Back to Main Page

Graphical and Numerical Comparisons in Support of Assessment Procedure

In order to reaffirm the uncertainty estimates of transition probabilities for the present compilation we made graphical and numerical comparisons of the results of different advanced calculations for as many transitions as possible, regardless of wavelength. Later we selected data for the Chandra spectral range 20 Å to 170 Å. To fit the data into systematic trends, or deviations from them, we found useful the theoretically predicted scaling of data along isoelectronic sequences. If available we always selected data from detailed configuration-interaction calculations with intermediate coupling. As usual these calculations are performed for transitions to the ground state or between low excited configurations. For the transitions involving high-lying configurations, only Opacity Project (OP) data are available. For the stronger transitions of many spectra, good agreement exists between the OP data and data from more detailed calculations that consider spin-orbit interactions.

Figure 1
Figure 1: Comparison of oscillator strengths of the individual lines for the B-like ion Mg VIII.
   Figure 1 shows a comparison of oscillator strengths of the individual lines for the B-like ion Mg VIII. The ratios of the OP [8] oscillator strengths in combination with LS coupling and selected calculations for individual lines to the [16,19,20] corresponding MCHF [10] values are plotted on a logarithmic scale versus the logarithm of the MCHF oscillator strengths. The dashed and dotted lines indicate bands of 10% and 25% around a perfect ratio of 1.00. For most transitions the agreement is better than 10% with calculations for the individual lines and better than 25% with the OP data.

The agreement among the OP calculations and different relativistic calculations gets worse for the weaker transitions and for transitions between those levels where one or both are appreciably mixed due to breakdown of LS coupling. Large disagreements are often observed for weaker transitions when calculations for the transition probabilities of weak transitions have encountered considerable problems due to appreciable cancellations of positive and negative components of the transition integral. A comparison of oscillator strengths for the C-like spectrum of Si IX is given in Fig. 2. The ratios of the CIV3 [13] oscillator strengths to the corresponding MCHF [10] values are plotted on a logarithmic scale versus the logarithm of the MCHF oscillator strengths. The dashed-dotted lines indicate a band of 50% around a perfect ratio of 1.00. Transitions 2s22p2-2s22p3s, 2s22p2-2s22p3d, 2p4-2s22p3s, and 2p4-2s22p3d in the Chandra range 54 Å to 125 Å are presented. This comparison shows good agreement for the transitions with large oscillator strengths but the scatter increases sharply for transitions with lower oscillator strengths.   Figure 2
Figure 2: Comparison of oscillator strengths for the C-like ion Si IX.

Figure 3
Figure 3: Comparison of oscillator strengths for the F-like ion Si VI.
 Figure 3 shows a comparison of oscillator strengths for the F-like ion Si VI in the range of 77 Å to 146 Å. The ratios of CIV3 [11] oscillator strengths to the corresponding MCHF [10] values are plotted on a logarithmic scale versus the logarithm of the MCHF oscillator strengths. The dotted and dashed-dotted lines indicate bands of 25% and 50% around a perfect ratio of 1.00. It is seen that agreements are better than 25% for most transitions. However for four transitions (spectroscopic designations for them are given on the plot) there is disagreement of more than 50%.

The numerical values associated with Fig. 3 along with wavelengths and transition classifications are given in Table 1. (i indicates the lower level and k the upper level.) The largest contribution to the eigenvector percentage composition of each upper level is also given. We do not provide these values for the lower levels, which are well described by LS coupling. From the table it is seen that the percentage contributions of the main terms to the upper levels belonging to transitions with large deviations are 2s22p4(3P)3d 2P3/2 68%; 2s22p4(1D)3d 2P1/2 85%; 2s22p4(1D)3d 2D5/2 77%; 2s22p4(3P)3d 2D5/2 86%. Thus, we may conclude for this case that when both levels have a main contribution of more than 85%, the agreement is better than 10%. If at least one level has a main contribution of less than 85%, the oscillator strengths may disagree up to 50%.

