ELECTRICAL CONDUCTIVITY OF METALS¹

John Bardeen, Minneapolis

I. INTRODUCTION

The classical theory of the electrical conductivity of metals was developed chiefly by Drude and Lorentz. Despite a number of successes, especially in connection with the explanation of the Wiedemann–Franz law, this theory led to substantial difficulties. These difficulties were almost completely overcome by the application of quantum mechanics, as was done by Sommerfeld, Houston, Bloch, Mott, and others. Before considering modern theories, it is useful to give a survey of the most important experimental data that must be explained by an adequate theory of conductivity, and also to give a brief survey of the old theories of conductivity, since they contain a number of correct statements. Indeed, the basic concepts have not changed greatly, although the formal aspect of the theory has taken on an entirely different form.

The basic idea of the theory of Drude and Lorentz, as of all the other theories, is that the carriers of current are electrons which have been torn away from the atoms forming the metal and can move more or less freely in it.

The most direct experimental proof of this fact was obtained only several years after the appearance of the theory. The experiments of Stewart and Tolman¹ and others showed for the first time that the carriers of current are particles of negative charge whose mass coincides with the mass of the electron. The experiment consisted in the following: a solenoid, rotating at high speed, was connected to a sensitive ballistic galvanometer. When the solenoid was suddenly stopped, the galvanometer registered a current impulse (since the kinetic energy acquired by the electrons during the rotation of the coil is dissipated owing to the resistance of the latter). Knowing the speed of rotation of the coil, the total charge carried by the current impulse, the resistance, and the dimensions of the solenoid, it was possible to find the ratio of charge to mass for the particles carrying the current. Stewart and Tolman found that the ratio

\[ \frac{e}{m} \]

has the following values:

for Cu	\(1.60 \cdot 10^7\)	CGSM
Ag	\(1.485 \cdot 10^7\)
Al	\(1.54 \cdot 10^7\)

¹ J. Appl. Physics, 11, 88, 1940, translated by S. V. Vonsovsky.

These values are rather close to the value of \(\frac{e}{m}\) for free electrons\(^{2}\): \(1.77\cdot 10^{7}\) CGSM. Later experiments were made with the aid of oscillating cylinders and led to the same results. These experiments indicate that the current is carried by electrons moving in the crystal lattice.

II. SUMMARY OF EXPERIMENTAL DATA

A. Conductors and insulators

Among the most important facts, the theory of conductivity must explain the sharp difference between the conductivity of metals and insulators. The resistance of metals is of the order of \(10^{-6}\) ohm·cm, while that of insulators is of the order of \(10^{12}\) ohm·cm. The classical theory gives no satisfactory explanation of this fact.

B. Temperature dependence

At high temperatures the resistance of pure metals is approximately proportional to the absolute temperature. It falls rapidly at lower temperatures and, at very low temperatures, is proportional to \(T^{5}\). On the other hand, the resistance of insulators increases as the temperature is decreased. This constitutes the most characteristic distinction between metals and nonmetals.

Fig. 1. Temperature variation of resistance for various metals. The curve represents the Bloch–Grüneisen function [equation (28)]. The numerical data are taken from Meissner (see bibliography).

Grüneisen\(^{3}\) showed that the resistance of most metals can be represented by a universal function of temperature:

\[ R = R_{\theta} f\left(\frac{T}{\theta}\right). \tag{1} \]

Here \(\theta\) is a characteristic temperature of the individual metal, which in general is rather close to the Debye characteristic temperature for heat capacities. The experimental points for a number of metals are shown in Fig. 1.

On this basis one may suppose that there is a close connection between the conductivity of pure metals and the thermal motion of the atoms, which is responsible for the heat capacity. V. Wien suggested that the resistance is proportional to the mean square of the amplitude of oscilla-

tion of ions. In this case the resistance must be proportional to the absolute temperature \(T\) at high temperatures and to \(T^4\) at very low ones. The theoretical curve in Fig. 1, used by Grüneisen, is based on the approximate theoretical expression given by Bloch. It will be considered below. According to Bloch, at very low temperatures the resistance should be proportional not to \(T^4\), but to \(T^5\), which agrees better with experiment. Both theory and experiment indicate that one of the most important factors determining the resistance of pure metals is the mean square amplitude of vibration of the atoms of the crystal.

C. Dependence on Pressure

The resistance of most metals decreases with increasing pressure. In Fig. 2 the resistance is shown as a function of pressure for several soft metals in the pressure range up to 30,000 atm. These curves were recently obtained by Bridgman.^4 The resistance of some metals (Li, Sr, Ca) increases with increasing pressure; other metals (Rb, Cs, Ba) at first decrease their resistance as pressure rises, and at comparatively small pressures their resistance reaches a minimum, after which it begins to increase. The curves for most anomalous metals are shown in Fig. 2; normal behavior is the continuous decrease of resistance with pressure. Grüneisen explains this normal decrease by the decrease in the thermal motion of atoms, since as pressure increases the atoms are more strongly “bound” to their equilibrium positions.

Fig. 2. Change of resistance with pressure for various metals. Data taken from Bridgman.^4

D. Matthiessen’s Rule

Every real metal contains some amount of impurities and therefore, when the temperature, decreasing, tends to absolute zero, the resistance tends not to zero, but to a certain constant value, which depends on the amount of impurities. The purer the metal, the smaller the residual resistance. Matthiessen pointed out that the resistance of a metal is the sum of two terms

\[ R = R_0 + R_T. \tag{2} \]

\(R_0\) is the constant term (not dependent on temperature), proportional to the amount of impurities; \(R_T\) is the term dependent on tempera-

tures characteristic of pure metals; it tends to zero when the temperature decreases to \(0^\circ\mathrm{K}\). This division of the resistance into two parts is called Matthiessen’s rule. It is in approximate agreement with experiments on the increase in the resistance of metals caused by the addition of small concentrations of a second metal in a solid solution. Scattering of electrons by atoms of a foreign metal embedded in the lattice of the base metal leads to an additional resistance. In the case of small concentrations, the resistances caused by impurities and by thermal motion are additive.

In determining the resistance of a pure metal, a correction is usually made for the residual resistance.

E. Resistance of Alloys

If an alloy is a mixture of microcrystals of separate pure metals, then the resistance is approximately equal to the mean value of the resistances of the components.

Fig. 3. Specific resistance of heterogeneous mixtures. Data taken from works cited by Meissner

In Fig. 3 curves are given for the resistance of Pb—Sn, Pb—Cd, Zn—Cd, and Zn—Sn as functions of concentration. All these metals are mutually insoluble in one another.

Fig. 4. Specific resistance of Ag—Au alloys. The curve for \(0^\circ\mathrm{K}\) was extrapolated from the results of Clay, cited by Meissner

On the contrary, if we have a solid solution, then its resistance is much greater than that of each of the components. Data for the Ag—Au alloy, which forms a solid solution at all concentrations, are given in Fig. 4. The resistance has a maximum at approximately 50 percent concentration. In this alloy there is no superstructure; the Ag and Au atoms are distributed randomly over the lattice sites.

Let us now consider what happens when an ordered structure is formed. In Fig. 5 data are given for the Cu—Au system, which has been studied in detail by many investigators. A quenched alloy, in which there is no superstructure, has a resistance curve of the same type as in the case of the Ag—Au system. If the alloy

annealed, then a superstructure is formed near the compositions 25% Cu, 75% Au and 50% Cu, 50% Au. The Cu and Au atoms are arranged more or less regularly at the lattice sites. From the graph it is clear that the resistance then drops sharply and, in magnitude, is close to the resistance of a pure metal. We shall consider this system in more detail later. For the moment we wish only to emphasize the fact that ordering of the atoms reduces the resistance.

The facts listed above indicate that the resistance of metals is due to irregularities in the lattice. These irregularities may be caused by: 1) the thermal motion of atoms; 2) impurities, and in the case of alloys—by a disordered solid solution. Another cause may be the disorder existing in a liquid or an amorphous solid.

Fig. 5. Specific resistance of Cu—Au alloys

a—quenched from 650° C, b—annealed at 200° C. Data of Johansson and Linde (Ann. Physik, 25, 1, 1936)

But there are a number of facts which the theory must explain. It must give absolute values of the conductivities of various pure metals and their dependence on temperature and pressure. For example, it must answer the question: why does copper have a low resistance, while iron has a comparatively high one?

III. OLD THEORIES

The old theories of Drude and Lorentz[^5] did not attempt to explain these facts in all their detail. Rather, they tried to explain the mechanism of conduction. It is very useful to consider Drude’s elementary theory, thanks to its simplicity and also because its basic assumptions were preserved in later theories. The fundamental idea of this theory is that current is carried by electrons, which can move more or less freely, while, however, undergoing collisions with the crystal lattice. Let us denote the mean time between two successive collisions by \(2\tau\) (\(\tau\) is the relaxation time). Suppose that the mean momentum of an electron disappears at each collision. Further, for the sake of simplicity, suppose that, despite the strong interaction between electrons, they move freely between collisions. Then the equation of motion has the form

\[ m \frac{dv_x}{dt} = - eF, \tag{3} \]

where \(-e\) and \(m\) are the charge and mass of the electron, and \(F\) is the field strength acting along the direction \(x\). The mean additional velocity in the direction of the field is therefore equal to

\[ v_d=\frac{1}{2}\cdot\left(\frac{-eF}{m}\right)\cdot 2\tau=-\frac{eF\tau}{m}. \tag{4} \]

If there are \(N\) electrons in a unit volume, then the current is

\[ J=-Nev_d \]

and the conductivity

\[ \sigma=\frac{J}{F}=\frac{Ne^2\tau}{m}. \tag{5} \]

Drude made the permissible simplifying assumption that all electrons move with the same mean velocity \(U\), and that this velocity can be obtained from the law of equipartition of energy over the degrees of freedom:

\[ \frac{1}{2}mU^2=\frac{3}{2}kT, \tag{6} \]

where \(k\) is Boltzmann’s constant. The mean free path \(l\) is determined by

\[ l=2\tau U. \tag{7} \]

Therefore

\[ \sigma=\frac{Ne^2l}{2mU}. \tag{8} \]

This formula gives an approximately correct order of magnitude for the conductivity at room temperatures if \(N\), in order of magnitude, is equal to the number of atoms in a unit volume, and \(l\) to the interatomic distance. However, in order to obtain the temperature dependence, it is necessary to admit that \(l\) increases very rapidly as the temperature decreases. Such specific behavior is very difficult to explain. Moreover, one might expect that, as the pressure increases, the mean distance between atoms decreases, \(l\) decreases, and consequently the conductivity should fall. As we have seen, this contradicts the behavior of normal metals. However, the greatest difficulty arose in connection with the question of heat capacity. According to the classical theory, an amount

\[ \frac{3R}{2}z \]

(\(z\) is the number of electrons per atom) should be added to the molecular heat capacity, this being the heat capacity of the electron gas. But the heat capacities of a metal are determined by the thermal motion of the atoms. Any addition from the electrons must be much smaller than \(R\). As is probably well known to most readers, this difficulty was removed by Sommerfeld, who applied Fermi–Dirac statistics to the electrons.

