Large enhancement in high-energy photoionization of Fe XVII and missing continuum plasma opacity
LLarge enhancement in high-energy photoionization of Fe xvii and missing continuumplasma opacity Sultana N. Nahar ∗ and Anil K. Pradhan † Department of Astronomy, The Ohio State University, Columbus, Ohio 43210.
Aimed at solving the outstanding problem of solar opacity, and radiation transport plasma modelsin general, we report substantial photoabsorption in the high-energy regime due to atomic corephoto-excitations not heretofore considered. In extensive R-Matrix calculations of unprecedentedcomplexity for an important iron ion Fe xvii (Fe ), with a wave function expansion of 99 Fe xviii (Fe ) LS core states from n ≤ complexes (equivalent to 218 fine structure levels), we find: i) upto orders of magnitude enhancement in background photoionization cross sections, in addition tostrongly peaked photo-excitation-of-core resonances not considered in current opacity models, andii) demonstrate convergence with respect to successive core excitations. The resulting increase inthe monochromatic continuum, and 35% in the Rosseland Mean Opacity, are compared with the"higher-than-predicted" iron opacity measured at the Sandia Z-pinch fusion device at solar interiorconditions. PACS number(s) :32.80.Aa,32.80.Fb,32.80.Zb,95.30.Ky
Radiation transport and light-matter interactions de-pend fundamentally on plasma opacity, which in turnentails an intricate interplay of atomic and plasma ef-fects under local conditions. In astrophysics, monochro-matic and mean opacities, averaged over photon-particledistributions, are crucial to the determination of notonly emergent spectra but also the chemical composi-tion of the source. Surprisingly, elemental abundances ofthe Sun have been called into question in recent years,and revised downwards by up to ∼
50% for common ele-ments such as C, N, O and Ne [1]. But that has posedan outstanding problem for stellar interior and helio-seismic models [2]. Recent laboratory measurements ofmonochromatic iron opacity at the Sandia Z-pinch ma-chine of plasma created at conditions similar to thosenear the convection and radiative zone boundary disagreeconsiderably with current opacity models by 30-400% inthe wavelength range of 7-13Å [3]. Discrepancies areevident not only at and between energies where discretetransitions occur but, more surprisingly, the measuredopacity is substantially higher in the high-energy feature-less bound-free continua (or photoionization) of a fewabundant iron ions Fe xvii , Fe xviii and Fe xix . The upward revision of opacities [3] tends to support the re-vised lower solar abundances. Indeed, the measured Zopacity for iron alone enhances the solar mixture opac-ity of all abundant elements by ∼ n complexes. The present coupledchannel calculations show such jumps and enhancementsin the ( e + ion ) continua, and that coupling of thephotoionizing level and the residual ion system requirescompleteness in terms of converged background up to suf-ficiently high energies. In this
Letter we report suchcompleteness in calculations for the coupled Fe xvii → (e + Fe xviii ) system using the Breit-Pauli R-Matrix(BPRM) methodology, and demonstrate the aforemen-tioned enhancement up to convergence and relevance toopacity models. Our study focuses on Fe xvii which isthe highest contributor to opacity in the range T ∼ × . − . K) and electron densi-ties N e ∼ − cm − , though the ionization fractionsof the three dominant ions Fe xvii , Fe xviii and Fe xix in local thermodynamic equilibrium (LTE) at the Ztemperature 182 eV (13.4 Ry) are 0.19, 0.38 and 0.29respectively [3, 4].Monochromatic plasma opacity κ ν largely depends onradiation absorption through bound-bound (bb) photo-excitation and bound-free (bf) photoionization as follows: κ bbν ( i → j ) = πe mc N i f ij φ ν ; κ bfν = N i σ P I ( ν ) (1)where f ij is the oscillator strength, σ P I is the photoion-ization cross section, N i is the ion density, and φ ν isa profile factor. While oscillator strengths are discrete, a r X i v : . [ a s t r o - ph . S R ] J un the cross sections form a continuum replete with reso-nant and background features. The contribution of σ P I to the opacity may be expressed as an effective oscillatorstrength integrated over a given energy range ∆ E< f > ∆ E = ∞ (cid:88) k (cid:90) ∞ (cid:15) = E k df ik d(cid:15) d(cid:15), (2)where df ik /d(cid:15) represents the photoionization continuathat are now