2 The Fe$^{11+} - {\rm e}^-$ scattering problem

2.1 19 term R-matrix computation

The R-matrix technique, implemented in the Queen's University Belfast R-matrix suite of programs (Berrington et al. 1995), has been used for the computation of the low partial wave contributions to the total collision strengths $\Omega\left(i\to j\right)\,=\,\sum_l \Omega_l\left(i\to j\right)$, $\Omega_l\left(i\to j\right)$ being the partial collision strength for transition $i\to\,j$, with l the orbital angular momentum of the scattered electron. For the calculation of collision strengths for electric dipole transitions within the R-matrix approach we included contributions coming from partial waves as high as $l_{\rm max}\,=\,16$,such that  

$$\Omega^{R-m}\,=\,\sum_{l=0}^{16}\,\Omega_l^{R-m}.$$ (1)

This will be shown to be a proper choice in order to ensure convergence between the low l R-matrix results and the high l Coulomb-Bethe (CBe) ones (see Sect.2.2). The expansion of the total wavefunction for the Fe¹¹⁺ target included the lowest 19 LS coupling terms 3s²3p^{3 4}S$^{\rm o}$, ²D$^{\rm o}$, ²P$^{\rm o}$, 3s3p^{4 4}P, ²D, ²P, ²S, 3s²3p²3d (³P)⁴F, (³P)⁴D, (¹D)²F, (¹D)²G, (³P)²P, (³P)⁴P, (¹S)²D (¹D)²D, (¹D)²S, (¹D)²P, (³P)²F, (³P)²D. In Table 1 we list all the fine-structure levels belonging to the 3s²3p³, 3s3p⁴, 3s²3p²3d configurations along with the relative observed energy values, taken from Corliss $\&$ Sugar (1982) and Jupen et al. (1993). Italic type in the column listing the observed energies indicates that experimental values are not yet available for those levels. In these cases theoretical energies computed with the atomic structure program SUPERSTRUCTURE (Eissner et al. 1974; Nussbaumer $\&$ Storey 1978) were empirically corrected with the procedure described in BMS. The theoretical energies listed in Table 1 have been obtained with a 24 configuration atomic model including those configurations with one electron in 4s, 4p, 4d and 4f orbitals of spectroscopic type (set3 in BMS). The energies for the 19 target LS coupling terms listed above were obtained as weighted averages over the fine-structure observed energies given in Table 1 and were listed in Table 5 of BMS. The 19 target terms were represented by configuration interaction (CI) expansions constructed with a common set of radial waves, describing the radial charge distribution of the 1s, 2s, 2p, 3s, 3p and 3d bound electrons. These radial functions were obtained with SUPERSTRUCTURE adopting the 12 configuration basis described in BMS. A total of 16 continuum orbitals was included in the calculation and the R-matrix "box'' boundary was set at 3.09 a.u. For the expansion of the $(N+1)-{\rm e}$ collision complex wavefunction we combined in all possible fashions one scattering electron having spin s=1/2 and $l\,=\,0,\dots ,16$with the LS coupling target terms listed above. In such a way we obtained a total of 121 intermediate states specified in Table 2. The calculation was performed with a very fine energy mesh of 1.59375 10^-3Ry in the closed channel region (2.55 to 5.5Ry), in order to delineate the resonance structure of the collision strength below the highest excitation threshold. In the open channel region (5.5 to 100Ry), on the contrary, a much coarser energy mesh was chosen, namely 0.5Ry between 5.5Ry and 20Ry, 1Ry between 20Ry and 50Ry, and 5Ry between 50Ry and 100Ry. This proved to be sufficient to describe the variation of the collision strengths as a function of energy.

