MAGNETISM AND SPECTROSCOPY*

A. Sommerfeld, Munich

Elementary considerations of classical electrodynamics show that to a moving electron possessing mechanical moment \(M\) one must also ascribe a magnetic moment \(\mu\), equal to:

\[ \mu=\frac{e}{2M}. \]

If, as quantum theory requires, the mechanical moment \(M\) is taken to be a multiple of the quantity \(\frac{h}{2\pi}\), then for the elementary magnetic moment we obtain the value:

\[ \mu=\frac{e}{m}\cdot\frac{h}{4\pi}. \]

Let us call, as is customary, the quantity \(N\) times larger than this (\(N\) is Avogadro’s number) the Bohr magneton:

\[ \mu_B=\frac{e}{m}\cdot\frac{h}{4\pi}\cdot N=5584. \]

In fact, as early as 1914, Niels Bohr drew the author’s attention to the circumstance that integral ratios of magnetons must, in the final analysis, be due to the integrality of the mechanical moments \(M\) (expressed in units of \(\frac{h}{2\pi}\)).

Still earlier, namely at the First Solvay Congress, Langevin, in connection with the author’s report, calculated the fundamental

* Report read at the Solvay Congress in October 1930.

a unit which differed from the Bohr magneton only by a numerical factor \(\left(\frac{1}{6}\right)\). At approximately the same time (in 1911) Gans proposed a calculation of the magneton, proceeding from the consideration that the kinetic energy of an electron moving in an orbit is equal to \(h\nu\) (\(\nu\) being the number of revolutions per second). This assumption is equivalent to

\[ M=\frac{h}{\pi} \]

and leads to a value twice as large as the Bohr magneton.

The original idea of the magneton goes back to P. Weiss. The experimentally determined value of the Weiss magneton is

\[ \mu_W = 1123\ \text{gauss cm}, \]

so that:

\[ \frac{\mu_B}{\mu_W}=4.97. \]

METHOD OF THE CLASSICAL QUANTUM THEORY

I. THE FIRST EXPRESSIONS FOR THE EXPLANATION OF THE RELATIONS

In an article published in 1920, W. Pauli* proceeded from such spatial quantization as leads to the normal Zeeman effect and which at that time was (incorrectly) applied to the hydrogen atom.

The axis of the angular momentum of the electron

\[ M = k \cdot \frac{h}{2\pi} \]

could have, with respect to the magnetic field, only \(2k\) allowed positions. These positions are such that the “magnetic quantum number,” i.e. the projection of \(\vec{k}\) onto the direction of the magnetic field, assumed the values:

\[ m = k, k-1, \ldots, 2, 1, -1, -2, \ldots -k. \]

The value \(m=0\) (when \(\vec{k}\) is perpendicular to the field) was excluded; all the remaining positions had the same quant—

* W. Pauli, Phys. Z. 21, 615, 1920.

… weight. The quantity \(\cos^2 \Theta\), which in Langevin’s theory is equal to \(\frac{1}{3}\), with such quantization receives the following values:

for \(k=1\):

\[ \cos^2 \Theta=\frac{1^2+1^2}{2}=1; \]

for \(k=2\)

\[ \cos^2 \Theta=\frac{1^2+\left(\frac{1}{2}\right)^2+\left(\frac{1}{2}\right)^2+1^2}{4}=\frac{5}{8}; \]

generally speaking,

\[ \cos^2 \Theta=\frac{1}{3}\frac{(k+1)\left(k+\frac{1}{2}\right)}{k^2}. \]

If the molar magnetic susceptibility is computed once in the usual way according to Langevin’s formula with paramagnetic moment \(p\mu_w\), and then according to quantum theory with \(k\mu\), while replacing \(\frac{1}{3}\) by \(\cos^2 \Theta\), we obtain:

\[ \chi=\frac{p^2\mu_w^2}{3RT}=\frac{k^2\mu_B^2}{RT}\cos^2\Theta; \tag{1} \]

thus

\[ \frac{p}{4.97}=k\sqrt{3\cos^2\Theta}, \]

whence

\[ \begin{aligned} p&=8.6 \quad \text{for } k=1 \quad \text{in NO, found } p\sim 9,\\ p&=13.6 \quad \text{for } k=2 \quad \text{in O}_2 \quad \text{found } p\sim 14, \end{aligned} \]

and in general:

\[ p=4.97\sqrt{(k+1)\left(k+\frac{1}{2}\right)}. \tag{2} \]

NO and \(\mathrm{O}_2\) were cited as the most important representatives of paramagnetic gases. Pauli attempted to extend his calculations also to solids and liquids. He also considered the difficulties that arise when applying space quantization to diatomic molecules; according to Pauli, the magnetic axes must be perpendicular to the axis of symmetry.

