On the Structure of Atoms1

Niels Bohr.

The General Picture of the Atom

The present state of atomic theory is characterized by the fact that we can not only regard the existence of atoms as unquestionably proven, but may even claim a thorough knowledge of the constituent parts of individual atoms. In the present case there is no possibility of giving a survey of the development of the science that led to such a result. I shall merely recall the discovery of electrons at the end of the last century; this discovery confirmed and finally clarified the ideas concerning the atomistic nature of electricity, ideas which had been slowly developing since Faraday’s discovery of the fundamental laws of electrolysis and Berzelius’s electrochemical theory; the brilliant triumph of these ideas was Arrhenius’s theory of electrolytic dissociation. The discovery of electrons and the elucidation of their properties was the result of the work of a large number of investigators, among whom Lenard and J. J. Thomson should especially be mentioned. The ingenious attempt by the latter investigator to develop ideas about the structure of atoms on the basis of the electron theory proved to be an extremely important step in the development of atomic theory. The conclusion of the preliminary stage in the development of our knowledge of the constituent parts of the atom was, however, the discovery of atomic nuclei, made by Rutherford, whose work on radioactive substances, discovered at the beginning of our century, enriched physics and chemistry in many respects.

According to our present ideas, the atom of an element consists of a nucleus, possessing a positive electric charge and serving as the location of almost the entire atomic mass, and of a certain number of electrons with equal negative charges and masses; the electrons move at such distances from the nucleus that are extraordinarily

large in comparison with the dimensions of the nucleus and the electrons. In such a picture, from the very first glance one is struck by the unusual similarity to a planetary system, like our solar system. The simplicity of the laws of celestial systems is closely connected with the fact that the dimensions of the individual luminaries are small in comparison with the dimensions of their orbits. The same relations in the structure of atoms make it possible directly to understand the essential features of natural phenomena, since they depend on the properties of the elements. These relations allow the properties of the elements to be divided into two sharply distinct classes. To the first class belong most of the ordinary physical and chemical properties of the elements: state, color, chemical capacity for reactions. These properties depend on the motion of the electronic system and on the type of changes in this motion that are caused by various external influences. Owing to the considerable mass of the nucleus in comparison with the mass of the electrons, and to the small dimensions of the nucleus in comparison with the dimensions of the electronic orbits, the motion of the electronic system depends very little on the mass of the nucleus and is determined with great accuracy by the total electric charge of the nucleus. In particular, the internal structure of the nucleus and the manner in which electricity and mass are distributed among the individual particles of the nucleus can exert only an insignificant influence on the features of the electronic system surrounding it. On the other hand, the structure of the nucleus determines the second class of properties of the elements, manifested in radioactivity. In radioactive processes we observe the explosion of the nucleus, in which positive and negative particles, the so-called $\alpha$ and $\beta$ particles, are ejected from it with colossal velocities. Our ideas about the structure of the atom thus give an immediate explanation of the complete absence of any connection between the two classes of properties of the elements. This, as is known, appears especially sharply in the existence of elements possessing, to a tremendous approximation, identical ordinary physical and chemical properties, but having different atomic weights and entirely different radioactive properties. Such elements, whose existence was first discovered by Soddy and other investigators in the study of the chemical properties of radioactive substances, are called isotopes, in accordance with the classification of substances by their ordinary physical and chemical properties. I have no need to set forth here1 how it later became apparent that isotopy is found not only among radioactive elements, but also among ordinary stable substances; a large number of elements that had until now been regarded as simple proved, on the basis of Aston’s well-known investigations, to be a mixture of isotopes with different atomic weights. The question of the internal structure of nuclei, which acquired new interest in connection

with these investigations, very little has so far been clarified, although the path of inquiry in this direction was indicated by Rutherford’s investigations on the disintegration of nuclei under bombardment by α-rays; regarding these investigations one may say that they opened a new epoch of natural science, since in them it proved possible for the first time artificially to transform one element into another. In what follows, however, we shall confine ourselves to a consideration of the ordinary physical and chemical

Fig. 1. Natural system of the elements.

properties of the elements and to an exposition of attempts to explain them on the basis of the indicated conception of the structure of atoms.

It is well known that, according to their physical and chemical properties, the elements can be arranged in the so-called natural system, revealing a peculiar kinship among the various chemical elements. Mendeleev and Lothar Meyer first showed that the chemical and physical properties of the elements display a distinct periodicity if they are arranged in a row coinciding in its main features with the row of successively increasing atomic weights. In Fig. 1 there is given a table of the natural, or periodic, system of the elements

the elements being arranged not as is usually done, but with a certain modification first indicated by the Danish chemist Julius Thomsen, who did much in this field. In the drawing the elements are denoted by their usual chemical symbols, and the various vertical columns correspond to the so-called periods. Elements in adjacent columns that possess homologous chemical and physical properties are connected by straight lines. The quadrangular frames in the higher periods around certain elements, whose properties show typical deviations from the indicated simple periodicity of the elements of the first periods, have a significance to which we shall return later.

In the course of the development of our ideas about the structure of atoms, the characteristic features of the natural system received a remarkably simple elucidation. We thus arrived at the conclusion that the number standing in the drawing next to the symbol of each element and indicating the place of the corresponding element in the system—the so-called atomic number—is exactly equal to the number of electrons moving, in a neutral atom, around the nucleus.

This simple law, though in an imperfect form, was first indicated by van den Broek, after determinations of the number of electrons in the atom by J. J. Thomson’s method and Rutherford’s investigations, which made it possible to carry out a direct measurement of the charge of the atomic nucleus, had rendered this law highly plausible.

As we shall see, the law just named has received convincing confirmations from various quarters, in particular on the basis of Moseley’s famous investigations of the X-ray spectra of the elements. I may perhaps also recall that the simple connection between the atomic number and the charge of the nucleus leads to a direct understanding of the law manifested in changes of the chemical properties of radioactive substances—changes accompanying the emission of $\alpha$ and $\beta$ particles—and expressed so simply in the so-called radioactive displacement law.

Stability of the Atom and Electrodynamic Theory

In attempting to establish a close connection between the properties of the elements and the structure of atoms, we encounter, however, profound difficulties; despite the analogy indicated earlier, there is an essential difference between the atom and a planetary system. The motions of bodies in a planetary system, obeying the general law of gravitation, are not fully determined by this law, but depend essentially on the previous history of the system. Thus, for example, the length of the year is determined not only by the masses of the sun and the earth, but also by the relations that existed at the formation of the solar system and are unknown to us in detail.

If one fine day an alien celestial body were to pass through the solar system near the Earth, we would have to be prepared for the fact that, beginning from that day, the length of the year would start to differ substantially from its former value.

The situation is entirely different in atoms. The definite, unchanging properties of the elements require that the state of the atom should not undergo strong changes under the influence of external actions. As soon as the atom is again left to itself, its particles must arrange themselves and move in a manner that is completely determined by the electric charges and masses of the particles. The most weighty evidence for this is provided by spectra, i.e. by that special radiation which, under certain conditions, is emitted by a substance and can be investigated with comparatively great precision by suitable instruments. It is well known that the wavelengths of the spectral lines of the elements, which in many cases can be measured with an accuracy greater than

\[ \frac{1}{1{,}000{,}000}, \]

under identical external conditions remain the same within the limits of observational accuracy, quite independently of the preceding manipulations performed on the substance.

It is precisely on this that spectral analysis is based, being for chemists so invaluable an aid in detecting elements, and having made it possible to conclude that even on the most distant luminaries there are elements with exactly the same properties as on the Earth.

