4 Transition probabilities

In the present work we have calculated all the E1, E2 and M1 transitions within the n=2 complex for Z=6-28; this amounts to 2254 transitions. The E2 and M1 transitions are of less importance than the considerably larger E1s, with the notable exception perhaps of the astrophysically relevant forbidden transitions within the $\sp2$P$\sp0_{\rm J}$ ground term. They are nevertheless listed for database purposes. The computed transition probabilities are listed in Tables 3-25.

Comparing the present A-values with previous datasets (see histograms in Fig. 4), it is found that DT list 272 E1 transitions for Z=6-26 of which 77% agree to the desired accuracy (10%), and 6 transitions show discrepancies larger than 50%; namely, the 2s2p$\sp2\ \sp2$S_1/2-2s$\sp2$2p$\sp2$P$\sp0_{3/2}$ spin-allowed transitions for Z=22 and Z=26 which are affected by the avoided crossing and are discussed

  

Figure 4: Histograms showing the percentage difference of the previously computed transition probability datasets with respect to the present results. a) DT, 272 A-values for Z=6-26 (excluding 6 A-values showing differences greater than 50%). b) MVGK, 686 A-values for Z=8-26 (excluding 32 A-values). c) CKD, 1100 A-values for Z=7-28 (excluding 134 A-values for the same reason). It can be concluded that the CKD dataset is of lower statistical reliability

in Sect. 4.1; the 2p$\sp3\ \sp2$D$\sp0_{\rm J}-$2s2p$\sp2\ \sp2$P$_{\rm J'}$ for Z=6 which are perturbed by intermixing n=3 states; and the 2s2p$\sp2\ \sp4$P_1/2-2s$\sp2$2p$\sp2$P$\sp0_{3/2}$ intercombination transition for Z=6 (see Sect. 4.2). MVGK list 686 E1 transitions for Z=8-26, of which 70% agree with present A-values to 10% and 32 are discrepant by more than 50%; namely, transitions affected by the avoided crossing (Z>17); 2p$\sp3\ \sp4$S$\sp0_{3/2}-$2s2p$\sp2\ \sp2$D$_{\rm J}$ intercombination transitions for Z<18; and transitions for Z=8. The dataset by CKD contains 1100 transitions for Z=7-28, mostly E1s but the E2 and M1 transitions within the $\sp2$P$\sp0_{\rm J}$ ground term are also listed; only 46% agree with the present dataset to within 10% and for 134 transitions the discrepancies are larger than 50%. In general, large discrepancies are found for: the low members of the sequence (Z<10); the transitions affected by the avoided crossing (e.g. 2s2p$\sp2\ \sp2$S_1/2-2s$\sp2$2p$\sp2$P$\sp0_{3/2}$,2p$\sp3\ \sp2$P$\sp0_{1/2}-$2s2p$\sp2\ \sp2$S_1/2 for Z>18); and the intersystem transitions involving the $\sp4$P$_{\rm J}$ and $\sp4$S$\sp0_{3/2}$levels for low and intermediate Z. From this comparison it is found that in the range Z=7-28, specially for low Z, the CKD dataset differs considerably from the DT, MVGK and present sets.

The present rates allow us to compute radiative lifetimes which can also be compared with recent experiments for C II (Reistad et al. 1986 and Nandi et al. 1996) and N III (Bengtsson et al. 1995). It is found that our theoretical lifetimes lie within the experimental error bars.

We now discuss the problems leading to the larger discrepancies in an attempt to estimate accuracy ratings.

4.1 Spin-allowed E1 transitions

Most of the present transition probabilities for spin-allowed transitions should be accurate to 10%, with the exception of (i) a few transitions for Z=6-7 where strong admixture with the low n=3 levels can lead to sensitive A-values and (ii) transitions in highly ionised members of the sequence that are affected by strong relativistic couplings such as those that lead to the avoided crossing of the 2s2p$\sp2\ \sp2$S_1/2 at $Z\sim 22$.For low Z, this assertion is supported by the excellent agreement ($\sim$10%) obtained with the detailed computations by Nussbaumer & Storey (1981), Lennon et al. (1985) and Froese Fischer (1994) on C II; Nussbaumer & Storey (1979), Bell et al. (1995) and Brage et al. (1995) on N III; and Brage et al. (1996) on O IV.

