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diff --git a/source/know/concept/ion-sound-wave/index.md b/source/know/concept/ion-sound-wave/index.md new file mode 100644 index 0000000..48a727d --- /dev/null +++ b/source/know/concept/ion-sound-wave/index.md @@ -0,0 +1,261 @@ +--- +title: "Ion sound wave" +date: 2021-10-31 +categories: +- Physics +- Plasma physics +- Plasma waves +- Perturbation +layout: "concept" +--- + +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. |