Cohesive Forces in Metals

N. F. Mott, Bristol*

In the last few years major advances have been achieved in the theoretical interpretation of the metallic state; these advances, however, have so far remained the province of a very limited circle of specialists. In the literature, meanwhile, one still quite often finds indications of our complete ignorance both of the nature of the metallic bond and of the role played in metals by the conduction electrons. This is not so, and at present we may assert that, for monovalent metals such as silver and the alkali metals, it is already possible to give almost as complete a theoretical “description” as was given in its time by Born and his collaborators for simple polar crystals analogous to rock salt.

These successes were achieved chiefly through the application of quantum mechanics to all questions concerning the behavior of electrons inside a metal. Historically, the first applications of the new theories were made to questions of the heat capacity and magnetic properties of conduction electrons. In particular, the well-known works of Sommerfeld¹, Pauli², and Heisenberg³ dealt with these questions. With regard to electrical conductivity, the theory at once achieved major successes on the question of the order of magnitude of the resistance of normal metals and of the causes of the high resistance and anomalous behavior of certain alloys; however, the very important phenomenon of superconductivity still remains without explanation.

In the present article we shall confine ourselves, first, to problems of a mainly chemical character that arise in connection with the nature of the metallic bond, and then to questions of the crystal structure of metals and alloys, as well as to the theoretical calculation of cohesive forces. These questions are extremely interesting in themselves and, moreover, one may think that their complete solution will prove very significant for questions of the strength of metallic crystals, recently touched upon by Andrade¹⁵.

The theory of the metallic bond must make it possible to calculate certain quantities that can be checked by experiment. Of such quantities the most important is the energy of the metal. From it we at once obtain the heats of sublimation. If the energy of the metal

* Science Progress, 31, 414, 1937; translated by N. V. Belov.

can be determined for various values of the interatomic distances, then we shall also obtain the coefficient of compressibility and the specific volume at zero pressure. If it is also possible to calculate the energy for a number of different crystalline structures, then the structure for which the energy has the lowest value will evidently be the one that characterizes the metal under the given conditions. We shall begin by considering the methods by which the energy of a metal is calculated.

A monovalent metal, for example sodium, may be represented as a packing of positive ions (\(\mathrm{Na}^{+}\)) with an equal number of free electrons, which move inside the lattice according to the law that is to be investigated. At absolute zero temperature the ions may be regarded as being at rest*, in corresponding positions of equilibrium, namely at the sites of a body-centered cubic lattice. In the alkali metals we may assume that the ions are not in contact with one another and that, consequently, those electrons of these ions which are packed into the inner closed shells of the noble-gas type do not interact with analogous electrons of neighboring ions. Under this condition the energy of the crystal is made up of the following algebraic terms:

1. The mutual electrostatic potential energy of the positive ions.

2. The electrostatic energy of the free electrons in the field of the positive ions.

3. The mutual electrostatic potential energy of the electrons.

4. The kinetic energy of the electrons.

The proposition that electrons possess kinetic energy even at absolute zero temperature acquired the right of citizenship already in the earliest forms of quantum theory, as, for example, in Bohr’s theory of the hydrogen atom, in which orbital electrons are assumed to be in a state of continuous motion. The kinetic energy of conduction electrons in metals plays, as we shall see below, an important role in explaining the cohesive forces.

First of all, however, we must consider the potential energy. Evidently, the energy of interaction of the electrons with the ions cannot be calculated until we establish how the electrons are distributed in the lattice. But the methods of quantum mechanics never give exact indications of the position of electrons either in gas atoms or in solids. They give only the probability of finding an electron in one place or another. This, however, is sufficient for calculating the energy of the crystal. If we denote by \(P(r) 4\pi r^{2}dr\) the probability that a free electron is at a distance between \(r\) and \(r + dr\) from a given ion,

* For our purposes the insignificant zero-point energy of the ions may be neglected.

then the energy of the electrons in the field of the given ion can be represented in the form

\[ -\int_0^\infty \frac{e^2}{r}\,P(r)\,4\pi r^2\,dr . \tag{1} \]

The expression for the potential energy, \(-\frac{e^2}{r}\), is thus averaged over all possible positions of the electron. It must be borne in mind that the region of integration of expression (1) is the entire volume of the metal, and the result will be the energy of all the free electrons in the field of the given ion.

