Electron Debye Length Calculator

Formula: Model and equations

For a weakly coupled, quasi-neutral plasma with an approximately Maxwellian electron population, the electron Debye length is

where $T_e$ is the electron temperature in kelvins, $n_e$ is the electron number density in m^-3, and ${\epsilon }_r$ is the relative permittivity of the background medium. At fixed temperature and density, a larger permittivity increases the Debye length because the medium weakens the electrostatic restoring field.

If you enter temperature in electronvolts, the calculator first converts it using

A second reported diagnostic is the number of electrons in a Debye sphere:

When $N_D$ ≫ 1, many particles participate in the shielding cloud and the classical collective picture is usually self-consistent. When $N_D$ ≲ 1, the simple Debye model should be treated with strong caution.

How to use it

1. Choose the temperature unit. Plasma work is often quoted in eV, while some laboratory and atmospheric settings use kelvins directly.
2. Choose the density unit. Both m^-3 and cm^-3 are common in plasma physics, so the calculator accepts either and converts internally to SI.
3. Enter relative permittivity. For most low-density plasmas, ${\epsilon }_r\approx 1$. Values above 1 are relevant only when the plasma is embedded in a dielectric medium.
4. Inspect the Debye-sphere diagnostic. The screening length alone is not enough. A tiny or huge value can both be mathematically correct while still falling outside the regime where the classical approximation is trustworthy.

Worked examples

Two useful checks are built into the default workflow:

• Standard plasma-formulary benchmark: $T_e=1\text{eV}$, $n_e={10}^{10}\text{cm}{}^{-3}$, ${\epsilon }_r=1$. The expected Debye length is about 74.3 µm.
• Warm laboratory plasma: $T_e=10000\text{K}$, $n_e={10}^{18}\text{m}{}^{-3}$, ${\epsilon }_r=1$. The calculator returns about 6.90 µm and $N_D$ ≈ 1.38 × 10³, comfortably inside the collective regime.

Interpreting the output

• Small ${\lambda }_D$ means rapid screening. This usually comes from high density, low temperature, or both.
• Large ${\lambda }_D$ means weaker screening and longer-ranged electrostatic influence. This is common in tenuous plasmas such as the solar wind.
• Large $N_D$ supports the classical plasma picture. The Debye sphere contains many particles and collective behavior is plausible.
• Small $N_D$ is a warning flag. The formula may still produce a number, but the assumptions behind Debye shielding are no longer secure.

Limitations and scope

• Electron Debye length only. This calculator does not implement the full multi-species screening sum needed for arbitrary ion mixtures.
• Weak-coupling assumption. Strongly coupled plasmas, dense matter, and some dusty-plasma regimes can violate the classical screening picture.
• Approximately Maxwellian electrons. Strongly nonthermal or beam-dominated distributions can require modified screening lengths.
• Small perturbations and near quasi-neutrality. Double layers, sheaths, and strongly nonlinear electrostatic structures are outside scope.
• No magnetic anisotropy correction. A single scalar Debye length does not capture all magnetized kinetic effects.