3. Results

In Tables 1 (click here), 2 (click here) and 3 (click here), we display the f-values obtained in the present work together with theoretical values reported by other authors, given that no experimental measurements have been found in the literature. In the figures we show the systematic trends that some of the studied oscillator strengths present for individual transitions along the isoelectronic sequence when they are plotted versus the inverse of Z.

For transition we have performed both RQDO and MCDF calculations. Inspection of Table 1 (click here) shows that our oscillator strengths compare generally well with the f-values calculated by Biémont (1986a, 1986b), who combined the self-consistent field (HXR) method of Cowan (1981), which explicitly accounts for configuration interaction and some relativistic effects for obtaining the wave functions with a semiempirical fitting procedure for obtaining the energy eigenvalues. Biémont reports f-values (1986a) for the first few ions of the sequence only. Our RQDO f-values take explicitly into account core-polarization effects. It should be born in mind that some of the levels involved in the transition can be subjected to perturbations by other levels, as pointed out by Biémont (1986a). These perturbations are explicitly accounted for in the MCDF calculations. In the RQDO procedure, the effect of perturbations is only implicitly included through the quantum defects extracted from experimental energies. Overall, the RQDO f-values agree reasonably well with the MCDF results and these by Biémont (1986a, 1986b), both expected to be correct. A source of discrepancy between the RQDO and HXR (Biémont 1986a) f-values is, quite possibly, the use of different energy data. Both the RQDO and MCDF results, as well as the HXR f-values, comply with the general feature which characterises an LS coupling scheme (Cowan 1981), in all ions comprised in Table 1 (click here), of reflecting that the strongest transitions are those corresponding to , in the two groups of triplet-triplet transitions which correspond, respectively, to the first three and last three columns of the table.

 

