Debye Length Calculator

The Debye length is the characteristic distance over which a plasma electrostatically screens out an external electric field or a local charge imbalance. Inside one Debye length of a charge, the plasma's own electrons and ions have not yet rearranged to cancel it out; beyond it, the plasma looks essentially neutral.

λ_D = √(ε₀ Te / (n e)), where ε₀ is the permittivity of free space, Te is the electron temperature in electronvolts, n is the electron number density, and e is the elementary charge. It grows with hotter, more energetic electrons and shrinks with denser plasma.

Plasma physicists almost universally quote temperature in electronvolts because it directly represents the typical kinetic energy of a particle (1 eV corresponds to about 11,600 K), making it easy to compare against ionization energies and other atomic-scale energies without unit conversion.

It defines the boundary of 'quasi-neutrality': on scales larger than the Debye length, a plasma looks electrically neutral because charges have rearranged to screen out imbalances, while on smaller scales individual particle charges are still visible. It is also the length scale used to define whether a system counts as a true, collective plasma at all.

A hot, dense fusion (tokamak) plasma has a Debye length of roughly 70 micrometres, while the tenuous solar wind, despite being far less dense, has a Debye length of several metres because its density is so much lower. Earth's ionosphere sits in between, at a few millimetres.

Within a Debye length of an inserted charge (like a probe or a spacecraft), the plasma has not fully screened the charge's electric field, so the field falls off exponentially rather than as the vacuum 1/r² Coulomb law. This exponentially-screened potential is called the Debye-Hückel or Yukawa potential.

The basic formula here uses only electron temperature and density, since electrons respond fastest and dominate the screening. A more complete treatment adds an ion contribution to the screening (especially important when ion and electron temperatures differ substantially), but the electron-only formula is the standard first approximation used throughout plasma physics.

The Debye sphere is the sphere of radius λ_D around any point in the plasma; the number of particles inside it, N_D, must be large for the plasma to behave collectively rather than through individual particle collisions. The related <a href="/science/plasma-physics/debye-sphere-particle-count-calculator/">Debye Sphere Particle Count Calculator</a> computes N_D directly.

The plasma parameter is the number of particles inside a Debye sphere, N_D = (4/3)π n λ_D³. A large plasma parameter (N_D >> 1) is one of the defining criteria for calling an ionized gas a true plasma rather than just a collection of charged particles.

Yes, for fixed temperature, λ_D shrinks as density increases, since more nearby charges are available to screen out any imbalance over a shorter distance. This is why dense laboratory and fusion plasmas have Debye lengths thousands of times shorter than the tenuous plasma of interplanetary space.

No, real plasmas (like the solar wind or a tokamak) have density and temperature that vary in space and time, so the Debye length varies correspondingly. This calculator gives the Debye length for a single, locally uniform temperature and density, the standard building block used in more detailed plasma models.