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----
-title: "Ion sound waves"
-firstLetter: "I"
-publishDate: 2021-10-31
-categories:
-- Physics
-- Plasma physics
-
-date: 2021-10-31T09:38:14+01:00
-draft: false
-markup: pandoc
----
-
-# Ion sound waves
-
-In a plasma, electromagnetic interactions allow
-compressional longitudinal waves to propagate
-at lower temperatures and pressures
-than would be possible in a neutral gas.
-
-We start from the [two-fluid model's](/know/concept/two-fluid-equations/) momentum equations,
-rewriting the [electric field](/know/concept/electric-field/) $\vb{E} = - \nabla \phi$
-and the pressure gradient $\nabla p = \gamma k_B T \nabla n$,
-and arguing that $m_e \approx 0$ because $m_e \ll m_i$:
-
-$$\begin{aligned}
-    m_i n_i \frac{\mathrm{D} \vb{u}_i}{\mathrm{D} t}
-    &= - q_i n_i \nabla \phi - \gamma_i k_B T_i \nabla n_i
-    \\
-    0
-    &= - q_e n_e \nabla \phi - \gamma_e k_B T_e \nabla n_e
-\end{aligned}$$
-
-Note that we neglect ion-electron collisions,
-and allow for separate values of $\gamma$.
-We split $n_i$, $n_e$, $\vb{u}_i$ and $\phi$ into an equilibrium
-(subscript $0$) and a perturbation (subscript $1$):
-
-$$\begin{aligned}
-    n_i
-    = n_{i0} + n_{i1}
-    \qquad
-    n_e
-    = n_{e0} + n_{e1}
-    \qquad
-    \vb{u}_i
-    = \vb{u}_{i0} + \vb{u}_{i1}
-    \qquad
-    \phi
-    = \phi_0 + \phi_1
-\end{aligned}$$
-
-Where the perturbations $n_{i1}$, $n_{e1}$, $\vb{u}_{i1}$ and $\phi_1$ are tiny,
-and the equilibrium components $n_{i0}$, $n_{e0}$, $\vb{u}_{i0}$ and $\phi_0$
-by definition satisfy:
-
-$$\begin{aligned}
-    \pdv{n_{i0}}{t} = 0
-    \qquad
-    \frac{\mathrm{D} \vb{u}_{i0}}{\mathrm{D} t} = 0
-    \qquad
-    \nabla n_{i0} = \nabla n_{e0} = 0
-    \qquad
-    \vb{u}_{i0} = 0
-    \qquad
-    \phi_0 = 0
-\end{aligned}$$
-
-Inserting this decomposition into the momentum equations
-yields new equations.
-Note that we will implicitly use $\vb{u}_{i0} = 0$
-to pretend that the [material derivative](/know/concept/material-derivative/)
-$\mathrm{D}/\mathrm{D} t$ is linear:
-
-$$\begin{aligned}
-    m_i (n_{i0} \!+\! n_{i1}) \frac{\mathrm{D} (\vb{u}_{i0} \!+\! \vb{u}_{i1})}{\mathrm{D} t}
-    &= - q_i (n_{i0} \!+\! n_{i1}) \nabla (\phi_0 \!+\! \phi_1) - \gamma_i k_B T_i \nabla (n_{i0} \!+\! n_{i1})
-    \\
-    0
-    &= - q_e (n_{e0} \!+\! n_{e1}) \nabla (\phi_0 \!+\! \phi_1) - \gamma_e k_B T_e \nabla (n_{e0} \!+\! n_{e1})
-\end{aligned}$$
-
-Using the defined properties of the equilibrium components
-$n_{i0}$, $n_{e0}$, $\vb{u}_{i0}$ and $\phi_0$,
-and neglecting all products of perturbations for being small,
-this reduces to:
-
-$$\begin{aligned}
-    m_i n_{i0} \pdv{\vb{u}_{i1}}{t}
-    &= - q_i n_{i0} \nabla \phi_1 - \gamma_i k_B T_i \nabla n_{i1}
-    \\
-    0
-    &= - q_e n_{e0} \nabla \phi_1 - \gamma_e k_B T_e \nabla n_{e1}
-\end{aligned}$$
-
-Because we are interested in linear waves,
-we make the following plane-wave ansatz:
-
-$$\begin{aligned}
-    n_{i1}(\vb{r}, t)
-    &= n_{i1} \exp\!(i \vb{k} \cdot \vb{r} - i \omega t)
-    \\
-    n_{e1}(\vb{r}, t)
