ equal to a fixed number
and let the other vary.
The circumstance that the frequency can be written as a difference between two
functions of entire numbers suggests an origin of the lines in the spectra in
question similar to the one we have assumed for hydrogen; i.e. that the lines
correspond to a radiation emitted during the passing of the system between two
different stationary states. For system containing more than one electron the
detailed discussion may be very complicated, as there will be many different
configurations of the electrons which can be taken into consideration as
stationary states. This may account for the difference sets of series in the
line spectra emitted from the substances in question. Here I shall only try to
show how, by help of the theory, it can be simple explained that the constant $
K$ entering in Rydberg's formula is the same for all substances.
Let us assume that the spectrum in question corresponds to the radiation emitted
during the binding of an electron; and let us further assume that the system
including the electron considered is neutral. The force on the electron, when at
a great distance apart the nucleus and the electrons previously bound, will be
very nearly the same as the above case of the binding of an electron by a
hydrogen nucleus.
The energy corresponding to one of the stationary states will therefore for
$\tau$ great be very nearly equal to
that given by the expression (3) on p. 5, if
we put $E = e$. For $\tau$ great we consequently get
$$
\lim[\tau^2 \cdot F_1(\tau)] = \lim [\tau^2 \cdot F_2(\tau)] = \ldots = \frac{2
\pi^2 me^4}{h^3},
$$
in conformity with Rydberg's theory.
\vspace{0.5cm}
\section*{
$\S~3$.{\it General Considerations Continued}}
\vspace{0.5cm}
We shall now return to the discussion (see p. 7) of the special assumptions used
in deducing the expression (3) on p. 5 for the stationary states of a system
consisting of an electron rotating round a nucleus.
For one, we have assumed that the different stationary states correspond to an
emission of a different number of \mbox{energy-qyanta}. Considering systems in
which the frequency is a function of the energy, this assumption, however, may
be regarded as improbable; for as soon as one quantum in sent out the frequency
is altered. We shall now see that we can leave the assumption used and still
retain the equation (2) on p. 5, and thereby the formal analogy with Planck's
theory.
Firstly, it will be observed that it has not been necessary, in order to account
for the law of the spectra by help of the expressions (3) for the stationary
states, to assume that in any case a radiation is sent out corresponding to more
than a single \mbox{energy-quantum}, $h \nu$. Further information on the
frequency of the radiation may be obtained by comparing calculations of the
energy radiation in the region of slow vibrations based on the above assumptions
with calculations based on the ordinary mechanics. As is known, calculations on
the latter basis are in agreement with experiments on the energy radiation in
the named region.
Let us assume that the ratio between the total amount of energy emitted and the
frequency of revolution of the electron for the different stationary states is
given by the equation
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
$W = f(\tau) \cdot h \omega$,
instead of by the equation
(2). Proceeding in the same way as above, we get in this case instead of (3)
$$
W = \frac{\pi^2 me E^2}{2h^2f^2(\tau)}, ~~~
\omega = \frac{\pi^2 me^2 E^2}{2h^3 f^3(\tau)}.
$$
Assuming as above that the amount of energy emitted during the passing of the
system from a state corresponding to $\tau = \tau_1$ to one for which
$\tau = \tau_2$ is equal to $h \nu$, we get instead of (4)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
$$
\nu = \frac{\pi^2 me^2 E^2}{2h^3} \cdot \left( \frac{1}{f^2(\tau_2)} -
\frac{1}{f^2(\tau_1)} \right).
$$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
We see that in order to get an expression of the same form as the Balmer series
we must put $f(\tau) = c \tau$.
