3 Atomic calculations

The Breit-Pauli R-matrix calculations are carried out using an eigenfunction expansion for Fe XXIV$\leftrightarrow$ (e + Fe XXV) containing 13 fine structure levels from configurations, $\rm 1s^2$, $\rm 1s2s$, $\rm 1s2p$, $\rm 1s3s$ and $\rm 1s3p$ of the core ion Fe XXV (Table 1). For Fe XXV$\leftrightarrow$ (e + Fe XXVI) the expansion contains 16 levels corresponding to the hydrogenic orbitals 1s, 2s, 2p, 3s, 3p, 3d, 4s, 4p, 4d, and 4f of Fe XXVI (Table 1). The orbital wavefunctions of the target are obtained using the atomic structure code SUPERSTRUCTURE (Eissner et al. 1974) which are input for the BPRM codes (Hummer et al. 1993). Table 1 lists the target level energies which are obtained from stage RECUPD which reconstructs the target states developed from SUPERSTRUCTURE.

 

Table 1: Energy Levels of Fe XXV and Fe XXVI in the eigenfunction expansion of Fe XXIV and Fe XXV respectively

The BPRM codes (also called the RMATRX1 codes; Berrington et al. 1995) from the IP, consisting of several stages similar to those in the OP codes (Berrington et al. 1987), are used for computations of the oscillator strengths for bound-bound levels. For each $J\pi$ of Fe XXIV, all possible combinations of doublets and quartets of Fe XXIV with L $\leq$ 7 and l $\leq$ 9 are included, and to those of Fe XXV, singlets and triplets of Fe XXV with L $\leq$ 5 and l $\leq$ 9 are included. The level energies of Fe XXIV and Fe XXV are obtained from the code STGB of the BPRM suite of codes. As the bound levels are scanned out by the effective quantum numbers $\nu$through the poles in the (e + ion) Hamiltonian, the mesh for $\nu$ should be fine enough to avoid any missing levels and to obtain accurate energies for the higher levels. For example, for Fe XXV, $\Delta \nu$ = 0.01 is adequately fine for the lower levels, but it is necessary to refine it to 0.001 in order to obtain a number of the higher energy levels for $J\geq$ 2.

The large number of bound levels yield very large sets of fine structure transitions for both Fe XXIV and Fe XXV. The code STGBB of the BPRM codes computes the gf-values for the bound-bound transitions. The transitions are identified by the "good'' quantum numbers $J\pi$ only. The datasets are processed, following the NIST format, for oscillator strengths (f-values), line strengths ($\cal S$-values) and transition probabilites (A-values) using a code BPRAD (Nahar, unpublished). The format is similar to that used for the OP data (OP 1995) in that the energies and transitions are labeled by level indices.

Identification of the calculated energy levels is a major task for BPRM calculations since a large number of energy levels are generated (and indeed exist). The BP Hamiltonian matrix yields energies corresponding to the different J-values. While STGB of the R-matrix codes from the Opacity Project is capable of sorting out the possible configurations contributing to a given LS term, the BP version of the code does not link the J-values with the corresponding LS terms and configurations, making the task of level identification difficult. The problem is particularly acute in the BPRM calculations since many levels may be very closely spaced within a small range of effective quantum number but corresponding to different interacting Rydberg series. This problem is not too severe for the highly charged ions under consideration in this work, since the Rydberg series can be relatively easily identified. For example, the total $J = 2^{\rm o}$ (odd parity) levels in Fe XXV correspond to the two Rydberg series 1snp and 1snf. For each $\nu$ there are two bound states; for example, for $\nu \approx 4.0$ there are the 1s4p $^3P^{\rm o} \ (J = 2)$ and 1s4f $^3F^{\rm o}$ (J = 2) levels. For more complicated atomic systems however, such as the ongoing BPRM intermediate coupling calculations for Fe V (e + Fe VI), the interacting Rydberg series problem is far more complex, and level identifications are much more difficult, since many levels corresponding approximately to the same $\nu$ lie close together (Nahar & Pradhan, in preparation).

The energy levels of ions in the present work have been identified by matching the J-values with the possible combination of total spin multiplicity and total orbital angular momentum of the ion whose LS term is derived, in turn, from combinations of the target term S_iL_i and angular momentun of the outer electron $\ell_i$. The target S_iL_i is determined from the spectroscopic configurations included in the eigenfunction expansion, and the n and l quantum numbers of the outer electron are determined from its effective quantum number. Hund's rule is followed to identify the positions of levels, e.g. the triplets lie below the singlets of same L, $\pi$ and configuration of He-like ions. The code BPRAD is written to process these data for complete identification of energy levels and corresponding transitions.