A Comparison between Debye-huckel Screening and the Stark Effect on the Determination of the Last Observable Spectral Line from a Hydrogen-like Plasma

Spectral linesemitted by transitions from levels tion of screening length D was determined as follows. near the continuum may disappear within a plasma Using the variational method described in ref. [4], and because of two effects: (I) the broadening of the lines, recalling that, for a given n, the state with the smallest causing closely lying states to merge together and (2) 1 has the largest binding energy, we have calculated the the displacement of the higher energy levels into the value of the critical screening length below which each continuum due to the fact that in a plasma the bound of the nine lowest s states disappears. These values electron moves in a screened potential rather than a were then used to determine the relation between ~D pure Coulomb potential. The line merging effect has and D. We found the relation been treated by Inglis and Teller [1] and others [2]. = 1.13 /b7~= /1 .27D/a 0 = /D/0.79a0, (1) The displacement of the upper energy levels into the contiuum has been considered by various authors [3, to be accurate to about 3%, where a0 is the Bohr radi4]. Since these two effects are both present in a plasus. In particular, we note that according to eq. (1) for ma, they must be considered simultaneously. This D < 0.79a0 there is no bound state,whereas our calwas first accomplished by Margenau and Lewis [5] culated variational result isD <0.84a0. Eq. (1) is in who reviewed the calculations of the maximum quanclose agreement with a similar result obtiained in ref. turn number predicted by each of the two effects and [6]. Since [7] then interpreted as the decisive effect that which preD = 6 9 (T/2N )i/ 2cm (2) dicted the smallest of the two maximum quantum numbers, appropriate to a specific density and temper. for a two component plasma of electrons and singly ature. charged ions (N 10~= Ne, where N denotes the number Here we present two improvements to the initial density) we have work of ref. [5]. First we obtain an improved value n = 2 97 (T/2N a~) 1J4. (3) of the greatest possible “principal” quantum numb r D e of a screened potential in the light of our variational Thus, results [4]. Then we combine this result with the results of Inglis and Teller to obtain analytically, for a /38.9 T\ 4 log ~D = log I I — logN (4) specified value of the plasma density, the critical tern\ a~ ,~ perature marking the boundary between the two different regimes where each of the effects dominate. Thelast deiscernable level for the two component The quantum number ~D of the last bound state singly charged plasma due to the line merging caused of a particle in the Debye-Huckel potential as a funcby the Stark effect has a quantum number n 5 deter-