ELECTRONIC THEORY OF METALS*

R. Fowler, Cambridge.

From the standpoint of the electronic theory, a metal is any solid body that conducts electricity well. The present essay is a brief survey of the current state of the electronic theory of metals. I cannot, of course, guarantee that everything said here fully corresponds to the present formulation of the problem. My task is only to outline the general position of the theory in the form it has assumed on the basis of quantum mechanics.

The Nature of the Metal and the Classical Theory

The founder of the electronic theory of metals was, in essence, Drude. In any case, its first real success was the brilliant explanation he gave of the relation between the coefficients of electrical and thermal conductivity. The theory was based on the simplifying assumption—retained also in the modern treatment—that a metal consists of a certain number of positive ions, departing comparatively little from their equilibrium positions in the crystal lattice, and of the electrons lost by them, moving freely in this lattice.

In the case of monovalent metals—such, for example, as the noble and alkali metals—it is most natural to suppose that each atom loses exactly one electron. The electrons and ions interact with one another according to the ordinary Coulomb law. In a first approximation one may assume that, as a result of this interaction, an electron moves in the metal as in a neutral region with a constant positive potential, so that in order to tear an electron out of the metal and transfer it to the outside—

* Suppl. to Nature, No. 3181. Translated by S. P. Shubin.

its space with zero potential, a certain amount of work must be expended. In this approximation, the free electrons behave like an ideal gas consisting of neutral particles of mass \(m\) (the mass of the electron). In the next approximation it is necessary to take into account the existence of a number of regularly situated points (ions), at which the motion of the electron changes more or less abruptly; classically, these changes are caused by collisions of the electron with the ions. The electrical resistance of a metal is determined by that fraction of the electrons moving in the direction of the electric field which, under the influence of collisions, change the direction of their motion. This mean coefficient of scattering of the momentum—\(p\)—depends on the mean free path \(\lambda\):

\[ \frac{dp}{dx}=-\frac{p}{\lambda}. \]

Classical theory even succeeded, by choosing comparatively natural values of \(\lambda\), in obtaining the correct order of magnitude for the electrical conductivity. However, it had to be rejected because of one decisive difficulty: an ideal electron gas, with the number of electrons equal to the number of atoms, ought to have a specific heat of \(\frac{3}{2}R\), i.e., 3 calories per gram-atom per degree. In fact, however, the entire heat capacity of the metal can quite well be reduced to the vibrations of the massive ions (Fig. 1), so that there is practically no need at all to speak of the heat capacity of the electron gas. Along with this there existed a number of other difficulties. Most of them could be circumvented by assuming that all the phenomena are caused by only a small number of free electrons, to which correspondingly large mean free paths had to be ascribed. The real existence of these large lengths was also indicated by the strange influence of negligible inhomogeneities of the metal on its resistance. But classically it was quite impossible to imagine that the mean free path could be appreciably greater than the distance between neighboring atoms in the lattice.

Changes introduced by quantum mechanics.

All the aforementioned and analogous difficulties were completely removed by quantum mechanics, applied to the same Drude model. It turned out that what was bad was not the physical picture of the metal, but the classical dynamics of the electron itself. The correct solution of the question is contained in some of the most subtle propositions of modern quantum mechanics. We shall first consider how, according to quantum mechanics, a single electron behaves when it is in such a force field as is determined by our picture of the metal.

As is known, quantum mechanics assigns to every such system a certain number of possible stationary states, whose energies are determined by the condition for the existence of solutions of a certain linear differential equation (Schrödinger’s equation), satisfying certain physically necessary requirements. If \(V\) is the potential energy of the electron, \(h\)—Planck’s constant, \(\Delta\)—the ordinary Laplace operator, then Schrödinger’s equation reads

\[ -\frac{h^2}{8\pi m}\Delta\psi + (V-W)\psi = 0, \]

where the possible values of the parameter \(W\) are the possible total energies of the electron. According to our picture, \(V\) must change very rapidly at the boundary of the metal, being equal to \(0\) inside the metal and to \(-C\) outside it. The transition layer probably has a certain thickness, which, however, is of no interest to us now, since it is easy to show,

