Since the
2019 redefinition of the SI base units, the only quantity in this list that does not have an exact value in
SI units is the electric constant (vacuum permittivity).
Alternative systems of units
The electrostatic
CGS system implicitly sets 4πε0 = 1, as commonly found in older physics literature, where the expression of the fine-structure constant becomes
As such, the fine-structure constant is just a quantity determining (or determined by) the
elementary charge: e = √4πα ≈ 0.30282212 in terms of such a natural unit of charge.
In the system of
atomic units, which sets e = me = ħ = 4πε0 = 1, the expression for the fine-structure constant becomes
This has a relative standard uncertainty of 1.5×10−10.[1]
This value for α gives µ0 = 4π × 1.00000000054(15)×10−7 H⋅m−1, 3.6 standard deviations away from its old defined value, but with the mean differing from the old value by only 0.54
parts per billion.
Historically the value of the
reciprocal of the fine-structure constant is often given. The 2018
CODATA recommended value is[11]
1/α = 137.035999084(21).
While the value of α can be determined from estimates of the constants that appear in any of its definitions, the theory of
quantum electrodynamics (QED) provides a way to measure α directly using the
quantum Hall effect or the
anomalous magnetic moment of the
electron.[12] Other methods include the A.C. Josephson effect and photon recoil in atom interferometry.[13]
There is general agreement for the value of α, 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.[13] The theory of QED predicts a relationship between the
dimensionless magnetic moment of the
electron and the fine-structure constant α (the magnetic moment of the electron is also referred to as the
electron g-factorge). One of the most precise values of α obtained experimentally (as of 2023) is based on a measurement of ge using a one-electron so-called "quantum cyclotron" apparatus,[12] together with a calculation via the theory of QED that involved 12672 tenth-order
Feynman diagrams:[14]
1/α = 137.035999166(15).
This measurement of α has a relative standard uncertainty of 1.1×10−10. This value and uncertainty are about the same as the latest experimental results.[15]
Further refinement of the experimental value was published by the end of 2020, giving the value
1/α = 137.035999206(11),
with a relative accuracy of 8.1×10−11, which has a significant discrepancy from the previous experimental value.[16]
Physical interpretations
The fine-structure constant, α, has several physical interpretations. α is:
The ratio of two energies:
the energy needed to overcome the
electrostatic repulsion between two electrons a distance of d apart, and
The ratio of the velocity of the electron in the first circular orbit of the
Bohr model of the atom, which is 1/4πε0e2/ħ, to the
speed of light in vacuum, c.[17] This is
Sommerfeld's original physical interpretation. Then the square of α is the ratio between the
Hartree energy (27.2 eV = twice the
Rydberg energy = approximately twice its ionization energy) and the
electronrest energy (511 keV).
is the ratio of the potential energy of the electron in the first circular orbit of the
Bohr model of the atom and the energy mec2 equivalent to the mass of an electron. Using the
virial theorem in the
Bohr model of the atom which means that Essentially this ratio follows from the electron's velocity being .
In the fields of
electrical engineering and
solid-state physics, the fine-structure constant is one fourth the product of the characteristic
impedance of free space, and the
conductance quantum, : The
optical conductivity of
graphene for visible frequencies is theoretically given by π/4G0, and as a result its light absorption and transmission properties can be expressed in terms of the fine-structure constant alone.[19] The absorption value for normal-incident light on graphene in vacuum would then be given by πα/ (1 + πα/2)2 or 2.24%, and the transmission by 1/(1 + πα/2)2 or 97.75% (experimentally observed to be between 97.6% and 97.8%). The reflection would then be given by π2α2/ 4 (1 + πα/2)2.
