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diff --git a/content/know/concept/langmuir-waves/index.pdc b/content/know/concept/langmuir-waves/index.pdc new file mode 100644 index 0000000..cd9b449 --- /dev/null +++ b/content/know/concept/langmuir-waves/index.pdc @@ -0,0 +1,260 @@ +--- +title: "Langmuir waves" +firstLetter: "L" +publishDate: 2021-10-30 +categories: +- Physics +- Plasma physics + +date: 2021-10-15T20:31:46+02:00 +draft: false +markup: pandoc +--- + +# Langmuir waves + +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 $m_i \gg m_e$), +and therefore no ion-electron momentum transfer, +the [two-fluid equations](/know/concept/two-fluid-equations/) +tell us that: + +$$\begin{aligned} + m_e n_e \frac{\mathrm{D} \vb{u}_e}{\mathrm{D} t} + = q_e n_e \vb{E} - \nabla p_e + \qquad \quad + \pdv{n_e}{t} + \nabla \cdot (n_e \vb{u}_e) = 0 +\end{aligned}$$ + +These are the electron momentum and continuity equations. +We also need [Gauss' law](/know/concept/maxwells-equations/): + +$$\begin{aligned} + \varepsilon_0 \nabla \cdot \vb{E} + = q_e (n_i - n_e) +\end{aligned}$$ + +We split $n_e$, $\vb{u}_e$ and $\vb{E}$ into a base component +(subscript $0$) and a perturbation (subscript $1$): + +$$\begin{aligned} + n_e + = n_{e0} + n_{e1} + \qquad \quad + \vb{u}_e + = \vb{u}_{e0} + \vb{u}_{e1} + \qquad \quad + \vb{E} + = \vb{E}_0 + \vb{E}_1 +\end{aligned}$$ + +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 $\vb{E}_0$ +by definition satisfy: + +$$\begin{aligned} + \pdv{n_{e0}}{t} = 0 + \qquad + \pdv{\vb{u}_{e0}}{t} = 0 + \qquad + \nabla n_{e0} = 0 + \qquad + \vb{u}_{e0} = 0 + \qquad + \vb{E}_0 = 0 +\end{aligned}$$ + +We insert this decomposistion into the electron continuity equation, +arguing that $n_{e1} \vb{u}_{e1}$ is small enough to neglect, leading to: + +$$\begin{aligned} + 0 + &= \pdv{(n_{e0} \!+\! n_{e1})}{t} + \nabla \cdot \Big( (n_{e0} \!+\! n_{e1}) \: (\vb{u}_{e0} \!+\! \vb{u}_{e1}) \Big) + \\ + &= \pdv{n_{e1}}{t} + \nabla \cdot \Big( n_{e0} \vb{u}_{e1} + n_{e1} \vb{u}_{e1} \Big) + \\ + &\approx \pdv{n_{e1}}{t} + \nabla \cdot (n_{e0} \vb{u}_{e1}) + = \pdv{n_{e1}}{t} + n_{e0} \nabla \cdot \vb{u}_{e1} +\end{aligned}$$ + +Likewise, we insert it into Gauss' law, +and use the plasma's quasi-neutrality $n_i = n_{e0}$ to get: + +$$\begin{aligned} + \varepsilon_0 \nabla \cdot \big( \vb{E}_0 \!+\! \vb{E}_1 \big) + = q_e (n_i - n_{e0} \!-\! n_{e1} ) + \quad \implies \quad + \varepsilon_0 \nabla \cdot \vb{E}_1 + = - q_e n_{e1} +\end{aligned}$$ + +Since we are looking for linear waves, +we make the following ansatz for the perturbations: + +$$\begin{aligned} + n_{e1}(\vb{r}, t) + &= n_{e1} \exp\!(i \vb{k} \cdot \vb{r} - i \omega t) + \\ + \vb{u}_{e1}(\vb{r}, t) + &= \vb{u}_{e1} \exp\!(i \vb{k} \cdot \vb{r} - i \omega t) + \\ + \vb{E}_1(\vb{r}, t) + &= \vb{E}_1 \:\exp\!(i \vb{k} \cdot \vb{r} - i \omega t) +\end{aligned}$$ + +Inserting this into the continuity equation and Gauss' law yields, respectively: + +$$\begin{aligned} + - i \omega n_{e1} = - i n_{e0} \vb{k} \cdot \vb{u}_{e1} + \qquad \quad + i \varepsilon_0 \vb{k} \cdot \vb{E}_1 = q_e n_{e1} +\end{aligned}$$ + +However, there are three unknowns $n_{e1}$, $\vb{u}_{e1}$ and $\vb{E}_1$, +so one more equation is needed. + + +## Cold Langmuir waves + +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: + +$$\begin{aligned} + m_e n_e \frac{\mathrm{D} \vb{u}_e}{\mathrm{D} t} + = q_e n_e \vb{E} +\end{aligned}$$ + +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: + +$$\begin{gathered} + m_e (n_{e0} \!