MODERN THEORY OF SOLIDS¹

F. Seitz and R. P. Johnson, New York

Part I
Introduction
I. Development of the Theory of Solids up to Quantum Theory
II. The Pauli–Sommerfeld Theory of Metals
III. Band Theory of Solids

Part II
IV. Five Types of Solids
V. Band Theory and Bulk Properties

Part III
VI. The Surface of a Crystal
VII. Influence of Impurities on Bulk Properties
VIII. Plasticity and Fracture of Crystals

Part I

INTRODUCTION

Until recently there was no unified theory explaining the properties of various solids. For example, in order to interpret the properties of copper, diamond, and rock salt, it was necessary to devise three entirely different pictures of their internal structure. Attempts to establish a connection between these pictures, or to relate them to the properties of isolated atoms, did not lead to satisfactory results. In modern theory the atomic nucleus and the electrons are regarded as primary particles, and it is assumed that their behavior is always governed by the same quantum laws, regardless of whether the particles are isolated, or combined into small groups forming an atom or a molecule, or collected in large numbers to form a macroscopic solid. The theory of the solid state, based on this point of view, is, of course, a unified theory. How satisfactorily it explains the properties of real solids will depend on the adequacy of the quantum laws for the theory of the solid, and on the accuracy with which these laws are ap—

¹ J. Appl. Phys., 8, 84, 186, 246, 1937. Translated by T. P. Kozlyakovskaya.

known, and on the possibility of drawing from them the required numerous conclusions. The only decisive test for this is the agreement of theoretical predictions with experimental observations. Therefore, one of our tasks will be to show how successfully quantum theory interprets many of the observed properties of solids that remain unexplained in the classical picture.

Finally, we wish to give, in nonmathematical exposition, a survey of the present state and development of this theory, in order to show where it is successful and where still larger gaps exist in our knowledge.

About half of the present article is devoted to a discussion of the bulk properties of solids, such as cohesion, order of magnitude of electrical conductivity, etc., determined by the joint action of the basic atoms composing the solid. It is precisely in this area that the modern theory is most satisfactory. Surface properties, including the phenomena of the photoelectric effect, thermoelectronic emission, adsorption, and so on, are discussed by us in the last part of the article. This class of properties has been studied comparatively less well than bulk properties.

Properties depending on the structure of the solid and determined by the presence of impurities or by small distortions in the crystal lattice are only just becoming objects of theoretical investigation. However, they are extremely important for technology; we shall note, as areas in which properties depending on structure were taken into account before others, fluorescence, photographic sensitivity, and the tensile strength of materials. In the last part we shall consider the modern explanation of these properties.

1. DEVELOPMENT OF THE THEORY OF SOLIDS BEFORE QUANTUM THEORY

I. Classification of Solids

The purpose of a general theory of solids is to provide an explanation of the numerous facts accumulated during the last two hundred years. These investigations (necessarily empirical) led to the division of different solids into groups. The classification was based on chemical constitution, on crystallographic symmetry, and on such (tensor) properties of solids as conductivity, viscosity, and dielectric behavior. On the basis of a consideration of all these properties it was found that solids can quite naturally be grouped into the following five main groups: 1) metals, 2) ionic crystals, 3) valence crystals, 4) semiconductors, 5) molecular crystals.

Metals are distinguished by high electrical and thermal conductivity. Most electropositive elements form solids of this group. Ionic crystals are characterized by good ionic conductivity at high temperature, strong absorption in the infrared part of the spectrum, and good cleavage. Practically all salts of the type NaCl, MgO, etc., fall into this group.

Valence crystals, such as, for example, diamond and carborundum, have low electrical conductivity, great hardness, and poor cleavage; they are formed from the lightest elements of the middle columns of the periodic table of the elements. Semiconducting crystals are distinguished chiefly by weak electronic conductivity, increasing with temperature; examples of this group are CuO, Cu₂O, ZnO. In cleavage, hardness, and lattice structure, semiconductors resemble valence crystals, but they do not always obey the rules of valence in reactions. Finally, molecular crystals constitute the class to which the majority of solid organic compounds belong. They have low melting and boiling temperatures, usually evaporate in the form of stable molecules, and, in short, behave as regular aggregates of weakly bound molecules.

There are, of course, a large number of solids whose properties lie in the interval between these five principal groups. However, in the further discussion we shall adhere to this classification.

2. Lorentz’s Theory of Metals

Before 1900 theoretical work on solids was, for the most part, phenomenological. To establish connections between various physical properties, thermodynamics and electromagnetic theory were used. The result of this stage in the development of the theory was summarized in the books of Drude¹, Voigt², and others. The few attempts made at that time to interpret the physical properties of solids on the basis of a picture of their internal structure³ are of interest chiefly as historical stages.

Fig. 1.
a — distribution by energies at various temperatures \((T_1 > T_2 > T_3 > 0^\circ\mathrm{K})\) of the free electron gas in a metal (Lorentz’s theory, Maxwell–Boltzmann statistics). The mean energy \(\frac{3}{2} kT\) is approximately \(0.04\ \mathrm{eV}\) at room temperature. b — dependence of the energy of electrons on their momentum \(p\).

The first work that we wish to discuss here is Lorentz’s theory of metallic conductivity, proposed by him in 1905⁴. Lorentz regarded the valence electrons in a metal as particles of an ideal gas, moving freely in the lattice and distributed in energy according to Maxwell’s law. The number of particles of a given energy and the change of energy as a function of momentum are shown in Fig. 1. The mean energy of an electron is equal to \(\frac{3}{2}kT\), where \(k\) is Boltzmann’s constant and \(T\) the absolute temperature. At \(T = 0\) all electrons have energy and velocity equal to zero. When to a metal

a potential difference is applied, the electron gas, according to Lorentz’s views, moves through the metal lattice under the action of the force which the field exerts on the individual particles, as a result of which an electric current is obtained. Resistance is interpreted as the result of elastic collisions of electrons with the ions of the lattice. Lorentz showed that the magnitude of the resistance and its temperature coefficient in the range of temperatures close to room temperature could be explained on the assumption that all valence electrons are free and that the mean distance they traverse between collisions is approximately equal to the lattice constant.

Fig. 2.

Thermoelectron emission according to Lorentz’s theory. Only a small number of electrons in the Maxwellian distribution (fig. a) can escape from the metal, namely those reaching the surface with energies \(E_x\) \(\left(E_x = \dfrac{p_x^2}{2m}\right)\) greater than the work function \(W\) (shown in fig. b).

1 — electrons capable of participating in thermoelectron emission, 2 — surface

This picture of the internal electron gas also agreed satisfactorily with observations of the emission of electrons from heated metals. If the potential energy of electrons outside the metal is taken as zero, then the total energy of the electrons inside the metal is \(E = W + \dfrac{p^2}{2m}\), where \(W\) is the negative internal potential energy and \(p\) is the momentum. At absolute zero temperature \(p = 0\), and \(W\) is equal to the work required to remove an electron from the metal.

At higher temperatures, electrons in the Maxwellian distribution that reach the surface (at \(x = 0\)) with such a component of momentum \(p_x\) that \(W + \dfrac{p_x^2}{2m}\) is a positive quantity can leave the metal; the corresponding thermoelectron current can then be measured. Richardson’s work\(^{5}\) shows that this simple picture is able to describe all the main features of thermoelectron emission.

However, the theory encounters a great difficulty in assigning a heat capacity of \(\dfrac{3}{2}k\) to each valence electron. The specific heat of most metals, as well as of other solids, is experimentally, to within a few percent, equal to \(3R\) per mole, which is connected only with the thermal motion of the ionic lattice. With respect to specific heat, the valence electrons in a metal do not behave as they would if they were all free,

but since, if only about one electron out of a hundred were free, then in order to avoid this difficulty it would have been possible to suppose that most of the valence electrons are bound to atoms and only a few remain free and conduct the electric current; but then, in order to explain the observed electrical conductivity, it is necessary that the mean free path between collisions increase from a value close to the lattice constant to a value roughly one hundred times greater. It had to be admitted that, although the theory gave a plausible interpretation of a whole series of basic phenomena, on the whole it was nevertheless unsatisfactory. Before the development of quantum theory no new studies were carried out that freed the theory of metals from the indicated contradiction.

3. Madelung’s ionic model

The next important step in the development of the theory of solids begins with Madelung’s attempt6 to calculate the binding energy of ionic crystals, treating them as a lattice system of positive and negative point charges and determining the electrostatic energy of such a configuration. Born, Karman, and many others used such an ionic model for various calculations over a long and fruitful period7. They were able to explain many properties of ionic crystals semi-quantitatively. Thus, for example, the frequencies of infrared absorption were related to elastic constants by considering the dependence of both on the assumed forces in the lattice. Madelung’s model was most satisfactory for considering monovalent alkali-halide crystals, but became untenable for divalent compounds such as, for example, MgO. As has now become clear, the reason for this shortcoming of the theory lies in the fact that the ionic-lattice model conveys the properties of a solid less and less correctly as the atoms of the solid approach the center of the periodic table of the elements.

4. Valence and the van der Waals bond

The development of the electronic theory of valence in the works of Lewis8, Langmuir9, and others in the period following 1916 naturally led to theories of the electronic structure of valence crystals. These theories of valence did not attempt to interpret the electron bond dynamically, but sought to explain the properties of a number of crystals by the “tendency of atoms to form closed groups.” What relation this “tendency” had to other properties of atoms, such as, for example, to their spectrum, was not known.

Quantum theory partially substantiated this hypothesis, but also showed that the rules of valence are not universal and that each substance requires special and, unfortunately, complicated consideration.

The bonding in semiconductors remained unexplained until 1925. Madelung’s model is not applicable to these bodies; to the rules of valence

they do not always obey them. Bonding in molecular crystals was also not known in sufficient detail. It was usually assumed,^10 that molecules are held near one another by the so-called van der Waals forces, first introduced to explain deviations in the behavior of real gases from the ideal gas. It was supposed that these forces are caused by the interaction of instantaneously induced dipole or quadrupole moments in neighboring molecules; no precise picture of the dynamic behavior of the valence electrons had been created.

5. The Problem of Electrical Conductivity

Before concluding this historical survey, it should be noted that in all classical pictures of nonconducting crystals the valence electrons were regarded as bound to certain atoms or molecules or to definite regions between neighboring atoms. The absence of conductivity was explained by the fact that the electrons could not move over large distances. On the other hand, Lorentz’s theory, despite its shortcomings, firmly supported the view that some fraction of the electrons in a metal can move comparatively freely through the lattice. In Table 1 we give, for comparison, the magnitudes of the conductivity for some metals and some typical nonconductors at room temperature.