The fine-structure constant gives the maximum positive charge of an atomic nucleus that will allow a stable electron-orbit around it within the Bohr model (element
feynmanium).[20] For an electron orbiting an atomic nucleus with atomic number Z the relation is mv2/r = 1/ 4πε0Ze2/r2. The Heisenberg
uncertainty principle momentum/position uncertainty relationship of such an electron is just mvr = ħ. The relativistic limiting value for v is c, and so the limiting value for Z is the reciprocal of the fine-structure constant, 137.[21]
The magnetic moment of the electron indicates that the charge is circulating at a radius rQ with the velocity of light.[22] It generates the radiation energy mec2 and has an angular momentum L = 1 ħ = rQmec. The field energy of the stationary Coulomb field is mec2 = e2/4πε0re and defines the classical electron radius re. These values inserted into the definition of alpha yields α = re/rQ. It compares the dynamic structure of the electron with the classical static assumption.
When
perturbation theory is applied to
quantum electrodynamics, the resulting
perturbative expansions for physical results are expressed as sets of
power series in α. Because α is much less than one, higher powers of α 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 α 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 (and the
positron) is the lightest charged object whose
quantum loops can contribute to the running. Therefore, 1/ 137.03600 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 effectiveα ≈ 1/127.[23]
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,[c]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.[d]
The first physical interpretation of the fine-structure constant α 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.[27]
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.[28]: 407
With the development of
quantum electrodynamics (QED) the significance of α 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 α/2π 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
Successive values determined for the fine-structure constant[29][e]
Physicists have pondered whether the fine-structure constant is in fact constant, or whether its value differs by location and over time. A varying α has been proposed as a way of solving problems in
cosmology and
astrophysics.[32][33][34][35]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 (not just α) actually vary.
In the experiments below, Δα represents the change in α over time, which can be computed by αprev − αnow . 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 α does change over time. So far, most experimental data is consistent with α being constant.
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
Oklonatural nuclear fission reactor. Their findings were consistent with no variation in the fine-structure constant between these two vastly separated locations and times.[36][37][38][39][40][41]
Improved technology at the dawn of the 21st century made it possible to probe the value of α 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 α.[42][43][44][45]
Using the
Keck telescopes and a data set of 128
quasars at
redshifts0.5 < z < 3, Webb et al. found that their spectra were consistent with a slight increase in α over the last 10–12 billion years. Specifically, they found that
In other words, they measured the value to be somewhere between −0.0000047 and −0.0000067. This is a very small value, but the error bars do not actually include zero. This result either indicates that α 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:[46][47]
However, in 2007 simple flaws were identified in the analysis method of Chand et al., discrediting those results.[48][49]
King et al. have used
Markov chain Monte Carlo methods to investigate the algorithm used by the UNSW group to determine Δα/α from the quasar spectra, and have found that the algorithm appears to produce correct uncertainties and maximum likelihood estimates for Δα/α for particular models.[50] This suggests that the statistical uncertainties and best estimate for Δα/α stated by Webb et al. and Murphy et al. are robust.
Lamoreaux and Torgerson analyzed data from the
Oklonatural nuclear fission reactor in 2004, and concluded that α has changed in the past 2 billion years by 45 parts per billion. They claimed that this finding was "probably accurate to within 20%". Accuracy is dependent on estimates of impurities and temperature in the natural reactor. These conclusions have to be verified.[51][52][53][54]
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.[55]
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 1⁄√t. The European
LOFARradio telescope would only be able to constrain Δα/α to about 0.3%.[55] The collecting area required to constrain Δα/α to the current level of quasar constraints is on the order of 100 square kilometers, which is economically impracticable at the present time.
Present rate of change
In 2008, Rosenband et al.[56]
used the frequency ratio of Al+ and Hg+ in single-ion optical atomic clocks to place a very stringent constraint on the present-time temporal variation of α, namely Δα/α = (−1.6±2.3)×10−17 per year. Note that any present day null constraint on the time variation of alpha does not necessarily rule out time variation in the past. Indeed, some theories[57]
that predict a variable fine-structure constant also predict that the value of the fine-structure constant should become practically fixed in its value once the universe enters its current
dark energy-dominated epoch.
