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WHY DOES THE SYSTEM OF CHEMICAL ELEMENTS HAVE PERIOD LENGTHS OF 2, 8, 8, 18, 18, 32?1
A. Landé.
Since Kossel’s attempt to interpret the chemical properties of the elements on the basis of the arrangement of their electrons, the riddle of the periodic system—and, in particular, the question of the origin of periods of 2, 8, 8, 18, 18, and 32 elements—has become a special question of the quantum theory of atomic structure. At present this problem may be regarded as solved in its main outlines, especially thanks to the investigations of Bohr, Stoner, and Pauli, as well as on the basis of certain results concerning spectral lines (the structure of multiplets and the Zeeman effect). An astonishingly simple relation is found between the period numbers \(2 = 2 \cdot 1^2\), \(8 = 2 \cdot 2^2\), \(18 = 2 \cdot 3^2\), \(32 = 2 \cdot 4^2\), and the quantum numbers of the electrons bound in the atom—numbers that can be established from the spectra of the elements.
For these quantum numbers spectroscopists usually use the symbols \(n, K, J, m\). Let us recall their meaning from the point of view of the atomic model and show how they are related to the period numbers. We shall see that the selection restricting these quantum numbers:
\[ n = 1, 2, 3 \ldots \infty ; \quad J = K \pm \frac{1}{2} \tag{1} \]
\[ K \le n - \frac{1}{2} \qquad |m| \le J \cdot \frac{1}{2} \]
is equivalent to the law of the period numbers, so that (1) may be regarded as the “formula of the periodic system.”
\(n\). The orbit of each electron participating in the structure of the atom may, with a certain approximation, be regarded as an ellipse. According to Bohr, the choice of such ellipses is in reality restricted: namely, the major semiaxis \(a\) of the elliptical orbit can have only discrete values \(a_1, a_2, a_3, \ldots\); in other words, \(a\) can be equal only to \(a_n\), where
\[ n = 1, 2, 3 \ldots \infty \tag{2} \]
(More precisely, these chosen values \(a_n\) are given by the formula
\[ a_n = 0.53 \cdot \frac{n^2}{z'} \cdot 10^{-8}\ \text{cm}, \]
where \(z'\) denotes the “effective” charge of the nucleus, i.e. the charge whose action on the electron under consideration is equivalent to the combined action of the nucleus and the remaining negative electrons belonging to the atom, which “screen” the nucleus.) The number \(n\), which thus determines the magnitude of the semimajor axis of the ellipse \(a_n\), is called the principal quantum number of the electron orbit under consideration; in this way, orbits with all possible principal quantum numbers are possible.
The question arises how many electrons with principal quantum number \(n=1\) can be contained in an atom; likewise, how many electrons can have \(n\) equal to 2, \(n\) equal to 3, and so on. We shall now see that an atom can have at most 2 electrons with \(n=1\), no more than 8 electrons for which \(n=2\), no more than 8 with \(n=3\), no more than 18 with \(n=3\), and no more than 32 for which \(n=4\).
Thus far we have spoken of the quantum determination of only the major semiaxis of the ellipse \((a_1, a_2 \ldots a_n \ldots a\infty)\). But the minor semiaxis in each electron orbit also assumes (according to Sommerfeld) only definite quantum values, which we shall denote by \(b_k\). Just as the minor semiaxis of the ellipse \(b_k\) is always smaller than, in the limiting case equal to, its major semiaxis \(a_n\), so likewise the quantum number \(K\) is always smaller (in the limiting case equal) to the quantum number \(n\). The study of spectra shows, in particular, that \(K\) assumes the values
\[ K = \frac{1}{2},\ \frac{2}{3},\ \frac{5}{2}\ldots \left(n-\frac{1}{2}\right) \]
or, more briefly,
\[ K \leq n-\frac{1}{2}. \]
The value of the minor semiaxis \(b_k\) corresponding to the quantum number \(K\) is
\[ b_k = 0.53 \cdot \frac{K^2}{z'} \cdot 10^{-8}\ \text{cm} \]
(compare above the formula for \(a_n\)).
