THE PERIODIC SYSTEM, CHEMICAL BONDS, AND CRYSTAL STRUCTURE1

A. Sommerfeld.

For Bohr’s theory of the periodic system2, the inequality of the subdivisions of the atomic electron shells \(K, L, M, N\ldots\), discovered by Stoner and Main Smith, is of great importance; in Bohr’s theory these subdivisions had been taken to be equal. The details of the new scheme are given in the table, which indicates the subgroups of the shells together with the numbers of electrons in each of them.

Table of X-ray levels.

\(K\)—a single shell, \(L\)—triple, \(M\)—quintuple, etc., in agreement with the data obtained from X-ray spectra (three absorption quanta \(L\), etc.). Of course, the principal quantum number \(n\) is 1 for the \(K\)-shell, 2 for \(L\), etc.

In the upper row of the table are placed the designations of the principal quantum numbers which we consider necessary for the classification of X-ray spectra. In addition to the principal quantum number \(n\), two subordinate quantum numbers are indicated in the form of subscripts. They are denoted by the symbols \(k_1\) and \(k_2\) or \(k\) and \(j\), where \(k_1=k\), and \(k_2\)

\[ = j+\frac{1}{2}; \]

\(j\) is the so-called inner quantum number.

All the numbers in the table are equal to \(2k_2 = 2j+1\)¹). But \(2j+1\) is the quantum weight, or the number of orientations of the angular momentum \(j\) in a magnetic field. Therefore Manne, Smith, and Stoner equate the number of electrons in a given level to the quantum weight \(2j+1\) of this level.

In the lower row of the table are given the numbers \(n_k\), indicating the type of orbits. We must pay special attention to the fact that each of the pairs of shells forming a “relativistic doublet” in the X-ray spectrum belongs to one and the same orbital type. This circumstance completely contradicts the original point of view on the basis of which the formula for the relativistic doublet was derived.

Replacing \(n_k\) by \(n_{kj}\), we encounter in the domain of X-ray spectra the same problem as for visible spectra. The orbit of one electron would seem to have to be completely determined by two quantum numbers \(n\) and \(k\); the third number \(m\) gives the orientation of the orbit in space; accordingly, for the three degrees of freedom of the rotating electron we should have in all three quantum numbers. But already in the case of hydrogen, in addition to \(n\) and \(k\), a further \(j\) is required to characterize the orbit itself; for a complete description of the orbit, including its position in space, four quantum numbers are required. It is not surprising that the same problem also arises for X-ray spectra; to describe one orbit here as well, \(n_{kj}\) are needed, and not \(n_k\). Can one hope that the resolution of the difficulty will be found within the framework of the new quantum mechanics proposed by Heisenberg and developed, for example, by Dirac?

The new theory frees us from the old difficulties with half-quantum numbers and with the various details of the anomalous Zeeman effect, but it is hardly able to elucidate the new degree of freedom for the orbital electron. In all probability, a new hypothesis will have to be introduced into the Hamiltonian function of the system. In this connection we must point to the interesting proposal of Goudsmit concerning the electron-top (a hypothesis in some respects expressed even earlier by Parson), though without a definite connection with quan-

¹) The systems of quantum numbers used by different authors differ somewhat from one another. This explains the apparent discrepancy between the number of possible orientations given by Sommerfeld and that indicated in the preceding article by Swinne. — Ed.

tovye numbers); perhaps here also lies the explanation of the missing degree of freedom1.

The number \(j\) of an individual electron must not be confused with the \(j\) of the whole atom, whose outer shell, generally speaking, consists of several orbits of the type under consideration. We shall denote by \(\bar j\) the \(j\) belonging to an atom, placing a bar above it; moreover, \(\bar j\) refers to the atom in the basic, unexcited state. In other words, \(\bar j\) is the inner quantum number of the basic term. In 1925, in Physikalische Zeitschrift, I formulated two theorems concerning \(\bar j\) of the basic term and its relation to the position of the element in the periodic system:

1. Every completed subgroup is characterized by the fact that \(\bar j = 0\).
2. The element immediately following or preceding an element with a completed subgroup has a value of \(\bar j\) identical with the \(j\) of the subgroup to which the element belongs.

The following scheme gives examples of these theorems.

Roman numerals indicate the number of electrons in the outer shell of a given series in the various cases. The counting begins from the initial stage of the shell and is applied in the same way to the \(N\)-, \(M\)-, and \(L\)-shells, if they contain a sufficient number of electrons.

I can touch only very briefly on the spectroscopic evidence for this scheme and must omit all details: \(\bar j=0\) in column II has been found for helium and the alkaline earths; in column IV, for tin and lead and, probably, also for ionized nitrogen (\(N^+\)). In column VIII \(\bar j=0\) has been established for neon; in column XII this is doubtful, and column XVIII (nickel, palladium, platinum) we shall consider below; \(\bar j=\frac{1}{2}\) in column I denotes the \(s\)-term of the alkalis, and in column III the \({}^{2}P_1\)-term of aluminum, gallium, indium, and thallium, etc., for others.