Spatial variation – Australian dipole
Researchers from Australia have said they had identified a variation of the fine-structure constant across the observable universe.[58][59][60][61][62][63]
These results have not been replicated by other researchers. In September and October 2010, after released research by Webb et al., physicists
C. Orzel and
S.M. Carroll separately suggested various approaches of how Webb's observations may be wrong. Orzel argues[64]
that the study may contain wrong data due to subtle differences in the two telescopes[65]
a totally different approach; he looks at the fine-structure constant as a scalar field and claims that if the telescopes are correct and the fine-structure constant varies smoothly over the universe, then the scalar field must have a very small mass. However, previous research has shown that the mass is not likely to be extremely small. Both of these scientists' early criticisms point to the fact that different techniques are needed to confirm or contradict the results, a conclusion Webb, et al., previously stated in their study.[61]
Other research finds no meaningful variation in the fine structure constant.[66][67]
Anthropic explanation
The
anthropic principle is an argument about the reason the fine-structure constant has the value it does: stable matter, and therefore life and intelligent beings, could not exist if its value were very different. One example is that, if modern grand unified theories are correct, then α needs to be between around 1/180 and 1/85 to have proton decay to be slow enough for life to be possible.[68]
Numerological explanations
As a dimensionless constant which does not seem to be directly related to any
mathematical constant, the fine-structure constant has long fascinated physicists.
Arthur Eddington argued that the value could be "obtained by pure deduction" and he related it to the
Eddington number, his estimate of the number of protons in the universe.[69]
This led him in 1929 to conjecture that the reciprocal of the fine-structure constant was not approximately but precisely the
integer137.[70]
By the 1940s experimental values for 1/α deviated sufficiently from 137 to refute Eddington's arguments.[28]
Physicist
Wolfgang Pauli commented on the appearance of
certain numbers in physics, including the fine-structure constant, which he also noted approximates the prime number
137.[71] This constant so intrigued him that he collaborated with psychoanalyst
Carl Jung in a quest to understand its significance.[72] Similarly,
Max Born believed that if the value of α differed, the universe would degenerate, and thus that α = 1/137 is a law of nature.[73][g]
Richard Feynman, one of the originators and early developers of the theory of
quantum electrodynamics (QED), referred to the fine-structure constant in these terms:
There is a most profound and beautiful question associated with the observed coupling constant, e – the amplitude for a real electron to emit or absorb a real photon. It is a simple number that has been experimentally determined to be close to 0.08542455. (My physicist friends won't recognize this number, because they like to remember it as the inverse of its square: about 137.03597 with an uncertainty of about 2 in the last decimal place. It has been a mystery ever since it was discovered more than fifty years ago, and all good theoretical physicists put this number up on their wall and worry about it.)
Immediately you would like to know where this number for a coupling comes from: is it related to pi or perhaps to the base of natural logarithms? Nobody knows. It's one of the greatest damn mysteries of physics: a magic number that comes to us with no understanding by humans. You might say the "hand of God" wrote that number, and "we don't know how He pushed His pencil." We know what kind of a dance to do experimentally to measure this number very accurately, but we don't know what kind of dance to do on the computer to make this number come out – without putting it in secretly!
Conversely, statistician
I. J. Good argued that a numerological explanation would only be acceptable if it could be based on a good theory that is not yet known but "exists" in the sense of a
Platonic Ideal.[h]
Attempts to find a mathematical basis for this dimensionless constant have continued up to the present time. However, no numerological explanation has ever been accepted by the physics community.
For historical reasons, α is known as the fine structure constant. Unfortunately, this name conveys a false impression. We have seen that the charge of an electron is not strictly constant but varies with distance because of quantum effects; hence α must be regarded as a variable, too. The value 1/137 is the asymptotic value of α shown in Fig. 1.5a.[76]
The mystery about α is actually a double mystery: The first mystery – the origin of its numerical value α ≈ 1/137 – has been recognized and discussed for decades. The second mystery – the range of its domain – is generally unrecognized.
^CODATA 2018, published in May 2019, takes into accounts measurements published up to 2018. Morel et al. (2020) have claimed a measurement with a relative uncertainty below 10−10; their value also differs significantly from the 2018 recommended value, to an agreement of merely 9×10−10.
