Application of the Doctrine of Quanta to the Theory of Spectral Series.

P. Ehrenfest ¹).

I. Section.

INTRODUCTION.

From 1896 to 1902 Max Planck concentrated all his powers on the theory of radiation. With unparalleled consistency and energy he first created the fundamental foundations of this theory and, at the turn of the century, came to the conviction that ordinary mechanics and Maxwell–Lorentz electrodynamics were insufficient for grounding the doctrine of radiation. The consistent application of these classical principles under all circumstances leads to such a law of distribution of energy (the so-called Rayleigh–Jeans distribution) which, in the region of short waves, stands in sharp contradiction with the data of experience. Hence Planck was faced with the necessity of introducing into the theory of radiation a new proposition, alien to mechanics and electrodynamics. In 1901 he succeeded, by a bold movement of thought, in finding the missing link in the chain of his conclusions. This link was the hypothesis of quanta, which not only solved all the riddles in the theory of radiation but, as subsequently turned out, took possession of all the remaining atomistic processes.

The application of the theory of quanta in atomistics has yielded, in recent years, a whole series of significant successes; and in this newest development of the theory Planck has likewise played an outstanding role. The aim of the present article is a review of the newest results of the theory of quanta, insofar as the matter concerns the theory of spectral series. Accordingly, at the beginning

¹) P. Ehrenfest’s article appeared in the issue of the journal “Die Naturwissenschaften” dedicated to M. Planck on the occasion of the sixtieth anniversary of his birth.

P. S. Ehrenfest is one of the prominent contemporary theorists, at present a privat-docent of the University of Zurich. At the beginning of his activity he was a privat-docent of Moscow University.

Translator.

are given, in brief, the fundamental principles of Planck’s doctrine of quanta (§§ 2—4), and the necessary factual material from the field of atomistics and spectroscopy is also set forth (§§ 5—6). The second division contains the first successful attempt to apply quantum theory to the atom, associated with the name of Niels Bohr, and culminating in the explanation of the simplest spectral series (§§ 7—10). In the third division, applications to systems with several degrees of freedom are considered in connection with those new conceptions that were created by Planck and Sommerfeld. Here, chiefly, the theory of the fine structure of lines similar to the hydrogen lines (wasserstoffähnliche), as well as the theory of the Stark phenomenon, is given (§§ 11—15). In conclusion, Planck’s views on the structure of phase space (Phasenraum) are considered.

§ 2. The Hypothesis of Energy Quanta.

In order to clarify the content of the quantum hypothesis, let us analyze in greater detail what was said in the introduction; in doing so we shall take as our point of departure the concept of a linear resonator. As such a resonator we may imagine an electron which is quasi-elastically (i.e., by a force proportional to the distance \(x\)) bound to its equilibrium position, as a result of which it performs, about this position, sinusoidal oscillations with a constant number of oscillations per second \(\nu\), characteristic of the given resonator:

\[ x=x_0\sin 2\pi\nu t\ .\ .\ .\ .\ .\ .\ .\ .\ .\ (1). \]

Here \(t\) denotes time, \(x_0\)—the amplitude.

According to the laws of electrodynamics, such a linear resonator must emit electromagnetic waves with the same number of oscillations \(\nu\), and conversely—under the influence of incident waves—perform forced oscillations. Thus, if within a perfectly reflecting space there is enclosed a large number of resonators with all possible \(\nu\), then they must be, with the radiation and among themselves, in a state of mobile equilibrium, in which each group of resonators (with a definite number of oscillations \(\nu\)) emits exactly the same amount of energy as it absorbs from the radiation incident upon it. This equilibrium radiation is called “black radiation,” and the problem posed by Planck consisted in finding the spectral distribution of the energy of black radiation. It turned out, however, that an incorrect law is inevitably obtained if one only admits that, in interacting with others, a resonator can receive all possible quantities of energy (and, consequently, amplitudes as well). Planck’s new, supremely bold assumption consisted in the fact that the energy \(A\) of a resonator

the resonator has an atomistic character and is an integral multiple of some elementary quantity \(\epsilon\),

\[ A=n\epsilon . \]

where \(n\) is an integer. On thermodynamic grounds (Wien’s displacement law), this elementary quantity of energy cannot be independent of the number of oscillations \(\nu\), but must be proportional to it: \(\epsilon=h\nu\). Thus, as an expression of the first formulation of Planck’s hypothesis, which is called the hypothesis of “energy quanta,” one obtains the formula

\[ A=nh\nu \ldots \ldots \ldots \ldots \ldots \ldots \ldots (2). \]

Here \(h\) is a universal constant whose dimension is that of action (energy times time), and which is therefore called Planck’s quantum of action (Wirkungsquantum). From the measurements of Kurlbaum, Lummer, and Pringsheim, Planck obtained for \(h\) the following numerical value

\[ h=6{,}55 \cdot 10^{-27}\ \mathrm{erg.\ sec.} \]

The expression for the energy of a linear resonator \(A\) will be obtained if one takes into account that at the instant of passage through the initial position \((t=0)\) there is only kinetic energy. If we denote the mass of the electron by \(\mu\), then

\[ A=\frac{\mu}{2}\left(\dot{x}\right)^2_{t=0}=2\mu(\pi\nu x_0)^2 \]

\[ \dot{x}=\frac{dx}{dt} \]

Relation (2) then gives

\[ 2\pi^2\mu\nu x_0^2=nh. \ldots \ldots \ldots \ldots \ldots \ldots (3). \]

Thus, quantum theory admits only certain selected amplitudes \(x_0\), namely those that satisfy relation (3).

§ 3. The Hypothesis of Quanta of Action.

Subsequently Planck gave his hypothesis another formulation, in which it becomes clearer that it is precisely action, and not energy, that is constant. In doing so he made use of the mechanical concept of impulse, or quantity of motion. As is known, if one uses Cartesian coordinates and considers conservative forces (i.e., such forces as have a potential), then the impulse corresponding to the coordinate \(x\) for a point with mass \(\mu\) is expressed as

\[ p_x=\mu\dot{x} \]

but even under more complicated circumstances one can in general determine the momentum \(p\) corresponding to the coordinate \(q\) of the system.

At the same time it is sufficient to know the values of all \(p\) and \(q\) at some instant in order thereby to determine the motion of the system completely and for any moment of time.

In the particular case where there is one degree of freedom, it is possible simply to express \(p\) as a function of \(q\).

For example, for a linear resonator, according to equation (1) we have

\[ \frac{p_x}{2\pi \nu \mu}=x_0\cos 2\pi \nu t \qquad\ldots\ldots\ldots\qquad (4). \]

Squaring (1) and (4) and adding them together, we obtain

\[ \left(\frac{x}{x_0}\right)^2+ \left(\frac{p_x}{2\pi \nu \mu x_0}\right)^2=1 \]

And this, as is known, is the equation of an ellipse with semiaxes \(a=x_0\) and \(b=2\pi\nu\mu x_0\). Thus, if \(p_x\) is taken as the ordinate and \(x\) as the abscissa, then for every particular value of \(x_0\) a certain ellipse will be obtained (Fig. 1), and for all possible \(x_0\)—a family of similar concentric ellipses.

Fig. 1.

In the general case of a system determined by the coordinate \(q\) and the corresponding momentum \(p\), by an analogous construction one can obtain another (non-elliptic) family of curves, each of which depicts the motion of the system for a definite value of the energy. On the basis of Planck’s assumption, according to equation (3), not all curves of the family are possible, but only certain completely

defined. Planck’s later formulation consists precisely in this: that the area enclosed between two consecutive curves admissible from the point of view of the quantum theory must be equal to the quantity of action $h$. Mathematically this is expressed as

$$ \iint dp\,dq = h $$

or

$$ \int (p_n - p_{n-1})\,dq = h $$

The integration is to be extended over the whole shaded region between the curves for $n=2$; $p_n$ and $p_{n-1}$ refer to the values of the momentum on the boundary curves.

Summing the expressions

$$ \int (p_1 - 0)dq = h,\quad \int (p_2 - p_1)dq = h,\ldots\quad \int (p_n - p_{n-1})\,dq = h, $$

we obtain, for the $n$-th motion admissible from the point of view of the quantum hypothesis, or, as we shall say, for the $n$-th “static form of motion”:

$$ \int (p_n - p_0)\,dq = nh $$

Here $p_0$ denotes the momentum on the innermost static curve with the least admissible value of the area.

For this $p_0$ one can expect a value different from zero only when the system under consideration, on mechanical grounds, admits a limiting state to which precisely the momentum $p_0$ corresponds. For the linear resonator this is not the case. Here minimal amplitudes and momenta, down to complete rest $(p_0 = 0)$, are mechanically possible. For such systems—and the majority of systems are just such—the quantum condition (5a) reduces simply to the following

$$ \int p\,dq = hn \ldots\ldots\ldots (5\text{b}), $$

i.e. the area bounded by the static curve is an integral multiple of the quantity of action $h$.

It is easy to see that this interpretation of Planck’s admissibility—which in this form is called the hypothesis of “quanta of action”—in the case of the linear resonator coincides with the hypothesis of energy quanta (2). Indeed, the area of the ellipse is equal to $\pi ab$, i.e.

$$ nh = \pi ab = 2\pi^2 \nu m x_0^2 $$

We have thus arrived at formula (3). One of the advantages presented by such a formulation of the quantum hypothesis was that the necessary changes in electrodynamics could be reduced to a minimum: at least the absorption of energy could be represented as continuous, and what was essentially new in this view could be reduced to the calculation of a statistical probability. And, since for what follows it is immaterial whether we regard the static forms of motion as the only possible ones or merely as those selected in a statistical sense, we shall return to this question only later (§ 16).

A further advantage of the second theory consists in the fact that in it the number of oscillations plays no role, and therefore one need not restrict oneself in advance solely to periodic motions. And indeed, subsequently it proved possible to apply this theory both to systems with several degrees of freedom and to certain known classes of nonperiodic motions.

§ 4. Development of the ideas about energy quanta.

It is remarkable that the conception of quanta of action, based on the concept of momentum, did not make the idea of energy quanta superfluous, but only shifted the domain of its application. And indeed this idea could be successfully applied to the explanation of many phenomena. In particular Einstein, to whom we owe the development of quantum theory in the most varied directions, persistently defended the point of view that, when radiation acts with a number of oscillations \(\nu\), an energy quantum \(h\nu\) must manifest itself. In this way he gave a picture of the quantitative relations in the photoelectric effect. According to his hypothesis, an electron emitted from a metal under the action of ultraviolet light of frequency \(\nu\) receives from the light the kinetic energy \(h\nu\), of which it expends some part \((P)\) on the work it performs in order to pass through the surface and overcome its bonds with the atoms of the metal. Hence, for the velocity \(v\) of the photoelectron one obtains Einstein’s equation (1905),

\[ \frac{\mu v^2}{2} + P = h\nu, \]

which is fully confirmed and which was recently used by R. Millikan (1916) for an exact determination of the value of \(h\).

The converse conception—the assumption that, in the process of emission of light, the number of oscillations, given by a mechanism unknown to us, can be determined by the available store of energy—we find for the first time

by Wien and Stark. Wien1 considers the process of the production of X-rays during the sudden braking of cathode rays at the anticathode, and puts forward the hypothesis that the number of oscillations of the emitted X-rays is determined by the kinetic energy \(T\) of the retarded electron from the relation

\[ T = ch\nu \; {}^{2}). \]

What is remarkable in this representation is that the quantity \(ch\nu\) in no case coincides with the energy radiated in the form of X-rays, for the latter amounts, as can be calculated3 and as follows from measurements4, to only an insignificant part (of the order of \(0.2\%\)) of the energy of the cathode rays. Thus condition (6) determines only the number of oscillations of the emitted radiation and says nothing about its energy. This is why we call this equation the “frequency condition” (Frequenzbedingung).

This relation has recently been verified (up to a voltage of 40,000 volts5). Moreover, it was confirmed with complete rigor that the greatest number of oscillations of X-rays produced by electrons with a definite kinetic energy satisfies equation (6).

