Categories: Perturbation, Physics, Plasma physics, Plasma waves.

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 B=0\vb{B} = 0, no ion motion ui=0\vb{u}_i = 0 (since mimem_i \gg m_e), and therefore no ion-electron momentum transfer, the two-fluid equations tell us that:

meneDueDt=qeneEpenet+(neue)=0\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 \qquad \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:

ε0E=qe(neni)\begin{aligned} \varepsilon_0 \nabla \cdot \vb{E} = q_e (n_e - n_i) \end{aligned}

We split nen_e, ue\vb{u}_e and E\vb{E} into a base component (subscript 00) and a perturbation (subscript 11):

ne=ne0+ne1ue=ue0+ue1E=E0+E1\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 ne1n_{e1}, ue1\vb{u}_{e1} and E1\vb{E}_1 are very small, and the equilibrium components ne0n_{e0}, ue0\vb{u}_{e0} and E0\vb{E}_0 are assumed to satisfy:

ne0t=0ue0t=0ne0=0ue0=0E0=0\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 decomposition into the electron continuity equation, arguing that ne1ue1n_{e1} \vb{u}_{e1} is small enough to neglect, leading to:

0=(ne0 ⁣+ ⁣ne1)t+((ne0 ⁣+ ⁣ne1)(ue0 ⁣+ ⁣ue1))=ne1t+(ne0ue1+ne1ue1)ne1t+(ne0ue1)=ne1t+ne0ue1\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 ni=ne0n_i = n_{e0} to get:

ε0(E0 ⁣+ ⁣E1)=qe(ne0+ne1ni)    ε0E1=qene1\begin{aligned} \varepsilon_0 \nabla \cdot \big( \vb{E}_0 \!+\! \vb{E}_1 \big) = q_e (n_{e0} + n_{e1} - n_i) \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:

ne1(r,t)=ne1exp(ikriωt)ue1(r,t)=ue1exp(ikriωt)E1(r,t)=E1exp(ikriωt)\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:

iωne1=ine0kue1 ⁣iε0kE1=qene1\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 ne1n_{e1}, ue1\vb{u}_{e1} and E1\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 Te=0T_e = 0, so that pe=0p_e = 0, leaving:

meneDueDt=qeneE\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 (ue1)ue1(\vb{u}_{e1} \cdot \nabla) \vb{u}_{e1} because ue1\vb{u}_{e1} is so small by assumption:

me(ne0 ⁣+ ⁣ne1)((ue0 ⁣+ ⁣ue1)t+((ue0 ⁣+ ⁣ue1))(ue0 ⁣+ ⁣ue1))=qe(ne0 ⁣+ ⁣ne1)(E0 ⁣+ ⁣E1)    qeE1=me(ue1t+(ue1)ue1)meue1t\begin{gathered} m_e (n_{e0} \!+\! n_{e1}) \bigg( \pdv{(\vb{u}_{e0} \!+\! \vb{u}_{e1})}{t} + \big( (\vb{u}_{e0} \!+\! \vb{u}_{e1}) \cdot \nabla \big) (\vb{u}_{e0} \!+\! \vb{u}_{e1}) \bigg) = 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:

iωmeue1=qeE1\begin{aligned} -i \omega m_e \vb{u}_{e1} = q_e \vb{E}_1 \end{aligned}

Solving this system of three equations for ω2\omega^2 gives the following dispersion relation:

ω2=ωne0ne1kue1=iωne0qeωmene1kE1=ine0ne1qe2iε0mene1=ne0qe2ε0me\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 ωp\omega_p, and describes the frequency of cold Langmuir waves, otherwise known as plasma oscillations:

ωp=n0eqe2ε0me\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 ω(k)=ωp\omega(k) = \omega_p, but that ωp\omega_p does not contain kk. This means that cold Langmuir waves do not propagate: the oscillation is stationary.

Warm Langmuir waves

Next, we generalize this result to nonzero TeT_e, in which case the pressure pep_e is involved:

mene0ue1t=qene0E1pe\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 pe\nabla p_e can be written as:

pe=γkBTene=γkBTe(ne0+ne1)=γkBTene1\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:

iωmene0ue1=qene0E1iγkBTene1k\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 ω2\omega^2 then gives:

ω2=ωne0ne1kue1=iωne0ωne0mene1k(qene0E1iγkBTene1k)=ne0qe2ε0meiωωmene1iγkBTene1(kk)\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 ωp2\omega_p^2, we therefore arrive at the Bohm-Gross dispersion relation ω(k)\omega(\vb{k}) for warm Langmuir waves:

ω2=ωp2+γkBTemek2\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 γ=3\gamma = 3 and k=kk = |\vb{k}|:

ω2=ωp2+3kBTemek2\begin{aligned} \omega^2 = \omega_p^2 + \frac{3 k_B T_e}{m_e} k^2 \end{aligned}

Unlike for Te=0T_e = 0, these “warm” waves do propagate, carrying information at group velocity vgv_g, which, in the limit of large kk, is given by:

vg=ωk3kBTeme\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, meaning that information travels at the thermal velocity for large kk.


  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.