Fig. 1. Specific heats of various solid bodies as a function of \(\dfrac{T}{\theta}\).

that the change in \(V\) may, without appreciable error, be regarded as a sudden jump. Further, since this jump is rather large (magnitudes of the order of 10 volts are already large for us), it may practically be regarded as infinitely large. Then the wave equation takes the form

\[ -\frac{h^2}{8\pi m}\Delta\psi - E\psi = 0, \]

where \(E\) is the kinetic energy. We must find its solution under the condition \(\psi = 0\) at the boundary and in the exterior region, where \(V\) is infinitely large.*

Fig. 2. Surface potential barrier.

This equation is easily solved in the case when the region in which the electrons are enclosed has the form of a parallelepiped with edges \(a, b, c\) (for practical applications the shape of this region is of no importance). The possible kinetic energies in such a parallelepiped are given by the formula

\[ E = \frac{h^2}{8m}\left(\frac{r^2}{a^2}+\frac{s^2}{b^2}+\frac{t^2}{c^2}\right), \]

where \(r, s, t\) are positive integers. The corresponding function \(\psi\) is

\[ \psi_{rst} = \sin\frac{\pi r x}{a}\sin\frac{\pi s y}{b}\sin\frac{\pi t z}{c}. \]

It is easy to see that, owing to the smallness of \(h\) and despite the smallness of \(m\), in a small box of ordinary dimensions, e.g. \(1\ \mathrm{mm}^3\), and even in a cube with edge \(0.5 \times 10^{-4}\ \mathrm{cm}\)—of the order of the wavelength of visible light—the energies of the possible stationary states lie so close to one another that the dis—

\[ \text{* To this our physical requirements reduce in the limiting case under consideration.} \]

their determination practically cannot be distinguished from the classical continuous distribution. Therefore, if one places in such a box as many electrons as there are approximately atoms in the corresponding volume of the metal, and if they in no way interfere with one another, then we again obtain the classical (Maxwellian) distribution over velocities, which is incapable of giving the real picture of the phenomena occurring in the metal.

However, it is precisely at this point (in its application to metals, this was first pointed out by Sommerfeld) that quantum mechanics introduces an essentially new element. The point is that even in the zeroth approximation—even while completely neglecting the mutual potential energy of the particles—it is impossible to assign to an aggregate of two or several electrons those states which are obtained by a simple combination of the states of a single electron. This fact (at least according to our present views) can in no way be reduced to simple Coulomb repulsion. This fact continues to hold also in that limiting case when this repulsion may be assumed to be neutralized. Nor can its cause be sought in those hypothetical forces which determine the existence of a finite volume of the electron. It has a far more fundamental significance and enters into quantum mechanics only because its equations are linear with respect to the wave function \(\psi\), in contrast to the equations of classical corpuscular mechanics, which, for example, with respect to momenta are essentially nonlinear. Mathematically, what we have here is a kind of interference caused by the superposition of waves, examples of which are often encountered in ordinary oscillatory systems. It should be noted, however, that this analogy must be used with extreme caution. Many have become accustomed to considering de Broglie waves associated with material particles as something analogous to electromagnetic waves in ordinary physical space-time. But, for example, for a pair of electrons the de Broglie wave is, strictly speaking, a wave in six-dimensional space, and the number of these dimensions for a larger number of electrons

accordingly increases. The value of electromagnetic and optical analogies is enormous, but in interpreting their results in ordinary space-time, the greatest caution must be observed.

