2 Atomic structure and scattering calculations

2.1 Bound-state calculations

The bound-state radial wavefunctions for Fe⁷⁺ were calculated using Froese Fischer's Hartree-Fock program (Froese Fischer [1991]). The 1s, 2s, 2p, 3s, 3p, and 3d orbitals were generated from a configuration-average Hartree-Fock (CAHF) calculation on the 3p⁵3d² configuration, while the 4s orbital was generated from a frozen-core CAHF calculation on 3p⁵3d4s configuration. Finally the 4p, 4d, and 4f orbitals were generated from frozen-core CAHF calculations on the 3p⁶4p, 3p⁶4d, and 3p⁶4f configurations, respectively. These orbitals were then employed within Breit-Pauli configuration-interaction (CI) calculations on the 5 even levels arising from the configurations 3p⁶3d, 3p⁶4s, and 3p⁶4d, and the 72 odd levels arising from the configurations 3p⁵3d², 3p⁵3d4s, 3p⁶4p, and 3p⁶4f.

In Table 1, we present the energies resulting from these CI calculations in comparison to available experimental data. The LS designations should be considered as labels only; for many of these levels, there is strong mixing due to the electrostatic interactions between LS terms with different parents, the spin-orbit interactions, and the interactions between configurations. For example, level 50 labeled as 3p⁵3d²(³F)²D_3/2 in Table 1 is actually: 0.845 $\times$ 3p⁵3d²(³F)²D_3/2 + 0.418 $\times$ 3p⁵3d²(¹D)²D_3/2 + 0.329 $\times$ 3p⁵3d²(³P)²D_3/2plus some smaller eigenvector components; this amount of mixing between different LS parents is typical of many of the levels of the 3p⁵3d²configuration.

We see that the agreement between experiment and theory is quite good for those few lower levels of the 3p⁵3d² configuration for which experimental measurements exist; however, the differences grow to an average of 3.2 eV for the 6 highest levels of this configuration, all of which are important because of their strong radiative rates to the ground-state configuration. On the other hand, the average differences between experiment and theory for the levels of 3p⁶4p, 3p⁶4f, and 3p⁵3d4s are only 1.11 eV, 1.64 eV, and 1.49 eV, respectively. In order to account for ground-state correlation, it would have been necessary to include such configurations as 3s⁰3p⁶3d³, 3s3p⁶3d², and 3s²3p⁴3d³ in our configuration-interaction expansion of the target; but then, in order to account for final-state correlation, one should also include the configurations formed from single and double promotions from the 3s subshell to the 3d subshell and double promotions from the 3p subshell to the 3d subshell for all excited configurations, and that would have made the problem prohibitively large. Finally, if the levels of 3s²3p⁶5$\ell$ and 3s3p⁶3d² (which are in the same energy range as some of the upper levels in our present calculation) had been included in our close-coupling expansion of the target, they would have some effect on our scattering calculations to the more highly excited states through both coupling, which would have tended to decrease the cross sections, and resonance contributions, which would have enhanced the cross sections. However, to include these configurations in our close-coupling expansion would have added 10 LS terms and 25 levels to an already large calculation.

 

 

Table 1: Energies in eV (relative to 3p⁶3d ²D_3/2) of the levels in the 33-term, 77-level R-matrix calculation on Fe⁷⁺

