3 Results and discussion

Given the semiempirical character of the RQDO method, quantum defects have to be known. The value of the ionization energy for Ni XVIII was derived by Edlén (1978) from core polarization theory applied to the nf series. The energy data required as input in our procedure are supplied by the critical compilation by Sugar & Corliss (1985). For two specific levels, $\rm 7d_{3/2}$ and $\rm 8f_{5/2}$, there are no energy data available from these authors (Sugar & Corliss 1985), and they have been taken from Feldman et al. (1971).

 

Table 4: Line strengths for the $n{\rm d}\to n'{\rm d}$ fine-structure and multiplet transitions

Transition	${\rm {}^2D_{3/2}-{}^2D_{3/2}}$	${\rm {}^2D_{3/2}-{}^2D_{5/2}}$	${\rm {}^2D_{5/2}-{}^2D_{3/2}}$	${\rm {}^2D_{5/2}-{}^2D_{5/2}}$	${\rm {}^2D-{}^2D}$
${\rm 3d \to 4d}$	0.382(-1)	0.163(-1)	0.165(-1)	0.658(-1)	0.137
	0.37(-1)	0.16(-1)	0.16(-1)	0.63(-1)	0.131
${\rm 3d \to 5d}$	0.276(-2)	0.118(-2)	0.118(-2)	0.474(-2)	0.987(-2)
	-	-	-	-	0.944(-2)
${\rm 3d \to 6d}$	0.696(-3)	0.299(-3)	0.298(-3)	0.120(-2)	0.249(-2)
	-	-	-	-	0.237(-2)
${\rm 3d \to 7d}$	0.276(-3)	0.121(-3)	0.118(-3)	0.483(-3)	0.997(-3)
	-	-	-	-	0.952(-3)
${\rm 3d \to 8d}$	0.143(-3)	0.616(-4)	0.612(-4)	0.246(-3)	0.512(-3)
	-	-	-	-	0.485(-3)
${\rm 4d \to 5d}$	0.408	0.174	0.176	0.702	0.146(+1)
	0.406	0.174	0.174	0.70	0.145(+1)
${\rm 4d \to 6d}$	0.285(-1)	0.122(-1)	0.122(-1)	0.489(-1)	0.102
	0.28(-1)	0.12(-1)	0.12(-1)	0.48(-1)	0.996(-1)
${\rm 4d \to 7d}$	0.684(-2)	0.297(-2)	0.293(-2)	0.119(-1)	0.246(-1)
	-	-	-	0.12(-1)	0.240(-1)
${\rm 4d \to 8d}$	0.271(-2)	0.116(-2)	0.116(-2)	0.463(-2)	0.966(-2)
	-	-	-	-	0.936(-2)
${\rm 5d \to 6d}$	0.237(+1)	0.101(+1)	0.102(+1)	0.408(+1)	0.849(+1)
	0.237(+1)	0.102(+1)	0.102(+1)	0.406(+1)	0.846(+1)
${\rm 5d \to 7d}$	0.157	0.672(-1)	0.672(-1)	0.269	0.561
	0.155	0.66(-1)	0.66(-1)	0.266	0.554
${\rm 5d \to 8d}$	0.365(-1)	0.157(-1)	0.156(-1)	0.627(-1)	0.131
	0.36(-1)	0.15(-1)	0.15(-1)	0.61(-1)	0.128
${\rm 6d \to 7d}$	0.101(+2)	0.420(+1)	0.435(+1)	0.169(+2)	0.355(+2)
	0.98(+1)	0.421(+1)	0.420(+1)	0.168(+2)	0.351(+2)
${\rm 6d \to 8d}$	0.619	0.265	0.266	0.106(+1)	0.221(+1)
	0.61	0.263	0.263	0.105(+1)	0.219(+1)
${\rm 7d \to 8d}$	0.318(+2)	0.136(+2)	0.142(+2)	0.563(+2)	0.115(+3)
	0.32(+2)	0.14(+2)	0.139(+2)	0.56(+2)	0.116(+3)

See footnotes in Table 1.

