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Analysis of discrepancies between quantal and semiclassical calculations of electron impact broadening in plasmas
Journal of Quantitative Spectroscopy & Radiative Transfer 65 (2000) 15}22
Analysis of discrepancies between quantal and semiclassical calculations of electron impact broadening in plasmas S. Alexiou, R.W. Lee*, S.H. Glenzer, J.I. Castor Lawrence Livermore National Laboratory, L 399, P.O. Box 808, Livermore, CA 94550, USA
Abstract In this work we analyse the serious discrepancies found between quantal calculations and non-perturbative semiclassical calculations of electron impact broadening in the case of B2`2s}2p. We use the Coulomb Bethe (CB) approximation to calculate both electron impact broadening and excitation of 2s}2p for comparison with the other methods. We "nd good agreement between CB and the non-perturbative semiclassical method for the line width contributions of individual partial waves, except for low ¸, where strong collision e!ects enter. We also "nd good agreement between CB and the R-matrix and Coulomb Born methods for the excitation cross section partial wave contributions, again except for low ¸. There is disagreement for the high partial wave cross sections between the non-perturbative semiclassical method and all of the quantum methods; this is resolved by applying a symmetrized method, for which we demonstrate excellent agreement with CB. The area in which the semiclassical, Coulomb}Born and R-matrix methods disagree has been reduced to the "rst three partial waves, and the disagreement must be due to the treatment of strong collisions. ( 2000 Elsevier Science Ltd. All rights reserved. Keywords: Coulomb}Bethe approximation; Coulomb}Born methods; Electronic impact broadening
1. Introduction The calculation of spectral line broadening in plasmas is important for diagnostics, radiative transfer, gain estimates in X-ray lasers and other applications involving a variety of plasma sources.
* Corresponding author. Tel.: 001-925-422-7209; fax: 001-925-423-6172. E-mail address: [email protected] (R.W. Lee) 0022-4073/00/$ - see front matter ( 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 2 - 4 0 7 3 ( 9 9 ) 0 0 0 5 1 - 5
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S. Alexiou et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 65 (2000) 15}22
These plasma sources include a wide range of physical conditions, elements, and ion stages for which a calculational capability must be developed. It is clear that one needs to verify the applicablility of the calculational procedures. In turn, this leads to the need for independent experiments and benchmarking techniques. As the processes of spectral line broadening in plasma concern the e!ects of the plasma environment on an emitter immersed in it, this "eld of study has largely evolved using semi-classical (SC) approaches [1]. This is due to computational di$culties and to the fact that the quantum mechanical (QM) formulations of line broadening are currently valid for a small subset of spectral lines * see comments below. Thus, although we need to verify our SC calculations against QM, this can only be achieved for an extremely small set of lines. Fortunately, there is an experimental result due to Glenzer and Kunze [2] where the simplest possible isolated line system, an Li-like 2s}2p, line pro"le (in this case for BIII) has been measured. This and similar experiments on isolated lines have led to an improvement in the SC calculational capability [3] and importantly to the opportunity to compare with QM calculations. Such QM calculations, based on the R-matrix version of close-coupling theory, and also the Coulomb}Born approximation with exchange, have been performed and reported by Griem, Ralchenko and Bray (GRB) [4]. The object of this paper is to report on progress in understanding the discrepancy between those QM calculations and the newest SC calculations. The outline of this paper is as follows. In Section 2 we discuss a completely independent approach, i.e., one found in neither GRB nor SC, namely Coulomb}Bethe (CB), that can treat part of this problem. Next, we show that for a wide range of partial waves ¸ this approach and the SC method agree for relevant contributions to the inelastic line width. Extending this comparison to include the predictions of the R-matrix code and the Coulomb}Born code is not possible, since we do not have the available data for individual partial waves of the thermally averaged cross sections for the latter codes. We do have partial wave cross sections at a single electon energy, and these have been compared with SC and CB, and with the symmetrized classical method that will be discussed. From these comparisons several observations can be made: (A) There is no evidence that the SC methods are invalid below ¸"2p. (B) Coulomb}Bethe and the SC agree for the width as a function of ¸. (C) Even though (B) is true, the SC method needs to be symmetrized in order to obtain the same large-¸ partial wave cross sections as the QM methods. (D) Symmetrized SC and CB agree within 6% for the partial wave line width contributions at ¸"1 and 2, and more closely than that for all other partial waves up to ¸"10. For ¸'10 the di!erence increases, but at this point the contributions are negligible. (E) The disagreement between QM * R-matrix and Coulomb}Born * calculations and nonperturbative semiclassical calculations for the line width can be attributed to the treatment of strong collisions. Finally, we point out that the issues involved here are clearly important for individuals using line shapes directly, for researchers employing line shapes only indirectly, for example in radiative transfer modeling, or for those involved in kinetics modeling, as both SC and quantal methods are used to compute rates for collisional-radiative calculations.
