in i in a self-consistent, uniðed manner, as explained in the Ðrst channel i

THE ASTOPHYSICAL JOUNAL SUPPLEMENT SEIES, 37:È7, November (. The American Astronomical Society. All rights reserved. Printed in U.S.A. ELECTON-ION ECOMBINATION ATE COEFFICIENTS AND PHOTOIONIZATION COSS SECTIONS FO ASTOPHYSICALLY ABUNDANT ELEMENTS. VI. Ni II SULTANA N. NAHA AND MANUEL A. BAUTISTA eceived January 3; accepted June ABSTACT We present the Ðrst detailed ab initio quantum mechanical calculations for total and state-speciðc recombination rate coefficients for e ] Ni III ] Ni II. These rates are obtained using a uniðed treatment for total electron-ion recombination that treats the nonresonant radiative recombination and the resonant dielectronic recombination in a self-consistent uniðed manner in the close-coupling approximation. Large-scale calculations are carried out using a 9 state wavefunction expansion from core conðgurations 3d8, 3d7s, and 3d6p that permits the inclusion of prominent dipole allowed core transitions. These extensive calculations for the recombination rates of Ni II required hundreds of CPU hours on the Cray T9. The total recombination rate coefficients are provided for a wide range of temperature. The state-speciðc recombination rates for 53 bound states of doublet and quartet symmetries, and the corresponding photoionization cross sections for leaving the core in the ground state, are presented. Present total recombination rate coefficients di er considerably from the currently used data in astrophysical models. In particular the enhancement in the low-temperature recombination rate coefficient should help address the long-standing problem of Nickel overabundance in several classes of astronomical objects. Subject headings: atomic data È ISM: general È planetary nebulae: general. INTODUCTION autoionizing resonances. The partial cross sections correspond to photoionization leaving the residual core in its Nickel is one of the most important iron peak elements. ground state. For example, radioactive decay Ni-Co-Fe is the primary power source for the light curves of supernovae. Dipole. THEOETICAL SUMMAY allowed and forbidden Ni II lines are often observed in the The calculations are carried out in the close-coupling interstellar medium and H II regions, di use nebulae, novae, approximation using the -matrix method essentially as in and supernova remnants. However, only a limited number the Opacity Project (Seaton 987; Seaton et al. 99). The of theoretical studies have been done for the atomic processes in Ni II. With a Fe-like core, the ion poses a computa- theoretical details for photoionization cross sections for Ni II can be found in Bautista (999). tional challenge with its strong electron-electron correlation The theory of uniðed treatment for total electron-ion e ects. Bautista (999) made the Ðrst detailed theoretical recombination in the CC approximation is brieñy outlined. study for the radiative processes for bound-bound and The treatment considers the inðnite number of states of the bound-free transitions in Ni II presenting total and partial recombined ion and incorporates both the radiative recombination () and the dielectronic recombination (D) pro- photoionization cross sections for the ground and excited states (Bautista 999). cesses in a self-consistent, uniðed manner. Details of the In this report we study electron-ion recombination, method are given in Nahar & Pradhan (99, 995) and in e ] Ni III ] Ni II, and present the total and the state-speciðc previous papers in the present series. recombination rate coefficients. We use the uniðed treatment of total electron ion recombination of Nahar & In the CC approximation the total wavefunction expansion, ((E), of the recombined ion, (N ] ) electron system, Pradhan (99, 99, 995). The method involves large scale is expanded in terms of the target (recombining ion) wavefunctions as computations in the close-coupling (CC) approximation using the -matrix method to (i) provide recombination rate coefficients that are fully consistent with the photoionization cross sections of the ion, which allows, for example, i i j j ((E) \ ; s h ] ; c ', () i j precise calculations of ionization balance, and (ii) treat nonresonant radiative and resonant dielectronic recombination and h is i where s is the target wavefunction in a speciðc state S L n the wavefunction for the (N ] )th electron i in i a i in a self-consistent, uniðed manner, as explained in the Ðrst channel i labeled as S L n k l (SL n); k being its kinetic paper of the present series (Nahar & Pradhan 997). energy. ' in the second i i summation i i i are i the bound channel Present results are the Ðrst detailed recombination rate functions j of the (N ] )-electron system accounting for coefficients, a (T ), for Ni II. State-speciÐc short-range correlations and the orthogonality between recombination rate coefficients for 53 continuum and bound orbitals. bound states of Ni II are presented. These are obtained from In the present approach the states of the recombined the partial photoionization cross sections including ion are divided into two groups, (A) low-n bound states, ranging from the ground to excited states with n ¹ n (typically n \ ), and (B) closely spaced high-n states, max Department of Astronomy, The Ohio State University, Columbus, OH 3. n \ n ¹ max O. max Centro de F sica, Instituto Venezolano de Investigaciones Cient Ðcas Group (A) bound states are treated via photorecombination using detailed balance with photoionization in the (IVIC), Altos de Pipe, Edo. Miranda, Venezuela.

