Study of the photoionization (PI) process of atomic systems provides the opportunity to investigate the dynamic interplay between many-body electron–electron correlations and the relativistic effects.^[1,2] Besides its fundamental significance, the PI cross section is also of vital importance as applied data in atmospheric physics, astrophysics, and plasma and tokomak researches.^[3–5] Much theoretical and experimental effort has been devoted for over three decades to the study of PI cross sections. Among these theoretical researches, the most comprehensive studies have been carried out for a large number of relatively light atoms and ions of astrophysical relevance by using the close-coupling R-matrix method in the frame work of the Opacity Project,^[6,7] the Iron,^[8] and the RmaX projects.^[9] However, many of those calculations have been performed in the framework of non-relativistic theory, and the relativistic effects, for example, the spin–orbit split cannot be included. Numbers of high-resolution and state-selective experimental measurements have been carried out with synchrotron-radiation sources,^[10,11] which provide accurate data on ionic structure and PI cross sections, especially the resonance peaks and guidance to the development of theoretical models. To explain these high-resolution experiments, both elaborate configuration interaction (CI) and fully relativistic effects should be considered.

C II is an important ion in astrophysical sources, such as the interstellar medium. Study of accurate features of C II is of considerable interest and importance for the accurate spectral analysis and astrophysical modeling. Over the last thirty years, a number of experiments and calculations have been performed for PI cross sections of C II. In theory, Yan et al.^[12,13] calculated the PI cross sections and oscillator strengths of C II, as one of the first ions studied in the Opacity Project, for the states of valence excitations with the effective quantum number n ≤ 10 and the total orbital angular momentum L ≤ 3 in the close-coupling approximation, by using the R-matrix method described by Berrington et al.^[14] These results have been revised by Lan et al.^[15] and later, within the Iron Project, have been calculated by Nahar et al.^[16] by using the same techniques such as those carried out within the Opacity Project, but a larger basis set and a finer energy resolution have been used. They also performed the calculations for the C²⁺ + e recombination process (the inverse of the C⁺ + hv PI process) in the close-coupling approximation by using the R-matrix method.^[17,18] In an experiment, Nicolosi and Recanatini et al.^[19,20] measured the photoabsorption spectrum of the C II ion by the dual laser plasma technique. They obtained both the discrete spectrum with a series of resonance lines converging to the ionization threshold, and the relative PI cross sections above the first limit from the ground state 2s²2p(²P^o) and the excited state 2s2p²(⁴P^e). Kjeldsen et al.^[21] reported the first absolute PI cross section with high accuracy by using a merged ion-photon beam setup with synchrotron radiation for C II for the region from threshold 24.38 eV up to 31 eV, where the photoionization of 2p valence electron and 2s²2p → 2s²p(³P^o)nl resonances are dominant. The extended experimental investigation has been performed subsequently by Kjeldsen et al.^[22] to include photon energies up to 105 eV. Above 31 eV, the photoionization of the sub-shell 2s valence electron is dominant, the spectral structures are not as pronounced as in the threshold region, and some resonances from two-electron transitions are observed. There is generally, the experimental PI cross sections^[21,22] and the various types of R-matrix calculations were in good agreement.^[12,16] However, some important deviations are also observed both with respect to the continuum cross section and to the resonance structure. In particular, it is found that the observation of the 2s²2p²P^o → 2s²p(³P)np²P^e resonances cannot be predicted by those norelativistic calculations.^[21,22] Using the Breit–Pauli R-matrix (BPRM) method, Nahar^[24] performed detailed calculations for the PI cross section of the ground 2s²2p(²P^o) and the excited 2s2p²(⁴P^e) states of C II near threshold and revealed the fine structure observed in the experiment, but his work was just limited to the photon energy region of 24 eV–31 eV. Therefore, the detailed relativistic R-matrix calculations are still needed for the region of higher photon energies (above 31 eV), where a difference of nearly up to 50% still exists between the experimental results and the previous calculations.

