electron number density — from formula anatomyElectron number density is the number of free electrons per unit volume in a material or plasma. It is a fundamental parameter in plasma physics, solid-state physics, and astrophysics, determining how a medium conducts electricity, reflects light, or supports electromagnetic waves. In metals, it is roughly the density of conduction electrons; in plasmas, it sets the plasma frequency above which the medium becomes transparent to radiation.
The concept was formalized in the early 20th century with the Drude model of metals (1900) and later in plasma physics, though the term 'electron density' became widespread with the development of ionospheric research in the 1920s.
The electron number density nₑ is typically measured in electrons per cubic meter (m⁻³) or per cubic centimeter (cm⁻³). It enters key formulas such as the plasma frequency ωₚ = √(nₑe²/ε₀mₑ), where e is the electron charge, ε₀ the vacuum permittivity, and mₑ the electron mass. In a metal, nₑ is on the order of 10²⁸ m⁻³, while in a laboratory plasma it might be 10¹⁶–10¹⁹ m⁻³, and in the interstellar medium as low as 10⁴–10⁶ m⁻³.
Electron number density is used to design semiconductor devices, to interpret plasma etching and fusion experiments, and to calibrate ionospheric radio propagation models.
In the Earth's ionosphere, the electron number density peaks around 300 km altitude at about 10¹² m⁻³, which is why long-distance radio communication relies on that layer reflecting signals.
| DM | variable | Dispersion measure — dispersion measure parsec per cubic centimetre |
| n_e | variable | Electron number density — electron number density per cubic metre |
| l | variable | Path length along line of sight — length metre |
| ∫ | operator | Integration along the line of sight — integral |
| n_e | variable | electron number density — electron number density per cubic metre |
| B_parallel | variable | magnetic field component along line of sight — magnetic flux density tesla |
| dl | variable | infinitesimal path element along line of sight metre |
| e | constant | elementary charge — elementary charge = 1.602176634e-19 coulomb |
| m_e | constant | electron rest mass — electron mass = 9.1093837015e-31 kilogram |
| c | constant | speed of light in vacuum — speed of light = 299792458 metre per second |
| ∫ | operator | summation over the line of sight — integral operator |
| n_e(z') | variable | electron number density at redshift z' — electron number density per cubic metre |
| z | variable | source redshift — redshift |
| z' | variable | integration variable redshift — redshift |
| c | constant | speed of light in vacuum — speed of light = 299792458 metre per second |
| ∫ | operator | summation over redshift from 0 to z — integral operator |