\errorcontextlines=10
\documentclass[11pt]{article}
\usepackage[dvips]{color}
\usepackage[dvips]{epsfig}
\usepackage[english]{babel}
%\RequirePackage[hyperindex,colorlink7s,backref,dvips]{hyperref}
\RequirePackage{hyperref}
\begin{document}
%\documentstyle[art14,fullpage]{article}
%\topmargin=-2cm
%\textheight=25.8cm
%\parindent=20pt
%\thispagestyle{empty}
%\begin{document}
N.~ Bohr, Philos. Mag. {\bf 26,} 476 \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}
\part*{
{\it Part II. -- Systems containing only a Single Nucleus}}\footnote{Part I was
published in Phil. Mag. XXVI. p. 1 (1913).}
\vspace{0.5cm}
\section*{
$\S~1$ {\it General Assumptions}}
\vspace{0.5cm}
Following the theory of Rutherford, we shall assume that the atoms of the
elements consist of a positively charged nucleus surrounded by a cluster of
electrons. The nucleus is the seat of the essential part of the mass of the
atom, and has linear dimensions exceedingly small compared with the distance
apart of the electrons in the surrounding cluster.
As in the previous paper, we shall assume that the cluster of electrons is
formed by the successive binding by the nucleus of electrons initially nearly at
rest, energy at the same time being radiated away. This will go on until, when
the total negative charge on the bound electrons is numerically equal to the
positive charge on the nucleus, the system will be neutral and no longer able to
exert sensible forces on electrons at distances from the nucleus great in
comparison with the dimensions of the orbits of the bound electrons. We may
regard the formation of helium from $\alpha$ rays as an observed example of a
process of this kind, an $\alpha$ particle on this view being identical with the
nucleus of a helium atom.
On account of the small dimensions of the nucleus, its internal structure will
not be of sensible influence on the constitution of the cluster of electrons,
and consequently will have no effect on the ordinary physical and chemical
properties of the atom. The latter properties on this theory will depend
entirely on the total charge and mass of the nucleus; the internal structure of
the nucleus will be of influence only on the phenomena of radioactivity.
From the result of experiments on \mbox{large-angle} scattering of
\mbox{$\alpha$-rays}, Ru\-ther\-ford\footnote{Comp.
also Geiger and Marsden, Phil. Mag. XXV.
p. 604 (1913).} found an electric charge on the nucleus corresponding per atom
to a number of electrons approximately equal to half the atomic weight. This
result seems to be in agreement with the number of electrons per atom calculated
from experiments on scattering of
R\"ontgen radiation.\footnote{Comp. C.G. Barkla,
Phil. Mag. XXI. p. 648 (1911).} The total experimental evidence supports the
hypothesis\footnote{Comp. A.v.d. Broek, Phys. Zeitschr. XIV. p. 32 (1913).} that
the actual number of electrons in a neutral atom with a few exceptions is equal
to the number which indicated the position of the corresponding element in the
series of element arranged in order of increasing atomic weight. For example on
this view, the atom of oxygen which is the eighth element of the series has
eight electrons and a nucleus carrying eight unit charges.
We shall assume that the electrons are arranged at equal angular intervals in
coaxial rings rotating round the nucleus. In order to determine the frequency
and dimensions of the rings we shall use the main hypothesis of the first paper,
viz.; that in the permanent state of an atom the angular momentum of every
electron round the centre of its orbit is equal to the universal value
$h/2 \pi$, where $h$ is Planck's constant.
We shall take as a condition of stability, that the total energy of the system
in the configuration in question is less than in any neighbouring configuration
satisfying the same condition of the angular momentum of the electrons.
If the charge on the nucleus and the number of electrons in the different rings
is known, the condition in regard to the angular momentum of the electrons will,
as shown in $\S~2$, completely determine the configuration of the system. i.e.,
the frequency of revolution and the linear dimensions of the rings.
Corresponding to different distributions of the electrons in the rings, however,
there will, in general, be more than one configuration which will satisfy the
condition of the angular momentum together with the condition of stability.
In $\S~3$ and $\S~4$ it will be shown that, on the general view of the formation
of the atoms, we are led to indications of the arrangement of the electrons in
the rings which are consistent with those suggested by the chemical properties
of the corresponding element.
In $\S~5$ will be shown that it is possible from the theory to calculate the
momentum velocity of cathode rays necessary to produce the characteristic
R\"ontgen radiation from the element, and that this is in approximate agreement
with the experimental values.
In $\S~6$ the phenomena of radioactivity will be briefly considered in relation
of the theory.
\vspace{0.5cm}
\section*{
$\S~2$ {\it Configuration and Stability of the System}}
\vspace{0.5cm}
Let us consider an electron of charge $e$ and mass $m$ which moves in a circular
orbit of radius $a$ with a velocity $v$ small compared with the velocity of
light. Let us denote the radial force acting on the electrons by
$e^2/a^2 F$; $F$ will in general be dependent on $a$. The condition of
dynamical equilibrium gives
$$
\frac{mv^2}{a} = \frac{e^2}{a^2}F.
$$
Introducing the condition of universal constancy of the angular momentum of the
electron, we have
$$
mva = \frac{h}{2 \pi}.
$$
From these two conditions we now get
\begin{equation}
a = \frac{h^2}{4 \pi^2 e^2 m} \cdot F^{-1} ~~~
\mbox{and} ~~~ v = \frac{2 \pi e^2}{h} \cdot F;
\end{equation}
and for the frequency of revolution $w$ consequently
\begin{equation}
\omega = \frac{4 \pi^2 e^2 m}{h^2} \cdot F^2.
\end{equation}
If $F$ is known, the dimensions and frequency of the corresponding orbit are
simply determined by (1) and (2). For a ring of $n$ electrons rotating round a
nucleus of charge $ne$ we have (comp. Part I., p. 20)????
$$
F = N - s_n, ~~~ \mbox{where} ~~~ s_n = \frac{1}{4} \cdot \sum^{s=n-1}_{s=1}
\mbox{cosec} \frac{s \pi}{n}.
$$
The values for $s_n$ from $n = 1$ to $n = 16$ are given in the table 1.
For systems consisting of nuclei and electrons in which the first are at rest
and the latter move in circular orbits with a velocity small compared with the
velocity of light, we have shown (see part I., p. 21)???? that the total kinetic
energy of the electrons is equal to the total amount of energy emitted during
the formation of the system from an original configuration in which all the
particles are at rest and at infinite distances from each other. Denoting this
amount of energy by $W$, we consequently get
\begin{equation}
W = \sum \frac{m}{2} v^2 = \frac{2 \pi^2 e^4m}{h^2} \sum F^2.
\end{equation}
Putting in (1), (2), and (3) $e = 4.7 \cdot 10^{-10}$, $\frac{e}{m} = 5.31 \cdot
10^{-17}$, and $h = 6.5 \cdot 10^{-27}$ we get
\begin{equation}
\begin{array}{ll}
a = 0.55 \cdot 10^{-8} F^{-1},&v = 2.1 \cdot 10^8 F,\\
\omega = 6.2 \cdot 10^{15} F^2,& W = 2.0 \cdot 10^{-11}
{\displaystyle \sum} F^2.
\end{array}
\end{equation}
In neglecting the magnetic forces due to the motion of the electrons we have in
Part I. assumed that the velocities of the particles are small compared with the
velocity of light. The above calculations show that for this to hold, $F$ must
be small compared with 150. As will be seen, the latter condition will be
satisfied for all the electrons in the atoms of elements of low atomic weight
and for a greater part of the electrons contained in the atoms of the other
elements.
