HYPERFINE STRUCTURE AND THE ATOMIC NUCLEUS*

G. Kalman (Berlin—Dahlem) and T. Schuler (Potsdam)

Introduction. — I. Experimental methods for determining hyperfine structure. — II. Magnetic splitting of spectral terms: a) elementary theory of the interaction of two magnetic moments; b) intensity laws; c) quantitative theory of the coupling between the nucleus and the electron shell; d) experimental results: 1) determination of hyperfine structures from the interval rule and intensity laws; 2) comparison of the results of experiment with the quantitative theory of coupling. — III. Questions of nuclear structure. — IV. Isotope shift. — V. Literature.

Introduction

Recently it has at last become possible to interpret and systematize the long-known hyperfine structures of spectral lines. It has turned out that there exist two types of hyperfine structures. One of them—the so-called “magnetic hyperfine structure,” first pointed out by Pauli\(^ {65}\), is based on the splitting of an atomic term as a result of magnetic interaction with the atomic nucleus. This type has recently acquired particular theoretical interest, since it gives the most accurate information on the mechanical and magnetic moment of the nucleus, and consequently also new information on the structure of the atomic nucleus.

The other type of hyperfine structure is due to the fact that the energy of the atom, and consequently also the work of ionization of the electron, is different for different isotopes of one and the same element. One and the same spectral term in different isotopes has several different values. This effect is called the “isotope shift.” Since the time of Sommerfeld’s first works it has been known that the binding energy of the electron with the atom, owing to the proper motion of the nucleus, also depends on the mass of the nucleus. However, this “ordinary” proper motion of the nucleus is insufficient to explain the cases considered here, for in heavy elements, where according to recent investigations (Schuler and Keyston\(^ {86}\)) the isotope effect appears especially clearly, it would give immeasurably small shifts.

The combination of magnetic splitting and the isotope effect leads to the structures of some spectral lines becoming extraordinarily complicated. Their deciphering became possible only thanks to precise investigation (determination of the splitting—

* Ergebnisse der exakten Naturwissenschaften B XI, translated by D. B. Gogeridze.

of states and intensities of individual lines of the hyperfine structure) of an entire series of spectral lines of one and the same element. On the basis of the known laws of magnetic splitting and the approximately known quantitative ratios of the isotopes, one can interpret individual structures by assigning definite mechanical moments to the atoms and nuclei of the isotopes. This interpretation may be regarded as unambiguous when, with its aid, it is possible to explain the separations and intensities of the hyperfine-structure components of all investigated spectral lines of the element. In what follows we shall first consider the experimental foundations for determining hyperfine structure, and then dwell separately on both effects of hyperfine structure.

I. Experimental methods for determining hyperfine structure

For an exhaustive analysis of hyperfine structure, in choosing an experimental method the following should be taken into account:

1. The spectroscopic instruments employed should, as far as possible, possess high resolving power, since here the question concerns line separations down to \(10^{-2}\ \mathrm{cm}^{-1}\).

2. The light sources employed should, at sufficient intensity, give sufficiently sharp spectral lines.

3. The light sources should be chosen in such a way that, as far as possible, there are no processes in them that mask the true intensity distribution in the fine-structure pattern; such processes include self-reversal, selective excitation, etc.

The spectroscopic instruments used are chiefly interferometric spectroscopes. By the very nature of these instruments, no universal apparatus can be specified for the region of interest to us. The choice of an instrument that solves the problem posed depends on the skill of the experimenter.

For the application of interferometric spectroscopes to the investigation of the fine structure of spectral lines, see the detailed review by G. Hansen\(^{37}\). Here it should be pointed out that certain fundamental improvements have been introduced into this method in recent times. It has proved possible to make new semitransparent mirrors for the Fabry–Perot interferometer with high reflectivity in the violet and ultraviolet regions, as a result of which the range of application of these instruments has greatly expanded (G. Hohheim, “I. G. Farbenindustrie,” Oppau).

An increase in resolving power has been achieved by means of the interferometric multiplex spectroscope (Gercke, Lau, and Ritter\(^{20,25}\)), on the one hand, and the reflecting echelon grating of G. Hilger (Williams\(^{95}\)) on the other. The resolving power that can be used at present is about

million. The use of gratings, which in luminosity and resolving power are inferior to the Fabry and Perot interferometer, is recommended only in especially favorable cases.

Alongside these improvements in interference spectroscopes, the achievement of substantial successes in this field was facilitated most of all by the improvement of the light sources used. The Doppler broadening of spectral lines, caused by temperature and often very troublesome, in the sources now employed is noticeably reduced by direct cooling of the discharge space with water, solid carbon dioxide, or liquid air.

Another important condition that light sources must satisfy is that the density of the atoms under study must be sufficiently small for interaction between the atoms to be excluded. A high vapor pressure in an arc of any type may cause broadening of the lines and a distortion of the intensity, impeding analysis of the spectral pattern as a result of adsorption. Therefore the best light source should be considered a glow discharge. But an ordinary glow discharge in the pure gas or metallic vapor being investigated also does not yet satisfy ideal conditions, since it still requires a noticeable gas density. This inconvenience is especially severe for light elements, since in working with them one has to use such a high temperature that the Doppler effect already interferes noticeably.

In special cases this density is inconvenient also because, when it is present, mutual perturbations of atoms have to be taken into account. All these difficulties, however, can be overcome if the substance under study is introduced as a small admixture into a noble gas in which a glow discharge occurs. With the aid of such a discharge one can obtain sharp spectral lines of sufficient brightness—the resonance line (Na, Schüler ^77).

When working with gases or easily evaporated substances, fulfillment of the indicated conditions for a glow discharge is easily achieved in ordinary discharge tubes (with external or internal electrodes); for all other elements (and this is the most frequent case), in order to obtain a glow discharge, other constructions must be used (Schüler ^77).

For the investigation of heavy elements (such as Tl and Pb), in which the Doppler effect is not of such great importance, one may use a glow discharge directly in the vapor of the metal under study, provided that a very large destructive force is not required ^76. This method is very convenient for studying weak central lines, since it gives a large light intensity.

Finally, it has been shown that, with a suitable form of glow discharge, the third condition is also satisfied—that of a relatively natural distribution of intensities; for this it is necessary to make the luminous part of the glow discharge as small as possible (Schüler lamp) and to work with small current strengths.

Experiments have shown that, when all these conditions are fulfilled, the theoretical intensities are observed, i.e., all the terms of the hyperfine structure are represented in accordance with their statistical weight.

II. Magnetic splitting of spectral terms of hyperfine structure

a) Elementary theory of the interaction of two magnetic moments

The magnetic splitting of spectral terms occurs because the electron shell at the place occupied by the atomic nucleus \((0)\) creates a magnetic field \(H(0)\), the nuclear moment being oriented in different ways.

In this magnetic field the magnetic moment of the nucleus \(\mu\) has an energy of the order \(\mu H(0)\), and this is precisely the energy that causes the splitting of the unperturbed electronic terms.

The magnetic field created by the electron shell at the position of the nucleus depends both on the orbital and on the intrinsic moments of the electrons. To determine its exact magnitude it is necessary to carry out calculations for each spectral term separately (see below). But exact knowledge of this quantity is necessary only when one wishes to determine the absolute magnitude of the splitting or to compare the splittings of different electronic terms. For a qualitative determination of the pattern of the splitting (the intensity and relative spacing of the hyperfine-structure lines), for a known value of \(\mu\) for the nucleus it is sufficient to know that the magnetic field at the nucleus depends on the external configuration of the electrons, whose total mechanical moment is determined by the quantum number \(J\).

Knowledge of \(J\), however, proves sufficient only when the energy of interaction between the nucleus and the electron shell is small in comparison with the energy of interaction between the individual orbital and spin moments of the electrons. If the atomic nucleus has an angular momentum with quantum number \(I\), then one can in the following way establish the pattern of splitting of the hyperfine structure for a definite electron shell (entirely analogously to the way this is done in determining the coupling between \(L\) and \(S\) in coarse multiplet splitting) (Goudsmit and Back \(^{29}\)).

A system consisting of \(I\) and \(J\) will have a total angular momentum \(F\), which can take all possible values, differing from one another by unity, between \(|J+I|\) and \(|I-J|\). Altogether, therefore, there can exist (for \(I \geq J\)) \(2J+1\), or (for \(J \geq I\)) \(2I+1\), systems with different total angular momenta. Since only systems with different \(F\) possess different energies, our coarse term splits either into

\[ (2J+1) \text{ for } I \geq J,\quad \text{or into } (2I+1) \text{ for } J \geq I \tag{1} \]

components. It is also possible to give the magnitude of the energy splitting. Suppose that the magnetic moment of the atomic nucleus is at the center of a circular current; the energy of the magnetic interaction of this system is proportional to the cosine of the angle formed by the corresponding rotation vectors. Similarly, the energy of the magnetic interaction of the nuclear moment, situated in the middle of the electronic system, with the electrons is proportional to \(\cos (I,J)\). As is known from quantum mechanics, the value of this cosine is equal to

\[ F \frac{(F+1)-I(I+1)-J(J+1)}{2IJ}. \]

The magnitude of the energy displacement of the term with total angular momentum \(F\) from the undisplaced position of the term, which in the case of a simple magnetic coupling coincides with the center of gravity of the split system, will therefore be equal to

\[ \Delta W=\mu H(0)\frac{F(F+1)-I(I+1)-J(J+1)}{2IJ}. \tag{2} \]

Here \(\mu\) is the magnetic moment of the nucleus with angular momentum \(I\), and \(H(0)\) is the magnitude of the magnetic field of the electrons at the site of the nucleus. It is clear that they depend both on \(J\) and on other quantum numbers of the atom. It follows further from (2) that the total splitting, i.e. the distance between the two levels with the largest and smallest \(F\), is equal to:

\[ \begin{aligned} \Delta W&=\mu H(0)\frac{2I+1}{I} && \text{for } (I \geq J) \tag{a}\\ \text{or }&=\mu H(0)\frac{2J+1}{J} && \text{for } (J \geq I). \tag{b} \end{aligned} \tag{3} \]

Moreover, from formula (2) it follows that the distances between two neighboring terms are in the ratio

\[ F_m:(F_m-1):(F_m-2):\ \text{etc.} \tag{4} \]

Here \(F_m\) denotes the largest value that \(F\) can assume at all, i.e. \(F_m=I+J\). Thus the internal subdivision of the splitting is determined by the expression \((I+J)\). This so-called Landé interval rule has special importance in the identification of spectra (see below).