More exact calculations by Lorentz, based on the same physical assumptions, emphasize still more sharply the difficulties of Drude’s theory.

A great success of the Drude–Lorentz theory is the explanation of the Wiedemann–Franz law, according to which the ratio of the thermal condu...

proportionality to the electrical conductivity for metals is proportional to the absolute temperature. On the basis of the very same assumptions that were used in the theory of electrical conductivity, Drude found that the thermal conductivity is equal to

\[ K = \frac{1}{2} NkUl, \tag{9} \]

and therefore

\[ \frac{K}{\sigma} = 3 \frac{k^2}{e^2} T. \tag{10} \]

The constant of proportionality contains only universal physical constants and agrees approximately with experimental values.

In the classical theories there was no possibility of calculating the absolute value of the conductivity of metals, or of explaining the differences in the conductivities of different metals.

IV. HALL EFFECT

One of the most important phenomena in the historical development of the theory of metals is the Hall effect. This effect consists in the action exerted by a magnetic field directed perpendicular to a plate through which a current flows. In this case a potential difference arises across the plate in a direction perpendicular to the current. The crude explanation is that the trajectories of the electrons are bent under the action of the electric field. In order that there be no component of the current in the transverse direction, a potential gradient must arise. The sign of the gradient depends on the sign of the charge carrying the current. For most metals this sign corresponds to electrons (negative charge), but some metals possess an anomalous sign (Zn, Cd). This fact led to the creation of the dualistic theory of conductivity, developed chiefly by Hall[^6]. He assumed that the current is carried not only by “free” electrons, but also by “bound” ones. An atom in a metal may be ionized, which leads to the creation of a “free” electron. The ion remains positively charged. A bound electron from a neighboring atom may pass to this ion. In this way the position of the positively charged ion may move in the metal (Fig. 6). The current created by such a displacement of charge leads to the anomalous Hall effect. Later we shall see that modern theory, to a certain extent, confirms this point of view.

Fig. 6. Illustration of the displacement of the charge of a bound electron according to Hall.

V. THE SOMMERFELD–BLOCH THEORY

The remaining part of the article will be devoted mainly to modern theories of electrical conductivity and their applications to particular problems. We shall dwell very briefly on the basic concepts of the general quantum theory of metals that are necessary for the consideration of questions of conductivity. The reader may find further details in a series of papers by Seitz and Johnson^7 or in the monographs cited at the end of the article.

According to modern views, the valence electrons (i.e., electrons outside closed shells) detach themselves from the atoms forming the metal and move freely in the crystal lattice. These conduction electrons are responsible not only for the metallic bond but also play the principal role in various electrical and magnetic properties. In our discussion we shall restrict ourselves only to these electrons. The ionic cores will usually be replaced by an effective potential field in which the conduction electrons move. This picture does not differ too greatly from the ideas of Drude and Lorentz. What, then, accounts for the very considerable success in applying quantum, rather than classical, mechanics to the theory of metals? Two fundamental propositions of quantum mechanics are most important for the problem under consideration: 1) the wave properties of electrons and 2) the Pauli principle.

A. Electron waves

In the direction of motion of each electron there propagates a wave associated with it. For free electrons (i.e., electrons not subject to the action of external forces) the wavelength \(\lambda\) is given by the de Broglie formula

\[ \lambda = \frac{h}{mv}, \tag{11} \]

where \(h\) is Planck’s constant, \(m\) is the mass of the electron, and \(v\) is its velocity. This relation was verified experimentally by Davisson and Germer, Thomson, and others. In the case of electrons moving in the periodic potential field of the crystal lattice, the relation between wavelength and velocity is no longer given by (11), but is described by a more complicated equation, which we shall give below [see (17)]. Instead of using wavelength or velocity to determine the state of an electron, it is more convenient to use the wave vector \(\mathbf{k}\), directed along the motion and of magnitude equal to \(\frac{2\pi}{\lambda}\). The energy of an electron \(E(k)\) is a function of the vector \(\mathbf{k}\). For free electrons

\[ E = \frac{1}{2}mv^2 = \frac{h^2 k^2}{8\pi^2 m}. \tag{12} \]

As is easy to see, this equality is a consequence of (11). The general expression for the energy of an electron moving in the periodic field of the lattice is more complicated.

The Pauli principle, as applied to the present problem, states that only two electrons can be in one and the same state, or, in other words, have the same wavelength and direction of motion. The factor 2 appears because of the two possible orientations of the electron spin. In an infinite crystal the possible wavelengths are distributed continuously, but in any finite crystal they form a discrete spectrum of values. As long as the wavelength is small in comparison with the dimensions of the crystal, the number of states between two wavelengths \(\lambda\) and \(\lambda+\Delta \lambda\) is proportional to the volume of the crystal and does not depend on its shape. Therefore the number of electrons per unit volume that can have wavelengths in the interval \(\lambda, \lambda+\Delta \lambda\) does not depend on the size or shape of the crystal, provided, of course, that the properties of the crystal themselves do not depend on size and shape.

B. Heat Capacity of the Electrons

One of the consequences of the Pauli principle is the fact that even at absolute zero temperature the energies of the various conduction electrons are distributed over a broad band, the width of which may reach several electron-volts.

Fig. 7. Distribution of the Fermi function, giving the probability that a given electronic state is occupied at a given temperature

Figure 7 shows a graph for the mean number of electrons in various states at \(T = 0^\circ\mathrm{K}\), and also at a certain higher temperature \(T_1\). At the temperature \(T = 0^\circ\mathrm{K}\) all states whose energy is below a certain maximum value \(E_{\max}\) are occupied; all states with higher energy are free. At higher temperatures a small number of electrons from states close to the top of the filled band are excited and pass into states with higher energies. The distribution thereby obtained has the form shown in Fig. 7. The fraction of excited electrons is, in order of magnitude, equal to the ratio

\[ \frac{kT}{E_{\max}}, \]

and the mean excitation energy of these electrons relative to \(E_{\max}\) is of the order of \(kT\). Therefore the mean thermal energy per electron is of the order

\[ \frac{(kT)^2}{E_{\max}}. \]

If there is one free electron per atom, then the heat capacity is, in order of magnitude, equal to

\[ \frac{kT}{E_{\max}} R, \]

where \(R\) is the gas constant. Since at room temperatures for most metals

\[ \frac{kT}{E_{\max}} \]

is less than \(0.01\), the heat capacity of the electrons may be neglected in comparison with the heat capacity due to the thermal vibrations of the lattice (\(\sim 3R\)). Only at very low (liquid He) temperatures can the electronic

heat capacity. Sommerfeld was the first to show that the Pauli principle and Fermi–Dirac statistics remove the difficulty with the heat capacity that had existed in the theories of Drude and Lorentz.

C. Sommerfeld’s Theory of Conductivity

In Sommerfeld’s theory, as in Drude’s theory, it is assumed that each electron moves freely between collisions, during which it is accelerated by the external field. The field acting on a given electron from the ions and the other electrons is neglected. Therefore Drude’s formula for the conductivity (8) may be applied. The essential difference lies in the fact that the mean free path

\[ l = 2\tau U \tag{13} \]

will be much larger, since \(U\)—the mean velocity of the electron—is much greater than that obtained from the theorem on the uniform distribution of energy among degrees of freedom (the kinetic energy is equal to \(\frac{3}{2}kT\)).

If one assumes that there is one free electron per atom, then for the mean free path at room temperatures one obtains an order of magnitude of 100 interatomic distances, instead of one, as followed from Drude’s theory.

The large mean free path is due to the wave properties of electrons. If it is difficult to imagine how a particle can move in a crystal lattice without collisions, the propagation of a wave without special obstacles raises no doubts. The situation here is the same as in the case of the propagation of light waves, or, rather, X-rays in a crystal. Individual waves are scattered by each atom, after which they interfere and continuously form a wave front. An estimate of the mean free path on the basis of this picture was first given by Houston.

D. Bloch’s Theory

The concept of electrons moving in an electrostatic field having the period of the crystal lattice was introduced by Bloch. Just as in the case of completely free electrons, to each electron one may assign a wave with a definite direction of propagation and wavelength. In this case, however, the wave will no longer be plane, but modulated with the period of the lattice. The wave function has the form

\[ \psi_k(x,y,z) = U_k(x,y,z)\ \exp[i(k_xx + k_yy + k_zz)]. \tag{14} \]

The vector \(\mathbf{k}\) with components \((k_x, k_y, k_z)\) is the propagation vector defined above. The second factor \(\exp[i(k_xx + k_yy + k_zz)]\) is a plane wave, while the modulating factor

\(U_k(x, y, z)\) is a periodic function with the period of the lattice. The energy \(E(\mathbf{k})\) of an electron in the state \(\mathbf{k}\) depends on the form of the field in which the electron moves.