If the velocity of the electrons in not small compared with the velocity of
light, the constancy of the angular momentum no longer involved a constant ratio
between the energy and the frequency of revolution. Without introducing new
assumptions, we cannot therefore in this case determine the configuration of the
systems on the basis of the consideration in Part I. Considerations given later
suggest, however, that the constancy of the angular momentum is the principal
condition. applying this condition for velocities not small compared with the
velocity of light, we get the same expression for $v$ as that given by (1),
while the quantity $m$ in the expressions for $a$ and $\omega$ is replaced by
$m/\sqrt{(1 - v^2/c^2)}$,
and in the expression for $W$ by
$$
m \cdot 2 \frac{c^2}{v^2} \cdot \left( 1 - \sqrt{1 - \frac{v^2}{c^2}} \right).
$$
As stated in Part I., a calculation based on the ordinary mechanics given the
result, that a ring of electrons rotating round a positive nucleus in general is
unstable for displacement of the electrons in the plane of the ring. In order to
escape from this difficulty, we have assumed that the ordinary principles of
mechanics cannot be used in the discussion of the problem in question, any more
than in the discussion of the connected problem of the mechanism of binding of
electrons. We have also assumed that the stability for such displacement is
secured through the introduction of the hypothesis of the universal constancy of
the angular momentum of the electrons.
As is easily shown, the latter assumption in included in the condition of
stability in $\S~1$. Consider a ring of electrons rotation round a nucleus, and
assume that the system is in dynamical equilibrium and that the radius of the
ring is $a_0$, the $v_0$, the total kinetic energy $T_0$, and the potential
energy $P_0$. As shown in Part i. (p. 21) we have $P_0 = - 2T_0$. Next consider
a configuration of the system in which the electrons, under influence of
extraneous forces, rotate with the same angular momentum round the nucleus in a
ring of radius $a = \alpha a_0$. In this case we have $P = \frac{1}{\alpha} P_
0$, and on account of the uniformity of the angular momentum $v = 1/\alpha
\cdot v_0$ and $T = 1/\alpha^2 \cdot T_0$.
Using the relation $P_0 = - 2T_0$, we get
$$
P + T = \frac{1}{\alpha} \cdot P_0 + \frac{1}{\alpha^2} T_0 =
P_0 + T_0 + T_0 \cdot \left( 1 - \frac{1}{\alpha} \right)^2.
$$
We see that the total energy of the new configuration is greater than in the
original. according to the condition of stability in $\S~1$ the system is
consequently stable for the displacement considered. In this connexion, it may
be remarked that in Part I. we have assumed that the frequency of radiation
emitted or absorbed by the systems cannot be determined from the frequencies of
vibration of the electrons in the plane of the orbits, calculated by help of the
ordinary mechanics. We have, on the contrary, assumed that the frequency of the
radiation is determined by the condition $h \nu = E$, where $\nu$ is the
frequency, $h$ Planck's constant, and $E$ the difference in energy corresponding
to two different ``stationary'' states of the system.
In considering the stability of a ring of electrons rotating round a nucleus for
displacements of the electrons perpendicular to the plane of the ring, imagine a
configuration of the system in which the electrons are displaced by
$\delta z_1, \delta z_2, \ldots \delta z_n$ respectively, and suppose that the
electrons, under influence of extraneous forces, rotate in circular orbits
parallel to the original plane with the same radial and the same angular
momentum round the axis of the system as before. The kinetic energy is unaltered
by the displacement, and neglecting powers of the quantities $\delta z_1, \ldots
\delta z_n$ higher than the second, the increase of the potential energy of the
system is given by
$$
\frac{1}{2} \cdot \frac{e^2}{a^3} \cdot N \sum (\delta z)^2 -
\frac{1}{32} \cdot \frac{e^2}{a^2} \cdot
\sum \sum \mid \mbox{cosec}^3 \frac{\pi(r - s)}{n} \mid \left( \delta z_r -
\delta z_s \right)^2,
$$
where $a$ is the radius of the ring, $Ne$ the charge on the nucleus, and $n$ the
number of electrons. According to the condition of stability in $\S~1$ the
system is stable for the displacement considered, if the above expression is
positive for arbitrary values of $\delta z_1, \ldots \delta z_n$. By a simple
calculation it can be shown that the latter condition is equivalent to the
condition
\begin{equation}
N > p_{n, 0} - p_{n, m},
\end{equation}
where $m$ denotes the whole number (smaller than $n$) for which $$
p_{n,k} = \frac{1}{8} \sum^{s=n-1}_{s=1} \cos 2k \cdot \frac{s \pi}{n}
\mbox{cosec}^3
\frac{s \pi}{n}
$$
has its smallest value. This condition is identical with the condition of
stability for displacements of the electrons perpendicular to the plane of the
ring, deduced by help of ordinary mechanical considerations.\footnote{Comp. J.W.
Nicholson, Month. Not. Roy. Astr. Soc. 72. p. 52 (1912).}
A suggestive illustration is obtained by imagining that the displacements
considered are produced by the effect of extraneous forces acting on the
electrons in a direction parallel to the axis of the ring. If the displacements
are produced infinitely slowly the motion of the electrons will at any moment be
parallel to the original plane of the ring, and the angular momentum of each of
the electrons round the centre of its orbit will obviously be equal to its
original value; the increase in the potential energy of the system will be equal
to the work done by the extraneous forces during the displacements we are led to
assume that the ordinary mechanics can be used in calculating the vibrations of
the electrons perpendicular to the plane of the ring -- contrary to the ease of
vibrations in the plane of the ring. This assumptions is supposed by the
apparent agreement with observations obtained by Nicholson in his theory of the
origin of lines in the spectra of the solar corona and stellar nebulae (see Part
I. pp. 6 \& 23).??????
In addition it will be shown later that the assumption seems to be in agreement
with experiments on dispersion.
The following table gives the values of $s_n$ and $P_{n,0}$ - $P_{n, m}$ from $n
= 1$ to $n = 16.$\\
\vspace{0.2cm}
Table 1.
\begin{center}
\begin{tabular}{c | c | c || c | c | c}
\hline
&&&&&\\
~~~~n~~~~&~~~~s$_n$~~~~&~~~~$p_{n,0} - p_{n,m}$~~~~
&~~~n~~~&~~~~$s_n$~~~~&~$p_{n,0} - p_{n,m}$~\\
&&&&&\\
\hline
&&&&&\\
1&0&0&9&3.328&13.14\\
2&0.25&0.25&10&3.863&18.13\\
3&0.577&0.58&11&4.416&23.60\\
4&0.957&1.41&12&4.984&30.80\\
5&1.377&2.43&13&5.565&38.57\\
6&1.828&4.25&14&6.159&48.38\\
7&2.305&6.35&15&6.764&58.83\\
8&2.805&9.56&16&7.379&71.65\\
\end{tabular}
\end{center}
\vspace{0.5cm}
We see from the table that the number of electrons which can rotate in a single
ring round a nucleus of charge $Ne$ increases only very slowly for increasing $
N$; for $N = 20$ the maximum value is $n = 10$; for $N = 13$; for $N = 60$, $n =
15$. We see, further, that a ring of $n$ electrons cannot rotate in a single
ring round a nucleus of charge $ne$ unless $n < 8$.
In the above we have suppose that the electrons move under the influence of a
stationary radial force and that their orbits are exactly circular. The first
condition will not be satisfied if we consider a system containing several rings
of electrons which rotate with different frequencies. If, however, the distance
between the rings is not small in comparison with their radii, if the ratio
between their frequency is not near to unity, the deviation from circular orbits
may be very small and the motion of the electrons to a close approximation may
be identical with that obtained on the assumption that the charge on the
electrons is uniformly distributed along the circumference of the rings. If the
ratio between the radii of the rings is not near to unity, the conditions of
stability on this assumption may also be considered sufficient.