S. Alexiou et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 65 (2000) 15}22
17
Fig. 1. Partial wave contributions to the total width (FWHM) for NPSC and Coulomb}Bethe. Ordinate: partial FWHM (As ); abscissa: total orbital angular momentum quantum number ¸. Solid: NPSC; dashed: Coulomb}Bethe; dotted: optimal symmetrized classical theory. The Coulomb}Bethe and optimal symmetrized classical curves cannot be distinguished on this plot.
2. Description of Coulomb}Bethe Since the QM and SC methods have been documented we brie#y outline the Coulomb}Bethe method: Coulomb}Bethe is a fully quantum-mechanical variant of the Coulomb}Born approximation. Brie#y, the Coulomb}Born approximation neglects the short-range distortion of the Coulomb scattering waves brought about by the detailed interaction between the target and the incident electron. It evaluates the reactance matrix in perturbation theory with or without exchange. The resulting transition matrix is found either by using its expansion in powers of the reactance matrix, or by resumming the expansion (unitarized version). A full multipole expansion is used for the interaction potential. Coulomb}Bethe is the special version of Coulomb}Born, where the additional approximations are made that multipole interactions other than dipole can be neglected, exchange can be neglected, and unitarization can be omitted. This method is valid for high partial waves of optically allowed transitions. Use of Coulomb}Bethe (CB) allows us to compare the CB FWHM with the FWHM derived from the non-perturbative semiclassical calculations (NPSC). In Fig. 1we show the FWHM contribution from each partial wave for the NPSC and CB. Since CB includes only the dipole interaction, this approximation was made in NPSC as well, for the comparison shown in Fig. 1.1 It is seen that NPSC and CB agree for high 1 The CB calculation uses Baranger's [5] formula, Eq. (3.1), omitting the elastic scattering term, and observes the excitation threshhold in forming the velocity integral for the excitation rate. NPSC uses Eqs. (3.3) and (3.4), for which the velocity integration is taken from v"0 to R.
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S. Alexiou et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 65 (2000) 15}22
partial waves. For small partial waves no agreement is expected, due to the di!erent approximations involved. (For example, NPSC is non-perturbative and perturbation theory is not valid for small partial waves.) As mentioned earlier, the line width contributions of individual partial waves, which must be averaged over a Maxwellian distribution of electron velocities, are not available for the GRB data.
3. Cross section comparison GRB computes the width by using the Baranger [5] formula:
P
=
P
dv vf (v)[p* (v)#p& (v)# d) D f *(),v)!f &(),v)D2], (3.1) */ */ 0 where N is the electron density and p is the total inelastic cross section, f (v) is the Max% */ well}Boltzmann velocity distribution and the superscripts i and f denote upper and lower states, respectively, for the transition in question. The functions f * and f & are elastic scattering amplitudes. This has the advantage of reducing the line-broadening problem to a (hopefully) computationally tractable problem, i.e., the computation of inelastic cross sections and elastic scattering amplitudes. Nevertheless, it should be realized that in addition to theoretical problems with this formula [6], excitation cross sections solely from one state to another are not exactly what is relevant for line broadening. For example, if virtual transitions are not included then we would "nd that an inelastic excitation cross section is strictly zero if the incident electron energy is below the excitation energy. On the other hand, in perturbation theory the most important processes are of the type FWHM"N %
% #0--*4*0/ % #0--*4*0/ DiT P D jT P DiT
(3.2)
