ZEEMAN EFFECT.

S. E. Frisch.

1. In 1895 H. A. Lorentz, while developing the electron theory of matter, came to the conclusion that spectral lines are emitted by electrons oscillating inside atoms. But a moving electron is like a current, and therefore a magnetic field must act upon it. Hence a magnetic field must also act upon spectral lines.

Lorentz was able to take into account the action of a magnetic field on spectral lines quite simply. Suppose that in an atom there is one electron performing simple harmonic oscillatory motion. The spectrum of such a hypothetical element will consist of one single ideally monochromatic line. In this case the light of an individual atom will prove to be linearly polarized. The light from a large number of differently oriented atoms, however, will be natural.

Let us place the atom in a magnetic field \(H\). Decompose the harmonic oscillatory motion into two components: one along the field \(H\), and another lying in a plane perpendicular to \(H\). According to the law of Biot and Savart, no magnetic force will act on the component along the field. To determine the action of the magnetic field on the component lying in the plane perpendicular to the field, decompose it into two circular motions taking place in opposite directions.

The period of revolution along the circles will remain equal to the period of the harmonic oscillatory motion \(T\). Denote the radius of the circle by \(R\), and the velocity of motion of the electron by \(v\). Harmonic oscillatory motion is motion taking place under the influence of a force directed toward the center and proportional to the displacement. Denoting the force by \(F\), the displacement by \(r\), and the coefficient of proportionality by \(f\), we write

\[ F = fr. \tag{1} \]

For \(r = R\) this force must be equal to the centripetal force outside the magnetic field, i.e.

\[ fR = \frac{mv^2}{R}. \tag{2} \]

If the period of the fundamental motion is \(T\), then the velocity is

\[ v=\frac{2\pi R}{T}. \]

Let us introduce the frequency

\[ \nu_0=\frac{1}{T} \]

then:

\[ v=2\pi\nu_0R. \]

Since the fundamental motion is harmonic, then by the well-known formula

\[ T=2\pi\sqrt{\frac{m}{f}}, \]

where \(m\), in the present case, is the mass of the electron. Hence

\[ \nu_0^2=\frac{1}{T^2}=\frac{1}{4\pi^2}\frac{f}{m}. \]

On the basis of the Biot–Savart law, the magnetic field acts on an electron moving in a circle clockwise with a force directed away from the center and equal to:

\[ F_H=\frac{evH}{c}, \]

where \(e\) is the charge of the electron, \(c\) is the speed of light. The motion is still performed in a circle. But now the force \(F_H\) is added to the force \(fR\). The force directed toward the center must again be balanced by the centrifugal force; therefore equality (2) will be replaced by:

\[ \frac{mv^2}{R}=fR+\frac{ev}{c}H. \tag{3} \]

The velocity \(v\) will now be different, and consequently the frequency will not be \(\nu_0\), but \(\nu_1\). Substituting into (3) \(v=2\pi\nu_1R\) and \(f=4\pi^2\nu_0^2m\), we find:

\[ \nu_1^2-\nu_0^2=\frac{e\nu_1H}{2\pi mc}. \tag{4} \]

Here \(\nu_0\) is the frequency of the electron’s oscillations before the appearance of the magnetic field, \(\nu_1\) in the magnetic field. Solving the obtained quadratic equation (4), we find the value of the new frequency \(\nu_1\). It is easy to see that, for practically attainable magnetic fields, the term \(\frac{eH}{4\pi mc}\) is very small; therefore, neglecting its square, we obtain from (4), approximately:

\[ \nu_1=\nu_0+\frac{1}{4\pi}\frac{e}{mc}H. \tag{5} \]

We shall likewise find, for an electron moving in a circle counterclockwise:

\[ \nu_2=\nu_0-\frac{1}{4\pi}\frac{e}{mc}H. \tag{6} \]

Equations (5) and (6) contain the solution of the question. Before the appearance of the magnetic field the electron emitted waves of one frequency \(\nu_0\). In the spectrum there was a single spectral line with wavelength

\[ \lambda_0=\frac{c}{\nu_0}. \]

In the magnetic field the harmonic oscillatory motion was replaced by three: an oscillatory motion along the field \(H\) with the former frequency \(\nu_0\), and two circular motions in the plane perpendicular to \(H\): a circular motion clockwise with frequency \(\nu_1\), and a circular motion counterclockwise with frequency \(\nu_2\).

We shall observe the light coming from the source along the field \(H\). The rectilinear component of the electron’s oscillations will not produce light waves propagating along the field, since light waves are transverse. Motions along circles will give two circularly polarized waves with frequencies \(\nu_1\) and \(\nu_2\), determined by equalities (5) and (6). Since the charge of the electron is negative, \(e/m\) is a negative quantity, whence \(\nu_1<\nu_0\), and the waves polarized clockwise will give a spectral line with a frequency smaller than the original one—a line shifted toward the red end of the spectrum. Similarly, the waves polarized counterclockwise will give a line shifted toward the violet end.

Fig. 1.

Finally, observing the light source in a magnetic field, along the field, we shall see instead of one line two (Fig. 1). They will be shifted symmetrically with respect to the original line by the amount:

\[ \Delta\nu=\frac{1}{4\pi c}\frac{e}{m}H \tag{7} \]

or, in wavelengths:

\[ \Delta\lambda=\frac{1}{4\pi c^2}\frac{e}{m}\lambda^2 H, \tag{7a} \]

The line shifted toward the red end of the spectrum will be circularly polarized clockwise; the line shifted toward the violet, counterclockwise.

If we observe the phenomenon in a direction perpendicular to the field, then the oscillation along the field will give a rectilinearly polarized

polarized light with the original frequency \(\nu_0\). Two circular oscillations are two shifted lines with frequencies \(\nu_1\) and \(\nu_2\). They will also be polarized rectilinearly, since when observed in a plane perpendicular to the direction of the field \(H\), their circular motions will be projected as rectilinear ones.

Thus, when observing perpendicular to the field, instead of one line we shall see three. The middle one will correspond to the original one. The other two, as in the longitudinal effect, will be displaced by the amount

\[ \Delta \nu = \frac{1}{4\pi c}\frac{e}{m}H \]

or

\[ \Delta \lambda = \frac{1}{4\pi c^2}\frac{e}{m}\lambda^2 H. \]

All three lines are polarized rectilinearly. The oscillations in the middle one take place along the lines of force of the field, and in the outer ones—perpendicular to them (Fig. 1).

The phenomenon described was in fact discovered by Zeeman in 1896 \([^{1}]\). Placing a light source between the poles of a powerful electromagnet and observing the spectral lines with a large diffraction grating, he was able to notice their splitting. When observed perpendicular to the field, the lines split into three components; when observed along the field—into two. (In the latter case it was necessary to drill through one of the cores of the electromagnet and observe the light source through the aperture.)

Zeeman’s work was accompanied by great difficulties. The displacement of the lines is very small. It is enough to point out that in a strong field of 25,000 gauss the difference in wavelengths between the components of the triplet for the line \(\lambda = 5000\,\text{\AA}\) will be about \(0.3\,\text{\AA}\), i.e., about one twentieth of the distance between the sodium lines \(D_1\) and \(D_2\). However, Zeeman succeeded not only in qualitatively verifying Lorentz’s theory. First of all he showed that the polarization agrees with that predicted. Particularly important was Zeeman’s observation that in the longitudinal effect it is precisely the line shifted toward the red end that is polarized clockwise. It followed from this that the light is indeed caused by oscillations of negatively charged particles. Moreover, Zeeman, having measured the displacement \(\Delta \nu\) in a given field \(H\), could, by formula (7), determine \(\frac{e}{m}\). Within the limits of observational error, the value obtained for \(\frac{e}{m}\) coincided with the value of the same quantity found for electrons by purely electrical measurements.