Therefore, for the individual principal quantum numbers \(n=1,2,3\ldots\), and so on, the following quantum numbers \(K\) of the elliptical orbit are possible:
\[ \begin{array}{ccc} n=1 & n=2 & n=3 \\[4pt] K=\frac{1}{2} & K=\frac{1}{2};\ \frac{3}{2} & K=\frac{1}{2};\ \frac{3}{2};\ \frac{5}{2}\ \text{and so on.} \end{array} \tag{3} \]
\(K\) is called the “subsidiary” or “azimuthal” quantum number, for it measures the azimuthal angular momentum of the rotation under consideration
the electron together with its orbit; namely, this component of the momentum is equal to \(\dfrac{K\cdot h}{2\pi}\), where \(h\) is Planck’s constant1.
The quantum numbers \(n\) and \(K\) determine the form of the allowed elliptical orbit, i.e. the value of the major and minor semiaxes \(a_n\) and \(b_k\), as well as the angular momentum of the electron \(\dfrac{K\cdot h}{2\pi}\) in the plane of the orbit. However, the plane of the electron’s orbit does not remain fixed in space, but performs a precessional motion (like the plane of rotation of a gyroscope). Thus, in addition to the angular momentum \(\dfrac{K\cdot h}{2\pi}\), corresponding to the rotation of the precessing orbit in its plane, there also acts an angular momentum \(\dfrac{J\cdot h}{2\pi}\), corresponding to the total motion, including the precessional one.
Here \(J\) denotes the “effective” quantum number of the electronic orbit, for this angular momentum too is subject to the quantum law, that is, it can act2 only in the form of a definite value selected in a quantum manner; investigations of complex spectra show that the effective quantum number is connected by the condition:
\[ J = K + \frac{1}{2} \qquad J = K - \frac{1}{2} \]
Therefore, for a given form of the elliptical orbit (cf. 3), there are still the following possibilities, determined by the action of the momentum
rotation of the orbit, specified by the acting quantum number \(J\) (which, moreover, cannot be equal to zero):
\[ \begin{array}{ccc} n=1 & n=2 & n=3\\[4pt] K=\dfrac{1}{2} & K=\dfrac{1}{2}\ \bigg|\ \dfrac{3}{2} & K=\dfrac{1}{2}\ \bigg|\ \dfrac{3}{2}\ \bigg|\ \dfrac{5}{2}\\[6pt] J=1 & J=1\ \bigg|\ 1,2 & J=1\ \bigg|\ 1,2\ \bigg|\ 2,3 \end{array} \tag{4} \]
\(m\): The axis of rotation (the axis of the top), with respect to which the angular momentum of rotation of the electronic orbit \(\dfrac{J\cdot h}{2\pi}\) acts (we shall call it the \(J\)-axis), may itself have various directions in space. If in space one can single out a definite preferred direction (for example, when a strong magnetic field is switched on, the direction of the lines of force will be the chosen direction in space), then the \(J\)-axis may make definite angles with this direction. \(\dfrac{Jh}{2\pi}\) is the angular momentum acting about the \(J\)-axis. With respect to the chosen direction, regarded as an axis inclined to the \(J\)-axis, there will act an angular momentum which we shall call \(\dfrac{m\cdot h}{2\pi}\), and which in absolute value must be smaller than, in the limiting case equal to, the angular momentum with respect to the \(J\)-axis. Spectroscopic investigations of the Zeeman effect further showed that
\[ |m|\leq J-\frac{1}{2} \]
or, in more detail:
\[ m=J-\frac{1}{2};\quad J-\frac{3}{2};\quad J-\frac{5}{2};\quad -\left(J-\frac{3}{2}\right);\quad -\left(J-\frac{1}{2}\right) \]
(negative values of \(m\) mean that the angle between the positive directions of the \(J\)-axis and the positive direction of the lines of force exceeds \(90^\circ\)). Thus an electronic orbit, characterized by the quantum numbers \(n, K, J\), may assume in space (for example, with respect to a strong magnetic field) the following additional positions, determined by the quantum number \(m\) (formula (5), compare also (2), (3), (4); see the table on p. 393).
