negative hydrogen and helium in a variety of debye plasmas

Negative Hydrogen and Helium in a Variety of Debye Plasmas

LI ZHANG and PETER WINKLER" Department of Physics, University of Nevada, Keno, Nevada 89557: e-mail for P. W.: [email protected]

Received Februa y 29, 1996; revised manuscript received April 1, 1996; accepted April 4, 1996

ABSTRACT m Aspects of the influence of a strong Debye plasma environment on the negative hydrogen ion and the neutral helium atom have been studied. Contrary to earlier work, in the present calculation all interactions have been screened. This increases the stability of the systems to the extent that neither the H- ion nor the ground state of helium will lose an electron by pressure ionization. It has been found that the charge distribution of H- remains remarkably constant over a vast range of values of the Debye parameter D. 0 1996 John Wiley & Sons, Inc.

Introduction

n the investigation of plasmas consisting of I atomic ions and electrons a good understand- ing of the interaction potential among those species is essential. Compared to the free-space case the force between the charges is modified due to short-range order and static and dynamical screening effects introduced by neighboring ions and fast electrons, respectively. These modifications effect one-electron properties (e.g., spectral lines which serve as key quantities in plasma diagnostics) as well as properties involving more than one elec-

*To whom correspondence should be addressed.

tron (e.g., dielectronic recombination rates which account for plasma losses). Extending the usual Debye-Hiickel treatment of screening to encom- pass also dynamic screening effects due to ion motions is an important research area in its own right. Such research is expected to be noticeably facilitated by the incorporation of the static screening effects into the calculation of atomic properties and processes right from scratch. Doing so, one can then formulate a procedure which allows the simultaneous evaluation of level broadening and of the lowering of the continuum threshold-both on the same footing.

In a recent study [11 the numerical pair function method of our earlier studies of the Debye potential [2] has been replaced by a variational calculation employing correlated wave functions. This change was needed in order to achieve high accu-

International Journal of Quantum Chemistry: Quantum Chemistry Symposium 30, 1643-1 650 (1 996) 0 1996 John Wiley & Sons, Inc. CCC 0360-8832 I96 I071 643-08

ZHANG AND WINKLER

racy in calculations of not only the ground-state energies of the negative hydrogen ion and the helium atom but also of some excited sttaes of the latter system, as well as the radical one-electron density function of such systems under various plasma conditions. In this investigation the attractive Coulomb potential between the electron and the nucleus has been replaced by a screened potential of the Debye type Zexp(-r/D)/~. Varying the Debye length D from infinity-which corresponds to no screening-to small values, different plasma conditions can be simulated. Since the De- bye parameter is a funtion of both the electron density n, and the electron temperature T,, a particular value of D corresponds to a range of plasma conditions. It is, however, meaningful to associate smaller values of D with stronger screening. The earlier studies, not including screening of the electron-electron interaction, indicated that the as- sumedly very fragile H- ion may in reality be more stable in weak-to-moderate plasma conditions than initially assumed. The inclusion of De- bye screening into the electron interaction in the present work corroborates these findings strongly. As expected, this modification leads to a further increase of stability for a H- ion embedded in a Debye plasma. Since screening in one form or the other is a fact on plasmas, the gross features of the present calculation, which is strictly based on the validity of the Debye model, may well serve as a general outline of trends under more general conditions. The detailed interpretation of the numerical results, however, requires caution: The Debye potential may not always be appropriate to model a real plasma. In particular, in the range of small values of the Debye parameter D, close to (or even smaller than) the average interparticle distance, the interpretation of the present results in terms of a plasma is not reasonable. However, even in those regions the results represent properties of the De- bye potential which have not been studied before. In particular, by extending a similar study by Rogers and co-workers [ 31 of the hydrogen atom to two-electron systems, it was possible to extract accurate ionization energies for the helium atom and, above all, detachment energies for H-. Over most of the range of D values covered in the present work, the results are indeed representative for typical effects of screening on the binding energy and the electron distribution of small atoms.

