Fermi level calculator
Copper, with about 8.47 × 10²⁸ conduction electrons per cubic metre, has a Fermi energy near 7 electronvolts. Enter an electron density per cubic metre to get the Fermi level in eV from the Sommerfeld electron-gas formula. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- Electron density
- 84,700,000,000,000,000,000,000,000,000
- Fermi level
- 7.0327613034 eV
- Fermi energy ev
- 7.0327613034
- Fermi wavevector
- 13,586,308,449.709108
The Fermi level of a metal is the energy of the highest occupied electron state at absolute zero in the free-electron picture: below that energy every state is filled and above it every state is empty. This calculator accepts the number density of those conduction electrons in SI units, electrons per cubic metre, and returns the Fermi energy in electronvolts from the Sommerfeld three-dimensional electron-gas formula. Students of solid-state physics, materials notebooks that compare simple metals, and exam generators all need the same closed-form value so homework, articles, and pipelines never disagree on Planck's constant, the electron mass, or the two-thirds power.
Input contract, SI units, and what happens when density is invalid
Zero density and negative density are likewise rejected: a Fermi sphere of zero or negative radius is not a physical input for this identity, and a non-positive n would make the two-thirds power ill-posed or complex. Keep units SI: if you tabulated carriers per cubic centimetre, multiply by one million before calling; the engine never converts cm^-3, angstroms to the minus three, or electrons per atom. Pair the electronvolt figure with an energy converter only if you later need joules.
Extremely large densities that overflow to Infinity are rejected rather than returned as a non-finite energy. The arithmetic is a handful of multiplies and one cube root; there is no table lookup and no network.
Where a Fermi level calculator earns its keep, and what it costs
Classrooms and tutoring platforms use a Fermi level calculator so learners can check copper, sodium, or a stated carrier density without dropping a digit in ħ or writing ten to the minus thirty-four by habit. Exam generators and LMS plugins need a deterministic oracle that rejects a non-positive density instead of silently returning zero or NaN that a buggy grader might treat as success. Condensed-matter notebooks that compare simple-metal Fermi energies call a tiny parse step instead of pulling a computer-algebra system for one closed-form power.
Failed validation does not consume balance. Pin the returned electronvolt value in tests, keep requests small, and you get reproducible Fermi levels for homework, explainers, and edge workers without standing up symbolic math infrastructure of your own.
What you can do with it
Classroom Fermi-energy checks
Let students submit a free-electron density in m^-3 and verify E_F in electronvolts against the Sommerfeld formula, including a clear error when density is zero or negative.
Simple-metal comparison notebooks
Compare copper, sodium, and aluminium Fermi levels from tabulated carrier densities without a computer-algebra library or a hand-typed ħ.
Exam generators and solid-state CMS widgets
Mark Fermi-energy items with a deterministic oracle, or embed a live Fermi level calculator on a free-electron-model article so readers see the electronvolt figure for a stated n.
FAQ
What happens if electron density is zero or negative?
A Fermi level requires a positive finite density; a non-positive n is rejected and is not billed.
Which physical constants does the calculator use?
Energy is converted to eV by dividing joules by e.
What units should I use for electron density?
SI electrons per cubic metre (m^-3). If your table is in cm^-3, multiply by 1e6 before calling. The engine does not convert cm^-3, Å^-3, or electrons per atom.
What Fermi energies do common metals have?
From tabulated electron densities, copper at about 8.47 × 10²⁸ electrons per cubic metre comes out near 7 electronvolts. Sodium, at about 2.65 × 10²⁸, and aluminium, at about 1.81 × 10²⁹, sit on the same curve.
How does the Fermi energy change with electron density?
It grows with density to the power two-thirds. Eight times more electrons per cubic metre therefore gives four times the Fermi energy, because the cube root of eight is two and two squared is four.
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/fermi-level \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"electron_density":8.47e+28}'const res = await fetch("https://api.kit.forhosting.com/phys/fermi-level", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"electron_density": 8.47e+28
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/phys/fermi-level",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"electron_density": 8.47e+28
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/phys/fermi-level", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"electron_density":8.47e+28}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"electron_density":8.47e+28}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/phys/fermi-level", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)Example request
{
"electron_density": 8.47e+28
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "phys.fermi_level",
"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.
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
A non-positive density is unphysical for this identity and is rejected as invalid_input; a value that is not a finite number is rejected the same way. Successful API calls cost $0.002; failed validation is not billed.
What the Fermi level is and how E_F follows from electron density
The Fermi energy of a three-dimensional free-electron gas is the energy of the highest occupied single-particle state at absolute zero. In the Sommerfeld model of a simple metal the conduction electrons are treated as non-interacting fermions in a box, so the occupied states fill a Fermi sphere in wavevector space whose radius k_F is fixed by the number of electrons: k_F equals the cube root of three pi squared times the electron number density n. Typical number densities sit near ten to the twenty-eight or twenty-nine electrons per cubic metre: copper is about 8.47 times ten to the twenty-eight, sodium about 2.65 times ten to the twenty-eight, and aluminium about 1.81 times ten to the twenty-nine, all on this same n to the two-thirds curve. The energy of that surface is E_F equals ħ squared times k_F squared over twice the electron mass, which is identical to E_F equals (ħ² / 2 m_e) times (3 π² n) to the two-thirds. This Fermi level calculator evaluates that identity and divides by the elementary charge so the answer is in electronvolts, the unit in which textbooks quote alkali and noble-metal Fermi energies. It does not apply a finite-temperature Fermi-Dirac chemical potential, a crystalline band-structure correction, an effective mass other than the free-electron mass, or the two-dimensional electron-gas formula used for quantum wells. Those are different models and different search intents.
Input contract, SI units, and what happens when density is invalid
POST a JSON body with one required field, electron_density, the free-electron number density in electrons per cubic metre. The value must be a finite positive number; it may arrive as a JSON number or as a numeric string such as 8.47e28. Aliases n, density, and n_e are accepted so lab notebooks and existing payloads map without renaming the key. Missing electron_density, an empty string, NaN, plus or minus Infinity, and any non-numeric token fail with invalid_input naming the field and are not billed. On success the body echoes electron_density, returns fermi_level and fermi_energy_ev as the same electronvolt value, reports fermi_wavevector in reciprocal metres, and includes formula, method equal to free_electron_gas, and unit equal to eV so clients can branch without sniffing shapes.
Where a Fermi level calculator earns its keep, and what it costs
Use the on-page tool for a single density at the desk; use the API when a grader, CMS plugin, or data pipeline needs the same contract under KIT balance. Do not send a semiconductor chemical-potential problem or a finite-temperature Fermi-Dirac integral and expect this endpoint to integrate it; this capability is the absolute-zero free-electron Fermi energy only.
Technical questions
How do I calculate the Fermi level from electron density?
Send a finite positive free-electron density in electrons per cubic metre as electron_density (aliases n, density, n_e). The API returns E_F = (ħ² / 2 m_e) (3π² n)^{2/3} in electronvolts.
What does the Fermi level calculator cost?
The browser tool is free. API requests use prepaid KIT balance at $0.002 per successful call; invalid input is not billed.