In the last line of formula (5) is listed how many possibilities an orbit characterized by the quantum number \(n\) (with major semiaxis \(a_n\)) has, if one takes into account its minor semiaxis (quantum number \(K\)), its acting angular momentum (quantum number \(J\)), and its position in space (quantum number \(m\)). We see that an electronic orbit with \(n=1\) has altogether 2 possib-
| \(n\) | \(K\) | \(J\) | \(m\) | Number of possibilities |
|---|---|---|---|---|
| 1 | \(\frac12\) | 1 | \(\pm \frac12\) | \(2 = 2\cdot 1^2\) |
| 2 | \(\frac12\) | 1 | \(\pm \frac12\) | \(2\) |
| 2 | \(\frac32\) | 1 | \(\pm \frac12\) | \(2\) |
| 2 | \(\frac32\) | 2 | \(\pm \frac12,\ \pm \frac32\) | \(4\) |
| 2 | \(= 8 = 2\cdot 2^2\) | |||
| 3 | \(\frac12\) | 1 | \(\pm \frac12\) | \(2\) |
| 3 | \(\frac32\) | 1 | \(\pm \frac12\) | \(2\) |
| 3 | \(\frac32\) | 2 | \(\pm \frac12,\ \pm \frac32\) | \(4\) |
| 3 | \(\frac52\) | 2 | \(\pm \frac12,\ \pm \frac32\) | \(4\) |
| 3 | \(\frac52\) | 3 | \(\pm \frac12,\ \pm \frac32,\ \pm \frac52\) | \(6\) |
| 3 | \(= 18 = 2\cdot 3^2\) | |||
| 4 | \(\frac12\) | 1 | \(\pm \frac12\) | \(2\) |
| 4 | \(\frac32\) | 1 | \(\pm \frac12\) | \(2\) |
| 4 | \(\frac32\) | 2 | \(\pm \frac12,\ \pm \frac32\) | \(4\) |
| 4 | \(\frac52\) | 2 | \(\pm \frac12,\ \pm \frac32\) | \(4\) |
| 4 | \(\frac52\) | 3 | \(\pm \frac12,\ \pm \frac32,\ \pm \frac52\) | \(6\) |
| 4 | \(\frac72\) | 3 | \(\pm \frac12,\ \pm \frac32,\ \pm \frac52\) | \(6\) |
| 4 | \(\frac72\) | 4 | \(\pm \frac12,\ \pm \frac32,\ \pm \frac52,\ \pm \frac72\) | \(8\) |
| 4 | \(= 32 = 2\cdot 4^2\) |
\[ \tag{5} \]
possibilities: \(m=+\frac12\) and \(m=-\frac12\); an orbit for which \(n=2\) has 2 possibilities \(\left(m=+\frac12\right.\) and \(\left.m=-\frac12\right)\) when \(K=\frac12\), and \(2+4=6\) possibilities when \(K=\frac32\) (namely: \(m=\pm\frac12\) and \(m=-\frac12\) for \(J=1\), and \(m=\frac32,\ \frac12,\ -\frac12,\ -\frac32\) for \(J=2\)), and altogether 8 possibilities, etc. In formula (5) one obtains precisely the numbers \(2, 8, 18, 32 \ldots\), indicating the numbers of possibilities for electronic orbits with principal quantum numbers \(1, 2, 3, 4 \ldots\). These numbers of possibilities are consequences of the restrictions compared in formula (1).
Thus the construction of the periodic system, i.e. the successive formation of electronic orbits, each of which in a strong magnetic field is characterized by the quantum numbers \(n, K, J, m\), must be conceived in the following way1. First, one electron with quantum number \(n=1;\ K=\frac12,\ J=1\) and \(m\), for example \(+\frac12\), is joined to the nucleus (hydrogen). The second electron with \(n=1\) has, according to (5), only one further possibility of occupying an orbit characterized by: \(n=1;\ K=\frac12,\ J=1;\ m=-\frac12\) (helium).
