Magnetism and the Structure of Atoms

By Ig. Tamm.

1. Langevin’s theory and the Weiss magneton. — 2. The Bohr magneton. — 3. Quantization of orientation. — 4. Direct proof of the quantization of orientation. — 5. Quantum theory and experimental data on the Bohr magneton. — 6. Spectroscopic method for determining the magnetic moment of atoms. — 7. Magnetism and the periodic system of the elements. — 8. Difficulties of the quantum theory of magnetism. — 9. Literature.

The present state of development of the theory of atomic magnetism undoubtedly deserves the closest attention. Recent years have brought the quantum theory a number of very major successes in this field. Questions of atomic magnetism have proved to be directly connected with the question of the complex structure of spectral lines, with the internal structure of the atom, and so on. But what is especially interesting is that in the theory of magnetism the characteristic aspects of the entire present situation of quantum theory as a whole have been reflected with unusual vividness. On the one hand, one must acknowledge the astonishing success of assumptions that are simple to the point of naiveté; on the other hand, there is the uninterrupted accumulation of ever sharper internal contradictions. These contradictions, rooted in the application of the laws of classical physics to the stationary states of the atom, nowhere, perhaps, manifest themselves with such sharpness and definiteness as in the field under consideration here. It is precisely for this reason that the further development of the quantum theory of magnetism promises to be especially fruitful.

This theory has already grown so extensive at the present time that, of course, in a journal article it is impossible to give any systematic exposition of the questions connected with it. The present article will be devoted chiefly to the question of the resultant magnetic moment of the atom as a whole; questions of intra-atomic magnetism, so important for the theory of spectra, will be touched upon only insofar as is necessary for the main theme; finally, ferromagnetic phenomena, rooted in the interaction between atoms, are entirely excluded from consideration.

1. Langevin’s Theory and the Weiss Magneton

Before proceeding to an exposition of the modern theories of magnetism, it is necessary to recall the experimental foundations of the whole doctrine of atomic magnetism.

With the exception of the new spectroscopic method, which will be discussed below, there exists only one method for determining the magnitude of the magnetic moment of paramagnetic atoms—the method based on Langevin’s kinetic theory1. In view of the exceptional importance of this theory, I shall briefly recall its basic propositions.

Let us consider a paramagnetic gas whose molecules possess a magnetic moment \(m\). In the absence of an external magnetic field, the gas molecules are oriented at random. The appearance of a field leads to the alignment of the magnetic axes of the molecules along the direction of the field, which, however, is opposed by thermal motion. As a result a stationary state is established in which, according to the well-known theorem of statistical mechanics, the distribution of the axes over different directions is determined by Boltzmann’s formula:

\[ dn = c \cdot e^{-\frac{P}{kT}} \cdot d\omega . \]

Here \(dn\) denotes the number of molecules in a mole (gram-molecule) whose axes lie within the solid angle \(d\omega\); \(P\) denotes the potential energy corresponding to this orientation of the molecule; finally, \(k\) is Boltzmann’s constant, and \(T\) is the absolute temperature. If the angle between the magnetic axis of the molecule and the direction of the magnetic field \(H\) is equal to \(\varphi\), then, as is known, \(P = -mH \cdot \cos \varphi\).

Introducing the notation

\[ a = \frac{mH}{kT}, \]

we obtain

\[ dn = c \cdot e^{a\cos\varphi} d\omega . \]

In all cases of interest to us \(a\) is small, and therefore the quantity \(e^{a\cos\varphi}\) may be expanded in a series, retaining only the first terms of the expansion:

\[ dn = c(1 + a\cos\varphi)\cdot d\omega . \]

The coefficient of proportionality \(c\) is determined from the condition that the total number of molecules in a mole must be equal to Avogadro’s number \(N\). Carrying out the calculation, we find that \(c = \dfrac{N}{4\pi}\), and consequently,

\[ dn = \frac{N}{4\pi}(1 + a\cos\varphi)\cdot d\omega . \]

The total magnetic moment of the molecules \(dn\) is equal to \(m\,dn\); in the direction of the field \(H\) there corresponds to it a component of magnetization \(dG\):

\[ dG=m\cdot \cos\varphi\cdot dn=\frac{Nm}{4\pi}\cos\varphi(1+a\cos\varphi)\cdot d\omega . \]

Integrating over all possible directions of the axes, we obtain the total magnetization of one mole of gas:

\[ G=\frac{Nm}{4\pi}\int(\cos\varphi+a\cos^{2}\varphi)\cdot d\omega =\frac{Nma}{4\pi}\int\cos^{2}\varphi\cdot d\omega =Nma\cdot \overline{\cos^{2}\varphi}. \]

Here \(\overline{\cos^{2}\varphi}\) denotes, as usual, the mean value of \(\cos^{2}\varphi\) over all possible orientations of the molecule:

\[ \overline{\cos^{2}\varphi} =\frac{1}{4\pi}\int\cos^{2}\varphi\cdot d\omega =\frac{1}{4\pi}\iint\cos^{2}\varphi\cdot\sin\varphi\cdot d\lambda\cdot d\varphi =1/3. \]

We shall not, however, for the time being introduce this numerical value into the preceding formula, but shall substitute into it only the value of the quantity \(a\). We obtain:

\[ G=\frac{Nm^{2}H}{kT}\cdot \overline{\cos^{2}\varphi}. \]

Dividing both sides of the equality by \(H\), we obtain an expression for the magnetic susceptibility (when computed per gram-molecule):

\[ \chi=\frac{G}{H}=\frac{Nm^{2}}{kT}\overline{\cos^{2}\varphi}. \]

Multiplying both parts of the fraction by Avogadro’s number \(N\) and recalling that \(Nk=R\), where \(R\) is the gas constant, and introducing, moreover, the new notation

\[ Nm=M, \]

we finally obtain:

\[ \chi=\frac{M^{2}}{RT}\overline{\cos^{2}\varphi};\qquad \overline{\cos^{2}\varphi}=1/3. \tag{1} \]

Such is the final form of Langevin’s formula for the case \(a\ll 1\). This theoretical formula is in complete agreement with the empirical formula previously found by Curie (P. Curie):

\[ \chi=\frac{C}{T},\qquad C=\mathrm{const}, \tag{2} \]

which well represents the temperature dependence of the magnetic susceptibility of a whole series of paramagnetic substances. Comparing the right-hand sides of these formulae leads to the relation

\[ C=\frac{M^{2}}{R}\cdot \overline{\cos^{2}\varphi},\quad \text{whence}\quad M=\sqrt{\frac{RC}{\overline{\cos^{2}\varphi}}}. \tag{3} \]

Substituting here \(\cos^2\varphi = 1/3\), we finally obtain

\[ M=\sqrt{3RC}. \tag{3a} \]

Since the Curie constant \(C\) is determined from experimental data, this formula may serve for determining \(M\).

The Langevin theory set forth applies to paramagnetic gases; for paramagnetic solid and liquid bodies the temperature dependence is expressed not by Curie’s formula (2), but by a more complicated formula with two constants:

\[ \chi=\frac{C}{T-\theta},\quad C=\mathrm{const},\quad \theta=\mathrm{const}. \tag{2a} \]

Weiss succeeded in giving a theoretical interpretation of this more complicated formula as well. Let us suppose that upon magnetization an internal magnetic field \(H_i\) is created, proportional to the magnetization \(H_i=\lambda G\); and let us suppose that this internal field, in its turn, acts in an orienting manner on the molecules of the body; in other words, let us take into account the magnetic interaction of the molecules. Then it will be necessary to correct in the corresponding way the expression for \(P\) and for \(a\); obviously, in this case

\[ a=\frac{m(H+H_i)}{kT}=\frac{m(H+\lambda G)}{kT}. \]

Repeating now the previous chain of reasoning, we finally obtain a formula of type (2a), namely

\[ \chi=\frac{C}{T-\lambda C}, \]

where, as before,

\[ C=\frac{M^2}{R}\cos^2\varphi . \tag{3} \]

Thus formula (3) proves applicable to all paramagnetic bodies, irrespective of their state of aggregation.

Weiss applied this formula to a large number of paramagnetic bodies and came to the conclusion that the magnetic moments of different molecules are almost always in ratios of whole numbers. In other words, Weiss found that the magnetic moment of a molecule is always an integral multiple of a certain elementary magnetic moment which, by analogy with the electron, received the name magneton1. The magnitude of this elementary magnetic moment \(m_o\)

proved to be equal to \(18.6\cdot 10^{-22}\) CGS. It is customary, however, to consider not \(m_0\), but the quantity \(M_0=Nm_0=1123.5\) gauss \(\times\) cm, i.e., in other words, to calculate the magnetic moment per mole (gram-molecule) of a substance.

As is known, the magnetic properties of chemical compounds obey the law of additivity; in verifying this law, however, it is necessary to take into account the state of ionization of the atoms, which has a decisive influence on their magnetic properties. In other words, it is necessary to distinguish the magnetism of neutral atoms from the magnetism of the corresponding ions. On the other hand, the magnetic moment of an atom in a given state of ionization is almost entirely independent of the state of aggregation of the substance under study (solution, crystalline compound, etc.).