In any case, whatever the correct physical interpretation may be, the theory has such a form that the possible states of two or several electrons confined in one box constitute only a certain specifically selected group from among all possible combinations of the states of each individual electron. This selection can be most simply characterized by saying that two electrons cannot simultaneously be in one and the same stationary state. This rule is valid for any collection of electrons, for example, for a single free atom, for which it was first established semi-empirically by Pauli. At present it is generally known under the name of the Pauli principle. For a rigorous formulation of this principle it is necessary to take into account that each electron possesses a mechanical moment of its own rotation and a corresponding magnetic moment, which can assume two different orientations: along and opposite to the magnetic field. These two orientations may be regarded as two possible values of a new quantum number, so that together with \(r, s\) and \(t\) one obtains in all 4 quantum numbers for each electron. The rigorous formulation of the Pauli principle states that in any collection of electrons there cannot be even a single pair of electrons for which all four quantum numbers would be identical.

I have always emphasized the independence of the Pauli principle from the kind and character of the mutual potential energy of the electrons. I believe that, in the present state of the theory, such a point of view is the most correct one. But if Eddington’s recent investigations prove fruitful, it is possible that the Pauli principle and the Coulomb interaction will be merged into a single, more intelligible scheme.

Quantum mechanics cannot show that all elec-

ELECTRONIC THEORY OF METALS

electrons in fact must obey the Pauli principle; it asserts only that if they obeyed it in some initial state, then they continue to do so always. Apparently, this is how matters stand in reality. As has already been said, it was precisely the introduction of this principle into the theory of metals, for which we are indebted to Sommerfeld, that gave the final impetus to a new development. If the Pauli principle is taken into account, it turns out that the number of free electrons in a unit volume of a metal whose kinetic energy lies between \(E\) and \(E+dE\), instead of the usual Maxwell formula

\[ n(E)\,dE=A'e^{-\frac{E}{kT}}E^{\frac{1}{2}}\,dE, \tag{1} \]

is given by the slightly different Fermi–Dirac law

\[ n(E)\,dE=\frac{4\pi\,2(m)^{\frac{3}{2}}}{h^3}\cdot \frac{E^{\frac{1}{2}}\,dE}{1+\frac{e^{E/kT}}{A}}. \tag{2} \]

\(A\) and \(A'\) are constants determined by the total number of electrons in a unit volume; moreover, for monovalent metals one may assume that there is one electron per atom. We note that for small \(A\) equation (2) reduces to (1), whereas for large \(A\) there is an essential difference between them. When \(A\) is large, it can approximately be expressed by the formula

\[ A=e^{\frac{\overline{E}}{kT}}, \qquad \overline{E}=\frac{h^2}{8m}\left(\frac{3n}{\pi}\right)^{\frac{2}{3}}, \]

where \(n\) is the density of the electron gas.

It is precisely here that the small magnitude of the electron mass makes itself felt. Evidently, \(A\) depends on the mass, and for densities of the order of those which we have in a metal it proves to be so large that, for example, for copper even at \(1500^\circ\) abs the distribution is quite unlike the classical one. This is clearly seen from Fig. 3, due to Nordheim. At ordinary room temperatures, and even at higher ones, we have approximately the following distribution:

\[ n(E)\sim E^{\frac{1}{2}}(E<\overline{E}); \qquad n(E)\sim e^{-\frac{E}{kT}}E^{\frac{1}{2}}(E>\overline{E}). \]

This means that at all practically attainable temperatures almost every possible state for \(E<\overline{E}\) is occupied by one electron, whereas for \(E>\overline{E}\) almost all states are empty. The number of electrons present there is so small that they do not “crowd” one another and therefore can obey Maxwell’s law.

Fig. 3. Distribution law at \(0^\circ\) and \(1500^\circ\) abs.

