On the Current State of Gyromagnetic Research

O. Auger, Berlin*

1. Historical Review

Almost a century separates the emergence of Ampère’s ideas on molecular currents from the discovery of the gyroscopic nature of the atom. This long interval is explained not so much by the slow as by the late development of atomistic conceptions, which began to be successfully elaborated only in the twentieth century. Ampère’s idea was ahead of its time. However, already in the past century repeated attempts were made to detect a gyroscopic effect in coils carrying a current, and also in magnets; the basis of these investigations was Ampère’s idea and Weber’s idea of the inertia of electric charges. Well known is the unsuccessful experiment of Clerk Maxwell, dating from 1861, in which he attempted to detect the inertia of electricity in a coil carrying a current. After changes in atomistic conceptions (brought about by the development of physics), connected with the atomic models of Lenard, Rutherford, and Bohr, attempts at direct proof of the gyroscopic nature of the atom by observing magnetic and mechanical actions received a new impetus, which has continued up to the present time. It should be noted, however, that the conviction that Ampère’s idea was correct had long been so strong that attempts to consider the problem from the gyroscopic side, essentially speaking, never ceased.

In most handbooks only two gyromagnetic experiments are mentioned, and the years 1914 and 1915 are regarded as the years in which the gyromagnetic effect was finally discovered. In this connection it should be recalled that the idea itself and the attempts to prove its correctness are considerably older. Barnett himself, to whom we owe the first successful proof of the gyroscopic nature of magnetic atoms, pointed out that John Perry’s experiment, dating from 1890 and consisting in an unsuccessful attempt to change the magnetization of an iron rod by rotating it about its axis, became known to him in 1896 or 1897, when Barnett read his book “Spinning Tops.” In 1909 Barnett again had

* Naturwissensch., 23, 202, 1935. Translated from the German by N. N. Malov.

ON THE CURRENT STATE OF GYROMAGNETIC RESEARCH

this idea as well, and after lengthy investigations he succeeded in obtaining a result. If one disregards Schuster’s attempt (1911–1912) to explain the deviation of the earth’s magnetic pole from its geographical pole, then it may be regarded as fair to say that magnetization by rotation is usually called the Barnett effect.

The converse idea—the occurrence of rotation upon magnetization—was first put forward by Richardson in 1908. To him also belongs the unsuccessful attempt to detect this effect, as well as the development of its theory. Only in 1915 did Einstein and de Haas succeed in obtaining a positive result by means of an artificial method—the resonance method. Their results could not indicate the sign of the effect with precision and gave only qualitative agreement with theory. However, their numerical values were fairly close to the ideas then current concerning electron trajectories, which, as is known, proved to be erroneous.

In addition to these two—or, if Maxwell’s experiment is counted together with Barnett’s, these three—experiments, there is also known a fourth possibility for detecting the gyromagnetic effect, consisting in the fact that, instead of rotating the magnet in its own or in a stationary external field, one uses the rotation of the field around a stationary magnet. This experiment may be regarded as the magnetic analogue of Barnett’s mechanical experiment, in which the entrainment of the axes of gyroscopes is caused not by the connection of the latter with the body as a whole, but is due to the influence of the field on the individual elementary magnets.

This experiment was carried out in 1921 by Tonks (the work was not published) and, independently of him, by Fisher (in 1922); the results of both experiments were negative. However, these experiments, like Barnett’s investigations, have the merit that they try to detect the gyromagnetic effect of elementary magnets by an indirect route, by observing changes in magnetization. This artificial method makes it possible to find a way out of the hopeless situation that arose after Maxwell’s unsuccessful attempt to detect the gyroscopic effect on a magnet as a whole. Later we shall see why experiments with a rotating field (instead of a rotating body) lead to negative results.

Thus we can distinguish four separate gyromechanical experiments, of which two give positive results and two negative ones:

1) The macroscopic effect of rotation, produced and observed on the whole body as a whole (Maxwell, 1861).

2) The atomic gyroeffect of rotation of elementary magnets, produced by the rotation of the whole body as a whole and observed on the elementary magnets (Barnett, 1909–1914).

3) The atomic effect of orientation of the axes of elementary magnets, produced by action on the elementary magnets and observed on the whole body as a whole (Richardson, Einstein, de Haas, 1908–1915).

