Sommerfeld parameter

The Sommerfeld parameter η, named after Arnold Sommerfeld, is a dimensionless quantity used in nuclear astrophysics in the calculation of reaction rates between two nuclei and also appears in the definition of the astrophysical S-factor. It is defined as[1]

${\displaystyle \eta ={\frac {Z_{1}Z_{2}e^{2}}{4\pi \epsilon _{0}\hbar v}}=\alpha Z_{1}Z_{2}{\sqrt {\frac {\mu c^{2}}{2E}}}}$,

where e is the elementary charge, Z₁ and Z₂ are the atomic numbers of two interacting nuclides, v is the magnitude of the relative incident velocity in the center-of-mass frame, α is the unitless fine-structure constant, c is the speed of light, and μ is the reduced mass of the two nuclides of interest.

One of its best-known applications is in the exponent of the Gamow factor P (also known as the penetrability factor),

${\displaystyle P=\exp(-2\pi \eta )}$,

which is the probability of an s-wave nuclide to penetrate the Coulomb barrier, 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 S-factor.

One of the first articles in which the Sommerfeld parameter appeared was published in 1967.[2]