By way of illustration we give a table of values of the magnetic moment of the ion \(\mathrm{Mn}^{\prime\prime}\), obtained by various investigators1. In this table, as usual, the Weiss magneton is taken as the unit of magnetic moment.

TABLE I.

Compound	Anhydrous salt in the solid state	With water of crystallization \(4\mathrm{H}_2\mathrm{O}\)	In aqueous solution	Observers
\(\mathrm{Mn}^{\prime\prime}\mathrm{SO}_4\) . . .	29.04 (Th)    29.0 29.05 (Th)	29.2 O. 29.06 (single crystal) F.	29.33	Th — (Theodoridès, 1922) Theodoridès. H — (Honda, 1914) Honda.
\(\mathrm{Mn}^{\prime\prime}(\mathrm{NO}_3)_2\) .	—	—	29.33 C.	O — (K. Onnes & Oosterhuis, 1913) Kamerlingh-Onnes and Oosterhuis.
\(\mathrm{Mn}^{\prime\prime}\mathrm{Cl}_2\) . . .	28.45 (Th)    27.3 H	—	29.43 C.	F — (Foex, 1921) Foex.
\(\mathrm{Mn}^{\prime\prime}\mathrm{O}\) . . .	27.43 (Th) 30.2 26.43 (Th)	— —	— —	C — (Cabrera) Cabrera.

One of the best contemporary specialists in the magnetism of salts, the Spanish physicist Cabrera (P. Cabrera), considers it “almost certain” that the magnetic moment of the \(\mathrm{Mn}^{\prime\prime}\) ion is equal to 29.0 magnetons. To an unprejudiced person, writing down a zero after the decimal point seems

at least premature, since mutual agreement among different definitions is not sufficient to judge the decimal places. This example is quite typical for the Weiss magneton theory. The validity of this theory, which asserts that the magnetic moments of atoms are equal to an integral number of Weiss magnetons, has repeatedly aroused doubts^1). Attention was drawn to the well-known prejudice of its supporters in deriving mean values; it was also pointed out that some atoms and ions are assigned so large a number of magnetons (up to 30) that, given the low accuracy of the experimental data, the distinction between such large integers and intermediate fractional numbers lies within the limits of experimental error. There is no doubt, in any case, that the reality of the Weiss magneton cannot be considered definitively established. It must be admitted, however, that for certain substances measurements give magneton numbers that are indeed extremely close to integers. In any case, the great merit of the Weiss magneton theory is that it served as an impetus for a whole series of investigations that collected a large body of experimental material.

As is known, paramagnetism is encountered almost exclusively in those regions of the periodic system of elements in which a rearrangement of the inner electron shells of the atom takes place. Such are the elements of the triads of the eighth group and those immediately preceding them; such also are the rare earths. However, only the elements of the iron group from Ti (22) to Ni (28) have been investigated at all fully; in most cases, therefore, we shall have to confine ourselves to consideration of this group of elements. Table II gives data on the number of magnetons in various ions of this group. For the most part they are taken from Cabrera’s critical summary (Cabrera [2]); the data concerning \(Mn''\) and \(Mn'''\) are taken from Gerlach’s paper (a); finally, the number of magnetons in \(V''\) and \(V'''\) was determined by Pascal (Pascal) and is cited by us according to Weiss. In the last line of the table are indicated, for certain ions, those limits within which the results of different measurements fluctuate.

TABLE II.

Name of ion:	\(Cr''\)	\(Cr'''\)	\(Mn''\)	\(Mn'''\)	\(Mn''''\)	\(Fe''\)	\(Fe'''\)	\(Co''\)	\(Ni''\)	\(Ti'''\)	\(V''\)	\(V'''\)
Number of magnetons.	24.0	19.0	29.0	25	19	26.0	29.0	25.0	16.0	8.6	9.2	6.7
Limits of fluctuation.	—	—	26.5—32.2	—	—	26—29	28.7—29.15	24.0—25.06	13—16.9	—	—	—

2. THE BOHR MAGNETON. The Weiss “magneton theory” is of a purely empirical character. It can be given a theoretical justification only on the basis of definite conceptions concerning the structure

^1) At the present time the author of the theory himself has acknowledged that the number of magnetons in an atom may be semi-integral (a multiple of one half)! See Journ. d. Ph., 1924, 5, p. 129.

of the atom. In the matter of studying the structure of atoms, such progress has been made during the last decade that, for the theory of atomic magnetism, it would seem that the necessary prerequisites have already been created.

First, the experiments of Einstein and de Haas (Einstein und de Haas) and of Barnett and their continuators revealed the existence of magneto-mechanical effects, namely the magnetization of ferromagnetic metals and alloys under rapid rotation and, conversely, the appearance of rotating ponderomotive forces upon magnetization of these metals. These experiments brought with them the long-awaited proof of Ampère’s theory of molecular currents; at the present time it may be regarded as beyond doubt that the magnetism of atoms is due to the motion of electric particles inside the atoms.

On the other hand, Bohr’s doctrine of the structure of atoms makes it possible to calculate in advance the very magnitude of the magnetic moment of atoms and, almost without any additional assumptions, leads to the concept of the magneton. In other words, Bohr’s theory leads to the assertion that atomic moments must be integral multiples of a certain elementary magnetic moment, or magneton. However, despite the qualitative agreement between the magnetic theories of Bohr and Weiss, there exists between them a sharp quantitative contradiction: Bohr’s theoretical unit of magnetism—the “Bohr magneton”—is five times greater than the empirically found “Weiss magneton.”

The calculation itself of the magnitude of the Bohr magneton presents no difficulties. According to the quantum theory, the angular momentum (moment of quantity of motion) of an electron in an atom \(j\) must be an integral multiple of \(\frac{h}{2\pi}\):

\[ j=\mu[rv]=\frac{nh}{2\pi}, \qquad n=1,2,3\ldots \tag{4} \]

Here \(\mu\) is the mass of the electron, \(r\) its distance from the center of the atom, and \(v\) its velocity. It is known that the angular momentum \(j\) is equal to the areal constant, i.e. equal to twice the sectorial velocity of the electron multiplied by its mass:

\[ j=2\mu\frac{ds}{dt}, \tag{5} \]

where \(s\) is the area described by the radius vector.

On the other hand, in calculating the magnetic field of a moving electron, we may replace it by a closed current, whose strength \(i\) is determined from the condition

\[ i=\frac{\varepsilon}{T} \]

where \(\varepsilon\) is the charge of the electron, and \(T\) is the period of its revolution. The magnetic moment \(m\) of a closed current is equal, as is known, to \(iS\):

\[ m=iS, \]

where \(S\) denotes the magnitude of the area swept out by the current. It is obvious that

\[ S = I\,\frac{ds}{dt}, \]

and therefore

\[ m = iS = \varepsilon\,\frac{ds}{dt}. \]

Comparing this with formula (5), we obtain the important relation

\[ m=\frac{\varepsilon}{2\mu}\,j, \tag{6} \]

whence, on the basis of (4), we finally find

\[ m=n\,\frac{\varepsilon\cdot h}{2\mu\cdot 2\pi},\qquad n=1,2,3\ldots \tag{7} \]

Thus the magnetic moment due to the motion of the electron in the atom must always be an integral multiple of the elementary magnetic moment \(m_o\), called the Bohr magneton:

\[ m=nm_o,\qquad m_o=\frac{\varepsilon\cdot h}{2\mu\cdot 2\pi}. \tag{8} \]

Substituting into formula (8) the known values of the universal constants and recalculating, as usual, the magnitude of the magneton per gram-molecule, we obtain

\[ \text{Bohr magneton}=M_o=Nm_o=\frac{N\varepsilon h}{2\mu\cdot 2\pi}=5584\ \text{gauss}\times\text{cm.} \tag{9} \]

Equations (4), (6), (7) can be given a very simple form, if only for the measurement of the rotational and magnetic moments of the atom \(j\) and \(m\) one uses not C.G.S. units, but rational units, equal respectively to \(\dfrac{h}{2\pi}\) and \(\dfrac{\varepsilon\cdot h}{2\mu\cdot 2\pi}\), i.e. the Bohr magneton. Then these equations take the following simple form:

\[ j=n,\qquad n=1,2,3\ldots \tag{4a} \]

\[ m=j, \tag{6a} \]

\[ m=n,\qquad n=1,2,3.. \tag{7a} \]

In what follows we shall always use this rational system of units, except only in specially stipulated cases.