Thus, on the basis of our picture of the structure of atoms, it is impossible to account for the characteristic stability of atoms required for explaining the properties of the elements, so long as we rely only on the ordinary laws of mechanics. The situation does not become any more favorable if one has recourse to the well-known electrodynamic laws established by Maxwell on the basis of the great discoveries of Ørsted (Örsted) and Faraday, made in the first half of the last century.

Maxwell’s theory could not only explain the already known electrical and magnetic phenomena—its great triumph, as is well known, was Hertz’s discovery of electromagnetic waves predicted by the theory and now applied on so broad a scale in wireless telegraphy. For a time it also seemed that this theory, especially in the form developed by Lorentz and Larmor in connection with the atomistic interpretation of electricity, was destined to serve as the basis for a detailed explanation of the properties of the elements. It is enough for me merely to recall the general attention attracted by the natural and simple explanation given by Lorentz of the principal features of the phenomenon discovered by Zeeman, consisting in a special change of the spectral lines in the case when a luminous body is placed in a magnetic field. Lorentz supposed,

that the light of a spectral line is emitted by an electron performing a harmonic oscillation about some position of equilibrium, just as the electromagnetic waves of wireless telegraphy are emitted as a consequence of electrical oscillations in the antenna; he showed that the change in spectral lines observed by Zeeman corresponds exactly to changes in the motion of the oscillating electron caused by a magnetic field. It proved impossible, however, on this basis to give a detailed explanation of the spectra of the elements, or even an explanation of a general type for the laws that are fulfilled with great accuracy for the wavelengths of spectral lines; these laws were discovered in the well-known works of Balmer, Rydberg, and Ritz. In the picture of the structure of the atom adopted by us, these difficulties appear still more clearly, for, remaining on the ground of classical electrodynamic theory, we cannot at all understand how a spectrum consisting of sharp lines can arise. This theory is in general incompatible with the existence of atoms of the structure described, since the motions of the electrons must be accompanied by a continuous radiation of the atom’s energy until the electrons fall onto the nucleus.

The Origin of Quantum Theory.

A way out of the indicated difficulties was meanwhile found in considerations borrowed from the so-called quantum theory. This theory is equivalent to a complete break with the conceptions that had hitherto been used in attempts to explain natural phenomena. Its beginning, as is known, was laid in 1900 by Planck in his investigations of the law of thermal radiation; this law, because of its independence of the special properties of matter, is a touchstone for testing the applicability of the laws of classical physics to atomic processes. Planck considered the radiation equilibrium between a number of systems with the same properties as the system used by Lorentz in his theory of the Zeeman effect; in doing so he not only showed that classical electrodynamics cannot explain the phenomena of thermal radiation, but found that complete agreement with the law of thermal radiation is quite attainable if, in contradiction to classical theory, one assumes that the energy of an oscillating electron changes not continuously, but in such a way that the energy of the system is always equal to an integer number of the so-called “quanta” of energy. The magnitude of such a quantum must be proportional to the frequency of oscillation of the particle; with respect to this frequency it is also assumed, as in classical theory, that it is equal to the frequency of oscillation of the emitted light. The factor of proportionality, the so-called Planck constant, in accordance with the very character of the reasoning, must be regarded as a new universal constant of nature, like the speed of light and the charge and mass of the electron.

Planck’s unexpected result at first stood entirely apart in natural science; however, thanks to Einstein’s important work in this field, within a few years the indicated conclusion received manifold application. First of all, Einstein drew attention to the fact that the requirement that the values of the energy of the oscillations of particles be limited could be tested by investigating the heat capacity of crystalline bodies, since in these bodies one has to do with similar oscillations—not merely of a single electron, but of an entire atom about its position of equilibrium in the crystal lattice. Agreement with Planck’s theory, discovered in this case by Einstein, was confirmed, as is known, by very important work by other authors.

Next, Einstein also emphasized another consequence of Planck’s result: the radiant energy of an oscillating particle can be emitted and absorbed only in so-called “quanta of radiation,” whose magnitude is equal to the product of Planck’s constant and the frequency of the oscillations. Seeking to give a visual interpretation to this result, Einstein proposed the so-called “hypothesis of light quanta,” according to which radiant energy, contrary to Maxwell’s electromagnetic theory of light, must propagate not in waves, but in atoms of light of negligible size; each such atom must contain energy corresponding to a quantum of radiation. This conception led Einstein to the well-known theory of the photoelectric effect, which illuminated the phenomenon in question in an entirely new way, a phenomenon incomprehensible in classical theory; the predictions of Einstein’s theory have in recent years received such exact experimental confirmation that measurements of the photoelectric effect apparently provide the most accurate method of determining Planck’s constant. Despite the heuristic value of the hypothesis of light quanta, it stands in complete contradiction to the phenomena of interference and is unsuitable for clarifying the question of the nature of radiation. It is enough merely to recall that interference phenomena give us the only method of investigating the properties of radiation and make it possible to attach a definite meaning to the frequency of oscillations that determines the magnitude of the light quantum.

In the following years, attempts were made from various sides to apply the quantum point of view to questions of the structure of the atom, with the center of gravity being shifted now to one, now to another consequence obtained by Einstein from Planck’s result. Among the best-known attempts in this direction, which, however, yielded no clear results, I may name the works of Stark, Sommerfeld, Hasenöhrl, Haas, and Nicholson. To the same period belongs the work of the Danish chemist Bjerrum, although it was not directly connected with the question of atomic structure; nevertheless, it was significant for the development of quantum theory. In 1912 Bjerrum drew attention

attention to the fact that the rotation of gas molecules can be studied from changes in certain absorption lines when the temperature is changed. At the same time he pointed out that the effect should not consist in a continuous broadening of the lines, as would follow from the classical theory, which places no restriction on the rotational motion of molecules; in connection with the quantum theory Bjerrum suggested that the lines must split into a series of components, corresponding to a series of discrete rotational motions possible for the molecules. This prediction was confirmed in the most beautiful way a few years later by the experiments of the Swedish investigator Eva von Bahr; this phenomenon must still be regarded as one of the clearest pieces of evidence for the reality of the quantum theory, although from the modern point of view the original interpretation must be altered in important details.

Quantum theory of the structure of atoms.

The question of the detailed development of the quantum theory received new illumination as a result of Rutherford’s discovery of atomic nuclei (1911). We have already seen how, after this discovery, it became clear that classical conceptions do not make it possible to understand the most essential properties of atoms. Hence there arose a search for such a formulation of the principles of the quantum theory as would be adapted to the requirements of stability of the structure of atoms and to the properties of the observed radiation. Such a formulation was proposed in 1913 by the lecturer in the form of two postulates, whose content may be expressed as follows:

I. Among the conceivable states of motion of an atomic system there is a series of so-called stationary states, with respect to which it is assumed that the motion of the particles in these states, while obeying to a considerable extent the classical mechanical laws, is distinguished, however, by a peculiar mechanically inexplicable stability, as a result of which it follows that any residual change in the motion of the system must consist in a complete transition from one stationary state to another.

II. In the stationary states themselves, contrary to the classical electromagnetic theory, no radiation occurs; however, the process of transition between two stationary states may be accompanied by electromagnetic radiation possessing the same properties as the radiation emitted, on the basis of the classical theory, by an electric particle performing harmonic oscillations with a constant frequency. This frequency, however, is not found,

however, in a simple relation to the motion of the particles of the atom and is determined by the condition:

\[ h\nu = E' - E'' \]

where \(h\) is Planck’s constant, and \(E'\) and \(E''\) are the values of the atomic energy in two stationary states, forming the initial and final states of the radiation process. Conversely, illumination of the atom by electromagnetic waves of this frequency may lead to an absorption process, transferring the atom from the final state to the initial one.