The first application of similar considerations to the optical spectrum was made by Stark6. He considers the emission of lines of the mercury spectrum and proceeds from the assumption that a preliminary condition for this emission is the presence of ionized atoms and free electrons in mercury gas. The luminescence occurs, in his opinion, upon the recombination of an electron with an ionized atom. Since in this process there is a finite amount of energy in reserve, namely the energy \(T\), which must be expended in order, conversely, to remove the electron from the atom to infinity (the so-called ionizing potential), and which is known from direct measurements, Stark concluded that the boundary of the mercury spectrum, i.e. the greatest number of oscillations \(\nu\) whose radiation is possible, is deter-

is given by equation (6). The latest measurements of the ionizing potential of mercury5) have confirmed the quantum equation (6). However, the number of oscillations \(\nu\) which is thereby determined refers not to the boundary of the spectrum, but to the resonance radiation of mercury \((253.6\mu\mu)\).

§ 5. Structure of series formulae. Ritz’s combination principle.

Our aim is to give an overview of the successes that have been achieved in recent years in explaining spectral laws through the application of quantum theory to atomistics. It is therefore necessary, first of all, to say a few words about the structure of spectral formulae and about contemporary views on the constitution of the atom.

In the study of the lines of which a spectrum consists, the existence of special sequences of these lines has been established. These sequences are called series. All the lines of one and the same series possess an external similarity and uniform physical properties, from which one concludes their genetic homogeneity. The representation of spectral series by means of a formula was introduced as early as 1885 by Balmer, who gave for the hydrogen series, subsequently named after him, the following formula

\[ \nu = N\left(\frac{1}{2^2} - \frac{1}{m^2}\right),\quad m = 3,\ 4,\ 5 \ldots \tag{7} \]

Here \(N\) is a constant; its best value at the present time is considered to be 109677.69. From this Balmer series, the only series possessed by hydrogen in the visible part of the spectrum, 29 lines are known, designated in order of decreasing wavelength \(H_{\alpha}, H_{\beta}, H_{\gamma}\ldots\) In the spectra of hydrogen tubes, however, only the first 13 lines have been found; the rest are inaccessible to observation with terrestrial sources because of their weakness, but are known in the spectra of stars. Formula (7) gives, with an accuracy leaving nothing to be desired, the numbers of oscillations of all 29 lines if successive integers from 3 to 31 are substituted for \(m\). Recent investigations, however, have shown that the lines of the Balmer series are not simple, but can be resolved, at least, into two closely situated components.

In the same way, an astronomical series was ascribed to hydrogen, which was found in 1896 by Pickering in the spectrum of the star \(\zeta\) Puppis:

\[ \nu = N\left(\frac{1}{2^2} - \frac{1}{m^2}\right),\quad m = 1.5;\ 2.5;\ 3.5 \ldots \tag{8} \]

5) J. Frank und G. Hertz. Verh. d. Deutsch. Phys. Ges. 16, p. 512, 1914.

We shall see (§ 10) that in reality it belongs to the ionized helium.

It is characteristic of the structure of these formulas that the number of oscillations $\nu$ in them is the difference of two terms $\dfrac{N}{2^2}$ and $\dfrac{N}{m^2}$. A detailed analysis of the experimental material existing at that time showed the Swedish physicist Rydberg, in 1890, that many spectral series of various substances can be well represented by the generalized formula

\[ \pm \nu = \frac{N}{(m_1+\mu_1)^2} - \frac{N}{(m_2+\mu_2)^2}\ .\ .\ .\ .\ . \tag{9} \]

Here $N$ is the same constant that occurs in Balmer’s formula (7). It thereby acquires the significance of a universal constant and is called the “Rydberg constant.” $\mu_1$ and $\mu_2$ are also two constants characteristic of the given element, $m_1$ and $m_2$ are integers (“ordinal numbers”); the sign must be chosen so that $\nu$ is positive. Already with constant $m_1$ a series is obtained if $m_2$ runs through a sequence of integer values. If, however, $m_1$ is also variable, then by the single expression (9) several series can be represented.

However, with the increase in the accuracy of spectral measurements the Rydberg formula proved insufficient. The best of the formulas subsequently proposed belongs to W. Ritz (1903). Ritz preserved the external form of the formula in the respect that $\nu$ is represented as the difference of two terms, and modified only these terms themselves. An even greater service to spectroscopy belongs to this Swiss scientist, unfortunately deceased early, for the creation of the so-called “combination principle.” This principle furnished proof that the two terms whose difference is equal to the number of oscillations are not an accidental feature of the mathematical formulation, but, independently of it, have a real physical meaning, an objective existence.

The combination principle consists in the fact that one of the two terms which determine some line of a series can be combined with one of the terms corresponding to another line (of the same series or of some other series of the same element) in such a way that their difference gives a spectral line. In this way Ritz succeeded, on the one hand, in finding new series, and, on the other, in assigning to series such lines as had seemed to stand alone. For example, applying this principle to Balmer’s formula (7), Ritz predicted the existence of a hydrogen (line) series of the form

\[ \nu = N\left(\frac{1}{3^2} - \frac{1}{m^2}\right);\quad m = 4, 5, 6\ .\ .\ .\ .\ . \tag{7} \]

which must lie in the infrared region. And already in the same year (1908) this prediction was confirmed by Paschen’s measurements; he found the following values for the wavelengths in air and the frequencies of oscillation of the first two lines:

$m$	$\lambda$ in Å observed	calculated	$\nu$ observed	calculated
4	18751,3	18751,6	5331,58	5331,49
5	12817,6	12818,7	7799,70	7799,10

Somewhat earlier, Lyman had measured in the ultraviolet region the first members of the series

\[ \nu = N\left(\frac{1}{1^{2}}-\frac{1}{m^{2}}\right); \quad m=2,\,3\,4 \ldots \quad (7\text{в}). \]

from which, by the combination principle, there follow both the Balmer series and the Ritz–Paschen series.

Subsequently Paschen in particular devoted much work to the combination principle. To him we owe both the experimental substantiation of this principle, through the discovery and precise measurement of combination lines in all serial spectra, and the final bringing together of the various series of a chemical element into one system of series. From the standpoint of this systematics, and by analogy with other elements, it was possible to expect, alongside the Pickering series (8), which was ascribed to hydrogen, still another series

\[ \nu = N\left(\frac{1}{1{,}5^{2}}-\frac{1}{m^{2}}\right); \quad m=2,\,3\,4 \ldots \quad (8\text{a}). \]

This had already been pointed out by Rydberg.

The principal line of this series was identified with the line $\nu=4687{,}88$ (calc. $\nu=4687{,}90$) in the spectra of certain fixed stars. However, only in 1912 did Fowler succeed in observing this series in terrestrial radiation. In the light of an electric discharge through a Geissler tube filled with a mixture of hydrogen and helium, he obtained three lines of series (8), four of series (8a), and three lines of the ultraviolet series

\[ \nu = N\left(\frac{1}{1{,}5^{2}}-\frac{1}{m^{2}}\right); \quad m=2{,}5;\,3{,}5;\,4{,}5 \ldots \quad (8\text{b}). \]

The observed frequencies of oscillation were somewhat greater than those calculated from formulas (8a, в). It is remarkable that these lines could not be obtained in pure hydrogen; the addition of helium proved absolutely necessary.

§ 6. Rutherford’s Model of the Atom.

Ever since it became known that electric charges play an essential role inside the atom, many attempts have been made to construct a model of the atom. The most popular model for a long time was the so-called “Epinos atom,” proposed by Lord Kelvin: positive electricity is distributed continuously, with constant density, throughout the whole (spherical) atom; inside this sphere there are electrons in such a number that their total charge exactly neutralizes the positive charge of the sphere. This model has the advantage that in it the electrons possess static positions of equilibrium, so that there is no need first to consider their motion. Of course, here it remains unclear what forces hold the positive electricity of the sphere together. J. J. Thomson substantiated this model by various atomistic considerations. Of his results, the following is chiefly important for us: the number x of electrons in an atom is approximately equal to half the atomic weight M. Among the various physical phenomena that led Thomson to this result, special attention should be paid to the scattering of X-rays by various substances. Arguments based on this phenomenon require only one premise: that there are electrons in the atom, and therefore are independent of the character of the combination of the latter. Hence Thomson’s result on the number of electrons remains valid for any other model as well.

On the other hand, there was also the idea that the atom is built of discrete positive particles and electrons which act on one another by Coulomb forces and, like a planetary system, move around one another ¹); static equilibrium under forces acting inversely proportional to the square of the distance is impossible. And only a few years ago (1911) a single, apparently insignificant, experimental result enabled Rutherford to decide the question in favor of the latter class of models of the atom. At his suggestion Geiger and Marsden (1909) investigated the deflection of α-rays ²) from their straight-line path when passing through thin sheets of various substances. In doing so they showed that in a small number of cases even very considerable (greater than 90°) angles of deflection are observed as a result of the collision of an α-particle with one

¹) Even before the discovery of the electron, F. Richarz (1894) considered the molecule as a system of two planets (positive and negative ions) revolving around one another.

²) α-rays are emitted by radioactive substances and consist of helium atoms flying with great velocity and carrying a positive charge (equal to twice the elementary charge).

a single atom. In order to obtain so considerable a deflection, the α-particle, in passing through the atom, must experience a correspondingly strong (electric) repulsion. A discussion of the conditions existing in Kelvin’s model shows that the electric fields present there are too weak to explain this effect. Sufficiently considerable field strengths are possible only when the entire positive charge of the atom—which, from Thomson’s works mentioned above, is already approximately known—is concentrated in a very small region, in the so-called positive “nucleus.”

Proceeding from this assumption, Rutherford was able to calculate that the number of α-particles deflected through an angle φ from the straight line is proportional to the following quantities: 1) \(\mathrm{Sin}^{-4}\varphi\) (or \(\varphi^{-4}\) for small \(\varphi\)); 2) the number of atoms per unit volume of the scattering substance; 3) the thickness \(d\) of the layer of the latter (so long as \(d\) is small); 4) the square of the charge of the nucleus \(E\); 5) the reciprocal of the square of the kinetic energy of the α-particle. By means of result (4), it was possible, on the basis of the available experimental material, to calculate the charges of the nucleus \(E=\kappa e\). It then turned out, in agreement with Thomson, that \(\kappa\) is approximately equal to half the atomic weight \(M\):

\[ \kappa=\frac{M}{2} \]

In a new work by Geiger and Marsden (1913), Rutherford’s conclusions were checked in every detail and proved to coincide with the experimental results. The sudden deflection after a collision is especially conspicuous in gases. In this case, by Wilson’s method (C. T. R. Wilson) (Fig. 2), the paths of α-particles can be made visible and these paths photographed. At the end of the visible path of some particles, where their velocity has already decreased, one may observe a more or less sharp bend, which is the result of a central collision with an atom of the gas:

Fig. 2.

Thus we arrive at the so-called “nuclear” theory of the atom, which may also be called the planetary theory: atoms consist of a nucleus, in which all the charge and almost all the mass of the atom are concentrated, and of a cloud of electrons which, like planets, revolve around the nucleus in more or less distant orbits. The mutual relations of such solar systems to one another, i.e. the chemical properties of the atom, are determined above all by the distribution of the peripheral electrons; likewise, the optical spectrum depends essentially on the periphery. As for the inner electrons, revolving close to the very

nucleus, are given to us by the X-ray spectra of the elements, as we shall discuss further in § 14.

The difference between the atomic weights of neighboring elements of the periodic system is, on the average, two units. To this there corresponds, according to the approximate equality (10), an increase of the charge of the nucleus \(z\) by one elementary unit in passing from a given element to the next. Hence there naturally follows the idea that the position in the periodic system is determined not by the atomic weight, but by the charge of the nucleus, or, as is often said, by the ordinal number \((k)\) of the element. It is only necessary for this to assume that the charge of the nucleus already determines the whole distribution of the electrons around the nucleus, on which, as we have seen, all the chemical and physical properties of the atom depend. The atomic weight changes only approximately parallel to the ordinal numbers, and this explains the circumstance that at various places in the periodic system \((Ar — K, Co — Ni, Te — J)\) the sequence of atomic weights does not correspond to the chemical properties. However, an impeccable proof that the charge of the nucleus of an atom, in passing from one place of the periodic system to another, changes by one unit was obtained only later with the aid of X-ray spectra (§ 14). According to our present knowledge, hydrogen consists of a nucleus with one charge \((k = 1)\) and one electron, helium—of a nucleus with two charges \((k = 2)\) and two electrons, and so on up to uranium, to which there corresponds the ordinal number \(k = 92\). In all, six elements are still unknown to us (ordinal numbers 43, 61, 72, 75, 85, 87).

