---
title: "Debye length"
firstLetter: "D"
publishDate: 2021-10-18
categories:
- Physics
- Plasma physics

date: 2021-10-15T20:28:31+02:00
draft: false
markup: pandoc
---

# 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](/know/concept/maxwells-equations/)
for the [electric field](/know/concept/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](/know/concept/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](/know/concept/boltzmann-relation/)
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](/know/concept/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.



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    2021, unpublished.