P. S. Epstein* and W. Gerlach** at the same time extended Pauli’s calculations to paramagnetic ions of atoms, in particular to ions of the iron group. The values obtained from formula (2) (not integral, of course) lie close to the values experimentally found by Weiss, Cabrera, and their collaborators (with the exception of \(\mathrm{Ni}^{++}\) and \(\mathrm{Fe}^{++}\)). This also explains the jumps by 5 units in Weiss’s scale in the observed values, as well as the frequent occurrence among the observed \(p\) values of \(p \sim 9\) \((k=1)\), \(p \sim 14\) \((k=2)\), \(p=19\) \((k \sim 3)\), etc. Gerlach also points to the moments of Pd, Pt, and some further magneton numbers occurring in the iron group, which lie close to these. Kossel, as early as 1916, discovered that ions of the iron group with the same number of electrons, for example \(\mathrm{Fe}^{+++}\) and \(\mathrm{Mn}^{++}\), or \(\mathrm{Mn}^{+++}\) and \(\mathrm{Cr}^{++}\), also possess the same number of magnetons (“magnetic displacement law”), and that the removal of one electron from the above-mentioned ions reduces \(p\) by 5 units, which directly indicates that each individual electron must be assigned one Bohr magneton.

Fig. 1.

* P. Epstein, Science. 57, 532, 1923.
** W. Gerlach, Phys. Z. 24, 275, 1923, and “Ergebnisse der exakten Naturwissenschaften”, vol. II.

2. Introduction of the anomalous Zeeman effect. Spectroscopic numbers of magnetons

Calculations according to the rules of the normal Zeeman effect prove incorrect for most atoms, and the abolition of the zero level (see above) appears, from the contemporary point of view, to be arbitrary. On the contrary, considerably more material is provided by the study of the anomalous Zeeman effect, which leads to the following picture: to each spectroscopic term (level) it is necessary to assign a quantum number \(j\), which (roughly speaking) corresponds to the total angular momentum of the given state. \(j\) consists (again roughly speaking) of the angular momentum of the electronic orbit and the intrinsic moment of the electron. \(j\) is an integer or half-integer depending on whether the atomic number is even or odd. The Zeeman splitting makes it possible directly to determine the values of the magnetic moments corresponding to each \(j\) in a given system of terms.

This moment is not equal to \(j\mu_B\), as is the case in the normal Zeeman effect, but

\[ \mu = gj\mu_B, \tag{3} \]

where \(g\) is the so-called Landé splitting factor. All calculations may be carried out as if \(j\) were set in space in such a way that the projection of \(\vec{j}\) on the direction of the field, which we shall again denote by \(m\) (“spectroscopic magnetic quantum number”), were, like \(j\), an integer or half-integer:

\[ m = j,\, j-1,\, j-2 \ldots -j+1,\,-j. \]

Thus, to each energy level there correspond in all \(2j+1\) positions in space and, correspondingly, \(2j+1\) magnetic splittings. Each position has quantum weight equal to 1; the total quantum weight of the term as a whole is \(2j+1\). To compute the susceptibility \(\chi\), it is necessary, analogously to (1), to express

\[ \cos^2 \Theta \cdot j^2 = \sum_{-j}^{+j} \frac{j m^2}{(2j+1)j^2} = \frac{j(j+1)}{3}; \tag{4} \]

thus

\[ \cos^2 \Theta \cdot \mu^2=\frac{j(j+1)}{3}\cdot g^2\cdot \mu_B^2, \tag{4a} \]

\[ \chi=\frac{j(j+1)}{3RT}\,g^2\mu_B^2. \tag{5} \]

If \(\chi\) is defined in terms of the Weiss magnetons conventionally introduced,

\[ \chi=\frac{p^2\mu_B^2}{3RT}, \tag{5a} \]

then, comparing (5) with (5a), we obtain:

\[ p\mu_w=\sqrt{j(j+1)}\,g\mu_B,\qquad \frac{p}{4.97}=\sqrt{j(j+1)}\cdot g. \tag{6} \]

With a complete analysis of the spectroscopic term it is possible to determine not only \(j\), but also the quantum numbers \(l\) and \(s\) entering into \(g\) (which the author previously denoted by \(j_a\) and \(j_s\)). Thus the numbers of magnetons can also be calculated spectroscopically. When the author* developed this point of view of “spectroscopic magneton numbers,” the question of the analysis of complex spectra was still in an embryonic state; in particular, there was still poor orientation at that time in the question of the ground states of ions of the iron group.

From experimental data it is known that the magneton numbers of this group increase from zero (Ar or \(K^+\) or \(Ca^{++}\)) to the middle of the group (\(Mn^{++}\) or \(Fe^{+++}\)) approximately linearly, and then after this, in the second half of the group, somewhat less regularly fall again to zero (for diamagnetic \(Cu^+\) or \(Zn^{++}\)); see Fig. 2. The initially assumed rectilinear increase of the magneton numbers, extrapolated in an appropriate way, which was in good agreement with the already known ground terms of neutral Cr and Mn, as it later turned out, does not quite agree with the spectroscopic properties of the corresponding ions.

* A. Sommerfeld, Phys. Z. 24, 360, 1923; Z. Physik. 19, 201, 1923.

An ideal example, in its simplicity, of spatial quantization is furnished by the vapors of alkali metals. The principal term of an atom of an alkali metal is a doublet \(S\)-term with \(j=\frac{1}{2}\) and \(g=2\). Here there occur (as is directly proved by the Stern–Gerlach experiment) two possible positions: parallel and antiparallel to the magnetic field.