Index	Level	Theor.	Exp.		Index	Level	Theor.	Exp.
		Energy	Energy a				Energy	Energy a
1	3p63d 2D3/2	0.00	0.00		2	3p63d 2D5/2	0.24	0.23
3	3p53d2(3F)4D1/2	47.75	---		4	3p53d2(3F)4D3/2	47.86	---
5	3p53d2(3F)4D5/2	48.05	---		6	3p53d2(3F)4D7/2	48.34	---
7	3p53d2(3F)4G11/2	50.62	---		8	3p53d2(3F)4G9/2	50.82	---
9	3p53d2(3F)4G7/2	51.07	---		10	3p53d2(3P)4P5/2	51.20	---
11	3p53d2(3F)4G5/2	51.34	---		12	3p53d2(3P)4P3/2	51.57	---
13	3p53d2(3P)4P1/2	51.85	---		14	3p53d2(3F)4F3/2	52.57	---
15	3p53d2(3F)4F5/2	52.73	---		16	3p53d2(3F)4F9/2	52.80	---
17	3p53d2(3F)4F7/2	52.91	---		18	3p53d2(1D)2D5/2	53.34	---
19	3p53d2(1G)2F5/2	53.43	53.47		20	3p53d2(1D)2D3/2	53.56	---
21	3p64s 2S1/2	53.75	---		22	3p53d2(1G)2F7/2	53.88	53.88
23	3p53d2(1D)2P1/2	54.60	---		24	3p53d2(1D)2P3/2	55.27	---
25	3p53d2(1G)2H11/2	55.35	---		26	3p53d2(1D)2F7/2	55.62	55.50
27	3p53d2(3F)2G7/2	55.95	---		28	3p53d2(3F)2G9/2	56.30	---
29	3p53d2(1G)2H9/2	56.60	---		30	3p53d2(1D)2F5/2	57.09	56.95
31	3p53d2(3P)4D7/2	57.24	---		32	3p53d2(3P)4D5/2	57.41	---
33	3p53d2(3P)4D3/2	57.65	---		34	3p53d2(3P)4D1/2	57.88	---
35	3p53d2(3P)2D3/2	59.99	---		36	3p53d2(3P)2D5/2	60.48	---
37	3p53d2(3P)4S3/2	60.80	---		38	3p53d2(3P)2S1/2	60.80	---
39	3p53d2(1G)2G9/2	61.19	---		40	3p53d2(1G)2G7/2	61.28	---
41	3p53d2(1S)2P3/2	64.14	63.05		42	3p64p 2P1/2	64.28	63.27
43	3p64p 2P3/2	65.02	63.92		44	3p53d2(1S)2P1/2	65.69	64.57
45	3p53d2(3F)2F5/2	69.17	66.44		46	3p53d2(3F)2F7/2	69.86	67.17
47	3p53d2(3P)2P1/2	76.33	73.39		48	3p53d2(3P)2P3/2	76.70	73.79
49	3p53d2(3F)2D5/2	77.86	73.95		50	3p53d2(3F)2D3/2	77.88	74.03
51	3p64d 2D3/2	82.72	---		52	3p64d 2D5/2	82.77	---
53	3p64f 2F5/2	96.33	94.69		54	3p64f 2F7/2	96.34	94.70
55	3p53d(3P)4s 4P1/2	103.49	---		56	3p53d(3P)4s 4P3/2	103.79	---
57	3p53d(3P)4s 4P5/2	104.35	---		58	3p53d(3P)4s 2P1/2	105.14	103.86
59	3p53d(3P)4s 2P3/2	105.78	104.50		60	3p53d(3F)4s 4F9/2	105.84	---
61	3p53d(3F)4s 4F7/2	106.15	105.03		62	3p53d(3F)4s 4F5/2	106.48	105.37
63	3p53d(3F)4s 4F3/2	106.81	105.74		64	3p53d(3F)4s 2F7/2	107.36	106.00
65	3p53d(3F)4s 2F5/2	108.03	106.70		66	3p53d(3D)4s 4D7/2	110.02	108.45
67	3p53d(3D)4s 4D5/2	110.26	108.70		68	3p53d(3D)4s 4D3/2	110.38	108.79
69	3p53d(3D)4s 4D1/2	110.47	108.89		70	3p53d(1D)4s 2D5/2	110.75	108.98
71	3p53d(1D)4s 2D3/2	110.99	109.27		72	3p53d(1F)4s 2F5/2	111.22	109.64
73	3p53d(1F)4s 2F7/2	111.49	110.01		74	3p53d(3D)4s 2D3/2	111.94	110.24
75	3p53d(3D)4s 2D5/2	112.07	110.45		76	3p53d(1P)4s 2P3/2	129.51	---
77	3p53d(1P)4s 2P1/2	129.51	---

^a

Sugar & Corliss ([1985]).

In order to provide the radiative data necessary to do collisional-radiative modeling for Fe⁷⁺, we have calculated all dipole-allowed radiative rates between the 5 even levels and the 72 odd levels listed in Table 1. Of particular importance are the strong radiative rates from the highest 6 levels of the 3p⁵3d² configuration and the two levels of the 3p⁶4f configuration to the 3p⁶3d ground-state configuration. These transitions occur in the far vacuum UV, have been observed using existing satellites, and should be accessible, under higher resolution, with the low-energy transmission grating of the recently launched Chandra X-ray satellite observatory (Brickhouse [1999]). These particular radiative rates should be most strongly affected by configuration interaction with configurations formed from double electron promotions from the 3p subshell to the 3d subshell. For that reason, we have performed an expanded CI calculation of these radiative rates in which we also included those levels of the 3p⁴3d³ configuration that mix strongly with the levels of 3p⁶3d, those levels of the 3p³3d⁴ that mix strongly with the 6 highest levels of the 3p⁵3d² configuration, and those levels of the 3p⁴3d²4f configuration that mix strongly with the levels of the 3p⁶4f configuration.

