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Re: Plasma



I just noticed that John Denker asked another question I overlooked in
my last reply:

Meanwhile, there is also a whole lot of plasma
out there that is not very hot. Yet it is ionized
too. That was the point of my previous note.
This can be readily understood in terms of the
Saha equation (and the ideas behind it) but
not otherwise AFAIK.

It is a challenge to answer this question in conceptual terms, but I'll
try. I would really like to invoke a "golden rule", but that's not
elementary, so I'll just wave my hands a bit through that part. I think
it is worth trying, however, and so here goes. I have prefaced various
necessary qualifications below with "N.B." You may skip these in your
first read through.

In any box of hydrogen in thermodynamic equilibrium at a given
temperature and pressure there is always a number of ions present. As I
pointed out in an earlier posting, two types of reactions are going on
at all times. While several (actually many) other processes are
possible, I list below some of the more common ones:

ionization ( 2H -> p + e + H )
or ( 2H -> p + e + Y + H )

and

recombination ( p + e -> H + Y )
or ( p + e + H -> 2H + Y )

H is a hydrogen atom, p is a proton, e is an electron, and Y is a
cheesy gamma; it represents a photon.

At equilibrium the concentrations (numbers of particles per unit
volume) of all three species are constant. I shall represent these
concentrations by nH, np, and ne, and henceforth use these parameters
instead of pressure.

N.B. Pressure can be got from the temperature and concentrations using
the ideal gas law, adding the contributions from all species present.
(John agrees that this is close enough for astronomy and astrophysics.)

N.B. I have now swept under the rug a fourth species that will be
present at sufficiently low temperatures and high pressures, hydrogen
molecules. In the full mathematical treatment including these does not
make for complication; doing so in this elementary treatment will make
it unnecessarily complicated. Molecular hydrogen is rare in the ISM
(interstellar medium) and it is unknown in the IGM.

N.B. ne = np

In equilibrium the frequency of ionizations is equal to the frequency
of recombinations. The degree of ionization in our box of gas depends
upon both temperature and particle concentration. It is this latter
dependence that probably bothers John just as it usually bothers
students. While it is easy to see it explicitly in the Saha equation,
its simple physical origin is somewhat subtle. That is what I will
discuss here.

Consider the first noted process of recombination in hydrogen plasma.
An electron and a proton collide, and a neutral hydrogen atom (perhaps
in an excited state) is one of the products. Consider this collision in
the center of mass frame of reference. The total momentum of the
reactants is zero, so the total momentum of the products must be zero.
Energy must be conserved, so another product of recombination must be
an energy component equal to the sum of the reactant energies and the
energy by which the electron is bound to the proton. That is why
another particle, the photon, must be produced in recombination. If
other considerations do not interfere, this energy could appear as a
small kinetic energy of the atom and an oppositely directed photon
having the greatest part of the energy.

N.B. Two or more photons might also be produced, but this is a much
less probable outcome than the one photon process. That's one of the
places I wave my hands instead of invoking the golden rule.

Now consider the second noted recombination process. This requires a
three particle collision, but because the second hydrogen atom relaxes
the constraints on energy and momentum conservation (feel a breeze?)
this is actually a very important process contributing to recombination
at higher particle concentrations.

Now return to the matter of the relative frequencies of these
processes. In each process, independent of any other factors, the
number of reactions per unit volume per unit time is proportional to
the number of encounters in which all reactants are simultaneously
involved per unit volume per unit time. As I noted before, at
equilibrium the sum of the frequencies of all ionization processes is
exactly equal to the sum of the frequencies of all recombination
processes.

Here's where the frog jumps into the pond: for each atom per unit
volume the number of encounters with other atoms is proportional to the
number of atoms per unit volume. Therefore the frequency of ionization
processes by 2H processes per unit volume is proportional to (nH)^2.

N.B. This is not a hand wave. If you didn't follow that please reread
and think about it.

N.B. Higher order ionization processes (e.g. 3H) are being neglected
here.

The frequency of recombination per unit volume is the sum of the
frequencies for all recombination processes. For the two processes
listed above, the frequencies are proportional, respectively, to
(ne x np) and (ne x np x nH). Recall that the second process is
important.

Now consider what happens if the volume of the box is suddenly doubled.
("Suddenly" means that the wall or walls of the box move faster than
any of the particles during the expansion.) The kinetic energies of the
particles in the box, and therefore the temperature of the system,
remains constant. The particle concentrations, however, are all halved.
Initially the frequency of ionization decreases by a factor of 4. The
frequency of the first recombination process likewise decreases by a
factor of 4, but the frequency of the second process decreases by a
factor of 8. The overall frequency of recombination is initially lower
than than the frequency of ionization; the system is no longer in
equilibrium. Ionizations will continue to be more frequent than
recombinations until equilibrium is reestablished at a higher degree of
ionization.

That's about it. For a given temperature, the lower the density in a
gas, the higher the degree of ionization.

I hope that wasn't too long. It took me more than a couple of hours to
write it.

Leigh