Dielectric constant of a collisional plasma

We have now investigated electromagnetic wave propagation through two different media possessing free electrons: plasmas (see Sect. 9.8), and ohmic conductors (see Sect. 9.10). In the first case, we obtained the dispersion relation (1155), whereas in the second we obtained the quite different dispersion relation (1191). This leads us, quite naturally, to ask what the essential distinction is between the response of free electrons in a plasma to an electromagnetic wave, and that of free electrons in an ohmic conductor. It turns out that the main distinction is the relative strength of electron-ion collisions.

In the presence of electron-ion collisions, we can model the equation of motion of an individual electron in a plasma or a conductor as

$$m_e \frac{d{\bf v}}{dt} + m_e \nu {\bf v} = -e {\bf E},$$ (1201)

where ${\bf E}$ is the wave electric field. The collision term (i.e., the second term on the left-hand side) takes the form of a drag force proportional to $-{\bf v}$. In the absence of the wave electric field, this force damps out any electron motion on the typical time-scale $\nu^{-1}$. Since, in reality, an electron loses virtually all of its directed momentum during a collision with a much more massive ion, we can regard $\nu$ as the effective electron-ion collision frequency.

Assuming the usual $\exp(-{\rm i} \omega t)$ time-dependence of perturbed quantities, we can solve Eq. (1201) to give

$${\bf v} = -{\rm i} \omega {\bf r} = - \frac{{\rm i} \omega e {\bf E}}{m_e \omega (\omega+{\rm i} \nu)}.$$ (1202)

Hence, the perturbed current density can be written

$${\bf j} = - e n_e {\bf v} = \frac{{\rm i} n_e e^2 {\bf E}}{m_e (\omega+{\rm i} \nu)},$$ (1203)

where $n_e$ is the number density of free electrons. It follows that the effective conductivity of the medium takes the form

$$\sigma = \frac{{\bf j}}{{\bf E}} = \frac{{\rm i} n_e e^2}{m_e (\omega+{\rm i} \nu)}.$$ (1204)

Now, the mean rate of ohmic heating per unit volume in the medium is written

$$\langle P\rangle = \frac{1}{2} {\rm Re}(\sigma) E_0^{ 2},$$ (1205)

where $E_0$ is the amplitude of the wave electric field. Note that only the real part of $\sigma$ contributes to ohmic heating, because the perturbed current must be in phase with the wave electric field in order for there to be a net heating effect. An imaginary $\sigma$ gives a perturbed current which is in phase quadrature with the wave electric field. In this case, there is zero net transfer of power between the wave and the plasma over a wave period. We can see from Eq. (1204) that in the limit in which the wave frequency is much larger than the collision frequency (i.e., $\omega\gg \nu$), the effective conductivity of the medium becomes purely imaginary:

$$\sigma \simeq \frac{{\rm i} n_e e^2}{m_e \omega}.$$ (1206)

In this limit, there is no loss of wave energy due to ohmic heating, and the medium acts like a conventional plasma. In the opposite limit, in which the wave frequency is much less than the collision frequency (i.e., $\omega\ll\nu$), the effective conductivity becomes purely real:

$$\sigma \simeq \frac{n_e e^2}{m_e \nu}.$$ (1207)

In this limit, ohmic heating losses are significant, and the medium acts like a conventional ohmic conductor.

Following the analysis of Sect. 9.7, we can derive the following dispersion relation from Eq. (1202):

$$k^2 c^2 = \omega^2 - \frac{\omega_p^{ 2} \omega}{\omega + {\rm i} \nu}.$$ (1208)

It can be seen that, in the limit $\omega\gg \nu$, the above dispersion relation reduces to the dispersion relation (1155) for a conventional (i.e., collisionless) plasma. In the opposite limit, we obtain

$$k^2 = \frac{\omega^2}{c^2} + {\rm i} \frac{\omega_p^{ 2} ... ...,c^2} = \mu_0 \omega (\epsilon_0 \omega + {\rm i} \sigma).$$ (1209)

where use has been made of Eq (1207). Of course, the above dispersion relation is identical to the dispersion relation (1191) (with $\epsilon=1$) which we previously derived for an ohmic conductor.

Our main conclusion from this subsection is that the dispersion relation (1208) can be used to describe electromagnetic wave propagation through both a collisional plasma and an ohmic conductor. We can also deduce that in the low frequency limit, $\omega\ll\nu$, a collisional plasma acts very much like an ohmic conductor, whereas in the high frequency limit, $\omega\gg \nu$, an ohmic conductor acts very much like a collisionless plasma.