Propagation of Electromagnetic Waves in a Metallic Conductor and Their Significance for the Problem of Superconductivity

R. de L. Kronig

The propagation of a plane linearly polarized electromagnetic wave of frequency \(\nu\) in a metallic conductor can be considered phenomenologically on the basis of Maxwell’s equations, if one assumes:

a) that the density of the electric current, oscillating with frequency \(\nu\), at every point of the conductor has the direction of the electric field of the wave \(E\);

b) that it consists of two parts, respectively equal to

\[ \sigma E \quad \text{and} \quad \frac{\varepsilon - 1}{4\pi}\frac{dE}{dt}, \]

the first of which is in phase with \(E\), while the second differs in phase by \(90^\circ\) from \(E\). \(\sigma\) and \(\varepsilon\), being functions of \(\nu\), may be called the coefficient of electrical conductivity and the dielectric constant of the metal for frequency \(\nu\).

The refractive index \(n\) and the attenuation coefficient \(\chi\) of the metal are connected with them by the relations

\[ n^2 = \frac{1}{2}\left(\sqrt{\varepsilon^2 + 4\frac{\sigma^2}{\nu^2}} + \varepsilon\right), \]

\[ \chi^2 = \frac{1}{2}\left(\sqrt{\varepsilon^2 + 4\frac{\sigma^2}{\nu^2}} - \varepsilon\right). \]

If \(n\) and \(\chi\) are known from experiment, then with the aid of these relations one can determine \(\sigma\) and \(\varepsilon\).

The theory of metallic conductivity must give \(\sigma\) and \(\varepsilon\) as functions of \(\nu\). For metals in the nonsuperconducting state it was shown[^1] that the experimental results can be explained by means of the Bloch model of a metallic conductor, according to which the electrons are regarded as moving independently in the self-consistent periodic field of the crystal lattice, while the electrical resistance is explained by deviations from perfect periodicity caused by thermal vibrations of the metal ions. We shall find, however, that each volume element of the metal interacts with the electromagnetic field as a system of harmonic oscillators in the classical theory. If by \(f(\nu)\) one denotes the oscillator strength per unit volume and per unit frequency interval, then it turns out that this function has, in general form, the shape shown in Fig. 1. If for a constant field there were no resistance at all, then we would have a sharp resonance at frequency \(\nu = 0\); but owing to the collisions of electrons with the lattice, caused by the thermal excitation of the ions, the maximum broadens

Fig. 1.

PROPAGATION OF ELECTROMAGNETIC WAVES IN METALLIC CONDUCTORS

into a line of finite width (of the order of \(500\ \mathrm{cm}^{-1}\) at room temperature). In the region of high frequencies (the visible and ultraviolet regions), the oscillator distribution reveals continuous bands corresponding to the possibility of electron transitions between different energy bands in the Bloch model of a metallic conductor. \(\sigma\) and \(\varepsilon\) are related to \(f\) by the relations

\[ \sigma(\nu)=\frac{e^2}{4m}f(\nu), \qquad \varepsilon(\nu)-1=-\frac{e^2}{\pi m}\int_0^\infty \frac{f(\omega)\,d\omega}{\nu^2-\omega^2}. \]

Here \(e\) and \(m\) denote the charge and mass of the electron, and for the integral one must take the principal value.

For the alkali metals, according to Zener\(^{14}\), the optical properties in the visible and ultraviolet regions can be explained if it is assumed that the bands corresponding to electronic transitions are quite weak, so that the presence of a single resonance frequency \(\nu=0\) completely describes the optical properties of the metal. In silver and, to a lesser extent, in gold, as Kronig\(^{15}\) has shown, the broadened line at \(\nu=0\) is separated from the first band in the ultraviolet region by a clearly pronounced gap. In the infrared region, for wavelengths greater than \(2\,\mu\), in these cases the broadened line \(\nu=0\) has the predominant significance. Therefore we can make definite quantitative predictions concerning the optical constants\(^{16}\). The unpublished data obtained by Czerny and Lameris concerning the reflecting power of these metals in the region from 1 to \(13\,\mu\) at room temperature and at the temperature of liquid air fully confirm these theoretical predictions. In most other metals the continuous bands extend into the region of lower frequencies and, merging with the broadened line at \(\nu=0\), deprive us of the possibility, on the basis of theory alone, of making quantitative assertions.

Passing to metals in the superconducting state, we encounter the question of how the oscillator distribution should be modified. The experiments of de Haas and Bremmer\(^{17}\), in which a sharp decrease of the thermal conductivity was discovered when a metal passed into the superconducting state, effected by lowering the external magnetic field to a value below the critical one, give the idea that some fraction \(\alpha\) of the total number of ordinary conduction electrons passes into a new superconducting phase. \(\alpha\) may apparently be regarded as approximately equal to the number expressing the relative change of the thermal conductivity. From the data of de Haas and Bremmer it is seen that, as the temperature is lowered, \(\alpha\) increases, starting from the value 0 at the transition temperature (thus, for tin it reaches approximately \(35\%\) at \(2^\circ\mathrm{K}\), i.e. at a temperature lying \(2^\circ\) below the transition point). To obtain the oscillator distribution \(f(\nu)\) in the superconducting state we shall make the following quite plausible assumptions: a) the desired oscillator distribution is additively composed of two parts, corresponding respectively to the normally conducting and to the superconducting electrons, b) the first part may be obtained by multiplying the usual oscillator distribution function for nonsuperconducting electrons \(f(\nu)\) by the factor \((1-\alpha)\), c) the interaction of the superconducting electrons with electromagnetic waves may, on empirical grounds, be regarded as the same as for completely free electrons. All these assumptions agree both with the so-called theory of electron acceleration, from which follows a relation between current and field in the superconducting state, and with the new relations recently proposed by F. London and H. London\(^{18}\). This occurs because, for an electromagnetic wave, the scalar potential vanishes,

Thus both points of view lead to identical results. Denoting by \(\overline{\sigma}\) and \(\overline{\varepsilon}\) the constants referring to the superconducting and to the normal states, in accordance with the assumptions made above we obtain

\[ \overline{\sigma}(\nu)=(1-\alpha)\sigma(\nu),\qquad \overline{\varepsilon}(\nu)-1=(1-\alpha)[\varepsilon(\nu)-1]-\alpha \frac{Ne^{2}}{\pi m\nu^{2}}, \]

where \(N\) denotes the number of conduction electrons per \(1\ \mathrm{cm}^{3}\). If \(\alpha\) has a sufficiently large value—which can always be achieved by carrying out the experiment at a temperature considerably lower than the transition temperature—then, upon the transition from the nonsuperconducting state to the superconducting state, one should expect easily measurable changes in the quantities \(n\) and \(\chi\).

The phenomena considered could be investigated by studying both the passage of radio waves through very thin sheets of superconductors at temperatures above and below the transition point, and the changes in the reflection coefficient for infrared and visible light, as well as the polarization of a linearly polarized beam upon reflection at an angle, that occur under the influence of an applied magnetic field during the transition to the superconducting state. It is quite possible that, for the purpose of investigating the superconducting state, the study of the fine structure at the edge of the X-ray absorption bands may prove useful, since this fine structure is explained by the interaction of the lattice with slow electrons (having energies of the same order as the conduction electrons). When the metal becomes superconducting, the fine structure may undergo changes.

We would like once again to emphasize that the relations written above for \(\overline{\sigma}\) and \(\overline{\varepsilon}\) are purely empirical; however, under any other assumptions for the quantities \(\sigma\) and \(\varepsilon\), and consequently also for \(n\) and \(\chi\), changes of the same order should be expected.