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Elastic scattering of photons by bound electrons in hot and dense aluminum plasmas
J. Qumt. Spec~rosc. Rodiar. Transfer Vol. 40, No. 5, pp. 613417, Printed in Great Britain
1988
0022-4073/88
$3.00 + 0.00
Pergamon Press plc
ELASTIC SCATTERING OF PHOTONS BY BOUND ELECTRONS IN HOT AND DENSE ALUMINUM PLASMAS A. BECHLER,~ Y. KUANG, and R. H. PRATT Department of Physics and Astronomy, University of Pittsburgh, Pittsburgh, PA 15260, U.S.A. (Received
28 December
1987; received for publication 24 May 1988)
Abstract-We present results of an exploratory calculation of elastic cross sections for forward scattering of photons by bound electrons in hot (500eV) and dense (lo*’ and 1023cn-3) aluminum plasmas. We compare detailed configuration (DC), non-local-thermodynamicequilibrium average atom (non-LTE-AA), and LTE-AA (Thomas-Fermi) predictions. We study the photon energy region 0.3-S keV.
Extinction of radiation in hot and dense plasma is due mainly to absorption and scattering.’ The scattering of radiation is also important in the modelling of its transport in hot and dense plasma, and in the description of plasma properties. How does scattering in such an environment differ from scattering for isolated atoms and ions? In this paper, we present some results of calculations of cross sections for scattering of radiation in an aluminum plasma at a temperature of 500 eV and ion densities of 102’ and 1O23cmd3. Under these circumstances, the plasma consists mainly of hydrogen- and helium-like ions with a small amount of lithium-like species. We have performed a calculation of the scattering cross section for three models of hot and dense plasma: detailed configuration (DC), non-LTE average atom (AA) and the LTE Thomas-Fermi average model (TF). For our purpose of studying the difference between DC and AA scattering cross sections, we use the occupation numbers of various ionic states in these plasmas, as a function of ion temperature and density, obtained by Duston and Davis.2 In Tables 1 and 2, we give these numbers for the temperature and densities under consideration in this note. In our comparison of models, we shall examine DCA cross sections, averaged over the species present in the plasma (populations of ionic species in their various ground and excited states), in comparisons with the cross section in scattering from an average atom in such models. In general, one may expect that the results of DC and non-LTE-AA calculations will agree for higher energies and lower densities, particularly well above the K-shell ionization threshold. At such energies, the details of atomic structure are less important and, for forward scattering, what matters is the total number of bound electrons Z,. At low densities, only a few ionic species of neighboring charge will be present. Since shell occupation numbers for a non-localthermodynamic-equilibrium average atom (non-LTE-AA) are given as averages of DC configurations,’ the scattering cross sections for the two models will be comparable, because the spread about the average charge is small. At higher density, the spread will grow and also the difference between the models. For the LTE-TF-AA model, since the electron occupation numbers can be very different from DC and non-LTE-AA models at low density (e.g. 1020cme3), one would expect that, in such cases, the cross section will remain very different, even for higher energies. As the density increases, the TF electron occupation numbers approach the non-LTE-AA and DC occupation numbers, the two AA model cross sections will be closer to each other for higher energies at high density. On the other hand, for photon energies below the K-ionization edge of any of the ionic species contributing to scattering, we expect large differences between these three types of results. This is due to scattering resonances, which occur at the energies corresponding to bound-bound transitions, which for different ions occur at different photon energies. There will be many more resonances in the DC calculation. Therefore, for instance, for a photon energy which is close to resonance in a hydrogen-like ion, the DC calculation for the plasma will lead to a result which tPermanent
address: Institute of Physics, University of Szczecin, Wielkopolska 15 70-451 Szczecin, Poland. 613
A. BECHLER et al
614
Table 1. Occupation numbers of ions in DCA Al plasmas at T = 500 eV and densities of 10zo and 1O23 cmm3, as calculated by Duston and Davis (private communication, see Ref. 2). These densities are relative numbers with respect to the total ion density. Relative Occupation at Ion State 1oz3 cmd
totally ionized
0.014
0.375
1s
0.314
0.151
AI XIII
Al XII
Al XI
2s
4.3x10d
8.5x10d
3P
2sx10-
9.9x10d
1s’
0.660
0.241
ls2s
1.06x10J
1.098x10-’
ls2p
2.79x10-’
3.208x10-’
ls3p
8.21x10+
5.46ox1o-2
lsZ2s
1.61x10-’
1.28~10-~
1sz2p
4.26~10~’
3.67x10-’
ls23s
1.61x10-I
7.63x10-’
1s’3F
4.52~10~
2.26~10~’
ls’3d
6.37~10~
3.75x10-?
