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Debye Shielding

Source lecture(s): pc368_lec01_intro, pc368_lec02_debye

Intuition

Drop a charged test particle into a plasma, and the mobile electrons and ions will rearrange themselves so that outside a small radius the electric potential is almost zero. This is the plasma analog of electrostatic screening in metals, except that the “perfect conductor” is collective motion rather than conduction electrons.

Formal Definition

Debye shielding is the exponential attenuation of the electric potential produced by an external charge embedded in a Maxwellian plasma. The shielding length is the Debye length:

\[\lambda_D = \sqrt{\frac{\varepsilon_0 k_B T_e}{n_e e^2}}\]

Mathematical Formulation

Start from Poisson’s equation:

\[\nabla^2 \phi = -\frac{\rho_{\text{ext}}}{\varepsilon_0} - \frac{e(n_i - n_e)}{\varepsilon_0}\]

For a singly charged, stationary source \(\rho_{\text{ext}} = Ze \delta(\mathbf{r})\):

\[\nabla^2 \phi - \frac{\phi}{\lambda_D^2} = -\frac{Ze}{\varepsilon_0} \delta(\mathbf{r})\]

The Green’s function gives the shielded Coulomb potential:

\[\phi(r) = \frac{Ze}{4\pi\varepsilon_0 r} e^{-r/\lambda_D}\]

Derivation

  1. Assume the external charge \(Ze\) is fixed.
  2. Electrons respond via Boltzmann distribution: \(n_e = n_0 \exp(e\phi/k_B T_e)\).
  3. Ions remain essentially immobile because \(m_i \gg m_e\).
  4. Linearize for \(|e\phi| \ll k_B T_e\): \(n_e \approx n_0(1 + e\phi/k_B T_e)\).
  5. Insert into Poisson: \(\nabla^2 \phi - \phi/\lambda_D^2 = -Ze\delta/\varepsilon_0\).
  6. Fourier transform or use spherical symmetry to obtain the Yukawa potential.

Worked Example

A dust grain of charge \(Q = 10^4 e\) sits in a plasma with \(n_e = 10^{16}\,\text{m}^{-3}\) and \(T_e = 1\) eV. Find the potential at \(r = 10 \lambda_D\).

  1. \(\lambda_D \approx 2.3\times 10^{-5}\,\text{m}\).
  2. At \(r = 10\lambda_D\), shielding factor is \(e^{-10} \approx 4.5\times 10^{-5}\).
  3. Hence the external charge is effectively invisible to distant plasma particles.

Common Mistakes

  • Debye ≪ L is not a micro-scale phenomenon. It is a global condition; the system must be many Debye lengths across.
  • Singularities survive at \(r=0\). Per real charge is singular inside a Debye sphere; shielding only works outside.
  • Debye length depends on \(T_e\). Hotter plasma elects a stronger shield.

Quiz Questions

  1. Conceptual: What is the physical reason that a test charge cannot polarize a plasma beyond \(\sim \lambda_D\)?
  2. Computational: Derive \(\lambda_D\) for argon plasma with \(n=10^{18}\,\text{m}^{-3}\) and \(T=0.5\) eV.
  3. MCQ: In Debye shielding, which species dominates the restoring force?
  4. A) Ions
  5. B) Electrons
  6. C) Both equally
  7. D) Neither; it is magnetic

Further Reading

  • Francis F. Chen, Introduction to Plasma Physics, Chapter 1.