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

Ion sound wave

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 momentum equations, rewriting the electric field E=ϕ\vb{E} = - \nabla \phi and the pressure gradient p=γkBTn\nabla p = \gamma k_B T \nabla n, and arguing that me0m_e \approx 0 because memim_e \ll m_i:

miniDuiDt=qiniϕγikBTini0=qeneϕγekBTene\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 γ\gamma. We split nin_i, nen_e, ui\vb{u}_i and ϕ\phi into an equilibrium (subscript 00) and a perturbation (subscript 11):

ni=ni0+ni1ne=ne0+ne1ui=ui0+ui1ϕ=ϕ0+ϕ1\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 ni1n_{i1}, ne1n_{e1}, ui1\vb{u}_{i1} and ϕ1\phi_1 are tiny, and the equilibrium components ni0n_{i0}, ne0n_{e0}, ui0\vb{u}_{i0} and ϕ0\phi_0 are assumed to satisfy:

ni0t=0Dui0Dt=0ni0=ne0=0ui0=0ϕ0=0\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:

mi(ni0 ⁣+ ⁣ni1)D(ui0 ⁣+ ⁣ui1)Dt=qi(ni0 ⁣+ ⁣ni1)(ϕ0 ⁣+ ⁣ϕ1)γikBTi(ni0 ⁣+ ⁣ni1)0=qe(ne0 ⁣+ ⁣ne1)(ϕ0 ⁣+ ⁣ϕ1)γekBTe(ne0 ⁣+ ⁣ne1)\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 assumed properties of ni0n_{i0}, ne0n_{e0}, ui0\vb{u}_{i0} and ϕ0\phi_0, and discarding products of perturbations for being too small, we arrive at the below equations. Our choice ui0=0\vb{u}_{i0} = 0 lets us linearize the material derivative D/Dt=/t\mathrm{D}/\mathrm{D} t = \ipdv{}{t} for the ions:

mini0ui1tqini0ϕ1γikBTini10qene0ϕ1γekBTene1\begin{aligned} m_i n_{i0} \pdv{\vb{u}_{i1}}{t} &\approx - q_i n_{i0} \nabla \phi_1 - \gamma_i k_B T_i \nabla n_{i1} \\ 0 &\approx - 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:

ni1(r,t)=ni1exp(ikriωt)ne1(r,t)=ne1exp(ikriωt)ui1(r,t)=ui1exp(ikriωt)ϕ1(r,t)=ϕ1exp(ikriωt)\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:

iωmini0ui1=ikqini0ϕ1ikγikBTini10=ikqene0ϕ1ikγekBTene1\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 ne1n_{e1} and ne0n_{e0}:

ikγekBTene1=ikqene0ϕ1    ne1=qeϕ1γekBTene0\begin{aligned} i \vb{k} \gamma_e k_B T_e n_{e1} = - i \vb{k} q_e n_{e0} \phi_1 \qquad \implies \qquad 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 pe=nekBTep_e = n_e k_B T_e, so that γe=1\gamma_e = 1. Note that this yields the first-order term of a Taylor expansion of the Boltzmann relation.

At equilibrium, quasi-neutrality demands that ni0=ne0=n0n_{i0} = n_{e0} = n_0, so we can rearrange the above relation to n0=kBTene1/(qeϕ1)n_0 = - k_B T_e n_{e1} / (q_e \phi_1), which we insert into the ion equation to get:

iωmikBTene1qeϕ1ui1=iqikBTene1qeϕ1ϕ1kiγikBTini1k    ωmiTene1qeϕ1kui1=Tene1k2γiTini1k2\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 k\vb{k}, and used that qi/qe=1q_i / q_e = -1. In order to simplify this equation, we turn to the two-fluid ion continuity relation:

0=(ni0 ⁣+ ⁣ni1)t+((ni0 ⁣+ ⁣ni1)(ui0 ⁣+ ⁣ui1))ni1t+ni0ui1\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}

Into which we insert our plane-wave ansatz, and substitute ni0=n0n_{i0} = n_0 as before, yielding:

0=iωni1+ini0kui1    kui1=ωni1ni0=ωqeni1ϕ1kBTene1\begin{aligned} 0 = - i \omega n_{i1} + i n_{i0} \vb{k} \cdot \vb{u}_{i1} \qquad \implies \qquad \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 ω(k)\omega(\vb{k}):

ω2miTene1qeϕ1qeni1ϕ1kBTene1=ω2mini1kB=k2(Tene1γiTini1)    ω2=k2mi(kBTene1ni1γikBTi)\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 ne1/ni1n_{e1} / n_{i1}. It cannot be 11, because then ϕ1\phi_1 could not be nonzero, according to Gauss’ law. Nevertheless, some authors tend to ignore this fact, thereby making the so-called plasma approximation. We will not, and thus turn to Gauss’ law:

ε0E=ε02ϕ1=qiniqene=qe(ni1ne1)\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 ne1n_{e1} and ne0n_{e0} to substitute ϕ1=kBTene1/(qene0)\phi_1 = - k_B T_e n_{e1} / (q_e n_{e0}):

qe(ne1ni1)=k2ε0ϕ1=k2ε0kBTene1qene0    ni1=ne1+k2ε0kBTene1qe2ne0=ne1(1+k2λDe2)\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 λDe\lambda_{De} is the electron Debye length. We thus reach the following dispersion relation, which governs ion sound waves or ion acoustic waves:

ω2=k2mi(kBTe1+k2λDe2+γikBTi)\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 kλDe1|\vb{k}| \lambda_{De} \ll 1, which is often reasonable, in which case this dispersion relation reduces to:

ω2=k2mi(kBTe+γikBTi)\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 vsv_s of these waves, i.e. the speed of sound, is then given by:

vs=ωk=kBTemi+γikBTimi\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 in a neutral gas, this velocity is nonzero even if Ti=0T_i = 0, meaning that the waves still exist then. In fact, usually the electron temperature TeT_e dominates TeTiT_e \gg T_i, even though the main feature of these waves is that they involve ion density fluctuations ni1n_{i1}.


  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.