Debye Length Calculator (plasma screening length, Debye number and plasma frequency)

Understanding the Debye length calculator — concept and principle

In a plasma, free electrons crowd around any extra charge and screen its electric field. The Debye length is the distance over which that screening happens. This calculator finds the Debye length, the number of electrons in a Debye sphere and the electron plasma frequency.

Beyond about 23.5 µm the surrounding electrons screen out a charge’s field; with 54,000 electrons inside, it behaves as a plasma.

How it is calculated

Debye length λD = √(ε₀ kB Te ÷ (ne e²)) (electron plasma with stationary ions).
With Te in eV, kBTe = Te × e, so λD = √(ε₀ Te ÷ (ne e)).
Electrons in a Debye sphere ND = (4/3) π ne λD³.
Electron plasma frequency fpe = (1 ÷ 2π) √(ne e² ÷ (ε₀ me)).
1 eV of temperature = 11,604.5 K.

The principle

A charge placed in a plasma attracts opposite charges and repels like charges. Thermal motion fights that rearrangement, so the screening cloud has a finite size: hotter plasmas screen over longer distances, denser plasmas over shorter ones. Beyond a few Debye lengths the plasma looks neutral. A gas behaves as a plasma when it is much larger than λD and each Debye sphere contains many electrons (ND ≫ 1), so collective behaviour dominates.

Worked example

Default inputs: ne = 1 × 10¹⁸ per m³, Te = 10 eV.

λD = √(8.854 × 10⁻¹² × 10 ÷ (10¹⁸ × 1.602 × 10⁻¹⁹)) ≈ 2.351 × 10⁻⁵ m = 23.51 µm.
ND = (4/3) π × 10¹⁸ × (2.351 × 10⁻⁵)³ ≈ 54,420, so the plasma criterion is well satisfied.
fpe ≈ 8.979 GHz.
Te = 10 × 11,604.5 ≈ 116,000 K.

Each point is the calculator re-run with only “Electron density (per m³)” changed; the gold point is the default example. Change the inputs above and this chart follows your numbers.

Electron density (per m³)	Debye length (µm)
2.5e+17	47.02
5e+17	33.25
7.5e+17	27.14
1e+18	23.51
1.25e+18	21.03
1.5e+18	19.19
2e+18	16.62

How to read the result

Laboratory plasmas of this kind are typically millimetres to metres across, thousands of Debye lengths, so they behave collectively. Electromagnetic waves below the plasma frequency are reflected, which is why the ionosphere reflects certain radio frequencies. Probes and sheaths near walls have thicknesses of a few Debye lengths.

Common mistakes

Entering density per cm³ instead of per m³ (a factor of 10⁶).
Using temperature in kelvin while the unit is set to eV.
Forgetting that ion screening can also matter in some plasmas.
Treating any ionised gas as a plasma without checking ND.

Keep in mind: Simple formulas for an ideal, non-magnetised plasma. Research plasmas often need more detailed models.

Frequently asked questions

What is the Debye length?

The characteristic distance over which a plasma screens out an electric charge.

How does temperature affect it?

Higher temperature means longer Debye length (λD ∝ √T).

Why must ND be much greater than 1?

Screening is a collective effect that needs many particles within each Debye sphere.

Sources: Chen, F. F., Introduction to Plasma Physics and Controlled Fusion, 3rd ed. Springer · NRL Plasma Formulary, US Naval Research Laboratory

Checked 9 Nov 2026. Debye length λ_D = √(ε₀k_BT_e/(n_eq_e²)): PlasmaPy documentation, plasmapy.formulary.lengths.Debye_length (electron plasma with stationary ions). Electron plasma frequency — standard formula. k_B and e exact (2019 SI), CODATA 2022 ε₀ and mₑ.

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