E. Electron velocity

In order to obtain the velocity of an electron in the state \(\mathbf{k}\), one must use the expression for the group velocity of waves. In the case of waves propagating in one dimension, this expression is equal to

\[ v=\frac{d\nu}{d\frac{1}{\lambda}} . \tag{15} \]

In our case the frequency \(\nu\) is equal to \(\frac{E}{h}\), \(k=\frac{2\pi}{\lambda}\), therefore

\[ v=\frac{2\pi}{h}\frac{dE}{dk}. \tag{16} \]

In the general case of three-dimensional motion, the expression, for example, for the \(x\)-th component of the velocity has the form

\[ v_x=\frac{2\pi}{h}\frac{\partial E}{\partial k_x}. \tag{17} \]

The current produced by the electron is then equal to

\[ j_k=-ev_x=-\frac{2\pi e}{h}\frac{\partial E}{\partial k_x}. \tag{18} \]

The total current is obtained by summing (18) for the electrons over all occupied states. If more electrons move along some direction than along the opposite one, then we have a resultant component of current along this direction. Since the states we have considered are stationary, this current will not decrease with time. Therefore an ideal crystalline lattice has no resistance. Resistance is caused by irregularities produced by thermal motion or by the presence of foreign atoms, which disturb the periodicity of the lattice. This is precisely what experiment requires.

F. Energy states

It is convenient to represent each electronic state by a point in three-dimensional \(k\)-space. The coordinates of the point are the components \((k_x, k_y, k_z)\) of the wave vector \(\mathbf{k}\). This \(k\)-space coincides exactly with the reciprocal lattice, which is so often used in the theory of X-ray diffraction. The points representing possible states are uniformly distributed in \(k\)-space. The number of states in the element \(dk_x, dk_y, dk_z\) is equal to \(\frac{V}{8\pi^3}\,dk_x\,dk_y\,dk_z\), where \(V\) is the volume of the crystal.

The energy of an electron is not, in general, a continuous function of the wave vector. According to the theory of the motion of an electron in a periodic potential field, all of \(k\)-space is divided into regions or zones, in each of which the energy is a continuous function of \(k\). On the surfaces between two zones the energy has a discontinuity. Each zone corresponds to a band of possible energies. The zones may be separated from one another by a region of forbidden energies, or may overlap in such a way that the lowest level of the upper band may be lower than the highest level of the lower one.

Zones, or energy bands, may be compared with the states of electrons in a free atom. One may imagine that the lattice constant of the crystal increases continuously, so that, in the end, the atoms are so far removed from one another that the interaction between them becomes negligible. The energy bands then become narrower and narrower and, in the end, turn into the discrete levels of the free atom. It is therefore possible to speak of the \(s\)-band of a monovalent metal, which is produced by the \(s\)-level of the valence electron in the free atom. The \(d\)-levels of the transition elements are split into several \(d\)-bands of the metal. In a free atom there may be two \(s\)-electrons. Therefore the total number of states in an \(s\)-band corresponds to two electrons per atom. In just the same way the total number of states in the \(d\)-bands corresponds to ten electrons per atom.

G. Metals and Insulators

If every allowed state in a band is occupied by an electron, then the total current is zero, since to every electron moving in one direction there may be matched another moving in the opposite direction with the very same velocity. If the band is only partially filled, then the number of electrons moving in one direction may be greater than the number of those moving in the opposite direction, and this gives a resultant current. The first case is characteristic of insulators, the second of metals. In an insulator the bands containing electrons are completely occupied, and there is a region of forbidden energies separating the upper unoccupied bands from the lower filled ones. An insulator must have an even number of valence electrons per unit cell. A metal contains bands in which only part of the states are occupied. Divalent metals (Be, Ca, etc.) have just enough electrons to fill the lowest (\(s\)) band, i.e. two electrons per atom. But since they are good conductors, it is necessary to assume that a higher band overlaps with the lower one, so that the electrons fall into two different bands, each of which is only partially occupied.

H. Acceleration of Electrons

The configuration of electrons in a metal may be described by specifying the distribution over occupied states in \(k\)-space. If an electric field acts on the metal, then the distribution is no longer

more symmetric with respect to the origin and is displaced in the direction of the field, giving a resultant current. The new distribution is the result of statistical equilibrium between transitions of electrons from some states to others, caused by the accelerating action of the field and by scattering of electrons on irregularities in the lattice, which lead to the appearance of resistance.

The classical expression for the acceleration of an electron in an electric field \(F\) has the form

\[ m\frac{dv}{dt}=-eF. \tag{19} \]

If we use the relation \(mv=\dfrac{hk}{2\pi}\), which is valid for free electrons, then the formula for the rate of change of the vector \(\mathbf{k}\) with time will take the form

\[ \frac{dk}{dt}=-\,\frac{2\pi eF}{h}. \tag{20} \]

What effect will the periodic field of the crystal lattice have on the acceleration? It is not difficult to show that (20) remains valid even in the case when formula (19) can no longer be used and when the velocity is no longer proportional to \(k\).

The energy of an electron as a function of \(k\) along the direction of the applied field is schematically shown in Fig. 8, \(a\). The wave vector \(\mathbf{k}\), according to (20), increases linearly with time, so that the point representing the state of the electron will move to the right with constant speed. We have seen [see (18)] that the velocity of the electron itself is proportional to the derivative \(\dfrac{dE}{dk}\). The velocity of an electron whose state is represented by the point \(A_1\), where the curvature is positive, increases with time, since \(\dfrac{dE}{dk}\) increases with increasing \(k\). On the other hand, the velocity of an electron whose state is represented by \(A_2\) decreases with time, since in this region of the curve \(\dfrac{dE}{dk}\) decreases. In this case the electron behaves like a particle with negative mass. When the state reaches the point \(A_3\), the velocity is zero. At this point there is a very small probability that the electron will pass to the next, higher band. Normally the electron appears again at \(A_3'\) (this point represents the same state as \(A_3\)) and again repeats its path along the lower band.

Fig. 8
\(a\)—motion of an electron in a band under the influence of an applied electric field, \(b\)—motion of a positive hole in a band.

The electron at \(A_3\) undergoes Bragg reflection and begins to move in the opposite direction.

Let us now suppose that all states are occupied except one, which is represented by the point \(A_1\) in Fig. 8, \(b\). If all states were occupied, the resulting current would be zero; therefore the total current (in Fig. 8, \(b\)) is equal to the negative of the current that would be carried by an electron in the state \(A_1\). And this current is equal to the current carried by an electron with a velocity equal in magnitude, but opposite in sign, to that of the electron in the state \(A_1\), i.e. to the current of an electron in the state \(A'_1\). Under the influence of the electric field \(F\), the whole distribution of electrons, and hence also the position of the unoccupied states, or “holes,” will move to the right with constant velocity. Accordingly, the point \(A'_1\) will move to the left in the direction in which a particle with positive charge would move. If, as shown in the figure, the point \(A'_1\) is in a region of negative curvature, the velocity and the current will increase with time. Therefore “holes” in unfilled bands behave like positively charged particles with positive mass in a region of negative curvature, and like particles with positive charge and negative mass in a region of positive curvature near the bottom of the band. We shall see (Section IV) that this circumstance can be used to explain the anomalous sign of the Hall coefficient.

Fig. 9. Change in the distribution of electrons as a result of an applied field

In a normal metal all states with energies below a certain maximum value \(E_{\max}\) are occupied, and states with energies higher than \(E_{\max}\) are free. (For the moment we neglect the small smearing of Fig. 7 caused by thermal motion.) All these electrons will be accelerated in the opposite field, so that the whole distribution in \(k\)-space will move in the direction of the field. At first the electron density will change only in those states which lie near the surface of the Fermi distribution, i.e. have energies close to \(E_{\max}\). Fig. 9 shows the density of occupied states in \(k\)-space of a normal metal before and after the application for a short time of an electric field.

Let \(J\) be the current equal to the sum of the currents of the individual electrons \(j_k\) [see (18)] over all occupied states. We can define the effective number of free electrons per unit volume by means of the equality\(^8\)

\[ \frac{dJ}{dt} = - \frac{e^2 F N_{\mathrm{eff}}}{m}. \tag{21} \]

In the case where the lattice field disappears, i.e. for free electrons, \(N_{\mathrm{eff}}\) is equal to the actual number of electrons \(N\) per unit volume. In general \(N_{\mathrm{eff}}\) may be either greater or less than \(N\).

The expression for \(N_{\mathrm{eff}}\) is especially simple if the constant-energy surfaces are spheres, so that \(E\) depends only on the magnitude, but not on the direction, of \(\mathbf{k}\). In this case

\[ N_{\mathrm{eff}}=\frac{4\pi^{2}mN}{h^{2}}\left(\frac{1}{k}\frac{dE}{dk}\right)_{E=E_{\max}} . \tag{22} \]

Therefore \(N_{\mathrm{eff}}\) is large for a broad band and small for a narrow one. This is a measure of the relative ease with which an electron can pass in the crystal from one atom to another. If the band is completely occupied by electrons, then \(N_{\mathrm{eff}}=0\), since \(\frac{dE}{dk}\) vanishes at the surface of the band.

I. Conductivity

In a metal with finite resistance, the distribution in space will be displaced in the direction of the applied field until equilibrium is reached, in which the effect of the field is exactly compensated by collision processes. The equilibrium distribution will be exactly the same as would have been obtained if the electrons had been accelerated for a time \(t=\tau\) in the absence of resistance. \(\tau\) is called the relaxation time (see Section III). For fields of ordinary magnitude, the equilibrium distribution will differ only slightly from the normal distribution in zero field.

The conductivity can be expressed by the Drude–Lorentz formula (8), only instead of \(N\) one must substitute \(N_{\mathrm{eff}}\), determined by (21),

\[ \sigma=\frac{N_{\mathrm{eff}}e^{2}}{m}\tau . \tag{23} \]

The factors determining \(\tau\) will be considered below; first we shall give an explanation of the anomalous sign of the Hall coefficient observed in some metals.