NAHA & BAUTISTA Vol. 37 energy range where both the background and resonant recombinations are important. The partial photoionization cross sections leaving the core in the ground state are obtained for all these bound states. The recombination cross sections, p, are then obtained from the photoioniza- tion cross sections, C p, through the principle of detailed balance: PI p \ g +u i C g mvc p PI, () j where g and g are the statistical weight factors of the recombined i and j recombining states, respectively, v is velocity of the photoelectron, and u is the photon frequency. The recombination rate coefficient, a (T ), at a given temperature is obtained by averaging p (T ) over the Maxwellian dis- tribution of electrons, f (v), as, C a (T ) \ P = vf (v)pc dv \ g i g j kt cjnm3kt ] P = EpPI (v)e~(v@kt)dv, (3) where v is photoelectron energy, E \ +u \ v ] I is the p photon energy, and I is the ionization potential. As photo- p ionization cross sections include the detailed structures of autoionizing resonances, the sum of individual rates of the bound states corresponds to inclusion of both the and the D in a uniðed manner. The states in group (B) are dense and belong to small energy regions below target thresholds, n \ n ¹ O, max where D dominates. In this region, the outer electron interacts weakly with the core which decays through emission of a photon. As the photon energy approaches the excited core threshold, the radiative decay rate of the core remains constant but the autoionization rate for the autoionizing states decreases as l~3, while the D increases by the same order (where l is the e ective quantum number of the ydberg series belonging to the converging threshold). These states are treated through precise D theory by Bell & Seaton (985) as extended in Nahar & Pradhan (99) to obtain the D collision strengths, )(D). The corresponding contributions to the total recombination rates are then obtained through Maxwellian average over )(D). 3. COMPUTATIONS The Ðrst large-scale calculations are carried out for recombination of Ni II in the close-coupling approximation using the -matrix method. The eigenfunction expansion of the Ni II includes 9 states of core conðgurations 3d8,3d7s, and 3d6p (Bautista 999). The expansion (listed in Table ) allows for prominent dipole allowed core transitions, e.g., 3d8(3F) ] 3d7p(3Go). The d orbital is included in the correlation conðgurations 3d7d and 3p53d8d. The target or the core wavefunctions were obtained using the atomic structure code SUPESTUCTUE (Eissner et al. 97). Observed energies are used for the target states for more accurate resonance positions in photoionization cross sections. The second term in ((E) (eq. []) describing the short-range electron correlation for Ni II conðguration includes all possible conðgurations with maximum occupancies of 3d, p3, and d. The continuum wavefunctions within the -matrix boundary are represented by a basis set of 5 terms. Computations include partial waves with l \. max We obtained partial photoionization cross sections, p, PI leaving the core in the ground state for 53 low-n bound states with S ] \ and, and L ¹ 7, and n B, and l ¹ 9. Calculations for these cross sections are repeated for the present work at a Ðner energy mesh, and for values at high-energy regions. Calculations were carried out using the -matrix package of codes developed by the Opacity Project (Seaton 987; Berrington et al. 987) that were extended for the Iron Project (Hummer et al. 993; Berrington et al. 995). The recombination rate coefficients, equation (3), of individual bound states are obtained on averaging over p (T ) C using the code ECOMB (Nahar 996). The resonance structures in the photoionization cross sections appear up to photon energies corresponding to the highest core state, 3d7p(Do) (Table ), through couplings of channels. At higher photon energies, the cross sections are extrapolated through Ðttings or by KramerÏs rule (Nahar & Pradhan 99). The sum of these individual rates comprises the low-n TABLE THE 9 LS TEMS AND ENEGIES, E (in ryd), OF Ni III IN THE EIGENFUNCTION EXPANSION OF Ni II t Number Term E t Number Term E t Number Term E t... 3d8 3Fe. 8... 3d7s 3Fe.88557 35... 3d7p 3Go.95... 3d8 De.888 9... 3d7s Fe.9 36... 3d7p Ho. 3... 3d8 3Pe.3... 3d7p 5Fo. 37... 3d7p Fo.36... 3d8 Ge.5... 3d7p 5Do.975 38... 3d7p 3Po.66 5... 3d8 Se.6969... 3d7p 5Go.68 39... 3d7p 3Do.3933 6... 3d7s 5Fe.968 3... 3d7p 3Go.55... 3d7p 3Io.677 7... 3d7s 3Fe.558378... 3d7p 3Fo.575... 3d7p 3Go.696 8... 3d7s 5Pe.693 5... 3d7p 3Do.786... 3d7p So.983 9... 3d7s 3Ge.679739 6... 3d7p 5So.56 3... 3d7p 3Do.566... 3d7s 3Pe.75399 7... 3d7p 5Do.3398... 3d7p Io.6338... 3d7s 3Pe.7585 8... 3d7p 3So.836 5... 3d7p 3Fo.7... 3d7s Ge.733 9... 3d7p 3Ho.939 6... 3d7p 3So.778 3... 3d7s 3He.7397 3... 3d7p 3Fo.36 7... 3d7p Po.796... 3d7s 3De.756 3... 3d7p 5Po.3 8... 3d7p 3Ho.8986 5... 3d7s Pe.769 3... 3d7p 3Po.688 9... 3d7p Do.889 6... 3d7s He.773 33... 3d7p Go.589 7... 3d7s De.7858 3... 3d7p 3Do.76