In order to determine $c$ let us now consider the passing of the system between
two successive stationary states corresponding to $\tau = N$ and $\tau = N - 1$;
introducing $f(\tau) = c\tau$, we get for the frequency of the radiation emitted
$$
\nu = \frac{\pi^2 me^2 E^2}{2c^2h^3} \cdot \frac{2N - 1}{N^2(N - 1)^2}.
$$
For the frequency of revolution of the electron before and after the emission we
have
$$
\omega_N = \frac{\pi^2 me^2 e^2}{2c^3 h^3 N^3} ~~~ \mbox{and} ~~~
\omega_{N - 1} = \frac{\pi^2 me^2 E^2}{2c^3 h^3(N - 1)^3}.
$$
If $N$ is great the ratio between the frequency before and after the emission
will be very near equal to 1; and according to the ordinary electrodynamics we
should therefore expect that the ratio between the frequency of radiation and
the frequency of revolution also very nearly equal to 1. This condition will
only be satisfied if $c = 1/2$. Putting $f(\tau) = \tau/2$, we,
however, again arrive at the equation (2) and consequently at the expression
(3) for the stationary states.
If we consider the passing of the system between two states corresponding to
$\tau = N$ and $\tau = N - n$, where $n$ is small compared with $N$, we get with
the same approximation as above, putting $f(\tau) = \tau/2$,
$$
\nu = n \omega.
$$
The possibility of an emission of a radiation of such a frequency may also be
interpreted from analogy with the ordinary electrodynamics, as an electron
rotating round a nucleus in an elliptical orbit will emit a radiation which
according to Fourier's theorem can be resolved into homogeneous components, the
frequency of which are $n \omega$, if $\omega$ is the frequency of revolution of
the electron.
We are thus led to assume that the interpretation of the equation (2) is not
that the different stationary states correspond to an emission of different
numbers of \mbox{energy-quanta}, but that the frequency of the energy emitted
during the passing of the system from a state in which no energy is yet radiated
out to one of the different stationary states, is equal to different multiples
of $\omega/2$, where $\omega$ is the frequency of revolution of the
electron in the state considered. From this assumption we get exactly the same
expressions as before for the stationary states, and from these by help of the
principal assumptions on p. 7 the same expression for the law of the hydrogen
spectrum. Consequently we may regard our preliminary considerations on p. 5 only
as a simple from of representing the results of the theory.
Before we leave the discussion of this question, we shall for a moment return to
the question of the significance of the agreement between the observed and
calculated values of the constant entering in the expressions (4) for the Balmer
series of the hydrogen spectrum. From the above consideration it will follow
that, taking the \mbox{starting-point} in the form of the law of the hydrogen
spectrum and assuming that the different lines correspond to a homogeneous
radiation emitted during the passing between different, stationary states, we
shall arrive at exactly the same expression for the constant in question as that
given by (4), if we only assume (1) that the radiation is sent out in quanta $h
\nu$, and (2) that the frequency of the radiation emitted during the passing of
the system between successive stationary states will coincide with the frequency
of revolution of the electron in the region of slow vibrations.
As all the assumptions used in this latter way of representing the theory are of
what we may call a qualitative character, we are justified in expecting --- if
the whole way of considering is a sound one -- an absolute agreement between the
values calculated and observed for the constant in question, and not only an
approximate agreement. The formula (40 may therefore be of value in the
discussion of the results of experimental determinations of the constants $e$, $
m$, and $h$.
While there obviously can be no question of a mechanical foundation of the
calculations given in this paper, it is, however, possible to give a very simple
interpretation of the result of the calculation on p. 5 by help of symbols
taken from the ordinary mechanics. Denoting the angular momentum of the electron
round the nucleus by $M$, we have immediately for a circular orbit
$\pi M = T/\omega$, where $\omega$ is the frequency of revolution and $
T$ the kinetic energy of the electron; for a circular orbit we further have $T =
W$ (see p. 3) and from (2), p. 5, we consequently get
$$
M = \tau M_0,
$$
where
$$
M_0 = \frac{h}{2 \pi} = 1.04 \cdot 10^{-27}.
$$
If we therefore assume that the orbit of the electron in the stationary states
is circular, the result of the calculation on p. 5 can be expressed by the
simple condition: that the angular momentum of the electron round the nucleus in
a stationary state of the system is equal to an entire multiple of a universal
value, independent of the charge on the nucleus. The possible importance of the
angular momentum in the discussion of atomic systems in relation to Planck's
theory is emphasized by Nicholson.\footnote{J.W. Nicholson, loc. cit. p. 679.}
The great number of different stationary states we do not observe expect by
investigation of the emission and absorption of radiation. It most of the other
physical phenomena, however, we only observe the atoms of the matter in a single
distinct state, i,e., the state of the atoms at low temperature. From the
preceding considerations we are immediately led to the assumption that the ``
permanent'' state is the one among the stationary states during the formation of
which the greatest amount of energy is emitted. According to the equation (3) on
p. 5, this state is the one which corresponds to $\tau = 1$.
