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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.

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

The essential ingredient in describing ionization equilibrium is the chemical potentials for the various species. The chemical potential depends on the statistics: Boltzmann, Fermi-Dirac, or Bose-Einstein. Taking the chemical potentials to be those of ideal Boltzmann gases, we get the result called the Saha equation.

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.

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The Saha equation giving the ratio of ionized hydrogen to neutral hydrogen holds for weakly ionized plasmas which is probably the case for large parts of the chromosphere. An ionized hydrogen atom is a proton which is a fermion.There is, however, a small percentage of deuterium in natural hydrogen which has a deuteron (proton+neutron) as nucleus. Thus an ionized deuterium atom is a deuteron which has spin 1 and therefore is a boson.

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