# Is an ionized hydrogen a boson or fermion? Does that mean the Saha equation is capable of describing a compound of both bosons and fermions?

My teacher asked me to calculate the ratio of hydrogen atoms in the chromosphere $(T=6000K)$ at $n1, n2, etc$. energy level, so I think I'll have to use the Saha equation. Is my intuition is correct?

Is an ionized hydrogen a boson or fermion? Is the Saha equation capable of describing a compound of both bosons and fermions, and therefore is a generalization of both F-D and B-E distribution? Thanks.

• What is the Fermi temperature for the chromosphere? At the given thermodynamical temperature, is it relevant whether the particles are fermions or bosons? – Robin Ekman Oct 23 '16 at 14:19
• Oh you are right! Oh man I'm so stupid. Thank you :D – J. Doe Oct 23 '16 at 14:37
• @RobinEkman Perhaps that could be an answer, rather than a comment. – rob Oct 23 '16 at 14:51

The proton (ionized hydrogen) and the electron are both spin-$\frac 1 2$, thus fermions. From the rules for adding spins, the total spin of atomic hydrogen is either $0$ or $1$, and therefore atomic hydrogen is a boson.
Now, the Fermi temperature for electrons at the densities in the chromosphere is at least a factor $\approx 100$ lower than the thermodynamic temperature given, which means it doesn't matter if we use Fermi-Dirac or Boltzmann statistics. The Fermi temperature is inversely proportional to the mass, so for protons and atomic hydrogen, using Boltzmann statistics is a forteriori justified. Thus the Saha equation is applicable.
There certainly are plasma regimes where electron degeneracy can be important, i.e. $T \lesssim T_F$, but then the density is something like six orders of magnitude higher than in the chromosphere.