J. Quant. Spectrosc. Radtat Transfer Vol. 58. No 46, pp 637-643. 1997 0 1997 Published by Elsevw Saence Ltd. All rights reserved Pergamon

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DENSITY EFFECTS ON COLLISIONAL RATES AND POPULATION KINETICS

CARLOS A. IGLESIAS and RICHARD W. LEE Lawrence Livermore National Laboratory, P.O. Box 808, Livermore, CA 94551, U.S.A.

Abstract-Although detailed quantum kinetic theory calculations obtained large plasma effects on electron collisional rates, population kinetics calculations indicate that the main impact of the density on non-equilibrium plasma models is continuum lowering. Density effects beyond continuum lowering lead to significantly smaller corrections on level populations and are uncertain since usual approximations may introduce anomalous behavior. These kinetics calculations take advantage of simple approximations to the electron colhsional rates that compare reasonably well with the quantum kinetic theory results. Finally, the analysis indicates that since continuum lowering is already included in standard kinetics models, the expected impact on level populations from the quantum kinetic theory results may not be realized. 0 1997 Published by Elsevier Science Ltd. Ail rights reserved

1. INTRODUCTION

Over the years there has been interest in the atomic properties of dense plasmas motivated by astrophysics and laboratory experiments involving high energy concentration. Although such research has practical application, there is a need for a fundamental understanding of how the plasma environment modifies atomic processes. It seems natural to study density effects in photon absorption or emission spectra, e.g., line shifts and continuum lowering. Interestingly, the one distinct feature between the recombination spectrum described by the Coulomb potential (infinite number of bound states) and screened Coulomb potential is a red-shift in the spectral lines ‘. Thus, it is difficult to distinguish whether the absence of spectral lines in the spectrum is a result of continuum lowering or line broadening. Furthermore, the level shifts produced by the screened Coulomb potential 2 are inconsistent with experiments ‘. There is also experimental evidence 4 against a theoretically predicted transparency window 5. Not surprisingly experimental spectra can be reproduced using isolated atom oscillator strengths and photoionization cross-sections together with modified line broadening theories 4.6. Consequently, density effects on atomic processes are difficult to study in photon absorption or emission experiments.

As an alternative to spectral observations, population kinetics experiments may be more suitable to study density effects on atomic processes since calculations based on quantum kinetic (QK) theory ’ 9 show large plasma effects on collisional rates. In the QK approach 7~9 plasma effects are described using quasi-particle energies and effective interactions of the form ’

where V(q) is the Fourier transform of the Coulomb interaction and E(q, co) the dielectric function. It is difficult to address problems with I&; thus, as a first step it is replaced by the static limit 7,

K(4) = Kf&,~ = 0) = A2q2

P %I) 1 + n-q

where

2 = JkT/4m2(N, + NJ (3)

with N, and NP the number density of free electrons and protons, respectively, kT the temperature in energy units, and e the elemental charge.

637

638 Carlos A. Iglesias and Richard W. Lee

As mentioned above, the screened potential in Eq. (2) leads to level shifts ’ inconsistent with spectral line measurements 3. The reason is, in part, that the neglected dynamic contributions to line shifts tend to cancel the static results “I. Therefore, E gives an incorrect, or incomplete, description of atomic processes in a plasma. On the other hand, static screening may provide a reasonable description of continuum lowering ‘I. Nevertheless, because of their relative simplicity, statically screened potentials are extensively used in dense plasma models 12.

The purpose here is to show that the reported 7-9 density dependence in collisional rates has a simple interpretation. More importantly, population kinetics are found to be most sensitive to the number of bound states used in the model. Therefore, density effects on ionization and recombination collisional rates do not have the expected impact on population kinetics. For simplicity, the discussion is limited to hydrogen plasmas.

2. COLLISIONAL RATES

The electron collisional rates for a non-degenerate, ideal plasma are defined by I3

(w) = r dEf(E)va(E) L”

(4)

where v is the electron velocity, c the cross-section, f(E) the electron energy distribution in the plasma, and E,,,,, the threshold energy for the excitation. In this section ideal gas results are obtained from fits to numerical evaluations of Eq. (4) without plasma effects assuming a Maxwell-Boltzmann distribution and Born approximation for 0 13.

