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:

\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.

1. P.M. Bellan, Fundamentals of plasma physics, 1st edition, Cambridge.
2. 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.