No., ELECTON-ION ECOMBINATION ATE COEFFICIENTS 3 contributions to the total recombination rate coefficients, a (T ). The collision strengths for dielectronic recombination, )(D), for high-n states are obtained using the same 9 state wavefunction as used for photoionization cross sections. We consider the channels radiatively decaying to the core ground state, 3d83F, via dipole allowed transitions. There are such core transitions as listed in Table. The oscillator strengths for these transitions (in Table ) for )(D) were obtained from SUPESTUCTUE (Eissner et al. 97). The computations were carried out using the extended code STGFD (Nahar & Pradhan 99). Independent close-coupling calculations for (e ] Ni III) scattering, using the same 9 state wavefunction, are also carried out for the electron impact excitation (EIE) collision strength, )(EIE), at the target thresholds. These calculations meet some consistency checks. It is important to determine the contributions of higher order multipole potentials since the present theory of D collision strengths, based on multichannel quantum defect theory, neglects these contributions (Nahar & Pradhan 99). On the other hand, a proper choice of the -matrix boundary and size of -matrix basis set can compensate for the di erence introduced by these potentials. We compute )(EIE) with and without the contributions from multipole potentials by a switching parameter, ipert, in the program, STGF. Values of )(EIE) excluding the contributions (parameter ipert \ ), and including them (ipert \ ), are presented in Table. Good agreement between the two sets of )(EIE) with ipert \ and indicates the validity of the D calculations consistent with quantum defect theory and the -matrix calculations. Strong electron-electron correlations and channel couplings demanded extensive computational resources for this ion. The memory size and CPU time constraints on the Cray T9 forced us to compute cross sections for one symmetry at a time and in energy segments. For example, the time for computations of photoionization cross section for the F states with a total of 7 channels, a Hamiltonian matrix of 99 ] 99, and memory requirement of 5 MW, needed about CPU hours. For overall computation the largest number of channels was 36 and the largest Hamiltonian matrix size was 55 ] 55. On the other hand, the calculation of D collision strengths, )(D), in the small energy region below the core TABLE THE TANSITION POBABILITIES (A ) FOM THE EXCITED THESHOLDS TO ji THE GOUND a3f STATE OF TAGET Ni III thresholds, l \l¹ O, required less computational time (B CPU max hours in total) as no dipole matrix elements needed to be computed. The nonresonant background contributions from the high-n group (B) states to the total recombination is also included through a top-up ÏÏ scheme as explained in Nahar (996). This contribution is signiðcant at low temperatures.. ESULTS AND DISCUSSIONS The results of present calculations for Ni II and comparison with previous data are described in subsections below... Low-n States: Photoionization Cross Sections Partial photoionization cross sections (p ) leaving the core in the ground state are obtained for 53 PI bound states of Ni II. (The total cross sections are reported in Bautista 999.) The cross sections exhibit extensive autoionizing resonances that enhance the overall photoionization rates considerably. State-speciÐc recombination rate coefficients are obtained from the partial photoionization cross sections. The percentage contributions and relative weight of individual bound states to the total a varies with temperature due to resonance structures and enhancement in the background cross sections at di erent energies. Some important features of photoionization cross sections related to recombination are illustrated in Figures and. Figure presents p of Ðve states of Ni II. The ground PI state is an equivalent electron state, 3d9(D), and is one of log [σ PI ] (Mb) 3 - - - Ni II + hν -> Ni III + e (a) 3d 9 ( D)..6.8.. (b) 3d 8 s( F)..6.8.. (c) 3d 7 ( F)sp( G o ) -..6.8...6.8 (d) 3d 7 ( F)sp( G o ) Transition A (a.u.) ji S)(D)T )(EIE) )(EIE) a3fe^z3go....98([9).5 3.76 3.76 a3fe^z3fo....86([8) 6.3 6.5 6.5 a3fe^z3do....6([8) 3.6 3.5 3.68 a3fe^y3fo....7([8).6.5.5 a3fe^y3do....8([9).957.9.9 a3fe^y3go... 3.93([9)... a3fe^x3do....9([9).8.793.793 a3fe^x3go....([8).678.. a3fe^w3do....8([8) 3.5 3.9 3.9 a3fe^x3fo....53([9).98.79.79 NOTES.ÈThe last three columns list the peak values of D collision strengths, S)(D)T, and electron-impact excitation collision strengths, )(EIE) (: excluding contributions of multipole potentials) and )(EIE) (: including these contributions), at the these thresholds. - -.6.8...6.8 -..6.8...6.8 Photon Energy (y) (e) 3d 8 ( 3 F)d( G) FIG..ÈPartial photoionization cross sections, p, of the ground state, (a) 3d9(D), and excited states, (b) 3d8s(F), (c) PI 3d7(F)sp(Go), (d) 3d7(F)sp(Go), and (e) 3d8(3F)d(G) ofniii leaving the core in the ground state 3d8(3F).