In a recent paper, Bhatia & Drachman (1999) have obtained term energies for high Rydberg states of lithium-like C IV using a polarization model potential that had Edlén's (1964, 1978) as a starting point. However, further corrections of adiabatic and non adiabatic type, intended to account for the energy shift experienced by the outer electron upon interaction with the core, were added. These corrections proved to give a relativistic description of the outer electron's behaviour, and made the model rather more complete than Edlén's (1964, 1978). Nevertheless, for states of high angular momentum (L>3), accurate energies were obtained for C IV with a simplified asymptotic polarizability expansion that included nonadiabatic effects (Bhatia & Drachman 1999). In the present calculations, the adopted ionization energy value, as reported by Edlén's (1978), although it clearly leaves plenty of room for improvement, is expected to lead to sufficiently correct quantum defects when combined with the reliable energy data compiled by Sugar & Corliss (1985).

 

Table 5: Line strengths for the $n{\rm d}\to 5{\rm g}$ fine-structure and multiplet transitions

Transition	${\rm {}^2D_{3/2}-{}^2G_{7/2}}$	${\rm {}^2D_{5/2}-{}^2G_{7/2}}$	${\rm {}^2D_{5/2}-{}^2G_{9/2}}$	${\rm {}^2D-{}^2G}$
${\rm 3d \to 5g}$	0.128	0.143(-1)	0.537(+1)	0.552(+1)
${\rm 4d \to 5g}$	0.398(+1)	0.444	0.167(+3)	0.171(+3)
${\rm 5d \to 5g}$	0.766(+1)	0.851	0.319(+3)	0.328(+3)

No comparative data.

The line and multiplet S-values for the electric quadrupole transitions (E2), between doublets of Na-like Ni XVIII, object of the present study, are displayed in Tables 1 to 8. For each transition for which comparative data were available, two sets of S-values are given. One, corresponding to the first entry, obtained with the RQDO formalism. And the other, being the second entry, including results from other sources: The multiplet strengths reported by Tull et al. (1972), and the line strengths from the critical compilation of Fuhr et al. (1988), derived from the former multiplet values (Tull et al. 1972) by using the LS-coupling rules. As already mentioned, the S-values from Tull et al. (1972) have been calculated using Hartree-Fock orbital wavefunctions of frozen-core type. No more data of line or multiplet strengths are, to our knowledge, available in the literature.

Results for the $n{\rm s \,{}^2S} - n^{\prime} {\rm d\,{}^2D}$ transitions with n = 3 and 4 and $n^{\prime} = 3 - 8$ are displayed in Table 1. We find a fairly good agreement between the RQDO S-values and the ones by Tull et al. (1972) for most of the doublet-doublet transitions, the largest discrepancies being of about 10%. The same feature is observed for $n{\rm p \,{}^2P^o} - n^{\prime}{\rm s \,{}^2P^o}$ transitions with n = 3 - 5 and $n^{\prime} = 4 - 6$ (Table 2), where our data agree satisfactorily with the comparative ones (differing by less than 4%). An exception is the $\rm 3p-5p$ transition, but it may be due to a misprint in the article by Tull et al. (1972).

In Table 3, the S-values for the $n{\rm p \,{}^2P^o }- n{\rm f\, {}^2F^o}$ transitions with n = 3 - 6 and $n^{\prime} = 4 - 8$ are collected. Again, a general good accord between the set of RQDO S-values and those reported by Tull et al. (1972) is found here, with the only exception of a few ones. For the RQDO ${\rm 3p \,{}^2P^o - 6,7f \,{}^2F^o}$ multiplet S-values, the deviation relative to the comparative line strengths is about 15% and 27%, respectively. An even larger disagreement is found for the ${\rm 3p \,{}^2P^o - 8f \,{}^2F^o}$ multiplet, but this transition appears to be rather weak.

Line strengths for $n{\rm d \,{}^2D} - n^{\prime}{\rm d \,{}^2D}$ transitions are displayed in Table 4. An inspection of this table reveals good similarities in magnitude between our RQDO S-values and the comparative ones for all of the transitions, differing by less than 6%.

RQDO results for a few $n{\rm d \,{}^2D - 5g \,{}^2G}$ transitions are included in Table 5. No comparative data have been found for these. However, we expect our values to have the same quality as that for the preceding transitions since no additional complications are present here. In Table 6, line strengths for $n{\rm f \,{}^2F^o} - n^{\prime}{\rm p \,{}^2P^o}$ transitions are collected, up to the highest ${}^2{\rm P^o}$ level for which energy data was available.