\vspace{0.5cm}
\section*{
$\S~4$. {\it Absorption of Radiation}}
\vspace{0.5cm}
In order to account for Kirchhoff's law it is necessary to introduce assumptions
on the mechanism of absorption of radiation which correspond to those we have
used considering the emission. Thus we must assume that a system consisting of a
nucleus and an electron rotating round it under certain circumstances can
absorb a radiation of a frequency equal to the frequency of the homogenous
radiation emitted during the passing of the system between different stationary
states. Let us consider the radiation emitted during the passing of the system
between two stationary states $A_1$ and $A_2$ corresponding to values for $\tau$
equal to $\tau_1$ and $\tau_2,~\tau_1 > \tau_2$. As the necessary condition of
the radiation in question was the presence of systems in the state $A_1$, we
must assume that the necessary condition for an absorption of the radiation is
the presence of systems in the state $A_2$.
These considerations seems to be in conformity with experiments on absorption in
gases. In hydrogen gas at ordinary conditions for instance there is no
absorption of a radiation of a frequency corresponding to the
\mbox{line-spectrum} of this gas; such an absorption is only observed in
hydrogen gas in a luminous state. This is what we should expect according to the
above. We have on p. 9 assumed that the radiation in question was emitted during
the passing of the systems between stationary states corresponding to $\tau \geq
2$. The state of the atoms in hydrogen gas at ordinary conditions should,
however, correspond to $\tau = 1$; furthermore, hydrogen atoms at ordinary
conditions combine into molecules, i.e., into system in which the electrons have
frequencies different from those in the atoms (see Part III.) From the
circumstance that certain substances in a \mbox{non-lumimous} state, as, foe
instance, sodium vapour, absorb radiation corresponding to lines in the \mbox{
line-spectra} of the substances, we may, on the other hand, conclude that the
lines in question are emitted during the passing of the system between two
states, one of which is the permanent state.
How much the above considerations differ from an interpretation based on the
ordinary electrodynamic of perhaps most early shown by the fact that we have
been forced to assume that a system of electrons will absorb a radiation of a
frequency different from the frequency of vibration of the electrons calculated
in the ordinary way. It may in this connexion be of interest to mention a
generalization of the considerations to which we are led by experiments on the
\mbox{photo-electric} effect and which may be able to throw some light on the
problem in question. Let us consider state of the system in which the electron
is free, i.e., in which the electron possesses kinetic energy sufficient to
remove to infinite distances from the nucleus. If we assume that the motion of
the electron is governed by the ordinary mechanics and that there is no (
sensible) energy radiation, the total energy of the system -- as in the above
considered stationary states -- will be constant. Further, there will be perfect
continuity between the two kinds of states, as the difference between frequency
and dimensions of the system in successive stationary states will diminish
without limit if $\tau$ increases. In the following considerations we shall for
the sake of brevity refer to the two kinds of states in question as ``
mechanical'' states; by this notation only emphasizing the assumption that the
motion of the electron in both cases can be assumed for by the ordinary
mechanics.
Tracing the analogy between the two kinds of mechanical states, we might now
expect the possibility of an absorption of radiation, not only corresponding to
the passing of the system between two different stationary states, but also
corresponding to the passing between one of the stationary states and a state in
which the electron is free; and as above, we might expect that the frequency of
this radiation was determined by the equation $E = h \nu$, where $E$ is the
difference between the total energy of the system in the two states. As it will
be see, such an absorption of radiation is just what is observed in experiments
on ionization by \mbox{ultra-violet} light and by R\"ontgen rays. Obviously, we
get in this way the same expression for the kinetic energy of an electron
ejected from an atom by \mbox{photo-electron} effect as that deduced by
Einstein\footnote{A. Einstein, Ann. d. Phys. XVII. p. 146 (1905).}
i.e., $T = h \nu - W$, where $T$ is the kinetic energy of the electron ejected,
and $W$ the total amount of energy emitted during the original binding of the
electron.