discretized according to excitation into ioncore levels k from an initial level i . This is a non-trivialextension. In principle, all possible bound levels of thecore ion contributing to the total (resonant+background)bound-free continuum, or σ ik from an initial bound level i into a final ( e + ion ) continuum with reference tothe excited core levels k of the residual ion, should beincluded until convergence and no further enhancement.Explicitly including successive core excitations in σ P I , < f > ∆ E = 14 παa o E n (cid:48) (cid:88) E ( k ) (cid:90) ∞ (cid:15) ≥ E k σ ik d(cid:15) k , (3)where E denotes the ground state of the core ion and E n (cid:48) the highest level included in the coupled-channel cal-culations. We calculated σ ik in the close coupling (CC)approximation using the R-matrix method [4]. We con-sider all possible dipole allowed transitions, which aretypically the dominant ones, into states formed by con-figurations with the outer electron in n =2, 3, 4 shells.Fig. 1 shows the photo-coupled [Fe xvii -Fe xviii ] sys-tem with a schematic diagram of the energy level struc-ture of the core Fe xviii ion included in the current cal-culations. The OP photoionization calculations for Fe xvii included only the ground complex of Fe xviii , a 2LS term expansion s s p ( P o ) and s s p ( S ) (abbreviated as 2CC) [5]. The OP and other opaci-ties calculations include these excited configurations, andinter-channel couplings and resonance structures, pertur-batively. But that, for example, does not account forasymmetric autoionization profiles or coupled core exci-tations affecting background opacity. In a previous work[4] we considered 30 LS terms or 60 coupled fine structurelevels up to the n = 3 complex; we refer to it as 30CC.We carried out the present calculations with a consider-ably larger expansion of the basis set up to 99 LS termsor 99CC, corresponding to 218 fine structure levels up to n = 4 . The size of the Hamiltonian to be diagonalizedin R-Matrix calculations increases as the total number ofchannels, hence the present calculations are more thanan order of magnitude larger than in [4]. We calculatedphotoionization cross sections for all 283 bound LS statesof Fe xvii up to n = 10 and l = 9, corresponding to454 fine structure levels obtained in the previous work[4]. The OP calculation [5] has a much smaller number,181 bound LS terms. Without loss of generality, andto optimize computational resources, we restrict the de-tailed calculations to 99 LS-coupled terms rather thana 218-level fine structure expansion, and use a coarser n = 2 n = 3 n = 4 E < 0E = 0E > 0 Fe XVII1 s s p Fe XVIII1 s s p
93 Ry 57 Ry 91 Ry1 s s p nl p d D p p S p lnl p lnl n = 2 n = 3 n = 4 Figure 1: Schematic representation of (e + Fe xviii ) → Fe xvii bound and autoionizing levels included in the coupledchannel calculations. The three results compared herein referto calculations including 2 LS terms of n = 2 complex un-der the OP, previous 30 LS terms (60 fine structure levels)of n = 2 , [4], and the present 99 LS terms with n = 2 , , in the total wave function expansion. Illustrative results pre-sented for photoionization in Fig. 3 of the 2 excited bound Fe xvii levels are highlighted (longer solid lines): p p S and p d D o . energy mesh than [4] to demonstrate the importance ofenhanced background cross sections with broad PEC res-onances and high- n core excitations; lower resolution perse should not lead to significant inaccuracy.Fig. 2 compares the σ P I of the Fe xvii ground statefrom three different calculations and core excitations: (a)OP — 2CC, b) 30-CC, LS terms with n ≤ [4], and(c) the present 99-CC, LS terms with n ≤ . Exceptfor differing resolution of resonances, all three σ P I havesimilar background cross sections; (b) and (c) are foundto have small enhancements of 2% and 9% over the OP(a). Owing to fine structure splittings and higher resolu-tion, the 30CC or 60-level cross sections (b), show moreprominent resonances than the 99CC (c). In contrast tothe ground state cross section, very large enhancementsare found for excited state photoionization of Fe xvii ,exemplified in Figs. 3, 4. Fig. 3a (top panel) show a de-tailed comparison of the OP monochromatic iron opacitywith that measured at the highest temperature/ densityachieved at the Sandia Z-machine, T = . × K and n e = 3 . × cc [3]. The measured opacity spectrabetween 7-13Å corresponds to σ P I in lower panels in theenergy range ∼
