Fine-structure constant

In physics, the fine-structure constant (usually denoted α, the Greek letter alpha) is a fundamental physical constant, namely the coupling constant characterizing the strength of the electromagnetic interaction. The numerical value of α is the same in all systems of units, because α is a dimensionless quantity.

Definition

Three equivalent definitions of α in terms of other fundamental physical constants are:

$\alpha =\ {\frac {e^{2}}{(4\pi \varepsilon _{0})\hbar c}}\ =\ {\frac {e^{2}c\mu _{0}}{2h}}={\frac {k_{\mathrm {e} }e^{2}}{\hbar c}},$

where:

• e is the elementary charge;
• ħ = h/2π is the reduced Planck constant;
• c is the speed of light in vacuum;
• ε₀ is the electric constant or permittivity of free space;
  • or µ₀ is the magnetic constant or permeability of free space;
  • or k_e is the Coulomb constant.

In electrostatic cgs units, the unit of electric charge, the statcoulomb, is defined so that the Coulomb constant, k_e, or the permittivity factor, 4πε₀, is 1 and dimensionless. Then the expression of the fine-structure constant becomes the abbreviated

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

an expression commonly appearing in physics literature.

Measurement

According to 2006 CODATA, the defining expression and recommended value for α are:

$\alpha ={\frac {e^{2}}{\hbar c\ 4\pi \varepsilon _{0}}}=7.297\,352\,5376(50)\times 10^{-3}={\frac {1}{137.035\,999\,679(94)}}.$

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.

However, after the 2006 CODATA adjustment was completed, an error was discovered in one of the main data inputs. Nevertheless, the 2006 CODATA recommended value was republished in 2008. A revised standard value, taking recent research and adjustments to SI units into account, is expected to be published in 2010 or early in 2011.

The value of α can be estimated from the values of the constants appearing in any of its definitions. However, 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.

The theory of QED predicts a relationship between the dimensionless magnetic moment of the electron (or the "Lande g-factor", g) and the fine structure constant α. The most precise value of α obtained experimentally through the present is based on a new measurement of g using a one-electron so-called "quantum cyclotron" apparatus, together with a calculation via the theory of QED that involved 891 four-loop Feynman diagrams:

$\alpha ^{-1}=137.035\,999\,084(51).$

This measurement of α has a precision of 0.37 parts per billion. This uncertainty is just one-twentieth of those of the nearest alternative experimental methods, including atom-recoil measurements. Comparisons of the measured and the calculated values of g place stringent tests on the theory of QED, and they also place limits on any possible internal structure of electrons.

Physical interpretations

The fine structure constant α has several physical interpretations. α is:

• The square of the ratio of the elementary charge to the Planck charge

$\alpha =\left({\frac {e}{q_{\mathrm {P} }}}\right)^{2}$.

• The ratio of two energies: (i) the energy needed to overcome the electrostatic repulsion between two electrons when the distance between them is reduced from infinity to some finite d, and (ii) the energy of a single photon of wavelength $\lambda =2\pi d$ (see Planck relation):

$\alpha ={\frac {e^{2}}{4\pi \varepsilon _{0}d}}\times {\frac {\lambda }{hc}}={\frac {e^{2}}{4\pi \varepsilon _{0}d}}\times {\frac {2\pi d}{hc}}={\frac {e^{2}}{4\pi \varepsilon _{0}\hbar c}}.$

• The ratio of the velocity of the electron in the Bohr model of the atom to the speed of light. Hence the square of α is the ratio between the Hartree energy (27.2 eV = twice the Rydberg constant) and the electron rest mass (511 keV).
• The ratio of three characteristic lengths: the classical electron radius $r_{e}$, the Bohr radius $a_{0}$ and the Compton wavelength of the electron $\lambda _{e}$:

$r_{e}={\alpha \lambda _{e} \over 2\pi }=\alpha ^{2}a_{0}$

• In quantum electrodynamics, α is the coupling constant determining the strength of the interaction between electrons and photons. The theory does not predict its value. Therefore α must be determined experimentally. In fact, α is one of the about 20 empirical parameters in the Standard Model of particle physics, whose value is not determined within the Standard Model.
• In the electroweak theory unifying the weak interaction with electromagnetism, α 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. The strength of the electromagnetic interaction varies with the strength of the energy field.
• In the fields of electrical engineering and solid-state physics the ratio of the electromagnetic impedance of free space, 1/(c ε₀) = 377 Ω, to the quantum of resistance, h/e² = 25.8 kΩ, is 2α.

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 extremely 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.

According to the theory of the renormalization group, the value of the fine structure constant (the strength of the electromagnetic interaction) grows logarithmically as the energy scale is increased. The observed value of α 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 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 fundamental 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

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

The fine structure constant so intrigued the physicist Wolfgang Pauli that he even collaborated with the psychologist Carl Jung in an extraordinary quest to understand its significance.

