Categories: Physics, Plasma physics.

# Debye length

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:

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:

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 $\lambda_D$:

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:

Where $A$ and $B$ are scaling constants that depend on the problem at hand.

## References

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*Plasma physics: lecture notes*, 2021, unpublished.