Enter the screening-species temperature, number density, and relative permittivity to calculate the Debye length. For an electron plasma with stationary ions, use electron temperature and density with relative permittivity 1. Vacuum permittivity, Boltzmann’s constant, and elementary charge are fixed internally.

eV means kBT expressed in electronvolts.

Enter 1 for an electron plasma in vacuum.

Optional settings

Debye Length Formula

The calculator uses a classical single-mobile-species, singly charged screening approximation. Multiple mobile species require their separate density, charge, and temperature contributions:

λD = √((εᵣ ε₀ kB T) / (n e²))

Temperature and density inputs are converted before the formula is applied:

TK = TC + 273.15
TK = (TF - 32)(5) / (9) + 273.15
nm³ = ncm³ × 10⁶
  • λD = Debye length, in meters
  • εr = relative permittivity of the medium
  • ε0 = vacuum permittivity, 8.8541878188 × 10-12 F/m
  • kB = Boltzmann constant, 1.380649 × 10-23 J/K
  • T = absolute temperature, in kelvin
  • n = number density of the screening species, in particles per cubic meter
  • e = elementary charge, 1.602176634 × 10-19 C

The temperature unit selector converts Celsius, Fahrenheit, or thermal temperature in eV to kelvin. An eV input represents kBT in electronvolts. The density unit selector converts particles/cm³ to particles/m³ when needed. The relative permittivity field scales the result for the medium, so a higher relative permittivity gives a longer Debye length. The result is then shown in meters, millimeters, micrometers, and nanometers.

Common Relative Permittivity Values

Use the relative permittivity of the material or medium where the charged particles are located. These approximate values alone do not make the single-species model a complete electrolyte calculation.

Medium Typical relative permittivity, εr Notes
Vacuum 1 Use for ideal vacuum calculations.
Air About 1.0006 Often approximated as 1.
Water at room temperature About 78.5 Strongly temperature-dependent.
Ethanol About 24 Typical room-temperature value.

Particle Density Unit Reference

Input density Equivalent in particles/m³ Conversion
1 particle/cm³ 1 × 106 particles/m³ Multiply by 106
1 × 1012 particles/cm³ 1 × 1018 particles/m³ Common unit change for plasma or electrolyte calculations
1 × 1018 particles/m³ 1 × 1018 particles/m³ No conversion needed

Example Calculations

Example 1: Debye length in vacuum-like conditions

Suppose the temperature is 300 K, the particle number density is 1 × 1018 particles/m³, and the relative permittivity is 1.

λD = √(((1)(8.8541878188 × 10⁻¹²)(1.380649 × 10⁻²³)(300)) / ((1 × 10¹⁸)(1.602176634 × 10⁻¹⁹)²))

The Debye length is approximately:

  • 1.195 × 10-6 m
  • 1.195 µm
  • 1195 nm

Example 2: Debye length using particles/cm³

As a numerical single-species illustration, suppose the temperature is 25 °C, the screening-species number density is 1 × 1015 particles/cm³, and the relative permittivity is 78.5. This is not a full water-electrolyte model.

First convert the inputs:

  • 25 °C = 298.15 K
  • 1 × 1015 particles/cm³ = 1 × 1021 particles/m³

The Debye length is approximately:

  • 3.34 × 10-7 m
  • 0.334 µm
  • 334 nm

FAQ

What does Debye length mean?

Debye length is the distance over which electric fields are screened by mobile charged particles. A shorter Debye length means charges are screened over a smaller distance. A longer Debye length means the electric influence extends farther through the medium.

Why does increasing density reduce the Debye length?

Higher particle number density means more charged particles are available to rearrange and screen an electric field. In the formula, density is in the denominator, so increasing density decreases the Debye length.

Why must temperature be in kelvin?

The Debye length equation uses absolute temperature. Celsius and Fahrenheit are relative temperature scales, so they must be converted to kelvin before the calculation. A temperature at or below 0 K is not physically valid for this formula.