 

Table 6: Line strengths for the $n{\rm f}\to n{\rm p}$ fine-structure and multiplet transitions

Transition	${\rm {}^2F_{5/2}^o-{}^2P_{1/2}^o}$	${\rm {}^2F_{5/2}^o-{}^2P_{3/2}^o}$	${\rm {}^2F_{7/2}^o-{}^2P_{3/2}^o}$	${\rm {}^2F^o-{}^2P^o}$
${\rm 4f \to 5p}$	0.159	0.437(-1)	0.263	0.466
	0.165	0.47(-1)	0.282	0.494
${\rm 4f \to 6p}$	0.685(-2)	0.194(-2)	0.117(-1)	0.204(-1)
	-	-	0.12(-1)	0.218(-1)
${\rm 5f \to 6p}$	0.172(+1)	0.470	0.283(+1)	0.502(+1)
	0.175(+1)	0.499	0.299(+1)	0.524(+1)

See footnotes in Table 1.

The S-values collected in Table 7, corresponding to $n{\rm f\, {}^2F^o} - n{\rm f \,{}^2F^o}$ arrays, show a satisfactory agreement, differing by less than 2%. The RQDO line strengths for a few transitions starting from the ${\rm 5g \,{}^2G_{7/2,9/2}}$ levels are collected in Table 8, where no comparative data are available.

A general inspection of the tables reveals that our results comply with a feature that is characteristic of the LS coupling scheme: the strongest line within a given LS transition array is the one for which $\Delta J = J-J{'}$ coincides in value with $\Delta L = L{'}-L$, in particular when either J or J' is the maximum J-value in the array. We take this as a proof of correctness in the RQDO calculations.

A global view of the quality of our results is shown in Fig. 1. We have plotted the deviation of our RQDO data from those reported by Tull et al. (1972). Only the multiplet S-values are shown. The graph allows us to see the rather good agreement between the RQDO values and the values reported by Tull et al. (1972). Most of the results show a deviation less than 10 per cent.

In order to apply another test to the quality of our results, we have plotted the RQDO S-values, multiplied by $({n^{\prime \ast}})^3$, against $({n^{\prime \ast}})$, the efective quantum number.

Figure 1: Relative deviation of the RQDO line strengths for the multiplets

Figure 2: Regularities in the line strength along the spectral series ${\rm 3p {}^2P_{1/2}} - n{\rm f {}^2F_{5/2}}$, ${\rm 3p {}^2P_{3/2} }- n{\rm f {}^2F_{5/2}}$, and ${\rm 3p {}^2P_{3/2}} - n{\rm f {}^2F_{7/2}}$

Figure 3: Regularities in the line strength along the spectral series ${\rm 3d {}^2D_{3/2}} - n{\rm d {}^2D_{3/2}}$, ${\rm 3d {}^2D_{3/2}} - n{\rm d {}^2D_{5/2}}$, ${\rm 3d {}^2D_{5/2}} - n{\rm d {}^2D_{3/2}}$, and ${\rm 3d {}^2D_{5/2}} - n{\rm d {}^2D_{5/2}}$

Figure 4: Regularities in the line strength along the spectral series ${\rm 4f {}^2F_{5/2}} - n{\rm f {}^2F_{5/2}}$, ${\rm 4f {}^2F_{5/2}} - n{\rm f {}^2F_{7/2}}$, ${\rm 4f {}^2F_{7/2}}- n{\rm f {}^2F_{5/2}}$, and ${\rm 4f {}^2F_{7/2}} - n{\rm f {}^2F_{7/2}}$

It has long been established that for all spectral series of hydrogen or hydrogenlike species, the square of the radial integral, to which the line strength is proportional, diminishes as n'^-3, with n' being the principal quantum number of the upper state in the transition (Martin & Wiese 1996 and references therein). If the quantum defects are not small, as in atomic species other than hydrogenlike, n' should be replaced by the effective principal quantum number. The variation of the line strength with ${n^{\prime \ast}}$ as we advance in an unperturbed spectral series, i.e., for sufficiently high n', when the behaviour becomes hydrogenic, satisfies the following expression (Martin & Wiese 1996):

$$% S(nlj,n{'}l{'}j{'}) \propto \ ({n^{\prime \ast}})^{-3}.$$ (9)

In Figs. 2 to 4, the RQDO line strengths multiplied by the third power of the effective quantum number of the final state are plotted against ${n^{\prime \ast}}$, for several of the line-structure transitions object of the present work.