But let us return to the numerical value of the Bohr magneton. As eq. (9) shows, the Bohr magneton is indeed 5 times (more precisely, 4.98 times) larger than the Weiss magneton; consequently, according to Bohr’s theory, the number of Weiss magnetons in an atom of any substance must always be

multiples of five. This proposition is in a definite contradiction with the experimental data; moreover, the sharpness of this contradiction is in no way connected with the question of the reality of the Weiss magneton. For Weiss arrived at the concept of the magneton by a purely empirical route, and the Weiss magneton is, by definition, equal to the greatest common divisor of the magnetic moments of various atoms. It follows that, in any case, the magnetic moments of atoms cannot be whole multiples of a Bohr magneton five times as large.

How is this contradiction between theory and experiment to be explained? Have we not made some omissions in the preceding discussion? First, it may appear that we have taken into account the magnetic field of only the electrons and have forgotten the field of the positive charges, which perhaps also move in the atom. Taking into consideration, however, that formula (7) remains valid also for positive charges, but that the ratio \(\varepsilon/\mu\) for these charges is incomparably smaller than the same ratio for electrons, we shall be convinced that the magnetic field of the positive charges may safely be neglected.

A defect of the theory set forth might also be seen in the fact that, if the magnetic moment of each of the electrons moving in the atom is equal to an integral number of Bohr magnetons, then from this circumstance it is still impossible to draw any conclusions about the magnetic moment of the whole atom as a whole: for the resultant magnetic moment of an atom is composed vectorially of the moments of the electrons entering into its composition and therefore depends on the geometrical arrangement of the orbits of these electrons. However, from the point of view of quantum theory this objection also proves untenable. In order to clarify this question of primary importance, we shall have to dwell in somewhat greater detail on the so-called quantization of orientations.

3. Quantization of orientations (Raumquantelung). Generally speaking, quantum theory restricts only the numerical magnitude of the vector of angular momentum \(j\), requiring that this vector assume only integral values\(^{1}\). The direction of the vector \(j\), generally speaking, remains arbitrary.

However, if, among all possible spatial directions, some one direction is physically distinguished in some way (coinciding, for example, with the direction of an electric or magnetic field), then an additional restriction is introduced concerning the direction of the vector \(j\): not only may the vector itself assume only integral values, but also its projection onto the distinguished (ausgezeichnete)

\(^{1}\) Of course, under the condition of choosing a rational unit of angular momentum, equal to \(\dfrac{h}{2\pi}\). See eq. (4a).

a spatial direction can have only an integral value1. If this rule is applied to the question of the structure of the atom, as was first done by Landé (Landé) 3, then the “distinguished” direction must be taken to be the direction of the resultant angular momentum (the normal to the invariable plane of the atom). Consequently, the projections of the angular momentum of each of the electrons on the direction of the resultant must have integral values; hence this resultant too will have an integral value. In other words, the resultant magnetic moment of the atom as a whole must be equal to an integral number of Bohr magnetons.

Thus the objection to the quantum theory of magnetons raised at the end of the preceding § proves, from the standpoint of this theory, to be untenable, and the contradiction between theory and experiment remains in full force. This contradiction for a long time was a favorite argument of the opponents of Bohr’s theory, until finally, in 1920, W. Pauli jun. showed that it is rooted merely in an insufficiently consistent application of quantum theory.

Indeed, the magnitude of the magnetic moment of paramagnetic atoms and ions is computed, as we have seen, by Langevin’s formula,

\[ M=\sqrt{3RC}. \tag{3a} \]

In deriving this formula one has to determine the mean value of \(\cos^2\varphi\), and it is assumed that the axes of the atoms may take any direction in space. However, the theory of space quantization restricts the direction of the axes of atoms to a number of discrete possibilities, which, of course, may substantially change the numerical value of the quantity \(\cos^2\varphi\). According to the classical theory, \(\cos^2\varphi\) under all circumstances is equal to one third; according to the quantum theory, however, the value of \(\cos^2\varphi\) depends on the magnitude of the angular momentum \(j\). Since the projection of the vector \(j\) on the “distinguished” direction of the external magnetic field \(H\) must take only integral values, then for \(|j|=n\) (where \(n\) is an integer) \(\cos\varphi\) can have only one of the following values2:

\[ \cos\varphi=\pm\frac{1}{n},\ \pm\frac{2}{n},\ldots,\pm\frac{n-1}{n},\ \pm\frac{n}{n}. \]

Therefore

\[ \overline{\cos^2 \varphi} = \frac{1}{n} \left[ \left(\frac{1}{n}\right)^2 + \left(\frac{2}{n}\right)^2 +\cdots+ \left(\frac{n-1}{n}\right)^2 + \left(\frac{n}{n}\right)^2 \right]. \]

Carrying out the calculation, we obtain

\[ \overline{\cos^2 \varphi} = \frac{1}{3}\frac{(n+1)(2n+1)}{2n^2}. \]

Thus, \(\cos^2\varphi\) assumes its “classical” value \(1/3\) only in the limit as \(n=\infty\); generally speaking, formula (3a) must be replaced by the following (see Eq. 3):

\[ M' = \sqrt{\frac{RC}{\overline{\cos^2\varphi}}} = \sqrt{3RC}\sqrt{\frac{2n^2}{(n+1)(2n+1)}}, \tag{3b} \]

or

\[ M' = M\sqrt{\frac{2n^2}{(n+1)(2n+1)}}. \tag{3c} \]

Here \(M'\) denotes the magnetic moment of the gram-atom, calculated according to quantum theory, while \(M\) is the value of the same moment according to classical theory; \(n\) is equal to the number of Bohr magnetons in the atom.

Thus, from the quantum point of view, formula (3a), by which the magnetic moment of atoms is usually calculated, turns out to be incorrect. There is therefore nothing surprising in the fact that the fictitious values of the magnetic moments found with the aid of this formula did not prove to be multiples of the Bohr magneton. On the contrary, the quantity \(M'\), obviously, must be an integral multiple of the Bohr magneton, i.e. the following equality must hold:

\[ M' = n \times 5584\ \text{gauss} \times \text{cm}, \quad \text{where } n \text{ is an integer}. \]

What, then, should be the relation between the Bohr and Weiss magnetons? The number \(k\) of Weiss magnetons in the atom is, of course, calculated from the magnitude of its magnetic moment \(M\), determined by the “classical” formula (3a); in other words, \(k\) is determined from the following relation:

\[ M = k \times 1123.5\ \text{gauss} \times \text{cm}. \]

Substituting the last two equations into formula (3c) and reducing (approximately) by \(1123.5\), we obtain:

\[ 5n = k\sqrt{\frac{2n^2}{(n+1)(2n+1)}}; \]

or

\[ k = 5n\sqrt{\left(1+\frac{1}{n}\right)\left(1+\frac{1}{2n}\right)}. \tag{10} \]

Thus, although the magnitude of the Bohr magneton is almost exactly 5 times greater than that of the Weiss magneton, nevertheless the relation between the number \(k\) of Weiss magnetons in an atom (calculated according to the classical theory) and the number \(n\) of Bohr magnetons in the same atom (calculated according to the quantum theory) has a rather complicated character.

It is easiest to understand this dependence with the aid of Table III. The number \(k\) of “apparent” Weiss magnetons corresponding to one, two, etc. Bohr magnetons has been calculated from formula (10) and entered in the last line of this table.

TABLE III.

Number of Bohr magnetons \(n\)	1	2	3	4	5
Apparent number of Weiss magnetons \(k\)	8.7	13.7	18.7	23.7	28.7

This table makes it possible to carry out a test of the theory. Indeed, if the quantum theory is correct, then processing the experimental data according to the “classical” formula (3a) must necessarily lead to one of the following values of the number \(k\) of Weiss magnetons (in round numbers): 9, 14, 19, 24, etc.

Pauli considered it possible to carry out this test of the quantum theory only with respect to paramagnetic gases: in solids and liquids, the internal molecular forces are too large for one to be able to speak of quantization of orientation with respect to an external magnetic field.

Up to the present, only two paramagnetic gases are known, NO and \(O_2\). The apparent number of Weiss magnetons \(k\), according to the latest measurements, ranges for NO from 8.9 to 9.2, and for \(O_2\) from 13.9 to 14.12. The agreement of these numbers with the numbers in the preceding table (\(k=8.7\), \(k=13.7\)) is quite satisfactory; this gives us the right to suppose that in NO there are two, and in \(O_2\) one, Bohr magneton; in other words, in each oxygen atom there is one Bohr magneton.

However strongly this quantitative agreement speaks in favor of the quantum theory, it must nevertheless be noted that the Pauli theory set forth by us applies, strictly speaking, only to monatomic gases; for the case of the diatomic gases NO and \(O_2\), the thermal rotation of the molecules would also have to be taken into account. A priori it seems very probable that rotation will smooth out the influence of spatial quantization. However, the indicated agreement of the elementary theory with experiment proves that thermal rotation does not destroy the discrete quantum orientation of the magnetic axes. This fact can be explained only by making the improbable assumption that the magnetic axis of the NO and \(O_2\) molecule is directed perpendicular to its axis of symmetry (i.e. to the line joining the centers of the atoms).