Table 2. Occupation numbers of ions in AA Al plasmas al T = 500 eV and densities of 10” and 10z3 cme3. These represenl the number of bound electrons in a given shell of the average ion. Non-LTE-AA results are obtained from Table 1; the TF-AA results are the same as in Ref. 3. Slight differences between the shell occupation numbers for the density 1O23cmand averages of the occupation numbers of ions at the same density (Table I) are due to the fact that there are morz DC-states at higher density which have not been shown ir Table 1. Due to relatively small occupations of these states theil contribution to the scattering cross section in the DC model is
Numbers of Electrons at Model
Shell 10” cm-’
10” cmJ
non-LTE
n=l
1.653
1.06
AA
n=2
9.8x10-’
0.07
n=3
2.0x10d
0.16
1.665
1.290
n=l
0.189
1.26
n=2
0.039
0.22
n=3
0.046
0.18
0.274
1.66
total number of bound electrons
TF-AA
total number of bound electrons
greater than that of both the AA- and TF-calculations for scattering from the plasma at the same energy. The same is true for an energy which is resonant in the helium-like ion. There are also scattering resonances which occur at the energies corresponding to bound-bound transitions in the fictitious average atom, which are in general located at different energies than resonances in either of the DC-configuration species. We may therefore expect a complicated pattern of scattering cross sections below the K-ionization threshold. To obtain a reliable comparison between results for various plasma models, the finite width of these resonances should be taken into account. The numerical code used for the present calculation is not suitable for this purpose, for photon energies below the K-edge we could obtain quantitative estimates of the scattering cross section only for energies not very close to a resonant energy. Well below the K-resonance region and above the L-edge, we expect a situation somewhat like that above the K-edge, determined by the number of electrons bound in the L- and M-shells. Since most electrons of these ions are in the K-shell, the cross sections above the L-edge are much smaller. The real scattering amplitude of the K-shell rises toward the first real resonance and is opposite in sign to the higher shell amplitudes. For a certain energy, the total real amplitude vanishes, leading to a dip (Fano profile) in the cross section, in advance of the resonance region. In Fig. 1, we show the results of our calculations for the forward scattering cross section at an ion density p = 10” cme3, where the DC cross section is the average of the specific ion-state cross sections. We do not attempt to show the complex resonance region. The photon energy varies from 0.5 to 5 keV. Above the K-edge, the LTE-TF-AA results are several orders of magnitude smaller than the DC results, due to the much smaller occupation number of the K-shell. For low photon energies and well below the two-electron ion-scattering resonance, which occurs at the energy corresponding to the K-L transition, the DC and non-LTE-AA results are similar in shape, but relative differences between the two types of calculation are about 2&30%. These differences is much
Elastic scattering of photons
615
c5
c
I
I
?
ii
Al, I
T=50oeV,
p =~o~~cm-~.
*.........., ----Pm,_, -----___
*-*****- DC model
--
non-LTE-AAmodel
-
LTE-TF-AA
-------
H-like
-*?? -6g 10
I
0.5
z L
I 1.4
I
1 2.3
Photon
Isolated
I 3.2
I Energy
model
Ion in Plasma H-like
I
Ion
I 4.1
I
I 5.0
(keV)
Fig. 1. The forward scattering cross sections are shown in unit bam/sr @I/W)for an aluminum plasma as a function of photon energy for the DC, non-LTE-AA and LTE-TF-AA models, at T = 500 eV and p = 10rscm-3. Also shown for comparison are cross sections for an isolated hydrogen-like ion and a hydrogen-like ion in this plasma. The region of the resonance structure below this threshold is not shown.
