- 2.1 C
^{2+}target - 2.2 C
^{+}bound state energies - 2.3 C
^{+}bound-bound radiative transitions - 2.4 C
^{+}photoionization cross-sections

We calculate the the bound-bound and bound-free radiative data for
C ^{+} states from a C ^{2+}
scattering
problem. The starting point for the calculations is the C ^{2+}
target. In our photoionization calculations, as in the OP
calculations (Yu Yan & Seaton [1987]), the C ^{2+}
target given by Berrington et al. ([1977]) was used. It includes
the 2s^{2} ^{1}S^{e}, 2s2p ^{3}P^{o}, ^{1}P^{o}
and 2p^{2} ^{3}P^{e}, ^{1}D^{e}, ^{1}S^{e} target
eigenstates represented in terms of seven orthogonal basis orbitals and
pseudo-orbitals: 1s, 2s, 2p, s, p, d,
f. The radial parts of the basis orbitals were obtained using
the CIV3 code of Hibbert ([1975]).

Energy (Ry) | ||

State | Calculated | Experimental |

^{2}S^{e} |
||

2s2p^{2} |
-0.899095 | -0.912867 |

2s^{2}3s |
-0.725740 | -0.730205 |

2s^{2}4s |
-0.358483 | -0.359335 |

2s^{2}5s |
-0.212136 | -0.212489 |

2s^{2}6s |
-0.140152 | -0.140346 |

2s^{2}7s |
-0.099483 | -0.099622 |

2s^{2}8s |
-0.074279 | |

2s^{2}9s |
-0.057592 | |

2s^{2}10s |
-0.045989 | |

2s^{2}11s |
-0.037622 | |

2s^{2}12s |
-0.031477 | |

2s2p3p | -0.027270 | |

2s^{2}13s |
-0.024921 | |

2s^{2}14s |
-0.022077 | |

2s^{2}15s |
-0.019253 | |

^{2}P^{e} |
||

2s2p^{2} |
-0.767228 | -0.783844 |

2s2p3p | -0.131191 | -0.133305 |

^{2}D^{e} |
||

2s2p^{2} |
-1.10066 | -1.10937 |

2s^{2}3d |
-0.463932 | -0.465812 |

2s^{2}4d |
-0.259290 | -0.260092 |

2s^{2}5d |
-0.165203 | -0.165578 |

2s^{2}6d |
-0.114471 | -0.114731 |

2s^{2}7d |
-0.084449 | |

2s2p3p | -0.070199 | -0.073480 |

2s^{2}8d |
-0.062318 | |

2s^{2}9d |
-0.049872 | |

2s^{2}10d |
-0.040445 | |

2s^{2}11d |
-0.033420 | |

2s^{2}12d |
-0.028068 | |

2s^{2}13d |
-0.023903 | |

2s^{2}14d |
-0.020599 | |

2s^{2}15d |
-0.017934 | |

^{2}G^{e} |
||

2s^{2}5g |
-0.160169 | -0.160314 |

2s^{2}6g |
-0.111251 | -0.111320 |

2s^{2}7g |
-0.081734 | -0.081775 |

2s^{2}8g |
-0.062574 | -0.062612 |

2s^{2}9g |
-0.049437 | |

2s^{2}10g |
-0.040041 | |

2s^{2}11g |
-0.033089 | |

2s^{2}12g |
-0.027802 | |

2s^{2}13g |
-0.023688 | |

2s^{2}14g |
-0.020424 | |

2s^{2}15g |
-0.017791 | |

Energy (Ry) | ||

State | Calculated | Experimental |

^{2}P^{o} |
||

2s2p^{2} |
-1.78778 | -1.79184 |

2s^{2}3p |
-0.589943 | -0.591739 |

2s^{2}4p |
-0.310095 | -0.311142 |

2p^{3} |
-0.245582 | -0.254462 |

2s^{2}5p |
-0.190823 | -0.194769 |

2s2p3s | -0.166010 | -0.172032 |

2s^{2}6p |
-0.124169 | -0.124589 |

2s^{2}7p |
-0.090202 | -0.090391 |

2s^{2}8p |
-0.068281 | |

2s^{2}9p |
-0.053445 | |

2s^{2}10p |
-0.042958 | |

2s^{2}11p |
-0.035277 | |

2s^{2}12p |
-0.029484 | |

2s^{2}13p |
-0.025008 | |

2s^{2}14p |
-0.021479 | |

2s^{2}15p |
-0.018647 | |

^{2}D^{o} |
||

2p^{3} |
-0.413724 | -0.421035 |

^{2}F^{o} |
||

2s^{2}4f |
-0.252011 | -0.252308 |

2s^{2}5f |
-0.161223 | -0.161382 |

2s^{2}6f |
-0.111882 | -0.112070 |

2s^{2}7f |
-0.082144 | -0.082228 |

2s^{2}8f |
-0.062855 | |

2s^{2}9f |
-0.049638 | |

2s^{2}10f |
-0.040189 | |

2s^{2}11f |
-0.033202 | |

2s^{2}12f |
-0.027890 | |

2s^{2}13f |
-0.023758 | |

2s^{2}14f |
-0.020480 | |

2s^{2}15f |
-0.017836 | |

We have calculated ionization energies for all doublet states with total
orbital angular momentum quantum number
and with principal quantum
number .
These energies are compared with existing experimental
energies in Tables 1 and 2. Note that the main
2s^{2}*nl* series states are perturbed by states from the 2s2p3s and 2s2p3p
electron configurations. The upper principal quantum number limit ()
of the present calculation was chosen to include the 2s2p3p ^{2}S term
which lies energetically between the 12s ^{2}S
and 13s ^{2}S
terms. This
perturbation is not included in the OP calculation for C^{+}, which only
extends to *n* = 10. Such perturbations significantly modify the
photoionization cross-sections and hence recombination coefficients for nearby
states in the main series.

We have described most of the techniques that we use to calculate bound-bound
radiative data elsewhere (Kisielius et al. [1998]). Briefly, for ,
,
all radiative transition probabilities are obtained from the R-matrix
calculation referred to above. For most other transitions we use a
combination of the Coulomb approximation and hydrogenic data, the methods and
codes used having been described by Storey ([1994]). For transitions