Z ION M

19 KVI a. 0.1352 0.0468 0.0337 0.0569 0.0360 0.1041

b. 0.1724 0.0581 0.0433 0.0706 0.0436 0.1280

c. 0.1413 0.0482 0.0349 0.0621 0.0372 0.1099

20 CaVII a. 0.1337 0.0467 0.0332 0.0565 0.0362 0.1031

b. 0.1709 0.0577 0.0430 0.0698 0.0433 0.1267

c. 0.1380 0.0482 0.0333 0.0607 0.0364 0.1074

21 ScVIII a. 0.1306 0.0460 0.0323 0.0554 0.0361 0.1009

b. 0.1695 0.0573 0.0427 0.0690 0.0431 0.1253

c. 0.1349 0.0460 0.0318 0.0593 0.0364 0.1026

22 TiIX a. 0.1271 0.0451 0.0312 0.0541 0.0361 0.0982

b. 0.1678 0.0568 0.0423 0.0680 0.0427 0.1238

c. 0.1288 0.0450 0.0304 0.0566 0.0364 0.0980

23 VX a. 0.1235 0.0443 0.0300 0.0528 0.0362 0.0952

b. 0.1706 - 0.0430 0.0690 0.0435 0.1257

24 CrXI a. 0.1202 0.0434 0.0289 0.0516 0.0375 0.0921

b. 0.1697 0.0576 0.0429 0.0685 0.0434 0.1248

25 MnXII a. 0.1172 0.0427 0.0279 0.0504 0.0372 0.0888

b. 0.1683 - 0.0426 0.0677 0.0431 0.1234

26 FeXIII a. 0.1144 0.0420 0.0269 0.0493 0.0380 0.0853

b. - 0.0581 - 0.0664 - 0.1212

27 CoXIV a. 0.1119 0.0414 0.0260 0.0483 0.0390 0.0816

b. 0.1512 - 0.0384 0.0660 0.0390 0.1207

28 NiXV a. 0.1097 0.0408 0.0252 0.0473 0.0402 0.0778

b. - - - 0.0642 - 0.1196

29 CuXVI a 0.1078 0.0403 0.0245 0.0465 0.0414 0.0738

b. 0.1607 0.0551 0.0410 0.0636 0.0415 0.1162

30 ZnXVII a 0.1060 0.0399 0.0239 0.0456 0.0427 0.0698

b. 0.1586 0.0546 0.0406 0.0620 0.0412 0.1135

31 GaXVIII a 0.1044 0.0395 0.0233 0.0449 0.0439 0.0658

b. 0.1514 0.0523 0.0389 0.0588 0.0395 0.1077

32 GeXIX a 0.1030 0.0392 0.0228 0.0442 0.0451 0.0619

b. 0.1496 0.0518 0.0385 0.0579 0.0392 0.1060

33 AsXX a 0.1016 0.0389 0.0223 0.0435 0.0462 0.0582

b. 0.1496 0.0519 0.0386 0.0577 0.0392 0.1056

34 SeXXI a 0.1004 0.0387 0.0219 0.0429 0.0472 0.0548

b. 0.1486 0.0517 0.0385 0.0571 0.0391 0.1046

35 BrXXII a 0.0993 0.0385 0.0216 0.0424 0.0481 0.0516

b. 0.1511 0.0526 0.0392 0.0581 0.0398 0.1063

36 KrXXIII a 0.0982 0.0383 0.0213 0.0419 0.0489 0.0487

b. 0.1477 0.0516 0.0384 0.0565 0.0390 0.1033

37 RbXXIV a 0.0972 0.0382 0.0211 0.0414 0.0496 0.0461

b. 0.1353 0.0476 0.0354 0.0509 0.0360 0.0933

38 SrXXV a 0.0962 0.0381 0.0208 0.0409 0.0503 0.0437

b. 0.1338 0.0472 0.0351 0.0502 0.0357 0.0918

 

 

