Debye length calculator
A plasma at 10,000 K with 10¹⁶ electrons per cubic metre shields a charge over less than a tenth of a millimetre. Enter the electron temperature in kelvin and the density per cubic metre to get the screening length in metres. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- Creativity
- 10,000
- Density
- 10,000,000,000,000,000
- Debye length
- 0.000069009 m
- Debye length m
- 0.000069009
- K b
- 0
- Elementary charge
- 0
Temperature is the electron kinetic temperature in kelvin and density is the electron number density in inverse cubic metres; the Debye screening length is returned in metres, the SI length used in plasma physics, ionospheric models, fusion notebooks, and laboratory discharges. The constants are the exact SI 2019 Boltzmann constant and elementary charge together with the CODATA 2018 vacuum permittivity, so two clients never disagree about which handbook digits were baked in. The engine uses only finite arithmetic with no network calls, no sampling, and no date-dependent shortcuts.
How temperature and density are read and how λ_D stays reproducible
Optional leading plus signs and scientific notation on numeric strings are allowed when they parse as finite decimals; embedded spaces, hex prefixes, and non-numeric tokens are rejected with a clear message so a bad cell does not become a wrong screening length. Temperature must likewise be strictly positive, because a non-positive kinetic temperature is not a thermal plasma.
Computation multiplies the precomputed factor sqrt of ε₀ k_B over e squared by the square root of T over n, using the exact SI 2019 Boltzmann constant, the exact elementary charge, and the CODATA 2018 vacuum permittivity so two clients never disagree about which handbook digits were baked in. The well-known classroom check that λ_D is about sixty-nine times the square root of T over n follows at once from that prefactor.
What you can do with it
Homework and plasma-physics table checks
Compute λ_D from textbook electron temperature and density for a glow discharge, ionosphere, or fusion plasma and verify the screening length before sheath or collective-effect problems.
Education tools and laboratory notebooks
Normalize user-entered temperature and electron density into Debye length for interactive lessons, Langmuir-probe demos, and lab notebooks.
FAQ
What does the Debye length calculator cost?
It is free in your browser on this page.
What happens if density is zero or negative?
Debye screening length is undefined for a non-positive electron number density.
What units should I use?
Enter electron temperature in kelvin and electron number density in inverse cubic metres (m^-3). The Debye length is returned in metres, with k_B, e, and ε₀ in SI units.
Is this the same as a Compton wavelength or a classical wave λ = v/f?
No. Compton wavelength is h/(m c) from rest mass, and a classical wave uses speed over frequency. This tool only returns the plasma Debye screening length from temperature and electron density.
What does the Debye length tell me about a plasma?
It is the distance over which the plasma shields an electric charge. Within one Debye length a test charge is only partly screened, and a few Debye lengths away its potential falls much faster than it would in empty space.
Related tools
For developers — API access
Everything on this page is available programmatically. This section is for teams who want to wire it into their own systems; everyone else can just use the tool above.
API endpoint
Prefer to automate it? One authenticated POST creates the task; the result comes back by webhook or a signed link. The same capability also runs here on the web, by email and from Telegram — and soon from our app too.
Call it from your stack
curl -X POST https://api.kit.forhosting.com/phys/debye-length \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"temperature":10000,"density":10000000000000000}'const res = await fetch("https://api.kit.forhosting.com/phys/debye-length", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"temperature": 10000,
"density": 10000000000000000
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/phys/debye-length",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"temperature": 10000,
"density": 10000000000000000
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/phys/debye-length", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"temperature":10000,"density":10000000000000000}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"temperature":10000,"density":10000000000000000}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/phys/debye-length", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)Example request
{
"temperature": 10000,
"density": 10000000000000000
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "phys.debye_length",
"status": "queued",
"_links": {
"result": "/tasks/tsk_…/result"
}
}The API is asynchronous: the call returns a task_id immediately and the result arrives by webhook. Polling is capped at 1 req/s per task.
Pricing
Published price — no tokens, no invented credits. A failed task is never charged.
Limits
max_temperature | 1e+40 |
max_density | 1e+40 |
Errors
| HTTP | Code | Meaning |
|---|---|---|
401 | unauthorized | Missing or invalid API key. |
402 | insufficient_balance | Your balance doesn't cover the task price. |
404 | unknown_type | That task type doesn't exist. |
429 | rate_limited | Too many requests. Use the webhook instead of polling. |
Technical notes
This Debye length calculator returns the electrostatic screening length of a plasma from electron temperature and electron number density using the closed-form identity λ_D equals the square root of ε₀ k_B T over n_e e squared. Density must be strictly positive: a zero or negative number density is rejected with invalid_input because the screening length is undefined when there are no thermal electrons to rearrange around a test charge.
