Debye Length Calculator - Lambda D from T, n_e, or I

A debye length calculator returns the screening distance over which mobile charges cancel out an applied electric field. The plasma branch uses electron temperature and density in lambda_D = sqrt(epsilon_0 k_B T / (n_e e^2)), while the electrolyte branch uses temperature, relative permittivity, and ionic strength in lambda_D = sqrt(epsilon_r epsilon_0 k_B T / (2 N_A I e^2)), with I in mol/L converted to mol/m^3 by a factor of 1000. A tokamak edge plasma at 1e4 K and 1e19 m^-3 yields about 2.18 micrometres; a 1 M NaCl solution in water at 298 K yields about 0.30 nanometres.

• • Plasma physics and fusion work: Find the screening length of a tokamak edge, the solar wind, or the ionosphere to set the boundary between collective and single-particle behaviour.
• • Colloid and electrolyte chemistry: Size the diffuse layer around a charged surface or estimate the screening length in a salt solution for a Debye-Huckel calculation.
• • Semiconductor and electrolyte interfaces: Set the cutoff radius for charge screening at a semiconductor-electrolyte junction or inside a nanopore where lambda_D is comparable to the pore size.
• • Laboratory plasma setup: Estimate the sheath thickness of a probe, the dark space of a glow discharge, or the working distance in an inductively coupled plasma source.

For the companion quantum view of the same electron density, the de Broglie wavelength calculator converts mass and velocity into a wavelength so lambda_D and the de Broglie wavelength can be compared in the same plasma.

The calculator reads the temperature and the medium-specific density, converts each to SI base units, then applies the plasma or monovalent electrolyte Debye length formula with the exact 2019 SI values of epsilon_0, k_B, e, and N_A. The electrolyte formula carries an extra 1000 factor because ionic strength is entered in mol/L while the formula needs mol/m^3.

• T: Absolute temperature in Kelvin, converted from Kelvin, Celsius, or Fahrenheit.
• n_e: Electron number density in m^-3 (plasma mode), converted from m^-3, cm^-3, or mm^-3.
• epsilon_r: Relative permittivity of the solvent (electrolyte mode): 78.5 for water at 298 K, 72 for many KCl tables.
• I_M: Ionic strength in mol/L (electrolyte mode), converted to mol/m^3 by the 1000 factor.
• Constants: epsilon_0 = 8.8541878128e-12 F/m, k_B = 1.380649e-23 J/K, e = 1.602176634e-19 C, N_A = 6.02214076e23 mol^-1 (CODATA 2018 / 2019 SI).

According to Wikipedia Debye length, the Debye length of a plasma is lambda_D = sqrt(epsilon_0 k_B T / (n_e e^2)) and the Debye length of a monovalent electrolyte is lambda_D = sqrt(epsilon_r epsilon_0 k_B T / (2 N_A I e^2)).

Because the Debye length formula uses the same thermal energy k_B T that appears in the Maxwell-Boltzmann distribution, the Boltzmann factor calculator is a natural complement for showing how the population of the surrounding ions feeds the screening cloud.

Four ideas cover every number the debye length calculator returns.

When the screening length of a relativistic plasma becomes comparable to the electron Compton wavelength, the Compton wavelength calculator returns the Compton wavelength so the dimensionless ratio lambda_D / lambda_C is easy to read off.

Six short steps move from a temperature and density reading to a defensible Debye length in any medium.

1. 1 Pick the medium: Choose plasma for electron temperature and density, or electrolyte for solvent permittivity and ionic strength.
2. 2 Enter the temperature: Type the absolute temperature and choose Kelvin, Celsius, or Fahrenheit; the calculator converts to Kelvin.
3. 3 Enter the plasma density (plasma mode only): Type the electron number density and choose per cubic metre, centimetre, or millimetre; the calculator converts to m^-3.
4. 4 Enter the permittivity and ionic strength (electrolyte mode only): Water at 298 K is 78.5; seawater is about 0.7 M.
5. 5 Read the Debye length in five units: lambda_D is listed in metres, millimetres, micrometres, nanometres, and angstroms, so the same result works for a tokamak edge plasma and a 1 M salt solution.
6. 6 Check the supporting values: Plasma frequency or Debye wave number, plus the Debye sphere particle count, tell you whether the medium is collective or single-particle.

If the working temperature is quoted in Celsius, Fahrenheit, or electronvolts and the user wants to confirm the absolute value, the Kelvin converter turns the entered reading into Kelvin with one click.

A dedicated debye length calculator saves time and removes the unit-conversion errors that show up when the screening length is solved by hand.

• • Handles both plasma and electrolyte modes: The same form accepts electron temperature and density for a plasma, or temperature, permittivity, and ionic strength for a monovalent electrolyte.
• • Reports lambda_D in five useful units: Metres cover the solar wind, millimetres and micrometres cover laboratory and ionospheric plasmas, and nanometres and angstroms cover electrolyte and colloid chemistry.
• • Shows supporting collective quantities: Plasma frequency, Debye wave number, and Debye sphere particle count distinguish screened collective behaviour from single-particle physics.
• • Uses the exact 2019 SI constants: Vacuum permittivity, Boltzmann constant, elementary charge, and Avogadro number are hard-coded at their post-2019 values.
• • Accepts the units you already use: Temperature in K, C, or F; density in m^-3, cm^-3, or mm^-3; ionic strength in M, mM, or mol/m^3.

For the same plasma viewed through its thermal radiation rather than its electric field, the blackbody radiation calculator uses the same temperature T to return the peak emission wavelength and total radiated power, so the Debye screening length and the blackbody spectrum can be read from the same input plasma.

Four inputs drive the answer, and two limitations tell you when to expect a real medium to differ from the model.

• • The linearised Poisson-Boltzmann model breaks down when the electrostatic potential energy approaches k_B T, so the calculator is most accurate for weakly charged media.
• • The monovalent formula assumes a 1:1 salt; for CaCl2 or MgSO4 the ionic strength carries a z^2 factor, which the calculator does not currently support.

According to NIST CODATA 2018 Fundamental Constants, the vacuum permittivity is 8.8541878128e-12 F/m, the Boltzmann constant is exactly 1.380649e-23 J/K, the elementary charge is exactly 1.602176634e-19 C, and Avogadro's number is exactly 6.02214076e23 mol^-1, so the calculator hard-codes these constants.

According to Omni Calculator Debye Length, the solar wind plasma at T = 1e5 K and n_e = 1e6 m^-3 has a Debye length of 21.82 m, and 1 M KCl in water at 298 K and epsilon_r = 72 has a Debye length of 0.29 nm.

For an electrolyte flowing through a microchannel where lambda_D sets the effective wall-screening distance, the Reynolds number calculator confirms whether the flow stays laminar in the same system.