become larger with increasing energy, as the photon energy becomes closer to the two-electron ion K-L bound-bound transition at 1.508 keV. In the resonant region, the three different model results may differ by an order of magnitude; to perform a realistic calculation in this region, one must include finite resonance widths. Below the scattering resonance region, the cross sections dip (Fano profile) due to cancellation between resonance and background amplitudes, as discussed before. At low energies, the contribution from the K-shell decreases, and the L-shell starts to dominate the scattering. However, the average occupation number of the L-shell is small, which results in a small value of the cross section. For low density, p = 1020cm-3, the occupation number of the L-shell in the non-LTE-AA model is so small as compared to the K-shell occupation, that dominance of the L-shell contribution does not occur above the L-edge ionization threshold. As a result, the non-LTE-AA cross section is monotonic, contrary to the LTE-TF-AA result, which does show a dip in the low-density case because it has larger L-shell occupation numbers. By
ki
,
L
!
\
P
Al,
T = 500eV,
p = 1023crn-3
.
-‘-.-.-~-.-._._.x
X
: 3
2; LL
$6 0.3
1.1
non-LTE-AAmodel
-.-
LTE-TF-AA
2.7
1.9 Photon
DC model
-
Energy
:
model
3.5
4.3
( keV)
Fig. 2. As in Fig. 1, the forward scattering cross sections in a aluminum plasma are shown as a function of photon energy, for the DC, non-LTE-AA and LTE-TF-AA models, at T = 5OOeV but for p = Wcm-3.
616
A. BECHLER et al.
contrast, for high density (Fig. 2), a dip occurs above the L-shell ionization edge for both models, since the L-shell occupation numbers are relatively large. The DC and non-LTE-AA results become quite close to each other above the K-edge corresponding to the one electron ion. The difference between these two approaches does not exceed 6%. The LTE-TF-AA results are about two orders of magnitude smaller than DC and non-LTE-AA results in this region, since the electron occupation numbers are an order of magnitude smaller and the Rayleigh scattering cross section is proportional to Z,$ with Z, representing the number of bound electrons. We also show for comparison the scattering cross sections for one-electron ions in their ground state and in the low-density plasma, for analogous hydrogen-like isolated ions. Plasma effects at this density are not large for hydrogenic ions. We see from Tables 1 and 2 that, for an ion density of 10” cmp3, the only shell which contributes significantly to scattering is the once or twice filled K-shell, both in the DC and non-LTE-AA models, since the occupation numbers of other or excited ionic states for the two models are some orders of magnitude smaller. This situation changes for higher densities and, for instance, for the density 1O23cme3, the occupation numbers of excited states in two-electron ions are only one order of magnitude smaller than the occupation numbers for their ground state. Since there are many such states, their contribution to the scattering cross section can not be neglected. Also, a contribution from the lithium-like ion can not be neglected for higher densities (it contributes as much as 20% to the cross section in the DC calculation); although occupation numbers in this case are N lo-*, there are now more occupied states which contribute to scattering. Since the calculation for the DC model and for a density of 1O23cmp3 requires much computation time, we performed primarily the non-LTE-AA and LTE-TF-AA calculations, with two comparison DC points. These results are shown in Fig. 2. As expected in this higher density case, for photon energies above the K-threshold or below the K-L resonant energy, the cross sections of the LTE-TF-AA model are closer to that of non-LTE-AA than they were in the low ion density (10”) case, since the electron occupation numbers are much closer to each other than in the low-density case. As the density changes from lo*’ to 10z3cme3, the dominant DC and non-LTE-AA electron-occupation numbers do not change by more than a factor of two, while the TF occupation numbers increase almost an order of magnitude and actually become a little larger than in the non-LTE-AA case. However, in the resonant region, due to differences in resonance and threshold positions for these two models, the cross sections show complex variations, and the cross section differences can be orders of magnitude, as in the corresponding low-density case. Although we have not calculated the full curve of the cross section for the DC model