The electron plasma oscillation frequency

Imagine a broad uniform electron slab displaced a small distance $\xi$ toward positive x relative to a stationary positive background. The positive charge uncovered at the left and excess negative charge at the right produce an electric field toward positive x inside the overlapping region. Its magnitude is $n_0e\xi/\epsilon_0$. An electron has charge $-e$, so its force points toward negative x: the displacement produces a restoring force rather than an instability.

Newton's law then has exactly the harmonic-oscillator form. Density is the equilibrium electron number per cubic metre, and the ions provide a fixed neutralizing background during the fast motion. The slab construction ignores edge fringing and large displacements. An equivalent local derivation combines linear electron continuity, momentum, and Poisson equations. It gives the same cold longitudinal frequency in a uniform unmagnetized plasma without requiring the whole apparatus to translate rigidly.

For initial displacement $\xi_0$ and zero initial velocity, displacement is a cosine. The electric field stores energy when charge separation is largest; electron bulk kinetic energy is largest as the slab passes through neutrality. In this ideal model the total quadratic energy is conserved. Zero field at an instant therefore does not mean that the disturbance has disappeared: its energy can temporarily reside entirely in coherent particle motion.

The cold electrostatic dispersion relation has no wave-number dependence: $\omega=\omega_{pe}$. Consequently its group velocity vanishes in that idealization. A spatially patterned oscillation can exist without this model carrying a packet at a finite group speed. Finite temperature changes the dispersion; magnetic fields introduce additional branches. One must not use the cold electron formula as the frequency of every disturbance in a plasma, especially slow ion-supported waves.

For a singly charged electron–ion plasma, the ion plasma frequency is smaller by $\sqrt{m_e/m_i}$. On a few electron periods, ion inertia therefore makes the fixed-background approximation reasonable. It is not a statement that ions never move. On longer times, their density response can change the mode structure and screening. If both species are mobile in a cold relative-displacement calculation, the squared relative-mode frequency contains both species' plasma-frequency squares.

Weak electron collisions require $\nu_e\ll\omega_{pe}$ for many visible oscillations; neutral collisions can spoil this condition in a partially ionized gas. A warm Maxwellian high-phase-speed Langmuir wave has the approximate correction $\omega^2\simeq\omega_{pe}^2+3k^2k_BT_e/m_e$ when $k\lambda_{De}\ll1$. Collisionless damping depends on the velocity distribution and is not represented by the cold oscillator. A damping coefficient inserted for illustration must be distinguished from a kinetic derivation of that damping.