J. Hall Coefficient

If the conduction band is only partially filled, so that the states near the edge of the Fermi distribution lie in the region where the curvature \(\left(\frac{d^{2}E}{dk^{2}}\right)\) is positive, then the electrons behave normally, i.e., as particles with negative charge and positive mass. The sign of the Hall coefficient will be that required for electrons. Suppose now that the band is almost filled, so that the electrons near the edge of the Fermi distribution (which are important for conductivity) are in a region with negative curvature. It is therefore more convenient to regard the current as being produced by unoccupied states, or “holes,” near the edge of the band (see Section V, H). They behave as particles with positive charge and positive mass, and lead to the appearance of a Hall coefficient of anomalous sign. In many metals the electrons occupy two unfilled bands.

In one of them the current is produced by holes, and in the other by “ordinary” electrons. The sign will depend on which of these currents predominates. Conduction by means of “holes” is the quantum-mechanical analogue of the bound electrons considered by Hall (Section IV).

The magnitude of the Hall coefficient can be calculated; the order of magnitude obtained is in agreement with experiment. Numerical agreement has been obtained for the alkali metals.

VI. RESISTANCE OF PURE METALS

A. High temperatures

The resistance of pure metals is caused by the thermal motion of the atoms. The relaxation time \(\tau\) can be determined for collisions caused by thermal motion only at high temperatures. At low temperatures the quantization of the various frequencies of vibration of the lattice is essential. In each collision a quantum of vibrational energy \(h\nu\) (\(\nu\) is the frequency of vibration) is either taken from, or given to, the lattice. Thus the collisions are inelastic. If the energy of the quantum is small in comparison with \(kT\) (and this will occur if the temperature is above the Debye characteristic temperature for the metal), then quantization may be neglected and the vibrations treated classically. It then turns out that the probability of collision of an electron with the lattice and, consequently, the fraction

\[ \frac{1}{\tau} \]

is proportional to the mean square amplitude of the motion of the ions, in accordance with Wien’s hypothesis. The mean square amplitude is proportional to the absolute temperature and inversely proportional to the mass of the atom \(M\) and to the square of the vibration frequency \(\nu\), i.e.

\[ \overline{x^2}=\frac{kT}{4\pi^2 M\nu^2}. \tag{24} \]

Therefore the resistance must be proportional to the absolute temperature, in agreement with experiment. The frequency of vibration is proportional to the characteristic temperature \(\theta\), whence

\[ \frac{1}{\tau}\simeq \overline{x^2}\simeq \frac{T}{M\theta^2}. \tag{25} \]

The second important factor in \(\frac{1}{\tau}\) is equal to the density of states for energies close to the edge of the Fermi distribution. If the density is large, this means that there are many states into which electrons can be scattered, and the magnitude of \(\frac{1}{\tau}\) is correspondingly larger. The probability of scattering is directly proportional to the density. It is clear that the density is large for a narrow band and small in the case of a broad one.

Deviations from proportionality between the resistance and the absolute temperature at very high temperatures (\(\sim 1000^\circ\)

were explained by Mott¹) by a change in \(\theta\) as a result of thermal expansion. When a metal expands, \(\theta\) decreases; therefore \(\frac{R}{T}\) increases as the temperature is raised. This is normal behavior; however, for certain metals, in particular for metals of the transition groups, \(\frac{R}{T}\) decreases with increasing temperature. In this group of metals there is a lower \(d\)-band, and at high temperatures the density of states changes appreciably over an energy interval of width \(kT\) at the edge of the Fermi distribution. The effective density decreases with increasing temperature \(T\), which leads to a decrease in \(\frac{R}{T}\), as observed for these metals. In Fig. 10 graphs are given of \(\frac{R}{T}\) as a function of \(T\) for various metals.

Fig. 10. Variation of the relative resistance of various metals, divided by the absolute temperature \(T\), illustrating the deviation from proportionality of resistance to temperature at high temperatures. Data of Grüneisen and Meissner (see the literature).

B. Low Temperatures

In order to examine the temperature dependence of the resistance at low temperatures, it is necessary to investigate in greater detail the collisions of electrons with lattice vibrations²). Following Debye, the vibrations may be regarded as a system of independent waves. Very long waves are ordinary sound waves; shorter waves correspond to the vibration of neighboring atoms in opposite directions. Each wave is described by the propagation vector \(\mathbf{q}\). The direction of \(\mathbf{q}\) indicates the direction of propagation of the wave; the magnitude \(q\) is equal to \(\frac{2\pi}{\lambda}\), where \(\lambda\) is the wavelength. There are three independent vibrations for each \(q\), corresponding to the transverse and longitudinal sound waves. If the wave velocity is \(c\), then the frequency is

\[ \nu=\frac{c}{\lambda}=\frac{cq}{2\pi} \tag{26} \]

and the energy of the vibrational quantum is \(h\nu=\frac{hcq}{2\pi}\).

At each collision of an electron with the lattice, the latter either emits or absorbs a vibrational quantum. The selection rules for the transition of an electron from the state \(\mathbf{k}\) to the state \(\mathbf{k}'\) have the form:

\[ \begin{aligned} \mathbf{k}'&=\mathbf{k}+\mathbf{q} && \text{(emission),}\\ \mathbf{k}'&=\mathbf{k}-\mathbf{q} && \text{(absorption).} \end{aligned} \tag{27} \]

¹) See also A. A. Smirnov, ZhETF, 4, 229, 1934; Sow. Phys., 5, 529, 1934. Translator’s note.

²) Or, as it is customary to say, with phonons. Translator’s note.

These vector relations are shown graphically in Fig. 11. \(\theta\) is the angle between \(k\) and \(k'\), through which the electron is deflected in the collision. In addition to these rules there is the requirement of conservation of energy. Therefore, on absorption, the electron energy in the final state \(k'\) must be equal to the electron energy in the initial state minus the energy of the vibrational quantum.

At low temperatures only vibrations with low energies can be excited, i.e., with large wavelengths and therefore with small values of \(q\). The low value of the resistance occurs not only because the vibrations are small, but also because the electrons are deflected through small angles (see Fig. 11). A quantum can be absorbed by the lattice even if no quanta had previously been absorbed by it. It might therefore seem that a metal should have a resistance at absolute zero temperature. However, a quantum can be absorbed only in the case when the electron can give up an equivalent amount of energy. The average excitation energy of electrons near the edge of the Fermi distribution is of the order of \(kT\). The electron cannot give up more than this amount, because all lower states are occupied.

Fig. 11. Illustration of the vector relation in the scattering of an electron wave by lattice waves

The derivation of the formula for the temperature dependence of the resistance, given by Bloch, contains a number of assumptions and approximations: 1) Debye’s theory for thermal vibrations; 2) thermal equilibrium of the vibrations; 3) \(E(k)\) is a function of \(|k|\). In addition to this, certain assumptions are made concerning the form of the interaction between the electrons and the lattice vibrations. The function giving the distribution of electrons in \(k\)-space, which is used in the calculation of the conductivity, is only an approximate solution of the Boltzmann equation. The formula obtained gives that the resistance is proportional\(^9\)

\[ R \simeq \left(\frac{T}{\theta}\right)^5 \int_0^{\theta/T} \frac{x^5 e^x\,dx}{(e^x-1)^2}. \tag{28} \]

This formula was used by Grüneisen, and its graph is given in Fig. 1 together with the experimental points for a number of metals. The agreement is too good, especially if one takes into account the approximate character of the theoretical derivation. In this respect the theory is in a certain sense analogous to Debye’s theory of heat capacities.

The formula shows that the resistance at very low temperatures must be proportional to \(T^5\). This law can be derived independently and justified theoretically in a firm way. Most metals seem to agree well with this law.\(^{10}\) But there are some deviations. In particular, the resistance of metals

of the transition groups at very low temperatures \((<10^\circ \mathrm{K})\) seems to follow a \(T^2\) law. Baber\(^{11}\) found in the resistance a term proportional to \(T^2\) and caused by collisions between electrons. The effect is never large, but it should be most sharply expressed in the case of metals with a very narrow conduction band, such, for example, as the \(d\)-band in metals of the transition groups.

C. Change of Resistance with Pressure

We have seen that the conductivity can be expressed in the form

\[ \sigma=\frac{N_{\mathrm{eff}} e^2}{m}\tau, \tag{23} \]

and \(1/\tau\) is proportional: 1) to the mean square amplitude of the thermal motion of the atoms, which according to Debye’s theory is proportional to

\[ \frac{T}{M\theta^2}, \]

and 2) to the density of electronic states in energy at the Fermi distribution edge, which is inversely proportional to \(\dfrac{dE}{dk}\).

As was mentioned earlier, the normal decrease of resistance with increasing pressure is caused by an increase of \(\theta\) with pressure, which in turn is connected with an increase in the binding forces when the atoms are brought closer together. The relative change of resistance with pressure, caused by the change of \(\theta\), is equal to

\[ \frac{d\lg R}{dp}=-\frac{2d\lg \theta}{dp}. \tag{29} \]

The dependence of \(\theta\) on pressure can be obtained from Grüneisen’s formula\(^{12}\)

\[ \frac{\beta V_0}{K C_v}=-\frac{d\lg \theta}{d\lg V}, \]

where \(\beta\) is the coefficient of volume expansion, \(K\) is the compressibility\(^{1}\), \(V_0\) is the volume of 1 g of substance, and \(C_v\) is the heat capacity. Since the compressibility

\[ K=-\frac{d\lg V}{dp}, \]

then

\[ \frac{d\lg \theta}{dp}=\frac{\beta V_0}{C_v}. \tag{30} \]

The values of \(-\dfrac{2d\lg \theta}{dp}\), calculated from (30), are compared in Table 1 with the observed values of \(\dfrac{d\lg R}{dp}\) for a number of metals. The agreement of the numbers, at least in order of magnitude, is good. However, there are a number of clear deviations, as, for example, for Li, Ca, Sr, for which the resistance increases with pressure. Since \(\theta\), undoubtedly, increases with pressure, these anomalous values must be connected with anomalous behavior of the electrons. First we shall consider

\(^{1}\) To avoid misunderstandings, we point out that the symbol \(K\) is also used to denote thermal conductivity.

