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A dimensionless quantity, its numerical value is the same in all systems of units. At present, the best estimate of its value is:[1]

\alpha\ =\ \frac{e^2}{\hbar c \ 4 \pi \epsilon_0}\ =\ \frac{e^2 c \mu_0}{2 h}\ =\ 7.297\,352\,570(5) \times 10^{-3}\ =\ \frac{1}{137.035\,999\,070(98)} .

(numbers within parentheses are uncertainties in the final digits), where e is the elementary charge, \hbar = h/2\pi is the reduced Planck constant, c is the speed of light in a vacuum, ε0 is the electric constant, and μ0 is the magnetic constant or vacuum permeability, a defined conversion factor.

According to 2006 CODATA, as reported in the NIST reference on constants, units, and uncertainty, the defining expression and recommended value for α are:

\alpha = \frac{e^2}{\hbar c \ 4 \pi \epsilon_0} = 7.297\,352\,5376(50) \times 10^{-3} = \frac{1}{137.035\,999\,679(94)} .

However, after the 2006 CODATA adjustment was completed, an error was discovered in one of the input data, leading to the first value given above.[1]

The fine-structure constant is so called because its earliest use was in the theory of the fine structure of atomic energy spectra. However, its subsequent use has been far less specialized than its name would suggest.

Related definitions

The fine-structure constant can also be defined as:

\alpha = \frac{k_e e^2}{\hbar c} = \frac{e^2}{2 \epsilon_0 h c}

where k_e \, is the electrostatic constant of Coulomb's law, e \, is the elementary charge, \hbar = h/(2 \pi) \, is the reduced Planck constant, c \, is the speed of light in a vacuum, and \epsilon_0 \, is the electric constant.

In electrostatic cgs units, the unit of electric charge (the Statcoulomb or esu of charge) is defined so that the permittivity factor, 4 \pi \epsilon_0 \,, is the dimensionless constant 1. The fine-structure constant then becomes

\alpha = \frac{e^2}{\hbar c} .

Measurement

Two example eighth-order Feynman diagrams that contribute to the electron self-interaction. The horizontal line with an arrow represents the electron while the wavy-lines are virtual photons, and the circles represent virtual electron-positron pairs.

The definition of \alpha\, contains several measurable constants. However, quantum electrodynamics (QED) provides a way to measure \alpha\, directly using the quantum Hall effect or the anomalous magnetic moment of the electron.

QED predicts a relationship between the dimensionless magnetic moment of the electron (or the Lande g-factor, g \,) and the fine structure constant \alpha\,. The most precise value of \alpha\, obtained to date is based on a new measurement of g \, using a one-electron quantum cyclotron, together with a QED calculation involving 891 four-loop Feynman diagrams:[1]

\alpha^{-1} = 137.035\,999\,068(96)

This measurement has a precision of 0.70 ppb. This uncertainty is 10 times smaller than those of the nearest rival methods that include atom-recoil measurements. Comparisons of the measured and calculated values of g \, test QED very stringently, and limit the possible internal structure of the electron.

Physical interpretation

The fine-structure constant has a number of physical interpretations including:

  1. The square of the ratio of the elementary charge to the Planck charge;
  2. A ratio of certain energies;
  3. The velocity of the electron in the Bohr model of the atom divided by the speed of light;
  4. A constant representing the strength of the interaction between electrons and photons;
  5. The strength of the electromagnetic interaction, which may change, depending on the strength of the energy field.

By (1) above:

\alpha = \left( \frac{e}{q_P} \right)^2.

For any arbitrary length s \,, the fine-structure constant is the ratio of two energies: (i) the energy needed to overcome the electrostatic repulsion between two electrons, by reducing the distance between them from infinity to s \, and (ii) the energy of a single photon of wavelength equal to the same length scaled by 2π (i.e. 2 \pi s = \lambda = \frac{c}{\nu} \, where \nu \, is the radiation frequency associated with the photon):

\alpha = \frac{e^2}{4 \pi \epsilon_0 s} \div h \nu = \frac{e^2}{4 \pi \epsilon_0 s} \div \frac{h c}{2 \pi s} = \frac{e^2}{4 \pi \epsilon_0 \hbar c}.

The fine structure constant is also the ratio between the velocity of the electron in the Bohr atom and the speed of light. The square of alpha is the ratio between the electron rest mass (511 keV) and the Hartree energy (27.2 eV = 2 Ry).

In the theory of quantum electrodynamics, the fine-structure constant is the coupling constant for the strength of the interaction between electrons and photons. The theory does not predict its value; thus it must be determined experimentally. In fact, it is one of the 20-odd "external parameters" in the Standard Model of particle physics.