We have assumed in $\S~1$ that the electrons in the atoms rotate in coaxial
rings. The calculation indicated that only in the case of systems containing a
great number of electrons will the planes of the rings separate; in the case of
systems containing a moderate number of electrons, all the rings will be
situated in a single plane through the nucleus. For the sake of brevity, we
shall therefore here only consider the latter case.
Let us consider an electric charge $E$ uniformly distributed along the
circumference of a circle of radius $a$.
At a point distant $z$ from the plane of the ring, and at a distance $r$ from
the axis of the ring, the electrostatic potential is given by
$$
U = \frac{1}{\pi} \cdot E \int \limits^{\pi}_0 \frac{d \vartheta}
{\sqrt{a^2 + r^2 + z^2 - 2 ar \cos \vartheta}}.
$$
Putting in this expression $z = 0$ and $\frac{r}{a} = \mbox{tan}^2 \alpha$, and
using the notation
$$
K(\alpha) = \int \limits^{\pi/2}_0 \frac{d \vartheta}
{\sqrt{1 - \sin^2 \alpha \cos^2 \vartheta}},
$$
we get for the radial force exerted on an electron in a point in the plane of
the ring
$$
e \frac{\partial U}{\partial r} = \frac{Ee}{r^2} Q(\alpha),
$$
where
$$
Q(\alpha) = \frac{1}{\pi} \sin^4 \alpha (K(2 \alpha) - \mbox{cot}
\alpha \cdot K' (2 \alpha)).
$$
The corresponding force perpendicular to the plane of the ring at a distance $r$
from the center of the ring and at a small distance $\delta z$ from its plane is
given by
$$
e \frac{\partial U}{\partial z} = \frac{Ee \delta z}{r^3} R(\alpha),
$$
where
$$
R(\alpha) = \frac{2}{\pi} \sin^6 \alpha [K(2 \alpha) +
\mbox{tan}(2\alpha) \cdot K'(2 \alpha)].
$$
A short table of the functions $Q(\alpha)$ and $R(\alpha)$ is given
on p. 485.???
Next consider a system consisting of a number of concentric rings of electrons
which rotate in the same plane round a nucleus of charge $Ne$. Let the radial of
the rings be $a_1, a_2, \ldots$, and the number of electrons on the different
rings $n_1, n_2, \ldots$
Putting $a_r/a_s = \mbox{tan}^2 (\alpha_{r,s})$ we get for the radial
force acting on an electron in the $r$th ring $e^2/a^2_r F_r$ where
$$
F_r = N - s - \sum n_s Q(\alpha_{r,s}).
$$
the summation is to be taken over all the rings except the one considered.
If we know the distribution of the electrons in the different rings, from the
relation (1) on p. 478,???? we can, by help of the above, determine
$a_1, a_2, \ldots$. The calculation can be made by successive approximations,
starting from a set of values for the $\alpha$'$s$, and from them calculating
the $F$'$s$, and then redetermining the $\alpha$ $s$ by the relation (1) which
gives $F_s/F_r = a_r/a_s = \mbox{tan}^2 (\alpha_{r,s})$, and
so on.
As in the case of a single ring it is supposed that the systems are stable for
displacements of the electrons in the plane of their orbits. In a calculation
such as that on p. 480,????? the interaction of the rings
ought strictly to be taken
into account. This interaction will involve that the quantities $F$ are not
constant, as for a single ring rotating round a nucleus, but will vary with the
radii of the rings; the variation in $F$, however, if the ratio between the
radii of the rings is not very near to unity, will be too small to be of
influence on the result of the calculation.
Considering the stability of the systems for a displacement of the electrons
perpendicular to the plane of the rings, it is necessary to distinguish between
displacements in which the centres of gravity of the electrons in the single
rings are unaltered, and displacements in which all the electrons inside the
same ring are displaced in the same direction.
The condition of stability for the first kind of displacements is given by the
condition (5) on p. 481,???? if for every ring we
replace $N$ by a quantity $G_r$
determined by the condition that $e^2/a^3_r G_r \delta z$ is equal to the
component perpendicular to the plane of the ring of the force -- due to the
nucleus and the electrons in the other rings -- acting on one of the electrons
if it has received a small displacement $\delta z$. Using the same notation as
above, we get
$$
G_r = N - \sum n_s R (\alpha_{r,s}).
$$
If all the electrons in one of the rings are displaced in the same direction by
help of extraneous forces, the displacement will produce corresponding
displacements of the electrons in the other rings; and this interaction will be
of influence on the stability. For example, consider a system of $m$ concentric
rings rotating in a plane round a nucleus of charge $Ne$, and let us assume that
the electrons in the different rings are displaced perpendicular to the plane by
$\delta z_1, \delta z_2, \ldots, \delta z_m$ respectively. With the above
notation the increase in the potential energy of the system is given by
$$
\frac{1}{2}\cdot N \sum n_r \frac{e^2}{a^3_n} \left(\delta z_n \right)^2 -
\frac{1}{4} \cdot \sum \sum n_r n_s \frac{e^2}{a^3_r} R \left( \alpha_{r,
s}\right) \left( \delta z_r - \delta z_s \right)^2.
$$
The condition of stability is that this expression is positive for arbitrary
values $\delta z_1, \ldots \delta z_m$. This condition can be worked out simply
in the usual way. It is not of sensible influence compared with the condition of
stability for the displacements considered above, except in cases where the
system contains several rings of few electrons.
The following Table. containing the values of $Q(\alpha)$ and $R(\alpha)$ for
every fifth degree from $\alpha = 20^{\circ}$ to $\alpha = 70^{\circ}$, gives an
estimate of the order of magnitude of these functions: --\\
\vspace{0.2cm}
Table 2.\\
\begin{center}
\begin{tabular}{| c | c | c | c |}
\hline
&&&\\
~~~~~$\alpha$~~~~~&~~~~~$\mbox{tan}^2 \alpha$~~~~~&~~~~~$Q(\alpha)$~~~~~
&~~~~$R(\alpha)$\\
&&&\\
\hline
&&&\\
20&0.132&0.001&0.002\\
25&0.217&0.005&0.011\\
30&0.333&0.021&0.048\\
35&0.490&0.080&0.217\\
40&0.704&0.373&1.549\\
45&1.000&-&-\\
50&1.420&1.708&4.438\\
55&2.040&1.233&1.839\\
60&3.000&1.093&1.301\\
65&4.599&1.037&1.115\\
70&7.548&1.013&1.041\\
&&&\\
\hline
\end{tabular}
\end{center}
\vspace{0.3cm}
\noindent
$\mbox{tan}^2 \alpha$ indicated the ratio between the radii of the rings $\left(
\mbox{tan}^2(a_{r,s}) = \frac{a_r}{a_s} \right)$. The values of $Q(\alpha)$ show
that unless the ratio of the radii of the rings is nearly unity the effect of
outer rings on the dimensions of inner rings is very small, and that the
corresponding effect of inner rings on outer is to neutralize approximately the
effect of a part of the charge on the nucleus corresponding to the number of
electrons on the ring. The values of $R(\alpha)$ show that the effect of outer
rings on the stability of inner -- though greater than the effect on the
dimensions -- is small, but that unless the ratio between the radii is very
great, the effect of inner rings on the stability of outer is considerably
greater than to neutralize a corresponding part of the charge of the nucleus.
The maximum number of electrons which the innermost ring can contain being
unstable is approximately equal to that calculated on p. 482 for a single ring
rotating round a nucleus. For the outer rings, however, we get considerably
smaller numbers than those determined by the condition (5) if we replace $Ne$ by
the total charge on the nucleus and on the electrons of inner rings.
