Plasma Frequency Calculator

Calculator Inputs

Use the mode switch to solve for frequency or density.

Mode

Particle

Relative permittivity (εᵣ)

Use 1 for vacuum or air.

Number density (n)

Electron densities often range 10¹⁴–10²⁰ m⁻³.

Target frequency (f)

Frequency is converted internally to hertz.

Ion parameters

Z amu

Example: He²⁺ → Z=2, amu≈4.

Custom particle

|q| = × e

m kg

Charge uses magnitude; sign does not affect ωₚ.

Significant figures

Controls rounding in the results block.

Formula Used

The plasma angular frequency for a single species is: ω_p = √( n q² / (ε₀ εᵣ m) ).

n is number density (m⁻³).
q is charge magnitude (C).
m is particle mass (kg).
ε₀ is vacuum permittivity, and εᵣ is relative permittivity.

Convert to ordinary frequency using f_p = ω_p / (2π). When solving for density, rearrange to n = (ε₀ εᵣ m / q²) (2πf)².

How to Use This Calculator

Select a mode: compute frequency from density, or density from frequency.
Choose the particle model: electron, proton, or a custom option.
Enter εᵣ. Use 1 for vacuum or air.
Provide density or frequency with your preferred units.
Click Calculate. Results appear above this form.
Use the CSV and PDF buttons to save your output.

Example Data Table

Examples below assume electrons and εᵣ = 1.

n (cm⁻³)	n (m⁻³)	f_p (Hz)	f_p (GHz)
1.00000 × 10^6	1.00000 × 10^12	5.19767 × 10^-9	0.00000000000000000519767
1.00000 × 10^8	1.00000 × 10^14	5.19767 × 10^-8	0.0000000000000000519767
1.00000 × 10^10	1.00000 × 10^16	5.19767 × 10^-7	0.000000000000000519767
1.00000 × 10^12	1.00000 × 10^18	5.19767 × 10^-6	0.00000000000000519767

Higher density increases plasma frequency by the square root.

Plasma Frequency Guide

1) What Plasma Frequency Means

Plasma frequency is the natural oscillation rate of free charge carriers after a small displacement. In an electron plasma, electrons slosh against an almost stationary ion background. The characteristic angular frequency is ω_p, and the ordinary frequency is f_p = ω_p/(2π).

2) Why It Matters in Real Systems

If a wave’s frequency is below f_p, the plasma tends to reflect it; above f_p, waves can propagate with modified dispersion. This helps explain radio reflection in the ionosphere, shielding in dense discharges, and cutoff behavior in plasma‑filled devices. It is a cornerstone parameter in plasma diagnostics.

3) Electron vs Ion Plasma Frequency

Because ω_p ∝ 1/√m, electrons dominate the high‑frequency response. Ion plasma frequencies are far lower due to larger ion mass, but you can explore ion cases by changing the particle mass input and comparing how f_p scales.

4) Typical Density Ranges

Electron density varies widely. The ionosphere can be about 10¹⁰–10¹² m⁻³, giving plasma frequencies from tens of kilohertz to a few megahertz. Laboratory plasmas may reach 10¹⁴–10¹⁶ m⁻³, pushing f_p into the megahertz to hundreds‑of‑megahertz range. Dense fusion plasmas can exceed 10¹⁹ m⁻³, where f_p can reach tens of gigahertz.

5) Relative Permittivity and Environment

The model uses ε = ε₀ε_r. Many low‑pressure plasmas are close to ε_r ≈ 1, but an increased ε_r lowers ω_p by √ε_r. That shift changes the effective cutoff frequency for a given density.

6) Units, Conversions, and Quick Checks

Density is often reported in cm⁻³. Use 1 cm⁻³ = 10⁶ m⁻³. Doubling density increases f_p by √2, not by 2. The calculator also reports ω_p in rad/s, which is useful for dispersion relations and time‑domain simulations.

7) Practical Examples

For HF radio (3–30 MHz), a layer with f_p near 10 MHz can reflect signals at or below that frequency. In microwave probes and reflectometry, selecting a frequency above expected f_p avoids cutoff, improves penetration, and reduces measurement ambiguity.

8) Interpreting Results and Limitations

The basic plasma frequency assumes a uniform, collisionless, unmagnetized plasma with small perturbations. Collisions, gradients, and magnetic fields add damping and extra resonances (for example, cyclotron effects), so treat f_p as a baseline parameter for cutoff and response scale.

FAQs

1) What is the difference between ωp and fp?

ωp is the angular plasma frequency in rad/s. fp is the ordinary frequency in hertz. They are related by fp = ωp/(2π). Use ωp for differential equations and fp for cutoff comparisons in Hz.

2) What density units should I use?

You may enter density in m⁻³ or cm⁻³. If you have cm⁻³ values, multiply by 10⁶ to convert to m⁻³. The calculator also provides both representations in the results for quick verification.

3) Why does higher density increase plasma frequency?

A larger carrier density strengthens the restoring electric field when charges are displaced. Stronger restoring force makes the oscillation faster, so ωp and fp scale with √n rather than linearly with n.

4) How does particle mass affect the result?

Plasma frequency is inversely proportional to √m. Electrons give much higher frequencies than ions. If you switch to a heavier ion mass, the plasma frequency drops sharply, often by orders of magnitude.

5) What does εr do in the formula?

εr increases the permittivity of the medium. A larger εr lowers ωp by a factor of √εr, which reduces the cutoff frequency for the same density and mass.

6) Is plasma frequency the same as cyclotron frequency?

No. Plasma frequency comes from electrostatic restoring forces in a neutralizing background. Cyclotron frequency is the rotation rate of charges in a magnetic field. In magnetized plasmas, both can appear together in wave behavior.

7) When is the cutoff approximation not enough?

If collisions are strong, temperature is high, density varies sharply, or magnetic fields are significant, simple cutoff reasoning can fail. Use fp as a starting point, then apply the relevant dispersion relation for your geometry and conditions.