2. The calculation

The basic atomic theory employed in the IRON Project, including methodology and computer codes, is described in IP I. Collision strengths for fine structure transitions are calculated from collision data obtained in LS coupling by using an algebraic transformation to intermediate coupling, as described in Sect. 2.6 of IP I (see also Saraph 1978). This method neglects the fine structure splitting of the target terms.

2.1. The target

Here we take account of the autoionization processes that can affect the cross section of the fine structure transition in the ²P ground term at collision energies up to that required to excite the 28th target term . Thus the energy range covered by the present paper is twice that of IP IV for the lighter ions and ten times that for the iron group ions. The dominant configuration in terms 3 to 28 is , where 3l is a spectroscopic orbital with . We take the radial orbitals from Clementi & Roetti (1974) and calculate using Hibbert's (1975) program CIV3. The parameterised form of P_3l(r) is the one used by Clementi & Roetti (1974), namely

where

Our choice for the integers in Eqs. (1) and (2) is for l = 0, for l = 1 and for l = 2. The values we give in Table 2 for C_3lp and were obtained following the procedure adopted by Mohan & Hibbert (1991). That is to say, the parameters of each P_3l were varied in order to minimise the energies of selected terms: for 3s; for 3p; for 3d. Having obtained satisfactory n = 3 orbitals for singly ionized neon we then used the Ne⁺ parameters as initial trial values for Na⁺². The optimised parameters obtained in this way for Na⁺² were then used as trial values for Mg⁺³ and so on along the sequence as far as Ni⁺¹⁹. In all cases we used the minimization code MODDAV by setting the CIV3 parameter IDAVID = 0.

Mohan & Hibbert (1991) have obtained spectroscopic n = 3 orbitals with CIV3 for six of the ions dealt with here. Our orbitals are in satisfactory agreement with theirs, except for . We have no explanation why Mohan & Hibbert's parameters for these two ions deviate from the smooth behaviour exhibited lower down the sequence. Figures 1a and 1b, which make use of the isoelectronic fitting procedure proposed by Burgess et al. (1997a), illustrate this point.

Blackford & Hibbert (1994) have also studied fluorine-like ions using CIV3, but their n=3 orbitals are non-spectroscopic.

  
Figure 1: Full line curves are cubic spline fits to data in Table 2 with rms errors a) 0.5% and b) 0.01%. The starred points correspond to results from Mohan & Hibbert (1991)

Table 1 lists the configurations that we include for the construction of the target terms. The first 28 terms that can be constructed from these configurations were included in the close-coupling expansion for the scattering calculation. They have dominant configurations labelled 1 to 11 and their calculated energies are given in Table 3, where a scaling factor (Z-9)^-2 is applied for convenience.

 

Label Configuration Label Configuration

1 11

2 12

3 13

4 14

5 15

6 16

7 17

8 18

9 19

10 20 Correlation configurations

Table 1: Configurations that give rise to the 28 lowest terms of fluorine-like ions. Parent terms are shown and each configuration includes . Correlation configurations used in the target are also listed

 

 