The pattern of splitting of an individual spectral line, i.e. its hyperfine structure, is obtained from the pattern of splitting of the term by applying the selection rule for transitions between the various distances. In addition to \(\Delta J=0,\pm1\), and in complete analogy with it, the rule

\[ \left. \begin{aligned} \Delta F&=0,\pm1,\\ F&=0\to0\ \text{forbidden.} \end{aligned} \right\} \tag{5} \]

For a somewhat more detailed consideration of the quantum-mechanical laws of the magnetic interaction of two systems, one should pass directly to the problem of eigenvalues. The simplest case will be that in which the magnetic energy of interaction has practically no influence on the translational motion of the system. For the case in which the systems possess interaction energies proportional to the cosine between \(I\) and \(J\), and under the condition that the external magnetic field \(H\) is directed along the \(Z\) axis, the total perturbation energy \(E\) is equal to

\[ E = W\left(\hat{\delta}^{I}_{x}\hat{\delta}^{J}_{x} +\hat{\delta}^{I}_{y}\hat{\delta}^{J}_{y} +\hat{\delta}^{I}_{z}\hat{\delta}^{J}_{z}\right) -\mu^{I}_{0}\left(H_{z}\hat{\delta}^{I}_{z}\right) -\mu^{J}_{0}\left(H_{z}\hat{\delta}^{J}_{z}\right). \]

Here \(W\) is the magnetic interaction energy of the two systems; it depends on their mutual spatial arrangement and is proportional to the magnetic moments \(\mu^{I}_{0}\) and \(\mu^{J}_{0}\) of both systems with mechanical moments \(I\frac{h}{2\pi}\) and \(J\frac{h}{2\pi}\). \(\hat{\delta}^{I}_{x}, \hat{\delta}^{J}_{x}\), etc. are unit vectors of these moments. They are quantum-mechanical operators acting only on the spin coordinates corresponding to \(I\) and \(J\). Moreover the latter, as is known, can take only \((2I+1)\) or \((2J+1)\) discrete values. The expression given above for the operator makes it possible to act on the wave function of the whole system, which here depends only on the spin coordinates, and thus to obtain the eigenvalues and eigenfunctions of the system. For the case in which \(I\) and \(J = 1/2\), we obtain the following eigenvalues of the energy:

\[ \begin{aligned} E_{1} &= +W + H\left(\mu^{I}_{0}+\mu^{J}_{0}\right),\\ E_{2} &= -W + \sqrt{4W^{2}+H^{2}\left(\mu^{J}_{0}-\mu^{I}_{0}\right)^{2}},\\ E_{3} &= +W - H\left(\mu^{I}_{0}+\mu^{J}_{0}\right),\\ E_{4} &= -W - \sqrt{4W^{2}+H^{2}\left(\mu^{J}_{0}-\mu^{I}_{0}\right)^{2}}. \end{aligned} \]

The eigenvalues from \(E_{1}\) to \(E_{3}\) in the absence of a field \((\mu=0)\) are equal to \(W\), while \(E_{4}\) is equal to \(-3W\); thus there is one triply degenerate eigenvalue \(W\) and one nondegenerate value \(-3W\), in agreement with equations (1)–(4).

For the eigenfunctions \(\varphi\) one obtains the following: in the case \(I=\frac{1}{2}\), \(J=\frac{1}{2}\), they consist only of the following four quantities: \(\varphi(+,+)\), the value of \(\varphi\) in the case when \(J\) and \(I\) are both directed toward the positive \(Z\) axis;

\[ \begin{aligned} &\varphi(+,-), &&\text{when } J \text{ is in the direction } +Z,\ \text{and } I \text{ in the direction } -Z;\\ &\varphi(-,+), &&\text{when } J \text{ is in the direction } -Z,\ \text{and } I \text{ in the direction } +Z;\\ &\varphi(-,-), &&\text{when } J \text{ is in the direction } -Z,\ \text{and } I \text{ in the direction } -Z. \end{aligned} \]

In this case, as is known, \(|\varphi(+,+)|^{2}\) gives the probability that both \(J\) and \(I\) are directed toward \(+Z\), and, analogously, the same is indicated by the squares of the moduli of the other quantities.

For the eigenfunctions corresponding to the above eigenvalues, the following quantities are obtained:

\[ E_{1};\quad \varphi_{1}=\{\varphi(+,+)=1;\ \varphi(+,-)=0;\ \varphi(-,+)=0;\ \varphi(-,-)=0\}; \]

\[ E_{2};\quad \varphi_{2}=\left\{\varphi(+,+)=0;\ \varphi(+,-)= -\frac{A}{\sqrt{A^{2}+B^{2}}};\ \varphi(-,+)= \frac{B}{\sqrt{A^{2}+B^{2}}}, \right. \]

\[ \left. \varphi(-,-)=0\right\}; \]

\[ E_{3};\quad \varphi_{3}=\{\varphi(+,+)=0;\ \varphi(+,-)=0;\ \varphi(-,+)=0;\ \varphi(-,-)=1\}; \]

HYPERFINE STRUCTURE AND THE ATOMIC NUCLEUS

\[ E_4;\ \varphi_4=\{\varphi(+,+)=0;\ \varphi(+,-)=\frac{-A}{\sqrt{A^2+C^2}};\ \varphi(-,+)=\frac{C}{\sqrt{A^2+C^2}} \]
\[ \varphi(-,-)=0\}; \]
\[ A=2W B=\sqrt{4W^2+H^2(\mu_0^J-\mu_0^I)^2}+H(\mu_0^J-\mu_0^I), \]
\[ C=\sqrt{4W^2+H^2(\mu_0^J-\mu_0^I)^2}-H(\mu_0^J-\mu_0^I). \]

Here the eigenfunctions have already been normalized to unity in the usual way.

The eigenvalues and eigenfunctions completely determine the behavior of the coupled system in a magnetic respect, namely, the splitting in a magnetic field (the Paschen–Back effect and the Zeeman effect), its magnetic moment \(\left(-\frac{\partial E}{\partial H}\right)\) (the Stern–Gerlach effect), the orientation of the individual system in a magnetic field, etc.

Results analogous to the case \(I=J=1/2\) are obtained also when for \(I\) and \(J\) we take any other integral or half-integral values. Then, instead of the quantities \(\sigma_x^I,\sigma_x^J,\ldots\), one must substitute not the spin Pauli matrices, but other many-row matrices (see, for example, Fermi \(^{17}\)).

In the analysis of hyperfine structures the matter reduces to determining, from the totality of the lines of splitting of terms, with the aid of the laws given above (1) and (4), the unknown value of \(I\) for the nucleus. This determination of \(I\) is at present the most important result of the study of hyperfine structures. With a known splitting of a term, \(I\) can be calculated from the number of its components only when \(J>I\). For this it is necessary to establish that, in this case, the coarse term with a definite \(J\) has fewer components than \((2J+1)\). It is also possible to calculate \(J+I\) from the ratio of the intervals (4).

It is true that, for large \(I\), the interval rule will permit only approximate conclusions, since for large \(J+I\) the distances differ little from one another. A second difficulty is that expression (4) holds only in the case of a purely magnetic interaction and, in particular, when the interaction energy between the nucleus and the electrons is very small in comparison with the magnetic interaction of the individual electrons. In general, these conditions are fulfilled, and correspondingly, in individual cases, the interval rule in hyperfine structures proves to be excellently confirmed (see below).

However, for \(Li\), deviations from the rule were found. Here the interaction between the electrons (the ordinary multiplet splitting) is comparable in magnitude with the interaction between the nucleus and the electrons (the hyperfine-structure splitting) (Göttinger and Pauli \(^{35}\))*.

* Note added in proof: A further very interesting deviation from the interval rule and from the simplest laws of magnetic split—

Thus we see that, proceeding from the number of components and the interval rule, we do not always arrive at completely unambiguous conclusions.

There exist, however, two other ways of studying \(I\). On the one hand, this is the study of the splitting of a hyperfine-structure term in an external magnetic field. It is known that a term with total angular momentum \(F\) is split in an external magnetic field into \((2F+1)\) components. Thus, from the magnetic splitting (the number of components) of a hyperfine-structure term one can directly determine the value of \(F\), and consequently also \(I\). But this Zeeman-effect method is complicated and has so far been applied only in a few cases (Bi, Tl) (Back and Goudsmit \({}^{3}\)).

b) Intensity laws

Another method, which can also be used for determining \(I\), and which in practice is much more important than the preceding one, is the method of comparing the intensities of the different hyperfine-structure lines (Schüler and Keyston \({}^{84}\)). Consider, for an atom with known \(J\), the splitting of a line corresponding to a transition between two terms with quantum numbers \(J_1\) and \(J_2\). The number of components is then determined from (1) and from the selection rule (5), and in this way the splitting pattern is determined by \(I\) and \(J\). As we shall see, however, \(I\) and \(J_1\) or \(J_2\) also determine the relative intensity of the hyperfine-structure lines; in the same way, conversely, from the intensity law, by comparing intensities, one can determine the value of \(I\). The general intensity laws have the following form:

for

\[ \Delta J=\pm 1. \]

\[ I=\frac{P(F)P(F-1)}{4FJ}, \quad \text{when } \Delta F=-1; \]

\[ I=\frac{2F+1}{4FJ(F+1)}\,P(F)Q(F), \quad \text{when } \Delta F=0; \]

\[ I=\frac{Q(F)Q(F-1)}{4FJ}, \quad \text{when } \Delta F=+1; \]

for \(\Delta J=0\):

\[ I=\frac{(2J+1)P(F)Q(F-1)}{4JF(J+1)}, \quad \text{when } \Delta F=\pm 1; \]

\[ I=\frac{(2J+1)\cdot(2F+1)}{4JF(J+1)(F+1)}\,R^2(F), \quad \text{when } \Delta F=0, \]

splitting was found by Schüler and Jones for the terms \(6^1D_2\) and \(6^3D_1\) in Hg—I. Here the deviations depend on the fact that these two coarse terms are separated from one another only by a distance \(\Delta \nu = 3\ \text{cm}^{-1}\), whereas the total splitting of the hyperfine-structure term reaches barely \(0.7\ \text{cm}^{-1}\).

where

\[ \begin{aligned} P(F)&=(F-I+J)(F+J+I+1),\\ Q(F)&=(-F+I+J)(F-J+I+1),\\ R(F)&=F(F+1)+J(J+1)-I(I+1). \end{aligned} \]

In many cases it proves possible to use the simplest expressions following from (6). The sum of all intensities belonging to any one term of the hyperfine structure with a definite \(F_n\) is related to the sum of the intensities of another term of the hyperfine structure \((F=F_m)\) as

\[ (2F_n+1):(2F_m+1), \tag{7} \]

provided that both of these hyperfine terms belong to one and the same gross term. The sums of all intensities of the various terms of the hyperfine structure are related to one another in the same way. It should be emphasized that the hyperfine-structure terms being compared must belong to one and the same gross term. These simple summation rules are especially important when we are dealing with lines that split into a small number of components. The simplest case will be that in which we have a transition between two terms, one of which does not split at all, while the other splits into only two terms. Then the spectral line has only two components, and their intensities are related as \((2(I+J)+1):(2(I+J)-1)\). This case is especially important for determining \(I\) in the alkali metals. Their resonance lines are doublets, since their \(p\)-term practically does not split, while the \(S\)-term, owing to \(J=\tfrac12\), can be only double. Thus the laws (1) and (4) cannot be applied here, since we have only one interval and are forced to restrict ourselves to a comparison of intensities. For \(J=\tfrac12,\ \tfrac32,\ \tfrac52,\ \tfrac72\) the following intensity ratios are obtained: \(3:1;\ 5:3;\ 7:5;\ 9:7\).