The above considerations may further account for the result of some experiments
of R.W. Wood\footnote{R.W. Wood, Physical Optics, p. 513 (1911).} on absorption
of light by sodium vapour. In these experiments, an absorption corresponding to
a very great number of lines in the principal series of the sodium spectrum is
observed, and in addition a continuous absorption which begins at the head of
the series and extends to the extreme \mbox{ultra-violet}. This is exactly what
we should expect according to the analogy in question, and, as we shall see, a
closer consideration of the above experiments allows us to trance the analogy
still further. As mentioned on p. 9 the radii of the orbits of the electrons
will for stationary states, corresponding to high values for $\tau$ be very
great compared with ordinary atomic dimensions. This circumstance was used as an
explanation of the \mbox{non-appearance} in experiments with \mbox{vacuum-tubes}
of lines corresponding to the higher numbers in the Balmer series of the
hydrogen spectrum. This is also in conformity with experiments on the emission
spectrum of sodium; in the principal series of the emission spectrum of this
substance rather few lines are observed. Now in Wood's experiments the pressure
was not very low, the states corresponding to high values for $\tau$ could
therefore not appear; yet in the absorption spectrum about 50 lines were
detected. In the experiments in question we consequently observe an absorption
of radiation which is not accompanied by a complete transition between two
different stationary states. According to the present theory we must assume that
this absorption is followed by an emission of energy during which the systems
pass back to the original stationary state. If there are no collisions between
the different systems this energy will be emitted as a radiation of the same
frequency as that absorbed, and there will be no true absorption but only a
scattering of the original radiation; a true absorption will not occur unless
the energy in question is transformed by collisions into kinetic energy of free
particles. In analogy we may now from the above experiments conclude that a
bound electron -- also in cases in which three is no ionization -- will have an
absorbing (scattering) influence on a homogeneous radiation, as soon as the
frequency of the radiation is greater than $W/h$, where $W$ is the total amount
of energy emitted during the binding of the electron. This would be highly in
favour of a theory of absorption as the one sketched above, as there can in such
a case be no question of a coincidence of the frequency of the radiation and a
characteristic frequency of vibration of the electron. If will further be seen
that the assumption, that there will be an absorption (scattering) of any
radiation corresponding to a transition between two different mechanical states,
is in perfect analogy with the assumption generally used that a free electron
will have an absorbing (scattering) influence on light of any frequency.
Corresponding considerations will hold for the emission of radiation.
In analogy to the assumption used in this paper that the emission of \mbox{line-
spectra} is due to the re-formation of atoms after one or more of the
lightly bound electrons are removed, we may assume that the
homogeneous R\"ontgen
radiation is emitted during the setting down of the systems after one of the
firmly bound electrons escapes, e.g. by impact of cathode particles.\footnote{
Compare J.J. Thomson, Phil. Mag. XXIII. p. 456 (1912).}
In the next part in this
paper, dealing with the constitution of atoms, we shall consider the question
more closely and try to show that a calculation based on this assumption is in
quantitative agreement with the results of experiments: here we shall only
mention briefly a problem with which we meet in such a calculation.
