Plasma Frequency Calculator

Plasma frequency is the natural oscillation frequency of electrons in a plasma when they are displaced from their equilibrium position (relative to the ions) and pulled back by the resulting electric restoring force, then overshoot and oscillate, much like a mass on a spring. It is one of the most fundamental characteristic frequencies in plasma physics.

ω_pe = √(n e² / (ε₀ mₑ)), where n is the electron number density, e is the elementary charge, ε₀ is the permittivity of free space, and mₑ is the electron mass. The ordinary frequency is f_pe = ω_pe / (2π).

The restoring force on a displaced sheet of electrons comes purely from the electric field generated by the resulting charge separation, which depends only on how many electrons are displaced (the density), not on how fast they were already moving thermally. This makes plasma frequency a clean density-only diagnostic.

An electromagnetic wave with frequency below the local plasma frequency cannot propagate through the plasma, its electric field simply cannot push the electrons fast enough to keep up, so it is reflected instead. This is exactly how the ionosphere reflects shortwave (HF) radio signals back to Earth, enabling long-distance radio communication, while higher frequencies (like most satellite signals) pass straight through.

A dense tokamak fusion plasma has an electron plasma frequency around 90 GHz, in the microwave range, while the tenuous solar wind's plasma frequency is only tens of kilohertz. Earth's ionosphere sits in between, at a few megahertz, exactly the shortwave radio band it reflects.

Ions can oscillate too, with their own much lower 'ion plasma frequency' (since ions are far more massive and respond more sluggishly), but the electron plasma frequency is almost always the one meant by 'plasma frequency' without qualification, because electrons dominate the fastest, most important collective response of a plasma.

Plasma frequency scales as 1/√m, so lighter particles oscillate faster for the same density and charge. Since electrons are about 1836 times lighter than protons, the electron plasma frequency is about √1836 ≈ 43 times higher than the corresponding proton (hydrogen ion) plasma frequency at the same density.

No. Plasma frequency comes from the electrostatic restoring force of charge separation and exists even with no magnetic field present, while cyclotron frequency comes from the magnetic force on a moving charge and requires a magnetic field. The related <a href="/science/plasma-physics/cyclotron-frequency-calculator/">Cyclotron Frequency Calculator</a> computes the magnetic case.

In magnetic confinement fusion, microwave heating systems (like electron cyclotron resonance heating) must account for the plasma frequency because waves below the local plasma frequency cannot penetrate into the plasma core, they are reflected at the 'cutoff' layer where the wave frequency equals the local plasma frequency.

The electron plasma frequency itself depends only on electron density and the electron's own mass and charge, not on which ions are present. The ion species would matter for the separate ion plasma frequency, which uses the ion mass and charge in the same formula.