Table 1

Values of specific resistance for some metals and other solids (from International Critical Tables)

Solids	\(T^\circ\mathrm{C}\)	\(\rho\), ohm cm
Ag	20	\(1.6\cdot 10^{-6}\)
Al	20	\(1.8\cdot 10^{-6}\)
Be	20	\(10.1\cdot 10^{-6}\)
W	0	\(5\cdot 10^{-6}\)
B	0	\(1.8\cdot 10^{6}\)
C (diamond)	15	\(\sim 10^{14}\)
SiO\(_2\) (crystal)	20	\(\sim 10^{15}\)
Mica	—	\(10^{9}—10^{11}\)
Paraffin	—	\(10^{16}—10^{19}\)

It is obvious from Table 1 that the enormous difference in the values of conductivity between metals and other solids must be attributed to some fundamental difference between them. If the valence electrons are free in a metal and bound in nonconductors, then it seems reasonable to assume a strong difference in their other properties as well, apart from conductivity. However, in most other properties conductors and nonconductors are very similar to one another. In particular, in the various groups of solids (excluding molecular crystals) there are substances with both low and high melting points, and this compels one to suppose that the bonding forces in conductors and nonconductors are due to the same causes. The explanation of the sharp difference in the magnitude of conductivity in metals and nonconductors constitutes one of the most significant achievements of modern theory.

II. THE PAULI–SOMMERFELD THEORY OF METALS

The quantum laws for atomic systems were soon after their discovery applied to explaining the properties of solids. The first such application was made by Pauli, who succeeded in explaining the weak paramagnetism of metals, which had previously seemed puzzling.

Like Lorentz, Pauli assumed that all valence electrons in a metal are free, and took the energy \(E\) of each valence electron to be given by the classical relation

\[ E-\{V_0\}=\frac{p^2}{2m}, \tag{1} \]

where \(V_0\) is the potential energy, regarded as constant inside the metal, and \(p\) is the momentum. By analogy with the quantum laws of atomic systems, he assumed that not all values of the momentum vector \(\mathbf p\) are allowed, but only those for which the components \(p_x\), \(p_y\), and \(p_z\) satisfy the relations:

\[ p_x=\frac{n_x\hbar}{L_x},\quad p_y=\frac{n_y\hbar}{L_y},\quad p_z=\frac{n_z\hbar}{L_z}. \]

Here \(n_x\), \(n_y\), and \(n_z\) are arbitrary integers; \(L_x\), \(L_y\), and \(L_z\) are the lengths of the edges of the crystal specimen along the \(x\), \(y\), and \(z\) directions, respectively; and \(\hbar\) is Planck’s constant divided by \(2\pi\).

This means that the permissible energy values of electrons in a crystal of finite dimensions are distributed not continuously, as in Lorentz’s theory, but discretely. However, the difference in energy between neighboring energy levels turns out to be so small for crystals of size of the order of \(1\ \mathrm{cm}\) that the energy spectrum may always be considered continuous, except in the case of application of another quantum rule—the Pauli exclusion principle.

According to the Pauli principle, on an energy level characterized by the triple of numbers \(n_x\), \(n_y\), \(n_z\), there can be only two electrons with opposite directions of the spin vector. No more than two electrons in the entire crystal can have the same momentum vector and the same energy. At absolute zero temperature, in the absence of an external field, only the very lowest levels are filled \((n_x=n_y=n_z=0)\), and for a metal with one valence electron per atom we obtain the diagram of energy levels shown in Fig. 3. The filled band of levels is directly adjacent to empty quasi-continuous levels. The average energy of the electrons at \(T=0\) is not zero, as in Lorentz’s theory, but is equal to several electron-volts, and the average momentum of an electron accordingly proves to be greater than the classical value. In the filled band, for each triple of values—

Fig. 3. Distribution by energy at \(T=0\) in the case where free electrons obey the exclusion principle. Filled levels occupy a region of several electron-volts.

\(1\) — energy spectrum, \(2\) — occupied levels.

two electrons correspond to each state \(n\). The total magnetic moment of the crystal is zero, since each electron with spin in one direction is compensated by another with spin in the opposite direction.

In the presence of an external magnetic field of strength \(H\), the energy of a free electron can no longer be determined by the simple relation (1), but changes by the amount

\[ \pm \frac{e\hbar}{2mc}\cdot H, \]

where

\[ \frac{e\hbar}{2mc} \]

is the magnetic moment of the rotating electron. The sign of this quantity depends on whether the electron moment is parallel or antiparallel to the field \(H\). The energy level associated with a given momentum thus splits into two levels: one, corresponding to a moment parallel to the field, is lowered by

\[ \frac{e\hbar}{2mc}\cdot H, \]

while the other, corresponding to a moment antiparallel to the field, is raised by the same amount (Fig. 4). In the distribution that gives the minimum total energy, equal numbers of electrons no longer correspond to the two spin directions. The number of electrons with moment parallel to the field increases, the number of electrons with antiparallel moment decreases, and the crystal acquires a magnetic moment.

Fig. 4. Shift of energy states with different spin in a magnetic field (greatly exaggerated). In equilibrium both bands are filled to the same height, so that when a magnetic field is applied there are more electrons with magnetic moment parallel to the field than with antiparallel moment.

\(1\) — antiparallel moment, \(2\) — parallel moment

The magnitude of the magnetic susceptibility calculated from this model agrees, in order of magnitude (\(10^{-6}\) CGS), with the measured susceptibilities of simple metals. The discrepancies (within an order of magnitude) may be attributed chiefly to the electron interaction, which is not taken into account in this picture.

The success of the first application of quantum theory led Sommerfeld to the idea of revising Lorentz’s theory of metallic conductivity by using the new concepts. If the gas particles obey the exclusion principle, then the energy distribution changes with temperature according to the Fermi–Dirac function, not Maxwell’s. The difference between these two statistics becomes appreciable at low temperatures or in the case of high gas density. The density of the electron gas (valence electrons) in metals is so great (for example, for Na \(2.56\cdot 10^{22}\) particles/\(\mathrm{cm}^3\)) that Maxwell’s law is wholly inapplicable in this case. The Fermi–Dirac distribution (more correct for electrons in a metal) is shown in Fig. 5 for several temperatures. An increase in temperature changes the distribution function

compared with the function for \(T=0\) only near higher energies, and in such a way that the abrupt drop is replaced by an exponentially decreasing tail. Most free electrons have energies corresponding to the flat part of these curves. Although these electrons move freely through the lattice, in two respects they behave as if they were bound.

First, as is seen from Fig. 5, they do not contribute to the specific heat. When the temperature of the metal is raised, only electrons already located near the boundary of the distribution in energy can acquire still more energy. Indeed, by the exclusion principle, an electron located on one of the lower levels cannot increase its energy by small portions, since the nearest level to which it could pass is already occupied. Thus the main difficulty of the Lorentz theory is automatically eliminated.

Fig. 5.
\(a\)—distribution by energies for several temperatures \((T_2>T_1>T_0=0^\circ\mathrm{K})\) of a free electron gas obeying the exclusion principle (Fermi–Dirac statistics). The energy \(\varepsilon\) is equal to several electron-volts. \(b\)—dependence of the electron energy on the momentum (the same as in Lorentz’s theory—see Fig. 1).

Second, if an emf is applied to the metal (and this leads to the appearance of a current in the direction of the field), then only those electrons which are on the upper filled levels of the energy spectrum, near unoccupied states, can change their momentum. As Bloch showed, the probability that, under the action of the field, an electron will change its energy by an amount greater than some small fraction of a volt is vanishingly small, except in cases of extremely strong fields. Electrons located on the lower energy levels are not perturbed by the field and do not participate in the transport of current, since to each of them having momentum \(+\mathbf p\) there corresponds another with momentum \(-\mathbf p\).

Sommerfeld’s conductivity equation is very reminiscent of Lorentz’s equation, but it takes into account only the small number of electrons in states bordering on unoccupied levels, which play the same role as the free electrons in Lorentz’s theory.

In Sommerfeld’s equation, as in Lorentz’s equation, there is a parameter having the meaning of a mean free path. In order to satisfy the experimentally found values of the electrical conductivity, this quantity must be of the order of hundreds of interatomic lattice spacings, since only approximately \(10\%\) of the free electrons participate in electrical conductivity. Just as in Lorentz’s theory, in Sommerfeld’s theory such a large value of the free path must be adopted arbitrarily. From more exact calculations, taking into account the wave properties

electron (which we shall discuss shortly), it follows that such an order of magnitude for the mean free path must be expected in reality. It turns out, contrary to Lorentz’s opinion, that conduction electrons do not undergo collisions with the ions of the lattice as long as the latter are at rest. Bloch found that electrons undergo inelastic collisions with ions that are in a state of thermal vibration. The mean free path between such collisions is not directly connected with the interatomic distance and, at low temperatures, depends strongly on temperature. The temperature dependence obtained by Bloch is shown in Fig. 6. At room temperature the magnitude of the mean free path has the order of magnitude consistent with observations, but at low temperatures the agreement proves not entirely satisfactory. This disagreement, however, was more or less eliminated by subsequent calculations.

Fig. 6.
Temperature dependence of the mean free path of electrons in a metal \((T\) in °K), according to Bloch’s calculations. The dashed line shows the dependence predicted by Lorentz’s theory.

For thermionic emission the Pauli–Sommerfeld picture agrees with observations just as completely as Lorentz’s theory. The Fermi–Dirac and Maxwell distributions have the same exponential form for high energies, and only electrons of high energies take part in thermionic emission.

Thus the Pauli–Sommerfeld theory makes it possible to give an internally consistent interpretation of most properties of metals. The main question, however—what the fundamental difference is between a metal and a nonconductor—was not resolved by this theory.

III. BAND THEORY OF SOLIDS¹²

1. Electrons in a Periodic Field

In the last few years the general theory of the solid state has developed greatly; it has proved capable of explaining the distinction between conductors and nonconductors and has had considerable success in interpreting many other properties of solids. The modern theory differs from the Pauli–Sommerfeld theory by taking into account two important factors: the wave nature of the electron and the periodic distribution of the potential in the lattice of a solid.

The basic idea of wave mechanics may be formulated in a few words as follows: with every particle moving with momentum \(p\) there is associated a wave of wavelength \(\lambda = h/p\). Quad-

the square of the amplitude of this wave at each point of space is proportional to the probability of finding the particle there. For an electron of mass \(m\), moving with constant total energy \(E\) in a potential field \(V(x,y,z)\), the wave function \(\psi(xyz)\) is given by Schrödinger’s equation:

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

The reader is undoubtedly familiar with the success of these ideas in the interpretation of the atomic spectrum—with the quantization rules, the selection rules, and the uncertainty principle that naturally follows from it, as well as with the experiments of Davisson and Germer, Thomson, and others in the field of electron diffraction, which demonstrated the correctness of this equation.

Fig. 7.
Dependence of \(E\) on the wave number \(\sigma\):
\(a\)—for an electron in a homogeneous field, \(b\)—for an electron in a simple one-dimensional periodic field

We must first of all discuss the spectrum of possible energies of electron waves moving in a periodic potential field, and then the question of how these energy levels are filled.

The solutions of Schrödinger’s equation for an electron situated in a solid are most conveniently described with the aid of the wave vector \(\sigma\) (absolute value \(\sigma=1/\lambda\)). This quantity has the dimension of momentum divided by action. If an electron moves with constant total energy \(E\), in a constant potential field \(V_0\) (analogously to the way it is treated in the Lorentz and Pauli–Sommerfeld theories for the interior region of a metal), then

\[ E-V_0=\frac{p^2}{2m}=\frac{h^2}{2m\lambda^2}=\frac{h^2\sigma^2}{2m} \quad \text{and} \quad \mathbf{p}=h\sigma . \]

It follows from this that the energy \(E\) varies parabolically with the momentum \(p\), and consequently also with the quantity \(\sigma\) (Fig. 7).

