Debye Length Calculator
Put a test charge inside a plasma and the surrounding electrons and ions rearrange to hide it. The Debye length is the distance over which that hiding happens, and it decides everything from sheath thickness at a reactor wall to whether an ionised gas counts as a plasma at all.
This calculator takes electron temperature and electron density, in the units your source actually uses, and returns the Debye length together with the electron plasma frequency and the Debye number, the particle count that tells you whether the screening model is even allowed to apply.
Assumes a non-degenerate, quasi-neutral plasma with Maxwellian electrons. The formula breaks down for strongly coupled or degenerate regimes (dense cold matter, metals at low temperature).
Quick Answer
Key takeaways
- Debye length grows with the square root of temperature and shrinks with the square root of density: quadruple either one and the length only doubles or halves.
- Most wrong answers are unit answers. Density needs a 106 step between cm-3 and m-3, temperature a factor of 11,604.525 between eV and kelvin.
- The Debye number is the honesty check: screening is statistical, so ND has to be well above 1 before any of these numbers mean anything.
- The plasma frequency depends on density alone, which is why a density error shows up there undiluted, with no square root to soften it.
- Quick Answer
- How to use the Debye length calculator
- The screening formula and the 69 rule
- Worked example: 10,000 K at 10^18 per cubic metre
- Debye lengths in real plasmas, from the solar wind to a fusion core
- Electrolytes and semiconductors: where εr and Z earn their fields
- What competing calculators miss
- Frequently Asked Questions
- Add This Calculator to Your Website
- References
How to use the Debye length calculator
Start with the two numbers every plasma source reports: electron temperature and electron density. Type the temperature and pick K or eV to match your source, then type the density and pick m-3 or cm-3. Leave Z at 1 and εr at 1 unless you have a specific reason to change them: Z is for multiply charged ions, and εr is for screening inside a material rather than a free plasma.
Press Calculate and read the result in three layers. The headline is the Debye length, auto-scaled into metres, millimetres, micrometres or nanometres. Under it sit the plasma frequency in both rad/s and Hz, then the Debye number with a traffic-light verdict. Read the verdict before you quote the length: green means the model applies, amber means treat it as an estimate, red means the inputs describe a regime this formula was never built for. The formula line at the bottom shows the exact SI values used after conversion, so a unit slip is visible instead of silent. Outputs come out in scientific notation, and if that format slows you down, the convert to exponential form calculator works through the same notation step by step.
The screening formula and the 69 rule
The formula balances thermal motion against electrostatic pull. Hotter electrons wander further before the field of a test charge can organise them, so the screening cloud is bigger; more electrons per cubic metre means the cloud forms faster, so it is smaller.
Fold the constants together for the common case (Z = 1, εr = 1) and you get the rule plasma physicists carry in their heads: λD ≈ 69 × √(T[K] / n[m-3]) metres. The NRL Plasma Formulary prints the same physics in cgs dress, λD = 743 × √(T[eV] / n[cm-3]) cm. Both forms agree once the units are handled honestly, and the worked example below lands on the same number either way. Keep that 743 in mind: it is the coefficient behind the most common wrong answer on the web, as the competitor section shows.
Worked example: 10,000 K at 10^18 per cubic metre
The calculator is pre-filled with this case. Following the tool’s own arithmetic:
Convert. Inputs are already SI: T = 10,000 K, n = 1 × 1018 m-3, Z = 1, εr = 1. No conversion factor is applied, and the formula readout confirms it.
Screening length. λD = √(8.854×10-12 × 1.381×10-23 × 10,000 / (1018 × (1.602×10-19)2)) = 6.900898 × 10-6 m. The 69 rule agrees: 69 × √(10,000 / 1018) = 69 × 10-7 = 6.9 × 10-6 m.
Plasma frequency. ωpe = √(1018 × (1.602×10-19)2 / (8.854×10-12 × 9.109×10-31)) = 5.6415 × 1010 rad/s, so fpe = 8.9787 × 109 Hz, displayed as 8.9787 GHz.