From measurements of the deflection of \(\alpha\)-rays Rutherford was able to indicate an upper limit for the size of the radius of the nucleus, which turned out to be (for gold) \(3.10^{-12}\) cm.\(^1\) In comparison with the order of magnitude of the atom \((10^{-8}\) cm.) these dimensions are so small that, in calculating its field, the nucleus may be regarded as pointlike. However, radioactive phenomena compel us to ascribe to the nucleus of the heavy elements a more or less complex structure: as is known, a new element arises from its radioactive predecessor by the emission either of one \(\alpha\)-ray (helium nucleus), or of one \(\beta\)-ray (electron). The chemical properties of the resulting product are indicated by the Fajans–Soddy rule (1913), according to which in every \(\alpha\)-transformation the element is displaced to the left by two groups of the periodic system, and in every \(\beta\)-transformation it is displaced into the next group.

From the point of view just set forth this may be expressed as follows: in an \(\alpha\)-transformation the charge of the nucleus is decreased by two units; in a \(\beta\)-transformation it is increased by one unit. This phenomenon

\(^1\) Proceeding from less stringent assumptions, Darwin obtained for the upper limit of the radius of the nucleus in hydrogen and helium \(1.7 . 10^{-13}\) cm.

is easily explained if one admits that both α-rays (which carry two positive charges) and β-rays (with one negative charge) arise from the nucleus. Therefore the nucleus, at least in radioactive atoms, is a complex formation, which includes electrons and helium nuclei ²).

II. Department.

Systems with One Degree of Freedom.

§ 7. Application of the quantum of action to Rutherford’s model of the atom.

For the application of the doctrine of quanta to atomistics we are indebted to the young Danish scientist Niels Bohr (1913). In his theory he makes such ingenious use of the elements set forth in §§ 2–6, and comes so close to the truth, that this theory may be regarded as a turning point for all atomistics. Planck’s quantum theory found in it a new and extensive field of application, in which, despite the short time that has elapsed since the appearance of Bohr’s work, we possess a whole series of firmly established theoretical truths.

²) A number of works carried out recently by Rutherford and his pupils (Marsden. Phil. Mag. 27, p. 499 (1914), Marsden and Lantsberry. l. c. 30, p. 240 (1915), and especially the last four papers by Rutherford printed in one booklet of Philosophical Magazine 37, pp. 5–37–587, (1919)) showed that the nucleus of radioactive elements apparently must include, besides helium nuclei, also hydrogen nuclei. At least with respect to radium C it was possible to ascertain that, along with α-particles, it emits lighter particles, which Rutherford identified with complete certainty as hydrogen nuclei (H-particles). It is not excluded, however, that in this case these H-particles owe their origin to occluded hydrogen. But with respect to one of the light elements—nitrogen—quite unambiguous and astonishing results were obtained. Namely, by subjecting nitrogen to bombardment by α-rays, Rutherford here too discovered the appearance of H-particles. Thus he succeeded in bringing about the disintegration of an ordinary (i.e. non-radioactive) element and in showing that in these elements too the nucleus is a complex formation. Let us quote Rutherford’s own historic words: “...We must conclude that the nitrogen atom is broken up under the influence of the enormous forces developed in collision with the stream of particles, and that the hydrogen atom which is thereby liberated enters as a constituent part into the nitrogen nucleus.”

Rutherford’s works contain a mass of interesting and important data on the structure of the atomic nucleus. Some details about them may be found in the abstract by Acad. P. P. Lazarev in the present issue of Uspekhi, as well as in books translated and soon to appear:

J. Grech, The Modern Development of Atomic Theory.

K. Fajans. Radioactivity and the Modern Development of the Doctrine of the Chemical Elements.

Translator.

According to § 6, the hydrogen atom consists of one nucleus with one charge and one electron. We shall consider a somewhat more general formation: a nucleus with charge \(xe\), around which one electron revolves (Fig. 3a). Such a system is called hydrogen-like (wasserstoffähnlich); if \(x\) is different from unity, then the system does not correspond to the normal state of any atom, or, in order to be electrically neutral, it lacks \(x-1\) electrons.

a  Fig. 3.  b

Thus, this will be an atom from which \(x-1\) electrons have been removed, or, as it is customary to say, an atom ionized \(x-1\) times. Let us try, similarly to what we did for the linear resonator in §§ 2 and 3, to select from all mechanically possible motions of the electron only certain ones, admissible from the point of view of quantum theory.

It turns out, however, that there is an essential difference between the case of the linear resonator and the present one. Indeed, in both cases the oscillations performed by an electric charge lead to the radiation of energy. But whereas with a quasi-elastic bond the motion of the electron is performed with a constant number of oscillations, independent of the store of energy, in the case of Newtonian forces, as a consequence of the loss of energy (through radiation), the very dimensions of the orbit change, and with them all the other constants of the motion as well. To get around this difficulty, Bohr dared to proceed contrary to electrodynamics: he simply assumed that on the orbits selected according to the quantum relations (“static” orbits), which in the end are the only ones that matter to us, no radiation occurs. Here there is only the mutual attraction of the nucleus and the electron according to the laws of electrostatics. Despite the extraordinary boldness of this assumption, it was confirmed by the brilliant results to which it led.

From this point of view, the motion of a single electron around the nucleus is periodic, and it is not difficult to apply Planck’s condition (5) to the present case, if we reduce it to the case of a system with one degree of freedom, considering only circular orbits. Indeed, the position \(M\) of the planet on a circular orbit is determined by only one coordinate, for which we may choose the angle \(\varphi\) formed by the radius drawn to the electron with some fixed direction \(or\) (Fig. 4). In this case the mass of the nucleus so greatly exceeds the mass of the electron \((\mu)\) that only with sufficient approximation may the nucleus be regarded as an infinitely heavy and replaced by its immovable center of attraction. We shall confine ourselves to this approximation in this and the following section, and shall take into account the proper motion of the nucleus only in § 10.

Let \(a\) be the radius of the circular orbit which the electron describes around the fixed center, \(v\) the velocity in this motion; let, as before, \(-e\) be the (negative) charge of the electron, and the positive charge of the nucleus \(\kappa e\). In circular motion the centrifugal force must be exactly balanced by the Newtonian attraction, which leads to the following relation between the radius and the velocity:

\[ \frac{\mu v^{2}}{a}=\frac{\kappa e^{2}}{a^{2}} \quad\text{or}\quad \mu v^{2}=\frac{\kappa e^{2}}{a}\ . \ . \ . \ (11). \]

With the aid of this relation we obtain the value of the total energy \(A\) of the electron, which is composed of kinetic and potential energy,

\[ A=\frac{1}{2}\mu v^{2}-\frac{\kappa e^{2}}{2a}, \ . \ . \ . \ . \ (12). \]

and also the angular velocity

\[ \varphi=\frac{v}{a}=e\sqrt{\frac{\kappa}{\mu a^{3}}} \ . \ . \ . \ . \ . \ . \ (13). \]

Fig. 4.

According to ordinary mechanics all values of the radius \(a\) are possible and, consequently, according to equation (12), all values (always negative) of the energy. Let us now make use of Planck’s quantum condition

\[ \int p\,dq=hn \]

in order to make a selection among these orbits. The coordinate \(q\) in the present case is the angle \(\varphi\), and the corresponding momentum \(p_{\varphi}\), according to the rules of dynamics, will be the so-called angular momentum \(p_{\varphi}=\mu av\), i.e. a constant quantity. The integration must be extended over the whole range of the variable, i.e. from \(0\) to \(2\pi\). In the end one obtains

\[ hn=\int_{0}^{2\pi} p_{\varphi}\,d\varphi =2\pi p_{\varphi}=2\pi\mu av \ . \ . \ . \ . \ (13). \]

From this equation and from relation (11), on eliminating \(v\), one obtains

\[ a_{n}=\frac{h^{2}}{4\pi^{2}\kappa\mu e^{2}}\,n^{2} \ . \ . \ . \ . \ . \ . \ . \ (14). \]

This last quantity, when substituted in (12) and (13), gives

\[ \left. \begin{aligned} A_{n}&=-\frac{2\pi^{2}\kappa^{2}\mu e^{4}}{h^{2}}\cdot\frac{1}{n^{2}},\\ \Phi_{(n)}&=\frac{8\pi^{3}\kappa^{2}\mu e^{4}}{h^{3}}\cdot\frac{1}{n^{3}} \end{aligned} \right\} \ . \ . \ . \ . \ . \ . \ (15). \]

Thus, from all mechanically possible \(a\) and \(A\), we have selected a series of discrete ones, admissible from the quantum point of view. This means that the electron cannot rotate at just any distance from the nucleus, but can move only along static orbits determined by expression (14). We see that, as the quantum number \(n\) increases, the distances between neighboring orbits become ever larger (the first orbits are shown in Fig. 4). The inverse relation holds for the static energy levels (15); these latter, as \(n\) increases, lie ever closer together and cluster around the value \(A = 0\) (i.e., \(a = \infty\)).

§ 8. The condition for frequency.

The motion of the electron along one of the static orbits, which, according to what was said above, occurs without loss of energy, constitutes, in Bohr’s view, the normal or equilibrium state of the atom. If, however, the electron, under the influence of some perturbing cause, is removed from such an orbit, it immediately tends to jump to another in order to restore equilibrium. On this new orbit the energy will, of course, be less, since any system without external influences can only give up energy (by radiation), not acquire it. Bohr assumes that the atom radiates only during such a transition of the electron from one static path to another. The question arises: how is the wavelength of the resulting radiation determined? If one adopts the standpoint of Wien–Stark, set forth in § 4, then this wavelength must be determined by the energy present. Thus, if the energy of the initial and final paths is denoted respectively by \(A_m\) and \(A_n\), then, by analogy with equation (6), one obtains

\[ ch\nu = A_m - A_n \ldots \ldots \ldots \quad (16). \]

This is, in fact, the second hypothesis which—alongside Planck’s quantum condition—Bohr uses in his theory1. We shall henceforth call it the “Bohr condition for frequency.”

It is easy to see that this hypothesis agrees with Ritz’s combination principle (§ 5) and contains a natural explanation of that principle. Indeed, according to formula (16), the number of oscillations is represented as the difference of two terms which physically signify the energies of two static orbits. Since the transition of the electron must be possible between any two static orbits (in the direction of decreasing energy), any two terms can therefore be combined with one another.

§ 9. Explanation of the simplest laws of spectral series.

If we again turn to the case of the hydrogen-like atom, then there it will only be necessary to substitute into formula (16) the value already found for the energy, in order to obtain the general expression for the series mentioned in § 5,

\[ \nu=\frac{2\pi^2\kappa^2\mu e^4}{h^3c}\left(\frac{1}{n^2}-\frac{1}{m^2}\right) = N\kappa^2\left(\frac{1}{n^2}-\frac{1}{m^2}\right)\ldots (17). \]

If one substitutes \(n=2\), then Balmer’s formula is obtained. But, in addition, it turns out that, in numerical magnitude, the factor \(N\) also agrees with the Rydberg constant, if for the constants \(\mu, e, h\) one substitutes the most accurate modern values obtained from other phenomena. It is best, in this calculation, to express \(N\) in the following form:

\[ N=\frac{2\pi^2\mu e^4}{h^3c} =\frac{2\pi^2}{c}\cdot\frac{\mu}{e}\left(\frac{e}{h}\right)^3. \]

This is because the ratios \(\frac{e}{\mu}=5{,}2908\). (Fortrat, 1912) and \(\frac{h}{e}=1{,}370\cdot 10^{-17}\) (Warburg and Müller, 1915) can be determined considerably more accurately than the constants themselves. Hence the following numerical value is obtained for \(N\):

\[ N=110100. \]

This should have an accuracy of approximately \(1.5\%\); we see that it does indeed, within the indicated limits of accuracy, agree with the experimental value (§ 5) \(109677{,}69\). At the present time, conversely, optical data are used for the most accurate determination of the universal constants (cf. § 10).

