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Debye Shielding Lab

Learning goal

Turn the three defining plasma parameters — \(\lambda_D\), \(\omega_{pe}\), \(N_D\) — from formulas into a feel for scale. The same two sliders take you from a fluorescent tube to the core of an imploding fusion pellet, thirty orders of magnitude apart in density.

Things to try

  1. Start at the ionosphere preset. \(\lambda_D \approx 7\) mm and \(f_{pe} \approx 6\) MHz. That plasma frequency is not a coincidence in your life: it is why shortwave radio below ~6 MHz bounces off the ionosphere and travels around the world at night, and why anything above it — FM, TV, satellite uplinks — punches straight through. See EM waves in plasma.

  2. Compare the two fusion approaches. Tokamak core: \(n = 10^{20}\) m⁻³, \(\lambda_D \approx 74\) µm. ICF pellet: \(n \approx 10^{31.5}\) m⁻³, \(\lambda_D \approx 0.13\) Å — smaller than an atom. Both are "fusion plasma"; they have almost nothing else in common. Check the Lawson criterion to see how they buy the same triple product from opposite directions.

  3. Break the plasma. Hold the temperature fixed and slide the density up. Watch \(N_D\) fall through 1 and the verdict flip. Below \(N_D = 1\) there are not enough particles inside a Debye sphere to do the shielding — the Boltzmann-response derivation of \(\lambda_D\) silently assumed a smooth statistical cloud, and that assumption is now false. This is the boundary of the ideal plasma, and white-dwarf interiors sit on the wrong side of it.

  4. Watch the two length scales race. Compare \(\lambda_D\) with the mean interparticle spacing \(n^{-1/3}\). Their ratio is the plasma parameter: \(N_D \sim (\lambda_D n^{1/3})^3\). The ideal-plasma condition \(\lambda_D \gg n^{-1/3}\) says the shielding cloud must be much bigger than the spacing between the particles that form it.

  5. Notice what temperature does. Raising \(T\) raises \(\lambda_D\) — hotter particles are harder to hold in a shielding cloud, so the cloud has to be bigger. Raising \(n\) lowers it: more shielders, less distance needed. \(\lambda_D \propto \sqrt{T/n}\) is that sentence.

The number worth memorising

\[\lambda_D\,[\text{m}] = 7430\sqrt{\frac{T_e\,[\text{eV}]}{n_e\,[\text{m}^{-3}]}} \qquad \frac{\omega_{pe}}{2\pi}\,[\text{Hz}] \approx 8980\sqrt{n_e\,[\text{cm}^{-3}]}\]

Physicists quote plasma temperatures in electron-volts (1 eV ≈ 11 600 K) because these formulas are clean in those units and nothing else about a plasma cares about Celsius.

Debye shielding · Plasma frequency · Quasi-neutrality · Ideal plasma · Debye length (eq.)