Electronic Theory of Metals Based on Wave Statistics1

A. Sommerfeld, Munich.

The problems of metallic conductivity and the question of the nature of the metallic state interest the chemist as much as the physicist. The solution of these problems seemed hopeless only a few years ago: here everything was contradictory. Richardson asserted that the electrons inside a metal follow the Maxwellian distribution of velocities, like the molecules of a monatomic gas, since the “thermo-ions,” or, more correctly, the “thermo-electrons,” emerging from an incandescent metal obey this law. But if this is so, the specific heat of a metal must depend to a considerable extent on the number of electrons; 6 calories (per mole) must be assigned to the atoms of the metal; if the number of electrons in the metal were equal to the number of atoms, then such an electron gas would have to add another 3 calories to the indicated figure. Measurements of specific heat show, however, that in reality almost nothing remains for the electrons, and it would seem that one should assume that the number of electrons is considerably smaller than the number of metal atoms. But this is improbable, at least for such typical monovalent substances as silver or the alkali metals. Moreover, in explaining other thermo- and

magneto-electric phenomena involving the number of electrons has to be treated no less freely in order to adapt the theory to various observations. The Volta effect, with potential differences of several volts, requires monstrous differences in the number of electrons in different metals; conversely, to explain thermoelectricity, negligibly small differences in these numbers are needed. In short, here there is the most complicated tangle of contradictions. Physicists were also interested in the following difficulty in the Wiedemann–Franz law. In this law, as is known, a relation is established between thermal conductivity and electrical conductivity in metals. The ratio of these two quantities, divided by the absolute temperature, as experiment shows (Disselhorst and Eger, Grüneisen), has a universal value, the same for all materials. It seemed that the classical theory of electrons in Drude’s form gave the correct value for this quantity, which was regarded as one of the finest results of the classical theory of electrons. A more rigorous calculation by Lorentz showed, however, that Drude’s number must be corrected in the ratio \(2:3\), and thus the numerical agreement was destroyed.

After cathode rays became known to us, one can hardly doubt that in an electric current something really flows, and that what flows is electrons. In a discharge tube electrons emerge from the cathode, for which charge and mass can be determined; the electric current feeding the cathode is closed by the cathode rays, and therefore we must conclude that it too consists of electrons. Such a conclusion was drawn already by Wilhelm Weber, but only in somewhat different words. The convective character of the electric current is also proven directly experimentally by Tolman’s remarkable experiment, in which he measured the impact produced by moving and suddenly stopped electrons in a metal. Electrons in an electric current receive, as a result of the applied field, a certain preferential direction in their flow; hence the inevitable conclusion is that in the absence of a field electrons move uniformly in all directions, and this

ELECTRON THEORY OF METALS

the motion must be determined by the temperature of the metal. Thus there arises the notion of an “electron gas” in temperature equilibrium with the metal. There was no lack of attempts to restore the electron theory by replacing the supposedly crude notion of the electron gas with special assumptions about the nature of the metallic lattice and about the motion of electrons in it. I shall mention here in particular the names of J. J. Thomson, Haber, and Bridgman. Such notions, however, corrected only individual points of a broad field and, moreover, included certain arbitrary assumptions. The true means was found in the quantum theory, namely in its newest variant, wave mechanics. The way of solving the problem lay not in new assumptions about the electrons or ions of the metal, but in a new method of calculation, in the reform of statistics.

Since the time of Boltzmann we have known that the physical meaning of thermodynamics is statistical. Entropy is a measure of probability. But in calculating probability everything is based on the definition of equally probable cases. Let us consider the simplest example. Suppose it is required to place two balls (molecules or electrons) in two cells. According to classical statistics this can be done in four ways, represented in the following scheme:

\[ \begin{array}{c|c} ab & -\\ - & ab\\ a & b\\ b & a \end{array} \left\} \right. \]