Z nlp=301 nlp=302 nlp=303 nlp=311 nlp=312 nlp=321 nlp=322

10 8.095246 3.266739 1.163410 3.530236 0.923711 2.462823 0.687194

0.099730 1.036262 0.260780 0.031488 0.991278

11 8.919863 3.567821 1.515644 4.009371 1.256699 3.136974 1.046041

0.120510 1.074791 0.320611 0.049018 0.981038

12 9.690180 3.883820 1.850745 4.480893 1.580585 3.791358 1.403897

0.135930 1.115100 0.367303 0.058978 0.972880

13 10.42993 4.205650 2.183463 4.945376 1.900644 4.427342 1.758242

0.148996 1.156396 0.403524 0.063832 0.967407

14 11.14813 4.530462 2.515404 5.404015 2.218909 5.050042 2.109039

0.160455 1.197579 0.434109 0.065677 0.964041

15 11.85268 4.854268 2.847533 5.857134 2.536399 5.662291 2.456780

0.170602 1.238524 0.465932 0.065850 0.962103

16 12.54460 5.177604 3.179915 6.305809 2.853645 6.267704 2.802139

0.179721 1.278762 0.483432 0.065052 0.961141

17 13.22786 5.499258 3.512871 6.750348 3.170891 6.867074 3.145558

0.172211 1.310172 0.503925 0.063741 0.960803

18 13.90064 5.820202 3.846325 7.191539 3.488298 7.462762 3.487506

0.195803 1.357124 0.522634 0.062132 0.960898

19 14.56788 6.137855 4.180618 7.629962 3.805878 8.055196 3.828262

0.202908 1.395478 0.539956 0.060383 0.961273

20 15.22800 6.453572 4.515461 8.065201 4.123840 8.644899 4.167991

0.209521 1.433188 0.555645 0.058593 0.961825

21 15.88471 6.761167 4.852479 8.497378 4.442167 9.232155 4.506887

0.215631 1.472043 0.569899 0.056812 0.962487

22 16.53345 7.069813 5.189106 8.927572 4.760788 9.817833 4.845138

0.221396 1.509494 0.583375 0.055063 0.963224

23 17.17390 7.375341 5.526314 9.356097 5.079613 10.40225 5.182819

0.226692 1.546702 0.596156 0.053365 0.964003

24 17.81315 7.677423 5.864517 9.782190 5.398982 10.98535 5.520072

0.231902 1.584103 0.608036 0.051732 0.964801

25 18.44708 7.979157 6.202479 10.20699 5.718553 11.56755 5.856933

0.236923 1.620604 0.619405 0.050166 0.965604

26 19.08486 8.266246 6.544050 10.62908 6.038546 12.14856 6.193407

0.249244 1.699552 0.629756 0.048676 0.966397

27 19.70004 8.571291 6.880852 11.05077 6.358677 12.73139 6.526987

0.246277 1.694465 0.639960 0.047153 0.967271

28 20.32927 8.847834 7.223726 11.47005 6.679255 13.31948 6.864242

0.250387 1.736172 0.649441 0.046126 0.967818

Table 2: Fluorine sequence: radial orbital parameters. For each Z, nlp the upper (lower) number is the value of (C_nlp)

 

 

Label Term Z=12 Z=13 Z=14 Z=15 Z=16 Z=17

11 0.76509 0.57471 0.47113 0.40655 0.36263 0.33091

8 0.73354 0.55500 0.45745 0.39642 0.35476 0.32460

8 0.73145 0.55377 0.45670 0.39593 0.35444 0.32437

8 0.73326 0.55411 0.45618 0.39493 0.35317 0.32295

8 0.73162 0.55297 0.45534 0.39427 0.35263 0.32250

8 0.72757 0.54955 0.45239 0.39168 0.35033 0.32043

5 0.70069 0.53418 0.44236 0.38454 0.34492 0.31614

5 0.70167 0.53432 0.44211 0.38410 0.34440 0.31559

5 0.69690 0.53081 0.43948 0.38209 0.34283 0.31435

5 0.69673 0.53045 0.43905 0.38165 0.34239 0.31392

5 0.69430 0.52849 0.43741 0.38024 0.34116 0.31283

5 0.68862 0.52390 0.43357 0.37694 0.33827 0.31026

10 0.69524 0.52486 0.43264 0.37532 0.33641 0.30835

7 0.68138 0.51666 0.42700 0.37109 0.33304 0.30557

7 0.66177 0.50287 0.41651 0.36271 0.32611 0.29968

7 0.65414 0.49747 0.41238 0.35936 0.32332 0.29729

4 0.62893 0.48202 0.40165 0.35133 0.31698 0.29210

4 0.62893 0.48202 0.40165 0.35133 0.31698 0.29210

4 0.62533 0.47946 0.39968 0.34974 0.31565 0.29095

4 0.62949 0.48141 0.40067 0.35025 0.31590 0.29105

4 0.61927 0.47520 0.39643 0.34712 0.31347 0.28909

9 0.63268 0.48077 0.39878 0.34793 0.31347 0.28865

4 0.61116 0.46944 0.39200 0.34354 0.31048 0.28652

6 0.59817 0.45835 0.38264 0.33527 0.30317 0.27999

3 0.57145 0.44153 0.37057 0.32618 0.29589 0.27396

3 0.55856 0.43289 0.36418 0.32116 0.29179 0.27051

2 0.32916 0.21074 0.15127 0.11644 0.09361 0.07830

1 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000

Table 3: Theoretical term energies for the fluorine sequence in . Label refers to configurations given in Table 1