Modern spectroscopic technique makes it possible to make measurements with far greater accuracy than was available to Zeeman. According to recent careful observations by Fortrat [²], the quantity

\[ a_\lambda=\frac{\Delta\lambda}{H\lambda^2} =\frac{1}{4\pi c^2}\left(\frac{e}{m}\right) =0.4680\cdot 10^{-4}. \]

Whence the ratio of the charge of the electron to its mass is obtained as

\[ \frac{e}{m}=1.764\cdot 10^7. \]

According to Millikan’s measurements, carried out in an entirely different domain,

\[ \frac{e}{m}=1.769\cdot 10^7. \]

The agreement, as is seen, is complete.

2. Lorentz’s theory, brilliantly confirmed by Zeeman, was a triumph of the electromagnetic theory of light. It seemed beyond doubt that inside atoms there are oscillating electrons, radiating according to classical electrodynamics. But this theory came to a dead end before the fundamental regularity in spectra—before the spectral series. The spectral series found their explanation only in Bohr’s quantum theory.

According to Bohr, the atom consists of a heavy positively charged nucleus of small dimensions and of electrons moving around it. The motion of the electrons should be determined according to ordinary mechanics, and then, from all mechanically possible orbits, only those should be selected which satisfy the quantum conditions:

\[ \int p_i\,dq_i=n_i h, \tag{1} \]

where \(q_i\) and \(p_i\) are the generalized coordinates and momenta by means of which the motion is described, \(h\) is Planck’s constant, and \(n_i\) is an integer. The integral is extended over the whole range of variation of the coordinate \(q_i\). Only orbits satisfying conditions (1) will be allowed. The motion of an electron along one of such orbits is stable and, contrary to classical electrodynamics, takes place without radiation. The electron may jump from one of the allowed orbits to another; in doing so it loses energy in the form of monochromatic radiation of frequency \(\nu\), determined by the “frequency rule”:

\[ \nu=\frac{W_a}{h}-\frac{W_e}{h}, \tag{II} \]

where \(W_a\) is the energy of the atom when the electron is on the first orbit, and \(W_e\) is the energy of the atom when the electron is on the final orbit.

For the case of the simplest atoms (hydrogen, ionized helium), consisting of a nucleus and one electron, the problem can be solved completely. Accordingly, we obtain exhaustive information about the spectra of these elements—information that is clearly confirmed by experiment.

For the simplest atoms the theory can also be extended to the case of the existence of an external magnetic field \([3]\). In the absence of a field the electron moves along an ellipse, with the nucleus placed at one of its foci. In the elementary theory of Lorentz we saw: in a magnetic field the frequency of uniform motion along a circle, in a plane perpendicular to the magnetic force \(H\), is increased or decreased (depending on the direction of rotation) by the amount

\[ \Delta \nu = \frac{eH}{4\pi mc}. \tag{8} \]

In other words, we may say: the magnetic field causes the plane of the orbit to rotate about the direction of the magnetic lines of force with frequency \(\omega=\Delta\nu\). It can be shown that also in the case of an orbit oriented in space in an arbitrary manner, the action of the magnetic field is manifested in a uniform rotation of the plane of the orbit about the lines of force with the same frequency \(\omega\)¹). This is the so-called Larmor theorem. Larmor’s theorem remains valid also for elliptical motion (Fig. 2).

Fig. 2.

Thus, in a magnetic field the electron moves about the nucleus along an ellipse which itself slowly (in comparison with the revolution of the electron) rotates about the lines of force. This is the mechanical solution of the problem of the action of a magnetic field on the simplest atom. Now we must, in accordance with conditions (I), choose from all mechanically possible orbits those orbits allowed by the quantum theory. Usually the quantization is carried out in spherical coordinates \(z, \vartheta, \varphi\). Here the quantization with respect to the latitude \(\vartheta\) leads to the result that the atom cannot be oriented arbitrarily in space, but that the plane of the orbit can make only certain angles with the direction of the magnetic lines of force.

The theory carried through to the end correctly gives the distance between the components. But without the corresponding additions \([4]\) it gives neither the number of components (in general, an infinite series of equidistant

¹) It should be borne in mind that, dynamically, the motion along a circle taking place under the influence of the Coulomb force differs from the motion along a circle considered in Lorentz’s theory. Nevertheless, the influence of the magnetic field remains the same.

lines, instead of the three observed experimentally), nor their polarization. Below we shall use another route, indicated by Bohr [^5], namely a route based on the principle of correspondence.

The principle of correspondence finds its basis in the following considerations. The frequencies of the lines in the spectrum of hydrogen, as is known, are covered by Balmer’s formula:

\[ \nu = N \left\{ \frac{1}{n_2^2} - \frac{1}{n_1^2} \right\}. \tag{9} \]

According to Bohr’s theory, \(n_1\) is the quantum number of the initial orbit, \(n_2\) of the final one; the constant \(N\) is equal to:

\[ N = \frac{2\pi^2 e^4 m}{h^3}. \tag{10} \]

The numerical agreement of the values of \(N\), determined from experiment and calculated by formula (3), serves as one of the chief arguments in favor of Bohr’s theory.

In the hydrogen atom the electron moves along a Keplerian ellipse. According to Kepler’s third law the frequency of revolution of the electron \(\omega\) will be:

\[ \omega = \sqrt{\frac{2U^3}{\pi^2 e^4 m}}, \tag{11} \]

where \(e\) is the charge, \(m\) the mass of the electron, \(U=-W\) the absolute value of the work required to remove the electron from the nucleus to infinity. For orbits stable from the point of view of quantum theory,

\[ U = \frac{2\pi^2 m e^4}{n^2 h^2}, \]

where \(n\) is the quantum number characterizing the given orbit, and \(h\) is Planck’s constant. Substituting this value of \(U\) into formula (4), we obtain:

\[ \omega = \frac{1}{n^3}\cdot \frac{4\pi^2 m e^4}{h^3}. \tag{12} \]

As is known, for any periodic motion with number of revolutions \(\omega\), the displacement \(\xi\) of a particle in a definite direction can be represented as the sum of harmonic oscillatory motions:

\[ \xi = C_1 \cos 2\pi(\omega t + c_1) + C_2 \cos 2\pi(2\omega t + c_2) + C_3 \cos 2\pi(3\omega t + c_3) + \ldots \]

or, in abbreviated form:

\[ \xi = \sum_k C_k \cos 2\pi(k\omega t + c_k), \tag{13} \]

THE ZEEMAN EFFECT

where the summation extends over all positive integer values of \(k\). Formula (13) is called a Fourier series. Hence, according to classical electrodynamics, an electron, moving along one of the orbits permitted from the point of view of quantum theory, should emit light of frequency \(\nu\), equal to the frequency of revolution \(\omega\) and its overtones, i.e. frequencies that are multiples of \(\omega\):

\[ \nu = k\omega \]

or, by formula (12):

\[ \nu = k \frac{1}{n^3}\cdot \frac{4\pi^2me^4}{h^3}. \tag{14} \]

We shall call the integer \(k\) the ordinal number of the given oscillation.