  

Table 1: Energy levels (cm^-1) for the lowest three configurations in FeXII. For values in italic see text

  

Table 2: Intermediate states ($S\,L\,\pi$) included in the expansion of the $(N+1)-{\rm e}$ collision complex wavefunction

2.2 Top-up in l

A top-up procedure to account for the missing partial wave contributions to $\Omega$ coming from $l\,\gt\,16$ was necessary in the open channel region in order to provide final reliable results. The application of geometric series properties to the partial collision strengths, discussed in BMS, proved to be inappropriate to the case of strong electric dipole transitions. For these transitions we estimated the contributions from partial waves with $l=17\to \infty$ with the Coulomb-Bethe (CBe) approximation (van Regemorter 1960; Burgess et al. 1970). The CBe method implies the "long range'' approximation, where only the long range potential of the target is effective in the interaction with the scattering electron. As stated in van Regemorter (1960) the "long range'' approximation is only valid for sufficiently high values of l, i.e. "distant'' encounters between the target and the colliding electron. Burgess $\&$ Tully (1978) define "distant'' as those partial waves with  

$$l\ge \left(k^2\,r_0^2\,+\,2\,z\,r_0\right)^{1/2}\,-\,\frac{1}{2}$$ (2)

where z is the ion net charge, r₀ the radius of the spherical region containing the target charge distribution and k² the scattering electron energy in Ry. This formula gives a convenient criterion to estimate the lowest "safe'' partial wave contribution to the total collision strength which can be computed with the CBe technique, given a specific target model and an energy range where collision strengths are required. Our choice of $l\,\gt\,16$ for the CBe partial wave contributions satisfied Eq. (2) and the convergence of R-matrix and Coulomb-Bethe results for the strongest electric dipole transitions is shown in Table 3. The convergence of the two methods at $E\,=\,100$Ry, for $l\,=\,16$, is in all cases better than $1\%$. After topping-up the R-matrix collision strengths with the CBe contributions for $l\,\gt\,16$, great care has been taken in checking the correct convergence to the high energy Bethe limits of the topped-up reduced collision strengths, $\Omega_{\rm r}$, as a function of reduced energy, $E_{\rm r}$, as defined by Burgess $\&$ Tully (1992). Figures 2 to 5 show a few examples of plots of $\Omega_{\rm r}$ vs. $E_{\rm r}$, in the open channel energy region, for selected electric dipole transitions.

  

Figure 2: Reduced collision strength vs. reduced energy for the 3s23p32P$^{\rm o}_{1/2}$ $\to$ 3s3p42S1/2 optically allowed transition. The asterisk shows the high energy Bethe limit

  

Figure 3: Reduced collision strength vs. reduced energy for the 3s23p32P$^{\rm o}_{3/2}$ $\to$ 3s23p23d(3P)2D5/2 optically allowed transition. The asterisk shows the high energy Bethe limit

  

Figure 4: Reduced collision strength vs. reduced energy for the 3s23p34S$^{\rm o}_{3/2}$ $\to$ 3s3p42P1/2 intercombination transition. The asterisk shows the high energy Bethe limit

  

Figure 5: Reduced collision strength vs. reduced energy for the 3s23p32D$^{\rm o}_{3/2}$ $\to$ 3s23p23d(3P)4F3/2 intercombination transition. The asterisk shows the high energy Bethe limit

  

Table 3: Partial R-matrix (Rm) and Coulomb-Bethe (CBe) collision strengths $\Omega_l$ for $l\,=\,14,\,15,\,16$ at $E\,=\,100$Ry

2.3 Top-up in energy

The assumption of constant $\Omega$ at $E\,\gt\,100\,{\rm Ry}$ in integrating the collision strengths over a Maxwellian distribution of electron energies, well justified in BMS for the forbidden transitions, is not appropriate to the strong electric dipole transitions we are treating here. In fact Burgess $\&$ Tully (1992) classify these transitions as type1 and give a high energy limiting behaviour of the kind  

$$\Omega\,\sim\,{\rm const}\,\cdot\,\ln\,(E)$$ (3)

