THE RELATION BETWEEN THE FINE STRUCTURE OF SPECTRAL LINES AND THE ROTATION OF THE NUCLEUS

S. Frisch, Leningrad

Until very recently, the fine structure of spectral lines was understood to mean any such structure that could be detected only with the aid of instruments of high resolving power. No assumptions were made concerning the physical causes of this structure. At the present time, in connection with the development of spectral systematics, it has become clear that sometimes lines with fine structure are ordinary, only very narrow, serial multiplets, while sometimes, on the contrary, the fine structure cannot be explained by the existing theoretical scheme and, consequently, requires the introduction of new factors for its clarification.

Examples of fine structure reducible to ordinary serial multipletness can readily be given. It is well known, for instance, that the lines of the alkali metals form doublet series. Already in potassium, where the doublets arising from the doubleness of the \(P_j\)-terms are still very broad, the lines differing only in the values of the \(D_j\)-terms are so close to one another that they are resolved only with difficulty [1]. In lithium, however, even the doublets arising from the doubleness of the \(P_j\)-terms are so narrow that they can be resolved only by a large diffraction grating or another instrument of high resolving power. An analogous example is presented by helium: according to the general serial scheme, the orthohelium lines must be triplets; until recently, even when observed with the most powerful instruments, they appeared as doublets; only quite recently Paschen and Houston, by cooling the light source with liquid air and using microphotometric recording, succeeded in detecting the third component [2].

The fine structure of the lines of hydrogen and ionized helium is also of a “serial” character. According to modern quantum mechanics, the spectra of H and He\(^+\) are constructed analogously to the doublet spectra of the alkali metals, except that some of their lines lie anomalously close to one another.

But, as has already been indicated, not every fine structure can be explained by the ordinary series scheme. This includes, for example, the fine structure of the lines of mercury, or the fact, recently discovered by Schuler and simultaneously by L. N. Dobretzov and A. P. Terenin [3], that each of the sodium \(D\)-lines is, in turn, an unusually narrow doublet. Likewise, in other alkali metals—Cs and Rb [4]—the lines of the principal series are narrow doublets. All these cases of fine structure cannot be explained by the series scheme of the corresponding atoms. For fine structure of this kind the name “hyperfine” (hyperfein) has become established. Under this name should be grouped all those cases of fine structure which do not find their explanation in the ordinary series scheme and which, as we shall see below, apparently are explained by the presence of magnetic moments in atomic nuclei. As for the distances between the components of lines with hyperfine structure, although they are always very small, they nevertheless cannot serve as a distinctive sign of structure of this kind: we saw in the example of orthohelium that even in the case of fine structure of the “series type” the distances between the components may be so small that they stand at the limit attainable by modern spectroscopic technique. In what follows in the present article only the hyperfine structure, in the sense just indicated, of spectral lines will be considered, and, for brevity, it will be called simply fine structure.

Series multiplets find their formal explanation in the vector scheme developed by Hund, which was set forth in Advances in the Physical Sciences in the reviewer’s preceding article (see vol. 10, issue 1, 1930). Here we shall merely recall briefly that, according to this scheme, in accordance with the hypothesis put forward by Uhlenbeck and Goudsmit, each electron is assigned a mechanical moment \(\sigma\), numerically equal to \(\frac{1}{2}\)

\[ \left(\text{in units } \frac{h}{2\pi}\right) \]

and the corresponding magnetic moment, equal to one Bohr magneton; furthermore, to each electron orbit, along with the principal quantum number \(n_i\), there is assigned a moment \(l_i\), which may take, for different orbits, the values: \(0, 1, 2 \ldots\). For an atom with many electrons all the moments \(\sigma\) are added vectorially, with observance of the rules of spatial quantization, into the vector \(\sigma\), and the vectors \(l_i\)—into the vector \(l\); in turn, the vectors \(\sigma\) and \(l\) are combined into the resultant vector \(j\), which determines the state of the whole atom as a whole.