In this study, we perform comprehensive fully relativistic R-matrix calculations for the PI cross section of the ground state and first excited state state of CII using the Dirac atomic R-matrix code (DARC).^[25] The attention mainly focuses on the L-shell (2p, 2s) photoionization for photon energy from the threshold (24.38 eV) up to 41.5 eV. The results are discussed and compared with early theoretical calculations and experimental measurements. In Section 2, the theoretical method and some computational details are briefly outlined. In Section 3, our calculations are presented and discussed. Finally, in Section 4, we draw some conclusions from the present study.

The R-matrix method for electron-atom and photon–atom interactions has been discussed in great detail by Burke et al.^[26] In this method, the Dirac Hamiltonian for the (N + 1)-electron system can be written in atomic units as

The continuum wavefunctions are constructed from single-particle wavefunctions in the following form:^[25]

In the present calculations, to generate the wavefunction of the (N + 1)-electron system, the wavefunctions of the C III target state are described by 14 relativistic orbitals: 1s_1/2, 2s_1/2, 2p_1/2, 2p_3/2, 3s_1/2, 3p_1/2, 3p_3/2, 3d_3/2, 3d_5/2, 4s_1/2, 4p_1/2, 4p_3/2, 4d_3/2, and 4d_5/2. The GRASP2K package^[27] based on the multiconfiguration Dirac–Fock (MCDF) method^[28,29] is used to obtain these relativistic orbital wavefunctions. In the calculations, the 1s_1/2, 2s_1/2, 2p_1/2, and 2p_3/2 orbitals are treated as spectroscopic orbitals and optimszed by performing an extended optimal-level calculation for all levels from the configurations 1s²2s², 1s²2s2p, 1s²2p², 1s2s²2p, and 1s2p³, and the others are optmized as correlation orbitals with considering the additional configurations of 1s²2snl and 1s²2pnl (n = 3, 4; l = s, p, d). There are a total of 38 configurations, namely 1s²2s², 1s²2s2p, 1s²2p², 1s2s²2p, 1s2s2p², 1s2p³, 1s²3s², 1s²3p², and 1s²2snl, 1s²2pnl, 1s2s²nl, 1s2p²nl, and 1s2s2pnl (n = 3, 4; l = s, p, d) are included in the module DSTG2 of the DARC^[25] to better describe the configuration interaction effects. The first 113 low-lying target states of them are incorporated to generate the possible PI channels and to obtain the wavefunctions of the (N + 1)-electron system of C II. The R-matrix boundary radius is set to be 9.2 a.u. (atomic unit), which is sufficient to envelop all the n = 2 atomic orbitals of the residual C III ion, and thus ensuring that the wave function is completely wrapped within the R-matrix sphere. For each angular momentum, the continuum orbitals are expressed as a linear combination of 40 numerical basis functions. To resolve the fine resonant structure in the respective PI cross section, a suitably chosen fine energy mesh of 10⁻⁴ Ryd (1 Ryd = 13.6056923 eV) between the thresholds is adopted. Our calculated relativistic R-matrix target energies are shown in Table 1, and compared with previous R-matrix results of term energies calculated by Hasoglu et al.^[30] and Wang et al.,^[31] and the NIST values^[32] for some valence- and core-excited states of C III ion. For valence-excited states, it is found that our results and others are in good agreement with each other. Our calculated energies accord well with NIST data within 2.18 percent and the maximum error appears for the excited state 1s²2p²¹S. The energy levels calculated by Hasoglu et al.^[30] are closer to the NIST values than ours and Wang et al.’s^[31] calculations within 1.3 per cent except for the level 1s²2p^{2 1}S, where the difference is 2.36 percent. For core-excited states, there are no available NIST data for comparison. Our calculated energies are larger than those in Refs. [30] and [31], but the difference is less than 0.5%. The present calculated ionization potential (IP) values for the ground state 2s²2p ²P^o_1/2 and the first excited state 2s²2p ²P^o_3/2 are 1.7928 Ryd and 1.7921 Ryd respectively, they are in good agreement with the measured energies of 1.7921 Ryd and 1.7916 Ryd. The IP for the 2s²2p ²P term is 1.7977 Ryd calculated by Hasoglu et al.^[30] and 1.7843 Ryd calculated by Wang et al.^[31]

Table 1.