with density 1O23cmm3, since many different configurations are now important, as shown in Table 1, we have calculated two high-energy (well above the K-shell ionization threshold) cross sections. There are quite large differences between the DC and non-LTE-AA types of calculations in these cases. For example, the DC forward scattering cross section for a photon energy of 3.5 keV is 2.9 x 10-l (barn/sr), whereas the non-LTE-AA result 2.1 x 10-l (barn/sr) is smaller by about 30%. The LTE-TF-AA calculation gives 3.1 x 10-l (barn/sr), which is closer to the DC result. Generally, as the density increases, the difference between the DC and non-LTE-AA models at high photon energy will increase. This result may be explained as follows. When we calculate the DC-model scattering cross sections, we add amplitudes coherently for electrons belonging to a given atomic species. Next we calculate the weighted average of the individual species cross sections, with the weights given by (fractional) occupation numbers. Since the individual forward cross sections are proportional to Zi for a given ion, the DC model cross section is proportional to (Zi). On the other hand, in the non-LTE-AA calculation, we substitute a hypothetical average atom with fractional shell occupation numbers for all ions contributing to the scattering. To find the amplitude, we calculate the weighted average of amplitudes for each shell, with weights equal to the (fractional) shell occupation numbers. The amplitude is therefore proportional to the average atom charge (Z,), and the cross section is proportional to (Z,)‘. Thus, the differences between the forward scattering cross section for the DC and non-LTE-AA models are due to the differences between (Zi), proportional to the average of the cross sections of ionic charges Z,,, and (ZiJ2, proportional to the cross section of a species of averaged ionic charge. In general, (Zt) > (Z&*, as can be derived from the Cauchy-Schwarz-Buniakowsky inequality.4 Therefore, DC results will
Elastic scattering of photons
617
be larger than non-LTE-AA results. The difference between (Zt) and (Z$ will grow as the charges between significant populated states become more different.? In the low-density case, only hydrogen- and helium-like species contribute, and the difference between (Z,)* and (Zi) is only about 8%. In the high density case, all of the species are important and can not be neglected; the differences between (Zi) and (Z,,)’ are about 35%, which is about equal to the difference between the high-energy forward scattering cross sections. As the density increases, the non-LTE-AA results become smaller than the DC results. Thus, contrary to common wisdom, we can not expect that the non-LTE-AA and DC results will converge at high density. Also, as the density increases, the LTE-TF-AA population approaches the non-LTE-AA distribution. Consequently the LTE-TF-AA results will become closer to the non-LTE-AA results. In the high-density case (10z3cme3), the LTE-TF-AA population is actually larger than that of the non-LTE-AA and the LTE-TF-AA results are closer to the DC than the non-LTE-AA results. A similar trend has been observed for photoionization in the same plasmas: the LTE-TF-AA results became closer to the DC calculations with increasing density over the same range of energy.3 Acknowledgement-This work was supported, in part, by the National Science Foundation, under Grant PHY-8704088. tWe may understand this result by providing a derivation of the inequality specialized to our situation. If we let q be the relative population of ions of charge Z,, so that q_,ni = 1, then the high-energy forward cross section in the DCA plasma will be determined by (Z:) = Z:_,n,Z&, while the corresponding cross section of the corresponding AA plasma will be determined in terms of the charge of the average ion (ZJ = Z;=, niZb, as (ZJ2 = (Z;_,,niZ,)z. The difference between these two cross sections is thus determined by (Zi) - (Z$ = Z;=,niZL - (X:=,n,Z,)* = 1/2(8;_,4Z& + Z_,njZij) - Zl .=,n,n,ZbiZbl= l/2 Z:,_,(n,n,Z& + ninjZij) - X:,,,n,n,Z,,Z,, = l/2 qj_,ninj(Z,, - Zb1)2> 0. We see that, the greater the charge differences between ionic states with significant occupation, the greater the magnitude of the inequality. REFERENCES 1. H. F. Argo and W. F. Huebner, JQ.!MT 16, 1091 (1976). 2. D. Duston and J. Davis, Phys. Reo. A 21, 1664 (1980); 23, 2602 (1981). 3. D. Salzmann, R. Y. Yin, and R. H. Pratt, Phys. Rev. A 32, 3627 (1985). 4. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products,
New York, NY (1980).