The firmly established fundamental propositions of Bohr’s theory leave no room for doubt that the fine structure of spectral lines has as its immediate cause the splitting of the corresponding energy levels (or, what is the same thing, spectral terms) into separate closely spaced sublevels. Formally, such a splitting of levels

can be explained by the introduction of a new vector \(i\), which is added to the vector \(j\) to form a new resultant vector \(f\). If one makes the assumption that the vectors \(i\) and \(j\) are added in accordance with the same rules of spatial quantization to which the addition of the vectors \(\sigma\) and \(l\) is subject, then the numerical values of the resultant vector \(f\) will be:

\[ \begin{aligned} f&=(j+i),(j+i-1),\ldots(j-i) && \text{for } j>i \\ \text{and}\qquad f&=(i+j),(i+j-1),\ldots(i-j) && \text{for } i>j \end{aligned} \tag{1} \]

It follows from this that, if different values of \(f\) correspond to several different energies, then each level, simple according to the ordinary serial scheme (determined by a given \(j\)), must split into \(2i+1\) sublevels for \(j>i\) and into \(2j+1\) for \(j<i\).

As we have already noted, according to the Uhlenbeck and Goudsmit hypothesis the vectors \(\sigma\), which formally explain the occurrence of spectral multiplets, physically characterize the mechanical and the associated magnetic moment of the electrons. The fruitfulness of this hypothesis is confirmed by the brilliant successes achieved in recent times by spectral systematics. After this it is, of course, natural to assume that the new formally introduced vector \(i\) gives (in units of \(\frac{h}{2\pi}\)) the mechanical moment of the atomic nucleus and characterizes the magnetic moment associated with it. In that case the splitting of each level into separate sublevels, leading to the fine structure of spectral lines, will be explained by the different possible orientations of the resultant moment \(j\) of the entire electron shell of the atom with respect to the moment \(i\) of its nucleus.

The first successful attempt to substantiate this hypothesis was made by Goudsmit and Back using the example of the fine structure of bismuth lines [5]. The experimentally very carefully investigated fine structure of a large number of lines of this element could be unambiguously reduced to the splitting of its terms into separate sublevels, the number of which—

Table I

Bi, \(\lambda\) 4722.

which agrees with scheme (1). Thus, for example, the Bi line, \(\lambda 4722\ \text{\AA}\), \({}^{2}D_{3/2} - {}^{2}S_{1/2}\), according to Baka’s measurements, carried out with a large diffraction grating, consists of 6 components, the wavelengths and wave numbers of which are given in Table I.

Graphically, the fine structure of this line is shown in the upper part of Fig. 1, in the lower part of which the levels \({}^{2}D_{3/2}\) and \({}^{2}S_{1/2}\) and their splitting into sublevels are represented. Transitions between the individual sublevels occur according to the usual selection rule: allowed

Fig. 1

are those transitions for which the changes \(\Delta f\) of the resultant vector \(f\) are:

\[ \Delta f = \pm 1 \quad \text{or} \quad \Delta f = 0 \; (\text{except for the case } f_1 = f_2 = 0) \tag{2} \]

As is seen from the drawing, the level \({}^{2}D_{3/2}\) splits into four, and the level \({}^{2}S_{1/2}\) into two sublevels; since for the first of them \(j = \frac{3}{2}\), and for the second \(j = \frac{1}{2}\), in both cases the number of sublevels is equal to \(2j + 1\).

Hence, from scheme (1) one may conclude that \(i\) is not less than \(\frac{3}{2}\). Indirect considerations, on which we shall dwell below, compel us to take \(i = \frac{9}{2}\); thus the resultant vector \(f_2\) assumes for the term \({}^{2}D_{3/2}\) the values: 3, 4, 5, 6, and for the term \({}^{2}S_{1/2}\) the values: 4 and 5. These

the values are given on the right in the drawing, and the addition of the vectors \(j\) and \(i\) into the resultant vector \(f\) is also represented graphically there.

Thus, the attempt to explain the fine structure of spectral lines leads to a new and very important hypothesis concerning the existence in atomic nuclei of mechanical and associated magnetic moments. Until now the sources of our information about atomic nuclei have remained very scanty, and, of course, the possibility of determining from spectral observations new quantities characteristic of nuclei—their moments—may prove extremely significant for the further development of all the physics of atomic nuclei.

