Ion acoustic wave

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 times 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 species plasma, the waves are dispersionless ($\omega=v_sk$) with a speed (in the long wavelength limit) given by

$v_s = \sqrt{\frac{\gamma_{e}K_{B}T_e+\gamma_{i}ZK_{B}T_i}{M}}$

where $K_{B}$ is Boltzmann's constant, $M$ is the mass of the ion, $Z$ is its charge, $T_e$ is the temperature of the electrons and $T_i$ is the temperature of the ions. Normally γ_e 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 γ_i 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 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 $p_{s1} = \gamma_s T_{s0} n_{s1}$ for species s. Using the ion continuity equation, the ion momentum equation becomes

$(-m_i\partial_{tt}+\gamma_iT_i\nabla^2)n_{i1} = Z_ien_{i0}\nabla\cdot\vec E$

We relate the electric field $\vec E_1$ to the electron density by the electron momentum equation:

$n_{e0}m_e\partial_t\vec v_{e1} = -n_{e0}e\vec E_1 - \gamma_eT_e\nabla n_{e1}$

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

$\vec E_1 = - {\gamma_eT_e \over n_{e0}e}\nabla n_{e1}$

Since we have already solved for the electric field, we cannot also find it from Poisson's equation. The ion momentum equation now relates $n_{i1}$ for each species to $n_{e1}$:

$(-m_i\partial_{tt}+\gamma_iT_i\nabla^2)n_{i1} = -\gamma_e T_e \nabla^2 n_{e1}$

We arrive at a dispersion relation via Poisson's equation:

${\epsilon_0 \over e}\nabla\cdot\vec E = [\sum_i n_{i0}Z_i - n_{ne0}] + [\sum_in_{i1}Z_i - n_{e1}]$

The first bracketed term on the right is zero by assumption (charge-neutral equilibrium). We substitute for the electric field and rearrange to find

$(1-\gamma_e \lambda_{De}^2\nabla^2)n_{e1} = \sum_iZ_in_{i1}$.

$\lambda_{De}^2 \equiv \epsilon_0T_e/(n_{e0}e^2)$ defines the electron Debye length. The second term on the left arises from the $\nabla\cdot\vec E$ term, and reflects the degree to which the perturbation is not charge-neutral. If $k\lambda_{De}$ is small we may drop this term. This approximation is sometimes called the plasma approximation.

We now work in Fourier space, and find

$n_{i1} = \gamma_eT_eZ_i {n_{i0} \over n_{e0}} [m_iv_s^2-\gamma_iT_i]^{-1} n_{e1}$

$v_s=\omega/k$ is the wave phase velocity. Substituting this into Poisson's equation gives us an expression where each term is proportional to $n_{e1}$. To find the dispersion relation for natural modes, we look for solutions for $n_{e1}$ nonzero.

$\gamma_eT_e \sum_i Z_i^2f_i[m_iv_s^2-\gamma_iT_i]^{-1} = \bar Z(1+\gamma_ek^2\lambda_{De}^2)$

$n_{i1}=f_in_{I1}$ where $n_{I1}=\Sigma_i n_{i1}$, and $\bar Z = \Sigma_i Z_if_i$. In general it is not possible to further simplify this expression. If $k\lambda_{De}$ is small (the plasma approximation), we can neglect the second term on the term, and the wave is dispersionless $\omega = v_sk$ with $v_s$ independent of k.

Specific Examples

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

$v_s^2 = {\gamma_eZ_iT_e \over m_i}[{1 \over 1+\gamma_e(k\lambda_{De})^2} + {\gamma_iT_{i0} \over \gamma_eZ_iT_{e0}}]$

For any number of ion species, all of which are cold ($\gamma_iT_{i0} \ll m_iv_s^2$), we obtain

$v_s^2 = {\gamma_eT_{e0} \over 1+\gamma_e(k\lambda_{De})^2} \sum_i {Z_i^2f_i \over \bar Z m_i}$

A case of interest to nuclear fusion is an equimolar mixture of deuterium and tritium ions ($f_D=f_T=1/2$). Let us specialize to full ionization ($Z_D=Z_T=1$), equal temperatures ($T_{e0}=T_{i0}\equiv T_0$), polytrope exponents $\gamma_e=1, \gamma_i=3$, and neglect the $(k\lambda_{De})^2$ contribution. The dispersion relation becomes a quadratic in $v_s^2$, namely:

$2A_DA_Tu^4 - 7(A_D+A_T)u^2 + 24=0$

where $A_i=m_i/m_u$, $m_u$ is the atomic mass unit, and $u^2=m_uv_s^2/T_0$. Using $(A_D,A_T)=(2.01,3.02)$ we find the two roots are $u^2=(1.10,1.81)$.

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.

See also

• Waves in plasmas
• Sound
• Alfven wave
• Magnetosonic wave
• List of plasma (physics) articles

External links

• Various patents and articles related to fusion, IEC, ICC and plasma physics