70 - 130 Ry. The prominent transitionarrays in the measurement are due to three dominant Feionization states: Fe xvii , Fe xviii and Fe xix . Themain points from the comparison are: (A) the measuredbackground is consistently higher than OP throughoutthe energy range measured 7.5 ≤ λ (Å) ≤ s p p S and s p d D o (Figs. 3b-c) are compared with the ob- Photon Energy (Ry) σ P I ( M b ) -1 b) 30-CC :Nahar et al 2011n=2 n=3BG: 2% higher100 120 140 16010 -1 c) 99-CC :Presentn=2 n=3 nBG: 9% higher10 -1 a) 2CC :OPn=2 Fe XVII + h ν -> Fe XVIII + eGround state: 2s S Figure 2: Comparison of photoionization cross sections σ PI of the ground state of Fe xvii s p ( S ) , using three differ-ent wave function expansions for the core ion Fe xviii : (a)2CC, the Opacity Project ([5], (b) 30CC (equivalent to 60fine structure levels of [4]), and (c) the present 99CC. Arrowspoint to the highest thresholds for n = 2 and 3 excitations(the n = 4 thresholds up to 183.57 Ry are beyond the rangeshown). ’BG’ is the background continuum at high energies;percentage enhancement relative to OP (a) is shown. served features in opacity (Figs. 3a). All three points(A,B,C) raised above can be accounted for by the presentresults. Most revealing is the enhancement of the back-ground (BG) cross sections, which is much higher thanOP throughout the high energy range by up to factorsof 40 and 50 at the last continuum energy for the 30CC n ≤ and the 99CC n ≤ expansions respectively forthe σ P I (2 s p p S ) level (3b). An even larger back-ground enhancement is seen for σ P I (2 s p d D o ) , upto factors of 410 and 540 respectively compared to OP(3c). The enhancements are clearly related to the onsetof the n = 3 and the n = 4 core excitation thresholds ofFe xviii (c.f. Fig. 1). Furthermore, σ P I decreases onlymarginally, even up to high energies, in the entire energyrange corresponding to the Z-measurements (3a). To es-timate the enhancement quantitatively, we calculated theaveraged oscillator strength < f > ∆ E (Eq. 3) using the σ P I (e.g. [7]) in the energy range of 70-130 Ry for thethree states, the ground s p ( S e ) and excited states s p p ( S e ) , s p d ( D o ) . The 2CC results with noexcited n -complexes in the Fe xviii expansion, underesti-mate the effective oscillator strength by up to two ordersof magnitude (with the exception of the ground state);the < f > values for the two excited states are 0.06, 1.81,3.10 and 0.01, 1.86, 1.93 for 2CC, 30CC and 99CC calcu-lations respectively. Including n = 4 core levels raises the background even below the n = 3 levels. Resonances grow O p ac it y ( x10 c m / g )
13 12 11 10 9 8 7200040006000800010000 OPExpt a) Z Fe-OPACITY
70 80 90 100 110 120 13010 -4 -3 -2 -1 Photon Energy (Ry)n=3 30-CC BG: 410x higherc) PHOTOIONIZATION ofFe XVII (2p D o ) n=499-CC BG: 540x higherOP: 2-CC10 -3 -2 -1 (cid:109) P I ( M b ) b) PHOTOIONIZATION ofFe XVII (2s S e )30-CC BG: 40 x Higher99-CC BG: 50 x HigherOP: 2CC Figure 3: Comparison of σ PI (panels b,c) of the excited s p p S state of Fe xvii from the three sets of core ex-citations: 2CC OP [5] (blue curves), 30CC [4], and 99CC(present), with features in the opacity measurements at San-dia Z-pinch [3] (panel a). The energy range is the same, butin unit of Å for (a) and Ry for (b-c). Large enhancementfactors relative to the OP (a) are marked. weaker, leading to a smooth background at the highest n = 4 thresholds in Figs. 3b,c. Hence there is unlikelyto be any significant enhancement by including higher n > excitation thresholds (with far more cost), and thecross sections appear to have converged.Another prominent feature is autoionizing resonancesdue to strong dipole PEC transition arrays (2p → n(cid:96) and (2p → n(cid:96) in the core ion Fe xviii , with thespectator electron n(cid:96) . Such PEC or Seaton resonances(e.g. [7]) remain large in photoionization of Rydberg lev-els with increasing n , even as the background decreasesand σ P I starts at lower ionization potentials. Fig. 4shows a few of the computed σ P I of the Rydberg serieslevels: s p ( P o ) np P with n = 3,4,5. Whereas, thebackground enhancement is large for n = 3 p and n = 4 p ,it is much smaller for n = 5 , while the resonances at PECpositions are higher. Photoionization of Rydberg levels isoften taken to decrease with energy as a power-law. Butfor L-shell ions while it is approximately true at highenergies as one approaches the K-edge, [7], σ P I remainssubstantial as in the present calculations ( viz.
