Electron · R.A.V.E.N. Living Encyclopedia

The electron (e⁻) is a charged lepton of spin ½, electric charge −e, and rest mass m_e = 9.109 383 7139(28) × 10⁻³¹ kg (Mohr et al. 2025). It is an excitation of the Dirac field, and its particle and wave descriptions are the two limits of that single quantum object: in scattering and detection it is localised, while in diffraction and in bound states it is described by a wavefunction of de Broglie wavelength λ = h/p (Davisson & Germer 1927). In the hydrogen atom the electron has no trajectory; its state is a stationary wavefunction whose squared modulus gives the probability density, with the 1s state spherically symmetric and its radius of maximum radial probability equal to the Bohr radius a₀ = 52.917 721 0544(82) pm (Mohr et al. 2025). The electron is a source of the electromagnetic field, carrying a Coulomb field at rest and radiating when accelerated, and its coupling to that field is the fine-structure constant α = 1/137.035 999 177(21) (Mohr et al. 2025). It participates in the electromagnetic and weak interactions but not the strong interaction. Its magnetic moment differs from the Dirac value g = 2 by the anomaly a_e = 1.159 652 180 46(18) × 10⁻³, in agreement with quantum electrodynamics (Fan et al. 2023; Aoyama et al. 2019); no internal structure is resolved above 10⁻¹⁸ m (Bourilkov 2001), and the electron is stable, with a lifetime limit of 6.6 × 10²⁸ yr at 90% confidence (Agostini et al. 2015). In astrophysical plasmas it is the principal source of free–free, bound–free and synchrotron emission and of Compton scattering of radiation. It was identified by Thomson (1897) from the charge-to-mass ratio of cathode rays.

02 · Equations

Governing equations

The electron is described by the Dirac field coupled to the electromagnetic field within quantum electrodynamics (QED). Relations are written in SI units with electron mass m_e and elementary charge e; numerical values are from the CODATA 2022 adjustment (Mohr et al. 2025).

Master · QED Lagrangian

QED Lagrangian density

$$\mathcal L=\bar\psi\left(i\hbar c\,\gamma^\mu D_\mu-m_ec^2\right)\psi-\tfrac14F_{\mu\nu}F^{\mu\nu},\qquad D_\mu=\partial_\mu+\frac{ie}{\hbar}A_\mu,\qquad j^\mu=-e\,\bar\psi\gamma^\mu\psi$$

The free Dirac equation, the Coulomb field, the bound-state spectrum, the anomalous magnetic moment and the scattering cross-sections given below follow from this Lagrangian, either as exact limits or as perturbative expansions in α.

01 · Relativistic wave equation

Dirac equation

$$\begin{gathered}\left(i\hbar c\,\gamma^\mu\partial_\mu-m_ec^2\right)\psi=0,\\ E_\pm=\pm\sqrt{p^2c^2+m_e^2c^4}\end{gathered}$$

Spin ½, a magnetic moment of −2μ_B at tree level (g = 2), and negative-energy solutions subsequently identified with the positron follow from the requirement of Lorentz covariance (Dirac 1928).

02 · Classical field

Coulomb field and Lorentz force

$$\begin{gathered}\mathbf E(\mathbf r)=-\frac{e}{4\pi\varepsilon_0 r^2}\hat{\mathbf r},\\ \mathbf F=-e\left(\mathbf E+\mathbf v\times\mathbf B\right)\end{gathered}$$

The far-field electrostatic field of the electron and its equation of motion in external fields. The latter governs the cathode-ray deflection of Thomson (1897) and the confinement of single electrons in Penning traps (Hanneke et al. 2008).

03 · Bound state

Hydrogen atom

$$\begin{gathered}\left[-\frac{\hbar^2}{2m_e}\nabla^2-\frac{e^2}{4\pi\varepsilon_0 r}\right]\psi=E\psi,\\ E_n=-\frac{hcR_\infty}{n^2}=-\frac{13.6057\ \mathrm{eV}}{n^2}\end{gathered}$$

Non-relativistic Schrödinger spectrum; hcR_∞ = 13.605 693 122 990(15) eV (Mohr et al. 2025). Fine structure (order α²), the Lamb shift (order α³ ln α) and hyperfine structure enter as corrections and provide independent tests of bound-state QED.

04 · Magnetic moment

g-factor and the anomaly

$$\begin{gathered}\boldsymbol\mu=-g\,\frac{e}{2m_e}\mathbf S,\\ a_e=\frac{g-2}{2}=\frac{\alpha}{2\pi}-0.328\,478\ldots\left(\frac{\alpha}{\pi}\right)^{2}+\cdots\end{gathered}$$

The Dirac value g = 2 is modified by radiative corrections. The leading term α/2π is due to Schwinger (1948); the QED series is known through tenth order (Aoyama et al. 2019). The measured a_e (Fan et al. 2023) agrees with the Standard Model prediction at the 10⁻¹² level, the comparison being limited by independent determinations of α (Parker et al. 2018; Morel et al. 2020).

