THE PERIODIC SYSTEM OF CHEMICAL ELEMENTS IN LIGHT OF THE THEORY OF ATOMIC STRUCTURE1

R. Swinne.

Predecessors. Foundation of the system. Atomic number and lengths of the periods. Models of the atom. Quantum numbers of elliptical orbits. Quantization of the orientation of orbits. Periodicity of electronic properties. Principal groups of electrons. Subdivision of the principal groups. Characterization of subgroups. Completion of subgroups. Geochemical peculiarities of the elements. Filling and limitation of the system. Properties of transuranium elements. Conclusion.

1. Introduction.

The concept of the chemical element, in the course of the evolution of chemistry, has repeatedly changed its content. The classical concept of a chemical element may be reduced to the fact that in a given substance, after any reactions—provided only that they take place in a closed space—it is always possible to re-establish a certain complex of properties. Already in the early stages of the development of quantitative chemistry there arose the idea that relations must exist among these complexes of properties. A substantial role in the development of these ideas was also played by the physical hypothesis of the atomistic structure of matter.

At the end of the eighteenth century the technical chemist I. B. Richter, from one ordinary fact (the preservation of neutrality when neutral salt solutions are mixed), derived the quantitative conclusion that chemical transformations of salts, as well as their formation from acids and bases, always occur with the preservation of constant weight ratios, equal to the ratios of their equivalent or combining weights. Richter not only put forward the supposition that, if these equivalent weights are arranged according to their magnitude, mathematical relations can be established among them, but went much further: having established the presence of gaps in the regular series he had found, he tried to explain them by the existence of substances not yet discovered. But in this he did not obtain any tangible successes. The extension of Richter’s idea of combining weights to any

chemical compounds had been made by Dalton already within the framework of the atomistic hypothesis.

All these ideas of Richter retained their significance during the further development of our knowledge of the chemical elements and the relations among them throughout the whole of the nineteenth century and up to our own day. In this, however, development proceeded in close connection with atomistic theory—from Dalton’s first (inaccurate) relative atomic weights and Prout’s hypothesis, based on them, concerning hydrogen as the primary substance, down to the modern Rutherford–Bohr nuclear theory of the structure of the atom. True, the more accurate determinations of atomic weights by Berzelius and Stas discredited Prout’s hypothesis for several decades. On the other hand, the interrelations between the physical properties and chemical behavior of both the various groups of elements and all of them in general became increasingly clear, and at the same time the idea of the uniformity of their structure appeared more and more well founded.

2. The Foundation of the Periodic System

As a predecessor here one should mention Newlands, who at the meeting of the British Association in 1864 formulated his law of octaves and was the first to make use of the ordinal number, or atomic number, of an element. He arranged the elements, apart from a few minor transpositions, simply according to the magnitude of their equivalents; it turned out that analogous elements were arranged in identical rows, while the ordinal numbers of these analogous elements, generally speaking, differed by seven or by a multiple of seven. The ground, however, was not yet ready. Newlands remained incomprehensible; after his report he was asked whether he had not tried to find regularities by arranging the elements according to the initial letters of their names.

Greater success was achieved by Lothar Meyer, on the one hand, and D. I. Mendeleev, on the other, who five years later, independently of one another, established several periodic systems of the elements, using atomic (and therefore not equivalent) weights. At that time both these quantities were used interchangeably without any particular distinction; the concept of molecular weight had only by then been given definite form. Also an obstacle was the insufficient knowledge of the valency of many elements. Likewise, recognition of the significance of the law of Dulong and Petit on atomic heat capacities and of Mitscherlich’s isomorphism for the calculation of atomic weights had only by that time become the common property of science.

L. Meyer was the first to draw attention to the regular change in the valency of the elements when they are arranged according to atomic weights. As a consequence, he also took into account the change in physical properties,—

TABLE 1.

		Group I	Group II	Group III	Group IV	Group V	Group VI	Group VII	Group VIII, transitional to Group I
Typical elements		H = 1, Li = 7	Be = 9,4	B = 11	C = 12	N = 14	F = 19
First period	Series 1	Na = 23	Mg = 24	Al = 27,3	Si = 28	P = 31	S = 32	Cl = 35,5	Fe = 56, Co = 59 Ni = 59, Cu = 63
First period	Series 2	K = 39	Ca = 40	— = 44	Ti = 50?	V = 51	Cr = 52	Mn = 55	Fe = 56, Co = 59 Ni = 59, Cu = 63
Second period	Series 3	(Cu = 63)	Zn = 65	— = 68	— = 72	As = 75	Se = 78	Br = 80
Second period	Series 4	Rb = 85	Sr = 87	(?Yt = 88?)	Zr = 90	Nb = 94	Mo = 96	— = 100	Ru = 104, Rh = 104 Pd = 104, Ag = 108
Third period	Series 5	(Ag = 108)	Cd = 112	In = 113	Sn = 118	Sb = 122	Te = 128?	I = 127
Third period	Series 6	Cs = 133	Ba = 137	— = 137	Ce = 138?	—	—	—	—
Fourth period	Series 7	—	—	—	—	—	—	—
Fourth period	Series 8	—	—	—	—	Ta = 182	W = 184	—	Os = 199?, Ir = 198?, Pt = 197, Au = 197
Fifth period	Series 9	(Au = 197)	Hg = 200	Tl = 204	Pb = 207	Bi = 208	—	—
Fifth period	Series 10	—	—	—	Th = 232	—	U = 240	—
Highest saline oxide		R₂O	R₂O₂ or RO	R₂O₃	R₂O₄ or RO₂	R₂O₅	R₂O₆ or RO₃	R₂O₇	R₂O₈ or RO₄
Highest hydrogen compound				(RH₅?)	RH₄	RH₃	RH₂	RH	—

THE PERIODIC SYSTEM OF CHEMICAL ELEMENTS

his well-known curve of atomic volumes proved very useful. However, the especially comprehensive and bold use of the periodicity of the elements belongs to Mendeleev. Thus, he applied the periodic law to the systematics of the elements, to the determination of the atomic weights of little-studied elements such as indium and uranium, to the correction of atomic weights in certain cases, as, for example, in the platinum family, and, finally, to the supplementation of information on the forms of chemical compounds and the valency of the elements. The successful prediction of the properties of elements not yet discovered but assumed in order to fill the empty places in the system of elements brought Mendeleev particular fame, and already in the eighties played an outstanding role in the recognition of the system. In doing this, Mendeleev named the unknown elements after the subsequent analogue of the same vertical group, adding to the name of this element a numeral borrowed from Sanskrit (eka—1, dvi—2, tri—3, chatur—4, etc.). Mendeleev’s predicted ekaaluminium (Ea) coincided with gallium, discovered in 1875 by Lecoq de Boisbaudran; ekaboron—with scandium, discovered by Nilson (1879); and ekasilicon—with germanium, discovered by Cl. Winkler. For ekamanganese Mendeleev expected an atomic weight of 100; in view of the complete analogy of the rare earths with the other long periods, for ekaniobium he predicted \(A = 146\), for ekacesium \(A = 175\), and dvitellurium \(A = 220\), for trimanganese \(A = 190\), and for ekatantalum \(A = 235\) (see Table 1).

Already in the arrangements of L. Meyer and Mendeleev, attention is drawn to the double periodicity among the elements following potassium (in the so-called long periods), as well as to the difficulty of placing the rare earths, which at that time were little known. The associated ambiguity of placement led, in the following decade, to numerous modifications in the systems of these two investigators, which from the very beginning did not fully coincide with one another. Some of these modifications were aimed at imparting to the periodic system the greatest possible regularity, even at the cost of doing violence to experimental facts; other, less numerous modifications, on the contrary, took these facts especially into account and, as a result, produced changes in the original form of the system. The most recent development is undoubtedly proceeding in the direction of the latter tendency.

Here, in particular, Julius Thomsen should be mentioned, who in 1895 proposed an arrangement of the elements that differed from the one accepted at that time—an arrangement that may be called “trapezoidal” (see Fig. 3, p. 349, where Thomsen’s arrangement is presented in Bohr’s modern interpretation). Thomsen at the same time drew attention to a “curiosity,” which consisted in the fact that the number of elements (without the noble gases, still unknown at that time)

in the individual periods is 1, 7, 17, 31, and these numbers may be represented in the form:

\[ 1 + 2 \times 3 + 2 \times 5 + 2 \times 7. \]

In a second communication, Thomsen, in connection with the discovery of argon that had just been made, admitted the existence of a special group of chemically inactive elements, of an electrically indifferent character, which were to represent the transition from one period to the next. Further, he assumed, as the completion of the VII period, of which at that time only two elements were known (thorium and uranium) and in which he likewise assumed 31 members (as in the preceding periods), the existence of an inactive element, which may now be called eka-emanation and should have atomic number 118 (see Fig. 3).

The discovery of the noble gases at the end of the nineteenth century did not cause any particular difficulties for the periodic system, since they could very conveniently be placed as intermediate members between the strongly electropositive alkali metals and the strongly electronegative halides. The only point that did not agree with the arrangement of the elements by atomic weights was the place occupied by argon \((A = 39.9)\). The change in the order of the elements undertaken by Mendeleev in other cases as well was made here too; moreover, Mendeleev thought throughout his life that the irregularity in the determination of the atomic weight of argon was due to impurities. In any case, similar deviations had already been known earlier in other places of the system (Co—Ni, Te—I).

3. Atomic number and the length of the periods.

The discovery of radioactivity, however, at first created great difficulties for the periodic system, since the number of radioelements was too large. An exact investigation of radioactive transformations led, however, to the discovery of isotopy, i.e., the existence of elements with different atomic weight and different stability, but with completely identical other properties. Therefore there was no need to increase the number of vacant places in the periodic system. Once again the places were occupied of ekatellurium—polonium, ekaxenon—emanation, ekabarium—radium, ekalanthanum—actinium, and ekatantalum—brevium-protactinium (the designation of analogous elements does not correspond to the obsolete Table 1, but to Table 2).

At the same time the classical concept of an element underwent a modification, since the entire complex of properties characterizing an element proved capable of passing into the complex of properties characterizing another element. The periodic system of the elements was preserved, but atomic weight lost its fundamental significance, especially after Aston discovered isotopes in many non-radioactive

elements. The radioactive displacement laws connected different places in the periodic system with one another and encouraged the extension of analogous considerations to the whole system, especially if use was made of Aston’s isotopes. At the same time, atomic weight, from whose generally recognized constancy Dalton had derived his law, and which had served for L. Meyer and especially for Mendeleev as the leitmotif of the entire periodic system, lost its guiding role.

The place of atomic weight was now taken by the atomic number, the ordinal number of the elements. Nylands had already used it, but only Rydberg showed in 1897 the advantages it possesses in comparison with atomic weight. The starting point of Rydberg’s investigations was the desire to establish mathematical relations among the elements. This aspiration led him not only to the foundation of the theory of spectral series, but also to important discoveries concerning the periodic system. The sequence of atomic weights was subjected by Rydberg to analysis from various points of view. Here we are interested only in the consequence derived from considering the differences of the atomic weights of successive elements, and consisting in the fact that the atomic weights, in first approximation, are linear functions of their ordinal numbers. “In studying the periodic system one should use, instead of atomic weights, the ordinal numbers of the elements as independent variables.”

The true significance of these positive integers, the ordinal numbers of the atoms of the elements, still remained entirely unknown to Rydberg. Initially he, like Mendeleev, arranged the elements according to the magnitude of their atomic weights; however, upon closer study of the system he arrived at a different division—into periods—and, with it, partly at other atomic numbers.

According to Rydberg, the so-called small periods from He to Ne and from Ne to Ar undoubtedly contain 8 elements each; after them come 2 so-called large periods of 18 elements each, from Ar to Kr and from Kr to Xe, with the individual periods always separated from one another by noble gases of zero valence. In addition, throughout the system there is a period of two elements, in the form of a constant alternation of even and odd elements. Rydberg pointed out in 1906 that the three numbers: 2, 8, and 18 can be written in the form

\[ 2 = 2 \times 1^2;\quad 8 = 2 \times 2^2;\quad 18 = 2 \times 3^2. \]

It would be continued by

\[ 2 \times 4^2 = 32;\quad 2 \times 5^2 = 50\ \text{and so on.} \]

This led him to the conclusion that the period beginning with xenon and ending with emanation contains only 32 elements,

but not twice 18, i.e., 36. Rydberg was fully justified in emphasizing in 1913 that, as the further discovery of new rare-earth elements proceeded and as the atomic weights of these elements were determined more accurately, the impossibility of their analogy with the elements from V to Sr or from Nb to Ba became increasingly clear. An exact analysis of the differences in the atomic weights of the known rare earths and comparison with the differences corresponding to analogous elements of neighboring periods indicate that in the group of rare earths only two elements are missing: a trivalent one between Nd and Sm and a tetravalent one between Lu and Ta, so that the total number of elements between Xe and Em will be 32.