Table 1

Change of specific resistance with pressure for various metals

Metal	$\dfrac{d\lg R}{dp}\cdot 10^{12}\ \mathrm{CGSE}$	$\dfrac{2\beta V_0}{C_v}\cdot 10^{12}\ \mathrm{CGSE}$	Metal	$\dfrac{d\lg R}{dp}\cdot 10^{12}\ \mathrm{CGSE}$	$\dfrac{2\beta V_0}{C_v}\cdot 10^{12}\ \mathrm{CGSE}$
Li	−4.0	21	Rb	200	120
Na	73	40	Sr	−47
Mg	5.9	9	Mo	1.5	1.1
Al	4.8	6	Ag	4.0	4.8
K	190	91	Cs	220	157
Ca	−8.9		Ta	1.7	1.8
Fe	2.7	2.0	Pt	2.1	1.9
Co	1.1	2.0	Au	3.4	3.3
Ni	2.1	2.0	Pb	15.4	12.5
Cu	2.3	3.0

monovalent metals, and then, following Mott, we shall give a possible explanation of the anomalous behavior of certain divalent metals.

The graph of the change of the resistance of the alkali metals with pressure, according to Bridgman’s observations, is shown in Fig. 2. Na behaves normally; its resistance decreases with increasing pressure. The resistance of Li increases with pressure, while K, Rb, and Cs have a resistance minimum, which first decreases and then, with further increase of pressure, rises.

Fig. 12. Course of $a$ [equation (31)] for sodium and lithium as a function of volume

To a first approximation, the electrons in a monovalent metal may be regarded as free. Then the energy is equal to

$$ E=\frac{1}{2}mv^2=\frac{h^2k^2}{8\pi^2m}. $$

A better approximation for $E$ is

$$ E=\frac{ah^2k^2}{8\pi^2m}, \tag{31} $$

where $a$ is the effective number of free electrons per atom. The values of $a$, which can be calculated for Li and Na from the basic principles of the theory$^{14}$, are plotted as a function of atomic volume in Fig. 12. It is found that $a$ in the case of Li is equal to 0.65 and decreases with decreasing volume. It is approximately equal to unity for Na and increases slightly as the volume decreases. For a very rough calculation of the change of resistance with pressure, one may assume that the characteristic temperature $\theta$ is inversely proportional to the square root of

of compressibility \(K\) (Einstein’s formula). Since \(\tau\) is proportional not only to \(\theta^{2}\), but also to \(\dfrac{dE}{dk}\) or \(a\), for the resistance we find

\[ R \simeq N_{\mathrm{eff}}^{-2}\theta^{-2}\simeq \frac{K}{a^{2}} \tag{32} \]

or

\[ \frac{R}{R_{0}}= \frac{\left[\dfrac{K}{a^{2}}\right]_{p}} {\left[\dfrac{K}{a^{2}}\right]_{0}} . \tag{33} \]

The subscript zero indicates that the corresponding values are taken at zero pressure. The values of \(\dfrac{R}{R_{0}}\) for sodium, calculated from (33), are compared with the experimental data in Fig. 13. The agreement is quite good.

The same calculations may be made for lithium. The decrease of \(a\) with pressure should lead to an increase in resistance. Frank\(^{13}\) attributes the observed increase in resistance to a decrease of \(N_{\mathrm{eff}}\). However, the decrease of \(a\) shown in Fig. 12 is not sufficient to compensate for the increase of \(\theta\). Therefore, according to (33), we obtain as a result a decrease in resistance, which is in contradiction with experiment. However, the factors given in (32) are not the only ones determining the resistance. It is necessary also to take into account a term representing the interaction between the electrons and the lattice vibrations. Unpublished calculations by I. Wainer, on the basis of the author’s theory\(^{14}\), indicate that this term has only a small influence on the dependence of resistance on pressure in the case of sodium. It acts in the direction of increasing the resistance at high pressures in the case of lithium. However, this effect is not very large, and therefore the calculation still gives a small decrease of resistance with increasing pressure. There are some data indicating that the conduction electrons of lithium lie in two overlapping bands, and this must be taken into account in order to obtain complete agreement between theory and experiment.

Fig. 13. Comparison of the experimental and theoretical values of the relative change in the resistance of sodium with pressure.

Mott\(^{15}\) carried out a calculation of the conductivity of divalent metals taking account of Brillouin zones. An excellent qualitative survey of his general method is given in the introduction to the second of the indicated

articles[^16]. If there are two intersecting energy bands, which we shall denote by \(a\) and \(b\), then the conductivity is equal to

\[ \sigma=\frac{Ne^{2}}{m}\left(\alpha\tau_{a}+\beta\tau_{b}\right). \tag{34} \]

Here \(\tau_{a}\) and \(\tau_{b}\) are the relaxation times of the electrons in the bands \(a\) and \(b\), while \(\alpha\) and \(\beta\) are the corresponding effective numbers of free electrons per atom. Mott showed that \(\tau_{a}\) and \(\tau_{b}\) are of the same order of magnitude; but \(\alpha\) and \(\beta\) may be very different. If in band \(a\) the density of states is small, and in band \(b\) large, then \(\alpha\) is much larger than \(\beta\), and the greater part of the current is carried by the electrons of band \(a\). In this case the resistance is caused chiefly by transitions of electrons from band \(a\) to band \(b\), and the relaxation times \(\tau_{a}\) and \(\tau_{b}\) will be inversely proportional to the density of states at the edge of the Fermi distribution in band \(b\). If, with increasing pressure, the overlap of the bands increases, so that the density of states at the Fermi edge increases, then the resistance will increase, since \(\tau_{a}\) and \(\tau_{b}\) will decrease. This will occur if either \(a\) or \(b\) has a lower energy. A schematic diagram of two such overlapping bands is shown in Fig. 14.

Fig. 14. Occupied electron states in two overlapping bands. Above—zero pressure; below—at a pressure different from zero (schematic).
In the diagram: “density of states”; “energy \(\to\)”; \(p=0\); \(p\gg0\).

Manning and Krutter[^17] carried out approximate calculations of the energy bands of Ca. They found that a dense \(d\)-band lies higher and overlaps with the normal \(s\)-band (which contains approximately two electrons per atom). They also found that the overlap increases with pressure, so that the resistance should increase, in qualitative agreement with experiment.

D. Absolute Value of the Conductivity

We have seen that one of the basic factors determining the resistance of metals is the square of the amplitude of the vibrations of the atoms. For this reason alone the conductivity should have been proportional to \(M\theta^{2}\), where \(\theta\) is the Debye characteristic temperature, and \(M\) is the atomic weight. In Fig. 15 is shown the course of

\[ \frac{\sigma}{M\theta^{2}}, \]

which should depend on the number of electronic factors as a function of the atomic numbers.

elements. The most surprising facts are the relatively high values of \(\dfrac{\sigma}{M\theta^{2}}\) for monovalent metals and the small values

Fig. 15. Values of \(\dfrac{\sigma}{M\theta^{2}}\), plotted as a function of the atomic number. For rare-earth metals there are no data

for transition elements. One should also note the large difference in the values of \(\dfrac{\sigma}{M\theta^{2}}\) for neighboring pairs (Ni, Cu), (Pd, Ag), and (Pt, Au). In transition metals there is an unfilled \(d\)-band overlapping with an \(s\)-band, which is similar to the \(s\)-bands of monovalent metals (Fig. 16). The current is carried mainly by the electrons of the \(s\)-band, but the large resistance is caused by transitions from the \(s\)-band into the narrow \(d\)-band. The probability of transitions of this type is high because of the large density of states in the \(d\)-band. Such transitions cannot occur in monovalent metals, in which the \(d\)-band is completely occupied.

Fig. 16. Schematic representation of a broad \(s\)-band overlapping a narrow \(d\)-band

The comparatively high resistances of divalent metals are probably due to the small effective number of free electrons for these metals. This is even more true for such semimetallic elements as Bi, Sb, and As.

The calculation of the absolute value of the conductivity of metals is a very difficult problem, owing to the fact that we know very little

both about the electronic wave functions and about the vibrational spectrum of most metals. Attempts at such calculations have been made only for monovalent metals. In order to illustrate the degree of agreement with experiment, Table 2 gives the results of the author’s calculations[^18]. These calculations were based on the following assumptions: 1) the wave functions of the electrons are assumed, in most of the volume, to be almost the same as for free electrons. 2) Debye’s theory is used for the lattice vibrations. 3) The perturbing potential that causes the scattering of electrons by lattice vibrations is assumed to consist of two parts: a) the change in the potential of the ions and b) the change in the potential of the self-consistent field of the valence electrons when the ions are displaced from their equilibrium positions during lattice vibrations.

Table 2

Electrical conductivity of monovalent metals
\(T = 0^\circ\mathrm{C}\) \((\mathrm{ohm}^{-1}\,\mathrm{cm}^{-1}\cdot 10^{-4})\)

Metal	Experimental value	Calculated value
Li	11.8	28
Na	23.4	23
K	16.4	20
Rb	8.6	33
Cs	5.3	22
Cu	64	174
Ag	66	143
Au	49	142

The agreement with experiment is quite good for Na and K (for these metals the above assumptions are probably better satisfied), but for the other monovalent metals the calculated conductivities are too large.

E. Ferromagnetic Metals

The resistance of ferromagnetic metals begins to increase rapidly as the Curie temperature is approached. On the resistance–temperature curve there is a bend at the Curie point, above which the curve is almost the same as for normal metals. The curve for nickel, which has been studied more thoroughly than all the other ferromagnetic metals, is shown in Fig. 17. Gerlach[^19] indicated that the resistance near the Curie point can be represented as the sum of the normal resistance \(R_N\), which is given by the Bloch–Grüneisen function, and a second term \(R_F\), which is a function of the spontaneous magnetization:

\[ R = R_N + R_F = R_N + C(I_0^2 - I^2). \tag{35} \]

Here \(I_0\) is the value of the spontaneous magnetization at absolute zero, \(I\) is the magnetization at temperature \(T\), and \(C\) is a constant independent of temperature.