with
and
we
use the following further technique. Using the R-matrix approach, photoionization
cross-section data from a given lower term (*S*_{i} *L*_{i}) to a specific final
symmetry
(*S*_{f} *L*_{f}) can be computed for negative free electron energies,
enabling us to obtain absorption oscillator strengths from the given lower term to high
members of the 2s
^{2} *nl* (*S*_{f} *L*_{f}) series. This technique has the advantage
over other approximate methods, such as the Coulomb approximation, that
configuration interaction is fully accounted for in the wave function of the
lower term. Cascading from high Rydberg states to the low-lying terms of the
ion is therefore more accurately treated.

In many cases treated by the OP, the free electron energy mesh on which the
photoionization cross-sections are calculated, was too coarse to accurately
map narrow resonance features. This usually leads to significant
contributions to the recombination coefficient being missing, since the
resonance contribution is proportional to the area underneath the resonance.
The photoionization cross-sections for C ^{+} generated for the OP
(Yu Yan & Seaton [1987]) have poor resolution in some of the resonance
structure because they were based on a mesh in which points are equally
spaced in effective quantum number relative to the next threshold
(hereafter referred to as a quantum defect mesh). One hundred points
were provided per integer change in effective quantum number and
subsequently, this number was further reduced by removing any points
that could be linearly interpolated from two adjacent points within
some prescribed accuracy.

In contrast to the OP data, we use a variable step energy mesh for photoionization cross-sections that delineates all resonances to a prescribed accuracy. First, we employ quantum defect theory to determine resonance positions and widths using a coarse energy mesh. We then use this information to generate an electron energy mesh which maps the resonances to a chosen accuracy (Kisielius et al. [1998]). In the third step we calculate photoionization cross-sections using the new variable step mesh.

Figure 1:
Comparison of the photoionization cross-section (in Mb)
for 2s2p^{2} ^{2}D, calculated by the OP (top) and
this work (bottom) |

In Fig. 1, we demonstrate the difference in resolution between an OP cross-section and one calculated by our method. The OP cross section consists of 199 points based on a quantum defect mesh whilst the latter is based on an energy mesh of 3281 points. In the OP cross-section, the resonance at 1.135 Ry consists only of 3 points, and the next one is missing completely. There is a significant difference in the peak heights of the resonances too.

Figure 2:
Comparison of the area (in Mb
Ry) under
the 2s2p^{2} ^{2}D cross-section.
Top figure - OP data, bottom - present calculation |

The effects of correctly delineating the resonances are more clearly
demonstrated in Fig. 2. We show the area under the cross-sections
which is a more important measure of their contribution to the recombination
coefficient. The total area under the 2s2p^{2} ^{2}D
photoionization
cross-section from the OP is
,
whereas our calculation gives
for the same energy range.

This detailed treatment of the energy mesh was undertaken
for the energy region from the 2s^{2}(^{1}S)
limit up to
4/15^{2} Ry below the 2s2p(^{3}P^{o}) threshold,
since this region contains the main contribution to the recombination
at the temperatures of interest. Above the 2s2p(^{3}P^{o})
threshold, a quantum defect mesh was used. A very high degree of accuracy
is not required here so a step in effective quantum number was chosen that
provided 1000 points per integer change in quantum number.

Finally, we note that we deal with radiative damping of resonances in the same
way as described by Kisielius et al. ([1998]), in that any autoionizing
states for which the radiative width is greater than the autoionization width
are omitted from the list of resonances used to generate the high resolution
energy mesh. As a result such resonances do not appear in the photoionization
cross-sections and do not contribute to the recombination. In practice this
means omitting all resonances associated with 2s2p(^{3}P^{o})*n*h
electron configurations, as well as the ^{2}H^{o} terms from
the 2s2p(^{3}P^{o})*n*g configurations.

Copyright The European Southern Observatory (ESO)