Experiments on the phenomena of \mbox{X-rays} suggest that not only the emission
and absorption of radiation cannot be treated by the help of the ordinary
electrodynamics, but not even the result of a collision between two electrons of
which the one is bound in an atom. This is perhaps most early shown by some very
instructive calculations on the energy of \mbox{$\beta$-particles} emitted from
radioactive substances recently published by Rutherford.\footnote{E. Rutherford,
Phil. Mag. XXIV. pp. 453 \& 893 (1912).}
These calculations strongly suggest that
an electron of great velocity in passing through an atom and colliding with the
electrons bound will loose energy in distinct finite quanta. As is immediately
seen, this is very different from what we might expect if the result of the
collisions was governed by the usual mechanical laws. The failure of the
classical mechanics in such a problem might also be expected beforehand from the
absence of anything like equipartition of kinetic energy between free electrons
and electrons bound in atoms. From the point of view of the ``mechanical''
states we see, however, that the following assumption -- which is in accord with
the above analogy -- might be able to account for the result of Rutherford's
calculation and for the absence of equipartition of kinetic energy; two
colliding electrons, bound or free, will, after the collision as well as before,
be in mechanical states. Obviously, the introduction of such an assumption would
not make any alteration necessary in the classical treatment of a collision
between two free particles. But, considering a collision between a free and a
bound electron, it would follow that the bound electron by the collision could
not acquire a less amount of energy than the difference in energy corresponding
to successive stationary states, and consequently that the free electron which
collides with it could not lose a less amount.
The preliminary and hypothetical character of the above considerations needs not
to be emphasized. The intention, however, has been to show that the sketched
generalization of the theory of the stationary states possibly may afford a
simple basis of representing a number of experimental facts which cannot be
explained by help of the ordinary electrodynamics, and that assumptions used do
not seem to be inconsistent with experiments on phenomena for which a
satisfactory explanation has been given by the classical dynamics and the wave
theory of light.
\vspace{0.5cm}
\section*{
$\S~5$.{\it The permanent State of an Atomic System}}
\vspace{0.5cm}
We shall now return to the main object of this paper -- the discussion of the
``permanent'' state of a system consisting of nuclei and bound electrons. For a
system consisting of a nucleus and an electron rotating round it, this state is,
according to the above, determined by the condition that the angular momentum of
the electron round the nucleus is equal to $h/2 \pi$.
On the theory of this paper the only neutral atom which contains a single
electron is the hydrogen atom. The permanent state of this atom should
correspond to the values of $a$ and $\omega$ calculated on p. 5. Unfortunately,
however, we know very little of the behaviour of hydrogen atoms on account of
the small dissociation of hydrogen molecules at ordinary temperatures. In order
to get a closer comparison with experiments, it is necessary to consider more
complicated systems.
Considering systems in which more electrons are bound by a positive nucleus, a
configuration of the electrons which presents itself as a permanent state is in
which the electrons are arranged in a ring round the nucleus. In the discussion
of this problem on the basis of the ordinary electrodynamics, we meet-- apart
from the question of the energy radiation -- with new difficulties due to the
question of the stability of the ring. Disregarding for a moment this latter
difficulty, we shall first consider the dimensions and frequency of the systems
in relation to Planck's theory of radiation.
Let us consider a ring consisting of $n$ electrons rotating round a nucleus of
charge $E$, the electrons being arranged at equal angular intervals the
circumference of a circle of radius $a$.
The total potential energy of the system consisting of the electrons and the
nucleus is
$$
P = - \frac{ne}{a} \cdot \left( E - es_n \right),
$$
where
$$
s_n = \frac{1}{4} \sum^{s=n-1}_{s=1} \mbox{cosec} \frac{s \pi}{n}.
$$
For the radial force exerted on an electron by the nucleus and the other
electrons we get
$$
F = - \frac{1}{n} \cdot \frac{dP}{da} = - \frac{e}{a^2}
\cdot \left( E - es_n \right).
$$
Denoting the kinetic energy of an electron by $T$ and neglecting the
electromagnetic forces due to the motion of the electrons (see Part II), we get,
putting the centrifugal force on an electron equal to the radial force,
$$
\frac{2T}{a} = \frac{e}{a^2} \cdot \left( E - es_n \right),
$$
or
$$
T = \frac{e}{2a} \cdot \left( E - es_n \right).
$$
From this we get for the frequency of revolution
$$
\omega = \frac{1}{2\pi} \cdot \sqrt{\frac{e \left( E - es_n \right)}{ma^3}}.
$$
The total amount of energy $W$ necessary transferred to the system in order to
remove the electrons to infinite distances apart from the nucleus and from each
other is
$$
W = - P - nT = \frac{ne}{2a} \cdot \left( E - es_n \right) = nT,
$$
equal to the total kinetic energy of the electrons.