Is the fine structure constant actually constant?

Physicists have pondered for many years whether the fine structure constant is in fact constant, i.e., whether or not its value differs by location and over time. Specifically, a varying α has been proposed as a way of solving problems in cosmology and astrophysics. More recently, theoretical interest in varying constants (not just α) 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.

More recently, improved technology has 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. 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 α over the last 10–12 billion years. Specifically, they found that

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

In 2004, a smaller study of 23 absorption systems by Chand et al., using the Very Large Telescope, found no measureable variation:

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

However, in 2007 simple flaws were identified in the analysis method of Chand et al., discrediting those results. Nevertheless, systematic uncertainties are difficult to quantify and so the Webb et al.. results still need to be checked by independent analysis, using quasar spectra from different telescopes.

King et al. have used Markov Chain Monte Carlo methods to investigate the algorithm used by the UNSW group to determine $\Delta \alpha /\alpha$ from the quasar spectra, and have found that the algorithm appears to produce correct uncertainties and maximum likelihood estimates for $\Delta \alpha /\alpha$ for particular models. This suggests that the statistical uncertainties and best estimate for $\Delta \alpha /\alpha$ stated by Webb et al. and Murphy et al. are robust.

Lamoreaux and Torgerson analyzed data from the Oklo natural nuclear fission reactor in 2004, and concluded that α has changed in the past 2 billion years by 4.5 parts in 10⁸. 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 but provide an interesting direction of study.

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 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 European LOFAR radio telescope would only be able to constrain Δα/α to about 0.3%. 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 economicallly impracticable at the present time.

In 2008, Rosenband et al. used the frequency ratio of Template:Element and Template:Element in single-ion optical atomic clocks to place a very stringent constraint on the present time variation of α, namely Δα̇/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 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.

In September 2010 researchers from Australia said they had identified a dipole-like structure in the fine structure constant across the observable universe. According to studies on quasars using the Very Large Telescope, the fine structure constant appears to have been larger by one part in 100,000 in the southern direction, 10 billion years ago. Similarly, the constant appeared to have been smaller by a similar fraction in the northern direction, billions of years ago.

Anthropic explanation

The anthropic principle is a controversial explanation of why the fine-structure constant takes on the value it does: stable matter, and therefore life and intelligent beings, could not exist if its value were much different. For instance, were α to change by 4%, stellar fusion would not produce carbon, so that carbon-based life would be impossible. If α were > 0.1, stellar fusion would be impossible and no place in the universe would be warm enough for life.

The fine structure constant plays a central role in John Barrow's and Frank Tipler's broad-ranging discussion of astrophysics, cosmology, quantum physics, teleology, and the anthropic principle.

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. 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 about 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 man. 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!

— Richard P. Feynman (1985). QED: The Strange Theory of Light and Matter. Princeton University Press. p. 129. ISBN 0691083886.

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. This led him in 1929 to conjecture that its reciprocal was precisely the integer 137. Other physicists neither adopted this conjecture nor accepted his arguments but by the 1940s experimental values for 1/α deviated sufficiently from 137 to refute Eddington's argument. Attempts to find a mathematical basis for this dimensionless constant have continued up to the present time. For example, the mathematician James Gilson suggested (earliest archive.org entry dated December 2006 [1]), that the fine-structure constant has the value:

$\alpha ={\frac {\cos \left(\pi /137\right)}{137}}\ {\frac {\tan \left(\pi /(137\cdot 29)\right)}{\pi /(137\cdot 29)}}\approx {\frac {1}{137.0359997867}}$ ,

29 and 137 being the 10^th and 33^rd prime numbers. The difference between the 2007 CODATA value for α and this theoretical value is about 3 x 10⁻¹¹, about 6 times the standard error for the measured value.

Quotes

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.

— Malcolm H. Mac Gregor, M.H. MacGregor (2007). The Power of Alpha. World Scientific. p. 69. ISBN 9789812569615.

If alpha [the fine structure constant] 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, A.I. Miller (2009). Deciphering the Cosmic Number: The Strange Friendship of Wolfgang Pauli and Carl Jung. W.W. Norton & Co. p. 253. ISBN 9780393065329.

See also

• Electric constant
• Gravitational coupling constant
• Planck's constant
• Speed of light

External links

• "Introduction to the constants for nonexperts," adapted from the Encyclopedia Britannica, 15th ed. Disseminated by the NIST web page.
• CODATA recommended value of α, as of 2006.
• "Fine structure constant," Eric Weisstein's World of Physics website.
• John D. Barrow, and Webb, John K., "Inconstant Constants," Scientific American, June 2005.