The successful attempt of Goudsmit and Back confronted spectroscopists with the task of reviewing, from the point of view of the hypothesis of magnetic moments of atomic nuclei, all the empirical material available on the fine structure of spectral lines, and of acquiring new material. The supposition that the moment \(i\) belongs to the atomic nucleus and not to the electron shell can evidently be confirmed experimentally if it can be shown, on sufficiently extensive material, that for each given element the fine structure of its lines is determined by a moment \(i\) having one and the same value for all terms; this requirement must apply both to arc and to spark lines of the atom, i.e. both to the lines emitted by it in the neutral state and at various stages of ionization. Indeed, an atom in the neutral state and in the form of an ion has entirely different electron shells and, correspondingly, spectra of different multiplicity; if it could be proved that in all these states the atom emits lines with a fine structure that leads to the same numerical values of the vector \(i\), this would be sufficiently convincing proof that the moment \(i\) is related not to the electron shell but to the atomic nucleus. One may, of course, conversely try to discover whether series of isoelectronic atoms and ions, i.e. those having identical electron shells (for example: Na, Mg\(^+\), Al\(^{++}\), ...), have different moments \(i\).

However, it turned out that by no means all the available experimental material on the fine structure of spectral lines fits the proposed scheme. Moreover, in some cases this failure could be explained by insufficient accuracy of the observations. This was the situation until Schuler [6] put forward an extremely interesting

Table II

Cd \(2^3P_1 — 2^3S_1\), \(\lambda\) 4800.

hypothesis that different isotopes of one and the same element may have different moments of the atomic nuclei. Schuler put forward this hypothesis on the basis of an analysis of the fine structure of cadmium lines. As an example let us take the line Cd \(23P_1 - 23S_1\), \(\lambda\ 4800\); this line consists of five components, the middle of which is the brightest. The distances of the individual components from this middle one, expressed in wave numbers, are given in Table II.

The fine structure of this line is represented graphically in Fig. 2.

Fig. 2

If the middle line is excluded, the four remaining ones reveal two constant differences of wave numbers, \(\Delta \nu = 0.203\) and \(\Delta \nu = 0.395\), and are easily explained by the splitting of each of the terms \({}^{3}S_1\) and \({}^{3}P_1\) into two, which in scheme (1) leads to \(i = \dfrac{1}{2}\). In order to explain the presence of the fifth, “extra,” component, Schuler, in a joint work with Bak, made the supposition that some of the Cd isotopes (cadmium has six isotopes) have moments of the atomic nuclei \(i = \dfrac{1}{2}\), while the remaining part have no moments and, consequently, give a simple line. This assumption is also justified for the other cadmium lines.

The atomic weights of the Cd isotopes are: 110, 111, 112, 113, 114, 116. Thus

as the atomic number of Cd is \(Z = 58\), the number of protons and electrons in the Cd isotope nuclei is given by the following table:

Table III

Which of these isotopes should be assigned moment \(i = 0\), and which \(i = \frac{1}{2}\), is, of course, an open question. Schuler assumes that nuclei consisting of an even number of protons and electrons have moments equal to 0, while those consisting of an odd number have moments equal to \(\frac{1}{2}\). Schuler motivates this supposition, first, by the fact that it leads to the correct intensity ratio between the “extra” component and the group of the other components in the fine structure of the Cd line, and, secondly, by the fact that zinc, which has 4 isotopes with atomic weights 64, 66, 68, and 70, all consisting of an even number of protons and electrons (Table IV), shows no fine structure on its lines.

Table IV

Zn; \(Z = 30\).

If the hypothesis put forward by Schuler is justified, then an entirely new fact will thereby be established: that isotopes of one and the same element can differ not only in atomic weight, but also in the moment of their nuclei. At the same time it becomes possible to detect isotopes spectroscopically from their line spectra; until now, a spectroscopic distinction between isotopes could be established only in molecular spectra.

In addition to Cd, Schuler also investigated Tl, whose atomic weight (204.4) differs greatly from an integer, which compels one to assume the presence of isotopes in Tl.^1 Here, too, apparently, there is an “extra” component.

