THE LONDON, EDINBURGH, AND DUBLIN

PHILOSOPHICAL MAGAZINE
AND
JOURNAL OF SCIENCE.

[SIXTH SERIES.]

JULY 1913.

I. ON THE CONSTITUTION OF ATOMS AND MOLECULES.

By N. BOHR, Dr. phil. Copenhagen[1].

CONTENTS

Introduction.

IN order to explain the results of experiments on scattering of a rays by matter Prof. Rutherford[2] has given a theory of the structure of atoms. According to this theory, the atoms 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 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

rays[3].

In an attempt to explain some of the properties of matter on the basis of this atom-model we meet, however, with difficulties of a serious nature arising from the apparent instability of the system of electrons: difficulties purposely avoided in atom-models previously considered, for instance, in the one proposed by Sir J. J. Thomson[4]. According to the theory of the latter the atom consists of a sphere of uniform positive electrification, inside which the electrons move in circular orbits.

The principal difference between the atom-models proposed by Thomson and Rutherford consists in the circumstance that the forces acting on the electrons in the atom-model of Thomson allow of certain configurations and motions of the electrons for which the system is in a stable equilibrium; such configurations, however, apparently do not exist for the second atom-model. The nature of the difference in question will perhaps be most clearly seen by noticing that among the quantities characterizing the first 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 be 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, photo-electric effect, Röntgen-rays, &c. The result of the discussion of these questions seems to be a general acknowledgment of the inadequacy of the classical electrodynamics in describing the behaviour of systems of atomic size[5]. 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 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, reasons are given for a principal hypothesis on which the considerations contained in the following parts are based.

I wish here to express my thanks to Prof. Rutherford for his kind and encouraging interest in this work.

PART I.—BINDING OF ELECTRONS BY POSITIVE NUCLEI.

§1. General Considerations.

The inadequacy of the classical electrodynamics in accounting for the properties of atoms from an 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

and the major-axis of the orbit

will depend on the amount of energy

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

and

respectively and the mass of the electron by

, we thus get

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

. We see that if the value of

is not given, there will be no values of

and

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.

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 of 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 seem to have absolutely fixed dimensions and frequencies. Further, if we consider any molecular process, the result seems always to be that after a certain amount of energy characteristic for the systems in question is radiated out, the systems will again settle down in a stable state of equilibrium, in which the distances 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

in a single emission being equal to

, where

is an entire number, and

is a universal constant[6].

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 the 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 systems containing only a single electron.

Let us now assume that, during the binding of the electron, a homogeneous radiation is emitted of a frequency

, 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

, where

is Planck’s constant and

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

. 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 [§3].

Putting

we get by help of the formula (1)

If in these expressions we give

different values, we get a series of values for

,

, and

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

is greatest if

has its smallest value

. This case will therefore correspond to the most stable state 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

and

, and introducing the experimental values

we get

We see that these values are of the same order of magnitude as the linear dimensions of the atoms, the optical frequencies, and the ionization-potentials.

The general importance of Planck’s theory for the discussion of the behaviour of atomic systems was originally pointed out by Einstein[7]. 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[8], in an attempt to explain the meaning and the value of Planck’s constant on the basis of J. J. Thomson’s 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[9]. 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 nebulæ 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 wave-length 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

. 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 wave-lengths 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 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 does not seem to be able to account for the well-known laws of Balmer and Rydberg connecting the frequencies of the lines in the 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:

(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.

(2) That the latter process is followed by the emission of a homogeneous radiation, for which the relation between the frequency and the amount of energy emitted is the one given by Planck's theory.

The first assumption seems to present itself; for it is known that the ordinary mechanics 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 contrast 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 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 [§3]), also arrive at the expressions (3) for the stationary states by using assumptions of somewhat different form. We shall, therefore, postpone the discussion of the special 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 line-spectrum of hydrogen.

§2. Emission of Line-spectra.

