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diff --git a/source/know/concept/langmuir-waves/index.md b/source/know/concept/langmuir-waves/index.md index 0b7f2d7..7dc5dbf 100644 --- a/source/know/concept/langmuir-waves/index.md +++ b/source/know/concept/langmuir-waves/index.md @@ -13,8 +13,8 @@ layout: "concept" In plasma physics, **Langmuir waves** are oscillations in the electron density, which may or may not propagate, depending on the temperature. -Assuming no [magnetic field](/know/concept/magnetic-field/) $\vb{B} = 0$, -no ion motion $\vb{u}_i = 0$ (since $m_i \gg m_e$), +Assuming no [magnetic field](/know/concept/magnetic-field/) $$\vb{B} = 0$$, +no ion motion $$\vb{u}_i = 0$$ (since $$m_i \gg m_e$$), and therefore no ion-electron momentum transfer, the [two-fluid equations](/know/concept/two-fluid-equations/) tell us that: @@ -34,8 +34,8 @@ $$\begin{aligned} = q_e (n_e - n_i) \end{aligned}$$ -We split $n_e$, $\vb{u}_e$ and $\vb{E}$ into a base component -(subscript $0$) and a perturbation (subscript $1$): +We split $$n_e$$, $$\vb{u}_e$$ and $$\vb{E}$$ into a base component +(subscript $$0$$) and a perturbation (subscript $$1$$): $$\begin{aligned} n_e @@ -48,8 +48,8 @@ $$\begin{aligned} = \vb{E}_0 + \vb{E}_1 \end{aligned}$$ -Where the perturbations $n_{e1}$, $\vb{u}_{e1}$ and $\vb{E}_1$ are very small, -and the equilibrium components $n_{e0}$, $\vb{u}_{e0}$ and $\vb{E}_0$ +Where the perturbations $$n_{e1}$$, $$\vb{u}_{e1}$$ and $$\vb{E}_1$$ are very small, +and the equilibrium components $$n_{e0}$$, $$\vb{u}_{e0}$$ and $$\vb{E}_0$$ by definition satisfy: $$\begin{aligned} @@ -65,7 +65,7 @@ $$\begin{aligned} \end{aligned}$$ We insert this decomposistion into the electron continuity equation, -arguing that $n_{e1} \vb{u}_{e1}$ is small enough to neglect, leading to: +arguing that $$n_{e1} \vb{u}_{e1}$$ is small enough to neglect, leading to: $$\begin{aligned} 0 @@ -78,7 +78,7 @@ $$\begin{aligned} \end{aligned}$$ Likewise, we insert it into Gauss' law, -and use the plasma's quasi-neutrality $n_i = n_{e0}$ to get: +and use the plasma's quasi-neutrality $$n_i = n_{e0}$$ to get: $$\begin{aligned} \varepsilon_0 \nabla \cdot \big( \vb{E}_0 \!+\! \vb{E}_1 \big) @@ -110,7 +110,7 @@ $$\begin{aligned} -\! i \varepsilon_0 \vb{k} \cdot \vb{E}_1 = q_e n_{e1} \end{aligned}$$ -However, there are three unknowns $n_{e1}$, $\vb{u}_{e1}$ and $\vb{E}_1$, +However, there are three unknowns $$n_{e1}$$, $$\vb{u}_{e1}$$ and $$\vb{E}_1$$, so one more equation is needed. @@ -118,7 +118,7 @@ so one more equation is needed. We therefore turn to the electron momentum equation. For now, let us assume that the electrons have no thermal motion, -i.e. the electron temperature $T_e = 0$, so that $p_e = 0$, leaving: +i.e. the electron temperature $$T_e = 0$$, so that $$p_e = 0$$, leaving: $$\begin{aligned} m_e n_e \frac{\mathrm{D} \vb{u}_e}{\mathrm{D} t} @@ -126,8 +126,8 @@ $$\begin{aligned} \end{aligned}$$ Inserting the decomposition then gives the following, -where we neglect $(\vb{u}_{e1} \cdot \nabla) \vb{u}_{e1}$ -because $\vb{u}_{e1}$ is so small by assumption: +where we neglect $$(\vb{u}_{e1} \cdot \nabla) \vb{u}_{e1}$$ +because $$\vb{u}_{e1}$$ is so small by assumption: $$\begin{gathered} m_e (n_{e0} \!