4) The atomic gyroeffect of rotation of elementary gyromagnets, produced by the rotation of elementary magnets and observed on these same magnets (Tonks, 1921; Fisher, 1922).

TABLE 1

Foundations of the experiments giving direct proof of the gyroscopic nature of the atom by means of observations of mechanical and magnetic actions

Note. $K$ — action on the body, $E$ — on elementary magnets.

2. Gyroscopic Foundations of the Experiments

The scheme presented above makes it possible at once to conclude that there are still a number of possibilities that are conversions of those indicated above; however, all of them, although for different reasons, appear

TABLE 2

Supplement to Table 1

more or less hopeless. Let us note only four of them, relating to the Maxwell experiment and the Richardson—Einstein–de Haas experiment, namely:

5) The macroscopic gyro-effect of orientation of the body’s axis, produced and observed on the body as a whole.

6) The atomic gyro-effect of orientation of axes,* produced and observed on elementary magnets.

7) The atomic gyro-effect of orientation of axes, produced in the entire body and observed on elementary magnets.

8) The macroscopic gyro-effect of rotation, produced on elementary magnets and observed on the body as a whole.

In Table 1 all the possibilities mentioned above are compared. By the rules of permutations the complete scheme should contain 16 effects. Four of them are practically identical with those indicated above, while another four cannot be realized. In Table 2, where \(K\) corresponds to the entire body and \(E\) to an elementary magnet, these 8 effects are indicated. We may thus leave these effects out of consideration and regard Table 1 as sufficiently complete.

All 8 effects contained in Table 1 can be reduced to two basic gyroscopic experiments which, at first glance, have nothing in common with magnetic phenomena.

Let us imagine a gyroscope \(K\), fastened in its center of gravity in such a way that it can freely rotate about its axis \(A\),**

* Corresponding to rotation about a horizontal axis (see below).

** In a rigid body, strict fulfillment of this condition is impossible (see \(A—3\)), since in that case no action at all on the body as a whole will be possible. On the other hand, a very strong coupling in the limiting case will destroy the gyro-effect. It is important, however, for the subsequent relation of the mechanical moment to the magnetic one that it proves independent of the coupling magnitude, unknown in detail.

and also about an axis \(C\), perpendicular to \(A\) and passing through the center of gravity. In addition, let \(A\) be able to rotate in the plane of the drawing about the horizontal axis \(B\) so that the angle \(\Theta\) between \(A\) and \(C\) can vary. Finally, \(B\) can rotate about the axis \(C\) together with \(A\) (Fig. 1).

On such a gyroscope one can observe two phenomenologically distinct basic phenomena.

1st basic phenomenon. If one attempts (while the gyroscope \(K\) is rotating about the axis \(A\)) slowly to rotate \(A\) and \(B\) about the axis \(C\) (“rotation”), i.e., to produce a torque about the axis \(C\), which is fixed in space, then the axis of the top \(A\) will resist the change in its position in space associated with this attempt and will set itself in such a way (almost in the initial plane) that the angle \(\Theta\) decreases if the directions of rotation of the gyroscope about the axis \(A\) and of the forced rotation about the axis \(C\) are the same; if, however, the directions of rotation are mutually opposite, then the angle \(\Theta\) increases. Thus the axis of the gyroscope \(A\) will rotate about the horizontal axis \(B\) until, by the shortest path, it becomes parallel to the forced rotation about \(C\) (“orientation of axes”). Thus the gyroscope \(K\), under the action of a torque about \(C\), tends to set its axis of rotation in such a way that the angle between its axis and the axis of the forced rotation is minimal, and the directions of both rotations coincide (Foucault’s theorem on the “tendency to parallelism”).

Fig. 1. Model of a free gyroscope.

2nd basic phenomenon. If, on the other hand, one attempts forcibly to diminish the angle \(\Theta\), i.e., to produce a torque about the axis \(B\) of the top participating in the motion (“orientation of axes”), then this attempt likewise will not succeed; instead, the axis \(A\) will turn in the horizontal plane about the axis \(C\). In this case the force acting in the direction of increasing the angle \(\Theta\) will produce a rotation of the axes \(A\) and \(B\) about \(C\) that coincides with the direction of rotation of the top about the axis \(A\), and conversely: the axis of the top \(A\) will precess about \(C\) so long as a torque acts on \(B\), since parallelism of the two moments of rotation cannot be achieved because the axis \(B\), bound to the body, participates in the motion (“rotation”).