If system of rings rotating round a nucleus in a single plane is stable for
small displacements of the electrons perpendicular to this plane, there will in
general be no stable configurations of the rings, satisfying the condition of
the constancy of the angular momentum of the electrons, in which all the rings
are not situated in the plane. An exception occurs in the special case of
two rings containing equal numbers of electrons; in this case there may be a
stable configuration in which the two rings have equal radii and rotate in
parallel planes at equal distances from the nucleus, the electrons in the one
ring being situated just opposite the intervals between the electrons in the
other ring. The latter configuration, however, is unstable if the configuration
in which all the electrons in the two rings are arranged in a single ring is
stable.
\vspace{0.5cm}
\section*{
$\S~3$ {\it Constitution of Atoms containing very few Electrons}}
\vspace{0.5cm}
At stated in $\S~1$, the condition of the universal constancy of the angular
momentum of the electrons, together with the condition of stability, is in most
cases not sufficient to determine completely the constitution of the system. On
the general view of formation of atoms, however, and by making use of the
knowledge of the properties of the corresponding elements, it will be attempted
, in this section and the next, to obtain indications of what configurations of
the electrons may be expected to occur in the atoms. In these considerations we
shall assume that the number of electrons in the atom is equal to the number
which indicates the position of the corresponding element in the series of
elements arranged in order of increasing atomic weight.
Exceptions to this rule will be supposed to occur only at such places in the
series where deviation from the periodic law of the chemical properties of the
elements are observed. In order to show clearly the principles used we shall
first consider with some detail those atoms containing very few electrons.
Forsake of brevity we shall, by the symbol $N(n_1, n_2 \ldots)$, refer to a
plane system of rings of electrons rotating round a nucleus of charge $Ne$,
satisfying the condition of the angular momentum of the electrons with the
approximation used in $\S~2$. $n_1, n_2 \ldots$ are the numbers of electrons in
the rings, starting from inside. By $a_1, a_2, \ldots$ and $\omega_1, \omega_2
\ldots$ we shall denote the radii and frequency of the rings taken in the same
order. The total amount of energy $W$ emitted by the formation of the system
shall simply be denoted by $W[N(n_1, n_2, \ldots)]$.
\vspace{0.5cm}
$N = 1$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~{\it Hydrogen.}
\vspace{0.5cm}
In Part I. we have considered the binding of an electron by
a positive nucleus of
charge $e$, and have shown that it is possible to account for the Balmer
spectrum of hydrogen on the assumption of the existence of a series of
stationary states in which the angular momentum of the electron round the
nucleus is equal to entire multiplies of the value $h/2 \pi$, where $h$
is Planck's constant. The formula found for the frequencies of the spectrum was
$$
\nu = \frac{2 \pi^2 e^4 m}{h^3} \cdot \left( \frac{1}{\tau^2_2} -
\frac{1}{\tau^2_1} \right),
$$
where $\tau_1$ and $\tau_2$ are entire numbers. Introducing the values for $e$,
$m$, and $h$ used on p. 479, we get for the factor before the bracket $3.1 \cdot
10^{15}$; \footnote{This value is that calculated
in the first part of the paper.
Using the values $e = 4.78 \cdot 10^{-10}$ (see R.A. Millikan, Brit. Assoc. Rep.
1912, p. 410), $e/m = 5.31 \cdot 10^{17}$ (see P. Gmelin, Ann. d. Phys.
XXVIII. p. 1086 (1909) and A.H. Bucherer, Ann. d. Phys.
XXXVII p. 597 (1912)), and
$e/h = 7.27 \cdot 10^{16}$ calculated by Planck's theory from the
experiments of E. Warbung G. Leithauser, E. Hupka,
and C. Muller, Ann.d.Phys. XL.
p. 611 (1913)) we get $2\pi^2 e^4 m/h^3 = 3.26 \cdot 10^{15}$ in very
close agreement with observations.} the value observed for the constant in the
Balmer spectrum is $3.290 \cdot 10^{15}$.
For the permanent state of a neutral hydrogen atom we get from the formula (1)
and (2) in $\S~2$, putting $F = 1$,
$$
{\bf 1}(1):~~~~~~~ \alpha = \frac{h^2}{4 \pi e^2m} = 0.55 \cdot 10^{- 8}, ~~~
\omega = \frac{4 \pi^2 e^4 m}{h^3} = 6.2 \cdot 10^{15}, ~~~
$$
$$
W = \frac{2 \pi^2 e^4 m}{h^2} = 2.0 \cdot 10^{-11}.
$$
These values are of the order of magnitude to be expected. For $W/e$ we
get 0.043, which corresponds to 13 volts; the value for the ionizing potential
of a hydrogen atom, calculated by Sir J.J. Thomson from experiments on positive
rays, is 11 volt.\footnote{J.J. Thomson, Phil. Mag. XXIV. p. 218 (1912).}
No other definite data, however are available for hydrogen atoms. For sake of
brevity, we shall in the following denote the values for $a, \omega$ and $W$
corresponding to the configuration {\bf 1}(1) by $a_0, \omega_0$, and $W_0$.
At distance from the nucleus, great in comparison with $a_0$, the system {\bf 1}
(1) will not exert sensible forces on free electrons. Since, however, the
configuration:
$$
{\bf 1}(2)~~~~~ a = 1.33 a_0, ~~~~\omega = 0.563 \omega_0, ~~~ W = 1.13 W_0.
$$
corresponds to a greater value for $W$ than the configuration {\bf 1}(1), we may
expect that a hydrogen atom under certain conditions can acquire a negative
charge. This is in agreement with experiments on positive rays. Since $W[1(3)]$
is only 0.54, a hydrogen atom cannot be expected to be able to acquire a double
negative charge.
\vspace{0.5cm}
$N = 2$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~{\it Helium.}
\vspace{0.5cm}
As shown in Part I., using the same assumptions as for hydrogen, we must expect
that during the binding of an electron by a nucleus of charge $2e$, a spectrum
is emitted, expressed by
$$
\nu = \frac{2 \pi^2 me^4}{h^3} \cdot \left( \frac{1}{(\frac{\tau_2}{2})^2} -
\frac{1}{(\frac{\tau_1}{2})^2} \right).
$$
This spectrum includes the spectrum observed by Pickering in the star $xi$
Puppis and the spectra recently observed by Folwer in experiments with vacuum
tubes filled with a mixture of hydrogen and helium. These spectra are generally
ascribed to hydrogen.
For the permanent state of a positively charge helium atom, we get
$$
{\bf 2}(1) ~~~ a = \frac{1}{2} a_0, ~~~ \omega = 4 \omega_0, ~~~ W = 4 W_0.
$$
At distances from the nucleus great compared with the radius of the bound
electron, the system {\bf 2}(1) will, to a close approximation, act an an
electron as a simple nucleus of charge $e$. For a system consisting of two
electrons and a nucleus of charge $2e$, we may therefore assume the existence of
a series of stationary states in which the electron most lightly bound moves
approximately in the same way as the electron in the stationary states of a
hydrogen atom. Such an assumption has already been used in Part I. in an attempt
to explain the appearance of Rydberg's constant in the formula for the \mbox{
line-spectrum} of any element. We can, however, hardly assume the existence of a
stable configuration in which the two electrons have the same angular momentum
round the nucleus and move in different orbits, the one outside the other. In
such a configuration the electrons would be so near to each other that the
deviations from circular orbits would be very great. For the permanent state of
a neutral helium atom, we shall therefore adopt the configuration
$$
{\bf 2}(2)~~~ a = 0.571 a_0, ~~~ \omega = 3.06 \omega_0, ~~~ W = 6.13 W_0.
$$
Since
$$
W[{\bf 2}(2)] - W[{\bf 2}(1)] = 2.13 W_0,
$$
we see that both electrons in a neutral helium atom are more firmly bound than
the electron in a hydrogen atom. Using the values on p. 488,???? we get
$$
2.13 \cdot \frac{W_0}{e} = 27~ \mbox{×}, \mbox{É}~~ 2.13 \cdot \frac{W_0}{h} =
6.6 \cdot 10^{15} 1/\mbox{sec.}
$$
these values are of the same order of magnitude as the value observed for the
ionization potential in helium, 20.5 volt,\footnote{J.Franck u. G. Hertz, Verb.
d. Deutsch. Phys. Ges. XV. p. 34 (1913).} and the value for the frequency of
the \mbox{ultra-violet} absorption in helium determined by experiments on
dispersion
$5.9 \cdot 10^{15}~ 1/sec.$\footnote{C. and M. Cuthbertson, Proc. Roy. Soc.
