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Ion acoustic wave

where ${\displaystyle k_{\text{B}}}$ is the Boltzmann constant, ${\displaystyle M}$ is the mass of the ion, ${\displaystyle Z}$ is its charge, ${\displaystyle T_{e}}$ is the temperature of the electrons and ${\displaystyle T_{i}}$ 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.

We derive the ion acoustic wave dispersion relation for a linearized fluid description of a plasma with electrons and ${\textstyle N}$ ion species. We write each quantity as ${\displaystyle X=X_{0}+\delta \cdot X_{1}}$where subscript 0 denotes the "zero-order" constant equilibrium value, and 1 denotes the first-order perturbation. ${\displaystyle \delta }$ 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 ${\displaystyle \delta }$. The terms involving only subscript-0 quantities are all order ${\displaystyle \delta ^{0}}$and must balance, and terms with one subscript-1 quantity are all order ${\displaystyle \delta ^{1}}$and balance. We treat the electric field as order-1 (${\displaystyle {\vec {E}}_{0}=0}$) and neglect magnetic fields,

Each species ${\displaystyle s}$ is described by mass ${\displaystyle m_{s}}$, charge ${\displaystyle q_{s}=Z_{s}e}$, number density ${\displaystyle n_{s}}$, flow velocity ${\displaystyle {\vec {u}}_{s}}$, and pressure ${\displaystyle p_{s}}$. We assume the pressure perturbations for each species are a Polytropic process, namely ${\displaystyle p_{s1}=\gamma _{s}T_{s0}n_{s1}}$ for species ${\displaystyle s}$. To justify this assumption and determine the value of ${\displaystyle \gamma _{s}}$, 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 equation$${\displaystyle \partial _{t}n_{s}+\nabla \cdot (n_{s}{\vec {u}}_{s})=0}$$and the momentum equation

${\displaystyle \partial _{t}{\vec {u}}_{s}+{\vec {u}}_{s}\cdot \nabla {\vec {u}}_{s}={Z_{s}e \over m_{s}}{\vec {E}}-{\nabla p_{s} \over n_{s}}}$.

We now linearize, and work with order-1 equations. Since we do not work with ${\displaystyle T_{s1}}$due to the polytropic assumption (but we do not assume it is zero), to alleviate notation we use ${\displaystyle T_{s}}$for ${\displaystyle T_{s0}}$. Using the ion continuity equation, the ion momentum equation becomes

${\displaystyle (-m_{i}\partial _{tt}+\gamma _{i}T_{i}\nabla ^{2})n_{i1}=Z_{i}en_{i0}\nabla \cdot {\vec {E}}_{1}}$

We relate the electric field ${\displaystyle {\vec {E}}_{1}}$ to the electron density by the electron momentum equation:

${\displaystyle n_{e0}m_{e}\partial _{t}{\vec {v}}_{e1}=-n_{e0}e{\vec {E}}_{1}-\gamma _{e}T_{e}\nabla n_{e1}}$

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 ${\displaystyle (n_{e0}e^{2}/\epsilon _{0}m_{e})^{1/2}}$. This is a good approximation for ${\displaystyle m_{i}\gg m_{e}}$, such as ionized matter, but not for situations like electron-hole plasmas in semiconductors, or electron-positron plasmas. The resulting electric field is

${\displaystyle {\vec {E}}_{1}=-{\gamma _{e}T_{e} \over n_{e0}e}\nabla n_{e1}}$

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

${\displaystyle (-m_{i}\partial _{tt}+\gamma _{i}T_{i}\nabla ^{2})n_{i1}=-\gamma _{e}T_{e}\nabla ^{2}n_{e1}}$

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

${\displaystyle {\epsilon _{0} \over e}\nabla \cdot {\vec {E}}_{1}=\left[\sum _{i=1}^{N}n_{i0}Z_{i}-n_{e0}\right]+\left[\sum _{i=1}^{N}n_{i1}Z_{i}-n_{e1}\right]}$

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

${\displaystyle (1-\gamma _{e}\lambda _{De}^{2}\nabla ^{2})n_{e1}=\sum _{i=1}^{N}Z_{i}n_{i1}}$.

${\displaystyle \lambda _{De}^{2}\equiv \epsilon _{0}T_{e}/(n_{e0}e^{2})}$ defines the electron Debye length. The second term on the left arises from the ${\displaystyle \nabla \cdot {\vec {E}}}$ term, and reflects the degree to which the perturbation is not charge-neutral. If ${\displaystyle k\lambda _{De}}$ is small we may drop this term. This approximation is sometimes called the plasma approximation.

We now work in Fourier space, and write each order-1 field as ${\displaystyle X_{1}={\tilde {X}}_{1}\exp i({\vec {k}}\cdot {\vec {x}}-\omega t)+c.c.}$ We drop the tilde since all equations now apply to the Fourier amplitudes, and find

${\displaystyle n_{i1}=\gamma _{e}T_{e}Z_{i}{n_{i0} \over n_{e0}}[m_{i}v_{s}^{2}-\gamma _{i}T_{i}]^{-1}n_{e1}}$

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

${\displaystyle \gamma _{e}T_{e}\left\langle {Z_{i}^{2} \over m_{i}v_{s}^{2}-\gamma _{i}T_{i}}\right\rangle =\langle Z_{i}\rangle (1+\gamma _{e}k^{2}\lambda _{De}^{2})}$.

