Re: ionization temperature

[David Bowman <dbowman@tiger.gtc.georgetown.ky.us>]

Dan Wilkins wrote:

>        Got a question on Cosmology.  All the books say decoupling of
> radiation from matter took place quite suddenly when the expanding universe
> cooled below 3,000-4,000K;  at this point the protons captured the
> electrons,  the hydrogen ceased to be ionized, and neutral hydrogen is much
> more transparent.  How can this be?  kT at 4,000K is only 1/3 eV.  With
> that little energy, so far below 13.6 eV of the hydrogen ground state, how
> could collisions ionize a substantial part of the Hydrogen?  If we use the
> maxwell-boltzmann relation, only a fraction exp (-13.6/0.333)
> = 1.8x10^(-18) of the H-atoms should be ionized, an infinitesimal fraction.
> It's hard to believe that only 10^5 free electrons in a mole of atoms
> could make the gas opaque to E+M radiation.

Your calculation above only gives the relative probability for occupancy for
*one single* barely ionized state.  The ionized states form a continuum.  To
get (a crude estimate of) the probability of a given atom to be ionized you,
essentially, need to multiply this factor by the number of possible final
ionized states with this near-threshold energy.  The density of ionized
states is limited by the finite (nonzero) concentration of protons and
electrons in the environment and by the Pauli Principle.  (In the zero
density limit of a single atom in an infinite universe we have an infinite
number of possible final states and the atom would be spontaneously ionized
in equilibrium at *any* positive absolute temperature no matter how cold.
But, in this case, the expected lifetime of the atomic ground state would be
*very* long at low temperatures.)

David Bowman
dbowman@gtc.georgetown.ky.us