A. LXXXIV. p. 13 (1910). In a previous paper (Phil. Mag. Jan. 1913) the author
took the values for the refractive index in helium, given by M. and C.
Cuthbertson, as corresponding to atmosphere pressure; these values, however,
refer to double atmosphere pressure. Consequently the value there given for the
number of electrons in a helium atom calculated from Drude's theory has to be
divided by 2.)}
The frequency in question may be regarded as corresponding to vibrations in the
plane of the ring (see p. 480).???? The frequency of vibration of the whole ring
perpendicular to the plane, calculated in the ordinary way (see p. 482), is
given by $\nu = 3.27 \omega_0$. The fact that the latter frequency is great
compared with that observed might explain that the number of electrons in a
helium atom, calculated by help of Drude's theory from the experiments on
dispersion, is only about \mbox{two-thirds} of the number to be expected. (Using
$\frac{e}{m} = 5.31 \cdot 10^{17}$ the value calculated is 1.2.)
For a configuration of a helium nucleus and three electrons, we get
$$
{\bf 2}(3) ~~~~ a = 0.703 a_0, ~~~ \omega = 2.02 \omega_0,~~~ W = 6.07 W_0.
$$
Since $W$ for this configuration is smaller than for the configuration {\bf 2}(
2), the theory indicates that a helium atom cannot acquire a negative charge.
This is in agreement with experimental evidence, which shows that helium atoms
have no ``affinite'' for free electrons.\footnote{See J. Franck, Verh. d.
Deutsch. Phys. Ges. XII. p. 613 (1910).}
In a later paper it will be shown that the theory offers a simple explanation of
the marked in the tendency of hydrogen and helium atoms to combine into
molecules.
\vspace{0.5cm}
$N = 3$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~{\it Lithium.}
\vspace{0.5cm}
In analogy with the cases of hydrogen and helium we must expect that during the
binding of an electron by a nucleus of charge $3e$, a spectrum is emitted, given
by
$$
\nu = \frac{2 \pi^2 me^4}{h^3} \cdot \left( \frac{1}{(\frac{\tau_2}{3})^2} -
\frac{1}{(\frac{\tau_1}{3})^2} \right).
$$
On account of the great energy to be spent in removing all the electrons bound
in a lithium atom (see below) the spectrum considered can only be expected to be
observed in extraordinary cases.
In a recent note Nicholson\footnote{J.W. Nicholson, Month. Not. Roy. Astr. Soc.
LXXIII. 382 (1913).} has drawn attention to the fact that in the spectra of
certain stars, which show the Pickering spectrum with special brightness, some
lines occur the frequencies of which to a close approximation can be expressed
by the formula
$$
\nu = K \cdot \left( \frac{1}{4} - \frac{1}{(m \pm 1/3)^2} \right).
$$
where $K$ is the same constant as in the Balmer spectrum of hydrogen. From
analogy with the Balmer- and Pickering-spectra, Nicholson has suggested that the
lines in question are due to hydrogen.
It is seen that the lines discussed by Nicholson are given by the above formula
if we put $\tau_2 = 6$. The lines in question correspond to $\tau_1 = 10, 13$
and 14; if we for $\tau_2 = 6$ put $\tau_1 = 9, 12$ and 15, we get lines
coinciding with lines of the ordinary \mbox{Balmer-spectrum} of hydrogen. If we
in the above formula put $\tau = 1,2$, and 3, we get series of lines in the
\mbox{ultra-violet}. If we put $\tau_2 = 4$ we get only a single line in visible
spectrum, viz.: for $\tau_1 = 5$ which gives $\nu = 6.662 \cdot 10^{14}$, or a
\mbox{wave-length} $\lambda = 4.503 \cdot 10^{-8}$ cm closely coinciding with
the \mbox{wave-length} $4.504 \cdot 10^{-8}$ cm of one of the lines of unknown
origin in the table quoted by Nicholson. In this table, however, no lines occur
corresponding to $\tau_2 = 5$.
For the permanent state of a lithium atom with two positive charges we get a
configuration
$$
{\bf 3}(1)~~~~ a = \frac{1}{3} a_0, ~~~ \omega = 9 \omega_0, ~~~ W = 9W_0.
$$
The probably of a permanent configuration in which two electrons move in
different orbits around each other must for lithium be considered still less
probable than for helium, as the ratio between the radii of the orbits would be
still nearer to unity. For a lithium atom with a single positive charge we
shall, therefore, adopt the configuration:
$$
{\bf 3}(2)~~~~ a = 0.364 a_0, ~~~ \omega = 7.56 \omega_0, ~~~ W = 15.13 W_0.
$$
Since $W[{\bf 3}(2)] - W[{\bf 3}(1)] = 6.13 W_0$ we see that the first two
electrons in a lithium atom very strongly bound compared with the electron in a
hydrogen atom; they are still more rigidly bound than the electrons in a helium
atom.
From a consideration of the chemical properties we should expect the following
configuration for the electrons in a neutral lithium atom:
$$
\begin{array}{llll}
{\bf 3}(2,1)& a_1 = 0.362 a_0, &\omega_1 = 7.65 \omega_0, \\
&&&W = 16.02 W_0\\
&a_2 = 1.182 a_0, &\omega_2 = 0.716 \omega_0,\\
\end{array}
$$
This configuration may be considered as highly probable also from a dynamical
point view. The deviation of the outermost electron from a circular orbit will
be very small, partly on account of the great values of the ratio between the
radii, and of the ratio between the frequencies of the orbits of the inner and
outer electrons, partly also on account of the symmetrical arrangement of the
inner electrons. accordingly, it appears probable that the three electrons will
not arrange themselves in a single ring and from the system:
$$
{\bf 3}(3) ~~~~ a = 0.413 a_0, ~~~ \omega = 5.87 \omega_0, ~~~ W = 17.61 W_0,
$$
although $W$ for this configuration is greater than for {\bf 3}(2,1).
Since $W$ [{\bf 3}(2,1) - $W$[{\bf 3}(2)] = 0.89W$_0$, we see that the outer
electron in the configuration {\bf 3}(2,1) is bound even more lightly than the
electron in a hydrogen atom. the difference in the firmness of the binding
corresponds to a difference of 1.4 volts in the ionization potential. A marked
difference between the electron in hydrogen and the outermost electron in
lithium lies also in the greater tendency of the latter electron top leave
the plane of this orbits. The quantity $G$ considered in $\S~2$, which gives a
kind of measure for the stability for displacements perpendicular to this plane,
is thus for the outer electron in lithium only 0.55, while for hydrogen it is 1.
This may have a bearing on the explanation of the apparent tendency of lithium
atoms to take a positive charge in chemical combinations with other elements.
For a possible negatively charged lithium atom we may expect the configuration:
$$
\begin{array}{llll}
{\bf 3}(2,2)& a_1 = 0.362 a_0, &\omega_1 = 7.64 \omega_0, \\
&&&W = 16.16 W_0\\
&a_2 = 1.516 a_0, &\omega_2 = 0.436 \omega_0,\\
\end{array}
$$
it should be remarked that we have no detailed knowledge of the properties in
the atomic state, either for lithium or hydrogen, or for most of the electrons
considered below.