We do not wish here to consider in detail the application of these intensity laws. We wish only to note that the value of \(I\), determined with the aid of (6) and (7) from the measured intensities, always proves to coincide with the values of \(I\) determined from the number of components and the interval rule. It must, however, be noted that this intensity law can be applied only with a certain caution. First, these formulas are approximate, and they may be applied only under the same restrictions as formulas (5) and (6) (see above). In addition, in order to obtain these theoretical intensities, the fulfillment of certain experimental conditions is also necessary. Care must be taken that the relative intensities are not distorted by the conditions of the experiment. Among such requirements imposed on the experiment, we shall mention first of all that there must be a complete absence of self-absorption of the line being investigated.

and that the excitation of the term must be “natural,” i.e., there must be no selective excitation of any one of the terms of the fine structure. The available results indicate that, with an appropriate choice of apparatus, these ideal intensity conditions can indeed be fulfilled (see above).

c) Quantitative theory of the coupling between the nucleus and the electron shell

With the help of the preceding considerations one can obtain, from the hyperfine structure, the value of \(I\) for the atomic nucleus. However, in order to be able to say anything also about the magnitude of the magnetic moment of the nucleus \(\mu\), one must take one further step and introduce into consideration the absolute magnitude of the hyperfine splitting from formulas (2). Here \(\mu\) can be determined if the magnetic field \(H(0)\), produced by the electron shell at the site of the nucleus, is known (Güttinger \(^{34}\)). It is therefore necessary to calculate the quantity \(H(0)\). It can be obtained in the known way from the magnitude of the current associated with the electron shell and equal to
\[ \frac{1}{c}\int \left[\mathbf{i}\,\frac{-\mathbf{r}}{r^3}\right]\,dV, \]
where \(\mathbf{i}\) is the current density, which is given by the unperturbed eigenfunctions of the electrons. One may also proceed in this way to determine the absolute magnitude of the splitting of the term (Fermi \(^{17}\)): the energy of interaction between the nuclear moment and the electron (spin and orbital) is substituted into the Schrödinger or Dirac equations (for an \(S\)-term), and the eigenvalue problem is then solved by known approximate methods. The term to be inserted into the Schrödinger equation for one electron has the following form:
\[ \frac{e}{mcr^3}\,\mu\cdot M-\frac{\mu_0}{r^3}\,\delta\cdot\mu+\frac{3\mu_0}{r^5}(\mathbf{r}\cdot\delta)(\mathbf{r}\cdot\mu), \tag{8} \]
where \(M\) is the orbital angular momentum of the electron, \(\mu_0\) is its magnetic moment, \(\delta\) is the unit vector of the electron spin, and \(\mu\) is the magnetic moment of the nucleus. The first term expresses the interaction between the nuclear moment and the orbit of the electron; the second and third terms—the interaction with the spin; \(\delta\) and \(\mu\) should be regarded quantum-mechanically as operators, which act in a known way on the coordinates of the electrons and the nucleus contained in the eigenfunctions. In calculations it will always be assumed that only those groups of electrons can be magnetically active for which \(J\) does not vanish. Thus only the upper, not yet filled shells are in interaction with the nucleus. Filled shells will be regarded as magnetically inactive. They enter into the indicated calculations only insofar as they change the course of the eigenfunctions of the outer electrons.

The qualitative result that is obtained for the one-electron problem from these calculations is easy to foresee. The interaction will be the more significant, the closer the electron is to the nucleus.

Accordingly, one should expect that the greatest splitting will be shown by those terms whose eigenfunctions have the greatest value at the position of the nucleus. These are the \(S\)-terms. The unperturbed \(\psi\)-functions of the \(S\)-terms have, near the nucleus, a finite, nonvanishing value, whereas the eigenfunctions of the \(P\), \(D\), and other terms near the nucleus tend to zero. Consequently, for \(S\)-terms one may expect the greatest splitting, and then, with increasing quantum number, the splitting should decrease; in general, as we shall see below, this is also confirmed by experiment.

The exact formulas for the hyperfine splitting for one magnetically active electron with \(J=\frac12\), for an \(S\)-term, are as follows:

\[ \Delta W=\frac{8\pi}{3}\,\mu\mu_0\,\psi^2(0), \qquad F=I+\frac12; \]

\[ \Delta W=-\frac{8\pi}{3}\,\mu\mu_0\,\psi^2(0)\left(\frac{I+1}{I}\right), \qquad F=I-\frac12; \]

for the term \(P_{1/2}\):

\[ \Delta W=\frac{8}{3}\,\mu\mu_0\,\overline{\frac{1}{r^3}}, \qquad F=I+\frac12; \]

\[ \Delta W=-\frac{8}{3}\,\mu\mu_0\,\overline{\frac{1}{r^3}}\left(\frac{I+1}{I}\right), \qquad F=I-\frac12; \]

for the term \(P_{2/3}\):

\[ \Delta W=\frac{8}{5}\,\mu\mu_0\,\overline{\frac{1}{r^3}}; \quad \frac{8}{5}\,\mu\mu_0\,\overline{\frac{1}{r^3}}\left(\frac13-\frac1I\right); \]

\[ -\frac{8}{5}\,\mu\mu_0\,\overline{\frac{1}{r^3}}\left(\frac13+\frac{4}{3I}\right); \quad -\frac{8}{5}\,\mu\mu_0\,\overline{\frac{1}{r^3}}\left(1+\frac1I\right), \]

for the four possible values of \(F\). \(\mu_0\) is the Bohr magneton, \(\psi(0)\) expresses the value of the \(\psi\)-function of the outer electron at the position of the nucleus, \(\overline{\frac{1}{r^3}}\) denotes the mean value of \(\frac{1}{r^3}\), averaged over the unperturbed functions of the electron states under consideration. These formulas coincide with equation (2). The factors

\[ \frac{8\pi}{3}\,\mu_0\,\psi^2(0); \qquad \frac{8}{3}\,\mu_0\,\overline{\frac{1}{r^3}}, \qquad \frac{8}{5}\,\mu_0\,\overline{\frac{1}{r^3}} \]

give \(H(0)\), the magnitude of the magnetic field at the position of the nucleus. Thus, for the application of the theory it is necessary to determine these quantities. \(\psi(0)\) can be calculated directly from the atomic model (by Fermi’s statistical method). The quantities \(\overline{\frac{1}{r^3}}\) can be taken from the doublet fine structure, which is also determined by these quantities. We thus see that, for an \(S\)-term with a positive nuclear moment, the terms lying highest in energy are those for which the rotational moments of the nucleus and of the electron are parallel. An analogous phenomenon occurs also for a \(P\)-term. This latter is easy to understand if one turns

attention to the fact that for it the Schrödinger distribution of the electron charges does not have spherical symmetry. The terms \(P_{3/2}\) have, for example, a smaller splitting than the terms \(P_{1/2}\). This is explained by the fact that the orbit and the spin of the electron in the case of the \(P_{3/2}\)-term have different signs of the magnetic field at the site of the nucleus.

In a similar way it is possible to calculate the splitting also for other non-closed electron groups (singlet and triplet systems, etc.). We, unfortunately, cannot dwell here on the convenient method which Goudsmit applied for this purpose \(^{26}\). We shall only point out that the calculation of the splitting of terms for which \(J\) consists of a spin and an orbital part can, for several electrons, be carried out with the aid of the well-known summation rule from the splittings for the one-electron problem.

We then find that \(P\)- and \(D\)-terms also may have a considerable splitting, especially when the outer electron groups contain one unsaturated deep \(s\)-electron. Finally, it is still necessary to note that formulas (9) can be applied only for small values of the nuclear charge. For large values of the nuclear charge \(Z\), close to the nucleus the motion of the electrons is so rapid that calculations must be made with relativistic corrections taken into account. Calculations show that in this case one more factor, depending on \(Z\), enters into the formulas given above (Breit, \(^{8}\) Racah \(^{66}\)).

A check of these formulas is possible in the following way: from one term the unknown quantity \(\mu\) for the nucleus is calculated; then the other terms must give the same values of \(\mu\). Comparison with experiment shows the following: from all the terms it is found that the magnetic moment of the nucleus, in order of magnitude, reaches approximately \(0.001\) of the electron moment (for more detail see section \(d\)). A more exact determination of \(\mu\) in this way appears, unfortunately, to be impossible; on the contrary, it has been found that rather different values of \(\mu\) are obtained from different terms, so that there is no doubt that this theory in its present form is still imperfect (Goudsmit). All expressions that follow only from the fact of magnetic interaction are apparently correct (Racah \(^{66}\)), but as soon as we wish to calculate the absolute value of the splitting of the terms, we encounter difficulties, on which we shall dwell further below.

Unfortunately, in this way it proves impossible to derive a quantitative dependence for the corresponding terms of different elements having the same electronic configuration. This deprives us of the possibility of quantitatively comparing with one another the nuclear moments of different elements. We shall return, in discussing the experimental results, to the attempts that have been made in this direction.

There exists one comparatively simple case which has been calculated in great detail. This is the case of Li II. Here we have in all only two electrons, and in this way we can carry out the calculation

significantly more accurate. These calculations were carried out by Goudsmit and Pauli \(^{31,35}\) and gave very satisfactory agreement with experiment (Schüler \(^{75}\)).

d) Experimental results

1. Determination of hyperfine structures from the interval rule and the intensity rule.

We shall now proceed to consider the experimental results and try to show how well the theoretical constructions given above agree with experiment. Measurements directly give the difference in frequencies between the individual hyperfine components of one and the same spectral line, corresponding to a transition between two gross terms with known \(J\). From these differences of vibration frequencies one can, in a known manner, arrive at the differences of the energies of the terms of the hyperfine structure, if one considers a line with a single gross unsplit term \(J=0\). In this case the differences of the vibration frequencies of the hyperfine components for other terms of the gross structure which, by virtue of the selection rule, must have \(J=0\), directly give the differences of the energies of the hyperfine-structure terms. From the frequency differences of other lines originating from these terms thus found, one can then analyze the whole system of terms consecutively. If there is no term with \(J=0\), as, for example, is the case for doublet spectra, then, to determine the splitting of the terms, the known method of constant frequency differences is applied, i.e. one tries to establish whether equal magnitudes of frequency differences correspond to different spectral lines obtained from one and the same gross term. These recurring frequency differences give the energy splitting of the common term. If the splitting of the gross term has been established in this way, then it is necessary to establish the value \(F\) for the individual terms of the hyperfine structure, and from this the mechanical moment of the nucleus can already be obtained. The determination of \(F\), as was indicated above, is carried out with the aid of the law on the number of components of the gross term, the interval rule, or the intensity rule.

Fig. 1. Part of the split term Bi I—\(6p^{3}\,{}^{2}D_{5/2}\) (Goudsmit and Back \(^{29}\))

In order to test the reliability of these laws, let us consider, as an example, the hyperfine splitting of the Bi term \(6P^{32}\,D_{5/2}\) (Back and Goudsmit \(^{29}\)) (Fig. 1). From the Zeeman effect, as well as from the number of components, we obtain for Bi \(I=9/2\). It follows from this that the energy differences of the hyperfine structures for the above-mentioned

of the lower term are in the ratio \(7:6:5:4:3\). The actual energy differences are plotted in Fig. 1, and we see that their ratios deviate by no more than \(\pm 5\%\) from the expected ones. As for this term, so also for other terms, and likewise for other elements, the interval rule proves to be well fulfilled. Pr, La (White \(^{21,96}\)) Mn (Ritschl and White \(^{102}\)), Cs—I (Kopfermann \(^{51}\)).

Usually the situation is that first, from the ratio of the intervals of one term, \(I\) is found, and then it is established that the ratio of the splittings of other terms is in agreement with the theoretical assumptions. And from the fact that the interval rule is well obeyed in a very large number of cases, it apparently follows that, in the splittings found for hyperfine structures, it is the result of the magnetic interaction of the electron shell with the nucleus.

In those cases where the value of \(I\) was determined from the interval rule and from the number of components, there were no quantitative measurements of intensities; in the literature there are only statements that the experimentally measured intensities appear to be in agreement with the theoretically expected ones. This comparatively modest result is due to the fact that precise intensity measurements are associated with difficulties and, in the case under consideration, cannot yield anything essential. The situation is different for elements with a large number of isotopes; here intensity measurements are absolutely necessary for clarifying the picture of the hyperfine structure. This is explained by the fact that in these cases an element is studied which does not show a homogeneous hyperfine structure; one part of the emitting atoms shows no hyperfine-structure splitting at all (isotopes with an even number of protons), while others do show such a splitting (isotopes with an odd number of protons). We shall show this using the example of cadmium, which was studied by Schuler and Brück \(^{79}\).