with i denoting the upper or lower level and j denoting any level distinct from i. Hence, in line-broadening we never have isolated excitations and deexcitations, but always a pair of excitation and deexcitation processes. (In non-perturbative calculations with many levels participating, we can also have cascade-like excitations and deexcitations, but the point remains the same.) Thus, quantum mechanically, an incident electron energy smaller than the excitation energy can still contribute, by `borrowinga the energy it needs for the process DiTPD jT and giving it back in the deexcitation process D jTPDiT. SC accounts for such processes without needing to introduce the virtual states separately. The quantal methods must endeavor to include these transitions by ensuring these are in the original set of basis states. Indeed, in the Coulomb}Born with Exchange (CBE) method used in GRB these virtual states are not added, while in a close-coupling approach used in GRB these virtual states are included. We infer them to be calculationally unimportant for the case in point, based on the good agreement of the close-coupling and CBE results. NPSC does not use the Baranger form and does not compute cross sections per se. Instead, the half-width (HWHM) w and shift d are computed from the equation: w#id"+ f Q , q q q
(3.3)
S. Alexiou et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 65 (2000) 15}22
19
where f is the collision frequency and q Q "1!S S~1. (3.4) q ** && The matrices S and S are elastic scattering matrices for electrons with, respectively, the initial and ** && "nal state for the dipole line transition in question. In each case, the colliding electron velocity is v. This is convenient because it is an expression common for both QM and SC. It indicates a sum over all collisions of all types, with each type labelled by `qa. In QM, the collision type q is described by partial wave l and velocity v. In SC, it is described by impact parameter o and velocity v. Thus,
P P
P A B
= + f "2pN vf (v) dv o do, (SC) q 0 q = + 2 2l#1 , (QM), + "2pN f (v)v dv mv 2 0 l where the S-matrices are functions of (o, v) in SC and (l, v) in QM. Transforming to the collision axes we obtain
(3.5)
w#id"+ f + SJ m@ 1MDJ m@TSJ m 1MDJ m T & & * * & & * * q q M,MmN d @d @!SJ m@DS DJ m TSJ m DS~1DJ m@ T * * # * * & & # & & . (3.6) ] m*,m* m&,m& 2J #1 * The symbol S denotes the scattering matrix referred to the collision axes. The inner sum is taken # over M"!1, 0, 1 and over all permitted values of m , m , m@, m@ . * & * & For the present comparisions we have modi"ed NPSC to compute cross sections. In particular, the excitation cross section 2s}2p is interesting as an initial focus because if we limit the set of perturbing states to the states with principal quantum number n"2 (as in GRB and the NPSC calculations) the only two channels contributing are the dipole excitations 2sP2p and 3@2 2sP2p , at least in perturbation theory. 1@2 Fig. 2 shows a comparison of the NPSC cross sections against the GRB data for the closecoupling (CC) and Coulomb}Born-Exchange (CBE) calculation (see Fig. 3 of GRB). Also shown is a Coulomb}Bethe calculation and a calculation labeled `classicala which we will discuss below. Of special interest is the comparison at high partial waves where, according to the criteria stated in GRB, SC should be valid. The SC perturbative excitation cross section takes the simple form:
A
B
3 8p 3+f v e2 2 j p } " [a(m, e )!a(m, e )], (3.7) 2s 2p 2 3 2mu .*/ .!9 4ne +v 2s~j 0 where j"2p and f is the oscillator strength between the 2s and 2p , u the 2p !2s 3@2 j 3@2 2s~j 3@2 energy di!erence in angular frequency units, v the incoming (asymptotic) velocity, e and e the .*/ .!9 eccentricities corresponding to the minimum and maximum impact parameters o and o , .*/ .!9 respectively, m"su /v, *&
(3.8)
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S. Alexiou et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 65 (2000) 15}22
Fig. 2. Partial-wave excitation cross sections for 2s}2p at initial electron energy 10 eV. Solid: CC; dashed: CBE; dotted: NPSC; dash-dotted: CB; and dash-double-dotted: optimal symmetrized classical. CC and CBE are from Fig. 3 of GRB. Ordinate: p } in units 2s 2p of pa2; abscissa: total orbital angular momentum quantum number ¸. 0
with
A
B
(Z!1)e2 e2!1 ~1@2 s" " , 4pe mv2 o2 0 and with Z the spectroscopic charge number, and a(m,e)"!meenmK (me)K@ (me), *m *m
(3.9)
(3.10)
with
P
K (me)" *m
=
du e~me #04) ucos mu,
(3.11)
P
(3.12)
0
and K@ (me)"! *m
=
du e~me #04) ucosh u cos mu.
