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.