NAHA & BAUTISTA Vol. 37 log [σ PI ] (Mb) - - - - - 7d - - -3 - - 3d 8 nd( F) d d 9d 8d 6d 5d Ni II + hν -> Ni III + e 3 G o 3 D o3 D o 3 F o at the core thresholds, such as, 3Go, 3Do, 3Fo, etc. (listed in Table ) for dipole allowed transitions of the core ground state, 3F. PEC resonances become more distinct with higher n... High-n States: Collision Strengths for Dielectronic ecombination The D collision strengths, )(D), are obtained for recombination into the high-n states in the energy region l \l¹ O below each core state contributing to D. The max resonances in )(D) correspond to ydberg series of states S L n ll converging to the threshold S L n. )(D) is obtained t in t t two forms: with detailed energy t variation t t with resonances, and analytically averaged over resonances. The features of )(D) are illustrated in Figure 3, where the dotted curves are the detailed ones with resonances, and the solid curves are resonance averaged. The lower panel presents )(D) below the core states which are thresholds for allowed transitions from the ground state of Ni III, while the upper panel presents the expanded features for the Ðrst three thresholds. The excited target thresholds, given in Table, are marked by arrows in the Ðgure. As l increases, the resonances appear closer while the background rises (as seen in the solid curve) such that )(D) peaks sharply converging on to the threshold. That is, as l increases the outer electron remains loosely bound in - -...6.8...6 Photon Energy (y) FIG..ÈPartial photoionization cross sections, p, of the ydberg series of states, 3d8nd(F), 5d ¹ nd ¹ d, ofniii illustrating PI PEC resonances at energies indicated by arrows in the top panel. PECs are caused by dipole allowed transition in the core from the core ground state. the dominant contributors at all temperatures because of its relatively large background photoionization cross sections even at high photon energies, as seen in the top panel, Figure a. The state also exhibits high resonances at relatively higher energies. The Ðrst excited state, 3d8s(F), is also a dominant state which shows extensive resonances and enhanced background in the high-energy region (Fig. b). Cross sections of excited states, 3d7sp(Go) and 3d7sp(Go) in Figures c and d are examples of states that dominate the low-temperature region because of their near threshold background raised by resonances which decay at higher energies. Figure e illustrates the dominance of the state 3d8(3F)d(G) at higher temperature by its dense and wide resonances in relatively high-energy regions. It is found that the summed contribution of the quartets dominates the total a (T ) at lower temperature, while that of the doublets contributes more at higher temperature. The existence of photoexcitation-of-core (PEC) resonances (Yu & Seaton 987) at high energies is also a reason for increased total recombination rate coefficients at high temperatures. These relatively wide resonances are observed in the photoionization cross sections of excited states with a valence electron. PECs are basically the inverse process of D. The PEC resonances are introduced when the core is excited via dipole allowed transitions, while the outer electron remains a spectator. Figure illustrates the wide and prominent PEC resonances in the cross sections of ydberg series of states, 3d8nd(F), where 5d ¹ nd ¹ d,ofniii. The arrows point the PEC positions Ω(D) Ω(D) Ω(D) Ω(D) 8 6..3.. -..5.5.55.6.65.7.75.8 8 6 Ω EIE ( 3 F- 3 G o ) Ω EIE ( 3 F- 3 F o )..3.. -. e + Ni III -> Ni II (expanded) 3 G o 3 F o 3 D o e + Ni III -> Ni II 3 G o 3 F o 3 D o 3 F o 3 G o3 D o 3 D o 3 F o.5..5..5 [k( F)] (y) Ω EIE ( 3 F- 3 F o ) FIG. 3.ÈD collision strength, )(D), for recombination of e ] Ni III]Ni II: detailed with resonances (dotted curves) and resonance averaged (solid curves). The arrows indicate the energy positions of the target thresholds 3d7p(z3Go, z3fo, z3do, y3fo, y3do, y3go, x3do, x3go, w3do, x3fo), where D resonances are converging in the bottom panel, while the top panel presents an expanded detail of the Ðrst three core thresholds. The Ðlled circles are the excitation collision strength, )(EIE), at these thresholds.