The first postulate concerns the general stability of the atom, expressed in the chemical and physical properties of the element, while the second corresponds above all to the existence of spectra consisting of sharp lines. The quantum description contained in the latter postulate is the starting point for interpreting the empirical laws of spectra indicated above. The most general of these laws, the combination principle, established by Ritz, states that the frequency of oscillation \(\nu\) for each spectral line of a certain element may be represented by the formula:

\[ \nu = T'' - T' \]

where \(T''\) and \(T'\) are two so-called “spectral terms,” belonging to the totality of terms characteristic of the given element.

With the aid of our postulates this law is interpreted directly on the basis of the assumption that the spectrum is emitted in transitions between a series of stationary states in which the numerical values of the energy of the atom are equal to the values of the spectral terms multiplied by Planck’s constant. Such an interpretation of the combination principle differs from ordinary electrodynamical conceptions not only by the assumption that there is no simple connection between the motion of the atom and the light emitted; the difference between our considerations and the foundations on which the ordinary description of nature rests may become especially clear if attention is drawn to the fact that the appearance of spectral lines corresponding to combinations of some definite term with various others is interpreted by saying that the properties of the atom’s radiation depend not only on the state of the atom at the beginning of the radiation process, but also on the state into which the atom passes in this process. At first glance, one might perhaps think that the formal interpretation of the combination principle set forth here can hardly, for this reason, be connected with our data on the constituent parts of the atom, based on experiments interpreted with the aid of classical mechanical and electrodynamical laws. Closer investigation has shown, however, that it is possible to establish a close connection between the various spectra of the elements and the structure of atoms on the basis of the postulates presented.

The Spectrum of Hydrogen.

The spectrum of hydrogen is the simplest of all spectra known to us; the frequencies of vibration of the lines of this spectrum can, as is known, be represented with great accuracy by Balmer’s formula:

\[ \nu = K \left( \frac{1}{n''^{2}} - \frac{1}{n'^{2}} \right) \]

where \(K\) is a constant, and \(n'\) and \(n''\) are two integers. In this spectrum, therefore, we encounter a simple series of spectral terms of the form

\[ \frac{K}{n^{2}}, \]

which decrease regularly as the number \(n\) increases. In accordance with the postulates, we must therefore think that every line of the hydrogen spectrum is emitted in the process of transition between two stationary states of the hydrogen atom, which are members of a series of such states, the numerical value of the energy in the stationary states of hydrogen being equal to

\[ \frac{hK}{n^{2}}. \]

According to our picture of the structure of the atom, the hydrogen atom consists of a positive nucleus and one electron, which describes, with great approximation, a periodic elliptical orbit with the nucleus at one of the foci (since in this case the usual mechanical representations may be applied). A simple calculation shows that the major axis of the orbit is inversely proportional to the work that must be performed for the complete removal of the electron from the nucleus; in connection with what has been said above, we must suppose that this work in the stationary states is precisely equal to

\[ \frac{hK}{n^{2}}. \]

We arrive, consequently, at a collection of stationary states for which the axis of the electronic orbits assumes a series of discrete values proportional to the square of an integer. This is represented schematically in Fig. 2. For simplicity the electronic orbits in the stationary states are shown as circles, although in reality the theory imposes no restrictions on the eccentricity of the orbits and determines only the length of the major axis. The arrows symbolize the transition processes corresponding to the red and green hydrogen lines, \(H_{\alpha}\) and \(H_{\beta}\), whose vibration frequencies are determined by Balmer’s formula for \(n'' = 2\), \(n' = 3\) and \(4\). Further represe—

Fig. 2. Schematic representation of the stationary states of the hydrogen atom.

the transition processes corresponding to the first three lines of the series of ultraviolet lines discovered in 1914 by Lyman; their frequencies are given by Balmer’s formula for \(n''=1\). In addition, the transition process is depicted for the first line of the infrared series discovered by Paschen several years earlier; this series is determined by Balmer’s formula for \(n''=3\).

The interpretation of the hydrogen spectrum set forth above leads naturally to an interpretation of this spectrum as evidence of the process by which the electron is “bound” by the nucleus.

The largest spectral term, with number 1, corresponds to the final stage of the binding process; the smaller terms, determined by larger numbers, correspond to stationary states defining the initial stages of the binding process, where the electron’s orbits still have considerable dimensions and where the work required to remove the electron from the nucleus is still small. The final stage of the binding process we may call the “normal state” of the atom; it differs from the other stationary states by the property that it can be changed only by the expenditure of energy capable of transferring the electron into one of the orbits of larger dimensions, corresponding to an earlier stage of the binding process.

The magnitude of the electron orbit in the normal state, calculated on the basis of the interpretation presented above, almost coincides with the values for the sizes of atoms of the elements calculated, by means of the kinetic theory of gases, from the properties of gases. We must, however, suppose, as an immediate consequence of the stability of the stationary states required by the postulates, that the interaction between two atoms in a collision cannot be fully described by the classical mechanical laws; therefore a comparison of the size of the atom and the size of the orbit of the normal state cannot be carried through completely.

A closer connection between the spectrum and the atomic model can, however, be obtained by investigating motions in stationary states with large numbers, where the size of the electron orbit and the period of revolution change comparatively little in passing from one stationary state to the next. Namely, it could be shown that, in transitions between two stationary states whose difference of numbers is small in comparison with the magnitude of the numbers themselves, the frequency of the emitted light very nearly coincides with the frequency of one of the harmonic components of the oscillation into which the motion of the electron can be resolved, and also with the frequency of one of the wave systems of light emitted, according to the classical electrodynamical laws, as a result of the motion of the electron. The requirement of such a coincidence within the indicated limits, where the stationary states relatively

differ little from one another, is equivalent to saying that the constants of Balmer’s formula are expressed by the relation:

\[ K=\frac{2\pi^2 e^4 m}{h^3} \]

where \(e\) and \(m\) are the charge and mass of the electron, and \(h\) is Planck’s constant. The relation just given has in fact been confirmed with that considerable accuracy which is allowed by the experimental values of the quantities \(e\), \(m\), and \(h\) (especially on the basis of Millikan’s excellent investigations).

This result has not only the significance of proving the connection between the hydrogen spectrum and the model of the hydrogen atom—a connection which becomes all the more striking if one takes into account the difference between the postulates and the classical mechanical and electrodynamical laws. This result also indicates how quantum theory may be interpreted as a natural modification of the fundamental concepts of classical electrodynamics, despite the sharp difference noted above. We shall return below to this important question. First, however, let us set forth how the explanation of the hydrogen spectrum on the basis of the postulates proved useful for elucidating, in various ways, the similarity in the properties of different elements.

Similarity between elements.