The nucleus of the hydrogen atom carries one positive charge, and therefore for this gas in formula (17) one should put \(\kappa=1\), and then this formula for \(n=1\), \(n=2\), and \(n=3\) gives the three series (7a), (7b), and (7c) of hydrogen. That is, the lines of the infrared Paschen series are emitted when the electron jumps from some outer ring (\(m=2,3,4,\ldots\)) to the first one nearest the nucleus. The lines of the Balmer series are obtained in the transition to the second ring, the lines of the ultraviolet Lyman series—in the transition to the third ring.

If one takes into account such a mechanism for the origin of the lines, it becomes clear why in Geissler tubes only the first 12 lines of the Balmer series are observed: for the appearance of the line corresponding to the ordinal number \(m\), the preliminary condition must be fulfilled that, in some part of the atoms, the electrons revolve in the \(m\)-th orbits. But it is obvious that orbits with a large radius are formed the more easily, the lower the density of the gas, the more rarely

it became that the sphere of the atom is disturbed by neighboring atoms and molecules. It must therefore be assumed that the gas pressure in those parts of the stellar atmosphere which give the 29 lines is considerably less than the ordinary pressure in Geissler tubes.

§ 10. Proper motion of the nucleus.

In § 7 we have already mentioned that all the remaining atoms, except hydrogen, satisfy the proposed formulas (16) and (17) only in the ionized state. For example, these formulas are valid for singly ionized helium, i.e. for an atom which consists of a helium nucleus with two charges \((\kappa=2)\) and only one electron. In this case formula (17) gives

\[ \nu=4N\left(\frac{1}{n^2}-\frac{1}{m^2}\right), \]

which may also be written as

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

We see that this formula contains the series (8a) and (8b) of § 5, which were attributed to hydrogen. Indeed, for \(n=3\) one obtains an expression comprising Rydberg’s series (8a) and Fowler’s series (8b):

\[ \nu=N\left[\frac{1}{1.5^2}-\frac{1}{\left(\frac{m}{2}\right)^2}\right];\quad m=4,5,6\ldots \qquad (19). \]

For \(n=4\) we have the formula

\[ \nu=N\left[\frac{1}{2^2}-\frac{1}{\left(\frac{m}{2}\right)^2}\right];\quad m=5,6,7\ldots \qquad (20). \]

which, for integral \(m\), gives, besides the lines of Pickering’s series (8), a number of other lines coinciding with the hydrogen lines of the Balmer series (7) and therefore not mentioned by Pickering.

The circumstance that the observed series (8, 8a, 8b) owe their origin not to hydrogen, as was formerly thought, but to helium, as is required by the theory just set forth, has been fully confirmed. We have already mentioned that Fowler found the presence of a helium impurity in hydrogen to be absolutely necessary for the excitation of these lines; then, in the summer of 1914, Paschen and Bartels observed them in pure helium. But Bohr himself already pointed to one circum-

evidence, which furnished decisive proof of his objections: we already know that the lines of the Fowler series do not quite satisfy formula (8b); this deviation is fully explained if the calculation is carried out somewhat more rigorously and the proper motion of the nucleus is taken into account.

In reality the nucleus is not some immobile center; it has a finite mass \(M\), as a result of which the electron and the nucleus describe circles about their common center of gravity. The radii of these orbits, as is known, are inversely proportional to the masses \(\mu\) and \(M\). Consequently, in the quantum condition (11), instead of \(P_\varphi\) one must introduce the sum of the angular momenta of the nucleus and the electron, and, after some intermediate calculations analogous to the preceding ones, formula (17) is obtained, in which, however, the quantity \(N\) now also depends on the mass \(M\)

\[ N=\frac{2\pi^2\mu e^4}{h^3c}\cdot\frac{1}{1+\frac{\mu}{M}}\ldots\ldots\ldots\quad (21). \]

Let us denote the expression for \(N\) according to formula (17), which we obtain by assuming the mass \(M\) infinitely large in comparison with \(\mu\), by \(N\infty\); then

\[ N=\frac{N\infty}{1+\frac{\mu}{M}}\ldots\ldots\ldots\quad (21^1). \]

Therefore the Rydberg number, strictly speaking, is not a universal constant, but varies, although very slightly, from element to element. The greatest deviations from \(N\infty\) occur precisely for hydrogen and helium, since for both these elements the ratio \(\frac{\mu}{M}\) assumes its greatest value.

The most recent measurements of Paschen (1916) gave for the Rydberg number

\[ N_H=109677{,}69;\quad N_{He}=109722{,}14\ldots\ldots\ldots\ldots\quad (22). \]

Taking \(e\), \(h\), \(M_H\) as known, one can, with the aid of these numbers, calculate the quantities \(\frac{e}{\mu c}\), \(\frac{M_H}{\mu}\), and \(N\infty\). One obtains

\[ \frac{e}{\mu c}=1{,}76\cdot 10^7;\quad \frac{M_H}{\mu}=1844;\quad N_\infty=109737{,}16\ ^1) \]

\(^1\) The factor \(C\) \((3\cdot 10^{10})\) has been introduced because the author everywhere expresses \(e\) in electrostatic units, whereas it is generally customary to give the ratio \(\frac{e}{\mu}\) in electromagnetic units.

Translator

The best value for \(\frac{e}{\mu e'}\), found experimentally (from the Zeeman effect), is, as already mentioned, \(1.76 \cdot 10^7\). Thus the agreement is complete.

This new triumph of Bohr’s theory had a decisive influence on the opinion of scientists. Previously, the majority of them had regarded this theory with restraint. It was generally accepted that Bohr had succeeded in constructing the Rydberg constant from the universal quantities \(e\), \(\mu\), and \(h\), but it was thought that his model had played an accidental role in this, and its productivity was considered exhausted by these results. But the fact that, by increasing the accuracy of the calculations, it proved possible to obtain new important results showed that here we are dealing not merely with a superficial analogy. And this, in turn, prompted some physicists to try to apply celestial mechanics still more deeply to atomic theory.

Let us also briefly mention how, in Bohr’s opinion, the hydrogen molecule should be arranged. The latter (\(\mathrm{H}_2\)) consists of two atoms and consequently possesses two nuclei and two electrons. The arrangement of these constituent parts is shown in Fig. 3b; both electrons move along one (shown in the figure) circle around the line joining the two nuclei, which also serves as the axis. The dimensions of the molecule admissible according to quantum theory can be calculated in the same way as was done for the atom (§ 7). For the innermost circle that the electrons can describe, and which corresponds to the normal state of the non-luminous gas, one obtains

\[ a' = 0.504 \cdot 10^{-8}\ \text{cm} \qquad \ldots\ldots\ldots\ (23'), \]

whereas formula (14) gives for the atom a value differing only slightly,

\[ a = 0.528 \cdot 10^{-8} \qquad \ldots\ldots\ldots\ (23). \]

Half the distance between the nuclei is related to the radius \(a'\) as

\[ 1 : \sqrt{3}. \]

The value (23′) is in good agreement with the results of kinetic theory. A further confirmation of Bohr’s model of the molecule is due to Debye (1915), who investigated the dispersion of a gas consisting of such formations and established complete agreement with the experimental data for the dispersion in hydrogen.

III. Section.

Systems with Several Degrees of Freedom.

§ 11. Generalization of the quantum conditions to the case of several degrees of freedom.

The question of how Planck’s condition (5) can be generalized to systems with several degrees of freedom was raised by Poincaré at the Brussels congress on quantum theory in 1911. But only four years later was this question, to a certain extent, resolved simultaneously by Planck and Sommerfeld. In doing so, Planck proceeded from general statistical considerations, whereas Sommerfeld had above all in mind the application to Bohr’s model of the atom. Since in the following paragraphs we shall consider certain special cases in the study of which Sommerfeld’s theory received brilliant confirmation, we shall follow precisely his path. We shall return later to Planck’s theory, which differs from Sommerfeld’s only formally (§§ 16 and 17).

Sommerfeld started from the fact that the lines of the Balmer series are not simple, but, when investigated with spectral apparatus of very great resolving power, turn out to be at least double. Since, according to Bohr’s theory (§ 8), a spectral line is the result of a combination of two static orbits, Sommerfeld concluded that there must exist a larger number of orbits than is given by Bohr’s formula (17); and this prompted him to consider not only circular, but also elliptical orbits.

Under the influence of a Newtonian center of attraction, a body describes, generally speaking, an ellipse (Kepler’s ellipse), in one of whose foci this attracting center is situated. Thus the matter reduces to finding, among all possible elliptical paths which the electron may describe according to the laws of classical mechanics, orbits satisfying the requirements of quantum theory, static orbits. But an ellipse is determined, in its size and in its form, by two constants (for example, the major and minor semiaxes), and therefore two quantum conditions are also necessary for establishing it. Let the position of the electron in the plane of the orbit be determined by the polar coordinates \(r\) and \(\varphi\), if the nucleus is taken as the origin (which we again regard as a fixed center). Sommerfeld retains the quantum condition

\[ \int P_{\varphi}\cdot d\varphi = nh \ldots \ldots \ldots \ldots \ldots (24), \]

which was justified in Bohr’s case, and supplements it by the analogous relation

\[ \int P_r\,dr=n'h \ldots\ldots\ldots (24'). \]

where by \(P_r\) is meant, of course, the momentum corresponding to the radius vector \(r\) (what is true for \(\varphi\), we have the right to require also of \(r\)). Both integrals are to be extended over all points of the path, i.e., over the entire period of the motion.

In the more general case, if the system is determined by \(f\) coordinates, \(q_1, q_2,\ldots, q_f\) and by the \(f\) momenta corresponding to them \(p_1, p_2,\ldots,p_f\), then, according to Sommerfeld, \(f\) conditions of the form

\[ \int p_i dq_i=n_i h;\ i=1,2,3 \ldots\ldots\ldots (25). \]

must be imposed on it.

Since \(p_i\) and \(dq_i\) always have the same signs, it follows from this definition that the \(n_i\) are always positive integers.

The result to which conditions (24) and (24′) lead for the semiaxes of the stationary ellipses is as follows:

\[ a=\frac{h^2}{4\pi^2me^2}(n+n')^2;\quad b=a\frac{n}{n+n'} \ldots (26). \]

Thus, for a given sum of the quantum numbers \(n+n'\), \(a\) will be constant, \(b\) variable, and the ratio \(\frac{b}{a}\) will be a proper fraction with denominator \(n+n'\). For quantum sums \(n+n'=2,3,4\), the possible ellipses are shown in Fig. 5. In this, the path which is represented by a straight line traversed twice (out and back)—\(b=0,\ n=0\)—is shown by a dotted line. On this path the electron would have to collide with the nucleus, and therefore it should be regarded as impossible.

Fig. 5 a, b, c.

Fig. 6 a, b, c.

The number of orbits actually existing in each case is equal to \(n+n'\). It should be noted that the ellipses are arranged confocally with respect to the nucleus, and not concentrically, as is shown in Fig. 5 for simplicity.

Sommerfeld’s aim was achieved insofar as a larger number of stationary orbits was obtained. However, at first this result caused disappointment, since the increase in the number of orbits was not at all accompanied by an increase in the number of energy levels. In fact, the energy corresponding to a Keplerian ellipse is a func-

only on its major axis; consequently, for all the orbits of each of our figures \((a, b, c)\) it will be one and the same. It is expressed as

\[ A=-\frac{2\pi^2 k^2 \mu e^4}{h^2(n+n')^2}\ . . . . . . \ . \ (27). \]

which, for integral \(n\) and \(n'\), gives precisely the same discrete values as Bohr’s expression

\[ A=-\frac{2\pi^2 k^2 \mu e^4}{h^2 n^2}. \]

Thus Sommerfeld obtained the same simple series of lines that are contained in formula (17), but every line arises, according to his theory, in different ways as the result of the combination of several different pairs of stationary orbits. It contains, so to speak, several coincident degrees of freedom. Only by taking into account the variable mass of the electron as a function of its velocity, in accordance with the requirement of the theory of relativity, did Sommerfeld succeed in separating these hidden degrees of freedom from one another and in obtaining a brilliant agreement with experiment (§ 18).

§ 12. Conditionally Periodic Motions.

In Sommerfeld’s theory, developed in the preceding paragraph, several questions nevertheless remain open. Even in the consideration of periodic motions, where the integration obviously had to be extended over the whole period, it was unclear which of the many possible coordinate systems by means of which the motion can be described should be chosen. In non-periodic motions—even the very limits of integration are unknown. Therefore the next step forward was made when, independently of one another, Schwarzschild and the author (1916), borrowing from celestial mechanics the concept of “conditionally periodic motions,” transferred it to the domain of atomistics and rigorously established, for this generalized class of mechanical systems, Sommerfeld’s conditions for the choice of coordinates and of the limits of integration.