p. 1093, Academic Press,
1988
0022-4073/88
$3.00 + 0.00
Pergamon Press plc
ELASTIC SCATTERING OF PHOTONS BY BOUND ELECTRONS IN HOT AND DENSE ALUMINUM PLASMAS A. BECHLER,~ Y. KUANG, and R. H. PRATT Department of Physics and Astronomy, University of Pittsburgh, Pittsburgh, PA 15260, U.S.A. (Received
28 December
1987; received for publication 24 May 1988)
Abstract-We present results of an exploratory calculation of elastic cross sections for forward scattering of photons by bound electrons in hot (500eV) and dense (lo*’ and 1023cn-3) aluminum plasmas. We compare detailed configuration (DC), non-local-thermodynamicequilibrium average atom (non-LTE-AA), and LTE-AA (Thomas-Fermi) predictions. We study the photon energy region 0.3-S keV.
Extinction of radiation in hot and dense plasma is due mainly to absorption and scattering.’ The scattering of radiation is also important in the modelling of its transport in hot and dense plasma, and in the description of plasma properties. How does scattering in such an environment differ from scattering for isolated atoms and ions? In this paper, we present some results of calculations of cross sections for scattering of radiation in an aluminum plasma at a temperature of 500 eV and ion densities of 102’ and 1O23cmd3. Under these circumstances, the plasma consists mainly of hydrogen- and helium-like ions with a small amount of lithium-like species. We have performed a calculation of the scattering cross section for three models of hot and dense plasma: detailed configuration (DC), non-LTE average atom (AA) and the LTE Thomas-Fermi average model (TF). For our purpose of studying the difference between DC and AA scattering cross sections, we use the occupation numbers of various ionic states in these plasmas, as a function of ion temperature and density, obtained by Duston and Davis.2 In Tables 1 and 2, we give these numbers for the temperature and densities under consideration in this note. In our comparison of models, we shall examine DCA cross sections, averaged over the species present in the plasma (populations of ionic species in their various ground and excited states), in comparisons with the cross section in scattering from an average atom in such models. In general, one may expect that the results of DC and non-LTE-AA calculations will agree for higher energies and lower densities, particularly well above the K-shell ionization threshold. At such energies, the details of atomic structure are less important and, for forward scattering, what matters is the total number of bound electrons Z,. At low densities, only a few ionic species of neighboring charge will be present. Since shell occupation numbers for a non-localthermodynamic-equilibrium average atom (non-LTE-AA) are given as averages of DC configurations,’ the scattering cross sections for the two models will be comparable, because the spread about the average charge is small. At higher density, the spread will grow and also the difference between the models. For the LTE-TF-AA model, since the electron occupation numbers can be very different from DC and non-LTE-AA models at low density (e.g. 1020cme3), one would expect that, in such cases, the cross section will remain very different, even for higher energies. As the density increases, the TF electron occupation numbers approach the non-LTE-AA and DC occupation numbers, the two AA model cross sections will be closer to each other for higher energies at high density. On the other hand, for photon energies below the K-ionization edge of any of the ionic species contributing to scattering, we expect large differences between these three types of results. This is due to scattering resonances, which occur at the energies corresponding to bound-bound transitions, which for different ions occur at different photon energies. There will be many more resonances in the DC calculation. Therefore, for instance, for a photon energy which is close to resonance in a hydrogen-like ion, the DC calculation for the plasma will lead to a result which tPermanent
address: Institute of Physics, University of Szczecin, Wielkopolska 15 70-451 Szczecin, Poland. 613
A. BECHLER et al
614
Table 1. Occupation numbers of ions in DCA Al plasmas at T = 500 eV and densities of 10zo and 1O23 cmm3, as calculated by Duston and Davis (private communication, see Ref. 2). These densities are relative numbers with respect to the total ion density. Relative Occupation at Ion State 1oz3 cmd
totally ionized
0.014
0.375
1s
0.314
0.151
AI XIII
Al XII
Al XI
2s
4.3x10d
8.5x10d
3P
2sx10-
9.9x10d
1s’
0.660
0.241
ls2s
1.06x10J
1.098x10-’
ls2p
2.79x10-’
3.208x10-’
ls3p
8.21x10+
5.46ox1o-2
lsZ2s
1.61x10-’
1.28~10-~
1sz2p
4.26~10~’
3.67x10-’
ls23s
1.61x10-I
7.63x10-’
1s’3F
4.52~10~
2.26~10~’
ls’3d
6.37~10~
3.75x10-?