The considerations set forth above can be applied directly to the process of binding an electron by a nucleus with an arbitrarily specified charge. Calculation shows that in the stationary state corresponding to a given value of \(n\), the major axis of the orbit is inversely proportional to the charge of the nucleus, while the work required to remove the electron from the nucleus is directly proportional to the square of the charge of the nucleus. The spectrum emitted during the binding of one electron by a nucleus whose charge is \(N\) times greater than the charge of the hydrogen nucleus can be represented by the formula:

\[ \nu=N^2 K\left(\frac{1}{n''^2}-\frac{1}{n'^2}\right) \]

If in this formula \(N\) is set equal to 2, we obtain a spectrum containing a series of lines in the visible region of the spectrum; such a spectrum was observed in certain stars and was described by Rydberg as hydrogen on the basis of its close analogy with the series of lines represented by Balmer’s formula. It was never possible to obtain these lines in pure hydrogen; however, just before the appearance of the theory of the hydrogen spectrum, Fowler observed these lines by passing a strong discharge through a mixture of hydrogen and helium. This investigator regarded these lines as hydrogen lines, since at that time nothing was yet known of the possibility of the existence of different substances with very complex

properties. On the basis of the theory it became clear, however, that the indicated lines belong to the spectrum of helium, emitted, however, not by the simple neutral helium atom, but by ionized helium, having only one electron revolving around a nucleus with double charge. Hence a new feature of similarity between the properties of the elements became clear—a similarity of such a type as fully accords with our modern conceptions of the structure of atoms; according to these conceptions, the physical and chemical properties of an element are determined above all by the electric charge of the atomic nucleus.

Soon after this question had been clarified, the existence of a general similarity of this kind in the properties of the elements was discovered; this was achieved by Moseley in his well-known investigations of the characteristic X-ray spectra of the elements; the possibility of such an investigation was provided by Laue’s discovery of the interference of X-rays in crystals and by the subsequent investigations of the Braggs (W. H. and W. L. Bragg).

It turned out that the X-ray spectra of different elements possess a simpler structure and a greater similarity than the optical spectra of the elements: they change from element to element in a manner fully corresponding to the formula written above for the binding of an electron by the nucleus, if in this formula \(N\) is set equal to the atomic number of the corresponding element. This formula expresses, with great approximation, the frequencies of oscillation of the strongest X-ray lines, if small integers are substituted for \(n'\) and \(n''\).

This discovery was very important in many respects. First of all, the similarity of the X-ray spectra of different elements proved so simple that it became possible to determine unambiguously the atomic numbers for all known elements and thereby to predict the values of the atomic numbers for all those elements still unknown for which there is a place in the natural system. In Fig. 3, for two characteristic X-ray lines of the so-called \(K\)-group, possessing the greatest penetrating power, the square roots of the frequencies of oscillation are presented as functions of the atomic number.

With very great regularity the points lie on straight lines; the uniform course of these straight lines is due to the fact that, in addition to the atomic numbers corresponding to already known elements, empty places are left for elements still unknown, as, for example, between molybdenum (42) and ruthenium (44), just as in Mendeleev’s original representation of the natural system of the elements. Furthermore, the simple laws of X-ray spectra served as confirmation of general theoretical conceptions, both with respect to the fundamental character of the structure of atoms and in connection with the explanation of spectra. The resemblance of X-ray spectra to the spectrum emitted in

Fig. 3. Square roots of the frequencies of two characteristic X-ray lines as a function of atomic number.

the binding of a single electron by the atomic nucleus is based simply on the fact that, in the case of X-ray spectra, transitions occur between stationary states accompanied by changes in the motion of a single electron in the inner regions of the atom, where the influence of the attraction of the nucleus is considerably greater than the repulsive forces of the other electrons.

The similarity of the remaining properties of the elements is often expressed in a much more complicated way, owing to the fact that in these cases the processes concern the motions of electrons in the outer parts of the atom, where the forces of interaction between electrons are quantities of the same order as the attraction of the nucleus, and consequently this interaction plays an essential role.

A characteristic example of this may be the filling of space by the atoms of the elements. As is known, Lothar Meyer already pointed out the peculiar periodic change in the ratio of atomic weight to density, or the so-called atomic volume, in the system of the elements. An idea of this change may be obtained from Fig. 4, which shows the variation of atomic volume as a function of atomic number. It is difficult to imagine a greater contrast than that revealed by comparison of this figure with the preceding one.

X-ray spectra change with atomic number quite uniformly, whereas atomic volumes exhibit a sharp perio-

Fig. 4. Dependence of the atomic volumes of the elements on the atomic number.

the periodicity of the change, exactly corresponding to the change in the chemical properties of the elements, expressed in the natural system.

Exactly the same occurs also in the ordinary optical spectra of the elements. But, despite the great difference between these spectra, many years ago Rydberg succeeded in detecting a certain general similarity between the spectrum of hydrogen and the spectra of other elements. Although the spectral lines of elements with high atomic numbers are combinations of a considerably more complex aggregate of spectral terms, not subordinated to a series of integers, nevertheless the terms can be arranged in series, each of which displays great similarity to a series of terms of the hydrogen spectrum.

This similarity is manifested in the fact that the empirical expression for the terms of each series can, with great accuracy, be written in the form \(\dfrac{K}{(n+a_k)^2}\), where \(K\) is the same constant as in the spectrum of hydrogen (it is often called the Rydberg constant), \(n\) is a number, and \(a_k\) is a constant, different for different series. This similarity to the hydrogen spectrum leads us directly to the interpretation of the indicated spectra as corresponding to the last stage of the process of formation of a neutral atom by the successive capture and binding of electrons by the atomic nucleus. It is clear that the electron captured last, in the initial stage of the binding process, when its orbit is still large in comparison with the orbits of the electrons bound earlier, is subjected, on the part of the latter and of the nucleus, to the action of forces,

differing little from the forces that act on the electron in the hydrogen atom when it moves in orbits of the corresponding dimensions.

The spectra for which the Rydberg law is fulfilled are emitted by elements during electric discharges under ordinary conditions and are often called arc spectra; in the case where especially powerful discharges are passed through the elements, they emit so-called spark spectra, for which it had previously not been possible to discover regularities such as those for arc spectra. Soon after the publication of the indicated interpretation of the hydrogen spectrum, Fowler (1914) found, however, that for spark spectra one can establish empirical formulae quite similar to the Rydberg law, with the sole difference that the constant \(K\) had to be replaced by a constant four times as large. We have already seen that the constant of the spectrum emitted when an electron is bound by a helium nucleus is equal to \(4K\), whence it is clear that spark spectra are emitted by ionized atoms, and that they correspond to the penultimate stage in the formation of a neutral atom in the successive capture and binding of electrons.

Absorption and excitation of spectral lines.

On the basis of the interpretation of the origin of spectra set forth above, it further proved possible to explain the peculiar laws governing the absorption spectra of the elements.

Kirchhoff and Bunsen had already shown that there exists an exact connection between the selective absorption of elements and their emission spectra, and this served as an essential basis for the application of spectral analysis to celestial bodies. From the point of view of classical theory, however, it was incomprehensible why elements in vapor form absorb some lines of the emission spectrum and not others.

On the basis of our postulates, however, we arrive at the supposition that the absorption of radiation corresponding to a definite spectral line, emitted in a transition from one stationary state of the atom to a state of lower energy, occurs when the atom returns from this latter state back to the first upon receiving the necessary energy. From this it becomes directly intelligible that under ordinary conditions a vapor or gas exhibits selective absorption only for those spectral lines which arise in the transition from some state corresponding to an early stage of the binding process into the normal state. Only at high temperatures or under the action of electric discharges, when a considerable number of atoms leave the normal state, may one expect, in agreement with experience, absorption for other lines of the emission spectrum.