Both balls \(a\) and \(b\) are placed either in the first cell, or in the second, or else separately fall into each of the cells. The statistics of quantum theory is not such—at least, the statistics leading to Planck’s law is not such, if the latter is to be derived on the basis of the notion of light quanta. Light quanta of one and the same frequency are indistinguishable from one another. Therefore the last two cases, differing only in the names of the balls and not in the number of balls contained in each of the cells, are identi-

are natural. Such statistics were proposed by the Indian Bose for the theory of radiation and extended by Einstein to the theory of gases. The matter, however, is not exhausted by this; something new enters the scene, the “Pauli prohibition,” according to which two individuals can never be in one and the same quantum state. If our two cells correspond to two quantized states, then both of the first possibilities are, from this point of view, excluded; there remains only a single case: one ball falls into one cell, the other into the second. This last type of statistics was developed by the young Italian Fermi and the young Englishman Dirac. Our simple example shows how fundamentally different the three statistics are: according to Maxwell—Boltzmann there are 4 possibilities, according to Bose—Einstein 3, according to Fermi—Dirac only one.

In what follows we shall use only the Fermi—Dirac statistics. In order here, among chemists, to present in an acceptable form the Pauli principle on which this statistics is based, let us note that the principle indicated, in addition to its significance in the theory of spectra, to some extent completes the theory of the periodic system of the elements. The Pauli principle indicates at what point, in the successive construction of atoms from electrons, a new period must begin, i.e. a new quantum state of the elements. It turns out, for example, that the shells of the noble gases cannot contain more than 8 electrons, and that in the group of rare earths there cannot be more than 14 elements. The reason for this is that all the places in the corresponding shell or group are occupied, and a new electron must seek a place in another shell, i.e. in another group of the periodic system.

The new statistics is closely connected with wave mechanics and the wave nature of the electron. Schrödinger’s wave mechanics is adapted to atomic relations and is in this sense micromechanical; its statistical character is revealed in the fact that an individual electron in this theory spreads out into an entire charged-

ELECTRONIC THEORY OF METALS

...cloud, whose density indicates the probability of finding the electron at one or another place in the atom. De Broglie’s electron wave pertains to an entire swarm of electrons; a plane wave is obtained for a group of electrons moving in one direction, a spherical wave for a group of electrons spreading out from one center in all directions. In observations we are always dealing not with individual electrons, but with whole groups; an individual electron is not observed at all. The behavior of a swarm of electrons is determined by laws analogous to the laws of light waves. This is asserted by wave mechanics, and experience confirms the same. In Fig. 1 are shown striking photographs by G. P. Thomson, son of J. J. Thomson, recently published in the reports of the Royal Society. Cathode rays at 20–30 kV fell upon the thinnest films of gold or celluloid. Behind the films, on a photographic plate, Debye–Scherrer rings appeared, just as when X-rays pass through the same kind of films. Consequently, the deflection of electrons is determined by the same laws as the diffraction of X-rays. One may speak of diffraction and reflection of electron rays and carry out calculations based on Laue’s theory for the interference of X-rays. Electrons (horribile dictu!) interfere with one another like light waves; in other words, the probabilities interfere that the electron will more readily be deflected in one direction than in another. The voltage applied to the cathode tube determines the wavelength of the de Broglie waves associated with the electrons. At 20 kV the wavelength is approximately \(0.1\ \text{Å}\). Parallel experiment

Fig. 1.

with X-rays of the same wavelength and with the same gold film would have given the same interference rings. By means of electron waves one can determine the structure of the microcrystals of a film just as exactly as with X-rays.

Thomson’s experiments are the most effective, but not the only facts proving the wave nature of the electron (or, better said, of a swarm of electrons). The experiments of Davisson and Germer, carried out in the scientific laboratory of the Bell Telephone Company in New York, reveal the same diffraction phenomena in single-crystal nickel. Here the voltage was considerably smaller, only about 200 V; the de Broglie waves in this case coincide with very soft X-rays. As in the original experiment of Laue, Friedrich, and Knipping with single-crystal zinc blende, here, instead of the rings of microcrystalline films, sharp interference spots appear.

A certain distortion of the interference pattern of Davisson and Germer, in comparison with the X-ray photograph for the same wavelength, is explained by the fact that the crystal, for relatively slow electrons, is not “optically empty”; there is a certain index of refraction. This index of refraction is connected with the work function of an electron from metals, which will be discussed further on.

Experiments of this kind are now being repeated in various places, both with electron and with atomic rays. Thus the theory of electron diffraction, and likewise all of wave mechanics, may be regarded as empirically strengthened. We have the right to apply it also to the electrons of metals.

