Saha Equation

The Saha equation can be expressed as

Nq+1Ne / Nq  = [(2πmekT)3/2/h3][2uq+1(T)/uq(T)]exp(-Iq / kT),

where

Nq+1 = total number of particles per unit volume in the q+1th ionization state of the element under consideration, (Nq+1 Σ r Nq+1,r),
Nq = total number of particles per unit volume in the qth ionization state of the element under consideration, (Nq Σ r Nq,r),
Ne = total number of free electrons per unit volume,
uq+1(T) = Σ r gq+1,r exp(-χq+1,r / kT) is the partition function for the q+1th ionization state, the rth term of which is proportional to the number
of particles (per unit volume) in the rth level of excitation,

uq(T) = Σ r gq,r exp(-χq,r / kT) is the partition function for the qth ionization state, the rth term of which is proportional to the number of
particles (per unit volume) in the rth level of excitation,

gq,r = the stistical weight of the rth excited level of the qth ionization state,
χq,r = the excitation energy of the rth excited level with respect to the ground level of the qth ionization state,
Iq = the ionization energy of the q+1th ionization state with respect to the ground state of the qth ionization state,
me = the electron mass = 9.1093897
´ 10-28g,
h = Planck's constant = 6.6260755
´ 10-27erg s,

k = Boltzmann's constant = 1.380622 ´ 10-16 erg/K = 8.61779 ´ 10-5 eV/K, and
T = the temperature expressed in Kelvins.

Note that the Saha equation can also be written

Nq+1Pe / Nq  = [(2πme)3/2/h3](kT)5/2[2uq+1(T)/uq(T)]exp(-Iq / kT),

where Pe = NekT is the electron pressure in dyn cm-2.

The logarithmic form of the Saha Equation is

log (Nq+1 / Nq) = -θIq [eV] + 2.5 log T - 0.48 + log [2uq+1(T)/uq(T)] - log Pe,

where θ º 5040/T.