0 There are at least two excellent analytic approximations for the a-function [7,8]. The factor of 3 in Eq. (3.7) arises from the fact that the oscillator strength for j"2p is twice 2 3@2 that for j"2p . This simple SC cross section exceeds the CBE and CC cross sections by a factor 1@2 of "ve, even at a high partial wave, say ¸"12, where validity of the SC is not in question according to the GRB criteria. The NPSC, although getting the same FWHM as the CB method, does not agree with either CB or the QM cross sections at high L. To achieve agreement, it is essential to use
S. Alexiou et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 65 (2000) 15}22
21
symmetrization [9] on the SC cross section.2 There are ambiguities in the symmetrization procedure, which is therefore not unique. In the next paragraph we describe our symmetrization approach, and how the ambiguities are resolved by comparing the results with CB, which leads to the method referred to as `symmetrized classicala in Fig. 2. Brie#y, symmetrization implies using expressions for m and e in Eq. (3.10) that are symmetrical in the initial and "nal velocity and angular momentum. There is no ambiguity in the symmetrization of m:
A
B
e2(Z!1) 1 1 m,g !g " ! , (3.13) & * 4pe + v v 0 & * where v is the initial (asymptotic) velocity and v is the "nal (asymptotic) velocity after the * & excitation/deexcitation. Here g refers to the Coulomb parameter: e2(Z!1) g" . (3.14) 4pe +v 0 The ambiguity is in the symmetrization of the eccentricity. When we compute the contribution of the ¸th partial wave, we need to evaluate the eccentricity e at the minimum and maximum impact parameters ¸+/mv and (¸#1)+/mv. Thus, for example, with the minimum impact parameter we have an eccentricity of
S A B S
¸2 ¸+ 2 e " 1# " 1# . .*/ g2 mvs
(3.15)
One way of achieving symmetry, which is found to be successful in matching the classical and quantum results in the asymptotic regime, is to set g2"g g (3.16) * & in Eq. (3.15), with the subscripts i and f corresponding to v and v de"ned above. * & However, there are di!erent forms of the eccentricity variable, depending on which pair of values of ¸2 is associated with the ¸th partial wave. Some possibilities are: compute ¸2"l2, where l is an integer, and associate the range lPl#1 with the lth partial wave; the same, except associate this range with partial wave l#1/2; compute ¸2"l(l#1) and associate the range lPl#1 with the lth partial wave; and many others. Excellent agreement is found between Coulomb}Bethe and either of these two choices: ¸2"l2 and lPl#1 for the l#1/2 partial wave, or ¸2"l (l#1) and l!1Pl for the lth partial wave. The numerical results for these two choices are identical for all practical purposes. The "rst one is what is actually plotted as `optimal symmetrized classicala in Fig. 2, and it is seen to agree excellently with CB, both for the cross section (Fig. 2) and for the line width (Fig. 1). Note that the symmetrized classical results show no indication of disagreement with the corresponding quantal results for partial waves 42p. The disagreement shows up as the missing value for ¸"0, and this is due to the ¸$1 ambiguity caused by the quantization of ¸.
2 Energy and angular momentum are conserved for the classical orbits, so to describe an inelastic process such as 2s}2p excitation, the energy and angular momentum must be averages of the initial and "nal values.
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S. Alexiou et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 65 (2000) 15}22
4. Conclusions The results of our comparisons of the inelastic component of the line width show that the SC agrees with the `independenta Coulomb}Bethe (CB) results for the observable, i.e., the FWHM, as a function of partial wave, as shown in Fig. 1. Unfortunately, we do not currently have such a partial wave breakdown for the QM calculations. We have also found from a comparison of excitation cross sections that: (A) There is no evidence that the SC methods are invalid below ¸"2p. (B) Coulomb}Bethe and the SC agree for the width as a function of ¸. (C) Even though (B) is true, the SC method needs to be symmetrized in order to obtain the same large-¸ partial wave cross sections as the QM methods. (D) Symmetrized SC and CB agree within 6% for the partial wave line width contributions at ¸"1 and 2, and more closely than that for all other partial waves up to ¸"10. For ¸'10 the di!erence increases, but at this point the contributions are negligible. (E) The disagreement between QM * R-matrix and Coulomb}Born * calculations and nonperturbative semiclassical calculations for the line width can be attributed to the treatment of strong collisions. In conclusion, we point out that there is further work required to resolve this issue. It is now apparent that the di!erence between NPSC and the close-coupling and Coulomb}Born results is not due to the de"ciency of classical theory used in the NPSC, since Coulomb}Bethe, a quantum method, agrees substantially with NPSC. Rather, we should investigate the di!erent ways that the strong collisions are treated. Thus, a detailed new calculation of the QM method, performed in a manner that allows extensive detailed comparison, is being undertaken.
Acknowledgements We are grateful to H.R. Griem and Yu. Ralchenko for discussions and to Yu. Ralchenko for providing the raw data for the partial wave breakdown.
References [1] [2] [3] [4] [5] [6] [7] [8] [9]
Griem HR. Spectral line broadening by plasmas. New York: Academic Press, 1974. Glenzer SH, Kunze HJ. Phys Rev A 1996;53:2225. Alexiou S. Phys Rev Lett 1995;75:3406. Griem HR, Ralchenko Yu, Bray I. Phys Rev E 1997;56:7186. [GRB]. Baranger M. Phys Rev 1958;111:481. Baranger M. Phys Rev 1958;112:855. Seaton MJ. J Phys B 1988;21:3033. PoqueH russe A. Phys Lett 1976;59A:438. PoqueH russe A, Alexiou S. JQSRT 1999;61:209 Alder K, et al. Rev Mod Phys 1956;28:434.