Hence it is immediately clear that the difficulty with the heat capacity disappears. Indeed, Nordheim’s drawing shows that the dependence of the energy of the electron gas on temperature is very small. A detailed calculation confirms this and leads to the following expression for the total kinetic energy

\[ E_{kin}=\frac{\pi}{40}\cdot\frac{h^2}{m}\left(\frac{3n}{\pi}\right)^{5/3} +\frac{2\pi^3 m k^2 T^2}{3h^2}\left(\frac{3n}{\pi}\right)^{1/3}, \]

in which the term depending on the temperature is negligibly small. It turns out even to be too small, and Bloch’s more exact calculations make it somewhat larger, and in some region even noticeably larger (this makes it possible to give a theoretical explanation of certain interesting anomalies of the heat capacity discovered at low temperatures by Simon (Fig. 4) in tin and other metals). The constant term, on the contrary, is very large; the electrons possess a large “zero energy.” But it plays no role in the phenomena in the metal.

NEW THEORY OF CONDUCTIVITY. Up to now we have considered only equilibrium states; let us now see what the new theory gives when applied to the problem of conductivity, i.e. to phenomena in stationary currents.

As a preliminary work, Sommerfeld revised Lorentz’s calculations with the new expression for \(n(E)\) instead of the Maxwellian one and showed that the correct-

...the correct order of magnitude for the electrical conductivity is obtained only for very large mean free paths. As we have already seen, other experimental considerations also require the same thing. But classically a mean free path of the order of 10–100 atomic distances seems completely impossible. According to quantum mechanics, however, as Houston, Bloch, and Peierls have shown, precisely this is to be expected. Indeed, if one considers the motion of an electron not in a constant potential, but in a potential that varies periodically in space, it turns out that these periodic variations affect only the energies of the stationary states of the electron in the box, but do not at all hinder its free motion through the box in any direction.

Fig. 4. Specific heat of gray tin. The circles show the observed values. Curves \(a\) and \(a'\) are Debye curves, not coinciding with the observations; curve \(c = a + b\), where \(b\) is the specific heat of the electron gas.

Such a strictly periodic variation of the potential exists only in an ideally pure metallic single crystal at absolute zero, when all the ions of this crystal are at rest relative to one another. In such a crystal the mean free path of an electron with kinetic energy of several volts would be infinitely large, i.e. the conductivity would be infinite. We encounter a phenomenon of an analogous kind in physical optics. When light passes through an optically homogeneous medium, for example, through air

or through certain crystals, then in first approximation it is almost not scattered at all and is not absorbed, although each molecule individually can indeed be a scattering center. As Lord Rayleigh first showed, the scattering of light is caused by deviations from uniform density, which in air (apart from dust) are due simply to fluctuations of concentration, and in a crystal—to cracks and foreign impurities. In exactly the same way, de Broglie electron waves are scattered not by regular lattices, but only by inhomogeneities of these lattices, caused by thermal motion, deformations, or impurities.

Fig. 5. Comparison of the observed values of \(F\) with the theoretical ones. \(F\) is plotted as a function of \(\dfrac{\sin\theta}{\lambda}\) for \(\mathrm{Na}^{+}\) and \(\mathrm{Cl}^{-}\) ions; \(\theta\) is the angle of observation; \(\lambda\) is the wavelength in Å. On the curves are plotted the theoretical values of \(F\) for those charge distributions which are given by wave mechanics. Crosses denote the values of \(F\) derived from observations under the assumption of the existence of zero-point energy; circles denote the values of \(F\) derived under the assumption of the absence of zero-point energy.

However, on closer examination such a simple interpretation proves not entirely suitable. The point is that both theoretically and experimentally (with the aid of X-rays) it has been established that at absolute zero temperature the ions of a metallic crystal do not rest in their equilibrium positions, but oscillate about them; thus in the crystal there remains a “zero-point energy” of appreciable magnitude. This fact (see Fig. 5) at first sight destroys the whole theory and deserves more detailed consideration, since at absolute zero the conductivity does indeed tend to infinity, even in the absence of superconductivity. In order to understand in what

ELECTRONIC THEORY OF METALS

Here, it is necessary to investigate the processes of exchange of energy and momentum between the electron and the lattice, which take place at the beginning and at the end of the free path.