Table 1.

Table 1. Calculated energies (in unit Ryd) for the target C III ion relative to the ground state 1s22s2 1S0, and other available data in the literature,[30,31] with the NIST values for comparison.[32] . Config. Term Jtπt This Ref. [30] Ref. [31] NIST[32] Config. Term Jtπt This Ref. [30] Ref. [31] NIST[32] 1s22s2 1Se 0 + 1s2s(1S)2p2 3Pe 0 + 22.0673 21.9689 22.0311 1s22s2p 3Po 0 − 0.4759 0.4745 0.4781 0.4772 1 + 22.0677 1− 0.4761 0.4774 2+ 22.0685 2− 0.4766 0.4779 1s2s(3S)2p2 3De 1+ 22.0919 21.9997 22.0359 1s22s2p 1Po 1− 0.9530 0.9570 0.9446 0.9327 2+ 22.0920 1s22p2 3Pe 0+ 1.2690 1.2649 1.2549 1.2523 3+ 22.0921 1+ 1.2693 1.2526 1s2s(3S)2p2 1De 2+ 22.4011 22.3201 22.3655 2+ 1.2698 1.2530 1s2s(3S)2p2 3Se 1+ 22.4130 22.3269 22.3157 1s22p2 1De 2+ 1.3525 1.3630 1.3401 1.3293 1s2s(3S)2p2 3Pe 0+ 22.4877 1s22p2 1Se 0+ 1.7615 1.7881 1.6823 1.6632 1+ 22.4882 1s22s3s 3Se 1+ 2.1619 2.1743 2.1708 2+ 22.4890 1s22s3s 1Se 0+ 2.2661 2.2519 2.2524 1s2s(3S)2p2 1Pe 1+ 22.6677 22.5674 22.6143 1s22s3p 1Po 1− 2.3747 2.3596 1s2s(1S)2p2 1Se 0+ 22.7249 22.6511 22.6449 1s22s3p 3Po 0− 2.3777 2.3666 1s2p3 5So 2− 22.4636 22.4255 1− 2.3777 2.3667 1s2p3 3Do 1− 22.7497 22.6802 22.6956 2− 2.3778 2.3668 2− 22.7498 1s2s22p 3Po 0− 21.4635 21.3622 3− 22.7499 1− 214640 1s2p3 3So 1− 22.8781 22.7780 22.8319 2− 21.4651 1s2p3 1Do 2− 22.9576 22.8708 22.9039 1s2s22p 1Po 1− 21.6935 21.5849 21.6360 1s2p3 3Po 0− 23.0565 22.9926 22.9657 1s2s(3S)2p2 5Pe 1+ 21.5735 21.5283 1− 23.0566 2+ 21.5740 2− 23.0567 3+ 21.5748 1s2p3 1Po 1− 23.2669 23.1865 23.1750

Table 1. Calculated energies (in unit Ryd) for the target C III ion relative to the ground state 1s22s2 1S0, and other available data in the literature,[30,31] with the NIST values for comparison.[32] .

The L-shell photonization processes for the ground state and the first excited state of CII ion can be described in LSJ terms as