Fig. 3).Even for σ P I of a highly excited state s p p ( P ) ,the calculated < f > is found to be much higher, 1.03,compared to 0.0013 from the OP cross sections. Theenergies of the strongest dipole transitions 2p → → O p ac it y ( x10 c m / g )
13 12 11 10 9 8 7200040006000800010000 OPExpt a) Z Fe-OPACITY
70 80 90 100 110 120 13010 -4 -3 -2 -1 Photon Energy (Ry) d) P o
3p :PresentOP10 -3 -2 -1 σ P I ( M b ) c) P o
4p :PresentOP10 -3 -2 -1 b) P o
5p :PresentOPPHOTOIONIZATION of Fe XVII(2p np P e )Impact of Seaton Resonances Figure 4: Comparison of features in the measured opacityspectrum [3] (a), with those due to PEC resonances in σ PI of a series of Rydberg states s p np ( P o ) P with n =3, 4 and 5 (b-d). Transition energies corresponding to PECresonances (not considered under OP [5]) are pointed out byarrows. The PECs are larger and wider resonances at higherexcited states due to strong dipole core transitions that canenhance the background by orders of magnitude. The energyscale in panel (a) is in Å and in Ry in (b-d). resonances are marked with arrows in the Fig. 4. Thesetransition arrays correspond to the inverse process of di-electronic recombination via satellite lines (2p → ns,nd) n (cid:48) (cid:96) (cid:48) , with the spectator electron n (cid:48) (cid:96) (cid:48) . Interestingly, someof the PEC resonances in Fe xvii correspond to the en-ergy region in between line (or resonant) features wherethe Z-opacity is much higher than the OP opacity (Fig-3a); including more ionization stages of Fe would furtherfill in the opacity.Finally, we extend the atomic calculations to obtainmonochromatic Fe xvii opacity at the Z plasma con-ditions shown in Fig. 5. The measured opacity (5a) ishigher than the OP (5d), as is the present calculatedopacity (5b,c). The considerable structure obtained inthe present 99CC calculations (Fig. 5c) is mostly due tothe large and broad PEC resonances, including autoion-ization broadening in an ab initio manner. We carryout a point-by-point normalized Lorentzian convolutionover all 454 Fe xvii photoionization cross sections usingan algorithm that simulates temperature-density depen-dent electron impact damping [12]. It is seen that mostresonances dissolve into and raise the continuum opacity,filling in the windows in the OP data, as seen experimen-tally in 5a) [3]. That would also yield much greater con-tinuum lowering than existing opacity calculations. TheRosseland Mean Opacity of Fe xvii from the present re-sults is 170 cm /g, 35% higher than the 126 cm /g fromOP. Substituting the present Fe xvii opacity alone intoa solar mixture of all abundant elements from H to Ni,yields a 2.1% increase. However, Fe xvii is only one of thedominant ions under Z conditions with ionization fraction 0.195; Fe xviii ionization fraction is 0.39, twice as much.Given the generality of the bound-free opacity enhance-ment shown in this Letter , including other contributingFe ions such as Fe xviii and Fe xix , would be consis-tent with the 7% higher iron opacity from all iron ionsin the Z data (perhaps higher), and consequently withthe expected increase in total solar opacity that couldsolve the abundances problem. Furthermore, the currentapproach treats the structures and divisions between thebound-bound and bound-free opacities without unphysi-cal approximations.Opacity calculations involve several other importantatomic-plasma issues. First, plasma broadening signifi-cantly affects the opacity distribution (c.f. Figs. 5b-c).Second, the Boltzmann factors in the equation-of-state(EOS) for Fe xvii imply low level populations in excitedstates, but there are nearly 200 such levels beginning 53Ry above the ground state (Fig. 1) at the Z tempera-ture ∼ × K ( exp( − ∆ E/k B T ) = 0 . ), and withfractional population between 0.1-1.0% of the groundstate; augmented with orders of magnitude enhancementin cross sections that ensures a significant contributionto high-energy bound-free opacity. Third, the total oscil-lator strength sum-rule is sometimes invoked to rule outenhanced opacity as measured [13]. However, it is the partial , not the total, differential oscillator strength dfd(cid:15) inthe relevant energy region, and atomic species at a giventemperature-density, that determines the mean opacity.There would be substantial re-distribution of oscillatorstrength from the bound-bound, as in existing opacitymodels, to the bound-free once the atomic/plasma effectsdemonstrated herein are included. Fourth, the measuredFe ion fractions at the Z are close to LTE values [3]. Con-sidering the multitude of high- n levels with large statis-tical weights [8], there are marked differences in occupa-tion probabilities in the LTE EOS among different mod-els [11]; sample calculations show that OP values fromthe Mihalas-Hummer-Däppen [9] EOS (also employed inthis work) are lower than OPAL by 3% for n = 3 , afactor of 6 for n = 5 , and by 2 orders of magnitude for n = 9 [10]; therefore, an upward revision would furtherenhance the contribution of the present high- n σ P I toopacity. The quantitative results presented herein helpexplain the observed discrepancies and the missing opac-ity due to photoabsorption from excited levels and core-excitations (the aforementioned points A, B and C). Thepresent work also points to a basic feature of photoioniza-tion of any atomic system: the bound-free cross sectionswould be incomplete unless all contributing final states ofthe residual core ion are coupled. However, that greatlyenlarges the scope of photoionization calculations even onstate-of-the-art computational platforms. Generally, theenhancement and convergence of photoionization crosssections up to high energies should manifest itself in thebound-free plasma opacity of many astrophysical and lab-oratory sources (e.g. [14]).We would like to thank Werner Eissner for contribu-tions. This work was partially supported by the U.S.