2.1. Ionization

The electron collisional ionization rates for the ground state of hydrogen were computed by Schlanges and Bornath ‘,I4 using QK for a statically screened potential. Their results are reproduced

15 16

log (N, I kT) [electrons I K cm31

17

Fig. 1. Ionization rates for the ground state of hydrogen at two temperatures as a function of NJkT: quantum kinetic ‘.I4 theory (solid), a’(dash), and a from Eq. (5) at the plasma conditions where the indicated level merges into the continuum (circles). For the statically screened Coulomb potential a level

n! merges with the continuum at a given value of N./kT.

Density effects 639

in Fig. 1 for two temperatures and show large enhancements with increasing electron density compared to the ideal gas result.

The enhancement in the ionization rate has two main contributions. The first arises from ionization to levels that are no longer bound in K. This effect can be estimated by

where K’d and aid are the ideal gas rates for excitation and ionization, respectively, and the sum is over bound states with n, and e, denoting the principal and orbital quantum numbers of the last level to merge into the continuum *. Results from Eq. (5) for the ground state of hydrogen are presented in Fig. 1. The calculations were performed at the density for which the level, indicated in Fig. 1, merges with the continuum 2 (for n 2 6 the chosen density corresponds to the np level) and stop at the 2s level as the results become density independent.

Secondly, plasma screening shifts the level energies. This contribution is approximated by scaling the cross-sections with the screened energies where the level shifts are obtained from analytic fits “. The result of Eq. (5) using scaled rates, a”(N,), are plotted in Fig. 1 showing reasonable agreement with the QK calculations. Note that this simple scaling does not screen the electronelectron interaction and dynamic screening of this interaction is known to increase the QK result 16. Consequently, the overestimate by the present a”(N,) is partly due to static screening of the electron-electron interaction in Schlanges and Bornath ‘*16.

2.2. Excitation

Figure 2 presents 1s to 2p excitation rates from QK using the statically screened Coulomb potential ‘.14. For comparison, Fig. 2 also displays scaled calculations, P(N,), where the energies in the ideal gas cross-section were adjusted for plasma screening. The results compare well with the QK calculations suggesting that the main contribution from density effects are experimentally

T=30,000K

I 14

I I IS 16

log (Ne I kT) [electrons / K cm31

I I7

Fig. 2. Excitation rate from the ground state to the 2p level of hydrogen at two temperatures as a function of N,/kT: quantum kinetic ‘.I’ theory (solid) and K‘ (dash).

640 Carlos A. Iglesias and Richard W. Lee

unobserved level shifts. Note that this rate is not affected by dynamic screening of the electron-electron interaction 16.

3. POPULATION KINETICS

Although density effects on collisional rates is an interesting subject, experiments can more easily address population kinetics. The rate equations for the temporal evolution of the population densities of levels with principal quantum number k are given by

dNk -= dt - N,N, ak + ‘&k,,, + “r Kk,” N,N&. + 'i'N,K, f "r N&,,, (6)

m=I m=k+l m=l m=k+l

where CI, /?, and K are the ionization, recombination, and excitation (de-excitation) rates, respectively, and nmax is the largest principal quantum number considered in the model. For the high densities of interest radiation can be neglected in the rate equations. Electroneutrality guarantees that

N, = Np , (7)

and particle conservation yields

where NO is the number density of nuclei which is assumed constant. Specific examples of population time evolution in plasmas are considered using 3 models for the rates and nmax.

N,=N,+*rN,, k=l

(8)

17

16

13 T=30,000K

N =lO%d 0

-11 -10 -9 -8

log t [set]

Fig. 3. Time evolution of population densities for a recombining plasma at conditions indicated in the figure: Model I (dash), Model 2 (solid), and Model 3 (dot-dash). The curves are labeled by principal

quantum number. Initially the plasma is completely ionized.

3.1. Model 1

Density effects 641

This model assumes ideal gas rates and nmax = 15 regardless of the plasma conditions and represents calculations that completely ignore density effects. It is emphasized that results from this model depend critically on the choice for nmllx, and the chosen value is arbitrary.