Rydberg combines the double periods into one group and continues them upward and downward. Hence there follows the following division of the periodic system:

\[ \begin{array}{c|ccccc} \text{Group} & G_1 & G_2 & G_3 & G_4 & \ldots\ G_p \\[2mm] \hline \text{Number of elements in the whole group} & \begin{array}{c}4 \times 1^2\\4\end{array} & \begin{array}{c}4 \times 2^2\\16\end{array} & \begin{array}{c}4 \times 3^2\\36\end{array} & \begin{array}{c}4 \times 4^2\\64\end{array} & \begin{array}{c}\ldots\ 4 \times p^2\\p\end{array} \\[4mm] \text{Number of elements in half a group} & \begin{array}{c}2 \times 1^2\\2\end{array} & \begin{array}{c}2 \times 2^2\\8\end{array} & \begin{array}{c}2 \times 3^2\\18\end{array} & \begin{array}{c}2 \times 4^2\\32\end{array} & \begin{array}{c}\ldots\ 2 \times p^2\\p\end{array} \end{array} \]

We shall call the proposition according to which the number of elements of a period—half a group according to Rydberg—is equal to \(2p^2\), Rydberg’s rule.

TABLE 2.

[[unclear: reduced diagram of Rydberg’s periodic system, with groups \(G_1\), \(G_2\), \(G_3\), \(G_4\), column headings including \(0\), \(+1\), \(-1\), \(+2\), \(-2\), …, \(+8\), and many element symbols with atomic numbers; the individual entries are not fully legible in the page image.]]

At the head of the periodic system Rydberg places an electron with atomic number 0; he admits the existence of two more unknown elements between H and He, namely a noble gas with atomic number, atomic weight, and molecular weight equal to 2, and a monovalent element with atomic number and atomic weight equal to 3 and molecular weight equal to 6. These hypothetical elements were identified with the hypothetical gases of the spectroscopists: coronium (solar

...corona) and nebulae. Table 2 presents one of the three forms of the periodic system proposed by Rydberg, namely, an arrangement in the form of a continuous wave line.

4. Models of the Atom.

The physical interpretation of atomic numbers was the result of ideas about the structure of the atom that developed in connection with the study of the passage of corpuscular and X-rays through material bodies. From Lenard’s dynamids and Nagaoka’s Saturn-like atom, the line of development leads to Rutherford’s nuclear hypothesis (1911). The latter is now regarded as a well-founded theory, especially thanks to the application by Bohr (from 1913 onward) and his followers of Planck’s quantum theory to the motions of the electrons surrounding the positive nucleus. The properties of the simplest atom, the hydrogen atom, could be theoretically interpreted fairly completely, but for more complex atoms the theoretical interpretation still encounters difficulties, on the overcoming of which work is still proceeding. The connection of the atomic number with Rutherford’s theory was indicated soon after its appearance by van den Broek, who proposed that the charge of the atomic nucleus, and with it the number of electrons neutralizing this charge, is equal not to half the atomic weight, but to the atomic number.

Thus the question arose whether these electrons could be assembled into definite groups with similar properties. The answer proved to be affirmative.

Even earlier, the characteristic X-rays discovered by Barkla had been known, namely the \(K\)- and \(L\)-radiation (subsequently \(M\)- and \(N\)-radiation were also discovered). To excite these characteristic X-rays of the elements, a minimum velocity of the electrons, or a minimum hardness of the exciting X-rays, is necessary. Of special significance for the development of our knowledge of the periodic system was Moseley’s measurement, in 1913/14, of the wavelengths of these characteristic X-rays for most of the elements. As a result of these measurements and of their interpretation in the sense of Bohr’s theory in the X-ray region, serial spectra were discovered, and the electrons closest to the atomic nucleus proved to obey the same laws as the electrons responsible for chemical behavior and for ordinary serial spectra. X-ray spectral analysis of the chemical elements arose, making it possible to determine unambiguously the atomic number of unknown elements. Namely, if one plots graphically the dependence between the reciprocal of the square root of the wavelength \(\lambda\) of the characteristic X-rays and the atomic number \(Z\), then for the X-ray lines

of analogous origin only slightly curved straight lines are obtained. Thus the brightest line, the so-called \(K_\alpha\)-line of the series, can be represented approximately by the equation

\[ \frac{1}{\lambda}=\frac{3}{4}(Z-1)^2 R, \tag{1} \]

where \(R\) is Rydberg’s constant of the spectral series, and \(Z\) is the atomic number, but not Rydberg’s atomic number, but rather the number corresponding to Fig. 3.

From Moseley’s relation it is evident that the position of the X-ray lines is determined not by the absolute value of the atomic number \(Z\), but always by the quantity \(Z-\Delta\). However, these relative values are quite sufficient for establishing to which already occupied or still unoccupied place in the system the measured lines belong. In favor of the fact that \(\Delta\), for example, in equation (1) is equal to 1, or, what is the same thing, that between H and He there are no unoccupied places, there also speak various other arguments and measurements. Namely: 1) the absolute magnitude of the nuclear charge on the basis of measurements of the simple scattering of alpha particles by various elements, and also the properties of the alpha particle itself as a helium nucleus in its passage through matter; 2) the absolute number of electrons in elements with small atomic numbers, i.e. the spectral peculiarities of helium itself; 3) the number of electrons responsible for the occurrence of the \(K\)-series, which must be identical with the number of electrons of neutral He. If all these grounds should prove insufficient, then the point of view developed below turns out to be decisive, so that we, contrary to Rydberg’s supposition, assign to helium the atomic number 2. But then coronium and nebulium must already be known elements, existing in an as yet unknown excited or ionized state. And indeed, Panekuk several years ago attributed the lines of the corona to doubly ionized calcium and gave arguments in favor of such a supposition.

For the interpretation of X-ray spectral series, a view developed by W. Kossel proved very essential. This view amounts to the following: when an electron is removed to infinity from one of the groups, a vacancy arises in this latter, which is filled by an electron from a group lying closer to the periphery of the atom. The frequency that is then emitted is determined by Bohr’s frequency condition, i.e. it is equal to the difference of the energies of the jumping electron in the two electron groups, divided by Planck’s constant \(h\).

At first one confined oneself to the conception of plane electronic orbits and spoke of electron rings; subsequently

it became necessary to consider the spatial arrangement of orbits and to speak of electron shells. Finally, quite recently, orbits of a strongly elongated form have been employed, so that the electron, moving along them, enters the space occupied by the inner orbits.

5. Quantum numbers of elliptical orbits.

The aim pursued by the quantum theory of the atom consists in interpreting all the general properties of the atom manifested in spectral series, in magnetic peculiarities, in chemical bonding, etc. In order to show the significance of quantum theory for the problem of the periodic system, it is necessary first of all to become acquainted, in general outline, with the quantum characterization of electron orbits1.

Bohr used (1913) two quantum postulates, one of which makes it possible to characterize the stationary states of electrons by quantum numbers, while according to the other postulate, in the transition from one stationary state (orbit) to another, an energy \(h\nu\) is radiated, equal to the difference of the energies in the initial state (index \(a\)) and the final state (index \(e\)):

\[ h\nu = W_a - W_e \tag{2} \]

If one proceeds from the Coulomb attraction between the nucleus and the electron, then in the simplest case of the purely periodic motion of one electron along an elliptical orbit around a nucleus with effective charge \(Z_{eff}\varepsilon\), the following expressions are obtained for the energy \(W\) of the electron (charge \(\varepsilon\) and mass \(\mu\)) and for the major axis \(2a\):

\[ W = -\frac{2\pi^2\mu^2\varepsilon^4}{h^2}\cdot\frac{Z_{eff}^{\,2}}{n^2} = -Rh\frac{Z_{eff}^{\,2}}{n^2};\quad 2a = n^2\frac{h^2}{2\pi^2\varepsilon^2\mu}, \tag{3} \]

Here \(R\) is the so-called Rydberg constant, which appears in the laws of spectral series, and \(n\) is the so-called principal quantum number1, which can have only integral values \((1, 2, 3\ldots)\). Using postulate (2), we obtain for the frequency of the emitted light:

\[ \nu = \frac{1}{h}(W_a - W_e) = RZ_{eff}^{\,2}\left(\frac{1}{n_e^2} - \frac{1}{n_a^2}\right) \tag{4} \]

If in the atom, besides the one electron whose motion we are considering, there are also others, then by the action of these electrons

the charge of the nucleus $Z$ is decreased by some quantity $\Delta$ (the screening constant), giving as a result the effective charge $Z_{eff}$. This is what led Moseley to his interpretation of the formula for X-ray spectra (1). The theoretical interpretation of the simplest spectra $Z=1$ (hydrogen) and $Z=2$ (singly ionized helium) served as the foundation for the first successes of Bohr’s theory. Subsequently, however, it turned out that the fine structure of spectral lines, even in these simplest cases, cannot be explained theoretically if the orbit of the electron is characterized by one single quantum number $n$. The theory interpreted this in such a way that one quantum number is sufficient only so long as the motion under consideration is purely periodic. If this is not fulfilled, then new quantum conditions are necessary, and hence new quantum numbers as well.

Such conditions were first put forward by Sommerfeld in 1916; he took into account the relativistic dependence of the electron’s mass on velocity, and also investigated motion under the action of a non-Coulomb central force1. Under such conditions the orbit begins to rotate in its own plane, and upon the fundamental frequency corresponding to motion along a Keplerian ellipse there is superposed a second frequency—the rotation of the orbit itself; hence the necessity of a new quantum condition. Sommerfeld initially introduced for this case the radial quantum number $n'$ and the azimuthal quantum number $k$, defining them by the corresponding quantum conditions; here

\[ n = n' + k, \tag{5} \]

where $n'$ and $k$ are again integers.

This azimuthal quantum number, which determines the eccentricity and the minor semiaxis of the ellipse, Bohr called the subsidiary quantum number, since in the expression for the energy corresponding to some orbit, alongside $n$, it plays only a subordinate role, determining the finer details of the orbit. This theory led Sommerfeld to an explanation of the fine structure of spectral lines. Indeed, in deriving the exact formula for the energy $W$ corresponding to an elliptical orbit, Sommerfeld obtained an expression in which, to the term depending only on $n$ (equation 3), there are added further terms depending on $n$ and $k$. As a consequence of this, for a given $n$, instead of a single energy level there arise several neighboring ones, which account for the appearance of regularly connected, closely spaced spectral lines. Instead of the equation

Ed.

(3) for the case under consideration, in the first approximation, the formula is obtained:

\[ W=-Rh\frac{Z_{\mathrm{eff}}^{2}}{n^{2}}\left[1+\frac{\alpha Z_{\mathrm{eff}}^{2}}{n^{2}}\left(\frac{n}{k}-\frac{3}{4}\right)\right] \tag{6} \]

\[ \alpha=\frac{2\pi e^{2}}{hc} \tag{7} \]

where \(\alpha\) is Sommerfeld’s fine-structure constant and \(c\) is the velocity of light. At the same time, the frequency difference \(\Delta\nu\) of such “relativistic doublets” in the first approximation will be:

\[ \Delta\nu=R\frac{\alpha Z_{\mathrm{eff}}^{4}}{n^{4}}\left(\frac{n}{k}-\frac{3}{4}\right). \tag{8} \]

For hydrogen (\(\mathrm{H}=1\)) Paschen found from his investigations of the \(\mathrm{He}^{+}\) spectrum \(\Delta\nu=0.3645\pm0.0045\ \mathrm{cm}^{-1}\). In the X-ray series, according to Sommerfeld, owing to the correspondingly much larger value of the effective \(Z\), much larger separations of the doublet lines are obtained, since \(\Delta\nu\) is proportional to \(Z^{4}\). This at the same time gives a new way of determining \(Z=Z_{\mathrm{eff}}+\Delta\), since the screening constant \(\Delta\) depends only slightly on the ordinal number \(Z\).

Comparison of the calculated spectral lines with the observed ones led to the establishment of “selection rules,” “allowed quantum transitions,” which found, in Bohr’s correspondence principle (1918), a more precise and generalized formulation. With the aid of this principle it proved possible “to gain complete command of those peculiar rules which govern, at first sight capriciously, the appearance of combination lines, and one may say that quantum theory not only furnished a simple explanation of the combination principle, but also substantially helped to banish the mysticism which had so long enveloped the applications of this principle.” Thus the correspondence principle leads to the following condition, which changes of the azimuthal quantum number must satisfy:

\[ \Delta k=\pm1; \tag{9} \]

if, however, non-central forces are added, then for \(k\) the possibility of other changes also opens up.

6. Quantization of orientation.

A more detailed investigation of X-ray spectra, optical multiplets, and the splitting of spectral lines in a magnetic field showed, however, that one subsidiary quantum number \(k\), together with the principal quantum number \(n\), is quite insufficient for the interpretation of the entire variety of observed phenomena.