Fig. 2.

Thus

\[ \frac{\cos^2 \Theta}{\cos^2 \Theta}=1;\quad \cos^2 \Theta \cdot j^2=\frac{1}{4}, \]
corresponding to equation (4);

\[ \mu^2=\mu_B^2; \]
corresponding to equation (4a);

Owing to this, for the alkali metals the susceptibility is obtained simply:

\[ \chi=\frac{\mu_B^2}{kT}, \]
corresponding to equation (5). \(\tag{6a}\)

Since it is known from the Stern–Gerlach experiment that the magnetic moment of an alkali-metal atom is equal to the Bohr magneton, then according to Langevin, i.e. for a uniform distribu-

directions, one should expect a susceptibility three times smaller than by (6a). Gerlach’s observations* on vapors of the alkali metals confirm formula (6a) and thus show directly that a one-sided setting of the magnetic axes actually occurs. Equation (6a) coincides with the following formula, which follows from (6) for the alkali metals:

\[ p = 4.97 \cdot \sqrt{3} = 8.6 . \]

This is the very value that is calculated, according to Pauli, for \(k = 2\) (cf. above with respect to NO). Conversely, for the triplet \(S\)-term \(j = 1\), \(g = 2\) according to (6):

\[ p = 4.97 \cdot \sqrt{2} = 14.0 . \]

Here, therefore, there proves to be a certain deviation from Pauli’s formula for \(k = 2\), and our present value lies closer to that indicated above for \(O_2\) than to Pauli’s value. If, in general, we compare expressions (6) and (2), equating \(jg\) in (6) to \(k\) in (2) and putting \(g = 2\) (\(S\)-term), we obtain in (6) \(\sqrt{k(k+2)}\)

\[ \text{and in (2) } \sqrt{(k+1)\left(k+\frac{1}{2}\right)} \]

The difference between the two expressions is insignificant for all values of \(k\) and disappears completely for \(k = 1\) and \(k = \infty\), when the two expressions coincide. Therefore the data of Epstein and Gerlach concerning the distribution of the numbers of magnetons on the Weiss scale retain their significance also from our new point of view.

3. The Number of Magnetons of the Rare Earths

The application of the theory of “spectroscopic numbers of magnetons” to the iron group encountered difficulties, but, as was shown by F. Hund**, for the case of the rare earths, whose ions are better defined and better shielded from

* W. Gerlach, Como-Congress., 1927.
** F. Hund, Z. Physik, 33, 855, 1925.

of external influences, this theory leads to complete success. In what follows we shall go beyond Hund’s ideas, since we shall give a finished formula for a consecutive series of magneton numbers, or, more precisely, two such formulas, one of which is suitable for Stoner’s first subgroup, and the other for the second subgroup into which the family of the rare earths splits up*.

As is known, in the rare earths one is concerned with the filling of the \(N\)-shell (\(k=2\)) by \(f\)-electrons, i.e. electrons for which the azimuthal quantum number \(l=3\) (in the iron group, correspondingly, the discussion concerned \(d\)-electrons with \(l=2\)). In view of the fact that throughout we shall be interested in the trivalent positive ions of the rare earths, we shall not consider at all the \(P\)-shells, just as also the \(O\)-shells, which form a closed eight-electron group. The total number of \(f\)-electrons, according to the Pauli principle, is equal to \(2(2l+1)=14\). Let \(z\) denote the number of \(f\)-electrons in each ion, and let \(z'\) also denote the number of \(f\)-electrons lacking to make up the fourteen-electron group; thus:

\[ z+z' = 2(2l+1)=14; \]

\(z\) and \(z'\) are those independent variables as functions of which we shall construct our curve of the number of magnetons.

Hund’s rule for determining the fundamental term of a given electronic configuration, as is known, states: the quantum numbers \(s\) and \(l\) belonging to the individual electronic orbits (\(s\)—the axial quantum number, \(l\)—the orbital quantum number) are to be combined in such a way that the resulting “group quantum numbers,” first \(s\) and, second \(l\), assume maximum values. This combination must be carried out under the assumption of the presence of a magnetic field, through which the “magnetic quantum numbers” \(m_s\) and \(m_l\) are determined; then one calculates:

\[ \bar{s}=\sum m_s,\qquad \bar{l}=\sum m_l. \]

* Cf. the author’s article in Ber. Wiener Akad., January 1930.

