Electron excitation collision strengths for transitions in K II^{}
S. S. Tayal  O. Zatsarinny
1  Department of Physics, Clark Atlanta University, Atlanta,
GA 30314, USA
2 
Department of Physics and Astronomy, Drake University, Des Moines, IA, 50311, USA
Received 2 March 2009 / Accepted 30 November 2009
Abstract
Aims. Oscillator strengths and electron impact excitation
collision strengths for transitions between the 43 finestructure
levels of the ,
,
,
,
and
configurations in K II are calculated. Thermally averaged collision
strengths are presented as a function of electron temperature for
application to astrophysical plasmas.
Methods. The Bspline BreitPauli Rmatrix method is used to
investigate the electron impact excitation of forbidden and allowed
transitions in singly ionized potassium. The relativistic effects have
been incorporated through mass, Darwin and spinorbit onebody
operators in the BreitPauli Hamiltonian in the scattering calculation,
while in the calculation of oscillator strengths the onebody and
twobody relativistic operators are included. Flexible nonorthogonal
sets of spectroscopic and correlation radial functions are used to
obtain accurate description of K II levels and to represent the
scattering functions. The 43 finestructure levels of the ,
,
,
,
and
configurations have been considered in both the radiative and scattering calculations.
Results. The present cross sections for the excitation of the
lowest 4s and 3d states show reasonable agreement with the previous
Rmatrix calculations. The calculated excitation energies are in
excellent agreement with experiment and represents an improvement over
the previous calculation. The oscillator strengths for transitions from
the ground state normally compare very well with previous calculations,
but for transitions from the metastable state a large discrepancy was
found with other calculations. The effective collision strengths are
obtained by integrating total resonant and nonresonant collision
strengths over a Maxwellian distribution of electron energies and these
are presented over a wide temperature range.
Key words: atomic data  atomic processes  line: formation
1 Introduction
Oscillator strengths and electronion collision processes play an important role in the understanding of the physical processes and conditions in various types of plasmas. The singly ionized potassium has been detected in Io's atmosphere (Fegley & Zolotov 2000; Kuppers & Schneider 2000). Accurate oscillator strengths and electron excitation collision strengths for transitions in K II are important for the diagnostics of Io's atmosphere and plasma torus. The prominent emission lines in the spectrum of K II arise due to the  ,  ,  ,  and  finestructure transitions. These lines occur in the extreme ultraviolet (EUV) region of the spectrum and should be observable by the EUV spectrometers on board space missions.
The excitation energy levels, oscillator strengths and electron excitation cross sections for transitions in K II have been studied both theoretically and experimentally. The most recent measured excitation levels of K II have been reported by Peterson et al. (2007). Tayal & Zatsarinny (2008) calculated cross sections for excitation of the , , , and levels from the ground level using the Bspline BreitPauli Rmatrix (BSR) computer code (Zatsarinny 2006). The accurate description of K II target wave functions was obtained by taking a proper account of the strong term dependence of the valence orbitals and an adequate treatment of large correlation corrections and relaxation effects. Berrington et al. (2006) reported an intermediatecoupling frame transformation Rmatrix (icfRM) calculation for the excitation of the , , , and levels. The experimental study of excitation cross sections of K II were reported by Zapesochny et al. (1986) using the crossedbeams technique. Both the BSR and icfRM calculations predicted large discrepancies with experimental cross sections in magnitude as well as in near threshold resonance structure. The overall agreement between the two calculations was found to be reasonable for the excitation of the and states, but large discrepancies up to a factor of 3 were noted for the excitation to the higherlying , and states. The discrepancies between the BSR and icfRM calculations were mainly attributed to the differences in the description of target states and in the scattering models. The term dependence effects have been found to be most important for the and states. There are large correlation corrections for the state due to the strong penetration of the 3d electron in the core. We have extended Bspline BreitPauli Rmatrix calculation of Tayal & Zatsarinny (2008) to calculate effective collision strengths for all allowed and forbidden transitions among the 43 finestructure levels. We have also calculated oscillator strengths and transition probabilities for the allowed and intercombination lines among these levels.
2 Excitation energies, oscillator strengths and lifetimes
The K II wave functions exhibit large correlation corrections due to the corevalence interaction and strong term dependence of the oneelectron valence orbitals in the configurations. In addition, there is a strong interaction between the and Rydberg series together with strong spinorbit mixing of different terms. The K II target state wave functions have been constructed with nonorthogonal orbitals that are optimized for each atomic state separately to account for the term dependence. Our calculations are performed using the multiconfiguration HartreeFock (MCHF) method (Zatsarinny & Froese Fischer 2000; Froese Fischer 1991). The 43 K II bound levels of the , , , , and configurations have been considered in our work. These states show different correlation patterns. We first generated the spectroscopic 1s, 2s, 2p, 3s and 3p core orbitals from a HartreeFock (HF) calculation for the K^{2+} ground state. The term dependent excited orbitals 4s, 3d, 4p, 5s and 4d were then generated in separate HF calculations for each ^{1,3}L state. The HF wave functions were improved by the correlation functions ( , ; l=04) with a 3s, 3pexcited core. The correlation orbitals were optimized for each term separately.
Table 1: Energies (Ryd) and lifetimes of the excitation levels.
The spectroscopic and correlation functions are used to construct configurationinteraction (CI) expansions for different atomic states by allowing oneelectron and twoelectron excitations from all the basic configurations mentioned above. In the construction of CI expansions for finestructure levels with various J and we used configurations generated in this excitation scheme for the LS states and with insignificant configurations with coefficients less than 0.001 omitted from the expansions. The final CI expansions for the even and odd parity finestructure J levels contain from 26 to 167 configurations. We used 149 different nonorthogonal orbitals in our calculations. The excitation energies of finestructure levels relative to the ground level are reproduced in Table 1 from the work of Tayal & Zatsarinny (2008). Our results have been compared with measured values from Peterson et al. (2007) and the NIST compilation (http://physics.nist.gov). The agreement between our calculation and the experiment is excellent for the lowerlying levels of the and configurations and is very good for the higher levels. The accuracy of the higherlying levels is slightly less than that for the lowerlying levels. The term dependence effects are most strong for the and states. The present calculation represents an improvement over the calculation of Berrington et al. (2006) who used a set of the same orthogonal oneelectron orbitals to represent all target states. We believe that our wave functions correctly represent the main correlation corrections, the interactions between different Rydberg series and term dependence effects. The theoretical lifetimes calculated in length form for various finestructure levels are also displayed in last column of Table 1. We are not aware of any other theoretical or experimental lifetimes for the finestructure levels of K II.
Figure 1: Collision strengths for the intercombination  (15) transition as a function of electron energy between the and thresholds. Present results, upper panel; Berrington et al. (2006), lower panel. 

