Since we are using the same numerical method as NR, we have attempted
to refine their calculation by increasing the configuration expansion. It
is found that some excitation energies can be considerably improved
by this procedure, particularly for species with
. The two configurations involving 
orbitals
which are listed in Table 1 contribute towards a very accurate
level energy for all ions in the
sequence. This contrasts with the relatively large discrepancies found
for low Z by NR with respect to experiment. The 
level remains
virtually unaffected by the inclusion of 
orbitals in the
physical model. The inclusion of other 
orbitals was found
to be unimportant in all cases.
We finally settled for the 18-configuration representation listed in Table 1,
which appears to be more "realistic'' than the 11-configuration set of NR:
it contains all even configurations with one or two excitations in n= 3
and a closed 
plus the important 
and
configurations.
The resulting level excitation energies are given in Table 4 together
with the experimental, NR and FS1 data. The FS1 data were computed with the
Multiconfiguration Hartree-Fock method plus Breit-Pauli
relativistic corrections (MCHF + BP) and an extensive configuration expansion
including states with 
. Throughout the sequence, the present
separations agree better with experiment than the FS1 results.
For all other energies the agreement between the FS1 data, the present
findings and experiment is within 5%.
Table 4: Experimental and calculated excitation energies in cm
from
the 2s
2p
P
ground state for the carbon sequence.
Expt: experiment Edlén (1983, 1985). Pres: present results.
NR: Nussbaumer &
Rusca (1979). FS1: Froese Fischer &
Saha (1985)
The present total transition probabilities are listed in Table 5, together
with the NR and FS1 datasets. Note that for most spectral lines either the
M1 or the E2 type transition dominates. The level of agreement with NR is
good: 86% of the data agree to within 10% (see Fig. 1 (click here)a). Large
differences (up to 63%) are mostly found for the 
E2 transition, in particular
for Z=6-9 and Z=25. These transitions have relatively
small line strengths and are sensitive to the level separation
in Eq. (6 (click here)) and to the TECs. For low Z, the A-value is
also sensitive to CI and to the spin-spin relativistic interaction.
The differences for Z=6-9 are therefore explicable in terms of the
present improved CI expansion and perhaps of a more extensive treatment of the
spin-spin contributions, while that for Z=25 is due to an anomalously high
NR A-value. Note that this sensitive transition probability does not differ
very much in the present work whether or not the two configurations
containing 
orbitals are taken into account. This
illustrates that a small deficit in CI can be compensated by a
slightly larger TEC value. Thus, the validity of the TEC procedure is
confirmed.
Table 5: Transition probabilities in s
within the ground state
configuration of the carbon sequence.
Pres: present results. NR: Nussbaumer &
Rusca (1979). FS1: Froese Fischer
& Saha (1985).
FS1c: FS1 corrected with the experimental wavelengths.
denotes 
Table 5: continued
The comparison with the FS1 work (see Fig. 1 (click here)b) is less impressive as only
72% of the data agree to 10%, with big discrepancies (as large
as a factor of 9) for the sensitive transitions
in both lowly and highly ionised systems.
Since FS1 evaluated their A-values with computed energies, which can
result in large errors (see end of Sect. 2), their results were corrected
using the experimental transition wavelengths (see FS1c in Table 5 and
Fig. 1 (click here)c). Although the
large differences for the transitions are
significantly reduced, the number of cases lying within the required
% range
is not increased. It is found that, while the accord for some A-values is
improved by this procedure, for others it is actually made worse cancelling
out the positive changes.
At this stage it is worth considering Garstang's (1968)
classic study who computed radiative
transition probabilities for forbidden lines at the lower end of the
carbon and oxygen sequences. His Hamiltonian included terms arising
from the spin-orbit, mutual spin-orbit and spin-spin interactions,
and his Hartree-Fock (HF) wavefunctions were adjusted to fit experimental
energies. There is good agreement between his work and ours for the
problematic transition indicating a high
sensitivity to relativistic effects with emphasis on the spin-spin interaction.
There is an extensive study of astrophysically important transitions in
neutral carbon by Hibbert et al. (1993) who used
the code CIV3. They treated mainly
electric dipole (E1) transitions but they also give results for the
forbidden transitions in the ground configuration. Concerning the large
E2 transition probability, there is excellent
agreement between the CIV3, FS1c and present A-value. This
confirms the carbon abundance in the sun proposed by Biémont et
al. (1993). The NR result for this transition appears
to be underestimated by some 20%, just as the
probability reported by Nicolaides et al. (1971) who used
the "charge-distribution'' (CD) concept.
