Plasma Frequency

Definition

The plasma frequency is the natural angular frequency at which the electrons of a plasma oscillate when their density is disturbed:

ω_p = √( n·e² / (ε₀·m_e) )

where n is the electron number density (m⁻³), e = 1.602×10⁻¹⁹ C the electron charge, ε₀ = 8.854×10⁻¹² F/m the vacuum permittivity, and m_e = 9.109×10⁻³¹ kg the electron mass. The ordinary (cyclic) frequency is f_p = ω_p / 2π.

Notice what is not in the formula: temperature, magnetic field, container shape. In the simplest "cold plasma" picture, the plasma frequency depends on one thing only — how densely the electrons are packed. Double the density and ω_p rises by √2. That single number then governs a wave's fate: whether it is reflected or transmitted.

How it works — a slab that rings like a spring

Picture a neutral slab of plasma: a uniform sea of light electrons sitting on a uniform lattice of heavy, nearly motionless ions. Now grab the electron sheet and slide it sideways by a small distance x. You have just exposed a thin layer of bare positive ions on one face and piled up a thin layer of bare electrons on the other. That separation is exactly a parallel-plate capacitor, and it builds an electric field across the slab.

The field's magnitude is E = n·e·x / ε₀. It points so as to pull the displaced electrons straight back toward neutrality. Newton's second law for one electron reads:

m_e·(d²x/dt²) = −e·E = −(n·e²/ε₀)·x

This is the equation of a simple harmonic oscillator, ẍ = −ω_p²·x, with

ω_p² = n·e² / (ε₀·m_e)

So the electron sheet overshoots equilibrium, the field reverses, and it rings back and forth at ω_p. The ions barely move — they are ~1836 times heavier per nucleon, so the restoring dance is almost entirely the electrons'. This is the "Langmuir oscillation," named for Irving Langmuir, who coined the word plasma in the 1920s.

The cutoff: why waves reflect

Now send an electromagnetic wave of frequency ω at the plasma. The wave's oscillating field shakes the electrons, and their response feeds back into the wave. Working through the equations of motion gives the dielectric function of a collisionless free-electron gas:

ε(ω) = 1 − ω_p² / ω²

The refractive index is n = √ε. Two regimes fall out immediately:

• ω > ω_p: ε is positive, n is real, the wave propagates. The plasma is transparent. (As ω → ∞, ε → 1 — the electrons can't keep up, and the medium looks like vacuum.)
• ω < ω_p: ε is negative, so n is purely imaginary. A wave with imaginary index does not travel; its field decays exponentially over a skin depth and carries no net power forward. With nowhere to go, essentially all of the incident energy is reflected.

The plasma frequency is therefore a sharp cutoff — a wall in frequency space. Below it, mirror. Above it, window. Right at ω = ω_p, ε crosses zero, which is the resonance condition for a bulk plasma oscillation.

Worked example — numbers you can check

Let's compute three plasma frequencies that matter, using the engineering shortcut that drops out of the formula:

f_p = (1/2π)·√(n·e²/(ε₀·m_e)) ≈ 8.98 × √n  [Hz, with n in m⁻³]

1. The ionosphere. Daytime peak electron density is roughly n = 10¹² m⁻³.

f_p ≈ 8.98 × √(10¹²) = 8.98 × 10⁶ Hz ≈ 9 MHz

So waves below about 9 MHz — AM broadcast, shortwave — bounce off the ionosphere and skip over the horizon. FM at 100 MHz is more than ten times above the cutoff, so it sails through to space. This is why a shortwave ham in Tokyo can be heard in Toronto, while your local FM station vanishes at the horizon.

2. A copper-like metal. Free-electron density n ≈ 8.5×10²⁸ m⁻³.

f_p ≈ 8.98 × √(8.5×10²⁸) ≈ 2.6×10¹⁵ Hz   (ħω_p ≈ 10.8 eV, λ ≈ 115 nm)

That is deep in the ultraviolet. Every visible photon (1.6–3.1 eV) is far below cutoff, so it reflects — that is the mirror shine. Hit the metal with 115 nm vacuum-UV and it starts to go transparent.

3. The collisionless skin depth in that metal. Deep below cutoff (ω ≪ ω_p) the evanescent decay length collapses to δ = c/ω_p:

δ = c/ω_p = (3×10⁸) / (2π × 2.6×10¹⁵) ≈ 1.8×10⁻⁸ m ≈ 18 nm

A visible wave penetrates a copper mirror only ~18 nm before being turned around. That is why even a gold leaf a few hundred atoms thick is an excellent mirror.

