4 Results and discussions

The BPRM intermediate coupling calculations in principle, and the present work in particular, should yield all possible atomic energy levels. We have obtained 83 bound fine structure levels of Fe XXIV in the range of SLJ with total spin multiplicity (2S+1) = 2,4, total orbital angular momentum, L $\leq$ 7, with total angular momentum, $J = 1/2\ -\ 11/2$, even and odd parities, and 138 of Fe XXV in the range of SLJ with total (2S+1) = 1,3, L = 9, and J = 0 - 4, even and odd parities. These numbers far exceed the observed or previously calculated ones. Accuracy of the energies is checked against the observed values from NIST (Sugar & Corliss 1985). All 23 observed bound levels of Fe XXIV and 25 of Fe XXV have been identified in the calculated dataset and are compared in Table 2. The calculated energies of both Fe XXIV and Fe XXV agree very well with the observed ones, differing by less than 1% for all levels (the accuracy may not be quite so good for more complicated atomic systems). These are the most detailed close coupling calculations for the two ions. The complete energy levels of Fe XXIV and Fe XXV are presented in Tables 3 and 4 respectively where they are listed in terms of $J\pi$ quantum numbers.

 

Figure 1: Comparison between the length and the velocity forms of f-values for a) Fe XXIV and b) Fe XXV

 

Table 2: Comparison of Fe XXIV and Fe XXV level energies in Breit-Pauli approximation, $E_{\rm c}$, with the observed ones, $E_{\rm o}$ (Sugar & Corliss 1985)

 

Table 3: Energy levels of Fe XXIV in Breit-Pauli approximation. $N\rm _b$ is the total number of bound levels of the ion and TL is total number of bound levels of quantum number $J\pi$

 

Table 4: Energy levels of Fe XXV in Breit-Pauli approximation. $N\rm _b$ is the total number of bound levels of the ion and TL is total number of bound levels of quantum number $J\pi$

/TD>

We obtain the transition probabilities for 802 transitions in Fe XXIV and for 2579 transitions in Fe XXV. These correspond to both dipole allowed and intercombination transitions in intermediate coupling. The two forms of oscillator strengths, length ($f\rm _L$) and velocity ($f\rm _V$), show less than 10% difference for almost all transitions. Figures 1a,b display $f\rm _L$ versus $f\rm _V$ for Fe XXIV and Fe XXV respectively to show the close correlation between the two sets going down to $f \sim 10^{-7}$. For some transitions the velocity form were not obtained and are not included in the figures. One tolerence criterion for the R-matrix codes is that $f\rm _V$ is not calculated for transitions for which the transition energy is extremely small. There is almost no dispersion of $f\rm _L$ and $f\rm _V$ for Fe XXIV even for the very weak transitions of the order of 10^-6. Although there are some transitions in Fe XXV where the $f\rm _L$ and $f\rm _V$differ by about 10% or higher, most are in closer agreement with each other.

 

Table 5: Comparison of Fe XXIV f- and A-values in Breit-Pauli R-matrix (BPRM) approximation with other works

a)  Yan et al. (1998),   b)  Cheng et al. (1979),  c)  Johnson et al. (1996),   d)  Vainshtein & Safronova (1985), e)  Armstrong et al. (1977),   f)  Zhang et al. (1990),  g)  Burkhalter et al. (1978), h)  Doschek et al. (1972),   i)  Hayes (1979),   j)   Fuhr et al. (1988).

 

Table 6: Comparison of Fe XXV f-values in Breit-Pauli R-matrix (BPRM) approximation with other works

a)  Drake (1979),   b)  Vainshtein & Safronova (1985),  c)   Lin et al. (1977a), d)  Lin et al. (1977b), e)  Fuhr et al. (1988), f)  Johnson et al. (1995).