The motion of ions in the lattice may be decomposed into elastic waves, just as Debye does in his theory of specific heats. In order for the free path of an electron to be terminated, it is necessary: 1) that the energy of the electron \(E\) and its momentum \(p\) pass into \(E'\), \(p'\), and that the latter, according to the Pauli principle, correspond to an unoccupied stationary state; 2) that the vibrational state of the lattice pass into another state with absorption of the energy \(E - E'\) and momentum \(p - p'\).

At high temperatures it is comparatively easy to satisfy both these requirements, but at low temperatures the electron, as a consequence of the Pauli principle, can only acquire energy, whereas the lattice vibrations, which are almost all in the very lowest state, can no longer lose it. Thus, despite the existence of zero-point energy, the probability of exchange, and with it the resistance, decreases rapidly.

An exact quantitative investigation of this question has been carried out only for high temperatures. In this region it gives a strict proportionality between the resistance and \(T\), which is in excellent agreement with experiment; at low temperatures the investigation becomes considerably more difficult, and for the resistance one obtains quantities proportional to \(T^3\), \(T^4\), or \(T^5\). Although some of these results agree fairly well with experiment (see Fig. 6), it seems to me that the final theory here still lies ahead. We might, however, already be satisfied with what has been done, were it not for the open problem of superconductivity, in which as yet nothing has been accomplished. It is apparently connected with the theory of magnetic effects, about which we have learned so much from Kapitza’s experiments. The results obtained by him have still not been explained, and however interesting they may be, we cannot dwell on them now.

THERMOELECTRIC EFFECTS. Before finishing with conductivity, it should be noted that the theory gives

In general, a satisfactory explanation is obtained for reversible thermoelectric phenomena. At the same time the usual thermodynamic relations are preserved, and for the Thomson and Peltier coefficients even in Sommerfeld’s simplest theory one obtains the correct order of magnitude and the course of the temperature dependence. The refinements introduced by Bloch and Peierls could undoubtedly yield still more satisfactory results.

Fig. 6. Course of the change of the ideal resistance of lead near absolute zero, plotted on a logarithmic scale. In the region between \(15^\circ\) and \(20^\circ\) the resistance is proportional to \(T^3\); below \(10^\circ\) it begins to change more rapidly than \(T^5\).

Electron emission. Until now we have tacitly assumed that all electrons are at all times located in the metal, as would be the case for an infinitely high potential barrier. In the more general case, with a finite height of the barrier, almost everything stated above remains valid, with the sole difference that, according to formula (2), there will always be present in the metal a certain (temperature-dependent) number of electrons with energy sufficient for escape. Since the mean free paths are comparatively large, one may neglect the periodic potential of the lattice and consider simply the fall of an electron with kinetic energy \(E\) onto a potential barrier whose height is a function only of the distance from the boundary. It is easy to show that in this case the component of the velocity

the electron parallel to the surface of the metal does not change, so that we may confine ourselves to considering a one-dimensional problem. Thus the problem reduces to the investigation of the motion of a beam of electrons falling normally on the surface with energy \(W\). In Fig. 7 various (real and idealized) types of boundary potential barriers are shown.

Fig. 7. Various types of natural and idealized boundary potential barriers (\(a\)). The natural form of the potential barrier in the presence of an electropositive monomolecular layer (if the force of “mirror reflection” is neglected); \(b\)—the same barrier, somewhat idealized for convenience of calculation, but preserving all its principal features; \(c\)—an idealized barrier (of the gap type) in the presence of a strong external field tearing out electrons, and the same barrier with the force of mirror reflection taken into account (the first of these curves contains the theory of the ordinary cold discharge, the second—the theory of the Schottky effect); \(d\)—a barrier with two exits: internal and external, explaining the combined action of strong fields and monomolecular surface layers.

