Plasma dielectric function and dispersion

Take a spatially homogeneous unmagnetized equilibrium, use perturbations proportional to exp(ikx − iωt), and choose real positive k. For each mobile species, write its equilibrium reduced distribution as density times a probability density g with velocity integral one. Linearizing the Vlasov equation makes the perturbation proportional to the equilibrium velocity derivative divided by v minus phase velocity. Combining its charge-density integral with Poisson gives the longitudinal dielectric function.

The charge appears squared in each susceptibility because particles respond to a force proportional to their charge and then contribute charge again to Poisson. Oppositely charged species therefore do not simply cancel their susceptibilities. A stationary neutralizing background has no dynamical susceptibility in the model. Initially define the velocity integral for imaginary frequency in the upper half-plane, where the denominator has no real-axis singularity; its causal continuation is part of the response definition, not an optional adjustment afterward.

For a zero-drift Maxwellian, define vT as the square root of thermal energy divided by mass, equal to the standard deviation of one velocity component. Some references instead call square root of two times this value the thermal speed. Either convention works, but the Gaussian exponent and the dispersion-function argument must change together. Define Debye length using vT divided by plasma frequency. This makes the relation between thermal spread, shielding length, and frequency explicit.

Scaling velocity by square root of two times vT expresses the susceptibility through the plasma dispersion function Z. That compact special function retains the resonant information in an analytically continued integral. A numerically computed root is meaningful only when the same analytic branch is used throughout. Substituting an ordinary real principal-value integral for Z discards an imaginary contribution and can incorrectly predict that a warm collisionless wave has no damping.

For a cold stationary species, the velocity probability density becomes a delta distribution. Integration by parts gives susceptibility minus plasma frequency squared divided by frequency squared. Thus a cold electron plasma with fixed ions oscillates at the plasma frequency independent of wavenumber. A uniform drift Doppler shifts that frequency. These limits describe singular narrow distributions; they do not justify dropping resonant tails from an arbitrary finite-temperature state.

For a Maxwellian with phase speed much larger than vT, expand the nonresonant denominator in velocity divided by phase speed. The first nonzero thermal correction contains the second moment and yields the factor three in the Bohm–Gross formula. This is a long-wavelength, weak-damping approximation for the Langmuir branch, not a full dispersion relation at arbitrary Debye-scale wavelength. Its real frequency can be accurate while an exponentially small damping rate requires a separate resonant calculation.