Analysis of discrepancies between quantal and semiclassical calculations of electron impact broadening in plasmas S. Alexiou, R.W. Lee*, S.H. Glenzer, J.I. Castor Lawrence Livermore National Laboratory, L 399, P.O. Box 808, Livermore, CA 94550, USA
Abstract In this work we analyse the serious discrepancies found between quantal calculations and non-perturbative semiclassical calculations of electron impact broadening in the case of B2`2s}2p. We use the Coulomb Bethe (CB) approximation to calculate both electron impact broadening and excitation of 2s}2p for comparison with the other methods. We "nd good agreement between CB and the non-perturbative semiclassical method for the line width contributions of individual partial waves, except for low ¸, where strong collision e!ects enter. We also "nd good agreement between CB and the R-matrix and Coulomb Born methods for the excitation cross section partial wave contributions, again except for low ¸. There is disagreement for the high partial wave cross sections between the non-perturbative semiclassical method and all of the quantum methods; this is resolved by applying a symmetrized method, for which we demonstrate excellent agreement with CB. The area in which the semiclassical, Coulomb}Born and R-matrix methods disagree has been reduced to the "rst three partial waves, and the disagreement must be due to the treatment of strong collisions. ( 2000 Elsevier Science Ltd. All rights reserved. Keywords: Coulomb}Bethe approximation; Coulomb}Born methods; Electronic impact broadening
1. Introduction The calculation of spectral line broadening in plasmas is important for diagnostics, radiative transfer, gain estimates in X-ray lasers and other applications involving a variety of plasma sources.
* Corresponding author. Tel.: 001-925-422-7209; fax: 001-925-423-6172. E-mail address: [email protected] (R.W. Lee) 0022-4073/00/$ - see front matter ( 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 2 - 4 0 7 3 ( 9 9 ) 0 0 0 5 1 - 5
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S. Alexiou et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 65 (2000) 15}22
These plasma sources include a wide range of physical conditions, elements, and ion stages for which a calculational capability must be developed. It is clear that one needs to verify the applicablility of the calculational procedures. In turn, this leads to the need for independent experiments and benchmarking techniques. As the processes of spectral line broadening in plasma concern the e!ects of the plasma environment on an emitter immersed in it, this "eld of study has largely evolved using semi-classical (SC) approaches [1]. This is due to computational di$culties and to the fact that the quantum mechanical (QM) formulations of line broadening are currently valid for a small subset of spectral lines * see comments below. Thus, although we need to verify our SC calculations against QM, this can only be achieved for an extremely small set of lines. Fortunately, there is an experimental result due to Glenzer and Kunze [2] where the simplest possible isolated line system, an Li-like 2s}2p, line pro"le (in this case for BIII) has been measured. This and similar experiments on isolated lines have led to an improvement in the SC calculational capability [3] and importantly to the opportunity to compare with QM calculations. Such QM calculations, based on the R-matrix version of close-coupling theory, and also the Coulomb}Born approximation with exchange, have been performed and reported by Griem, Ralchenko and Bray (GRB) [4]. The object of this paper is to report on progress in understanding the discrepancy between those QM calculations and the newest SC calculations. The outline of this paper is as follows. In Section 2 we discuss a completely independent approach, i.e., one found in neither GRB nor SC, namely Coulomb}Bethe (CB), that can treat part of this problem. Next, we show that for a wide range of partial waves ¸ this approach and the SC method agree for relevant contributions to the inelastic line width. Extending this comparison to include the predictions of the R-matrix code and the Coulomb}Born code is not possible, since we do not have the available data for individual partial waves of the thermally averaged cross sections for the latter codes. We do have partial wave cross sections at a single electon energy, and these have been compared with SC and CB, and with the symmetrized classical method that will be discussed. From these comparisons several observations can be made: (A) There is no evidence that the SC methods are invalid below ¸"2p. (B) Coulomb}Bethe and the SC agree for the width as a function of ¸. (C) Even though (B) is true, the SC method needs to be symmetrized in order to obtain the same large-¸ partial wave cross sections as the QM methods. (D) Symmetrized SC and CB agree within 6% for the partial wave line width contributions at ¸"1 and 2, and more closely than that for all other partial waves up to ¸"10. For ¸'10 the di!erence increases, but at this point the contributions are negligible. (E) The disagreement between QM * R-matrix and Coulomb}Born * calculations and nonperturbative semiclassical calculations for the line width can be attributed to the treatment of strong collisions. Finally, we point out that the issues involved here are clearly important for individuals using line shapes directly, for researchers employing line shapes only indirectly, for example in radiative transfer modeling, or for those involved in kinetics modeling, as both SC and quantal methods are used to compute rates for collisional-radiative calculations.