No., ELECTON-ION ECOMBINATION ATE COEFFICIENTS 5 a highly excited state while the core decays radiatively. )(D) goes to zero beyond the threshold as the trapped electron Ñux in the closed channels below the threshold is released through excitation of the target state. We also note that )(D) is almost zero at the starting energy where l \., indicating negligible radiation damping below this energy. For the total recombination rate coefficients presented here, we choose contributions from the resonance averaged )(D), rather than the detailed one, because of higher numerical accuracy. In Figure 3, the Ðlled circles are the )(EIE), collision strengths for electron impact excitation (EIE), at the excited target thresholds, These are obtained from (e ] Ni III) scattering calculations in the close-coupling approximation, as mentioned before. The comparison of )(EIE) with S)(D)T provide a check of conservation of total electron-photon Ñux such that the trapped electron Ñux due to D resonances below a threshold equals that released due to EIE at the threshold, i.e., S)(D)T should equal )(EIE) (without the multipole potentials ipert \ ). Comparison between the two collision strengths given in Table show good agreement in general except at threshold x 3Go, where the di erence is large. The discrepancy can be explained by the presence of a near threshold resonance in )(EIE) and insuf- Ðcient energy resolution for both collision strengths..3. State-speciÐc and T otal ecombination ate Coefficients We present both the state-speciðc and the total recombination rate coefficients for the recombined ion, Ni II. The total rates are obtained from the sum of the state-speciðc rates along with the contributions from the high n states. The individual state recombination rates of the bound states are of importance in the determination of nonlocal thermodynamic equilibrium (NLTE) level population and recombination line intensities. State-speciÐc rates are obtained for 53 bound states of Ni II with n ¹ and l ¹ 9. Table 3 presents the dominant states in order of their contributions, for doublets and quartets separately, at temperatures T \,,,, and, K. The order of the contributing states varies with temperature depending on the resonance structures in p with energies. The reso- PI nances in the energy region from l[ to the next core threshold are averaged through Gailitis averaging (e.g., Nahar & Pradhan 99). However, the D contributions in this region are not included individually for each statespeciðc recombination rate; the summed D contributions are added for the total a. Hence, the state-speciðc rates at higher temperatures are somewhat underestimated. It should be noted that state-speciðc rate coefficients are different from e ective recombination rate coefficients, used in spectroscopic modeling, since the latter include contributions from higher bound states through radiative cascades. The complete table for the state-speciðc rates are available electronically. However, there may be some spectroscopic misidentiðcation of states. The strong electron-electron correlation in Ni II introduces some complications in the quantum defect analysis for LS term identi- Ðcation. There are some cases when the same state can correspond to a few possible spectroscopic identiðcations, rather than a unique one. We also noted several highly excited observed states of conðguration 3d7sp and one of 3d7s that are not present in the calculated set. We used an e ective quantum number mesh of *l \. to scan the poles in the Hamiltonian for the energy eigenvalues. Missing states indicate that a Ðner mesh was needed to locate them. Contributions from these states are needed for more accurate total recombination rates but may not be very signiðcant as they lie at high energies. Table presents the total electron-ion recombination rate coefficients, a (T ), of Ni II recombining from Ni III ground state. The rates are given at temperatures from to 7 K, with a temperature mesh of *log T \.. The solid curve in Figure shows log a versus log T and the basic features. As for other ions, the total (e ] ion) recombination rate for Ni II is maximum at low temperatures due to dominance by radiative recombination. a (T ) decreases by more than an order of magnitude as the temperature increases from log T (K) \ to log T (K) B., but rises again at higher temperature because of dominant contributions of D which peaks at about 5 K, above which it falls smoothly. There is a small bump ÏÏ in the rates, the low-temperature D bump, at around 3 K due to near threshold autoionizing resonances in the photoionization cross sections. This bump has been observed in several other ions (e.g., Nahar & Pradhan 997). We note that the contributions from the nonresonant background cross sections of high-n states, n \ n ¹ O, max to the total a (T ) is considerable at low temperatures when the mean electron energy is in general low for forming autoionizing resonances contributing to D. Present total a (T ) (solid curve) are compared with the recombination rate coefficients derived by Shull & van Steenberg (98) in Figure. They derived the radiative α (cm 3 sec - ) - - - -3 - -5 Total (Unified,present) (Shull & Steenberg) D (Shull & Steenberg) +D (Shull & Steenberg) e + Ni III -> Ni II 5 6 7 T (K) FIG..ÈTotal electron-ion recombination rate coefficients, a (T ), for e ] Ni III]Ni II of the present work (solid curve). The dashed curve presents the, dotted curve the D, and dot-dashed curve the total rates by Shull & van Steenberg (98).

