3 Results

The transition probabilities obtained from approximations A, A', B, C and D are tabulated in Tables 3-5. Approximation A' is the same as A but excludes the spin-spin contribution. In Fig. 1 we compare the differences of each approximation from the standard approximation A. It may be appreciated that the neglect of the spin-spin interaction (approximation A') leads to effects that decrease with Z. This finding clearly illustrates the conclusions reached by Jones (1970) regarding the character of the relativistic corrections. Namely, since the one-body and two-body (Breit) relativistic corrections respectively scale as $\alpha^2(Z-S)^4$ and $\alpha^2(Z-S)^3$, where $\alpha$ is the fine-structure constant and S is a screening constant, the Breit contribution decreases in importance as Z increases along the sequence. Although in the present study the spin-spin contribution increases $A(\sp5{\rm S}\sp{\rm o}_2\ -\ \sp3$P₁) by only $\sim$5% at low Z (see Fig. 1a), the reductions in $A(\sp5{\rm S}\sp{\rm o}_2\ -\ \sp3$P₂) and $A(\sp5{\rm S}\sp{\rm o}_2\ -\ \sp1$D₂) can be seen (Fig. 1b and Fig. 1c) to be as large as 20%. Therefore we conclude that the Breit interaction must be explicitly taken into account in the calculation of accurate radiative rates for these intercombination transitions.

 

Table 3: A-values (s$\sp{-1}$) for the $\sp5$S$\sp{\rm o}_2\ -\ \sp3$P₁ transition in the carbon sequence computed in approximations A, A', B, C and D. $a\pm b\equiv a\times 10^{\pm b}$

 

Table 4: A-values (s$\sp{-1}$) for the $\sp5$S$\sp{\rm o}_2\ -\ \sp3$P₂ transition in the carbon sequence computed in approximations A, A', B, C and D. $a\pm b\equiv a\times 10^{\pm b}$

 

Table 5: A-values (s$\sp{-1}$) for the $\sp5$S$\sp{\rm o}_2~-~\sp1$D₂ transition in the carbon sequence computed in approximations A, A', B, C and D. $a\pm b\equiv a\times 10^{\pm b}$

 

Table 6: Radiative lifetimes (ms) for the $\sp5$S$\sp{\rm o}_2$ metastable state of the carbon sequence resulting from approximations A, A', B, C and D. $a\pm b\equiv a\times 10^{\pm b}$

 

Table 7: Branching ratio $B=A(\sp5{\rm S}\sp{\rm o}_2\ -\ \sp3$P$_2)/A(\sp5{\rm S}\sp{\rm o}_2\ -\ \sp3$P₁) in the carbon sequence obtained from approximations A, A', B, C and D

 

Figure 1: Percentage difference of A-values computed in approximations A' (x), B (circles), C (squares) and D (asterisk) with respect to the standard approximation A. a) $\sp5{\rm S}_2\sp{\rm o}\ -\ \sp3$P1. b) $\sp5{\rm S}_2\sp{\rm o}\ -\ \sp3$P2. c) $\sp5{\rm S}_2\sp{\rm o}\ -\ \sp1$D2

Regarding electron correlation effects, it is shown in Fig. 1 that the contributions from configurations containing n=4 orbitals are only conspicuous for Z<10 and are very difficult to harness for the neutral (Z=6). By examining the differences resulting from approximations B, C and D at low Z, it is shown in Fig. 1 that the progressive increase of the configuration basis does not necessarily lead to increasingly accurate results. It is therefore essential to include the complete complex (i.e. approximation D). From the present study it is possible to select a "best" set of data which is believed to be stable to within 5%: approximation D for $Z\leq 8$ and approximation A for Z>8.

In Tables 6-7 we tabulate the $\sp5{\rm S}\sp{\rm o}_2$ radiative lifetimes and the B branching ratio as functions of Z for the different approximations. They are also compared in Fig. 2. In the case of lifetimes, it is seen that the exclusion of the spin-spin interaction only leads to small differences (less than 11%) for the whole series, and for Z>10 they are less than 5%. Similarly, the CI from n=4 configurations only makes differences greater than 5% for the specific case of Z=6. Regarding the branching ratio, it is seen that although the n=4 configurations make little difference (except again for the neutral) the inclusion of the spin-spin contribution causes a large decrease at low Z from those obtained by including only the one-body relativistic corrections (approximation A'), which as expected tend to the value of 3 at low Z discussed by Ellis & Martinson (1984). These findings fully support the earlier conclusion by Fleming & Brage (1997) regarding the sensitivity of B to the Breit interaction in O III. As discussed above, the relative magnitude of the spin-spin contribution decreases along the sequence, and by Z=20 its effects have been reduced to less than 5%.

 

Figure 2: Comparison of a) scaled $\sp5{\rm S}\sp{\rm o}_2$ radiative lifetimes (ms) and b) the branching ratios B for the different approximations considered. Filled circle: approximation A. x: A'. Circle: B. Square: C. Asterisk: D. The scaled ($\tau'$) and unscaled ($\tau$) radiative lifetimes are related by $\tau'=\tau(Z-4.0)^5$

 

Table 8: Comparison of the best present A-values (s$\sp{-1}$) for the $\sp5$S$\sp{\rm o}_2\ -\ \sp3$P₁ transition in the carbon sequence with other theoretical results. MCHF: Froese Fischer & Saha (1985). SSTR: Bhatia (1982), Bhatia et al. (1987), Bhatia & Kastner (1993), Bhatia & Doschek (1993a, 1993b, 1993c, 1995) and Mason & Bhatia (1978). MCDF: Cheng et al. (1979). CIV3: Aggarwal (1986) and Aggarwal et al. (1997a, 1997b)). SCIV3: Brage et al. (1997) and Fleming & Brage (1997). NS: Nussbaumer & Storey (1981). HB: Hibbert & Bates (1981). $a\pm b\equiv a\times 10^{\pm b}$