From these two basic experiments all the effects described above follow of themselves, if the elementary magnets are regarded as equivalent mechanical gyroscopes.

A. Four known experiments

1) Maxwell’s experiment is a direct reproduction of the first fundamental phenomenon, under the assumption that a coil carrying a current, or a magnet in which elementary Ampèrian currents flow, is, owing to the inertia of electricity, equivalent to a mechanical gyroscope. Therefore the rotation of a magnet or solenoid fastened as shown in Fig. 1 should entail the orientation of the magnet or solenoid as a whole. As is known, in 1861 Maxwell tried unsuccessfully to detect this effect. The impossibility of this experiment from the energetic point of view was proved by de Haas and de Haas–Lorentz in 1915.

2) The transfer of the same idea to elementary magnets, and the possibility of detecting an orientation of their axes equivalent to an increase in magnetization, was demonstrated by the Barnett effect in 1914.

3) Equally fruitful should be the inversion of the experiments described above, analogous to the second fundamental phenomenon, if there exists some connection between the precessional motion of elementary magnets and the body itself. This connection, about whose nature we still know nothing, is a necessary assumption in all gyromagnetic investigations, since otherwise no influence whatever of the elementary gyroscopes situated inside the body could be transmitted to the body as a whole. The positive result of the Richardson—Einstein—de Haas experiments, which revealed the possibility of a body’s rotation arising owing to the orientation of the axes of elementary magnets, i.e. upon magnetization, proves the existence of this connection, just as does the positive result of Barnett’s experiments.

4) In order to understand the negative result of the fourth experiment, we shall have to begin somewhat from afar. At first the thought seems obvious that it is a matter of complete indifference whether we set the axis \(A\) into rotation by rotating the whole body, because the axis is connected with the body (experiment 2), or whether we do this by means of a rotating magnetic field that carries along the axes of the elementary magnets (experiment 4); one might think that in both cases this will entail an orientation of the axis of rotation (fundamental phenomenon 1). However, as is known, one of the experiments gives a positive result, the other a negative one, as is proved not only by the experiments of Tonks and Fisher, but also by Barnett’s latest investigations (1933).

How is one to understand this apparent contradiction? Proceeding from ideas about the mechanism of rotation of a single elementary magnet, we shall not reach the goal. We shall have to bring into the discussion certain magnetic phenomena. First of all let us note that Fisher’s experiment, like all experiments based on fundamental phenomenon 1, was carried out exclusively on ferromagnetic substances; experiments based on the Richardson, Einstein, de Haas effect (of the type of fundamental phenomenon 2), on the other hand, were also carried out on other substances. Therefore we must take into account the pecu-*

features of the ferromagnetic state. As is known, a ferromagnetic body in the unmagnetized state consists of a multitude of spontaneously magnetized elementary regions, whose vectors are distributed in all directions in an arbitrary manner, whereas each separate elementary region consists of a multitude of elementary magnets whose vectors are parallel to one another; in other words, each elementary region is magnetized to saturation. Each of the vectors can be resolved into three mutually perpendicular components (Fig. 2); the components parallel to the axis $A$ do not affect the results of the experiment if the magnetic field $\mathbf{H}$ rotates about the axis $A$, since their direction will not change in doing so. Thus we need investigate only the distribution of the vectors in a layer parallel to the plane $BC$. Let us suppose that an arbitrary distribution of vectors is characterized

Fig. 2. Arrangement of Fischer’s experiment.

Fig. 3. a) Ferromagnetic model of Fischer’s experiment.
b) Transition of the magnetization vectors from the plane $BC$ into the direction of the axis $A$ along the shortest path (indicated by the dotted arrow).

by four possible directions (Fig. 3, where the direction of the arrow corresponds to clockwise rotation); let us further assume that the magnetic field rotates from $C$ to $B$ (the direction of the arrow beyond the plane of the drawing, parallel to $A$ in Fig. 2). In this case all our vectors will try to rotate in the planes $CA$ and $BA$ in such a way as to align themselves parallel to the forced rotation along the shortest path (principal phenomenon 1). Therefore, irrespective of the disposition of the vectors in the plane $CB$, magnetization in the direction of the axis $A$ must arise. Thus we arrive at the conclusion that the Fischer effect should give a positive result. The reason for the failure of the corresponding experiments must lie deeper.