The quantity \(eP(r)\), therefore, determines the mean charge density in the metal at any distance \(r\) from the given ion, excluding, of course, those charges that belong to the electrons in the closed shells of the ions themselves. Calculations of the mean charge density in atoms were carried out by Hartree\(^{4}\) and his collaborators and are well known. The methods of determining the charge density in a metal are quite analogous. Fig. 1 shows the result of a calculation\(^{5}\) of the charge density in metallic sodium as a function of the distance along a line joining two neighboring ions. It is easy to see that, with the exception of the region immediately adjoining each ion, the charge density is almost constant. Thus the model of a monovalent metal will be a packing of positive ions floating in an almost uniformly distributed negative electricity.

Fig. 1. Distribution of charge density in metallic sodium (with the charge of the ions subtracted). The dashed lines denote the corresponding charge density in a free atom.

The next step in calculating the energy is the determination of the energy of interaction of the electrons with one another. For this we need the relative positions of the electrons. Wigner’s work\(^{6}\) showed that electrons tend, as far as possible, to move away from one another. Again we can speak only of probable positions, and it is in this sense that one may assume that, while moving through the lattice, an electron is always located inside a certain sphere, the probability of finding another electron inside it being very small.

At first glance, the summation of all the corresponding terms of the potential energy seems a very complicated task; however, as Wigner and Seitz showed, it is, on the contrary, very simple. If, for example, we take the body-centered cubic lattice, which characterizes all alkali metals, and draw in it planes bisecting the lines connecting each atom with its (eight) nearest neighbors, as well as with the (six) atoms of the second-nearest zone, then we divide the whole volume of the lattice into polyhedra, with one and only one atom inside each such polyhedron. Such a polyhedron (cubooctahedron) is shown in Fig. 2.

Fig. 2. Elementary polyhedron

The potential energy of the whole lattice, obviously, may be composed of the potential energy of charges of opposite sign located inside each individual polyhedron plus the mutual energy of the polyhedra themselves. The first term is calculated very simply. As was said above, each electron is surrounded by a sphere within which there cannot be (with very small probability) another electron, i.e., at any given moment there cannot be two electrons in one polyhedron, and therefore we may always neglect the mutual energy of two electrons within one polyhedron. Thus, in order to calculate the potential energy of the charges inside a polyhedron, we have to consider only the energy of one single electron in the field of the ion; this energy is easily calculated by formula (1). The energy of interaction of the polyhedra with one another may be neglected, for according to (1) this energy must be calculated as if the electronic charge were distributed over the whole polyhedron; and since all polyhedra are electrically neutral and, moreover, very close in shape to a sphere, the field outside each of them is very small. It would be exactly equal to zero if, instead of a polyhedron, we had a sphere, but even without this condition the corresponding energy does not exceed 1% of the total binding energy of the metal.

Thus the force that determines the cohesion of the metal is simply the attraction between each positive ion and the electron that has entered the given polyhedron according to the laws of probability; all other terms in the expression for the potential energy may be regarded, to a good approximation, as mutually canceling. At first glance it seems unclear in what way these electrically neutral polyhedra are bound to one another; it must be borne in mind, however, that, owing to their rapid motion through the metal, the electrons turn out to be more or less uniformly distributed, and therefore, if the crystal were somehow expanded, the electrons on average would find themselves farther from the ions.

The next question will be how the metal ions are held at some definite distance from one another,

The answer will be that the kinetic energy of the electrons increases greatly upon compression of the metal. Experimental proof that conduction electrons in metals have a kinetic energy much more significant than that given by the corresponding values of the kinetic theory of gases was provided in the work of O’Bryan and Skinner^8 on ultrasoft X-ray emission bands. Thus, they found that for the \(K\)-radiation of lithium the band has a width of \(4.2\ \mathrm{eV}\), instead of the sharp line that would correspond to the gas-kinetic value.

From the theoretical point of view, an electron moving with velocity \(v\) is associated with a wave (de Broglie wave), whose length is equal to \(\frac{h}{mv}\). If an electron is locked in a block of metal of length \(L\), then the de Broglie wave will be a standing wave and, as such, must have one of the possible lengths \(2L,\ 2\frac{L}{2},\ 2\frac{L}{3},\ldots,\ 2\frac{L}{n},\ldots\), etc. The corresponding velocities are expressed as

\[ \frac{nh}{2mL}. \]

and the corresponding energies as

\[ \frac{n^2h^2}{8mL^2}. \tag{2} \]

According to the Pauli principle, no more than two electrons can be in one and the same quantum state. As is well known, this principle has proved to be of exceptional importance for explaining the periodic table and the X-ray-optical levels of atoms. In our case it leads to the requirement that in our entire block of metal of length \(L\) there be only two electrons with energy \(\frac{1^2h^2}{8mL^2}\), then only two electrons with energy \(\frac{2^2h^2}{8mL^2}\), and so on. The kinetic energy of each electron, and consequently also their total kinetic energy, is inversely proportional to \(L^2\), i.e. varies proportionally to \(V^{-2/3}\), if the volume of the metal is denoted by \(V\).