6 NAHA & BAUTISTA Vol. 37 TABLE 3 INDIVIDUAL STATE ECOMBINATION ATE COEFFICIENTS (IN UNITS OF cm3 s~) FO e ] Ni III ] Ni II AT TEMPEATUES T \,,,, AND, K K K K K State a State a State a State a Doublets 3d7s He.99[3 3d8 3Pes Pe 6.69[3 3d7s 3Fep Go.7[3 3d9 De.8[3 3d8 Gep Fo.99[3 3d7s Pe.39[3 3d7s 3Fep Do.6[3 3d8 3Fed Ge.[3 3d8 3Fep Do.85[3 3d8 3Pep Do.88[3 3d7s 3Fep Fo 8.63[ 3d7s 3Fep Go.6[3 3d9 De.3[3 3d8 3Pep Po.9[3 3d8 3Pes Pe 8.5[ 3d7s 3Fep Do 7.9[ 3d8 3Fep Fo.33[3 3d8 Dep Po.[3 3d8 3Fep Go 6.[ 3d7s 3Fep Fo 5.53[ 3d8 3Fep Go.37[3 3d8 Ges Ge.[3 3d7s Pe 5.66[ 3d8 3Fep Do.7[ 3d8 Ges Ge.35[3 3d7s He 9.6[ 3d8 3Fep Do 5.65[ 3d8 3Fep Go.3[ 3d8 3Fes Fe.35[3 3d8 3Fep Do 8.76[ 3d8 3Pep Do.[ 3d8 3Pes Pe 3.33[ 3d8 3Pep Po 8.[ 3d8 3Fep Fo 7.5[ 3d9 De.8[ 3d8 3Fep Fo.88[ 3d8 3Fe5p Go 7.78[ 3d9 De 7.9[ 3d8 3Fep Fo 3.75[ 3d8 3Pep Do.6[ 3d8 3Pep Do 7.5[ 3d8 3Pe5p Do 5.[ 3d8 3Pep Po.38[ 3d7s Pe.9[ 3d7s 3Fep Go 6.8[ 3d8 3Fep Go.3[ 3d8 Dep Po.66[ 3d7s 3Gep Go.9[ 3d7s De 6.[ 3d8 3Fes Fe.35[ 3d8 Gep Go.5[ 3d8 3Fes Fe.7[ 3d8 3Fe5p Do 6.[ 3d7s 3Pep Po 3.97[ 3d8 Ges Ge.9[ 3d8 3Pep Po.8[ 3d8 3Fe5p Fo 5.6[ 3d7s 3Fep Go 3.8[ 3d8 3Fed Ge.3[ 3d8 Gep Fo 8.99[5 3d8 Dep Po 5.9[ 3d8 Gep Fo 3.[ 3d8 3Fes Fe.9[ 3d8 Gep Go 8.88[5 3d8 3Fe5f Io.[ 3d8 Ded Pe.7[ 3d8 Gep Fo.[ 3d8 Dep Po 8.39[5 3d8 3Fed He.35[ 3d8 De5d Pe.69[ 3d8 Dep Do.5[ 3d8 3Fef Io 8.3[5 3d7sp Ho.7[ 3d8 3Fe5p Go.[ 3d7s He 9.35[5 3d7s Fe 8.3[5 3d8 3Fe5d He.[ 3d7s De.7[ 3d7s 3Gep Go 8.3[5 3d8 Ge5p Go 7.8[5 Sum.67[.7[ 9.5[3.5[ Total 5.9[.85[.8[ 3.69[ % Contribution 5% 3% % 3% Quartets 3d7s 5Fep Go.[ 3d7s 5Fep Go.38[ 3d7s5Fep Go.8[3 3d7s 5Fep Go.9[3 3d7s 5Fep Fo 6.68[ 3d7s 5Fep Fo.69[ 3d7s 5Fep Fo.6[3 3d7s 5Fep Fo.95[3 3d8 3Fep Go 3.36[ 3d8 3Fep Go.6[ 3d7s 5Fep Do.73[3 3d7s 5Fep Do.35[3 3d8 3Fep Fo.[ 3d8 3Fep Fo 7.75[3 3d8 3Fep Do.5[3 3d8 3Fep Go.3[3 3d8 3Fep Do 7.7[3 3d8 3Fep Do 6.93[3 3d8 3Fep Go.95[3 3d7s 3Fep Fo.3[3 3d7s 5Fep Do.8[3 3d7s 5Fep Do.56[3 3d8 3Fep Fo.3[3 3d8 3Fep Do.5[3 3d8 3Fe5p Go.9[3 3d8 3Fe5p Fo 6.[ 3d8 3Pep Do 8.7[ 3d8 3Fep Fo 9.9[ 3d8 3Fed He.[3 3d8 3Fe5p Go 6.36[ 3d7s 5Fef Do 8.7[ 3d8 3Fes Fe 7.8[ 3d8 3Fef Io.[3 3d8 3Pep Po 6.[ 3d7s 3Gep Fo 5.55[ 3d7s 3Gep Fo 6.9[ 3d8 3Fe5p Fo 9.[ 3d8 3Pep Do 5.7[ 3d8 3Fes Fe.6[ 3d7s 3Fep Go 6.5[ 3d8 3Fe5f Io 8.8[ 3d8 3Fed He 3.[ 3d7s 3Fep Fo 3.99[ 3d7s 3Gep Go 5.5[ 3d8 3Fe5d He 8.8[ 3d8 3Fef Io 3.5[ 3d7s 3Gep Go 3.8[ 3d7s 3Fep Fo 5.7[ 3d7s 3Hep Ho 8.53[ 3d8 3Fe6p Go.7[ 3d7s 3Fep Fo 3.65[ 3d7s 5Fef Do.7[ 3d8 3Fed Ge 7.9[ 3d8 3Fe5d He.69[ 3d7s 5Pep Do 3.8[ 3d8 3Pep Do.53[ 3d8 3Fe5f Ho 7.67[ 3d8 3Fe5f Io.69[ 3d7s 3Fep Go.9[ 3d8 3Fef Ho.6[ 3d8 3Fe6p Go 7.8[ 3d8 3Fe5p Do.6[ 3d8 3Pep Po.7[ 3d7s 5Pep Do 3.6[ 3d8 3Fe5p Do 7.[ 3d7s 3Hep Ho.58[ 3d7s 3Fep Do.[ 3d7s 3Gep Ho 3.59[ 3d8 3Fe5d Ge 7.5[ 3d8 3Fed Ge.[ 3d7s 3Gep Ho.[ 3d7s 3Fep Go 3.[ 3d8 3Fef Go 6.68[ 3d8 3Fe5f Ho.3[ 3d8 3Fe5p Fo.99[ 3d7s 3Fep Do 3.8[ 3d7s Fe 6.66[ 3d8 3Fe5d Ge.6[ 3d7s 3Fep Go.95[ 3d7s 3Hep Io.3[ Sum.58[.