We are therefore not worried, for instance, with the 35% and 40% discrepancies found with DT for the spin-allowed transitions 2p$\sp3\ \sp4$S$\sp0_{3/2}-$2s2p$\sp2\ \sp4$P$_{\rm J}$ in N III and 2s2p$\sp2\ \sp2$P_3/2-2s$\sp2$2p$\sp2$P$\sp0_{1/2}$in O IV, respectively. As mentioned before, for high Z a notably problematic transition is 2s2p$\sp2\ \sp2$S_1/2- 2s$\sp2$2p$\sp2$P$\sp0_{3/2}$ which, as seen in Fig. 5a, is strongly perturbed by the avoided crossing. Due to the good overall agreement between DT, MVGK and present data, our A-values for Z<17 can be expected to be well within 10%, but for $Z\sim 25$an accuracy rating would be uncertain. It would have been useful to have A-values for this transition for Z=23,25,27 from DT and MVGK, as these would have provided a better picture of the troublesome region. A similar situation is found for the 2p$\sp3\ \sp2$P$\sp0_{1/2}-$2s2p$\sp2\ \sp2$S_1/2 (Fig. 5b) and 2p$\sp3\ \sp2$P$\sp0_{1/2}-$2s2p$\sp2\ \sp2$P_1/2 (Fig. 5c) transitions, where the destructive perturbation caused by the avoided crossing leads to poor accuracy ratings for the range 17<Z<22. The accuracy of the A-value at Z=6 for the former transition (see Fig. 5b) is also questionable as it seems to be perturbed by a state belonging to the n=3 configurations. A similar situation is encountered in C II for the transitions 2p$\sp3\ \sp2$D$\sp0_{\rm J}-$2s2p$\sp2\ \sp2$P$_{\rm J'}$.As shown in Fig. 5d, destructive interference is also found for $Z\sim 28$in the 2s2p$\sp2\ \sp2$D_3/2-2s$\sp2$2p$\sp2$P$\sp0_{3/2}$ transition, causing the accuracy of the rates for Z>26 to be of lower quality. For most of these difficult transitions (see Figs. 5a-c) the data by CKD are consistently discrepant (differences are greater than 20%) with DT, MVGK and the present dataset throughout the sequence. Also the data by MVGK begin to show significant differences from the present ones for Z<12 where correlation effects are more conspicuous and the present CI method is expected to perform with higher reliability.

  

Figure 5: A-values (s$\sp{-1}$) for troublesome spin-allowed transitions as a function of Z. a) ${\rm 2s2p}\sp2\ \sp2{\rm S}_{1/2}- {\rm 2s}\sp2{\rm 2p}\ \sp2{\rm P}\sp0_{3/2}$.b) ${\rm 2p}\sp3\ \sp2{\rm P}\sp0_{1/2}- {\rm 2s2p}\sp2\ \sp2{\rm S}_{1/2}$.c) ${\rm 2p}\sp3\ \sp2{\rm P}\sp0_{1/2}- {\rm 2s2p}\sp2\ \sp2{\rm P}_{1/2}$.d) ${\rm 2s2p}\sp2\ \sp2{\rm D}_{3/2}- {\rm 2s}\sp2{\rm 2p}\ \sp2{\rm P}\sp0_{3/2}$.Filled squares: present results. Open circles: DT. Open triangles: CKD. Open squares: MVGK

4.2 Spin-forbidden E1 transitions

In contrast to the good rating assigned to most spin-allowed transitions, the accuracy for the intercombination transitions is noticeably poorer. The typical situation is illustrated in Fig. 6 with the important 2s2p$\sp2\ \sp4$P$_{\rm J}-$2s$\sp2$2p$\sp2$P$\sp0_{\rm J'}$transitions. With regard to the DT, CKD, MVGK and present datasets, differences larger than 10% begin to show up for these transitions for Z<20, and for Z<10 the situation becomes critical. In order to assign ratings to this transition array, we compare in Table 26 present rates for Z=6-8 with DT, CKD, MVGK, detailed single-system calculations and recent experiments. It may be appreciated that differences between the present data and DT are as large as a factor of 2 for C II and diminish to $\sim 40$% for N III and O IV. The situation with respect to CKD is even considerably worse as the differences reach an order of magnitude. Discrepancies with MVGK for O IV are generally small except for $\sp4{\rm P}_{3/2}-\sp2{\rm P}\sp0_{1/2}$ where they reach a factor of 2. A comparison of the present results with the detailed calculations and experiment allows us to assign a realistic 20% rating to the present A-values for Z=6-8. Furthermore, a recent measurement of the lifetime of the $\sp4{\rm P}_{5/2}$ level in Fe XXII by Hutton et al. (1997) of 14.8$\pm$1.0 ns is in excellent agreement with those given by DT (14.2 ns), CKD (14.3 ns), MVGK (13.4 ns) and the present work (13.7 ns). Therefore the 20% rating for these transitions can be safely extended to the whole sequence.

The situation with the ${\rm 2p}\sp3\ \sp2{\rm L}\sp0_{\rm J}-{\rm 2s2p}\sp2\ \sp4{\rm P}_{\rm J'}$ transitions is even more serious since, as shown in Fig. 7, the rates can be strongly perturbed by the avoided crossing. Taking into account the good agreement between the present data and those by Bell et al. (1995) for Z=7 we are encouraged to also maintain the 20% rating for such transitions except for Z=6 and around the perturbation ($18\leq Z\leq 24$). For the intersystem transitions involving the ${\rm 2p}\sp3\ \sp4{\rm S}\sp0_{3/2}$, in spite of very large differences found with CKD for Z<20, we find good agreement with MVGK ($\sim20$%) for Z>12. However, as shown in Fig. 8, the situation is complicated for Z<9 where very significant differences are encountered with the detailed calculations. For this range we do not attempt a rating assignment.