The third quantum condition must be invoked in the case when the influence of a weak magnetic field on the stationary orbits of electrons is being studied. In this case the plane of the orbit does not retain its position unchanged, but the normal to the plane of the orbit performs a uniform precessional motion about the magnetic lines of force. This motion is restricted by the third quantum condition, which also contains the third quantum number. This latter was called by Sommerfeld the internal quantum number and denoted by the symbol \(j\); in turn, this \(j\), in comparison with \(n\) and \(k\), plays a lesser role: it determines the spatial orientation of the orbit of the optical electron relative to the skeleton of the atom. The correspondence principle, in complete agreement with experiment, requires, alongside (9), the fulfillment of one more condition restricting quantum transitions:

\[ \Delta j=\pm 1 \ \text{and}\ \Delta j=0. \tag{10} \]

In establishing the quantum theory of the influence of a strong magnetic field on the stationary orbits of electrons—according to Landé (1923)—one has to use a fourth quantum condition, since here the symbols \(n\), \(k\), and \(j\) are no longer sufficient. As a result a fourth—magnetic—quantum number \(m\) is introduced, which directly gives the energy of the atom in the magnetic field, for it is precisely this quantum number, and not the one that characterizes the moment of momentum, that determines the magnitude of the magnetic moment of the optical electron. The permissible changes of this magnetic quantum number are restricted by the condition:

\[ \Delta m=\pm 1 \ \text{and}\ \Delta m=0. \tag{11} \]

Landé, to whom the quantum theory of the Zeeman effect and of multiplets owes many substantial successes, uses somewhat different quantum symbols\(^{1}\), namely: instead of \(k\) he introduces \(K\), instead of \(j\)—\(J\). The quantities \(K\) differ from the corresponding values of \(k\) by \(+\frac{1}{2}\) (cf. Table 9). The following relations empirically connect the quantum symbols \(n\), \(K\), \(J\), and \(m\) with one another:

\[ \left. \begin{aligned} n&=1,\ 2,\ 3\ldots \infty\\ K&\leq n-\frac{1}{2}\\ J&=K\pm\frac{1}{2}\\ |m|&\leq J-\frac{1}{2} \end{aligned} \right\} \tag{12} \]

\(^{1}\) See the article by Landé mentioned in the note on p. 339.

Ed.

From the standpoint of the atom model, \(K\) determines, according to Landé, the angular momentum of the optical electron, so that the angular momentum of the electron is equal to \(K\dfrac{h}{2\pi}\). However, the plane of the orbit does not remain fixed in space, but, like the plane of a top, performs a precessional motion. Outside the atom, along with the moment \(K\dfrac{h}{2\pi}\) of the precessing elliptical orbit, there also acts the angular momentum of the non-precessing orbit, \(J\dfrac{h}{2\pi}\). The axis of the top, with respect to which this angular momentum acts externally (the \(J\)-axis), may, moreover, occupy various positions in space. If a preferred direction can be singled out in space—for example, when a strong magnetic field is switched on, the preferred direction will be that of the lines of force—then the \(J\)-axis may make various angles with this preferred direction. Whereas \(J\dfrac{h}{2\pi}\) denotes the angular momentum acting with respect to the \(J\)-axis, with respect to the direction of the lines of force (at an angle to the \(J\)-axis) a smaller angular momentum \(m\dfrac{h}{2\pi}\) acts.

From relations (12) follows the compatibility of the values \(K, J, m\) with the known values of \(n\). The cases where \(n=1, 2, 3, 4\) are collected in Table 3. From it one sees that the number of possibilities for the spatial orientation of the angular momentum of one electron for \(K=\dfrac{1}{2}\) is

TABLE 3.

\(n\)	\(K\)	\(J\)	\(m\)	Number of possibilities
1	\(\dfrac{1}{2}\)	1	\(\pm\dfrac{1}{2}\)	\(2 = 2\cdot 1^2\)
2	\(\dfrac{1}{2}\)	1	\(\pm\dfrac{1}{2}\)	2
2	\(\dfrac{1}{2}\)	1	\(\pm\dfrac{1}{2}\)	2
2	\(\dfrac{3}{2}\)	2	\(\pm\dfrac{1}{2}, \pm\dfrac{3}{2}\)	\(4\)
2	—	—	—	\(2+2+4=8=2\cdot 2^2\)
3	\(\dfrac{1}{2}\)	1	\(\pm\dfrac{1}{2}\)	2
3	\(\dfrac{1}{2}\)	1	\(\pm\dfrac{1}{2}\)	2
3	\(\dfrac{3}{2}\)	2	\(\pm\dfrac{1}{2}, \pm\dfrac{3}{2}\)	4
3	\(\dfrac{3}{2}\)	2	\(\pm\dfrac{1}{2}, \pm\dfrac{3}{2}\)	4
3	\(\dfrac{5}{2}\)	3	\(\pm\dfrac{1}{2}, \pm\dfrac{3}{2}, \pm\dfrac{5}{2}\)	6
3	—	—	—	\(2+2+4+4+6=18=2\cdot 3^2\)
4	\(\dfrac{1}{2}\)	1	\(\pm\dfrac{1}{2}\)	2
4	\(\dfrac{1}{2}\)	1	\(\pm\dfrac{1}{2}\)	2
4	\(\dfrac{3}{2}\)	2	\(\pm\dfrac{1}{2}, \pm\dfrac{3}{2}\)	4
4	\(\dfrac{3}{2}\)	2	\(\pm\dfrac{1}{2}, \pm\dfrac{3}{2}\)	4
4	\(\dfrac{5}{2}\)	3	\(\pm\dfrac{1}{2}, \pm\dfrac{3}{2}, \pm\dfrac{5}{2}\)	6
4	\(\dfrac{5}{2}\)	3	\(\pm\dfrac{1}{2}, \pm\dfrac{3}{2}, \pm\dfrac{5}{2}\)	6
4	\(\dfrac{7}{2}\)	4	\(\pm\dfrac{1}{2}, \pm\dfrac{3}{2}, \pm\dfrac{5}{2}, \pm\dfrac{7}{2}\)	8
4	—	—	—	\(2+2+4+4+6+6+8=32=2\cdot 4^2\)

only 2. For \(K=\dfrac{3}{2}\) it is already equal to 6, for \(K=\dfrac{5}{2}\)—10, and for \(K=\dfrac{7}{2}\)—14.

For \(n=1\) the number of possible orientations will be only 2, but

\[ \begin{array}{ll} \text{for } n=2 & 2+6=8,\\ \text{” } n=3 & 2+6+10=18,\\ \text{” } n=4 & 2+6+10+14=32. \end{array} \]

In connection with this last result it should be noted that already Sommerfeld (1916), in his investigations of the Zeeman effect in hydrogen, took into account the number of possible orientations. Sommerfeld’s theory makes it possible to predict, for each increase in \(k\), the increase in the number of possible orientations. However, the significance of these numbers for the problem of the structure of the atom and of the periodic system has been understood only recently, in connection with the work of Stoner and W. Pauli, Jr., to whom this field in general owes substantial advances.

It should also be mentioned that experimental confirmation of spatial quantization was provided by the experiments of Stern and Gerlach, who discovered the discreteness of the magnetic moments of individual atoms1.

7. Periodicity of electronic properties.

The study of the valency of the chemical elements laid the foundation for the conception of definite groupings of the electrons forming the atom. Already in the first forms of the periodic system (cf. Table 1, according to Mendeleev) a regular increase of O- or H-valency is observed in passing from vertical column to vertical column. Abegg went considerably further at the beginning of the present century. According to Abegg, each element possesses a positive and a negative valency, which together always give the number 8, the former always being equal to the number of the vertical row. Abegg considered the smaller valency \((<4)\) to be the normal one, and therefore also the stronger one; the contravalency is the numerically larger, but weaker, and polar-opposite to the first. In this way the following distribution of valency is obtained (cf. also the upper line of Table 2):

Vertical rows	1	2	3	4	5	6	7
Normal valencies	\(+1\)	\(+2\)	\(+3\)	\(\pm 4\)	\(-3\)	\(-2\)	\(-1\)
Contravalencies	\((-7)\)	\((-6)\)	\((-5)\)		\(+5\)	\(+6\)	\(+7\)

PERIODIC SYSTEM OF CHEMICAL ELEMENTS

Abbott himself had already tried to interpret this result in the spirit of the electronic theory; a very definite conception was then expressed by Drude, who considered Abbott’s positive valence (both for normal valence and for contravalence) to be equal to the number of loosely bound electrons. Conversely, negative valence, according to Drude, must be equal to the number of electrons which the given atom can tear away from other atoms or, at least, attach to itself more firmly. This conception was based on the fact,

Fig. 1.

established by Drude, that the number of loosely bound electrons obtained from dispersion is equal to or less than Abbott’s positive valence.

The electronic theory of valence served as the starting point for Kossel (1916), who came to the conclusion that the groupings of electrons in atoms of the noble gases are especially stable. Atoms strive toward these stable groupings by giving up electrons in the case of positive valence, or by attaching them in the case of negative valence. Fig. 1 shows that this special position of the noble gases with respect to the surrounding elements extends throughout the entire periodic system. According to Kossel, the arrangements of the outer electrons realized in the noble gases have

a well-known analogy with states of equilibrium. If one passes from one noble gas to the next in the direction of increasing atomic number, then the gradually added electrons represent easily detachable valence electrons, whose number is equal to the positive valence of Abegg and Drude.

What, then, happens next to the completed electronic group of a noble gas? Since Moseley established a uniform change of the frequencies of the K- and L-series with increasing atomic number, Kossel was inclined to admit, for the inner groups of electrons, the absence of a periodic change. Therefore the electronic groupings of the noble gases are preserved in the elements that follow them as inner groups, while the newly added valence electrons are arranged outside. The difficulties encountered in the formation of the large periods led Kossel to undertake the division of electrons into groups only for the first 25 elements (up to Mn). In contrast to Bohr’s somewhat earlier attempt, Kossel here took the correct path, as is shown by Table 4, where the figures printed in bold type refer to valence electrons, and the noble gases represent at the same time the end and the beginning of periods.

TABLE 4.

Groups of electrons according to Kossel (1916).

H	H	H	H	H	H	H	H	H
(1)	(1)	(1)	(1)	(1)	(1)	(1)	(1)	(1)
He (2, 0) Ne (2, 8, 0) A (2, 8, 8, 0)	Li (2, 1) Na (2, 8, 1) Na (2, 8, 8, 1)	Be (2, 2) Mg (2, 8, 2) Ca (2, 8, 8, 2)	B (2, 3) Al (2, 8, 3) Sc (2, 8, 8, 3)	C (2, 4) Si (2, 8, 4) Ti (2, 8, 8, 4)	N (2, 5) P (2, 8, 5) V (2, 8, 8, 5)	O (2, 6) S (2, 8, 6) Cr (2, 8, 8, 6)	F (2, 7) Ce (2, 8, 7) Mn (2, 8, 8, 7)	Ne (2, 8) A (2, 8, 8)

The difficulties connected with the large periods consist in the fact that the eighth element \((Z = 26)\) after argon, as is known, is not a noble gas, although a system with 28 electrons, according to Kossel’s scheme, should be stable. Only the eighteenth element after argon is likewise a noble gas (krypton, \(Z = 36\)). The same is true for the next period from Kr to Xe \((Z = 54)\), and especially for the period from Xe to Em \((Z = 86)\), where the rare earths are located. All these large periods are characterized by the fact that they are rows of elements with a constant minimal valence: from Ti to Ni, from Zr to Pd, from Ta to Pt are situated elements that are minimally divalent; further, the rare earths—almost all (with the excep-

—with the exception of Sm and Eu)—are at least trivalent. These elements form paramagnetic and colored ions; moreover, they are all situated at the minima of the curve of atomic volumes. These features prompted Ladenburg, 5 years ago, to assume in such elements a constant—

Fig. 2.

Text in the figure: vertical axis: “Atomic volume / atomic weight / density”; horizontal axis: “Atomic number.”
Symbols: + paramagnetic; hatched areas—colored ions; ++ unfilled intermediate shells with loosely bound electrons.

—number of outer, easily detachable electrons. The remaining valence electrons must form an intermediate shell between the outer and inner groups of electrons; these intermediate electrons determine the color and paramagnetism of the ions (cf. Fig. 2).

8. Main groups of electrons.

Already J. J. Thomson assumed the presence, in the noble gases, of especially stable groupings of electrons. Kossel used not only data concerning heteropolar compounds and electro-

valence, but also the results of X-ray spectroscopy in order to substantiate the constancy of already completed groups of electrons (shells). According to Kossel, for the noble gases the following groups are obtained: for He \(=2\); for Ne \(=2+8\); for A \(=2+8+8\) electrons.

At approximately the same time (1916), the author was occupied with this question and assumed that in the noble gases, in passing from each of them to the neighboring one with a higher atomic number, the number of electron groupings that remain unchanged increases by one. At the same time the author assumed that, for such a completed group of electrons, the characteristic principal quantum number (then called the quantum sum) increases by one as each new completed group is added (in agreement with the data of X-ray spectroscopy). For the noble gases following A, the author adopted the following electron groups:

\[ \begin{aligned} \mathrm{Kr} &= 2+8+8+18 \\ \mathrm{X} &= 2+8+8+18+18 \\ \mathrm{Em} &= 2+8+8+18+18+32 . \end{aligned} \]

Ladenburg introduced (1919), for the elements of the secondary families, the concept of an intermediate group, which in these elements is in the process of completion, while Vegard had already earlier (1917) expressed an analogous supposition concerning the rare earths. Thus, in the noble gases Kr, X, and Em, in all cases an identically filled group of 8 electrons was assumed. Therefore the first three of the named noble gases should have the following groups:

\[ \begin{aligned} \mathrm{Kr} &= 2+8+8+10+8 \\ \mathrm{X} &= 2+8+8+10+8+10+8 \\ \mathrm{Em} &= 2+8+8+10+8+10+8+14+10+8 \end{aligned} \]

electrons.