^
"Among other substances [that were] tried in the preliminary experiments, were thallium, lithium, and hydrogen. ... It may be noted, that in [the] case of the red hydrogen line, the interference phenomena disappeared at about 15,000 wave-lengths, and again at about 45,000 wave-lengths: So that the red hydrogen line must be a double line with the components about one-sixtieth as distant as the sodium lines."[25](p430)
^"Wir fügen den Bohrschen Gleichungen (46) und (47) die charakteristische Konstante unserer Feinstrukturen(49) α = 2πe2/chhinzu, die zugleich mit der Kenntnis des Wasserstoffdubletts oder des Heliumtripletts in §10 oder irgend einer analogen Struktur bekannt ist." ———
(We add, to Bohr's equations (46) and (47), the characteristic constant of our fine structures (49) α = 2πe2/ch which is known at once from knowledge of the hydrogen doublet or the helium triplet in §10 or any analogous structure.)[26](p
91)
^Numbers in parentheses (e.g. the "(11)" appearing at the end of the value "137.035999206(11)") give its
standard uncertainty referred to the least significant preceding digit.
^This is not an experimentally measured value; instead it is a value determined by the current theory from an experimentally determined value of the
electron magnetic moment.
^"If alpha were bigger than it really is, we should not be able to distinguish matter from ether [the vacuum, nothingness], and our task to disentangle the natural laws would be hopelessly difficult. The fact however that alpha has just its value 1/137 is certainly no chance but itself a law of nature. It is clear that the explanation of this number must be the central problem of natural philosophy." –
Max Born[73]
^"There have been a few examples of numerology that have led to theories that transformed society: See the mention of
Kirchhoff and
Balmer in
Good (1962) p. 316 ... and one can well include
Kepler on account of
his third law. It would be fair enough to say that numerology was the origin of the theories of electromagnetism, quantum mechanics, gravitation. ... So I intend no disparagement when I describe a formula as numerological. When a numerological formula is proposed, then we may ask whether it is correct. ... I think an appropriate definition of correctness is that the formula has a good explanation, in a Platonic sense, that is, the explanation could be based on a good theory that is not yet known but 'exists' in the universe of possible reasonable ideas." —
I. J. Good (1990)[74]
^
Barrow, J.D. (2002). The Constants of Nature: From Alpha to Omega – the Numbers That Encode the Deepest Secrets of the Universe.
Vintage.
ISBN978-0-09-928647-9.
^
Fujii, Yasunori (2004). "Oklo constraint on the time-variability of the fine-structure constant". Astrophysics, Clocks, and Fundamental Constants. Lecture Notes in Physics. Vol. 648. pp. 167–185.
doi:
10.1007/978-3-540-40991-5_11.
ISBN978-3-540-21967-5.
^
King, J. A.; Mortlock, D. J.; Webb, J. K.; Murphy, M. T. (2009). "Markov chain Monte Carlo methods applied to measuring the fine structure constant from quasar spectroscopy". Memorie della Societa Astronomica Italiana. 80: 864.
arXiv:0910.2699.
Bibcode:
2009MmSAI..80..864K.
^The asymptotic value of αfor larger observation distances, is intended here. Caption: Fig 1.5. Screening of the (a) electric charge and (b) the color charge in quantum field theory. Graph of Electron charge versus Distance from the bare e- charge. From: Halzen, F.; Martin, A.D. (1984). Quarks and Leptons: An Introductory Course in Modern Particle Physics. John Wiley & Sons. ISBN 978-0-471-88741-6, p. 13.
Physicists Nail Down the ‘Magic Number’ That Shapes the Universe (Natalie Wolchover, Quanta magazine, December 2, 2020). The value of this constant is given here as 1/137.035999206 (note the difference in the last three digits). It was determined by a team of four physicists led by Saïda Guellati-Khélifa at the Kastler Brossel Laboratory in Paris.