(dispgen)

${\displaystyle n_{i1}=f_{i}n_{I1}}$ where ${\displaystyle n_{I1}=\Sigma _{i}n_{i1}}$, so the ion fractions satisfy ${\displaystyle \Sigma _{i}f_{i}=1}$, and ${\displaystyle \langle X_{i}\rangle \equiv \Sigma _{i}f_{i}X_{i}}$ is the average over ion species. A unitless version of this equation is

${\displaystyle {\gamma _{e} \over \langle Z_{i}\rangle }\left\langle {Z_{i}^{2}/A_{i} \over u^{2}-\tau _{i}}\right\rangle =1+\gamma _{e}k^{2}\lambda _{De}^{2}}$

with ${\displaystyle A_{i}=m_{i}/m_{u}}$, ${\displaystyle m_{u}}$ is the atomic mass unit, ${\displaystyle u^{2}=m_{u}v_{s}^{2}/T_{e}}$, and

${\displaystyle \tau _{i}={\gamma _{i}T_{i} \over A_{i}T_{e}}}$

If${\displaystyle k\lambda _{De}}$ is small (the plasma approximation), we can neglect the second term on the right-hand side, and the wave is dispersionless ${\displaystyle \omega =v_{s}k}$ with ${\displaystyle v_{s}}$ 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 ${\displaystyle u^{2}}$. All of the roots should be real-positive, since we have neglected damping. The two signs of ${\displaystyle u}$ correspond to right- and left-moving waves. For a single ion species,

${\displaystyle v_{s}^{2}={\gamma _{e}Z_{i}T_{e} \over m_{i}}{1 \over 1+\gamma _{e}(k\lambda _{De})^{2}}+{\gamma _{i}T_{i} \over m_{i}}={\gamma _{e}Z_{i}T_{e} \over m_{i}}\left[{1 \over 1+\gamma _{e}(k\lambda _{De})^{2}}+{\gamma _{i}T_{i} \over Z_{i}\gamma _{e}T_{e}}\right]}$

We now consider multiple ion species, for the common case ${\displaystyle T_{i}\ll T_{e}}$. For ${\displaystyle T_{i}=0}$, the dispersion relation has N-1 degenerate roots ${\displaystyle u^{2}=0}$, and one non-zero root

${\displaystyle v_{s}^{2}(T_{i}=0)\equiv {\gamma _{e}T_{e}/m_{u} \over 1+\gamma _{e}(k\lambda _{De})^{2}}{\langle Z_{i}^{2}/A_{i}\rangle \over \langle Z_{i}\rangle }}$

This non-zero root is called the "fast mode", since ${\displaystyle v_{s}}$ is typically greater than all the ion thermal speeds. The approximate fast-mode solution for ${\displaystyle T_{i}\ll T_{e}}$is

${\displaystyle v_{s}^{2}\approx v_{s}^{2}(T_{i}=0)+{\langle Z_{i}^{2}\gamma _{i}T_{i}/A_{i}^{2}\rangle \over m_{u}\langle Z_{i}^{2}/A_{i}\rangle }}$

The N-1 roots that are zero for ${\displaystyle T_{i}=0}$ are called "slow modes", since ${\displaystyle v_{s}}$ 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 (${\displaystyle f_{D}=f_{T}=1/2}$). Let us specialize to full ionization (${\displaystyle Z_{D}=Z_{T}=1}$), equal temperatures (${\displaystyle T_{e}=T_{i}}$), polytrope exponents ${\displaystyle \gamma _{e}=1,\gamma _{i}=3}$, and neglect the ${\displaystyle (k\lambda _{De})^{2}}$ contribution. The dispersion relation becomes a quadratic in ${\displaystyle v_{s}^{2}}$, namely:

${\displaystyle 2A_{D}A_{T}u^{4}-7(A_{D}+A_{T})u^{2}+24=0}$

Using ${\displaystyle (A_{D},A_{T})=(2.01,3.02)}$ we find the two roots are ${\displaystyle u^{2}=(1.10,1.81)}$.

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 ${\displaystyle \gamma _{e}=1}$ and ${\displaystyle \gamma _{i}=3,T_{i}=T_{e}/2}$ for both ion species, and charge states Z=5 for boron and Z=50 for gold. We leave the boron atomic fraction ${\displaystyle f_{B}}$ unspecified (note ${\displaystyle f_{Au}=1-f_{B}}$). Thus, ${\displaystyle {\bar {Z}}=50-45f_{B},\tau _{B}=0.139,\tau _{Au}=0.00761,F_{B}=2.31f_{B}/{\bar {Z}},}$ and ${\displaystyle F_{Au}=12.69(1-f_{B})/{\bar {Z}}}$.

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.

• Waves in plasmas
• Sound
• Alfvén wave
• Magnetosonic wave

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