“Conditionally periodic” is the name given to systems for the determination of which one can choose coordinates in such a way that they execute monotonic oscillations between two constant limits, or, as it is customary to say, perform librations. The simplest example of such a motion is the resultant of two sinusoidal oscillations perpendicular to one another,

\[ X=X_0\sin(\omega_x t+\sigma_x),\quad Y=Y_0\sin(\omega_y t+\sigma_y), \]

where \(X_0,\ Y_0,\ \omega\), and \(\sigma\) are constants. It is perfectly clear that \(X\) in the course of time \(t\) runs through all values from \(-X_0\) to \(+X_0\), while \(Y\)—all values

from \(-Y_0\) to \(+Y\). If the frequencies \(\omega_x\), \(\omega_y\) are incommensurable, then the resulting curve (Fig. 7) comes arbitrarily close to any point of the rectangle enclosed between the limits of libration, or, to use the mathematical term, fills the rectangle with uniform density. Let us consider another example, important for the theory of Sommerfeld. We have already mentioned that the mass of the electron, strictly speaking, is not constant, but depends on its velocity. The form of this dependence is established by the theory of relativity. If this circumstance is taken into account, then motion under the influence of a Newtonian center of attraction undergoes a certain modification: the orbit remains elliptical as before; however, this ellipse does not preserve its position, but its major axis rotates with an insignificant angular velocity in the plane of the orbit, about the focus, with the same period as the electron in its motion along the ellipse. The curve described in this case by the electron is shown in Fig. 8. Here again the coordinate \(r\) varies between two constant limits \(r_1\) and \(r_2\), and the cyclic variable \(\varphi\)—from 0 to \(2\pi\). The annular region \(r_1 \leqq r \leqq r_2\), generally speaking, is filled by the orbit with uniform density.

Fig. 7.          Fig. 8.

From the mathematical point of view, the characteristic feature of conditionally periodic motions consists in the fact that, when the coordinates are chosen in the manner just described, the momentum \(p_i\) corresponding to any coordinate \(q_i\) depends only on the variable \(q_i\) and does not depend on the other \(q\)'s,

\[ p_i = p_i(q_i) \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots (25), \]

where \(p_i\) vanishes at the boundaries of the libration \((p_i = a_i,\ q_i = b_i)\).

The quantum theory for conditionally periodic systems, in the form in which it was formulated by the author, consists in the fact that the quantum integral (25), extended between two successive contacts with one and the same boundary of libration, or, what is the same thing, the integral taken twice from one boundary of libration to the other, is equal to \(n_i h\), i.e.

\[ 2\int_{a_i}^{b_i} p_i\,dq_i = n_i h \qquad \ldots\ldots (29). \]

If the coordinate is cyclic, as for example \(\varphi\) in the case of the ellipse, then the integration must be carried out from \(0\) to \(2\pi\)

\[ \int_{0}^{2\pi} p_\varphi dq_2 = n_2 h \qquad \ldots\ldots (29'). \]

It can be proved in general that these conditions, insofar as all degrees of freedom are subordinated to them, completely determine the energy as a function of the quantum number.

§ 13. Fine structure of the hydrogen lines.

As a first application of the rules established for conditionally periodic systems, let us consider the already mentioned example in the preceding section of motion in an ellipse \(^{1}\). The conditions (29) and (29′), with the notation of Fig. 8, respectively take the form

\[ 2\int_{r_1}^{r_2} p_r\,dr = n'h;\qquad \int_{0}^{2\pi} p_\varphi d\varphi = nh \qquad \ldots\ldots (30). \]

The theory of relativity introduced only insignificant changes into the conditions that led us to the elliptical orbits considered in § 11. Thus, the quantum relations continue to select, from all mechanically possible orbits, a series of “stationary” ones, which, with sufficient approximation, are represented by the ellipses of the figure.

However, in contrast to that case, the energy levels corresponding to different ellipses of one and the same figure are not strictly equal to one another, but differ slightly from each other. Indeed, the approximate \(^{2}\) expression for the energy has the following form

\[ A = -\frac{Nhc\alpha^{2}}{(n+n')^{2}} -\frac{Nhc\alpha^{4}a^{2}}{(n+n')^{4}} \left[\frac{1}{4}+\frac{n}{n'}\right] \qquad \ldots\ldots (31). \]

The letter \(\alpha\) here denotes a constant which depends in the following way on the universal constants \(e\), \(h\), and \(c\)

\[ \alpha = \frac{2\pi e^{2}}{hc} \]

\(^{1}\) In fact, this case had been investigated by Sommerfeld even before the establishment of the general rules (29) and (29′) for conditionally periodic systems.

\(^{2}\) An exact expression for the energy can also be given, but for our purposes formula (31) is more transparent.

As for the physical meaning of this number, it represents the velocity (measured relative to the velocity of light) possessed by the electron in the first static orbit \((n=1)\) of the hydrogen atom \((\kappa=1)\). The numerical value of the square that interests us is as follows:

\[ \alpha^2 = 5.316 \cdot 10^{-5}, \]

and therefore the correction term introduced by the theory of relativity is always small in comparison with the first term of expression (31).

This correction term evidently has a twofold effect: first, it increases the absolute value of the energy for all ellipses with one and the same major axis by the amount

\[ \frac{Nhc\kappa^4\alpha^2}{4(n+n')^4}; \]

second, it increases the energy of different ellipses by amounts which differ from one another and are proportional to the ratio \(\dfrac{n'}{n}\):

\[ \frac{n'}{n}\cdot \frac{Nhc\kappa^4\alpha^2}{(n+n')^4}\cdots \]

Taking into account Bohr’s condition

\[ hcv=A_1-A_2, \]

we obtain the frequency of the light emitted in the transition of the electron from one static path \((m,\ m')\) to another \((n,\ n')\) in the following form:

\[ \nu = N\kappa^2 \left[ \frac{1}{(n+n')^2} - \frac{1}{(m+m')^2} \right] + \frac{N\alpha^2\kappa^4}{4} \left\{ \frac{1}{(n+n')^4} \left( 1+\frac{4n'}{n} \right) - \frac{1}{(m+m')^4} \left( 1+\frac{4m'}{m} \right) \right\}. \tag{33} \]

Owing to the small magnitude of the nuclear charge \(\kappa\), this entire expression differs only very slightly from its first term, which, according to § 9, gives the expression for the Balmer series or for any other similar series. Therefore the oscillation numbers (33), for given values of the quantum numbers, are closely grouped around the number obtained from formula (17) and create the fine structure of the hydrogen line.

Fig. 5 shows that for a given system of numbers \((m,\ m',\ n,\ n')\) there exist \(m+m'\) possible initial orbits and \(n+n'\)—final ones. Hence it follows that a hydrogen line may consist of \((m+m')(n+n')\) components. However, the number of components actually observed is apparently smaller. In order to introduce a restriction, Sommerfeld was guided by the principle that the quantum numbers

are essentially positive numbers; therefore it is natural to suppose that, when electrons jump, not only the sum of these numbers must decrease, but that they themselves, taken separately, cannot increase. That is, not only must the inequality \(m+m' > n+n'\) hold, but also

\[ m \geq n;\quad m' \geq n' \ldots\ldots\ldots\ldots\ldots (34). \]

Let us take, for example, the principal line of the Balmer series \((H_{\alpha})\)

\[ \nu = N\left(\frac{1}{2^2} - \frac{1}{3^2}\right); \]

\(m+m' = 3;\ n+n' = 2\). Thus here a priori \(2 \cdot 3 = 6\) modes of transition would have been possible, which, however, by the inequalities (34) are reduced to four according to the following scheme:

\[ \begin{array}{rcl} m=3,\ m'=0 & \searrow & \\[-2mm] m=2,\ m'=1 & \to & n=2,\ n'=0 \\[-1mm] m=1,\ m'=2 & \searrow & n=1,\ n'=1 . \end{array} \]

Subsequently, however, it turned out that Sommerfeld’s inequalities are not strict, but only approximate: under certain circumstances (depending on the manner of excitation of the Geissler tube), lines appear which contradict them, though always only with weak intensity.

Sommerfeld’s theory also provides grounds for estimating the intensity of the components. From statistical considerations it follows with great probability that the probability of an elliptical path is proportional to the ratio of the axes of this ellipse \(\left(\dfrac{n'}{n+n'}\right)\). The probability of a transition from one orbit to another will then be proportional to the product of the corresponding numbers for the initial and final orbits, i.e.

\[ \frac{n'}{n+n'} \cdot \frac{m}{m+m'} . \]

In reality, the intensity of the components approximately follows this rule when the luminescence of the gas is produced by a spark discharge; whereas with a constant current, apparently, the situation is somewhat different.

In Fig. 9 the components of the first Balmer lines \((H_{\alpha}\) and \(H_{\beta})\) are shown according to position and intensity. Here the unreal lines that contradict the conditions (34) are shown either by a dotted line or marked with a short arrow; the theoretically expected intensities,

for clarity, are represented by the length of the corresponding lines. Characteristic for the arrangement of the lines is the constancy of the distances between different pairs of lines. This occurs because, for one of the possible final paths (e.g. \(n=2,\ n'=0\)), formula (33) gives \(m+m'\) possible values of \(\nu\), depending on the choice of the initial path. For another final path one obtains again \(m+m'\) values of \(\nu\), which differ from the corresponding values of the first group only by the difference of the correction terms depending on \(n\) and \(n'\), i.e. by a constant quantity.

Fig. 9.

\[ \Delta \nu_H=\frac{N\alpha^2}{16} \]

(as is not difficult to verify by making the appropriate substitutions in formula (33). Transl.). (This quantity \(\Delta \nu_H\) Sommerfeld calls the difference of oscillations of the hydrogen doublet; its numerical value is obtained theoretically on the basis of the data in §§ 5 and 13 for the quantities \(N\) and \(\alpha^2\):

\[ \Delta N_H=0{,}364\ \mathrm{cm}^{-1}. \]

In comparing with observations it should be remembered that the structures shown in Fig. 9 cannot be fully resolved even with the aid of the most powerful optical means and appear as simple doublets. Indeed, in hydrogen the components are rather broad and blurred, while the distances are so small that neighboring lines easily merge with one another. What can be directly measured as the width of the doublet is the distance \(\Delta \nu_H\) between the centers of gravity of the two close groups of lines. If one uses the theoretical intensities, then for \(H_\alpha\) one obtains

\[ (\lambda=6562{,}8) \]

\[ \Delta \nu'_H=0{,}842,\quad \Delta \nu_H=0{,}307\ \mathrm{cm}^{-1} \]

or, in wavelengths,

\[ \Delta \lambda'_H=0{,}132\ A^0 \]

Experimentally, the following values were found:

	\(\Delta \lambda_H\)	\(\Delta \nu_H\)
Michelson . . .	\(0{,}140\,A\)	\(0{,}33\ \mathrm{cm}^{-1}\)
Fabry and Buisson.	\(0{,}132\)	\(0{,}307\)
Meissner . . . .	\(0{,}124\)	\(0{,}289\)

The agreement with the theoretical value is brilliant.

More favorable circumstances for testing the theory occur in the series of ionized helium \((z = 2)\). First, because the helium lines are sharper; second, because, owing to the presence of \(z\) in the correction term of formula (33), the structure of the helium lines is sixteen times more distinct than that of the corresponding hydrogen lines. The corresponding measurements, with all the accuracy at present attainable, were made by Paschen (in 1916). Fig. 10 gives the theoretical picture and the experimental results for two different methods of excitation for the principal line of the Fowler series \((\lambda = 4686)\)

\[ \nu = 4N\left(\frac{1}{3^2} - \frac{1}{4^2}\right) \]

Fig. 10.

If we examine the results shown in the figure, we shall see that they correspond to the theoretical expectations in every detail. Unforeseen components are absent; there are missing only, under constant current, the lines III \(a, b\), contradicting inequalities (34), and the line I \(d\)—under spark discharge; line II \(c\) was not measured, since on the photographic plate it is impossible to establish a weak line between two bright ones. Individual pairs of lines lying in close proximity merge with one another. The agreement between theory and experiment illustrated by this figure constitutes a truly brilliant page, the greatest triumph of the quantum theory.

