Gamow factor

Remarks

The Gamow factor approximates the probability of tunnelling through the Coulomb barrier when $E$ is much smaller than the height of the barrier. With the relative speed $v$ and $E = m_r v^2 / 2$, it equals $e^{-2 \pi \eta}$, where $\eta = Z_a Z_b \alpha c / v$ is the Sommerfeld parameter; indeed $(2 \pi \eta)^2 = 2 m_r c^2 (\pi \alpha Z_a Z_b)^2 / E$. Example: for two protons, $m_r$ is half the proton mass, so $m_r c^2 \approx 469 \, \mathrm{MeV}$ and $E_G \approx 2 \cdot 469 \cdot (\pi / 137)^2 \, \mathrm{MeV} \approx 0.49 \, \mathrm{MeV}$; at $E = 1.3 \, \mathrm{keV}$, typical in the core of the Sun, $P_G \approx e^{-19.5}$, about $3.5 \times 10^{-9}$. Non-example: a neutron and a nucleus have $Z_a Z_b = 0$, which the definition excludes; there is no Coulomb barrier, and no Gamow factor applies. Multiplied by the Maxwell–Boltzmann weight $e^{-E / (k_B T)}$, the Gamow factor gives the Gamow peak, whose maximum lies at $E_0 = (E_G (k_B T)^2 / 4)^{1/3}$.