However, in the true lattice of a solid the potential field in which the given electron moves is not homogeneous, but is a complicated function of the position of the electron. It depends not only on how the atomic nuclei are arranged, but also on how all the other electrons move. The distribution of the potential, the wave functions \(\psi\) of the individual electrons, and the charge distribution must be calculated simultaneously by approximate methods, into which we shall not go. It is, however, quite clear that the distributions of potential and electric charge must have the same periodicity and symmetry as the lattice. Strutt, Morse, Peierls, Brillouin\({}^{13}\), and others found that in such a periodic potential-offsetof

field the electron energy \(E\) is not proportional to \(\sigma^2\), as in a homogeneous field, but depends on the configuration of the field, as well as on the direction and magnitude of \(\sigma\). For a given direction of \(\sigma\), \(E(\sigma)\) has the form shown in Fig. 7, \(b\). At certain values of \(\sigma\) the energy \(E\) has discontinuities; some energy regions prove to be forbidden for the electron if its wave vector has the corresponding direction. For different directions of the vector \(\sigma\) these discontinuities

Fig. 8.

\(a\)—\(E(\sigma)\) for three typical directions in a cubic lattice [for example, for the directions (111), (110), and (100)]. Discontinuities occur at different points: \(\pm\sigma_a\), \(\pm\sigma_b\), and \(\pm\sigma_c\); the allowed regions overlap so that there are no completely forbidden energy regions. \(b\)—the discontinuities are so wide that there is no complete overlap and certain energy regions remain forbidden.

will, generally speaking, determine different forbidden energy regions. If the forbidden energy region for one of the directions of \(\sigma\) is not completely overlapped by the allowed regions for other directions, then in the energy spectrum there exists a gap into which no electron can fall, whatever momentum it may have. To find whether overlap of the allowed regions can occur, it is necessary to investigate \(E(\sigma)\) for each of the possible directions of \(\sigma\). Fortunately, this question can be answered by calculating \(E(\sigma)\) for a selected direction in the lattice, since it may be expected that the energy function behaves in this direction in the most characteristic way. Figure 8 shows two such possible cases. The allowed regions for the three directions of \(\sigma\) in Fig. 8, \(a\), overlap in such a way that absolutely forbidden energy values do not exist. In Fig. 8, \(b\), the allowed regions do not overlap completely, and forbidden regions exist.

Figure 9 presents the same energy curves for an electron moving in a “simple two-dimensional periodic potential field.” The discontinuities in the function \(E(\sigma_x,\sigma_y)\) occur on the sides of regular polygons with center at the point \(\sigma_x=\sigma_y=0\). For

of a three-dimensional periodic field the corresponding equal-energy surfaces are complex surfaces in $\sigma$-space and have discontinuities at the boundaries of regular polyhedra with centers at the origin.

In the Pauli–Sommerfeld theory, the energy spectrum was a practically continuous set of discrete levels—discrete because the components of momentum $p_x$, $p_y$, and $p_z$ were quantized. In the new picture discrete levels still exist, but the quantization is now obtained automatically, as a result of the wave nature of the electron. It turns out that the energy levels are no longer arranged so uniformly. If we have a crystal containing $N$ unit cells, then the energy spectrum of the electrons must be divided into groups of $N$ levels. Within each such group the $N$ levels are always so closely spaced that, for all practical purposes, they form a continuous band, just as in the Pauli–Sommerfeld picture. We shall call such a band of $N$ levels a zone. In the one-dimensional case (Fig. 7, $b$), each continuous region of energy between discontinuities of the function $E(\sigma)$ contains $N$ energy levels compressed into one zone; the same also holds in the three-dimensional case. The distribution of the discrete levels by energy thus depends directly on how the energy changes with the wave number. If the forbidden energy regions for one direction $\sigma$ are overlapped by allowed regions for other directions, then one may say that the zones overlap. If the allowed regions are not overlapped in this way, then the zones are separated by energy gaps in which there is not a single allowed level.

Fig. 9. Equal-energy lines (drawn thinly) for electrons in a simple two-dimensional periodic field. Discontinuities of energy occur at the boundaries of regular polygons (2 squares are shown). For a three-dimensional periodic field the discontinuities occur on the surfaces of regular polyhedra.

Which levels in this type of zone of the energy spectrum are occupied by electrons? Each level in a zone is doubly degenerate in the sense that it can contain two electrons with oppositely directed spins. The occupancy of a given level, according to the exclusion principle, is limited only to these two electrons; thus the electrons are distributed among the levels according to the Fermi–Dirac function, as in the Sommerfeld model. At $T = 0$ the very lowest levels are all filled, and each by two electrons, while all the levels above them are completely empty. At higher temperatures some of these empty levels may, under the influence of thermal excitation, be filled by electrons that have passed from the lower-lying levels.

It should be noted that if the forbidden energy regions are narrow (Fig. 8, a), the curve \(E(\sigma)\) everywhere, except in the immediate vicinity of the discontinuities, has an almost parabolic form, analogous to that for free electrons (i.e., for electrons in a homogeneous field). Accordingly, we may expect that the electrons move essentially as if, instead of the periodic one-dimensional potential field, there were a potential field. Conversely, we may expect fewer similarities with the behavior of free electrons if the allowed zones are narrow and the forbidden ones broad. This situation is depicted, for example, in Fig. 8, b.

There is another way of representing the relation between \(E\) and \(\sigma\), which does not emphasize the deviations from the behavior of free electrons but has compensating advantages. We shall illustrate it using the example of the one-dimensional case and then show the result of its generalization to an actual three-dimensional lattice. The curve \(E(\sigma)\) for a one-dimensional periodic potential field is shown in Fig. 10, a. The discontinuities occur at

\[ \sigma = \pm a, \pm 2a, \pm 3a \pm \cdots \pm na. \]

If we now shift the curve between \(a\) and \(2a\) horizontally back into the region \(-a > \sigma > 0\), then shift the segment between \(-2a\) and \(-a\) horizontally into the region \(0 > \sigma > a\), and do the same for all continuous pieces between discontinuities, we obtain the scheme of Fig. 10, b. \(E\) is then a multivalued function in the region \(-a > \sigma > a\), and each branch of it is continuous. Each branch corresponds to one zone, and we can renumber these branches consecutively, denoting them: zone I, II, and so on. We shall call this course of the curves \(E(\sigma)\) the reduced zone scheme, in contrast to the extended zone scheme discussed up to now.

Fig. 10.
Continuous regions of the curves \(E(\sigma)\) for the extended zone scheme a are shifted horizontally in order to obtain the more compact reduced zone scheme b.

Exactly the same process can be carried out in the two- and three-dimensional cases. In two dimensions the discontinuities occur on the boundaries of regular polygons centered about \(\sigma = 0\). A simple case is shown in Fig. 9. The area between any one of these polygons and the next, larger one is always equal to the area of the first smallest polygon; thus, the energy function for each such region can be transferred into this first polygon. Then \(E\) will be a multivalued function in the first polygon.

polyhedron, and the corresponding method of transfer gives discontinuous energy surfaces, similar to discontinuous curves in the one-dimensional case. In three dimensions discontinuities occur at the boundaries of the regular polyhedron centered about $\sigma = 0$; the volume between successive neighboring polyhedra is equal to the volume of the first, and all the energy functions of this region can be transferred into the first polyhedron in the form of a set of continuous functions $E_1(\sigma)$, $E_2(\sigma)$, etc. with $\sigma$ distributed only over this polyhedron. If this is done, curves of the type shown in Fig. 8 may be replaced by the system of Fig. 11. It turns out that two neighboring energy functions are in general not equal for any identical value of $\sigma$, which means that, generally speaking, they do not intersect. However, in special, degenerate, cases, they may be equal at $\sigma = 0$ or for all values of $\sigma$ in a given direction, as, for example, along an axis of symmetry of the lattice. In nondegenerate cases they are often equal at different values of $\sigma$; precisely this sort of equality corresponds to overlapping bands in the energy spectrum.

Fig. 11. Reduced band curves for the corresponding extended band schemes of Figs. 8, a and 8, b. In order to show the character of the behavior of the upper bands, a larger energy region has been taken. In the lower example the second strip contains 3 bands. Two of them are degenerate in the directions a and b, so that only 2 curves are shown. In direction c there is no degeneracy. This case is like that of LiF.

The reduced band schemes have certain advantages, in particular compactness. This advantage becomes appreciable if one has to depict four or five bands. The selection rules for the absorption of light are also expressed very simply with the aid of this scheme.

Let us emphasize here that we are considering a solid as a large molecule containing, for example, $8.5 \cdot 10^{22}$ atoms if the crystal specimen consists of copper (Cu) and has faces of $1\ \mathrm{cm}$. Each valence electron is in a definite energy state, just as in an isolated atom; such an electron moves freely through the solid, just as an electron in an isolated atom can move in the potential field of the nucleus and of the other electrons.

The probability of finding an electron at a certain point of the lattice depends on the configuration of the potential field and has the periodicity of this field (Fig. 12). This method of description was first applied to the study of metals by Bloch.

We have assumed up to now that each electron moves in the averaged potential field of the nuclei and all the other electrons, and the zonal structure of the energy spectrum is based on this assumption. A more detailed investigation shows that such a picture is not entirely correct.

Fig. 12.
The lower curve schematically depicts the potential field in the lattice; the upper curve, the corresponding periodic distribution of electronic charge

From the classical point of view, around each electron there is a region in which the probability of finding another electron is small because of their electrostatic repulsion from one another. The exclusion principle likewise forbids two electrons to approach too closely in space; the magnitude of the corresponding virtual repulsion depends on how close the momenta of the two electrons are to one another and on whether their spins are parallel or antiparallel. Both of these electron interactions have analogues in the wave-mechanical picture, and both were neglected in the zone theory described above. It turns out, however, that for many purposes the zone picture is a very good approximation for all valence electrons, in metals and in nonconductors (for example, even for the eight valence electrons in NaCl). We shall use the zone picture, keeping the indicated limitations in mind. Electrons in closed atomic cells, information about which is obtained in the study of X-ray spectra, remain in any picture of the solid firmly bound to individual atoms. We shall leave these electrons out of consideration.

2. Conductivity from the point of view of zone theory

Since in nonconductors the valence electrons move freely through the lattice, we must seek an explanation of the difference between conductors and nonconductors in the zone structure of the electronic energy levels. Let us examine several cases in detail.

Let us suppose, first, that the zones do not overlap, so that there is one lowest zone, containing \(N\) levels, separated by an interval of several volts from the next nearest zone (Fig. 13, a). Suppose that the solid under consideration has one valence electron per unit cell (for example, an alkali metal).

The lower band in this case will be half filled (Fig. 13, b). Electrons move in all directions in the lattice, and there is no resultant electric current. If a field is applied, some of the electrons with the highest energy pass into higher energy states lying immediately above the initial ones; statistical equilibrium is disturbed and an electric current appears. This is precisely the case considered by Sommerfeld. Thus it is evident that solids made of monovalent alkali elements with one atom in the unit cell must be metals.

Now suppose that each unit cell of the solid under consideration contains two valence electrons, while the bands are still separated by a wide forbidden interval. Then the first band will be completely filled (Fig. 13, c). If the applied electric field is not very large, then (since the lowest unoccupied levels in this case are very far from the highest occupied levels) practically no electron will be able to reach them. Statistical equilibrium is almost not disturbed, and electric current is practically absent.

