Ion-acoustic waves and electron response

For an isothermal electron response with negligible inertia along the wave, electric force balances the electron pressure gradient. Integrating gives electron density proportional to the exponential of positive e times potential divided by electron thermal energy. The positive sign is physical: a positive potential attracts electrons and raises their density. Linearization requires potential energy small compared with thermal energy. The response is an approximation along connected electron trajectories, not a universal law in a sheath or trapped-electron population.

A collisionless kinetic electron population can produce this leading real response when phase speed is much less than its thermal speed. Its small imaginary response still carries electron Landau damping and is discarded by the simplest Boltzmann closure. Thus Boltzmann electrons do not imply that all electrons have no inertia in every situation, nor do they establish a collisional Maxwellian at every instant. The equilibrium distribution, connectivity, frequency, and perturbation amplitude determine when the closure is appropriate.

Take singly charged cold ions with no equilibrium flow. Linear continuity connects ion density to velocity, and the ion momentum equation connects velocity to electric acceleration. For a Fourier disturbance these give density proportional to potential times wavenumber squared divided by frequency squared. Combining that ion response with the Boltzmann electron response in Poisson yields an acoustic frequency divided by a Debye-length correction. Quasineutrality is therefore a controlled long-wavelength limit, rather than exact equality imposed before all field dynamics are considered.

At wavelengths long compared with the electron Debye length, phase speed approaches the electron-pressure sound speed. At shorter wavelengths, charge separation reduces the phase speed in this cold-ion model and frequency approaches the ion plasma frequency. That mathematical limit does not ensure that neglected kinetic or electron-inertia effects remain small for a chosen experiment. A warm-ion pressure term adds a closure-dependent correction; the familiar factor three corresponds to one-dimensional adiabatic compression in an appropriate high-phase-speed limit.

Define each species' one-component thermal speed as square root of thermal energy divided by mass. Weakly damped ion-acoustic propagation typically needs phase speed much greater than the ion thermal speed and much smaller than the electron thermal speed. The first requirement suppresses resonant ions, while the second permits the nearly Boltzmann electron response. A sufficiently large electron-to-ion temperature ratio creates this window in the ordinary electron–ion mass hierarchy.

If electron and ion temperatures are comparable, the acoustic phase speed is near the ion thermal population and ion Landau damping can be substantial. A real fluid sound frequency then does not prove that a well-defined weakly damped mode exists. Electron drift relative to ions can instead provide free energy and destabilize the branch, requiring the full shifted distributions. Distinguish a closure calculation, a weak-damping kinetic expansion, and a measured propagating signal before assigning the same label to their results.