Let us first return to the statistical starting point. If one carries out the calculation for a monatomic gas according to Maxwell–Boltzmann, one obtains the Maxwellian distribution law:

\[ f = A \cdot e^{-\frac{\varepsilon}{kT}}, \tag{1} \]

where \(f\) is the number of gas atoms flying in a definite direction with kinetic energy \(\varepsilon=\dfrac{mv^2}{2}\)—more precisely, the number of atoms whose energy lies between \(\varepsilon\) and \(\varepsilon+1\), \(T\) is the absolute temperature, \(k\) is the gas constant for an individual atom, i.e. \(k=\dfrac{R}{N}\) (\(N\) is Loschmidt’s number), and \(A\) is a constant. If the dependence of \(f\) on \(v\) is represented graphically, one obtains the well-known bell-shaped curve (Fig. 2), corresponding to Gauss’s law of errors. But if the calculation is carried out according to Fermi statistics, one obtains:

\[ f=\frac{1}{\dfrac{1}{A}\cdot e^{\frac{\varepsilon}{kT}}+1}. \tag{2} \]

Fig. 2.

If \(A\) is small in comparison with 1, then (2) coincides with (1). But if \(A\) is large in comparison with 1, then \(f\) will be equal to 1 for all velocities lying below a certain boundary. This agrees with the requirement of the Pauli principle: each quantum state is encountered once and only once. The drop to zero occurs only at a certain velocity, which we shall denote by \(\bar v\). In this case \(\bar v\) turns out to be independent of temperature. The mean velocity of the gas atoms \(v_m\) is somewhat less than \(\bar v\), but is proportional to \(\bar v\), namely

\[ v_m=\sqrt{\frac{5}{3}}\,\bar v . \]

Consequently, this mean velocity \(v_m\) is also independent of temperature. In classical statistics, as is known, in contrast to this the mean velocity is proportional to \(\sqrt{T}\).

If the total energy of the gas (Fig. 3), i.e. chiefly the square of the velocity, is represented as a function of \(T\), then the following picture is obtained: in classical statistics this

will be a straight line passing through zero on the temperature scale; in Fermi statistics the curve over some extent must not depend on the temperature and then, gradually, at high temperatures begins to coincide with the straight line of classical statistics. At absolute zero there must exist a certain energy \(E_0\) and pressure \(p_0\). Qualitatively the change of pressure may be represented by the same Fig. 3. States in which the energy does not depend on temperature we call degenerate. The gas equation in this region already ceases to be expressed by the formula \(pv = RT\) and becomes \(p = p_0 = \mathrm{Const}\). The electron gas in a metal, if the number of electrons is put equal to the number of atoms of the metal, is completely degenerate even at temperatures of many thousands of degrees. This was already noted by Pauli, and Einstein concluded from this that the heat capacity of the electron gas must be equal to zero, since the specific heat is nothing other than the change of the total internal energy when the temperature is raised by one degree. This change over the whole region of the degenerate state is equal to zero, at least in the first approximation. Thus one of the chief difficulties of the former electron theory is removed.

Fig. 3.

What is the situation with the Maxwellian distribution of velocities, which supposedly must exist inside the metal and is, as it were, confirmed by observations on thermoelectrons? We see that for the overwhelming majority of electrons the energy distribution is in no way Maxwellian: it does not depend on temperature. Only the extreme branch of the distribution curve behaves according to Maxwell and depends on temperature. This is shown by the dotted curve in Fig. 2. As the temperature is lowered, the decline of the distribution curve according to Fermi is considerably steeper than according to Maxwell. In the Richardson effect, however, we are dealing precisely with the extreme part of the distribution curve,

where the two laws practically coincide. Only the fastest electrons can break out through the “lattice” of the metal surface, as we shall discuss in more detail below. Let \(\xi_0\) (Fig. 4, p. 776) be the velocity which an electron must possess along the \(x\)-axis in order to pass through the surface. From observations one can prove that \(\xi_0\) must be greater than our limiting velocity \(v\); consequently, in Fig. 2, \(\xi_0\) must lie to the right of \(v\), and this means that the few electrons overcoming the surface and indicated in Fig. 2 by the shaded area are in a region having a practically Maxwellian character. Thus another contradiction noted by us above is satisfactorily removed.

