Categories: Physics, Plasma physics.

If a charged object is put in a plasma, it repels like charges and attracts opposite charges, leading to a **Debye sheath** around the object’s surface with a net opposite charge. This has the effect of **shielding** the object’s presence from the rest of the plasma.

We start from Gauss’ law for the electric field \(\vb{E}\), expressing \(\vb{E}\) as the gradient of a potential \(\phi\), i.e. \(\vb{E} = -\nabla \phi\), and splitting the charge density into ions \(n_i\) and electrons \(n_e\):

\[\begin{aligned} \nabla^2 \phi(\vb{r}) = - \frac{1}{\varepsilon_0} \Big( q_i n_i(\vb{r}) + q_e n_e(\vb{r}) + q_t \delta(\vb{r}) \Big) \end{aligned}\]

The last term represents a *test particle*, which will be shielded. This particle is a point charge \(q_t\), whose density is simply a Dirac delta function \(\delta(\vb{r})\), and is not included in \(n_i\) or \(n_e\).

For a plasma in thermal equilibrium, we have the Boltzmann relations for the densities:

\[\begin{aligned} n_i(\vb{r}) = n_{i0} \exp\!\bigg( \!-\! \frac{q_i \phi(\vb{r})}{k_B T_i} \bigg) \qquad \quad n_e(\vb{r}) = n_{e0} \exp\!\bigg( \!-\! \frac{q_e \phi(\vb{r})}{k_B T_e} \bigg) \end{aligned}\]

We assume that electrical interactions are weak compared to thermal effects, i.e. \(k_B T \gg q \phi\) in both cases. Then we Taylor-expand the Boltzmann relations to first order:

\[\begin{aligned} n_i(\vb{r}) \approx n_{i0} \bigg( 1 - \frac{q_i \phi(\vb{r})}{k_B T_i} \bigg) \qquad \quad n_e(\vb{r}) \approx n_{e0} \bigg( 1 - \frac{q_e \phi(\vb{r})}{k_B T_e} \bigg) \end{aligned}\]

Inserting this back into Gauss’ law, we arrive at the following equation for \(\phi(\vb{r})\), where we have assumed quasi-neutrality such that \(q_i n_{i0} = q_e n_{e0}\):

\[\begin{aligned} \nabla^2 \phi &= - \frac{1}{\varepsilon_0} \bigg( q_i n_{i0} - n_{i0} \frac{q_i^2 \phi}{k_B T_i} + q_e n_{e0} - n_{e0} \frac{q_e^2 \phi}{k_B T_e} + q_t \delta(\vb{r}) \bigg) \\ &= \bigg( \frac{n_{i0} q_i^2}{\varepsilon_0 k_B T_i} + \frac{n_{e0} q_e^2}{\varepsilon_0 k_B T_e} \bigg) \phi - \frac{q_t}{\varepsilon_0} \delta(\vb{r}) \end{aligned}\]

We now define the **ion** and **electron Debye lengths** \(\lambda_{Di}\) and \(\lambda_{De}\) as follows:

\[\begin{aligned} \boxed{ \frac{1}{\lambda_{Di}^2} \equiv \frac{n_{i0} q_i^2}{\varepsilon_0 k_B T_i} } \qquad \quad \boxed{ \frac{1}{\lambda_{De}^2} \equiv \frac{n_{e0} q_e^2}{\varepsilon_0 k_B T_e} } \end{aligned}\]

And then the **total Debye length** \(\lambda_D\) is defined as the sum of their inverses, and gives the rough thickness of the Debye sheath:

\[\begin{aligned} \boxed{ \frac{1}{\lambda_D^2} \equiv \frac{1}{\lambda_{Di}^2} + \frac{1}{\lambda_{De}^2} = \frac{n_{i0} q_i^2 T_e + n_{e0} q_e^2 T_i}{\varepsilon_0 k_B T_i T_e} } \end{aligned}\]

With this, the equation can be put in the form below, suggesting exponential decay:

\[\begin{aligned} \nabla^2 \phi(\vb{r}) &= \frac{1}{\lambda_D^2} \phi(\vb{r}) - \frac{q_t}{\varepsilon_0} \delta(\vb{r}) \end{aligned}\]

This has the following solution, known as the **Yukawa potential**, which decays exponentially, representing the plasma’s **self-shielding** over a characteristic distance \(\lambda_D\):

\[\begin{aligned} \boxed{ \phi(r) = \frac{q_t}{4 \pi \varepsilon_0 r} \exp\!\Big( \!-\!\frac{r}{\lambda_D} \Big) } \end{aligned}\]

Note that \(r\) is a scalar, i.e. the potential depends only on the radial distance to \(q_t\). This treatment only makes sense if the plasma is sufficiently dense, such that there is a large number of particles in a sphere with radius \(\lambda_D\). This corresponds to a large Coulomb logarithm \(\ln\!(\Lambda)\):

\[\begin{aligned} 1 \ll \frac{4 \pi}{3} n_0 \lambda_D^3 = \frac{2}{9} \Lambda \end{aligned}\]

The name *Yukawa potential* originates from particle physics, but can in general be used to refer to any potential (electric or energetic) of the following form:

\[\begin{aligned} V(r) = \frac{A}{r} \exp\!(-B r) \end{aligned}\]

Where \(A\) and \(B\) are scaling constants that depend on the problem at hand.

- P.M. Bellan,
*Fundamentals of plasma physics*, 1st edition, Cambridge. - M. Salewski, A.H. Nielsen,
*Plasma physics: lecture notes*, 2021, unpublished.

© Marcus R.A. Newman, a.k.a. "Prefetch".
Available under CC BY-SA 4.0.