 

 

Label Term Z=18 Z=19 Z=20 Z=21 Z=22 Z=23

11 0.30696 0.28826 0.27327 0.26099 0.25075 0.24208

8 0.30176 0.28388 0.26951 0.25772 0.24787 0.23952

8 0.30160 0.28376 0.26943 0.25766 0.24782 0.23949

8 0.30010 0.28225 0.26791 0.25616 0.24636 0.23805

8 0.29972 0.28191 0.26761 0.25589 0.24611 0.23783

8 0.29784 0.28019 0.26604 0.25444 0.24479 0.23656

5 0.29432 0.27723 0.26348 0.25220 0.24278 0.23479

5 0.29377 0.27669 0.26297 0.25171 0.24231 0.23434

5 0.29276 0.27586 0.26227 0.25112 0.24180 0.23390

5 0.29236 0.27547 0.26191 0.25077 0.24147 0.23359

5 0.29137 0.27458 0.26109 0.25002 0.24077 0.23293

5 0.28907 0.27249 0.25918 0.24826 0.23914 0.23142

10 0.28720 0.27070 0.25746 0.24662 0.23758 0.22994

7 0.28484 0.26865 0.25566 0.24501 0.23613 0.22862

7 0.27972 0.26412 0.25161 0.24136 0.23281 0.22556

7 0.27762 0.26226 0.24994 0.23984 0.23142 0.22428

4 0.27326 0.25851 0.24667 0.23695 0.22882 0.22194

4 0.27326 0.25851 0.24667 0.23695 0.22882 0.22194

4 0.27226 0.25762 0.24586 0.23622 0.22816 0.22132

4 0.27226 0.25756 0.24576 0.23609 0.22802 0.22117

4 0.27063 0.25618 0.24457 0.23504 0.22708 0.22033

9 0.26993 0.25534 0.24365 0.23407 0.22610 0.21935

4 0.26838 0.25418 0.24277 0.23342 0.22560 0.21896

6 0.26248 0.24881 0.23785 0.22886 0.22136 0.21501

3 0.25735 0.24436 0.23392 0.22536 0.21820 0.21214

3 0.25438 0.24176 0.23161 0.22328 0.21632 0.21041

2 0.06693 0.05831 0.05157 0.04617 0.04176 0.03810

1 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000

 