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The oscillator strengths for the dipoleallowed transitions from the ground and first metastable levels were compared by Tayal & Zatsarinny (2008) with the CI calculation of Berrington et al. (2006), the relativistic configurationinteraction (RCI) calculation of Beck (2002), the results calculated by Smirnov & Shapochkin (1979) from lifetime measurements and the measured values of Henderson et al. (1997). A rather reasonable agreement between different calculations was noted for transitions from the ground level except for the  and transitions where noticeable discrepancies were noted. For oscillator strengths from the first metastable level, the CI calculations of Tayal & Zatsarinny (2008) and Berrington et al. (2006) showed good agreement with each other, but large discrepancy was noted with the calculation of Smirnov & Shapochkin (1979). In the present paper, we present oscillator strengths and transition probabilities for all allowed and intercombination lines between the 43 finestructure levels in Table 2. The initial and final levels of a transition are given in Cols. 1 and 2 and in Col. 3 we have listed the corresponding indices of these levels from Table 1. The relativistic effects were included through the spinorbit, spinotherorbit, spinspin, mass and Darwin BreitPauli operators. The values of oscillator strengths for intercombination lines are usually much smaller than those for the allowed transitions. The intercombination lines are induced by the spinorbit interaction by causing mixing between different LS symmetries with the same set of quantum numbers J and . The agreement between the length and velocity forms of oscillator strengths may to some extent indicate the accuracy of wave functions and convergence of CI expansions. However, it is not a sufficient condition for the accuracy of results. The convergence of results is an important accuracy criterion. There is normally a very good agreement between the present length and velocity forms of oscillator strengths.
3 Collision strengths
The scattering calculations were carried out with a new general Rmatrix code (Zatsarinny 2006), in which nonorthogonal orbitals are used to describe both the target levels and the Rmatrix continuum basis functions. In our work the scattering orbitals were constructed orthogonal to the 1s, 2s, 2p, 3s and 3p core orbitals and to the spectroscopic valence 4s, 3d, 4p, 5s and 4d orbitals. No orthogonality constraints to the correlated orbitals have been imposed. Thus the bound part of the closecoupling expansions contained only a limited number of configurations. We chose 87 continuum basis functions for each angular momentum and a Rmatrix box of radius a = 30 au to contain all atomic orbitals. The relativistic effects in the scattering calculations have been incorporated in the BreitPauli Hamiltonian through the use of Darwin, mass correction and spinorbit operators. The scattering parameters are then found by matching the inner solution at r = ato the asymptotic solutions in the outer region. The calculations include all scattering symmetries with and these are estimated to give converged results for many forbidden transitions. The higher partial waves contributions ( ) for the dipoleallowed transitions are calculated using the CoulombBethe approach and a topup procedure based on geometric series approximation for the nonallowed transitions.
Figure 2: Collision strengths for the forbidden  (117) transition as a function of electron energy between the and thresholds. Present results, upper panel; Berrington et al. (2006), lower panel. 