This phenomenon was considered theoretically by Mott[^20]. The resistance of nickel is due to a significant extent to transitions of electrons from the \(s\)- to the \(d\)-band. Each state in the \(d\)-band can be occupied by two electrons with opposite spins. At low temperatures, when the spontaneous magnetization reaches

saturation, all states in the \(d\)-band are occupied by electrons with spins parallel to the direction of magnetization. The remaining electrons have antiparallel spins. However, there are not enough of them to fill the band. The probability that an electron will change its spin during a collision is rather small, so that only half of the electrons in the \(s\)-band, namely those whose spin is antiparallel to the magnetization, can pass into the \(d\)-band. At higher temperatures, with incomplete magnetization, electrons with either spin direction may take part in the transitions, which increases the resistance. Curves \(a\) and \(c\) in Fig. 17 give the resistance of hypothetical paramagnetic nickel and of nickel magnetized to saturation. The three points give the resistance calculated by Mott for the actual magnetization at the given temperatures. The calculations are rather complicated and do not have the simple form indicated by Gerlach. The agreement with the experimental curves is fairly good. The resistance decreases when the specimen is magnetized by an external magnetic field. It is not yet clear whether this effect can be explained quantitatively simply by a change in magnetization.

Fig. 17. Specific resistance of nickel as a function of temperature.
\(a\)—curve calculated for hypothetical paramagnetic nickel, \(b\)—experimental curve, \(c\)—curve calculated for magnetization at \(0^\circ\mathrm{K}\), \(\odot\)—values calculated from the observed magnetizations at the given temperatures (according to Mott \(^{60}\)).

VII. CONDUCTIVITY OF ALLOYS

A. Dilute solutions

We have already considered Matthiessen’s rule (Section II, D), which gives the increase of resistance for small additions of a foreign metal in a solid solution. In the quantum theory of metals this rule receives a simple explanation. An ideal periodic lattice has no resistance. In a pure metal the resistance is caused by thermal motion, which disturbs the periodicity of the lattice. In solid solutions the electron waves can also be scattered by impurity atoms. At small impurity concentrations the resistances caused by these two causes are additive. Matthiessen’s rule is applicable in the case when the part of the resistance that depends on temperature and is due to thermal vibrations of the lattice does not depend on concentration.

The increase in resistance caused by the dissolved metal is usually very large. At room temperatures the resistance

metals can be doubled by the addition of one atomic percent of impurities. Measurements of resistance are often used to check the purity of specimens.

The influence of various dissolved metals may vary over wide limits and in most cases is not amenable to theoretical treatment. However, Norbury\(^{21}\) pointed out an interesting circumstance. Comparing the increase in resistance caused by one atomic percent of various metals dissolved in a common solvent, he observed a definite dependence on the valency of the solvent and of the dissolved metals. In general, the greater the difference in valencies (or in the horizontal position in the periodic table), the greater the additional resistance. The effect is most striking for metals dissolved in Cu, Ag, and Au. In a later paper Linde\(^{22}\) found that for these metals the increase in resistance varies approximately in proportion to the square of the difference in valencies. If \(Z+1\) is the number of electrons outside the filled \(d\)-shell (so that \(Z=0\) for Cu, \(Z=1\) for Zn, \(Z=2\) for Ga, etc.), then the additional resistance varies as \(Z^2\) (Fig. 18). A simple

Fig. 18. Increase in resistance caused by one atomic percent of various metals dissolved in Cu, Ag, and Au. The abscissae are proportional to the square of the difference in valencies of the solvent and dissolved metals. Exactly the same results are obtained if the dissolved metals belong to other columns of the periodic table (data after Linde\(^{22}\)).

explanation was given by Mott\(^{23}\). All electrons outside the filled \(d\)-shells, both in the dissolved metals and in the solvents, are conduction electrons, which move freely in the metal. The ionic cores of the dissolved atoms have a charge \(Ze\) exceeding the charge of the atoms of the solvent. The electrons are scattered by the field of this additional charge. By Rutherford’s law the scattering is proportional to the square of the charge \((Ze)\), and therefore the additional resistance is proportional to \(Z^2\). Quantitative calculations, taking into account the screening action of the conduction electrons on the additional charge, give the correct order of magnitude of the effect.

Mott\(^{24}\) also investigated the additional resistance caused by one atomic percent of a metal \(A\) dissolved in a metal \(B\), in comparison with the resistance caused by dissolving one atomic percent of \(B\) in \(A\). He found that under certain conditions these resistances should be approximately equal.

B. Disordered Solid Solutions

In this article we shall consider only the conductivity of simple homogeneous phases. (Many alloys consist of a simple mechanical mixture of two phases; in this case the conductivity is approximately equal to the mean value of the conductivities of the individual phases.) Let us first consider the case of a disordered alloy, in which the atoms of the metals forming the alloy are distributed randomly over the lattice sites. Ordered alloys will be considered in the following paragraph. The resistance of an alloy can be expressed as the sum of two terms

![Figure 19 and Figure 20]

Fig. 19. Specific resistance of the alloys K—Rb, Pt—Pd, and In—Pb as a function of atomic concentration

$T = 25^\circ\mathrm{C}$ for all curves. Data from International Critical Tables

Fig. 20. Specific resistance of the alloys Cu—Pd, Ag—Pd, and Au—Pd as a function of the atomic percent of Pd. Cu—Pd and Ag—Pd according to Svensson (Ann. Physik 14, 699, 1932). Au—Pd according to Gebel (International Critical Tables)

\[ R = R_0 + R_T . \tag{36} \]

The term \(R_T\), which depends on temperature, is caused by the thermal motion of the atoms. It is analogous to the resistance of a pure metal. \(R_0\) is the resistance at absolute zero. It is caused by the disturbance of the periodicity of the lattice due to the random distribution of the atoms forming the alloy.

The theory of the resistance of binary mixtures on the basis of Bloch’s theory was given by Nordheim. He found that the resistance \(R_0\) is proportional to \(x(1-x)\), where \(x\) is the concentration of atoms \(A\), and \(1-x\) that of atoms \(B\). The graph of \(R_0\) as a function of the concentration \(x\) has the well-known form of a bell-shaped curve with a maximum at a concentration of \(50\%\). As we have seen (Fig. 4), this is precisely the type of curve that was found experimentally for Ag—Au alloys. Similar curves were found for other binary alloys whose metallic components are mutually soluble in all proportions and which exist in one phase, for example K—Rb; Pt—Pd; In—Pb (Fig. 19). A somewhat different type of curve is obtained for alloys of noble metals with transition metals, as shown in Fig. 20 for the case of alloys of Cu, Ag, and Au with Pd. Maxi-

the resistance maximum is displaced toward higher Pd concentrations. The explanation given by Mott is, in brief, as follows. As we have already mentioned, the relatively high resistance of the transition metals, including Pd, is caused by the presence of an unfilled $d$-shell, which makes possible transitions from the $s$-band of the current-carrying electrons. In the alloys under consideration the $d$-band is not filled for all Pd concentrations above 40%. The anomalously high resistance values of alloys above this concentration are caused by $s-d$ transitions.

Fig. 21. Specific resistance of Cu—Ni alloys; at 0, 250 and 500°C—according to Svensson. (Ann. Physik, 25, 263, 1936); at −273°C—according to Kruzhkovskii and de Haas (Comm. Leiden, N 194, 1, 1928)

In Fig. 21 is shown the course of the resistance of Cu—Ni alloys at various temperatures according to the measurements of Svensson. These alloys are ferromagnetic for Ni concentrations above 40 atomic percent. Constantan, with an approximate composition CuNi, belongs to this type of alloy. Svensson attempted to express the results of his experiments in the form of a sum of three terms: 1) $R_T$—the resistance caused by thermal motion; 2) $R_0$—the resistance caused by the disordered distribution of different types of atoms, and 3) $R_F$—the ferromagnetic part of the resistance, which is a function of the magnetization. A qualitative theoretical treatment of this question was given by Mott$^{20,a}$.

C. Ordering Alloys$^{25}$

In Fig. 5 was shown the resistance curve of Cu—Au alloys as a function of concentration. Quenched alloys give a simple bell-shaped curve, similar to the curve for silver-gold alloys. If the alloys are annealed at a temperature below 400°, the resistance falls sharply near the compositions CuAu and Cu$_3$Au. On the basis of chemical and electrical measurements Tammann showed that the atoms at these concentrations assume a more or less regular arrangement. The ordered structure was later proved for Cu$_3$Au by Johansson and Linde$^{26}$ with the aid of analysis of “superstructure” lines on X-ray photographs. After this many other ordered alloys were discovered. The measurement of resistance plays an important role in the discovery and study of these alloys, supplementing the data of X-ray analysis and heat capacities. Copper-gold alloys have been studied more thoroughly than any other system; therefore we shall confine ourselves to considering the alloy Cu$_3$Au as a typical example.

The face-centered lattice of the copper–gold system may be regarded as the superposition of four mutually interpenetrating simple cubic lattices. At high temperatures the atoms are distributed randomly over the lattice sites. But at temperatures roughly below \(400^\circ\), the gold atoms of the alloy \(\mathrm{Cu}_3\mathrm{Au}\) tend to arrange themselves in one of these simple cubic lattices, forming an ordered structure. In the composition \(\mathrm{CuAu}\), below a certain critical temperature, the atoms of copper and gold are distributed in alternating planes. The distance between these planes changes, and the cubic structure becomes tetragonal.