  

Figure 6: Scaled A-values (s$\sp{-1}$) for the intercombination transitions ${\rm 2s2p}\sp2\ \sp4{\rm P}_{\rm J}-{\rm 2s}\sp2{\rm 2p}\ \sp2{\rm P}\sp0_{\rm J'}$as a function of Z showing the large scatter for low Z. a) $\sp4{\rm P}_{3/2}-\sp2{\rm P}\sp0_{1/2}$.b) $\sp4{\rm P}_{5/2}-\sp2{\rm P}\sp0_{3/2}$.Filled squares: present results. Open circles: DT. Open triangles: CKD. Open squares: MVGK. Crosses: Froese Fischer (1994), Brage et al. (1995) and Brage et al. (1996). Star: Nussbaumer & Storey (1984, 1979). Asterix: Lennon et al. (1985) and Bell et al. (1995)

  

Figure 7: A-values (s$\sp{-1}$) for the intercombination transition ${\rm 2p}\sp3\ \sp2{\rm L}\sp0_{\rm J}-{\rm 2s2p}\sp2\ \sp4{\rm P}_{\rm J'}$ plotted as a function of Z. a) $\sp2{\rm D}\sp0_{3/2}-\sp4{\rm P}_{1/2}$ showing the perturbation caused by the avoided crossing at $Z\sim 22$. b) $\sp2{\rm P}\sp0_{3/2}-\sp4{\rm P}_{1/2}$ showing large differences with MVGK for the whole sequence. Filled squares: present results. Open triangles: CKD. Open squares: MVGK. Open circles: Bell et al (1995)

  

Figure 8: A-values (s$\sp{-1}$) for the ${\rm 2p}\sp3\ \sp4{\rm S}\sp0_{3/2}-{\rm 2s2p}\sp2\ \sp2{\rm L}_{\rm J}$intercombination transitions plotted as a function of Z. a) $\sp4{\rm S}\sp0_{3/2}-\sp2{\rm S}_{1/2}$.b) $\sp4{\rm S}\sp0_{3/2}-\sp2{\rm D}_{3/2}$.Filled squares: present results. Open triangles: CKD. Open squares: MVGK. Open circles: Bell et al. (1995). Asterix: Nussbaumer & Storey (1981). It may be seen that the agreement for Z<9 is poor and that large discrepancies exist with CKD for even fairly high Z

4.3 Forbidden transitions

The only forbidden transitions that have been previously considered for the B-like ions are the E2 and M1 transitions within the 2s$\sp2$2p$\sp2$P$\sp0_{\rm J}$ term. In this context present results are in excellent agreement (better than 10%) with those by Froese Fischer (1983). Taking this outcome as a reference, the A-values listed by CKD are found to be highly unreliable for Z<12.

  

Table 3: Transition probabilities (s^-1) for C II

  

Table 4: Transition probabilities (s^-1) for N III

  

Table 5: Transition probabilities (s^-1) for O IV

  

Table 6: Transition probabilities (s^-1) for F V

  

Table 7: Transition probabilities in s^-1 for Ne VI

  

Table 8: Transition probabilities (s^-1) for Na VII

  

Table 9: Transition probabilities (s^-1) for Mg VIII

  

Table 10: Transition probabilities (s^-1) for Al IX

  

Table 11: Transition probabilities (s^-1) for Si X

  

Table 12: Transition probabilities (s^-1) for P XI

  

Table 13: Transition probabilities (s^-1) for S XII

  

Table 14: Transition probabilities (s^-1) for Cl XIII

  

Table 15: Transition probabilities (s^-1) for Ar XIV

  

Table 16: Transition probabilities (s^-1) for K XV

  

Table 17: Transition probabilities (s^-1) for Ca XVI

™

  

Table 18: Transition probabilities (s^-1) for Sc XVII

  

Table 19: Transition probabilities (s^-1) for Ti XVIII

  

Table 20: Transition probabilities (s^-1) for V XIX

  

Table 21: Transition probabilities (s^-1) for Cr XX

  

Table 22: Transition probabilities (s^-1) for Mn XXI

  

Table 23: Transition probabilities (s^-1) for Fe XXII

  

Table 24: Transition probabilities (s^-1) for Co XXIII

  

Table 25: Transition probabilities (s^-1) for Ni XXIV

  

Table 26: Comparison of present A-values (s$\sp{-1}$) for the ${\rm 2s2p}\sp2\ \sp4{\rm P}_{\rm J}-{\rm 2s}\sp2{\rm 2p}\ \sp2{\rm P}\sp0_{\rm J'}$intercombination transitions in Z=6-8 with other theoretical results and experiment. a) DT. b) Nussbaumer & Storey (1981). c) Lennon et al. (1985). d) Froese Fischer (1994). e) Nussbaumer & Storey (1979). f) Bell et al. (1995). g) Brage et al. (1995). h) CKD. i) Brage et al. (1996). j) MVGK. Experiment by Fang et al. (1993a,b); the experimental uncertainties are given by the quantities in brackets. From the present comparison, a 20% accuracy rating is assigned to the present data