In plasma physics, an ion acoustic wave is one type of longitudinal oscillation of the ions and electrons in a plasma, much like acoustic waves traveling in neutral gas.
However, because the waves propagate through positively charged ions, ion acoustic waves can interact with their electromagnetic fields, as well as simple collisions.
They commonly govern the evolution of mass density, for instance due to pressure gradients, on time scales longer than the frequency corresponding to the relevant length scale.
Ion acoustic waves can occur in an unmagnetized plasma or in a magnetized plasma parallel to the magnetic field.
For a single ion species plasma and in the long wavelength limit, the waves are dispersionless (
Normally γe is taken to be unity, on the grounds that the thermal conductivity of electrons is large enough to keep them isothermal on the time scale of ion acoustic waves, and γi is taken to be 3, corresponding to one-dimensional motion.
In collisionless plasmas, the electrons are often much hotter than the ions, in which case the second term in the numerator can be ignored.
We derive the ion acoustic wave dispersion relation for a linearized fluid description of a plasma with electrons and
To linearize, we balance all terms in each equation of the same order in
The terms involving only subscript-0 quantities are all order
We assume the pressure perturbations for each species are a Polytropic process, namely
, one must use a kinetic treatment that solves for the species distribution functions in velocity space.
The polytropic assumption essentially replaces the energy equation.
due to the polytropic assumption (but we do not assume it is zero), to alleviate notation we use
: We arrive at a dispersion relation via Poisson's equation: The first bracketed term on the right is zero by assumption (charge-neutral equilibrium).
We substitute for the electric field and rearrange to find
term, and reflects the degree to which the perturbation is not charge-neutral.
We now work in Fourier space, and write each order-1 field as
We drop the tilde since all equations now apply to the Fourier amplitudes, and find
Substituting this into Poisson's equation gives us an expression where each term is proportional to
To find the dispersion relation for natural modes, we look for solutions for
is small (the plasma approximation), we can neglect the second term on the right-hand side, and the wave is dispersionless
independent of k. The general dispersion relation given above for ion acoustic waves can be put in the form of an order-N polynomial (for N ion species) in
is typically greater than all the ion thermal speeds.
can be comparable to or less than the thermal speed of one or more of the ion species.
A case of interest to nuclear fusion is an equimolar mixture of deuterium and tritium ions (
Another case of interest is one with two ion species of very different masses.
An example is a mixture of gold (A=197) and boron (A=10.8), which is currently of interest in hohlraums for laser-driven inertial fusion research.
for both ion species, and charge states Z=5 for boron and Z=50 for gold.
The Landau damping occurs on both electrons and ions, with the relative importance depending on parameters.