From 3170fc4b5c915669cf209a521e551115a9bd0809 Mon Sep 17 00:00:00 2001
From: Prefetch
Date: Wed, 20 Oct 2021 11:50:20 +0200
Subject: Expand knowledge base

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+---
+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.
-- 
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