The equilibrium degree of order depends on temperature. The distribution above the critical temperature, \(T_c\), is random. Ordering begins to grow continuously as the temperature falls below the critical one. Many definitions of the degree of order in a crystal have been proposed; the best of them is the definition of Bragg and Williams\({}^{27}\). Let us denote the two types of atoms in the alloy by \(A\) and \(B\). In complete order, the \(A\) atoms occupy a sublattice; the positions in this lattice are called \(\alpha\)-positions. In exactly the same way the \(B\) atoms will occupy \(\beta\)-positions. Let, in a partially ordered crystal, the probability that an \(A\) atom will occupy an \(\alpha\)-position be equal to \(p_A\). And let, finally, \(r_A\) be the value of \(p_A\) for a completely disordered distribution (i.e. \(r_A\) equals \(\frac{1}{4}\) for \(\mathrm{Cu}_3\mathrm{Au}\), if \(A\) refers to gold). We define the degree of (long-range) order as

\[ S=\frac{p_A-r_A}{1-r_A}. \tag{37} \]

\(S=1\) for complete order and \(S=0\) in a disordered alloy. This definition has been criticized, since it does not take into account order among neighbors, which may also exist. Attempts to take short-range order into account were made by Bethe\({}^{28}\), Peierls\({}^{29}\), Kirkwood\({}^{30}\), and others. There is as yet no fully suitable definition of order for the consideration of resistance measurements. In all probability the degree of order is connected with two critical distances in a metal: the electronic wavelength (\(\sim 10\ \text{Å}\)) and the mean free path (\(\sim 100\ \text{Å}\)).

Fig. 22. Theoretical curves of long-range order \(S\) as a function of \(\dfrac{T}{T_c}\) for an alloy of type \(A_3B\)

\(a\)—according to Bragg and Williams, \(b\)—according to Peierls (for a body-centered cubic lattice)

The theoretical curves of Bragg and Williams and of Peierls, giving the degree of order \(S\) as a function of temperature for an alloy of type \(A_3B\), are shown in Fig. 22. The degree of long-range order changes discontinuously

at the critical temperature, and therefore a heat of transition occurs. This is unlike the behavior of alloys of the type \(AB\), for which there is no heat of transition, and the degree of order, beginning from zero at the transition point, increases continuously as the temperature is lowered.

Figure 23 gives curves of the temperature dependence of the resistance for \(\mathrm{Cu_3Au}\), from the observations of Sykes and Evans\(^{31}\). Curve \(a\) refers to the equilibrium state, which is obtained after prolonged annealing. For curve \(b\) the cooling rate is \(30^\circ\) per hour. Curve \(c\) was obtained by extrapolating the data obtained for temperatures above the critical one for an alloy quenched from \(450^\circ\), and represents the resistance curve for a completely disordered alloy.

Fig. 23. Specific resistance as a function of temperature for \(\mathrm{Cu_3Au}\)

\(a\)—quenched alloy; \(b\)—cooled at a rate of \(30^\circ\mathrm{C}/\mathrm{hr}\); \(c\)—slow cooling and holding in equilibrium above \(350^\circ\mathrm{C}\). Branch \(OA\) is discussed in the text. Data of Sykes and Evans\(^{31}\).

Owing to the relatively small rate at which the different atoms of the crystal exchange places, the rate of approach to the equilibrium configuration is small, except at temperatures close to the transition point. At room temperatures the rate is so small that the alloy may remain for as long as desired in a metastable state. Measurements of resistance as a function of temperature for different rates of cooling or heating are very useful for determining the rates of approach to equilibrium.

Unfortunately, measurements of resistance cannot be used for exact quantitative determinations, since the resistance cannot simply be related to the degree of order. According to the general theory considered above, we should expect the resistance to be equal to the sum of two terms: a term depending on the temperature, \(R_T\), which is due to the thermal motion of the atoms, and a second term \(R_0\), representing the resistance due to the disordered distribution of atoms. \(R_0\) depends on the temperature indirectly, through its dependence on the degree of order. For the sake of simplification Bragg and Williams assumed that the second term is a linear function of the degree of order \(S\) and vanishes for complete order. As can be seen from a comparison of Figs. 22 and 23, there is some justification for such a choice of the dependence of \(R_0\) on \(S\). This question was considered theoretically by Mott\(^{32}\), who extended the Nordheim treatment of disordered alloys to the case of ordering alloys. He found a quadratic dependence of \(R_0\) on \(S\). However, he did not take into account the influence of order at near-neighbor distances, which may occur.

In a series of very interesting papers, Sykes and Jones33 investigated the formation of nuclei of ordering and their influence on the resistance of Cu$_3$Au. A quenched alloy was rapidly heated to 350° and the resistance rose to the point $O$ (Fig. 23), corresponding to the disordered alloy. Ordering did not have time to become established. If the alloy was held at this temperature, the resistance fell continuously along the line $OA$, until at last it reached its equilibrium value. In each crystal the gold atoms may be distributed over one of four simple cubic lattices, which form a face-centered structure. Nuclei of ordering may arise in any one of them quite at random and grow until they meet one another. Therefore the crystal will consist of a large number of small regions, in each of which the structure is ordered, but there are discontinuities in phase at the boundaries. Some of these regions will gradually grow at the expense of their neighbors, thereby increasing their size. If such growth occurs, the resistance decreases. Sykes and Jones determined the mean sizes of the nuclei at various stages of this process from the width of superstructure lines on X-ray photographs. Various annealing times were taken, corresponding to approximately equal changes in resistance, and then these alloys were quenched in water. In this way a whole series of metastable states was obtained, corresponding to different stages of growth of the nuclei, which were then used to study the resistance and its dependence on the size of the nuclei.

Fig. 24. Specific resistance as a function of the apparent sizes of nuclei
Annealing temperatures: $a$—376° C; $b$—346° C; $c$—298° C. Data according to Jones and Sykes22

Figure 24 gives the behavior of the resistance as a function of the apparent sizes of the nucleus for three different annealing temperatures, as obtained by Jones and Sykes. The true sizes of the nuclei are approximately one half of the apparent $\varepsilon$, determined by X-ray methods. The difference between the three curves is caused by the difference in the degree of order in the individual nuclei at different annealing temperatures.

The curves pass into straight lines for values of $\frac{1}{\varepsilon}$ smaller than $10^{-2}$ ($\varepsilon$ in Å). In this region the sizes of the nuclei are large compared with the thickness of the boundary, and it may be assumed that the resistance is caused by reflection of electrons from the boundaries. Assuming that the effective number of free electrons per atom is equal to unity, Jones and Sykes found that the reflection coefficient at the boundary is equal to...

approximately 0.1. In fact it may be somewhat larger than this value.

The anomaly of the heat capacity in a quenched alloy heated between 100 and 200° is observed long before there is a noticeable effect on the electrical resistance. This may be explained by a very early stage of nucleus formation. Probably a large part of the ordering at short distances requires relatively large changes in energy, but it occurs in such small volumes that it has little effect on the resistance.

The resistance of an ordered alloy changes appreciably under cold working. Fig. 25 gives Dahl’s data on the effect of plastic deformation on the resistance of the alloy Cu$_3$Au. With sufficient deformation, the resistance of the annealed alloy approaches that of the quenched alloy. At the same time the superstructure lines also disappear on the X-ray photographs, which indicates a return to a disordered structure. A large change in resistance after cold working is often regarded as an indication (though not a very reliable one) of the presence of ordering in the alloy.

Fig. 25. Effect of plastic deformation on the resistance of the alloy Cu$_3$Au (data according to O. Dahl, Z. Metallkunde, 28, 133, 1936)

Fig. 26
Top—resistance of the β-phase (51.25 atomic % Cu, 48.75 atomic % Zn) as a function of temperature. Data according to V. Webby (Phys. Rev., 55, 297, 1939). Bottom—course of $1-S^2$, where $S$ is the degree of long-range order according to the theories of Bragg and Williams and of Bethe.

As a second example of the influence of order on resistance, Fig. 26 gives the resistance of the β-phase (CuZn) as a function of temperature. In this case we are dealing with the structure of a body-centered cube; in the ordered state the Cu and Zn atoms tend to distribute themselves over two interpenetrating simple cubic lattices, which form the structure. According to Muto’s theory$^{32}$, the resistance due to disorder, $R_0$, should be proportional to $(1-S^2)$, where $S$ is the degree of order (37). The course of $(1-S^2)$ as a function of temperature according to the theories of Bragg and Williams and of Bethe is shown in Fig. 26.

D. Strengthening of Alloys upon Aging34

If a supersaturated alloy is quenched from a high temperature, then one or more phases may begin to decompose in the solid solution either at room temperature or after annealing. This causes a gradual change in the physical properties of the alloy. The hardness usually increases with time, reaching a maximum, after which it gradually falls. The electrical and magnetic properties also change during the quenching process. The resistance sometimes decreases, but often at first it increases. The maximum resistance usually occurs earlier than the maximum hardness.

The resistance, of course, depends on the distribution of atoms in the lattice. A disordered distribution leads to a high resistance, whereas a conglomerate of crystals of separate pure metals or compounds has a comparatively lower resistance (see Figs. 3 and 4). Therefore one might expect the resistance to decrease when a metal is precipitated from a solid solution. However, as Mott35 pointed out, a disordered distribution may not be the only cause leading to the highest resistance. If the atoms are distributed in very small volumes, whose average dimensions are of the order of the electron wavelength (\(\sim 10\ \text{Å}\)), then they may scatter waves more strongly than volumes of larger or smaller dimensions. Therefore the maximum resistance may correspond to a very early stage of decomposition.

VIII. THE WIEDEMANN–FRANZ LAW

It is not our task to consider thermal conductivity or such phenomena as thermoelectricity and the various galvanomagnetic and thermomagnetic effects in metals, although they are very closely connected with the electrical conductivity studied here. However, we shall make some remarks concerning the Wiedemann–Franz law. As is well known, in a metal heat can be carried either by conduction electrons or by the lattice itself with the aid of thermal vibrations. In non-conducting crystals the second mode of thermal conductivity is essential. Conversely, in metals the dominant role in thermal conductivity is played by conduction electrons.