If, in considering the hyperfine structure of Cd, we proceed as is done on the following page, then very soon we encounter a sharp contradiction between the interval rule and the intensity law. Consider (Fig. 2) \(\lambda = 4678\), the final term of which is \(5^{3}P_{0}\); we find 3 hyperfine-structure lines with intensities \(2:10:1\). Since here only the upper term (\(J=2\)) can be split, no more than three components should be expected, whose intensity ratios should be \(3:2:1\), and the intensities should decrease in this order. Thus the fact that the middle component is the most intense and that the ratio of intensities is \(1:10\) is completely incompatible with the intensity laws of magnetic splitting. If, therefore, we accept the validity of the intensity laws for magnetic splitting—and we shall see below that they are valid not only qualitatively but also quantitatively—then we arrive at the conclusion that, in this case, the pattern of hyperfine

structures cannot be explained only by the magnetic splitting of the term \({}^3S_1\), but that here there is something fundamentally new. Schüler and Brück introduced this new point in the form of the assumption that there exist isotopes with different nuclear moments, namely that all odd isotopes have the same magnetic and mechanical moments, while isotopes with an even number of protons must have vanishingly small nuclear moments. On the basis of this assumption one can conclude from the splitting pattern that the nuclear moment of the odd Cd isotopes must be \(I=\frac{1}{2}\); the outer components belong to them. This assumption can be checked on the same line in the following additional way: from the intensity rule we find that, first, the two outer components must be related to one another as \(2:1\), and, second, the unsplit line must lie at the center of gravity* of the split system. Both requirements are sufficiently well satisfied. But still more can be extracted from the splitting of these lines. If all odd isotopes do indeed have moment \(1/2\), and all even ones—0, then the intensities of the middle components must be related to the sums of the intensities of the two outer components as the abundance of the even isotopes is to the abundance of the odd ones. For the case of cadmium this result has not yet been obtained by other methods, and one must be content with establishing that this ratio must be constant for all Cd lines, as indeed proved to be the case for all lines studied so far (Fig. 2, \(a\) and \(b\)). We see, therefore, that already from the analysis of a single line we arrive at very reliable results, which are further confirmed by the fact that with their aid it is possible to carry out, without contradictions, the analysis of other lines as well. We have carried out the analysis of this hyperfine-structure pattern in such detail specifically in order to show that it is based not on hypotheses or arbitrary assumptions, but is a direct consequence of measurements. At the same time it also becomes clear what enormous importance the interval rule and the intensity law have for obtaining these results.

We would also like to point out here that the intensity law for the hyperfine-structure splitting of Cd has been subjected to a careful test, and good agreement between experiment and theory was obtained. Thus, for example, for \(\lambda=4678\) (Fig. 2\(c\)) the theoretical intensity ratio \(a:b\) is equal to \(2:1\), while experiment gives \(2.08:1\). Here the intensity measurements were carried out by the known method using intensity standards (“blackening marks”). In other cases, when only a qualitative comparison is needed, for lines with a small difference in intensities—

* The center of gravity of a split system is determined by the expression \(\sum aJ=0\), where \(a\) is the distance of the component from the center of gravity and \(J\) is the theoretical intensity. The position of the center of gravity of the splitting pattern, as follows from the splitting and intensity rules (1)—(7), does not depend on the coupling strength.

it proves sufficient to determine the intensities by comparing the exposure times (Schüler and Jones \(^{81}\)). With the aid of methods for deciphering hyperfine structures, similar to those considered here in the example of cadmium, the hyperfine structures of Hg, Tl, Pb, and Ba were also analyzed. To be sure, there is here the additional complication that the unsplit lines of the even isotopes do not coincide, but have different wave numbers (and, correspondingly, the centers of gravity of the splitting patterns of the odd isotopes also do not coincide). This so-called isotope-shift effect we shall consider in the second part. Despite these complications, however, the hyperfine structures of these elements too can be unambiguously analyzed by means of the above-mentioned methods.

Fig. 2. Structure patterns of the CdI triplet \(5^3P_{0,1,2}-6^3S_1\), Schüler and Brück \(^{79}\), Schüler and Keyston \(^{84}\).

The fact that the distinction between magnetic hyperfine structure and isotope shifts is not an arbitrary interpretation of the structures under study is proved by Konfermann’s work \(^{52}\), who, using different kinds of lead (U—Pb and Th—Pb), was able directly to show which line should be assigned to which isotope.

We have already indicated above how well the interval rule and the law of intensities are confirmed by observation, and in general what an important role these laws play in deciphering hyperfine structures. The nuclear moments determined by this method are given in Table 4, and we shall now proceed to discuss them.

SUPERFINE STRUCTURE AND THE ATOMIC NUCLEUS

TABLE 1

Splitting of the terms of Hg I and Tl II for isotopes

Term	099 ($I = 1/2$) Hg I	203,205 ($I = 1/2$) Tl II	Remarks
$6s\cdot 6s\cdot {}^{1}S_{0}$	0	0
$6s\cdot 7s\cdot {}^{1}S_{0}$	0	0
$6s\cdot 8s\cdot {}^{1}S_{0}$	0	—
$6s\cdot 9s\cdot {}^{1}S_{0}$	0	—
$6s\cdot 7s\cdot {}^{3}S_{1}$	$+1070$	$+4980\ (S)\ (M)$	$+2250$: Pb III $(M)$
$6s\cdot 8s\cdot {}^{3}S_{1}$	$+1045$	—
$6s\cdot 9s\cdot {}^{3}S_{1}$	$S\ +1030$	$+4520\ (M)$
$6s\cdot 6p\cdot {}^{1}P_{1}$	$-181$	$+1010\ (M)$	Formerly $6s\cdot 8p\,{}^{1}P_{1}$ — now $5d^{9}\cdot 6s^{2}\cdot mp\,{}^{1}P_{1}$; formerly $6s\cdot 9p\cdot {}^{1}P_{1}$ — now $6s\cdot 8p\cdot {}^{1}P_{1}$ see Shenstone and Russell $^{91}$
$6s\cdot 7p\cdot {}^{1}P_{1}$	—	$-1270$
$(6s\cdot 8p\cdot {}^{1}P_{1})$	$-167$	—
$(6s\cdot 9p\cdot {}^{1}P_{1})$	$-386$	—
$6s\cdot 6p\cdot {}^{3}P_{0}$	0	—
$6s\cdot 7p\cdot {}^{3}P_{0}$	—	0
$6s\cdot 6p\cdot {}^{3}P_{1}$	$+727$	—
$6s\cdot 7p\cdot {}^{3}P_{1}$	—	$+4020\ (M)$
$6s\cdot 6p\cdot {}^{3}P_{2}$	$+758$	$+3210\ (M)$
$6s\cdot 7p\cdot {}^{3}P_{2}$	—	$+3472\ (S)\ (M)$
$6s\cdot 6d\cdot {}^{1}D_{2}$	$+860$	$+834\ (S)\ (M)$
$6s\cdot 7d\cdot {}^{1}D_{2}$	$+496$	$+620\ (M)$
$6s\cdot 8d\cdot {}^{1}D_{2}$	—	$+100\ (M)$
$6s\cdot 6d\cdot {}^{3}D_{1}$	—	$-2121\ (S)\ (M)$
$6s\cdot 7d\cdot {}^{3}D_{1}$	—	$-2210\ (M)$
$6s\cdot 8d\cdot {}^{3}D_{1}$	—	$-2250\ (M)$
$6s\cdot 6d\cdot {}^{3}D_{2}$	$-470$	$+550\ (S)\ (M)$
$6s\cdot 7d\cdot {}^{3}D_{2}$	—	$+780\ (M)$
$6s\cdot 8d\cdot {}^{3}D_{2}$	—	$+1660\ (M)$
$6s\cdot 6d\cdot {}^{3}D_{3}$	—	$+3330\ (M)$
$6s\cdot 7d\cdot {}^{3}D_{3}$	—	$+3390\ (M)$
$6s\cdot 8d\cdot {}^{3}D_{3}$	—	$+3450\ (M)$
$6s\cdot 5f\cdot {}^{1}F_{3}$	—	$+1480\ (S)\ (M)$
$6s\cdot 6f\cdot {}^{1}F_{3}$	—	$+2950\ (M)$
$6s\cdot 5f\cdot {}^{3}F_{2}$	—	$-2477\ (S)\ (M)$
$6s\cdot 6f\cdot {}^{3}F_{2}$	—	$-2580\ (M)$
$6s\cdot 5f\cdot {}^{3}F_{3}$	—	$-675\ (S)\ (M)$
$6s\cdot 6f\cdot {}^{3}F_{3}$	—	$-2240\ (M)$
$6s\cdot 5f\ {}^{3}F_{4}$	—	$+3310\ (M)$
$6s\cdot 6f\cdot {}^{3}F_{4}$	—	$+3350\ (M)$
$X_{2}$	—	$+642\ (S)$	$X_{2}$, probably $5d^{9}\cdot 6s^{2}\cdot mp$, formerly tentatively $6s\cdot 7p\cdot {}^{1}P_{1}$;
$Y_{1}$	—	$-203\ (S)$	$Y_{1}$ origin unknown

Data for Hg I: Schüler and Keyston $^{88}$; Schüler and Jones $^{81}$

Data for Tl II: Schüler and Keyston $^{86}$; MacLennan and Crawford $^{59}$

we want only first to briefly discuss the question of how well the quantitative theory of the interaction between the nuclear moment and the electron shell agrees with experiment.

2. Comparison of experimental results with the quantitative coupling theory. We have already noted that such a comparison is possible only to a very incomplete extent. In order to clarify the picture, the accompanying tables give the splittings of hyperfine structures for all terms so far investigated in Hg I, Tl I and II, Pb I and II; the material in the tables is arranged in such a way that each of them gives spectra belonging to identical electron configurations. For example, in Table 1 (p. 423) Hg I and Tl II, etc. The splitting of terms is given in units of \(10^{-3}\ \mathrm{cm}^{-1}\). Zero (0) denotes the absence of splitting, a dash (—) denotes that no investigation was carried out, and a plus (+) denotes that the energetically lowest term has the smallest value of \(F\); for an \(S\)-term this means that the magnetic moment of the nucleus depends on the rotation of positive charge. Generally speaking, spark spectra should show a larger splitting than the corresponding arc spectra, which is in fact essentially the case. This occurs because the density of optical electrons for one and the same term near the nucleus should be greater for an ion than for a neutral atom. Further, as was to be expected, generally speaking, especially large splittings are shown by those terms which contain one deep \(s\)-electron. This is found for the resonance lines of the alkali metals, where in general the \(P\)-term is barely split, whereas the \(S\)-term, on the contrary, shows considerable splitting. The same follows also from Table 2 (p. 425), where the values for Pb II and Bi III are given. For each element the term \(7^{2}S_{1/2}\) has a splitting \(4\frac{1}{2}\) times larger than \(7^{2}P_{1/2}\). For Tl I the comparisons are not so good, since the term \(7^{2}P_{1/2}\) is unknown, while the term \(6^{2}P_{1/2}\) has, apparently, an anomalously large splitting, of which we shall speak further below. If one compares the term \(2^{2}S_{1/2}\) with the term \(2^{2}P_{3/2}\), we shall see that here also it is substantially larger than for the term \(2^{2}P_{3/2}\). Analogous results, as also follows from theory, are obtained for the system with two electrons. Here, for Hg I and Tl II, the term \(7^{3}S_{1}\) shows the larger splitting. Thus, as is evident, these theoretical predictions are qualitatively confirmed experimentally.

