0.0072973526
137.035999
0.00729735
0.000000004823
eV
122.252935
0.000000000053
m
13.605693123
eV
2.817940326200e-15
m
18,778.865045
0.0072973526
137.035999
0.00729735
0.000000004823
eV
122.252935
0.000000000053
m
13.605693123
eV
2.817940326200e-15
m
18,778.865045
The fine structure constant α = e²/(4πε₀ℏc) ≈ 1/137.036 = 7.2973525693 × 10⁻³ is the dimensionless coupling constant of electromagnetism — the fundamental measure of how strongly charged particles interact via the electromagnetic force. It is one of the most mysterious and celebrated numbers in physics, appearing in atomic spectra, quantum electrodynamics, and the stability of matter itself.
The name 'fine structure constant' comes from the splitting of atomic spectral lines that Arnold Sommerfeld explained in 1916. The fine structure of hydrogen's spectral lines (splitting of single lines into closely-spaced pairs) is proportional to α²/n³, where n is the principal quantum number. The relative size of the hydrogen fine structure is α² ≈ 5.3 × 10⁻⁵ — tiny compared to the gross structure but measurable with spectrographs.
The key relationships involving α reveal its ubiquity: the Bohr radius a₀ = ℏ/(m_e c α); the Rydberg energy R_∞ = ½ m_e c² α² = 13.606 eV (the ionization energy of hydrogen); the classical electron radius r_e = α²a₀; and the fine structure energy of hydrogen levels ΔE = R_∞ α²/n³. These relations show that α determines the scale of atomic structure, chemical bond energies, and all of chemistry.
One of the deepest puzzles in physics is why α ≈ 1/137. This dimensionless number is determined by three fundamental constants (e, ℏ, c) that each have seemingly arbitrary values in SI units, yet their combination yields this particular dimensionless number. Richard Feynman called α 'a magic number that comes to us with no understanding.' The precision to which α is known — α = 1/137.035999084(21) — makes it one of the best-measured constants in nature.
In quantum electrodynamics (QED), α is the expansion parameter for perturbation theory. Each additional photon exchange in a Feynman diagram contributes an additional factor of α ≈ 1/137, making higher-order corrections small. The anomalous magnetic moment of the electron ae = (g-2)/2 has been calculated to 10th order in α (thousands of Feynman diagrams), giving 12 significant figures of prediction — the most precise agreement between theory and experiment in the history of science.
Select the calculation: display α and derived quantities (Bohr radius, Rydberg energy), compute hydrogen fine structure energy splitting for level n, or see how α runs with energy scale (α is not truly constant but increases at higher energies). α = 7.2973525693 × 10⁻³ = 1/137.035999084 (CODATA 2018).
α ≈ 1/137 means electromagnetic interactions are relatively weak — alpha particles in QED perturbation theory converge quickly. If α were 1, all perturbative QED would fail. The value α ≈ 1/137 is believed to be necessary for stable atoms and chemistry to exist.
Inputs
Results
α = 1/137.036. Bohr radius a₀ = 0.529 Å. Rydberg = 13.606 eV (ionization energy of ground-state hydrogen). All atomic energy and length scales derive from α, m_e, ℏ, and c.
Inputs
Results
At the Z boson mass (91.2 GeV), α has 'run' from 1/137 at low energies to approximately 1/128. This energy dependence (running coupling) is a fundamental prediction of quantum field theory confirmed at LEP.
The value α ≈ 1/137 determines the strength of electromagnetic interactions. If α were much larger, atoms would be unstable (electrons would fall into nuclei). If much smaller, stars could not fuse hydrogen (Coulomb barriers would be too low to allow tunneling). Many anthropic arguments suggest that life-permitting universes require α in a narrow range around its observed value.
A dimensionless constant has the same value regardless of what units you use. This makes α a truly fundamental number in nature, not an artifact of human measurement choices. The values of e, ℏ, and c individually depend on your unit system (SI, Gaussian, Planck units), but their combination α = e²/(4πε₀ℏc) is always 1/137.036.
In quantum field theory, the effective coupling 'runs' with energy scale due to vacuum polarization. At low energies (eV scale), α ≈ 1/137. At the Z boson mass (91.2 GeV), α ≈ 1/128. This running is a measurable prediction of QED, confirmed by precision measurements at LEP. At the GUT scale (~10¹⁶ GeV), the three gauge couplings are predicted to unify.
The most precise measurement uses the anomalous magnetic moment of the electron: ae = (g-2)/2. QED predicts ae as a power series in α: ae = α/(2π) - 0.3285 × (α/π)² + ... The experimental value ae = 1.15965218073 × 10⁻³ (measured via single electron in a Penning trap) gives α⁻¹ = 137.035999206(11), with 0.08 ppb uncertainty — the most precise determination.
Astronomical evidence from quasar absorption spectra has been used to search for variation in α over cosmic time. Some studies claimed a 10⁻⁵ level variation, but more recent analyses using multiple systems show no variation at the 10⁻⁶ level. Laboratory clock comparisons (optical transitions sensitive to α) set a limit on dα/dt of less than 10⁻¹⁸ per year — consistent with no variation.
QED is the most precisely tested theory in physics because α ≈ 1/137 is small, making perturbation theory (expansion in powers of α) rapidly convergent. Corrections of order α, α², α³ etc. get smaller and smaller. The electron magnetic moment has been predicted to 10th order in α, involving thousands of Feynman diagrams, and agrees with experiment to 12 significant figures.
In hydrogen: E_n = -R_∞/n² = -13.6 eV/n² (Bohr levels). The fine structure correction: ΔE_fs = -R_∞ α² (relativistic + spin-orbit) / (n³(j+½)). The Lamb shift (QED correction): ΔE_Lamb ≈ +R_∞ α³ ln(1/α)/n³. Each successive correction is smaller by a factor of α² ≈ 5 × 10⁻⁵, testing QED to ever higher orders.
The Standard Model has three fundamental coupling constants: α ≈ 1/137 (electromagnetism), αW (weak force), and αs ≈ 0.12 (strong force at 1 GeV). Grand Unified Theories (GUTs) predict these three couplings converge to a single value at energies ~10¹⁶ GeV. The Minimal Supersymmetric Standard Model (MSSM) achieves this unification more precisely than the Standard Model alone.
Sommerfeld's 1916 relativistic quantum mechanics model of hydrogen predicted that the energy levels depend on both the principal quantum number n and the orbital angular momentum quantum number k, with corrections of order α². The formula E = -m_e c²[1 - 1/√(1 + (α/(n-k+√(k²-α²)))²)] explained the observed doublet splitting of spectral lines before quantum mechanics was fully developed.
If α = 1, the electromagnetic force would be as strong as the strong nuclear force. Electrons would have ~137 times more energy at each orbital, atoms would be ~137 times smaller than they are, and chemical bonds would be orders of magnitude stronger. More critically, perturbation theory in QED would fail completely, requiring entirely different mathematical techniques. It is widely believed that stable atoms and chemistry as we know it would be impossible.
Roboculator Team
The Roboculator Team explains calculations, planning tools, and practical formulas in clear language for real-life situations.
How helpful was this calculator?
Be the first to rate!
Energy Converter
Physical Constants & Unit Converters
Speed of Light Calculator
Physical Constants & Unit Converters
Planck Constant Calculator
Physical Constants & Unit Converters
Gravitational Constant Calculator
Physical Constants & Unit Converters
Coulomb Constant Calculator
Physical Constants & Unit Converters
Electron Mass Calculator
Physical Constants & Unit Converters