 

Table 7: Line strengths for the $n{\rm f}\to n^{\prime } {\rm f}$ fine-structure and multiplet transitions

Transition	${\rm {}^2F_{5/2}^o-{}^2F_{5/2}^o}$	${\rm {}^2F_{5/2}^o-{}^2F_{7/2}^o}$	${\rm {}^2F_{7/2}^o-{}^2F_{5/2}^o}$	${\rm {}^2F_{7/2}^o-{}^2F_{7/2}^o}$	${\rm {}^2F^o-{}^2F^o}$
${\rm 4f \to 5f}$	0.482	0.803(-1)	0.806(-1)	0.671	0.132(+1)
	0.481	0.80(-1)	0.80(-1)	0.67	0.131(+1)
${\rm 4f \to 6f}$	0.273(-1)	0.456(-2)	0.455(-2)	0.380(-1)	0.744(-1)
	0.27(-1)	-	-	0.38(-1)	0.740(-1)
${\rm 4f \to 7f}$	0.595(-2)	0.993(-2)	0.991(-2)	0.827(-2)	0.162(-1)
	-	-	-	-	0.160(-1)
${\rm 4f \to 8f}$	0.218(-2)	0.362(-3)	0.363(-3)	0.301(-2)	0.591(-2)
	-	-	-	-	0.585(-2)
${\rm 5f \to 6f}$	0.351(+1)	0.583	0.586	0.487(+1)	0.954(+1)
	0.352(+1)	0.59	0.59	0.489(+1)	0.958(+1)
${\rm 5f \to 7f}$	0.205	0.342(-1)	0.342(-1)	0.285	0.558
	0.205	0.34(-1)	0.34(-1)	0.284	0.557
${\rm 5f \to 8f}$	0.444(-1)	0.739(-2)	0.740(-2)	0.615(-1)	0.121
	0.44(-1)	-	-	0.61(-1)	0.120
${\rm 6f \to 7f}$	0.157(+2)	0.261(+1)	0.263(+1)	0.218(+2)	0.428(+2)
	0.158(+2)	0.263(+1)	0.263(+1)	0.219(+2)	0.429(+2)
${\rm 6f \to 8f}$	0.908	0.151	0.152	0.126(+1)	0.247(+1)
	0.91	0.151	0.15	0.126(+1)	0.247(+1)
${\rm 7f \to 8f}$	0.541(+2)	0.904(+1)	0.903(+1)	0.755(+2)	0.148(+3)
	0.54(+2)	0.90(+1)	0.90(+1)	0.75(+2)	0.147(+3)

See footnotes in Table 1.

 

Table 8: Line strengths for the ${\rm 5g}\to n{\rm d}$ fine-structure and multiplet transitions

Transition	${\rm {}^2G_{7/2}-{}^2D_{3/2}}$	${\rm {}^2G_{7/2}-{}^2D_{5/2}}$	${\rm {}^2G_{9/2}-{}^2D_{5/2}}$	${\rm {}^2G-{}^2D}$
${\rm 5g \to 6d}$	0.414	0.457(-1)	0.172(+2)	0.176(+2)
${\rm 5g \to 7d}$	0.229(-1)	0.248(-2)	0.934	0.959
${\rm 5g \to 8d}$	0.481(-2)	0.531(-3)	0.200	0.205

No comparative data.

It can be observed that the trend followed by our data is such that a constant value is reached as soon as the upper state is suficiently excited to acquire a near-hydrogenic character. This is another proof of the correctness of the RQDO wavefunctions, given that the studied spectral series are free from perturbations by others.

On the basis of the general good agreement between the RQDO and the comparative S-values, together with the compliance of the present results with the above mentioned feature of the LS coupling scheme, as well as with the expected systematic trends along different spectral series, we are confident in the potential usefulness of the RQDO procedure for supplying data relative to electric quadrupole transitions, that may be required in astrophysics and fusion plasma research.