Debye number. ND = 1018 × (4π/3) × (6.900898×10-6)3 = 1.3766 × 103. Roughly 1,377 particles per Debye sphere, comfortably above 1.
| Output | Tool result | Reading |
|---|---|---|
| Debye length λD | 6.900898 × 10-6 m | Fields are screened within about 7 micrometres |
| Electron plasma frequency ωpe | 5.6415 × 1010 rad/s | Electron oscillation rate at this density |
| Plasma frequency fpe | 8.9787 GHz | Same rate in cycles per second |
| Debye number ND | 1.3766 × 103 | Genuine plasma |
Debye lengths in real plasmas, from the solar wind to a fusion core
The same two inputs span twelve orders of magnitude across real systems, which is why no single intuition covers this quantity. Each row below was run through this calculator; enter the temperature and density yourself to reproduce it.
| System | Temperature | Density (m-3) | λD | ND |
|---|---|---|---|---|
| Solar wind near Earth | 100,000 K | 5 × 106 | 9.7593 m | 1.95 × 1010 |
| Gas discharge, 1 eV electrons | 1 eV | 1 × 1016 | 7.4339 × 10-5 m | 1.72 × 104 |
| Worked example above | 10,000 K | 1 × 1018 | 6.900898 × 10-6 m | 1.38 × 103 |
| Fusion core, 10 keV | 10 keV | 1 × 1020 | 7.4339 × 10-5 m | 1.72 × 108 |
Two patterns are worth noticing. First, the gas discharge and the fusion core land on the same length, 74 micrometres, by opposite routes: the discharge is cold and thin, the core is ten thousand times hotter and ten thousand times denser, and the square roots cancel. Second, the Debye number separates them completely, 104 against 108, which is why quoting λD without ND tells only half the story. In a semiconductor processing plasma the length also sets the sheath scale at the wafer: a capacitively coupled argon reactor at 4.2 eV and 3.8 × 1016 m-3 gives λD ≈ 7.82 × 10-5 m and ND ≈ 7.60 × 104 in this tool, the starting point for any sheath-width estimate.
Electrolytes and semiconductors: where εr and Z earn their fields
Debye’s original 1923 problem was not a plasma at all; it was screening in an electrolyte, and the same length controls the electrical double layer at an electrode and the screening of dopant charges in a semiconductor. That is why this tool carries two fields most plasma calculators drop. Raise εr to the value for your medium and the length stretches by √εr, because a polarisable medium weakens the field doing the organising. Set Z above 1 when the mobile ions carry multiple charges, as multivalent salts do, and the length contracts.
One honest boundary: in an electrolyte the density that belongs in the formula is the total mobile ion number density, and if your starting figure is a molar concentration you need the conversion through Avogadro’s number before it belongs here. Working the concentration side carefully with a concentration calculator first prevents the classic mistake of feeding a molarity straight into a formula that expects particles per cubic metre.
What competing calculators miss
Five Debye tools were checked against the worked example above while preparing this page. The formula itself is standard; the failures are in units, labelling and arithmetic.
Miss 1: a worked example out by a factor of 108. Calculator Ultra’s example states that 10,000 K and 1 × 1020 m-3 give 7.43 × 10-5 m. Run those inputs here and you get 6.9009 × 10-7 m. The 7.43 is the NRL cgs coefficient, 743, surfacing in an SI calculation where it does not belong, the fingerprint of an eV/kelvin and cm-3/m-3 mix-up.
Miss 2: a second example that matches nothing. Brave Calculator’s example puts 10,000 K and 1 × 1018 m-3 at about 2.35 × 10-5 m. The correct figure, verified above step by step, is 6.900898 × 10-6 m, roughly 3.4 times smaller. An example you cannot reproduce is worse than no example.