In any case, whatever the difficulties connected with the further development of the theory may have been, one thing is beyond doubt: by pointing out the incorrectness of the Langevin formula from the point of view of quantum theory, Pauli thereby proved the complete untenability of those objections to the quantum theory of magnetism which were based on an apparent contradiction between this theory and the “direct” measurements of atomic magnetic moments.

In Pauli’s small paper under discussion (1920), strictly speaking, all the fundamental propositions of the modern theory of atomic magnetism are already contained; the subsequent development of the theory consisted in the elaboration and experimental verification of these propositions.

4. Direct proof of the quantization of orientation. We have convinced ourselves that the fundamental premise of the quantum theory of magnetism is the hypothesis of the quantum, discrete character of the laws of orientation of atoms. It is natural, therefore, that direct experimental proof of the quantization of orientation should have enormous significance for this entire theory. The way to such a proof was indicated by O. Stern in 1921; the experiment itself was successfully carried out by him in collaboration with W. Gerlach in 1922.

This experiment is striking in its simplicity and is almost the most direct of all experimental confirmations of the quantum theory known so far. Its persuasive force is so great that, chiefly under the influence of the success of precisely this experiment, M. Planck, in the 5th edition (1923) of his famous book Theorie der Wärmestrahlung, abandoned the second variant of his quantum theory of radiation1.

In view of the wide renown which the Stern and Gerlach experiment2 has managed to acquire, in describing it I shall permit myself to confine the account to only the most essential points. The problem that confronted Stern was to measure the magnetic moment of an atom or, more precisely, the component of this moment in the direction of the external magnetic field. Stern pointed out that for this it is sufficient to measure the mechanical action of the field on the atom, provided that this field is non-uniform. Indeed, in a non-uniform magnetic field, besides the couple of forces tending to turn the atom and set its axis in the direction of the field, there also acts on the atom a resultant force applied at its center of gravity. Let us suppose, for simplicity, that

the direction of the gradient \(\dfrac{\partial H}{\partial s}\) coincides with the direction of the field \(H\) itself, then, obviously, the resultant force applied to the atom will be

\[ F = m \cdot \frac{\partial H}{\partial s} \cdot \cos (m, H), \]

where the magnetic moment \(m\) must, of course, be expressed not in rational but in ordinary CGS units.

Thus, the magnitude of this force \(F\) depends on the projection of \(m\) onto the direction of the field \(H\), i.e. on the quantity \(m \cos (m, H)\). By measuring the force \(F\) and the gradient \(\dfrac{\partial H}{\partial s}\), it became possible to measure also \(m \cos (m, H)\).

According to classical theory, any values of \(\cos (m, H)\) from 0 to 1 are possible, whereas according to quantum theory all atoms are divided into several sharply defined groups, and to each group there corresponds one definite value of the projection \(m \cos (m, H)\). Observations of the deflection of atoms in an inhomogeneous magnetic field under the influence of the force \(F\) must decide which of these theories is correct.

Fig. 1.            Fig. 2.

In practice this experimentum crucis was carried out as follows. In a high vacuum1 silver was boiled; from the stream of atoms escaping from the surface of the molten silver, two diaphragms with an aperture of \(0.5 \times 0.05\) mm cut out a narrow beam of atoms; this “atomic ray” passed along a wedge-shaped pole of an electromagnet at a distance of several tenths of a millimeter from it and was then caught by a cooled glass plate. Near the pole the gradient \(\dfrac{\partial H}{\partial s}\) of the magnetic field reached \(150{,}000\ \dfrac{\text{gauss}}{\text{cm}}\), the direction of the gradient coinciding with the direction of the field. Fig. 1 shows a microphotograph, enlarged 40 times, of the silver deposit obtained on the glass plate with the electromagnet switched off; Fig. 2 shows the deposit (the trace of the beam) obtained in the presence of the magnetic

fields. In the latter case the silver beam was sharply split in two; one part of the atoms was attracted, the other was repelled with the same force from the pole of the magnet.

Thus, all the atoms were divided into two and only two groups: in one group the axes are directed along the field, in the other—in the directly opposite direction. Intermediate orientations are entirely absent.

Thus this experiment proves, first, the quantization of orientation in a magnetic field and, second, the presence in the silver atom of one magneton. The latter follows from the circumstance that, as was indicated on p. 114, the number of possible orientations of the axis of an atom is twice the number of magnetons present in it.

Moreover, the experiment of Stern and Gerlach makes it possible to measure the very magnitude of the magnetic moment of silver, i.e. the magnitude of the Bohr magneton. For this it is evidently necessary only to measure the magnitude of the deflection of the beam in the magnetic field and the magnitude of the gradient \(\dfrac{\partial H}{\partial s}\), as well as the time during which the atom was subjected to the action of the deflecting force. The latter is determined from the length of the path traversed in the magnetic field and from the velocity of flight of the atoms (this velocity had been directly measured by Stern under the same experimental conditions as early as 1920). The mean value of the magnetic moment of the silver atom from two different experiments proved to be (calculated per gram-atom):

\[ mN=M=5475\pm 5\% \ \text{gauss}\times \text{cm}, \]

which agrees excellently with the theoretical value of the Bohr magneton,

\[ M_o=5584 \ \text{gauss}\times \text{cm}. \]

Thus, the experiments of Stern and Gerlach brought with them a brilliant confirmation of the quantum theory of the magneton.

In connection with this success of the experiment, a number of questions naturally arise. In what way does the process of setting the axis of the atom into the quantum-allowed direction take place and, in particular, what happens when the direction of the external field is changed? Does the axis of the atom follow the direction of the field continuously or not? In an extremely interesting joint work, Einstein and Ehrenfest considered a number of possible assumptions and arrived at the following rather discouraging conclusions.

Every change in the orientation of the magnetic axis of an atom in an external magnetic field must be accompanied by the emission or absorption of the corresponding energy. If this radiation and absorption of energy occurred according to the classical laws, then for the quantum setting of the axis of the atom an interval of time would be required exceeding by \(10^{14}\) times the time of flight of the atom in the experiment of Stern and Gerlach. One may renounce the classical views and suppose that, upon a change ...

the direction of the field the direction of the atom’s axis at first lags behind it, and that the setting of the axis then occurs by a jump, with the release (absorption) of excess (deficient) energy taking place according to quantum laws. However, such a conception would lead to the necessity of making an entirely incomprehensible fundamental distinction between systems capable of radiating (charged) and those not capable of radiating (uncharged). The introduction of such a distinction sharply contradicts our knowledge and conceptions of the heat capacity of solid and gaseous bodies.

If, finally, one assumes that the axis of the atom instantaneously follows every change in the direction of the field, then this will lead us into contradiction with the laws of mechanics.

Thus, the great success of the quantum theory entailed the emergence of a whole series of perplexing and unresolved questions.

1. The Bohr and Weiss Magneton. Despite the indicated difficulties of theoretical interpretation, the success of the experiment of Stern and Gerlach extraordinarily strengthened the position of Pauli’s theory, which we set forth in § 3. Among the shortcomings of this theory one could have counted the extreme narrowness of the domain of its direct application (two paramagnetic gases). Pauli did not venture to apply the formulae he had derived to liquid and solid bodies, considering that the interatomic forces in these bodies, the hydration of ions in solutions, etc., must alter beyond recognition the character of the orientation of elementary magnets. These considerations seemed so obvious and indisputable that Sommerfeld, as late as January 1923 (12), while discussing Cabrera’s survey (see § 1) and mentioning Pauli’s theory, deemed it necessary to emphasize the inapplicability of this theory to solid and liquid bodies. Meanwhile, it was only necessary to compare the survey of experimental data (Table 2) with the theoretical Table 3 in order to be convinced of the undoubted applicability of the quantum theory to paramagnetic bodies of any aggregate state1.

According to the quantum theory of magnetism, one, two, etc., real Bohr magnetons should correspond (in round numbers) to 9, 14, 19, 24, 29, etc., apparent Weiss magnetons. Experimentally, however, for ions of the iron group the following values of the number \(k\) of Weiss magnetons have been found (also in round numbers): twice 9, twice 19, once \(k = 24\), and twice values close to this value \(k = 25\), finally, twice \(k = 29\). All these 7 values fully correspond to the theoretical predictions. The remaining 3 values

\[ k = 6,7\ (V^{''''}),\quad k = 16\ (Ni^{''})\ \text{and}\ k = 26\ (Fe^{''}). \]

The data for Fe are in fact among the least reliable: separate determinations fluctuate between 26 and 29; finally, the data for \(V^{\prime\prime\prime\prime}\) have been taken by me from the old and comparatively inaccurate measurements of Pascal1. This agreement of the results of measurement with the theoretical predictions also holds for other paramagnetic substances, as is evident from Table IV, in which only ferromagnetic metals2, complex compounds, and rare earths have not been included. Here in the second column stand the theoretically calculated values of the number \(k\), corresponding to one, two, etc. Bohr magnetons; in the next column the same values are rounded off; finally, in the fourth column are indicated the experimentally found values of \(k\) for various substances. These values only in the three cases already considered differ at all appreciably from those theoretically permissible.