Further, \(m_s\) can take only the values \(\pm \dfrac{1}{2}\); thus \(s\) then and only then attains its maximum value when all \(m_s\) are equal to one another and equal to \(+\dfrac{1}{2}\). The corresponding maximum value is

\[ s=\frac{z}{2}. \]

Of the four separate quantum numbers \(n, l, m_s, m_l\) to which the Pauli principle applies, the first three for \(f\)-electrons are the same for all \(z\), namely

\[ n=4,\quad l=3,\quad m_s=+\frac{1}{2}. \]

Therefore, according to the Pauli principle, all \(m_l\) must be different. But \(m_l\) can take only the following values:

\[ m_l=l,\; l-1,\; l-2,\ldots (l-1),\; -l. \]

The maximum value is obtained when one chooses in succession and adds the largest values of \(m\), i.e.

\[ l=l+(l-1)+(l-2)+\cdots +(l-z+1)= \]

\[ =\frac{z}{2}(2l-z+1). \tag{7a} \]

The fact that only \(2l+1\) different values of \(m\) can exist shows directly that the calculations made up to now are valid only for the case when \(z \leq 2l+1\), i.e. for the first half of the rare-earth group. For \(z>2l+1\), on the contrary, one must take into account not the number of electrons present, but the number of missing ones, i.e. replace \(z\) by \(z'\); then we obtain

\[ s=\frac{z'}{2},\quad l=\frac{z'}{2}(2l-z'+1),\quad z'\leq 2l+1. \tag{7b} \]

It is known that within a given multiplet \(j\) can take values from

\[ j_{\min}=|l-s| \quad \text{to} \quad j_{\max}=l+s. \]

As a supplement to Hund’s rule we must therefore assume that, in the first half of the group, the lowest energy level belongs to the inner quantum number \(j_{\min}\) (normal multiplet), and in the second half to \(j_{\max}\) (inverted multiplet). From formulas (7) and (7a) we obtain for the quantities \(z\) and \(j\) entering equation (6) the following values:

\[ z<2l+1 \quad j=j_{\min}=|\bar l-\bar s|=\bar l-\bar s=\frac{z}{2}(2l-z); \tag{8} \]

\[ z\geq 2l+1,\quad j=j_{\max}=\bar l+\bar s=\frac{z'}{2}(2l-z'+2). \]

On the other hand, the factor \(g\) entering equation (6) has the following value:

\[ g=1+\frac{j(j+1)+\bar s(\bar s+1)-\bar l(\bar l+1)}{2j(j+1)} = \frac{3}{2}+\frac{1}{2}\frac{(\bar s-\bar l)(\bar s+\bar l+1)}{j(j+1)}. \]

Substituting the corresponding values from (7) into this formula, we obtain:

\[ \text{for } z<2l+1,\quad g(j+1)=\frac{z}{2}(2l-z-1)+1; \tag{8a} \]

\[ \text{for } z\geq 2l+1,\quad gj=\frac{z'}{2}(2l-z'+3). \tag{8b} \]

Taking (8) into account, we obtain from (6):

\[ z<2l+1; \]

\[ \frac{P}{4.97} = \sqrt{\frac{z(2l-z)}{z(2l-z)+z}} \cdot \left[ \frac{z}{2}(2l-z-1)+1 \right]; \tag{9a} \]

\[ z\geq 2l+1;\quad \frac{P}{4.97} = \sqrt{\frac{z'(2l-z'+2)+2}{z'(2l-z'+2)}} \cdot \frac{z}{z'} \left[ 2l-z'+3 \right]. \tag{9b} \]

According to (9a), \(p=0\) for \(z=0\) La (lanthanum) and for \(z=2l\) Eu (europium), and has a maximum between them; for \(z>2l\) it becomes imaginary. Therefore in Fig. 1 we cut off the curve at the value \(z=2l\). According to (9b), \(p=0\) for \(z'=0\)

Cp (cassiopeium); for \(z=2l+1\), Gd (gadolinium), the right-hand part (9b) becomes equal to \(\sqrt{(2l+1)(2l+3)}=\sqrt{63}\), after it has reached a maximum between dysprosium (Dy) and holmium (Ho). Therefore we have drawn the curve, beginning with \(z'=2l+1\), as a dotted line, in order to show that it is not real here.

The agreement with the observations of Stefan Meyer and Cabrera, as is already known from Hund’s work, is satisfactory over the entire length of the curve, with the exception of La and Eu (for Eu one may assume the presence of an admixture of strongly magnetic Gd). In any case, the division of the curve into two parts, corresponding to the two Stoner subgroups, is very characteristic; and it can hardly be doubted that the spectroscopic theory in the rare earths embraces the most characteristic magnetic properties.

4. Difficulties in the case of the iron group. Broad and narrow multiplets. Dependence on temperature

One might have expected the same laws to hold also for the iron group (with \(l=2\) instead of 3). The fact that this is actually not so can be explained by the following differences between these groups:

The multiplet levels of the rare earths are widely developed:

\[ h\cdot \Delta \nu \gg kT. \]

In the iron group, however, they are closely spaced:

\[ h\cdot \Delta \nu < kT \text{ or } \sim kT. \]

Therefore we must accept that, in the rare earths, only the lowest levels of the multiplets play an essential role in calculating the susceptibility, whereas in the iron group the susceptibility will be determined by all, or at least by many, levels.

It is clear that when one speaks of broadly and narrowly developed levels, it should be understood that they are referred to the temperature at which they are being studied. Instead of speaking of a large or small \(\Delta \nu\), one may speak of a small or large \(T\).