We see that the only difference in the above formula and those holding for the
motion of a single electron in a circular orbit round a nucleus is the exchange
of $E$ for $E - es_n$. It is also immediately seen that corresponding to the
motion of an electron in an elliptical orbit round a nucleus, there will be a
motion of the $n$ electrons in which each rotates in an elliptical orbit with
the nucleus in the focus, and the $n$ electrons at any moment are situated at
equal angular intervals on a circle with the nucleus as the centre. The major
axis and frequency of the orbit of the single electrons will for this motion be
given by the expressions (1) on p. 3 if we replace $E$ by $E - es_n$ and $W$ by
$W/n$. Let us now suppose that the system of $n$ electrons rotating in a
ring round a nucleus is formed in a way analogous to the one assumed for a
single electron rotating round a nucleus. It will thus be assumed that the
electrons, before the binding by the nucleus, were at a great distance apart
from the latter and possessed no sensible velocities, and also that during the
binding a homogeneous radiation is emitted. As in the case of a single electron,
we have here that the total amount of energy emitted during the formation of the
system is equal to the final kinetic energy of the electrons. If we now suppose
that during the formation of the system the electrons at any moment are situated
at equal angular intervals on the circumference of a circle with the nucleus in
the centre, from analogy with the considerations, on p. 5 we are here led to
assume the existence of a series of stationary configurations in which the
kinetic energy per electron is equal to $\tau h \omega/2$, where $\tau$
is an entire number, $h$ Planck's constant, and $\omega$ the frequency of
revolution. The configuration in which the greatest amount of energy is emitted
is, as before, the one in which $\tau = 1$. This configuration we shall assume
to be the permanent state of the system if the electrons in this state are
arranged in a single ring. As for the case of a single 3electron we get that the
angular momentum of each of the electrons is equal to $h/2 \pi$. It may
be remarked that instead of considering the single electrons we might have
considered the ring as an entity. This would, however, lead to the same result,
for in this case the frequency of revolution $\omega$ will be replaced by the
frequency $n \omega$ of the radiation from the whole ring calculated from
ordinary electrodynamics, and $T$ by the total kinetic energy $nT$.
There may be many other stationary states corresponding to other ways of forming
the system. The assumption of the existence of such states seems necessary in
order to account for the \mbox{line-spectra} of systems containing more than one
electron (p. 11); it is also suggested by the theory of Nicholson mentioned on
p. 6, to which we shall return in a moment. The consideration of the spectra,
however, gives, as far as I can see, no indication of the existence of
stationary states in which all the electrons are arranged in a ring and which
correspond to greater values for the total energy emitted than the one we above
have assumed to be the permanent state.
Further, there may be stationary configurations of a system of $n$ electrons and
a nucleus of charge $E$ in which all the electrons are not arranged in a single
ring. The question, however, of the existence of such stationary configurations
is not essential for our determination of the permanent state, as long as we
assume that the electrons in this state of the system are arranged in a single
ring. Systems corresponding to more complicated configurations will be discussed
on p. 24.?????
Using the relation $T = h \omega/2$ we get, by help of the above
expressions for $T$ and $\omega$, values for $a$ and $\omega$ corresponding to
the permanent state of the system which only differ from those given by the
equations (3) on p. 5, by exchange of $E$ for $E - es_n$.
The question of stability of a ring of electrons rotating round a positive
charge is discussed in great detail by Sir. J.J. Thomson\footnote{Loc. cit.} An
adaption of Thomson's analysis for the case here considered of a ring rotating
round a nucleus of negligibly small linear dimensions is given by
Nicholson.\footnote{Loc. cit.} The investigation of the problem in question
naturally divides in two parts: one concerning the stability for displacements
of the electrons on the plane of the ring; one concerning displacements
perpendicular to this plane. As Nicholson's calculations show, the answer to the
question of stability differs very much in the two cases in question. While the
ring for the latter displacements in general is stable if the number of
electrons is not great; the ring is in no case considered by Nicholson stable
for displacement of the first kind.
