Fine-Structure Constant Calculator
Compute the fine-structure constant α = e²/(4πε₀ħc) and related quantities (inverse α, Bohr radius, Compton wavelength).
Input Data
Results
At a glance:The fine-structure constant α ≈ 1/137.036 is a dimensionless fundamental constant measuring the strength of electromagnetic interaction: α = e²/(4πε₀ħc) = e²/(4πε₀ħc). It sets the scale of atomic structure — the splitting of spectral lines (fine structure), the binding of electrons, and the ratio of electron speed to light in hydrogen (v/c = α). Related lengths: Bohr radius a₀ = 4πε₀ħ²/(m_e e²) = ħ/(m_e c α) ≈ 5.29×10⁻¹¹ m, and Compton wavelength λ_C = h/(m_e c) ≈ 2.43×10⁻¹² m. Inverse α ≈ 137.036. It is dimensionless, so its value is the same in every unit system — a deep puzzle in physics.
Formula
α = e²/(4πε₀ħc)
a₀ = ħ/(m_e c α)
λ_C = h/(m_e c)
1/α ≈ 137.036
$$\alpha = \frac{e^2}{4\pi\varepsilon_0 \hbar c} \approx \frac{1}{137.036}, \quad a_0 = \frac{\hbar}{m_e c \alpha}, \quad \lambda_C = \frac{h}{m_e c} = 2\pi a_0 \alpha$$How to Use
- Enter the electron charge e and mass m_e (defaults are CODATA values).
- The calculator returns α, inverse α, Bohr radius and Compton wavelength.
Case Studies
Hydrogen scale
α ≈ 1/137, m_e = 9.11×10⁻³¹ kg.
a₀ ≈ 5.29×10⁻¹¹ m (atomic size).
v/c = α ≈ 0.0073 (non-relativistic, good for Bohr model).
FAQ
Why is α called dimensionless?
It is a pure number (~1/137) with no units, combination of e, ħ, c and ε₀. Its value is independent of the unit system — a genuine property of nature.
What does it mean physically?
It quantifies EM coupling strength: the electron's speed in hydrogen is α·c, and it controls atomic energy splittings and the size of atoms relative to the Compton wavelength.
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References
Content review: Calculatorism Science Team. Results are for reference only; please refer to the relevant authorities for the official figures.