Ion Acoustic Wave

Source lecture(s): pc368_lec09_wave1, pc368_lec10_wave2, pc368_lec11_streaming_ins

Intuition

Ion acoustic waves are the ion-scale analog of sound waves in a plasma. Ions provide the inertia, while pressure gradients and electron Boltzmann response provide the restoring force.

Formal Definition

The ion acoustic dispersion relation is

\[\\omega^2 = \\frac{k^2 c_s^2}{1 + k^2\\lambda_{De}^2}\]

where \(c_s = \\sqrt{(\\gamma_e k_B T_e + \\gamma_i k_B T_i)/m_i}\) is the ion sound speed.

Mathematical Formulation

From two-fluid equations with Boltzmann electrons (\(n_e = n_0 \\exp(e\\phi/k_B T_e)\)) and cold ions (\(T_i \\ll T_e\)), linearizing gives

\[\\omega^2 = \\frac{k^2 k_B T_e/m_i}{1 + k^2\\lambda_{De}^2}\]

Derivation

Linearize ion continuity and momentum.
Use Boltzmann response for electrons to close Poisson.
Eliminate density and potential.
The ion inertial term \(m_i\) in the denominator makes these waves slow compared to Langmuir waves.

Worked Example

For \(T_e = 10\) eV, \(T_i = 1\) eV, \(m_i = m_p\): - \(c_s \\approx \\sqrt{k_B T_e/m_i} \\approx 2.5\\times 10^3\\,\\text{m/s}\) - At \(k\\lambda_{De} \\ll 1\), \(\\omega \\approx k c_s\).

Common Mistakes

Assuming \(T_i = T_e\). Strong ion acoustic waves require \(T_e \\gg T_i\).
Ignoring electron inertia. It is negligible because \(m_e \\ll m_i\).

Quiz Questions

Why does the ion acoustic speed depend on electron temperature, not ion temperature?
What happens when \(k\\lambda_{De} \\gg 1\)?