Spectrum of Hydrogen.—General evidence indicates that an atom of hydrogen consists simply of a single electron rotating round a positive nucleus of charge

[10]. 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

, we get for the total amount of energy radiated out by the formation of one of the stationary states,

The amount of energy emitted by the passing of the system from a state corresponding to

to one corresponding to

, is consequently

If now we suppose that the radiation in question is homogeneous, and that the amount of energy emitted is equal to

, where

is the frequency of the radiation, we get

and from this

We see that this expression accounts for the law connecting the lines in the spectrum of hydrogen. If we put

and let

vary, we get the ordinary Balmer series. If we put

, we get the series in the ultra-red observed by Paschen[11] and previously suspected by Ritz. If we put

and

, we get series respectively in the extreme ultra-violet and the extreme 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

we get

The observed value for the factor outside the bracket in the formula (4) is

The 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 [§3] return to consider the possible importance of the agreement in question.

It may be remarked that the fact, that it has not been possible to observe more than

lines of the Balmer series in experiments with vacuum tubes, while

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

. For

the diameter is equal to

, or equal to the mean distance between the molecules in a gas at a pressure of about

; for

the diameter is equal to

, corresponding to the mean distance of the molecules at a pressure of about

. 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 tilled 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 [§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[12] in the spectrum of the star

Puppis, and the set of series recently found by Fowler[13] 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

and two electrons. Now considering the binding of a single electron by a helium nucleus, we get, putting

in the expressions (3) on [page 5], and proceeding in exactly the same way as above,

If we in this formula put

or

, we get series of lines in the extreme ultra-violet. If we put

, and let

vary, we get a series which includes

of the series observed by Fowler, and denoted by him as the first and second principal series of the hydrogen spectrum. If we put

, we get the series observed by Pickering in the spectrum of

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

, we get series, the strong lines of which are to be expected in the 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 as 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.

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[14]—the frequency corresponding to the lines of the spectrum of an element can be expressed by

where

, and

, are entire numbers, and

,

,

,.... are functions of

which approximately are equal to

,

,....

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

, or

, 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 systems 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 different 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 simply explained that the constant

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 from the nucleus and the electrons previously bound, will be very nearly the same as in 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

great be very nearly equal to that given by the expression (3) on [p. 5], if we put

. For

great we consequently get

in conformity with Rydberg’s theory.

[§3. General Considerations continued.]

We shall now return to the discussion (see [p. 7]) of the special assumptions used in deducing the expressions (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 energy-quanta. 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 is 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 energy-quantum,

. 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

, instead of by the equation (2). Proceeding in the same way as above, we get in this case instead of (3)

Assuming as above that the amount of energy emitted during the passing of the system from a state corresponding to

to one for which

is equal to

, we get instead of (4)

We see that in order to get an expression of the same form as the Balmer series we must

.

In order to determine

let us now consider the passing of the system between two successive stationary states corresponding to

and

; introducing

, we get for the frequency of the radiation emitted

For the frequency of revolution of the electron before and after the emission we have

If

is great the ratio between the frequency before and after the emission will be very near equal to

; and according to the ordinary electrodynamics we should therefore expect that the ratio between the frequency of radiation and the frequency of revolution also is very nearly equal to

. This condition will only be satisfied if

. Putting

, 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

and

, where

is small compared with

, we get with the same approximation as above, putting

,

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 frequencies of which are

, if

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 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

where

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 form 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 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

, 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 (4) may therefore be of value in the discussion of the results of experimental determinations of the constants

,

, and

.

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

, we have immediately for a circular orbit

, where

is the frequency of revolution and

the kinetic energy of the electron; for a circular orbit we further have

(see [p. 3]) and from (2), [p. 5], we consequently get

where

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[15].

The great number of different stationary states we do not observe except by investigation of the emission and absorption of radiation. In 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

.

[§4. Absorption of Radiation.]