Several years ago, in order to overcome the energetic difficulties connected with considering the idea of a simultaneous change of the poles of an entire elementary region, Akulov introduced the idea of “inversion.” According to this idea, the magnetization vector $J$ of an elementary region, during its reversal of magnetization, does not rotate, as Ewing assumed, while preserving its magnitude unchanged, but continuously decreases its value from $+J$ through zero to $-J$, while its direction remains unchanged.

This idea, which we are compelled to accept thanks to

the existence of a reversible permeability in weak fields seems incompatible with the gyroscopic model of the atom, since a gyroscope of atomic dimensions changes the direction of its rotation, of course, by rotating the axis through 180° at an unchanged speed of rotation, and not by a gradual decrease of the speed of rotation to zero and a subsequent increase of it to its initial magnitude with the opposite direction of rotation. However, this representation can be reconciled with the gyroscopic nature of an individual atom if one recalls that the vector of an elementary region is the resultant of all the atomic vectors of the region under consideration, and that only the rotation of this resultant of the multitude of atomic vectors is accessible to observation. The contradiction between inversions and the gyroscopic nature of the atom was clarified by Bloch and Becker in 1932. These authors proposed the idea of “boundary displacement,” asserting that the transition from one direction of the vector to another always occurs at the boundary of two separate regions magnetized to saturation, with the elementary region that is more favorably oriented with respect to the external field expanding at the expense of the region oriented less favorably, i.e. its boundary is displaced. Therefore, in weak fields we must not reason as was done in the discussion of Fig. 3, where the vectors of individual elementary regions were depicted; rather, it is necessary to examine in detail how the boundaries of our four regions will move (Fig. 4). It is easy to see that in these regions the boundaries will be displaced in the direction of the dotted lines if the external field H rotates in the direction indicated by the arrow, since each time the region whose vector makes the smallest angle with the vector of the external field H* will increase. But the transition from one direction to another when the boundary is displaced may occur either statistically or—which is more probable—along the shortest path, since it proceeds under the action of two strong fields of the nearest vectors, which, as the external field rotates, must energetically increase or decrease. Therefore there should be no direct rotation of our vectors in the direction of the vector of the external field. If one analyzes the rotations in detail, it is pos-

Fig. 4. “Boundary displacement” when the direction of the vector of the external field changes. The initial boundary is shown by a solid line, the final one by a dotted line.

* It should be noted that in the literature on magnetism the magnetic moment is interpreted either as a gyroscopic moment or as a dipole moment. In this respect the gyroscopic representation has indisputable advantages.

** If the atomic gyroscope is considered as a combination of two connected tops, like that shown in Fig. 6, then the process of inversion can be reconciled with the gyroscopic model of the atom.

occurring in our model, it is easy to see that they arise in the following way (Fig. 5).

The curved arrow in the plane of the drawing indicates the direction of rotation of the boundary-layer vector; the arrow perpendicular to the plane of the drawing determines the resulting component of magnetization that arises thereby. From the figure it is immediately clear that the directions of rotation of the resultants are mutually opposite. Consequently, the Fischer effect in weak fields must be negative.

Completely incomprehensible from the purely gyroscopic point of view, the negative results of Fischer’s experiments may be regarded as a confirmation of the correctness of the idea of boundary displacement, if one assumes that the hypothesis is valid of a transition in the boundary layer from one direction to another, occurring statistically or along the shortest path, but in any case independently of the direction of rotation of the external field.*

Fig. 5. Displacement of magnetization vectors at “boundaries.”

