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Diffstat (limited to 'source/know/concept/ion-sound-wave')
| -rw-r--r-- | source/know/concept/ion-sound-wave/index.md | 58 |
1 files changed, 29 insertions, 29 deletions
diff --git a/source/know/concept/ion-sound-wave/index.md b/source/know/concept/ion-sound-wave/index.md index 622605c..cb86c04 100644 --- a/source/know/concept/ion-sound-wave/index.md +++ b/source/know/concept/ion-sound-wave/index.md @@ -16,9 +16,9 @@ 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 $m_e \ll m_i$: +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 $$m_e \ll m_i$$: $$\begin{aligned} m_i n_i \frac{\mathrm{D} \vb{u}_i}{\mathrm{D} t} @@ -29,9 +29,9 @@ $$\begin{aligned} \end{aligned}$$ Note that we neglect ion-electron collisions, -and allow for separate values of $\gamma$. -We split $n_i$, $n_e$, $\vb{u}_i$ and $\phi$ into an equilibrium -(subscript $0$) and a perturbation (subscript $1$): +and allow for separate values of $$\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 @@ -47,8 +47,8 @@ $$\begin{aligned} = \phi_0 + \phi_1 \end{aligned}$$ -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 $\phi_0$ +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: $$\begin{aligned} @@ -65,9 +65,9 @@ $$\begin{aligned} Inserting this decomposition into the momentum equations yields new equations. -Note that we will implicitly use $\vb{u}_{i0} = 0$ +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: +$$\mathrm{D}/\mathrm{D} t$$ is linear: $$\begin{aligned} m_i (n_{i0} \!+\! n_{i1}) \frac{\mathrm{D} (\vb{u}_{i0} \!+\! \vb{u}_{i1})}{\mathrm{D} t} @@ -78,7 +78,7 @@ $$\begin{aligned} \end{aligned}$$ Using the defined properties of the equilibrium components -$n_{i0}$, $n_{e0}$, $\vb{u}_{i0}$ and $\phi_0$, +$$n_{i0}$$, $$n_{e0}$$, $$\vb{u}_{i0}$$ and $$\phi_0$$, and neglecting all products of perturbations for being small, this reduces to: @@ -118,7 +118,7 @@ $$\begin{aligned} \end{aligned}$$ The electron equation can easily be rearranged -to get a relation between $n_{e1}$ and $n_{e0}$: +to get a relation between $$n_{e1}$$ and $$n_{e0}$$: $$\begin{aligned} i \vb{k} \gamma_e k_B T_e n_{e1} @@ -131,12 +131,12 @@ $$\begin{aligned} Due to their low mass, the electrons' heat conductivity can be regarded as infinite compared to the ions'. In that case, all electron gas compression is isothermal, -meaning it obeys the ideal gas law $p_e = n_e k_B T_e$, so that $\gamma_e = 1$. +meaning it obeys the ideal gas law $$p_e = n_e k_B T_e$$, so that $$\gamma_e = 1$$. Note that this yields the first-order term of a Taylor expansion of the [Boltzmann relation](/know/concept/boltzmann-relation/). -At equilibrium, quasi-neutrality demands that $n_{i0} = n_{e0} = n_0$, -so we can rearrange the above relation to $n_0 = - k_B T_e n_{e1} / (q_e \phi_1)$, +At equilibrium, quasi-neutrality demands that $$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} @@ -148,8 +148,8 @@ $$\begin{gathered} = T_e n_{e1} |\vb{k}|^2 - \gamma_i T_i n_{i1} |\vb{k}|^2 \end{gathered}$$ -Where we have taken the dot product with $\vb{k}$, -and used that $q_i / q_e = -1$. +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: @@ -160,7 +160,7 @@ $$\begin{aligned} \end{aligned}$$ Then we insert our plane-wave ansatz, -and substitute $n_{i0} = n_0$ as before, yielding: +and substitute $$n_{i0} = n_0$$ as before, yielding: $$\begin{aligned} 0 @@ -172,7 +172,7 @@ $$\begin{aligned} \end{aligned}$$ Substituting this in the ion momentum equation -leads us to a dispersion relation $\omega(\vb{k})$: +leads us to a dispersion relation $$\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}} @@ -184,8 +184,8 @@ $$\begin{gathered} = \frac{|\vb{k}|^2}{m_i} \Big( k_B T_e \frac{n_{e1}}{n_{i1}} - \gamma_i k_B T_i \Big) \end{gathered}$$ -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, +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**. @@ -199,8 +199,8 @@ $$\begin{aligned} \end{aligned}$$ One final time, we insert our plane-wave ansatz, -and use our Boltzmann-like relation between $n_{e1}$ and $n_{e0}$ -to substitute $\phi_1 = - k_B T_e n_{e1} / (q_e n_{e0})$: +and use our Boltzmann-like relation between $$n_{e1}$$ and $$n_{e0}$$ +to substitute $$\phi_1 = - k_B T_e n_{e1} / (q_e n_{e0})$$: $$\begin{gathered} q_e (n_{e1} - n_{i1}) @@ -213,7 +213,7 @@ $$\begin{gathered} = n_{e1} \big( 1 + |\vb{k}|^2 \lambda_{De}^2 \big) \end{gathered}$$ -Where $\lambda_{De}$ is the electron [Debye length](/know/concept/debye-length/). +Where $$\lambda_{De}$$ is the electron [Debye length](/know/concept/debye-length/). We thus reach the following dispersion relation, which governs **ion sound waves** or **ion acoustic waves**: @@ -224,7 +224,7 @@ $$\begin{aligned} } \end{aligned}$$ -The aforementioned plasma approximation is valid if $|\vb{k}| \lambda_{De} \ll 1$, +The aforementioned plasma approximation is valid if $$|\vb{k}| \lambda_{De} \ll 1$$, which is often reasonable, in which case this dispersion relation reduces to: @@ -233,7 +233,7 @@ $$\begin{aligned} = \frac{|\vb{k}|^2}{m_i} \bigg( k_B T_e + \gamma_i k_B T_i \bigg) \end{aligned}$$ -The phase velocity $v_s$ of these waves, +The phase velocity $$v_s$$ of these waves, i.e. the speed of sound, is then given by: $$\begin{aligned} @@ -245,11 +245,11 @@ $$\begin{aligned} \end{aligned}$$ Curiously, unlike a neutral gas, -this velocity is nonzero even if $T_i = 0$, +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 $T_e \gg T_i$, +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 $n_{i1}$. +is that they involve ion density fluctuations $$n_{i1}$$. |