Z ION M

39 YXXVI a 0.0953 0.0380 0.0207 0.0405 0.0508 0.0416

b. 0.1388 0.0490 0.0365 0.0523 0.0370 0.0956

40 ZrXXVII a 0.0944 0.0380 0.0205 0.0400 0.0514 0.0397

b. 0.1473 0.0518 0.0387 0.0560 0.0391 0.1023

41 NbXXVIII a 0.0936 0.0380 0.0204 0.0396 0.0519 0.0380

b 0.1481 0.0521 0.0389 0.0563 0.0394 0.1027

42 MoXXIX a. 0.0928 0.0380 0.0203 0.0393 0.0524 0.0365

b 0.1478 0.0521 0.0389 0.0554 0.0393 0.1010

44 RuXXXI a. 0.0914 0.0382 0.0202 0.0386 0.0534 0.0339

b 0.1213 0.0436 0.0325 0.0435 0.0330 0.0795

45 RhXXXII a. 0.0907 0.0383 0.0201 0.0383 0.0538 0.0328

b. 0.1220 0.0439 0.0328 0.0435 0.0332 0.0794

46 PdXXXIII a. 0.0900 0.0384 0.0201 0.0380 0.0543 0.0318

b. 0.1239 0.0446 0.0333 0.0439 0.0337 0.0802

47 AgXXXIV a. 0.0894 0.0385 0.0201 0.0377 0.0547 0.0309

b. 0.1280 0.0458 0.0342 0.0450 0.0347 0.0821

48 CdXXXV a. 0.0887 0.0387 0.0202 0.0374 0.0552 0.0301

b. 0.1312 0.0471 0.0352 0.0463 0.0355 0.0842

49 InXXXVI a. 0.0882 0.0389 0.0202 0.0372 0.0557 0.0294

50 SnXXXVII a. 0.0876 0.0391 0.0202 0.0369 0.0562 0.0287

51 SbXXXVIII a. 0.0870 0.0393 0.0203 0.0367 0.0567 0.0281

52 TeXXXIX a 0.0865 0.0395 0.0204 0.0364 0.0572 0.0275

53 IXL a 0.0860 0.0398 0.0205 0.0362 0.0577 0.0270

54 XeXLI a 0.0855 0.0401 0.0206 0.0360 0.0583 0.0265

In Table 2 (click here), we compare our two sets of f-values for the transition with a few theoretical results, which seem to be the only data available in the literature. For the first four ions, it is possible to compare our oscillator strengths with data reported by Biémont, using the same HXR method as in the above transition (Biémont 1986a), and with the f-values obtained with the length and velocity forms of the dipole transition operator by Nahar & Pradham (1993) within the Opacity Project. Their method of calculation, the close coupling (CC) aproximation using the R-matrix method, explicitly accounts for configuration interaction. For these ions, our f-values appear to be in general good agreement with the other theoretical results, again expected to be correct. For the last several ions we have used the energies obtained in our MCDF calculations for extracting quantum defects to be used in RQDO procedure. This has lead to no breackage in the steadily decreasing trend of the RQDO f-values with increasing Z, which can be taken as a sign of consistency of the RQDO procedure. One possible feature that may explain the discrepancies between the RQDO oscillator strengths and those obtained by Biémont (1986a) (which are, in no case, greater than 8%) is his reported perturbations between the and configurations, which have a non-negligible influence on the transition probabilities. The mixing between the and terms of the configuration, as detected by Biémont (1986a) from his analysis of the energy data (Corliss & Sugar 1979a, b; Sugar & Corliss 1979, 1980) may also be responsible for the aforementioned discrepancies.

 

Z ION Other theoretical values

19 KVI 0.1348 0.1401

20 CaVII 0.1258 0.1357

21 ScVIII 0.1180 0.1309

22 TiIX 0.1113 0.1262

23 VX 0.1117 0.1217

24 CrXI 0.1068 0.1174

25 MnXII - 0.1133

26 FeXIII 0.0976 0.1094

27 CoXIV 0.0937 0.1057

28 NiXV 0.0911 0.1021

29 CuXVI 0.0860 0.0986

30 ZnXVII 0.0803 0.0952

31 GaXVIII 0.0720 0.0919

32 GeXIX 0.0697 0.0888

33 AsXX 0.0691 0.0858

34 SeXXI 0.0676 0.0830

35 BrXXII 0.0697 0.0804

36 KrXXIII 0.0660 0.0781

37 RbXXIV 0.0523 0.0760

38 SrXXV 0.0478 0.0742

39 YXXVI 0.0436 0.0725

40 ZrXXVII 0.0399 0.0711

41 NbXXVIII 0.0366 0.0698

42 MoXXIX 0.0335 0.0688

44 RuXXXI 0.0288 0.0670

45 RhXXXII 0.0267 0.0664

46 PdXXXIII 0.0249 0.0659

47 AgXXXIV 0.0234 0.0654

48 CdXXXV 0.0220 0.0651

49 InXXXVI 0.0232 0.0648

50 SnXXXVII 0.0179 0.0647

51 SbXXXVIII 0.0190 0.0646

52 TeXXXIX 0.0182 0.0645

53 IXL 0.0175 0.0646

54 XeXLI 0.0165 0.0647

a) RQDO with polarization, this work.
b) MCDF, this work.
c) HXR (Biémont 1986a).
d) O.P., length and velocity forms, respectively (Nahar & Pradhan 1993).  

  
Figure 1: Oscillator strengths versus the reciprocal of the nuclear charge for the transition in the silicon isoelectronic sequence. *: RQDO, this work. : MCDF, Huang (1985). : HXR, Biémont (1986a, b)

  
Figure 2: Systematic trends of the oscillator strength of the transition along the silicon isoelectronic sequence. *: RQDO, this work. : MCDF, Huang (1985). : HXR, Biémont (1986a, b)

In Table 3 (click here) we include only our RQDO f-values given that an extense MCDF calculation, using Declaux's code (1975), has been reported by Huang (1985). For the transition, our RQDO results, which explicitly account for core polarization effects, are close in magnitude to the HXR f-values reported by Biémont (1986a, b), by Bromage (1980) and Bromage et al. (1978), as well as other theoretical values, which are claimed by their authors to be correct. However, the MCDF f-values from Huang (1985) show very large discrepancies with the rest of the theoretical results displayed in Table 3 (click here). These discrepancies increase with atomic number, being the MCDF oscillator strengths too small. When comparing these MCDF data with those of our own MCDF calculations (not shown in the Table 3 (click here)), the latter present the same feature roughly the same characteristics as Huang's (1985). This leads us to suggest that the MCDF procedure is not adequate for this particular transition, possibly due to cancellation effects in the transition integral. Biémont (1986b) also attributes the large discrepancies observed between Huang's f-values and his HXR f-values to an incorrect assignment of the energy levels by Huang (1985). We once more explain the much less important discrepancies observed between the RQDO oscillator strengths and the HXR ones (Biémont 1986a, b) to their explicit inclusion of configuration mixing. This seems to be particularly important between level corresponding to the final state, and the level and more so as Z increases (Biémont 1986a), as in the previous transition.

