Quasiclassical Kinetics of Highly Excited Atomic States Population in Laboratory and Astrophysical Plasmas

M.G. Levashova, V.S. Lisitsa
NFI RRC “Kurchatov Institute”, Kurchatov sq. 1, Moscow, 123182, Russia
Abstract. The results of two-dimensional (in energy and angular momentum) numerical modeling of the steady-state
classical radiative-collisional kinetics of highly excited atomic electrons in plasma are presented. The highly excited state
electron is subjected to collisions with plasma particles and radiative cascade to the lower atomic states, with the atomic
states being populated by various sources. The results obtained here for photorecombination source of population are
shown to be in agreement with the limiting case of purely radiative cascade between lower atomic states where the
collisions with plasma particles are not essential.
Keywords: Kinetic Equation, Populations, Transition Probabilities, Radiative Transitions, Quantum Numbers,
Radiative Cascade
PACS: 03.65.Sq, 02.60.Cb
1. INTRODUCTION
The calculation of radiative cascade between highly exited atomic states is required by various physical
applications. Examples include calculations of the populations and line intensities of hydrogen and ionized He II in
interstellar gas plasmas; spectral line calculations for highly stripped ions in hot rarefied plasmas; level population
calculations for atoms excited by stepwise laser transitions, atoms in edge tokamak plasma etc.
Quantum mechanical calculations of atomic level population require a direct solution of a system of kinetic
equations. It becomes rather complicated for large principal quantum number values n, of the order of 102 or larger.
In reality the quantity of matrix elements for a particular radiative transition between Rydberg atomic states is of the
order of n2 x n’2 = 104 x 104 = 108 for n → n’ transitions between atomic levels with n, n’ ≈ 102. This quantity must
be multiplied by the n2
numbers of the matrix elements responsible for population sources. Consequently, the direct
calculation becomes tedious even in the case of hydrogen-like atomic states where all matrix elements are well
known. On the other hand, for n >> 1 and l >> 1 the semi-classical method should describe the radiative transition
probabilities and the bound electronic states quite well, as soon as such an approach was confirmed by agreement
with quantum mechanical calculations [4].
So, the classical approach is more preferable here since the problem reduces to finding a solution of only one
differential equation and gives a transparent picture of the system behavior.
General method of classical description of radiative cascade between Rydberg atomic states with allowance for
collisions with particles in plasma was employed in [1]. It is describing the 3D populations (in energy, angular
momentum and its projection) of highly excited states of plasma ions for three-body recombination source of
population from the continuum.
Purely radiative cascade was analytically described in [2] (both in classical and quasi-classical approaches) for
arbitrary source of population.
We present in this paper the numerical modeling of the steady-state classical radiative-collisional kinetics for
photorecombination source of population. For photorecombination source, the radiative transitions determine both
the population source and the radiative cascade.
2. CLASSICAL KINETIC EQUATION
Classical kinetic equation for three-body recombination source of population from the continuum was obtained in
[1].
This equation includes:
1. radiative cascade (energy and angular momentum loss due to radiation emission by highly excited, bound electron
in the field of ion in the approximation of classical electrodynamics):
2. pair distant Coulomb collisions of free electrons in plasmas (Landau kinetic equation, i.e. neglect of electron’s
trapping in the atom/ion):
3. sources of population: three-body recombination source corresponds to setting of boundary condition at energy
E=0, other external sources – q ( e.g. photorecombination).
We consider here only the steady-state case with photorecombination source of atom (or ion) levels population.

FIGURE 4. Electron distribution function in atom/ion with account of Coulomb collisions of electrons for photorecombination
source of atomic levels population. Ne ~ 104 cm-3, Te ~ 1 eV.
6. CONCLUSION
This work gives a numerical value of electron distribution function in an atom or ion with account of collisions
with plasma particles employing semi-classical approach.
The electron distribution function is a result of two process “competition”: collisions of free electrons in plasmas
and radiation emission. As a result of this “competition” the distribution function experiences distortion depending
on the process dominating at the given conditions.
REFERENCES
1. Belyaev S.T., Budker G.I., In the book: Plasma Physics and Nuclear Fusion Research, under the editorship of M.A.
Leontovich, Moscow: Academy of Sciences of the USSR publisher, 1958 vol.3, p.46.
2. Kukushkin A.B., Lisitsa V.S., Sov. Phys. JETP, 61 (1985) 937.
3. J. Adams, “MUDPACK: Multigrid Fortran Software for the Efficient Solution of Linear Elliptic Partial Differential
Equations,” Applied Math. and Comput. vol.34, Nov 1989, pp.113-146.
4. V.I. Kogan, A.B. Kukushkin, Zh. Eksp. Teor. Fiz. 87 1164 (1984) [Sov. Phys. JETP 60, 665 (1984)]
5. M.A. Gordon, R.L. Sorochenko, Radio Recombination Lines: Their Physics and Astronomical Applications, Kluwer
Academic Publishers, 2002.

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