Fig. 13.
a — typical energy spectrum for valence electrons in a crystal. The hatched regions are allowed, the white gaps are forbidden. b — the lower band is half occupied; the crystal has an odd number of valence electrons in the unit cell. This crystal is a good conductor. c — the lowest band is completely filled (an even number of valence electrons per unit cell) and is separated by a wide gap from the next allowed band. This crystal is a good insulator. d — the allowed bands overlap. This crystal is a good conductor, regardless of whether the unit cell contains an even or an odd number of valence electrons. e — the lowest band itself is occupied, but the forbidden band above it is narrow. Electrons are thermally excited into the upper band, and the “holes” that remain from them in the lower band give the crystal a weak electronic conductivity at ordinary temperatures.

Simple calculations show that, in order to obtain a measurable current, fields of the order of \(10^6 \,\mathrm{V/cm}\) are required. Although the valence electrons move freely through the lattice, this substance is a good insulator. In reality, some substances with two valence electrons in the unit cell, such as, for example, Ca, are good conductors. We come to the conclusion that in this case the bands are not separated, but overlap (Fig. 13, d). Calculations show that the bands do in fact overlap both in the alkaline-earth and in the alkali metals. It is clear that the alkali metals would be conductors even if their bands were separated, whereas for the conductivity of the alkaline-earth metals the overlap of the bands is essential. If the filled and unfilled bands overlap, the substance will be a conductor regardless of whether the unit cell contains an even or an odd number of electrons.

On the other hand, all valence and ionic crystals have an even number of valence electrons in the unit cell. Since they are good insulators, the filled bands in them are separated from the unfilled ones. In order that there be no overlap, the allowed bands must be narrow; in fact, it turns out that precisely this narrowness of the bands characterizes the difference between nonconductors and metals.

If the energy gap is small (Fig. 13, e), then electrons from the filled lower band can be thermally transferred to the upper band, so that the solid acquires a measurable electronic conductivity, increasing with temperature.

3. Bands and atomic energy levels

Between the electronic states of isolated atoms and bands in a solid there are certain interesting correspondences. In many of the simpler solids each band can be associated with a certain definite atomic state; this association becomes clear if one depicts the behavior of the bands as the interatomic distances of the lattice are continuously increased while its symmetry is preserved. The bands then become narrower and narrower and, finally, reduce to atomic levels.

In isolated atoms the energy levels are very narrow, and the individual valence electrons undoubtedly occupy “orbits” around their nuclei. Similarly, if the energy bands in a solid are narrow and if each band can, under such an expansion of the lattice, be assigned to the atomic level from which it was formed, then the individual valence electrons are located on a kind of quasi-atomic orbit around one or another nucleus of the lattice. Up to this point modern theory is in agreement (for some substances, as, for example, for alkali-halide compounds) with the classical conception that the “atomic character” of the constituent atoms is less disturbed in the formation of an insulating solid than in the formation of a metal. In no case, however, can it thereby support the classical point of view that definite valence electrons in insulators are bound to individual atoms.

If the formation of the lattice strongly affects the electronic orbits, as is the case in practically all metals, then the bands may become very broad and overlap in a complex way; in this case the association of orbits with atomic states has more of a symbolic than a real character. If the overlap of bands brings filled and unfilled levels into close proximity, then the solid is a conductor. Such overlap becomes very probable if the atomic states are situated very closely, as occurs in almost all atoms with a large atomic number.

In alkali-halide compounds it is simplest to relate the bands to the energy states of isolated ions. According to this conception, the \(3s\) and \(3p\) bands of the \(\mathrm{Cl}^{-}\) ion can be unambiguously distinguished. These bands are narrow at the equilibrium lattice dimensions, and the atomic (ionic) character of the electronic orbits is preserved to a considerable degree.

If the atomic states are widely separated, as, for example, in carbon, it may happen that, when the lattice dimensions are reduced, the bands meet, intersect, and then separate into groups (Fig. 14), with the different components of each group being able to originate from different atomic levels[^14]. Such a solid cannot conduct current if the filled levels extend to the boundaries of the forbidden region. Cases of this kind we shall discuss more fully in the next part.

In constructing molecular crystals, we may proceed from one of the following two assumptions. Observing the electron-energy states of the molecules composing the lattice as the dimensions of the latter are decreased, we expect that, at the true intermolecular distance for solids, these states have not yet broadened so much as to overlap. If one starts from the energy states of atoms, then one may expect (in particular, if we are dealing with carbon) that the bands will overlap in a complicated way and, at the actual interatomic distances, will be separated into groups. The final distribution of the bands will, of course, be one and the same, whatever starting point we choose.

Fig. 14. Semiquantitative representation of the band scheme of diamond as a function of the lattice dimensions

4. Bond energy

The bond energy of a solid—the work necessary to separate one gram-molecule into its constituent atoms—is the algebraic sum of the energies of the electrons in occupied states and the energy of nuclear repulsion. The electrons, taken together, have a smaller energy in the solid than in the isolated atoms, and therefore the term corresponding to the bond energy favors bonding. Nuclear repulsion, of course, increases when the distance between atoms is decreased, so that the corresponding term reduces cohesion. The sum of these terms is small in comparison with each of them separately; therefore, in calculating these terms, care must be taken so that their sum deserves confidence. The energy of nuclear repulsion can be calculated comparatively easily, as can the energy of the electrons in closed atomic shells, on which the formation of the solid has little effect. The difficulty is caused by the valence electrons, moving along complex orbits in the lattice. If we have their energy spectrum (the band diagram) at the true lattice dimensions, then, in the first approximation, the total energy of all valence electrons is the sum of the energies of the individual

electrons in occupied levels. But to this there must always be added a correction that takes into account the electronic interactions which, as we noted above, band theory neglects.

The correction term obtained by taking into account the virtual repulsion introduced by the exclusion principle is usually called the exchange term. The term due to the fact that two electrons tend not to be close together as a result of their mutual electrostatic repulsion is called the correlation term. Both terms usually favor binding, and both are of the same order of magnitude as the binding energy itself; therefore they must be taken into account in order to obtain the correct value of the binding energy. If they are neglected, the calculated value of the binding energy usually has the wrong sign, so that the solid turns out to be highly unstable at \(T=0\), contrary to the fact that substances form solids at sufficiently low temperatures.

We emphasize that there is nothing mysterious in these exchange and correlation energies. Basically, the cohesive energy of a solid is obtained from the familiar Coulomb attraction1 between the positive nucleus and the electrons. From this must be subtracted the energy of ion–ion repulsion and that part of the Coulomb energy of electron repulsion which is not retained in isolated atoms. The exchange and correlation energies are important because, although they are relatively small, they balance the other large terms. In reality they are only corrections to the energy of repulsion of electrons from one another, since this energy is for the most part included in the simple band picture, and these corrections must be introduced only because the band picture by itself incompletely describes the repulsion of electrons from one another.

Can the band picture be modified so as to include these detailed electronic interactions completely? Unfortunately, it cannot without losing its most important practical features. If we assign an electron to a particular energy level in a band system, we are, as it were, assigning it to a definite orbit in the solid and assuming that it remains on this orbit, even though its path may be changed by some external field. If we consider the detailed electronic interactions—“electronic collisions”—then it becomes more and more impossible to ascribe to a given electron a definite constant orbit. Instead of energy levels for individual electrons, we must then speak only of the fact that \(n\) electrons in the aggregate occupy \(n\) levels. Exactly the same problem undoubtedly arises in considering the electronic structure of complex atoms. The approximation that we have used in band theory, and which corresponds to each electron moving in the mean field of the nuclei and the other electrons, is called the Hartree approximation. The band picture is useful and corres—

serves its purpose if certain general properties of the solid are being discussed, but we must not, in doing so, look too deeply into the exact character of the motion of the individual electrons.

It must be noted that binding energies are computed in the same way, at least in principle, for all solids. The great difference in the observed values among different solids results from differences in the details of the distributions of the energy states and from the way in which these states are occupied by electrons. In what follows we shall note cases in which particular features of the distribution of electronic levels can explain the observed peculiarities in the cohesive properties of crystals.

We shall begin the next part with a discussion of particular types of solids and shall see how the band picture explains many of their properties.

Part II

IV. Five Types of Solids

1. Pure Metals[^15]

The theory set forth in the preceding part of the article was applied to a quantitative study of the alkali metals, and the accuracy of the band spectrum obtained was checked by direct calculation of the heats of sublimation. The curves of the change of energy as a function of the wave number for Na and Li are shown in Fig. 15, and the spatial distribution of the electrons in Fig. 16. The lattice of these metals is body-centered, but the result would be almost the same if they were face-centered. In both substances the discontinuities in the curves \(E(\sigma)\) are very small, and there is considerable overlap of bands. The charge distribution inside the lattice accordingly proves to be completely

Fig. 15. The curves \(E(\sigma)\) for Na and Li show the first series of discontinuities for three directions of the lattice. Along the abscissa is plotted \(\sigma \cdot d\), where \(d\) is the lattice constant (\(d\) is the edge of the cube, \(4.23\ \text{Å}\) for Na and \(3.46\ \text{Å}\) for Li). The curves for Na lie nearer to the free-electron parabola than the corresponding curves for Li. The lower portions (up to 0.6 on the abscissa) determine the filled levels. The energy is everywhere expressed in electron volts, and the zero of potential is taken outside the metal.

uniform and, thus, these substances correspond fully to Sommerfeld’s picture: the periodicity of the lattice field does not have a great influence on the motion of the electrons. In Fig. 17 it is shown schematically how zones are formed from the energy levels of atoms and how they overlap when the interatomic distances decrease. Such behavior is characteristic of all monovalent metals, including the noble metals Cu, Ag, and Au[^16].

This behavior does not change greatly on going to the second column of the periodic table of the elements. The energy states of isolated atoms in this case are separated by wider intervals, especially for the light elements, so that one cannot expect the overlap of zones to occur to the same extent. On going to the third column, we find that, as follows from experiments, B is a semiconductor, and we conclude from this that, at equilibrium lattice dimensions, its zones either do not overlap at all or else separate again after overlapping. The study of diamond, as will be described below, shows that the latter supposition is more probable. We shall return to consideration of this later. The crystal structure of B has not been determined, but one can predict in advance that each of its unit cells contains an even number of atoms, since each B atom contains three valence electrons and, at the same time, this solid does not have the properties of metals, which would inevitably be manifested with incompletely filled zones. Al has one atom and, consequently, an odd number of electrons in the unit cell; thus, it is a metal irrespective of how its zones overlap. In reality one may expect that the arrangement of its zones is similar to that in B.

Fig. 16. Relative distribution of the charge of valence electrons between neighboring atoms in Na and Li. The flattening of the curves in the middle part indicates an approximately uniform distribution over the entire unit cell.

The light elements of the remaining columns of the periodic table are nonmetals; we shall discuss their properties later. On the other hand, the heavy elements of these columns practically all turn out to be metals. This is quite understandable, since, as we have already said, the energy states of the atoms of the heavier elements are situated with smaller intervals, and upon formation of the lattice the overlap of zones in this case is almost inevitable.