\vspace{0.5cm}
$N = 4$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~{\it Beryllium.}
\vspace{0.5cm}
For reasons analogous to those considered for helium and lithium we may for the
formation of a neutral beryllium atom assume the following states:
$$
\begin{array}{llll}
{\bf 4}(1)& a = 0.25 a_0, &\omega = 16 \omega_0,& W = 16W_0,\\
{\bf 4}(2)& a = 0.267 a_0, & \omega = 14.06 \omega_0,& W = 28.13W_0,\\
{\bf 4}(2,1)& a_1 = 0.263 a_0,& \omega_1 = 14.46 \omega_0,\\
&&&W = 31.65W_0,\\
&a_2 = 0.605 a_0,& \omega_2 = 2.74 \omega_0,\\
{\bf 4}(2,2)& a_1 = 0.262 a_0,& \omega_1 = 14.60 \omega_0,\\
&&&W = 33,61 W_0,\\
&a_2 = 0.673 a_0,&\omega_2 = 2.21 \omega_0,\\
\end{array}
$$
although the configurations:
$$
\begin{array}{llll}
{\bf 4}(3)& a = 0.292 a_0, &\omega = 11.71 \omega_0,&W = 35.14 W_0, \\
{\bf 4}(4)& a = 0.329 a_0, &\omega = 9.26 \omega_0,&W = 37.04 W_0, \\
\end{array}
$$
correspond to less values for the total energy than the configuration {\bf 4}(
2,1) and {\bf 4}(2,2).
From analogy we get further for the configuration of a possible negatively
charged atom,
$$
\begin{array}{llll}
{\bf 4}(2,3)& a_1 = 0.263 a_0, &\omega_1 = 14.51 \omega_0, \\
&&&W = 33.66 W_0\\
&a_2 = 0.803 a_0, &\omega_2 = 1.55 \omega_0,\\
\end{array}
$$
Comparing the outer ring of the atom considered with the ring of a helium atom,
we see that the presence of the inner ring of two electrons in the beryllium
atom markedly charges the properties of the outer ring; partly because the outer
electrons in the configuration adopted for a neutral beryllium atom are more
lightly bound than the electrons in a helium atom, and partly because the
quantity $G$, which for helium is equal to 2, for the outer ring in the
configuration {\bf 4}(2,2) is only equal 1.12.
Since $W$[{\bf 4}(2,3)] - $W$[{\bf 4}(2,2)] = 0.05W$_0$, the beryllium atom
will further have a definite, although very small affinity for free electrons.
\vspace{0.5cm}
\section*{
$\S~4$ {\it Atoms containing greater numbers of electrons}}
\vspace{0.5cm}
From the examples discussed in the former section it will appear that the
problem of the arrangement of the electrons in the atoms is intimately connected
with the question of the confluence of two rings of electrons rotating round a
nucleus outside each other, and satisfying the condition of the universal
constancy of the angular momentum. apart from the necessary conditions of
stability for displacements of the electrons perpendicular to the plane of the
orbits, the present theory gives very little information on this problem. It
seems, however, possible by the help of simple considerations to throw some
light on the question.
Let us consider two rings rotating round a nucleus in a single plane, the one
outside the other. Let us assume that the electrons in the one ring act upon the
electrons in the other as if the electric charge were uniformly distributed
along the circumference of the ring, and that the ring with this approximation
satisfy the condition of the angular momentum of the electrons and stability for
displacements perpendicular to their plane.
Now suppose that, by help of suitable imaginary extraneous forces acting
parallel to the axis of the rings, we pull the inner ring slowly to one side.
During this process, on account of the repulsion from the inner ring, the outer
will move to the opposite side of the original plane of the rings. During the
displacements of the rings angular momentum of the electrons round the axis of
the system will remain constant, and the diameter of the inner ring will
increase while that of the outer will diminish. At the beginning of the
displacement the magnitude of the extraneous forces to be applied to the
original inner ring will increase but thereafter decrease, and at a certain
distance between the plane of the rings the system will be in a configuration of
equilibrium. This equilibrium, however, will not be stable. If we let the rings
slowly return they will either reach their original position, or they arrive at
a position in which the ring, which originally was the outer, is now the inner,
and {\it vise versa}.
If the charge of the electrons were uniformly distributed along the
circumference of the rings, we could by the process considered at most obtain an
interchange of the rings, but obviously not a junction of them. Taking, however,
the discrete distribution of the electrons into account, it can be shown that in
the special case when the number of electrons on the two rings are equal, and
when the rings rotate in the same direction, the rings will unite by the
process, provided that the final configuration is stable. In this case the radii
and the frequency of the rings will be equal in the unstable configuration of
equilibrium mentioned above. In reaching this configuration the electrons in the
one ring will further be situated just opposite the intervals between the
electrons in the outer, since such an arrangement will correspond to the
smallest total energy. If now we let the rings return to their original plane,
the electrons in the one ring will pass into the intervals between the electrons
in the other, and from a single ring. Obviously the ring thus formed will
satisfy the same condition of the angular momentum of the electrons as the
original rings.
If the two rings contain unequal numbers of electrons the system will during a
process such as that considered behave very differently, and, contrary to the
former case, we cannot expect that the rings will flow together, if by help of
extraneous forces acting parallel to the axis of the system they are displaced
slowly from their original plane. It may in this connexion be noticed that the
characteristic for the displacements considered is not the special assumption
about the extraneous forces, but only invariance of the angular momentum of the
electrons round the centre of the rings; displacements of this kind take in the
present theory a similar position to arbitrary displacements in the ordinary
mechanics.
The above considerations may be taken as an indication that there is greater
tendency for the confluence of two rings when each contains the same number of
electrons. Considering the successive binding of electrons by a positive
nucleus, we conclude from this that, unless the charge on the nucleus is very
great, rings of electrons will only join together if they contain equal numbers
of electrons; and that accordingly the numbers of electrons on inner rings will
only be 2, 4, 8, $\ldots$. If the charge of the nucleus is very great the rings
of electrons first bound, if few in number, will be very close together, and we
must expect that the configuration will be very unstable, and that a gradual
interchange of electrons between the rings will be greatly facilitated.
This assumption in regard to the number of electrons in the rings is strongly
supported by the fact that the chemical properties of the elements of low atomic
weight vary with a period of 8. Further, it follows that the number of electrons
on the outermost ring will always be odd or even, according as the total number
of electrons in the atom is add or even. This has a suggestive relation to the
fact that the valency of an element of low atomic weight always is odd or even
according as the number of the element in the periodic series is odd or even.
For the atoms of the elements considered in the former section we have assumed
that the two electrons first bound are arranged in a single ring, and, further,
that the two next electrons are arranged in another ring. If
{\bf $N \geq$} 4 the
configuration {\bf $N$} (4) will correspond to a smaller value for the total
energy than the configuration {\bf $N$}(2,2). The greater the value of $N$ the
closer will the ratio between the radii of the rings in the configuration {\bf
$N$}(2,2) approach unity, and the greater will be the energy emitted by an
eventual confluence of the rings. The particular member of the series of the
elements for which the four innermost electrons will be arranged for the first
time in a single ring cannot be determined from the theory. From a consideration
of the chemical properties we can hardly expect that it will have taken place
before boron ({\bf $N$} = 5) or carbon
({\bf $N$} = 6), on account of the observed
trivalency and tetravalency respectively of these elements; on the other hand,
the periodic system of the elements strongly suggests that already in neon
({\bf $N$} = 10) an inner ring of eight electrons will occur. Unless
{\bf $N$} $>$ 14 the
configuration {\bf $N$}(4,4) corresponds to smaller value for the total energy
that the configuration {\bf $N$}(8); already for
{\bf $N$} $\geq$ 10 the latter
configuration, however, will be stable for displacements of the electrons
perpendicular to the plane of their orbits. A ring of 16 electrons will not be
stable unless
{\bf $N$} is very great; but in such a case the simple considerations
mentioned do not apply.