Near absolute zero, where in the distribution law a sharp angle is formed at the velocity \(v\), all the electrons of the metal, although not at rest, are bound in their motions to definite quantum centers. All quantum states here are occupied, the metal is overpopulated; there are “homeless dwellers.” This is a consequence of the Pauli principle, which permits each quantum state to be occupied by only one inhabitant. At high temperatures there are likewise not many possibilities for exchanging quantum states; only the richest in energy electrons, the few chosen thousands, can avail themselves of the luxury of the Maxwellian distribution; only they can afford excursions into the most remote regions of the domain of degeneracy. They must, as was said, have a normal component of velocity greater than \(\xi_0\). The energy corresponding to this velocity,

\[ W_a=\frac{m\xi_0^{\,2}}{2}, \]

they lose on leaving the metal; this is a kind of tax for traveling abroad.

What are the forces that hold the majority of electrons in the metal and release only the fastest electrons, i.e. what is the origin of the work of emission \(W_a\)? In an ordinary gas enclosed in a vessel, the elastic forces

forces of the vessel walls hinder the escape of molecules. In an electron gas these forces are electrical and, owing to the magnitude of the elementary charge, especially considerable. The electron gas is enclosed in an electrified lattice. Inside the metal the positive metallic ions are, on the average, neutralized by free negative electrons. When an electron leaves the metallic lattice, it is acted upon by the attraction of the positive ions, which on the outer side are no longer neutralized; and only those electrons whose velocity exceeds $\xi_0$ can escape.

It is possible to create conditions under which electrons will leave the metal in considerable numbers even at ordinary temperature. Millikan and his pupils obtained strong fields, about a million volts per centimeter near the emitting surface; at this, at room temperature, a noticeable stream of electrons was observed, not increasing appreciably when the temperature was raised; in any case, when the temperature was raised the number of electrons did not increase uniformly over the whole surface of the metal, but only in separate small regions. It must be thought that in these places the work function $W_a$ is lowered as a result of impurities, or as a result of geometrical irregularities (the action of points); the field gradient in these places is increased. We obtain the following picture: in an electrified lattice in which the electrons are confined there are small apertures. If the electrons are strongly “sucked out” from outside, then they can get out under the joint action of the external suction and the internal pressure. The pressure practically does not depend on the temperature and is equal to the pressure at absolute zero; likewise, suction does not depend on the temperature. The electron stream also is almost independent (in Millikan’s experiments) of the temperature—this is a “cold electron current.”

This is sufficient for a clear representation of the conditions inside and at the surface of a metal. We must imagine a lattice of positive metal ions, neutralized by negative electrons that have lost their bond with

substances—the ions, and moving in the metal with a certain kinetic energy. At the same time, in their motions they are restricted by the Pauli exclusion principle: two electrons cannot be in one and the same state of motion. Occasionally, in exceptional cases, electrons may fly out of the ionic lattice.

Let us now turn to the fundamental problem of electrical conductivity. In the absence of a field, through any cross-section in the metal there move as many electrons in the forward direction as in the reverse direction; when a field is applied, however, a preferential direction is obtained. The excess gives a current; if the potential drop is divided by the current, one obtains the specific resistance \(\rho\). From quite elementary kinetic considerations Drude derived for it the following expression:

\[ \rho=\frac{mv^{2}}{e^{2}ln}, \tag{3} \]

\(l\) is here the mean free path of the electrons, i.e. the average path between two successive collisions with the ions of the metal, \(n\) is the number of electrons per unit volume, \(v\) is the mean velocity, and \(e\) and \(m\) are the charge and mass of the electron. This formula is correct up to a certain numerical factor. The same formula is also obtained in the new statistics, with the indicated numerical factor equal to 1, if \(\bar v=v\). As we have seen, \(v\) does not depend on the temperature; therefore the resistance should not vary with temperature if the mean free path \(l\) did not vary with temperature. An increase in temperature is manifested by an increase in the disorder in the arrangement of the metal ions, in their deviations from the ideal lattice. How will this increasing disorder affect the mean free path? The investigation of this question on the basis of wave mechanics had already been prepared by Debye’s theory (1914) of the influence of temperature on Laue’s phenomenon. Here before you is a second remarkable example showing the wave nature of the electron—or, better, of the swarm of electrons; it is almost as instructive for us as the forego-