Table 1. Oscillator Strengths of Si VI
lambda (Å)   Config.
(i)		   Config.
(k)		   Terms
i-k		   Ji   Jk   LS principal
contribution
to upper
level k		   f (MCHF)   f (CIV3)/
f (MCHF)		   Agreement %
96.018 2s22p5 2s22p4(1D)3s 2P°-2D 3/2 3/2 98% 4.19E-03 0.97 -3
77.412 2s22p5 2s22p4(1S)3d 2P° -2D 3/2 3/2 98% 6.95E-03 1.05 5
91.370 2s22p5 2s22p4(1S)3s 2P° -2S 3/2 1/2 97% 1.23E-02 0.89 -11
83.284 2s22p5 2s22p4(3P)3d 2P° -2P 3/2 1/2 87% 1.55E-02 1.01 1
91.797 2s22p5 2s22p4(1S)3s 2P° -2S 1/2 1/2 97% 1.55E-02 0.90 -10
83.005 2s22p5 2s22p4(3P)3d 2P° -2P 3/2 3/2 68% 1.82E-02 0.48 -52
99.096 2s22p5 2s22p4(3P)3s 2P° -2P 3/2 1/2 98% 2.08E-02 0.99 -1
99.966 2s22p5 2s22p4(3P)3s 2P° -2P 1/2 3/2 96% 3.28E-02 1.00 -1
80.490 2s22p5 2s22p4(1D)3d 2P° -2P 3/2 1/2 85% 3.33E-02 1.58 58
80.395 2s22p5 2s22p4(1D)3d 2P° -2D 3/2 3/2 91% 3.39E-02 1.18 18
83.639 2s22p5 2s22p4(3P)3d 2P° -2P 1/2 1/2 87% 4.57E-02 0.93 -7
96.023 2s22p5 2s22p4(1D)3s 2P° -2D 3/2 5/2 98% 6.39E-02 0.97 -3
77.429 2s22p5 2s22p4(3P)3d 2P° -2D 3/2 5/2 98% 6.81E-02 1.06 6
99.599 2s22p5 2s22p4(3P)3s 2P° -2P 1/2 1/2 98% 7.66E-02 0.99 -1
96.490 2s22p5 2s22p4(1D)3s 2P° -2D 1/2 3/2 98% 7.80E-02 0.97 -3
81.031 2s22p5 2s22p4(1D)3d 2P° -2S 1/2 1/2 95% 8.32E-02 1.25 25
83.257 2s22p5 2s22p4(3P)3d 2P° -2D 3/2 3/2 68% 9.98E-02 1.10 9
99.460 2s22p5 2s22p4(3P)3s 2P° -2P 3/2 3/2 96% 1.00E-01 1.00 -1
77.718 2s22p5 2s22p4(1S)3d 2P° -2D 1/2 3/2 97% 1.01E-01 1.10 10
80.909 2s22p5 2s22p4(1D)3d 2P° -2P 1/2 3/2 90% 1.11E-01 1.11 11
80.698 2s22p5 2s22p4(1D)3d 2P° -2S 3/2 1/2 95% 1.35E-01 0.89 -11
83.128 2s22p5 2s22p4(3P)3d 2P° -2D 3/2 5/2 86% 1.80E-01 2.16 116
80.449 2s22p5 2s22p4(1D)3d 2P° -2D 3/2 5/2 77% 1.90E-01 1.76 76
83.611 2s22p5 2s22p4(3P)3d 2P° -2D 1/2 3/2 68% 1.99E-01 0.81 -19
83.358 2s22p5 2s22p4(3P)3d 2P° -2P 1/2 3/2 68% 2.03E-01 1.20 20
80.577 2s22p5 2s22p4(1D)3d 2P° -2P 3/2 3/2 90% 2.67E-01 1.09 9
80.821 2s22p5 2s22p4(1D)3d 2P° -2P 1/2 1/2 85% 2.73E-01 0.96 -4
80.725 2s22p5 2s22p4(1D)3d 2P° -2D 1/2 3/2 91% 3.68E-01 1.14 14

Problems concerning large discrepancies in transition probabilities for fluorine-like spectra between the OP [8] and CIV3 [11] results were discussed earlier in [21]. At that time extended relativistic calculations for individual lines were available only from CIV3 [11] calculations. Last year new MCHF [10] data became available. The dependence of accuracy on the purity of LS coupling is illustrated by an example for the fluorine-like ion S VIII. The following three plots show detailed comparisons of oscillator strengths for allowed transitions of S VIII.

In Fig. 4 the ratios of OP [8] and CIV3 [11] oscillator strengths to the MCHF [10] values are plotted on a logarithmic scale against the logarithm of the MCHF oscillator strength. The dashed lines indicate a band of 50% around a perfect ratio of 1.00. Some large disagreements are observed with the OP data, even for the stronger lines. The agreement between MCHF and CIV3 is clearly better, but for many transitions the agreement is still not good.   Figure 4
Figure 4: Comparison of oscillator strengths for the F-like ion S VIII.

In studying these transitions for which the agreement is not good, we found that for almost all of them, one or both of the levels involved in the transition could be considered as mixed. By mixed we mean that the main contribution to the wave function of the level is less than 80%. Correspondingly, a pure level here means that that the main contribution to the wave function composition of this level is more than 80%.

Figure 5

Figure 5: Comparison of oscillator strengths of allowed transitions between mixed levels for the F-like ion S VIII.

   Figure 5 shows a comparison of oscillator strengths of allowed transitions between mixed levels for the F-like ion S VIII. The ratios of CIV3 [11] oscillator strengths to the corresponding MCHF [10] values are plotted on a logarithmic scale versus the logarithm of the MCHF oscillator strength. The dashed lines indicate a band of 50% around a perfect ratio of 1.00. It is seen that for most transitions the agreement is better than 50%, which is within the range of data listed in the NIST reference tables.
Figure 6 shows the comparison of oscillator strengths of allowed transitions between pure levels for the F-like ion S VIII. The ratios of CIV3 [11] oscillator strengths to the corresponding MCHF [10] values are plotted on a logarithmic scale against the logarithm of the MCHF oscillator strength. Out of 33 transitions, 31 have agreements between the CIV3 and MCHF calculations of better than 10%. The other two agree within 20%.   Figure 6
Figure 6: Comparison of oscillator strengths of allowed transitions between pure levels for the F-like ion S VIII.

The study of such dependence on LS coupling is important when we have transition probabilities available only from one source and need to estimate their accuracy on the basis of extrapolation from comparisons with other sources in overlapping areas.

Introduction   |   Comparison   |   Table Arrangement   |   References
Back to Main Page