AVERAGE ELECTROSTATIC POTENTIAL IN A CRYSTAL1

The theory of the state of electrons in a crystal, in its applications to questions of electron diffraction, metallic conductivity (dependence on temperature, on an external magnetic field), etc., makes use of the assumption that the potential of the spatial lattice possesses threefold periodicity in its oscillations about a certain mean value. At present, with regard to this mean value of the potential it is customary to assume that, first, it is a specific constant of the substance and, second, it can be directly calculated from the distribution of charges inside the unit cell. The author of the article shows that both of these propositions are not justified.

To calculate the potential one must compute the values of \(\int E\,dl\) (where \(E\) is the electric-field strength) from infinity to the observation point. The path of integration must pass over the surface of the crystal, and the potential therefore depends on the values of the field at the surface. One must not allow (although, because of the polarizability of the structural units, the conditions at the surface are determined by the field inside the crystal lattice) that the influence of the surface conditions on the value of the lattice potential is a special problem of the dynamics of the crystal lattice.

A correctly chosen structural unit of a bounded crystal (for an unbounded crystal, as is known, structural units can be chosen with the greatest variety of symmetry) must possess charge equal to zero, and in a crystal with a center of symmetry it must, in addition, possess a dipole moment equal to zero. The structural unit always has a quadrupole moment, whose six components form a symmetric tensor

\[ Q_{ik}=\int_v x_i x_k \rho\,dv, \]

where \(v\) is the volume of the structural unit, \(\rho\) is the electron density, and \(x_i\) and \(x_k\) are coordinates whose origin is immaterial.

It can be shown that the mean value of the potential in the boundary layer, whose normal has direction \(n\), differs from the mean value of the potential outside the layer by

\[ -2\pi \frac{Q_{nn}}{v}. \]

The mean value of the potential (from the microscopic point of view) is the macroscopic potential. Therefore the same fact can be expressed in words as follows: at the surface of the crystal the potential undergoes a jump, or, in other words, at the surface of the crystal there is a double layer with moment

\[ \frac{1}{2}\frac{Q_{nn}}{v}. \]

It can also be shown that linear charges must be situated on the edges of the crystal.

It is obvious that, in the presence of double layers on the surfaces and linear charges, the macroscopic potential, or, in other words, the mean value of the microscopic potential, cannot be constant.

However, special cases are possible when the mean value of the potential inside the lattice is constant. In the case of a cubic crystal the tensor has spherical symmetry, and all \(Q_{nn}\) are equal to one another. The surface of the crystal corresponds to a closed homogeneous double layer; as is known, in electrostatics the potential inside this system is constant. Approximately the same is true for a needle-shaped crystal elongated in the direction of the axis of symmetry. In this case the tensor \(Q_{ik}\) has axial symmetry, and its components that play a role are equal to one another. Finally, the mean value of the lattice potential will be constant if the crystal is built of neutral atoms possessing spherical symmetry. The distortion of the lattice at the edges (additional polarizability of the surface atoms) is not taken into account in this calculation; the significance of this factor is quite considerable. In conducting crystals, the motion of charges ensures the constancy of the mean value of the potential. However, it is completely unclear on what basis one can suppose that this mean value does not depend on the magnitude and orientation of the faces of the crystal.

The experimental facts in this field are still very meager and do not help in resolving the indicated difficulties.

A. Kitaigorodskii, Moscow

Literature

1. M. v. Laue, Naturwiss., 28, 516, 1940; Z. Krist., 103, 1940.