Table 2. Occupation numbers of ions in AA Al plasmas al T = 500 eV and densities of 10” and 10z3 cme3. These represenl the number of bound electrons in a given shell of the average ion. Non-LTE-AA results are obtained from Table 1; the TF-AA results are the same as in Ref. 3. Slight differences between the shell occupation numbers for the density 1O23cmand averages of the occupation numbers of ions at the same density (Table I) are due to the fact that there are morz DC-states at higher density which have not been shown ir Table 1. Due to relatively small occupations of these states theil contribution to the scattering cross section in the DC model is
Numbers of Electrons at Model
Shell 10” cm-’
10” cmJ
non-LTE
n=l
1.653
1.06
AA
n=2
9.8x10-’
0.07
n=3
2.0x10d
0.16
1.665
1.290
n=l
0.189
1.26
n=2
0.039
0.22
n=3
0.046
0.18
0.274
1.66
total number of bound electrons
TF-AA
total number of bound electrons
greater than that of both the AA- and TF-calculations for scattering from the plasma at the same energy. The same is true for an energy which is resonant in the helium-like ion. There are also scattering resonances which occur at the energies corresponding to bound-bound transitions in the fictitious average atom, which are in general located at different energies than resonances in either of the DC-configuration species. We may therefore expect a complicated pattern of scattering cross sections below the K-ionization threshold. To obtain a reliable comparison between results for various plasma models, the finite width of these resonances should be taken into account. The numerical code used for the present calculation is not suitable for this purpose, for photon energies below the K-edge we could obtain quantitative estimates of the scattering cross section only for energies not very close to a resonant energy. Well below the K-resonance region and above the L-edge, we expect a situation somewhat like that above the K-edge, determined by the number of electrons bound in the L- and M-shells. Since most electrons of these ions are in the K-shell, the cross sections above the L-edge are much smaller. The real scattering amplitude of the K-shell rises toward the first real resonance and is opposite in sign to the higher shell amplitudes. For a certain energy, the total real amplitude vanishes, leading to a dip (Fano profile) in the cross section, in advance of the resonance region. In Fig. 1, we show the results of our calculations for the forward scattering cross section at an ion density p = 10” cme3, where the DC cross section is the average of the specific ion-state cross sections. We do not attempt to show the complex resonance region. The photon energy varies from 0.5 to 5 keV. Above the K-edge, the LTE-TF-AA results are several orders of magnitude smaller than the DC results, due to the much smaller occupation number of the K-shell. For low photon energies and well below the two-electron ion-scattering resonance, which occurs at the energy corresponding to the K-L transition, the DC and non-LTE-AA results are similar in shape, but relative differences between the two types of calculation are about 2&30%. These differences is much
Elastic scattering of photons
615
c5
c
I
I
?
ii
Al, I
T=50oeV,
p =~o~~cm-~.
*.........., ----Pm,_, -----___
*-*****- DC model
--
non-LTE-AAmodel
-
LTE-TF-AA
-------
H-like
-*?? -6g 10
I
0.5
z L
I 1.4
I
1 2.3
Photon
Isolated
I 3.2
I Energy
model
Ion in Plasma H-like
I
Ion
I 4.1
I
I 5.0
(keV)
Fig. 1. The forward scattering cross sections are shown in unit bam/sr @I/W)for an aluminum plasma as a function of photon energy for the DC, non-LTE-AA and LTE-TF-AA models, at T = 500 eV and p = 10rscm-3. Also shown for comparison are cross sections for an isolated hydrogen-like ion and a hydrogen-like ion in this plasma. The region of the resonance structure below this threshold is not shown.