According to quantum theory, the frequency of the emitted light is not at all equal to the frequency of revolution, but is determined by the rule:

\[ \nu = \frac{\Delta W}{h}, \]

where \(\Delta W\) is the difference of the energies of two stationary states. This rule, applied to the hydrogen atom, gives Balmer’s formula:

\[ \nu = \frac{2\pi^2me^4}{h^3}\left\{\frac{1}{n_2^2}-\frac{1}{n_1^2}\right\}, \]

which we shall rewrite as:

\[ \nu = (n_1-n_2)\frac{2\pi^2me^4}{h^3}\frac{n_1+n_2}{n_1^2 n_2^2}. \]

If both numbers \(n_1\) and \(n_2\) are large, and the difference between them \(\Delta n = n_1-n_2\) is small, then approximately one may write:

\[ \nu = \Delta n \frac{1}{n^3}\cdot \frac{4\pi^2me^4}{h^3}. \tag{15} \]

Comparing this expression with expression (7) and taking into account that \(\Delta n\) is an integer, we see: for distant orbits the classical and quantum theories give identical frequencies of the emitted light.

It is natural to suppose that for distant orbits the agreement between the two theories concerns not only the frequencies, but is complete. Classical theory must, for distant orbits, give both the correct frequencies and the correct amplitudes and polarization. We shall call this hypothesis the “correspondence principle.”

If we pass to more inner orbits, then it is obvious that classical theory will not immediately begin to give results completely different from the results of quantum theory. It will give solutions which, though incorrect, nevertheless somewhat resemble the correct ones.

From this, Bohr, extending the “principle of coincidence,” formulates: to each quantum transition there corresponds a certain frequency, computed according to the classical theory, namely that frequency whose ordinal number \(k\) coincides with the change of the quantum number \(\Delta n\). If, according to the classical theory, one computes the amplitude and polarization of the partial oscillation corresponding to this frequency and transfers them to the spectral line obtained by virtue of the corresponding quantum transition, then the intensity and polarization of the spectral line come out quite correctly for infinitely large quantum numbers and approximately correctly for medium quantum numbers. This hypothesis may be called the “principle of correspondence.”

The application of the principle of correspondence becomes especially important when, in formula (13), \(\xi\) is equal to zero both for the initial and for the final orbit. In this case the line must be absent. There are no such spectral lines whose corresponding frequencies do not occur in the series (13). The combination principle turns into a “selection principle.” It begins to indicate which differences of energies of stationary states should not be taken into account when counting emitted lines.

For what follows it will be important for us to consider the application of the principle of correspondence to purely periodic motions, such as, for example, that performed by Planck’s oscillator. Planck’s oscillator oscillates harmonically. Aligning its motion with the \(x\)-axis, we have:

\[ \xi_x = C \cos 2\pi(\nu t + c). \]

Here the Fourier series reduces to a single term; \(|k|\) has only one value, \(|k| = 1\). Owing to its special simplicity, the principle of correspondence becomes the principle of coincidence. The quantum number \(n\) can change only by \(\pm 1\). The spectrum of the Planck oscillator consists of a single line corresponding to the quantum transition \(\Delta n = \pm 1\); the line is polarized along the \(x\)-axis1. The frequency of the line coincides with the frequency determined by the classical theory. The energy of the stationary states of Planck’s oscillator is an integral multiple of \(h\nu\), where \(\nu\) is the mechanical frequency of its oscillations.

We saw above that the effect of a magnetic field on the motion of an electron manifests itself in a uniform rotation of the plane of the orbit about the magnetic lines of force with frequency \(\omega\). Thus the perturbation is of a purely periodic character. Before the appearance of the magnetic field the atom possessed a series of stationary states. Now each of the stationary states is perturbed by an additional rotation and, according to quantum theory, must split into a series of new stationary sta-

states. By analogy with the definition of the special values of the energy of Planck’s oscillator, one may expect that the energy difference between two different stationary states corresponding to one and the same stationary state of the unperturbed system is simply an integral multiple of the perturbation frequency \(\omega\), multiplied by \(h\). Thus we arrive directly at the following expression for the energy of the stationary states of the perturbed system:

\[ W = W_n + mh\omega, \tag{16} \]

where \(W_n\) is the energy of the stationary states of the unperturbed system, \(m\) is a new quantum number, which may take positive and negative values; it is called the magnetic quantum number.

The frequencies emitted by the unperturbed atom, according to Bohr’s frequency rule, are equal to:

\[ \nu = \frac{1}{h}(W'_n - W''_n). \]

The frequencies emitted by an atom situated in a magnetic field are:

\[ \nu + \Delta \nu = \frac{1}{h}\left\{(W'_n + m'h\omega) - (W''_n + m''h\omega)\right\}, \]

or

\[ \nu + \Delta \nu = \frac{1}{h}(W'_n - W''_n) + (m' - m'')\omega. \]

Whence the change of frequency of a spectral line in a magnetic field is:

\[ \Delta \nu = (m' - m'')\omega \]

or, substituting for \(\omega\) its value (8), p. 85:

\[ \Delta \nu = (m' - m'')\frac{eH}{4\pi mc}, \]

where

\[ (m' - m'') = 0, \pm 1, \pm 2, \pm 3,\ldots \]

For the time being we obtain an unlimited series for \(m' - m''\), i.e. instead of one spectral line an unlimited series of lines at a true distance

\[ \Delta \nu = \frac{eH}{4\pi mc} \]

from one another. But the correspondence principle removes this misunderstanding.

The magnetic field causes the uniform rotation of the orbit indicated above. Hence we simply find that every elliptic-harmonic component with frequency \(k\nu\), appearing in the expansion of the unperturbed motion, splits in the magnetic field into three harmonic-

...components: a rectilinear one, along the line of force, with frequency \(k\nu\), and two circular ones with frequencies \(k\nu+\omega\) and \(k\nu-\omega\), oscillating in opposite directions in the plane perpendicular to the field. Consequently, the motion will be represented by the formula:

\[ \xi=\sum C_{k\chi}\cos 2\pi\{\,t(k\nu+\chi\omega)+C_{k\chi}\,\}, \]

where \(\chi\) can take the values \(\chi=0\), or \(\chi=\pm1\). Hence, by virtue of the correspondence principle, \(m'-m''\) can be equal only to \(0\), or to \(\pm1\). In this way the superfluous components are eliminated. Further, for \(m'-m''=0\), a component must arise which is polarized rectilinearly along the field; for \(m'-m''=\pm1\), two components polarized circularly with the plane of oscillations perpendicular to the direction of the magnetic lines of force. Thus the polarization too is determined in agreement with experiment.

Fig. 3a.

The quantum theory of the splitting of spectral lines may be illustrated by the following drawing.

