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 () with a speed given by (see derivation below)
where is Boltzmann's constant, is the mass of the ion, is its charge, is the temperature of the electrons and 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.
We derive the ion acoustic wave dispersion relation for a linearized fluid description of a plasma with multiple ion species. A subscript 0 denotes constant equilibrium quantities, and 1 denotes first-order perturbations. We assume the pressure perturbations for each species (electrons and ions) are a Polytropic_process, namely for species s. Using the ion continuity equation, the ion momentum equation becomes
We relate the electric field 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. 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 for each species to :
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
defines the electron Debye length. The second term on the left arises from the term, and reflects the degree to which the perturbation is not charge-neutral. If is small we may drop this term. This approximation is sometimes called the plasma approximation.
We now work in Fourier space, and find
is the wave phase velocity. 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 nonzero and find:
where , and . A unitless version of this equation is
with , is the atomic mass unit, , and
If is small (the plasma approximation), we can neglect the second term on the right-hand side, and the wave is dispersionless with independent of k.
To illustrate some features of ion acoustic waves, we can consider some specific examples of the general dispersion relation given above. First, for a single ion species, we find
For any number of ion species, all of which are cold (), we obtain
A case of interest to nuclear fusion is an equimolar mixture of deuterium and tritium ions (). Let us specialize to full ionization (), equal temperatures (), polytrope exponents , and neglect the contribution. The dispersion relation becomes a quadratic in , namely:
Using we find the two roots are .
Two Ion Species: Fast and Slow Modes
As seen in Eq. coldTi above, for cold ions there is a single root to the ion acoustic wave dispersion relation (with + and - vs corresponding to right and left moving waves). In the general dispersion relation Eq. dispgen, each ion species contributes a term on the left-hand side, and gives an additional root. This applies, for instance, to the D-T example considered above. 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 and for both ion species, and charge states Z=5 for boron and Z=50 for gold. We leave the boron atomic fraction unspecified (note ). Thus, and .