In attempts at a quantitative treatment, as all work carried out so far has shown, one encounters severe contradictions, which we shall examine using a particularly typical example. According to the theory, the ratio of the splitting of the terms \(2^{2}P_{3/2}\) to the term \(2^{2}P_{1/2}\) should depend little on the nuclear charge and be independent of the principal quantum number. Thus, for the spectra Tl I and Pb II we should obtain practically the same ratio, which, when the relativistic correction is taken into account, equals

for an electron located near the nucleus, \(5:1\). For Bi III, since there \(I\) has a larger value than \(J\), we should have obtained a somewhat different value, namely \(10:3\). Experimentally, however, the following values were found (see Table 2).

TABLE 2

Tl I, Pb II, Bi III

Term	Tl I	Pb II	Bi III
\(6s^{2}\cdot 7s\cdot {}^{2}S_{1/2}\)	\(+403\)	\(+352\,(S)\)	\(+2360\)
\(6s^{2}\cdot 6p\cdot {}^{2}P_{1/2}\)	\(+707\)	—	—
\(6s^{2}\cdot 7p^{2}\cdot {}^{2}P_{1/2}\)	—	\(+77\,(S)\)	\(+520\)
\(6s^{2}\cdot 8p\cdot {}^{2}P_{1/2}\)	\(\sim +15\)	—	—
\(6s^{2}\cdot 9p\cdot {}^{2}P_{1/2}\)	\(\sim +11\)	—	—
\(6s^{2}\cdot 6p\cdot {}^{2}P_{3/2}\)	\(+8\)	—	—
\(6s^{2}\cdot 7p\cdot {}^{2}P_{3/2}\)	—	\(+22\,(S)\)	\(+310\)
\(6s^{2}\cdot 8p\cdot {}^{2}P_{3/2}\)	\(0\)	—	—
\(6s^{2}\cdot 9p\cdot {}^{2}P_{3/2}\)	\(0\)	—	—
\(6s^{2}\cdot 10p^{2}\cdot {}^{2}P_{3/2}\)	\(0\)	—	—
\(6s^{2}\cdot 6d\cdot {}^{2}D_{3/2}\)	\(0\)	—	—
\(6s^{2}\cdot 6d\cdot {}^{2}D\)	\(0\)	\(+713\,(K)\)	—
\(6s^{2}\cdot 5f\cdot {}^{2}F\)	—	\(0\,(K)\)	—
\(6s^{2}\cdot 6f\cdot {}^{2}F\)	—	\(0\,(K)\)	—
\(6s^{2}\cdot 8g\cdot {}^{2}G\)	—	\(0\,(K)\)	—
\(6s^{2}\cdot 9g\cdot {}^{2}G\)	—	\(0\,(K)\)	—
\(6s\cdot 6p^{2}\cdot {}^{2}D^{3}_{2}\)	—	\(\sim +950\,(S)(K)\)	—
\(6s\cdot 6p^{2}\cdot {}^{2}D^{3}_{1/2}\)	—	\(+956\,(S)(K)\)	—
\(6s\cdot 6p^{2}\cdot 1_{1/2}\)	—	—	\(+7500\)
\(6s\cdot 6p^{2}\cdot 2_{5/2}\)	—	—	\(+12500\)
\(6s\cdot 6p^{2}\cdot 3_{2}\)	—	—	\(+9300\)

Data for Tl I: Schuler and Keyston\(^{8}\)

Data for Pb II: Conferman\(^{52}\); Schuler and Jones\(^{82}\)

Data for Bi III: Fisher and Goudsmit\(^{18}\)

for Tl I \(90:1\), for Pb II \(3.5:1\), and for Bi III \(1.7:1\). Thus there can be no question of agreement, and because of this the question arises whether it makes sense to compare the hyperfine-structure splittings of different elements with one another and, from them, to judge the magnetic moments of nuclei—for example, by calculating the magnetic moments of Tl, Pb, and Bi from the splitting of the term \(7\,S\), Tl I and Pb II. If

if one nevertheless does this, it turns out that the nuclear moment of Pb is substantially smaller than that of Tl. From a comparison of the splittings Tl I and Tl II with Bi III and Bi V, MacLennan, McLay, and Crawford concluded\(^{60}\) that the latter have a magnetic moment 3 to 4 times larger than Bi. From a comparison of Tl II and Pb III they further concluded that Pb has a magnetic moment 4 to 5 times smaller than Tl.

Rose and Granath \(^{70a}\) compared Tl I, Pb I, and Bi I. From the splittings for \(P_{1/2}\) determined with the aid of Goudsmit’s theory (see p. 418), it follows that the ratio of the magnetic moment to the mechanical moment for the nuclei Bi and Pd is approximately the same, while for Tl it is almost twice as large. However, as has already been indicated, all these remarks should be regarded as purely qualitative. The extent to which caution is needed here is shown by the case of the term \(^{1}P_{1}\) of Hg I. The term \(6^{1}P_{1}\) has a splitting of 181; the terms hitherto designated as \(8^{1}P_{1}\) and \(9^{1}P_{1}\) show splittings of 167 and 386, and, generally speaking, as the orbital quantum number increases the splitting decreases. Here, however, for the term \(9^{1}P_{1}\) we suddenly find a splitting 2.3 times larger. This indicates that we are dealing with a perturbed term, and, in fact, Shenstone and Rense\(^{91}\) showed that the term hitherto designated as \(8^{1}P_{1}\) is in reality a complex term of the configuration \(5d^{9}6s^{2}mp\), which especially perturbs the neighboring terms.

The last splittings of the Bi III terms given in Table 2 are unusually large, which occurs because here we are also dealing with complex terms, the large splitting of which apparently depends on a single deeply lying unsaturated \(S\)-electron. We cannot enter into a more detailed discussion of the material reported here within the scope of this article, all the more so since its results would not be very fruitful. The qualitative conclusions about the magnitude of the magnetic moment of the nucleus that can be drawn on the basis of the measurements considered will be discussed in detail in the following chapter.

III. QUESTIONS OF NUCLEAR STRUCTURE

The measurements of hyperfine structures carried out so far directly give us three quantities characterizing the nucleus. Namely: the quantum number of the angular momentum of the nucleus \(J\), the sign of the magnetic moment of the nucleus, and its approximate magnitude.

If one assumes that the nucleus is built from protons and electrons and that the moments of the individual constituent parts (spin and orbit) combine in the same way as occurs for the outer electron shell of the atom, then one should expect that, at least for those nuclei which contain an even number of protons and an odd number of electrons, there exists a magnetic moment caused by the electrons.*

* Recently, in connection with the discovery of new particles—neutrons and positrons—the views on the structure of the atomic...

(On the contrary, with an even electron number, the electrons could be mutually saturated.) There exist four such elements: H 2, Li 6, B 10, and N 14. Of these, only lithium has been investigated. In this case the hyperfine structure was not detected at all. The magnetic moment of Li 6, as can be calculated from a comparison with Li 7 (Göttinger and Pauli[^35]), is at least five thousand times smaller than the Bohr magneton of the electron. This indicates that the electron in this nucleus does not manifest itself magnetically. However, on the basis of the theory of relativity, one should not have expected that the magnetic moment of an electron in the nucleus would have the same magnitude as that of a free electron; nevertheless, one could not have expected it to be 5000 times smaller, on the basis of the laws that govern the outer electron shell.*

All the remaining non-radioactive elements with an odd number of electrons also have an odd number of protons. For them, on the basis of the ordinary laws, it may happen that the proton and the electron saturate each other in such a way that the mechanical moment is equal to zero. Such a system also has no magnetic moment. This follows from the fact that a magnetic field that is not too strong cannot break the bond between the proton and the electron. This bond leads to the fact that, in such a field, the electron is in a stationary state with probability \(1/2\) in the direction of the field and with the same probability in the opposite direction.

Thus the state is non-magnetic (this can also be obtained directly from consideration of the eigenfunctions). However, experiment has shown that this case does not occur in nature, since these elements do not have a magnetic moment \(I\) equal to zero; nevertheless, in these nuclei the magnetic moment of the electron is not detected.

If we further note that, for the majority of nuclei, the sign of the magnetic moment is determined in such a way as if this moment depended only on the rotation of the positive charges and had a magnitude of the order of the proton moments, then we come to the conclusion that the electron spin in the nucleus is not magnetically detected.**

nucleus. In particular, the view is considered almost generally accepted that within the nucleus there are no electrons, but only neutrons and protons. Therefore all the material presented here at present needs substantial clarification. Translator’s note.

* The fact that the magnetic moment of the nucleus is many times smaller than the corresponding moment for the electron apparently proves beyond doubt that there are no electrons in the nucleus itself. Translator’s note.

** This circumstance excellently confirms the point of view already mentioned by us, that in the nucleus there are no electrons at all, but that it consists only of neutrons and protons. In doing so, we, of course, leave aside the question whether the neutron, for its part, is not an entity made up of a proton and an electron. Incidentally, the circumstance that, in the case that the neutron had such a structure, the binding forces between the consti-

In Table 4 (p. 430) are given the values of \(I\) known up to the present time. Very many of them have been determined from the measurement of hyperfine structures \((A)\). Those values that have been determined from measurements in band spectra are denoted by \((B)\). We must note that these values contain a specific inaccuracy. The method of band spectra assumes that the nuclei of some isotope of one element are all homogeneous (are in the same state). If this does not correspond to reality, then from measurements of band spectra no conclusion at all can be drawn concerning \(I\) (Heitler\(^{40}\)).

Up to now, however, nothing speaks in favor of the existence of such an inhomogeneity in nuclei. The determination of \(I\) from hyperfine structures seems to us comparatively reliable. Only a value of the moment equal to zero cannot be determined accurately by this means. Indeed, \(I=0\) means that no hyperfine structure has been found. But this may mean both that \(I\) is actually (Table 4) equal to zero and that the magnetic moment of the nucleus is only very small. In fact, in some elements, such as Al, Cl, P, K\(^{93,78}\), no hyperfine structure has been found; however, there are grounds permitting one to think that these elements have a mechanical moment.

Measurements with band spectra \(^{16,50,75}\) have also shown the presence of a magnetic moment. Thus one must assume that these nuclei have an especially small magnetic moment.

Table 4 further shows that all elements investigated up to now with an even number of protons have no splitting at all; consequently, either for them the value \(I=0\), or else they have a considerably smaller magnetic moment than elements with an odd number of protons. For elements with a large value of the nuclear charge (Hg, Pb), measurements for these hypothetical moments give a value no greater than 1% of the magnetic moment of isotopes with an odd number of protons.

From this one might conclude that for them the value is actually equal to zero. This, however, is contradicted by the fact that for N 14 the value \(I=1\) has been determined from band spectra. Up to now this is the only case of an element with an even number of protons for which a value of \(I\) different from zero has been determined.

Further, the table shows that the values of \(I\) for elements with an odd number of protons are half-integral. From this one may perhaps conclude that the nuclei of all elements with an even number of protons have values of \(I\) equal either to zero or to an integer.

These results are very significant, even if one relies

on its constituent parts would be comparable in magnitude with those forces which take place in the collision of a neutron with matter, and, consequently, there would have to exist a finite probability of destruction of the neutron in such processes, which in fact is not observed experimentally. Analogous considerations apply also to protons. Translator’s note.