Just as good an agreement is observed also for the second line of the Pickering series; however, we shall not dwell on this case, but shall pass to the fine structure of the third line (Fig. 11). Corresponding to the three possible orbits, the components are arranged here, as in Fig. [[unclear: missing figure number]], into three groups I, II, and III. But whereas in the first case groups I and II overlap one another, here they are completely separated and so narrow that each of them appears as a single broad line. The components \(f\), apparently, are too weak and inaccessible to observation.

Fig. 11.

From the measurement of all these fine structures one obtains the following best experimental value of the difference \(\Delta \nu_H\)

\[ \Delta \nu_H = 0.3645 \pm 0.0045. \]

Thus spectroscopy gives us three quantities which, according to the theory, are constructed from the universal constants \(e, \mu, h\), namely: the Rydberg number (§ 8), the change of the Rydberg number from element to element, due to the proper motion of the nucleus (§ 10), and the difference of the oscillations of the hydrogen doublet \(\Delta \nu_H\). From three equations one can numerically determine the three unknowns \(e, \mu, h\). With the present accuracy of spectroscopic measurements, this method already now gives results identical with all other methods.

§ 14. Spectra of X-rays.

The discovery by Laue, Friedrich and Knipping made it possible to apply spectral analysis in the region of X-rays. We owe to Moseley the first detailed investigations of the wavelengths of these rays for a series of elements. The measured vibration numbers can be arranged in several series, which are called, in order of decreasing vibration numbers (hardness), respectively: the \(K\)-series, the \(L\)-series, etc. (the lines \(K\alpha\), \(K\beta\), etc.). Fig. 12 gives Moseley’s photographs of the \(K\alpha\) and \(K\beta\) lines for a series of elements. The last member of this series relates to brass (Brass). The fact that here the two zinc lines simply join the two \(Cu\) lines of the preceding photograph indicates that X-ray spectra are a purely additive property of the atom.

The analogous lines in this figure are arranged along a parabola. Moseley succeeded in representing the vibration numbers of the \(K\alpha\)-lines by the formula

\[ \nu_K = N(\kappa - 1)^2 \left[ \frac{1}{1^2} - \frac{1}{2^2} \right], \]

and of the \(L\alpha\) lines by

\[ \nu_L = N(\kappa - 7.4)^2 \left[ \frac{1}{2^2} - \frac{1}{3^2} \right], \]

where \(\kappa\) denotes the number of charges of the nucleus of the corresponding element.

From this result two consequences follow: first, the idea set forth in § 6 is confirmed, that the charge of the nucleus, in passing from one element to another (in the periodic system), increases by one unit. Secondly, the X-ray lines \(K\alpha\) and \(L\alpha\) turn out to be similar to the hydrogen lines, with the small modification that what is effective is not the whole charge \(\kappa\), as in the Balmer series (7) or (17), but only a part of it.

Upon more precise investigation it turns out that these lines are not simple, but possess a fine structure. Theoretically, with complete similarity to hydrogen, one might have expected the lines \(La\) to have exactly the same structure as the line \(Ha\) (Fig. 9), with the only difference that all distances in the first case would have to be \((\kappa - 7.4)^4\) times greater than in the second. In reality one obtains only a simple doublet, whose difference of oscillations is found to be in the theoretically correct ratio to the difference of oscillations for hydrogen.

\[ \Delta \nu = (\kappa - 7.4)^4 \Delta \nu_H \]

Thus, the similarity to hydrogen, evidently, owing to the complexity of the excitation conditions, is limited. As for \(Ka\), its structure fully satisfies the requirements of the theory. Here there is one possible final orbit \((n + n' = 1\) or \(n = 1,\ n' = 0)\), but two initial ones. Therefore the sought structure is a simple doublet, broadened in comparison with \(\Delta \nu_H\) by \((\kappa - 1)^4\) times.

Fig. 12.

The measurements made by Ziegban and his collaborators for all elements, insofar as this was technically possible, confirmed the requirements of the theory. For illustration we give the following drawing (Fig. 13), which contains all the experimental material for the \(K\)-series from sodium \((\kappa = 1)\) to neodymium \((\kappa = 60)\). Along the ordinate axis are plotted \(\nu \cdot 10^{-4}\); along the abscissa axis, \(\kappa\); \(\alpha_1\) and \(\alpha_2\) are components \(Ka\), \(\beta_1\) and \(\beta_2\) are Ziegban’s designations for \(K\beta\) and \(K\gamma\). As for the differences of oscillation numbers, according to precise measurements their course coincides with Sommerfeld’s predictions, and this is an excellent confirmation of the theory, if one takes into account that \(\Delta \nu\), for example for \(\kappa = 92\) for uranium, is \((84.4)^4\), i.e. in round numbers 150 million times greater than for hydrogen. Few theories exist that allow such extrapolation!

The circumstance that the X-ray spectra are partly similar to the spectrum of hydrogen is explained as follows. If an electron rotates close to the nucleus, then the remaining electrons, which move at a greater distance, exert on it a very slight influence, which may be neglected. The matter thus reduces, in essential features, to the case where there is only a nucleus with \(\kappa\) charges and one single electron, as is also assumed

in the theory developed in §§ 7, 8, and 13. How, however, can one explain that part of the charge of the nucleus is masked? A possible answer to this question was already indicated by Bohr: let us imagine that not one electron, but several,

Fig. 13.

arranged in a circle, rotate around the nucleus (Fig. 14). Then the attraction of the nucleus must counteract the repulsion between the electrons, which is manifested precisely in the fact that instead of \(\kappa^2\) the formula contains the smaller factor \((\kappa-\sigma)^2\). At the same time, one could also consider the case where several, say \(p\), electrons are closer to the nucleus than those whose motion we are considering; then the complex consisting of the nucleus with \(\kappa\) positive charges and \(p\) electrons at certain distances acts approximately like a nucleus with \(\kappa-p\) positive charges. According to the work of Debye and the as yet unpublished investigations of Kroo and Sommerfeld, the matter apparently stands as follows: the atom is constructed, as Bohr supposed, from a nucleus with several concentric rings occupied by electrons. The innermost is called the \(K\)-ring, the next the \(L\)-ring, and so on. The emission of X-rays occurs in the following way: under the influence of an external cause (a cathode ray) an electron pe-

Fig. 14.

jumps from the inner ring to the outer one, e.g. from the \(K\)-ring to the \(L\)-ring. Then the former will have one electron fewer than it should, while in the latter there will be one extra electron. When the electron returns to its normal position on the inner ring—this process is accompanied by the radiation of energy—an X-ray wave is emitted, in accordance with Bohr’s condition (16).

A consistent development of this point of view has not yet been completed, and therefore one still cannot say exactly what number of electrons on the rings corresponds to the normal state. Apparently, as a preliminary matter, one may speak respectively of three and eight electrons on the two inner rings; however, it is by no means excluded that the numbers 2, 8, 8, 18, 18, etc., which follow from chemical considerations (the periodic law), will be confirmed.¹

§ 15. Theory of the Stark Effect.

If a radiating atom is introduced into an electric field, then under the influence of this field the character of the radiation changes: spectral lines which, in the absence of the field, appeared simple, in many cases split under the action of the electric forces into several components. We owe this remarkable discovery to J. Stark (1913), and for this reason the phenomenon itself bears his name. Although the influence of even moderate field strengths is already quite distinct, and although no especially delicate optical means at all were required for observing the phenomenon, this discovery was very difficult to make by chance, since it is extremely difficult to obtain, in a luminous medium, even a moderate potential drop. Only Stark, who systematically sought this phenomenon, succeeded in overcoming the aforementioned difficulties by means of an ingenious arrangement: he excited luminescence in a layer of rarefied gas between the plates of a capacitor, placed inside a discharge tube, by canal rays entering it.²

¹ J. J. Thomson, Corpuscular Theory of Matter, chap. 6; N. Bohr, Phil. Mag. 26, p. 857, 1913; W. Kossel, Ann. d. Phys. 49, p. 229, 1916.

² When luminescence is excited in a discharge tube under the influence of a field, then at the same time, owing to the ionization of the gas, a conduction current is produced. Therefore the potential difference immediately falls, and it becomes impossible to subject the luminous molecules to the influence of large field strengths. On the other hand, if at high rarefaction, when the free path is measured in centimeters, the electrodes are placed very close to one another (at a distance of several millimeters), then on the path between them no ionization by impact occurs, but at the same time no luminescence occurs either. The whole ingenuity of Stark’s method consists in the fact that into this narrow space, throughout which a large potential drop could be created, he introduced luminous molecules from outside. These luminous molecules were supplied to him by particles of canal rays.—Stark’s tube had three electrodes: an anode, a cathode with an ot-

Thus he was able to investigate a whole series of substances. For our purpose, however, especially important are the very careful measurements which he made on the “fine electrical decomposition” of the first four lines of the Balmer series of hydrogen \((H_\alpha, H_\beta, H_\gamma, H_\delta)\). It turned out that the components into which these lines are decomposed are arranged symmetrically with respect to the normal position, and that the splitting increases in proportion to the electric force. Observations were also made perpendicular to the direction of the field; in this case the resolved components proved to be linearly polarized. In Tables I–IV Stark’s results are compared under the heading “measured.” The letter \(p\) denotes a component whose electric vector is directed parallel to the field; the letter \(S\), a component with electric vector perpendicular to the field. \(\Delta\lambda\) is the distance between the corresponding components (as well as between those symmetric with them) in Ångström units; under the heading “int.” is placed the relative intensity, these intensities being compared with one another only within a single column. The electric force to which the given values of \(\Delta\lambda\) refer is known with some uncertainty and is estimated by Stark at \(104{,}000\) volt/cm. In Fig. 15 the pattern of the splitting is represented graphically.

Fig. 15.

Stark’s theory of the phenomenon reduces to considering the motion of the hydrogen atom in a homogeneous electric field. This motion—if one neglects the correction term introduced on the basis of the theory of relativity—is conditionally periodic. In order that

substances and a third electrode placed very close to the cathode. Between the latter two electrodes a strong field was excited, and it was into this field that the particles of the canal rays entered.

Translator.

to reduce the expressions for the momenta to the form (28), one has to use parabolic coordinates1. The coordinate surfaces are obtained by rotating the drawing about the straight lines \(\xi=0\) and \(\eta=0\), as axes. Thus there are obtained two families of confocal paraboloids of revolution (\(\xi=\mathrm{const}\) and \(\eta=\mathrm{const}\)) and meridian planes \(\varphi=\mathrm{const}\) (where \(\varphi\) denotes the angle between the rotating plane and some fixed position of it). A point of space is therefore determined by the coordinates \(\xi\), \(\eta\), and \(\zeta\).

We shall not take into account the proper motion of the nucleus; we shall regard it as a fixed center. Then the system of coordinates must be oriented so that the nucleus is the origin, and the straight line \(\xi=0\) coincides with the direction of the field. The solution of the problem shows that the motion of the electron is enclosed in an annular space, which is obtained by rotating the curvilinear quadrilateral \(ABCD\), Fig. 16 (about the axis \(\xi=0\), \(\eta=0\)). At the same time one must imagine that the electron moves along the curve shown inside the quadrilateral, and that simultaneously the whole drawing rotates (with variable angular velocity) about the aforementioned axis.

Fig. 16.

The position of the boundaries of libration (cf. § 12) \(\xi=\xi_1\), \(\xi=\xi_2\), \(\eta=\eta_1\), \(\eta=\eta_2\) of course depends on the initial position and the initial velocity of the electron; moreover, for each particular case the motion will be different, so that, by a corresponding choice of initial conditions, one can assign to the quantities \(\xi_1\), \(\xi_2\), \(\eta_1\), and \(\eta_2\) any value. In the interest of § 17 let us consider three limiting cases. First, suppose that \(\eta_1\) coincides with \(\eta_2\), and \(\xi_1\) with \(\xi_2\); then the quadrilateral contracts into a single point and the electron moves in a circular orbit situated perpendicular to the direction of the field. Next, we may put \(\xi_1=0\), and \(\eta_1=0\); then the path of the electron fills the region obtained by rotating the triangle \(B^1CD^1\). Now let also \(\xi_2\) continually decrease and finally become equal to zero; then the motion of the electron will be confined to a straight-line segment: it will oscillate along the ray \(\xi=0\) between the points \(O\) and \(B^1\), as a result of which we shall call this orbit pendulum-like. We obtain the third special case if we suppose that not \(\xi_2\), but \(\eta_2\), decreases to zero; here there is obtained a completely analogous “pendulum-like” orbit along the ray \(\eta=0\) between

points \(O\) and \(D^1\). When the orbit (without an electric field) is a Kepler ellipse (§ 11)—we reject the case of a rectilinear path as improbable, and this prediction is justified by the experimental study of the fine structure. Pendulum-like paths appear a priori just as improbable; it is remarkable, however, that the components corresponding to them appear in the Stark phenomenon, though with very weak intensity.