It is customary to depict the stationary states of an atom in the form of energy levels, and the appearance of spectral lines by means of arrows connecting the initial state with the final one. In drawing (3a) the levels corresponding to the quantum numbers \(n'\) and \(n''\), with energies \(W'\) and \(W''\), are shown. In the transition from one level to another there arises a line with frequency

\[ \nu_0=\frac{W'}{h}-\frac{W''}{h}. \]

In a magnetic field each of the levels \(W\) is replaced by three levels \(W-h\omega\), \(W\), \(W+h\omega\), corresponding to the three values \(\chi=0,\pm1\), or to the three values of the magnetic quantum number \(m\), \(m\pm1\). This is shown in drawing (3b). There, by arrows, the occurrence of all three components is also shown: \(\nu=\nu_0-\omega\); \(\nu=\nu_0\); \(\nu=\nu_0+\omega\). The forbidden transitions, to which a change of \(\chi\) by \(\pm2\) would correspond, are shown by dotted arrows.

Fig. 3b.

The same thing may be represented by yet another scheme. Let us write the energy values in the initial and final states of the atom in two rows, so that the values with one and the same quantum number \(m\) fall one below another:

\[ \begin{array}{ccc} W'-h\omega & W' & W'+h\omega\\[4pt] \downarrow\ \ \searrow\nearrow & \downarrow\ \ \searrow\nearrow & \downarrow\\[-2pt] W''-h\omega & W'' & W''+h\omega \end{array} \qquad \left. \begin{array}{c} \\[18pt] \end{array} \right\} \tag{17} \]

THE ZEEMAN EFFECT

Forming the differences of the quantities connected by arrows and dividing them by \(h\), we obtain the frequencies of the emitted lines. But we are interested not in the frequencies themselves, but only in their changes in the magnetic field. Therefore we may put \(W'=0\) and \(W''=0\) in (17). Moreover, in order that the differences should give us at once the changes of frequency, and not the changes of energy, we divide both lines by \(h\); then

\[ \begin{array}{ccc} -\omega & 0 & +\omega\\ \downarrow \ \searrow \swarrow & \downarrow & \searrow \swarrow \ \downarrow\\ -\omega & 0 & +\omega \end{array} \left\} \right. \tag{18} \]

In this scheme the differences of the quantities connected by vertical arrows give the undisplaced component \(\Delta\nu=0\), polarized along the field or, as we shall denote it briefly, the \(\pi\)-component. The differences of the quantities connected by oblique arrows will give the two displaced components

\[ \Delta\nu=\pm\omega=\pm\frac{eH}{4\pi mc}, \]

polarized perpendicular to the field\(^1\); these components we shall denote as \(\sigma\)-components.

If we measure the splitting in units of \(\omega\), then scheme (11) is replaced by the still simpler one:

\[ \begin{array}{ccc} -1 & 0 & +1\\ \downarrow \ \searrow \swarrow & \downarrow & \searrow \swarrow \ \downarrow\\ -1 & 0 & +1 \end{array} \left\} \right. \tag{19} \]

\[ \Delta\nu=0,\ \pm 1 \quad \text{(in units of } \omega\text{).} \]

The rule for polarization is written as:

\[ \Delta m= \begin{cases} 0 & \pi\text{-component},\\ \pm 1 & \sigma\text{-component}. \end{cases} \]

We shall make repeated use of this scheme in what follows.

Concluding the exposition of the quantum theory of the normal Zeeman effect, we can indicate why the classical theory was also able to arrive at the correct result. First of all, owing to the simplicity of the perturbation, the correspondence principle here passes over (as also for the Planck oscillator) into the principle of coincidence. Therefore both theories give the same number of lines and the same polarization. Further, owing to the accidental cancellation of Planck’s constant \(h\) in the final formula, the numerical values of the displacements of the spectral lines are the same in both theories. This is not the case in the serial formulas; Rydberg’s constant (see formula (10), p. 86) contains explicitly the quantum constant \(h\); it cannot be obtained numerically correctly by any classical theory. The situation is analogous with the splitting

\(^1\) In what follows it will always be understood that the splitting of the lines is observed in a direction perpendicular to the field.

of spectral lines in an electric field—with the Stark effect—where classical theory proves powerless.

Bohr’s theory, being more general, explains both what could be explained by classical theory and what presented insurmountable difficulties for classical theory.

3. The quantum theory of the Zeeman effect set forth above applies to the simplest atomic systems—to hydrogen and ionized helium. The lines of these elements do indeed split into triplets. The distance between the components agrees with that predicted by the theory. But the majority of the lines of other elements split in a magnetic field into a larger number of components. In contrast to this splitting, as compared with the theoretical “normal” splitting, it is customary to speak of the anomalous, or complicated, Zeeman effect.

Modern spectroscopic technique makes it possible to study this phenomenon in sufficient detail. Large diffraction gratings are used, chiefly Rowland concave reflection gratings, Michelson echelon gratings, Lummer plates, etc. These instruments resolve two lines separated from one another by hundredths of an ångström. Powerful water-cooled electromagnets give fields of 50,000 gauss and more.

The light sources present great difficulties. Ordinary sources giving intense lines—the voltaic arc and the spark—are of little use for the following reasons: first, they give excessively broad and diffuse lines (especially the arc); second, the magnetic field acts strongly upon them and, if it does not extinguish them altogether, greatly diminishes their intensity. To obtain the spectra of gases, special Geissler tubes are used, for the most part made of quartz. For other elements Back [6] has recently used a very ingenious light source, the so-called “interrupted” arc. One of the electrodes of this arc is made of the metal under investigation; the other is a tungsten rod. The latter is set into rapid vibration by a special mechanism and makes with the other electrode about ten contacts and breaks of contact per second. Each time, at the moment of break, an arc flashes up; before the field extinguishes it, a new contact occurs, and the whole process begins anew. The entire arc is placed in a vessel filled with hydrogen at low pressure, owing to which the spectral lines are obtained very fine. If one recalls that the distance between the poles of the electromagnet, where the arc is placed, is only a few millimeters, it becomes clear what experimental skill is required to develop such a light source.

Experimental data obtained for the anomalous Zeeman effect made it possible to establish two important laws: Runge’s law and Preston’s law.

THE ZEEMAN EFFECT

Runge’s law [7] is as follows: in complex splittings, the distances of the components, reckoned from the middle of the splitting, are rational fractions of the normal splitting \(\Delta\nu_0\):

\[ \delta\nu=\frac{q}{z}\,\Delta\nu_0, \tag{20} \]

where \(q\)—the “Runge numerator” and \(z\)—the “Runge denominator”—are small integers. For a given line \(z\) is constant, while \(q\) takes a series of values: \(q=0,\pm1,\pm2,\ldots\). The rationality of the fractions in Runge’s law \(\left(\frac{1}{2},\frac{1}{3},\frac{1}{6},\text{ etc.}\right)\) has been established with all the accuracy afforded by spectral instruments of high resolving power. Usually the error does not exceed \(0.1\%\).

Fig. 4.

As an example of Runge’s law one may take the types of splitting of the yellow sodium lines \(D_1\) and \(D_2\), \(\lambda=5896\,\text{\AA}\), \(\lambda=5890\,\text{\AA}\). The line \(D_1\) is split into four components, the line \(D_2\)—into six. Below are given the distances of the components, reckoned from the middle of the splitting:

\[ D_1\ldots\ \delta\nu=\pm\frac{2}{3}\Delta\nu_0;\ \pm\frac{4}{3}\Delta\nu_0, \tag{21} \]

\[ D_2\ldots\ \delta\nu=\pm\frac{1}{3}\Delta\nu_0,\ \pm\frac{3}{3}\Delta\nu_0,\ \pm\frac{5}{3}\Delta\nu_0. \tag{22} \]

All the components are polarized parallel or perpendicular to the field. Figure 4 shows both the splitting and the polarization of the lines.

