2 Method

We employ the Breit-Pauli R-matrix method (Scott & Burke 1980) to calculate the photoionization cross sections. This method has the advantage of including the effects of resonances which converge onto the LSJ target states of Fe XIX which are included in the calculation. The present work is based on including in the eigenfunction expansion the following 11 Fe XIX LS target states, $\Phi_i$: 2s²2p^{4 3}P, 2s²2p^{4 1}D, 2s²2p^{4 1}S, 2s2p^{5 3}P$^\circ$, 2s2p^{5 1}P$^\circ$, 2p^{6 1}S, 2s²2p³3s ³S$^\circ$, 2s²2p³3s ³P$^\circ$, 2s²2p³3s ³D$^\circ$, 2s²2p³3s ¹D$^\circ$ and 2s²2p³3s ¹P$^\circ$. Each of these target states are represented by configuration interaction wavefunctions of the form  

$$\Phi \left( LS \right) = \sum_{n=1}^M a_n \phi_n \left( \alpha_n LS \right)$$ (1)

where ${\phi_n~n=1,...,M}$ represents a set of configuration state functions which possess the same total $LS\pi$ symmetry and are constructed from a set of one-electron orbitals whose radial part is given by a linear combination of Slater-type orbitals  

$$P_{nl}\left(r\right) = \sum_{j=1}^{k}c_{jnl}\left[ \frac{\le... ...right)! } \right]^{\frac12} r^{I_{jnl}}{\rm e}^{-\zeta_{jnl}r}.$$ (2)

The configuration state functions are chosen to be those configurations generated by considering the addition of one electron from the orbital set $\{$1s, 2s, 2p, 3s, 3p, 3d$\}$ to the basis distributions 2s²2p³, 2s2p⁴ and 2p⁵. A total of 68 configurations were thus considered in representing the target states. The 1s, 2s and 2p orbitals used in the representation of these wavefunctions were taken to be the Hartree-Fock orbitals of the ground state of Fe XIX determined by Clementi & Roetti (1974) while the remaining orbitals were generated by the configuration interaction code CIV3 (Hibbert 1975) in the following manner. Treating I_jnl as fixed, the c_jnl and $\zeta_{jnl}$ parameters of Eq. (2) are then treated as variational parameters. The 3s orbital was thus obtained by varying these parameters in order to minimize the energy of the 2s²2p³(⁴S)3s ³S$^\circ$state. Similarly the 3p and 3d orbitals were optimized using the energies of the 2s²2p³(⁴S)3p ³P and 2s²2p³(⁴S)3d ³D$^\circ$ states respectively where in each case only a single configuration was used in the wavefunction expansion. The resulting parameters for these orbitals are given in Table 1.

  

Table 1: Radial function parameters for oxygen-like Fe XIX orbitals

The energies obtained for these target states were calculated in LS-coupling using the CIV3 configuration interaction code and in LSJ-coupling using the RECUPD module of the R-matrix codes where the same orbitals and level of correlation were used in each case. The results of both calculations are presented in Table 2 and compared with the experimental results tabulated by Corliss & Sugar (1982) and the relativistic calculations of Dasgupta (1995) (using the non-relativistic operators plus the spin-orbit operator, mass correction operator and the Darwin term). (Note that a comparison between the LS calculation and the data of Corliss & Sugar requires consideration of a weighted average of the ground state provided by Corliss & Sugar). Clearly some discrepancy exists between the LS-coupling calculation and the previous studies. However, attempts at introducing further correlation into this calculation had minimal effects on these energies. Threshold positions and thus resonance positions in general would have been of insufficient accuracy if a solely LS-coupling calculation had been performed. Thus, it is clearly essential to include relativistic effects, in that the introduction of interaction between different $LS\pi$ symmetries is highly relevant to the accuracy of the target state energies. The results of the LSJ-coupling calculation are highly satisfactory and examination of the eigenvectors verifies that significant mixing between different $LS\pi$ symmetries possessing the same $J\pi$ value does indeed occur particularly in the ground state terms. (For example in the LS calculation the ground state, ³P, is reasonably pure but in the relativistic calculation 11% of the ³P₂ ground state is made up of the 2s²2p^4 1D₂ configuration. This is enough to lower the ground state by as much as 8 a.u.).

  

Table 2: Target state energies (in a.u.) relative to the ground state of Fe XIX. Previous data are from the tabulation by Corliss & Sugar (1982) and from the relativistic calculation of Dasgupta (1995)

The (N+1)-electron wavefunction, $\Psi$ is expanded in the following manner  

$$\Psi = \sum_k A_{Ek}\psi_k$$ (3)

where A_Ek are energy dependent coefficients and $\psi_k$ are states which form a basis for the total wavefunction, are energy independent and are given by the expansion

$$\begin{eqnarray} \lefteqn{\psi_k\left(x_1,...,x_{N+1}\right) =} \cr & & {\cal A}... ... \cr & & + \sum_j d_{jk} \chi_j \left( x_1,......,x_{N+1} \right) \end{eqnarray}$$ (4)

where $\overline {\Phi}_i$ are the channel functions obtained by coupling the target states $\Phi_i$ with the angular and spin functions of the continuum electron, u_ij, to form states of total angular momentum and parity. ${\cal A}$ is the antisymmetrization operator which accounts for electron exchange between the target electrons and the free electron while $\chi_i$represents the quadratically integrable (L²) functions (or (N+1)- electron configurations) which are formed from the bound orbitals and are included to ensure completeness of the total wavefunction. These configurations are generated by the addition of two electrons from the orbital basis set into the basis distributions 2s²2p³, 2s2p⁴ and 2p⁵. This approach ensures that a measure of equilibrium is attained between the target and (N+1)-electron systems by treating the Fe XVIII states as bound states of the (Fe XIX + e^-) system. Since we are interested in photoionization of both the Fe XVIII ground state (2s²2p^5 2P$^\circ_{\frac32}$) and the first excited state 2s²2p^5 2P$^\circ_{\frac12}$) the following $J\pi$ symmetry transitions interest us: $\rm {\frac32}^\circ \rightarrow {\frac12}^e, {\frac32}^e, {\frac52}^e$ and $\rm {\frac12}^\circ \rightarrow {\frac12}^e, {\frac32}^e$. The entire range of LS matrices which contribute to the above $J\pi$ symmetries are included in the calculation and are recoupled in the manner of Scott & Burke (1980) to give the ${\frac32}^\circ$, ${\frac12}^\circ$, $\rm {\frac12}^e$, $\rm {\frac32}^e$and $\rm {\frac52}^e$ Hamiltonian matrices.

Using an R-matrix radius of 2.2 a.u., 25 continuum orbitals for each value of $l \leq 6$ and a free electron energy mesh of $1\ 10^{-3}$ Ryd, the R-matrix codes (Berrington et al. 1987; Seaton 1987) are then utilized to calculate the photoionization cross sections. The ionization energy of the Fe XVIII ground state was found to be 99.954 Ryd, in excellent agreement with the experimental value of $100.108\ \pm\ 0.294$ Ryd tabulated by Corliss & Sugar (1982). The 2s²2p^{5 2}P$^\circ_{\frac12}$ Fe XVIII bound state was found to be at 0.96068 Ryd relative to the ground state, compared with the value 0.93541 Ryd given by Corliss & Sugar (a difference of only 2.6%), suggesting that the approach used in the calculation of bound states is of sufficient accuracy.