In 1921 Bohr made a further step forward, eliminating the shortcomings of the scheme just given. Bohr’s procedure consisted in considering how, from the point of view of quantum theory, an atom can be formed by the successive capture and binding of individual electrons in the field of the nucleus \(Ze\). Bohr assumed here that these \(Z\) electrons split up into a certain number of principal groups, which are determined by one and the same value \(n\). The final filling of a principal group of electrons, characterized by the principal quantum number \(n\), is reached when the number of electrons in this group becomes equal to \(2n^2\), i.e. for \(n=1\), \(Z_{\max}=2\); for \(n=2\), \(Z_{\max}=8\); for \(n=4\), \(Z_{\max}=32\). Bohr thus obtained the following schemes of numbers of electrons, corresponding to different \(n\), for the noble gases (cf. Table 5).

TABLE 5.

\(Z\)	\(n \to\)	1	2	3	4	5	6
2	He	2
10	Ne	2	8
18	A	2	8	8
36	Kr	2	8	18	8
54	X	2	8	18	18	8
86	Em	2	8	18	32	18	8

Bohr did not give an explanation or interpretation of these maximum numbers of electrons in a group as a function of the various values of \(n\). This has been accomplished only quite recently (1925), thanks to the work of Stoner and Pauli the younger, with the use of Landé’s quantum schemes (Table 15). We have already seen that the number of possibilities for the spatial orientation of the acting angular momentum of the electron orbit

Fig. 3.

in a strong magnetic field for \(n=1\) there will be only 2; for \(n=2\)—already 8; for \(n=3\)—18, and for \(n=4\)—32. And so Pauli asserts that in an atom there cannot exist two or a greater number of electrons for which, in strong magnetic fields, the four quantum numbers \(n, k, j, m\) would coincide. If in an atom there is an electron for which these quantum numbers have (in an external field) definite values, then the state corresponding to these values of the quantum numbers will already be occupied, and the next electron must move in such an orbit for which at least one of the quantum numbers will differ from the preceding ones. Thus the number of electrons in a group is explained, for for a given value of \(n\) there exist exactly as many electrons of one and the same group as there are different values of \(m\) corresponding to this value of \(n\) (see Table 3). At the same time, in a completely filled group of electrons, to each system of values \(n, K, J, m\) there corresponds one single electron.

Bohr proposed that in the Ladenburg intermediate groups of the subsidiary families of elements there occurs the filling of the still incompletely filled main groups of electrons. Thus, in the elements Sc—Ni there occurs the filling of the main group with \(n=3\) from 8 to 18 electrons; in Y—Pd—the main group with \(n=4\) from 8 to 18 electrons; in the rare earths Ce—Yb—the filling of the main group from 18 to 32 electrons, and in Cp—Pt—the main group with \(n=5\) from 8 to 18 electrons.

Very graphic is the form of the periodic system used by Bohr and representing a modification of the table proposed in 1895 by Julius Thomsen (Fig. 3). Here analogous elements are connected by strokes, and those elements in which the intermediate groups are in the process of filling are enclosed in frames. Cf. also Ladenburg’s curve of atomic volumes (Fig. 2).

9. Subdivision of the Main Groups.

In his quantum theory of the structure of the atoms of elements, Bohr assumed a subdivision of the main groups of electrons with the same principal quantum number into subgroups with different values of the subsidiary quantum number \(k\). At the same time the number of such groups of electrons with different values of \(k\), according to Bohr, must be equal to the quantity \(n\), i.e. 1 group for \(n=1\); 2 subgroups for \(n=2\); 3 subgroups for \(n=3\), etc. This number of subgroups also corresponds to the number of subsidiary quantum numbers according to Landé (Table 3).

Further, Bohr admitted that the maximum number of electrons in all subgroups for one and the same \(n\) is always the same. Thus, for the electrons in the noble gases the following subdivision into subgroups was adopted (Table 6). Subsequently Bohr himself drew attention

TABLE 6.

\(Z\)	Element	\(n=1,\ k=1\)	\(n=2,\ k=1\)	\(n=2,\ k=2\)	\(n=3,\ k=1\)	\(n=3,\ k=2\)	\(n=3,\ k=3\)	\(n=4,\ k=1\)	\(n=4,\ k=2\)	\(n=4,\ k=3\)	\(n=4,\ k=4\)	\(n=5,\ k=1\)	\(n=5,\ k=2\)	\(n=5,\ k=3\)	\(n=5,\ k=4\)	\(n=5,\ k=5\)	\(n=6,\ k=1\)	\(n=6,\ k=2\)	\(n=6,\ k=3\)	\(n=6,\ k=4\)	\(n=6,\ k=5\)	\(n=6,\ k=6\)	\(n=7,\ k=1\)	\(n=7,\ k=2\)	\(n=7,\ k=3\)
2	He	2
10	Ne	2	[4	4]
18	A	2	4	4	[4	4]	—
36	Kr	2	4	4	6	6	6	[4	4]	—	—
54	X	2	4	4	6	6	6	6	6	6	—	[4	4]	—	—	—
86	Em	2	4	4	6	6	6	8	8	8	8	6	6	6	—	—	[4	4]	—	—	—	—
118?	?	2	4	4	6	6	6	8	8	8	8	[8	8	8	8]	—	[6	6	6]	—	—	—	[4	4]	—

to the irregularity of such a distribution of electrons into subgroups in elements with a completed and a still incomplete principal group.

In Tables 6 and 7 such completion of the principal groups is indicated by placing the corresponding number of electrons in square brackets. This symmetrical subdivision of the principal groups into equally filled subgroups, however, in the case of such completion causes a certain difficulty. Namely, according to Bohr, in passing from the preliminary completion of a subgroup to the final one, a complete regrouping must take place. We had such a case in the first half of the first large period, in the transition of the third principal group from the electron grouping \((4,4)\)—in argon—to the grouping \((6, 6, 6)\) in Cu; analogously in Kr—Ag and in X—Au. The same occurs in the transition of the fourth principal group from the \((6, 6, 6)\) grouping to the grouping \((8, 8, 8, 8)\)—in Cp (cf. Table 7).

It has already been pointed out above that, for a complete characterization of the stationary orbits of electrons, the two quantum numbers \(n\) and \(k\) are not sufficient. Better agreement with experimental data is obtained by introducing a third, internal quantum number \(j\). The development of X-ray spectrometry, especially by Siegbahn and his pupils, led to the same result. Through these experimental and theoretical works—in the latter respect, alongside Cossel, special credit belongs to Sommerfeld, Bohr, Koster, Smekal, and Wentzel—the number of electron groupings was established: for the \(K\)-series 1, for the \(L\)-series 3, for the \(M\)-series 5, and for the \(N\)-series 7.

Making use of all these results and, furthermore, somewhat extending the argumentation, Stoner in 1924 undertook a new distribution

R. SVINNE

TABLE 7.

\(Z\)	\(\dfrac{n}{K}\)	\(1_1\)	\(2_1\)	\(2_2\)	\(3_1\)	\(3_2\)	\(3_3\)	\(4_1\)	\(4_2\)	\(4_3\)	\(4_4\)	\(5_1\)	\(5_2\)	\(5_3\)	\(5_4\)	\(5_5\)	\(6_1\)	\(6_2\)	\(6_3\)	\(6_4\)	\(6_5\)	\(6_6\)	\(7_1\)	\(7_2\)
1	H	1
2	He	2
3	Li	2	1
4	Be	2	2
5	B	2	2	1
—	—	—	—	—
10	Ne	2	[4	4]
11	Na	2	4	4	1
12	Mg	2	4	4	2
13	Al	2	4	4	2	1
—	—	—	—	—	—	—
18	A	2	4	4	[4	4]
19	K	2	4	4	4	4	—	1
20	Ca	2	4	4	4	4	—	2
21	Sc	2	4	4	4	4	1	(2)
22	Ti	2	4	4	4	4	2	(2)
—	—	—	—	—	—	—	—	—
29	Cu	2	4	4	[6	6	6]	1
30	Zn	2	4	4	6	6	6	2
31	Ga	2	4	4	6	6	6	2	1
—	—	—	—	—	—	—	—	—	—
36	K	2	4	4	6	6	6	[4	4]
37	Rb	2	4	4	6	6	6	4	4	—	—	1
38	Sr	2	4	4	6	6	6	4	4	—	—	2
39	Y	2	4	4	6	6	5	4	4	1	—	(2)
40	Zr	2	4	4	6	6	6	4	4	2	—	(2)
—	—	—	—	—	—	—	—	—	—	—	—	—
47	Ag	2	4	4	6	6	6	[6	6	6]	—	1
48	Cd	2	4	4	6	6	6	6	6	6	—	2
49	In	2	4	4	6	6	6	6	6	6	—	2	1
—	—	—	—	—	—	—	—	—	—	—	—	—	—
54	X	2	4	4	6	6	6	6	6	6	—	[4	4]
55	Ca	2	4	4	6	6	6	6	6	6	—	4	4	—	—	—	1
56	Ba	2	4	4	6	6	6	6	6	6	—	4	4	—	—	—	2
57	La	2	4	4	6	6	6	6	6	6	—	4	4	1	—	—	(2)
58	Ce	2	4	4	6	6	6	6	6	6	1	4	4	1	—	—	(2)
59	Pr	2	4	4	6	6	6	6	6	6	2	4	4	1	—	—	(2)
—	—	—	—	—	—	—	—	—	—	—	—	—	—	—	—	—	—
71	Cp	2	4	4	6	6	6	[8	8	8	8]	4	4	1	—	—	(2)
72	Hf	2	4	4	6	6	6	8	8	8	8	4	4	2	—	—	(2)
—	—	—	—	—	—	—	—	—	—	—	—	—	—	—	—	—	—
79	Au	2	4	4	6	6	6	8	8	8	8	[6	6	6]	—	—	1
80	Hg	2	4	4	6	6	6	8	8	8	8	6	6	6	—	—	2
81	Tl	2	4	4	6	6	6	8	8	8	8	6	6	6	—	—	2	1
—	—	—	—	—	—	—	—	—	—	—	—	—	—	—	—	—	—	—
86	Em	2	4	4	6	6	6	8	8	8	8	6	6	6	—	—	[4	4]
87	—	2	4	4	6	6	6	8	8	8	8	6	6	6	—	—	4	4	—	—	—	—	1
88	Ra	2	4	4	6	6	6	8	8	8	8	6	6	6	—	—	4	4	—	—	—	—	2
89	Ac	2	4	4	6	6	6	8	8	8	8	6	6	6	—	—	4	4	1	—	—	—	(2)
90	Th	2	4	4	6	6	6	8	8	8	8	6	6	6	—	—	4	4	2	—	—	—	(2)
—	—	—	—	—	—	—	—	—	—	—	—	—	—	—	—	—	—	—	—	—	—	—	—
118?	—	2	4	4	6	6	6	8	8	8	8	[8	8	8	8]	—	[6	6	6]	—	—	—	[4	4]

of the main groups, bringing it into connection with the internal quantum numbers. The groupings of electrons for the noble gases obtained in this way are given in Table 8.

TABLE 8.

\(Z\)	\(\dfrac{n}{k}\)	1/1	2/1	2/2	3/1	3/2	3/3	4/1	4/2	4/3	4/4	5/1	5/2	5/3	5/4	5/5	6/1	6/2	6/3	6/4	6/5	6/6
2	He	2
10	Ne	2	2	6
18	A	2	2	6	2	6
36	Kr	2	2	6	2	6	10	2	6	10
54	X	2	2	6	2	6	10	2	6	10		2	6
86	Em	2	2	6	2	6	10	2	6	10	14	2	6	10			2	0

This step by Stoner removed the difficulties which Bohr had encountered in filling the previously completed subgroups. For, according to Stoner, no rearrangement of the latter need take place at all; what actually occurs is the addition of the next subgroup. Thus, Stoner permits the completion of a group of ten electrons in the elements of the iron, palladium, and platinum families, and of a group of fourteen electrons in the rare earths. In this representation, Ladenburg’s intermediate shells receive the simplest interpretation; at the same time the general scheme of the periodic system (Fig. 3) and of the main groups of electrons in the noble gases, according to Bohr (Table 5), underwent not the slightest change.

Simultaneously with Stoner, an analogous asymmetric distribution was proposed by Main Smith, who made use, along with chemical data, also of spectroscopic data.

10. CHARACTERIZATION OF A SUBGROUP.

Since Stoner uses three quantum conditions, each subgroup of electrons is determined by three quantum symbols \(n, k, j\). The maximum number of electrons in a subgroup, according to Stoner, is equal to \(2j\). The physical meaning of this assertion reduces to the fact that the number of possible states of motion of the electrons in the subgroup is also equal to \(2j\). These various possible orbits of the electrons differ in their orientation with respect to the atom as a whole. New electrons can enter into each subgroup until those

TABLE 9.