Moreover, it may undoubtedly be expected that the azimuthal quantum number of the electron bound last will be the same as \(k\)

of the periodic system, i.e., coincides with the first index in \(n_{kj}\). On this basis, in spectroscopic terms, the theory of the periodic system is Bohr’s theory. But this \(k\), in the case of atoms with several valence electrons, differs from the “group quantum number” \(l\), introduced (though with another notation) in the work of Russell and Saunders for the alkaline earths. The apparent exceptions to Bohr’s scheme, for example for iron and titanium, are based on a mixing of \(k\) and \(l\), as was first indicated by Russell and Saunders.

We now turn to chemical applications.

The fundamental principle of chemical bonding may be expressed as a striving toward the completion of a subgroup. A completed group of 8 electrons has long been familiar, and it is well known how the simplest binary compounds approach such a configuration of the inert gases from both sides. Next in importance is the “paired shell” (two-shell), of which helium is the typical example; here two electrons rotate in opposite directions, so that their moments mutually cancel. An example may be furnished by \( \mathrm{LiH} = \mathrm{Li}^{+}\mathrm{H}^{-} = \mathrm{He}_{3}\mathrm{He}_{1} \)1. The elements zinc, cadmium, and mercury belong to this same class; their two outer electrons are bound in a similar way. We know many very stable compounds that approach this configuration. I shall point only to a few, for example, \(\mathrm{PbO}\) (or \(\mathrm{PbS}\)), in which lead is divalent; in other words, lead here acquires the “paired shell” of mercury, giving two electrons to oxygen or sulfur. Indeed, \(\mathrm{PbO}\) is more stable than \(\mathrm{PbO}_{2}\), although in the latter compound lead has, as is said, the “regular” valence. One may also point to compounds of monovalent thallium and of trivalent arsenic, antimony, and bismuth, in which there is a tendency toward a paired shell of the helium type. Thus, the work of Main Smith and Stoner will undoubtedly lead chemists in the future to the conclusion that, in speculations about chemical affinity, one must take into account not only the octuple shell, but, with equal right, also the paired shell.

Let us now consider the so-called eighteen-electron shell, which is assumed to be completed at the ends of the triads by nickel, palladium, and platinum. There is no doubt that the elements copper, zinc, silver, cadmium, gold, and mercury, in their chemical compounds, often figure with eighteen-electron shells. But is this shell actually completed, like the shell of an inert gas? What are the spectroscopic data on this question? For palladium the situation is clear and indisputable: the fundamental term here is \({}^{1}S_{0}\), a term lying considerably below the other levels. Thus, palladium has a completed

PERIODIC SYSTEM

shell. The situation is otherwise with respect to nickel or platinum. In nickel the fundamental term is \({}^3F\), with the term \({}^3D'\) lying somewhat higher; in platinum the fundamental term is \({}^3D'\). This is reflected in the chemical and spectroscopic properties of the following elements: copper, silver, and gold. Silver in all cases, without exception, is monovalent; when its single valence electron is removed it passes into the stable configuration of palladium. Copper is mono- and divalent, gold mono- and trivalent. Consequently, here, besides the valence electron, one or more electrons may be removed from the inner parts of the atom. The inner part of the atom in copper and gold is not complete, which agrees with the properties of nickel and platinum.

The same result is obtained from the spectroscopic properties of these elements. Silver has a simple spectrum, similar to the alkali spectra, whereas in copper and gold, besides the system of alkali doublets, a large number of extraneous lines is found in the spectrum. It is very characteristic that Stücklein (Fr. Stücklein, ZS. f. Phyz. 34, 562, 1925) found for the most stable state of copper not the \(s\)-term of doublets, but one of the terms probably belonging to the system of quartets. To this exactly corresponds the incomplete shell of nickel. Thus, the position of an element in the periodic system gives us reliable keys to the understanding of its spectrum.

The properties of nickel and platinum and the closely connected properties of copper and gold must be regarded as a partial exception to our rule concerning the resulting moments of the moment \(j\); the condition \(j=0\) is a sign of a completed subgroup, but in certain cases the last electron, the one that has just completed the subgroup, may, under the conditions of energy, more easily adjust itself somewhere in another place.

We now turn to the final members of those subgroups which differ from the following subgroups by different values not of \(k\), but of \(j\); they belong to one and the same orbital type \(n_k\), as do the following elements. It is clear that a close separation is less distinct and less noticeable chemically than the completion of subgroups considered above. Let us first consider the difference between \(n_{43}\) and \(n_{44}\) (for \(n=4\)) in Table I; it occurs in the rare earths and appears in the difference between the cerium and ytterbium earths, especially in the magnetic respect. The curve of paramagnetism of the ions, according to Cabrera and S. Meyer (Cabrera, Stefan Meyer), rises to a maximum and then falls almost to zero at the end of the cerium earths, reaches an even greater maximum among the ytterbium earths, didymium and holmium, and finally falls to zero at cassiopeium (atomic number 71).