For the singlet-singlet transition, we have calculated oscillator strengths using our semiempirical procedure RQDO only. In Table 3 (click here) our f-values are compared with other theoretical results. The most extense calculation is the MCDF one performed by Huang (1985). This author has accounted for electron correlation by including all relativistic configurations in the n=3 complex.

 

19 KVI 0.4467 0.0762 1.0901 1.061

20 CaVII 0.3977 0.0826 0.9873 0.9549

21 ScVIII 0.3850 0.0862 0.8956 0.8640

22 TiIX 0.3679 0.0876 0.8160 0.7864

23 VX 0.3480 0.0874 0.7385 0.7197

24 CrXI 0.3067 0.0861 0.6809 0.6617

25 MnXII 0.2948 0.0839 0.6297 0.6106

26 FeXIII 0.2935 0.0810 0.5862 0.5652

27 CoXIV 0.3008 0.0775 0.5474 0.5244

28 NiXV 0.2692 0.0735 0.5143 0.4878

29 CuXVI 0.2375 0.0691 0.4840 0.4547

30 ZnXVII 0.2781 0.0644 0.4578 0.4251

31 GaXVIII 0.2603 0.0593 0.4398 0.3988

32 GeXIX 0.2398 0.0540 0.4194 0.3756

33 AsXX 0.2189 0.0462 0.3998 0.3555

34 SeXXI 0.1991 0.0419 0.3831 0.3382

35 BrXXII 0.1775 0.0348 0.3654 0.3234

36 KrXXIII 0.1635 0.0267 0.3532 0.3109

37 RbXXIV 0.1511 0.0175 0.3505 0.3002

38 SrXXV 0.1362 0.0083 0.3419 0.2912

39 YXXVI 0.1224 0.0021 0.3340 0.2836

40 ZrXXVII 0.1087 6.05(-5) 0.3271 0.2770

41 NbXXVIII 0.0956 0.0004 0.3208 0.2713

42 MoXXIX 0.0834 0.0016 0.3151 0.2662

44 RuXXXI 0.0512 0.0041 0.3115 0.2576

45 RhXXXII 0.0428 0.0048 0.3070 0.2537

46 PdXXXIII 0.0349 0.0050 0.3029 0.2501

47 AgXXXIV 0.0274 0.0048 0.2993 0.2466

48 CdXXXV 0.0209 0.0041 0.2960 0.2433

49 InXXXVI 0.0148 0.0031 0.2931 0.2401

50 SnXXXVII 0.0095 0.0020 0.2905 0.2370

51 SbXXXVIII 0.0054 0.0011 0.2882 0.2339

52 TeXXXIX 0.0023 4.44(-4) 0.2863 0.2309

53 IXL 0.0004 1.15(-4) 0.2846 0.2280

54 XeXLI 0.0000 2.24(-6) 0.2832 0.2252

a) RQDO, this work; b) MCDF (Huang 1985); c) HXR (Biémont 1986a); d) O.P., (Nahar & Pradhan 1993); e) HXR (Biémont 1986b); f) HXR (Bromage 1980); g) HXR (Bromage et al. 1978); h) Flower & Nussbaumer (1974); i) Fuhr et al. (1981).  

Our semiempirical f-values are in better agreement than in previous transitions with the values obtained with the quite much more complex procedures, even with the results obtained using the ample Multiconfigurational Dirac-Fock method, possibly due to the less delicate level perturbation situation here.

Finally, in Figs. 1 (click here) and 2 (click here) we compare the oscillator strengths corresponding to the last two transitions, respectively, in graph form. For that purpose, we have plotted the f-value for the entire group of isoelectronic ions against the reciprocal of the atomic number Z. These figures also serve the purpose of reflecting systematic trends in the individual f-values along the isoelectronic sequence, which have long been considered (Wiese & Weiss 1968) as a qualitative proof of correctness and allows, when they are regular enough, the interpolation or extrapolation of non-calculated data.