^1 By Aston’s method Tl has so far not been investigated, owing to experimental difficulties.

The Schuler hypothesis, that only those Cd isotopes which consist of an odd number of protons and electrons possess a moment \(i=\dfrac{1}{2}\), leads to yet another very remarkable consequence concerning the mechanism of the construction of nuclei. According to Table III, isotopes having moment \(i=\dfrac{1}{2}\) are obtained from the preceding ones, which have no moment, by the addition of one proton and one electron. But the moment of a free electron is equal to \(\dfrac{1}{2}\); likewise a free proton apparently possesses moment \(\dfrac{1}{2}\)—this is confirmed by the existence of two varieties of hydrogen molecules, para- and orthohydrogen, discovered by Bonhoeffer and Harteck [7], and by the structure of the molecular spectrum of hydrogen. Consequently, the addition to the nucleus of one proton and one electron, it would seem, ought to change the moment of the nucleus by unity, or leave it unchanged (if the moments of the proton and the electron mutually compensate each other). In order to escape this difficulty, Schuler assumes that the electron inside the nucleus loses the moment inherent in it [8].

In the preceding part of our exposition we touched only upon the question of the number of components of lines with fine structure. Obviously, no less essential is the question of the distances between the components. According to empirical observations, the distances between sublevels depend strongly on the type of the electron shell. The fine structure on spark lines, as a rule, turns out to be considerably wider than the fine structure of arc lines of the same element. Thus we have already noted that the lines of the principal series of Cs are narrow doublets, which is explained by the splitting of its normal term \(({}^2S_{1/2})\) into two close sublevels; the splittings of its \({}^2P_i\)-terms, however, are already so insignificant that they cannot be resolved. According to the observations of E. F. Gross and A. N. Filippov [9], however, a whole series of lines of ionized cesium \((\mathrm{Cs}^+)\) exhibits a rather complicated fine structure. This fact is expressed still more sharply in lithium: the arc lines of lithium do not exhibit components not embraced by the series scheme, whereas, according to Schuler, the spark lines of lithium \((\mathrm{Li}^+)\) give a fine structure which can be explained by the fact that the lithium isotope with atomic weight 7 \((\mathrm{Li}_7)\) has moment \(i=\dfrac{1}{2}\) (the second lithium isotope \((\mathrm{Li}_6)\), apparently, has no moment). Likewise, according to Urey’s observations on \(\mathrm{P}_2\), and those of Wood and Kimura on I, the spark lines of these elements exhibit fine structure, while the arc lines, at practically attainable resolving powers, appear simple [10].

Theoretically, the question of the magnetic moments of atomic nuclei was considered by Hargreaves and Fermi (11), and these authors succeeded in...

show that it can be brought within the circle of considerations of modern quantum mechanics. Fermi attempted to connect the width of the splitting of the Cs levels with the absolute value of the magnetic moment of its nucleus. According to the hypothesis of Uhlenbeck and Goudsmit, the electron possesses a mechanical moment

\[ \sigma = \frac{1}{2} \left( \text{in units of } \frac{h}{2\pi} \right) \]

and a magnetic moment

\[ \mathfrak{m}=1 \quad (\text{in units of the Bohr magneton}). \]

Thus the magnetic moment of the electron, measured in units of the Bohr magneton, is twice as large as its mechanical moment, measured in units of \(\frac{h}{2\pi}\). The question of the relation of the mechanical moment \(i\) of an atomic nucleus to its magnetic moment \(\mathfrak{m}\) remains a priori open, and for its resolution an analysis of the available experimental material from the point of view of quantum mechanics is required. The circumstance that the experimentally numerical value of the moment \(i\) for Cs remains undetermined does not allow unequivocal conclusions to be drawn from Fermi’s calculations.