Label Term Z=24 Z=25 Z=26 Z=27 Z=28

11 0.23465 0.22822 0.22261 0.21762 0.21321

8 0.23235 0.22613 0.22071 0.21587 0.21159

8 0.23233 0.22612 0.22071 0.21587 0.21160

8 0.23093 0.22476 0.21939 0.21460 0.21036

8 0.23073 0.22457 0.21920 0.21443 0.21020

8 0.22954 0.22345 0.21815 0.21343 0.20926

5 0.22793 0.22199 0.21680 0.21218 0.20810

5 0.22750 0.22158 0.21641 0.21181 0.20774

5 0.22712 0.22124 0.21612 0.21155 0.20750

5 0.22682 0.22095 0.21584 0.21128 0.20725

5 0.22621 0.22038 0.21530 0.21077 0.20677

5 0.22480 0.21905 0.21404 0.20959 0.20564

10 0.22338 0.21771 0.21274 0.20836 0.20447

7 0.22217 0.21659 0.21170 0.20739 0.20356

7 0.21935 0.21397 0.20926 0.20510 0.20141

7 0.21817 0.21286 0.20823 0.20413 0.20049

4 0.21603 0.21090 0.20641 0.20244 0.19892

4 0.21603 0.21090 0.20641 0.20244 0.19892

4 0.21546 0.21037 0.20591 0.20198 0.19848

4 0.21530 0.21021 0.20576 0.20182 0.19832

4 0.21454 0.20952 0.20512 0.20123 0.19777

9 0.21356 0.20855 0.20416 0.20030 0.19686

4 0.21327 0.20833 0.20401 0.20019 0.19679

6 0.20957 0.20485 0.20071 0.19707 0.19383

3 0.20693 0.20241 0.19845 0.19496 0.19185

3 0.20534 0.20094 0.19709 0.19368 0.19066

2 0.03500 0.03236 0.03008 0.02809 0.02634

1 0.00000 0.00000 0.00000 0.00000 0.00000

2.2. Collision strength

The results for from the extended target join on fairly smoothly at the threshold to those from the 2-term calculation (IP IV). This is because, as predicted in IP IV, resonances caused by the 26 higher terms start to enhance the collision strength at energies well above for ions with . The 26 terms arising from configurations with , lie energetically close together and this means that the associated resonances form a dense "forest'' of spikes covering a broad energy region. The widths of these spikes and the windows between them are very narrow compared to that of the electron velocity distribution function. Since it is our aim to calculate effective collision strengths it is not necessary, at these high excitation energies, to identify individual resonances, or to obtain their positions to high accuracy. Instead we obtain the collision strength to sufficient detail so that the average is correct. In any case, measurements that might allow one to replace calculated term energies by experimental values are incomplete for the higher terms and therefore we could not correct the target term energies empirically as was done in IP IV. At low temperatures, it will be recalled, the width of the velocity distribution function and those of the resonances are comparable and therefore the correct positioning of the low energy resonances due to the term dramatically affects the low temperature effective collision strength of several ions.

We illustrate the complicated energy dependence of our collision strengths by plotting in Figs. 2 to 10 as a function of from threshold (i.e. ) to a value just above that necessary to excite the 28th term. The collision strength is obtained using different steplengths in four distinct energy bands. From threshold up to S we use the collision strength obtained in IP IV. Then at energies between the second and third terms, i.e. between S and P, the collision strength is calculated using an energy mesh based on equal steps in effective quantum number , see IP IV. This mesh is ideal for delineating the resonance structure in this interval and about 3600 mesh points are used for each ion. Partial wave contributions are summed up to J=7. The energy range from just below the 3rd term and up to the 28th term was scanned at constant steps in energy : between the 3rd and 17th terms and between the 17th and 28th terms , where x varies between x= 3 for the lighter ions and x=1 for the heavier ones. We found that the resonances were in all cases less prominent at energies between terms 17 and 28 which is why we could use a coarser mesh. From the 3rd threshold onwards, i.e. for , partial waves were summed up to J = 10. At energies above all 28 thresholds collision strengths were obtained on a very coarse mesh but particular attention was paid to the convergence with respect to angular momentum J. The contributions from the two highest partial waves (J= 9 and 10) were taken to fit a geometric series and thus used to estimate a top-up to the collision strengths. Near the energy of the 28th term this top-up amounted to about 2% of the total and it gradually increased to 10% at around four times that energy. Tabulation was stopped at this point and, for the purpose of calculating effective collision strengths, the high energy results were then spline fitted including a value at obtained in the Born approximation (Burgess et al. 1997b). These limiting Born values are given in Table 4.