 

Table 9: Comparison of the best present A-values (s$\sp{-1}$) for the $\sp5$S$\sp{\rm o}_2\ -\ \sp3$P₂ transition in the carbon sequence with other theoretical results. Reference keys as in Table 8. $a~\pm~ b \equiv a\times 10^{\pm b}$

 

Table 10: Comparison of the best present A-values (s$\sp{-1}$) for the $\sp5$S$\sp{\rm o}_2\ -\ \sp1$D₂ transition in the carbon sequence with other theoretical results. Reference keys as in Table 8. $a~\pm~ b \equiv a\times 10^{\pm b}$

The best present A-values are compared with other calculations in Tables 8-10 and in Fig. 3. In the case of $A(\sp5{\rm S}\sp{\rm o}_2\ -\ \sp3$P$_{\rm J})$,differences greater than 5% between the MCHF dataset and the present are found for Z< 12, growing to $\sim 20$% for Z<8. However, the excellent agreement (better than 3%) between present data and those by Brage et al. (1997) and Fleming & Brage (1997) for Z=7,8 in the SCIV3 method gives us confidence in the accuracy of the present A-values for these transitions even at low Z. Still, significant discrepancies are found with the other datasets (MCDF, SSTR and CIV3) throughout the sequence. Regarding the relatively smaller A-values for the $\sp5{\rm S}\sp{\rm o}_2~-~\sp1$D₂ transition, differences greater than 20% are found with MCHF throughout the sequence reaching a factor of 3 for Z=6. Such large discrepancies are difficult to explain. In relation to other theoretical datasets, differences larger than 20% are found with MCDF, SSTR and CIV3 for Z< 20. This comparison seems to indicate that the present A-values for this transition are probably not accurate to better than 20%.

 

Figure 3: Percentage difference of other theoretical A-values with respect to the best present results (approximation D). a) $\sp5{\rm S}_2\sp{\rm o}~ -~\sp3$P1 b) $\sp5{\rm S}_2\sp{\rm o}~ -~\sp3$P2. c) $\sp5{\rm S}_2\sp{\rm o}~ - ~\sp1$D2 Circle: MCHF. Triangle: SSTR. x: MCDF. Cross: CIV3. Square: SCIV3. Asterisk: Nussbaumer & Storey (1981). Rhombus: Hibbert & Bates (1981)

 

Table 11: Comparison of the best present radiative lifetimes (ms) for the $\sp5$S$\sp{\rm o}_2$ state with other theoretical and experimental results. Keys for theoretical results: MCHF, Froese Fischer & Saha (1985); SSTR, Bhatia (1982), Bhatia et al. (1987), Bhatia & Kastner (1993), Bhatia & Doschek (1993a,b,c 1995) and Mason & Bhatia (1978); MCDF, Cheng et al. (1979); CIV3, Aggarwal (1986), Aggarwal et al. (1997a,b) and Bell et al. (1995); SCIV3, Brage et al. (1997) and Fleming & Brage (1997). Keys for measurements: TWP, Träbert et al. (1998); CJ, Calamai & Johnson (1991); JSKP, Johnson et al. (1991); K, Knight (1982). The experimental error is given by the quantity in brackets. $a\pm b\equiv a\times 10^{\pm b}$

 

Table 12: Comparison of the best present results for the branching ratio $B=A(\sp5{\rm S}\sp{\rm o}_2~-~\sp3$P$_2)/A(\sp5{\rm S}\sp{\rm o}_2~-~\sp3$P₁) with other theoretical and experimental results. Keys for theoretical results as in Table 11. Keys for experimental results: BWG, Bridges et al. (1996); MBDW, Musielok et al. (1996); CGL, Curry et al. (1997). The experimental uncertainty is given by the digits in parentheses

Radiative lifetimes and branching ratios computed with the best present transition probabilities are compared with other theoretical results in Tables 11-12 and in Fig. 4. Recent measurements are also included in the tabulations. The best agreement (1%) is found with the SCIV3 results for N II and O III. Discrepancies larger than 5% are found with MCHF for Z<12, but they increase up to 30% for Z<8. By examining the branching ratios (Fig. 4b), it is apparent that MCDF and SSTR did not include the Breit interaction in their computations, and the former displays a questionable departure for Z=8. Perhaps for this same reason their lifetimes are significantly higher than MCHF and present results for Z<12. The CIV3 dataset contains data for Z=8 that lead to a comparable branching ratio but a noticeably higher lifetime. From this outcome and a further comparison with the experimental results (see Tables 11-12), and in spite of the scatter found in the measured values, we are confident in assigning a 5% rating to the present results for Z>8 and 10% otherwise.

 

Figure 4: Comparison of the best present a) scaled $\sp5{\rm S}\sp{\rm o}_2$radiative lifetimes (ms) and b) branching ratio B with other theoretical results. Filled circle: present work. Circle: MCHF. Triangle: SSTR. x: MCDF. Cross: CIV3. Square: SCIV3. Asterisk: Nussbaumer & Storey (1981). Rhombus: Hibbert & Bates (1981). The scaled ($\tau'$) and unscaled ($\tau$) radiative lifetimes are related by $\tau'=\tau(Z-4.0)^5$