Waves in metals
The theory we have worked out in this chapter for solid materials can also be applied to good conductors, like metals, with very little modification. In metals some of the electrons have no binding force holding them to any particular atom; it is these “free” electrons which are responsible for the conductivity. There are other electrons which are bound, and the theory above is directly applicable to them. Their influence, however, is usually swamped by the effects of the conduction electrons. We will consider now only the effects of the free electrons.  If there is no restoring force on an electron—but still some resistance to its motion—its equation of motion differs from Eq. (32.1) only because the term in $\omega_0^2x$ is lacking. So all we have to do is set $\omega_0^2=0$ in the rest of our derivations—except that there is one more difference. The reason that we had to distinguish between the average field and the local field in a dielectric is that in an insulator each of the dipoles is fixed in position, so that it has a definite relationship to the position of the others. But because the conduction electrons in a metal move around all over the place, the field on them on the average is just the average field $\FLPE$. So the correction we made to Eq. (32.8) by using Eq. (32.28) should not be made for conduction electrons. Therefore the formula for the index of refraction for metals should look like Eq. (32.27), except with $\omega_0$ set equal to zero, namely, \begin{equation} \label{Eq:II:32:38} n^2=1+\frac{Nq_e^2}{m\epsO}\,\frac{1}{-\omega^2+i\gamma\omega}. \end{equation}  This is only the contribution from the conduction electrons, which we will assume is the major term for metals. Now we even know how to find what value to use for $\gamma$, because it is related to the conductivity of the metal. In Chapter 43 of Volume I we discussed how the conductivity of a metal comes from the diffusion of the free electrons through the crystal. The electrons go on a jagged path from one scattering to the next, and between scatterings they move freely except for an acceleration due to any average electric field (as shown in Fig. 32–2). We found in Chapter 43 of Volume I that the average drift velocity is just the acceleration times the average time $\tau$ between collisions. The acceleration is $q_eE/m$, so \begin{equation} \label{Eq:II:32:39} v_{\text{drift}}=\frac{q_eE}{m}\,\tau. \end{equation}  This formula assumed that $E$ was constant, so that $v_{\text{drift}}$ was a steady velocity. Since there is no average acceleration, the drag force is equal to the applied force. We have defined $\gamma$ by saying that $\gamma mv$ is the drag force [see Eq. (32.1)], which is $q_eE$; therefore we have that \begin{equation} \label{Eq:II:32:40} \gamma=\frac{1}{\tau}. \end{equation}  Although we cannot easily measure $\tau$ directly, we can determine it by measuring the conductivity of the metal. It is found experimentally that an electric field $\FLPE$ in a metal produces a current with the density $\FLPj$ proportional to $\FLPE$ (for isotropic materials): \begin{equation*} \FLPj=\sigma\FLPE. \end{equation*}  The proportionality constant $\sigma$ is called the conductivity. This is just what we expect from Eq. (32.39) if we set \begin{equation} j=Nq_ev_{\text{drift}}.\notag \end{equation}  Then \begin{equation} \label{Eq:II:32:41} \sigma=\frac{Nq_e^2}{m}\,\tau. \end{equation}  So $\tau$—and therefore $\gamma$—can be related to the observed electrical conductivity. Using Eqs. (32.40) and (32.41), we can rewrite our formula for the index, Eq. (32.38), in the following form: \begin{equation} \label{Eq:II:32:42} n^2=1+\frac{\sigma/\epsO}{i\omega(1+i\omega\tau)}, \end{equation}  where \begin{equation} \label{Eq:II:32:43} \tau=\frac{1}{\gamma}=\frac{m\sigma}{Nq_e^2}. \end{equation}  This is a convenient formula for the index of refraction of metals.