+\! n_{e1}) \Big( \pdv{(\vb{u}_{e0} \!+\! \vb{u}_{e1})}{t} + + \big( (\vb{u}_{e0} \!+\! \vb{u}_{e1}) \cdot \nabla \big) (\vb{u}_{e0} \!+\! \vb{u}_{e1}) \Big) + = q_e \big( n_{e0} \!+\! n_{e1} \big) \big( \vb{E}_0 \!+\! \vb{E}_1 \big) + \\ + \implies \qquad + q_e \vb{E}_1 + = m_e \Big( \pdv{\vb{u}_{e1}}{t} + \big(\vb{u}_{e1} \cdot \nabla \big) \vb{u}_{e1} \Big) + \approx m_e \pdv{\vb{u}_{e1}}{t} +\end{gathered}$$ + +And then inserting our plane-wave ansatz yields +the third equation we were looking for: + +$$\begin{aligned} + -i \omega m_e \vb{u}_{e1} = q_e \vb{E}_1 +\end{aligned}$$ + +Solving this system of three equations for $\omega^2$ +gives the following dispersion relation: + +$$\begin{aligned} + \omega^2 + = \frac{\omega n_{e0}}{n_{e1}} \vb{k} \cdot \vb{u}_{e1} + = \frac{i \omega n_{e0} q_e}{\omega m_e n_{e1}} \vb{k} \cdot \vb{E}_1 + = \frac{i n_{e0} n_{e1} q_e^2}{i \varepsilon_0 m_e n_{e1}} + = \frac{n_{e0} q_e^2}{\varepsilon_0 m_e} +\end{aligned}$$ + +This result is known as the **plasma frequency** $\omega_p$, +and describes the frequency of **cold Langmuir waves**, +otherwise known as **plasma oscillations**: + +$$\begin{aligned} + \boxed{ + \omega_p + = \sqrt{\frac{n_{0e} q_e^2}{\varepsilon_0 m_e}} + } +\end{aligned}$$ + +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: + +$$\begin{aligned} + m_e n_{e0} \pdv{\vb{u}_{e1}}{t} + = q_e n_{e0} \vb{E}_1 - \nabla p_e +\end{aligned}$$ + +From the two-fluid thermodynamic equation of state, +we know that $\nabla p_e$ can be written as: + +$$\begin{aligned} + \nabla p_e + = \gamma k_B T_e \nabla n_e + = \gamma k_B T_e \nabla (n_{e0} + n_{e1}) + = \gamma k_B T_e \nabla n_{e1} +\end{aligned}$$ + +With this, insertion of our plane-wave ansatz +into the electron equation results in: + +$$\begin{aligned} + -i \omega m_e n_{e0} \vb{u}_{e1} = q_e n_{e0} \vb{E}_1 - i \gamma k_B T_e n_{e1} \vb{k} +\end{aligned}$$ + +Which once again closes the system of three equations. +Solving for $\omega^2$ then gives: + +$$\begin{aligned} + \omega^2 + = \frac{\omega n_{e0}}{n_{e1}} \vb{k} \cdot \vb{u}_{e1} + &= \frac{i \omega n_{e0}}{\omega n_{e0} m_e n_{e1}} \vb{k} \cdot \Big( q_e n_{e0} \vb{E}_1 - i \gamma k_B T_e n_{e1} \vb{k} \Big) + \\ + &= \frac{n_{e0} q_e^2}{\varepsilon_0 m_e} - \frac{i \omega}{\omega m_e n_{e1}} i \gamma k_B T_e n_{e1} \big(\vb{k} \cdot \vb{k}\big) +\end{aligned}$$ + +Recognizing the first term as the plasma frequency $\omega_p^2$, +we therefore arrive at the **Bohm-Gross dispersion relation** $\omega(\vb{k})$ +for **warm Langmuir waves**: + +$$\begin{aligned} + \boxed{ + \omega^2 + = \omega_p^2 + \frac{\gamma k_B T_e}{m_e} |\vb{k}|^2 + } +\end{aligned}$$ + +This expression is typically quoted for 1D oscillations, +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: + +$$\begin{aligned} + v_g + = \pdv{\omega}{k} + \to \sqrt{\frac{3 k_B T_e}{m_e}} +\end{aligned}$$ + +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 $k$. + + + +## 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. |