Proceeding from this point of view, Laporte and the author* attempted to clarify the question of the magneton numbers for the iron group. First of all, it is striking that in the middle of the group \(z = 2l + 1 = 5\) (\(\mathrm{Fe}^{+++}\) or \(\mathrm{Mn}^{++}\)), where, according to (7a), \(l = 0\), i.e. where we are dealing with an \(S\)-term, the value of \(p\) calculated from (6) agrees well with the measured value 29.5 (according to (7) and (8), here \(j = \dfrac{5}{2}\); sextet \(S\)-term). In fact, the distinction between strongly and weakly split levels for an \(S\)-term disappears, since an \(S\)-term is always simple. As a consequence, we are compelled to attribute the deviations which occur in the iron group to the multiplicity of the terms.

If it is assumed that the number of ions “occupying” a given energy level is determined by Boltzmann’s law, i.e. if we put

\[ N_j = (2j + 1)e^{-\frac{h\nu_j}{kT}} \]

(where \(2j + 1\) is equal to the statistical weight), then instead of (5) we obtain for the susceptibility

\[ \chi = \frac{\sum (j)\, N_j j(j+1)g^2}{\sum (j)\, N_j} \cdot \frac{\mu_B^2}{3R}, \]

and, correspondingly, instead of (6):

\[ \frac{p}{4.97} = \sqrt{ \frac{\sum N_j j(j+1)g^2}{\sum N_j} }. \tag{10} \]

The summation here and in what follows is carried out over the range from \(j = |l-s|\) to \(j = l+s\). Dividing the numerator and denominator of the fraction by the Boltzmann factor for the ground level, and measuring all \(\Delta \nu_j\) from this level, the latter being more conveniently expressed not in numbers of oscillations but in wave numbers (multiplying the frequency by the velocity of light \(c\)), we must henceforth understand by \(N_j\):

\[ N_j = (2j+1)e^{-\frac{hc\Delta\nu_j}{kT}}. \tag{10a} \]

* A. Sommerfeld u. O. Laporte, Z. Physik, 40, 333, 1931.

For very widely separated multiplet levels, (10) passes, of course, into (6), and all \(N_i\) vanish, with the exception of the \(N_i\) belonging to the ground level. For very closely spaced multiplet levels, on the contrary, all the exponential Boltzmann factors may be set equal to unity, and then we obtain:

\[ \frac{p}{4.97} = \sqrt{ \frac{\sum (2j+1)j(j+1)g^2}{\sum (2j+1)} }. \tag{11} \]

Both limiting cases (6) and (11) are represented in Fig. 2; we shall turn to the clarification of the meaning of curve (13) in the next paragraph. Curve (6) has the same character as for the rare earths and, consequently, is asymmetric with respect to the middle; curve (11) is symmetric, since the asymmetry is eliminated by summation over \(j\). Owing to this symmetry, (11) approaches the observed values for those numbers of magnetons which in the iron group are not due to Stońer’s division of the whole group into two subgroups and, at least approximately, must be symmetric with respect to the middle of the group. Following our reasoning, one might have expected that the experimental points would lie somewhere between the two curves (6) and (11). This is the case for the first half of the curve, but it does not hold for its second half (see \(\mathrm{Ni}^{++}\), \(\mathrm{Co}^{++}\)). A consequence of our figure, which requires experimental verification, is the following: in the first half of the iron group the Curie constant, or, what is the same thing, \(p\), according to Weiss, must have a positive temperature coefficient, and in the second half a negative one, because with increasing \(T\) the extreme curve should approach (11), and because (11) in the first half of the group lies below (6), while in the second half it lies above it. However, we shall see in the next paragraph that from entirely different premises one may expect a somewhat smaller, but always positive, temperature coefficient.

In deriving (11) it was assumed that \(l\) and \(s\)* are “normally”

* In what follows we shall omit the bars over \(l\) and \(s\).

interacted with \(j\) and that \(j\) was oriented in the magnetic field as required by quantum theory. But for very weakly split multiplets it is perhaps more correct to assume that there is no interaction between \(l\) and \(s\) at all, and that therefore \(l\) and \(s\) can separately orient themselves independently in the magnetic field. The latter assumption makes it possible to replace (11) by the simpler expression

\[ \frac{p}{4.97}=\sqrt{4s(s+1)+l(l+1)}. \tag{11a} \]

This can be proved in the following way. To each of the angular momenta \(s\) and \(l\) there corresponds a magnetic moment \(\mu_s=2s\mu_B\) and \(\mu_l=l\mu_B\), respectively. Projecting them onto the direction of the magnetic field, we obtain

\[ \mu_H=\mu_s\cos\Theta_s+\mu_l\cos\Theta_l. \tag{12} \]

To calculate the susceptibility we must, just as we did in (1) and (4), write the expression for \(\overline{\mu_H^2}\)

\[ \overline{\mu_H^2}=4s^2\cdot\overline{\cos^2\Theta_s}+4s\,\overline{\cos\Theta_s}\,l\,\overline{\cos\Theta_l}+l^2\overline{\cos^2\Theta_l}, \tag{12a} \]

where the magnetic quantum numbers \(m_s\) and \(m_l\) may take all values from \(-s\) to \(+s\) (or from \(-l\) to \(+l\)), differing from one another by unity. Therefore

\[ \overline{\mu_H^2}= \left( 4\sum_{-s}^{+s}\frac{m_s^2}{2s+1} + \sum_{-l}^{+l}\frac{m_l^2}{2l+1} \right)\mu_B^2. \tag{12b} \]