  
Figure 2: :

  
Figure 3: :

  
Figure 4: :

  
Figure 5: :

  
Figure 6: :

  
Figure 7: :

  
Figure 8: :

  
Figure 9: :

  
Figure 10: :

 

Ion (a) (b) (c)

F 4.040 3.609 0.00368

Ne⁺ 2.678 4.600 0.00711

Na⁺² 1.922 4.999 0.01245

Mg⁺³ 1.452 5.205 0.02031

Al⁺⁴ 1.137 5.375 0.03138

Si⁺⁵ 0.9156 5.543 0.04642

P⁺⁶ 0.7540 5.706 0.06628

S⁺⁷ 0.6317 5.857 0.09191

Cl⁺⁸ 0.5373 5.991 0.12432

Ar⁺⁹ 0.4628 6.111 0.16462

K⁺¹⁰ 0.4027 6.217 0.21402

Ca⁺¹¹ 0.3537 6.312 0.27379

Sc⁺¹² 0.3133 6.396 0.34532

Ti⁺¹³ 0.2793 6.471 0.43008

V⁺¹⁴ 0.2508 6.538 0.52963

Cr⁺¹⁵ 0.2262 6.599 0.64562

Mn⁺¹⁶ 0.2052 6.654 0.77982

Fe⁺¹⁷ 0.1870 6.704 0.93408

Co⁺¹⁸ 0.1711 6.750 1.11036

Ni⁺¹⁹ 0.1572 6.792 1.31071

Table 4: (a) High energy limiting values of obtained by spline fitting the Born results of Burgess et al. (1997b) in the manner proposed by Burgess et al. (1997a). (b) Logarithm of the temperature of maximum coronal abundance obtained by spline fitting the data given in Arnaud & Rothenflug (1985), see Burgess et al. (1997a). (c) The fine structure transition energy in Ry from Edlén's (1969) fit to observed intervals

 

2.3. Effective collision strength

Seaton (1953) defines the thermally averaged, or effective, collision strength for a transition to be the integral

Since our calculation is in LS coupling it ignores the fine structure energy splitting. This means that the incident and final collision energies E_i, E_f are identical. In Table 5 we tabulate as a function of log, where is a scaled temperature that is convenient to use for the fluorine isoelectronic sequence. The present results agree with those in IP IV except at the highest temperature for each ion considered in IP IV where the earlier results are usually a few percent lower.

 

