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 E\vb{E}, expressing E\vb{E} as the gradient of a potential ϕ\phi, i.e. E=ϕ\vb{E} = -\nabla \phi, and splitting the charge density into ions nin_i and electrons nen_e:

2ϕ(r)=1ε0(qini(r)+qene(r)+qtδ(r))\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 qtq_t, whose density is simply a Dirac delta function δ(r)\delta(\vb{r}), and is not included in nin_i or nen_e.

For a plasma in thermal equilibrium, we have the Boltzmann relations for the densities:

ni(r)=ni0exp ⁣( ⁣ ⁣qiϕ(r)kBTi)ne(r)=ne0exp ⁣( ⁣ ⁣qeϕ(r)kBTe)\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. kBTqϕk_B T \gg q \phi in both cases. Then we Taylor-expand the Boltzmann relations to first order:

ni(r)ni0(1qiϕ(r)kBTi)ne(r)ne0(1qeϕ(r)kBTe)\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 ϕ(r)\phi(\vb{r}), where we have assumed quasi-neutrality such that qini0=qene0q_i n_{i0} = q_e n_{e0}:

2ϕ=1ε0(qini0ni0qi2ϕkBTi+qene0ne0qe2ϕkBTe+qtδ(r))=(ni0qi2ε0kBTi+ne0qe2ε0kBTe)ϕqtε0δ(r)\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 λDi\lambda_{Di} and λDe\lambda_{De} as follows:

1λDi2ni0qi2ε0kBTi1λDe2ne0qe2ε0kBTe\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 λD\lambda_D is defined as the sum of their inverses, and gives the rough thickness of the Debye sheath:

1λD21λDi2+1λDe2=ni0qi2Te+ne0qe2Tiε0kBTiTe\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:

2ϕ(r)=1λD2ϕ(r)qtε0δ(r)\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 λD\lambda_D:

ϕ(r)=qt4πε0rexp ⁣( ⁣ ⁣rλD)\begin{aligned} \boxed{ \phi(r) = \frac{q_t}{4 \pi \varepsilon_0 r} \exp\!\Big( \!-\!\frac{r}{\lambda_D} \Big) } \end{aligned}

Note that rr is a scalar, i.e. the potential depends only on the radial distance to qtq_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 λD\lambda_D. This corresponds to a large Coulomb logarithm ln(Λ)\ln(\Lambda):

14π3n0λD3=29Λ\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:

V(r)=Arexp(Br)\begin{aligned} V(r) = \frac{A}{r} \exp(-B r) \end{aligned}

Where AA and BB 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.