-    &= n_{e1} \exp\!(i \vb{k} \cdot \vb{r} - i \omega t)
-    \\
-    \vb{u}_{i1}(\vb{r}, t)
-    &= \vb{u}_{i1} \exp\!(i \vb{k} \cdot \vb{r}  - i \omega t)
-    \\
-    \phi_1(\vb{r}, t)
-    &= \phi_1 \,\,\exp\!(i \vb{k} \cdot \vb{r} - i \omega t)
-\end{aligned}$$
-
-Which we then insert into the momentum equations for the ions and electrons:
-
-$$\begin{aligned}
-    - i \omega m_i n_{i0} \vb{u}_{i1}
-    &= - i \vb{k} q_i n_{i0} \phi_1 - i \vb{k} \gamma_i k_B T_i n_{i1}
-    \\
-    0
-    &= - i \vb{k} q_e n_{e0} \phi_1 - i \vb{k} \gamma_e k_B T_e n_{e1}
-\end{aligned}$$
-
-The electron equation can easily be rearranged
-to get a relation between $n_{e1}$ and $n_{e0}$:
-
-$$\begin{aligned}
-     i \vb{k} \gamma_e k_B T_e n_{e1}
-     = - i \vb{k} q_e n_{e0} \phi_1
-     \quad \implies \quad
-     n_{e1}
-     = - \frac{q_e \phi_1}{\gamma_e k_B T_e} n_{e0}
-\end{aligned}$$
-
-Due to their low mass, the electrons' heat conductivity
-can be regarded as infinite compared to the ions'.
-In that case, all electron gas compression is isothermal,
-meaning it obeys the ideal gas law $p_e = n_e k_B T_e$, so that $\gamma_e = 1$.
-Note that this yields the first-order term of a Taylor expansion
-of the [Boltzmann relation](/know/concept/boltzmann-relation/).
-
-At equilibrium, quasi-neutrality demands that $n_{i0} = n_{e0} = n_0$,
-so we can rearrange the above relation to $n_0 = - k_B T_e n_{e1} / (q_e \phi_1)$,
-which we insert into the ion equation to get:
-
-$$\begin{gathered}
-    i \omega m_i \frac{k_B T_e n_{e1}}{q_e \phi_1} \vb{u}_{i1}
-    = - i q_i \frac{k_B T_e n_{e1}}{q_e \phi_1} \phi_1 \vb{k} - i \gamma_i k_B T_i n_{i1} \vb{k}
-    \\
-    \implies \qquad
-    \omega m_i \frac{T_e n_{e1}}{q_e \phi_1} \vb{k} \cdot \vb{u}_{i1}
-    = T_e n_{e1} |\vb{k}|^2 - \gamma_i T_i n_{i1} |\vb{k}|^2
-\end{gathered}$$
-
-Where we have taken the dot product with $\vb{k}$,
-and used that $q_i / q_e = -1$.
-In order to simplify this equation,
-we turn to the two-fluid ion continuity relation:
-
-$$\begin{aligned}
-    0
-    &= \pdv{(n_{i0} \!+\! n_{i1})}{t} + \nabla \cdot \Big( (n_{i0} \!+\! n_{i1}) (\vb{u}_{i0} \!+\! \vb{u}_{i1}) \Big)
-    \approx \pdv{n_{i1}}{t} + n_{i0} \nabla \cdot \vb{u}_{i1}
-\end{aligned}$$
-
-Then we insert our plane-wave ansatz,
-and substitute $n_{i0} = n_0$ as before, yielding:
-
-$$\begin{aligned}
-    0
-    = - i \omega n_{i1} + i n_{i0} \vb{k} \cdot \vb{u}_{i1}
-    \quad \implies \quad
-    \vb{k} \cdot \vb{u}_{i1}
-    = \omega \frac{n_{i1}}{n_{i0}}
-    = \omega \frac{q_e n_{i1} \phi_1}{k_B T_e n_{e1}}
-\end{aligned}$$
-
-Substituting this in the ion momentum equation
-leads us to a dispersion relation $\omega(\vb{k})$:
-
-$$\begin{gathered}
-    \omega^2 m_i \frac{T_e n_{e1}}{q_e \phi_1} \frac{q_e n_{i1} \phi_1}{k_B T_e n_{e1}}
-    = \omega^2 m_i \frac{n_{i1}}{k_B}
-    = |\vb{k}|^2 \big( T_e n_{e1} - \gamma_i T_i n_{i1} \big)
-    \\
-    \implies \qquad
-    \omega^2
-    = \frac{|\vb{k}|^2}{m_i} \Big( k_B T_e \frac{n_{e1}}{n_{i1}} - \gamma_i k_B T_i \Big)
-\end{gathered}$$
-
-Finally, we would like to find an expression for $n_{e1} / n_{i1}$.