Direct confirmation of our interpretation of series spectra on the basis of the postulates is obtained further in experiments concerning the excitation of spectral lines and the ionization of atoms by collisions with free electrons possessing prescribed velocities. A certain success in this field was first achieved in the well-known experiments of Franck and Hertz (1914). On the basis of these experiments it turned out that it is impossible, by means of an electron impact, to impart to an atom an arbitrary amount of energy; one can impart only the energy required to transfer the atom from the normal state into one of the other stationary states, whose existence we learn of from the spectra and whose energy is determined by the magnitude of the spectral terms. Subsequently it was possible to obtain decisive proof of the independence of the processes which, according to the postulates, lead to the emission of different lines of the spectrum; atoms thus brought into a stationary state of higher energy can, as direct experiment shows, return to the normal state while emitting light consisting of only a single spectral line. Further investigation of electron impacts, in which many physicists took part, confirmed in detail the assumptions concerning the origin of series spectra. In particular, it was possible to show that, for the ionization of atoms by electron impacts, an energy is required which corresponds exactly to the work theoretically necessary for removing from the atom the last bound electron; this work is directly determined as the product of Planck’s constant and the spectral term corresponding to the normal state, which, according to what has been said above, serves as the limiting value of the frequencies of oscillations of the spectral series associated with selective absorption.

The quantum theory of multiply periodic systems.

Relying directly on the fundamental postulates of quantum theory, it was thus possible to take into account, in general outline, the properties of the elements; for a more detailed explanation of these properties, further development of the theory was required. In recent years, through the development of a formal method, a broader theoretical foundation has been created, making it possible to determine the stationary states of electronic motions of a more general type. For purely periodic motion of the simple harmonic oscillator and, at least in the first approximation, for the motion of a single electron around a positive nucleus, the totality of stationary states can be simply subordinated to a certain sequence of integers. The stationary states of motions of the more general type just mentioned, the so-called multiply periodic motions, however form a more complicated totality, in which each state, by means of the indicated formal methods, is characterized by several integers, so-called—

called “quantum numbers.” Many investigators took part in the development of the theory, and the first use of several quantum numbers can be found in the works of Planck himself. A reason for decisive further success in the study of the atom was the explanation of the fine structure of the hydrogen lines, observed with spectroscopes of high resolving power; this explanation was given by Sommerfeld in 1915. The fine structure arises from the fact that already in the hydrogen atom we are dealing not with purely periodic motion. The electronic orbit performs a slow precessional motion in its own plane as a result of the change in the mass of the electron, depending on the velocity of motion, as required by the theory of relativity. Owing to this, the motion becomes doubly periodic, and, in order to determine stationary states, besides the number characterizing the terms in Balmer’s formula, which we shall call the “principal quantum number,” since it primarily determines the energy of the atom, one more quantum number is required, which we shall call the “subsidiary quantum number.”

Fig. 5. Electronic orbits in stationary states of the hydrogen atom, taking into account the change of the electron mass with velocity.

The motions in stationary states determined in this way are presented in Fig. 5, which shows the relative magnitude and form of the electronic orbits. Each orbit is denoted by the symbol \(n_k\), where \(n\) is the principal quantum number and \(k\) the subsidiary one. All orbits with one and the same principal quantum number have, to a first approximation, one and the same major axis, while orbits with the same value of \(k\) possess the same length of the parameter, i.e. the smallest chord passing through the focus of the orbit. The energy values for different states with the same \(n\), but different \(k\), differ little from one another; therefore, to each hydrogen line corresponding to definite values \(n'\) and \(n''\) in Balmer’s formula there corresponds a series of different transition processes, for which the frequencies of the vibrations of the emitted light, calculated on the basis of the second postulate, are somewhat different. Sommerfeld was able to prove that the components calculated in this way for each hydrogen line agree, within the accuracy of experiment, with observations of the fine structure of the hydrogen spectrum. The arrows in the figure denote tra-

transitions corresponding to the components of the red and green lines of the hydrogen spectrum, whose oscillation frequencies are obtained from Balmer’s formula for \(n'' = 2\) and \(n' = 3\) or \(4\).

In considering the figure, however, one must not forget that the depiction of the orbits is incomplete, since on this scale it was impossible to indicate the slow precession. This precession is so slow that even in the most rapidly rotating orbits the electron manages to complete 40,000 revolutions before the perihelion completes one revolution. Nevertheless, this precession is the sole basis for the properties of the set of stationary states characterized by the azimuthal quantum number. If, for example, the hydrogen atom is under the influence of small external forces disturbing the regular precession, then the electronic orbit in the stationary states will assume forms quite different from those shown in the figure. At the same time the fine structure will become blurred, but the spectrum of hydrogen will still consist of lines determined, to a good approximation, by Balmer’s formula, which is connected with the preservation of the approximately periodic character of the motion. The spectrum will begin to undergo significant changes only in the case when the perturbing forces are so large that already within the time of a single revolution the orbit is substantially distorted.

Therefore the frequently encountered view that the introduction of two quantum numbers is a necessary condition for explaining Balmer’s formula is a misunderstanding and is caused by a failure to grasp the essence of the theory.

Sommerfeld’s theory was able to explain not only the fine structure of the hydrogen lines, but also the fine structure of the lines of the spark spectrum of helium, analogous to the spectrum of hydrogen; the separation between the components of the lines in this case, owing to the higher velocities of the electrons, is considerably greater than in hydrogen, and could be measured with greater precision. It even proved possible to take into account certain aspects of the fine structure of X-ray spectra, where one has to deal with frequency differences reaching values more than a million times greater than the corresponding frequency differences of the components of the hydrogen lines.

Soon after this result was found, Epstein and Schwarzschild (1916) simultaneously succeeded, by analogous considerations, in explaining the details of the characteristic changes undergone by hydrogen lines in an electric field and discovered in 1914 by Stark. An explanation of the essential features of the Zeeman effect in hydrogen lines was given simultaneously by Sommerfeld and Debye (1917). The application of the postulates in this case led to the conclusion that only a definite orientation of the atom in a magnetic field is permissible; this peculiar consequence of quantum theory received

recently (1922) received direct confirmation in the excellent experiment of Stern and Gerlach on the deflection of rapidly moving silver atoms in an inhomogeneous magnetic field.

The Correspondence Principle.

The stage in the development of the theory of spectra described above was based on the elaboration of formal methods for determining stationary states. In recent years, in connection with the important works of Ehrenfest and Einstein, the speaker has succeeded in illuminating the theory from a new point of view, tracing a peculiar formal connection between quantum theory and the classical electrodynamic theory, already revealed in the spectrum of hydrogen. As a result, the so-called “correspondence principle” was established, according to which the occurrence of transitions between stationary states, accompanied by radiation, is connected with the harmonic components of the oscillation in the motion of the atom, which in classical theory determine the properties of the radiation emitted as a consequence of the motion of the particle. Thus, according to this principle, it is assumed that every process of transition between two stationary states is connected with the corresponding harmonic component of the oscillation in such a way that the probability of the occurrence of the transition depends on the amplitude of the oscillation, while the polarization of the radiation is determined by the more detailed properties of the oscillation, just as the intensity and polarization of the radiation in the system of waves emitted by an atom, according to classical theory, as a consequence of the occurrence of the indicated components of the oscillation, are determined by the amplitude and other properties of the latter. With the aid of the correspondence principle it became possible to deepen and extend the results cited above. It proved possible to develop a complete quantum explanation of the Zeeman effect in the hydrogen lines, and this explanation possesses a profound similarity to the explanation proposed by Lorentz on the basis of classical theory, despite the essential difference in the character of the premises of the two theories. The existence of the Stark effect, before the explanation of which classical theory remained completely helpless, receives, with the aid of the correspondence principle, a quantum explanation; moreover, it is possible also to elucidate the polarization of the various components of the splitting of lines in an electric field, as well as the characteristic distribution of intensities among the components. The latter question was investigated more closely by Kramers, and the appended figures may give an idea of how complete is the explanation of the phenomenon under consideration. Fig. 6 is one of Stark’s well-known beautiful photographs relating to the splitting of hydrogen lines. The figure clearly shows how complexly and peculiarly the intensities change from component to component; the middle part of the figure corresponds to components polarized perpendicular to the direction

field; the upper part—to components polarized parallel to the field. Fig. 7 gives a schematic representation of the experimental and theoretical results for the line \(H_{\gamma}\), whose frequency of oscillation is determined by Balmer’s formula for \(n''=2\) and \(n'=5\). The vertical lines denote the components of the splitting, with the components polarized parallel being shown on the right, and the perpendicularly polarized ones on the left.