In Table 2, we compare the radiative rates that result from the two CI calculations for these particular transitions. For both calculations, we adjusted the energies to the experimental values before calculating the rates. As can be seen, this extra CI reduces the radiative rates for the transitions from the 6 highest levels of the 3p⁵3d² configuration by an average of 21%. On the other hand, the radiative rates for the transitions from the 3p⁶4f configuration are reduced by less than 10%.

 

 

Table 2: Dipole radiative rates from the six highest levels of the 3p⁵3d²configuration and the two levels of the 3p⁶4f configuration to the two levels of the 3p⁶3d ground-state configuration

Transition	$A_{\rm r}$(s-1) a	$A_{\rm r}$(s-1) b	Ratio c
3p53d2(3F)2F $_{5/2} \rightarrow$ 3p63d 2D3/2	1.32 1011	0.98 1011	0.742
3p53d2(3F)2F $_{5/2} \rightarrow$ 3p63d 2D5/2	6.68 109	4.49 109	0.672
3p53d2(3F)2F $_{7/2} \rightarrow$ 3p63d 2D5/2	1.42 1011	1.04 1011	0.732
3p53d2(3P)2P $_{1/2} \rightarrow$ 3p63d 2D3/2	2.79 1011	2.27 1011	0.814
3p53d2(3P)2P $_{3/2} \rightarrow$ 3p63d 2D3/2	2.81 1010	2.50 1010	0.890
3p53d2(3P)2P $_{3/2} \rightarrow$ 3p63d 2D5/2	2.53 1011	2.05 1011	0.810
3p53d2(3F)2D $_{5/2} \rightarrow$ 3p63d 2D3/2	2.25 1010	1.67 1010	0.742
3p53d2(3F)2D $_{5/2} \rightarrow$ 3p63d 2D5/2	3.59 1011	3.05 1011	0.850
3p53d2(3F)2D $_{3/2} \rightarrow$ 3p63d 2D3/2	3.48 1011	2.93 1011	0.842
3p53d2(3F)2D $_{3/2} \rightarrow$ 3p63d 2D5/2	3.78 1010	3.12 1010	0.825
3p64f 2F $_{5/2} \rightarrow$ 3p63d 2D3/2	1.81 1011	1.64 1011	0.906
3p64f 2F $_{5/2} \rightarrow$ 3p63d 2D5/2	1.32 1010	1.19 1010	0.902
3p64f 2F $_{7/2} \rightarrow$ 3p63d 2D5/2	1.96 1011	1.78 1011	0.908

^a

Calculated from CI calculations involving the 77 levels listed in Table 1, with the energies adjusted to the experimental values.

^b

Calculated from CI calculations involving the 77 levels listed in Table 1 plus configuration-interaction with the levels of 3p⁴3d³ that mix strongly with 3p⁶3d, the levels of 3p³3d⁴ that mix strongly with 3p⁵3d², and the levels of 3p⁴3d²4f that mix strongly with the levels of 3p⁶4f. Again, the energies were adjusted to the experimental values.

^c

The ratio of the rates calculated with the extra CI to the rates calculated without it.

2.2 Excitation calculations

Our collisional excitation calculations were performed using the intermediate-coupling frame transformation (ICFT) method, which is described in detail in Griffin et al. ([1998]). It is based on the application of multi-channel quantum defect theory, in which unphysical K-matrices are first generated from an R-matrix close-coupling calculation in pure LS coupling and are then transformed to intermediate coupling. The physical K matrices are then determined from a simple frame transformation. This method has been shown (Griffin et al. [1998] and Griffin et al. [1999a]) to eliminate many of the problems associated with the transformation of the physical LS S- or K-matrices to intermediate coupling and is capable of producing accurate level-to-level cross sections in far less time than required for a full Breit-Pauli R-matrix calculation.