The completion of the subgroup \(n_{32}\) is not noticeable spectroscopically; on the contrary, the completion of \(n_{21}\) is revealed by the fact that \(j=0\) in tin and lead,

and, probably, also in carbon and silicon. But how does the completion of subgroups manifest itself chemically? Are there compounds that tend toward a fourth shell, just as there is a tendency toward eight- and eighteen-electron shells? The answer is of great importance for the understanding of chemical compounds, and I can point out the following in this connection, on the basis of Grimm’s (H. G. Grimm) observations concerning the structure of the crystals of certain diamond-like compounds (ZS. f. Phys., 1926).

The character of the structure of diamond is known from the work of Sir William Bragg. Diamond is simply built of tetrahedra in such a way that each carbon atom is surrounded by a tetrahedron of other carbon atoms. Zinc sulfide crystallizes in the same arrangement in the form of zinc blende: each zinc atom is surrounded by a tetrahedron of sulfur atoms, and conversely. But in wurtzite zinc sulfide crystallizes differently. Here there are still mutually interpenetrating tetrahedral systems, but the structure is hexagonal, not cubic; here the tetrahedra are arranged differently with respect to one another than in diamond. We classify both types as tetrahedral systems. The crystal of carborundum (CSi), which is of great technical importance, is likewise tetrahedral, with various modifications differing from one another in the manner of alternating diamond and wurtzite structures along the hexagonal \(C\)-axis. As a result of such alternation, the repeating distance here is unusually large—almost \(40 \overset{\circ}{A}\) (true, only in one of the modifications).

It is now necessary to draw attention to the following: such a tetrahedral structure is found not only in the 4th column of the periodic system (i.e., in carbon, silicon, carborundum, germanium, tin), but also in compounds of neighboring elements, with the neighbors being equally distant on both sides from the fourth column. Indeed, one can establish a definite theorem that the tetrahedral structure occurs only in those binary compounds whose two components are at most three cells removed from the four-electron shell, and the removals for both components must be equal.

        I.       II.       III.       IV.       V.       VI.       VII.

                 Be        Al         C         N        O

        Ag       Zn                   -Si                S         J

It is very natural to suppose that in all these compounds the mechanism of the bonds is the same as in diamond; it is clear that here the nonpolar

bond, since it joins two identical atoms; hence we conclude that in zinc sulfide, too, the bond is nonpolar, i.e. the compound does not have the form $\mathrm{Zn}^{++}\mathrm{S}^{--}$. This is confirmed by the measurement of the intensity of X-rays, which, according to Ott (H. Ott), definitely does not correspond to $\mathrm{Zn}^{++}$ and $\mathrm{S}^{--}$, but probably pertains to neutral Zn and S. The idea of a four-electron shell even suggests the opposite assumption, $\mathrm{Zn}^{--}$ and $\mathrm{S}^{++}$. The sulfur atom would have to give up two electrons in order to obtain the four-electron shell of silicon, while zinc would have to capture two electrons in order to resemble germanium with its fourfold shell. However, the intensity ratios rather point to neutral Zn and S. Similarly, beryllium oxide, unlike magnesium oxide and calcium oxide, is at the very least not a polar salt $\mathrm{Be}^{++}\mathrm{O}^{--}$. In CaO and $\mathrm{CaCO}_3$ the tetrahedral structure has not been observed, nor can it be expected theoretically, since at a distance of two cells from Ca there is no four-electron shell.

I prefer for the time being not to discuss the mechanism of mutual bonding in such four-electron shells; even in the simplest case—diamond—we know nothing about it. It is most probable that the bond is effected by pairs of electrons, each of which revolves around two carbon atoms, as is assumed in Lewis’s theory (G. N. Lewis) of shared bonding electrons.

In conclusion, one may say that the theory of the periodic system indicates in which elements the completion of subgroups may be expected. In addition to the eight-electron shell of the inert gases, there is a sixteen- and even a two-electron shell, toward which, as a goal, the elements strive in stable compounds. On the other hand, the tetrahedral structure of crystals testifies to the existence of a four-electron shell determining the combination of atoms of the same type as in diamond. This bond is effected not by electrostatic attractions of ions, but is probably created by neutral atoms. There is reason to hope that the great chemical problem of nonpolar bonds will come nearer to solution through further study of the tetrahedral structures of crystals.

1. \(\mathrm{He}_{3}\) denotes a helium configuration with a triple nuclear charge \((\mathrm{Li}^{+})\); \(\mathrm{He}_{1}\) corresponds to a helium configuration with the same nuclear charge. ↩↩↩

2. An exposition of this theory may be found in the books: 1) Bohr, Three Articles on Spectra and the Structure of Atoms. 2) Kramers and Holst, The Structure of the Atom and Bohr’s Theory. State Publishing House, 1926. 3) Sommerfeld, Atomic Structure and Spectra. State Publishing House, 1926. See also the accompanying article by Swinne. Translator’s note. ↩