The third electron already finds all the possibilities for \(n=1\) occupied by the first two electrons and therefore can join only by entering an orbit with \(n=2\); it thus begins the second period of the system; the first period (\(n=1\)) contains, consequently, only two elements (H and He). Beginning with the third and up to the tenth,
electrons are arranged in orbits corresponding to \([2+(2+4)]=8\) possibilities, which formula (5) opens up for \(n=2\); these eight elements therefore form the second period of the system:
\[ \mathrm{Li,\ Be,\ B,\ C,\ N,\ O,\ F,\ Ne.} \]
The eleventh electron already finds exhausted all the possibilities for \(n=1\) and \(n=2\), and can attach itself only to an orbit with \(n=3\), i.e., it begins the third period. Beginning with the eleventh and up to the eighteenth, the electrons successively occupy those \([2+(2+4)]=8\) orbits with \(n=3\) which correspond to \(K=\frac12\) and \(K=\frac32\) (from Na to A) (cf. (5), and also the table of electronic orbits for the noble gases, which presents the filled groups \(n,\ K\)).
TABLE OF THE NUMBERS OF ELECTRONS IN THE ORBITS \(n,\ K\).
| \(n=1\) \(K=\frac12\) |
\(n=2\) \(K=\frac32,\ \frac12\) |
\(n=3\) \(K=\frac12,\ \frac32,\ \frac52\) |
\(n=4\) \(K=\frac12,\ \frac32,\ \frac52,\ \frac72\) |
\(n=5\) \(K=\frac12,\ \frac32,\ \frac52,\ \frac72,\ \frac92,\ \frac12\) |
\(n=7\) \(K=\frac32,\ \frac52,\ \ldots\) |
|
|---|---|---|---|---|---|---|
| 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 | ||
| 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 6 |
The nineteenth electron does not occupy the orbit \(n=3,\ K=\frac52\), since it proves to be energetically less stable than certain orbits with \(n=4\), into one of which it is placed, beginning the fourth period. Only somewhat later are the orbits \(n=3,\ K=\frac52\) also filled, so that ultimately the electrons, from the nineteenth to the thirty-sixth (K to Kr), fill 2 orbits \(n=4,\ K=\frac12\), \((2+4)=6\) orbits with \(n=4,\ K=\frac32\), and \((4+6)=10\) orbits with \(n=3,\ K=\frac52\).
The thirty-seventh electron now begins the fifth period, placing itself in an orbit \(n=5\), which ends with the fifty-fourth electron (Rb to X); these eighteen electrons fill: 2 orbits \(n=1,\ K=\frac12\); \((2+4)=6\) orbits \(n=5,\ K=\frac32\), and \((4+6)=10\) orbits \(n=4,\ K=\frac52\).
The fifty-fifth electron begins the sixth period (\(Cs\) to \(Em\)), containing 32 elements; in this period the electrons occupy: 2 orbits \(n=6,\ K=\frac{1}{2}\), \((2+4)=6\) orbits \(n=6,\ K=\frac{3}{2}\); \((4+6)=10\) orbits \(n=5,\ K=\frac{5}{2}\), and \((6+8)=14\) orbits \(n=4,\ K=\frac{7}{2}\).
The eighty-seventh electron begins the seventh period with the orbit \(n=7\), which, however, is interrupted in the middle by the ninety-second element (\(U\)).
Thus formula (1), containing electron orbits of various form and orientation [represented in greater detail by formula (5)], makes it possible to regard the lengths of the periods in the system of elements as a direct consequence of the quantum laws, according to which the capture of electrons must take place while satisfying both the integral (or, correspondingly, half-integral) requirements of these laws.
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The quantum numbers \(n\) are considered, of course, for the normal unexcited atom. Translator’s note. ↩↩↩
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To comprehend the manner of this “action” is one of the most difficult, still unresolved problems of atomic physics. Objections may be raised against the conception presented here and the related question of the meaning of the quantum number \(J\) (as well as \(K\), \(n\), and \(m\)); however, there is no need here to dwell in greater detail on these questions. ↩