In physics, the fine-structure constant, also known as Sommerfeld's constant,
commonly denoted α, is a fundamental physical constant characterizing the
strength of the electromagnetic interaction between elementary charged
particles. It is related to the elementary charge e, which characterizes
the strength of the coupling of an elementary charged particle with the
electromagnetic field, by the formula 4πε0ħcα = e2. Being a dimensionless
quantity, it has the same numerical value in all systems of units. Arnold
Sommerfeld introduced the fine-structure constant in 1916.
Definition Some equivalent definitions of α in
terms of other fundamental physical constants are:
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;
µ0 is the magnetic constant or permeability of free space;
ke is the Coulomb constant; Z is the atomic number;
RK is the von Klitzing constant. The definition reflects the relationship
between α and the electromagnetic coupling constant e, which equals
√4παε0ħc. = In non-SI units=
In electrostatic cgs units, the unit of electric charge, the statcoulomb, is
defined so that the Coulomb constant, ke, or the permittivity factor, 4πε0, is
1 and dimensionless. Then the expression of the fine-structure constant, as
commonly found in older physics literature, becomes
In natural units, commonly used in high energy physics, where ε0 = c = ħ = 1,
the value of the fine-structure constant is
As such, the fine-structure constant is just another, albeit dimensionless,
quantity determining the elementary charge: e = √4πα ≈ 0.30282212 in terms
of such a natural unit of charge. Measurement
The 2014 CODATA recommended value of α is
This has a relative standard uncertainty of 0.32 parts per billion. For reasons
of convenience, historically the value of the reciprocal of the fine-structure
constant is often specified. The 2014 CODATA recommended value is given by
While the value of α can be estimated from the values of the constants
appearing in any of its definitions, the theory of quantum electrodynamics
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 and the
fine-structure constant α. The most precise value of α obtained
experimentally is based on a measurement of g using a one-electron so-called
"quantum cyclotron" apparatus, together with a calculation via the theory of QED
that involved 12,672 tenth-order Feynman diagrams:
This measurement of α has a precision of 0.25 parts per billion. This value and
uncertainty are about the same as the latest experimental results.
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 The ratio of two energies: the energy
needed to overcome the electrostatic repulsion between two electrons a
distance of d apart, and the energy of a single photon of wavelength :
The ratio of the velocity of the electron in the first circular orbit of
the Bohr model of the atom to the speed of light in vacuum. This is Sommerfeld's
original physical interpretation. Then the square of α is the ratio between the
Hartree energy and the electron rest mass.
The two ratios of three characteristic lengths: the classical electron radius ,
the Compton wavelength of the electron , and the Bohr radius :
In quantum electrodynamics, α is directly related to 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.
Given two hypothetical point particles each of Planck mass and elementary
charge, separated by any distance, α is the ratio of their electrostatic
repulsive force to their gravitational attractive force.
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, Z0 = µ0c, and the conductance quantum, G0 = 2e2/h:
Fine structure constant gives the maximum positive charge of the central
nucleus that will allow a stable electron-orbit around it. For electron
around the nucleus, mv2/r = 1/4πε0. 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 reciprocal of fine structure constant 137.
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=
According to the theory of the renormalization group, the value of the
fine-structure constant 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
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 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. Is the fine-structure constant actually
constant? 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. 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.
= 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. 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 0.1, stellar fusion
would be impossible and no place in the universe would be warm enough for life
as we know it. 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. 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. The fine-structure constant so intrigued
physicist Wolfgang Pauli that he collaborated with psychiatrist Carl Jung
in a quest to understand its significance. Similarly, Max Born
believed if the value of alpha were any different, the universe would be
degenerate, and thus that 1/137 was a law of nature.
Richard Feynman, one of the originators and early developers of the theory of
quantum electrodynamics, 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. 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!
Conversely, statistician I. J. Good argued that a numerological explanation
would only be acceptable if it came from a more fundamental theory that also
provided a Platonic explanation of the value.
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 community. 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. See also
Hyperfine structure Electric constant
Gravitational coupling constant Dimensionless physical constant
Planck's constant Speed of light
References External links
Stephen L. Adler, "Theories of the Fine Structure Constant α"
FERMILAB-PUB-72/059-T "Introduction to the constants for
nonexperts", adapted from the Encyclopædia Britannica, 15th ed.
Disseminated by the NIST web page. CODATA recommended value of α, as of
2006. Quotes About Fine Structure Constant
"Fine Structure Constant", Eric Weisstein's World of Physics website.
John D. Barrow, and John K. Webb, "Inconstant Constants", Scientific
American, June 2005. Eaves, Laurence. "The Fine Structure
Constant". Sixty Symbols. Brady Haran for the University of Nottingham.