3.2. Model 2

This model has q,,,, I 15 where continuum lowering and number of bound states are determined from Stewart and Pyatt “. In addition, the ionization cross-sections are adjusted for the reduction in ionization energy, AZ, so that the ionization rates include corrections beyond the simple factor exp{AZ/kT} ‘*. As a result, the recombination rates also contain density effects ‘. Ideal gas results are used for the bound-bound excitation and de-excitation rates. This model is similar to commonly used kinetics approaches 19.

3.3. Model 3

This model chooses nmax as in Model 2 above. However, ideal gas results are used for all the rates. Note that this model does not conserve the number of states (formally bound levels merge with the continuum and are not destroyed) and is introduced to isolate the level population dependence on the allowed number of bound states.

4. NUMERICAL EXAMPLES

The examples below were chosen to provide contact with those previously considered by Ebeling and Leike 8. In the first case the relaxation process is started assuming a completely ionized plasma with Maxwellian electrons at T = 30,OOOK. The results are displayed in Fig. 3 and Fig. 4 for two densities. Figure 3 shows small differences between the models at the lower density. At the higher density in Fig. 4, however, there is a large discrepancy between Model 1 and either Model 2 or 3. The difference between the latter two models is relatively small suggesting that the main effect is the allowed number of bound levels. That is, the large difference in the ionization and

21

19

18

1’

T=30,00( )K

N =102’~rn-3 0

-14 -13 -12 -1 I -10

log t [set]

Fig. 4. Same as Fig. 3 at a higher density.

JQSRT 58/4-f-I

642 Carlos A. lglesias and Rtchard W. Lee

14 -10

I I

-9

log t [set]

Fig. 5. Time evolution of population densities for an ionizing plasma at conditions indicated in the figure where initially the bound levels have an equilibrium distribution corresponding to kT= 5OOOK: Model

1 (dash) and Model 2 (solid). Note that Models 2 and 3 are indistinguishable.

recombination rates (Model 2 contains density effects but not Model 3) do not translate to large differences in the population kinetics.

The second example is an ionizing plasma with Maxwellian electrons at T = 30,00OK, where initially the bound states have an equilibrium distribution appropriate to T = 5000K. Figure 5 shows that results from Models 2 and 3 are indistinguishable. Results from Model 1 are also identical except at late times where the populations approach thermal equilibrium.l_ This difference is a direct consequence of the allowed bound levels in the models. The agreement, except at late times, is occurring in spite of the large differences in the ionization and recombination rates.

5. CONCLUSION

On the one hand the results corroborate earlier conclusions 7.8 that the main density effect on collisional ionization is continuum lowering where the Mott transition changes a bound state to a delocalized state. On the other hand, the verified importance of density effects in the kinetic equations are already included in standard kinetics calculations I!‘. This suggests that although the quantum kinetic approach ‘. ‘.I6 may provide a unified treatment of atomic processes in plasmas, there may not be experimental consequences. To wit:

1. Present non-equilibrium models usually approximate plasma effects on the ionization and recombination rates; that is, scaling of the collisional cross-section 20.

2. Calculations for ionizing and recombining hydrogen plasmas show that the dominant effect on population kinetics is the allowed number of bound levels rather than density effects on the collisional rates.

3. Density effects that go beyond continuum lowering affect collisional rates via the details of the assumed screened potential. That is, the predicted density effects for the collisional excitation rates between bound states in quantum kinetic theory 7.8.‘6 are mostly due to experimentally

YFigure 4 of Ref. 8 apparently has not reached the long-time limit (i.e., thermal equilibrium)

Density effects 643

unobserved level shifts. Consequently, additional corrections beyond continuum lowering are uncertain.

It is clear from recent comparisons of population kinetics calculations I9 that there is an incomplete understanding of atomic processes in plasmas. Therefore, in spite of the predicted absence of clear experimental signatures for dense plasma effects on atomic processes, the continuation of experimental and theoretical studies of non-equilibrium plasma models at high densities are essential for a fundamental comprehension of this complex subject.

Acknowledgements-It IS pleasure to recognize valuable discussions with Manfred Schlanges, Thomas Bornath, and Albert Osterheld. Thanks are also due to Manfred Schlanges and Thomas Bornath for their numerical results. Work performed under the auspices of the US Department of Energy by Lawrence Livermore National Laboratory under Contract W-7405-Eng-48.

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