Preston’s law [8] states: first, all lines belonging to one and the same spectral series have qualitatively and quantitatively \(\left(\text{the fraction } \frac{q}{z}\right)\) one and the same splitting. Secondly, lines of different elements, but having the same series membership, also give the same type of splitting.

Let us give an example: the lines \(D_1\) and \(D_2\) of sodium constitute the first doublet of the principal series; their series symbol is: \(1s — 2p_1\) for \(D_2\) and \(1s — 2p_2\) for \(D_1\) \({}^{1}\)). The type of splitting of these lines is indicated in Fig. 4. According to Preston’s rule, all sodium doublets of the principal series must give the same splitting. And in all elements that have a principal series of doublets, the first line of the doublet, counted from the red end of the spectrum, must split into a quadruplet, the second into a sextet, with the same values \(\dfrac{80}{H}\).

In short: all lines with the series symbol \(sp_2\) (we deliberately write \(sp_2\), and not \(1s — 2p_2\), in order to show the independence of the type of splitting from the coefficient 1 at \(s\) and 2 at \(p\)) give a quadruplet, with the indicated values of the Runge fraction \(\dfrac{q}{z}\), while lines with the series symbol \(sp_1\)—a sextet.

Thus each series may be assigned its own type of splitting. In brief, the laws of Runge and Preston may be formulated as follows: the series symbol determines the type of magnetic splitting of the line.

\({}^{1}\)) Let us remind the reader of the basic principles of series symbolism needed for what follows. As is known, all series of the hydrogen spectrum can be covered by the formula

\[ \nu = R\left(\frac{1}{i^2} - \frac{1}{k^2}\right), \tag{1} \]

where \(R\) is a universal constant (the Rydberg constant, equal to \(1.097 \cdot 10^{-5}\)), and \(i\) and \(k\) may have different integer values. Here \(i\), for a given series, has a constant value, so that the first term of formula (1), \(\dfrac{R}{i^2}\), is a constant quantity (a constant term); the second term is a variable term. Indeed, for \(n = 2\) and \(k = 3, 4, 5\ldots\) we obtain the usual Balmer series; for \(n = 1\), \(k = 2, 3, 4\ldots\)—the ultraviolet Lyman series; for \(n = 3\), \(k = 4, 5\ldots\)—the infrared Paschen series (for more on the spectral series of hydrogen, see P. Epstein’s article, Uspekhi Fizicheskikh Nauk, vol. II).

Rydberg showed that all spectral series in general can be represented by a general formula of the type

\[ \nu = \psi(i) - \varphi(k) \left\{ \begin{array}{l} i = 1,\,2,\,3\ldots\\ k = 2,\,3,\,4\ldots, \end{array} \right. \tag{2} \]

where

\[ \psi = \frac{R}{(i+\alpha)^2}, \qquad \varphi = \frac{R}{(k+\beta)^2}, \]

where \(R\), as before, is the universal Rydberg constant, and \(\alpha\) and \(\beta\) are empirical parameters characteristic of the given element and of the given series. If \(i\) is assigned a constant value (for example, \(i = 1\)), while \(k\) is made to run through all possible integer values, then \(\psi\) will be a constant term and \(\varphi\) a variable one. For the so-called principal series \(i = 1\); we introduce the following designations: \(\alpha = s\), \(k = m\), \(\beta = p\), so that

\[ \nu = \frac{R}{(1+s)^2} - \frac{R}{(m+p)^2} \tag{2a} \]

At the present time the type of magnetic decomposition has been established for a large number of series. It turns out that lines belonging to singlet series always give a normal splitting. Lines that are components of a series doublet or triplet decompose in a complicated way. Especially complicated is the decomposition of lines of diffuse series. For example, the lines \(2p_1 - md_1\) of triplet series split into 15 components. The elements at the end of the periodic system, \(Mn\), \(Cr\), \(Fe\), have recently discovered quartet, quintet, etc. series and, correspondingly, a very complicated Zeeman effect. There are lines on which the magnetic field does not act at all, for example, \(Fe\), \(\lambda = 3767\) Å. With rare exceptions, the lines of complex splitting, like those of normal splitting, are strictly polarized either along the field or perpendicular to it. In Fig. 5 the types of decomposition of the lines of the principal series of triplets are given.

Fig. 5.

Deviations from Preston’s law are explained by the Paschen–Back effect \([^{9}]\): if \(\delta\nu\) is close to the distance between the components of a doublet or triplet \(\nu_1 - \nu_2\), then the required type of splitting begins to become simplified, and with increasing field \(H\) the whole group tends toward nor-

Since \(R\) is a universal constant, the first term of formula (2a), for brevity, is usually denoted simply as: \((1s)\), and the second term as: \((mp)\). Hence the symbolic formula for the principal series (Prinzipal serie) will be written

\[ \nu = 1s - mp \qquad m = 2, 3, 4\ldots, \]

for the so-called first subordinate, or diffuse, series the symbolic formula will be

\[ \nu = 2p - md \qquad m = 3, 4, 5\ldots, \]

for the second subordinate or sharp (scharfe) series:

\[ \nu = 2p - ms \qquad m = 2, 3, 4\ldots, \]

for the so-called Bergmann series:

\[ \nu = 3d - mb \qquad m = 4, 5, 6\ldots \]

Further, a series may consist either of simple lines (singlets), or of doublets, triplets, etc. This too is expressed in the series notation. For simple lines the letters will always be capital \(S, P, D, B\) (for example, \(1S - mP\)); for multiple lines the variable term is written in general in the form \(mp_i\): for doublets—\(mp_1, mp_2\), for triplets—\(mp_1, mp_2, mp_3\), etc. It should also be noted that the term \(s\) is always simple, i.e. \(s_1 = s_2\); \(s_1 = s_2 = s_3\). Ed.

...to the normal triplet. This effect was first established by Paschen and Back on the narrow oxygen triplet \(1s—3p_i\), \(\lambda = 3947,438;\ 3947,626;\ 3947,371\). In weak fields each of the lines begins to split according to Fig. 5. As the field is increased, when the components of one line overlap the components of another, a series of anomalies begins. The lines shift relative to one another; some of them lose intensity and disappear altogether. Finally, in a strong field—about 32,000 gauss—only three lines remain from the whole group, at the normal separation \(\Delta \nu_0\) and with almost regular polarization.

A normal triplet should give lines belonging to one of the series of singlets, for example, \(1s—3p\) (\(p\) simple). It may therefore be said that in strong magnetic fields the serial terms become simplified.

The Paschen–Back effect, schematically, on lines of the sodium \(D_1, D_2\) type, is shown in Fig. 6.

Fig. 6.