...the diffraction photographs given above. The average velocity \(v\) that interests us is of the order of magnitude (depending on the number of free electrons) of about \(1000\ \text{km/sec}\); the corresponding de Broglie wave, according to wave mechanics, has a length of about \(5\ \text{\AA}\). It is therefore analogous to very soft X-rays obtained in a tube with a voltage of \(4\ V\). This wavelength is greater than the distance between atoms, for example in a copper crystal. Hence it follows that in this case true Laue interference spots cannot be obtained. There will remain only general scattering, well known from X-ray photographs in the form of a continuous background; this scattered radiation increases with rising temperature, i.e. with increasing irregularities of the lattice. Scattering means the deflection of X-rays from their initial direction, or, if we pass to electrons, the deflection of electron waves, i.e. the collision of electrons with the atoms of the metal.

Fig. 4.

In the present state of the theory we cannot analyze an individual act of impact, but wave mechanics gives us a reliable method for the statistical analysis of the process; it is possible to calculate what percentage of the number of electrons encountering an ion of the metal is deflected in a given direction. The method consists in replacing the electron by an electronic wave and treating the atom in accordance with Schrödinger’s wave mechanics. In this way we arrive at a rational statistical formula for the mean free path, or for the frequency of collisions causing the deflection of electrons from their original direction. According to Houston or according to Debye this formula is written as follows:

\[ \frac{1}{l}\sim \frac{1}{a^{2}}\int_{0}^{x}\frac{\xi\cdot d\xi}{e^{\xi}-1}, \qquad x=\frac{\theta}{T}. \tag{4} \]

The formula is illustrated by Fig. 5. Here \(\theta\) is the characteristic temperature, equal to \(\dfrac{h\nu}{k}\) (\(\nu\) is the number of natural oscillations of an atom in the lattice). Since the resistance \(\rho\), according to formula (3), is proportional to \(\dfrac{1}{l}\), at large \(T\) the resistance will be proportional to the temperature; at small \(T\) the resistance must fall rapidly, which agrees with the observations of Grüneisen. Thus for the first time there appeared a rational theory of electrical resistance! On our curve for the resistance, however, the abrupt fall of the resistance characteristic of superconductivity is not visible. On the other hand, the wave theory explains well the dependence of resistance on pressure, according to Bridgman’s observations, and also makes it possible to calculate absolute values of the resistance; it even explains the anisotropy of resistance in the hexagonal single crystals zinc and cadmium, respectively, by the difference of diffraction phenomena from the symmetry of the crystal lattice. Extremely interesting is the question of the number of free electrons per atom of the metal. According to Houston, for polyvalent atoms (Pb, Fe, Pt) this number is apparently greater than 1.

Fig. 5.

Similarly to electrical resistance, the thermal resistance of a metal (i.e. the quantity inverse to thermal conductivity) depends on the mean free path. If one forms the ratio of these resistances, then the mean free path cancels, and it remains proportional to \(T\) with a universal coefficient, just as required by the Wiedemann–Franz law. The numerical factor in this law according to the new statistics is equal to \(\dfrac{\pi^2}{3}\), in contrast to Drude’s factor, equal to 3. The first factor agrees with observations still

better than in Drude’s theory, and thus one more shortcoming of the classical theory of electrons is removed.

We finally turn to the much-discussed Volta effect, i.e. the potential difference that arises between two pure, dry metals. Chemists in particular have never fully believed in this effect and have sought to explain it by chemical processes in liquid films on the surface of the metals. In school we were taught that Volta’s series of voltages is characteristic of the arrangement of the metals themselves, with alkalis and zinc at the positive end of the series, and the noble metals and carbon at the negative end. It is known that in the voltaic pile, discovered in 1800, not the pure Volta effect is manifested, since here moist, acidified spacers act electrochemically. However, there are old and new experiments that fully confirm the existence of the pure Volta effect. In particular, Millikan showed that the Volta potential difference is determined by the difference of the constants governing the emission of electrons from the metal in the Richardson effect. The explanation of the Volta effect given by electron statistics is in complete agreement with this. Namely:

\[ e V_{12}= C_2 - C_1, \tag{5} \]

where the constant of the Richardson effect has the following value:

\[ C = W_a - W_i . \tag{6} \]