TABLE 3

Pb I and Bi II

Term	Pb I	Bi II
\(6s^{2}\cdot 6p^{2}\cdot {}^{1}S_{0}\)	\(0\,(S)\)	—
\(6s^{2}\cdot 6p\cdot 7p\cdot {}^{3}S_{1}\)	—	\(-360\)
\(6s^{2}\cdot 6p^{2}\cdot {}^{3}P_{0}\)	\(0\,(K)\)	—
\(6s^{2}\cdot 6p\cdot 8p\cdot {}^{3}P_{0}\)	\(0\,(S)\)	—
\(6s^{2}\cdot 6p\cdot 9p\cdot {}^{3}P_{0}\)	\(0\,(S)\)	—
\(6s^{2}\cdot 6p^{2}\cdot {}^{3}P_{1}\)	\(-117\,(K)\)	—
\(6s^{2}\cdot 6p\cdot 8p\cdot {}^{3}P_{1}\)	\(-155\,(S)\)	\(-140\)
\(6s^{2}\cdot 6p\cdot 9p\cdot {}^{3}P_{1}\)	\(\sim -150\,(S)\)	—
\(6s^{2}\cdot 6p^{2}\cdot {}^{3}P_{2}\)	\(+225\,(K)\,(S)\)	—
\(6s^{2}\cdot 6p^{2}\cdot {}^{1}D_{2}\)	\(+65\,(S)\)	—
\(6s^{2}\cdot 6p\cdot 7p\cdot {}^{1}D_{2}\)	—	\(+780\)
\(6s^{2}\cdot 6p\cdot 7p\cdot {}^{3}D_{1}\)	—	\(+1020\)
\(6s^{2}\cdot 6p\cdot 7p\cdot {}^{3}D_{2}\)	—	\(+2500\)
\(6s^{2}\cdot 6p\cdot 7p\cdot {}^{3}D_{3}\)	—	\(+370\)
\(6s^{2}\cdot 6p\cdot 7s\cdot {}^{1}P_{1}\)	\(-60\,(S)\)	\(-530\)
\(6s^{2}\cdot 6p\cdot 7s\cdot {}^{3}P_{0}\)	\(0\,(K)\,(S)\)	\(0\)
\(6s^{2}\cdot 6p\cdot 7s\cdot {}^{3}P_{1}\)	\(+444\,(K)\,(S)\)	\(+3910\)
\(6s^{2}\cdot 6p\cdot 8s\cdot {}^{3}P_{1}\)	\(+290\,(S)\)	—
\(6s^{2}\cdot 6p\cdot 7s\cdot {}^{3}P_{2}\)	—	\(2180\)
\(6s^{2}\cdot 6p\cdot 6d\cdot {}^{3}D_{1}\)	—	\(-1650\)
\(6s^{2}\cdot 6p\cdot 6d\cdot {}^{3}D_{1}\)	\(+255\,(K)\)	\(+2550\)
\(6s^{2}\cdot 6p\cdot 6d\cdot {}^{3}F_{2}\)	\(-130\,(K)\)	\(+1980\)
\(\bullet\,6s^{2}\cdot 6p\cdot 6d\cdot {}^{3}F_{3}\)	\(\sim +250\,(K)\)	—
\(6s^{2}\cdot 6p\cdot 5f\cdot {}^{3}F_{2}\)	—	\(-160\)
\(6s^{2}\cdot 6p\cdot 5f\cdot {}^{3}F_{3}\)	—	\(-700\)
\(6s^{2}\cdot 6p\cdot 5f\cdot {}^{3}G_{3}\)	—	\(+1950\)
\(6s\cdot 6p^{3}\cdot {}^{3}D_{2}\)	—	\(+8200\)

Data for Pb I: Kopfermann \(^{53}\); Schüler and Jones \(^{82}\).

Bi II: Fisher and Goudsmit \(^{18}\).

TABLE 4

Summary of firmly established nuclear moments

Z	Element	Isotopes	Nuclear moment	Author
1	H	1	\(1/2\)
2	He	4	0
3	Li	6	0	Schuler \((A)^{75}\), Harvey and Jenkins \((B)^{39}\)
3	Li	7	\(3/2\)	Schuler \((A)^{75}\), Harvey and Jenkins \((B)^{39}\)
7	N	14	1	Kronig \((B)^{53}\)
8	O	16	0
9	F	19	\(1/2\)	Gallo and Monk \((B)^{19}\)
11	Na	23	\(5/2\) \((3/2?)\)	Schuler \((A)^{85}\)
15	P	31	\(1/2\)	Jenkins and Ashleigh \((B)^{50}\)
17	Cl	35	\(5/2\)	Elliott \((B)^{16}\)
25	Mn	55	\(5/2\)	White and Ritschl \((A)^{102}\)
29	Cu	63, 65	\(3/2\)	Ritschl \((A)^{69}\) and Shenstone \((A)^{90}\)
31	Ga	69, 71	\(3/2\)	Jackson \((A)^{48}\)
33	As	75	\(3/2\)	Tolansky \((A)^{94}\)
35	Br	79, 81	\(3/2\)	de Bruin \((A)^{11}\) and Tolansky \((A)^{92}\)
37	Rb	85; 87	\(3/2?\)	Jackson \((A)^{49}\)
48	Cd	111, 113	\(1/2\)	Schuler and Brook \((A)^{79}\)
48	Cd	110, 112, 114, 116	0	Schuler and Brook \((A)^{79}\)
49	In	115	\(5/2\)*	Campbell and Bacher \((A)^{13}\)
51	Sb	121, 123	\(3/2?\)	Loewenthal \((A)^{56}\)
53	J	127	\(9/2\)	Tolansky \((A)^{92}\)
55	Cs	133	\(7/2?\)	Kopfermann \((A)^{51}\) and Jackson \((A)^{44}\)
56	Ba	137	\(3/2?\)	Ritschl and Sawyer \((A)^{70**}\) \(1/2\)
56	Ba	136, 138	0	Ritschl and Sawyer \((A)^{70**}\) \(1/2\)
57	La	139	\(5/2\)	White \((A)^{97}\)
59	Pr	141	\(5/2\)	Gibbs, White and Ruedy \((A)^{21}\)
75	Re	187, 189	\(5/2\)	Gremmer and Ritschl \((A)^{33}(S)\)
79	Au	197	\(3/2?\)	Ritschl \((A)^{69}\)
80	Hg	199	\(1/2\)	Schuler and Keyston \((A)^{88**}\)
80	Hg	201	\(3/2\)	Schuler and Keyston \((A)^{88**}\)
80	Hg	198, 200, 202, 204	0	Schuler and Keyston \((A)^{88**}\)
81	Tl	203, 205	\(1/2\)	Schuler and Brook \((A)^{78}\)
82	Pb	207	\(1/2\)	Kopfermann \((A)^{52}\)
82	Pb	204, 206, 208	0	Kopfermann \((A)^{52}\)
83	Bi	209	\(9/2\)	Goudsmit and Back \((A)^{29,3}\)

\((A)\) indicates that the nuclear moment was determined from atomic lines.
\((B)\) that it was determined from band spectra.
\(*\) One of the authors (Schuler) and Jones disputes the value found by Ritschl and Sawyer for Ba II, and considers that, on the basis of the center-of-gravity law, proceeding from the observed intensities and using comparison with other doublet spectra, it is more probable that Ba\(_{137}\) has a nuclear moment equal to \(1/2\).
\((S)\) Zeeman, Gisolph and de Bruin \(^{104}\) investigated the Zeeman effect and established the value I.
\(**\) Murakawa \(^{26}\) investigated the Zeeman effect for \(\lambda_{4047}\) and established the value I.
\(*\) Note added in proof: In Phys. Rev. 40, 1040, 1932, Campbell corrects the value \(9/2\) to \(11/2\).

only to cases that have so far been firmly established experimentally. According to the usual conceptions, one would have expected that it was not the number of protons, but the total number of the particles composing the nucleus, electrons and protons, that would determine the integrality (even total number) or half-integrality (odd total number) of the moment \(I\). But this is evidently not so. This can be seen especially clearly in the cases of Cd, Hd, and Pb. Their isotopes with an odd number of protons also have an odd number of electrons and, consequently, should have had an integral value of \(I\). Yet there is no doubt at all that the values of \(I\) for them are half-integral. This discrepancy occurs also for nitrogen. Nitrogen has 14 protons and 7 electrons, i.e., an odd number of constituent particles, and, despite this, it has an integral value of \(I\). And if the absence of hyperfine structure in \(\mathrm{Li}^6\) is explained by the fact that for it the value \(I=0\), then here too the same discrepancy should have appeared, since \(\mathrm{Li}^6\) has 6 protons and 3 electrons. Thus, with respect also to the determination of the mechanical moments of nuclei, the situation is as if the electron spin in the nucleus were imperceptible. We would arrive at similar results if we began to investigate whether atomic nuclei obey Bose or Fermi statistics. Nuclei with an even number of constituent particles should have obeyed Bose statistics, and those with an odd number Fermi statistics (Ehrenfest and Oppenheimer\({}^{15}\)). Analysis of band spectra, however, has shown that nitrogen nuclei, although they consist of an odd number of constituent particles, obey Bose statistics. Thus here too everything appears as if the electrons in the nucleus should not be taken into account.

Consequently, the determinations of magnetic moments, spins, and the statistics of nuclei have yielded a result which leaves no doubt that the laws of coupling between electrons and protons in their usual form are inapplicable to the nucleus.

To explain the results obtained, one may take the point of view that, in determining the properties of the nucleus, electrons cannot be regarded as independent individuals. Then all the results known so far can be explained in the following way.

The basis for such an explanation is, roughly speaking, the following: the energy of an electron \(mc^2 = 0.5\) million volts. Compared with the binding energies present in the nucleus, this is not a very appreciable quantity. Under such circumstances it seems difficult to regard an electron in the nucleus as a separate individual.

To explain the discrepancies mentioned, however, one may also take a completely different point of view. The ordinary laws of interaction of individual particles are valid in the nucleus as well. It is not these laws that are erroneous, but our conception of the elementary constituent parts of the atomic nucleus. According to this, the nucleus must consist not only of protons and electrons, but also, at least, of yet another kind of elementary particles, namely neutrons (Pauli); a hypothesis which recently, thanks to the discovery of neutrons in atomic disintegration, has acquired great weight.

probability. In order to understand the indicated results it would be necessary to assume that the neutron has a spin equal to \(\frac{1}{2}\frac{h}{2\pi}\) and obeys Fermi statistics. Within the framework of this survey we can point only to the existing possibilities, without entering into a detailed analysis of these questions. We would like to note here also that one can speak of an “explanation” only in the case where the neutron is a new elementary particle. If, however, the neutron consists of an electron and a proton, then all the difficulties mentioned remain in force as before.

Let us now proceed to the discussion of the question of the nature of the mechanical and magnetic moments of the nucleus. The simplest assumption would be that they are made up of the orbital moment and the spin of the protons. For the light elements, Briden \(^{12}\) and White \(^{100}\) tried to construct, by analogy with the outer electron shell, a model of a proton shell, in favor of which, as it seems to them, the values of \(I\) known up to now speak. However, in this way a systematics of all the values of \(I\) known up to the present has not yet been achieved. Experimentally the following has been established: mechanical moments occur from zero to \(9/2\). Elements with an odd number of protons, as has already been indicated, have half-integral spins. For greater clarity we give in Table 5 (p. 433) how the spin values are distributed over the various isotopes. It is striking that elements with even and odd isotopes do not have large values of \(I\), and that always (with the exception of Hg 199 and 201) the odd isotopes of one element have identical values of \(I\). In addition, no clearly expressed regularity is seen in passing from element to element; on the contrary, the impression remains of a certain disorder. This is clearly seen, for example, for the nuclei 207 and 208, \(Z=82\), and 209, \(Z=83\): 207 has spin \({}^{1}/_{2}\), 208 spin zero, and 209 suddenly spin \({}^{9}/_{2}\).