Designation according to Bohr and Koster	Term designation according to Sommerfeld	Term designation according to Koster	X-ray term designation according to Sommerfeld	Number of electrons according to Stoner	Stoner quantum numbers: \(n\)	Stoner quantum numbers: \(k\)	Stoner quantum numbers: \(j\)	Landé quantum numbers: \(n\)	Landé quantum numbers: \(k\)	Landé quantum numbers: \(j\)	Landé quantum numbers: \(m\)	Number of possible arrangements according to W. Pauli
K	K	\(1_1 b\)	K	\(1s\)	2	1	1	1	1	\(\frac12\)	1	\(\pm \frac12\)
\(L_{\mathrm I}\)	\(L_3\)	\(2_1 b\)	\(L_{11}\)	\(2s\)	2	2	1	1	2	\(\frac12\)	1	\(\pm \frac12\)
II	2	\(2_1 a\)	21	\(2p_1\)	2	2	2	1	2	\(\frac32\)	1	\(\pm \frac12\)
III	1	\(2_2 a\)	22	\(2p_2\)	4	2	2	2	2	\(\frac32\)	2	\(\pm \frac12, \pm \frac32\)
\(M_{\mathrm I}\)	\(M_5\)	\(3_1 b\)	\(M_{11}\)	\(3s\)	2	3	1	1	3	\(\frac12\)	1	\(\pm \frac12\)
II	4	\(3_1 a\)	21	\(3p_1\)	2	3	2	1	3	\(\frac32\)	1	\(\pm \frac12\)
III	3	\(3_2 a\)	22	\(3p_2\)	4	3	2	2	3	\(\frac32\)	2	\(\pm \frac12, \pm \frac32\)
IV	2	\(3_2 b\)	32	\(3d_2\)	4	3	3	2	3	\(\frac52\)	2	\(\pm \frac12, \pm \frac32\)
V	1	\(3_3 b\)	33	\(3d_3\)	6	3	3	3	3	\(\frac52\)	3	\(\pm \frac12, \pm \frac32, \pm \frac52\)
\(N_{\mathrm I}\)	\(N_7\)	\(4_1 b\)	\(N_{11}\)	\(4s\)	2	4	1	1	4	\(\frac12\)	—	\(\pm \frac12\)
II	6	\(4_1 a\)	21	\(4p_1\)	2	4	2	1	4	\(\frac32\)	—	\(\pm \frac12\)
III	5	\(4_2 a\)	22	\(4p_2\)	4	4	2	2	4	\(\frac32\)	—	\(\pm \frac12, \pm \frac32\)
IV	4	\(4_2 b\)	32	\(4d_2\)	4	4	3	2	4	\(\frac52\)	—	\(\pm \frac12, \pm \frac32\)
V	3	\(4_3 b\)	33	\(4d_3\)	6	4	3	3	4	\(\frac52\)	—	\(\pm \frac12, \pm \frac32, \pm \frac52\)
VI	2	\(4_3 a\)	43	\(4f_3\)	6	4	4	3	4	\(\frac72\)	—	\(\pm \frac12, \pm \frac32, \pm \frac52\)
VII	1	\(4_4 a\)	44	\(4f_4\)	8	4	4	4	4	\(\frac72\)	—	\(\pm \frac12, \pm \frac32, \pm \frac52, \pm \frac72\)
\(O_{\mathrm I}\)	\(O_5\)	\(5_1 b\)	\(O_{11}\)	\(5s\)	2	5	1	1	5	\(\frac12\)	1	\(\pm \frac12\)
II	4	\(5_1 a\)	21	\(5p_1\)	2	5	2	1	5	\(\frac32\)	1	\(\pm \frac12\)
III	3	\(5_2 a\)	22	\(5p_2\)	4	5	2	2	5	\(\frac32\)	2	\(\pm \frac12, \pm \frac32\)
IV	2	\(5_2 b\)	32	\(5d_2\)	4	5	3	2	5	\(\frac52\)	2	\(\pm \frac12, \pm \frac32\)
V	1	\(5_3 b\)	33	\(5d_3\)	6	5	3	3	5	\(\frac52\)	8	\(\pm \frac12, \pm \frac32, \pm \frac52\)
\(P_{\mathrm I}\)	\(P_3\)	\(6_1 b\)	\(P_{11}\)	\(6s\)	2	6	1	1	6	\(\frac12\)	1	\(\pm \frac12\)
II	2	\(6_1 a\)	21	\(6p_1\)	2	6	2	1	6	\(\frac32\)	1	\(\pm \frac12\)
III	1	\(6_2 a\)	22	\(6p_2\)	4	6	2	2	6	\(\frac32\)	2	\(\pm \frac12, \pm \frac32\)

until all these possible orbits have been occupied. The circumstance that the number of possible orientations of electron orbits is indeed equal to \(2j\) is confirmed by data concerning the number of stationary states of peripheral electrons in a strong external magnetic field.

Stoner considered the case of the alkali metals in greater detail. These arguments of his were then refined by W. Pauli, who succeeded in obtaining an explanation of why the maximum number of electrons in the principal group is equal to \(2n^2\). We shall return to this below.

When, in the further development of the periodic system, these initially peripheral electrons become internal ones, the numerical values of the quantum symbols characterizing them remain unchanged. This is supported by the fact that the optical multiplets of peripheral electrons can be characterized by the same quantum symbols as the X-ray multiplets of internal electrons. The transition from the ordinary optical series to the Barkla–Moseley X-ray series is represented by the vacuum-ultraviolet lines discovered by Millikan in atoms ionized to various degrees; at the same time, for the multiplets in this region, the very same regularities are found as in the remaining parts of the spectrum.

Table 9 gives a summary of the designations of electron subgroups used by various investigators and their characterization by quantum numbers. Along with the characterization of these subgroups by the corresponding numbers of electrons, the question arose of determining the levels or stages of energy of these electron subgroups. One way of doing this consists in determining the limiting wavelength \(\lambda_A\) of the X-rays exciting the given X-ray series, i.e. the so-called absorption edge, or the minimum velocity \(v_A\) (corresponding to this wavelength), or else the minimum potential \(V_A\) of the electrons exciting the X-rays. Both quantities, as is known, are related by

\[ eV_A = h \frac{c}{\lambda_A}, \tag{13} \]

and also

\[ \frac{1}{2}mv_A^2 = h \frac{c}{\lambda_A}, \tag{14} \]

if the dependence of mass on velocity does not yet introduce a substantial correction.

Another way consists in the magnetic analysis of secondary electrons excited by monochromatic X-rays (after de Broglie). When illuminated by such X-rays of wavelength \(\lambda_n\), in addition to the normal photoelectric effect (which gives electrons with velocity \(v_n\)), there is also a selective effect, conditioned—

excited by excitation of individual subgroups of the radiator’s electrons; let these secondary electrons exhibit the velocity \(v_s\). The difference in kinetic energies of these and other electrons (of the normal and selective photo-effect) is precisely equal to the energy \(A\) of the excited group of electrons. Since the correction for the dependence of \(m\) on velocity is insignificant, we have:

\[ \frac{1}{2} m v_s^2=\frac{1}{2} m v_n^2-A=h\frac{c}{\lambda_n}-A. \tag{15} \]

Fig. 4. Square roots of the spectral terms of X-ray spectra as a function of atomic number.

This method is especially important for determining the energy of those subgroups to which there correspond X-ray wavelengths considerably shorter than those accessible to determination by means of a crystal lattice.

The magnitudes of the energies of individual subgroups, determined by different methods, agree well with one another. The most abundant material is provided by spectrographic data processed by means of the combination principle; in such spectroscopic data the energy is usually expressed through the wave number divided by the Rydberg constant \(\left(\frac{1}{\lambda R}\right)\).

If one compares the various energy values corresponding to a series of elements, then these values, as \(Z\) decreases, decrease and, for known elements, tend to zero. The analogous situation holds for the wave numbers of the lines of X-ray series connected with a given energy level. If, however, one considers the relative brightness of these lines within the periodic system, or the magnitude of the jump in intensity at the absorption edge of X-ray absorption series, then these quantities remain constant within fairly wide limits;

then, beginning with a certain value of \(Z\), they begin to undergo an ever stronger weakening and, finally, become inaccessible to measurement. The explanation here reduces to the fact that the maximum number of electrons of the corresponding subgroup at \(Z\), beginning with which a decrease in intensity is observed,—that this number of electrons begins to decrease and, for zero line intensity, becomes zero.

11. Completion of Subgroups.

The Stoner–Pauli theory, explaining the maximum number of electrons in individual groups, is insufficient to explain why the known groupings of electrons possess the properties of the noble gases. For there exist noble gases with only a small number of completely occupied subgroups, from among those which appear possible from the quantum point of view and which are shown in Table 10. The latter are obtained by the successive addition of electrons to already filled subgroups, each subgroup developing only when all the preceding ones, i.e. the subgroups with smaller \(n\), are completely occupied. The elements placed in parentheses, in certain terms, exhibit a magnetic moment equal to zero. Whether an “electronic isomer” with a structure corresponding to Table 10 can be realized for Ni, we shall for the present leave open. To this deviation of the actual

TABLE 10.

\(Z\)	\(\dfrac{n}{k}\)	1/1	2/1	2/2	3/1	3/2	3/3	4/1	4/2	4/3	4/4	5/1	5/2	5/3	5/4	5/5	6/1	6/2
2	He	2
4	(Be)	2	2
10	Ne	2	2	6
12	(Mg)	2	2	6	2
18	A	2	2	6	2	6
28	[Ni]	2	2	6	2	6	10
30	(Zn)	2	2	6	2	6	10	2
36	Kr	2	2	6	2	6	10	2	6
46	[Pd]	2	2	6	2	6	10	2	6	10
60	[Nd]	2	2	6	2	6	10	2	6	10	14
62	[Sm]	2	2	6	2	6	10	2	6	10	14	2
68	[Er]	2	2	6	2	6	10	2	6	10	14	2	6
78	[Pt]	2	2	6	2	6	10	2	6	10	14	2	6	10
92	[U]	2	2	6	2	6	10	2	6	10	14	2	6	10	14
110		2	2	6	2	6	10	2	6	10	14	2	6	10	14	18
112		2	2	6	2	6	10	2	6	10	14	2	6	10	14	18	2
118		2	2	6	2	6	10	2	6	10	14	2	6	10	14	18	2	6

TABLE 11.

		K_I	L_I	L_II	L_III	M_I	M_II	M_III	M_IV	M_V	N_I	N_II	N_III	N_IV	N_V	N_VI	N_VII	O_I	O_II	O_III	O_IV	O_V—O_IX	P_I	P_II	P_III	P_IV	P_V—P_XI	Q_I	Q_II
n		1	2	2	2	3	3	3	3	3	4	4	4	4	4	4	4	5	5	5	5	5	6	6	6	6	6	7	7
k		1	1	2	2	1	2	2	3	3	1	2	2	3	3	4	4	1	2	2	3	3—5	1	2	2	3	3—6	1	2—
j		1	1	1	2	1	1	2	2	3	1	1	2	2	3	3	4	1	1	2	2	3—5	1	1	2	2	3—6	1	1—
H	1	1
He	2	2
Li	3	2	1
Be	4	2	2
B	5	2	2	1
C	6	2	2	2
N	7	2	2	2	1
O	8	2	2	2	2
F	9	2	2	2	3
Ne	10	2	2	2	4
Na	11	2	2	2	4	1
Mg	12	2	2	2	4	2
Al	13	2	2	2	4	2	1
Si	14	2	2	2	4	2	2
P	15	2	2	2	4	2	2	1
S	16	2	2	2	4	2	2	2
Cl	17	2	2	2	4	2	2	3
A	18	2	2	2	4	2	2	4
K	19	2	2	2	4	2	2	4			1
Ca	20					2	2	4			2
Sc	21					2	2	4	(1)		(2)
Ti	22					2	2	4	(2)		(2)
V	23					2	2	4	(3)		(2)

TABLE 12.

		KI	LI	LII	LIII	MI	MII	MIII	MIV	MV	NI	NII	NIII	NIV	NV	NVI	NVII	OI	OII	OIII	OIV	OV—OIX	PI	PII	PIII	PIV	PV—PXI	QI	QII
$n$		1		2				3						4						5					6			7
$k$		1	1	2	2	1	2	2	3	3	1	2	2	3	3	4	4	1	2	2	3	3—5	1	2	2	3	3—6	1	2—
$j$		1	1	1	2	1	1	2	2	3	1	1	2	2	3	3	4	1	1	2	2	3—5	1	1	2	2	3—6	1	1—
Cr	24					2	2	4	(4)		(2)
Mn	25					2	2	4	(3)	(2)	(2)
Fe	26					2	2	4	(3)	(3)	(2)
Co	27					2	2	4	(3)	(4)	(2)
Ni	28					2	2	4	(3)	(5)	(2)
Cu	29					2	2	4	4	6	(1)
Zn	30					2	2	4	4	6	2
Ga	31					2	2	4	4	6	2	1
Ge	32					2	2	4	4	6	2	2
As	33					2	2	4	4	6	2	2	1
Se	34					2	2	4	4	6	2	2	2
Br	35					2	2	4	4	6	2	2	3
Kr	36	2	2	2	4	2	2	4	4	6	2	2	4
Rb	37	2	2	2	4	2	2	4	4	6	2	2	4					1
Sr	38										2	2	4					2
Y	39										2	2	4	(2)				(2)
Zr	40										2	2	4	(2)				(2)
Nb	41										2	2	4	(3)				(2)
Mo	42										2	2	4	(4)				(2)
Ma	43										2	2	4	(4)	(1)			(2)
Ru	44										2	2	4	(4)	(2)			(2)
Rh	45										2	2	4	(4)	(3)			(2)
Pd	46										2	2	4	(4)	(4)			(2)

TABLE 13.