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Figure 3: Collision strengths for the forbidden  (119) transition as a function of electron energy between the and thresholds. Present results, upper panel; Berrington et al. (2006), lower panel. 

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We chose a fineenergy grid of 10^{5} Ryd in the closedchannel energy region up to 2.1 Ryd. This allowed us to delineate important resonance structures almost completely in the collision strengths. At higher energies where all channels are open and there are no resonances, the collision strengths show smooth behavior and we use an energy grid of 0.1 Ryd. The collision strength calculations are performed in the energy range from threshold to 30.0 Ryd. The collision strengths for the intercombination  (15) and forbidden  (117) and  (119) transitions have been displayed as a function of incident electron energy from threshold to the threshold at 1.7 Ryd in Fig. 1 and from threshold to threshold around 2.0 Ryd in Figs. 2 and 3. Our results (upper panel) have been compared with the calculation of Berrington et al. (2006) (lower panel). The resonance structures are almost completely resolved with a very fine energy mesh in our calculation, whereas Berrington et al. (2006) appear to use a coarse energy mesh. The resonance structures are very complex and quite dense due to many overlapping Rydberg series converging to various thresholds. The background collision strengths away from resonances appear to be very similar from the two calculations, but there are differences in magnitude and position of resonances. The differences are mostly caused by the wave functions used in target descriptions and by the scattering models. The resonances make substantial enhancements in collision strengths for many transitions. Our calculation adequately account for the shortrange electron correlation to ensure correct position of resonances in the low energy region.
Figure 4: Collision strengths for the  (15) ( upper curves),  (117) ( lower curves) and  (119) ( middle curves) transitions as a function of electron energy in all open channels region. Present results, solid curve; Berrington et al. (2006), dashed curve. 

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We have displayed collision strengths for the  (15) (upper curves),  (117) (lower curves) and  (119) (middle curves) transitions in the energy region from 2.5 to 30.0 Ryd in Fig. 4 where all channels are open and collision strengths are expected to show smooth variation. Our results are compared with the calculation of Berrington et al. (2006). There is a reasonable agreement between the two calculations, particularly for the weaker forbidden  (117) and  (119) transitions. It may be noted that our collision strengths show a smooth behavior with energy, but the results of Berrington et al. (2006) contain pseudoresonance structures in the energy region up to about 14.0 Ryd. Our results for the  (15) transition are higher than the calculation of Berrington et al. (2006) by 10% to 16% at incident electron energies above 14.0 Ryd. The discrepancies between the two calculations for the  (15) transition perhaps have been caused by the differences in target wave functions, scattering models and the number of partial waves. The inclusion of pseudostates and additional (N+1) terms in the calculation of Berrington et al. (2006) may account for coupling to higher autoionizing and continuum levels and thus reduce the collision strength at intermediate energies. The cause of differences for 127 transition appears to be the same. It may be noted that Berrington et al. (2006) included up to J = 14.5 partial waves and applied a topup. The nonresonant background collision strengths for the allowed transitions are normally larger than for the forbidden and intercombination transitions, while the resonance enhancement in collision strengths for the forbidden and intercombination transitions are generally larger compared to allowed transitions. We present collision strengths for all transitions between the 43 levels at seven incident electron energies 2.1, 4.0, 6.0, 8.0, 12.0, 16.0 and 20.0 Ryd in Table 3. The indices of initial and final levels of a transition (ij) are given in the first column of Table 3 and these are taken from Table 1. These energies are above the highest excitation threshold at 2.066 Ryd where collision strength varies smoothly with incident electron energies. The collision strengths for different types of transitions generally show expected energy behavior in the higher energy region. The collision strengths for dipoleallowed transitions exhibit logarithmic behavior whereas collision strengths for intercombination transitions falloff rapidly with increasing energy.
4 Effective collision strengths
We have calculated effective collision strengths by integrating the collision strengths over a Maxwellian distribution of electron energies. We have presented effective collision strengths for all transitions between the lowest 43 finestructure levels considered in our work in Table 4. The indices of lower and upper levels of a transition are given in Table 1. The results are presented at 11 temperatures in the range K suitable for modeling of astrophysical plasmas. The results for transitions to higher excitation levels may be somewhat uncertain because of the neglect of coupling with exciting levels above level 43. The effective collision strength for the allowed transitions increases with increasing temperature and the effective collision strength for the intercombination transitions decreases with increasing temperature in the hightemperature regime. The intercombination transitions can occur only through electron exchange.
Figure 5: Effective collision strengths for the  (15) (solid curve),  (117) (dashed curve) and  (119) (dashdotted curve) transitions as a function of electron temperature. 