With regard to magnetic moments the following has been established: only nuclei with an odd number of protons have a noticeable magnetic moment. It is unknown whether the absence of a magnetic moment in nuclei with an even number of protons is due to the absence of spin altogether, or depends on the very small magnetic action of the moment of rotation. The sign of the magnetic moment is for the most part positive, and only Cd 111 and 113 and Hg 201 have a negative magnetic moment. Since this is established from the \(S\) terms of these elements, the opposite sign of their moments cannot be inferred from an anomaly in the electron shell. This is especially noticeable for Hg 199. Here Hg 199 has a positive sign, and Hg 201 a negative one. Experimentally this is manifested in the fact that the terms of 201 lie reversed with respect to the terms of 199.

The absolute magnitude of the magnetic moment is known only very approximately, since the theory of the interaction between the nucleus and the electron shell, which has to be used, is quantitatively unsatisfactory, with the exception of \(\mathrm{Li}^{+}\). If one assumes that the theory gives for the \(S\)-term, at least, the correct

TABLE 5

Nuclear Moment and Isotopes

Nuclear moment	Simple elements	Elements with two odd isotopes	Elements with even and odd isotopes¹
$I = {}^{9}/_{2}$	Bi 209 I 127	—	—
$I = {}^{7}/_{2}$	Cs 133	—	—
$I = {}^{5}/_{2}$	Pr 141 La 139 In 115² Mn 55 Na 23	Re 189, 187 Cl 37, 35 (39)	—
$I = {}^{3}/_{2}$	Au 197 As 75	Sb 123, 121 (?) Rb 87, 85 Br 81, 79 Ga 71, 63 Cu 65, 63	Hg 201 (?) Ba 137 Li 7
$I = {}^{1}/_{2}$	P 31 F 19	Tl 205, 203	Pb 207 Hg 199 H 1

1. Even isotopes have $I = 0$ ($\mathrm{H}_2$?; $I = 0$?).
   Continuation of the table of Schüler and Keyston.
2. Note: for In, Campbell has recently discussed the possible values ${}^{9}/_{2}$ or ${}^{11}/_{2}$.

order of magnitude, then for the nuclear moment one obtains, generally speaking, a quantity of the same order as the proton moment. For Li$^{+}$, where the calculations are considerably more reliable, we arrive at a value approximately 1:600 of the electron moment.

In some cases, however, we evidently also arrive at very small values of the magnetic moment (Al, K, Cl, P). In general, the impression remains that nuclei with large $I$ do not necessarily have a very large value of $\mu$. This, perhaps, indicates that large values of $I$ are obtained not as a result of the parallel orientation of the spins of several protons; the same is probably also suggested by the fact that no splitting has yet been found in even isotopes.

A special discussion is required by the comparison of the values of $\mu$ for isotopes. It has been shown that all odd isotopes have not only identical $I$, but also approximately equal $\mu$; the only exception is Hg 199 and Hg 201. Hg 199 has spin ${}^{1}/_{2}$, 201—${}^{3}/_{2}$, and their magnetic moments are in the inverse ratio. Since these are isotopes, it may be assumed that their electronic

shells are practically identical, and therefore the magnetic moments of Hg 199 and 201 can be compared with one another. The splitting of the terms of Hg 199 and 201 is given in Table 6. The total splitting is obtained

TABLE 6

Total splitting of spectral terms, Schüler and Jones81

Term	Total splitting	In \(10^{-3}\ \mathrm{cm}^{-1}\)
\(6^1S_0\) to \(9^1S_0\)	0	0	—
\(7^3S_1\)	1070	\(-1070\)	\(-0.89\)
\(8^3S_1\)	1045	\(-1035\)	\(-0.89\)
\(6^1P_1\)	\(-181\)	165	\(-0.98\)
\((8^1P_1)\)	\(-167\)	172	\(-0.86\)
\((9^1P_1)\)	\(-386\)	385	\(-0.89\)
\(6^3P_0\)	0	0	—
\(6^3P_1\)	727	\(-725\)	\(-0.89\)
\(6^3P_2\)	758	\(-880\)	\(-0.86\)
\(6^1D_2\)	860	\(-795\)	\(-1.03\)
\(7^1D_2\)	496	\(-531\)	\(-0.93\)
\(6^3D_2\)	\(-470\)	507	\(-0.93\)

from formula (4)

\[ \Delta W=\mu H^{(0)}\frac{2J+1}{J}\quad \text{for } J\gg I, \]

and

\[ \mu H^{(0)}\frac{2I+1}{I}\quad \text{for } I\gg J. \]

From this the quantity

\[ \frac{\mu_{199}}{\mu_{201}} . \]

can be calculated. The values obtained in this way are given in the fourth column of Table 6. It is seen from it that, for all terms, approximately

\[ \frac{\mu_{199}}{\mu_{209}}=0.9 . \]

For individual terms slightly different results are obtained. The differences found for \(6^1P_1\) and \(6^1D_2\) lie beyond the limits of experimental error and, consequently, are undoubtedly real. Thus it is clear from this that the foundations of the theory are only approximately correct*. It is important to establish that the absolute values of \(\mu\) 199 and 201 are approximately equal, while their spins are respectively \(1/2\) and \(3/2\). Especially noteworthy is the result that, up to now, all odd isotopes of any element have the same absolute value of \(\mu\). This could be understood if one always had to deal with a spin equal to \(1/2\), which could be

Note added in proof.* For \(6^1D_2\) we have a perturbation of the term (see the note on p. 413) and, consequently, a deviation from the law of magnetic interaction. The behavior of \(6^1P_1\) has not yet been clarified.

in such a case assign to the proton. But isotopes with large \(I\) also have identical \(I\) and \(\mu\). In going from one element to some other neighboring one, the value of \(I\), and also, apparently, \(\mu\), often changes.

The idea of constructing the nuclear moment exclusively from the spins of protons seems to us to be refuted by experiment. Starting from it, it would be impossible to explain the existence of negative moments. One would also have to expect that, if it were correct, \(\mu\) would be proportional to \(I\), which is not observed. We wish here to examine somewhat more closely the possibility that the rotation of the nucleus as a whole is combined with the spin of the proton.

Let us denote the orbital quantum number of the nucleus by \(l_k\); in that case this orbit should be assigned a magnetic moment of magnitude

\[ \mu m l_k = m l_k \frac{e h Z}{4\pi c M}, \tag{10} \]

where \(M\) is the mass of the nucleus and \(Z\) is the charge of the nucleus. As for the orbits of positive and negative charge, it is necessary to take into account that these charges may be arranged in various ways relative to one another. It is therefore necessary to multiply the right-hand side of (10) by a quantity \(g(l_k)\), which may be either positive or negative (when the electrons lie very far away). In addition, it is also necessary to take into account the moment which arises from the spins of the protons. It is equal to

\[ \mu m s_k = 2 m s_k \frac{e h}{4\pi M_0 c}, \tag{11} \]

where \(s_k\) is the geometrical sum of the spins of the protons, and \(M_0\) is the mass of one proton. Composing from \(s_k\) and \(l_k\) the total nuclear moment \(I\), we obtain the total moment from the \(g\)-formula of Landé,

\[ \eta = \frac{e h m}{4\pi M_0 c} \left( \frac{g(l_k) Z \frac{M_0}{M}\,[I(I+1)+l_k(l_k+1)-s_k(s_k+1)]}{2I(I+1)} + \frac{2[I(I+1)+s_k(s_k+1)-l_k(l_k+1)]}{2I(I+1)} \right). \tag{12} \]

From this formula it is seen that negative nuclear moments can also arise in the case when \(g(l_k)\) is positive, but arbitrarily small, and when, moreover, \(I\) and \(s_k\) are smaller than \(l_k\). This may be expressed vividly by saying that the moment is negative when \(s_k\) is directed oppositely to \(l_k\) and when the magnetic moment \(l_k\) is smaller than \(s_k\). This gives the possibility of explaining the change of sign in passing from Hg 199 to Hg 201. For this it is necessary to substitute for \(l_k\) the value \(2\), and for \(s_k\), \(1/2\). For \(I=3/4\) in this case one obtains a negative magnetic moment of magnitude \(3/5\) of the proton moment. It is assumed here that the moment \(l_k\) is vanishingly small. It must be emphasized that such attempts

explain nuclear moments must be regarded with the greatest caution. As we see, here one has to encounter such difficulties that the above arguments may be regarded not as a substantiated theory, but only as a discussion of possibilities.

IV. ISOTOPIC SHIFT

We now proceed to the study of the phenomenon of the so-called isotopic shift, i.e., the fact that the corresponding

Fig. 3. Separation of the isotopes of Hg, Tl, and Pb by means of isotopic shift.

(a) Isotopes—Hg. Violet. Red.
\(\lambda 6072\); 5 mm standard. Schuler and Keyston (86).
\(\lambda_{204}: 6.65\%\), \(\lambda_{202}: 29.97\%\), \(\lambda_{200}: 23.57\%\), \(\lambda_{198}: 9.89\%\), \(\lambda_{199} + \lambda_{201}: 13.94\%\).

(b) Isotopes—Tl. Red. Violet.
\(Tl_{205}\), \(Tl_{203}\).
\(\lambda 5351\); 10 mm standard. Schuler and Keyston (86).

(c) Isotopes—Pb.
Ordinary lead. Uranium lead.
\(\lambda 4058\); 8 mm standard. Kopfermann (52).

(d) Isotopes—Pb. Red. Violet.
\((\alpha)\) Weakly illuminated; \((\beta)\) Strongly illuminated.
\(Pb_{204}\), \(Pb_{206}\), \(Pb_{208}\), \(Pb_{207}\).
Without components of the transverse group of fringes.
\(\lambda 5201\); 15 mm standard. Schuler and Jones (82).

the spectral lines of different isotopes of one element have different frequencies. Before turning to the consideration of this phenomenon, let us give several photographs of spectral lines in order to show what this shift looks like when a Fabry and Perot standard is used (Fig. 3). In Fig. 4, with typical examples of Hg and Pb lines, the relative arrangement is shown of the structural pattern of the even isotopes and the centers of gravity of the splitting pattern of the odd ones. Up to the present time there is still no theory that would

explained the isotope shift in isotopes with large ordinal numbers. We shall therefore have to confine ourselves here chiefly to reporting the experimental data on the determination of the isotope shift and to indicating their relation to individual terms.

The fact that a spectral line characterizing a transition between two definite states has, for different isotopes of one and the same element, different frequencies has been known for a long time. It is also known that this dependence of the frequencies on the mass

Fig. 4. Example of the asymmetric superposition of the center of gravity of odd isotopes between even ones. Schuler and Keiton[^88]; Schuler and Jones[^82]

of the nucleus cannot be explained by the Bohr–Sommerfeld theory, which takes into account the motion of the nucleus.