The Wiedemann–Franz law can be derived from quite general assumptions on the basis of Bloch’s theory. It states that the ratio of the thermal conductivity \(K\) to the electrical conductivity \(\sigma\) is equal to

\[ \frac{K}{\sigma} = \frac{\pi^2}{3}\frac{k^2}{e^2}\,T. \]

The numerical factor is somewhat larger than that obtained in Drude’s theory \(\left(\frac{\pi^2}{3}\text{ instead of }3\right)\), and is in better agreement with experimental values. In deriving the law, only the conduction electrons are taken into account; the thermal conductivity of the lattice is neglected. This is legitimate when the resistance is due to disorder (in alloys) or

by the thermal motion of the atoms, but in this latter case only for temperatures above the Debye characteristic temperature. In addition, it is assumed that \(kT\) is small in comparison with the conduction band; therefore, for example, in the case of transition metals at high temperatures deviations may be expected.

IX. CONCLUSION

This brief survey does not claim to be complete. We wished only to give illustrative examples of some problems connected with the conduction of electric current in metals and alloys and, insofar as this was possible, to establish a connection between them on the basis of Bloch’s theory. At the same time, a number of questions have remained untouched: 1) the dependence of the resistance on crystallographic directions, 2) the resistance of liquid metals, 3) the effect of polymorphic transformations, 4) the influence of an external magnetic field on the resistance, 5) the temperature coefficient of resistance of alloys, 6) structure-sensitive properties, similar to the effect of cold working, and 7) the resistance of thin films.

With a deeper understanding of the fundamental causes of resistance in metals and alloys, resistance measurements will be able to play a major role in the study of other physical properties. This applies especially to problems connected with the distribution of various types of atoms in an alloy. There is a great need for further experimental and theoretical investigations, and if this article serves as some stimulus to further activity in this direction, then the goal pursued by the author will have been achieved.

Before concluding, it is worthwhile to point out certain phenomena for which Bloch’s theory could not provide even a qualitative explanation. The best known of these is, of course, superconductivity. A second is the existence, discovered by de Haas and collaborators \(^{36}\), of a minimum in the curve of the temperature dependence of the resistance of gold at liquid-helium temperatures. At extremely low temperatures \((<1^\circ \mathrm{K})\) a very rapid increase of the resistance begins. The curve indicates that the resistance may become infinitely large at absolute zero. The position of the minimum depends on the purity of the gold; the minimum shifts to lower temperatures when the amount of impurities is reduced. No explanation of this phenomenon has yet been given \(^{37}\). As a third example one may point to the existence of a group of semiconductors with a partially filled \(d\)-band \(^{38}\). We have already mentioned (Sec. V, G) that a necessary condition for an insulator is an even number of valence electrons per unit cell. If their number is odd, the crystal must be a metal. Substances such as MnO, CoO, \(\mathrm{Mn}_3\mathrm{O}_4\), etc., fall outside this rule, being semiconductors rather than metals.

Bloch’s theory is based on the assumption that each electron moves independently in a periodic potential field. The electrostatic interaction of electrons is neglected, except for the fact that it is assumed that the electron moves in a periodic

in the potential field created by the ions and by all the other electrons1). Nevertheless, the theory is extraordinarily valuable, since with the aid of a simple physical picture it explains (in many cases quantitatively) a large range of phenomena.

It should be noted, however, that in some cases it may prove necessary to go beyond the Bloch picture in order to explain phenomena that depend to a greater degree on electron interaction or on the collective behavior of a large number of electrons.

LITERATURE

a) Monographs

The references given below are only an illustration for the various phenomena considered in the article. A complete survey of the experimental literature up to 1934 is given by Meissner in Handbuch der Experimentalphysik, vol. XI/2. Other sources of experimental data may be found in:

1. G. Grüneisen, Handb. d. Physik, 24, 1933.
2. G. Borelius, Handb. d. Metallphysik, 1/1, Leipzig, 1935.
3. W. Hume-Rothery, The Metallic State, Oxford, 1931.

The following books and articles are devoted chiefly to questions of theory:

1. A. Sommerfeld and H. Bethe, Handb. d. Physik, 24/2, 1933.
   Russian translation: Sommerfeld and Bethe, Theory of Metals.
2. N. F. Mott and H. Jones, Theory of the Properties of Metals and Alloys, Oxford, 1936.
3. A. H. Wilson, Theory of Metals, Cambridge, 1936.
4. H. Fröhlich, Elektronentheorie der Metalle, Berlin, 1936.
5. A. H. Wilson, Semi-Conductors and Metals, Cambridge, 1939.
6. Nordheim, Uspekhi fizich. nauk, 15, 570, 675, 779, 939, 1935. See also an excellent survey of the theory of the electrical conductivity of metals by L. Landau and A. Kompaneets, Electrical Conductivity of Metals, ONTI, Kharkov, 1935.

b) Journal articles

1. R. D. Tolman and T. D. Stewart, Phys. Rev., 8, 97, 1916; 9, 164, 1917.
2. Darwin showed that the experiment must give the value \(e/m\) for the free electron; no “effective” mass is introduced.
   G. G. Darwin, Proc. Roy. Soc., A 154, 61, 1936.
3. E. Grüneisen, Verh. d. D. Phys. Ges., 15, 186, 1913; Leipziger Vorträge, 46, 1930; Ann. Physik, 16, 530, 1933.
4. P. W. Bridgman, Proc. Amer. Acad., 72, 157, 1938.
5. P. Drude, Ann. Physik, 1, 566, 1900; 3, 370, 869, 1900; 7, 687, 1902; 14, 936, 1904. H. A. Lorentz, Elektronentheorie der Metalle, Leipzig, 1909.
6. The theory is summarized in the recently published book: E. H. Hall, A Dual Theory of Conduction in Metals, Cambridge, 1938.
7. F. Seitz and R. P. Johnson, J. Appl. Physics, 8, 84, 186, 246, 1937.
   For a Russian translation see Uspekhi fizich. nauk, 23, 89, 293, 1940.
8. See Mott and Jones, op. cit., p. 96.
9. F. Bloch, Z. Physik, 59, 208, 1930. See also Grüneisen, reference 3.
   A simple derivation was given by F. Sauter, Naturwiss., 7, 109, 1930.

1. W. J. de Haas and G. J. Van den Berg, K. Onnes Lab. Leiden Comm. Nos. 241—251, Suppt. No. 82 A, 1936, measured the specific resistances of a number of metals at temperatures from \(20^\circ\ \mathrm{K}\) to \(1^\circ\ \mathrm{K}\). The results can be expressed by the formula \(CT^b\), where \(b\) usually varies from 4 to 5 and only for Pt \(b = 2\).

2. W. G. Baber, Proc. Roy. Soc., A 158, 383, 1937.

3. See, for example, J. K. Roberts, Heat and Thermodynamics, p. 437, Blackie, 1933.

4. N. H. Frank, Phys. Rev., 47, 282, 1935.

5. J. Bardeen, J. Chem. Phys., 6, 369, 1938.

6. N. F. Mott, Proc. Phys. Soc. Lond., 46, 680, 1934; Proc. Roy. Soc., A 153, 699, 1936. See also the theoretical discussion by A. H. Wilson, Proc. Roy. Soc., A 167, 580, 1938.

7. The already cited Mott and Jones, p. 265.

8. M. F. Manning and H. M. Krutter, Phys. Rev., 51, 761, 1937.

9. J. Bardeen, Phys. Rev., 52, 688, 1937. Analogous calculations were made by E. L. Peterson and L. W. Nordheim, Phys. Rev., 51, 355, 1937.

10. W. Gerlach, Z. Physik, 59, 847, 1930.

11. a) N. F. Mott, Proc. Roy. Soc., A 153, 699, 1936; b) A 156, 368, 1936. See also A. H. Wilson, reference 15.

12. A. L. Norbury, Trans. Farad. Soc., 16, 570, 1921.

13. J. O. Linde, Ann. Physik, 10, 52, 1931; 14, 353, 1932; 15, 219, 1932.

14. N. F. Mott, Proc. Camb. Phil. Soc., 32, 281, 1936.

15. N. F. Mott, Proc. Phys. Soc. Lond., 46, 680, 1934; see also reference 22.

16. For a general review of this question, see F. C. Nix and W. Shockley, Rev. Mod. Phys., 10, 1, 1938. (For the Russian translation see Uspekhi fizicheskikh nauk, 20, 536, 1938.) See also G. Borelius, Proc. Phys. Soc., 49E, 77, 1937.

17. C. H. Johansson and J. O. Linde, Ann. Physik, 78, 439, 1925.

18. W. L. Bragg and E. J. Williams, Proc. Roy. Soc., A 145, 699, 1934; A 151, 540, 1935.

19. H. A. Bethe, Proc. Roy. Soc., A 150, 552, 1935.

20. R. Peierls, Proc. Roy. Soc., A 154, 207, 1936.

21. J. G. Kirkwood, J. Chem. Phys., 6, 70, 1938.

22. C. Sykes and H. Evans, J. Inst. Metals, 58, 255, 1936.

23. T. Muto, Sci. Pap. Inst. Phys. Chem. Res., 30, 99, 1936; 31, 153, 1937.

24. C. Sykes and F. W. Jones, Proc. Roy. Soc., A 157, 213, 1936; A 166, 376, 1938.

25. For a brief review of this question see C. H. Desch, Proc. Phys. Soc. Lond., 49E, 103, 1937.

26. The discussion is based on the paper by M. L. V. Gayler, J. Inst. Metals, 60, 55, 1937.

27. W. J. de Haas, H. B. Casimir and G. J. Van den Berg, Physica, 5, 225, 1938. References to earlier works may be found there.

28. S. V. Vonsovskii, ZhETF, 9, 154, 1939; Journ. Phys. USSR, 2, 113, 1940.

29. J. H. de Boer and E. J. W. Berwey, Proc. Phys. Soc. Lond., 49E, 59, 1937.

1. Self-consistent-field method. Translator’s note. ↩