2 Method

The precise approach adopted to compute effective collision strengths for the present system is a legacy of the experience gained in our previous treatment of the excitation by electron impact of Na-like Fe XVI (Eissner et al. 1999). Following the common policy of the IRON Project, the method is based on the close-coupling approximation of Burke & Seaton (1971) as implemented in the R-matrix package of Burke et al. (1971) and Berrington et al. (1974, 1978), with developments in the asymptotic region by Seaton (1985) and Berrington et al. (1987). In this method the wavefunction of the target-electron system is expanded in terms of the target eigenfunctions which are discussed in Sect. 3. Previous work by Dufton et al. (1990) follows a similar method but with a reduced target representation (8-state). The calculations by CNP, BMB and BM have been carried out in the distorted wave approximation that neglects channel coupling.

Collision strengths for the fine-structure levels are calculated including relativistic effects by means of a Breit-Pauli version of the R-matrix codes (Scott & Burke 1980; Scott & Taylor 1982) within an intermediate coupling scheme that leads to intermediate states of the total system with angular momentum and parity quantum numbers $J\pi$.In the work by CNP, BMB, BM and Dufton et al. (1990), collision strengths for the fine-structure levels are computed by an algebraic recoupling of the LS reactance matrices, including relativistic effects in the target by means of term-coupling coefficients (Hummer et al. 1993). This approach, although computationally less involved, is not expected to perform as reliably as a full Breit-Pauli approximation (Eissner et al. 1999).

Due to the important contributions to the collision strengths of optically allowed transitions from the long-range coupling of non-coulombic potentials in the asymptotic region (see Eissner et al. 1999), the present computations are performed taking into account this interaction throughout the partial wave expansion. The occasional appearance of artificially high resonances caused by numerical instabilities is managed by comparing with calculations in the resonance region that exclude this effect, and trimming down any resonance that differs by a factor larger than 5.

The high-l top-up of the collisional strength for optically allowed transitions is computed for $J\gt J^{\rm oa}_{\rm max}$ with a procedure based on the Coulomb-Bethe approximation (Burgess 1974) as discussed within the context of the close-coupling approximation by Burke & Seaton (1986). The intermediate coupling implementation of this top-up procedure in the R-matrix code was developed by one of us and tested in the study of Fe XVI. The corresponding top-up for non-allowed transitions is approximated for $J\gt J^{\rm na}_{\rm max}$ with a geometric series sum. Similarly to our earlier work on the Na-like ion, it has also been found that for this system the Coulomb-Born regime in allowed transitions and in some quadrupole transitions is only reached at the high energies (E> 100 Ryd, say) when l is very high. It is therefore necessary to take into account an extended partial wave range in the R-matrix calculation in order to ensure a reasonable degree of reliability in the top-up procedures for all the slow converging transitions. In the present work we have settled for $J^{\rm oa}_{\rm max}=J^{\rm na}_{\rm max}=40.5$. This model contrasts with those adopted by CNP, BMB and BM where a dipole top-up was introduced at l>11 and with that by Dufton et al. (1990) at l>8. Moreover, in these previous calculations the quadrupole top-up has been generally neglected, except by BMB and BM for transitions involving the ground level.

As shown by Eissner et al. (1999) the energy-mesh step size in the resonance region must be carefully considered due to the complicated structure of narrow features. After some experimentation a step of $\delta e/z^2=10^{-5}$ Ryd was selected where z=14 is the effective charge of the system. In the region of all channels open above the highest excitation threshold a broader step of $\delta E/z^2=10^{-2}$ Ryd was regarded as adequate. The mesh adopted in previous work was generally coarse, since the resonance contribution to the rates has been assumed to be negligible in the temperature range of interest.

Collision strengths and effective collision strengths are analysed with the scaling techniques developed by Burgess & Tully (1992), in particular the convergence of the partial wave expansion. The collision strength $\Omega(E)$ is mapped onto the reduced form $\Omega_{\rm r}(E_{\rm r})$, where the infinite energy E range is scaled to the finite $E_{\rm r}$ interval (0,1). For an allowed transition the scaling is given by the relations

$$\begin{eqnarray} E_{\rm r} &=& 1-\frac{\ln(c)}{\ln(E/\Delta E+c)}\\ \Omega_{\rm r}(E_{\rm r}) & =& \frac{\Omega(E)}{\ln(E/\Delta E+{\rm e})}\end{eqnarray}$$ (1) (2)

with $\Delta E$ being the transition energy, E the electron energy with respect to the reaction threshold and c is an adjustable scaling parameter. For an electric dipole transition the important limit points are

$$\begin{eqnarray} \Omega_{\rm r}(0) &=& \Omega(0) \\ \Omega_{\rm r}(1) &=& \frac{4gf}{\Delta E}\end{eqnarray}$$ (3) (4)

where gf is the weighted oscillator strength for the transition. This method can also be extended to treat the effective collision strength

$$\Upsilon(T)=\int_0\sp\infty\Omega(E) \exp(-E/\kappa T){\rm d}(E/\kappa T)$$ (5)

through the analogous relations

$$\begin{eqnarray} T_{\rm r} &=& 1-\frac{\ln(c)}{\ln(\kappa T/\Delta E+c)}\\ \Upsi... ...(T_{\rm r}) & =&\frac{\Upsilon(T)}{\ln(\kappa T/\Delta E+{\rm e})}\end{eqnarray}$$ (6) (7)

where T is the electron temperature and $\kappa$ the Boltzmann constant; the limit points remain

$$\begin{eqnarray} \Upsilon_{\rm r}(0) &=& \Omega(0)\\ \Upsilon_{\rm r}(1) &=& \frac{4gf}{\Delta E} \ .\end{eqnarray}$$ (8) (9)

Similarly, for a forbidden transition the scaling relations are given by

$$\begin{eqnarray} E_{\rm r} &=& \frac{E/\Delta E}{E/\Delta E+c}\\ \Omega_{\rm r}... ...kappa T/\Delta E+c}\\ \Upsilon_{\rm r}(T_{\rm r}) &=& \Upsilon(E)\end{eqnarray}$$ (10) (11) (12) (13)

with the following limit points:

$$\begin{eqnarray} \Omega_{\rm r}(0) &=& \Omega(0)\\ \Omega_{\rm r}(1) &=& \Omega_... ...{\rm r}(0) &=& \Omega(0)\\ \Upsilon_{\rm r}(1) &=& \Omega_{\rm CB}\end{eqnarray}$$ (14) (15) (16) (17)

where $\Omega_{\rm CB}$ is the Coulomb-Born high-energy limit.

Caution is required in the use of Eq. (5) to calculate effective collision strengths when $\Omega(E)$ varies rapidly due to the presence of resonances. To ensure the proper behaviour at low temperatures, the integration technique presented by Burgess & Tully (1992) was adopted.