05 · Coupling strength

Fine-structure constant

$$\alpha=\frac{e^2}{4\pi\varepsilon_0\hbar c}=\frac{1}{137.035\,999\,177(21)}$$

The dimensionless coupling of the electron to the photon; value from Mohr et al. (2025). Atom-recoil determinations (Parker et al. 2018; Morel et al. 2020) compared with the value inferred from a_e (Fan et al. 2023) constitute the most stringent test of QED.

06 · Photon scattering

Compton shift

$$\begin{gathered}\lambda'-\lambda=\frac{h}{m_ec}\left(1-\cos\theta\right),\\ \lambda_C=\frac{h}{m_ec}=2.426\ \mathrm{pm}\end{gathered}$$

The wavelength shift depends only on the scattering angle and the electron mass, establishing the quantum nature of X-ray scattering from free electrons (Compton 1923); λ_C from Mohr et al. (2025).

07 · Wave–particle

de Broglie wavelength

$$\begin{gathered}\lambda=\frac{h}{p}=\frac{h}{\sqrt{2m_eT}}\;\;(T\ll m_ec^2),\\ \lambda(100\ \mathrm{eV})=0.123\ \mathrm{nm}\end{gathered}$$

At 100 eV the electron wavelength, computed from the constants of Mohr et al. (2025), is comparable to interatomic spacings; electron diffraction from a nickel crystal confirmed the relation (Davisson & Germer 1927).

08 · Charge probe

Mott cross-section

$$\frac{d\sigma}{d\Omega}=\left(\frac{Z\alpha\,\hbar c}{4E}\right)^{2}\sin^{-4}\!\frac{\theta}{2}\,\left(1-\beta^2\sin^2\tfrac{\theta}{2}\right)$$

Cross-section for elastic scattering of a relativistic electron from a point nucleus of charge Ze. The final factor is the spin correction to the Rutherford formula; departures from it measure nuclear charge form factors.

03 · Numbers

Measured properties

Recommended values from the CODATA 2022 adjustment (Mohr et al. 2025) and the exact 2019 SI definition of the elementary charge. Parenthesised digits give the standard uncertainty in the final digits shown.

−1.602 176 634 × 10⁻¹⁹ C

Charge · exact · SI 2019

0.510 998 950 69 MeV

Rest energy m_ec² · Mohr et al. 2025

g = −2.002 319 304 360 92

g-factor · 1.3 × 10⁻¹³ relative · Fan et al. 2023

r < 10⁻¹⁸ m

Size · upper limit · Bourilkov 2001

Quantity	Value	Status	Meaning & convention
Electric charge	−1.602 176 634 × 10⁻¹⁹ C	exact · SI 2019	The SI fixes e by definition (exact since 2019); the electron carries exactly −e (Mohr et al. 2025).
Mass	9.109 383 7139(28) × 10⁻³¹ kg	measured · 3.1 × 10⁻¹⁰	Equivalently 5.485 799 090 441(97) × 10⁻⁴ u (Mohr et al. 2025).
Rest energy	0.510 998 950 69(16) MeV	derived · m_ec²	Energy of one electron at rest; the energy of each photon from e⁺e⁻ annihilation at rest (Mohr et al. 2025).
Spin	½ ℏ	exact · representation	Intrinsic angular momentum; a fermion, obeying the Pauli exclusion principle (Navas et al. 2024).
Magnetic moment	−9.284 764 6917(29) × 10⁻²⁴ J T⁻¹	measured	Antiparallel to the spin because the charge is negative; \|μ_e\| = 1.001 159 652 μ_B (Mohr et al. 2025).
g-factor	−2.002 319 304 360 92(36)	measured · 1.8 × 10⁻¹³	Measured to a relative precision of 1.3 × 10⁻¹³ (Fan et al. 2023; Mohr et al. 2025).
Magnetic anomaly a_e	1.159 652 180 46(18) × 10⁻³	derived · (\|g\|−2)/2	The radiative correction; the leading term α/2π = 1.161 × 10⁻³ is due to Schwinger (1948); value from Mohr et al. (2025).
Charge-to-mass ratio	−1.758 820 008 38(55) × 10¹¹ C kg⁻¹	derived · −e/m_e	Value from Mohr et al. (2025); first determined by Thomson (1897) from cathode-ray deflection.
Compton wavelength	2.426 310 235 38(76) pm	derived · h/m_ec	Length scale at which pair creation becomes relevant; not a spatial extent (Mohr et al. 2025).
Classical radius	2.817 940 3205(13) fm	derived · e²/4πε₀m_ec²	Sets the Thomson cross-section σ_T = 8πr_e²/3; not a spatial extent (Mohr et al. 2025).
Bohr radius	52.917 721 0544(82) pm	derived · ℏ/m_ecα	Radius of maximum radial probability in hydrogen 1s; the mean radius is 1.5 a₀ (Mohr et al. 2025).
Rydberg energy hcR_∞	13.605 693 122 990(15) eV	measured · 1.1 × 10⁻¹²	Hydrogen ground-state binding energy in the infinite-nuclear-mass limit (Mohr et al. 2025).
Electric dipole moment	\|d_e\| < 4.1 × 10⁻³⁰ e·cm	limit · 90 % CL	Roussy et al. (2023), HfF⁺ ions. A non-zero value would violate time-reversal symmetry.
Lifetime	> 6.6 × 10²⁸ yr	limit · 90 % CL	Agostini et al. (2015). Decay is forbidden by charge conservation, the electron being the lightest charged particle.