In order to obtain an exact formula, it is of course necessary to take into account that the electrons in the metal can move in any directions. For simplicity we may consider \(N\) electrons moving in all directions inside a cubic box with edge \(L\); then the wave representing the electron is expressed by the formula

\[ \sin \frac{\pi}{L}(n_1x+n_2y+n_3z)\cos 2\pi\nu t \]

and the corresponding energy, analogously to (2), is expressed as

\[ E=\frac{h^2}{8mL^2}(n_1^2+n_2^2+n_3^2), \]

where \(n_1, n_2, n_3\) are integers. If we take Cartesian coordinates and assign \(n_1\) to the \(x\)-axis, \(n_2\) to the \(y\)-axis, etc., then it is easy to calculate that the number of quantum states with energies less than \(E\) is one eighth of the volume of a sphere of radius \(\sqrt{8mL^2E/h^2}\), equal to

\[ \frac{\pi}{6}\left(\frac{8mL^2E}{h^2}\right)^{\frac{3}{2}}. \]

Since in each state there can be two electrons, the maximum energy \(E_{\max}\) which an electron can possess will be obtained if this expression is set equal to \(\frac{1}{2}N\):

\[ \frac{1}{2}N=\frac{\pi}{6}\left(\frac{8mL^2E_{\max}}{h^2}\right)^{\frac{3}{2}}. \]

Putting \(L^3\) equal to the volume \(V\), we obtain

\[ E_{\max}=\frac{h^2}{8m}\left(\frac{3N}{\pi V}\right)^{\frac{2}{3}} \tag{3} \]

—the formula which was obtained at the time by Sommerfeld\(^1\). The corresponding wavelength—the smallest in the available set of electrons—will be

\[ \lambda_{\min}=2\left(\frac{\pi V}{3N}\right)^{\frac{1}{3}}. \tag{4} \]

It is easy to show that the total kinetic energy is equal to \(\frac{3}{5}NE_{\max}\). Thus, as before, we see that the kinetic energy of the electrons is proportional to \(V^{-\frac{2}{3}}\).

Formula (3), originally derived by Sommerfeld\(^1\), is in good agreement with the width of the spectral bands found by O’Bryan and Skinner.

This model, in which each electron is regarded as moving freely through the lattice, is, of course, only an approximation. In reality each electron undergoes numerous collisions with other electrons. A detailed analysis shows, however, that such collisions do not have any appreciable effect on the total kinetic energy.

Conduction electrons, therefore, determine two principal terms in the energy of a metal: the potential energy of the electrons in the field of the ions, which, since it is inversely proportional to the mean distance of the electrons from the ions, we may write in the form \(-A V^{-\frac{1}{3}}\), and the kinetic energy, which may be written in the form \(B V^{-\frac{2}{3}}\). The sum of both terms as a function of the different values of \(V\) is given in the form of a curve with a minimum in Fig. 3. The position of this minimum determines the corresponding specific volume of the metal, its depth gives the lattice energy, and, finally, the curvature gives the compressibility of the metal.

Calculations carried out in this way for the alkali metals\(^{5,7}\) have led to very good agreement with experiment. Moreover, by calculating the changes that occur in the expression for the electrostatic potential energy under deformation of the lattice, it proved possible also to calculate the elastic constants of single crystals of the alkali metals\(^{9}\). The results obtained are in good agreement with recent experimental determinations of these quantities\(^{10}\).

Fig. 3. Energy of electrons in the metallic lattice as a function of atomic volume

Similar calculations for copper, silver, and gold, however, led to values of the compressibility coefficient that were much too large. The cause of the discrepancy is evidently the circumstance that in these metals the positive ions themselves are much larger than the ions of the alkali metals. This leads to the fact that, much earlier than the minimum of the curve (Fig. 3) is reached, the ions come into contact with one another. From Born’s theory of polar crystals, analogous to NaCl, it is well known that both the sodium ion \(\mathrm{Na}^{+}\) and the chlorine ion \(\mathrm{Cl}^{-}\) strongly repel one another upon mutual contact. Evidently the same should be expected of the ions of copper and silver. Calculation\(^{9}\) shows that the corresponding repulsive forces in metals do indeed increase much more sharply than simple Coulomb repulsive forces.