It is likewise possible to make a prediction regarding the binding energy. The overlap of closely lying levels causes a higher density of electronic states in solids made of heavy elements than in monovalent and divalent metals. Therefore

for valence electrons a large number of low-energy states is allowed without violating the exclusion principle. Fig. 18 shows the typical behavior of the bands of different elements in one of the long periods of the Mendeleev table.

The \((n)d\) band overlaps the \((n+1)s\) bands and the \((n+1)p\) bands. These \(s\), \(p\), and \(d\) bands contain, respectively, one, three, and five zones. If the atomic \(p\)-level lies low, we may roughly expect that half the levels of these nine zones lie below the atomic \(s\)- and \(d\)-levels; but if the \(p\)-level is high, there may be fewer of them. If the \(p\)-level is very high, the minimum number of lower zones will be about three, i.e. half of the six \(s\) and \(d\) zones. Since

Fig. 17.
Schematic representation of the behavior of atomic energy levels as the lattice constant of alkali metals is compressed. At the equilibrium distance \(A\), below the lower curve the spectrum is “continuous”

Fig. 18.
Schematic representation of the lowest band, containing 9 zones, for the case of a solid formed by an element from one of the long periods. The binding energy depends on the degree of filling of the band and has a maximum when this band is filled approximately up to the height of the atomic \(s\)- and \(d\)-levels

electrons fill the lower-lying zones, we may expect that the binding in this case increases; but if the filled levels lie higher than the midpoint between the \(s\)- and \(d\)-states of the atom, it may be expected that the binding decreases.

Consequently, we expect that the binding energy of metals in one of the long periods at first increases with atomic number, reaches a maximum for atoms containing from six to nine valence electrons, and then falls again. This agrees with the experimental observations that, in the first long period, iron, which has eight valence electrons, possesses the greatest binding energy, whereas in the third period tungsten, with six valence electrons, possesses the greatest binding energy. The data for the second period are still not sufficiently complete.

2. Ionic crystals17

The band structure and cohesion have been investigated quantitatively for the simplest monovalent halide compounds. The most

it is convenient to regard these bands as arising from the energy levels of the electrons of isolated ions, rather than of atoms.

Fig. 19 shows the behavior of the lower \(s\)- and \(p\)-valence levels, which form the lattice as the ions are brought closer together (the positive ion is denoted by \(I^{+}\) and the negative by \(I^{-}\)). The electronic levels of the positive ion rise, since the field of the neighboring negative ions becomes effective, whereas the levels of the negative ion fall owing to the influence of the field of the neighboring positive ions. As a result one obtains a lower-lying band \(s_I\), arising from the \(s\)-level of the negative ion, and a higher \(s_{II}\), arising from the \(s\)-level of the positive ion. Between these two bands there is a band \(p_I\), arising from the \(p\)-level of the negative ion. \(s_{II}\) and \(p_{II}\) are apparently overlapped by bands arising from higher ionic levels, and form a continuum for high energies; \(s_I\) and \(p_I\) contain one and three bands, respectively, and are filled with the eight valence electrons in the unit cell.

Fig. 19. Qualitative behavior of ionic energy levels upon compression of an ionic crystal to its equilibrium dimensions

The curves \(E(\sigma)\) and the charge distribution for LiF at the actual lattice dimensions are shown in Fig. 20. In this case \(s_I\) and \(p_I\)

Fig. 20.
\(a\) — allowed and forbidden energy regions for the three most important directions of LiF; \(b\) — distribution of the valence charge between neighboring F and Li nuclei (scales relative)

at the equilibrium distance, corresponding to \(r=A\) in Fig. 19, do not yet intersect. It must be noted that the greater part of the valence electronic charge is located near the negative ion; such behavior is typical of all monovalent halides. In other words, quantum calculations lead approximately to the same average charge distribution that was adopted in the Madelung–Born theory. The discrepancy between the classical and quantum pictures is connected

with the width of the energy bands, since it measures the degree to which the bond becomes similar to the bond in metals (and, as we shall see below, in valence crystals). In the classical picture this width would be zero. The binding energy, calculated on the basis of band theory, contains terms depending on the band width, in addition to the ionic terms, which alone were taken into account by Madelung, Born, and others. The additional terms compensate one another, which is probably what explains the success of the classical calculations.

Detailed calculations taking the new picture into account have not been carried out for divalent ionic crystals of the MgO type. Born’s results for the binding energy of these crystals often deviate quite substantially from experimental observations. We believe that the terms connected with the band width, which classical theory does not take into account and which practically compensate one another in alkali-halide crystals, are not compensated so successfully in the case of divalent solids. The equilibrium lattice dimensions probably correspond to the distance \(B\) in Fig. 19, where the bands \(s_{\mathrm{I}}\) and \(p_{\mathrm{I}}\) overlap. These crystals are, of course, nonconductors, since both bands are filled, but the width of the combined band is large and the bond therefore is to some extent similar to the bond of a metal or of a valence crystal. The electronic charge is also distributed nonuniformly, i.e., for the most part around O rather than around Mg. Therefore these crystals possess ionic properties, for example, ionic conductivity at high temperatures.

3. Valence Crystals

A semiquantitative study of the electronic states was carried out for diamond\(^18\); its atomic configuration \(2s^2 2p^2\) indicates that there are four valence electrons per atom and eight per unit cell. The atomic level \(2s\) gives rise to two bands in the energy spectrum of the solid, while the \(2p\) level forms six bands, altogether giving eight lower-lying bands. Figure 14 shows how they change with the distance between nearest-neighbor atoms or ions. At large distances the \(2s\)-bands form a band separated from the \(2p\)-band, which contains six bands. When the atoms approach one another, these two bands intersect, and at small distances the eight bands are divided into two groups of four bands each, one group lying higher and the other lower. This is the difference from the band intersection in monovalent halide salts, since the \(s\)- and \(p\)-bands there remain comparatively well separated. Evidently, the most stable state of the entire system will occur when the lowest band, consisting of four bands, is filled and the distance between nearest atoms corresponds approximately to the minimum of this lower band. In this state the occupied bands do not overlap the unoccupied ones, and the solid proves to be a nonconductor.

The energy spectrum of diamond differs from the spectrum of metals in that in it there occurs an astonishing separation of the bands after their

overlap. From the point of view of wave theory, this separation presupposes a concentration of the electronic charge along the direction between nearest neighbors. In other words, the distribution of electrons in valence crystals is less isotropic than in metals. Such localization of electrons was noticed very early in the history of valence theory and gave rise to the very useful concept of bound electron pairs. In reality, directed localization is never perfect, even in diamond.

If the most stable state is obtained at large distances before the eight zones overlap and split into groups of four, then in this case the solid naturally turns out to be a metal. This gives an explanation of the metallic properties of graphite. In the graphite lattice the atoms in a given layer are brought very close together (1.42 Å), but the layers are situated rather far from one another (3.69 Å). Conversely, in diamond each atom has four nearest neighbors, situated at the vertices of a regular tetrahedron. The band system of graphite will thus correspond roughly to its mean interatomic distance, which, as shown in Fig. 14, implies overlap of all eight zones; as a result of the overlap, conductivity appears.

Si has the same electronic structure as carbon, and all its modifications crystallize in the same lattice as diamond. We should expect that the system of bands arising from the \(3s\)- and \(3p\)-atomic states will be qualitatively the same as in diamond. However, the atomic levels are more closely spaced in Si. Accordingly, we cannot expect that the two bands into which the eight zones split will be as far apart in Si as in diamond. Indeed, in agreement with the first of these predictions, silicon at very low temperatures is a very poor conductor. At higher temperatures a weak electronic conductivity appears, increasing with temperature, which agrees with the second prediction.

Carborundum is one of the substances similar to diamond and silicon, differing from them only in that its unit cell contains two different atoms. The atomic states of silicon \(3s\) and \(3p\) lie above the \(2s\)- and \(2p\)-states of carbon. When the lattice is compressed, the bands arising from the \(2s\)- and \(3s\)-states merge and then separate again, forming two bands \(s_{\mathrm{I}}\) and \(s_{\mathrm{II}}\), one of which lies lower and the other higher than in the atom, as in ionic crystals. The levels \(2p\) and \(3p\) combine in a similar way, forming bands \(p_{\mathrm{I}}\) and \(p_{\mathrm{II}}\), each of which contains three zones. \(s_{\mathrm{I}}\) and \(p_{\mathrm{I}}\) overlap, forming a band corresponding to the lowest one in diamond and containing four completely filled zones. This overlap of \(s_{\mathrm{I}}\) and \(p_{\mathrm{I}}\) is practically the same as is obtained in MgO, BeO, etc. (distance \(B\) in Fig. 19). We see that there is no great difference between ionic and valence crystals. For example, one may expect that BeO has some directed localization of electronic charge and that SiC has some ionic properties. Since the atomic levels of carbon are situated lower,

than the corresponding levels in silicon, we must expect that the electron concentration will be greater near carbon. The unit cell will therefore have a dipole moment, as in ionic crystals. Strong absorption in the infrared part of the absorption spectrum is a qualitative confirmation of the existence of this dipole moment.

4. Semiconductors¹)

The interpretation of semiconductors on the basis of band theory is obvious: these are solids in which the distances between bands are so small that an appreciable thermal excitation of electrons from a filled band into an unfilled one is possible. From this point of view the electrical conductivity \(\sigma\) depends on \(E\)—the energy gap between the filled and unfilled bands—and on the temperature \(T\) in the following way:

\[ \sigma = C e^{-\frac{E}{2kT}}, \tag{2} \]

which can easily be shown; here \(k\) is Boltzmann’s constant and \(C\) is a coefficient that varies comparatively little with temperature. It is clear that the excited electrons are practically not restricted by the exclusion principle, since they are in a band with a large number of free states. As a result, when an electric field is applied they can immediately carry current. It should be noted that the almost filled band, from which these electrons have been removed by thermal excitation, can also participate in conduction, since in it there is formed a certain number of vacant states not occupied by electrons. The factor 2 in the denominator of the exponent in expression (2) is obtained as a consequence of this additional conductivity²). Expression (2) is similar to the expression for ionic conductivity of solids. In the case of ionic conductivity the energy \(\frac{E}{2}\) in expression (2) corresponds to the binding energy of an ion to a definite site in the lattice. For some semiconductors these two energies may be of one and the same order. In principle, ionic conductivity can be distinguished from electronic conductivity by electrolytic decomposition of the crystal.

Since the magnitude \(E\) is always finite, it follows from equation (2) that all nonconductors at sufficiently high temperatures must possess electronic conductivity. The magnitude of the energy gap \(E\) at room temperature varies greatly for different semiconductors. In fact, there is a continuous gradation

¹) Wilson¹⁹ and Gudden²⁰ are among the principal investigators in the theory of semiconductors. They assumed that the majority of semiconductors possess electrical conductivity owing to the presence of impurities in them. We shall discuss this point of view in the third part of the article. It does not refute the general position of this section.

²) The coefficient 2 is obtained in the expression for the number of excited electrons, calculated from the conditions of statistical equilibrium, and not as a consequence of the participation of holes in electrical conductivity. Translator’s note.

between insulators and metals, and, generally speaking, a sharp distinction is impossible; it can be established only by an arbitrary definition.

