The Sommerfeld parameter , named after

Arnold Sommerfeld Arnold Johannes Wilhelm Sommerfeld, (; 5 December 1868 – 26 April 1951) was a German theoretical physicist who pioneered developments in atomic and quantum physics, and also educated and mentored many students for the new era of theoretica ...

, is a

dimensionless quantity A dimensionless quantity (also known as a bare quantity, pure quantity, or scalar quantity as well as quantity of dimension one) is a quantity to which no physical dimension is assigned, with a corresponding SI unit of measurement of one (or 1) ...

used in

nuclear astrophysics Nuclear astrophysics is an interdisciplinary part of both nuclear physics and astrophysics, involving close collaboration among researchers in various subfields of each of these fields. This includes, notably, nuclear reactions and their rates as ...

in the calculation of reaction rates between two nuclei and also appears in the definition of the astrophysical

S-factor In nuclear physics, the astrophysical S-factor is a rescaling of a nuclear reaction's total cross section to account for the Coulomb repulsion between the charged reactants. It determines the rates of nuclear fusion reactions that occur in the cor ...

. It is defined as :

\eta = \frac = \alpha Z_1 Z_2 \sqrt

, where is the

elementary charge The elementary charge, usually denoted by is the electric charge carried by a single proton or, equivalently, the magnitude of the negative electric charge carried by a single electron, which has charge −1 . This elementary charge is a fundame ...

, and are the

atomic number The atomic number or nuclear charge number (symbol ''Z'') of a chemical element is the charge number of an atomic nucleus. For ordinary nuclei, this is equal to the proton number (''n''p) or the number of protons found in the nucleus of every ...

s of two interacting nuclides, is the magnitude of the relative incident

velocity Velocity is the directional speed of an object in motion as an indication of its rate of change in position as observed from a particular frame of reference and as measured by a particular standard of time (e.g. northbound). Velocity is a ...

in the center-of-mass frame, is the unitless

fine-structure constant In physics, the fine-structure constant, also known as the Sommerfeld constant, commonly denoted by (the Greek letter ''alpha''), is a fundamental physical constant which quantifies the strength of the electromagnetic interaction between ele ...

, is the speed of light, and is the

reduced mass In physics, the reduced mass is the "effective" Mass#Inertial mass, inertial mass appearing in the two-body problem of Newtonian mechanics. It is a quantity which allows the two-body problem to be solved as if it were a one-body problem. Note, how ...

of the two nuclides of interest. One of its best-known applications is in the exponent of the

Gamow factor The Gamow factor, Sommerfeld factor or Gamow–Sommerfeld factor, named after its discoverer George Gamow or after Arnold Sommerfeld, is a probability factor for two nuclear particles' chance of overcoming the Coulomb barrier in order to undergo nu ...

(also known as the penetrability factor), :

P = \exp(-2 \pi \eta)

, which is the probability of an

s-wave __NOTOC__ In seismology and other areas involving elastic waves, S waves, secondary waves, or shear waves (sometimes called elastic S waves) are a type of elastic wave and are one of the two main types of elastic body waves, so named because th ...

nuclide to penetrate the

Coulomb barrier The Coulomb barrier, named after Coulomb's law, which is in turn named after physicist Charles-Augustin de Coulomb, is the energy barrier due to electrostatic interaction that two nuclei need to overcome so they can get close enough to undergo a ...

, according to the WKB approximation. This factor is particularly helpful in characterizing the nuclear contribution to low-energy nucleon-scattering cross-sections - namely, through the astrophysical

. One of the first articles in which the Sommerfeld parameter appeared was published in 1967.{{cite journal , last=Breit , first=G. , year=1967 , title=Virtual Coulomb Excitation in Nucleon Transfer , url=http://www.pnas.org/content/57/4/849.full.pdf , journal=Proceedings of the National Academy of Sciences of the United States of America , volume=57 , issue=4 , pages=849–855 , doi= 10.1073/pnas.57.4.849, pmid=16591541 , pmc=224623 , access-date=27 January 2015, bibcode = 1967PNAS...57..849B , doi-access=free

References

Nuclear physics Astrophysics