TABLE IV.

\(n\).	\(k\).
1	8.7	9	Ag \((n=1)\); Cu\({}^{\prime\prime}\) \(k=9—10\); V\({}^{\prime\prime\prime\prime}\) 9.2; V\({}^{\prime\prime\prime\prime}\) 9; Ti\({}^{\prime\prime\prime}\) 8.6; NO 9.2; Pt 8—9; Pd \(\sim 8\).
2	13.7	14	O\(_2\) 14, V\({}^{\prime\prime\prime\prime}\) 14, Ni\({}^{\prime\prime}\) 16 (?)
3	18.7	19	Cr\({}^{\prime\prime\prime}\) 19, Mn\({}^{\prime\prime\prime\prime}\) 19.
4	23.7	24	Co\({}^{\prime\prime}\) 24, Cr\({}^{\prime\prime}\) 24, Mn\({}^{\prime\prime}\) 25, Fe\({}^{\prime\prime}\) 26 (?).
5	28.7	29	Fe\({}^{\prime\prime\prime}\) 29, Mn\({}^{\prime\prime\prime}\) 29.

Taking into account the low accuracy of most of the measurements, the agreement of theory with experiment cannot but be regarded as more than satisfactory3. This agreement is all the more surprising and unexpected because we have not taken into account the forces of intermolecular interactions, wh—

which, of course, cannot fail to distort the results of the quantization of orientation in an external magnetic field.

To account for this unexpected success of an admittedly incomplete theory, it remains only to point out that all magnetometric measurements were carried out in very strong fields; it is obvious that, for the quantization of orientations, the determining factor is the ratio of the strength of this field to the strength of the molecular fields.

In this connection it is interesting to recall the fact that in many crystals, even in comparatively weak fields, a distinct Zeeman splitting of the absorption spectrum and of the fluorescence spectrum is observed; in this, undoubtedly, the predominance of the influence of the external field over the internal fields is manifested. In any case, it cannot be mere chance that all the measured values of the number \(k\) are quite definitely grouped around the five definite numbers predicted by the theory. It must also be noted that the quantum theory assigns to the measured atoms a small number (up to 5) of magnetons, whereas, according to Weiss, for many atoms the number of magnetons exceeds \(20—25\). Obviously, such large values of the number \(k\) considerably diminish the persuasiveness of Weiss’s theory: after all, with the aid of sufficiently large integers one can approximate any sequence of any numbers as closely as desired.

Finally, it must be taken into account that the quantum theory theoretically precomputed the magnitude of the magneton, without making any use of experimental data and proceeding only from universal constants \((e, \mu \text{ and } h)\).

It remains only once more to state that the simplest and, it would seem, improbable assumptions (the applicability of quantization to bodies of any state of aggregation) are by no means always erroneous.

The persuasiveness of the quantum theory is, undoubtedly, further strengthened when one considers the dependence of the magnetic moment of ions on the number of their outer (valence) electrons. We shall, however, postpone discussion of this question until we have become acquainted with another, completely independent method for determining the number of magnetons in an atom—the spectroscopic method.

6. Spectroscopic method for determining the magnetic moment of atoms. Questions of atomic magnetism have in recent years acquired decisive importance for the further development of the theory of spectra. At the center of attention of this theory there now stand two questions to which elementary quantum theory was powerless to give an answer. These are the question of the characteristic “multiplicity” or “order” (Multiplizität) of the majority of spectral lines (doublets, triplets, etc.), and the question of the anomalous character of the splitting of these lines

in a magnetic field (the anomalous Zeeman effect1). We now know that the impotence of the theory in these questions is explained by the fact that they are directly connected with the magnetic properties of atoms.

At the present time it may be regarded as established that all magnetic properties of atoms are anomalous in character2. This is evidently explained by the fact that the ordinary laws of electromagnetism are inapplicable even to the stationary states of the atom. The necessity of modifying the fundamental laws of mechanics and electrodynamics is established with complete definiteness by the whole modern development of the quantum theory, but the unsuitability of classical views perhaps nowhere appears with such sharpness as precisely in the theory of magnetism (see Landé [^18]).

It is therefore understandable that, up to now, all attempts to elucidate the internal “magnetic” mechanism of atoms have ended in failure; but in recent years, chiefly thanks to the works of Landé, Sommerfeld, and Heisenberg, it has been possible to create a purely formal theory, or rather, a consistent numerical scheme, which embraces all these complex phenomena as a whole. This theory is so far splendidly justified by experiment, and a whole series of its predictions has already received experimental confirmation. In particular, only thanks to this theory has it at last been possible to decipher such complex spectra as, for example, the spectra of Mn, Cr, Fe, and so on. However, this theory is still only in the stage of development; moreover, owing to its formal character, it admits a number of different interpretations. Up to the present, the exposition of the theory by different authors differs not only in substance, but even in the choice of the fundamental quantities and in their notation. We do not intend to expound this theory in any detail, but we shall nevertheless have to dwell on it briefly, for it leads to a new spectroscopic method of determining the magnetic moment of atoms. In doing so we shall adhere to that form of the theory which Sommerfeld uses, first, because it is somewhat simpler than Landé’s theory, and secondly because it was Sommerfeld who applied his theory to the question of the magneton3 [^15].

The multiplicity of spectral lines is evidently conditioned by the multiplicity of the stationary states of the atom. Generally speaking, the difference between the states of an atom lies in the difference of the orbits of the outer electron (Leucht-

electron); and the internal energy of the atom depends primarily on the character of these orbits. Each orbit is characterized by two quantum numbers: the so-called principal (Hauptquantenzahl) and azimuthal numbers (\(n\) and \(k\) in Bohr’s notation). However, only in the very simplest cases (H and \(\mathrm{He}^+\)) are these two numbers sufficient for an exhaustive determination of the state of the atom; generally speaking, several more or less closely spaced energy levels may correspond to one and the same pair of quantum numbers \(n\) and \(k\). The existence of these finer subdivisions of the energy levels is revealed in the multiple structure of spectral lines.

The cause of the splitting lies in the possibility of different orientations of the orbit of the outer electron (or electrons) with respect to the “core” of the atom (Atomrumpf, as the aggregate of the nucleus and the inner electrons is called). It is assumed that the atomic core is a more or less bound whole, and that the axis of the core can take various orientations with respect to the plane of the orbit of the outer electron, while obeying the laws of quantization. Different orientations of the core correspond to different potential energies of it in the magnetic field of the outer electron. Thus, in order to characterize fully the state of an atom, it is necessary, in addition to the two quantum numbers \(n\) and \(k\), which determine the orbit of the outer electron, also to specify the angle between the plane of this orbit and the axis of the core. For this it is sufficient to specify the resultant angular momentum of the atom, which is geometrically composed of the angular momentum of the outer electron and the angular momentum of the atomic core, and which, consequently, depends on the orientation of the core axis. The number expressing the magnitude of this resultant angular momentum in rational units \(\left(1 = \dfrac{h}{2\pi}\right)\) is called the inner quantum number and is denoted by the letter \(j\). Unlike ordinary quantum numbers, it can take not only integral but also half-integral values (multiples of one half). Thus, to one and the same value \(n_k\) there may correspond a series of different inner numbers, and consequently a series of different energy levels of the atom. For example: to the yellow sodium doublet \(D_1\) and \(D_2\) there correspond the initial orbits \(2p_2\) and \(2p_1\); the first of these is characterized by the quantum numbers \(n=2,\ k=2,\ j=1/2\), while the second by the numbers \(n=2,\ k=2,\ j=3/2\)¹).

Let us now turn to the Zeeman effect.—In the absence of an external field, the direction of the axis of the atom, in other words, the direction of its resultant angular momentum \(j\), is completely arbitrary. When an—

¹) The values of the quantum numbers corresponding to the various states of the atom are determined on the basis of the totality of spectral data. We cannot enter into the details here; in what follows we shall regard these numbers as given.

MATHEMATISM AND THE STRUCTURE OF ATOMS

When an external magnetic field arises, the laws of quantization of orientation come into their own, and the direction of the atom’s axis is restricted to a number of discrete possibilities. The orientation of the atom is characterized by the angle between the direction of the field \(H\) and the direction of the vector of angular momentum \(\bar j\). The quantization rule reduces to the fact that the projection \(j_H\) of the vector \(\bar j\) on the direction \(H\) \([j_H=j\cos(jH)]\) must either be equal to \(\pm j\), or may differ from it by an integral number of units. Thus, for example, for \(j=3/2\), 4 values of the projection \(j_H\) are possible:

\[ \pm 3/2 \quad \text{and} \quad \pm 1/2. \]

Depending on the orientation of the atom, its potential energy in the external magnetic field also changes; this energy is equal to

\[ \Delta E=-mH\cos(m,H), \tag{11} \]

where \(m\) is the magnetic moment of the atom, expressed this time not in rational, but in ordinary \(CGS\) units.