The classical solution of this problem is somewhat rigid and does not cover the whole variety of observed phenomena. According to the classical theory, an electron always leaves the metal when \(W>B\) (\(B\) is the height of the highest point of the barrier)

and can never emerge from it when \(W < B\). In quantum mechanics the matter is different: an electron cannot emerge from a metal only if \(W < C\), where \(C\) is the finite height of the barrier, but it always has a definite probability of escaping when \(W > C\), even if \(W < B\). If we denote this probability by \(D(W)\), and the number of electrons incident per unit time on a unit surface, whose energy lies between \(W\) and \(W + dW\), by \(N(W)\, \(dW\), then the total saturation current \(I\) from a unit surface of the metal will be equal to

\[ I = e \int_{0}^{\infty} N(W)D(W)\,dW . \]

The quantity \(N(W)\) may be taken from Sommerfeld’s theory (Fig. 8), which for the present purpose is quite a sufficient approximation.

Fig. 8. Number of electrons \(N(W)\) incident on the surface of a metal with that normal component of velocity which corresponds to the kinetic energy \(W\), at \(0^\circ\) and \(1500^\circ\) abs.

Further, one can calculate the coefficient \(D(W)\), and with it the entire thermionic current \(I\). It seems to me that the result of these calculations may be regarded as quite successful. It shows that \(D(W)\) depends essentially on the form of the barrier, i.e. on the properties of the surface layer, which may often be a layer of another substance, specially deposited on the given metal or accidentally fallen upon it. The theoretical expression for the current has the form

\[ I = AT^{2} e^{-\frac{x}{kT}}, \]

where \(A\) and \(x\) are constants characterizing the given metal. This formula agrees with the well-known empirical dependence. But, in addition, the theory makes it possible to explain also the relation between \(A\) and \(x\) (the work function), on the basis of the properties of potential barriers. It seems to me one may safely say that, under the most natural assumptions about the form of potential barriers, the theory

quite satisfactorily, as a first approximation, explains all the usual phenomena of electron emission at high temperatures, under the action of strong electric fields, and under the influence of light.

It is interesting to note that, according to quantum mechanics, an electron can also pass through such regions in which its kinetic energy should have been negative. The probability of this passage is comparatively large if the region under consideration is very narrow, and decreases exponentially as its thickness increases. From the point of view of the wave nature of the electron, as a group of de Broglie waves, this phenomenon is completely analogous to a well-known phenomenon from physical optics. When a ray of light falls on the boundary between two media at an angle of incidence greater than the critical one, total internal reflection occurs. But if the second medium is a very thin layer, beyond which the first medium again follows, then in this third region a weak transmitted beam is nevertheless obtained.

Magnetism and Ferromagnetism. Up to now we have throughout ignored the presence of the ordinary Coulomb interaction between every pair of electrons and ions. Heisenberg showed that if this interaction is taken into account and, in addition, the intrinsic magnetic moments of the electrons are considered, then it is possible, even if only in the crudest way, to explain ferromagnetic phenomena.

The first step in this field was made by Pauli. It still belonged to that domain of approximation in which the Coulomb interaction may be neglected. As is known, the study of atomic spectra leads to the conclusion that each electron possesses an intrinsic angular momentum of rotation equal to

\[ \frac{h}{4\pi}, \]

i.e. to \(1/2\) quantum, and an intrinsic magnetic moment equal to one Bohr magneton. A Bohr magneton is the magnetic moment of any electronic orbit in any atom whose mechanical moment is equal to an integral quantum, i.e.

\[ \frac{h}{2\pi}. \]

Thus, for the intrinsic rotation of the electron, the ratio of the magnetic moment to the mechanical one turns out to be twice as large as for orbital rota-

tion. This fact receives a brilliant interpretation in the latest version of relativistic quantum mechanics, which we owe to Dirac. Since each of the free metallic electrons carries with it one Bohr magneton, whose axis may be oriented along the magnetic field or opposite to it, the alkali and noble metals should be strongly paramagnetic. The dependence of this paramagnetism on temperature would be determined by the classical Langevin formula, if here again the Pauli principle did not enter the scene, which at ordinary temperatures forbids any large accumulation of electrons in states corresponding to orientation of the magnetic moment along the field.