The Calculations

The nonrelativistic Hamiltonian

Z exp( - r 2 / D ) exp( - yI2/D) + , (1) -

y2 r12

describing two-electron systems in a Debye plasma characterized by the parameter D, has been diago- nalized in a basis of correlated functions of the following type:

x e - Y m r l * . (2)

The variables y1 and y2 are the radial coordinates of the two electrons and the variable yI2 stands for the distance between them, while the quantity Nklm is a normalization factor. Expansions of this type have previously been shown to be very efficient in obtaining highly accurate variational wave functions for S states of two-electron atoms [4]. The two signs correspond to singlet and triplet states, respectively. The nonlinear parameters ( Y k ,

P I , and 'y, have been preselected so as to approxi- mate a definite integral optimally as described in the earlier reference which introduced these functions as a particular type of integral transform wave functions. The linear parameters are deter- mined in the usual variational procedure by solv- ing the generalized eigenvalue equation

( H - S E ) @ = 0. ( 3 )

Here H and S stand for the Hamiltonian and overlap matrices, respectively. Since the main goal of this study has been to evaluate all of the binding energy of the H- ion embedded in a pure Debye plasma without overestimating the effect, a comparatively large basis set of 220 functions of the type given in Eq. ( 2 ) has been used. The use of correlated wave functions containing the interelec- tronic distance yI2 as a dynamical variable in con- nection with the Debye potential is very conve- nient because all integrals can be expressed in closed form modifying the closed-form expressions given in [4].

The radial one-electron density function is obtained by integrating the two-electron density ex-

1644 QUANTUM CHEMISTRY SYMPOSIUM NO. 30

HYDROGEN AND HELIUM DEBYE PLASMAS

pression over the coordinate of one of the particles as well as over the angular variables of the other, which in the present case of spherically symmetric states reduces to applying a factor of 4rr:

In the present study we focus on the lowest bound state to obtain information about the spatial extension of the systems in various plasma environments. Hence, in Eq. (4) the quantity @ stands for the ground-state wave function but could be replaced by the eigenfunction of any excited state to evaluate the density of that state.

Although an extension of the correlated wave function approach to atoms with more than two electrons may not be practical, the treatment of systems of particles interacting via Yukawa forces by an appropriate modification of more standard methods of quantum chemistry or atomic theory is possible. In fact, systems of particles interacting via Yukawa forces have been studied early on in nuclear physics, e.g., by Swiatecki [5]. This reference contains also the formulas for the interaction integrals of particles described by wave functions

0

-0.05

-0.1

I .- c -0.15 3 0 $ z -0.2 C

r .-

F 15 -0.25

-0.3

-0.35

-0.4

of Gaussian type, i.e., exp(-ar2), or interacting via Yukawa forces. The adaptation to systems with both attractive and repulsive screened interactions and the interpretation of the results as meaningful for ions embedded in a plasma, however, is new and promises interesting results, in particular, for the further sudy of atomic processes in plasmas.

All calculations have been performed in extended precision on the Hewlett-Packard 9000 735/125 computer system of the Physics Depart- ment of the University of Nevada at Reno.

Results

RESULTS FOR THE ENERGIES

The results for the ground-state energies of both the negative hydrogen ion and the neutral hydrc- gen atom, which have been reported before [ 11, are summarized in Figure 1. Although the results for large values of the Debye parameter D are not displayed here for the sake of readability in the low D region, the results for large but finite values of D are represented directly as detachment energies in Figure 2. For D + M, the energy curves for

\\Energy of Neutral Hydrogen

Energy of the Negative Ion -- -'-, ------.

I I I I I

1 2 3 4 5 6 D in Atomic Unit

FIGURE 1. The energy of the negative hydrogen ion vs. the screening parameter D compared to the ground-state energy of the hydrogen atom under the same plasma conditions. The ionic system is bound over the whole range of D values in which the neutral system is also capable of supporting at least one bound state.

INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY 1645

ZHANG AND WINKLER

0.03

0.029

0.028

0.027

0.026 + x 0.025 P C .-

2 0.024

0.023

0.022

0.021

0.02

/ I I

I I

I

70 80 90 100 10 20 30 40 50 60 D in a.u.

FIGURE 2. Detachment energy for the negative hydrogen ion vs. the Debye length D.

H- and H approach smoothly the corresponding vacuum values of - 0.52775 and - 0.5 a.u., respectively.