That \alpha \, is clearly less than 1 allows the use of perturbation theory in quantum electrodynamics. Physical results in this theory are expressed as power series in \alpha \,, with higher powers of \alpha \, becoming increasingly unimportant. In contrast, the large value of the corresponding factors in quantum chromodynamics makes calculations involving the strong force extremely difficult.

In the electroweak theory unifying the weak interaction with electromagnetism, the fine-structure constant is absorbed into two other coupling constants associated with the electroweak gauge fields. In this theory, the electromagnetic interaction is treated as a mixture of interactions associated with the electroweak fields.

According to the theory of the renormalization group, the value of the fine-structure constant (the strength of the electromagnetic interaction) depends on the energy scale. In fact, it grows logarithmically as the energy is increased. The observed value of \alpha \, is associated with the energy scale of the electron mass; the electron is a lower bound for this energy scale because it (and the positron) is the lightest charged object whose quantum loops can contribute to the running. Therefore, we can say that 1/137.036 is the value of the fine-structure constant at zero energy. Moreover, as the energy scale increases, the strength of the electromagnetic interaction approaches that of the other two interactions, a fact important for grand unification theories. If quantum electrodynamics were an exact theory, the fine-structure constant would actually diverge at an energy known as the Landau pole. This fact makes quantum electrodynamics inconsistent beyond the perturbative expansions.

History

The fine-structure constant was introduced in 1916 by Arnold Sommerfeld, in his theory of the relativistic deviations of atomic spectral lines from the predictions of the Bohr model.

Historically, the first physical interpretation of the fine-structure constant, \alpha \,, was as the ratio of the velocity of the electron in the first circular orbit of the relativistic Bohr atom to the speed of light in the vacuum[2]. Equivalently, it was the quotient between the maximum angular momentum allowed by relativity for a closed orbit, and the minimum angular momentum allowed for it by quantum mechanics. It appears naturally in Sommerfeld's analysis, and determines the size of the splitting or fine-structure of the hydrogenic spectral lines.

Is the fine structure constant really constant?

Physicists have pondered for many years whether the fine structure constant is truly constant, i.e., whether or not its value differs by location and over time. Specifically, a varying \alpha \, has been proposed as a way of solving problems in cosmology and astrophysics.[3][4][5] More recently, theoretical interest in varying constants (not just \alpha \,) has been motivated by string theory and other such proposals for going beyond the Standard Model of particle physics. The first experimental tests of this question examined the spectral lines of distant astronomical objects, and the products of radioactive decay in the Oklo natural nuclear fission reactor. The findings were consistent with no change.[6][7][8][9][10][11]

More recently, improved technology has made it possible to probe the value of \alpha \, at much larger distances and to a much greater accuracy. In 1999, a team lead by John K. Webb of the University of New South Wales claimed the first detection of a variation in \alpha \,.[12][13][14][15] Using the Keck telescopes and a data set of 128 quasars at redshifts 0.5<z<3, Webb et al. found that their spectra were consistent with a slight increase in \alpha \, over the last 10-12 billion years. Specifically, they found that

\frac{\Delta \alpha}{\alpha} \ \stackrel{\mathrm{def}}{=}\ \frac{\alpha _\mathrm{then}-\alpha _\mathrm{now}}{\alpha_\mathrm{now}} = \left( -0.57\pm 0.10 \right) \times 10^{-5}.

A more recent and smaller study of 23 absorption systems by Chand et al., using the Very Large Telescope, found no measureable variation:[16][17]

\frac{\Delta \alpha}{\alpha_\mathrm{em}}= \left(-0.6\pm 0.6\right) \times 10^{-6}.

Although the Chand et al. result disagrees with Webb et al., systematic uncertainties are difficult to quantify. There are ongoing efforts to collect and analyze more data. All other astrophysical findings to date are consistent with no variation.[18]

Very recently, Khatri and Wandelt of the University of Illinois at Urbana-Champaign realized that the 21 cm hyperfine transition in neutral hydrogen in the early Universe leaves a unique absorption line imprint in the cosmic microwave background radiation.[19] They proposed using this effect to measure the value of α during the epoch before the formation of the first stars. In principle, this technique provides enough information to measure a variation of 1 part in 109 (4 orders of magnitude better than the current quasar constraints). However, the constraint which can be placed on α is strongly dependent upon effective integration time, going as t − 1 / 2. The LOFAR telescope would only be able to constrain Δα / α to ~0.3%[20]. The collecting area required to constrain Δα / α to the current level of quasar constraints is on the order of 100km2, which is impracticable at present.

   
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