Thus metals such as copper and silver may be regarded as a packing of hard spheres (the ions \(\mathrm{Cu}^{+}\), \(\mathrm{Ag}^{+}\)) immersed in a sea of negative charge. This negative charge, as a result of interaction with the positive ions, tends to contract and thereby packs the hard spheres densely. This is the reason why these metals are characterized by a structure corresponding to cubic close packing, i.e., such a structure in which the largest possible number of spheres is contained in a given volume. The purely electrostatic energy, as can be shown\(^{11}\), proves to be minimal not for

face-centered (closest-packed cubic), but for a body-centered structure, which thus is also characteristic of the alkali metals.

The electrostatic forces just discussed lead to energies which, in order of magnitude, are equal to the binding energies of metals, i.e. about 100 kg cal per 1 g atom. In order to arrive at the crystal structure of metals and alloys, it is necessary to calculate much smaller energy differences corresponding to the various possible structures. In many cases these calculations can be carried out without first calculating the total energy of the metal. As was first shown by Jones¹², in this way one may arrive at the well-known Hume-Rothery rule¹³ concerning the structure of alloys. The essence of this rule reduces to the following: each of the alloys such as Cu—Zn, Cu—Al, Cu—Sn, etc., as the concentration of the second component in it is increased, successively passes through one and the same more or less constant series of structures (phases). If one assigns to copper one valence electron, to zinc two, to aluminum three, and to tin four, then it turns out that the phase boundaries for all these systems correspond to one and the same ratio of the number of electrons to the number of atoms. Thus, for example, the α-phase, which is characterized by the same face-centered lattice as pure copper, becomes unstable when the ratio of the number of valence electrons to the number of atoms exceeds 1.35—1.4. The β-phase (body-centered cubic) is stable within narrow limits, close to the ratio of the number of electrons to the number of atoms \(3/2\). Finally, a very complex γ-structure arises at the indicated ratio, equal to \(21/13\).

We saw above that inside a metal the electrons move through the lattice with various velocities up to a certain maximum. To each velocity there corresponds a definite wavelength. These lengths form a series of decreasing values down to a certain minimum, and since, as was indicated above, to each wave state there can correspond no more than two electrons, it follows that the more electrons there are in a given volume, the smaller will be the corresponding minimum wavelength; namely, the corresponding minimum is given by the formula

\[ \lambda_{\min}=2\left(\frac{\pi}{3}\right)^{\frac{1}{3}}\left(\frac{V}{N}\right)^{\frac{1}{3}}, \tag{5} \]

where \(N\) is the number of electrons in the volume \(V\).

If we increase the concentration of (divalent) zinc in its alloy with (monovalent) copper, we thereby increase the number of electrons and lower the minimum wavelength. As a result, we shall approach the point at which the fastest electrons will have so small a wavelength that they will fall under

under the condition of Bragg’s law of reflection when they are incident normally on certain nodal planes of the lattice. The crystal, if one may put it so, avoids such a state for the following reasons.

The simple relation between energy and wavelength

\[ E=\frac{1}{2}\frac{h^2}{m\lambda^2}, \]

which can be derived for free electrons from both equations

\[ E=\frac{1}{2}mv^2 \quad \text{and} \quad \lambda=\frac{h}{mv}, \]

in this case ceases to be valid. Calculation shows that if one plots the energy \(E\) of an electron moving normally to a series of parallel lattice planes as a function of the reciprocal wavelength, one obtains the curve in Fig. 4. \(\lambda_c\) is here the critical wavelength, determined by Bragg’s law of reflection, and at precisely this wavelength we have a discontinuity in the energy curve. The width \(\Delta E\) of this discontinuity, as can be shown, is approximately proportional to the intensity of reflection of the X-ray beam from the corresponding series of parallel nodal planes. The electron cannot have an energy that would be characterized by an ordinate ending within \(\Delta E\); electrons with such energies, upon entering the crystal from outside, would undergo total reflection.

Fig. 4. Energy of electrons in a metal as a function of wavelength

The existence in the energy spectrum of finite intervals for which total reflection takes place in the lattice was demonstrated directly experimentally by Davisson and Germer in the reflection of cathode rays from a nickel crystal, and was interpreted by Bethe[^14] in the spirit of this theory. These forbidden energy intervals are analogous to the finite reflection intervals in Ewald’s theory of X-ray reflection.

