Debye Length Calculator
What is Debye length?
The Debye length is a characteristic screening distance in a plasma or electrolyte. It sets the scale over which electric fields are significantly reduced by the collective response of mobile charges (electrons and ions). Beyond a few Debye lengths from a charge perturbation, the net electric potential is strongly suppressed and the plasma appears nearly electrically neutral.
This calculator estimates the Debye length from temperature, electron density, and relative permittivity. It is useful in plasma physics, space physics, fusion research, semiconductor processing, and electrochemistry, wherever understanding electric field screening is important.
Debye length formula
For a simple, singly ionized, quasi-neutral plasma dominated by electrons, a common expression for the electron Debye length is
where:
- λ is the Debye length (m).
- ε0 is the vacuum permittivity (F/m).
- εr is the relative permittivity of the medium (dimensionless).
- kB is Boltzmann’s constant (J/K).
- T is the absolute temperature (K).
- n is the electron number density (m−3).
- e is the elementary charge (C).
In vacuum or a dilute plasma where the background medium is effectively free space, the relative permittivity is approximately εr ≈ 1. In an electrolyte such as water, εr can be much larger (around 80 at room temperature), shortening the Debye length for the same temperature and number density.
How to interpret the Debye length
The Debye length measures how far electrostatic disturbances extend before being screened:
- Distances much smaller than λ: Coulomb interactions between particles are relatively long-ranged, and charge separations can persist.
- Distances comparable to λ: Electric potentials are partially screened. Many plasma wave and instability behaviors depend on this regime.
- Distances much larger than λ: Net potentials from local charge imbalances are strongly suppressed. The plasma appears quasi-neutral on these scales.
A small Debye length indicates strong screening: the plasma or electrolyte quickly rearranges charges to cancel electric fields. This typically occurs at high densities and/or in media with high permittivity.
A large Debye length indicates weak screening: charges can influence each other over longer distances. This is typical for low-density plasmas, such as the solar wind or interstellar medium.
Worked example
Consider a laboratory plasma with the following properties:
- Temperature: T = 10,000 K
- Electron density: n = 1 × 1018 m−3
- Relative permittivity: εr = 1 (approximately vacuum)
Using the calculator, you enter 10,000 for temperature, 1e18 for electron density, and 1 for relative permittivity. The output Debye length will be on the order of tens of micrometers. This implies that any localized charge imbalance in the plasma is screened out over tens of micrometers, and on millimeter scales the plasma appears nearly neutral.
By contrast, if you lower the density to n = 1 × 1012 m−3 while keeping the same temperature, the calculator will return a Debye length many orders of magnitude larger, indicating much weaker screening.
Typical Debye lengths in different environments
The table below provides indicative values to help you compare the Debye length you compute with common physical situations. Values are approximate and intended for order-of-magnitude intuition.
| Environment | Temperature T (K) | Electron density n (m−3) | Relative permittivity εr | Typical Debye length λ (m) |
|---|---|---|---|---|
| Laboratory glow discharge plasma | 104 | 1016 – 1018 | ≈ 1 | 10−5 – 10−4 |
| Solar wind near Earth | 105 | 106 – 108 | ≈ 1 | 0.1 – 10 |
| Earth ionosphere (F-region) | 103 | 1011 – 1012 | ≈ 1 | 10−3 – 10−2 |
| Dense fusion plasma (tokamak core) | 108 | 1019 – 1020 | ≈ 1 | 10−6 – 10−5 |
| Aqueous electrolyte (room temperature) | 300 | effective ion densities 1025 – 1027 | ≈ 80 (water) | 10−10 – 10−8 |
Use these ranges only as rough guides. Real systems can vary substantially, and more detailed models may be needed for precision work.
Using the Debye Length Calculator
To use the calculator effectively:
- Enter the temperature T (K). This is the kinetic temperature of the electrons (or the species you are modeling). Typical values range from a few hundred kelvin in electrolytes to 103–108 K in plasmas.
- Enter the electron density n (m−3). This is the number of electrons per cubic meter. You can convert from cm−3 using 1 cm−3 = 106 m−3.
- Enter the relative permittivity εs. For a plasma in vacuum or low-density gas, use εr ≈ 1. For an electrolyte in water, values around 80 at room temperature are common; for other solvents, consult material data.
- Compute the Debye length. The tool returns λ in meters. You may wish to convert to millimeters, micrometers, or kilometers depending on context.
Limitations and assumptions
The simple Debye length model used here relies on several important assumptions:
- Quasi-neutral plasma or electrolyte. The analysis assumes that on scales larger than the Debye length, the net charge density is nearly zero and any perturbations are small.
- Linear response and small perturbations. The formula comes from linearizing the Poisson–Boltzmann or Vlasov–Poisson equations. Strong, highly nonlinear fields or double layers may not be well described.
- Maxwellian velocity distributions. The standard Debye length assumes particles follow approximately Maxwellian distributions. Strongly nonthermal plasmas may require modified screening lengths.
- Classical, non-relativistic regime. At extremely high temperatures (relativistic electrons) or extremely high densities (degenerate matter), quantum or relativistic corrections become important.
- Weak coupling. The Debye picture assumes the plasma is weakly coupled: the potential energy of Coulomb interactions is small compared with particle thermal energy. Strongly coupled plasmas (e.g., dusty plasmas or dense astrophysical plasmas) can deviate significantly.
- Isotropic, unmagnetized or weakly magnetized conditions. Strong magnetic fields can produce anisotropic screening and other collective effects that are not captured by a single scalar Debye length.
For regimes where these assumptions fail, more sophisticated kinetic or fluid models are needed. The calculator is best suited for quick estimates, order-of-magnitude checks, and educational purposes.
Related concepts
The Debye length is one of several important spatial scales in plasmas and electrolytes:
- Plasma frequency: the natural oscillation frequency of electrons about quasi-neutrality. Together with the Debye length, it characterizes many wave phenomena.
- Mean free path: the average distance a particle travels between collisions. It is a collisional property and can be much larger or smaller than the Debye length.
- Skin depth: the distance over which electromagnetic waves are attenuated in a conductor or plasma. It depends on frequency and conductivity rather than purely on thermodynamic variables.
Comparing the Debye length with these other scales helps determine whether a plasma behaves more like a collisionless medium, a conductive fluid, or a strongly damped medium for a particular phenomenon.