The Paschen–Back effect can explain apparent deviations from the laws of Preston and Runge. For example, all alkali metals have series of doublets, but the smaller the atomic weight, the narrower the doublets. The doublets are especially narrow in lithium. In all alkali metals the principal series decomposes according to Fig. 4. In lithium, however, at those fields at which observation usually becomes possible, there is a normal triplet. At first glance this may appear to be a deviation from Preston’s law. In fact, however, Kent \([^{10}]\), using an instrument with very high resolving power, showed that in weak fields the decomposition occurs according to Preston’s law. Only in stronger fields does it pass over into the normal triplet. An analogous case was investigated independently by Fortrat \([^{2}]\) and Back \([^{11}]\) on the principal series of sodium. According to Preston’s law all the lines of this series should split in the same way. But in reality only the first member of the principal series \(1s—2p_i(D_1, D_2)\) is so broad that each of its lines gives a regular decomposition. The next doublet, \(1s—3p_i\), \(\lambda = 3302,47;\ 3303,07\), is much narrower and in strong fields reveals an anomaly. The doublet \(1s—4p_i\),

\(\lambda = 2852.828;\ 2853.031\) is so narrow that in a field of about 50,000 gauss it may be transformed almost into a normal triplet.

Simplifications in strong magnetic fields occur only when the nearby lines are components of one serial doublet or triplet. If the lines are not connected with one another in the serial sense, then no simplifications take place. Thus, Back \([^{11}]\) observed the resolution of the lines of the diffuse series \(S_2,\ 2p_1 — 3d_j\). Near one of the lines, namely \(2p_2 — 3d_3,\ \lambda = 4876.235\), there is a very close line, \(\lambda = 4876.479\), but one not connected with it serially. The resolution of this line does not change as the field increases, whereas the resolution of the line \(2p_2 — 3d_3\) is spoiled by the vicinity of the more distant, but serially connected, line \(2p_2 — 3d_2,\ \lambda = 4872.660\).

1. A somewhat rigorous theory of the complex Zeeman effect does not yet exist. But according to Bohr’s model, in this case too \(\delta \nu\) must be due to the splitting of two orbits:

\[ \delta \nu = \frac{\Delta W'}{h} - \frac{\Delta W''}{h} \]

or

\[ \delta \nu = \delta_1 - \delta_2 . \tag{23} \]

From the point of view of Bohr’s theory, the primary role is played not by the frequencies of spectral lines, but by their serial terms, which are proportional to the energies of the atom in the separate stationary states. All laws concerning spectral lines must refer not to frequencies, but to terms. In the case of the complex Zeeman effect it is important to establish not the splitting of the individual lines, but the splitting \(\delta_1\) and \(\delta_2\) of the separate serial terms.

According to Runge’s rule, \(\delta \nu = \dfrac{q}{z}\Delta \nu_0\). Hence both \(\delta \nu_1\) and \(\delta \nu_2\) must be rational fractions of \(\Delta \nu_0\):

\[ \delta \nu_1 = \frac{q_1}{z_1}\Delta \nu_0;\quad \delta \nu_2 = \frac{q_2}{z_2}\Delta \nu_0, \]

\[ \delta \nu = \frac{q_1}{z_1}\Delta \nu_0 - \frac{q_2}{z_2}\Delta \nu_0 = \frac{q}{z}\Delta \nu_0, \]

whence

\[ z = z_1 \cdot z_2;\quad q = q_1 z_2 - q_2 z_1 . \tag{24} \]

Equalities (24) indicate the connection of the observed Runge denominator and numerator of a spectral line with the Runge numerators and denominators of the separate terms. But they do not make it possible to compute unambiguously, from the experimentally determined \(\dfrac{q}{z}\), the splittings of the terms \(\dfrac{q_i}{z_j}\).

Landé [12] assumed that the complex splitting of a spectral line is determined by the splitting of its terms according to the same scheme as normal splitting is determined by the energy splitting of the initial and final states of the atom [see (19)]. Since the decomposition of spectral lines is always symmetric with respect to the middle, the splitting of each term must also be symmetric. In units \(\Delta\nu_0\), for some term \(x\), it will be either:

\[ \ldots - \frac{q_2}{z},\ - \frac{q_1}{z},\ 0,\ + \frac{q_1}{z} + \frac{q_2}{z}, \ldots \]

or:

\[ \ldots - \frac{q_2}{z},\ - \frac{q_1}{z},\ + \frac{q_1}{z},\ + \frac{q_2}{z}, \ldots . \]

Analogously, the splitting of some other term \(y\) is represented. According to scheme (18), p. 91, the type of decomposition of the line \(\nu = nx - my\) is obtained either from the expression:

\[ \left. \begin{array}{cccccc} \ldots - \dfrac{q_2}{z}, & - \dfrac{q_1}{z}, & 0, & + \dfrac{q_1}{z}, & + \dfrac{q_2}{z}, \ldots \\ \downarrow & \searrow\!\swarrow & \downarrow & \searrow\!\swarrow & \downarrow \\ \ldots - \dfrac{q'_2}{z'}, & - \dfrac{q'_1}{z'}, & 0, & + \dfrac{q'_1}{z'}, & + \dfrac{q'_2}{z'}, \ldots \end{array} \right\} \tag{25} \]

or from the expression:

\[ \left. \begin{array}{cccc} \ldots - \dfrac{q_2}{z}, & - \dfrac{q_1}{z}, & + \dfrac{q_1}{z}, & + \dfrac{q_2}{z}, \ldots \\ \downarrow & \searrow\!\swarrow & \downarrow & \searrow\!\swarrow & \downarrow \\ \ldots - \dfrac{q'_2}{z'}, & - \dfrac{q'_1}{z'}, & + \dfrac{q'_1}{z'}, & + \dfrac{q'_2}{z'}, \ldots \end{array} \right\}. \tag{26} \]

By analogy with the theory of the normal Zeeman effect, we assume that to each fraction \(\dfrac{q_i}{z}\) there corresponds a magnetic quantum number \(m\). Along the rows, from fraction to fraction, it changes by one, and numbers in different rows standing one below another have identical quantum numbers. In the case of scheme (26), the magnetic quantum number \(m\) must be assigned not integral values \(m = 0, \pm 1, \pm 2, \ldots\), but fractional ones:

\[ m = \pm \frac{1}{2},\ \pm \frac{3}{2},\ \pm \frac{5}{2}, \ldots \]

The differences of quantities connected by vertical arrows give the components polarized along the field, and those connected by oblique arrows give the components polarized perpendicular to the field.

Landé showed that, using the experimental data concerning the splitting of lines and the schemes (25) and (26), one can unambiguously choose the quantities \(\dfrac{g_i}{z_j}\) for all spectral terms. For example, to the term \(s\) of doublets we assign the splitting \(\pm 1\); to the term \(p_2\) of doublets, \(- \pm \dfrac{1}{3}\); to the term \(p_1\), \(- \pm \dfrac{2}{3}, \pm \dfrac{6}{3}\). Then the splitting of the lines \(sp_1\) and \(sp_2\) is obtained from the schemes:

\[ \begin{array}{ccccc} & sp_2 \\ & s & -1 && +1 \\ && \downarrow & \searrow\!\!\swarrow & \downarrow \\ p_2 & -\dfrac{1}{3} &&& +\dfrac{1}{3} \\ \pi\text{-comp.}: & \pm \dfrac{2}{3} \\ \sigma\text{-comp.}: & \pm \dfrac{4}{3} \end{array} \qquad \begin{array}{ccccccc} & sp_1 \\ & s & -1 && +1 \\ && \swarrow & \downarrow & \searrow\!\!\swarrow & \downarrow & \searrow \\ p_1 & -\dfrac{6}{3} & -\dfrac{2}{3} && +\dfrac{2}{3} & +\dfrac{6}{3} \\ \pi\text{-comp.}: & \pm \dfrac{1}{3} \\ \sigma\text{-comp.}: & \pm \dfrac{3}{3},\ \pm \dfrac{5}{3}. \end{array} \]

The results, in complete agreement with the experimental data shown in Fig. 4, are finally represented by the following table for the splitting of the terms of singlets, doublets, and triplets:

TABLE I.