deduced from the Bethe approximation. By using the scaling rules in Eqs.(5) and (6) of Burgess $\&$ Tully (1992) and Eq.(7) of Burgess $\&$ Tully (1978), it can be shown that, in the limit of high E, $\Omega_{\rm r}$ is a linear function of $E_{\rm r}$. We therefore scaled our collision strengths as suggested in Burgess $\&$ Tully (1992) for type1 transitions and performed linear interpolation of $\Omega_{\rm r}$between $\Omega_{\rm r}\,(100\,{\rm Ry})$ and $\Omega_{\rm r}\,(\infty)$, given by the Bethe limit $4\,\omega_i\,f_{ij}/E_{ij}$, where $\omega_i$ is the statistical weight of the initial level, f_ij the oscillator strength and E_ij the excitation energy of the transition $i\to\,j$. By applying the inverse transformation to these interpolated values we obtained a set of $\Omega$ values between 100Ry and 9900Ry, with an energy mesh of 100Ry, which takes proper account of the logarithmic increase of $\Omega$ with E. The integration of the extended set of collision strengths is discussed in the following section.

2.4 Results

Topped-up final collision strengths $\Omega\left(i\to\,j\right)$ for all electric dipole fine-structure transitions between the ground 3s²3p³ and the first excited 3s3p⁴ configurations are compared with previous calculations in Table 4 at two energy values in the open channel region. Table 5 shows a similar comparison for all electric dipole transitions between the ground 3s²3p³ and the second excited 3s²3p²3d configurations. The complete set of collision strength values at all electron energies considered in our scattering calculation (see Sect.2.1), in the closed as well as open channel energy region, can be obtained in electronic format via anonymous ftp from the IRON Project data bank at iron.am.qub.ac.uk. Effective, or thermally averaged collision strengths, given by  

$$\Upsilon\left(i\to j\right)=\int_{0}^\infty \Omega\left(i\to... ..._{\rm e}}\right) {\rm d}\left(\frac{E_j}{kT_{\rm e}}\right)\\$$ (4)

where E_j is the colliding electron kinetic energy after excitation, have been obtained by integrating the collision strengths using the linear interpolation technique described in Burgess $\&$ Tully (1992). The extended set of original plus interpolated $\Omega\left(i\to\,j\right)$ values was used in the integration process and it was assumed $\Omega\left(9900\,{\rm Ry}\,<\,E\,<\,\infty\right)\,=\,\Omega\left(9900\,{\rm Ry}\right)$.The exponential factor in Eq. (4) drops off rapidly with increasing energy so that the contribution to $\Upsilon\left(i\to\,j\right)$ coming from the energy region $E_{\rm min}\,<\,E\,<\,\infty$, where $\Omega$ is taken constant, decreases rapidly with increasing $E_{\rm min}$. For example, for $T_{\rm e}\,=\,10^7$ K, this contribution was as high as $25\%$ for $E_{\rm min}\,=\,100\,{\rm Ry}$, for most electric dipole transitions we studied, dropping down to a typical $10^{-65}\%$ (!) when the extended set of $\Omega$'s is integrated ($E_{\rm min}\,=\,9900\,{\rm Ry}$). Effective collision strengths for all electric dipole 3s²3p³$\to$3s3p⁴ fine-structure transitions are given in Tables 6 and  7 for two different temperature ranges. Similar data are presented in Tables 8 and 9 for all electric dipole 3s²3p³$\to$3s²3p²3d fine-structure transitions.

  

Table 4: Collision strengths for all electric dipole fine-structure transitions between the ground 3s²3p³ and the first excited 3s3p⁴ configurations in FeXII. ^a see text

  

Table 5: Collision strengths for all electric dipole fine-structure transitions between the ground 3s²3p³ and the second excited 3s²3p²3d configurations in FeXII. ^a and ^b see text

  

Table 6: Effective collision strengths for all electric dipole fine-structure transitions between the ground 3s²3p³ and the first excited 3s3p⁴ configurations in FeXII. Temperatures in the range 410⁵ K - 310⁶ K. ^a, ^b and ^c see text

  

Table 7: Effective collision strengths for all electric dipole fine-structure transitions between the ground 3s²3p³ and the first excited 3s3p⁴ configurations in FeXII. Temperatures in the range 410⁶K - 10⁷K