Thus, one and the same system of values \(n\), \(k\), and \(j\) may correspond to a number of different values of the atom’s energy [depending on the angle \(\cos(m,H)\)]. In other words, in a magnetic field each spectral term (energy level) splits into a number of terms close to one another, corresponding to different orientations of the magnetic axis of the atom; this also explains the magnetic splitting of spectral lines (the Zeeman phenomenon).

The magnitude of the splitting of the terms \(\Delta E\) can be measured spectroscopically. On the other hand, the direction of the vector \(m\) of course coincides with the direction of the vector \(\bar j\), i.e.

\[ \cos(m,H)=\cos(j,H). \]

Thus, in equation (11) the quantities \(\Delta E\) and \(H\) can be measured directly, \(\cos(j,H)\) is determined by the rules of quantization, and, consequently, this equation makes it possible, on the basis of measurements of the Zeeman effect, to calculate the magnitude of the atomic magnetic moment \(m\).

It may seem that we have quite unnecessarily complicated a simple problem. For we have already repeatedly referred to the well-known relation between the angular and magnetic moments

\[ m=\frac{\varepsilon}{2\mu}\,j \quad \text{(eq. 4)}. \]

We have already pointed out that if \(m\) and \(j\) are measured in rational units (respectively equal to \(\dfrac{h}{2\pi}\) and the Bohr magneton \(\dfrac{\varepsilon h}{2\mu\cdot 2\pi}\)), then equality (4) reduces to the equality of the numerical values of the angular and magnetic moments of the atom:

\[ m=j. \tag{4a} \]

The quantity \(j\) entering into this formula is nothing other than the internal quantum number; consequently, the number of magnetons in the atom \(m\) must simply be equal to the internal quantum number \(j\).

Thus it might seem that, in order to determine the magnetic moment \(m\), there is no need at all to resort to measuring the Zeeman effect.

Unfortunately, however, in reality the matter is much more complicated: this is indicated first of all by the anomaly of the Zeeman effect for the majority of spectral lines. The elementary theory, based on equality (4a), requires a normal Zeeman triplet for all lines. In order to explain the totality of the experimental data, one has to admit that the relation between the magnetic and rotational moments is much more complicated than was assumed by the classical theory, and that equality (4a) must be replaced by the relation

\[ m = gj. \tag{12} \]

The factor \(g\) entering into this formula is called the “splitting factor” (Aufspaltungsfaktor) and is a rather complicated function of the azimuthal quantum number \(k\), the internal quantum number \(j\), and, finally, still a third number \(r\), which characterizes the so-called maximum multiplicity of the terms of the given spectral series1.

Thus, in order to determine \(m\), it is no longer sufficient to know only \(j\); rather, one has to resort to the method indicated above, based on measuring the splitting of the terms \(\Delta E\). This method, as has already been said, makes it possible to determine \(m\) directly; then, by formula (12), one can also calculate \(g = m/j\).

It must be admitted that, by replacing the equality \(m = j\) (4a) with the relation \(m = gj\), the quantum theory of magnetism is, strictly speaking, pulling the ground out from under its own feet: for equality (4a) follows from the fundamental laws of electromagnetism, and the very concept of the Bohr magneton rests upon it.

Here we must again note the process, so characteristic of the modern development of physics, of the growth of deep internal contradictions in new, fruitful theories. In spite of these contradictions, however, in many cases these theories lead us

...to the discovery of new regularities—regularities so simple and convincing that they undoubtedly correspond to the true nature of things.

Such is the situation in the case we are considering. The factor \(g\) was introduced by Landé1 only in order to explain the anomaly of the Zeeman effect in doublet and triplet spectral lines (doublets and triplets). The natural generalization of this theory then made it possible to predict the complex structure of multiple lines (multiplets) and the character of their splitting in a magnetic field. These theoretical predictions for the first time made it possible to bring order into an extremely complex and confused field of spectral analysis, and were brilliantly confirmed by experiment. Recently almost every month has brought new successes in this field1. This merit of the theory alone is extraordinarily great. But, in addition, it turned out that the very same theory is capable, almost without any additional assumptions, of “explaining” the anomaly of the magneto-mechanical effect2.

Finally—and this is for us at present the most important point—on the basis of the same theory one can determine the number of magnetons in the atom, and the results thus obtained prove to be in complete agreement with the results of direct magnetic measurements.

As has already been indicated, in order to determine the magnetic moment of the atom one must resort to formula (12). We shall not describe the manner in which the values of the quantities \(g\) and \(j\), corresponding to the given state of the atom, are found, for this would lead us too far afield.

We shall mention only one simple rule which in a number of cases makes it possible to determine the value of \(m\) directly.

Of all the quantum numbers characterizing the state of an atom, the azimuthal number \(k\) has the greatest importance. In what follows we shall speak of one-quantum, two-quantum, etc., states, meaning thereby the value of the number \(k\), for the value of the principal quantum number \(n\) has no effect either on the angular momentum or on the magnetic moment of the atom. Let us also mention that in spectroscopy it is customary to use the letters \(s, p, d\), etc., to denote orbits corresponding to \(k=1\), \(k=2\), \(k=3\), etc.

Having agreed on the terminology, we can formulate Sommerfeld’s rule in the following way: in the normal one-quantum state \(s\) \((k=1)\), the number of magnetons in the atom is one less than the maximum multiplicity of its energy levels1. Thus, for example, for an atom emitting simple lines, e.g. Ca, the maximum multiplicity of the lines is \(=1\); consequently the number of magnetons \(m\) is equal to \(m=1-1=0\). For atoms emitting doublets (e.g. Na), the maximum multiplicity is \(=2\); consequently \(m=2-1=1\); for atoms emitting triplets (e.g. the same Ca) the maximum multiplicity is equal to 3; consequently

\[ m=3-1=2, \]

and so on.

In particular, calcium atoms can emit both simple lines and triplets; consequently, they occur in two different states. In spectroscopy, in order to distinguish these states, it is customary to designate them respectively by capital and small letters: \(S, P, D\ldots\) and \(s, p, d,\) etc. In the state \(S\) (simple lines) the number of magnetons is equal to zero; in the state \(s\) (triplets) the number of magnetons is equal to two. Of course, the indicated simple rule is applicable only to normal one-quantum states of the atom. In the excited state of the atom its angular momentum \(j\) changes, and consequently so does the magnetic moment \(m\).

Finally, it is necessary to note that some substances even under ordinary conditions are in many-quantum states \((k>1)\), which thus are normal for them. Such are, for example, Al and Tl vapors (normally \(2p\)), Fe vapors (normally \(3d\)), etc. To these substances the indicated simple Sommerfeld rule is not applicable.

7. Magnetism and the periodic system of the elements.

One of the best confirmations of the quantum theory of magnetism is the complete agreement of the results obtained by entirely different methods (spectroscopic and direct magnetic—

metric). Figure 3 gives Sommerfeld’s diagram, on which are plotted the results of measurements of that group of elements which has been best studied magnetically. Along the abscissa axis is plotted the number of “external” electrons1 in the atom or ion; along the ordinate axis, the number of magnetons in it. Above the number of external electrons stands the total number of electrons in the atom and the name of the corresponding

Fig. 3.

neutral atom. The results of spectroscopic measurements are underlined; the remaining quantities were found by the usual magnetometric method (see the summary in Table IV). The number of magnetons is rounded off to integral values, which may raise doubt only with respect to

\[ \mathrm{Ni}^{\prime\prime}\ \text{and}\ \mathrm{Fe}^{\prime\prime}\,.[^2] \]

The diagram has been supplemented with some data which were not included in it by Sommerfeld (\(\mathrm{K}\), \(\mathrm{V}^{\prime\prime}\), \(\mathrm{V}^{\prime\prime\prime}\), \(\mathrm{Mn}^{\prime\prime\prime}\), \(\mathrm{Fe}^{\prime}\), \(\mathrm{Cr}\)); the number of magnetons for Fe was calculated by me on the basis of a new spec-

spectroscopic work of Hilde Gieseler and V. Grotrian \([^{16}]^{1})\). For some atoms (Ca, Cr, Mn) two different values are given, since these atoms may be in two different one-quantum states \(S\); this circumstance is manifested in the presence of two different systems of spectral lines. Thus, for example, the Ca atom may emit both singlet lines and triplets, Cr—quintets and septets, etc.

At first glance at the diagram, the strict regularity appearing in it immediately stands out, testifying to complete agreement between the results obtained by two entirely independent methods (magnetometric and spectroscopic).