Figure 1(b) shows the calculated total PI cross section of the states (solid curve) and (dashed curve) of CII in the near threshold regions, namely 24 eV–31 eV of photon energy. As a comparison, the nonrelativistic R-Matrix results for the ground term 2s²2p(²P^o) obtained by Nahar^[16] in LS coupling are plotted in Fig. 1(a). It is found that the fine structure features are different and more extensive resonance structures are introduced due to automatically including the relativistic effects (mainly the spin–orbital and spin-spin interactions) in the present calculations, besides the resonance positions that have about 0.047 eV minor shifts toward lower energy when compared with the nonrelativistic calculations.^[16] The cross section in regions below 2s2p ³P^o limit (30.88 eV)^[22] is mainly characterized by 2s → np resonances embedded in continua of the type 2p → εl transition. The most distinct resonances observed in Figs. 1(a) and 1(b) should be due to 2s²2p²P^o → 2s2p(³P)np ²P^e, ²D^e, and ²S^e transitions (n ≥ 4). For a given autoionizing resonance configuration, such as 2s2p(³P)4p complex, the first three resonances in cross section belong to the 2s2p(³P)4p ²P^e, ²D^e and ²S^e states respectively in Fig. 1(b), and the fourth resonance, which can be identified as the combination of the 2s2p(¹P)4p ²P^e, ²D^e, ²S^e states, is common for the whole Rydberg series. It is obvious that the 2s²2p²P^o → 2s2p(³P)np ²P^e resonances are not predicted by the non-relativistic calculations in Nahar’s work^[16] (Fig. 1(a)), because the autoionizing states 2s2p(³P)np ²P^e cannot decay via a radiationless transition to the 2s continuum with the same symmetry ²P^e, but are allowed in relativistic intermediate coupling. Figure 1(c) displays that the present relativistic PI cross sections after the convolution of the calculated resonance with a Gaussian distribution function of the full width at half maximum (FWHM) of 35 meV representing the instrumental resolution, are compared with the measured absolute PI cross sections of CII (see Fig. 1(d)) by Kjeldson et al.^[21,22] by using a merged ion-photon beam setup with synchrotron radiation. It is found that, in general, the resonance features of the present calculations (Fig. 1(c)) and the experimental results (Fig. 1(d)) are in very good agreement. However, some differences are also noted for the higher resonance complexes, and the calculated continuum cross section is approximately 10% larger than the measurement. The PI cross sections for the levels of the ground state , and of the metastable state of CII in the region 24 eV–31 eV have also been calculated by Nahar^[24] including 20 fine-structure levels target states by using the BPRM method. The present results are in good agreement with the BPRM calculations by Nahar^[24] for the states. It demonstrates that the relativistic effects should be incorporated into the PI cross section calculations, even for light atomic systems, such as C II.

Fig. 1.

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Fig. 1. (color online) Photoionization cross sections of CII from the photon energy region 24 eV–31 eV. (a) Nonrelativistic R-Matrix results for ground term 2s22p(2Po) in LS coupling by Nahar;[16] (b) DARC results for fine structure levels of the ground state 2s22p (solid curve) and the first excited state (dashed curve); (c) present theoretical data convolved with a Gaussian FWHM of 35 meV; (d) experimentally measured cross sections by Kjeldsen et al.[21,22] The unit 1 barn = 10−24 cm2.

Figures 2(a) and 2(b) show the partial and total PI cross sections for the fine structure levels of and in the region 31.4 eV–37.4 eV, where the photon energies are above the 2s2p³P^o limit (30.88 eV), and the 2s → np resonances, and 2s → ε l continua phtoionization spectrum is dominated. For the level, the partial cross section from the Jπ = 3/2^e transition contributes most to the total PI cross section, while for the state, the most important partial cross section is due to the J^π = 5/2^e transition, and the partial cross sections, which come from J^π = 1/2^e and 3/2^e, are also important for the resonances. In Fig. 2(c), we compare the total cross sections for the and levels; it is found that both the continuum and resonance features are very similar for those two states because of the very small split of the energy, though the partial cross sections are quite different for them. Thus, it demonstrates that the influences of relativistic effects on PI cross sections of different fine structure levels derive from the same term in CII are small and can be neglected.

Fig. 2.

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	Fig. 2. (color online) Partial and total photoionization cross section of the and states of C II from the region 31.4 eV–37.4 eV (a) for the state, (b) for the state; (c) comparison of the total cross sections between (solid curve) and (dotted curve).