It is not difficult to calculate the susceptibility of such an electron gas. It turns out that it must possess a small, temperature-independent paramagnetic susceptibility of the same order of magnitude as the ordinary diamagnetic susceptibility. If, in addition, one takes into account the normal diamagnetic effect of the ions, then one can satisfactorily explain all the paramagnetic effects observed in the light alkali metals. Since these effects are extremely small, we shall not take them into account in what follows; that is, we shall assume that an idealized metal, in which Coulomb interactions are absent, is magnetically neutral. Let us note that throughout we are dealing here with inert ions, such as the monovalent ions of the alkali and noble metals, which contain only closed groups of electrons.

In expounding his theory of ferromagnetism, Heisenberg first of all points out that the entire domain of ferromagnetic phenomena, from the formal point of view, can be quite satisfactorily explained by the ordinary Weiss theory. According to the latter, the total internal energy of any ferromagnetic substance contains a large additional term, which depends in a definite way on the degree of magnetization. If one assumes the existence of this so-called Weiss molecular field, then everything follows—

... proceeds successfully. The whole difficulty had hitherto consisted in somehow giving a rational justification for its existence, since the actual magnetic energies with which we are dealing are tens of thousands of times smaller than the energy of this hypothetical field. This difficulty was removed by Heisenberg on the basis of quantum mechanics.

Heisenberg first of all draws attention to the gyromagnetic anomaly. When a metallic rod is magnetized, it acquires a certain mechanical angular momentum which, for a sufficiently mobile rod, can be measured. Since for all electronic orbits the ratio of the magnetic moment to the mechanical one is equal to one and the same constant value \(\frac{e}{2mc}\), one should expect that the same constant ratio will be obtained for our rod as well, whence the ratio \(\frac{e}{m}\) can be determined. Experiment shows that the observed value of this ratio does indeed come out the same for all ferromagnets, but the value of \(\frac{e}{m}\) calculated from it proves to be exactly twice as large as expected. This is precisely the anomaly mentioned above. Thus the situation is as if all magnetization were determined exclusively by the orientation of the intrinsic moments of the electrons themselves, and not of electronic orbits. Heisenberg* therefore assumes that the entire magnetism of ferromagnets is due to the orientation of the “spins” of weakly bound or free electrons. In this connection it turns out that the true nature of Weiss’s field lies in ordinary Coulomb interactions, i.e. in electrostatic attractions and repulsions, which until now had not been taken into account in our theory. At this point, however, we encounter one of the most elegant subtleties of quantum mechanics, which I shall try to explain by the simplest possible example.

In order to approach the problem of interaction,

* Here \(c\) is the velocity of light, and \(\frac{e}{m}\) is the ratio of the charge of the electron to its mass.

Heisenberg idealizes the metal somewhat differently than we have done up to now. He considers that the crystal lattice of the metal consists not of ions, but of atoms situated at rather large distances from one another. At first glance it seems that this model sharply contradicts the one we have used up to now. But in reality this disagreement is only external, since Bloch has shown (to me personally, at least, his arguments seem convincing) that effects analogous to Heisenberg’s can also be obtained by introducing the Coulomb interaction into our old model. However, with such a method of treatment the technique of the calculations becomes considerably more complicated, so that in what follows we shall follow Heisenberg, keeping in mind that in his approximate picture there is nothing that would contradict our former model, even if one assumes that ferromagnetism is due precisely to the conduction electrons (as recent experiments indicate). Thus our problem is reduced to the investigation of the possible stationary states and energies of a large number of correctly arranged identical atomic systems, in each of which there is one or several electrons, so weakly bound that their interaction plays a noticeable role. The fact that there are very many of these atoms complicates only the details, but does not affect the essence of the problem; we may therefore restrict ourselves to the case of two atoms, for example two hydrogen atoms in the normal state.