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 homogeneous 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

and

corresponding to values for

equal to

and

,

. As the necessary condition for an emission of the radiation in question was the presence of systems in the state

, we must assume that the necessary condition for an absorption of the radiation is the presence of systems in the state

.

These considerations seem 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 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

. The state of the atoms in hydrogen gas at ordinary conditions should, however, correspond to

; furthermore, hydrogen atoms at ordinary conditions combine into molecules, i. e. into systems in which the electrons have frequencies different from those in the atoms (see [Part III].). From the circumstance that certain substances in a non-luminous state, as, for instance, sodium vapour, absorb radiation corresponding to lines in the 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 electrodynamics is perhaps most clearly 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 photo-electric effect, and which may be able to throw some light on the problem in question. Let us consider a 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 systems in successive stationary states will diminish without limit if

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 accounted 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

, where

is the difference between the total energy of the system in the two states. As it will be seen, such an absorption of radiation is just what is observed in experiments on ionization by ultra-violet light and by Röntgen rays. Obviously, we get in this way the same expression for the kinetic energy of an electron ejected from an atom by photo-electric effect as that deduced by Einstein[16], i. e.

, where

is the kinetic energy of the electron ejected, and

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[17] 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 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 trace 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

be very great compared with ordinary atomic dimensions. This circumstance was used as an explanation of the non-appearance in experiments with 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, and the states corresponding to high values for

could therefore not appear; yet in the absorption spectrum about

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 there is no ionization—will have an absorbing (scattering) influence on a homogeneous radiation, as soon as the frequency of the radiation is greater than

, where

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. It 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 line-spectra is due to the reformation of atoms after one or more of the lightly bound electrons are removed, we may assume that the homogeneous Röntgen radiation is emitted during the settling down of the systems after one of the firmly bound electrons escapes, e. g. by impact of cathode particles[18]. In the next part of 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 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 clearly shown by some very instructive calculations on the energy of

-particles emitted from radioactive substances recently published by Rutherford[19]. 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 slates, 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 the 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.

§5. The permanent State of an Atomic System.

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

.

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

and

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 one 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

electrons rotating round a nucleus of charge

, the electrons being arranged at equal angular intervals around the circumference of a circle of radius

.

The total potential energy of the system consisting of the electrons and the nucleus is

where

For the radial force exerted on an electron by the nucleus and the other electrons we get

Denoting the kinetic energy of an electron by

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,

or

From this we get for the frequency of revolution

The total amount of energy

necessary transferred to the system in order to remove the electrons to infinite distances apart from the nucleus and from each other is

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

for

. 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

electrons in which each rotates in an elliptical orbit with the nucleus in the focus, and the

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

by

and

by

. Let us now suppose that the system of

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

, where

is an entire number,

Planck’s constant, and

the frequency of revolution. The configuration in which the greatest amount of energy is emitted is, as before, the one in which

. 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 electron we get that the angular momentum of each of the electrons is equal to

. 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

will be replaced by the frequency

of the radiation from the whole ring calculated from the ordinary electrodynamics, and

by the total kinetic energy

.

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 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

electrons and a nucleus of charge

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

we get, by help of the above expressions for

and

, values for

and

corresponding to the permanent state of the system which only differ from those given by the equations (3) on [p. 5], by exchange of

for

.

The question of stability of a ring of electrons rotating round a positive charge is discussed in great detail by Sir J. J. Thomson[20]. 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[21]. The investigation of the problem in question naturally divides in two parts: one concerning the stability for displacements of the electrons in 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 displacements 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 amount 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 the 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 of the enormous rarefaction of the matter there may be 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 form for the laws connecting the lines discussed by Nicholson and those connecting the ordinary line-spectra considered in this paper.

Proceeding to consider systems of a 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 potential 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 configurations of the system will exist in which the kinetic energy of every electron is equal to the frequency of revolution multiplied by

where

is an entire number and

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

for every electron is equal to

. Considering that the ratio of kinetic energy to frequency for a particle rotating in a circular orbit is equal to

times the angular momentum round the centre of the orbit, we are therefore led to the following simple generalization of the hypotheses mentioned on p[p. 15] and [22].