We find, for example, that the conductivity of monatomic solids belonging to one and the same valence group improves with increasing atomic number. Thus, for example, diamond is a nonconductor, silicon is a semiconductor, germanium is a fairly good conductor and is classified as a metal, while tin is an undoubted metal. The atomic energy states are arranged more and more closely in passing to the heavier elements of any valence group, and the overlap of the corresponding energy bands in solids becomes more and more sharply expressed.

Diatomic compounds behave in the same way as their constituent elements when the atomic weight of the latter is increased, and for the same reason. Thus, for example, BeO and CuO are good insulators, whereas CdS, Cu₂O, and ZnO are increasingly good semiconductors. Finally, alloys of metals either have a metallic character or are good semiconductors. A qualitative prediction of the magnitude of the electronic conductivity of a substance can be made on the basis of an investigation of the distances between the energy levels of the constituent atoms. If this distance is large, the substance, generally speaking, will be a poor conductor (except in cases where the unit cells contain an odd number of valence electrons), whereas if it is small, the metallic properties will be relatively clearly expressed.

Table 2

Magnitude of the energy gap \(E\) for certain elements and compounds (in electron-volts)

C (diamond)	7	CdS	0.7
B	2	Cu₂O	0.6–0.2
S	0.7	CuO	0.5
P (white)	1.2	ZnO	0.4–0.0
P (black)	0.2	WO₃	0.3
		UO₂	0.25
		CuS	metal

Table 2¹ gives some numerical values for characteristic groups of monatomic and diatomic solids. In all cases \(C\) is of the order of \(1\ \mathrm{ohm}^{-1}\,\mathrm{cm}^{-1}\); a more exact determination of it is difficult.

It remains to consider one more important property of semiconducting crystals, namely their frequent violation of the valence rule. We first note that in substances such as diamond the arrangement of the levels is characterized by the separation of a band consisting of four zones from the rest of the system, so that eight electrons can be firmly bound in their levels. Such a separation results from the large distance between the energy levels in the atoms and is connected with the localization of the electronic charge along certain directions in the solid. Since in all light elements the distances between atomic levels are sufficiently large, we may in general expect the presence of four low-lying zones in the re-

¹ Some of these data are taken from Gudden’s paper²⁰; the rest have been calculated from the conductivity values given in the International Critical Tables.

lattice with equidistant neighboring atoms, and consequently the fulfillment of the “rule of eight.” It may happen that, under certain conditions, the atoms form a lattice of another type (for example, graphite instead of diamond), in which the four levels are not separated from the rest of the spectrum. If such a modification exists, the “rule of eight” is violated. In elements with a large atomic number (characterized by a denser energy spectrum), in which the overlap of the bands is more pronounced, the separation of the four lower bands is less noticeable, and the “rule of eight” gradually loses its significance. In other words, in a stable configuration there may be more or fewer than eight electrons per unit cell, since the number of low-lying bands is not equal to four. One may expect that the tendency toward the formation of such “anomalous” lattices will be more evident when the overlap of the energy bands is greater, i.e., in metals. And indeed, the “combination relations” of alloy systems are difficult to interpret on the basis of the usual rules of valence. The significance of the Hume-Rothery rule will be noted by us later (see V, 3).

5. Molecular Crystals

Above we ascribed the insulating properties of these crystals to the narrowness (at equilibrium dimensions) of the bands obtained when a lattice is compressed from the electronic energy levels of molecules; of course, it is assumed here that the bands are completely filled with electrons, and this assumption is indeed guaranteed by the fact that such molecules have saturated valences.

In the expression for the binding energy of molecular crystals, obtained on the basis of band theory, the correction terms turn out to be the most important (see III, 4). The correlation term varies inversely as the sixth power of the intermolecular distance and is precisely the van der Waals interaction energy. The exchange term, which in this case favors repulsion, increases very rapidly as the distance decreases, and as a result equilibrium occurs at such lattice dimensions for which the energy bands are very narrow. All the remaining terms in the expression for the electron interaction energy, owing to the narrowness of the bands, prove in the crystal to be practically the same as in isolated molecules.

Let us note that the valence rules for molecules are interpreted from the quantum point of view in the same way as for solids (see IV, 4) and have the same limits of applicability. For molecules, as for solids, the rule of eight is not strictly fulfilled if the atoms composing the molecule have a continuous energy spectrum. In a crystal composed of molecules containing such complex atoms, other terms in the expression for the binding energy may become significant—the forces between molecules may become comparable with the forces between atoms in the molecule. In short, in a solid the molecules to a considerable extent lose their “molecular character.”

Let us emphasize once again that the classification of types of solids used by us is to a high degree arbitrary. In nature there exist gradations between any of these types. For example, in the group of solids NaF, MgF₂, AlF₃, and SiF₄ there is a transition from a “purely ionic” lattice with a high melting temperature, good electrolytic conductivity, etc., to an easily fusible crystal with typical molecular properties1. If, in accordance with this, the theory is applied successively to the members of such a group, then the relative importance of the various features of the general theory gradually changes.

IV. ZONAL THEORY AND VOLUMETRIC PROPERTIES

1. Heat of sublimation; thermal properties; phase transformations

The difficulties connected with calculating the heat of sublimation have already been discussed by us in the first part of the review (see III, 4). If the use of the “variational method” gives good agreement between observed and calculated quantities, this means that the picture of solids used for the calculation is sufficiently good, and the agreement of the results obtained cannot be explained by chance. Similar calculations have been made for the simplest metals and ionic crystals²². The results (for characteristic examples see Table 3) include, of course, exchange and correlation corrections to the zonal picture of electron motion, and the agreement obtained with experiment indicates that zonal theory is a good first approximation. These calculations are, generally speaking, extremely laborious; therefore their extension to more complex substances proceeds slowly. A good beginning was made by calculations for diamond and the noble metals—copper, silver, and gold.

Table 3

Characteristic values of the heat of sublimation (kg cal/mole)

Substance	Calculated values	Observed values
NaCl	175	183
Li	23	26
Na	34	39

In determining the thermal properties of solids (such as, for example, specific heat), the energy states of nuclear vibrations play a more important role than the electronic energy states. If both these kinds of energy states are known, then the entropy, free energy, etc., can be calculated by the methods of wave mechanics. Einstein, Debye, and others achieved significant results in this direction even before the creation of modern quantum theory, neglecting the participation of electrons in these phenomena. In recent investigations the corresponding correction has been introduced, and the vibrational energy spectrum of the solid has also been considered more accurately²³.

Formally, vibrational waves in a lattice are similar to electron waves, and the band method of description proves suitable also for vibrations. The modern study of the vibrational spectrum, in method and in the character of its results, is an extension of Debye’s investigations in approximately the same degree as band theory is an extension of Sommerfeld’s picture of the electron gas.

Two kinds of phase transformations are distinguished. Transformations of the first kind (such as ordinary melting, for example) occur abruptly when a definite temperature is reached. The corresponding explanation of the sharp transition temperature is as follows. Suppose that the crystal is regarded as a single atomic system and that the two phases \(A\) and \(B\) are regarded as two separate states of this system, with energies \(E_A\) and \(E_B\) and entropies \(S_A\) and \(S_B\), respectively. Then, according to the general theorem of statistical mechanics, the probability \(P_A\) that the system is in state \(A\), referred to the probability \(P_B\) that it is in state \(B\), is determined by the equation:

\[ \frac{P_A}{P_B} = e^{-\frac{\{(E_A-S_A T)-(E_B-S_B T)\}}{kT}}, \]

where \(k\) is Boltzmann’s constant. The numerator of the exponent (the difference of the free energies in the two states) is of the order of several calories, except for the narrow temperature region in which its sign changes. The denominator \(kT\) is only of the order of \(10^{-20}\) cal at ordinary temperatures. Correspondingly, \(P_A/P_B\) is practically either zero or infinity, except for a small temperature interval in which the differences between the free energies are of the order of \(10^{-20}\) cal (Fig. 21). The use of Boltzmann’s constant in this interpretation of the sharpness of the temperature transition includes the essential principle of band theory, which assumes that the crystal is a single atomic system.

Fig. 21. Above—a schematic representation of the free energies for two phases \(A\) and \(B\) in the region of the order of several degrees around the transition temperature \(T'\). Below—the corresponding changes of \(\lg \frac{P_A}{P_B}\); \(\frac{P_A}{P_B}\) is approximately equal to unity only in a very narrow region around \(T'\).

The second kind of thermal transitions (for example, the disappearance of ferromagnetism and of secondary structure in alloys) proceeds continuously up to the “Curie temperature,” at which the transition is completely completed (Fig. 22);

Fig. 22. Decrease of order in an alloy from complete order \((R=1)\) at \(T=0^\circ\mathrm{K}\) to complete disorder \((R=0)\) at the Curie point \(\theta\).

in some cases there are gradual changes, ending in a sudden termination at the Curie point. According to the view that has only just been developed, these transitions pass through a series of intermediate states, and the free energy of each state differs only slightly from the free energy of the immediately preceding or following state. Recent theoretical investigations of these “order–disorder” transitions[^24] relate them to a change in the entropy of a solid when its energy changes.

2. Elastic constants

The calculation of elastic constants is carried out by the same method and encounters the same difficulties as the calculation of binding energy. The quantity necessary for the calculations is the change in the total energy when the crystal is displaced from the equilibrium state. The simplest perturbation of a crystal from the standpoint of theoretical treatment is one in which all interatomic distances are changed by one and the same factor; from the corresponding change in energy the compressibility can be calculated.

Table 4

Characteristic values of compressibilities
\((\kappa^{-1}\ \mathrm{cm}^{2})\)

Solid	Calculated values	Observed values
Na	16	\(9\cdot 10^{-6}\)
Cu	0.69	\(0.7\cdot 10^{-6}\)
NaCl	8.4	\(4.24\cdot 10^{-12}\)

Good agreement between the values calculated by this method and the observed values was obtained for NaCl[^26] and for certain metals[^25] (Table 4).

The classical theory of ionic crystals assumes that the elastic forces are static forces of interaction between pairs of lattice points. For crystals having central symmetry, this assumption leads to definite equalities between the coefficients of elasticity relating the components of stress to the components of strain.

Experimentally these Cauchy–Poisson relations are not even approximately correct, with the exception of monovalent ionic crystals; and even in this case a discrepancy[^26] of about \(10\%\) is obtained. Their failure in the general case is easy to predict on the basis of band theory, since the interactions determining the total energy as a function of the lattice dimensions are not static but dynamic, involving moving electrons.

On the other hand, modern theory, as we have already seen from the foregoing, partly confirms the classical picture of ionic crystals; the discrepancy between these pictures, associated with the width of the electronic energy bands, may influence the elastic constants of alkali-halide salts, changing them by the above-mentioned \(10\%\).

3. Crystalline Structure

Usually a solid crystallizes in that structure for which the free energy is minimal. The simultaneous existence of allotropic forms is evidently caused by the fact that the time required for transition into the state with the lowest free energy may be very large. Therefore diamond and graphite exist simultaneously at room temperature, although one of them should be energetically more stable than the other. The explanation of such cases brings into consideration the absolute rate of chemical reaction, but this theory[^27] has so far not been developed sufficiently fully for it to be readily applied.