In the present case, we first performed an LS R-matrix calculation with exchange, which included all 33 terms arising from the 3p⁶3d, 3p⁵3d², 3p⁵3d4s, 3p⁶4s, 3p⁶4p, 3p⁶4d, and 3p⁶4f configurations in the close-coupling expansion and all LS$\Pi$ partial waves from L = 0 to 14. This allowed us to use our ICFT transformation method to generate the contributions between the 77 levels listed in Table 1 for the J$\Pi$ partial waves from J = 0 to 12. However, this partial-wave expansion must be extended to much higher values of J in order to provide accurate data for collisional-radiative modeling calculations. Therefore, we also performed a no-exchange LS R-matrix calculation for all LS$\Pi$ partial waves from L = 11 to 50; this allowed us to generate intermediate-coupling results for all J$\Pi$ partial waves from J = 13 to 48. These high J contributions were then topped-up for the dipole-allowed transitions using a method originally described by Burgess [1974] for LS coupling and implemented in our ICFT program for intermediate coupling; additionally, the non-dipole transitions were topped-up assuming a geometric series in J.

For all our calculations, the size of the R-matrix box was set to 8.02 a.u., 27 basis orbitals were used to represent the continuum orbitals for each value of the angular momentum; this was more than sufficient to allow us to carry out our calculations to a maximum energy of 600 eV. The long-range multipole contributions were included in the asymptotic region. For the asymptotic part of the calculation, we found that an energy mesh with a separation between adjacent energy points of 0.0165 eV allowed us to resolve the vast majority of narrow resonances in the energy range up through the highest threshold.

In Figs. 1, 2, 3, and 4, we show our calculated cross section from the two levels of 3p⁶3d to the highest six levels of the 3p⁵3d² configuration and to the two levels of the 3p⁶4f configuration. As can be seen, the non-dipole allowed transitions 3p⁶3d ²D_3/2 $\rightarrow$ 3p⁵3d²(³F)²F_7/2 and 3p⁶3d ²D_5/2 $\rightarrow$ 3p⁵3d²(³P)²P_1/2 are dominated by resonance contributions in the low-energy region. In addition, resonant contributions are important in the low-energy region for the dipole-allowed transitions 3p⁶3d ²D_3/2 $\rightarrow$ 3p⁵3d²(³F)²F_5/2, 3p⁶3d ²D_5/2 $\rightarrow$ 3p⁵3d²(³F)²F_5/2, and 3p⁶3d ²D_5/2 $\rightarrow$ 3p⁵3d²(³F)²F_7/2, but are of less importance for the other transitions shown.

Figure 1: Electron-impact excitation cross sections for transitions from the levels of 3p63d 2D to the levels of 3p53d2(3F)2F in Fe7+

Figure 2: Electron-impact excitation cross sections for transitions from the levels of 3p63d 2D to the levels of 3p53d2(3P)2P in Fe7+

Figure 3: Electron-impact excitation cross sections for transitions from the levels of 3p63d 2D to the levels of 3p53d2(3F)2D in Fe7+

Figure 4: Electron-impact excitation cross sections for transitions from the levels of 3p63d 2D to the levels of 3p64f 2F in Fe7+

As discussed in the last subsection, the effects of configuration interaction with configurations formed from double electron promotions from the 3p subshell to the 3d subshell reduce the radiative rates for transitions from these same levels of the 3p⁵3d² configuration to the ground-state configuration by approximately 20% and the transitions from the levels of 3p⁶4f to the levels of the ground-state configuration by about 10%. If we had been able to include these correlations in our scattering calculations, they would have also reduced the cross sections for the strong-dipole allowed transitions from the ground state to these levels. Furthermore, as we have already mentioned, coupling to more highly excited bound states would affect these cross sections, especially those for the excitations to the 3p⁶4f levels. Finally, it has been shown from extensive pseudo-state calculations (see for example, Bartschat & Bray [1997]; Marchalant et al. [1997], and Griffin et al. [1999b]) that coupling between the bound states and the target continuum will tend to further reduce the excitation cross sections, and should be more important for excitations to the 3p⁶4f levels than for the excitations to the levels of 3p⁵3d²; however, these effects should be less than 20% for a seven-times ionized species (Griffin et al. [1999b]). Thus, we might expect that our calculated cross sections for the strong dipole-allowed excitations to these levels to be uncertain at the 20% to 30% level, and that in general, our calculations will tend to overestimate these cross sections. The accuracy of the non-dipole transitions, that are often dominated by resonances at lower energies, is much more difficult to estimate.