S. Alexiou et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 65 (2000) 15}22
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Fig. 1. Partial wave contributions to the total width (FWHM) for NPSC and Coulomb}Bethe. Ordinate: partial FWHM (As ); abscissa: total orbital angular momentum quantum number ¸. Solid: NPSC; dashed: Coulomb}Bethe; dotted: optimal symmetrized classical theory. The Coulomb}Bethe and optimal symmetrized classical curves cannot be distinguished on this plot.
2. Description of Coulomb}Bethe Since the QM and SC methods have been documented we brie#y outline the Coulomb}Bethe method: Coulomb}Bethe is a fully quantum-mechanical variant of the Coulomb}Born approximation. Brie#y, the Coulomb}Born approximation neglects the short-range distortion of the Coulomb scattering waves brought about by the detailed interaction between the target and the incident electron. It evaluates the reactance matrix in perturbation theory with or without exchange. The resulting transition matrix is found either by using its expansion in powers of the reactance matrix, or by resumming the expansion (unitarized version). A full multipole expansion is used for the interaction potential. Coulomb}Bethe is the special version of Coulomb}Born, where the additional approximations are made that multipole interactions other than dipole can be neglected, exchange can be neglected, and unitarization can be omitted. This method is valid for high partial waves of optically allowed transitions. Use of Coulomb}Bethe (CB) allows us to compare the CB FWHM with the FWHM derived from the non-perturbative semiclassical calculations (NPSC). In Fig. 1we show the FWHM contribution from each partial wave for the NPSC and CB. Since CB includes only the dipole interaction, this approximation was made in NPSC as well, for the comparison shown in Fig. 1.1 It is seen that NPSC and CB agree for high 1 The CB calculation uses Baranger's [5] formula, Eq. (3.1), omitting the elastic scattering term, and observes the excitation threshhold in forming the velocity integral for the excitation rate. NPSC uses Eqs. (3.3) and (3.4), for which the velocity integration is taken from v"0 to R.
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S. Alexiou et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 65 (2000) 15}22
partial waves. For small partial waves no agreement is expected, due to the di!erent approximations involved. (For example, NPSC is non-perturbative and perturbation theory is not valid for small partial waves.) As mentioned earlier, the line width contributions of individual partial waves, which must be averaged over a Maxwellian distribution of electron velocities, are not available for the GRB data.
3. Cross section comparison GRB computes the width by using the Baranger [5] formula:
P
=
P
dv vf (v)[p* (v)#p& (v)# d) D f *(),v)!f &(),v)D2], (3.1) */ */ 0 where N is the electron density and p is the total inelastic cross section, f (v) is the Max% */ well}Boltzmann velocity distribution and the superscripts i and f denote upper and lower states, respectively, for the transition in question. The functions f * and f & are elastic scattering amplitudes. This has the advantage of reducing the line-broadening problem to a (hopefully) computationally tractable problem, i.e., the computation of inelastic cross sections and elastic scattering amplitudes. Nevertheless, it should be realized that in addition to theoretical problems with this formula [6], excitation cross sections solely from one state to another are not exactly what is relevant for line broadening. For example, if virtual transitions are not included then we would "nd that an inelastic excitation cross section is strictly zero if the incident electron energy is below the excitation energy. On the other hand, in perturbation theory the most important processes are of the type FWHM"N %
% #0--*4*0/ % #0--*4*0/ DiT P D jT P DiT
(3.2)