The general fact that the energy of a stable lattice differs very little from the energy of other configurations makes prediction of the lattice structure from general considerations still quite impossible. With approximate calculation methods the cohesion energy is rarely calculated, by band theory, with an accuracy better than 5 kcal/mol, while the difference in energies between stable and quite unstable lattices may be much smaller than this quantity. The best that can be expected here is the correct sign of this difference.

In the classical theory of ionic crystals[^28] of Madelung–Born, the lattice energy was determined exclusively by static interactions between pairs of ions that are centers of force. To obtain the energy, two terms were used: a Coulomb term of the type $\dfrac{e_1 e_2}{r}$, attractive for unlike ions and repulsive for like ions, and a term of the type $\dfrac{1}{r^n}$, repulsive independently of the sign of the ion pair. This latter term corresponds to the impenetrability of ions, and $n$ was chosen sufficiently large to make this term negligibly small for large $r$ and very large for $r$ smaller than the observed value of the lattice constant. As we have seen, band theory indicates that the lattice energy must also include other terms, arising from the finite width of the energy bands occupied by valence electrons; these terms do not depend at all on the distance between lattice sites. In alkali-halide compounds these terms have the smallest magnitude, and therefore the assumption of central forces may be approximately correct for this case. However, the classical theory cannot explain why the halide salts of Li, Na, and K are face-centered, whereas the halide salts of cesium are body-centered. From this we must conclude that even in monovalent ionic crystals small energy terms, dependent on noncentral forces, play an important role in determining the stable configuration. These terms amount to approximately 10% of the total binding energy, but, unfortunately, their quantitative investigation has so far been carried out only very crudely. Thus, qualitatively, modern theory removes the difficulties that arose in connection with the problem of lattice energy, but the necessary quantitative calculations have still not been made.

In the simplest metals the energy of the valence electrons depends hardly at all on the structure of the lattice, but depends strongly on the volume of the unit cell. This fact follows from the relative isotropy of the distribution of electronic charge, associated with a sharply expressed overlap of bands. Making use of this, one can immediately make two predictions: first, that the actual structure will be simple, since the anisotropy of the individual atoms is small (which does not favor the formation of an anisotropic lattice), and, second, that the energy of the stable lattice will differ only slightly from the energy of other simple configurations. Recently Fuchs^29, using band theory, found that in the alkali and noble metals (Cu, Au, and Ag) the structure is determined by small deviations in the laws of interaction of ions from the Coulomb law of repulsion. His calculations indicate the greater stability of the face-centered lattice in the noble metals. However, the body-centered structure of the alkalis has still not yet been explained.

The empirical Hume-Rothery rule for many alloys establishes that, for one and the same ratio of the number of valence electrons to the number of atoms in the lattice, the same phase structure is obtained. Jones^30 showed how this rule must be explained from the point of view of the band model. The number of valence electrons in a crystal is determined by the composition of the alloy, while the corresponding energy spectrum is determined by the arrangement of the atoms in the lattice—the phase structure—independently of what kind of atoms occupy each of these sites. The most stable phase structure is the one that gives an energy spectrum with states low enough to accommodate all the valence electrons. The stable phase structure is determined by the number of valence electrons in the whole crystal. The number of atoms in the unit cell and, therefore, the number of atoms in the crystal, is fixed. Accordingly, identical phase structures are obtained for a definite ratio of the number of valence electrons to the number of atoms.

The possibility of a metallic modification of solid hydrogen has recently been discussed in a number of works^31 in calculations of the energy of the simplest types of monatomic lattice. The results show that metallic hydrogen is less stable than the molecular crystal at ordinary pressures, having an excess energy of approximately 1 eV per atom. Whether this difference changes sign at higher pressures is still unknown.

4. Mechanical properties

Such properties as hardness, malleability, and strength are not entirely bulk properties, and may depend to a very large degree on the purity of the specimen, its polycrystalline structure, and the conditions on its surface. Therefore most of the known experimental facts are hopeless to interpret in the present state of the theory, which deals with an ideal lattice. We shall make only a general comparison of idealized materials.

The high malleability of pure metals and, in particular, the softness and plasticity of the alkali metals apparently follow naturally from the fact that the electronic energy of simple metals depends more strongly on the volume of the unit cell than on its shape. Thus, deformations can occur with small changes in energy.

The difficulty of cleaving an ideal crystal evidently depends on the energy of the new surface thereby formed. Classical theory predicts, in agreement with experimental observations, that face-centered ionic crystals should cleave most readily along the planes \((100)\), less readily along the planes \((110)\), and still more difficultly along the planes \((111)\). The reason for this is seen from Fig. 23.

The ions of different sign on one side of the \((100)\) plane alternate in both directions along the surface, and the attractive forces of the two sides of the plane are, to a considerable degree, compensated by repulsive forces. In the \((110)\) plane the ions alternate in only one direction, whereas in the \((111)\) plane all ions on one side have the same sign. This simple explanation is accepted, in the main, by modern theory as well, with the limitation that it applies only to such crystals (monovalent and, possibly, also to some divalent ionic solids) whose binding energy is determined chiefly by the Coulomb interaction of the ions. Its application, for example, to metals evidently requires caution. We should not expect good cleavage in such crystals as diamond, since in it many valence bonds intersect even the most favorable plane in the lattice, and in reality cleavage is determined by random defects in which stress is concentrated. The observation that the electronic part of the cohesive energy is localized depending on the distribution of electronic charge is confirmed if we recall that charge in the crystal is distributed so as to ensure a minimum of energy; in order for the distribution to change somewhat (which must occur when crystals are split), some work must be done.

Fig. 23. Arrangement of ions on opposite sides of simple planes of a crystal analogous to rock salt. The difficulty of cleavage increases in the order \((100)\), \((110)\), \((111)\).

The first stage of deformation of a pure metallic single crystal under tension is usually slip along certain lattice planes\(^2\). Since, as a result of slip, a new surface appears, the surface energy must be taken into account in determining which planes are slip planes; the magnitude of the surface energy has not been calculated because of mathematical

difficulties. We shall consider the effect of the accumulating irregularities of structure, which accompany slip and make it irreversible, together with other phenomena that accompany cold working.

5. Optical absorption

A solid absorbs light: 1) causing vibrations of the lattice and 2) exciting electrons into states of higher energy.

The first kind of absorption usually occurs in the infrared part of the spectrum and is observed in those cases where some possible type of lattice vibration leads to the appearance of a dipole moment. Thus, for example, an NaCl crystal acquires a dipole moment when the Na ions as a whole are displaced relative to the Cl ions, and in so doing it absorbs light corresponding to vibrations of this kind. Conversely, in diamond all atoms are equivalent in position and in charge, and no regular displacements of pairs lead to the formation of a dipole moment; in a pure specimen, therefore, no absorption in the infrared region can be expected1. A careful prediction[^236] of the structure of the absorption spectrum in the infrared region requires a complete investigation of lattice vibrations by means of quantum theory, but so far no quantitative results have been obtained in this direction.

From the existence of practically continuous bands of energy levels of valence electrons in a solid, one may legitimately expect the presence of continuous bands also in the spectrum of optical absorption, corresponding to the excitation of valence electrons. Such bands have in fact been observed in the visible and ultraviolet regions of the spectrum. Their structure can be predicted by applying quantum selection rules, if the distribution and population of the energy levels are known. The selection rules for volume absorption restrict it very severely: an electron from a given state cannot be transferred into some other state in that very same band, but only into one of the states in some other band, and moreover into one that has the same wave vector $\sigma$ as the initial state. This means, roughly speaking, that the momentum of the electron is conserved in the process of absorption, since the momentum of the photon $\frac{h\nu}{c}$ is negligibly small. The energy-conservation equation $h\nu = \Delta E$, of course, must also be satisfied. In the band scheme considered, the only allowed transitions are vertical ones (Fig. 24); the smallest quantum that can be absorbed is equal to the smallest difference in energy along the vertical between occupied and unoccupied states in different bands. Therefore, in a metal whose bands overlap, the lowest frequency of volume absorption is not equal to zero, since the nearest energy values corresponding to one and the same value of $\sigma$ are not equal to one another. If the bands are arranged—

arranged as in Fig. 25, a (this is typical for simple metals), then the electrons with the highest energy will be excited first, while in the case shown in Fig. 25, b, the electrons with the lowest energy will be excited first. The theoretical threshold for Na and Li is of the order of 2 eV.

In simple metals the energy states are most dense near the top of the lower band and near the bottom of the upper band.

Fig. 24.
Vertical transitions of electrons in the reduced-zone scheme, allowed in the case of excitation of electrons by absorbed light

Fig. 25.
The shaded regions show the occupied levels of the lower bands.

a—optical absorption near the threshold A leads to the transition of electrons to higher energy levels; b—the first to undergo transition are electrons with low energy. In Fig. a many transitions of type B are possible. In such a substance a small absorption peak near the threshold should occur.

(Fig. 25, a), therefore there should be an absorption maximum for quanta of some energy B, close to the absorption threshold A. With increasing frequency of the incident light, absorption, owing to transition into the first unoccupied band, begins to decrease and reaches a minimum (possibly even—a region of transparency of the substance appears) until the conditions necessary for the transition of electrons into the next unoccupied band are again attained. The bands become very dense in the region of high energies, since the high levels in an atom are closely spaced; therefore the absorption minima become progressively lower and less sharply expressed as the frequency of the light increases. The spectrum of volume absorption predicted in this way has, in general outline, the form shown in Fig. 26 (solid curve).

The observed absorption spectrum at small energies differs somewhat from that shown in Fig. 26 owing to weak surface absorption. The dotted curve (greatly enlarged) shows the usual form of this surface component, which is important for the photoelectric effect. Wood^34 found that in the alkali metals there is a region of transparency up to the frequency corresponding to point A in Fig. 26. This transparency cannot readily be detected in

in visible light because of the high reflecting power of metals. Insulators and metals should have approximately the same kind of volume-absorption spectrum, which is indeed observed in reality. The absorption threshold in insulators usually lies farther in the ultraviolet, since in them the filled and unfilled bands are separated by a wider energy gap.

Fig. 26.
The theoretically predicted spectrum of volume absorption of a metal. Peaks I, II, III, etc. correspond to transitions into the first, second, third, etc. unfilled bands. The dashed curve (enlarged approximately 500 times) represents surface absorption.

At very short wavelengths (below approximately 100 Å) excitation of electrons of the outer closed shell of the ionic residue begins to prevail over excitation of valence electrons, and a simple investigation by means of the band model is no longer applicable.

6. Photoconductivity23

Many insulators, after absorbing light, acquire a weak conductivity for some time. This property can be explained directly on the basis of band theory as follows: a small number of valence electrons, being optically excited, pass into the nearest free bands, where they are little affected by the exclusion principle and where they can carry current. Optical excitation in photoconductors simply replaces thermal excitation in semiconductors, and conductivity will exist as long as optical absorption exists. This explanation, however, contradicts the observation that if an insulating substance is purified, its photoconductivity becomes small, although the absorption remains almost unchanged. This fact has long been known for halides, and the latest experimental data23 show that, apparently, it also holds for diamond, which was considered by us earlier

Fig. 27.
Possible behavior of volume absorption in alkali-halide salts near the absorption threshold, as proposed in the work of Slater and Shockley. Electrons excited by absorption in peak A are associated with “positive holes” left by them in the lower band. The excitation corresponding to peak B is accompanied by such a pronounced coupling. Absorption of type A does not always bring the crystal into a photoconducting state.

as a typical “idiochromatic” crystal, photoconducting in the pure state.