Sources · NIST CODATA 2022 (physics.nist.gov/constants) · PDG 2024 electron listing S003 · Fan et al. PRL 130, 071801 (2023) · Roussy et al. Science 381, 46 (2023) · Agostini et al. PRL 115, 231802 (2015)

04 · Situations

Representative configurations

Six configurations computed from CODATA 2022 constants: the hydrogen ground state, cyclotron motion with spin precession, Compton scattering, e⁺e⁻ annihilation, cathode-ray deflection, and conduction in copper. Each scene states any scale factor applied to the rendering.

05 · Latest research

Recent literature

Preprints and papers retrieved at page load, ordered by submission date. arXiv: categories physics.atom-ph, hep-ex, hep-ph, quant-ph and astro-ph.HE, abstracts matching the electron magnetic moment, electric dipole moment, fine-structure constant, Penning-trap or positron-annihilation work. INSPIRE-HEP: the same terms in titles of high-energy-physics records. Dates are arXiv submission dates and INSPIRE record dates; no publisher issue dates are used.

Querying arXiv · INSPIRE-HEP …

Date	Source	Title and authors	Reference
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06 · References

References

Declared representations

Spin representationThe renderings draw a spin axis only; no rotating surface is implied. Spin ½ is a quantum number.

Bound statesThe hydrogen scene samples the exact 1s density; no trajectories are drawn, as none are defined in the theory.

Spatial extentThe rendered glow is a display convention. The Compton wavelength and classical radius are interaction lengths, not sizes.

Declared scalingWhere a physical ratio is below display resolution (the g−2 advance; the drift velocity), the applied scale factor is stated in the scene.

Sources

J. J. Thomson (1897). Cathode rays. Phil. Mag. 44, 293.

N. W. Ashcroft & N. D. Mermin (1976). Solid State Physics. Holt, Rinehart and Winston; Table 1.1 (n) and Table 2.1 (v_F) for copper.

T. Siegert et al. (2016). Gamma-ray spectroscopy of positron annihilation in the Milky Way. Astron. Astrophys. 586, A84.

S. Navas et al., Particle Data Group (2024). Review of particle physics. Phys. Rev. D 110, 030001.

R. A. Millikan (1913). On the elementary electrical charge and the Avogadro constant. Phys. Rev. 2, 109.

A. H. Compton (1923). A quantum theory of the scattering of X-rays by light elements. Phys. Rev. 21, 483.

C. Davisson & L. H. Germer (1927). Diffraction of electrons by a crystal of nickel. Phys. Rev. 30, 705.

P. A. M. Dirac (1928). The quantum theory of the electron. Proc. R. Soc. A 117, 610.

J. Schwinger (1948). On quantum-electrodynamics and the magnetic moment of the electron. Phys. Rev. 73, 416.

D. Bourilkov (2001). Hint for axial-vector contact interactions in the data on e⁺e⁻ → e⁺e⁻(γ) at centre-of-mass energies 192–208 GeV. Phys. Rev. D 64, 071701.

D. Hanneke, S. Fogwell & G. Gabrielse (2008). New measurement of the electron magnetic moment and the fine structure constant. Phys. Rev. Lett. 100, 120801.

R. H. Parker et al. (2018). Measurement of the fine-structure constant as a test of the Standard Model. Science 360, 191.

T. Aoyama, T. Kinoshita & M. Nio (2019). Theory of the anomalous magnetic moment of the electron. Atoms 7, 28.

X. Fan et al. (2023). Measurement of the electron magnetic moment. Phys. Rev. Lett. 130, 071801.

T. S. Roussy et al. (2023). An improved bound on the electron’s electric dipole moment. Science 381, 46.

L. Morel et al. (2020). Determination of the fine-structure constant with an accuracy of 81 parts per trillion. Nature 588, 61.

M. Agostini et al., Borexino (2015). Test of electric charge conservation with Borexino. Phys. Rev. Lett. 115, 231802.

P. J. Mohr et al. (2025). CODATA recommended values of the fundamental physical constants: 2022. Rev. Mod. Phys. 97, 025002 · physics.nist.gov/constants · PDG electron listing.