with i denoting the upper or lower level and j denoting any level distinct from i. Hence, in line-broadening we never have isolated excitations and deexcitations, but always a pair of excitation and deexcitation processes. (In non-perturbative calculations with many levels participating, we can also have cascade-like excitations and deexcitations, but the point remains the same.) Thus, quantum mechanically, an incident electron energy smaller than the excitation energy can still contribute, by `borrowinga the energy it needs for the process DiTPD jT and giving it back in the deexcitation process D jTPDiT. SC accounts for such processes without needing to introduce the virtual states separately. The quantal methods must endeavor to include these transitions by ensuring these are in the original set of basis states. Indeed, in the Coulomb}Born with Exchange (CBE) method used in GRB these virtual states are not added, while in a close-coupling approach used in GRB these virtual states are included. We infer them to be calculationally unimportant for the case in point, based on the good agreement of the close-coupling and CBE results. NPSC does not use the Baranger form and does not compute cross sections per se. Instead, the half-width (HWHM) w and shift d are computed from the equation: w#id"+ f Q , q q q
(3.3)
S. Alexiou et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 65 (2000) 15}22
19
where f is the collision frequency and q Q "1!S S~1. (3.4) q ** && The matrices S and S are elastic scattering matrices for electrons with, respectively, the initial and ** && "nal state for the dipole line transition in question. In each case, the colliding electron velocity is v. This is convenient because it is an expression common for both QM and SC. It indicates a sum over all collisions of all types, with each type labelled by `qa. In QM, the collision type q is described by partial wave l and velocity v. In SC, it is described by impact parameter o and velocity v. Thus,
P P
P A B
= + f "2pN vf (v) dv o do, (SC) q 0 q = + 2 2l#1 , (QM), + "2pN f (v)v dv mv 2 0 l where the S-matrices are functions of (o, v) in SC and (l, v) in QM. Transforming to the collision axes we obtain
(3.5)
w#id"+ f + SJ m@ 1MDJ m@TSJ m 1MDJ m T & & * * & & * * q q M,MmN d @d @!SJ m@DS DJ m TSJ m DS~1DJ m@ T * * # * * & & # & & . (3.6) ] m*,m* m&,m& 2J #1 * The symbol S denotes the scattering matrix referred to the collision axes. The inner sum is taken # over M"!1, 0, 1 and over all permitted values of m , m , m@, m@ . * & * & For the present comparisions we have modi"ed NPSC to compute cross sections. In particular, the excitation cross section 2s}2p is interesting as an initial focus because if we limit the set of perturbing states to the states with principal quantum number n"2 (as in GRB and the NPSC calculations) the only two channels contributing are the dipole excitations 2sP2p and 3@2 2sP2p , at least in perturbation theory. 1@2 Fig. 2 shows a comparison of the NPSC cross sections against the GRB data for the closecoupling (CC) and Coulomb}Born-Exchange (CBE) calculation (see Fig. 3 of GRB). Also shown is a Coulomb}Bethe calculation and a calculation labeled `classicala which we will discuss below. Of special interest is the comparison at high partial waves where, according to the criteria stated in GRB, SC should be valid. The SC perturbative excitation cross section takes the simple form:
A
B
3 8p 3+f v e2 2 j p } " [a(m, e )!a(m, e )], (3.7) 2s 2p 2 3 2mu .*/ .!9 4ne +v 2s~j 0 where j"2p and f is the oscillator strength between the 2s and 2p , u the 2p !2s 3@2 j 3@2 2s~j 3@2 energy di!erence in angular frequency units, v the incoming (asymptotic) velocity, e and e the .*/ .!9 eccentricities corresponding to the minimum and maximum impact parameters o and o , .*/ .!9 respectively, m"su /v, *&
(3.8)
20
S. Alexiou et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 65 (2000) 15}22
Fig. 2. Partial-wave excitation cross sections for 2s}2p at initial electron energy 10 eV. Solid: CC; dashed: CBE; dotted: NPSC; dash-dotted: CB; and dash-double-dotted: optimal symmetrized classical. CC and CBE are from Fig. 3 of GRB. Ordinate: p } in units 2s 2p of pa2; abscissa: total orbital angular momentum quantum number ¸. 0
with
A
B
(Z!1)e2 e2!1 ~1@2 s" " , 4pe mv2 o2 0 and with Z the spectroscopic charge number, and a(m,e)"!meenmK (me)K@ (me), *m *m
(3.9)
(3.10)
with
P
K (me)" *m
=
du e~me #04) ucos mu,
(3.11)
P
(3.12)
0
and K@ (me)"! *m
=
du e~me #04) ucosh u cos mu.