become larger with increasing energy, as the photon energy becomes closer to the two-electron ion K-L bound-bound transition at 1.508 keV. In the resonant region, the three different model results may differ by an order of magnitude; to perform a realistic calculation in this region, one must include finite resonance widths. Below the scattering resonance region, the cross sections dip (Fano profile) due to cancellation between resonance and background amplitudes, as discussed before. At low energies, the contribution from the K-shell decreases, and the L-shell starts to dominate the scattering. However, the average occupation number of the L-shell is small, which results in a small value of the cross section. For low density, p = 1020cm-3, the occupation number of the L-shell in the non-LTE-AA model is so small as compared to the K-shell occupation, that dominance of the L-shell contribution does not occur above the L-edge ionization threshold. As a result, the non-LTE-AA cross section is monotonic, contrary to the LTE-TF-AA result, which does show a dip in the low-density case because it has larger L-shell occupation numbers. By
ki
,
L
!
\
P
Al,
T = 500eV,
p = 1023crn-3
.
-‘-.-.-~-.-._._.x
X
: 3
2; LL
$6 0.3
1.1
non-LTE-AAmodel
-.-
LTE-TF-AA
2.7
1.9 Photon
DC model
-
Energy
:
model
3.5
4.3
( keV)
Fig. 2. As in Fig. 1, the forward scattering cross sections in a aluminum plasma are shown as a function of photon energy, for the DC, non-LTE-AA and LTE-TF-AA models, at T = 5OOeV but for p = Wcm-3.
616
A. BECHLER et al.
contrast, for high density (Fig. 2), a dip occurs above the L-shell ionization edge for both models, since the L-shell occupation numbers are relatively large. The DC and non-LTE-AA results become quite close to each other above the K-edge corresponding to the one electron ion. The difference between these two approaches does not exceed 6%. The LTE-TF-AA results are about two orders of magnitude smaller than DC and non-LTE-AA results in this region, since the electron occupation numbers are an order of magnitude smaller and the Rayleigh scattering cross section is proportional to Z,$ with Z, representing the number of bound electrons. We also show for comparison the scattering cross sections for one-electron ions in their ground state and in the low-density plasma, for analogous hydrogen-like isolated ions. Plasma effects at this density are not large for hydrogenic ions. We see from Tables 1 and 2 that, for an ion density of 10” cmp3, the only shell which contributes significantly to scattering is the once or twice filled K-shell, both in the DC and non-LTE-AA models, since the occupation numbers of other or excited ionic states for the two models are some orders of magnitude smaller. This situation changes for higher densities and, for instance, for the density 1O23cme3, the occupation numbers of excited states in two-electron ions are only one order of magnitude smaller than the occupation numbers for their ground state. Since there are many such states, their contribution to the scattering cross section can not be neglected. Also, a contribution from the lithium-like ion can not be neglected for higher densities (it contributes as much as 20% to the cross section in the DC calculation); although occupation numbers in this case are N lo-*, there are now more occupied states which contribute to scattering. Since the calculation for the DC model and for a density of 1O23cmp3 requires much computation time, we performed primarily the non-LTE-AA and LTE-TF-AA calculations, with two comparison DC points. These results are shown in Fig. 2. As expected in this higher density case, for photon energies above the K-threshold or below the K-L resonant energy, the cross sections of the LTE-TF-AA model are closer to that of non-LTE-AA than they were in the low ion density (10”) case, since the electron occupation numbers are much closer to each other than in the low-density case. As the density changes from lo*’ to 10z3cme3, the dominant DC and non-LTE-AA electron-occupation numbers do not change by more than a factor of two, while the TF occupation numbers increase almost an order of magnitude and actually become a little larger than in the non-LTE-AA case. However, in the resonant region, due to differences in resonance and threshold positions for these two models, the cross sections show complex variations, and the cross section differences can be orders of magnitude, as in the corresponding low-density case. Although we have not calculated the full curve of the cross section for the DC model with density 1O23cmm3, since many