# Fermi Level Calculator (Free-Electron Model)

URL: https://speedycalculator.com/calculators/physics/fermi-level/
Last updated: 2026-10-30

> The Fermi level calculator finds the Fermi wave vector, Fermi energy, Fermi temperature and Fermi velocity of a metal from its free-electron density, and the Fermi–Dirac occupation probability of a state at a given energy and temperature.

## Inputs

- Metal
- Free-electron density (m⁻³)
- Energy E (for the occupation) (eV, keV, MeV, J, zJ)
- Temperature T (K, °C, °F)

## Results

- Fermi wave vector k_F
- Fermi energy E_F
- Fermi temperature T_F
- Fermi velocity v_F
- Occupation probability f(E)

## Formula

```
k_F = (3π²n)^{1/3},  E_F = ħ²k_F²/(2mₑ),  T_F = E_F/k_B,  v_F = ħk_F/mₑ,  f(E) = 1/(e^{(E − E_F)/(k_BT)} + 1)
```

## Explanation

Electrons in a metal fill energy levels up to the Fermi energy – several electronvolts, equivalent to tens of thousands of kelvins. Only electrons near the Fermi level take part in conduction and heat capacity.

### How to use the Fermi Level Calculator

- **Metal**
- **Free-electron density (m⁻³)**
- **Energy E (for the occupation)**
- **Temperature T**

Results update as you type, every field has its own unit menu, and everything is calculated in your browser.

### How it works

Choose a metal (or enter the electron density), and optionally an energy and temperature for the Fermi–Dirac occupation.

### Inputs explained

FieldWhat to enterMetalChoose one: Lithium, Sodium, Potassium, Copper …Free-electron density (m⁻³)A numberEnergy E (for the occupation)Units: eV, keV, MeV, J, zJTemperature TUnits: K, °C, °F

### What each result means

- **Fermi wave vector k_F** = (3×pi^2×[n])^(1÷3)
- **Fermi energy E_F** = 1.054571817e-34^2×[Fermi wave vector k_F]^2÷(2×9.1093837015e-31)
- **Fermi temperature T_F** = [Fermi energy E_F]÷1.380649e-23
- **Fermi velocity v_F** = 1.054571817e-34×[Fermi wave vector k_F]÷9.1093837015e-31
- **Occupation probability f(E)** = 1÷(exp(([Energy E (for the occupation)] - [Fermi energy E_F])÷(1.380649e-23×[Temperature T])) + 1)

### Example values

How the main result changes when **Free-electron density** changes and everything else stays at the example values above:

Free-electron density (m⁻³)Fermi energy E_F (eV)4.25e287.03286.38e287.03288.50e287.03281.06e297.03281.28e297.03281.70e297.0328

## Worked example

Inputs — Metal: Copper; Energy E (for the occupation): 7 eV; Temperature T: 300 K.

Results — Fermi wave vector k_F = 13,586,308,449.70909 m⁻¹; **Fermi energy E_F = 7.03276 eV**; Fermi temperature T_F = 81,611.80589 K; Fermi velocity v_F = 1,572.85481 km/s; Occupation probability f(E) = 0.780274.

## Key terms

- **Fermi level**: Energy at which occupation probability is ½.

## FAQ

**What is the Fermi energy of copper?**
About 7.0 eV.

**Why is the Fermi temperature so high?**
E_F is far larger than thermal energy at room temperature.

**What is the occupation at E = E_F?**
Exactly ½ at any temperature.

**Is the free-electron model accurate?**
Good for simple metals like Na, K, Cu, Ag.


## Sources

- [OpenStax – University Physics Volume 3](https://openstax.org/details/books/university-physics-volume-3)
- [NIST – Fundamental physical constants](https://physics.nist.gov/cuu/Constants/)