The isotope shift of a spectral line indicates that the binding energy of the electron in a definite state is different for different isotopes. We shall denote this difference in binding energies as the isotope shift of a term. This notation is not unambiguous, since it is also necessary to indicate which particular electron has broken its bond with the atom. Thus, one may expect that we shall obtain a different value of the isotope shift in the case where we detach a weakly bound electron from the nucleus than in the case where we remove a strongly bound one. In what follows, by the isotope shift of a term we shall always mean the difference between the ionization works of the most weakly

of the bound electron. In those cases where other electrons are also removed from the atom (complex terms), we shall make a special reservation. But even this definition of the isotope shift is not entirely unambiguous. As is known, there exist states with several ionization potentials (in them the ion may remain in different states, for example Pb). In such a case it is also necessary to specify which particular ionization is being considered. In these cases, however, the distinction in isotope shift does not appear, since terms with equal \(n\) and \(l\), but with different \(j\), apparently have approximately the same isotope shift.

How is the isotope shift of a term to be determined? Its determination would be extremely simple if the effect consisted only in the energy values being multiplied by a constant factor (as in the ordinary theory that takes into account the motion of the nucleus). In that case, if the shift of some one line were known, the shift of the whole term would immediately be obtained from it.

But this is not so. It is therefore necessary to determine spectroscopically the ionization potentials and to establish their differences for the isotopes. In practice one starts from the term with the greatest possible principal quantum number (Schüler and Keyston \(^{88}\)). For it one can already theoretically expect that it will possess a vanishingly small isotope shift. This assumption can also be confirmed experimentally, since it can be established that all lines originating from this term and leading to other terms with high quantum numbers show no shift effect. From these high terms one then passes successively to lower ones. If it is found that a spectral line whose initial term is not shifted has a shift, then this shift must be attributed to the deeper final term. From this term, with its now known shift, one may pass to further terms and thus determine the isotope shift of all terms. In this way the tables presented have been obtained. We now turn to their discussion.

We shall begin with the isotope shift of Hg \(^{88,91}\). Hg has four spectroscopically detected even isotopes, 198, 200, 202, and 204, and two odd isotopes, 199 and 201. In Fig. 5 the shift of terms for the even isotopes is given. It may be noticed that in the isotope of smallest mass the electron is bound most strongly.

The quantities given in Fig. 5 mean the following: it is assumed that the \(8\) and \(9\,^{1}S_0\) terms show no shift, which means that to remove an electron from these levels to infinity the same energy is required for all isotopes. If there were some energy difference here, it would enter as an additive constant into the shift of all terms. We see that, relative to the term \(9\,^{1}S_0\), the terms \(8\,^{1}S_0\), \(6\) and \(7\,^{1}D_2\), \(6\,^{3}D_{2}\), \(6\,^{3}P_{0,1,2}\), and \(6\,^{1}P_1\) have no shift; since it is very improbable that all of them have a shift equal to the shift of the term \(9\,^{1}S_0\), it is natural to suppose that this term too is not shifted. Under this assumption, for the term \(6\,^{1}S_0\) one obtains—

there is a displacement of \(160\cdot 10^{-3}\) cm. \({}^7S_0\) and \({}^7S_0\) both have a somewhat smaller displacement, approximately about \(30\cdot 10^{-3}\) cm. We see, therefore, that with an increase of the quantum number there is a strong decrease in the displacement. Terms lying energetically below \(7^3S_1\), namely

Fig. 5. Effect of isotopic displacement in Hg I. Schuler and Keyston \(^{83}\)

\(6^1P_1\) and \(6^3P_{0,1,2}\), show no displacement. The term \(8^2S_0\) is given here with a displacement of about 20, while for the term \(8^1S_0\) it has been set equal to zero. The displacement of the term \(8^3S_1\) was not measured directly, but only determined from the distance between the centers of gravity of the odd isotopes, so that a comparison between \(8^1S_0\) and \(8^3S_1\), apparently, is not permissible.

There remain two more terms showing a noticeable isotopic displacement, namely the terms \(5d_9, 6s^2mp\) and the higher term in the \(1^p_1\) series. Formerly these terms were denoted as 8 and \(9^1P_1\). It was extremely strange that

these terms showed a large displacement, whereas \(6^{1}P_{1}\) showed none; an analogous anomaly in the behavior of these terms was also established by us already on p. 417 in considering the magnetic splitting. The explanation of this is that these terms are perturbed. (This had already long been known, because for them the energy values do not agree with those expected for the \(^{1}P_{1}\) series.) Recently Shenstone and Russell \(^{91}\) have shown that the term formerly denoted as \(8^{1}P_{1}\) is a complex term, while that formerly denoted as \(9^{1}P_{1}\) is the perturbed \(8^{1}P_{1}\).

Therefore the isotope displacement of these terms cannot be explained solely by the removal of one electron from the level \(P\); it is also caused by the transition of an electron from the \(6s^{2}\) shell to the \(5d^{2}\) shell. The isotope displacement of this term means, consequently, that more energy is needed to transform the isotope 198 into the normal ion than is needed for the heavier isotopes. The same must be assumed for the neighboring perturbed terms.

If, for the anomalous term \(5d^{9},\ 6S^{2}mp\), one may assume, for removal of the electron \(mp\), that the isotope displacement, as in the normal \(6^{1}P_{1}\)-term, is equal to zero, then it would depend on the transition from the \(6s^{2}\) shell to the deeper \(5d^{9}\) shell, and thus the energy released in this process would be smallest for 198. We shall return to this once more in the discussion.

Fig. 6. Effect of isotope displacement in Hg [Schüler and Jones \(^{83}\)].

In general, one may say (Schüler and Jones \(^{83}\)) that the displacements between two different even isotopes for one and the same term are approximately equal. The centers of gravity of the magnetically split terms of the odd isotopes do not, however, lie midway between two even ones; rather, the term of an isotope always lies closer to the nearest smaller even isotope. The distance between the centers of gravity of the terms of two odd isotopes is approximately equal to the distance between two even isotopes. The asymmetry resulting from this is visible in Fig. 4, where several examples of spectral lines of different isotopes are given. In this case, for odd iso-

of isotopes, instead of the magnetic splitting of the spectral lines, their centers of gravity are plotted.

Let us now turn to the effect of isotopic displacement (Fig. 6) for HgII\(^{83}\). Four terms \(6^2S_{1/2}\), \(6^2P_{3/2}\), \(7^2S_{1/2}\), \(7^2P_{3/2}\) have, as was found experimentally\(^{83}\), approximately equal isotopic displacements. It may apparently be assumed that their displacements are equal to zero. The two other terms that have been measured are complex terms: the deeper of them, which shows a very large isotopic displacement, is the term \(5d^9, 6s^2\); the higher one, denoted by \(7^2P_{3/2}\), must, according to McLennan, represent an analogous complex term. If this explanation is accepted, then for the isotopic displacement the following is obtained: for the removal of an electron from the shell \(6s^2\), considerably more energy is required for isotope 198 than for the heavier ones, quite analogously to what takes place in the \(6s^2\) shell of HgI. Further, we find that in the transition of an electron from the shell \(6s^2\) to the shell \(5d^9\), the energy liberated is greater for isotopes of smaller mass, analogously to what takes place for the complex term considered above. If this explanation is correct, then it means that for the removal of an electron from the shell \(6s^2\) more energy is required for the isotope of smaller mass, regardless of whether the electron passes into one of the higher terms or only into the \(5d\) shell.

Fig. 7. Effect of isotopic displacement in TlI. Schuler and Keyston\(^{86}\), Jackson\(^{44}\)

Fig. 7 gives us the isotopic displacement of TlI\(^{86}\). Here both terms \(6^2P_{1/2,\;3/2}\) show approximately the same displacement; it must, however, be pointed out that the energy required for the liberation of the electron is in this case greater for the isotope of larger mass (the reverse of what takes place in Hg).

In the scheme of terms of TlII (Fig. 8), whose outer configuration is identical with HgI, isotopic displacement is found only in two complex terms\(^{36}\). Of these, these are again the terms \(5d^9 = 6s^2 \cdot 6p\), i.e., a term analogous to the complex terms of HgI and showing displacement in the same direction as in the case of HgI. The fact that for

The TlII displacement direction in the opposite sense, perhaps, can be explained by the fact that the displacement is determined by the configuration of the terms. It is remarkable that for the outer terms no displacement is noticeable. Meanwhile, by analogy with HgI one would have expected it, at least for the terms \(7^1S_0\) and \(7^3S_1\). In any case, a displacement comparable with the displacement for HgI is not observed here, unless one makes

Fig. 8. Effect of isotope shift in TlII. Schuler and Keyston \(^{86}\)

the highly improbable assumption that a whole series of quite different terms have the same displacement.

Fig. 9 gives the isotope shift of PbI \(^{82,52}\). Here the terms with one \(6p\)-electron show the greatest displacement, namely all four of these terms are the same (\(\Delta \nu = 90\)), and only the \(^{1}S_0\)-term shows a somewhat smaller one (\(\Delta \nu = 72\)). Further, the terms with one \(8p\)-electron show a further displacement \(\Delta \nu = 10\)—15. For the terms with one \(9p\)-electron the displacement is assumed to be zero. This also gives, for the terms with one \(7\)—\(8s\)-electron, a displacement equal to zero, and only one term with one \(7s\)-electron shows a small

effect. Finally, terms with one \(6d\)-electron also have an isotope shift. For PbI the direction of the shift is opposite to that which Hg has.

We find an analogous direction also in the spectrum of PbII \(^{52, 82}\), which is analogous to TlI. Unfortunately, an exact comparison cannot be carried out with this spectrum, since for some of the terms the shift

Fig. 9. Effect of isotope shift in Pb I. Schuler and Jones \(^{82}\), Kopfermann \(^{52}\)

with respect to the absolute value has not been established, for these terms are not connected with others by any transitions. In Fig. 10 these terms are denoted by the sign \(+x\).

It must also be pointed out that the designations of terms are in many cases not entirely reliable.

For other elements the isotope shift has been established for H, Li, Ne, Cl, K (see \(^{83}\)), but since these observations are for the most part

were made only on single lines, it is impossible here to assign the shifts to definite terms.

For the isotope effect, which differs from the ordinary Bohr–Sommerfeld effect of nuclear motion, there exists the following explanation, given by Hughes and Eckart[^42]. It is based on the fact

Fig. 10. The effect of isotopic shift in Pb II. Kopfermann[^52], Müller and Jones[^82], and unpublished measurements.

that, for a problem with many electrons, allowance for the motion of the nucleus must be made differently than for a problem with one electron. In this case an additional part, depending on the interaction of the electrons, is added to the usual effect. The isotope shift found in Li—only for which Hughes and Eckart gave a theory—apparently is indeed approximately explained by this theory.

It could have been supposed that in other elements as well the isotope shift could be interpreted in an analogous way. But up to now, especially for heavy elements, it has not been possible to obtain an unambiguous explanation of the phenomenon in this way. According to this supposition, the difference in the displacement of one term of two isotopes ought to have been proportional to \(\frac{1}{M^2}\), whence for heavy isotopes a very small effect should have resulted.

It must be noted, however, that in heavy elements the number of electrons also increases considerably, and that according to the theory of Urey and Eckart one would have expected this effect to be the greater, the greater the number of electrons entering into interaction. Nevertheless, it still seems difficult to explain in this way the order of magnitude found experimentally. On the basis of this theory it is also quite unclear why the centers of gravity of odd isotopes lie asymmetrically with respect to those of even ones, as according to it one would have expected.

Such difficulties led to another cause being proposed for isotope shifts in heavy isotopes.

An attempt was made to explain this effect by the assumption of non-Coulomb forces of interaction between the nucleus and the electrons. This attempt at explanation still encounters many difficulties; the calculations carried out so far by Bartlett ⁴ and Racah ⁶⁷ have not yet led to any satisfactory results.

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