different configurations are now important, as shown in Table 1, we have calculated two high-energy (well above the K-shell ionization threshold) cross sections. There are quite large differences between the DC and non-LTE-AA types of calculations in these cases. For example, the DC forward scattering cross section for a photon energy of 3.5 keV is 2.9 x 10-l (barn/sr), whereas the non-LTE-AA result 2.1 x 10-l (barn/sr) is smaller by about 30%. The LTE-TF-AA calculation gives 3.1 x 10-l (barn/sr), which is closer to the DC result. Generally, as the density increases, the difference between the DC and non-LTE-AA models at high photon energy will increase. This result may be explained as follows. When we calculate the DC-model scattering cross sections, we add amplitudes coherently for electrons belonging to a given atomic species. Next we calculate the weighted average of the individual species cross sections, with the weights given by (fractional) occupation numbers. Since the individual forward cross sections are proportional to Zi for a given ion, the DC model cross section is proportional to (Zi). On the other hand, in the non-LTE-AA calculation, we substitute a hypothetical average atom with fractional shell occupation numbers for all ions contributing to the scattering. To find the amplitude, we calculate the weighted average of amplitudes for each shell, with weights equal to the (fractional) shell occupation numbers. The amplitude is therefore proportional to the average atom charge (Z,), and the cross section is proportional to (Z,)‘. Thus, the differences between the forward scattering cross section for the DC and non-LTE-AA models are due to the differences between (Zi), proportional to the average of the cross sections of ionic charges Z,,, and (ZiJ2, proportional to the cross section of a species of averaged ionic charge. In general, (Zt) > (Z&*, as can be derived from the Cauchy-Schwarz-Buniakowsky inequality.4 Therefore, DC results will
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be larger than non-LTE-AA results. The difference between (Zt) and (Z$ will grow as the charges between significant populated states become more different.? In the low-density case, only hydrogen- and helium-like species contribute, and the difference between (Z,)* and (Zi) is only about 8%. In the high density case, all of the species are important and can not be neglected; the differences between (Zi) and (Z,,)’ are about 35%, which is about equal to the difference between the high-energy forward scattering cross sections. As the density increases, the non-LTE-AA results become smaller than the DC results. Thus, contrary to common wisdom, we can not expect that the non-LTE-AA and DC results will converge at high density. Also, as the density increases, the LTE-TF-AA population approaches the non-LTE-AA distribution. Consequently the LTE-TF-AA results will become closer to the non-LTE-AA results. In the high-density case (10z3cme3), the LTE-TF-AA population is actually larger than that of the non-LTE-AA and the LTE-TF-AA results are closer to the DC than the non-LTE-AA results. A similar trend has been observed for photoionization in the same plasmas: the LTE-TF-AA results became closer to the DC calculations with increasing density over the same range of energy.3 Acknowledgement-This work was supported, in part, by the National Science Foundation, under Grant PHY-8704088. tWe may understand this result by providing a derivation of the inequality specialized to our situation. If we let q be the relative population of ions of charge Z,, so that q_,ni = 1, then the high-energy forward cross section in the DCA plasma will be determined by (Z:) = Z:_,n,Z&, while the corresponding cross section of the corresponding AA plasma will be determined in terms of the charge of the average ion (ZJ = Z;=, niZb, as (ZJ2 = (Z;_,,niZ,)z. The difference between these two cross sections is thus determined by (Zi) - (Z$ = Z;=,niZL - (X:=,n,Z,)* = 1/2(8;_,4Z& + Z_,njZij) - Zl .=,n,n,ZbiZbl= l/2 Z:,_,(n,n,Z& + ninjZij) - X:,,,n,n,Z,,Z,, = l/2 qj_,ninj(Z,, - Zb1)2> 0. We see that, the greater the charge differences between ionic states with significant occupation, the greater the magnitude of the inequality. REFERENCES 1. H. F. Argo and W. F. Huebner, JQ.!MT 16, 1091 (1976). 2. D. Duston and J. Davis, Phys. Reo. A 21, 1664 (1980); 23, 2602 (1981). 3. D. Salzmann, R. Y. Yin, and R. H. Pratt, Phys. Rev. A 32, 3627 (1985). 4. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products,
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p. 1093, Academic Press,