Z=12 Z=13 Z=14 Z=15 Z=16 Z=17

3.0 3.579^-1 5.136^-1 3.014^-1 2.570^-1 2.408^-1 2.691^-1

3.2 3.605^-1 4.895^-1 3.793^-1 2.600 ^-1 2.674^-1 2.859^-1

3.4 3.679^-1 4.703^-1 4.176^-1 2.751 ^-1 2.883^-1 2.881^-1

3.6 3.827^-1 4.637^-1 4.351^-1 2.923^-1 2.968^-1 2.799^-1

3.8 4.045^-1 4.636^-1 4.340^-1 3.045^-1 2.974^-1 2.725^-1

4.0 4.289^-1 4.625^-1 4.241^-1 3.147^-1 2.996^-1 2.734^-1

4.2 4.482^-1 4.549^-1 4.081^-1 3.193^-1 2.986^-1 2.721^-1

4.4 4.518^-1 4.328^-1 3.796^-1 3.071^-1 2.824^-1 2.552^-1

4.6 4.334^-1 3.937^-1 3.369^-1 2.764^-1 2.500^-1 2.233^-1

4.8 3.960^-1 3.444^-1 2.872^-1 2.362^-1 2.103^-1 1.855^-1

5.0 3.491^-1 2.941^-1 2.393^-1 1.967^-1 1.724^-1 1.503^-1

Z=18 Z=19 Z=20 Z=21 Z=22 Z=23

3.0 4.206^-1 1.418^-1 1.577^-1 1.852^-1 2.168^-1 1.000^-1

3.2 3.572^-1 1.602^-1 1.702^-1 1.800^-1 1.916^-1 1.107^-1

3.4 3.093^-1 1.723^-1 1.734^-1 1.698^-1 1.697^-1 1.143^-1

3.6 2.736^-1 1.779^-1 1.719^-1 1.609^-1 1.567^-1 1.183^-1

3.8 2.541^-1 1.866^-1 1.769^-1 1.630^-1 1.590^-1 1.312^-1

4.0 2.511^-1 2.010^-1 1.889^-1 1.737^-1 1.694^-1 1.475^-1

4.2 2.488^-1 2.082^-1 1.939^-1 1.779^-1 1.717^-1 1.531^-1

4.4 2.321^-1 1.976^-1 1.821^-1 1.664^-1 1.585^-1 1.426^-1

4.6 2.014^-1 1.724^-1 1.573^-1 1.431^-1 1.348^-1 1.215^-1

4.8 1.656^-1 1.420^-1 1.286^-1 1.165^-1 1.087^-1 9.786^-2

5.0 1.327^-1 1.138^-1 1.026^-1 9.273^-2 8.576^-2 7.700^-2

Z=24 Z=25 Z=26 Z=27 Z=28

3.0 1.004^-1 1.062^-1 1.091^-1 1.033^-1 9.624^-2

3.2 1.073^-1 1.083^-1 1.048^-1 9.679^-2 8.698^-2

3.4 1.081^-1 1.055^-1 9.811^-2 9.036^-2 8.019^-2

3.6 1.112^-1 1.080^-1 9.694^-2 9.112^-2 8.175^-2

3.8 1.235^-1 1.221^-1 1.055^-1 1.012^-1 9.280^-2

4.0 1.382^-1 1.380^-1 1.165^-1 1.123^-1 1.044^-1

4.2 1.419^-1 1.417^-1 1.187^-1 1.137^-1 1.064^-1

4.4 1.308^-1 1.299^-1 1.088^-1 1.034^-1 9.695^-2

4.6 1.105^-1 1.088^-1 9.165^-2 8.644^-2 8.103^-2

4.8 8.845^-2 8.638^-2 7.317^-2 6.862^-2 6.432^-2

5.0 6.920^-2 6.695^-2 5.715^-2 5.341^-2 5.004^-2

Table 5: Fluorine sequence: thermally averaged collision strength as a function of the atomic number Z and the logarithm of the scaled temperature

 

 

Z 1,2 2,3 3,28

14 3.0 100 0 0 0

4.0 85 14 1 0

5.0 28 32 9 31

20 3.0 100 0 0 0

4.0 48 50 1 1

5.0 12 54 5 29

26 3.0 99 1 0 0

4.0 31 67 1 1

5.0 7 61 4 27

Table 6: Showing how percentage contributions to from four energy bands vary with temperature. The first 3 bands are identified by pairs of indices which label initial and final term energies. For example, the column headed is for integration from to . Term energies are measured relative to the ground term, i.e. . The last column is for the band that extends from to .  is the scaled temperature defined by

 

It is important to understand the validity of approximations made in the current scattering calculations. First, one establishes the range of temperatures over which data are required. Next, one determines the range of energies for which accurate collision data need to be calculated. Table 6 shows the relative contributions to the total effective collision strength from the four different energy bands defined in Sect. 2.2 as a function of temperature. As discussed in IP IV, the isolated resonances in the first band must be delineated to the best possible accuracy because the velocity distribution function emphasizes these structures in a very selective way. Resonances in the second band are narrower relative to the width of the distribution function. Consequently the exact position is no longer so crucial but good delineation is still very important. This is ensured by the use of a steplength that is a function of effective quantum number calculated relative to the third target term . The step size therefore decreases as , where is the energy separation from the third target term. The third band would pose considerable computational problems if one wanted to cover it using a fixed steplength because of the many overlapping resonances from different channels. However, their mean contribution to the total effective collision strength can be obtained by using energy sampling at a moderately small steplength (see Saraph & Storey 1996). The last (i.e. fourth) band contributes very little although it is infinite in size. In order to include all collision channels here one would have to increase the close coupling expansion further and further, including also continuum states beyond the ionization threshold. Such calculations are very demanding on computer time (see Pelan & Berrington 1997), but fortunately they are not necessary for the present purposes. The ions considered in this paper have maximum coronal abundances at temperatures between 10⁵ K and 10⁷ K, and the present calculations are accurate for these temperatures because the higher energy bands, for which this calculation is rather crude, contribute relatively little.

We note that the present high temperature results tend to the Born limit where . Values of the limit for neutral fluorine and all ions in the sequence as far as are given in Table 4.