Ion Sound Waves

If we now take ion dynamics into account then the dispersion relation (1007), for electrostatic plasma waves, generalizes to

$$1 + \frac{e^2}{\epsilon_0\,m_e\,k}\int_{-\infty}^{\infty}\fr... ...nfty}\frac{\partial F_{0\,i}/\partial u}{\omega-k\,u}\,du = 0:$$ (1032)

i.e., we simply add an extra term for the ions which has an analogous form to the electron term. Let us search for a wave with a phase velocity, $\omega/k$, which is much less than the electron thermal velocity, but much greater than the ion thermal velocity. We may assume that $\omega\gg k\,u$ for the ion term, as we did previously for the electron term. It follows that, to lowest order, this term reduces to $-\omega_{p\,i}^{~2}/\omega^2$. Conversely, we may assume that $\omega\ll k\,u$ for the electron term. Thus, to lowest order we may neglect $\omega$ in the velocity space integral. Assuming $F_{0\,e}$ to be a Maxwellian with temperature $T_e$, the electron term reduces to

$$\frac{{\omega}_{p\,e}^{~2}}{k^2}\frac{m_e}{T_e} = \frac{1}{(k\,\lambda_D)^2}.$$ (1033)

Thus, to a first approximation, the dispersion relation can be written

$$1 + \frac{1}{(k\,\lambda_D)^2} - \frac{\omega_{p\,i}^{~2}}{\omega^2} = 0,$$ (1034)

giving

$$\omega^2 = \frac{\omega_{p\,i}^{~2}\,k^2\,\lambda_D^{~2}}{1+... ...a_D^{~2}} = \frac{T_e}{m_i} \frac{k^2}{1+k^2\,\lambda_D^{~2}}.$$ (1035)

For $k\,\lambda_D\ll 1$, we have $\omega=(T_e/m_i)^{1/2}\,k$, a dispersion relation which is like that of an ordinary sound wave, with the pressure provided by the electrons, and the inertia by the ions. As the wave-length is reduced towards the Debye length, the frequency levels off and approaches the ion plasma frequency.

Let us check our original assumptions. In the long wave-length limit, we see that the wave phase velocity $(T_e/m_i)^{1/2}$ is indeed much less than the electron thermal velocity [by a factor $(m_e/m_i)^{1/2}$], but that it is only much greater than the ion thermal velocity if the ion temperature, $T_i$, is much less than the electron temperature, $T_e$. In fact, if $T_i\ll T_e$ then the wave phase velocity can lie on almost flat portions of the ion and electron distribution functions, as shown in Fig. 37, implying that the wave is subject to very little Landau damping. Indeed, an ion sound wave can only propagate a distance of order its wave-length without being strongly damped provided that $T_e$ is at least five to ten times greater than $T_i$.

Of course, it is possible to obtain the ion sound wave dispersion relation, $\omega^2/k^2 = T_e/m_i$, using fluid theory. The kinetic treatment used here is an improvement on the fluid theory to the extent that no equation of state is assumed, and it makes it clear to us that ion sound waves are subject to strong Landau damping (i.e., they cannot be considered normal modes of the plasma) unless $T_e\gg T_i$.

Figure 37: Ion and electron distribution functions with $T_i\ll T_e$.