We have omitted the middle term, since both \(\overline{\cos\Theta_s}\) and \(\overline{\cos\Theta_l}\) are equal to zero. But just as in (4), the following relations also hold here:

\[ \sum_{-s}^{+s}\frac{m_s^2}{2s+1}=\frac{s(s+1)}{3}, \qquad \sum_{-l}^{+l}\frac{m_l^2}{2l+1}=\frac{l(l+1)}{3}, \]

whence

\[ \overline{\mu_H^2}=\bigl[4s(s+1)+l(l+1)\bigr]\frac{\mu_B^2}{3}. \]

The last expression is equivalent to (11a), since the 3 in the denominator cancels with the 3 in the Langevin formula used to determine \(p\).

Van Vleck established quantitatively that the difference between (11) and (11a) is insignificant. As a result, we may use our curve (11) in Fig. 2 also for representing (11a). It should be noted that, in the case of very slightly split multiplets, wave mechanics leads directly to equation (11a) even without a special consideration of the question of the mode of interaction between \(l\) and \(s\).

The distinction between strongly and weakly split multiplets, although introduced quite naturally, still cannot remove the difficulties that occur in the iron group. This was shown in detail by Laporte, who calculated \(\Delta \nu\) for ions for which these quantities are not known directly from experiment, using a fairly reliable relativistic doublet formula (with appropriate screening numbers). The curves thus obtained for the magneton numbers for the first half of the group lie too low, and for the second—too high, in comparison with the experimentally observed points. Furthermore, these calculations lead to the conclusion that Kossel’s law—the “law of magnetic displacement”—should not be valid: the curve for trivalent ions lies somewhat below the corresponding curve for divalent ions, whereas according to Kossel’s law they ought to intersect.

Generally speaking, the transfer of values of \(\Delta \nu\) calculated for the vapor-like state to solutions and crystals is rather doubtful. Joos* noted that the coloration of solutions in the iron group, which was initially connected by Ladenburg with paramagnetic properties, proves to be incomprehensible if one proceeds from data for ions in the vapor-like state, because in the spark spectrum of these vapors there is not a single absorption line in the visible region; the corresponding \(\lambda\)’s are in general \(< 1700\) Å. From this Joos concludes that, apparently, the coloration is due to complex formations in which—

* G. Joos. Ann. d. Phys. 81, 1076, 1926.

...form weakly bound compounds. We cannot agree with the conclusion that the paramagnetic properties, too, must be attributed not to the cation but to the complex salt.

5. Weak and strong interaction with the surrounding medium

Along with the distinction between strongly and weakly split multiplets, Stoner* also introduced the distinction between weakly and strongly perturbed multiplets. The energy levels of the rare earths are not only strongly split, but also weakly perturbed, since their electrons, which determine the magnetic properties, belong to the \(N\)-shell and are shielded from the outside by a completed \(O\)-shell of electrons, which does not affect the magnetic properties. The situation is different in the iron group, where the \(M\)-shell is not “saturated.” Here the two electrons of the \(N\)-shell which, beginning with neutral calcium, are situated outside, are absent in the ions of the subsequent elements, thereby completely exposing the \(M\)-shell. Thus, here the “perturbation” of the electrons active in the magnetic respect by the surrounding medium is large. Stoner assumes that this perturbation exerts an essential influence not on the moment of the electron itself, \(s\), but on the orbital moment \(l\), and that this perturbation manifests itself in the fact that the number \(l\) decreases. Therefore in equation (11a), which, as has already been mentioned, differs essentially from (11) not at all, he puts \(l = 0\), assuming that an extremely strong interaction with the surrounding electrons takes place. And if equation (11a) can be retained for the case of weakly split and weakly perturbed multiplets, then in the case of weakly split but strongly perturbed multiplets the following equation \((l = 0)\) must hold instead of the indicated equation:

\[ \frac{p}{4.97} = \sqrt{4s(s+1)}. \tag{13} \]

* Stoner, Phil. Mag. 8, 250, 1929.

According to equations (7) and (7a), we can write, for the first and second halves of the period respectively:

\[ \frac{p}{4.97}= \left\{ \begin{array}{cc} \sqrt{z(z+2)} & z\\ \sqrt{z'(z'+2)} & z' \end{array} \right\}<2l+1 . \tag{13a} \]

The new curve obtained in this way, like the previous one (11), is symmetric with respect to the middle. The region between it and curve (11) in Fig. 2 is shaded. In this region all the experimentally observed points should lie, and, depending on the degree of interaction, closer to or farther from (13), and moreover closer to (13) than to (11), since in the iron group, in general, one should expect a very strong perturbation. This, as Fig. 2 shows, is in fact the case. Stoner further puts forward the very simple consideration that at low temperatures curve (13) should be valid, and at high temperatures—(11). The temperature dependence, according to the considerations indicated above, should be somewhat weaker than that which we obtained in the preceding paragraph; the temperature coefficient in both halves of the iron group is positive. It is clear that in the middle and at the ends of the group equations (11) and (13) must coincide, since here we always have an \(S\)-term, where \(l=0\).