“In any molecular system consisting of positive nuclei and electrons in which the nuclei are at rest relative to each other and the electrons move 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

, where

is Planck’s constant”[22].

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 foundation of it from another point of view.

April 5, 1913.

[1] Communicated by Prof. E. Rutherford, F.R.S.

[2] E. Rutherford, Phil. Mag. xxi. p. 669 (1911).

[3] See also Geiger and Marsden, Phil. Mag. April 1913.

[4] J. J. Thomson, Phil. Mag., vii. p. 237 (1904).

[5] See f. inst., ‘Théorie du rayonnement et les quanta.’ Rapports de la reunion à Bruxelles, Nov. 1911. Paris, 1912.

[6] See f. inst., M. Planck, Ann. d. Phys. xxxi. p. 758 (1910); xxxvii. p. 642 (1912); Verh. deutsch. Phys. Ges. 1911, p. 138.

[7] A. Einstein, Ann. d. Phys. xvii. p. 102 (1905): xx. p. 199 (1906); xxii. p. 180 (1907).

[8] 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. Zeitschr. xii. p. 409 (1911), Verh. deutsch. Phys. Ges. 1912, p. 431; F. A. Lindemann, Verh. deutsch. Phys. Ges. 1911, pp. 482, 1107; F. Haber, Verh. deutsch. Phys. Ges. 1911, p. 1117.

[9] J. W. Nicholson, Month. Not. Roy. Astr. Soc. lxxii. pp. 49, 139, 677, 693, 729 (1912).

[10] See f. inst. N. Bohr, Phil. Mag. xxv. p. 24 (1913). The conclusion drawn in the paper cited is 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 loss of more than one electron (comp. Phil. Mag. xxiv. p. 672 (1912)).

[11] F. Paschen, Ann. d. Phys. xxvii. p. 565 (1908).

[12] E. C. Pickering, Astrophys. J. iv. p. 369 (1896); v. p. 92 (1897).

[13] A. Fowler, Month. isot. Kov. Astr. Soc. lxxiii. Dec. 1912.

[14] W. Ritz, Phys. Zeitschr. ix. p. 521 (1908).

[15] J. W. Nicholson, loc. cit. p. 679.

[16] A. Einstein, Ann. d. Phys. xvii, p. 146 (1905).

[17] R. W. Wood, Physical Optics, p. 518 (1911). Phil. Mag. S. 6. Vol. 26. No. 151. July 1913.

[18] Compare J. J. Thomson, Phil. Mag. xxiii. p. 456 (1912).

[19] E. Rutherford, Phil. Mag. xxiv. pp. 453 & 893 (1912).

[20] Loc. cit.

[21] Loc. cit.

[22] Communicated by the Author.

PART II.—SYSTEMS CONTAINING ONLY A SINGLE NUCLEUS [23] [24].

[§1. General Assumptions.]

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 distances 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

rays as an observed example of a process of this kind, an

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 large-angle scattering of

-rays, Rutherford[25] 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 Röntgen radiation[26]. The total experimental evidence supports the hypothesis[27] that the actual number of electrons in a neutral atom with a few exceptions 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. 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 where

, where

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 [§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 [§3] and [§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 [§5] it will be shown that it is possible from the theory to calculate the minimum velocity of cathode rays necessary to produce the characteristic Röntgen radiation from the element, and that this is in approximate agreement with the experimental values.

In §6 the phenomena of radioactivity will be briefly considered in relation to the theory.

[§2. Configuration and Stability of the Systems.]