First of all, attention is drawn to the straight line passing through the origin of coordinates at an angle of \(45^\circ\); it is entirely filled with mutually coincident points. The magnetic moments of all 15 atoms and ions lying on this straight line obey the following simple and convincing rule: the number of magnetons in an atom is equal to the number of its outer (valence) electrons. The loss of electrons by atoms upon ionization entails at the same time the loss of magnetons. Example: Mn has 7 electrons and 7 magnetons (\(m=7\)); for \(\mathrm{Mn}''\), \(m=5\); for \(\mathrm{Mn}'''\), \(m=4\); finally, for \(\mathrm{Mn}^{''''}\), \(m=3\). This rule is confirmed with particular force by the coincidence of the number of magnetons in ions of entirely different origin but with the same number of outer electrons. Example: \(m=1\) for K, \(\mathrm{Ca}'\), \(\mathrm{Ti}'''\), and \(\mathrm{V}^{''''}\).

Let us now turn to the next straight line, lying somewhat below the first, on which there are also quite a few points\(^{2}\). The number of magnetons at the corresponding points of these two straight lines differs by exactly two. Examples: for Ca, \(m=2\) and \(m=0\); for Cr, \(m=6\) and \(m=4\); for Mn, \(m=7\) and \(m=5\).

Without attempting to enter into the details of the internal mechanism, one may interpret it as follows. The maximum number of magnetons in an atom is equal to the number of its outer electrons and corresponds to the rotation of all these electrons in one and the same direction. If, however, one of the electrons begins to rotate in the opposite direction, then its magnetic field neutralizes the field of one of the direct electrons, and the resultant magnetic moment of the atom is decreased by two. Example: Ca possesses two valence electrons; when both electrons rotate in the same direction, \(m=2\); when the directions of rotation are opposite, \(m=0\). Another example: \(\mathrm{Fe}''\) has 6 electrons, and \(m=4\); consequently one of the electrons rotates in the opposite direction. This reverse electron is evidently the least firmly bound to the atom and—

\(^{1}\) See also the note by Angerer and Joos in Naturwissenschaften of 15/II 1924.

\(^{2}\) This straight line is absent from Sommerfeld’s diagram, but the considerations set forth by me were developed by Sommerfeld in his latest articles.

this first of all is torn away upon further ionization. Thus, the ionization of Fe″ leads not to a decrease, but to an increase of the resultant magnetic moment of the atom: in Fe‴ \(m=5\) (all 5 remaining electrons rotate in one direction).

There may obviously be not one, but several reverse electrons in an atom. For example, in Ni″ the total number of outer electrons is 8, while \(m=2\); one must suppose that in it three “reverse” electrons neutralized three “direct” ones.

In any case, however one may regard these attempts at a visual interpretation, from the diagram considered the following proposition undoubtedly follows: the number of magnetons in an atom may either be equal to the number of its outer electrons, or be less than it by an even number of units (mutually compensating electrons drop out in pairs). The consequence is that in atoms with an even number of electrons the number of magnetons is also even, and conversely.

The sole exception to this rule is the Co″ ion, possessing 25 electrons and 4 magnetons. It should also be noted that the magnitude of the magnetic moment of the Ni″ ion can be equated to two magnetons only under a certain tension (see Table IV).

With these exceptions, the rule just stated is confirmed not only for the elements of the iron group, but also for all the generally known material on the magnetism of ions and monatomic gases. We may mention, for example, the diamagnetism of the noble gases, the result of Stern and Gerlach’s experiment on silver, the spectroscopic laws of displacement and the alternation law1 (Verschiebungssatz und Wechselsatz), etc.

Strongly pronounced paramagnetism occurs only in definite places of the periodic system of the elements (the triads of the 8th group and the rare earths); in other words, only in these places are the maximum possible numbers of magnetons, equal to the number of the atom’s outer electrons, encountered. This is probably connected with the circumstance that it is precisely in these places of the periodic system that the filling of the inner shells of the atom with new electrons takes place. In general, the inner state of the atom is established at the minimally possible value of the magnetic moment for it (one magneton in the odd and zero magnetons in the even columns of the periodic system).

Thus, if the question of Co″ is left aside, all the remaining experimental data fit completely into Sommerfeld’s coherent scheme, striking and captivating in its extreme simplicity.

And yet, precisely because of its simplicity, this scheme appears completely incomprehensible.

In fact, Sommerfeld’s rule on the number of magnetons in an atom seems so convincing because it is involuntarily associated with the idea of an extremely simple intra-atomic mechanism. This rule, evidently, should be understood to mean that in the normal state of the atom all its outer electrons move in one-quantum orbits (one-quantum in the sense of the azimuthal quantum number, the orbits \(n_1\)), so that to each electron there corresponds one magneton. Moreover, the orbits of all the electrons are situated in one plane, so that the geometrical addition of the magnetic moments of the individual electrons is replaced by an arithmetical one. However, not one of these assumptions withstands the slightest criticism from the point of view of our present knowledge of the structure of the atom; on the contrary, the extreme complexity of the orbits of the valence electrons may be regarded as firmly established; the planes of these orbits are in any case not parallel; finally, some of the valence electrons undoubtedly move not in one-quantum orbits.

Nevertheless, however, Sommerfeld’s rule on the number of magnetons in an atom cannot be regarded as a purely mnemonic rule without any physical content. The extreme simplicity and persuasiveness of Sommerfeld’s scheme, and its complete confirmation by experiment, serve as a guarantee that in this scheme some new essential regularity of atomic structure has found expression. We are unable to understand this quantum regularity, discrete in its very essence, only because in its present state quantum theory rests upon an internally contradictory foundation. The search for regularities similar to those considered by Sommerfeld should make it possible to place beneath quantum theory some new and firm foundation.

8. Difficulties of the quantum theory of magnetism. Modern quantum theory bears the stamp of internal inconsistency; every success of it gives rise to new difficulties for it. Such, in particular, is the fate of the theory of magnetism. We have already had occasion to mention the theoretical difficulties that arose in connection with the success of the Stern and Gerlach experiment. Let us now turn to difficulties of another kind.

We have until now passed over in silence the following fact. According to the now generally accepted model of helium (Landé’s “crossed” model), the helium atom must possess a magnetic moment, and consequently paramagnetic properties. Meanwhile, in reality helium is distinguished by a sharply expressed diamagnetism. How is one to get out of this contradiction? For this there are two ways. First, one may suppose that, contrary to all the laws of electrodynamics, the helium atom nevertheless does not possess a magnetic moment; thus, for example, Bohr expresses the idea of the possibility of electron orbits that are “dead” in the magnetic respect (magnetisch

tot). To this, in essence, the formal theory of Sommerfeld also reduces, assigning to helium an internal quantum number \(j\) equal to zero (the angular momentum is equal to zero).

On the other hand, without denying the presence of a magnetic moment in the helium atom, one may suppose that in a magnetic field the axis of the helium atom is established not along the field, but perpendicular to the field (this supposition, of course, also contradicts the laws of electrodynamics). This point of view is held by Landé, to a brief examination of whose theory we shall now turn (see [\(^{11}\)]).

In § 6 we said that from the spectroscopically measurable magnitude of the magnetic splitting of the energy levels of an atom,

\[ \Delta E = -mH\cos(mH) \tag{11} \]

one can determine the magnetic moment \(m\) of the atom. However, the solution of this equation is not unique. From experiment one can directly find only

\[ m\cos(mH)=-\frac{\Delta E}{H}, \]

whereas the manner of resolving the product \(m\cos(mH)\) into factors remains arbitrary; the magnitude \(m\) can be determined only by applying the rules concerning the quantization of orientation. It is precisely in the formulation of these rules that Landé’s theory differs from Sommerfeld’s theory. Without going into details, we shall note only some of the final conclusions of Landé’s theory.

According to Landé’s theory, in spatial quantization an orientation of the axis along the field is forbidden, so that the axis of the atom is always inclined to the direction of the field. For the helium atom (for which, according to Landé, the internal quantum number \(j=\tfrac{1}{2}\)) only one single position proves possible, namely the perpendicular one, which is thus in complete agreement with the diamagnetism of this gas\(^{1}\). The Stern and Gerlach experiment is interpreted by Landé in the following way: the silver atom possesses not one, but two magnetons (\(j=1,\ g=2,\ m=gj=2\)); in a magnetic field the axis of the atom is set at an angle of \(60^\circ\), so that

\[ \cos\varphi=\pm\frac{1}{2}. \]

Consequently, the projection of the magnetic moment onto the direction of the field turns out to be equal to

\[ m\cos\varphi=2\times\left(\pm\frac{1}{2}\right)=\pm1, \]

\(^{1}\) According to an oral communication by P. S. Ehrenfest, Oskar Klein, calculating the action of a magnetic field on the hydrogen atom, came to the conclusion that the axis of the hydrogen atom is always established perpendicular to the field. If Klein’s calculations prove correct, they will constitute a serious argument in favor of Landé’s views.

which is in full agreement with the results of the Stern and Gerlach experiment. It is curious that in this way even this extremely simple experiment requires, in the opinion of one of the best experts in this field, a rather complicated and not very convincing interpretation. If, according to Landé’s theory, one calculates what number \(k\) of “apparent” Weiss magnetons corresponds to one, two, etc., Bohr magnetons, then one obtains the very same values \(8.7\), \(13.7\), etc., as according to the Pauli–Sommerfeld theory (see Table V). As we have seen, this prediction of both theories is well confirmed by experiment. The only difference between the theories in this question cannot be tested experimentally; this difference consists in the different character of the correspondence between the numbers \(k\) and \(n\), and is easily seen from Table V.