For collisional-radiative modeling calculations, we require rate coefficients, rather than cross sections. However, the effective collision strength, first introduced by Seaton (Seaton [1953]), is much more convenient for input to such calculations because it has a much more gradual variation with the electron temperature than the rate coefficient. It is defined by the equation:

$$\Upsilon_{ij} = \int ^\infty_0 \Omega(i \rightarrow j)\exp ... ...}}\right) {\rm d} \left(\frac{\epsilon_j}{kT_{\rm e}}\right),$$ (1)

where $\Omega_{ij}$ is the collision strength for the transition from level ito level j and $\epsilon_j$ is the continuum energy of the final scattered electron. The rate coefficients for collisional excitation $q_{i \rightarrow j}$ and de-excitation $q_{j \rightarrow i}$ can then be determined from the equations

$$q_{i \rightarrow j} = \frac{2\sqrt{\pi}\alpha c a_0^2}{\omega... ...{\left(-\frac{\Delta E_{ij}}{kT_{\rm e}}\right)}\Upsilon_{ij},$$ (2)

and

$$q_{j \rightarrow i} = \frac{\omega_i}{\omega_j} \ \exp{\left(\frac{\Delta E_{ij}}{kT_{\rm e}}\right)}\ q_{i \rightarrow j},$$ (3)

where ${2\sqrt{\pi}\alpha c a_0^2}$ = 2.1716 10^-8 cm³ s^-1, $I_{\rm H}$ = 13.6058 eV, $\Delta E_{ij}$ is the threshold energy for the transition from level i to level j, and $\omega_i$ and $\omega_j$ are the statistical weights of level i and level j, respectively.

Our table of collision strengths for the 2926 transitions among the 77 LSJ levels listed in Table 1 is far too large to be shown here. However, in Table 3, we show the effective collision strengths for the same transitions for which we presented cross sections in Figs. 1, 2, 3, and 4. It is the transitions to the four highest levels of the 3p⁵3d² configuration and the two levels of the 3p⁶4f configuration that are of primary importance for the line-emission intensity ratios discussed in the next section.

 

 

Table 3: Effective collision strengths from the two levels of the 3p⁶3d ground-state configuration to the six highest levels of the 3p⁵3d²configuration and the two levels of the 3p⁶4f configuration

	Electron temperature (K)
Transition	6.40 104	1.28 105	3.20 105	6.40 105	1.28 106	3.20 106
3p63d 2D3/2 $\rightarrow$ 3p53d2(3F)2F5/2	4.44	4.47	4.62	4.99	5.72	7.24
3p63d 2D3/2 $\rightarrow$ 3p53d2(3F)2F7/2	0.268	0.215	0.143	0.100	0.069	0.042
3p63d 2D3/2 $\rightarrow$ 3p53d2(3P)2P1/2	2.04	2.08	2.18	2.39	2.77	3.55
3p63d 2D3/2 $\rightarrow$ 3p53d2(3P)2P3/2	0.454	0.451	0.462	0.498	0.571	0.721
3p63d 2D3/2 $\rightarrow$ 3p53d2(3F)2D5/2	0.464	0.475	0.500	0.546	0.634	0.810
3p63d 2D3/2 $\rightarrow$ 3p53d2(3F)2D3/2	4.59	4.71	4.98	5.49	6.41	8.26
3p63d 2D3/2 $\rightarrow$ 3p64f 2F5/2	0.878	0.892	0.954	1.10	1.35	1.89
3p63d 2D3/2 $\rightarrow$ 3p64f 2F7/2	0.128	0.118	0.097	0.087	0.066	0.051
3p63d 2D5/2 $\rightarrow$ 3p53d2(3F)2F5/2	0.567	0.488	0.404	0.371	0.374	0.422
3p63d 2D5/2 $\rightarrow$ 3p53d2(3F)2F7/2	6.35	6.36	6.55	7.05	8.06	10.2
3p63d 2D5/2 $\rightarrow$ 3p53d2(3P)2P1/2	0.055	0.039	0.026	0.021	0.017	0.014
3p63d 2D5/2 $\rightarrow$ 3p53d2(3P)2P3/2	3.71	3.78	3.97	4.34	5.03	6.42
3p63d 2D5/2 $\rightarrow$ 3p53d2(3F)2D5/2	7.19	7.39	7.82	8.61	10.1	13.0
3p63d 2D5/2 $\rightarrow$ 3p53d2(3F)2D3/2	0.525	0.538	0.565	0.619	0.718	0.918
3p63d 2D5/2 $\rightarrow$ 3p64f 2F5/2	0.202	0.193	0.175	0.168	0.171	0.194
3p63d 2D5/2 $\rightarrow$ 3p64f 2F7/2	1.34	1.36	1.44	1.64	2.01	2.78