Analogous ideas may also be applied to irreversible displacements of the boundary (Barkhausen jumps), since they are entirely independent of the velocities of boundary displacement. Only in strong fields can rotations of the resultant vectors arise, and therefore one may think that Fischer’s experiments in very strong fields will give a positive result. However, such investigations appear at present impossible, since with transverse magnetization fields of 5 to 10 thousand oersteds are required, because the demagnetizing factor is very large. Meanwhile, for example, Barnett, who worked with fields of about 15 oersteds, worked in the region of initial permeability. The same applies to the field of the order of 100 oersteds used by Fischer.

The explanation set forth here for the negative results of Fischer’s experiment was given by Barnett; but Barnett replaced the rotation of the vectors by a periodic change in the polarity of the vector components situated in the plane of the section.

B. Four Further Effects

In what follows we shall briefly consider four effects, denoted by the numbers 5—8. These effects are theoretically conceivable, but experimentally there is as little hope of detecting them as there is of obtaining a positive result in Maxwell’s experiment.

5) Just as Maxwell’s experiment is a reproduction of the first fundamental phenomenon, experiment 5, i.e. the precession of a magnet during

* A model of “reversible” rotations in the preferred directions determined by the crystalline structure also makes it possible to explain the negative result of Fischer’s experiments in weak fields.

change in its position, is a reproduction of the basic phenomenon 2. Apparently, Maxwell did not attempt to carry out this experiment, whose energetic possibilities are just as negligible as those of Maxwell’s experiment.

6) Just as the precession of the basic phenomenon 2 is revealed in the Richardson—Einstein—de Haas effect as a mechanical action on the entire body as a whole, it must manifest itself in the same way as a magnetic effect acting on individual atoms or elementary regions: precession is equivalent to an additional diamagnetism. At Gans’s suggestion, Čerlinskij unsuccessfully tried to detect this effect. In the region of saturation it should appear as a decrease in the intensity of magnetization proportional to the field. It should be noted that these arguments have an essential gap, since the parallel setting of all axes with respect to the direction of the external field cannot arise without the presence of some mechanism transmitting energy. Since this mechanism is unknown to us, it cannot be decided a priori whether the energy of the parallel setting of the axes at the constant angular velocity of precession

\[ \omega=-\frac{e}{2mc}\mathbf{H} \tag{1} \]

(i.e., diamagnetism) is obtained only at the expense of the energy of the external field, or whether it may also be obtained at the expense of a decrease in the energy of precession. From this point of view, experiment 6 concerns not so much the absolute magnitude of the magnetic moment of unit volume as its occurrence in time; separating the latter from the effect of eddy currents, however, must be very difficult.

7) In experiment 7, in which one attempts to achieve an orientation of the axes of elementary magnets by rotating the whole body as a whole about the horizontal axis \(B\), one must bear in mind that the axis \(B\) is in this case fixed in space. If, as in the basic phenomenon 2, it were connected with the body, then upon rotation of a body possessing residual magnetization there should arise a diamagnetic effect superposed on the residual ferromagnetism and causing an increase of the magnetic moment, i.e., a positive diamagnetic effect; this is explained by the fact that the resulting increase of the angle \(\Theta\) (Fig. 1) must also be neutralized by the decrease of it formed according to the sine law. In reality, however, in experiment 7 the horizontal axis \(B\) is fixed in space, so that this experiment is formally analogous to Barnett’s experiment with the axis turned through \(90^\circ\).

Thus, when the body is rotated, a transverse component of magnetization must be formed. The difficulty of this experiment lies above all in the need to create a space free from the field. Theoretically, however, it is of no interest, since it contains nothing new in comparison with Barnett’s experiment, and experimentally it is less convenient (for example, owing to the presence of a demagnetizing factor).

8) Finally, the Fischer experiment should reveal a mechanical action on the whole body as a whole, since, when the axes of the elementary magnets (magnetization in the direction of the axis of rotation of the magnetic field) associated with the body are oriented, the latter must be brought into a certain, though very weak, rotation. Since Fischer’s experiment with ferromagnetic substances, for the reasons indicated above, proves unsuccessful, experiment 8 must also be without result. In strong fields, however, and with paramagnetic substances, it may give positive results.

