Ion acoustic wave

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.

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