[.6[.6[ Total 5.9[.85[.8[ 3.69[ % Contribution 9% 59% 53% % NOTES.ÈThe Ðrst dominant doublets and quartets are listed in order of their contributions. Their sum and percentage to the total are speciðed below. Notation a[b means a ] ~b. recombination rates (dashed curve) by extrapolation along isoelectronic sequences and obtained the dielectronic recombination rates (dotted curve) using the Burgess General Formula (Burgess 965). Their and D rates have been added for the total (dot-dashed curve). These values underestimate Ni II recombination rates in the lowtemperature region since the near threshold resonance structures were not considered in their data sources. The high temperature rates on the other hand are overestimated because the Burgess formula does not include (i) the interference e ect of the resonant D and continuum background, which is considerable for multielectron systems, and (ii) the autoionization into excited core states in addition to the ground state. This latter process reduces D considerably if the channel couplings are strong, as in the present case. However, present total recombination rates are expected to be somewhat higher at very high temperatures, above 5 K due to 3dÈf transitions in the core

No., ELECTON-ION ECOMBINATION ATE COEFFICIENTS 7 TABLE TOTAL ECOMBINATION ATE COEFFICIENTS, a (T ), IN UNITS OF cm3 s~, FO e ] Ni III ] Ni II log T a log T a log T a....78e[ 3....6E[ 5....5E[....57E[ 3....E[ 5.3... 3.9E[....38E[ 3.3....E[ 5....85E[.3....E[ 3....E[ 5.5....7E[....7E[ 3.5... 8.93E[ 5.6....77E[.5... 9.5E[ 3.6... 7.66E[ 5.7....36E[.6... 8.36E[ 3.7... 6.57E[ 5.8....3E[.7... 7.E[ 3.8... 5.6E[ 5.9... 7.68E[3.8... 6.59E[ 3.9....87E[ 6... 5.69E[3.9... 5.9E[....8E[ 6....9E[3... 5.9E[... 3.86E[ 6... 3.8E[3....78E[... 3.66E[ 6.3....9E[3....3E[.3... 3.69E[ 6....59E[3.3... 3.96E[... 3.9E[ 6.5....5E[3... 3.6E[.5....35E[ 6.6... 8.8E[.5... 3.8E[.6....8E[ 6.7... 5.97E[.6....97E[.7... 5.3E[ 6.8....3E[.7....67E[.8... 5.7E[ 6.9... 3.9E[.8....38E[.9... 5.7E[ 7....E[.9....E[ 5... 5.E[ 3....85E[ 5....75E[ NOTE.ÈThe temperature is given in K. which are not included. The oscillator strengths for higher l ] transitions are usually larger than those for the lower l [ transitions and can introduce prominent PEC resonances in the photoionization cross sections. However, for Ni III, the 3dÈi transitions are about twice as high, about ryd, than the 3dÈp ones at about ryd. So the e ect of f PECs will be smaller than p PECs by about a factor of 6, the ratio of exp ([E/kT ) att \ 5 K. Taking into account that the f contributions may still be comparable, the total recombination rates above T \ 5 K may lie somewhere between the present and the Shull & van Steenberg (98) rates. The accuracy of the present total recombination rate coefficients, a (T ), for Ni II may be estimated to be