In the general case, we must write the quantum conditions (29) and (29′) in the following form:

\[ 2\int_{\xi_1}^{\xi_2} p_\xi(\xi)\,d\xi=n_1h;\qquad 2\int_{\eta_1}^{\eta_2} p_\eta(\eta)\,d\eta=n_2h;\qquad \int_0^{2\pi} p_\varphi\,d\varphi=n_3h. \]

From these three relations one obtains, with sufficiently good approximation, the following expression for the energy:

\[ A=-\frac{N\pi^2hc}{(n_1+n_2+n_3)^2} +\frac{3h^2E}{8\pi^2\mu e}\,(n_1+n_2+n_3)(n_1-n_2). \]

Thus, the electric field, like the dependence of mass on velocity, exerts such an influence that the number of energy levels increases, and therefore the degrees of freedom (possibilities of occurrence) already existing in the spectral line but superposed on one another are separated and become visible.

Fig. 17.

From Bohr’s condition (16) we obtain the following value for the change in the number of vibrations in comparison with the normal one:

\[ \Delta\nu=-\frac{3h}{8\pi^2H\mu ec}\,EZ \]

\[ Z=(m_1+m_2+m_3)(m_1-m_2)-(n_1+n_2+n_3)(n_1-n_2)\quad (37). \]

As before, the numbers \(m\) and \(n\) refer to the initial and final orbit, respectively, and for components of one and the same line

\[ m_1+m_2+m_3=\mathrm{const}\quad\text{and}\quad n_1+n_2+n_3=\mathrm{const}. \]

From the very structure of expression (37) there immediately follow two important properties of the Stark phenomenon in hydrogen lines: first, the splitting is proportional to the field strength \(E\); second, it is symmetrical. Indeed, if we interchange the numerical values \(m_1\) and \(m_2\) and simultaneously \(n_1\) and \(n_2\), \(Z\) changes its sign; hence for every positive \(\Delta\nu\) there exists an equal negative one. We have already mentioned that Stark did in fact establish precisely these properties for hydrogen lines.

The numerical value of the coefficient in formula (37) for hydrogen \((H=1)\) is the following: \(6{,}43\cdot 10^{-5}\), if \(E\) is expressed in \(\dfrac{\mathrm{volt}}{\mathrm{cm}}\). Therefore, for the displacement of a component in wavelengths one may write

\[ \Delta\lambda=\lambda^{2}\Delta\nu=6{,}43\cdot 10^{-5}\lambda^{2}EZ\ \mathrm{cm}. \tag{38} \]

The choice of possible values of \(Z\) is restricted by Sommerfeld’s inequalities, which in the present case read

\[ m_{1}\geq n_{1};\quad m_{2}\geq n_{2};\quad m_{3}\geq n_{3}\ .\ .\ . \tag{39} \]

The calculated values of \(\Delta\lambda\) are compared in the following tables. In doing so, the field strength was taken to be, for \(H_{\alpha}\), \(106.000\); for \(H_{\gamma}\), \(109.000\); and for \(H_{\delta}\), \(110.000\ \dfrac{\mathrm{volt}}{\mathrm{cm}}\).

Table I (\(H_{\alpha}\)-line).

\(m_{1}+m_{2}+m_{3}=3\)  \(\lambda=6562{,}8\ \text{Å}\)

\(Z\)	Calculated: \(m_{3}-n_{3}=2l\), \(\Delta\lambda\)	Calculated: \(m_{3}-n_{3}=2l\), \(Q_m\)	Calculated: \(m_{3}-n_{3}=2l+1\), \(\Delta\lambda\)	Calculated: \(m_{3}-n_{3}=2l+1\), \(Q_m\)	Measured: \(p\)-Comp., \(\Delta\lambda\)	Measured: \(p\)-Comp., Int.	Measured: \(s\)-Comp., \(\Delta\lambda\)	Measured: \(s\)-Comp., Int.
5	14,7	*			—
4	11,7	1			11,5	1,2
3	8,8	1			8,8	1,1
2	5,9	1	5,9	*	6,2	1	—
1	2,9	*	2,9	(1)	—		2,6	1
0	—		0	(1)			0	2,6

Table II (\(H_{\beta}\)-line).

\(m_{1}+m_{2}+m_{3}=4\)  \(\lambda=4861{,}3\ \text{Å}\)

\(Z\)	Calculated: \(m_{3}-n_{3}=2l\), \(\Delta\lambda\)	Calculated: \(m_{3}-n_{3}=2l\), \(Q_m\)	Calculated: \(m_{3}-n_{3}=2l+1\), \(\Delta\lambda\)	Calculated: \(m_{3}-n_{3}=2l+1\), \(Q_m\)	Measured: \(p\)-Comp., \(\Delta\lambda\)	Measured: \(p\)-Comp., Int.	Measured: \(s\)-Comp., \(\Delta\lambda\)	Measured: \(s\)-Comp., Int.
12	19,4	*	[19,4]	3	19,4	1	19,3	1
10	16,1	2	[16,1]	3	16,3	11,5	16,4	1,1
8	12,9	2	12,9	*	13,2	9,1	13,2	1,3
6	9,7	2	9,7	1	10,0	4,8	9,7	9,7
4	6,5	*	6,5	1	6,7	1	6,6	1·2,6
2	3,2	(2)	3,2	1	3,3	1,2	3,4	3,3
0	0	(2)	0	1	0	1,4	0	1,4

Table III (Hγ line).

$m_1 + m_2 + m_3 = 5$          $\lambda = 4340.5\ \text{Å}$

$Z$	Calculated: $m_3-n_3=2l$, $\Delta\lambda$	Calculated: $m_3-n_3=2l$, $Q_m$	Calculated: $m_3-n_3=2l+1$, $\Delta\lambda$	Calculated: $m_3-n_3=2l+1$, $Q_m$	Measured: $p$-Comp., $\Delta\lambda$	Measured: $p$-Comp., Int.	Measured: $s$-Comp., $\Delta\lambda$	Measured: $s$-Comp., Int.
21	28.0	*	—		29.4	1 doub.
20	—		[26.6]	4			26.3	1
18	23.9	3	—		23.9	10.8
17	—		[22.7]	4			22.8	1.1
16	—	.	21.3	*			—
15	20.0	3	—		19.9	7.2
13	—		17.3	2			17.3	6.1
12	16.0	3	—		15.9	2.0
11	14.4	*	—		—
10	—		13.3	2			13.3	4.3
9	12.0	*	—		—
8	10.6	2	—		10.6	1
7	—		9.3	2			9.7	1.2
6	—		8.0	*			—
5	6.7	2	—		6.6	1.5
4	—		5.3	*			—
3	—		4.0	(3)			3.9	3.6
2	2.7	2	—		2.6	1.6
1	1.3	*	—		—
0	—		0	(3)			0	7.2

From these tables it is evident that the agreement between the computations and the theory is brilliant. The bright components are located precisely at the computed positions, and only for weak lines, whose positions are very difficult to determine, are the deviations somewhat larger. The components marked with an asterisk are those which owe their origin to pendulum-like orbits. We have already indicated that these orbits are very improbable; from the tables it is evident that they give only the weakest—

both components, whereas for the lines \(H\alpha\) and \(H\beta\) the corresponding components could not be taken at all. If one disregards these lines, there remains only one more line—predicted theoretically and yet not present in Stark’s photographs \((H\delta, Z=0)\). On the other hand, there are three lines that contradict the inequalities (39). It is interesting that the first two inequalities are satisfied strictly, while the third, which, according to (36), pertains to the azimuth \(\varphi\), is apparently valid only in broad outline. A similarly isolated position of the azimuthal quantum number was also established by Sommerfeld for the case of fine structure under spark excitation of luminescence.

As for polarization, the following highly remarkable empirical rule is confirmed: an even difference of the azimuthal quantum numbers \(m_3-n_3\) leads to parallel \((p)\) polarization, an odd one to perpendicular polarization.

This rule holds in all cases without exception, although we have absolutely no way of understanding it. It even seems that, in general, it is impossible to explain the state of polarization from the orientation of the initial and final orbit with respect to the electric field; for in the transition of the electron between the two already mentioned circular orbits, which are always situated perpendicular to the field, according to this rule the lines \(H\alpha\) and \(H\gamma\) give \(S\)-components, while the lines \(H\beta\) and \(H\delta\) give \(p\)-components. As for the occurrence of great intensity, the author considers a possibly large value of one of the differences \(m_1-n_1,\ m_2-n_2,\ m_3-n_3\) to be a necessary (but not sufficient) condition for it. Therefore the largest of these three differences is given in the tables under the heading \(Qm\).

Be that as it may, the presented results of the theory of the Stark phenomenon, together with Sommerfeld’s theory of fine structure, provide one of the most decisive proofs in favor of the doctrine of quanta and of the application of this doctrine to atomistics, which was made by Bohr.

IV. Section

The Structure of Phase Space.

§ 16. The significance of Planck’s hypothesis for statistics.

In the preceding exposition we proceeded from the interpretation of the quantum conditions which was given by Sommerfeld and developed in detail by Schwarzschild and the author (§§ 11 and 12). Let us now consider the form of these conditions given by Planck himself for the case of several degrees of freedom. To this end, we shall first of all supplement the exposition of the hypo-

thesis of “quanta of action” (see § 3), with certain considerations concerning its statistical significance.

The reason that forced Planck to abandon the idea of energy quanta (§ 2) was the contradiction between the electrodynamic and the statistical parts of his first theory of black radiation. Whereas the interaction between the resonators and the radiation field had to obey the laws of electrodynamics, and consequently a resonator had to be capable of possessing any store of energy, in considering the question of the distribution of energy among the individual resonators the assumption was made that only discrete energy levels are possible, namely multiples of \(h\nu\).

We can represent this graphically in a manner similar to that already done in Fig. 1. We know that the state of a linear resonator is completely determined by the position of the electron \(x\) (or \(q\)) and its momentum \(p\). Therefore, to every instantaneous state of the system—which, following Gibbs, is called the phase of the system—there corresponds a point of the \(C(p,q)\)-diagram, or, as it is called, the “phase plane.” In the course of time the point representing the state of the resonator describes a certain curve, the “phase curve,” which, in the case when no energy is supplied from outside, will be an ellipse (§ 3). If there is a large number of resonators, then at any moment each of them corresponds to a point of the diagram, and the distribution of these “phase” points gives the distribution of energy. This will become perfectly clear to us if we recall that the energy at a certain point is determined by the constants of the ellipse from the family of concentric ellipses passing through it (cf. §§ 2 and 3

\[ A=\frac{b^2}{2\mu} \]
).

The purpose of the statistical method consists in determining the density with which the points are distributed on the diagram (“phase density”)—for the most probable state of the system of all resonators, the state in which the resonators are in equilibrium with one another and with the radiation. According to the first electrodynamic part of Planck’s investigation, the energy can take any values, and therefore the phase density must be continuous, i.e. the phase points may lie at any distance from the origin. Further, considerations based on the theory of probability on the one hand and on mechanics and electrodynamics on the other tell us even more. Namely, from these considerations it follows with certainty that the phase density must be a continuous function of position, a function which, taking constant values on the elliptical curves, changes continuously in passing from the origin of coordinates to ever more distant points of the plane. The course of this function is investigated in detail in the second, statistical, part of Planck’s work. But here he was compelled to adopt

that the points in the phase plane are distributed discontinuously and can be located only on discrete curves satisfying the relation \(A=h\nu\) (shown in Fig. 1). For only in this way was it possible to arrive at the correct law of radiation.