-It cannot be $1$, because then $\phi_1$ could not be nonzero,
-according to [Gauss' law](/know/concept/maxwells-equations/).
-Nevertheless, authors often ignore this fact,
-thereby making the so-called **plasma approximation**.
-We will not, and therefore turn to Gauss' law:
-
-$$\begin{aligned}
-    \varepsilon_0 \nabla \cdot \vb{E}
-    = - \varepsilon_0 \nabla^2 \phi_1
-    = q_i n_i - q_e n_e
-    = - q_e (n_{i1} - n_{e1})
-\end{aligned}$$
-
-One final time, we insert our plane-wave ansatz,
-and use our Boltzmann-like relation between $n_{e1}$ and $n_{e0}$
-to substitute $\phi_1 = - k_B T_e n_{e1} / (q_e n_{e0})$:
-
-$$\begin{gathered}
-    q_e (n_{e1} - n_{i1})
-    = |\vb{k}|^2 \varepsilon_0 \phi_1
-    = - |\vb{k}|^2 \varepsilon_0 \frac{k_B T_e n_{e1}}{q_e n_{e0}}
-    \\
-    \implies \qquad
-    n_{i1}
-    = n_{e1} + |\vb{k}|^2 \varepsilon_0 \frac{k_B T_e n_{e1}}{q_e^2 n_{e0}}
-    = n_{e1} \big( 1 + |\vb{k}|^2 \lambda_{De}^2 \big)
-\end{gathered}$$
-
-Where $\lambda_{De}$ is the electron [Debye length](/know/concept/debye-length/).
-We thus reach the following dispersion relation,
-which governs **ion sound waves** or **ion acoustic waves**:
-
-$$\begin{aligned}
-    \boxed{
-        \omega^2
-        = \frac{|\vb{k}|^2}{m_i} \bigg( \frac{k_B T_e}{1 + |\vb{k}|^2 \lambda_{De}^2} + \gamma_i k_B T_i \bigg)
-    }
-\end{aligned}$$
-
-The aforementioned plasma approximation is valid if $|\vb{k}| \lambda_{De} \ll 1$,
-which is often reasonable,
-in which case this dispersion relation reduces to:
-
-$$\begin{aligned}
-    \omega^2
-    = \frac{|\vb{k}|^2}{m_i} \bigg( k_B T_e + \gamma_i k_B T_i \bigg)
-\end{aligned}$$
-
-The phase velocity $v_s$ of these waves,
-i.e. the speed of sound, is then given by:
-
-$$\begin{aligned}
-    \boxed{
-        v_s
-        = \frac{\omega}{k}
-        = \sqrt{\frac{k_B T_e}{m_i} + \frac{\gamma_i k_B T_i}{m_i}}
-    }
-\end{aligned}$$
-
-Curiously, unlike a neutral gas,
-this velocity is nonzero even if $T_i = 0$,
-meaning that the waves still exist then.
-In fact, usually the electron temperature $T_e$ dominates $T_e \gg T_i$,
-even though the main feature of these waves
-is that they involve ion density fluctuations $n_{i1}$.
-
-
-
-## References
-1.  F.F. Chen,
-    *Introduction to plasma physics and controlled fusion*,
-    3rd edition, Springer.
-2.  M. Salewski, A.H. Nielsen,
-    *Plasma physics: lecture notes*,
-    2021, unpublished.