Terms	Terms	Splitting	Terms	Terms	Splitting
Singlets	\(S\)	\(0\)	Triplets	\(s\)	\(0,\ \pm 2\)
Singlets	\(P\)	\(0,\ \pm 1\)	Triplets	\(p_1\)	\(0,\ \pm \dfrac{3}{2},\ \pm \dfrac{6}{2}\)
Singlets	\(D\)	\(0,\ \pm 2\)	Triplets	\(p_2\)	\(0,\ \pm \dfrac{3}{2}\)
Doublets	\(s\)	\(\pm 1\)	Triplets	\(p_3\)	\(0\)
Doublets	\(p_1\)	\(\pm \dfrac{2}{3},\ \pm \dfrac{6}{3}\)	Triplets	\(d_1\)	\(0,\ \pm \dfrac{8}{6},\ \pm \dfrac{16}{6},\ \pm \dfrac{24}{6}\)
Doublets	\(p_2\)	\(\pm \dfrac{1}{3}\)	Triplets	\(d_2\)	\(0,\ \pm \dfrac{7}{6},\ \pm \dfrac{14}{6}\)
Doublets	\(d_2\)	\(\pm \dfrac{3}{5},\ \pm \dfrac{9}{5},\ \pm \dfrac{15}{5}\)	Triplets	\(d_3\)	\(0,\ \pm \dfrac{3}{6}\)
Doublets	\(d_3\)	\(\pm \dfrac{2}{5},\ \pm \dfrac{6}{5}\)

The correctness of the scheme presented can be verified in many ways. Until recently only the types of arrangement of the principal, sharp, and diffuse series had been established. The so-called

combination lines seemed not to obey any law analogous to Preston’s law. At the present time, knowing the splitting of the individual terms \(s, p_i, d_j\), we can predict the splitting of any of their combinations, i.e. the Zeeman effect of any spectral line for which only its serial affiliation is known.

As an example let us consider the spectrum of mercury \([^{13}]\). Mercury has two sets of series—the singlet series, with terms \(S, P, D\), and the triplet series with terms \(s, p_1, p_2, p_3, d_1, d_2, d_3\). There exist lines whose serial symbol contains one term from the singlet group and another from the triplet group. For example, the famous resonance line \(\lambda = 2537\,\text{\AA}\) has the serial symbol \(1S—2p_2\). In addition, the lines \(Dp_i\), \(d_2P\), etc. are known. The types of their splitting are obtained from the schemes:

\[ \begin{array}{c} S \qquad\qquad 0 \\[-2mm] \qquad\qquad \swarrow\quad \downarrow\quad \searrow \\[-1mm] p_2 \qquad -\dfrac{3}{2}\qquad 0 \qquad +\dfrac{3}{2} \end{array} \]

\[ \pi\text{-components: }0;\qquad \sigma\text{-components: }\pm \dfrac{3}{2}. \]

\[ \begin{array}{c} D \qquad -2 \quad -1 \quad 0 \quad +1 \quad +2 \\[-1mm] \qquad\quad \searrow\ \downarrow\ \swarrow\quad \searrow\ \downarrow\ \swarrow\quad \searrow\ \downarrow\ \swarrow \\[-1mm] p_2 \qquad -\dfrac{3}{2}\qquad 0 \qquad +\dfrac{3}{2} \end{array} \]

\[ \pi\text{-components: }0,\ \pm \dfrac{1}{2}. \]

\[ \sigma\text{-components: }\pm \dfrac{1}{2},\ \pm \dfrac{2}{2},\ \pm \dfrac{3}{2}. \]

\[ \begin{array}{c} P \qquad -1 \qquad 0 \qquad +1 \\[-1mm] \qquad \swarrow\ \downarrow\ \searrow\quad \swarrow\ \downarrow\ \searrow\quad \swarrow\ \downarrow\ \searrow \\[-1mm] d_2 \qquad -\dfrac{14}{6}\quad -\dfrac{7}{6}\qquad 0\qquad +\dfrac{7}{6}\quad +\dfrac{14}{6} \end{array} \]

\[ \pi\text{-components: }0,\ \pm \dfrac{1}{6} \]

\[ \sigma\text{-components: }\pm \dfrac{6}{6},\ \pm \dfrac{7}{6},\ \pm \dfrac{8}{6}. \]

The splitting of the lines \(p_iD\) was confirmed by Miller \([^{14}]\), \(d_2P\) by Loman \([^{15}]\), \(Sp_2\) by Paschen \([^{16}]\).

Of particular interest is the application of the indicated scheme to the appearance of forbidden lines in diffuse series. The diffuse series \(2p_i — md_j\) have the peculiarity that both of their terms are compound, whereas in the principal and sharp series only the term \(p_i\) is compound. From the doublet terms \(2p_i\) and \(md_j\), \(i=1,2,\ j=1,2\), for a given \(m\) one can form four combinations. The expression \(2p_i — md\) represents four lines.

ZEEMAN EFFECT

In reality only the following three are observed (the so-called complete Rydberg doublet):

\[ \underline{2p_1 - md_1}; \]

\[ 2p_1 - md_2;\qquad \underline{2p_2 - md_2}. \]

The formally possible line \(2p_2 - md_1\) is absent. The most intense lines are underlined. In most elements \(d_2 - d_1 < p_2 - p_1\), and therefore the line \(2p_1 - md_2\) appears as a weak satellite near the line \(2p_1 - md_1\). Similarly, for triplets, instead of 9 formally possible lines only the following 6 lines are observed:

\[ \underline{2p_1 - md_1}; \]

\[ 2p_1 - md_2;\qquad \underline{2p_2 - md_2}; \]

\[ 2p_1 - md_2;\qquad 2p_2 - md_3;\qquad \underline{2p_3 - md_3}. \]

Here again the most intense lines are underlined. The absence of lines possible according to the combination principle in diffuse series may be explained by some selection principle [17], analogous to the selection principle indicated on p. 97. In Fig. 7 the complete doublet of Cs is shown schematically. The unobserved line is indicated by a dotted line.

In connection with the complex structure of the lines of diffuse series, peculiarities must also be observed in the Zeeman effect. When producing the magnetic splitting of the lines, one may take such a strong field that the components of the lines \(2p_1 - 3d_1\) and \(2p_1 - 3d_2\) (see Fig. 7) overlap one another, but will still be far from the components of the line \(2p_2 - d_2\). Then, by virtue of the Paschen–Back effect, simplifications should begin, but only partial simplifications. The type of splitting will pass from the type \(2p_i - md_j\) to the type \(2p_i - md\), where the term \(d\) is simple. If it were possible to increase the field so much that \(\delta \nu\) became greater than \(\Delta p_i\), then the whole type would pass into a normal triplet.

Fig. 7.

