4. Reaction equilibria
4.1 The Saha equation
If particles and radiation are in equilibrium with themselves and each other, we call their state
Thermodynamic Equilibrium, and the spectra are given by a Planckian for the radiation and
a Fermi distribution (a Maxwellian if the density is not too high) for the particles. If the
particles are only in equilibrium with themselves, but not with the radiation, we speak of Local
Thermodynamic Equilibrium (LTE), in which the radiation spectrum is not a priori known,
but the particle spectrum is still a Fermi, resp. a Maxwellian, distribution.
In Local Thermodynamic Equilibrium the distribution of atoms with ionization state
i
over the
various energy states
m
is proportional to exp(
−
χ
i,m
/kT
), where
χ
i,m
is the excitation energy
of state
m
relative to the ground state. Then
N
i,m
N
i,
1
=
g
i,m
g
i,
1
exp(
−
χ
i,m
kT
)
(4
.
1)
where
g
i,m
are the statistical weights of state (
i, m
), that is the number of independent ar-
rangements of the atomic electrons that give the same ionization state and energy level. After
summing over all
m
we ±nd
N
i
=
s
m
N
i,m
=
N
i,
1
g
i,
1
s
m
N
i,m
g
i,m
exp(
−
χ
i,m
kT
) =
N
i,
1
g
i,
1
u
i
(
T
)
(4
.
2)
⇒
N
i,m
N
i
=
g
i,m
u
i
(
T
)
exp(
−
χ
i,m
kT
)
where
u
i
(
T
) =
s
m
g
i,m
exp(
−
χ
i,m
kT
)
(4
.
3)
is called the
level partition function
. We can derive the
Saha
equation that, using the level
partition sums, can be written as an equation for the ionization equilibrium.
N