Present f- and A-values for Fe XXIV and Fe XXV are compared with the best previous calculations and experiments in Tables 5 and 6 repectively. Most of the previous values have been compiled by the NIST (Fuhr et al. 1988; Shirai et al. 1990). In Table 5 most of the BPRM f-values for Fe XXIV agree quite well with those in the NIST compilation (Fuhr et al. 1988), obtained by several investigators, such as by Cheng et al. (1979), Armstrong et al. (1976), Doschek et al. (1972). Although the NIST rating for the accuracy of these transitions varies from B+ to D (< 10% - 30%), nearly all of the available f-values agree to better than 10% with the present ones. As mentioned earlier, Yan et al. (1998) have calculated the level energies and oscillator strengths for lithium like ions up to Z = 20 using Hylleras type variational method including finite nuclear mass effects. They extend the results to higher Z ions including relativistic corrections. Present f-values for Fe XXIV compare very well with their values obtained for the transitons, $\rm 2s(^2S_{1/2})-2p(^2P^{\rm o}_{1/2,3/2})$. Present A-values agree with those by Johnson et al. (1996) obtained from relativistic third-order many-body perturbation theory to about 5% for the two transitions $\rm 2s(^2S_{1/2})-2p(^2P^{\rm o}_{1/2,3/2})$, and by less than 1% for the two transitions, $\rm 2s(^2S_{1/2})-3p(^2P^{\rm o}_{1/2,3/2})$.We also find very good agreement with most of the transition probabilities, A-values, by Vainshtein & Safronova (1985) obtained using the Z^-1-expansion method, which yields more accurate A-values with increasing Z. Present f-values for the transitions 2s(²S_1/2) - 2p(²P$^{\rm o}_{1/2,3/2})$ agree within error bars with the measured values of Buchet et al. (1984).

The BPRM f-values for Fe XXV are compared with the previous calculations in Table 6. Present f-values agree within 5% with detailed calculations by Drake (1979) and Johnson et al. (1995) for the dipole allowed and within 1% with Drake (1979) for the intercombination transitions, $\rm 1s^2(^1S_0)-1s2p(^{1,3}P^{\rm o}_1)$. We agree very well with Vainshtein & Safronova (1985) for the dipole allowed transition who employed the Z^-1-expansion method. Very good agreement is obtained of the present f-values with those by Lin et al. (1977a) for all the transitions compared in Table 6. Of the transitions, $\rm 1s^2(^1S_0)-1s(3,4,5)p(^{1,3}P^{\rm o}_1)$, the dipole allowed ones were calculated by Lin et al. (1977a), and agree quite well with the present values. However, f-values for the intercombination transitions were obtained by Fuhr et al. (NIST 1988) through extrapolation of the data by Johnson & Lin (1976). The present values differ by more than 10% with the extrapolated NIST values. Since our results are in better than 10% agreement with the actual calculated results from other investigators, it appears that the NIST data for these intercombination transitions might not be accurate. The agreement between the present results and those by Lin et al. (1977b) is better than 10% for all transitions except the two transitions $\rm 1s2s(^3S_1)-1s2p(^3P^{\rm o}_{0,1})$ whose f-values are of the order of 10^-3. However, for the stronger intercombination transition $\rm 1s2s(^3S_1)-1s2p(^3P^{\rm o}_2)$ the agreement with Lin et al. is about 3%, and also in good agreement with the measured value of Buchet et al. (1984). Present f-values agree to a similar degree with those by Vainshtein & Safronova (1985) for the dipole allowed as well as the intercombination transitions.

Owing to the large volume of the present data and the number of transitions computed, the complete set of data will be made available electronically. The tables include: transition probabilities A, oscillator strengths f, and line strengths S for all the fine structure transitions. These electronic files will include the calculated level energies also for level identifications. Samples of these data are presented in Tables 7a,b for Fe XXIV and Fe XXV, respectively. Indices "i" and "k" correspond to the two levels with the even/odd parity total $J\pi$ symmetries specified in the column headings. Transition probabilities can also be identified from the energy Tables 3 and 4. Negative values of f_ik imply E_i > E_k (emission), and positive values imply E_i < E_k. The format of the tables follow closely that of the OP (1995), with the main exception that the present results are in intermediate coupling with $J\pi$ as the defining quantum numbers instead of $SL\pi$.

 

Table 7a: Sample of the complete table for A_ik, f_ik and S values for fine structure transitions in Fe XXIV. $N_{\rm f}$ is the total number of bound-bound transitions for the ion and $T_{\rm f}$ is the number of transitions for a pair of JJ'

 

Table 7b: Sample of the complete table for A_ik, f_ik and S values for fine structure transitions in Fe XXV. $N_{\rm f}$ is the total number of bound-bound transitions for the ion and $T_{\rm f}$ is the number of transitions for a pair of JJ'