		K_I	L_I	L_II	L_III	M_I	M_II	M_III	M_IV	M_V	N_I	N_II	N_III	N_IV	N_V	N_VI	N_VII	O_I	O_II	O_III	O_IV	O_V—O_IX	P_I	P_II	P_III	P_IV	P_V—P_XI	Q_I	Q_II
n		1	2	2	2	3	3	3	3	3	4	4	4	4	4	4	4	5	5	5	5	5	6	6	6	6	6	7	7
k		1	1	2	2	1	2	2	3	3	1	2	2	3	3	4	4	1	2	2	3	3—5	1	2	2	3	3—6	1	2—
j		1	1	1	2	1	1	2	2	3	1	1	2	2	3	3	4	1	1	2	2	3—5	1	1	2	2	3—6	1	1—
Ag	47										2	2	4	4	6			1
Cd	48										2	2	4	4	6			2
In	49										2	2	4	4	6			2	1
Sn	50										2	2	4	4	6			2	2
Sb	51										2	2	4	4	6			2	2	1
Te	52										2	2	4	4	6			2	2	2
I	53										2	2	4	4	6			2	2	3
X	54	2	2	2	4	2	2	4	4	6	2	2	4	4	6			2	2	4
Cs	55	2	2	2	4	2	2	4	4	6	2	2	4	4	6			2	2	4			1
Ba	56																	2	2	4			2
La	57																	2	2	4	(1)		(2)
Ce	58															1		2	2	4	(1)		(2)
Pr	59															2		2	2	4	(1)		(2)
Nd	60															3		2	2	4	(1)		(2)
—	61															4		2	2	4	(1)		(2)
Sa	62															5		2	2	4	(1)		(2)
Eu	63															(5)	(1)	2	2	4	(1)		(2)
Gd	64															6	1	2	2	4	(1)		(2)
Tb	65															6	2	2	2	4	(1)		(2)
Dy	66															6	3	2	2	4	(1)		(2)
Ho	67															6	4	2	2	4	(1)		(2)
Er	68															6	5	2	2	4	(1)		(2)
Tu	69															6	6	2	2	4	(1)		(2)

TABLE 13 (continued).

Element	\(Z\)	\(K_{\mathrm I}\)	\(L_{\mathrm I}\)	\(L_{\mathrm{II}}\)	\(L_{\mathrm{III}}\)	\(M_{\mathrm I}\)	\(M_{\mathrm{II}}\)	\(M_{\mathrm{III}}\)	\(M_{\mathrm{IV}}\)	\(M_{\mathrm V}\)	\(N_{\mathrm I}\)	\(N_{\mathrm{II}}\)	\(N_{\mathrm{III}}\)	\(N_{\mathrm{IV}}\)	\(N_{\mathrm V}\)	\(N_{\mathrm{VI}}\)	\(N_{\mathrm{VII}}\)	\(O_{\mathrm I}\)	\(O_{\mathrm{II}}\)	\(O_{\mathrm{III}}\)	\(O_{\mathrm{VI}}\)	\(O_{\mathrm V}\!-\!O_{\mathrm{IX}}\)	\(P_{\mathrm I}\)	\(P_{\mathrm{II}}\)	\(P_{\mathrm{III}}\)	\(P_{\mathrm{IV}}\)	\(P_{\mathrm V}\!-\!P_{\mathrm{XI}}\)	\(Q_{\mathrm I}\)	\(Q_{\mathrm{II}}\)
		1	2	2	2	3	3	3	3	3	4	4	4	4	4	4	4	5	5	5	5	5	6	6	6	6	6	7	7
		1	1	2	2	1	2	2	3	3	1	2	2	3	3	4	4	1	2	2	3	3–5	1	2	2	3	3–6	1	2—
		1	1	1	2	1	1	2	2	3	1	1	2	2	3	3	4	1	1	2	2	3, 5	1	1	2	2	3–6	1	1—
Ad	70															6	7	2	2	4	(1)		(2)
Cp	71															6	8	2	2	4	(1)		(2)
Hf	72															6	8	2	2	4	(2)		(2)
Ta	73															6	8	2	2	4	(3)		(2)
W	74															6	8	2	2	4	(4)		(2)
Re	75															6	8	2	2	4	(4)	(1)	(2)
Os	76															6	8	2	2	4	(4)	(2)	(2)
Ir	77															6	8	2	2	4	(4)	(3)	(2)
Pt	78															6	8	2	2	4	(4)	(4)	(2)
Su	79															6	8	2	2	4	4	6	(1)
Hg	80															6	8	2	2	4	4	6	(2)
Tl	81															6	8	2	2	4	4	6	2	1
Pb	82															6	8	2	2	4	4	6	2	2
Bi	83															6	8	2	2	4	4	6	2	2	1
Po	84															6	8	2	2	4	4	6	2	2	2
—	85															6	8	2	2	4	4	6	2	2	3
Em	86	2	2	2	4	2	2	4	4	6	2	2	4	4	6	6	8	2	2	4	4	6	2	2	4
—	87	2	2	2	4	2	2	4	4	6	2	2	4	4	6	6	8	2	2	4	4	6	2	2	4			1
Ra	88																						2	2	4			2
Ac	89																						2	2	4	(1)		(2)
Th	90																						2	2	4	(2)		(2)
Pa	91																						2	2	4	(3)		(2)
U	92																						2	2	4	(4)		(2)

of the noble gases from the “ideal from the quantum point of view” corresponds to the deviation of the actual lengths of the periods \(2, 8, 8, 18, 18, 32\) from the “quantum-ideal” \(2, 8, 18, 32\).

Bohr attempted to find an explanation for these discrepancies by bringing into consideration the energetic side of the matter. The electron which is added after the completion of an actual noble gas is bound in an orbit with a quantum number one unit larger, since the energy of such binding is less than the energy of binding in the still incomplete subgroup with the principal quantum number of the peripheral electrons of the noble gas. At the same time, in the normal state those orbits must be filled to which the lower energy corresponds.

The formation of electron groups in the elements of the first three periods, so far as one can judge, takes place entirely “quantum-ideally” (see Table 11). Bohr justified this assertion by spectroscopic considerations. His method, as is known, consisted in considering the binding of all new electrons in the field of a nucleus with charge \(+Ze\). The addition of one electron takes place by its binding in an orbit with quantum numbers \(n=1,\ k=1,\ j=1\), corresponding to the normal state (the term of the Lyman series); this case is characterized, according to Landé, by a value of \(m\) approximately equal to \(+\dfrac{1}{2}\). The second electron, in such a case, can be bound only in the orbit \(n=1,\ k=1,\ j=1\) and \(m=-\dfrac{1}{2}\), corresponding to the normal state of helium. In the metastable state of helium, alongside the orbit \((n=1,\ k=1,\ j=1)\), there also exists the orbit \((n=2,\ k=1,\ j=1)\).

Since, according to Stoner–Pauli, only two orbits \(1_1\) are possible (i.e. \(n=1,\ k=1\)), the first period comprises only two elements; the third electron is added in the orbit \(2_{11}\) (i.e. \(n=2,\ k=1,\ j=1\)), as is evidenced by the arc spectrum of lithium. This orbit \(2_{11}\), owing to its eccentricity, penetrates far inside the orbits \(1_1\). Thus, according to Bohr, an intimate connection is effected between the orbits of electron groups characterized by different quantum numbers, in contrast to the greater independence of the binding of an electron of one and the same group. The fourth electron is likewise bound in the orbit \(2_{11}\), whereby this group proves to be filled, since at our disposal there is a choice of only two values of \(m\). The fifth electron is added in the orbit \(2_2\), which is also confirmed by the spark spectrum of C, according to Fowler; in the same orbit are also bound the 6th, 7th, 8th, 9th, and 10th electrons; moreover, with the last electron all possibilities of orientation for the orbits \(2_2\), according to Landé, are exhausted, and the period ends.

The eleventh electron begins the third period with a new type of orbit, just as this occurred for the third and fifth elec-

trons. These orbits \(3_{11}\) are even more sharply expressed “comet-like” orbits than the \(2_{11}\) orbits in lithium. The conditions for binding the twelfth electron correspond completely to the conditions for binding the fourth; likewise, the conditions for binding the electrons from the thirteenth to the eighteenth correspond completely to the conditions in the capture of the fifth and so on up to the tenth electron. According to Stoner, the division of the subgroup with \(k=2\) into two parts with two (\(j=1\)) and four (\(j=2\)) electrons is indicated by the peculiarities of the electro-valence of these elements. Such are: the positive four- and, in part, divalence of Si, the positive tri- or pentavalence of P, the positive di-, tetra-, and hexavalence of S, the positive 3-, 5-, and in part 7-valence of J and of the homologues of these elements.

With the addition of the nineteenth electron the peculiarities of the large periods begin to make themselves felt (cf. Table 12), for the interpretation of which Bohr had to put forward a new, energetic point of view. The nineteenth and twentieth electrons (K and Ca) are bound in \(4_{11}\) orbits, which, owing to their very great eccentricity, differ strongly from the \(4_1\) orbits in H. If we now pass to elements with a larger nuclear charge, then at scandium (\(Z=21\)) the moment arrives when the binding in the \(3_3\) orbit for the nineteenth electron surpasses the binding in the \(4_1\) orbit. This tendency is already evident in the spark spectrum of Ca, which reveals a characteristic difference from the arc spectrum of K, despite the fact that both correspond to the binding of the nineteenth electron. With a further increase of the nuclear charge one may expect an increase in the number of electrons in \(3_3\)-orbits already in the normal state of the atom. But this signifies the development of an inner group of electrons, the “intermediate” group according to Ladenburg. Only after it has been filled does the change in properties with increasing nuclear charge begin that is usual for the small periods. This begins for the first time with Cu, whose spark spectrum is analogous to the spark spectrum of Na, although the existence of \(Cu^{''}\) compounds indicates an insufficiently strong binding of the \(3_3\)-electrons. In the following element, Zn, there is already a constant chemical divalence, which is explained correspondingly by the firm binding of all the \(3_3\)-electrons. The subsequent elements display properties quite analogous to the properties of their homologues from the small periods.

An entirely analogous picture holds for the elements of the fifth or second large period. Namely: first in Rb and Sr there is a filling of the very eccentric \(5_{11}\) orbits, then a gradual filling of the \(4_3\) orbits, after which follows a further filling of the \((5_1\) and) \(5_2\) orbits. In the sixth period, at the beginning, there occurs something analogous to what was described above, i.e. the filling of the \(6_{11}\) orbits in Cs and Ba. But now not only the \(5_3\) orbits, but also the \(4_4\) orbits, prove not to be completely filled, so that with increasing nuclear charge the moment arrives when the binding in these orbits becomes stronger. Since,

however, a successive filling of these orbits takes place and, along with it, the completion of an inner grouping of electrons—it is to be expected that elements with similar properties will result. This is what we have in the group of the rare earths. After the filling of the corresponding groups of electrons, the development of the quantum orbits of the electrons begins to proceed in complete analogy with both large periods (cf. Table 13).

It should be noted that Tables 12 and 13 were constructed by the author in accordance with the views of Stoner, Sommerfeld, and Hund.

Men Smith gave in 1924 a scheme for the formation of elements with intermediate groups of electrons. The difference between Smith’s scheme and that of the author consists in the fact that the former assumes that, for the same \(k\), orbits with larger values of \(j\) are filled earlier than orbits with smaller values of \(j\). However, the precise determination of the X-ray terms in the rare earths carried out by Nishina speaks in favor of the author’s scheme.

Corresponding to this quantum subdivision of the main electron groups into \(2 + 6 + 10 + 14\), the following scheme of the periodic system is convenient (Table 14).

An exact characterization of the electron orbits by means of the quantum symbols \(n\), \(k\), \(j\)—especially for elements with intermediate groups—still requires much work. For such elements the author introduced the concept of “electronic isomerism,” indicating the existence of several possibilities for groupings of electrons.

TABLE 14.

Ia	IIa	IIIb	IVb	Vb	VIb	VIIb	VIII	VIII	VIII	Ib	IIb	IIIa	IVa	Va	VIa	VIIa	O
1 H 3 Li 11 Na 19 K 37 Rb 55 Cs 87 —	4 Be 12 Mg 20 Ca 38 Sr 56 Ba 88 Ra	21 Sc 39 Y 57—71 La etc. 89 Ac	22 Ti 40 Zr 72 Hf 90 Th	23 V 41 Nb 73 Ta (Pa)	24 Cr 42 Mo 74 W (U)	25 Mn 43 Ma 75 Re —	26 Fe 44 Ru 76 Os —	27 Co 45 Rh 77 Ir —	28 Ni 46 Pd 78 Pt —	29 Cu 47 Ag 79 Au —	30 Zn 48 Cd 80 Hg —	5 B 13 Al 31 Ga 49 In 81 Tl —	6 C 14 Si 32 Ge 50 Sn 82 Pb —	7 N 15 P 33 As 51 Sb 83 Bi —	8 O 16 S 34 Se 52 Te 84 Po —	9 F 17 Cl 35 Br 53 I 85 —	2 He 10 Ne 18 A 36 Kr 54 X 86 Em —

While the absolute quantity of an element in our stellar system is a property of its nucleus, the distribution of this element in the various layers of the earth depends on its peripheral electrons. It is assumed here that our earth constitutes a closed whole and can neither lose elements—for example, as a result of the velocity of molecular motion overcoming the attraction of the earth—nor acquire them, for example, in the form of radioactive recoil atoms from our sun. In attempts to fill the empty places of the periodic system, it is highly important not only what the probable abundance is, but also with what substances this hypothetical element occurs together.