In Fig. 3, the averaged total PI cross sections for the state are compared with the measured cross sections by Kjeldson et al.,^[21,22] together with the nonrelativistic cross sections by Nahar et al.^[16] In the experiment by Kjeldson et al.,^[21,22] in addition to the C II ground state, there was also a 5% metastable-state fraction giving additional resonances not included in our calculations. The theoretical cross sections by Nahar et al.^[16] have been convoluted with Gaussian distribution functions of FWHM of 55 meV representing the instrumental resolution in Ref. [21]. To resolve the more detailed resonances of the measurements, the FWHM of 40 meV was taken into account in the present calculations. From Fig. 3, we can find that the position of the resonance and the intensity of the peak obtained from the present calculations and their corresponding measurements are in good agreement. In particular, we note that the observed double peaks due to the transitions 2s²2p²P^o → 2s2p(¹P^o)5p , are clearly revealed in the present calculations, which have not been shown in the results of Nahar et al.^[16] By analyzing the partial transition contributions to the total photoionization cross section for the states and , respectively, we find that detailed resonant structures of the observed double peaks are associated with the autoionizing state in lower energy, and the autoionizing state in higher energy. The resonance from the 2s2p(¹P)5p²S_1/2^e autoionizing state is very weak compared with those from and , so it cannot be observed in the experimental PI spectra. For the other 2s²2p²P^o → 2s2p(¹P)np n ≥ 6 Rydberg series, the situation is similar to the scenario for the transition 2s²2p²P^o- , , but the double peaks are not very obvious, especially for the higher-n Rydberg series. The resonances from the and , in Fig. 3 show broader features than from the other members of 2s²2p²P^o-2s2p(¹P)np resonance series. Besides the above mentioned resonances from the one-electron transitions, the two-electron transitions of 2s²2p²P^o → 2p²(³P)3s, 2p²(³P)3d ²P^e, ²D^e, ²S^e are also very prominent in the region 31 eV–37 eV of photon energy. It should be noted that our results are still approximately 20% larger than the measured absolute PI cross sections in the region, whereas their agreement is good near the threshold.

Fig. 3.

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	Fig. 3. (color online) Total photoionization cross section of the 2s22p 2P term of CII from the region 31.4–37.4 eV. Solid black curve: experimental data measured by Kjeldsen et al.;[22] solid blue curve: the present data convolved with a Gaussian FWHM of 40 meV and multiplied by 0.8; solid gray curve: norelativistic R-matrix results by Nahar.[16]

The PI cross section of the and of C II in the region 36.5 eV–41.5 eV are shown in detail in Fig. 4. The resonance peaks in the region are dominated by two-electron transitions, which are more difficult to calculate and to identify. We only identify a few of them following the CI calculations performed by Kjeldson et al.^[22] The relativistic cross section for the levels and show many overlapping resonances. The main resonance structures above the 2s2p(¹P) limit are due to 2s²2p²P^o → 2p²(¹D)nd, 2p²(³P)ns, nd transitions, which mix together, thus complicating the analysis. Comparing the present convolved cross sections with experimentally measured cross sections by Kjeldsen et al.^[22] and norelativistic R-matrix calculations by Nahar,^[16] good agreement is obtained for the resonance peaks both in positions and shapes, but the calculations predict that the cross section is 15% larger than the experimental values.

Fig. 4.

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Fig. 4. (color online) PI cross section of the ground state and the excited state of C II ions from the region 36.5 eV–41.5 eV of photon energy, showing (a) detailed resonances for the ground state (solid curve) and the excited state , (b) theoretical data convolved with Gaussian FWHM of 40 meV and multiplied by 0.85 (magenta line), and compared with experimentally measured cross sections (black line) by Kjeldsen et al.[22] and norelativistic R-matrix results (green line) by Nahar.[16]

Using the fully relativistic R-Matrix code DARC, we have calculated the L-shell (2p, 2s) photoionization cross section of the ground state and the first excited state of C II ions. It is found that the relativistic effects are very important even for light atomic systems such as C II, which account for nearly all experimentally observed resonant features that may not be obtained in the nonrelativistic LS coupling calculations. The present fully relativistic results agree with the relativistic BPRM calculations and reveal the fine structure observed for the experimental measurement of near the threshold PI cross sections. Good agreement is also found between our results and the experimental resonance peaks for the inner 2s subshell PI cross sections, however, there are still about 10%–20% deviations observed for the continua phtoionization spectra that need to be further studied in the future.