Curve (13) runs approximately rectilinearly, because for large values of \(z\) and \(z'\) the twos under the radical in equation (13a) may be neglected. This also explains the integral dependence, noted earlier by the author, between the numbers of magnetons for ions of different valences (cf. note to § 5). The meaning of equation (13) may also be expressed in the following way. With a strong interaction of the electrons with the surrounding medium, the numbers of magnetons behave as if the state of the molecules were always determined by its \(S\)-term.

Stoner’s considerations relate to solutions. The solid crystalline state was investigated by Bethe.* He also established that a strong interaction of neighboring atoms

* Bethe, Ann. d. Phys. 3, 133, 1929.

of the crystal causes a stronger perturbation of the orbital moments than of the axial ones. This is confirmed by the character of the Zeeman effect (Becquerel).

Bose’s work leads to the same result as Stoner’s, and becomes physically comprehensible in the light of Stoner’s hypothesis on the existence of a strong interaction at not very high temperatures. Bose succeeded, with the aid of (13) and (13a), in expressing the observed values for the iron group.

6. Generalization to the Case of Diatomic Molecules

The formulae belonging here were derived by Van Vleck from quantum mechanics. We shall present them here only in translation into the language of the old theory. One must imagine that the orbital moment \(l\) is connected with the atom of the molecule, i.e. with the geometrical axis of the molecule (the line joining the nuclei of the atoms), and, under the influence of the electric forces, can precess about the direction of the field, approximately as happens in the Stark effect. Thus the component of \(l\) perpendicular to the geometrical axis changes in this process; the component parallel to the geometrical axis, which we shall denote by \(\lambda\), remains constant and quantized. The character of the terms is determined by the value of \(\lambda\): if \(\lambda=0\), then an \(S\)-term; \(\lambda=1\), a \(P\)-term, etc.

Conversely, the intramolecular electric forces have little influence on the axial mechanical moment. It is oriented in space in such a way that not only \(S\), but also its components along the geometrical axis, which we shall call \(\delta\), and which, simultaneously with \(S\), are integral or half-integral, are quantized.

There are possible \(2S+1\) different positions of \(S\) with respect to the geometrical axis, differing by the values \(\sigma=S, S-1,\ldots,-S\); hence their \(S\)-multiplicity of the given term is determined: for \(S=0\) a singlet, \(S=\tfrac12\) a doublet, \(S=2\) a triplet, etc. Integral values of \(S\) give odd, and half-integral \(S\) give even multiplicity. Just as for atoms, an odd number of electrons in molecules (the sum

electrons in both atoms) entails an integer value of \(S\), i.e., an even multiplicity, and conversely. The total angular momentum (the total mechanical moment) of the molecules is composed of \(S\), \(\lambda\), and the rotational moment, directed perpendicular to the geometrical axis. The latter, even at small rotational velocities, is sufficiently large owing to the large moment of inertia of the molecule. The total mechanical moment (but not its component perpendicular to the geometrical axis) is quantized; its quantum number \(n\) distinguishes the line of the rotational spectrum.

The multiplet is determined by the constant value of \(\lambda\) for different positions of \(S\) relative to the geometrical axis, i.e., by the variable \(\delta\). Within a given rotational band the multiplet structure is repeated. We shall again have to distinguish broad and narrow multiplets.

For broad multiplets, from the magnetic point of view only the lowest levels are to be considered. In this case \(\delta=\pm S\), depending on whether one is dealing with a normal or an inverted multiplet. In both cases \(S\) coincides with the direction of the geometrical axis (in the same or in the opposite direction, as does \(\lambda\)). The magnetic moment has the same direction and is equal to

\[ \mu=(2\delta+\lambda)\frac{e}{m}\cdot\frac{h}{4\pi}. \tag{14} \]

Since, owing to the large moment of inertia of the molecule, the geometrical axis of the molecule cannot itself orient in the magnetic field, while all its positions relative to the magnetic field are equally probable, we have \(\cos^{2}\Theta=\frac{1}{3}\), and therefore Langevin’s formula holds exactly.

Hence for the number of Weiss magnetons we obtain:

\[ \frac{P}{4.97}=2\delta+\lambda \tag{14a} \]

As an application of this, let us consider the molecule NO. The number of electrons is \(7+8=15\), i.e., odd. The ground state is a \({}^{2}P\)-term, with \(\Delta\nu=121\ \mathrm{cm}^{-1}\), and moreover—a normal

\({}^{2}P_{1/2}\) with \(\sigma=-\frac{1}{2},\ \lambda=1\) lies lower than the level \({}^{2}P_{3/2}\) with \(\sigma=+\frac{1}{2},\ \lambda=1\).