E (Rydbergs) l og κ ( M b ) λ (A)
13 12 11 10 9 8 7−3−2−10 d) FeXVII OP κ −3−2−10 c) FeXVII Detailed κ −3−2−10 b) FeXVII Convolved κ O pa c i t y ( x c m / g ) a) Z Total Fe−OPACITY l og κ ( M b )
13 12 11 10 9 8 7−3−2−10 d) FeXVII OP −3−2−10 c) FeXVII Detailed κ −3−2−10 b) FeXVII Convolved κ O pa c i t y ( x c m / g )
70 80 90 100 110 120 130200040006000800010000
Figure 5: a): Total iron opacity measured at Z and OP atT = 2.1 × K and n e = 3 . × cm − . Present Fe xvii opacity b)-c): detailed autoionization resonance structures; c)convolved over electron impact broadening Lorentzian profile[12], which also fills in several windows in opacity compared toOP results d). Present values in b)-c) fall off more slowly thanthe OP d), and are higher in the high-energy region towards7 Å. Department of Energy (de-sc0012331) and and NationalScience Foundation (AST-1409207). The computationalwork was carried out at the Ohio Supercomputer Centerin Columbus, Ohio. ∗ [email protected], † [email protected] [1] M. Asplund, N. Grevesse, J. Sauval and P. Scott, Annu.Rev. Astro. Astrohys., 47, 481 (2009).[2] J. Christensen-Dalsgaard, M. P. Di Mauro, G. Houdek,and F. Pijpers, Astron. Astrophys. , 494, 205 (2009).[3] J. E. Bailey, T. Nagayama, G. P. Loisel, G. A. Rochau,C. Blancard, J. Colgan, Ph. Cosse, G. Faussurier, C. J.Fontes, F. Gilleron, I. Golovkin, S. B. Hansen, C. A. Igle-sias, D. P. Kilcrease, J. J. McFarlane, R. C. Mancini,S. N. Nahar, C. Orban, J.-C. Pain, A. K. Pradhan,M. Sherill & B. G. Wilson, Nature, 517, 56-59 (2015).References to other radiation transport codes ATOMIC,OPAS, RCO-RCG, OPAL are given are given in this pa-per; some of them include more transition arrays thanthe OP, and the integrated mean opacities are similar.[4] S. N. Nahar, A. K. Pradhan, G.-X. Chen & W. Eissner,
Phys. Rev. A , 83, 053417 (2011).[5] M. P. Scott; data at http:cdsweb.u-strasbg.fr/topbase/topbase.html[6] M. J. Seaton, Y. Yu, D. Mihalas, and A. K. Pradhan,
Mon. Not. R. astr. Soc.
The Opacity Project , Institute of PhysicsPublishing, Vol 1, (1995); Vol. 2 (1996); OP opacities athttp: opacities.osc.edu.[7] A. K. Pradhan and S. N. Nahar,
Atomic Astrophysicsand Spectroscopy , Cambridge University Press (2011).[8] J.-C. Pain and F. Gilleron, High Energy Density Physics,15, 30-42 (2015).[9] D. Mihalas, D. G. Hummer and W. Däppen,
Astrophys.J.
J. Phys. B , 36, 4367(2003).[11] R. Trampedach, W. Däppen and V. A. Baturin,
Astro-phys. J. , 646, 560-578 (2006).[12] A new algorithm has been developed for numerical simu-lation of temperature-density dependent electron impactdamping of autoionizing resonances in photoionizationcross sections, tabulated at thousands of energies for eachof the 454 bound Fe xvii levels. Point-by-point convolu-tion with a Lorentzian functional is extremely CPU in-tensive, and would be described elsewhere.[13] C. A. Iglesias,
High Energy Physics , 15, 4 (2015).[14] R. P. Drake,