Matters stand much more favorably in the more particular question—the question of the relative distances between components. Here the simple assumption is sufficient that the additional energy caused by the presence of the moment of the atomic nucleus \(i\) is proportional to the numerical value of the moments \(i\) and \(j\) and to the cosine of the angle between them (12), i.e.:

\[ \Delta W \sim i, j, \cos(i,j). \]

Replacing the cosine by the expression

\[ \frac{f(f+1)-i(i+1)-j(j+1)}{2ij}, \]

as is required by modern quantum mechanics, we obtain:

\[ \Delta W \sim i\cdot j \left[ \frac{f(f+1)-i(i+1)-j(j+1)}{2ij} \right], \]

whence the distance between two sublevels characterized by the values of the resultant vector \((f+1)\) and \(f\) is proportional to:

\[ ij \left[ \frac{(f+1)(f+2)-i(i+1)-j(j+1)}{2ij} \right] - ij \left[ \frac{f(f+1)-i(i+1)-j(j+1)}{2ij} \right] = f+1, \]

whence the distances between rows of sublevels into which, owing to the presence of the moments of the atomic nuclei, the levels split according to the simple general serial scheme, will be related to one another as \((f+1):(f+2)\ldots\) Since the magnitude of the spectral terms, expressed in wave numbers,

proportional to the energies, then the same relation is preserved for intervals expressed in wave numbers:

\[ \Delta\nu_1:\Delta\nu_2:\ldots=(f+1):(f+2). \tag{2} \]

This rule is excellently confirmed on a large number of lines of ionized Pr investigated by Yaitom; the lines of this element split into 6 very characteristically arranged components: the distances between the components decrease monotonically in one direction. The lines investigated are a combination of various high terms with the quintet term \({}^{5}K_7\), i.e., a term characterized by the value of the vector \(j=7\); their fine structure is explained by the splitting of the term \({}^{5}K_7\) into 6 sublevels.1 Hence one may directly conclude that the moment of the atomic nucleus Pr is \(i=\frac{5}{2}\), and that the resultant vector \(\mathbf f\) assumes, for the term \({}^{5}K_7\), the values: \(19/2,\ 17/2,\ 15/2,\ 13/2,\ 11/2,\ 9/2\). Then, according to rule (2), the distances between the sublevels, and consequently also between the components of the fine structure, must be related to one another as:

\[ \Delta\nu_1:\Delta\nu_2:\ldots=19:17:15:13:11. \]

In Table V, the second column gives the distances measured on spectrograms, in \(\mathrm{cm}^{-1}\), between the components of certain lines of \(\mathrm{Pr}^{++}\), whose wavelengths are given in the first column; the third column gives the ratios between these distances. As can be seen, these ratios are quite close to the ratio \(19:17:15:13:11\).

Table V

Rule (2) may also be used, so to speak, in the opposite direction, i.e., for establishing the numerical value of the vector \(i\), when it cannot be established directly from the number of components. On the basis precisely of this rule it may be asserted that the moment of the atomic nucleus Bi is equal to \(\frac{9}{2}\). Indeed, from drawing 1 it is seen that the level \({}^{2}D_{3/2}\) of bismuth splits into four sublevels, located from one another at distances: \(0.152;\ 0.198;\ 0.255\ \mathrm{cm}^{-1}\); if \(i=\frac{9}{2}\), then \(f\) assumes for these sublevels respectively the values

1 The splittings of higher terms are apparently so narrow that they cannot be detected.

3, 4, 5, 6 and, consequently, the distances between them must be in the ratio \(4:5:6\), which is also fulfilled with considerable accuracy.

Finally, an extremely interesting confirmation of the indicated points of view is the work of Back and Goudsmit, devoted to the Zeeman effect on the bismuth lines (13). In a strong external magnetic field the moment \(i\) must be oriented independently of the resultant moment of the electron shell \(j\). According to the rules of space quantization, the moment \(i\) can then be oriented in \((2i+1)\) different ways with respect to the external magnetic field. Since several different values of the additional energy will correspond to these different orientations, it follows that, in a strong external magnetic field, each component of the ordinary Zeeman splitting must consist of \((2i+1)\) closely spaced components. For bismuth \(i=\dfrac{9}{2}\), \((2i+1)=10\), and, indeed, Back succeeded in observing that in a strong magnetic field each of the components of the ordinary magnetic splitting consists of 10 components.

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