Fig. 6. The Stark effect for the hydrogen lines \(H_{\delta}\), \(H_{\gamma}\), \(H_{\beta}\).

The experimental results are presented in the upper part of the diagram. The distance of the lines from the dashed line corresponds to the measured displacement of the components; the length of the lines is proportional to the relative intensity of the components, which was estimated by Stark from the blackening of the photographic plate. In the lower part of the diagram, for comparison, the theoretical results are shown according to the drawing from Kramers’ article. The symbols written beneath the lines \((n'_s-n''_s)\) indicate the processes of transitions between stationary states of the atom in an electric field in which the given components are emitted. In addition to the principal quantum number \(n\), the stationary states are characterized by the subsidiary quantum number \(s\), which may be either positive or negative, and which has an entirely different significance from the quantum number \(k\) in the theory of the relativistic fine structure of hydrogen lines, which determines the form of the electronic orbit of the unperturbed atom. Under the action of the electric field, both the form and the position of the orbit undergo profound changes, but certain properties of the orbits remain unchanged: they are described by the subsidiary quantum number \(s\). The position of the components in the figure corresponds to the frequencies calculated for the various transitions, and the length of the lines is proportional to the probability of the various transitions, which can be estimated on the basis of the correspondence principle, just as

Fig. 7. The Stark effect in the hydrogen line \(H_{\gamma}\). Comparison of the results of observation (upper part) and theory (lower part). On the right are components polarized parallel to the field, on the left perpendicular.

and the polarization of the radiation corresponding to the transitions. We see that the theory conveys all the chief features of the experimental results, and on the basis of the correspondence principle we may say that the Stark effect reflects, in the minutest details, the action exerted by the electric field on the electronic orbits in the hydrogen atom, although, in contrast to the Zeeman phenomenon, the splittings in this case are so complicated that, on the basis of the classical theory of electromagnetic radiation, we could hardly understand the motion in the atom.

Interesting results have also been obtained for the series spectra of elements with higher atomic numbers, whose explanation was considerably advanced thanks to Sommerfeld’s work in connection with the introduction of several quantum numbers for describing electronic orbits. With the aid of the correspondence principle it proved possible fully to elucidate the peculiar rules governing, at first sight, the capricious absence or presence of lines arising from the combination principle; one may say that quantum theory has given not only a simple explanation of the combination principle but, moreover, has done very much to remove the mystique that for a long time prevailed in applications of this principle.

The same point of view proved fruitful in the investigation of the so-called band spectra (Bandenspektren). These spectra arise not from atoms but from molecules, and the large number of lines in these spectra is based on the complexity of the motion produced by the vibrations of atomic nuclei with respect to one another and by the rotation of the molecule as a whole. The first to apply our postulates to this problem was Schwarzschild, but the theory was developed by the Swedish physicist Heurlinger, who, by his important works, clarified much concerning the structure and origin of band spectra. His conclusions are directly connected with Bjerrum’s theory of the influence of molecular rotation on the infrared absorption lines in gases, which we mentioned at the beginning of the report. We cannot suppose, however, that rotation is reflected in the spectrum in the way required by classical electrodynamics, and we imagine that the components of the lines are due to transitions between stationary states differing with respect to rotational motion. The fact that essential features foreseen by the classical theory are preserved in the phenomenon is a typical consequence of the regularity expressed in the correspondence principle.

The Natural System of the Elements

The views concerning the explanation of spectra developed above served as the basis for a theory of the structure of the atoms of the elements, which proved suitable for elucidating, in general outline, the properties of the elements that find their expression in the natural system. This theory descri-

is based above all on considerations concerning the formation of the atom by the successive capture and binding of electrons by the nucleus. As we have seen, the optical spectra of the elements provide us with data on the course of the last stage of this process of atom creation. An idea of the character of these data, obtained on the basis of the nearest study, may be formed from Fig. 8, which is a schematic representation of the orbits in stationary states corresponding to the emission of the arc spectrum of potassium. The curves depict the forms of the orbits described by the last captured electron in the potassium atom in stationary states that are stages of the process in which the 19th electron is bound, after the first 18 electrons have already been bound in normal orbits. In order not to complicate the drawing, we have not attempted to depict in any way the orbits of these inner electrons, marking only by a dotted circle the region within which they move. Generally speaking, in an atom with several electrons the orbits take on a complex character. Owing to the symmetrical nature of the force field surrounding the nucleus, the motion of each electron can be described approximately as a plane periodic motion, upon which is superposed a uniform rotation in the plane of the orbit.

Fig. 8. Electron orbits in the stationary states of the potassium atom corresponding to the emission of the arc spectrum.

Every electron orbit will therefore, to a first approximation, be doubly periodic and will be determined by two quantum numbers, similarly to the stationary states of the hydrogen atom when allowance is made for the precession arising from the variation of the mass of the electron with velocity.

Therefore, just as in Fig. 5, the electron orbits in Fig. 8 are denoted by the symbol \(n_k\), where \(n\) is the principal quantum number and \(k\) the subsidiary quantum number. In the initial stages of the binding process, when the quantum numbers are large, the orbit of the last captured electron passes entirely outside the region of the previously captured electrons; in the final stages, however, the situation is different. Thus, in the potassium atom, electron orbits with subsidiary quantum numbers 2 and 1 penetrate, as is evident from the figure, into the inner region. As a consequence, the orbits will differ extremely strongly from simple Keplerian motion; they will consist of a series of successive one after another

loops of identical size and shape, each of which is turned through a considerable angle relative to the preceding one. In the figure only one such outer loop is shown; each of them by itself closely coincides with a part of a Keplerian ellipse; they are connected, as indicated, by inner loops of a complex character, along which the electron comes very close to the nucleus. This applies in particular to the orbit with subsidiary quantum number 1, which, as the nearest investigation shows, approaches the nucleus more closely than any of the previously bound electrons. Such penetration into the inner region leads to the fact that, despite the corresponding electronic orbits running over the greater part of their path in a force field of the same character as the field of the nucleus of the hydrogen atom, the force with which the electron in this orbit is held by the atom is considerably greater than the force holding an electron in the hydrogen atom in an orbit with the same principal quantum number; at the same time the maximum distance of the electron from the nucleus during its revolution is considerably less than in a similar orbit in the hydrogen atom. As we shall see, this feature of the binding of the electron by atoms with a large number of electrons is essential for understanding the distinctive periodicity of the change of the properties of the elements with atomic number, manifested in the natural system.

