Gamow factor

The Gamow Factor or Gamow-Sommerfeld Factor,^[1] named after its discoverer George Gamow, 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, but 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,

$P_{g}(E)\equiv e^{{-{\frac {E_{g}}{E}}^{{1/2}}}}$^[2]

Where $E_{g}$ is the Gamow Energy.

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

Here, $m_{r}$ is the reduced mass of the two particles.

$m_{r}={\frac {m_{1}m_{2}}{m_{1}+m_{2}}}$

The constant $\alpha$ is the fine structure constant, and the constants $Z_{a}$ and $Z_{b}$ are the respective atomic numbers of each particle.

The probability of overcoming the barrier increases rapidly with increasing particle energy, but at a given temperature the probability of a particle having a high energy falls off rapidly, following the Maxwell–Boltzmann distribution. Gamow found that, taken together, these effects mean that for any given temperature, the particles that actually fuse are mostly in a temperature-dependent narrow range of energies known as the Gamow window.^[3]

References

1. Yoon, Jin-Hee; Wong, Cheuk-Yin (February 9, 2008). "Relativistic Modification of the Gamow Factor". arXiv:nucl-th/9908079.
2. "Nuclear reactions in stars" (PDF). Dept. Physics & Astronomy University College London.
3. "Temperature and Pressure in Stars". Dept. Physics & Astronomy, University of Tennessee.