Important investigations in this direction have been carried out in recent years, connected with ideas about the structure of our earth.

12. Geochemical Characteristics of the Elements.

Among the unsolved problems of the periodic system are the stability of the atomic nucleus and the limited number of elements and isotopes. Questions connected with the nucleus are touched upon only slightly in this survey, since its main subject is the electron shells of the atom. In order, however, to consider the question of the existence of still-undiscovered elements, we must become somewhat acquainted with the laws governing the distribution of the elements in nature. At the beginning of our century, along with the Al-rich silicate rocks of the earth’s surface (the corresponding layer was therefore called “sial,” or “sial”), still richer Mg deep silicate rocks were distinguished (therefore called “sima”) and, in addition, a nickel-containing iron core (called “nife”). The transition from “sima” to “nife” was supposed to take place by means of gradually increasing nife veins in the sima. V. M. Goldschmidt (1922) considers such a mixture of two substances of very different density (approximately 3.6 and 7.6) in the strong gravitational field of the earth to be unstable. He therefore admits the existence of an intermediate sulfide layer, which consists chiefly of iron sulfide and magnetite. Taking into account also the earth’s atmosphere, Goldschmidt accordingly distinguishes four groups of elements according to the place where they occur: atmophiles (from “atmosphere”), lithophiles (silicate shell), chalcophiles (sulfide shell), and siderophiles (iron core).

We learn about the distribution of the elements in all these four principal groups from analyses of air, ocean, silicate rocks, and various meteorites; further indications are provided by metallurgical experience and special experimental investigations. Hence there arises the possibility, on the basis of the relative abundance of the elements composing the earth’s crust, of drawing conclusions about the composition of the earth’s core. It is assumed here that originally there existed a homogeneous

a liquid mass in which, upon cooling, a separation of phases gradually took place in such a way, however, that the separate liquid phases (silicate, sulfide, metallic) during separation were not in equilibrium.

In agreement with these ideas of Goldschmidt are Tammann’s theoretical and experimental works (1923/1924) on the structure of the earth’s core. Tammann derives his conclusions indirectly, on the basis of a number of physicochemical considerations; thus,

Fig. 5.

from the relative electropositivity of metals in comparison with iron, which frequently enters already into the silicate shell of the earth, one may conclude that it has accumulated in the center of the earth.

The distribution of the elements among the four geochemical groups mentioned leads to certain simple regularities, if one considers the relative frequency of the distribution of the elements in connection with the curve of atomic volumes (as a function of atomic number, Fig. 5). All siderophile elements fall at the minima of the curve; all chalcophile elements lie on the ascending branches of the curve; all lithophile elements lie on the descending branches of the curve; to these elements

also include the halides and those elements in which the atomic ions have the character of noble gases with a completed electron shell. The atmophilic elements include all the noble gases, \(\mathrm{H}_2\) and \(\mathrm{N}_2\).

13. Completion and Limitation of the System.

The refinement of X-ray spectroscopy and the detailed study of the X-ray spectra of the elements made possible the discovery and isolation of element \(Z\ 72\), hafnium. Already in 1913 Rydberg had ascribed normal tetravalence to the element preceding gallium, in contrast to the other rare earths (not counting Ce and Pr). Kossel held the same view. But only Bohr, in connection with his theory of the periodic system, defended this assertion with all his energy and gave impetus to the experimental work of Hevesy and Coster, which ended in confirmation of Bohr’s views.

In 1925 the missing homologues of manganese were discovered—masurium (\(Z\ 73\)) and rhenium (\(Z\ 75\)). The discovery was made by O. Berg, W. Noddack, and I. Tacke as the result of a very energetic investigation of the question, after a 1000-fold enrichment of the ore. Finally, quite recently (1926), the long and vainly sought element \(Z\ 61\)—illinium—was discovered.

At the end of the periodic system there is a series of radioelements, beginning immediately after Tl \(Z\ 81\) and extending to uranium \(Z\ 92\). The question arises whether, among the elements Po \(Z\ 84\), Em \(Z\ 86\), Ra \(Z\ 88\), Ac \(Z\ 89\), Pa \(Z\ 91\) (and among the still unknown—eka-iodine \(Z\ 85\) and eka-caesium \(Z\ 87\)), of which only more or less rapidly disintegrating isotopes are so far known, there do not exist more stable isotopes. In connection with the view expressed by Rutherford, according to which radioactivity is based on “excited” states of the nucleus, the posing of such a question seems quite appropriate. The possibility of the existence of “isotopes of higher orders” was already discussed in 1918 by Stef. Meyer, who considered cases in which nuclei, with the same charge and the same atomic weight, may possess different arrangements of structural units and, as a consequence of this, different stability.

As early as 1914 the author pointed out that such isotopes of higher orders could explain the change in the rate of decay in \(\alpha\)- and \(\beta\)-transformations of nuclei possessing the same charge but different mass. He thereby obtained a rule which is a modification and generalization of the rule previously established by Fajans. This rule consists in the fact that, for \(\alpha\)-emitters with a given nuclear charge, the rate of transformation, with decreasing atomic weight, at first increases to a certain maximum and then falls very rapidly.

(Example: Po isotopes, \(Z\ 84\).) If one now constructs such curves for different groups of elements, then the envelope of all these curves, passing through their maxima, has the same character as the curve constructed for the same atomic number and different atomic weights. For \(\beta\)-emitters the curves have the opposite course.

This rule of the author makes it possible to predict, for unknown isotopes, at a given nuclear charge, the probability of their decay with the emission of \(\alpha\)- or \(\beta\)-rays. It may be thought that this rule is not limited to the known radioelements enclosed between Tl and U, but that it can also be traced for smaller atomic weights, proceeding from the idea of the radioactivity of all elements. At the same time it makes it possible to make predictions concerning the rate of decay of elements which might be situated beyond uranium and which might be called “trans-uraniums.”

Thus the author came to the conclusion that both eka-iodine and eka-caesium must have radioactive isotopes in the form of \(\alpha\)-emitters with a very short decay period. Widdowson and Russell (1924) expected that both these elements should be radioactive descendants of the Pa isotope with atomic weight \(A = 233\), and that \(Z\ 87\) should have atomic weight \(A = 221\), while \(Z\ 85\), \(A = 215\), and both should be \(\alpha\)-emitters. According to Goldschmidt, the monovalent ion of eka-iodine should be lithophilic, whereas the nonionic compounds of eka-iodine are chalcophilic; eka-caesium, according to Goldschmidt, should be lithophilic to a high degree.

The series of known elements breaks off at uranium, \(Z\ 92\). Do “trans-uraniums” in fact exist? This break in the periodic system at uranium could be explained by three causes: 1. The instability of the corresponding nuclei. 2. The instability of the corresponding atoms, caused by the interaction between the nucleus and the electron surrounding it: as a result of this interaction the electrons fall onto the nucleus and do not allow the nuclear charge to increase. 3. The absence of these trans-uraniums in the rocks accessible to us, caused by the electronic (chemical) properties of the atom. All these possibilities have already been subjected to discussion.

With regard to the instability of the nucleus, a conclusion may be drawn on the basis of the author’s rule mentioned above. It turns out that the elements immediately following \(Z\ 92\) must have a short lifetime. Elements with a long lifetime should occur only between \(Z\ 98\) and \(Z\ 102\), and then again at \(Z\ 108\) and \(Z\ 110\); in the intervals there should be rapidly decaying radioactive elements.

Instability caused by the electrons surrounding the nucleus, in the first years of the existence of the Rutherford–Bohr theory, was not taken into account at all. Only after Schrödinger

and Bohr introduced the concept of orbits with a large eccentricity (Tauchbahnen), i.e. of such orbits of peripheral electrons along which the electron, in part of its path, comes close to the inner orbits,—only after this did they begin to take account of the above-mentioned possibility of instability. Rosseland brought in these orbits to explain the radioactivity of elements with large \(Z\) and to explain the boundedness of the periodic system. Bohr himself, soon after Rosseland, pointed out that the electron must fall onto the nucleus if the quantum numbers of its orbit satisfy the relation

\[ \frac{Z_{eff}}{k}=\frac{hc}{2\pi e^2}=137, \]

i.e. if the smallest value of \(k\) is 1, then the boundary of the periodic system lies at \(Z_{eff}=137\); but if \(k_{mn}=\frac{1}{2}\) (see Table 3), then \(Z_{eff}=68.5\), assuming the admissibility of such simplified calculations. Sommerfeld (1924) rightly pointed out that the problem of the periodic system will be solved only when not only the length of the individual periods is explained, but also the boundary of the entire system.

The third possibility is connected with the physicochemical properties of the elements following U. V. M. Goldschmidt, who expressed the opinion (1924) that the elements \(Z\,94\), \(Z\,95\), and \(Z\,98\) should be higher homologues of the platinum group. He proposed for these elements the name “neptunium group” and believed that, even with contemporary methods and means, one could find at least \(Z\,94\) and \(Z\,96\) in the platinum series or in iridous osmium, assuming that these elements are sufficiently long-lived. At the same time, according to Goldschmidt, the periodic system ends, only in an apparently seeming way, at \(Z\,92\), for this is the last of the known lithophile elements, while the following elements are undoubtedly definitely siderophile. Only the element \(Z\,119\), whose existence is, of course, highly problematic, should again be typically lithophile. These assumptions about the possible geochemical features of as-yet unknown elements are a simple extrapolation along Goldschmidt’s curve of atomic volumes (see Fig. 5), on the assumption of a known homology of peripheral electronic (especially chemical) properties.

14. Properties of the trans-uraniums.

In order to proceed to the search for those of the trans-uraniums which presumably possess a sufficiently long lifetime, it is necessary to try to draw a picture of their possible properties. These properties, above all, depend in the trans-uraniums on the length of the seventh period, following emanation (\(Z\,86\)). If, as the author assumes, this series has 18 elements, then the trans-uraniums—as

shows already at a glance at Table 14—must be the higher homologues of the elements of the period Cs—Em. In that case Eka Em will be \(Z\ 104\).

If, however, this seventh period contains not 18 but 32 elements, as Rydberg (1913) and later Bohr (1922) already supposed, then the eka-emanation must already be \(Z\ 118\), and in this series there must somewhere be room for 14 elements, analogous to the 14 rare-earth elements following lanthanum, which are very similar to one another chemically. It would be convenient to combine this group of 14 elements and give it a common name by adding the ending “-ides” to the name of the element immediately preceding it; it is precisely in this way that Goldschmidt (1925) constructed the name he proposed for the rare earths—“lanthanides.” Bohr admits in his theory of the periodic system a similar attachment of a group of 14 electrons shortly after uranium, but not immediately at this element, rather somewhere between \(Z\ 94\) and \(Z\ 107\). Bohr gives no arguments in favor of this supposition. It is therefore necessary to examine in more detail the attachment of such a group. The elements themselves—since the place where their attachment begins is not yet sufficiently clear—we shall call “trans-uranides,” without thereby prejudging the question of whether their development does not begin even before uranium.

From the point of view of the quantum theory of the periodic system set forth above, this “attachment of trans-uranides” means the filling of the electron group \(O_{\mathrm{VI,VII}}\) with \(n=5\) and \(k=4\). The appended Table 15 gives the arrangement of electrons for the two possibilities mentioned: Eka Em \(Z\ 104\) or \(Z\ 118\).

TABLE 15.

	\(O_{\mathrm{I}}\)	\(O_{\mathrm{II}}\)	\(O_{\mathrm{III}}\)	\(O_{\mathrm{IV}}\)	\(O_{\mathrm{V}}\)	\(O_{\mathrm{VI}}\)	\(O_{\mathrm{VII}}\)	\(P_{\mathrm{I}}\)	\(P_{\mathrm{II}}\)	\(P_{\mathrm{III}}\)	\(P_{\mathrm{IV}}\)	\(P_{\mathrm{V}}\ldots\)	\(Q_{\mathrm{I}}\)	\(Q_{\mathrm{II}}\)	\(Q_{\mathrm{III}}\)
				\(5\)						\(6\)				\(7\)
	\(1\)		\(2\)		\(3\)		\(4\)	\(1\)		\(2\)		\(3\)	\(1\)	\(2\)
\(Z\ 104\)	2	2	4	4	6			2	2	4	4	6	2	2	4
\(Z\ 118\)	2	2	4	4	6	6	8	2	2	4	4	6	2	2	4

In other places of the periodic system, this attachment of electrons for the filling of an intermediate group occurs after the third element of each period: Sc—Ti; Y—Zr, La—Ce, Ac—Th. Here, in the case of the uranium period, provided Rydberg’s rule is valid, there is a delay in the filling of the not yet completely occupied subgroups \(O_{\mathrm{VI,VII}}\). Perhaps this is expressed in the “aftereffect” of the influence of the atomic volume of the elements following the rare earths, caused by the filling of the subgroups \(N_{\mathrm{VI,VII}}\)

PERIODIC SYSTEM OF THE CHEMICAL ELEMENTS

in these elements. With complete homology of the elements between Eka Ta and Eka Em to the series of elements from Ta to Em, for the former the following distribution of electrons over the subgroups P and Q is obtained (cf. Table 16).