What the Debye screening length means and what the response contains
The Debye length is the characteristic distance over which a plasma screens an electrostatic perturbation. Peter Debye and Erich Hückel introduced the scale for electrolytes; in a weakly coupled plasma the same Poisson–Boltzmann balance between thermal kinetic energy and Coulomb energy yields a closed form for the electron Debye length: λ_D equals the square root of vacuum permittivity times Boltzmann’s constant times electron temperature, divided by the product of electron number density and the elementary charge squared. Inside one Debye length a test charge is only partially screened; beyond a few Debye lengths the potential of a point charge in a thermal plasma falls much faster than the vacuum Coulomb law. This calculator returns the electron temperature you supplied, the electron number density you supplied, the Debye screening length in metres as both debye_length and debye_length_m, the Boltzmann constant k_b, the elementary charge, the vacuum permittivity epsilon_0 used in the product, a unit string m, the method tag debye_length, and a short formula reminder. There is no Monte Carlo sampling of particle orbits and no table lookup of named plasmas: you supply T in kelvin and n_e in inverse cubic metres and the engine either builds a finite positive λ_D inside the supported magnitude range or rejects the input with invalid_input. Rejected cases include missing temperature or density, non-finite values, magnitudes above the documented ceilings, non-positive temperature, and any density that is zero or negative, because Debye length is undefined for a non-positive number density.
How temperature and density are read and how λ_D stays reproducible
The preferred fields are temperature for the electron kinetic temperature in kelvin and density for the electron number density in inverse cubic metres, but aliases T, te, temperature_k, and electron_temperature are accepted for temperature, and n, ne, n_e, electron_density, and number_density are accepted for density, so spreadsheets, query strings, and typed forms can reuse the same contract. Density must be strictly positive: if density is zero, negative, or negative zero, the request fails with invalid_input because Debye screening length is undefined for a non-positive number density. Pair the free browser run on this page with the priced API when you need the identical mathematical result inside an automated pipeline without re-implementing the Debye formula yourself, and keep temperature in kelvin and density in inverse cubic metres so the published length fields stay in metres without a hidden unit conversion.
Where Debye length fits in study, research, and software
Teachers can pre-generate answer keys for modest temperature and density values without trusting opaque calculator apps that silently switch from metres to centimetres or drop scientific notation. Developers call the same contract from scripts to seed deterministic test vectors for plasma modules, to normalize user-entered temperature and density before sheath or Langmuir-probe demos, or to satisfy programming-challenge statements that demand λ_D rather than a hand-waved seventy-times-sqrt shortcut with undocumented units. Laboratory notebooks use the length to decide whether a probe tip, a grid spacing, or a simulation cell is smaller than the screening cloud. Space-physics pipelines compare ionospheric and magnetospheric Debye lengths from measured electron temperature and density. Students meet the Debye length when a plasma-physics course moves from single-particle orbits to collective shielding, when a contest problem asks for λ_D of a glow discharge, an ionospheric layer, or a fusion core from T and n_e, or when they compare electrostatic scales without a full kinetic-theory course. Product and education tools use the formula when a learner types T and n_e and the UI must show the screening length in metres next to the constants that produced it. On this page the work runs entirely in your browser; through the API each successful lookup costs $0.002, failed validation is not charged, and nothing is stored after the response returns. That split — free interactive exploration and a priced, identical API — matches the rest of the KIT tier-A catalog and lets you move from a classroom demo to an automated pipeline without changing the mathematical result. Pair it with cyclotron-frequency or Compton-wavelength tools when you need related plasma or particle scales, and with other physics endpoints when you are building a small lab of free, deterministic science helpers.
Integration use cases
Deterministic test fixtures
Generate reproducible Debye lengths for unit tests, property checks, and CI assertions on plasma, ionospheric, or particle-in-cell modules.
Technical questions
How is the Debye screening length defined here?
For electron temperature T and electron number density n_e, λ_D = sqrt(ε₀ k_B T / (n_e e²)) with k_B = 1.380649×10⁻²³ J/K (exact SI 2019), e = 1.602176634×10⁻¹⁹ C (exact SI 2019), and ε₀ = 8.8541878128×10⁻¹² F/m (CODATA 2018). The SI unit is metres.