Fine-Structure Constant

The fine-structure constant, denoted by $\alpha$, is a dimensionless physical constant that characterizes the strength of electromagnetic interactions. Its familiar low-energy value is approximately $1/137.036$. Originally introduced to describe small splittings in atomic spectral lines, it is now understood as the coupling parameter of quantum electrodynamics (QED), the quantum theory of interactions between electrically charged particles and photons. The conventionally tabulated constant refers to the limit of zero momentum transfer; effective electromagnetic couplings at higher energies differ from this value. (pml.nist.gov)

Definition and numerical value

In the International System of Units (SI),

where $e$ is the elementary charge, $\varepsilon_0$ is the vacuum permittivity, $\hbar=h/(2\pi)$ is the reduced Planck constant, and $c$ is the speed of light in vacuum. The dimensions cancel, leaving a pure number. The 2022 CODATA recommended values are

and

Parentheses indicate the standard uncertainty in the final quoted digits. The relative standard uncertainty is approximately $1.6\times10^{-10}$, or 0.16 parts per billion. The frequently quoted expression “$1/137$” is therefore an approximation, not an exact mathematical identity. (physics.nist.gov)

Because $\alpha$ is dimensionless, its value does not change when metres are replaced by centimetres or when a different consistent system of physical units is adopted. Its expression in terms of dimensional constants does change. In rationalized natural units, with $\hbar=c=1$, the electromagnetic coupling is commonly written

Here $e$ denotes the charge parameter in that convention, rather than its numerical value in coulombs. Unit conventions can simplify the formula but cannot turn the physical value of $\alpha$ into unity. (physics.nist.gov)

Physical meaning

The constant compares electromagnetic strength with the quantum-relativistic scale set by $\hbar c$. For two particles carrying charges of magnitude $e$, the magnitude of their Coulomb potential energy at separation $r$ is

Thus, $\alpha$ specifies the electromagnetic interaction energy relative to a characteristic quantum energy at the same length scale. This relation connects Coulomb’s law with the description of electromagnetism in quantum field theory. (damtp.cam.ac.uk)

Its small value makes perturbation theory particularly useful in QED: many observables can be calculated as expansions in powers of $\alpha$, often accompanied by factors of $\pi$. Successive orders describe increasingly detailed quantum corrections. Nevertheless, $\alpha$ is not a universal probability of photon emission or absorption; probabilities also depend on the process, particle energies, geometry, and available final states. Nor does small $\alpha$ make every electromagnetic problem simple: bound states and other circumstances can require reorganized or nonperturbative calculations. (damtp.cam.ac.uk)

Atomic structure and spectral fine structure

The name refers to fine structure, the small splitting of atomic energy levels caused by relativistic effects and electron spin. In hydrogen-like systems, the relevant expansion parameter is $Z\alpha$, where $Z$ is the atomic number. To leading nonrelativistic order, a one-electron ion with an infinitely massive point nucleus has energies

where $m_e$ is the electron mass and $n$ is the principal quantum number. Leading fine-structure corrections are of order $m_ec^2(Z\alpha)^4$, making them smaller than the principal binding-energy scale by a factor of order $(Z\alpha)^2$. (damtp.cam.ac.uk)

In the nonrelativistic expansion of the Dirac equation, these corrections include the relativistic kinetic-energy correction, spin–orbit coupling, and the Darwin term. Their combined effect depends on the total electronic angular momentum. Fine structure must be distinguished from hyperfine structure, which involves nuclear properties, and from radiative energy shifts such as the Lamb shift. (faculty.washington.edu)

The same constant sets basic atomic length and energy scales. For example, the Bohr radius is

and the Hartree energy is $E_{\mathrm h}=\alpha^2m_ec^2$. Consequently, $\alpha$ enters the theoretical interpretation of atomic spectroscopy as well as the calculation of binding energies. These formulas describe characteristic scales; precision predictions also require nuclear mass, nuclear size, and quantum corrections. (physics.nist.gov)

Historical development

Arnold Sommerfeld introduced $\alpha$ in 1916 while extending the Bohr model of the atom to account for hydrogen’s spectral fine structure. His treatment incorporated elliptical orbits and relativistic corrections. In the first circular Bohr orbit of hydrogen, the model gives

providing an early interpretation of the constant as a velocity ratio. This is a property of the historical orbit model, not a claim that modern quantum mechanics assigns the electron a definite circular trajectory. (pml.nist.gov)

Sommerfeld’s theory reproduced important spectral features but did not incorporate electron spin. The relativistic electron theory introduced by Paul Dirac in 1928 supplied a more complete account of hydrogenic fine structure. With the development of QED, $\alpha$ acquired its broader interpretation as an electromagnetic coupling constant rather than merely a parameter of atomic spectra. (pml.nist.gov)

Experimental determination

Two especially important routes to $\alpha$ use electron magnetic measurements and atomic recoil.

