Ionization occurs when an electron is supplied with enough energy that allows it to severe its bond to the nucleus. Usually, this occurs at high enough temperatures. However, for degenerate gas, it can also happen when the pressure is high enough so the Fermi energy exceeds the ionization energy $ \varepsilon_f \ge u_i $. According to the uncertainty principle

$ \Delta x \Delta p \ge h $

where $ h $ is the Planck constant. The uncertainty is approximated as the average distance between the particles $ \Delta x \approx n^{-1/3} $. The uncertainty in momentum is approximated as the Fermi momentum $ \Delta p \approx p_f $. Hence,

$ p_f = h n^{1/3} $.

The Fermi energy is therefore given by

$ e_f \approx \frac{p_f^2}{m} $

where $ m $ is the mass of the electrons. The pressure is approximately given by

$ p_f \approx e_f n \approx \left( \frac{e_f^5 m^3}{h^6} \right)^{1/2} $

For hydrogen, this predicts that ionization occurs at a pressure of 0.2 Mbar.