TABLE V.

	0	1	2	3	4
True number of Bohr magnetons \(n\) . . .	0	1	2	3	4
Apparent number of Weiss magnetons \(k\) according to Pauli . . .	0	8.7	13.7	18.7	23.7
Apparent number of Weiss magnetons \(k\) according to Landé . . .	0	0	8.7	13.7	18.7

In general it must be said that, despite the different interpretation of the experimental facts, all of them are explained equally well by both theories. An experimentum crucis between them is impossible, for these theories are as yet, in essence, only very fruitful formal schemes systematizing the experimental material.

In comparison with Landé’s theory, the original Pauli–Sommerfeld theory was distinguished by greater simplicity; subsequently, however, it proved necessary to introduce into this theory as well certain complicating corrections.

The point is that in § 3, in deriving formula (3b) and in calculating Table III, we based ourselves on the equality (in rational units)

\[ m=j. \tag{6a} \]

Meanwhile, in § 6 we became acquainted with the need to complicate this formula by introducing the splitting factor \(g\):

\[ m=gj. \tag{12} \]

This complication entails the necessity of a corresponding change in formula (3b), since it is the vector \(j\), and not the vector \(m\), that is subject to quantization of orientation. Let us explain this by an example.

Suppose that \(m=3\); then, according to Pauli’s original theory, likewise \(j=3\), and for the axis of the atom there are 6 possible different orientations, corresponding to

\[ \cos \varphi=\pm 1,\ \pm \frac{2}{3},\ \pm \frac{1}{3}. \]

Therefore

\[ \overline{\cos^2 \varphi}=\frac{14}{27}=0.52. \]

When using the corrected formula, one must first make a definite assumption concerning \(g\). If the atom is in a normal \(S\)-state, then, as we already know, \(g=2\); hence, for \(m=3\), \(j=\frac{3}{2}\). Thus for the axis of the atom there are 4 possible different orientations, corresponding to

\[ j_H=j\cdot \cos \varphi=\pm \frac{3}{2},\ \pm \frac{1}{2},\ \text{i.e.}\ \cos \varphi=\pm 1,\ \pm \frac{1}{3}, \]

for, according to the quantization rule, the projection \(j_H\) must either be equal to \(j\), or differ from it by an integral number of units1. Consequently,

\[ \overline{\cos^2 \varphi}=\frac{5}{9}=0.56. \]

Thus the results of the corrected and of the original theory are very close to one another. For a more complete comparison of the two theories we give Table VI.

Comparing the numbers in Table VI with the experimental data given in Table IV, it is easy to see that the agreement of the theory with experiment is not only not impaired by the introduction into it of the indicated correction, but, perhaps, is even improved.

At the present time the introduction of this correction into the original theory is recognized as necessary both by Sommerfeld and by Pauli himself.

TABLE VI.

	1	2	3	4	5
Number of Bohr magnetons \(n\) . . . . . . .	1	2	3	4	5
Apparent number of Weiss magnetons \(k\) according to the original theory . . . . . . .	8.7	13.7	18.7	23.7	28.7
The same, according to the corrected theory (for \(g=2\)) . .	8.7	14.1	19.2	24.4	29.4

Thus the success of the initial theory is explained only by a more or less accidental circumstance: the small influence of the correction factor \(g\) on the final result of the calculation.

Thus, the quantum theory of magnetism has undergone the fate of all quantum (and not only quantum) theories: an initially simple idea becomes extraordinarily complicated, encumbered by an abundance of details and reservations. But still incomparably more serious is the fact that this theory turns out to be torn by internal contradictions. For the entire quantum theory of the atom, the calculation of stationary orbits, and the very calculation of the magnitude of the Bohr magneton rest upon the application of classical electrodynamics to the stationary states of the atom. By abandoning the relation

\[ m=j, \]

and introducing an entirely incomprehensible, classically inadmissible, splitting factor \(g\), we thereby deprive the whole theory of any theoretical foundation. Lack of space does not allow me, unfortunately, to dwell on still more serious paradoxes, in comparison with which even “the anomaly of the splitting factor \(g\) appears merely as an insignificant violation of the laws of mechanics”1 (Landé [^17]).

How, then, should one regard this complex and internally contradictory theory? Half a century ago such a theory could have found for itself neither recognition nor response. But since then much has changed. Then there prevailed a proud confidence that the basic outlines of the physical picture of the world were known, and that it remained only to fill in the details of this picture. Naturally, the chief criterion of the suitability of a new theory was then its logical harmony and simplicity. Now we have become convinced that the laws governing elementary, intra-atomic processes are still completely unknown to us. The intricacy and contradictoriness of contemporary theories—let us recall Bohr’s correspondence principle (Korrespondenzprinzip), let us recall the application of astronomical perturbation theory to the calculation of electron orbits—the complexity of these theories is explained by the fact that we are trying to interpret the atomic processes of the microcosm on the basis of the “classical” laws of the macrocosm, which are alien to it. These theories must be approached first of all not from the point of view of their internal harmony and completeness—for in any case these theories are doomed to demolition—but chiefly from the point of view of their fruitfulness in seeking new simple facts and regularities, which will serve as the foundation for the simple and harmonious theory of the future; I say “simple” because the conviction in pro-

of the laws of nature has always lain at the foundation of all scientific activity. From this point of view, the quantum theory of atomic magnetism, as we have seen, has to its credit such major achievements as the Stern and Gerlach experiment; a simple interpretation of the experimental magnetometric material; Sommerfeld’s simple scheme, connecting the chemical and magnetic properties of the elements; and finally, a scheme not yet amenable to interpretation, but in essence extraordinarily simple and harmonious, embracing together three magnetic “anomalies” (the multiplicity of spectral lines, the Zeeman effect, and the magneto-mechanical effect). All these achievements are indisputable and enduring values, which not only justify the very existence of the theory, but also allow one to hope that at its foundation lies the right idea, and that its further development will lead to new successes and will help create a fundamentally new, coherent theory of intra-atomic processes.

Literature

Reviews:
a) W. Gerlach. Magnetismus u. Atombau, in the collection “Ergebnisse der exakten Naturwissenschaften,” Vol. II, p. 124, 1923.
b) A. Landé. Fortschritte beim Zeemaneffekt. Ibid., p. 147.

Articles:

1) P. Weiss. Phys. Ztschr., 12, 935, 1911, and Arch. f. Elektrotechnik, 2, 1, 1913.
2) B. Cabrera. Journ. d. Phys., 3, 443, 1922.
3) A. Landé. Phys. ZS. 20, 228, 1919 and Verh. d. D. Phys. Ges. 21, 585, 1919.
4) W. Pauli. Phys. ZS. 21, 615, 1920.
5) O. Stern. ZS. f. Phys., 7, 249, 1921.
6) W. Gerlach u. O. Stern. ZS. f. Phys. 8, 110, 1921, 9, pp. 349 and 353, 1922.
7) A. Einstein u. P. Ehrenfest. ZS. f. Phys., 11, 31, 1922.
8) A. Sommerfeld. Ann. d. Phys., 70, 32, 1923.
9) P. Epstein. Science, 1923, p. 532.
10) W. Gerlach. Phys. ZS. 24, 276, 1923.
11) A. Landé. ZS. f. Phys., 5, 231, 1921; 15, 192, 1923; 19, 112, 1923.
12) A. Sommerfeld. Ann. d. Phys. 63, 112, 1920. ZS. f. Phys., 8, 257, 1922, Ann. d. Phys., 70, 132, 1923, Ann. d. Phys., 1924, Heft 3/4.
13) W. Heisenberg. ZS. f. Phys., 8, 273, 1922.
14) A. Sommerfeld. Phys. ZS. 24, 360, 1923; Ann. d. Phys., 1924, Heft 3/4.
15) A. Sommerfeld. ZS. Phys., 19, 221, 1923.
16) H. Gieseler u. W. Grotrian. ZS. f. Phys., 22, 245, 1924.
17) A. Landé. Naturwissenschaften, p. 725, 1923.
18) A. Landé. Phys. ZS. 24, 441, 1923.

1. Landé gave a very interesting survey of those fundamental difficulties with which one has to contend in the contemporary theory of atomic magnetism (Landé [^18]). ↩↩↩↩↩↩↩↩↩↩↩↩↩↩↩↩↩

2. See § 5. Gerlach \([^{12}]\) assumes for \(\mathrm{Fe}^{\prime\prime}\) \(n = 5\), and not \(n = 4\), as we do. ↩↩↩↩↩

3. Some propositions of Sommerfeld’s first works, which in part entered the third edition of his well-known book Atombau und Spektrallinien, were subsequently somewhat modified by him. ↩↩↩