Here again the linearity of the wave equation introduces an essentially new point. It turns out that, if the electrostatic interaction is taken into account, there can be no such stationary state in which one electron would remain all the time in one atom, and the other electron in the other. If such a state did exist at the initial moment, then after the lapse of a certain time, depending on their mutual distance, the electrons would have exchanged places. When the atoms are comparatively close to one another, this exchange takes place extremely often, and the corresponding term in the expression

energy—the so-called electrostatic energy of exchange—becomes very large and almost comparable in magnitude with the unperturbed energies. The actual stationary states of the system (there are two of them) may be regarded as the result of a superposition of the unperturbed states (i.e. those in which each electron is in its own atom).

These two states possess entirely different energies, depending in different ways on the interatomic distance (see Fig. 9). In one of them the electronic “spins” mutually neutralize one another, so that the system as a whole has no magnetic moment. In the other, these “spins” are, on the contrary, parallel to one another, so that the system has a moment of two Bohr magnetons. The difference in the energy levels of these states, caused by the presence of electrostatic exchange, is considerably greater (at least when the atoms are comparatively close to one another) than the magnetic energy itself.

Fig. 9. Two different possible values of the mutual potential energy of two colliding hydrogen atoms.

Let us now suppose that a certain number of such pairs of atoms is placed in a magnetic field (each pair being comparatively far removed from the others). This field will change the distribution of all the electronic magnets (to calculate this change is one of the tasks of Heisenberg’s theory); in particular, it will change the ratio of the number of neutralized pairs to the number of pairs possessing a magnetic moment, as well as the orientation of the latter. In other words, it will change the distribution of the pairs over their stationary states, i.e. the total ener-

of the system. This change in energy, depending on the degree of magnetization, will be large in comparison with the magnetic energies themselves.

Herein lies the true origin of Weiss’s molecular field! An atom in a real metallic lattice, each with its “free” electron, constitutes a system completely analogous to the one analyzed above, although in this case, for the convenient carrying out of the calculations, the aid of the abstract theory of groups is required. Here we shall state only the result of these calculations. Heisenberg showed that for a metal with one “active” electron per atom the magnetic permeability is the root of an equation that, in the main, coincides with Weiss’s. Heisenberg took into account only the interaction between neighboring atoms; therefore in his final equations there enters the quantity \(I_0\), characterizing the splitting of the energy levels caused by this interaction. In order that a metal be a ferromagnet at all, \(I_0\) must be positive; and in order that its Curie point be at a comparatively high temperature, \(I_0\) must be large. Heisenberg showed in the most general form that both these conditions are comparatively difficult to satisfy, so that ferromagnets must be comparatively rare and must belong to metals with small atomic volume (as is indeed the case in reality). This conclusion is a brilliant triumph of the theory!1

Conclusions. It should not be thought that, in assessing the present state of the theory of metals, I am a blind optimist. Much in this field still remains unfinished. I may be considered an optimist only insofar as I am convinced that quantum mechanics, in its present state, is a wholly adequate instrument for the electronic theory of metals, as also for any other physical theory in which the velocity of light may be regarded as infinitely large—

…tive. But the theory of metals itself is far from complete. In particular, I think that a complete theory of metals must be able to explain, on the basis of the known properties of atoms and the principles of quantum mechanics, why an aggregate of copper atoms with a definite store of energy forms a metal and not, for example, a gas. Further, it is necessary to clarify what simplifications may be made in the subsequent treatment of the problem of conductivity.

In the works of which I have spoken here, these more fundamental problems were not touched upon. In my opinion, a very promising step in this direction has been made by Slater. It is quite probable that in a more complete theory the simple model that we have used up to now will be considerably modified. But my faith in quantum mechanics rests in part on the fact that, for any physically acceptable model, it gives results at least qualitatively in agreement with experiment. Therefore the further advance of theory and experiment may proceed with the assurance that we are on the right path and are moving toward a sure, not very distant, success.

1. Bloch (ZS. Physik 61, 206, 1930) gave a new development of the theory, refining it for low temperatures, with the aid of a very important new method introduced by Slater (Phys. Rev., March 1930). ↩