Ion acoustic wave

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. In plasmas, ion acoustic waves are frequently referred to as acoustic waves or even just sound waves. 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 (formula_1) with a speed given by (see derivation below)
where formula_3 is Boltzmann's constant, formula_4 is the mass of the ion, formula_5 is its charge, formula_6 is the temperature of the electrons and formula_7 is the temperature of the ions. 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.
Derivation.
We derive the ion acoustic wave dispersion relation for a linearized fluid description of a plasma with electrons and formula_8 ion species. We write each quantity as formula_9where subscript 0 denotes the "zero-order" constant equilibrium value, and 1 denotes the first-order perturbation. formula_10 is an ordering parameter for linearization, and has the physical value 1. To linearize, we balance all terms in each equation of the same order in formula_10. The terms involving only subscript-0 quantities are all order formula_12and must balance, and terms with one subscript-1 quantity are all order formula_13and balance. We treat the electric field as order-1 (formula_14) and neglect magnetic fields, 
Each species formula_15 is described by mass formula_16, charge formula_17, number density formula_18, flow velocity formula_19, and pressure formula_20. We assume the pressure perturbations for each species are a Polytropic process, namely formula_21 for species formula_15. To justify this assumption and determine the value of formula_23, one must use a kinetic treatment that solves for the species distribution functions in velocity space. The polytropic assumption essentially replaces the energy equation.
Each species satisfies the continuity equationformula_24and the momentum equation
formula_25.
We now linearize, and work with order-1 equations. Since we do not work with formula_26due to the polytropic assumption (but we do "not" assume it is zero), to alleviate notation we use formula_27for formula_28. Using the ion continuity equation, the ion momentum equation becomes
We relate the electric field formula_30 to the electron density by the electron momentum equation:
We now neglect the left-hand side, which is due to electron inertia. This is valid for waves with frequencies much less than the electron plasma frequency formula_32. This is a good approximation for formula_33, such as ionized matter, but not for situations like electron-hole plasmas in semiconductors, or electron-positron plasmas. The resulting electric field is
Since we have already solved for the electric field, we cannot also find it from Poisson's equation. The ion momentum equation now relates formula_35 for each species to formula_36:
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
formula_40 defines the electron Debye length. The second term on the left arises from the formula_41 term, and reflects the degree to which the perturbation is not charge-neutral. If formula_42 is small we may drop this term. This approximation is sometimes called the plasma approximation.
We now work in Fourier space, and write each order-1 field as formula_43 We drop the tilde since all equations now apply to the Fourier amplitudes, and find 
formula_45 is the wave phase velocity. Substituting this into Poisson's equation gives us an expression where each term is proportional to formula_36. To find the dispersion relation for natural modes, we look for solutions for formula_36 nonzero and find:
formula_48 where formula_49, so the ion fractions satisfy formula_50, and formula_51 is the average over ion species. A unitless version of this equation is
with formula_53, formula_54 is the atomic mass unit, formula_55, and
If formula_42 is small (the plasma approximation), we can neglect the second term on the right-hand side, and the wave is dispersionless formula_58 with formula_59 independent of k.
Dispersion relation.
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 formula_60. All of the roots should be real-positive, since we have neglected damping. The two signs of formula_61 correspond to right- and left-moving waves. For a single ion species,
We now consider multiple ion species, for the common case formula_63. For formula_64, the dispersion relation has N-1 degenerate roots formula_65, and one non-zero root
This non-zero root is called the "fast mode", since formula_59 is typically greater than all the ion thermal speeds. The approximate fast-mode solution for formula_68 is
The N-1 roots that are zero for formula_64 are called "slow modes", since formula_59 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 (formula_72). Let us specialize to full ionization (formula_73), equal temperatures (formula_74), polytrope exponents formula_75, and neglect the formula_76 contribution. The dispersion relation becomes a quadratic in formula_77, namely:
Using formula_79 we find the two roots are formula_80.
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 a concrete example, consider formula_81 and formula_82 for both ion species, and charge states Z=5 for boron and Z=50 for gold. We leave the boron atomic fraction formula_83 unspecified (note formula_84). Thus, formula_85 and formula_86.
Damping.
Ion acoustic waves are damped both by Coulomb collisions and collisionless Landau damping. The Landau damping occurs on both electrons and ions, with the relative importance depending on parameters.