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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.

References

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 M. Salewski, A.H. Nielsen, Plasma physics: lecture notes, 2021, unpublished.

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