inf i-1
__ __
\ \
dni/dt = -ne n1 Si - ne ni / Cij - ni / Aij
-- --
j<>i j=1
inf inf (8)
__ __
\ \
+ ne / nj Cji + / nj Aji + { Bi + ai ne } ne n+
-- --
j<>i j=i+1
There is such an equation for each and every discrete level i = 1, 2, 3,..inf.
Thus, we obtain an infinite number of coupled differential equations describing
the population densities of all the discrete levels.We can reduce the problem to a finite set of levels by taking into account that excited states with large quantum numbers correspond to physicaly larger wavefunctions. Thus ions with arbitrarily large quantum numbers are in principle not possible in a relatively dense plasma. Each ion has only enough room within the crowded plasma for excited states up to a certain level, beyond which the outer electron is so perturbed by neighboring ions that it is no longer significantly influenced by the weak attration of the parent ion, and is considered part of the continuum (ionized).
For large quantum number, collision processes play an essential role, because atoms in very high quantum states have collisional cross-sections higher by many powers of ten than the lowest states with small orbital radii of the electrons. The wavefunctions of highly excited quantum states are larger and more weakly bound than the more tightly bound lower states, free electrons have a greater influence on them. Levels for which the collisional rates dominate over the radiative rates can, to a first approximation, be considered in local thermodynamic equilibrium, greatly reducing the number of levels to solve in equation (8).
The QSS approximation: ...
After some algebra we obtain two easily solved 32 X 32 matrix equation:
r
__
\
/ Bij rj(k) = Ri(k) i = 2,3, ..., r and k = 0 or 1
--
j=1
Bij = ne Cij + (Zj/Zi)Aji for i < j
Bij = ne Cij for i > j
s i-1
__ __
\ \
Bij =-ne Si - ne / Cij - / Aij for i = j
-- --
j< > i j=1
s
__
\
Ri(0)= / { ne Cij - (Zj/Zi)Aji } - (Bi + ai ne)/Zi
--
j> r
Ri(1)= -ne Ci1
The partition function for each level i can be evaluated as :
Zi= (gi/2g+) (h^2/2 pi me kTe)^(3/2) exp( Xi / kTe)
gi = statistical weight of level i
g+ = statistical weight of ion level
Xi = ionization potential of level i
me = electron mass
pi = 3.1415926535
Te = electron temperature
k = Boltzmann's constant
h = Planck constant
Once the matrix Bij is inverted, the problem is basically solved and the
population density can be expressed as follows
ni = Zi [ ri(0) n+ ne + ri(1) n1/Z1 ]The only sticky detail in the simulation is the evaluations as a function of electron density and electron temperature of the collisional radiative rates Aij, Cij, ai, Bi and Si which must be calculated from quantum mechanical approximations because present day atomic databases deal with very few of the rates required in this simulation. Recently there have been attempts to provide electronic access to this data.
Don't let the sophistication of the collisional-radiative model intimidate you; it has proven to accurately reproduce the behavior of laboratory experiments on recombining plasma lasers.