3. Consideration of the Experiments from the Point of View of the Theory of the Atom

Up to now, basing ourselves on the notions of Ampere’s old idea of molecular currents and Weber’s hypothesis of the inertia of electricity, we have reasoned, however, in such a way as though the gyroscopic nature of the atom were known to us. And indeed Procopiu, basing himself on Bohr’s model of the atom, calculated the magnitude of the magneton, i.e. the unit of magnetic moment in “unit poles × centimeters” or in “ergs/oersted.” However, this calculation proves the reality of the gyroscopic nature, which can be discovered only by experiment, to just as small a degree as does the atomic model itself.

From Weber’s definition of the electromagnetic mass of a current in CGS units, for the magnetic moment \(\mu\) the expression may be obtained:

\[ \mu = e \nu \pi r^2 \tag{2} \]

(where \(e\) is the charge of the electron in electromagnetic units, and \(\nu\) is the frequency of its rotation at a distance \(r\) from the nucleus of the atom).

Under these same conditions the mechanical moment \(p\) is equal to:

\[ p = m \nu 2\pi r^2, \tag{3} \]

where \(m\) is the mass of the electron.

From (2) and (3) the ratio of the mechanical moment to the magnetic moment is determined directly:

\[ \frac{p}{\mu} = \frac{2m}{e} = 1.13 \cdot 10^{-7}. \tag{4} \]

Since, according to quantum theory, the mechanical moment must be an integral multiple of \(h/2\pi\), it follows from (4) that the magnetic moment must be the same. For the quantum number 1 we find:

\[ \mu_1 = \frac{he}{4\pi m} = 9.174 \cdot 10^{-21}\ \text{erg/oersted} \tag{5} \]

or, for a mole:

\[ \mu_B = 5564\ \text{unit poles} \times \text{cm}, \tag{6} \]

which corresponds to one Bohr magneton.

It follows from equation (4) that an atom having a mechanical moment equal to zero should likewise not possess a permanent magnetic moment, i.e. it should be diamagnetic.

Since an ordinary gyroscope with mechanical moment equal to zero cannot exist, then, wishing to draw an analogy between an elementary magnet and a gyroscope, we necessarily come to the conclusion that a diamagnetic atom must consist of at least two gyroscopes, in some way connected with one another (Fig. 6). This conclusion corresponds to the experimental fact that diamagnetism is observed only in atoms with completed electron shells, the lowest symmetry of which is equal to two.

However, experiment does not confirm that the ratio \(p/\mu = 1.13 \cdot 10^{-7}\). For ferromagnetic substances this ratio is close to one half of the indicated value. To reconcile our reasoning with experiment, equation (4) should be written in a more general form:

\[ p/\mu = \frac{2}{g}\frac{m}{e}, \tag{7} \]

where \(g\) denotes the Landé splitting factor, known from the theory of spectra:

\[ g = \frac{2s + l}{s + l}. \tag{8} \]

Here \(l\) is the orbital quantum number, \(s\) is the spin quantum number; equation (8) determines the ratio of the magnetic moment to the mechanical one in units of \(\frac{e}{2m}\).

Fig. 6. Example of connected gyroscopes possessing inertial resistance only with respect to the axis \(A\). (Parallel lines between the gyroscopes symbolize the presence of a connection.)

From this formula there follows the remarkable conclusion that the total angular momentum of an atom can never coincide with the angular momentum of the electrons, which, as is known, was tacitly assumed in all considerations. This difficulty can be overcome if a screw-like motion is ascribed to the electron. The electron must not only rotate in an orbit (with quantum number \(l\)) around the nucleus, but must also rotate about its own axis (with quantum number \(s\)). This hypothesis of the electron spin is conditioned not only by the results of the investigation of spectra (Uhlenbeck and Goudsmit, 1925), but is also proved by any gyromagnetic experiment, which is nothing other than a determination of the factor \(g\). At the same time the gyroscopic nature of the atom is also proved, since the adoption of this hypothesis gives results consistent with experiment.

In the \(S\)-state of the atom, where \(l = 0\), only the spin of the electron has significance, equal, for one electron, according to empirical data, to \(s = \frac{1}{2}\).

Since for \(l = 0\) and \(s = \frac{1}{2}\) one obtains \(g = 2\), it follows that

\[ \frac{p}{\mu} = 0.5 \cdot 1.13 \cdot 10^{-7}; \tag{9} \]

but this value is very close to the quantity \(P_\mu\), found for all ferromagnetic substances.