According, however, to the point of view taken in this paper, the question of
stability for displacements of the electrons in the plane of the ring is most
intimately connected with the question of the mechanism of the binding of the
electrons, and like the latter cannot be treated on the basis of the ordinary
dynamics. The hypothesis of which we shall make use in the following is that the
stability of a ring of electrons rotating round a nucleus is secured through the
above condition of the universal constancy of the angular momentum, together
with the further condition that the configuration of the particles is the one by
the formation of which the greatest of energy is emitted. As will be shown, this
hypothesis is, concerning the question of stability for a displacement of the
electrons perpendicular to the plane of the ring, equivalent to that used in
ordinary mechanical calculations.
Returning to the theory of Nicholson on the origin of lines observed in the
spectrum of the solar corona, we shall now see that the difficulties mentioned
on p. 7 may be only formal. In the first place, from the point of view
considered above the objection as to the instability of the systems for
displacements of the electrons in the plane of the ring may not be valid.
Further, the objection as to emission of the radiation in quanta will not have
reference to the calculations in question, if we assume that in the coronal
spectrum we are not dealing with a true emission but only with a scattering of
radiation. This assumption seems probable if we consider the conditions in the
celestial body in question: for on account comparatively few collisions to
disturb the stationary states and to cause a true emission of light
corresponding to the transition between different stationary states; on the
other hand there will in the solar corona be intense illumination of light of
all frequencies which may excite the natural vibrations of the systems in the
different stationary states. If the above assumption is correct, we immediately
understand the entirely different from for the laws connecting the lines
discussed by Nicholson and those connecting the ordinary \mbox{line-spectra}
considered in this paper.
\vspace{0.5cm}
Proceeding to consider systems of more complicated constitution, we shall make
use of the following theorem, which can be very simply proved; --
``In every system consisting of electrons and positive nuclei, in which the
nuclei are at rest and the electrons move in circular orbits with a velocity
small compared with the velocity of light, the kinetic energy will be
numerically equal to half the principal energy.''
By help of this theorem we get -- as in the previous cases of a single electron
or of a ring rotating round a nucleus -- that the total amount of energy
emitted, by the formation of the systems from a configuration in which the
distances apart of the particles are infinitely great and in which the particles
have no velocities relative to each other, is equal to the kinetic energy of the
electrons in the final configuration.
In analogy with the case of a single ring we are here led to assume that
corresponding to any configuration of equilibrium a series of geometrically
similar, stationary configuration of the system will exist in which the kinetic
energy of every electron is equal to the frequency of revolution multiplied by
$\tau/2 h$ where $\tau$ is an entire number and $h$ Planck's constant. In
any such series of stationary configurations the one corresponding to the
greatest amount of energy emitted will be the one in which $\tau$ for every
electron is equal to 1. Considering that the ratio of kinetic energy to
frequency for a particle rotating in a circular orbit is equal to $\pi$ times
the angular momentum round the center of the orbit, we are therefore led to the
following simple generalization of the hypotheses mentioned on
pp. 15 and 22. ??????
{\it ``In any molecular system consisting of positive nuclei and electrons in
which the nuclei are at rest relatire to each other and the electrons more in
circular orbits, the angular momentum of every electron round the centre of its
orbit will in the permanent state of the system be equal to $h/2 \pi$,
where $h$ is Planck's constant.''}\footnote{In the considerations leading to
this hypothesis we have assumed that the velocity of the electrons is small
compared with the velocity of light. The limits of the validity of this
assumption will be discussed in Part II.}
In analogy with the considerations on p. 23, we shall assume that a
configuration satisfying this condition is stable if the total energy of the
system is less than in any neighbouring configuration satisfying the same
condition of the angular momentum of the electrons.
As mentioned in the introduction, the above hypothesis will be used in a
following communication as a basis for a theory of the constitution of atoms and
molecules. It will be shown that it leads to results which seem to be in
conformity with experiments on a number of different phenomena.
The foundation of the hypothesis has been sought entirely in its relation with
Planck's theory of radiation; by help of considerations given later it will be
attempted to throw some further light on the formation of it from another point
of view.
\vspace{0.5cm}
{\it April 5, 1913}\\
\end{document}