22P088-R
Electronic State Density and Average Ionization Based on the Plasma Microfield
プラズママイクロフィールドに基づく電子状態数密度と電離度	 
Takeshi Nishikawa
西川	  亘Graduate School of Natural Science and Technology, Okayama University
3-1-1, Tsushimanaka, Kita-ku, Okayama 700-8530, Japan
岡山大学大学院自然科学研究科	  〒700-8530	  岡山市北区津島中3-1-1工学部3号館
In addition to numbers of bound state densities, free state density in a plasma is computed by the atomicmodel based on the plasma microfield. In the model, we assume that ions are immersed in the statistically
distributed electric microfield in a plasma by Holtsmark and the large electric field components of which
are due to a binary collision of two atoms. Continuous changes on state densities from bound to freestates for various number densities of ions, and contour of average ionization degree of hydrogenic plasma
are shown as typical results.

1. Introduction
Numerical computations of average ionization
degree are old, but still a difficult problem onplasma physics. Once we choose a set of thestatistical weight of bound state and the state
density of free state, the average ionization
degree seems to be able to be computed simply by
the Saha-Boltzmann relation⎛ I ⎞ 2N Z +1 N e U Z +1=exp⎜− p ⎟ 2NZUZ⎝ k BTe ⎠ π=€⎛∞⎞∫ E exp⎜⎝− k ET ⎟⎠dE0⎛ I ⎞ ⎛ k T ⎞
U Z +1exp⎜− p ⎟2⎜ B e ⎟UZ⎝ k BTe ⎠ ⎝ 2π ⎠

B e

3/2

(1)

where N Z and N e are the number density ofZ-charged ion and free electron, respectively.
U Z and Ip are the number of state and theionization potential of Z-charged ion, respectively.
Te is the electron temperature. Atomic units areused through out in this article.
For hydrogenic case, number of bound state ofwhich principal quantum number is n is 2n2 andstate density of free state is proportional to thesquare root of energy of free electron. The
problem is which maximum principal quantum
number of the bound states has to be used in thecomputation. It is not simple. In Fig. 1, theproblem is described using the simplesthydrogenic case.
All lines are contours of average ionization
degree of 0.5, but maximum principal quantum
number taken into account in the computations
are different; maximum principal quantum
numbers are up to n = 5, 10, 30, respectively.
Generally, average ionization degree becomes
smaller as the density becomes large, since freestate density becomes relatively smaller to boundstate. For low temperature cases, three curves
converge into one. In the low temperature,almost all bound electrons exist in the groundstate so that number of bound state does not affectin the average ionization degree computation.
On the other hand, three curves spread out into
two-order of magnitude range width in ion
number density as the temperature becomes
higher. In the high temperature, number ofbound electron in excited states cannot be
neglected
inaverageionization
degreecomputation. As a result, three curves of theaverage ionization degree of 0.5 separate into.
This shows the importance of choice of themaximum principal quantum number of boundstate in the atomic process modeling.

Fig. 1. Contour of average ionization degree equal to 0.5
for various choice of maximum principal quantum
numbers of n = 5, 10, and 30 in the atomic processmodeling.

2. Electronic State density by the plasma
microfield and average ionization degree
Modeling on the bound state based in theplasma microfield by Holtsmark[2] had alreadybeen developed[1]. In the model, numbers ofbound states are expressed simple analyticalexpression. In addition to the bound state, wehave tried to make a simple expression of state
density of free electron based on the plasma
microfield. In the model, we rewrite the relation
between the electron’s energy and the number ofstate density of

p2 = E +€€Z− Fr cosθr(2)where F is the electric field at position (r, θ )inside the ion sphere R0 defined by
34 πR0 N i / 3 = 1. N i is the number density of ions.
The state densities are used those from quantum
mechanics, while the potentials are from classicalelectro-magnetic theory.
Sorry to say, thecomputed free state density by the simple state
density
has
been
shown
inappropriate.
Especially, the inappropriate overestimation infree state density has occurred in relatively large
electric field. The values of such electric field
are larger than what arises from the nearest
neighbor atom set inside the ion sphere. In orderto remove the difficulty, we assume that the large
electric field components are due to binary
collisions of two atoms. Therefore, the potentialfrom the electric field has been computed as thebinary symmetry on the saddle point as if two
nucleus exist inside the ion sphere. Using theimprovement, physically appropriate state densities
of free has been able to be obtained. In Fig. 2, thestate densities of free and bound states for various
number densities of ions are shown. In order toshow the continuous change of state densities frombound to free state, the bound states are shown asboxes of which integral values have 2n2. The
boxes have spread over between intermediate
energy levels of nearest larger and smaller levels.
For low ion number density of Ni = 1016 cm-3,
number of bound state almost fully exists up to n =8 and gradually disappears as the energy of the state
becomes larger. Alternatively, free state appear
even the energy of electron is less than zero. As
the number density of ion becomes larger, free state
density becomes smaller as proportional to its
number density and disappearance of the boundstates density begins at the lower bound state.

Fig. 2 Numbers of bound state densities and states
densities of free states for various ion number densities.

Using the number of bound state and free state
density as a function of electron energy describedabove, average ionization degree can be computed byeq. (1).
Equation (1) only shows the relation
between the ground states of different charge,
contribution of the excited states has also been
included in the computation. In Fig. 3, contour of
average ionization degree of hydrogenic plasmas areshown. Solid curve shows the average ionization
degree of 0.5.
At the low-temperature andlow-density case, the average ionization degree agrees
with the result in Fig. 1. Even in the low temperature,
average ionization degree becomes larger as the ion
density becomes large due to the effect of thedisappearance of the bound states. This effect has
been called ‘Pressure ionization’. The present model
naturally can introduce the effect.

Fig. 3 Average ionization degree of hydrogenic plasma.
References[1] T. Nishikawa: ApJ 532 (2000) 670.
[2] H. Griem: Spectral Line Broadening by plasmas(Academic Press, New York, 1974), Chap. II, p.17.