Plasma Frequency Squares from Density - Condensed Matter and Plasma Physics Step by Step

The plasma frequency omega_p = sqrt(n_e * e^2 / (epsilon_0 * m_e)) sets the natural oscillation rate of displaced electrons, scaling as the square root of electron density because the restoring field ties displacement to charge density. The arithmetic is exact only for stated electron density, mass, and charge in the ideal cold, uniform plasma model. Real plasmas add collisions, temperature spread, boundaries, nonuniform density, diagnostics calibration, and measurement/source uncertainty.

The collective oscillation scale is a square-frequency ledger. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

highlighted = computed this step

The frequency square is a normalized density budget

Normalized constant 12 per second squared times density ratio 3 over mass ratio 6 gives omega squared 6 per second squared.

\omega^{2}=\frac{12\cdot 3}{6}=6\ \text{s}^{-2}

Density raises the collective oscillation scale

The result is omega squared 6 per second squared for the whole displaced electron slab.

\omega^{2}=6\ \text{s}^{-2}