0 There are at least two excellent analytic approximations for the a-function [7,8]. The factor of 3 in Eq. (3.7) arises from the fact that the oscillator strength for j"2p is twice 2 3@2 that for j"2p . This simple SC cross section exceeds the CBE and CC cross sections by a factor 1@2 of "ve, even at a high partial wave, say ¸"12, where validity of the SC is not in question according to the GRB criteria. The NPSC, although getting the same FWHM as the CB method, does not agree with either CB or the QM cross sections at high L. To achieve agreement, it is essential to use
S. Alexiou et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 65 (2000) 15}22
21
symmetrization [9] on the SC cross section.2 There are ambiguities in the symmetrization procedure, which is therefore not unique. In the next paragraph we describe our symmetrization approach, and how the ambiguities are resolved by comparing the results with CB, which leads to the method referred to as `symmetrized classicala in Fig. 2. Brie#y, symmetrization implies using expressions for m and e in Eq. (3.10) that are symmetrical in the initial and "nal velocity and angular momentum. There is no ambiguity in the symmetrization of m:
A
B
e2(Z!1) 1 1 m,g !g " ! , (3.13) & * 4pe + v v 0 & * where v is the initial (asymptotic) velocity and v is the "nal (asymptotic) velocity after the * & excitation/deexcitation. Here g refers to the Coulomb parameter: e2(Z!1) g" . (3.14) 4pe +v 0 The ambiguity is in the symmetrization of the eccentricity. When we compute the contribution of the ¸th partial wave, we need to evaluate the eccentricity e at the minimum and maximum impact parameters ¸+/mv and (¸#1)+/mv. Thus, for example, with the minimum impact parameter we have an eccentricity of
S A B S
¸2 ¸+ 2 e " 1# " 1# . .*/ g2 mvs
(3.15)
One way of achieving symmetry, which is found to be successful in matching the classical and quantum results in the asymptotic regime, is to set g2"g g (3.16) * & in Eq. (3.15), with the subscripts i and f corresponding to v and v de"ned above. * & However, there are di!erent forms of the eccentricity variable, depending on which pair of values of ¸2 is associated with the ¸th partial wave. Some possibilities are: compute ¸2"l2, where l is an integer, and associate the range lPl#1 with the lth partial wave; the same, except associate this range with partial wave l#1/2; compute ¸2"l(l#1) and associate the range lPl#1 with the lth partial wave; and many others. Excellent agreement is found between Coulomb}Bethe and either of these two choices: ¸2"l2 and lPl#1 for the l#1/2 partial wave, or ¸2"l (l#1) and l!1Pl for the lth partial wave. The numerical results for these two choices are identical for all practical purposes. The "rst one is what is actually plotted as `optimal symmetrized classicala in Fig. 2, and it is seen to agree excellently with CB, both for the cross section (Fig. 2) and for the line width (Fig. 1). Note that the symmetrized classical results show no indication of disagreement with the corresponding quantal results for partial waves 42p. The disagreement shows up as the missing value for ¸"0, and this is due to the ¸$1 ambiguity caused by the quantization of ¸.
2 Energy and angular momentum are conserved for the classical orbits, so to describe an inelastic process such as 2s}2p excitation, the energy and angular momentum must be averages of the initial and "nal values.
22
S. Alexiou et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 65 (2000) 15}22
4. Conclusions The results of our comparisons of the inelastic component of the line width show that the SC agrees with the `independenta Coulomb}Bethe (CB) results for the observable, i.e., the FWHM, as a function of partial wave, as shown in Fig. 1. Unfortunately, we do not currently have such a partial wave breakdown for the QM calculations. We have also found from a comparison of excitation cross sections that: (A) There is no evidence that the SC methods are invalid below ¸"2p. (B) Coulomb}Bethe and the SC agree for the width as a function of ¸. (C) Even though (B) is true, the SC method needs to be symmetrized in order to obtain the same large-¸ partial wave cross sections as the QM methods. (D) Symmetrized SC and CB agree within 6% for the partial wave line width contributions at ¸"1 and 2, and more closely than that for all other partial waves up to ¸"10. For ¸'10 the di!erence increases, but at this point the contributions are negligible. (E) The disagreement between QM * R-matrix and Coulomb}Born * calculations and nonperturbative semiclassical calculations for the line width can be attributed to the treatment of strong collisions. In conclusion, we point out that there is further work required to resolve this issue. It is now apparent that the di!erence between NPSC and the close-coupling and Coulomb}Born results is not due to the de"ciency of classical theory used in the NPSC, since Coulomb}Bethe, a quantum method, agrees substantially with NPSC. Rather, we should investigate the di!erent ways that the strong collisions are treated. Thus, a detailed new calculation of the QM method, performed in a manner that allows extensive detailed comparison, is being undertaken.
Acknowledgements We are grateful to H.R. Griem and Yu. Ralchenko for discussions and to Yu. Ralchenko for providing the raw data for the partial wave breakdown.
References [1] [2] [3] [4] [5] [6] [7] [8] [9]
Griem HR. Spectral line broadening by plasmas. New York: Academic Press, 1974. Glenzer SH, Kunze HJ. Phys Rev A 1996;53:2225. Alexiou S. Phys Rev Lett 1995;75:3406. Griem HR, Ralchenko Yu, Bray I. Phys Rev E 1997;56:7186. [GRB]. Baranger M. Phys Rev 1958;111:481. Baranger M. Phys Rev 1958;112:855. Seaton MJ. J Phys B 1988;21:3033. PoqueH russe A. Phys Lett 1976;59A:438. PoqueH russe A, Alexiou S. JQSRT 1999;61:209 Alder K, et al. Rev Mod Phys 1956;28:434.