At very low temperatures \(kT \ll hc\,\Delta\nu\) the doublet may be regarded as broad. Application of equation (14) then gives \(2\delta+\lambda=0,\ p=0\), i.e. NO at very low temperatures must approach the diamagnetic state. For multiplets that are only slightly separated we must take into account all levels obtained for the different positions of \(S\) for a given \(\lambda\). In doing so we shall use the representation that was introduced in deriving equation (12), with the only difference that instead of “position relative to the magnetic field” we shall have to say: position relative to the geometrical axis. Thus, analogously to (12) and (12a), we write

\[ \mu_F=(2S\cdot \cos\Theta_s+\lambda)\cdot\mu_B, \tag{15} \]

where \(\Theta_s\) is the angle between \(S\) and the direction of the geometrical axis, while \(\lambda\) coincides with the latter:

\[ \overline{\mu_F^{\,2}}=(4S^2\cos^2\Theta_s+4S\cos\Theta_s\cdot\lambda+\lambda^2), \tag{15a} \]

where the middle term on the right-hand side is equal to zero and \(S\cos\Theta_s=\sigma\):

\[ \overline{\mu_F^{\,2}} = \left(4\sum_{-s}^{+s}\frac{\sigma^2}{2S+1}+\lambda^2\right)\mu_B^2 = \left[\frac{4S(S+1)}{3}+\lambda^2\right]\mu_B^2. \tag{15b} \]

In addition to the component \(\sigma\) in the direction of the geometrical axis, it is also necessary to take into account the component \(S\), perpendicular to the direction of the geometrical axis \(S_n\), which is not destroyed by rapid precession, as is the case with the normal component \(l\) (see above). Let \(\Theta'\) be the angle between \(S_n\) and the magnetic field, and \(\Theta\) the angle between the geometrical axis and the magnetic field; then:

\[ \mu_{\mathrm{H}}=2S_n\cos\Theta' + \mu_F\cos\Theta \]

\[ \overline{\mu_{\mathrm{H}}^{\,2}}=\frac{1}{3}\left(4S_n^{2}+\mu_F^{2}\right). \tag{15c} \]

The factor \(\frac{1}{3}\) appeared because the geometric axes and the components \(S_n\) normal to them can occupy all positions in space with equal probability.

Next we have:

\[ S_n^2 = |S|^2 - \sigma^2;\qquad \overline{S_n^2}=|S|^2-\delta^2 \]

\(\sigma^2\), as has already been used in (15), is equal to \(\frac{S(S+1)}{3}\).

\[ S^2=S(S+1) \]

(follows from quantum mechanics), whence

\[ S_n^2=S(S+1)-\frac{S(S+1)}{3}=\frac{2}{3}S(S+1). \]

Substituting this and (15b) into (15c), we obtain:

\[ \mu_{\mathrm{H}}^2=\frac{1}{3}\,[4S(S+1)+\lambda^2]\,\mu_{\mathrm{B}}^2. \tag{16} \]

Conversion into Weiss magnetons gives the formula analogous to (11a)

\[ \frac{P}{4.97}=\sqrt{4S(S+1)+\lambda^2} \tag{16a} \]

The difference is that \(l\) enters here only through its quantized component \(\lambda\), instead of \(l(l+1)\). In the case of the NO molecule, (16a) gives at high temperatures, when \(kT > h\cdot\Delta\nu\), where both doublets appear with their respective weights, the upper limit

\[ \frac{P}{4.97}=2. \]

Thus NO does not obey Curie’s law; its number of magnetons \(p\) increases from zero at \(P=0\) to 10 at \(P=\infty\). The value calculated by Pauli (see § 1), \(p=8.6\), is valid only for room temperatures and by no means has general significance.

Another example is given by \(\mathrm{O}_2\). Its ground state is the \({}^{3}S\)-term* \((l=0,\ \lambda=0,\ S=1,\ \sigma=+1\ \text{and}\ 0,\ \text{i.e. three levels})\). Here (16a) is valid down to the very lowest temperatures and gives

* R. S. Mulliken, Phys. Rev. 32, 880, 1928. It is noteworthy that this character of the term was predicted in advance by van Vleck from equation (16).

\[ \frac{p}{4.97}=2\sqrt{2}, \qquad p=14.0, \]

which agrees well with the experimental data and with strict observance of Curie’s law. Comparison of this with equation (6a) for the atomic \(^{3}S\)-term gives the same result; the reason for this is that for a single \(S\)-term \(l\) and \(\lambda\) become equal to zero.

For NO, Van Vleck calculated a transition formula for intermediate temperatures, which in our notation is written as follows:

\[ \frac{p}{4.97} = 2\sqrt{ \frac{1-e^{-x}+xe^{-x}}{x(1+e^{-x})} } \qquad x=\frac{h\cdot\Delta\nu}{kT}; \qquad \Delta\nu=121\ \mathrm{cm}^{3}. \]

It is analogous (only quantum-mechanically improved and adapted for NO) to our former equation (10). From this there follow the limiting cases already determined earlier for

\[ \begin{aligned} T&=0, & x&=\sim, & p&=0,\\ T&=\sim, & x&=0, & p&=10. \end{aligned} \]

For certain intermediate temperatures (16) was very well confirmed by Aharoni and Scherrer.*

* Zs. Physik 68, 749, 1929.