Gamow factor

The Gamow factor, Sommerfeld factor or Gamow–Sommerfeld factor,^[1] 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 nuclear reactions, for example in nuclear fusion. By classical physics, there is almost no possibility for protons to fuse by crossing each other's Coulomb barrier at temperatures commonly observed to cause fusion, such as those found in the Sun. When George Gamow instead applied quantum mechanics to the problem, he found that there was a significant chance for the fusion due to tunneling.

The probability of two nuclear particles overcoming their electrostatic barriers is given by the following equation:^[2]

${\displaystyle P_{\text{G}}(E)=e^{-{\sqrt {{E_{\text{G}}}/{E}}}}}$

where ${\displaystyle E_{\text{G}}}$ is the Gamow energy,

${\displaystyle E_{\text{G}}\equiv 2m_{\text{r}}c^{2}(\pi \alpha Z_{\text{a}}Z_{\text{b}})^{2}}$

Here, ${\displaystyle m_{\text{r}}={\frac {m_{\text{a}}m_{\text{b}}}{m_{\text{a}}+m_{\text{b}}}}}$ is the reduced mass of the two particles. The constant ${\displaystyle \alpha }$ is the fine-structure constant, ${\displaystyle c}$ is the speed of light, and ${\displaystyle Z_{\text{a}}}$ and ${\displaystyle Z_{\text{b}}}$ are the respective atomic numbers of each particle.

While the probability of overcoming the Coulomb barrier increases rapidly with increasing particle energy, for a given temperature, the probability of a particle having such an energy falls off very fast, as described by the Maxwell–Boltzmann distribution. Gamow found that, taken together, these effects mean that for any given temperature, the particles that fuse are mostly in a temperature-dependent narrow range of energies known as the Gamow window.