The confluence of two rings of equal number of electrons, which rotate round a
nucleus of charge $Ne$ outside a ring of $n$ electrons already bound, must be
expected to take place more easily than the confluence of two similar rings
rotating round a nucleus of charge $(N - n) \cdot e$;
for the stability of the rings
for a displacement perpendicular to their plane will (see $\S~2$) be smaller in
the first than the latter case. This tendency for stability to decrease for
displacements perpendicular to the plane of the ring will be especially marked
for the outer rings of electrons of a neutral atom. In the latter case we must
expect the confluence of rings to be greatly facilitated and in certain cases it
may even happen that the number of electrons in the outer ring may be greater
than in the next, and that the outer ring may show deviations from the
assumption of 1, 2, 4, 8 electrons in the rings, e.g. the configurations {\bf
5}(2,3) and {\bf 6}(2,4) instead of the configuration {\bf 5}(2,2,1) and {\bf
6}(2,2,2). We shall here not discuss further the intricate question of the
arrangement of the electrons in the outer ring. In the scheme given below the
number of electrons in this rings is arbitrary put equal to the normal valency
of the corresponding element; i.e. for electronegative and electropositive
elements respectively the number of hydrogen atoms and twice the number of
oxygen atoms with which one atom of the element combines.
Such an arrangement of the outer electrons is suggested by considerations of
atomic volumes. As is well known, the atomic volume of the elements is a
periodic function of the atomic weights. If arranged in the usual way according
to the periodic system, the elements inside the same column have approximately
the same atomic volume, while this volume changes considerably from one column
to another, being greatest for columns corresponding to the smallest valency 1
and smallest for the greatest valency 4. An approximate estimate of the radius
of the outer ring of a neutral atom can be obtained by assuming that the total
forces due to the nucleus and the inner electrons is equal to that from a
nucleus of charge $ne$, where $n$ is the number of electrons in the ring.
Putting $F = n - s_n$ in the equation (1) on p. 478, ??????
and denoted the value of $
a$ for $n = 1$ by $a_0$, we get for $n = 2$, $a = 0.41 a_0$; and for $n = 4,$ $a
= 0.33 a_0$. According the arrangement chosen for the electrons will involve a
variation in the dimensions of the outer ring similar to the variation in the
atomic volumes of the corresponding elements. It must, however, be borne in mind
that the experimental determinations of atomic volumes in most cases are deduced
from consideration of molecules rather that atoms.
From the above we are led to the following possible scheme for the arrangement
of the electrons in light atoms: --\\
\begin{center}
\begin{tabular}{lll}
{\bf 1}(1)~~~~~&{\bf 9}(4,4,1)~~~~~&{\bf 17}(8,4,4,1)\\
{\bf 2}(2)~~~~~&{\bf 10}(8,2)~~~~~&{\bf 18}(8,8,2)\\
{\bf 3}(2,1) ~~~~~&{\bf 11}(8,2,1)~~~~~&{\bf 19}(8,8,2,1)\\
{\bf 4}(2,2)~~~~~&{\bf 12}(8,2,2)~~~~~&{\bf 20}(8,8,2,2)\\
{\bf 5}(2,3)~~~~~&{\bf 13}(8,2,3)~~~~~&{\bf 21}(8,8,2,3)\\
{\bf 6}(2,4)~~~~~&{\bf 14}(8,2,4)~~~~~&{\bf 22}(8,8,2,4)\\
{\bf 7}(4,3)~~~~~&{\bf 15}(8,4,3)~~~~~&{\bf 23}(8,8,4,3)\\
{\bf 8}(4,2,2)~~~~~&{\bf 16}(8,4,2,2)~~~~~&{\bf 24}(8,8,4,2,2)\\
\end{tabular}
\end{center}
Without any fuller discussion it seems not unlikely that this constitution of
the atoms will correspond to properties of the elements similar with those
observed.
In the first place there will be a marked periodicity with a period of 8.
Further, the binding of the outer electrons in every horizontal series of the
above scheme will become weaker with increasing number of electrons per atom,
corresponding to the observed increase of the electropositive character for an
increase of atomic weight of the elements in every single group of the periodic
system. A corresponding agreement holds for the variation of the atomic volumes.
In the case of atoms of higher atomic weight the simple assumptions used do not
apply. A few indications, however, are suggested from consideration of the
variations in the chemical properties of the elements. At the end of the 3rd
period of 8 elements we meet with the \mbox{iron-group}. This group takes a
particular position in the system of the elements, since it is the first time
that elements of neighbouring atomic weight show similar chemical properties.
This circumstance indicates that the configurations of the electrons in the
elements of this group differ only in the arrangement of the inner electrons.
The fact that the period in the chemical properties of the elements after the
\mbox{iron-group} is no longer 8, but 18, suggests that elements of higher
atomic weight contain a recurrent configuration of 18 electrons in the innermost
rings. The deviation from 2, 4, 8, 16 may be due to a gradual interchange of
electrons between the rings, such as is indicated on p. 495. Since a ring of 18
electrons will not be stable the electrons may be arranged in two parallel rings
(see p. 486). ???????
Such a configuration of the inner electrons will act upon the
outer electrons in very nearly the same way as nucleus of charge
$(N - 18) \cdot e$. It
might therefore be possible that with increase of $N$ another configuration of
the same type will be formed outside the first, such as is suggested by the
presence of a second period of 18 elements.
On the same lines, the presence of the group of the rare earths indicates that
for still greater values of $N$ another gradual alteration of the innermost
rings will take place. Since, however, for elements of higher atomic weight than
those of this group, the laws connection the vibration of the chemical
properties with the atomic weight are similar to these between the elements of
low atomic weight, we may conclude that the configuration of the innermost
electrons will be again repeated. The theory, however, is not sufficiently
complete to give a definite answer to such problems.
\vspace{0.5cm}
\section*{
$\S$~5 {\it Characteristic R\"ontgen Radiation}}
\vspace{0.5cm}
According to the theory of emission of radiation given in Part I., the ordinary
\mbox{line-spectrum} of an element is emitted during the reformation of an
atom when one or more of the electrons in the other rings are remover. In
analogy it may be supposed that the characteristic R\"ontgen radiation is sent
out during the setting down of the system if electrons in inner rings are
removed by some agency, e.g. by impact of cathode particles. This view of the
origin of the characteristic R\"ontgen radiation has been proposed by Sir. J.J.
Thomson.
Without any special assumption in regard to the constitution of the radiation,
we can from this view determine the minimum velocity of the cathode rays
necessary to produce the characteristic R\"ontgen radiation of a spacial type by
calculating the energy necessary to remove one of the electrons from the
different rings. Even if we know the numbers of electrons in the rings, a
rigorous calculation of this momentum energy might still be complicated, and
the result largely dependent on the assumptions used; for, as mentioned in
Part I., p. 19, ??????????
the calculation cannot be performed entirely on the basis of the
ordinary mechanics. We can, however, obtain very simply an approximate
comparison with experiments if we consider the innermost ring and as a first
approximation neglect the repulsion from the electrons in comparison with the
attraction of the nucleus. Let us consider a simple system consisting of a bound
electron rotating in a circular orbit round a positive nucleus of charge $Ne$.
From the expressions (1) on p. 478 ???????
we get for the velocity of the electron, putting $F = N$,
$$
v = \frac{2 \pi e^2}{h} N = 2.1 \cdot 10^8 \cdot N.
$$
The total energy to be transferred to the system in order to remove the
electron to an infinite distance from the nucleus is equal to the kinetic energy
of the bound electron. If, therefore, the electron is removed to a great
distance from the nucleus by impact of another rapidly moving electron, the
smallest kinetic energy possessed by the latter when at a great distance from
the nucleus must necessarily be equal to the kinetic energy of the bound
electron before the collision. The velocity of the free electron therefore
must be at least equal to $e$.