In fact, our findings indicate that the negative hydrogen ions survives increasing plasma conditions as well as neutral hydrogen does and that both species lose their binding property simultane- ously as a function of decreasing values of D. The smallest value of D for which binding of the two-electron system has still been obtained (although with a tiny value of the detachment energy of 0.68 X l op4 a.u.1 is D = 0.860 a.u. This value is so close to the value obtained by Rogers and co-workers [31 for the limit beyond which not even neutral hydrogen is able to support a bound state that it is safe to state as our main finding the fact that, in a pure Debye potential, there is no way for the negative ion to lose its second electron due to pressure ionization. Any detachment that occurs is due to the action of one-particle operators such as collisions with other particles or photons. As a function of decreasing values of D the detachment energy of H- decreases initially very slowly from its vacuum value of 0.0277 a.u. and holds still a sizable 0.0234 a.u. at D = 6. From then on its drops rapidly.

The corresponding results for the helium atom in comparison to the ground-state energy of the

helium ion are presented in Figure 3. Again it has been found that both systems lose their capability to support bound states at the same value of D. More precisely, we calculated this to happen at D = 0.4311 a.u. for the helium ion. The critical point of no binding for the first two excited singlet S states of helium is noticeably different from this value: D = 0.460 a.u. for the (ls2s) state and D =

0.4957 a.u. for the (ls3s) state. Since the excited states of helium are pushed into the continuum in a region of D values where the positive helium ion still carries a bound ground-state-though barely so-the corresponding curves in Figure 3 do intersect. For higher excited states this crossing will occur at values of D which still represent plasma situations. In these regions, pressure ionization of one Rydberg electron is possible even in a real Debye plasma. In Figure 4 the ionization energies of the neutral helium atom are plotted versus the Debye length D.

RESULTS FOR THE ONE-ELECTRON RADIAL DENSITY

The one-electron radial density has been calculated for the negative hydrogen ion under various plasma conditions mainly in order to study the spatial extension of this system as it becomes less



0.5

0

-0.5

v)

C 3 .- V -1 E 3

c .-

C ._ $ -1.5 0

I5 -2

-2.5

-3

. . ~ . . . . ................... ....... ...... .... ................................................................................

He(ls1s)

0 1 2 3 4 5 6 7 8 9 10 D in Atomic Units

FIGURE 3. The energy of the two lowest 'S states of the helium atom vs. the screening parameter D compared to the ground-state energy of the positive helium ion under the same plasma conditions. While the ground state of He is bound over the whole range of D values for which there is a bound state of the ion, the curve of the excited He state crosses the ionic curve, although this is not noticeable in the graph.

1

0.95

0.9

: C .-

0.85 F Q

W

0.8

0.75

0.7

I I I I I I , I I

I

10 20 30 40 50 60 70 80 90 loo D in a.u.

FIGURE 4. Ionization energy for the neutral helium atom vs. the Debye length D.

INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY 1647

ZHANG AND WINKLER

and less bound with decreasing value of D. This effect is frequently addressed as ”lowering of the continuum threshold.” The expectation is that H- increases in spatial extension with the reduction of the binding energy. This is indeed the case as shown in Figure 5. However, this increase in size occurs in a region of small D values which is not representative of a plasma. The most prominent feature of these results is rather the high degree of stability the ion exhibits in various plasma conditions. There is very little change in the density from D = 20,000 a.u. to D = 4 a.u. A small shift of electron density from the inner regions toward the tail of the distribution is noticeable. The most striking feature, however, is the fact that we do not see more of it. Even at D = 2 a.u. the curve retains its maximum at roughly the same position. From then on, however, the charge distribution changes noticeably. The lowest curve, corresponding to D = 0.860 a.u., is so extended that within a region of Y I 80 a.u. not more than 25% of the total charge is collected. The relative constancy of both the detachment energy and the electron distribution of H- accounts well for its property to

enhance the opacity of plasmas. The unexpectedly high degree of stability of the ion points toward its presence even in high-density laboratory plasmas as opposed to just in astrophysical plasmas.

SOME SPECULATNE RESULTS

While the Debye theory postulates a uniform screening potential between two charges, it seems plausible to assume that local fluctuations of the screening are not only possible but rather the rule. If that is so, it may be argued that the electrons, being light particles, can be more efficient in tak- ing advantage of such local fluctuations than an electron-ion pair which has one less mobile part- ner. In particular, it may be argued that electron pairs will preferably move into regions with higher screening potential (i.e., lower values of D> because this will lower the energy of the system and that electron pairs are more efficient to do so than electron-ion pairs. In order to explore the rudi- ments of an emerging theory of such ideas, we have studied cases in which the values of the Debye parameter for the electron-electron interac-