Paschen and Back [18] sought partial simplification on the lines Al, Zn, Ca, and Cb. They did not succeed in obtaining fields strong enough to transform the term \(d_j\) into a simple term \(d\), but they obtained an entirely new and unexpected effect. In a sufficiently strong field several new lines appeared, which, with further increase of the field, shifted proportionally to its strength and, apparently, represented the components of some new line, split-

by a field. The middle of the decomposition corresponded to the place where the forbidden line \(2p_i-md_j\) was supposed to be located. The splitting of the forbidden line can be calculated from Lande, and, knowing from the serial combination its original wavelength, one can calculate the position of the components for the given field. The wavelengths calculated in this way coincided with the observed ones. According to Lande (see Table 1), the forbidden line of the doublets \(p_2d_1\) should give the splitting:

\[ \begin{array}{c} p_2 \qquad -\dfrac{1}{3}+\dfrac{1}{3} \\[0.5em] \begin{array}{ccccccc} & & \swarrow & \downarrow & \searrow & \swarrow & \downarrow & \searrow \end{array} \\[-0.2em] d_1\quad -\dfrac{15}{5} & -\dfrac{9}{5} & -\dfrac{3}{5} & +\dfrac{3}{5} & +\dfrac{9}{5} & +\dfrac{15}{5} \end{array} \]

\[ \text{near the }\pi\text{-components: }\pm\dfrac{4}{15}; \qquad \sigma\text{-components: }\pm\dfrac{14}{15},\ \pm\dfrac{22}{15}. \]

Paschen and Back observed the aluminum doublet \(2p_i-3d_j\):

\[ \begin{aligned} 2p_1-3d_1,\quad &\lambda=3092{,}710; \qquad &(2p_2-3d_1,\quad \lambda=3082{,}026),\\ 2p_1-3d_2,\quad &\lambda=3092{,}836; \qquad &2p_2-3d_2,\quad \lambda=3082{,}152. \end{aligned} \]

In parentheses is enclosed the line not observed experimentally under ordinary conditions. Each of the lines \(2p_1-3d_1\), \(2p_1-3d_2\), \(2p_2-3d_2\) was resolved according to the existing scheme. In a field of about 20000 gauss, however, new lines appeared, coinciding with the components calculated according to Lande for \(2p_2-3d_1\):

TABLE II.

\[ H=21900\ \text{gauss.} \]

Polariz.	Observed.	Calculated.
\(\sigma\)	3082,148	\(+\dfrac{22}{15}=3082{,}169\)
\(\sigma\)	—	\(+\dfrac{14}{15}=3082{,}117\)
\(\pi\)	3082,029	\(+\dfrac{4}{15}=3082{,}052\)
\(\pi\)	3082,000	\(-\dfrac{4}{15}=3082{,}000\)
\(\sigma\)	3081,912	\(-\dfrac{14}{15}=3081{,}935\)
\(\sigma\)	3081,861	\(-\dfrac{22}{15}=3081{,}883\)

Line \(+\frac{14}{15}\) could not be measured, since it coincided with a bright line constituting one of the other lines. The observed numbers almost coincide with the theoretical ones; only the whole group is shifted by \(0.022\,\text{\AA}\) toward shorter \(\lambda\). As the field increases the group continues to shift toward shorter wavelengths (from \(p_2d_2\)). This displacement is explained by the incipient Paschen–Back effect. The same result is obtained for the doublet \(Ca\, 2p_i—4d_j\) and the zinc triplet \(2p_i—3d_j\), where all three forbidden lines are resolved in the field.

A partial Paschen–Back effect can be observed on those lines of the diffuse series where \(\Delta d\) is practically equal to zero. Back investigated the triplet line \(Mg,\ 2p_i—3d\) [¹⁹] and the doublet \(Na,\ 2p_i—3d\) [⁶]. The latter doublet was also investigated by the referee [²³]. Theoretically this case was considered by Sommerfeld and Heisenberg [²⁴].

As we have seen, in the region of the complex Zeeman effect there exists a series of strict empirical regularities. There is still no theory, despite the numerous works of Landé, Sommerfeld, Pauli, and others [²⁴]. The authors named succeeded in establishing a number of formal rules, but did not succeed in constructing a model of the phenomenon. The region of the complex Zeeman effect remains as obscure as the related region of the magnetic properties of atoms.

Apparently, atomic quantum mechanics, as given by Bohr, proves insufficient here. The integral laws of the splitting of spectral lines at present seem to us as cabalistic as the series formulae seemed before Bohr’s theory. Let us hope, however, that the consequences drawn from them will prove as important and fruitful as the consequences drawn from Balmer’s formula.

LITERATURE.

1) An account of all works concerning the Zeeman effect carried out up to 1914 may be found in the book: P. Zeeman, Researches in Magneto-optics. London, 1913. There is a German translation: P. Zeemann, Magnetooptische Untersuchungen. Leipzig, 1914. In Russian on the Zeeman effect: Khvolson. Course of Physics, vol. V, p. 578. Van Logeizen. The Zeeman phenomenon. Questions of Physics. 1913, p. 363.

2) Fortrat. Ann. de Phys. 9, III, p. 282, 1915.

3) Sommerfeld. Atombau u. Spektrallinien. 3 Auflage, p. 366, 1922.

4) Rubinowicz. Phys. Zeitschr. 19, p. 441, 1918.

5) N. Bohr. Zeitschr. f. Phys. 2, p. 423, 1920. Russian translation: Advances in the Physical Sciences, III, p. 29, 1922; also in the book: N. Bohr. Three Articles on Spectra and the Structure of Atoms. Moscow, 1923.

6) Back. Ann. d. Phys. 70, p. 333, 1923.

7) Runge u. Paschen. Abh. Akad. d. Wiss. Berlin, 1902.

8) Preston. Trans. Rog. Soc. Dublin (2), 7, p. 7, 1899.

9) Paschen u. Back. Ann. d. Phys. 39, p. 897, 1912.

10) Kent. Astrophys. J. 40, p. 313, 1914.

11) Back. Zur Prestonischen Regel. Dissert., Tübingen, 1921.

12) Landé. Zeitschr. f. Phys. 5, p. 231, 1921; 7, p. 393, 1921.

13) Landé. Phys. Zeitschr. 22, p. 417, 1921.

14) Mieler. Ann. d. Phys. 24, p. 105, 1907.

15) Lohmann. Dissert., Halle, 1907.

16) Paschen. Ann. d. Phys. 35, p. 877, 1911.

17) Sommerfeld. Ann. d. Phys. 62, p. 221, 1920.

18) Paschen u. Back. Physica, 1, 8—10, p. 261, 1921.

19) Back. Naturwissenschaften, 9, Heft 12, p. 199, 1921.

20) Fourth Congress of Russian Physicists in Leningrad, p. 75; 1924.

21) Sommerfeld. Zeitschr. f. Phys. 8, p. 257, 1922.
Heisenberg. Zeitschr. f. Phys. 8, p. 273, 1922.
Sommerfeld. Atombau u. Spektrallinien, 3rd edition, p. 483, 1922.

22) Sommerfeld u. Heisenberg. Zeitschr. f. Phys. 11, p. 131, 1922.
Landé. Zeitschr. f. Phys. 15, p. 189, 1923.
Pauli Jr. Zeitschr. f. Phys. 16, p. 155, 1923.

1. By the plane of polarization we everywhere mean the plane of oscillations of the light vector, and not the plane perpendicular to it, as is usually done. ↩