Electron magnetic moment

The electron’s anomalous magnetic moment measures the departure of its magnetic response from the leading Dirac prediction. Its leading QED contribution is

Precision measurements, combined with higher-order QED calculations and small contributions from other interactions, therefore determine $\alpha$. Conversely, inserting an independently measured value of $\alpha$ into the calculation tests QED. The distinction matters: using the same magnetic-moment measurement both to determine the constant and to test the prediction would not constitute an independent test. (physics.nist.gov)

Atomic recoil

An atom changes momentum when it absorbs or emits photons. Atom interferometry can measure the ratio $h/m_{\mathrm{atom}}$, which is related to $\alpha$ through

where $R_\infty$ is the Rydberg constant. Precision spectroscopy and mass-ratio measurements supply the other quantities. A 2018 caesium-recoil experiment reported $\alpha^{-1}=137.035\,999\,046(27)$, while a 2020 rubidium experiment reported a relative uncertainty of 81 parts per trillion. These measurements provide independent input for comparisons with electron magnetic-moment theory. (tsapps.nist.gov)

CODATA recommended values are obtained from a self-consistent adjustment of experimental and theoretical data, rather than by selecting one experiment. The 2022 adjustment documented inconsistencies among important inputs and enlarged relevant input uncertainties to accommodate their scatter. Its quoted uncertainty therefore reflects the adjustment procedure as well as individual measurement precision. (arxiv.org)

Running with energy scale

In quantum field theory, vacuum polarization changes the effective electromagnetic interaction. Charged-particle fluctuations screen electric charge; probing at different momentum-transfer scales resolves this screening differently. The resulting scale dependence is described through renormalization and a running coupling $\alpha(Q^2)$. (pml.nist.gov)

The tabulated value near $1/137$ is the low-momentum limit, usually denoted $\alpha(0)$. At electroweak energy scales, the effective coupling is approximately $1/128$. Precise values depend on the scale and the definition of the coupling, including the renormalization scheme. Contributions from quarks and hadrons complicate the calculation and introduce additional uncertainty. (pdg.lbl.gov)

This established energy dependence is not equivalent to a change in the laws of physics over time. Measurements at different energies probe different effective couplings within the same theory. Searches for temporal or spatial variation instead compare the coupling at a specified physical scale across different times or locations. (pdg.lbl.gov)

Relation to SI definitions

Since May 20, 2019, the SI has assigned exact defining values to $e$ and $h$, while $c$ also remains exact. This does not make $\alpha$ exact: vacuum permittivity and permeability are no longer fixed by definition. Rearranging the defining relation gives

Their uncertainties consequently depend on the measured uncertainty of $\alpha$. This illustrates the distinction between choosing exact dimensional constants to define units and experimentally determining a dimensionless property of nature. (bipm.org)

Searches for variation

Comparisons of atomic clocks exploit transitions with different sensitivities to $\alpha$. Changes in their frequency ratios can constrain slow drifts or oscillations of the constant. Such measurements also test models in which new fields, including certain proposed dark matter fields, modify electromagnetic interactions. (arxiv.org)

Astronomical spectra probe $\alpha$ in distant environments, while the cosmic microwave background constrains its value during early cosmic evolution. Each method has different observational systematics and model dependencies. These searches address possible variation of the underlying low-energy constant, rather than the ordinary running of the coupling with momentum scale. (arxiv.org)

References

1. 2022 CODATA adjustmentphysics.nist.gov
2. Current advances: The fine-structure constantpml.nist.gov
3. CODATA recommended values of the fundamental physical constants: 2018physics.nist.gov
4. Fine structure of hydrogenic atomsfaculty.washington.edu
5. CODATA recommended values of the fundamental physical constants: 2022physics.nist.gov
6. CODATA Recommended Values of the Fundamental Physical Constants: 2022arxiv.org
7. Measurement of the fine-structure constant as a test of the Standard Modelarxiv.org
8. Determination of the fine-structure constant with an accuracy of 81 parts per trillionnature.com
9. Electroweak Model and Constraints on New Physicspdg.lbl.gov
10. Physical Constantspdg.lbl.gov