From these results, illustrated by Fig. 4, it follows that electrons whose velocity is close to the critical velocity but smaller than the latter undergo a decrease in energy, whereas electrons that have a somewhat greater velocity, on the contrary, increase it still more. Thus a structure in which not one of the electrons has a velocity sufficient for reflection will have a low total energy and therefore will be stable, whereas if the number of electrons increases so much that the wavelengths of some of them become smaller than \(\lambda_c\), their energy rises sharply, and thus the structure proves unstable if another structure is possible in which, by a different arrangement of the nodal planes, this circumstance can be avoided.

Let us consider the case of closest cubic packing. The earliest reflection here can occur from the planes (111), which in this structure are the planes most widely separated from one another. The corresponding wavelength is

\[ \lambda_c=\frac{2}{\sqrt{3}}a, \]

where \(a\) is the lattice constant. Since there are four atoms in the unit cell, this expression may be rewritten in the form

\[ \lambda_c=\frac{2}{\sqrt{3}}\left(\frac{4V}{N_A}\right)^{\frac{1}{3}}, \]

where \(N_A\) is the number of atoms in the volume \(V\). Comparing with (5), we see that reflection will first take place when the number of atoms becomes such that

\[ 2\left(\frac{\pi}{3}\right)^{\frac{1}{3}}\left(\frac{V}{N}\right)^{\frac{1}{3}} = \frac{2}{\sqrt{3}}\left(\frac{4V}{N_A}\right)^{\frac{1}{3}}, \]

i.e.,

\[ \frac{N}{N_A}=\frac{\pi\sqrt{3}}{4}=1.362. \]

This quantity coincides almost exactly with the value derived above for the ratio of the number of electrons to the number of atoms at which the \(\alpha\)-phase actually becomes unstable. The experimental values are given below.

In the \(\beta\)-phase (body-centered cubic), reflection first occurs from the planes (110). Accordingly we obtain

\[ \lambda_c=\frac{2}{\sqrt{2}}a. \]

With two atoms in the elementary cell this corresponds to

\[ \frac{N}{N_A}=\frac{\pi\sqrt{2}}{3}=1.480. \]

We thus see that the body-centered cubic lattice in fact does not yet reach the limit of stability at the time when it has already been reached (in number of electrons per atom) for the face-centered lattice. However, the \(\beta\)-phase also becomes unstable upon a further increase of the ratio of the number of electrons to the number of atoms beyond \(3/2\). Table 1 contains

TABLE 1

a summary of the experimentally established phase boundaries in various systems.

By a completely similar calculation, Jones showed how the γ-phase arises. In this case, however, a complication arises in that one must take into account not the sequence of parallel planes which first begin to reflect the electron waves, but those planes for which the intensity of the X-ray reflection is greatest.

The two examples given serve as an illustration of how quantum mechanics makes it possible to provide an electronic theory of metals and their alloys. Up to now it has been possible to carry out calculations of the cohesive forces only for copper, silver, and the alkali metals. The theory has not yet been extended to metals like zinc and aluminum, which have more than one valence electron per atom, nor to the more complex transition metals, like nickel. The theoretical justification of the phase diagrams of various alloys given above is, of course, only qualitative in character. Exact values of the energy difference between the various phases could not be given, and, moreover, there is a large number of phase diagrams of other types, the calculation of which has so far not appeared possible at all. In any case, one may think that calculations of this kind will, in time, make it possible to arrive at an understanding of the nature of the metallic bond in all the most important metals and their alloys.

References

1. Sommerfeld, Z. Physik, 47, 1, 1928.
2. Pauli, Z. Physik, 41, 81, 1926.
3. Heisenberg, Z. Physik, 49, 619, 1928.
4. Hartree, Proc. Camb. Phil. Soc., 24, 89, 1928.
5. Seitz, Phys. Rev., 47, 400, 1935.
6. Wigner, Phys. Rev., 46, 1002, 1934.
7. Wigner and Seitz, Phys. Rev., 43, 804, 1933.
8. O’Bryan and Skinner, Phys. Rev., 43, 370, 1934.
9. Fuchs, Proc. Roy. Soc., A 153, 522, 1936.
10. Bender, Naturwiss. (in press).
11. Fuchs, Proc. Roy. Soc., A 151, 585, 1935.
12. Jones, Proc. Roy. Soc., A 147, 396, 1934.
13. Hume-Rothery, The Metallic State, Oxford, 1931.
14. Bethe, Ann. d. Physik, 87, 55, 1928.
15. Andrade, Science Progress, 30, 593, 1936.