Let us consider an electron of charge

and mass

which moves in a circular orbit of radius

with a velocity

small compared with the velocity of light. Let us denote the radial force acting on the electrons by

;

will in general be dependent on

. The condition of dynamical equilibrium gives

Introducing the condition of universal constancy of the angular momentum of the electron, we have

From these two conditions we now get

and for the frequency of revolution

consequently

If

is known, the dimensions and frequency of the corresponding orbit are simply determined by (1) and (2). For a ring of

electrons rotating round a nucleus of charge

we have (comp. Part I., [p. 20])

The values for

from

to

are given in the table on [p. 32].

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. 24]) 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

Putting in (1), (2), and (3)

,

, and

we get

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,

must be small compared with

. 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 is not small compared with the velocity of light, the constancy of the angular momentum no longer involves 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 considerations 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

as that given by (1), while the quantity

in the expressions for

and

is replaced by

and in the expression for

by

As stated in [Part I.], a calculation based on the ordinary mechanics gives the result, that a ring of electrons rotating round a positive nucleus in general is unstable for displacements 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 displacements 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 is included in the condition of stability in [§1]. Consider a ring of electrons rotating round a nucleus, and assume that the system is in dynamical equilibrium and that the radius of the ring is

, the velocity of the electrons

the total kinetic energy

, and the potential energy

. As shown in Part I. ([p. 21]) we have

. 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

. In this case we have

, and on account of the uniformity of the angular momentum

and

. Using the relation

, we get

We see that the total energy of the new configuration is greater than in the original. According to the condition of stability in [§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

, where

is the frequency,

Planck’s constant, and

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

,

,....

, respectively, and suppose that the electrons, under influence of extraneous forces, rotate in circular orbits parallel to the original plane with the same radii 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

, ....

, higher than the second, the increase of the potential energy of the system is given by

where

is the radius of the ring,

the charge on the nucleus, and

the number of electrons. According to the condition of stability in [§1] the system is stable for the displacements considered, if the above expression is positive for arbitrary values of

,....

. By a simple calculation it can be shown that the latter condition is equivalent to the condition

where

denotes the whole number (smaller than

) for which

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[28].

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. From such considerations 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 case of vibrations in the plane of the ring. This assumption is supported 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 nebulæ (see Part I. p[p. 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

and

from

to

.

We see from the table that the number of electrons which can rotate in a single ring round a nucleus of charge

increases only very slowly for increasing

; for

the maximum value is

; for

,

; for

,

. We see, further, that a ring of

electrons cannot rotate in a single ring round a nucleus of charge ne unless

.

In the above we have supposed 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, and if the ratio between their frequencies 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 obtained on this assumption may also be considered as sufficient.

We have assumed in [§1] that the electrons in the atoms rotate in coaxial rings. The calculation indicates 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

uniformly distributed along the circumference of a circle of radius

.

At a point distant

from the plane of the ring, and at a distance

from the axis of the ring, the electrostatic potential is given by

Putting in this expression

and

, and using the notation

we get for the radial force exerted on an electron in a point in the plane of the ring

where

The corresponding force perpendicular to the plane of the ring at a distance

from the centre of the ring and at a small distance

from its plane is given by

where

A short table of the functions

and

is given on [p. 35.]

Next consider a system consisting of a number of concentric rings of electrons which rotate in the same plane round a nucleus of charge

. Let the radii of the rings be

,

,...., and the number of electrons on the different rings

,

,....

Putting

, we get for the radial force acting on an electron in the

th ring

where

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. 28], we can, by help of the above, determine

,

, .... The calculation can be made by successive approximations, starting from a set of values for the

’s, and from them calculating the

’s, and then redetermining the

’s by the relation (1) which gives

, 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. 30.], the interaction of the rings ought strictly to be taken into account. This interaction will involve that the quantities

are not constant, as for a single ring rotating round a nucleus, but will vary with the radii of the rings; the variation in

, 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. 31.], if for every ring we replace

by a quantity

, determined by the condition that

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

. Using the same notation as above, we get

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

concentric rings rotating in a plane round a nucleus of charge

, and let us assume that the electrons in the different rings are displaced perpendicular to the plane by

,

,....

respectively. With the above notation the increase in the potential energy of the system is given by

The condition of stability is that this expression is positive for arbitrary values of

,....