4. Quantitative Estimate of the \(g\)-Factors of Para- and Ferromagnetic Substances

We know very little, or even nothing at all, about the quantum state of an atom in a solid. In these cases the study of optical spectra gives no results, with very rare exceptions. The only possible means is the study of the magnetism of binary mixed crystals, which has barely begun but has already made it possible to draw some conclusions about the ionization of atoms in solid solutions. Another direct path of investigation is the study of gyromechanical phenomena, on the assumption that equation (7) corresponds to reality. However, in this direction too only the first steps have been taken, and up to the present time there is almost a complete absence of systematic study of series of alloys of binary systems, which is very important for investigating changes of ionization with concentration.

This same path makes it possible to make progress with regard to the study of the physics of the metallic state. *

The data of Table 3 show that in normal ferromagnetic substances the value of \(g\) is close to two, but is undoubtedly somewhat less than two. This is understandable, since the atoms of a cubic ferromagnetic lattice are in a quasi-\(S\)-state, so that upon magnetization there occurs predominantly an orientation of the spin moments of the electrons. However, deviations from the value \(g = 2\) show that, in addition to this phenomenon, a certain, albeit insignificant, influence is also exerted by the orbital moment, since otherwise the condition \(g = 2\) would have to be strictly satisfied. A classical example is pyrrhotite, which has \(g \simeq 0.63\); this value of the factor \(g\) is undoubtedly closely connected with the circumstance that pyrrhotite is ferromagnetic only in one of the crystallographic planes, while in all other directions it proves to be paramagnetic. Enz (1934) proposed a simple model to explain this value of \(g\); in his opinion, the orbital moment is antiparallel to the spin moment and is coupled with it. For \(l = 2\) and \(s = -\dfrac{1}{2}\), according to equation (8) we find:

\[ g = \frac{-1 + 2}{-\dfrac{1}{2} + 2} = 0.66, \]

which agrees well with the observed value (Gorter, 1933).

* Apart from all anomalous crystals, of special interest are bimagnetic hematite crystals, which are paramagnetic in the direction of the principal crystallographic axis and diamagnetic in the perpendicular direction.

In all the foregoing considerations, however, one should remember that the gyromechanical effect gives little information about the true rotational moments.

TABLE 3
Values of $g$ for ferromagnetic substances

TABLE 4
Values of $g$ for paramagnetic substances

These moments may fail to coincide with the full moment of the atom that it would have if it were in the same state and were free, but determine only that moment whose polarity can be changed by the action of an external magnetic field on a substance in the solid state.

TABLE 5
Values of $g$ for some rare earths

These moments may sometimes coincide with the moments proper to free atoms and observed in spectroscopic investigations, as is proved by the satisfactory agreement of the calculated and observed values for some rare earths (Table 5).

But this coincidence should not lead to the conclusion that the indicated area of physics is at present illuminated sufficiently fully. Research here is only beginning. If at the present time we have succeeded in clarifying the so-called “gyromagnetic anomaly,” i.e., the contradiction between the previously expected value of the orbital number and the actually found spin number, by using the hypothesis of the Compton electron possessing spin, it should nevertheless be pointed out that in the foundations of the modern theory there still remain many obscurities, such as, for example, the questions of the transfer of angular momentum from elementary particles to the body as a whole, questions concerning the forbidden orientations of gyroscopic axes in the direction of the external field, and also the circumstance that the theoretical value \(\frac{p}{\mu}\) for the electron is a function of the model of the electron adopted as the basis of the reasoning; that is, this ratio will have different values depending on whether we use the conception of a “surface charge,” a “volume charge,” a vortex ring, or else consider the electron from the point of view of wave mechanics.

Despite the apparent successes in this area of physics, it should be noted that the experimental methods, despite the enormous demands placed on the skill of the experimenter and the strict criticism of experimental errors, have to a considerable extent outstripped the theoretical explanation of the results of the experiments; this observation applies especially to gyromechanical research. Just as, at the beginning of the development of measurement physics, the apparent “anomalies” it revealed posed new tasks for theoretical physics, so too the further investigation of the values of \(g\) in the solid and metallic state will in the future present theoretical physics with a whole series of new problems.

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