Variants and regimes

Regime / refinement	What changes	Result
Cold plasma (ideal)	No thermal pressure, no field	ω = ω_p exactly, independent of wavelength — oscillation in place
Warm plasma (Bohm–Gross)	Add electron thermal velocity v_th	ω² = ω_p² + 3·k²·v_th² — a true travelling Langmuir wave
EM wave in plasma	Transverse light, not longitudinal	ω² = ω_p² + c²·k² — propagation only above cutoff
Magnetized plasma	Add field B → cyclotron ω_c = eB/m_e	Upper-hybrid frequency ω_uh = √(ω_p² + ω_c²)
Ion plasma frequency	Use ion mass and charge	ω_pi = √(n·Z²e²/(ε₀·m_i)) — far lower, drives ion-acoustic waves
Surface plasmon	Oscillation at a metal/vacuum interface	ω_sp = ω_p/√2 — confined, tunable, basis of plasmonics
Drude model (with collisions)	Add damping rate γ	ε(ω) = 1 − ω_p²/(ω² + iγω) — real metals, finite reflectivity

Common pitfalls and misconceptions

• "It's a wave that propagates." The cold plasma oscillation does not propagate — it has the same frequency ω_p at every wavelength, so its group velocity is zero. You need thermal pressure (Bohm–Gross) to make it travel. Don't confuse the longitudinal oscillation with the transverse light wave that obeys ω² = ω_p² + c²k².
• "Reflection means the wave is absorbed." Below cutoff the field is evanescent — it stores energy in the surface layer but, in a collisionless plasma, dissipates none. The energy is returned: the plasma is a perfect mirror, not a sponge. Absorption only appears once you add collisions (the Drude γ term).
• "Plasma frequency depends on temperature." The ideal ω_p depends only on density. Temperature enters through dispersion (the v_th term) and through Debye shielding, but the bare resonance is set by n alone.
• "Heavier ions oscillate too." They do, but at the ion plasma frequency ω_pi, which is smaller by roughly √(m_e/m_i) — about 43× lower for hydrogen. When people say "the" plasma frequency, they almost always mean the electron one.
• "Metals are transparent to UV because the photons are energetic enough to ionize." No — it's purely the dielectric cutoff. Once ω exceeds ω_p, ε turns positive and the metal becomes a transmitting medium, no ionization required.
• "f_p and ω_p are interchangeable." They differ by 2π. Quoting "the plasma frequency is 9 MHz" means f_p; the angular ω_p is 2π times larger. Mixing them up throws every cutoff calculation off by 6.28×.

Where plasma frequency shows up

• Radio over the horizon. Shortwave and AM bounce off the ionosphere because they sit below its few-MHz cutoff; FM and satellite uplinks pass through. The ionogram a space-weather station records is literally a plot of reflection versus frequency.
• The shine of metals. The UV plasma cutoff makes silver, aluminium, and gold mirrors across the entire visible band — the working principle of every telescope coating and household mirror.
• GPS and satellite ranging. The ionosphere's frequency-dependent index delays signals; receivers use the ω_p-driven dispersion between two GPS frequencies to subtract the error.
• Fusion and lab plasmas. Microwave interferometry measures plasma density by reading off the phase shift, which is set by ω_p; reflectometry uses the cutoff layer as a movable mirror.
• Plasmonics and biosensing. Surface plasmons at ω_p/√2 confine light below the diffraction limit and shift with adsorbed molecules — the basis of label-free biosensors and gold-nanoparticle stained glass.
• Solar and astrophysical radio. Type III solar bursts drift in frequency as an electron beam climbs through the corona's falling density, sweeping ω_p downward — a direct readout of the Sun's density profile.
• Metamaterials and ENZ optics. "Epsilon-near-zero" devices engineer an effective plasma frequency so that ε ≈ 0 at a chosen optical wavelength, enabling exotic phase and tunneling effects.

Performance / derivation analysis

The whole edifice rests on a two-line derivation, which is why it is so robust. Start from the displaced-slab restoring force, m_e·ẍ = −(n·e²/ε₀)·x. The coefficient on the right is ω_p², full stop — no approximations beyond "ions don't move" and "density change is small." That is the cold, collisionless limit, and it is astonishingly accurate for the bulk resonance.

To get the optical cutoff, drive the same electron with a wave field E·e^(−iωt). The steady-state displacement is x = eE/(m_e·ω²), giving a polarization P = −n·e·x and hence the dielectric function:

ε(ω) = 1 + P/(ε₀·E) = 1 − ω_p²/ω²

Add the Drude collision rate γ and it becomes ε(ω) = 1 − ω_p²/(ω² + iγω). The imaginary part is what makes a real metal absorb a few percent rather than reflect 100%. For a good conductor γ is ~10¹³–10¹⁴ s⁻¹, far below the optical ω, so visible reflectivity stays above ~95% — exactly what a clean silver mirror delivers.

The dispersion relations follow the same logic. A transverse EM wave obeys ω² = ω_p² + c²k², so its phase velocity v_φ = ω/k exceeds c while its group velocity v_g = c²/v_φ stays below c — no causality violation. At cutoff (k → 0) the wave's group velocity goes to zero: it piles up at the reflection layer, which is why reflectometry can pin the cutoff position to within millimeters in a fusion device.

One sanity check worth keeping: the plasma frequency in MHz is just 9·√(n in 10¹² m⁻³). Memorize that and you can estimate any ionospheric cutoff, fusion diagnostic, or metal mirror in your head — the formula scales as the square root of density across more than sixteen orders of magnitude, from the solar wind (~10⁷ m⁻³, f_p ~ 30 kHz) to a metal (~10²⁹ m⁻³, f_p ~ PHz).