In the appended table are given the results obtained by the author concerning the structure of the atoms of the elements by considering successive captures and bindings of electrons by the atomic nucleus. The numbers standing beside the symbols of the elements are the atomic numbers, indicating the total number of electrons in the neutral atom. The numbers in the columns give the number of electrons in orbits corresponding to the principal and subsidiary quantum numbers placed above. According to the generally accepted notation, we shall, for brevity, call an orbit with principal quantum number \(n\) an \(n\)-quantum orbit. The electron bound first moves in the orbit corresponding to the normal state of the hydrogen atom and denoted \(1_1\). In the hydrogen atom there is only one electron; with respect to atoms of other elements, however, we assume that the next electron too is bound in an orbit of the same type \(1_1\). The following electrons, as is seen from the table, will be bound in two-quantum orbits. At first the binding takes place in the \(2_1\)-orbit; subsequently, however, the electrons are bound in \(2_2\)-orbits until, after the binding of the first ten electrons, the completed configuration of the two-quantum orbits is reached, for which we assume that it contains four orbits of both types. A neutral atom with such a configuration is encountered for the first time in neon, which completes the second period of the system of elements. The further electrons are bound in three-quantum orbits; after completion of the third period, in the elements of the fourth period we find for the first time electrons in four-quantum orbits, and so on.

TABLE I

Electronic groups in the normal states of atoms of the elements

This picture of the structure of the atom retains many features noted in earlier works by other authors. Attempts to explain the periodic system by assuming a group division of the electrons in the atom go back to the works of J. J. Thomson in 1904; later the same point of view was developed by Kossel (1916), who placed the group division in close connection with the regularities discovered by subsequent investigations in the field of X-ray spectra. Further, Lewis and Langmuir attempted to explain the similarity of the properties of the elements on the basis of group division. These authors suppose that the electrons do not move around the nucleus, but maintain positions of equilibrium. In this way, however, it is impossible to establish a connection between the properties of the elements and the experimental results obtained concerning the constituent parts of the atom. Static equilibrium configurations for the electrons are impossible, since the forces between the particles of the atom obey, at least approximately, the laws of attraction and repulsion of electric charges. The possibility of a detailed account of the properties of the elements, based on the above-mentioned laws, is characteristic of a picture of the structure of the atom founded on the theory of quanta. As regards this picture, the idea of connecting group division with a classification of electronic orbits according to increasing quantum numbers was natural after Moseley’s discovery of the laws of X-ray spectra and Sommerfeld’s work on the fine structure of these spectra. This was also noted by Vegard, who several years ago proposed a group division of the electrons in the atoms of the elements in connection with investigations of X-ray spectra; his division in many respects resembles that given in the table. The basis for the detailed development of the picture set forth has been created only recently by the study of the processes of binding of electrons in the atom, of which we learn from experience with optical spectra; the characteristic features of these processes could be illuminated by the correspondence principle. It is essential here that the restriction of the binding process, manifested in the presence of multi-quantum orbits in the normal state of the atom, can be naturally connected with the general condition for the occurrence of radiation processes in transitions between stationary states, a condition formulated in the principle just named. Another essential feature of the theory is the influence on the strength of binding and on the dimensions of orbits exerted by the penetration of later-captured electrons into the region of earlier-bound ones; we saw an example of this in the exposition of the origin of the spectrum of potassium. This circumstance may be regarded as the actual cause of the sharp periodic change in the properties of the elements; it entails that the atomic dimensions and chemical properties of homologous substances in different periods, such as, for example, the alkali metals, possess a considerably greater similarity than one might have thought on the basis of a direct comparison of the orbit

of the last captured electron with orbits having the same quantum number in the hydrogen atom.

The indicated increase of the principal quantum number for the last electron captured in the atom—an increase that we encounter as we move along the series of elements—gives a direct explanation of the characteristic deviations from simple periodicity in the natural system. These deviations are marked by the framing of certain rows of elements in the natural system in Fig. 1; we encounter them for the first time in the fourth period, and the reason for this can be simply explained on the basis of Fig. 8, which depicts the orbits of the last captured electron in potassium, the first element of the fourth period. Here, for the first time in the series of elements, we have a case in which the principal quantum number of the orbit of the last captured electron in the normal state of the atom is greater than in the earlier stages of the binding process. The normal state here corresponds to a \(4_1\)-orbit, in which the binding force of the electron, owing to its penetration into the inner region, is considerably stronger than in the four-quantum orbit of the hydrogen atom, and even stronger than in the two-quantum orbit of that atom. The binding force of the electron is therefore more than twice as great as in the circular \(3_3\)-orbits, in which the electron remains outside the inner region all the time and in which the binding force differs little from the binding in the three-quantum orbit of the hydrogen atom. The situation does not, however, remain so when we pass to consideration of the binding of the 19th electron in substances with higher atomic numbers, since here there is a considerably smaller relative difference between the force field outside and inside the region in which the first 18 electrons are bound. Investigation of the spark spectrum of calcium shows that already here the binding force of the electron in the \(4_1\)-orbit is only slightly stronger than in the \(3_3\)-orbit, and in passing to scandium we must suppose that the 19th electron in the normal state will already be located in the \(3_3\)-orbit, since in it, in this case, the binding force will be greater than in the \(4_1\)-orbit. The electron group on two-quantum orbits is finally completed at the end of the second period; the development that the group of electrons on three-quantum orbits receives during the third period, however, must be regarded only as a preliminary completion; as indicated in the table, this group develops further, capturing into three-quantum orbits electrons in the framed elements of the 4th period. This leads to new relationships, since the development of the electron group on four-quantum orbits, so to speak, is halted until the three-quantum electron group has been definitively completed. We are not yet in a position to elucidate the course of the gradual development of the three-quantum electron group in all details; nevertheless, on the basis of quantum theory the appearance for the first time in the 4th period of the system of a series of substances with properties as similar as in the family of the iron metals becomes directly intelligible; it is even possible to understand why these ele-

elements possess certain paramagnetic properties. The idea of a connection between the chemical and magnetic properties of the indicated elements and the development of an inner group of electrons in the atom had already been expressed by Ladenburg, independently of quantum theory.

I do not propose to go into further details; I shall only note that the relations which we encounter in the 5th period receive the same explanation as in the 4th period; the properties of the elements discussed in this period, as follows from the table, are determined by a certain stage in the development of the electron group on four-quantum orbits, beginning with the capture of electrons into \(4_3\)-orbits. In the 6th period, however, we encounter new relations. Besides the development of five- and six-quantum orbits, we are concerned here also with the final completion of the development of the electron group of four-quantum orbits, beginning with the appearance of orbits of type \(4_4\) in the normal state of the atom. This development is expressed in a characteristic way in the appearance of a peculiar family of rare earths in the 6th period. As is known, these elements, in their chemical properties, are even more similar than the elements of the family of the iron metals, which is connected with the development of an electron group deep within the atom. It is interesting to note that the theory gives a natural explanation of the strong difference in the magnetic properties of these elements, which are so similar in other respects. The thought that the appearance of the rare earths is connected with the development of an inner group of electrons in the atom has been proposed from various sides. It was expressed, for example, by Vegard, and, simultaneously with the work of the speaker, was considered more closely by Bury in connection with Langmuir’s static model of the atom, in the study of the systematic dependence between chemical properties and group division in the atom.

But whereas until now there has been no sufficient theoretical basis for understanding the development of an inner electron group, quantum theory gives such a direct explanation of this fact that it will hardly be an exaggeration to say that, if the existence of the rare earths had not been established by chemical investigation, the presence of such a family in the 6th period of the natural system of elements could have been predicted theoretically.

In the 7th period of the system we encounter seven-quantum orbits for the first time, and we must expect, in essential features, the same relations as in the 6th period; before the completion of the initial stage in the development of seven-quantum orbits, a further completion of the groups of six- and five-quantum orbits must take place.

It is not possible, however, to obtain direct confirmation of this, since only a few elements from the beginning of the 7th period are known; this is connected, as may be supposed, with the instability of nuclei with large charge, expressed in the radioactivity of elements with high atomic numbers.