Almost all the experiments were restricted to wavelengths above \(2500 \,\text{\AA}\), for which high-intensity light sources could be used.

The absorption of light without photoconductivity in presumably pure crystals leads to the conclusion that this absorption is caused either by imperceptible impurities and lattice defects, or by other factors that do not lead to the formation of free electrons, or, finally, that the application of band theory to the interpretation of absorption in the near ultraviolet leads to large errors. The second possible explanation of this phenomenon was discussed by Sletёr and Shockley\(^{36}\), who attempted to derive the conclusion that optical absorption can have a sharp peak near the absorption threshold without accompanying photoconductivity (Fig. 27).

After such absorption, the excited electron and the “positive hole” that it left in the lower band are considered to move together in the lattice. The stability of such a neutral pair in an electric field has not been investigated. Further theoretical work is needed, as well as extension of these experiments to the region of shorter wavelengths.

7. Metallic Conductivity

In the preceding part of the article we saw that the basic difference between conductors and insulators is a natural consequence of an energy spectrum of the band type. In the study of metallic conductivity, modern theory gained two important advantages over the theories of Lorentz and Sommerfeld: it was able to explain the temperature dependence of electrical conductivity and the anomalous galvanomagnetic effect.

Experience shows that the electrical conductivity of the purest metals near room temperature varies as \(\frac{1}{T}\), but increases more rapidly with further decrease of temperature and at very low temperature varies approximately as \(\frac{1}{T^5}\). This kind of dependence was difficult to explain by the theories of the electron gas. They led for the conductivity to an expression of the following form:

\[ \sigma = \text{const}\,\frac{e}{m}\,\frac{n \cdot l}{v_{\text{средн}}}, \]

where \(n\) is the number density of free electrons; \(l\) is the mean free path (partly determined by elastic collisions with ions of the lattice); \(v_{\text{средн.}}\) is the mean thermal velocity, and \(\frac{e}{m}\) has its usual meaning. According to Lorentz’s theory \(v_{\text{средн}}\) varies proportionally to \(T^{1/2}\); according to Sommerfeld’s theory it is practically independent of \(T\). Observations required that either \(n\) or \(l\) vary sharply when the temperature is lowered, but neither of these possibilities seemed-

acceptable. On the other hand, in the wave picture an ideal lattice must have a resistance equal to zero, since the “mean free path” in it is infinite. In reality, resistance is caused by the scattering of electron waves at places where the periodicity of the lattice is disturbed. At some temperature above \(0^\circ\)K, the thermal vibrations of the lattice ions disturb its ideal periodicity. In a quantitative treatment of this problem of electron scattering, the observed change of resistance with temperature is well confirmed. Let us note that impurity atoms in the lattice also disturb its ideal periodicity even at \(0^\circ\)K and thus cause the appearance of scattering of electron waves and, consequently, electrical resistance. This gives an explanation of the high resistance of certain alloys and of Matthiessen’s rule, according to which the resistance of a dilute solid solution is the sum of the normal resistance, depending on temperature, and another term, which increases with the concentration of the dissolved substance and is almost independent of temperature. Obviously, the explanation of the insignificant differences in the values of the specific resistivities of different metals requires a detailed study of the energy spectrum of the electrons and the vibrational spectrum of the lattice.

Of the galvanomagnetic phenomena, the Hall effect is best known. If a metallic strip through which a current flows in the direction \(x\) is placed in a magnetic field directed along the \(y\)-axis, then in the direction of the \(z\)-axis a small electric field arises—the Hall gradient. In some metals, for example Zn, Cd, and Pb, the direction of this gradient is such as if the current were carried not by electrons, but by positive particles. This anomalous Hall effect proves to be characteristic of metals in which the highest zone containing electrons is almost, but not quite, filled. This is a consequence of the fact that the energy states near the top of the zone correspond to a smaller group velocity (the group velocity of a wave is a measure of the velocity of the electron considered as a particle) than do states with lower energies in the very same zone. A detailed study of the behavior of electrons in an almost filled zone establishes the very convenient fact that, in all interactions with the electric and magnetic fields, the metal behaves as if a small number of unoccupied states near the top of this zone were filled with positive particles having a mass comparable with the mass of the electron, while the lower filled (by electrons) states were free (i.e., not filled with “holes”). In the anomalous Hall effect these positive “holes” move toward the negative electrode; since their charge and velocity have signs opposite to the corresponding quantities for a free electron, their path in the magnetic field is curved as the path of an electron would be curved, which also causes the appearance of the reverse Hall gradient.

We shall not attempt here to give a survey of the remarkable properties of the superconducting state\(^37\), into which many metals pass in a narrow temperature region characteristic of a given metal (about

to \(1^\circ\) K). In part, the phenomena of superconductivity can be investigated with the aid of thermodynamics and electromagnetic theory, but a complete satisfactory picture has not yet been found. Quite recently Slater\(^{38}\) proposed a point of view offering hopes for the resolution of this question. He considers the energy states of the metal as a whole and finds an indication of the existence of groups of discrete states corresponding to the superconducting phase. This idea cannot be adequately discussed within the band approximation.

8. Magnetic behavior

We can only briefly indicate the basic points of view of the modern theory on the magnetic behavior of solids. As in the classical theory, diamagnetism is explained by the orbital motion of electrons. Paramagnetic behavior is attributed to the magnetic spin moment of “unpaired” electrons (see II). In a solid these two tendencies counteract one another. Where the electron spins are constantly paired, as is the case in insulators with filled bands, only diamagnetism exists. The strong paramagnetism of rare-earth salts is explained by the presence of unfilled inner \(f\)-shells in the atoms. The problem of ferromagnetism has been investigated on the basis of Heisenberg’s assumption that the spontaneous arrangement of elementary magnets is a result of the exclusion principle. However, it seems more probable to suppose that a complete, unobjectionable solution of this question will not be found until the methods of approximate calculations have been substantially improved. For a more detailed acquaintance with the magnetic behavior of solids we refer the reader to the book by Mott and Jones\(^{39}\) (Ch. VI), where there is a fairly complete discussion of the modern interpretation of magnetic phenomena in various metals.

(Conclusion in the next issue)

LITERATURE

1. Drude, Lehrbuch der Optik, Leipzig, 1906.
2. Voight, Kristallphysik, Leipzig, 1910.
3. See, for example, Lorentz, Verh. d. K. Akad. v. Wet. Amsterdam, 18, 1879.
4. Lorentz, Theory of Electrons, Leipzig, 1923.
5. Richardson, Emission of Electricity from Hot Bodies, N. Y., 1916.
6. Madelung, Physik. Z., 11, 898, 1910.
7. Born u. Goeppert-Mayer, Handb. d. Physik, Berlin, XXIV/2, 1933.
8. Lewis, Valence and Structure of Atoms and Molecules, N. Y., 1923.
9. Langmuir, J. Am. Chem. Soc., 41, 868, 1909.
10. See the review by Kronig, Handb. d. Physik, XXIV/2, 272, 1933.
11. See the review by Sommerfeld and Bethe, Handb. d. Physik, XXIV/2, and also Brillouin, Quantenstatistik, Berlin, 1930.
12. This subject has been discussed in several review articles and books. See.

literature in Mott and Jones, Theory of Properties of Metals and Alloys, Oxford, 320, 1936.

1. Strutt, Ann. Physik, 86, 319, 1928; Morse, Phys. Rev., 35, 1310, 1930; Peierls, Ann. Physik, 4, 121, 1930; Brillouin, J. d. Physique, 1, 377, 1930, and also Quantenstatistik.

2. Kimball, J. Chem. Phys., 3, 560, 1935.

3. Wigner and Seitz, Phys. Rev., 43, 804, 1933; 46, 509, 1934; Seitz, Phys. Rev., 47, 400, 1935; Slater, Phys. Rev., 45, 794, 1934, Rev. Mod. Phys., 6, 210, 1934; Millman, Phys. Rev., 47, 286, 1935; a good survey is given in Mott and Jones, Theory of Metals and Alloys, Oxford, 1936, and in Froehlich, Elektronentheorie der Metalle, Berlin, 1936.

4. Krutter, Phys. Rev., 43, 654, 1935; Fuchs, Proc. Roy. Soc., A 151, 585, 1935.

5. Shockley, Phys. Rev., 50, 754, 1936; Eving and Seitz, Phys. Rev., 50, 760, 1936.

6. Kimball, J. Chem. Phys., 3, 560, 1935.

7. Wilson, Proc. Roy. Soc., A 133, 458, 1931; 134, 277, 1932.

8. Gidden, Erg. exakt. Naturwiss., 13, 223, 1934.

9. Grimm u. Wolff, Handb. d. Physik, XXIV/2.

10. Ionic crystals: a) Hylleraas (LiH), Z. Physik, 63, 771, 1930. b) Landshoff (NaCl), Z. Physik, 102, 201, 1936. Landshoff’s results were somewhat corrected in Table 1 of our review. Metals: the review is contained in the book by Mott and Jones, ch. 15, IV.

11. a) Blackman, Proc. Roy. Soc., A 143, 365, 1935. b) Barnes, Brattain and Seitz, Phys. Rev., 48, 532, 1935.

12. Bragg and Williams, Proc. Roy. Soc., A 145, 699, 1934; Bethe, Proc. Roy. Soc., A 150, 552, 1935.

13. Mott and Jones; Fuchs, Proc. Roy. Soc., A 151, 585, 1935.

14. Balamuth, Phys. Rev., 45, 715, 1934.

15. See the review in Eyring, Chem. Rev., 17, 65, 1935.

16. Born u. Goeppert-Mayer, Handb. d. Physik, XXIV/2.

17. Fuchs, Proc. Roy. Soc., A 153, 622, 1936.

18. Jones, Proc. Roy. Soc., A 147, 396, 1934. Discussion of this and other problems of alloys: see Bragg, J. Inst. Metals, 56, 275, 1935, and Hume-Rothery, Structure of Metals and Alloys, London, 1936.

19. Wigner and Huntington, J. Chem. Phys., 3, 764, 1935.

20. Schmid u. Boas, Kristallplastizität, Berlin, 1935.

21. Robertson, Fox and Martin, Phil. Trans. Roy. Soc., A 232, 463, 1934; Proc. Roy. Soc., A 157, 579, 1936.

22. Wood, Phys. Rev., 44, 353, 1933.

23. Reviews are devoted to this subject: Nix, Rev. Mod. Phys., 4, 725, 1932, and Hughes, Rev. Mod. Phys., 8, 294, 1935.

24. Slater and Shockley, Phys. Rev., 50, 705, 1936.

25. See the summary in Smith and Wilhelm, Rev. Mod. Phys., 7, 237, 1935.

26. Slater, Phys. Rev., 51, 195, 1937. See also Van Vleck, Electric and Magnetic Susceptibilities, Oxford, 1932.

27. Stoner, Magnetism and Matter, London, 1934; Bozorth, Elec. Eng., 54, 1151, 1935; Slater, Phys. Rev., 49, 537, 1936.

1. Robertson, Fox, and Martin[^33] found structure-dependent absorption in the infrared region, apparently arising from lattice imperfections. ↩↩↩