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.

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