Debye Length
\[\lambda_D = \sqrt{\frac{\varepsilon_0 k_B T_e}{n_e e^2}}\]
Source lecture(s): pc368_lec01_intro, pc368_lec02_debye
Physical Meaning
The Debye length is the distance over which a charge disturbance in a plasma is screened by the rearrangement of electrons and ions. It is the plasma analog of the Thomas–Fermi screening length in metals, but here the “screening” is collective and thermal.
Variable Definitions
| Symbol | Definition | SI Units |
|---|---|---|
| \(\varepsilon_0\) | Vacuum permittivity | F m\(^{-1}\) |
| \(k_B\) | Boltzmann constant | J K\(^{-1}\) |
| \(T_e\) | Electron temperature | K (or eV × 11605) |
| \(n_e\) | Electron number density | m\(^{-3}\) |
| \(e\) | Elementary charge | C |
Assumptions
- Plasma is quasi-neutral on scales \(L \gg \lambda_D\).
- Electrons obey a Boltzmann distribution; ions are immobile.
- The perturbation is small: \(|e\phi| \ll k_B T_e\).
- The medium is unmagnetized (or B-fields do not affect radial shielding).
Derivation
Start from Poisson’s equation for a purely electrostatic potential \(\phi(r)\) in spherical symmetry:
\[\frac{1}{r^2} \frac{d}{dr}\Bigl(r^2 \frac{d\phi}{dr}\Bigr) = -\frac{\rho}{\varepsilon_0}\]
For a pure electron plasma responding to a positive test charge \(+Ze\):
\[\rho = e\bigl(Z\delta(\mathbf{r}) - n_e\bigr)\]
with Boltzmann response \(n_e = n_0 \exp(e\phi/k_B T_e)\). Linearizing:
\[n_e \approx n_0\Bigl(1 + \frac{e\phi}{k_B T_e}\Bigr)\]
Insert into Poisson and assume \(r > 0\) (outside the singular source):
\[\frac{1}{r^2} \frac{d}{dr}\Bigl(r^2 \frac{d\phi}{dr}\Bigr) - \frac{\phi}{\lambda_D^2} = 0\]
The Green’s function is the Yukawa potential:
\[\phi(r) = \frac{Ze}{4\pi\varepsilon_0 r} e^{-r/\lambda_D}\]
Applications
- Quasi-neutrality: Any macroscopic plasma region \(L \gg \lambda_D\) is effectively neutral.
- RF shielding: High-frequency electromagnetic waves reflect at critical densities \(n_c > \varepsilon_0 m_e \omega^2/e^2\) inside \(\sim \lambda_D\).
- Collisionless damping: The Landau length scale is \(k\lambda_D \sim 1\).
Connections to Other Equations
- Plasma Frequency: \(\omega_{pe}^2 = n_e e^2/(m_e \varepsilon_0)\), and \(\lambda_D = v_{te}/\omega_{pe}\).
- Vlasov Equation: Linearization uses \(\lambda_D\) as the small parameter.
- Langmuir Waves: \(\omega^2 = \omega_{pe}^2 + 3 k^2 v_{te}^2\).