At the same time, although this table corresponds to both distributions of the electrons of Eka Em indicated in Table 15, the filling of the group with fourteen electrons, i.e. of the subgroups \(O_{\mathrm{VI,VII}}\), must proceed in accordance with Table 17, where the 14 trans-uranides are denoted by the letters of the Greek alphabet from \(\alpha\) to \(\xi\). The variable valence already of the series of elements from Ta to Eka V permits one to expect the same for their higher homologues as well, as Table 16 also shows. The same is true also for the trans-uranides corresponding to these elements; among them some, such as \(\alpha\), \(\beta\), \(\varepsilon\), \(\zeta\), and \(\delta\), may also exhibit other valences. For the rare earths, generally speaking, distinguished by constant trivalence, other valences appear only in the first elements, where the filling of the subgroup begins, and in the last elements, corresponding to the final filling of the first subgroup. Namely: Ce and Tb may also be tetravalent, Sm and Eu divalent, Pr tetravalent and pentavalent. Therefore the filling of the fourteen-electron group under consideration may perhaps already begin after Th (\(Z\ 92\)) or after Pa (\(Z\ 93\)), i.e. Pa and U may be the first thorides with a valence differing from Th, while uranium could be the first protactinide with a valence differing from Pa. The available data concerning the structure of U are insufficient for making a choice between these possibilities.

If we now turn to the possible geochemical features of the trans-uranes, then the assumption of Goldschmidt set out above concerning the elements \(Z\ 94\) and \(Z\ 96\), as a group of ekaplatinum, from our point of view will appear to be one of several possibilities. If U is a thoride or a protactinide (which, however, presupposes a change in the quantum numbers of the electron groups indicated in Table 13), or if the trans-uranides are in fact uranides, then the series of lithophile elements extends no farther than \(Z\ 106\), assuming sufficient stability of the elements. If, however, the trans-uranides begin between Tri Mn and Eka Pt., then the series of siderophile elements begins no earlier than \(Z\ 93\) and ends no farther than \(Z\ 110\). Finally, if the trans-uranides begin only after Eka Au, then the beginning of the chalcophile elements lies already at \(Z\ 97\), and the end can lie only at \(Z\ 117\); dvi-iodine could already be clearly metallic and chalcophile. In investigations of rocks for trans-uranes one should keep in mind the points of view set out here. It is possible that the partly lithophile ekaplatinum character of Tri Mn, and also of the group (to a considerably lesser degree) in comparison with Dvi Mn and the platinum group before the addition of the trans-uranides is due to an increase in electropositivity.

TABLE 16.

\(Z\)		\(P_{\mathrm I}\)	\(P_{\mathrm{II}}\)	\(P_{\mathrm{III}}\)	\(P_{\mathrm{IV}}\)	\(P_{\mathrm V}\ldots P_{\mathrm{XI}}\)	\(Q_{\mathrm I}\)	\(Q_{\mathrm{II}}\)	\(Q_{\mathrm{III}}\ldots\)
	\(n\)				6			7
	\(k\)	1	2		3…	6	1	2…	7
87	Eka Cs	2	2	4			1
88	Ra	2	2	4			2
89	Ac	2	2	4	(1)		(2)
90	Th	2	2	4	(2)		(2)
	Eka Ta	2	2	4	(3)		(2)
	Eka W	2	2	4	(4)		(2)
	Tri Mn	2	2	4	4	1	(2)
	Eka Os	2	2	4	4	(2)	(2)
	Eka Ir	2	2	4	4	(3)	(2)
	Eka Pt	2	2	4	4	(4)	(2)
	Eka Au	2	2	4	4	6	(1)
	Eka Hg	2	2	4	4	6	(2)
	Eka Tl	2	2	4	4	6	2	1
	Eka Pb	2	2	4	4	6	2	2
	Eka Bi	2	2	4	4	6	2	2	1
	Eka Po	2	2	4	4	6	2	2	2
	Dwi I	2	2	4	4	6	2	2	3
	Eka Em	2	2	4	4	6	2	2	4

TABLE 17.

	\(O_{\mathrm I}\)	\(O_{\mathrm{II}}\)	\(O_{\mathrm{III}}\)	\(O_{\mathrm{IV}}\)	\(O_{\mathrm V}\)	\(O_{\mathrm{VI}}\)	\(O_{\mathrm{VII}}\)
\(n\)				5
\(k\)	1	2		3		4
	2	2	4	4	6
\(\alpha\)	2	2	4	4	6	1
\(\beta\)	2	2	4	4	6	2
\(\gamma\)	2	2	4	4	6	3
\(\delta\)	2	2	4	4	6	4
\(\varepsilon\)	2	2	4	4	6	5
\(\zeta\)	2	2	4	4	6	(5)	(1)
\(\eta\)	2	2	4	4	6	6	1
\(\vartheta\)	2	2	4	4	6	6	2
\(\iota\)	2	2	4	4	6	6	3
\(\kappa\)	2	2	4	4	6	6	4
\(\lambda\)	2	2	4	4	6	6	5
\(\mu\)	2	2	4	4	6	6	6
\(\nu\)	2	2	4	4	6	6	7
\(\xi\)	2	2	4	4	6	6	8
	2	2	4	4	6	6	8

From the preceding considerations it follows that the elements \(Z\ 107\)—\(Z\ 110\) must, in any case, be siderophile. This could fail to be the only case if the series of trans-uranides began only after Eka-Au, which, however, is hardly probable, since this would mean a displacement of the group of fourteen elements by at least seven places. Further, the extrapolation carried out by the author shows that in the region between \(Z\ 108\) and \(Z\ 110\) one should expect the existence of stable elements which, as was shown, could occur in the earth’s core and in iron meteorites. Several years ago (1919), in connection with the hypothesis he had advanced concerning the origin of “penetrating radiation,” the author turned to the study of the so-called “polar dust.” The latter is dust with a high iron content, found by Nordenskiöld on the Greenland ice, to which Nordenskiöld attributed a cosmic origin. Thanks to the kindness of the Allied trans-Greenland expedition, it was possible to examine one of the collected samples both for radioactivity and by X-ray methods. The investigation gave an indication of the presence of substances volatile on red heat and possessing a weak radioactivity varying with time, which made it possible to infer the presence of at least three elements. The X-ray investigation gave an uncertain indication of the presence of \(Z\ 108\). Owing to the insufficient quantity of material, these conclusions could not be confirmed; nevertheless they—especially the latter, from the point of view of the theoretical premises set forth—are by no means impossible. It is to be hoped that improved methods of investigation will settle the question of the existence of trans-uraniums, and at the same time will shed new light on the question of the limits of the periodic system of the elements.

LITERATURE.

§§ 1 and 2. G. Rudorf. Das periodische System. Hamburg and Leipzig, 1904.

W. Ostwald. Leitlinien der Chemie. Leipzig, 1906. Das natürliche System der chemischen Elemente. Ostwalds Klassiker Nr. 68. 2. Aufl. Leipzig, 1913.

Julius Thomsen. Zeitschr. f. anorg. Chem. 9 (18. 5), 190/3, 283, 8.

§ 3. K. Fajans. Radioaktivität und die neueste Entwickelung der Lehre von den chemischen Elementen. 4. Aufl. Braunschweig, 1922.

R. Swinne. Die Naturwiss. 8. 727—34. (1920.)

J. R. Rydberg. Zeitschr. f. anorg. Chem. 14. 66—102. (1897.) Lunds Universitäts Årskrift, N. F. Abt. II, 9. Nr. 18 (1913).

§ 4. N. Bohr. Abhandlungen über Atombau, Braunschweig, 1921.

N. Bohr. Drei Aufsätze über Spektren und Atombau. 2. Auflage. Braunschweig, 1924.

N. Bohr. Über den Bau der Atome. 2. Aufl. Berlin, 1924.

W. Kossel. Valenzkräfte und Röntgenspektren. 2. Aufl. Berlin, 1924.

A. Landé. Fortschritte der Quantentheorie. Dresden and Leipzig, 1922. 2. Aufl. 1926.

A. Sommerfeld. Atombau und Spektrallinien. 4. Aufl. Braunschweig, 1924.

E. Back and A. Landé. Zeemaneffekt und Multiplettstruktur der Spektrallinien. Berlin, 1925.

§ 5. A. Pannekoek. Bull. Astron. Inst. Netherlands. No. 21, II, VII, 1922.

§ 6. A. Landé. Ergebnisse d. exakt. Naturwiss. 2 (1923), 147—162. Die Naturwiss. 13, 604—606.

W. Gerlach and O. Stern. Zeitschr. f. Phys. 8, 40 (1021), 9, 349 (1922).

§ 7. W. Kossel. Ann. d. Phys. (4) 49, 229—362 (1916).

R. Ladenburg. Die Naturwiss. 8, 5—11 (1920). Zeitschr. f. Elektrochem. 26, 262—274 (1920).

R. Swinne. Phys. Zeitschr. 17, 481—4?8 (1916).

L. Vegard. Verh. d. D. Phys. Ges. 19, 344—353 (1917).

E. C. Stoner. Phil. Mag. (6) 48, 719—736 (1924).

W. Pauli, jr. Zeitschr. f. Phys. 31, 765—783 (1925).

§ 8. J. D. Main Smith. Chemistry and Atomic Structure. London, 1924, especially Chaps. XIII and XIV, Tables 12 to 16.

§ 11. R. Swinne. Zeitschr. f. Elektrochem. 31 (1925). Wiss. Veröff. d. Siemens-Konzerns. 5, No. 1.

F. Hund. Zeitschr. f. Phys. 33, 345—371, 385—889 (1925).

F. Paneth. Ergebn. d. exakt. Naturwiss. 2, 173 (1923).

§ 12. R. Swinne. Die Naturwiss. 7, 529?0 (1919).

V. M. Goldschmidt. Zeitschr. f. Elektrochem. 28, 411—421 (1922). Die Naturwiss. 10, 918—920 (1922). Geochemische Verteilungssätze der Elemente, 1; Vid. Selsk. Skriftes 1 Mat.-nat. Kl. No. 3, Kristiania, 1923; II ib., No. 4, 1924; III (with L. Thomassen), ib., No. 5, 1924; IV (with F. Ulrich and T. Barth), Skrifter Norske Vid. Academie, 1 Mat.-nat. Kl. No. 5, Oslo 1925; V (with T. Barth and G. Lunde), ib., No. 5, 1925.

G. Tamman. Zeitschr. f. anorg. u. allg. Chemie 13, 96—100 (1923); 134, 263—276 (1924).

G. Berg. Zeitschr. f. angew. Chem. 37, 352—355 (1924).

§ 13. G. v. Hevesy. Recherches sur les propriétés du Hafnium. Danske Vid. Selsk. Math.-phys. Medd. VI, 7, 1924.

W. Noddack, J. Tacke and O. Berg. Sitzber. d. Preuss. Acad. d. Wiss. Berlin, 1925, 400—409; Die Naturwiss. 1, 567—401 (1925).

R. Swinne. Zeitschr. f. techn. Phys. 6, 464/5 (1925).

P. Günther and Stranski. Zeitschr. f. phys. Chem. 106, 433—441 (1923).

W. Prandtl and A. Grimm. Zeitschr. f. anorg. u. allg. Chem. 136, 283—288 (1924).

E. Rutherford. J. Roy. Soc. Arts. 73, 389—402; Ref. Phys. Abstr. 1925, No. 1677.

St. Meyer. Sitzber. Acad. Wien. VI a, 127, 1283 (1918); Zeitschr. f. phys. Chem. 95, 417 (1920).

R. Swinne. Lecture, 19. VI. 1914, Chem. Ges. Heidelberg. Ref. Zeitschr. f. angew. Chem. 27, 596, VII (1914).

K. Fajans. Le Radium. 10, 171 (1913); Phys. Zeitschr. 14, 950 (1913).

W. P. Widdowson and A. Russell. Phil. Mag. (6) 48, 293—306 (1924).

S. Rosseland. Nature. 111, 387 (1923).

N. Bohr. Ann. d. Phys. (4) 71, 266 (1921); Die Naturwiss. 11, 620 (1923).

A. Sommerfeld. Atombau und Spektrallinien. 4th ed., pp. 465—470, 1924.

§ 14. N. Bohr. Drei Aufsätze über Spektren und Atombau. 2nd ed. Braunschweig, 1924, pp. 126—127.

V. M. Goldschmidt. Geochem. Verteilungsgesetze. II, pp. 23, 28, 1924; see pp. 10, 53—54, 1925.

A. E. Nordenskjöld. Öfvers. Vet. Ak. Förhandling. Stockholm 1870, No. 10; 1888, No. 4.

R. Swinne. Die Naturwiss. 7, 529—530 (1919).

1. Cf. the articles by I. Tamm, “Magnetism and the structure of atoms,” Uspekhi Fizicheskikh Nauk, vol. V, nos. 1–2, and by N. N. Semenov, “Molecular beam,” ibid. — Ed. ↩↩↩↩↩