. 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

and

for every fifth degree from

to

, gives an estimate of the order of magnitude of these functions:—

indicates the ratio between the radii of the rings

. The values of

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

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 without being unstable is approximately equal to that calculated on [p. 32.] 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

by the total charge on the nucleus and on the electrons of inner rings.

If a 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.

[§3. Constitution of Atoms containing very few Electrons.]

As stated in [§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.

For sake of brevity we shall, by the symbol

, refer to a plane system of rings of electrons rotating round a nucleus of charge satisfying the condition of the angular momentum of the electrons with the approximation used in [§2].

,

,... are the numbers of electrons in the rings, starting from inside. By

,

,... and

,

,... we shall denote the radii and frequency of the rings taken in the same order. The total amount of energy

emitted by the formation of the system shall simply be denoted by

.

N=1. Hydrogen.

In [Part I.] we have considered the binding of an electron by a positive nucleus of charge

, 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 multiples of the value

, where

is Planck’s constant. The formula found for the frequencies of the spectrum was

where

and

are entire numbers. Introducing the values for

,

, and

used on [p. 29], we get for the factor before the bracket

[29]; the value observed for the constant in the Balmer spectrum is

.

For the permanent state of a neutral hydrogen atom we get from the formula (1) and (2) in [§2], putting

These values are of the order of magnitude to be expected. For

we get

, which corresponds to

; the value for the ionizing potential of a hydrogen atom, calculated by Sir J. J. Thomson from experiments on positive rays, is

[30]. No other definite data, however, are available for hydrogen atoms. For sake of brevity, we shall in the following denote the values for

,

, and

corresponding to the configuration

by

,

and

.

At distances from the nucleus, great in comparison with

, the system

will not exert sensible forces on free electrons. Since, however, the configuration:

corresponds to a greater value for

than the configuration

, 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

is only

, a hydrogen atom cannot be expected to be able to acquire a double negative charge.

N=2. Helium.

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

a spectrum is emitted, expressed by

This spectrum includes the spectrum observed by Pickering in the star

Puppis and the spectra recently observed by Fowler 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 charged helium atom, we get

At distances from the nucleus great compared with the radius of the bound electron, the system

will, to a close approximation, act on an electron as a simple nucleus of charge

. For a system consisting of two electrons and a nucleus of charge

, 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 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

Since

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. 38], we get

these values are of the same order of magnitude as the value observed for the ionization potential in helium,

assume[31], and the value for the frequency of the ultra-violet absorption in helium determined by experiments on dispersion

[32].

The frequency in question may be regarded as corresponding to vibrations in the plane of the ring (see [p. 30]). The frequency of vibration of the whole ring perpendicular to the plane, calculated in the ordinary way (see [p. 32]), is given by

. 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 two-thirds of the number to be expected. (Using

the value calculated is

.)

For a configuration of a helium nucleus and three electrons, we get

Since

for this configuration is smaller than for the configuration

, 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 “affinity” for free electrons[33].

In a later paper it will be shown that the theory offers a simple explanation of the marked difference in the tendency of hydrogen and helium atoms to combine into molecules.

N=3. Lithium.

In analogy with the cases of hydrogen and helium we must expect that during the binding of an electron by a nucleus of charge

, a spectrum is emitted, given by

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[34] 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

where

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

. The lines in question correspond to

; if we for

put

, we get lines coinciding with lines of the ordinary Balmer-spectrum of hydrogen. If we in the above formula put

, we get series of lines in the ultra-violet. If we put

we get only a single line in visible spectrum, viz.: for

which gives

, or a wave-length

closely coinciding with the wave-length

of one of the lines of unknown origin in the table quoted by Nicholson. In this table, however, no lines occur corresponding to

.

For the permanent state of a lithium atom with two positive charges we get a configuration

The probability 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:

Since