According to Whiddington's
experiments\footnote{R. Whiddington, Proc. Roy. Soc. A.
LXXXV. p. 323 (1911).} the velocity of cathode rays just able to produce the
characteristic R\"ontgen radiation of the \mbox{so-called}
\mbox{$K$-type-the}
hardest type of radiation observed--from an element of atomic weight $A$ is
for elements from Al to Se approximately equal to $A \cot 10^8$ cm/sec. As seen
this is equal to the above calculated value for $r$, if we put $N =
A/2.$
Since we have obtained approximate agreement with experiment by ascribing the
characteristic R\"ontgen radiation of the \mbox{$K$-type} to the
innermost ring,
it is to be expected that no harder type of characteristic radiation will exist.
This is strongly indicated by observations of the penetrating power of
$\gamma$ rays.\footnote{Comp. E. Rutherford, Phil. Mag. XXIV. p. 453 (1912).}
It is worthy of remark that the theory gives not only nearly the right value
for the energy required to remove an electron from the outer ring, but also
the energy required to remove an electron from the innermost ring. The
approximate agreement between the calculated and experimental values is all the
more striking it is recalled that the energies required in the two cases for an
element of atomic weight 70 differ by a ratio of 1000.
In connexion with this it should be emphasized that the remarkable
homogeneity of the characteristic R\"ontgen radiation -- indicated
by experiments
on absorption of the rays, as well as by the interference observed in recent
experiments on diffraction of R\"ontgen rays in crystals -- is in agreement
with
the main assumption used in part I. (see p. 7) in considering the emission of
\mbox{line-spectra}, viz. that the radiation emitted during the passing of the
systems between different stationary states is homogeneous.
Putting in (4) $F = N$, we get for the diameter of the innermost ring
approximately $2a = 1/N \cdot 10^{-8}$ cm. For $N = 100$ this gives $2a
= 10^{-10}$ cm, a value which is very small in comparison with ordinary atomic
dimensions but still very great compared with the dimensions to be expected for
the nucleus. according to Rutherford's calculation the dimensions of the
latter are of the same order of magnitude as $10^{-12}$ cm.
\vspace{0.5cm}
\section*{
$\S~6$ {\it Radioactive Phenomena}}
\vspace{0.5cm}
According to the present theory the cluster of electrons surrounding the
nucleus is formed with emission of energy, and the configuration is determined
by the condition that the energy emitted is a maximum. The stability involved
by these assumptions seems to be in agreement with the general properties of
matter. It is, however, in striking opposition to the phenomena of
radioactivity, and according to the theory the origin of the latter phenomena
may therefore be sought elsewhere than in the electronic distribution
round the nucleus.
A necessary consequence of Rutherford's theory of the structure of atoms is
that the \mbox{$\alpha$-particles} have their origin in the nucleus. On the
present theory it seems also necessary that the nucleus is the seat of the
expulsion of the \mbox{high-speed} \mbox{$\beta$-particles.} In the first place,
the spontaneous expulsion of a \mbox{$\beta$-particle} from the cluster of
electrons surrounding the nucleus would be something quite foreign to the
assumed properties of the system. further, the expulsion of an
\mbox{$\alpha$-particle} can hardly be expected to
produce a lasting effect on the
stability of the cluster of electrons. The effect of the expulsion will be of
two different kinds. Partly the particle may collide with the bound electrons
during its passing through the atom. This effect will be analogous to that
produced by bombardment of atoms of other substances by \mbox{$\alpha$-rays} and
cannot be expected to give rise to a subsequent expulsion of
\mbox{$\beta$-rays.} Partly the expulsion of the particle will involve an
alteration in the configuration of the bound electrons, since the charge
remaining on the nucleus is different from the original. In order to consider
the latter effect let us regard a single ring of electrons rotating round a
nucleus of charge $Ne$, and let us assume that an \mbox{$\alpha$-particle} is
expelled from the nucleus in a direction perpendicular to the plane of the ring.
The expulsion of the particle will obviously not produce any alteration in the
angular momentum of the electrons; and if the velocity of the
\mbox{$\alpha$-particle} is small compared with the velocity of the electrons --
as it will be if we consider inner rings of an atom of high atomic
weight -- the
ring during the expulsion will expand continuously, and after the expulsion
will take the position claimed by the theory for a stable ring rotating round a
nucleus of charge $(N - 2) \cdot e$.
The consideration of this simple case strongly
indicates that the expulsion of an \mbox{$\alpha$-particle} will not have a
lasting effect on the stability of the internal rings of electrons in
the residual atom.
The question of the origin of \mbox{$\beta$-particles} may also be considered
from another point of view, based on a consideration of the chemical and
physical properties of the radioactive substances. As is well known, several of
these substances have very similar chemical properties and have hitherto
resisted every attempt to separate them by chemical means. There is also some
evidence that the substances in question show the same
\mbox{line-spectrum.}\footnote{see A.S. Russel and R. Rossi,
Proc. Roy. Soc. A.
LXXXVII. p. 478 (1912).} It has been suggested by several writers that the
substances are different only in \mbox{radio-active} properties and atomic
weight but identical in all other physical and chemical respects. according to
the theory, this would mean that the charge on the nucleus, as well as the
configuration of the surrounding electrons, was identical in some of the
elements, the only difference being the mass and the
internal condition of the
nucleus. From the considerations of $\S~4$ this assumption is already strongly
suggested by the fact that the number of radioactive substances is greater than
the number of places at our disposal in the periodic system. If, however, the
assumption is right, the fact that two apparently identical elements emit
\mbox{$\beta$-particles} of different velocities, shows that the
\mbox{$\beta$-rays} as well as the \mbox{$\alpha$-rays} have their origin in the
nucleus.
This view of the origin of $\alpha$- and \mbox{$\beta$-particles} explains very
simply the way in which the change in the chemical properties of the
radioactive substances is connected with the nature of the particles emitted.
The results of experiments are expressed in the two rules:--\footnote{See A.S.
Russell, Chem. News, CVII. p. 49 (1913); G.v. Hevesy, Phys.
Zeitschr. XIV. p. 49
(1913); K. Fajaus, Phys. Zeitschr. XIV. pp. 131 \& 136 (1913);
Verh. d. deutsch. Phys. Ges. XV. p. 240 (1913); F. Soddy,
Chem. News, CVII. p. 97 (1913).}
1. Whenever an \mbox{$\alpha$-particles} is expelled the group in
the periodic system to which the resultant product belongs is two
units less than that to which the parent body belongs.
2. Whenever a \mbox{$\beta$-particle} is expelled the group of
the resultant body is 1 unit greater than that of the parent.
As will be seen this is exactly what is to be expected according
to the considerations of $\S~4$.
In escaping from the nucleus, the \mbox{$\beta$-rays} may be
expected to collide with the bound electrons in the inner rings. This
will give rise to an emission of a characteristic radiation of the
same type as the characteristic R\"ontgen radiation emitted from elements
of lower atomic weight by impact of cathode-rays. The assumption that
the emission of \mbox{$\gamma$-rays} is due to collisions of
\mbox{$\beta$-rays} with bound electrons is proposed by
Rutherford\footnote{E. Rutherford, Phil. Mag. XXIV. pp. 453\&893
(1912).} in order
to account for the numerous groups of homogeneous \mbox{$\beta$-rays}
expelled from certain radioactive substances.\\
In the present paper it has been attempted to show that the
application of Planck's theory of radiation to Rutherford's
\mbox{atom-model} through the introduction of the hypothesis of the
universal constancy of the angular momentum of the bound electrons,
leads to results which seem to be in agreement with experiments.
In a later paper the theory will be applied to systems containing
more than one nucleus.
\end{document}