X-ray Spectra and the Structure of the Atom

The center of gravity in our exposition of ideas about atomic structure up to now has been the question of the creation of the atom by the successive capture of electrons. The exposition would remain, however, very incomplete if we did not point to that support for the theory which is provided by investigations of X-ray spectra. Since the cessation of Moseley’s fundamental investigations, caused by his premature death, the study of X-ray spectra has been continued with remarkable skill by Professor Siegbahn in Lund. On the basis of the large amount of material collected by him and his collaborators in recent times, it has been possible to create a classification of X-ray spectra that admits a direct interpretation with the aid of quantum theory, in which the above-mentioned works of Kossel and Sommerfeld were of great importance. First of all it proved possible, just as in optical spectra, to represent the frequencies of the vibrations for each line of the X-ray spectrum in the form of the difference of two spectral terms, the totality of which is characteristic for the given element. Further, it was possible to achieve a direct connection with the theory of the atom, assuming that the product of each such spectral term by Planck’s constant is equal to the work which must be performed in order to remove from the atom one of the inner electrons. According to the considerations set forth above concerning the formation of the atom by the capture of electrons, the removal of an inner electron in the completed atom must lead to transition processes in which

Fig. 9. Square roots of spectral terms of X-ray spectra as functions of atomic number.

the place of the removed electron will be taken by one of the electrons belonging to one of the least tightly bound electron groups in the atom; as a result, after the transition, one electron in these latter groups will disappear. X-ray lines must, therefore, be regarded as the manifestation of a process during which a reorganization of the atom takes place, owing to a disturbance in its inner region. According to our views on the stability of electron configurations, such a disturbance must consist in the complete removal of an electron from the atom, or at least in its transfer from its normal orbit to an orbit with a higher quantum number than that which corresponds to completed groups; this circumstance is clearly revealed in the characteristic difference between selective absorption in the X-ray region and absorption in the optical region.

The classification of X-ray spectra just mentioned has recently made it possible, by a complete study of the variation of the terms of X-ray spectra in connection with the atomic number, to obtain very direct confirmation of certain theoretical conclusions concerning the structure of the atom. In Fig. 9 the abscissa corresponds to atomic numbers, while the ordinate is proportional to the square roots of the spectral terms; the symbols $K$, $L$, $M$, $N$, $O$ attached to the individual terms refer to the characteristic discontinuities in the selective absorption of X-rays by the elements, first discovered by Barkla and providing a means for the detailed investigation of X-ray spectra even before the discovery of the interference of X-rays in crystals.

Although the curves in the figure run, on the whole, very uniformly, they nevertheless show a number of deviations from this uniformity; these deviations have been elucidated especially by the new investigations of Coster, who worked for several years in Siegbahn’s laboratory. The deviations were discovered only after the publication of the theory of atomic structure set forth above; they correspond exactly to what is to be expected on the basis of this theory. At the bottom of the figure, vertical strokes indicate those places where, according to the theory, one should first expect, in the normal state of the atom, the appearance of a certain $n_k$-orbit. We see that it is possible to connect the appearance of a definite spectral term with the presence of an electron in an orbit of a definite type; the removal of this electron from the atom corresponds to the given term. The circumstance that, in general, more than one curve corresponds to each type of orbit $n_k$ is an expression of a certain complication in the spectra, which must be ascribed to deviations of the electron orbits from the simple types of motion described earlier—deviations caused by the interactions of different electrons within one group; an account of this question would take us too far afield. Those intervals in the system of the elements in which, as a consequence of the appearance of electron orbits of definite types, the further development of a certain inner electron group takes place are marked at the bottom.

of the figure by horizontal lines bounded by vertical strokes with the corresponding quantum symbols. We see that the development of the inner group is everywhere reflected in a characteristic way in the curves. In particular, the course of the \(N\)- and \(O\)-curves may be regarded as a direct expression of that stage in the development of the four-quantum orbits which is the cause of the appearance of the rare earths. Although an essential feature of Moseley’s discovery was the circumstance that the more complicated similarity of most of the other properties of an element apparently had no effect at all on the X-ray spectra, we now find, thanks to the progress of research in recent years, a close connection between the X-ray spectra and the general relations of similarity among the elements in the natural system.

Before concluding my report, I should like to touch upon one more point where X-ray spectroscopic investigations have confirmed the theory. I am speaking of the properties of the element, still unknown until now, with atomic number 72. On this question opinions were divided with regard to conclusions about the similarity of this element to others in the natural system; in many representations of the system it was assigned a place in the family of the rare earths. But already in Julius Thomsen’s representation of the periodic system, as also in ours in Fig. 1, this hypothetical element is assigned a place homologous with titanium and zirconium. The same assumption is also expressed in our table of the classification of electronic orbits in the atoms of various elements with respect to the element with atomic number 72. Berry comes to the same conclusion on the basis of the considerations indicated above concerning the systematic connection between the group division of electrons in the atom and the properties of the elements. Half a year ago, however, a report appeared by Dauvillier on the observation of several weak lines in the X-ray spectrum of a preparation containing rare earths; these lines were ascribed to the element with atomic number 72, which was identified with one of the elements of the rare-earth family; its existence in the given preparation had already been supposed several years earlier by Urbain. This report naturally aroused doubts about some details of the theoretical conclusions. A more recent investigation showed, however, that the assumption that the element with atomic number 72 belongs to the family of the rare earths would require a change in the strength of the electronic bond with changes in atomic number, which is incompatible with the requirements of quantum theory. In this connection, quite recently Dr. Coster and Professor Hevesy, both of whom were at that time in Copenhagen, undertook a renewed study of the question, examining by X-ray spectroscopy preparations made from materials containing zirconium; and I can report that in these days they have succeeded in detecting in the minerals studied a considerable quantity of the ele-

element with atomic number 72; its chemical properties are very similar to those of zirconium and differ substantially from the properties of the rare earths1.

I hope that in this communication I have succeeded in giving a survey of the most important results obtained in recent years in the field of atomic theory, and I should like, in conclusion, to make a few general remarks concerning the point of view from which one ought to approach the assessment of the results found, in particular concerning the extent to which one may speak of these results as an explanation, in the usual sense of the word. By a theoretical explanation of the phenomena of nature one generally understands the classification of a certain domain of observations by means of analogies borrowed from other domains, where it is assumed that simpler phenomena occur; the most that can be demanded of a theory is the extension of the indicated classification in such a way that it leads to a broadening of the domain of observation through the prediction of new phenomena. Turning to atomic theory, we see that we are in a peculiar situation: on the one hand, there can be no question of explanation in the indicated sense, since we have to deal with phenomena that, by the very nature of things, are simpler than the phenomena of any other domain, which are conditioned by the interaction of a large number of atoms; we are therefore obliged to be more modest in our demands and must be satisfied with conceptions that are formal in the sense that they do not possess the vivid intuitiveness which we are accustomed to demand of the conceptions with which natural-scientific theories operate. In particular, bearing this in mind, I have tried to give you an idea of the other side of the theory, namely that its results, at least to a certain extent, correspond to the expectations that may be entertained with respect to any theory: I have sought to show how the development of atomic theory has led to the classification of extensive domains of observation and has indicated a way of supplementing this classification by predicting new facts. It hardly needs emphasizing, however, that the theory is still to a considerable extent in its initial stage, and that there are still many fundamental questions awaiting an answer.

Translated by S. Vavilov.

1. Addition after the lecture. Coster and Hevesy proposed naming the new element hafnium (Hafnium) in honor of the city of Copenhagen (Köbenhavn), where the discovery was made. French investigators call the new element celtium (celtium) and attribute the honor of its discovery to Dauvillier. Translator’s note. ↩↩↩