In conclusion the discrepancies between the FS1 data and the SUPERSTRUCTURE results (NR and present), appear to be due to real
differences in the line strengths. This is most probably linked to the
use of the TEC procedure in the latter calculations.
Further conclusions will be drawn after the analysis of the data for the
oxygen sequence.
The 12-configuration representation selected for the oxygen sequence
is listed in Table 1. It may be seen that, in contrast to the calculation
for the carbon sequence, the correlation orbital was found
to be important. A comparison of the present theoretical energy separations
with experiment, BZ and FS2 is given in Table 6. BZ used the CI program
CIV3 with BP relativistic corrections and their physical model includes
extensive correlation within . FS2 used the MCHF+BP method with
correlation configurations up to . It is found that the agreement
between theory and experiment is very good (within 5%) except for
the level in neutral oxygen (Z=8). Generally speaking,
BZ obtained the best accord with measurements.
Present A-values for transitions within the ground configuration of this
sequence are compared with BZ and FS2 in Table 7 and Fig. 2 (click here). Excellent
agreement is found with BZ as 94% of the data agree to within 10% (see
Fig. 2 (click here)a). Differences larger than 10% are found at low Z ()
for the E2 transitions which,
similar to the homologous case in the C sequence, have small line strengths
and are sensitive to CI and the treatment of relativistic effects. The
worst case is in O I where the discrepancy is as large as 41%. With regard
to FS2, only 72% of the data agree with the present results to within 10%
(see Fig. 2 (click here)b), and large differences (up to a factor of 2) are noted for
the difficult transitions. The very large
difference (a factor of 10) found for the
transition for Z=13 is believed to be due to a typographical error. In
contrast
with the C sequence, substantially improved accord is found if the FS2
A-values are corrected by substituting
the experimental wavelengths (see FS2c in Table 7 and
Fig. 2 (click here)c): 83% of the data now agree to the desired accuracy (10%), and the
discrepancies in the aforementioned E2 transition go down to the 30% level.
The comparison between the BZ and FS2 datasets is interesting:
although the 10% agreement level applies only to 71% of the data,
this accuracy rating is reached by 93% of the
A-values in FS2c (see Fig. 2 (click here)d). Suprisingly, the larger differences
() are not found in the transitions,
as in the above comparisons with present results, but in other weak E2
transitions, i.e. at low Z.
Note that BZ present a detailed comparison of their results with those
of Cheng et al. (1979). The discussion will not be repeated here.
The conclusion drawn by BZ is still valid: a "fully'' relativistic calculation
with little CI is reliable for high Z but not towards the neutral end of
a given isoelectronic sequence.
In O I, the CD A-value calculated for transition -
by Nicolaides et al. (1971) is close to the present result.
Using many-body perturbation theory (MBPT), Gaigalas et
al. (1994) obtained good agreement with the line strengths
of FS2 and BZ. Note that in their Tables X and XII the last two columns have
been interchanged. The A-values recommended by BZ are in the column
marked .
For O I, Nicolaides et al. (1971) and BZ discuss some
experimental results on the ratio :
(Le Blanc et al. 1966), (McConkey et
al. 1966), on : 1.0 s (McConkey &
Kernahan 1969), on the lifetime of the
state: (Omholt 1956), 0.76 s
(Corney & Williams
1972). The corresponding present values are respectively
14.2, 1.124, 0.83, consistent with the experimental results. It should be
noted that BZ can claim better overall agreement. However, some of the
experiments mentioned above are not accurate enough to draw definitive
conclusions and there is a clear need for further measurements.
Figure 1a:Percentage difference histograms for the
A-value data sets under
consideration for the carbon sequence.
a)
NR and present excluding 1 A-value
outside the 50% range;
b) FS1 and present excluding 14 A-values;
c) FS1c (wavelength corrected) and present excluding 7 A-values and
d) FS1c (wavelength corrected) and NR excluding 6 A-values. It can be seen
that the agreement with the FS1 data set is not greatly improved by the
wavelength correction apart from reducing the number of A-values outside the
50% range
3.2. The oxygen isoelectronic sequence
Figure 2: Percentage difference histograms for the A-value data sets
under
consideration for the oxygen sequence. a) BZ and present; b)
FS2 and
present excluding 4 A-values outside the 50% range; c) FS2c
(wavelength
corrected) excluding 1 A-value; d) BZ and FS2c (wavelength corrected)
excluding
1 A-value. The agreement with FS2 is significantly improved by the
wavelength correction making its overall accuracy rating as reliable
as the BZ and present datasets
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