To overcome these difficulties and at the same time reduce to a minimum the necessary changes in electrodynamics, Planck modified his hypothesis, allowing that although the absorption of a linear resonator proceeds according to the laws known to us, the resonator, however, cannot radiate, i.e., it does not possess “damping.”

As for radiation, it is much easier for us to imagine that it occurs precisely in a discontinuous manner. It is therefore assumed that radiation can (but need not) occur at those moments when the energy of the resonator is an integral multiple of \(h\nu\). At that moment the resonator gives up all its energy at once. Since here one must in advance abandon the usual laws of electrodynamics, the conclusion that the phase density changes discontinuously is no longer obligatory. However, the mechanism of the phenomenon of absorption nevertheless compels us to regard this density as continuous. Therefore, instead of the assumption of a discontinuous distribution of probability, Planck introduces a new one—that the probability is distributed, although continuously, yet not in an uninterrupted manner within annular regions whose area is equal to \(\Delta g=h\), and which are bounded by the selected curves of Fig. 1; the probability—or, what is the same thing, the phase density—must be constant, and only on the boundary curve, upon passing into the neighboring region, does it decrease (or increase) in a stepwise fashion. Under these assumptions, for black radiation one obtains the same spectral distribution as under the former ones.

The quantity of action \(h\) thereby acquires the meaning of the numerical magnitude of an elementary region of probability, since statistics makes no distinction between the different points of this region \(\Delta g\).

According to the new view, the “statistical” phase curves no longer represent the only possible motions, but are selected only insofar as they bound elementary regions. An interesting consequence of this hypothesis consists in the fact that one cannot extract all of their energy from resonators by bringing them into equilibrium with radiation of very low intensity (i.e., of very low temperature). For when all the phase points, each of which represents a resonator, are located in the very innermost elementary region, any further decrease of energy is already impossible, since, according to the preceding, radiation occurs only at energy values \(A=nh\nu\) (i.e., for static curves). Therefore even at absolute zero of temperature the system possesses the so-called zero-point

energy, the mean value of which for the resonator is \(\frac{h\nu}{2}\).

If, turning away from the linear resonator, we consider other systems, then everywhere we shall see an analogous picture: the static curves can always be regarded either as the only possible ones, or else the existence of all mechanically admissible orbits may be accepted and the static ones regarded merely as those selected from the standpoint of probabilities. The latter view, whose champion is Planck, in no case leads to contradiction with experiment; on the contrary, precisely from it one may expect the clarification of certain still obscure problems (the specific heat of diatomic gases). The apparent difficulty which existed in one particular case (the theory of the so-called “rotational spectrum”3) has recently been overcome by Planck in an astonishing manner. In exactly the same way, Bohr’s theory (§§ 7 and 8) is fully compatible with this point of view, if only the condition for the hypothesis is somewhat modified: one need only assume that under ordinary conditions radiation does not occur; it occurs only when the electron undergoes a perturbation and jumps to a new orbit; in this case the emitted frequency, according to condition (16), is determined by the energies at the boundaries of the old and the new elementary region.

§ 17. Structure of Phase Space.

In the light of the considerations set forth in the preceding paragraph, the points of view which underlie the interpretations of (5) and (5a), as well as (5b), of the quantum conditions can be rigorously formulated as follows. In the first case, the double integral is extended over all phase points which correspond to different stores of energy of the linear resonator, i.e. the static curves are derived from all the states which the resonator can assume under all possible conditions. In the second case, the integration is extended only over the individual motions of the resonator, occurring—in the absence of external perturbations. Accordingly, when considering systems of many dimensions, one may have recourse either to the one point of view or to the other. Sommerfeld followed the second path, whereas Planck proposed the first.

In the general case—\(f\) degrees of freedom with variables \(q_i\) and momenta \(p_i\)—to every instantaneous state, or to every phase of the system

Translator

there corresponds its own system of values of the \(2f\) quantities \(p_i, q_i\). The aggregate of all phases already forms not a plane, but a manifold of \(2f\) dimensions, an ideal \(2f\)-dimensional “phase space,” if \(p_i, q_i\) are interpreted as rectangular coordinates. Here, too, Planck poses the question of the elementary region \(\Delta g\) of probability, or of the “structure of phase space.” Since every separate product \(p_i q_i\) already has the dimension of action, it must be assumed that the volume of the elementary region (of \(2f\) dimensions) is

\[ \Delta g = h^f. \]

How are the boundaries of the elementary region now to be found? Planck does this in such a way that he seeks \(f\) expressions of the general motion which take place in one dimension and which, therefore, can be interpreted as in § 3. From the solutions of these separate cases the general solution is then constructed.

Whether in every case motions expressed with the required qualities can be found has not yet been investigated. However, this requirement is in any case satisfied for one group of systems, for conditionally periodic systems (§ 12); for this class of motions Planck’s solution can be carried out completely. For example, for the motions studied in § 11 (\(f=2\)) one would have to consider the circular orbit (\(r=\mathrm{const}\)) and the rectilinear one, as a limiting case of an ellipse with the ratio of axes 0 (\(\varphi=\mathrm{const}\)); for the Stark phenomenon (\(f=3\))—the three particular cases considered on p. 49 (the circle and the two pendulum-like orbits). It can be shown, in general, that for a conditionally periodic system the expression for the volume of the elementary region decomposes, generally speaking, into \(f\) factors,

\[ \Delta g = \Delta g_1 \cdot \Delta g_2 \cdot \ldots \cdot \Delta g_f, \]

of which each represents the area of a portion of the corresponding coordinate plane \((p_i,q_i)\), equal in magnitude to \(h\); further, that the static motions of the system coincide with the motions determined by the conditions (29) and (29′) of § 121. Therefore in Planck’s theory the quantum conditions (29) for conditionally periodic systems are already contained implicite, as well as those conclusions which were derived from these conditions for the particular cases in §§ 13 and 15.

Within the limits of this report we cannot dwell on the interesting interpretation of these conditions given by Einstein2. However, we shall briefly set forth here that extremely important point of view which is connected with the idea that static orbits are the only possible ones, and which was put forward by Ehrenfest (1916). Let us consider the change experienced by a system when one of the external

parameters (but not the variables \(p\)) is subjected to a slow action, for example, when in the Stark phenomenon the external field is gradually intensified. Such a process is called an “adiabatic change of state,” for in it the energy of the system changes through the mediation of an external parameter, and not by a direct increase of the vis viva; quite analogously to the way in which, in the adiabatic compression of a gas, the change in the store of energy depends on the work expended, and not on direct heat exchange. In the initial state the system performs a certain static motion. If one starts from such a motion and subjects the system to an infinitely slow adiabatic process, then, on the one hand, one may, completely ignoring the quantum conditions, pose the question of the form of motion which is obtained in purely mechanical fashion from the initial motion. For the initial motion in this process undergoes a continuous modification, so that at every moment there corresponds to it an entirely unambiguous new form. On the other hand, for every value of the parameter one could write down the quantum conditions and determine the corresponding static orbits. The question now arises: do mechanics and quantum theory continually contradict one another, or do they consistently lead to one and the same motions? The essence of Ehrenfest’s hypothesis (“Adiabatenhypothese”) consists in taking the latter proposition: motions admissible from the point of view of quantum theory pass, by means of an infinitely slow (purely mechanical) process, into other motions likewise admissible from the point of view of quantum theory. Hence it follows that those constants of motion which, according to conditions (25), are assumed equal to the universal constant \(nh\), cannot change during an adiabatic change, or, as one says, are “adiabatic invariants.” Verification of this hypothesis for periodic (Ehrenfest) and conditionally periodic (Burgers) systems has shown that the quantum integrals of §§ 11 and 12 are indeed adiabatic invariants.

Ehrenfest’s hypothesis has already helped to clarify certain questions that are fundamentally difficult; its importance becomes obvious if one recalls that by means of an adiabatic process one can pass from simple systems to more complex ones. Of course, the same hypothesis can also be interpreted from another point of view, namely, that static motions are not the only possible ones, but merely those selected from the standpoint of the theory of probabilities. In that case, in its formulation, the word “admissible” must be replaced by the word “selected.”

§ 18. Conclusion.

From the preceding exposition, especially from Sections II and III, the reader has probably gained the impression that Planck’s theory, in its new

field of its application, in atomistics and spectroscopy, has already yielded a number of results which, without exaggeration, may be regarded as great successes, and which justify the attempt to make this subject accessible to a broad circle of readers. However, it is well known to everyone who works in this field that we are still far here from lawful clarity: many fundamental and computational difficulties still have to be overcome; many provisional conceptions still have to be modified or replaced by new ones.

First of all, one may pose the question of how, for such general mechanical systems as conditionally periodic ones, it is possible to find the motions admissible according to quantum theory. There already exist separate attempts to answer this question: we have already mentioned that Planck’s method is applicable also in the domain of conditionally periodic motions, although it would not be easy to give unambiguous instructions for its application. On the other hand, Burgers1 pointed to a method introduced into celestial mechanics by Delaunay (1860), which consists in the fact that a given system is approximated as closely as desired by means of a consecutive series of conditionally periodic systems chosen in a definite way. The author, independently of Delaunay, in investigating one particular case drew attention to the possibility of such an approximation. However, the calculations carried out by him by this method for the spectrum of neutral helium (the three-body problem), unfortunately, apparently do not allow great hopes to be placed on this method. Finally, Ehrenfest’s hypothesis, set forth in the preceding §, could probably be used as a heuristic method.

However, all these are purely practical questions, the solution of which perhaps the very near future will bring us. More serious are the difficulties connected with the fundamental side of Bohr’s assumptions (§§ 7 and 8). We have already mentioned several times that quantum theory is in a certain contradiction with electrodynamics, and permits only a limited application of the latter within the atom. How, then, must electrodynamics be modified in order to remove this contradiction? That we indeed lack a synthesis of the two theories is clear from the following circumstance: the Stark phenomenon finds a complete explanation in quantum theory, whereas it does not yield to a purely electrodynamic explanation; the opposite picture we apparently have in the related phenomenon—in the Zeeman phenomenon (the splitting of spectral lines in a magnetic field): here, for the time being, one can get farther with electrodynamics than with quantum theory (although a complete explanation still cannot be obtained). Why, in motion along a static orbit, does no radiation occur, and what

is accomplished when the electron jumps from one static orbit to another? The few facts that are known to us concerning the latter process are, in essence, Bohr’s condition for frequencies (§ 8) and the polarization rules in the Stark effect (§ 15). But both these laws, in their present formulation, have to some extent a teleological character, one altogether unacceptable to a naturalist, so that many scholars are rightly indignant at these “Bauern-Regeln.” Indeed, it would be highly desirable to review the process of radiation in all its details and to find an explanation for the aforementioned laws. If this demand is too high for the present time, nevertheless it may perhaps even now be possible, with greater hope of success, by introducing the degrees of freedom of the ether, to bring the condition for the frequency to the same form (25) as the quantum conditions for matter have1.

Thus, although in quantum theory many gaps still have to be filled—perhaps even such gaps as occur at the very beginning—we may be satisfied with the rate at which our knowledge of the atom is growing, a growth that we owe to this theory. Reviewing quite objectively the course of development of quantum theory in recent years and the successes it has achieved, one must acknowledge that Max Planck’s work has advanced us one step along the path toward the ultimate goal of exact natural science: the knowledge of the structure of matter and the ether—one step which, as it seemed only a few years ago, even under the boldest expectations, would have had to lead us into vast and indefinite distances.

Translated by E. V. Shpolsky.

1. As A. Rubinowicz has shown (Phys. Zs; 18, p. 96, 1917), this can be done for the theory of black radiation; the author likewise owes the last remark to Mr. Rubinowicz. ↩↩↩↩↩↩

2. A. Einstein. Verh. d. Deutsch. phys. Ges. 19, S. 82, 1917. ↩

3. This is the name given to the infrared absorption spectra of gases observed by Bjerrum, Rubens, and others. In explaining them, Bjerrum assumed that the charged atoms in the molecule execute oscillations, while the molecule itself, in addition, rotates. Hence the origin of the name itself. ↩↩

4. W. Wien. Ann. d. Phys. 18, p. 991; 1905. ↩

5. Duane and Hunt, Phys. Rev. 6, p. 166, 1915. ↩

6. W. Stenling. Phys. Zs. 10, p. 991, 1905. ↩