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Figure 6: Effective collision strengths for the forbidden  (118) (solid curve),  (121) (dashed curve) and  (125) (dashdotted curve) transitions as a function of electron temperature. 

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We have plotted effective collision strengths for the  (15),  (117) and  (119) transitions as a function of electron temperature in the range K in Fig. 5. To the best of our knowledge there are no other previous effective collision strengths available to compare with our results. The high quality wave functions that take adequate account of the term dependence of valence orbitals and of relaxation effects have been used in our calculation. However, we have not checked on the coupling effects from higher lying excitation levels. The effective collision strengths for the forbidden  (118),  (121) and  (125) transitions are shown in Fig. 6 as a function of electron temperature. The effective collision strengths for the forbidden transitions generally display a slow falloff with increasing temperature. The effective collision strengths for the forbidden  (118) transition decrease more rapidly than for the  (121) and  (125) transitions.
5 Discussion and conclusions
There are generally two major sources of error in any closecoupling scattering calculation: target description and convergence of closecoupling expansion. Based on a comparison of calculated excitation energies and oscillator strengths with experiments and other reliable calculations as well as the fact that we used a well tested sound approach, our wave functions appear to be of high quality to yield an accurate target description. The excitation energies of the target levels shown in Table 1 demonstrate that our theory deviates from experiment by less than 0.5% for all but 7 excited levels (1216, 26 and 29) where the percentage deviation is generally less than 0.6%. The worst agreement is for the level which deviates from the experiment by 0.65%. The excitation energies depend to first order on the accuracy of wave functions, while oscillator strengths depend to second order and thus can provide a better test of the wave functions. A detailed comparison of oscillator strengths from the ground and first metastable levels with other calculations and experiment was presented by Tayal & Zatsarinny (2008) who noted a reasonable agreement with other reliable calculations. Our radiative parameters are estimated to be accurate to about 5%.
A sufficient number of target states in the closecoupling expansion are required to achieve convergence for the transitions of interest. This factor may introduce varied amount of uncertainties in different transitions. The collision strengths for transitions between the levels of , , and configurations are well converged in our calculation. The transitions from the ground level to the , and levels should be most accurate (15%) and somewhat less accurate (20%) for transitions between excited levels. For higher levels of the configuration there may be significant coupling effects from higher autoionizing and continuum states and thus our results for transitions involving levels may be less accurate (25% or worse). It may also be noted that we used a very fineenergy mesh (10^{5} Ryd) in the threshold energy region to resolve almost all resonances and performed scattering calculation up to 30.0 Ryd. The Bspline Rmatrix calculations were carried out for partial waves up to J = 50.5. The topup procedures based on the CoulombBethe approach for the allowed transitions and geometric progression approach for nonallowed transitions were used to estimate the contributions for . These scattering parameters were considered sufficient and not to cause any further significant uncertainties in our results.