We have already spoken in detail above about the “external work of escape” \(W_a\). It must be diminished by a certain “internal work of escape” \(W_i\), which corresponds to the tendency of electrons to escape outward owing to the high internal pressure (the pressure at absolute zero). The Volta contact potential difference is not the potential difference of the inner parts of the metal; the latter is rather determined by the difference of the two external works of escape \(W_a\) with the opposite sign. The Volta difference is equal to the potential of the electric field formed between the two metallic surfaces when their contact is imperfect—

…phenomena (counting from the outer side of one surface to the outer side of the other), as Eckart justified in connection with my own work. The name “electricity on contact” in this sense may be somewhat misleading. In complete contact, which we may call the “place of soldering,” the external work function is excluded, as is proved by the facts of thermoelectricity.

The chemist, in most cases, is not directly interested in the Volta effect, since the electromotive force of a galvanic cell can be calculated from other data. Recently, however, M. Corbino has convincingly shown that here, too, the Volta effect plays a role, and in the Daniell cell even the predominant role, since the heat of solution of zinc and the heat of condensation of copper almost exactly compensate one another.

I have come to the end: among all substances with which the chemist has to deal, metals occupy a special, interesting place. Characteristic of metals, as before and now, we consider to be the presence of free electrons. According to calculations recently published by Herzfeld, one can well understand why, when metallic atoms are brought closer together, their outer electrons enter unstable states and are torn away from the atom. In this way one can directly calculate which atoms in the solid state must exhibit a metallic character. To understand the behavior of free electrons inside the metal and at the surface, we must resort to the methods of quantum theory in its newest wave interpretation. These methods describe statistically the behavior of electrons as fully and simply as the wave theory of light describes the phenomena of optics and of X-rays. As I have heard, the Society for the Advancement of German Science (Notgemeinschaft) intends, with its characteristic breadth and energy, to revive the study of the metallic state. On the basis of what has been set forth, it would seem that the time chosen for carrying out this plan is the right one, and that in this hitherto dark field much will soon become clearer.

Our new atomic physics has everywhere been based on the achievements of chemistry. That chemists, too, are interested in the latest results of physics is eloquently attested by your kind invitation to give the present lecture.

Some References

1. E. Fermi. Z. f. Phys., 36, 902, 1926.
2. G. P. Thomson. Proc. Roy. Soc., 117, 600, 1928.
3. C. Davisson and G. H. Germer. Nature, 119, 538, 1927; Phys. Rev., 30, 705, 1927.
4. W. Pauli jr. Z. f. Phys., 41, 81, 1927.
5. A. Einstein. Sitzb. Preuss. Ak. XXII, 261, 1924; I, 3, III, 18, 1925.
6. A. Sommerfeld. Naturwissenschaften, 15, 825, 1927; Z. f. Phys., 47, 1, 1928.
7. R. A. Millican and G. Eyring. Phys. Rev., 27, 51, 1926.
8. R. A. Millican and Lauristen. Proc. Nat. Ac. Science, 14, 45, 1928.
9. W. V. Houston. Z. f. Phys., 47, 33, 1928.
10. R. H. Fowler. Proc. Roy. Soc., 117, 549, 1928.
11. R. H. Fowler and L. Nordheim. Proc. Roy. Soc., 119, 173, 1928.
12. W. V. Houston. Z. f. Phys., 48, 449, 1928.
13. P. Debye. Verh. D. Phys. Ges., 15, 678, 1913; Ann. d. Phys., 43, 49, 1914.
14. J. Frenkel. Z. f. Phys., 47, 819, 1928.
15. K. Eckart. Z. f. Phys. I, 47, 38, 1928.
16. M. Corbino. Phil. Mag., 4, 436, 1927.
17. K. F. Herzfeld. Phys. Rev., 29, 701, 1927.

1. Report delivered at the German Chemical Society on April 28, 1928, and printed in Berichte d. Deutschen Chemischen Gesellschaft, 61, 1171, 1928, Part B. ↩