0.25 1 I I I I I 1

FIGURE 5. The radial one-electron density of the negative hydrogen ion in various plasma environments. The top curve (taken at the maximum) represents the density at a value of D = 2000 a.u., while the lower curves correspond to a series of values of D = 4, 2, 1, 0.875, and 0.860 a.u., respectively.



tion was chosen different from those of the electron-nucleus potential. In practice, we have aug- mented the last term of the Hamiltonian Eq. (1) by a scaling factor y to read exp( - y ~ ~ ~ / D ) / r , , while the other terms remain unchanged. By changing y one can modify the screening of the electron interaction relative to the other interactions. Results of these calculations are presented in Table I. The most interesting feature of this scenario is the possibility of excited states of H-. Although here only states of S symmetry have been included, it is likely that bound states of higher angular mo- mentum do exist. Then one would expect low- energy electromagnetic transitions, which, if iden- tifiable, might serve diagnostic purposes. Astro- physical spectra usually have lots of broadened

low-energy transition lines so that the identifica- tion of such transitions may be difficult. Labora- tory plasma have a much better chance for the detection.

Summary

While in previous work the screening effects were applied only to the electron-ion interaction, the present study extends Debye screening also to the electron-electron interaction. A variational calculation of the ground-state energy of the negative hydrogen ion has been performed for various plasma conditions characterized by different val-

TABLE I Binding energies of the negative hydrogen ion are listed for various values of the screening parameter D and various values of the scaling factor y by which the screening of the electron -electron interaction can be modified without affecting the screening of the electron nuclear potential. For comparison, the values for neutral hydrogen (Is) are given in column 2. Multiple entries in columns 2 - 5 indicate the existence of excited states of 'S symmetry

Negative Hydrogen Ion

H Atom y = 1.0 y = 1.1 y = 1.5 y = 2.0

D = 200.0

D = 1'00.0

D = 50.0

D = 30.0

D = 20.0

D = 15.0

D = 10.0

D = 8.0

D = 6.0

D = 4.0

0.49502

0.49007

0.48030

0.46748

0.451 82

0.43653

0.40706

0.38588

0.35226

0.29092

0.52276

0.51 780

0.50795

0.49499

0.47904

0.46338

0.43295

0.41 092

0.37568

0.31 074

0.52325 0.4951 7

0.51876 0.49033

0.50978 0.48060

0.49785 0.46772

0.48303 0.451 92

0.46834

0.43942

0.41821

0.38390

0.31960

0.52520 0.49642 0.49578 0.52254 0.49237 0.491 09 0.51 693 0.48361 0.48131 0.50894 0.471 27 0.46804 0.49829 0.45545 0.451 88 0.48706 0.43960

0.46340 0.40867 0.44499 0.38656 0.41 372

0.351 35

0.52760 0.49801 0.49684 0.5271 7 0.49501 0.49275 0.52554 0.48761 0.48319 0.52203 0.4761 8 0.46968 0.51 590 0.46066 0.45284 0.50827 0.44461 0.43673 0.48975 0.41 252 0.47387 0.38929 0.44507 0.35312 0.38351

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ZHANG AND WINKLER

ues of the Debye parameter D. This results in a considerable increase in stability of the ion as compared to the case of an unscreened interaction (i.e., with only the electron-ion potential screened).

As an example of related research, we refer to a study by Scheibner, Weisheit, and Lane [6] of plasma screening effects on proton-impact excita- tion of positive ions. In this work two screened ions interact via a Debye force.

ACKNOWLEDGMENT

This work was supported by the Division of Chemical Sciences, Office of Basic Energy Sciences,

Office of Energy Research, U.S. Department of Energy.

References

1. P. Winkler, Brief Report, Phys. Rev. E, to appear. 2. Z . Wang and P. Winkler, Phys. Rev. A 52, 216 (1995).

3. F. J. Rogers, H. C. Graboske, Jr., and D. J. Hanvood; Phys.

4. P. Winkler and R. N. Porter, J. Chem. Phys. 61, 2038 (1974).

5. W. J. Swiatecki, Proc. Roy. SOC. A205, 238 (1951).

6. K. Scheibner, J. C. Weisheit, and N. F. Lane, Phys. Rev. A 35,

Rev. Al, 1577 (1970).

1252 (1987).

1650 QUANTUM CHEMISTRY SYMPOSIUM N0.30

negative hydrogen and helium in a variety of debye plasmas

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