Miss 3: broken output and a slipped Debye number. The TrueGeometry widget for a comparable problem returns a symbolic fragment containing √(-1) instead of a number. A long engineering treatment of a 4.2 eV processing plasma gets λD right at 78.2 μm, then reports ND = 7,610; its own numbers give 7.60 × 104, ten times more. This page shows the full formula readout with the converted SI inputs, so every figure can be audited by hand.
Miss 4: no plasma check. Most tools return a length for any inputs at all, including densities where fewer than one particle sits in a Debye sphere and the concept has collapsed. Returning ND with a plain verdict is the difference between a number and an answer.
Frequently Asked Questions
λD = √(εrε0 kB T / (n e2 Z2)), with ε0 = 8.854 × 10-12 F/m, kB = 1.381 × 10-23 J/K and e = 1.602 × 10-19 C. The quick check is λD ≈ 69 × √(T[K] / n[m-3]) metres for Z = 1 and εr = 1.
The formula is SI throughout: temperature in kelvin, density in m-3, and the result in metres. This calculator also accepts temperature in eV and density in cm-3, the units plasma papers actually use, and converts both before computing.
Multiply by 106: 1 cm-3 = 106 m-3. The same density typed as 1012 cm-3 or 1018 m-3 gives the identical Debye length in this tool, which is the fastest way to check you have converted the right way.
The Debye length is a distance: how far an electric field reaches before the plasma screens it. The electron plasma frequency ωpe = √(n e2 / ε0 me) is a rate: how fast electrons oscillate back when displaced. One comes mostly from temperature and density together, the other from density alone.
ND = n(4π/3)λD3, the number of particles inside a sphere one Debye length in radius. Screening is a statistical effect, so it only works when many particles share the job. ND well above 1 means the model applies; ND near or below 1 means a strongly coupled regime where this formula should not be trusted.
Check the units first. Forgetting the 106 step between cm-3 and m-3 moves the answer by a factor of 1,000, and forgetting the 11,604.525 step between eV and kelvin moves it by a factor of about 108. After units, the physics is simple: hot, thin plasmas screen over long distances, cold, dense ones over short distances.
Yes. In this tool Z enters squared in the denominator, so the length scales as 1/Z: a plasma of doubly charged ions (Z = 2) at the same electron density has half the Debye length of a singly charged one. Leave Z at 1 for hydrogen and for most textbook problems.
When ND approaches 1 (strongly coupled plasmas), when the electrons are degenerate (metals and dense cold matter, where Fermi statistics replace the Maxwellian assumption), and when a strong magnetic field, not electrostatics, controls the particle motion. The verdict line under the result flags the first of these for you.
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References
- US Naval Research Laboratory, “NRL Plasma Formulary” (Huba, revised edition), the standard tabulated source for the Debye length, plasma frequency and typical plasma parameters used here. NRL Plasma Formulary (University of Maryland mirror). Accessed October 2026.
- National Institute of Standards and Technology, CODATA value for the Boltzmann constant, 1.380649 × 10-23 J/K. NIST: Boltzmann constant. Accessed October 2026.
- National Institute of Standards and Technology, CODATA value for the elementary charge, 1.602176634 × 10-19 C. NIST: elementary charge. Accessed October 2026.
- National Institute of Standards and Technology, CODATA value for the vacuum electric permittivity, 8.854 × 10-12 F/m. NIST: vacuum electric permittivity. Accessed October 2026.
- National Institute of Standards and Technology, CODATA value for the electron mass, 9.109 × 10-31 kg, used in the plasma frequency. NIST: electron mass. Accessed October 2026.
- University of Rochester plasma astrophysics lecture notes, “Lecture 20: Plasma Effects,” deriving Debye shielding and the conditions (system size far above λD, ND far above 1) for the plasma picture to hold. University of Rochester: Plasma Effects lecture notes. Accessed October 2026.
- Lawrence Berkeley National Laboratory plasma theory chapter on Debye shielding, including the 69 × √(T/n) practical form and sheath formation at plasma boundaries. LBNL: Debye shielding chapter. Accessed October 2026.