Fine-structure constant

Value of

In physics, the fine-structure constant, also known as the Sommerfeld constant, commonly denoted by , is a fundamental physical constant that quantifies the strength of the electromagnetic interaction between elementary charged particles.
It is a dimensionless quantity, independent of the system of units used, which is related to the strength of the coupling of an elementary charge with the electromagnetic field, by the formula. Its numerical value is approximately, with a relative uncertainty of
The constant was named by Arnold Sommerfeld, who introduced it in 1916 when extending the Bohr model of the atom. quantified the gap in the fine structure of the spectral lines of the hydrogen atom, which had been measured precisely by Michelson and Morley in 1887.
Why the constant should have this value is not understood, but there are a number of ways to measure its value.

Definition

In terms of other physical constants, may be defined as:
where
Since the 2019 revision of the SI, the only quantity in this list that does not have an exact value in SI units is the electrical permittivity of space; the others are all defining constants.

Alternative systems of units

The electrostatic CGS system implicitly sets, as commonly found in older physics literature, where the expression of the fine-structure constant becomes
A normalised system of units commonly used in high energy physics selects artificial units for mass, distance, time, and electrical charge which cause ; in such a system of "natural units" the expression for the fine-structure constant becomes
As such, the fine-structure constant is chiefly a quantity determining the elementary charge: in terms of such a natural unit of charge.
In the system of atomic units, which sets, the expression for the fine-structure constant becomes

Measurement

The CODATA recommended value of is
This has a relative standard uncertainty of
This value for gives the following value for the vacuum magnetic permeability :, with the mean differing from the old defined value by only 0.13 parts per billion, 0.8 times the standard uncertainty of its recommended measured value.
Historically, the value of the reciprocal of the fine-structure constant is often given. The CODATA recommended value is
While the value of can be determined from estimates of the constants that appear in any of its definitions, the theory of quantum electrodynamics provides a way to measure directly using the quantum Hall effect or the anomalous magnetic moment of the electron. Other methods include the A.C. Josephson effect and photon recoil in atom interferometry.
There is general agreement for the value of, as measured by these different methods. The preferred methods in 2019 are measurements of electron anomalous magnetic moments and of photon recoil in atom interferometry. The theory of QED predicts a relationship between the dimensionless magnetic moment of the electron and the fine-structure constant . One of the most precise values of obtained experimentally is based on a measurement of using a one-electron so-called "quantum cyclotron" apparatus, together with a calculation via the theory of QED that involved tenth-order Feynman diagrams:
This measurement of has a relative standard uncertainty of. This value and uncertainty are about the same as the latest experimental results.
Further refinement of the experimental value was published by the end of 2020, giving the value
with a relative accuracy of, which has a significant discrepancy from the previous experimental value.

Physical interpretations

The fine-structure constant,, has several physical interpretations. is:
When perturbation theory is applied to quantum electrodynamics, the resulting perturbative expansions for physical results are expressed as sets of power series in. Because is much less than one, higher powers of are soon unimportant, making the perturbation theory practical in this case. On the other hand, the large value of the corresponding factors in quantum chromodynamics makes calculations involving the strong nuclear force extremely difficult.

Variation with energy scale

In quantum electrodynamics, the more thorough quantum field theory underlying the electromagnetic coupling, the renormalization group dictates how the strength of the electromagnetic interaction grows logarithmically as the relevant energy scale increases. The value of the fine-structure constant is linked to the observed value of this coupling associated with the energy scale of the electron mass: the electron's mass gives a lower bound for this energy scale, because it is the lightest charged object whose quantum loops can contribute to the running. Therefore, is the asymptotic value of the fine-structure constant at zero energy.
At higher energies, such as the scale of the Z boson, about 90 GeV, one instead measures an effective ≈ 1/127.
As the energy scale increases, the strength of the electromagnetic interaction in the Standard Model approaches that of the other two fundamental interactions, a feature 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 undermines the consistency of quantum electrodynamics beyond perturbative expansions.

History

Based on the precise measurement of the hydrogen atom spectrum by Michelson and Morley in 1887,
Arnold Sommerfeld extended the Bohr model to include elliptical orbits and relativistic dependence of mass on velocity. He introduced a term for the fine-structure constant in 1916.
The first physical interpretation of the fine-structure constant 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.
Equivalently, it was the quotient between the minimum 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. This constant was not seen as significant until Paul Dirac's linear relativistic wave equation in 1928, which gave the exact fine structure formula.
With the development of quantum electrodynamics the significance of has broadened from a spectroscopic phenomenon to a general coupling constant for the electromagnetic field, determining the strength of the interaction between electrons and photons. The term is engraved on the tombstone of one of the pioneers of QED, Julian Schwinger, referring to his calculation of the anomalous magnetic dipole moment.

History of measurements

The CODATA values in the above table are computed by averaging other measurements; they are not independent experiments.

Potential variation over time

Physicists have pondered whether the fine-structure constant is in fact constant, or whether its value differs by location and over time. A varying has been proposed as a way of solving problems in cosmology and astrophysics. String theory and other proposals for going beyond the Standard Model of particle physics have led to theoretical interest in whether the accepted physical constants actually vary.
In the experiments below, represents the change in over time, which can be computed by _past − _now . If the fine-structure constant really is a constant, then any experiment should show that
or as close to zero as experiment can measure. Any value far away from zero would indicate that does change over time. So far, most experimental data is consistent with being constant, up to 10 digits of accuracy.

Past rate of change

The first experimenters to test whether the fine-structure constant might actually vary examined the spectral lines of distant astronomical objects and the products of radioactive decay in the Oklo natural nuclear fission reactor. Their findings were consistent with no variation in the fine-structure constant between these two vastly separated locations and times. The most recent constraint from Oklo, from Davis & Hamdan, set an upper limit of 11 ppb difference at 95% confidence level, a constraint comparable in strength to that from atomic-clock measurements.
Improved technology at the dawn of the 21st century made it possible to probe the value of at much larger distances and to a much greater accuracy. In 1999, a team led by John K. Webb of the University of New South Wales claimed the first detection of a variation in.
Using the Keck telescopes and a data set of 128 quasars at redshifts, Webb et al. found that their spectra were consistent with a slight increase in over the last 10–12 billion years. Specifically, they found that
In other words, they measured the value to be somewhere between and. This is a very small value, but the error bars do not actually include zero. This result either indicates that is not constant or that there is experimental error unaccounted for.
In 2004, a smaller study of 23 absorption systems by Chand et al., using the Very Large Telescope, found no measurable variation:
However, in 2007 simple flaws were identified in the analysis method of Chand et al., discrediting those results.
King et al. have used Markov chain Monte Carlo methods to investigate the algorithm used by the UNSW group to determine from the quasar spectra, and have found that the algorithm appears to produce correct uncertainties and maximum likelihood estimates for for particular models. This suggests that the statistical uncertainties and best estimate for stated by Webb et al. and Murphy et al. are robust.
In 2007, Khatri and Wandelt of the University of Illinois at Urbana-Champaign realized that the 21 cm hyperfine transition in neutral hydrogen of the early universe leaves a unique absorption line imprint in the cosmic microwave background radiation. They proposed using this effect to measure the value of 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 . However, the constraint which can be placed on is strongly dependent upon effective integration time, going as. The European LOFAR radio telescope would only be able to constrain to about 0.3%. The collecting area required to constrain to the current level of quasar constraints is on the order of 100 square kilometers, which is economically impracticable at present.