In conclusion, we have presented accurate oscillator strengths and effective collision strengths for K II lines among the 43 finestructure levels. We used nonorthogonal orbitals for the construction of CI wave functions for target levels and for the representation of scattering functions. The checks on excitation energies and oscillator strengths provide us confidence that our target wave functions are likely to be accurate to yield reliable collision strengths from the Bspline BreitPauli Rmatrix scattering calculations. The present target wave functions contain both the valence and corevalence correlation, together with the relaxation effects due to the strong penetration of the electron in the core. We have attempted to account for important physical effects such as electron correlation, relativistic, and channel coupling effects. The Rydberg series of resonances converging to several excited levels are found to enhance collision strengths substantially. The effective collision strengths are presented over a wide range of electron temperatures suitable for use in astrophysical plasmas modeling. We have also reported accurate radiative rates for E1 transitions using nonorthogonal wave functions in multiconfiguration HartreeFock approach which are of considerable interest in astrophysical applications.
AcknowledgementsThis research work is supported by NASA grant NNX09AB63G from the Planetary Atmospheres program. The authors would like thank Professor K. A. Berrington for sending their collision strength data in electronic form.
References
 Beck, D. R. 2002, J. Phys. B, 35, 4155 [NASA ADS] [CrossRef] (In the text)
 Berrington, K. A., Nakazaki, S., & Murakami, Y. 2006, J. Phys. B, 39, 4733 [NASA ADS] [CrossRef] (In the text)
 Fegley, B., & Zolotov, M. Y. 2000, Icarus, 148, 193 [NASA ADS] [CrossRef] (In the text)
 Froese Fischer, C. 1991, Comput. Phys. Commun., 64, 369 [NASA ADS] [CrossRef] (In the text)
 Henderson, M., Curtis, L. J., Matulioniene, R., Ellis, D. G., & Li, Y. 1997, Phys. Rev. A, 55, 2733 [NASA ADS] [CrossRef] (In the text)
 Kuppers, M., & Schneider, N. M. 2000, Geophys. Res. Lett., 27, 513 [NASA ADS] [CrossRef] (In the text)
 Peterson, K., Ekberg, J. O., Martinson, I., & Reader, J. 2007, Phys. Scr., 75, 702 [NASA ADS] [CrossRef] (In the text)
 Smirnov, Y. M., & Shapochkin, M. B. 1979, Opt. Spectrosc., 47, 6 [NASA ADS] (In the text)
 Tayal, S. S., & Zatsarinny, O. 2008, Phys. Rev. A, 78, 012713 [NASA ADS] [CrossRef] (In the text)
 Zapesochny, A. I., Imre, A. I., Aleksakhin, I. S., Zapesochny, I. P., & Zatsarinny, O. I. 1986, Sov. Phys. JETP, 63, 1155 (In the text)
 Zatsarinny, O. 2006, Comput. Phys. Commun., 124, 247 [NASA ADS] [CrossRef] (In the text)
 Zatsarinny, O., & Froese Fischer, C. 2000, Comput. Phys. Commun., 124, 247 [NASA ADS] [CrossRef] (In the text)
Footnotes
 ... II^{}
 Tables 24 are only available in electronic form at the
CDS via anonymous ftp to cdsarc.ustrasbg.fr (130.79.128.5) or via
http://cdsweb.ustrasbg.fr/cgibin/qcat?J/A+A/510/A79
All Tables
Table 1: Energies (Ryd) and lifetimes of the excitation levels.
All Figures
Figure 1: Collision strengths for the intercombination  (15) transition as a function of electron energy between the and thresholds. Present results, upper panel; Berrington et al. (2006), lower panel. 

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In the text 
Figure 2: Collision strengths for the forbidden  (117) transition as a function of electron energy between the and thresholds. Present results, upper panel; Berrington et al. (2006), lower panel. 

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In the text 
Figure 3: Collision strengths for the forbidden  (119) transition as a function of electron energy between the and thresholds. Present results, upper panel; Berrington et al. (2006), lower panel. 

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In the text 
Figure 4: Collision strengths for the  (15) ( upper curves),  (117) ( lower curves) and  (119) ( middle curves) transitions as a function of electron energy in all open channels region. Present results, solid curve; Berrington et al. (2006), dashed curve. 

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In the text 
Figure 5: Effective collision strengths for the  (15) (solid curve),  (117) (dashed curve) and  (119) (dashdotted curve) transitions as a function of electron temperature. 

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In the text 
Figure 6: Effective collision strengths for the forbidden  (118) (solid curve),  (121) (dashed curve) and  (125) (dashdotted curve) transitions as a function of electron temperature. 

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In the text 
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