+\! n_{e1}) \Big( \pdv{(\vb{u}_{e0} \!+\! \vb{u}_{e1})}{t} @@ -147,7 +147,7 @@ $$\begin{aligned} -i \omega m_e \vb{u}_{e1} = q_e \vb{E}_1 \end{aligned}$$ -Solving this system of three equations for $\omega^2$ +Solving this system of three equations for $$\omega^2$$ gives the following dispersion relation: $$\begin{aligned} @@ -158,7 +158,7 @@ $$\begin{aligned} = \frac{n_{e0} q_e^2}{\varepsilon_0 m_e} \end{aligned}$$ -This result is known as the **plasma frequency** $\omega_p$, +This result is known as the **plasma frequency** $$\omega_p$$, and describes the frequency of **cold Langmuir waves**, otherwise known as **plasma oscillations**: @@ -169,16 +169,16 @@ $$\begin{aligned} } \end{aligned}$$ -Note that this is a dispersion relation $\omega(k) = \omega_p$, -but that $\omega_p$ does not contain $k$. +Note that this is a dispersion relation $$\omega(k) = \omega_p$$, +but that $$\omega_p$$ does not contain $$k$$. This means that cold Langmuir waves do not propagate: the oscillation is "stationary". ## Warm Langmuir waves -Next, we generalize this result to nonzero $T_e$, -in which case the pressure $p_e$ is involved: +Next, we generalize this result to nonzero $$T_e$$, +in which case the pressure $$p_e$$ is involved: $$\begin{aligned} m_e n_{e0} \pdv{}{\vb{u}{e1}}{t} @@ -186,7 +186,7 @@ $$\begin{aligned} \end{aligned}$$ From the two-fluid thermodynamic equation of state, -we know that $\nabla p_e$ can be written as: +we know that $$\nabla p_e$$ can be written as: $$\begin{aligned} \nabla p_e @@ -203,7 +203,7 @@ $$\begin{aligned} \end{aligned}$$ Which once again closes the system of three equations. -Solving for $\omega^2$ then gives: +Solving for $$\omega^2$$ then gives: $$\begin{aligned} \omega^2 @@ -213,8 +213,8 @@ $$\begin{aligned} &= \frac{n_{e0} q_e^2}{\varepsilon_0 m_e} - \frac{i \omega}{\omega m_e n_{e1}} i \gamma k_B T_e n_{e1} \big(\vb{k} \cdot \vb{k}\big) \end{aligned}$$ -Recognizing the first term as the plasma frequency $\omega_p^2$, -we therefore arrive at the **Bohm-Gross dispersion relation** $\omega(\vb{k})$ +Recognizing the first term as the plasma frequency $$\omega_p^2$$, +we therefore arrive at the **Bohm-Gross dispersion relation** $$\omega(\vb{k})$$ for **warm Langmuir waves**: $$\begin{aligned} @@ -225,16 +225,16 @@ $$\begin{aligned} \end{aligned}$$ This expression is typically quoted for 1D oscillations, -in which case $\gamma = 3$ and $k = |\vb{k}|$: +in which case $$\gamma = 3$$ and $$k = |\vb{k}|$$: $$\begin{aligned} \omega^2 = \omega_p^2 + \frac{3 k_B T_e}{m_e} k^2 \end{aligned}$$ -Unlike for $T_e = 0$, these "warm" waves do propagate, -carrying information at group velocity $v_g$, -which, in the limit of large $k$, is given by: +Unlike for $$T_e = 0$$, these "warm" waves do propagate, +carrying information at group velocity $$v_g$$, +which, in the limit of large $$k$$, is given by: $$\begin{aligned} v_g @@ -244,7 +244,7 @@ $$\begin{aligned} This is the root-mean-square velocity of the [Maxwell-Boltzmann speed distribution](/know/concept/maxwell-boltzmann-distribution/), -meaning that information travels at the thermal velocity for large $k$. +meaning that information travels at the thermal velocity for large $$k$$. |