about 3% in most of the temperature range up to 5 K. The uncertainty is higher at very high temperatures where (in addition to missing contributions from the 3dÈf excitation) the cross sections are extrapolated, and at very low temperatures where resonances may require higher resolution. The accuracy estimate is based on the general agreement between the calculated and the observed energies, use of observed target energies for accurate resonance positions, the general accuracy of the CC method for photoionization cross sections, electron scattering, and D collision strengths, and previous comparison between results of the uniðed method and experiment (e.g., Zhang, Nahar, & Pradhan 999). Some e ects not included here could improve the results. First, the relativistic e ects could be important for this heavy, many-electron ion. Although the uniðed treatment for (e ] ion) recombination has been extended to include relativistic e ects in the Breit-Pauli -matrix approximation (Zhang & Pradhan 997; Zhang et al. 999) and employed for highly charged Li-like and He-like C and Fe (Nahar, Pradhan, & Zhang, ) we do not expect relativistic e ects to be too important for Ni II. Also, use of a larger wavefunction expansion that includes states for core transitions at higher energies, especially those belonging to 3dÈf transitions, could enhance the recombination rates as discussed above. 5. CONCLUSION Total and state-speciðc electron-ion recombination rate coefficients, a (T ), are presented for e ] Ni III ] Ni II. The calculations were carried out in the close-coupling approximation employing a uniðed treatment. To our knowledge, this is the Ðrst detailed and accurate study for the recombination of Ni II. The partial photoionization cross sections of a large number of bound states are also presented. Present recombination rates, along with the total photoionization cross sections obtained with the same close-coupling expansion (Bautista 999), will provide self-consistent atomic data for accurate calculations of ionization balance for plasmas in photoionization or coronal equilibria. The total recombination rate coefficients di er considerably from the currently used values. All data may be obtained from the Ðrst author via e-mail (nahar=astronomy.ohio-state.edu). This work was supported partially by the NSF (AST 987-89) and the NASA Astrophysics Data Program. The computations were carried out at the Center for Computational Science at NASA Goddard Space Flight Center, and at the Ohio Supercomputer Center in Columbus, Ohio. Bautista, M. A. 999, A&AS, 37, 59 Bell,. H., & Seaton, M. J. 985, J. Phys. B, 8, 589 Berrington, K. A., Burke, P. G., Butler, K., Seaton, M. J., Storey, P. J., Taylor, K. T., & Yan, Yu. 987, J. Phys. B,, 6379 Berrington, K. A., Eissner, W., & Norrington, P. 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