2 Method

In this work physical processes are described by the Hamiltonian  

$$H^{\rm BP}=H^{\rm nr}+H^{\rm mass}+H^{\rm Dar}+H^{\rm so} ,$$ (1)

where Breit-Pauli (BP) contributions of order $\alpha\sp2Z\sp4$Ry are added to the non-relativistic Hamiltonian $H\sp{\rm nr}$; namely the mass-velocity, one-body Darwin and ordinary spin-orbit terms. The two-body components of relative BP-order 1/Z are neglected, which is adequate in our case. Each term of (1) involves N target electrons and a colliding electron, and $H\sp{\rm nr}$ includes a central potential due to an atomic nucleus with charge number Z. Thus expression (1) can be expanded in terms of states of total angular momentum and parity $J\pi$when taking the entire Hamiltonian, and of states of total spin, orbital angular momentum and parity $SL\pi$ when the contribution $H^{\rm so}$ is neglected. We examine results obtained in either of these coupling schemes; they are connected by vector-coupling and term-coupling coefficients (TCC) and the transformation can be carried out at different stages of the calculation. This in turn affects computer time and accuracy.

2.1 R-matrix techniques

When collision-type trial functions (Burke & Seaton 1971) are used, the wavefunction for the total system of target and colliding electron is given by the expansion

$$\Psi^{J\pi}={\cal A}\sum_i\chi_i{F_i(r)\over r} + \sum_jc_j\Phi_j\ ,$$ (2)

where ${\cal A}$ is the antisymmetrization operator, $\chi_i$ are vector coupled products of the target eigenfunctions and the angular part of the incident-electron functions, and the F_i(r) are the radial parts of the latter. The functions $\Phi_j$ are bound-type functions of the total system introduced to compensate for orthogonality conditions imposed on the F_i(r) and to render short-range correlations. The Kohn variational principle yields coupled integro-differential (ID) equations for the radial functions F_i(r) for each angular symmetry.

R-matrix techniques (Burke et al. 1971; Berrington et al. 1974, 1978) are employed to solve the ID equations inside the R-matrix box (for $r\leq a$, say).

2.2 Relativistic effects

One way to take relativistic effects into account is to employ the BP extensions which were first introduced in the R-matrix method by Scott & Burke (1980) and Scott & Taylor (1982). The target Hamiltonian is diagonalized in intermediate coupling. We make use of the resulting TCCs when recoupling the prior systems with respective $SL\pi$ symmetries to $J\pi$ symmetries, and we add the spin-orbit contribution from $H\sp{\rm so}$ due to the colliding electron.

To match to physical asymptotic boundary conditions a pair coupling scheme is adopted where the target angular momentum j is added to the orbital angular momentum l and spin s of the incident electron through an intermediate quantum number K (see, for example, Saraph 1972):  

$$\mbox{\boldmath$j$}+\mbox{\boldmath$l$}=\mbox{{\boldmath$K$}... ...d\mbox{\boldmath$K$}+\mbox{\boldmath$s$}=\mbox{\boldmath$J$}\,.$$ (3)

Since spin-orbit coupling is negligible when $r\geq a$,no kinetic effects enter between the auxiliary states K and total angular momentum J. We make use of this fact when deriving expressions for contributions from high angular momenta.

The other recipe, computationally cheaper, is to first solve the coupled ID equations in the collisional $SL\pi$ symmetry. Then, following Saraph (1978), collision strengths between fine-structure levels are obtained by algebraic recoupling of the scattering matrices from LS to pair coupling. The method allows for term mixing via TCCs. So the relativistic effect of term mixing through spin-orbit interaction is obtained. However channel energies are degenerate for fine-structure levels associated with the same term. This approach will be referred to as the TCC method.

2.3 Long range potentials

The asymptotic region ($r\geq a$) is treated by the perturbative method described by Seaton (1985) and Berrington et al. (1987). In this region the ID equations reduce to the ordinary coupled differential equations

$$\begin{eqnarray} \lefteqn{\left({{\rm d}^2\over {\rm d}r^2}-{l_i(l_i+1)\over r^2... ...ber}\hspace{4cm} \\ & & \mbox{} -\sum_{i'}V_{ii'}(r)F_{i'}(r)=0\,,\end{eqnarray}$$ (4)

where z=Z-N is the residual charge of the ion, l_i and $\epsilon_i$ are the channel orbital angular momentum and energy of the colliding electron (in Rydberg units) and the quantities V_ii' are long-range multipole potentials. Since asymptotically |V_ii'(r)| $\ll 2z/r$, these potentials can be treated by perturbation techniques. The asymptotic codes devised for the IP work allow for two options. Option 1: the potentials V_ii' are included to first order, and Option 0: the potentials V_ii' are neglected. Since computations with Option 0 are considerably less time consuming, we investigate its range of validity.

2.4 Contributions from high angular momenta

The high-l contribution to the collision strength (top-up) for optically allowed (oa) transitions is computed for $J\gt J^{\rm oa}_{\max}$ with a procedure based on the Coulomb-Bethe approximation (Burgess 1974) and formulated by Burke & Seaton (1986) in the context of the R-matrix coding.

$$\begin{eqnarray} \sum_{l=\lambda+1}\sp\infty\Omega_{l,l-1}+\Omega_{l-1,l}=\biggl... ...\lambda-1,\lambda}\biggr]\bigg/ \bigl[\epsilon_1-\epsilon_2\bigr]\end{eqnarray}$$ (5)

where $\Omega_{l,l'} = \Omega(i_1l,i_2l')$.The formulation requires just two partial collision strengths for each dipole allowed transition (i₁,i₂) at some $\lambda\approx L\sp{\max}$ (n.b. i stands for ${\mit\Gamma}_iS_iL_i$and $S_2\!=\!S_1$ for these transitions). This procedure has been extended to intermediate coupling and details will be published elsewhere. For non-allowed (na) transitions we approximate the top-up for $J\gt J^{\rm na}_{\max}$ with a geometric series. Here $J^{\rm na}_{\max}$ is the highest J-value for which the ID equations are solved.

In order to assess the convergence of the partial wave expansion the resulting collision strengths are analyzed by means of the scaling techniques developed by Burgess & Tully (1992). The collision strength $\Omega(E)$ is mapped onto the reduced form $\Omega_{\rm r}(E_{\rm r})$, where the infinite energy range is scaled to the finite interval (0,1). For an allowed transition the scaled parameters $E_{\rm r}$ and $\Omega_{ \rm r}$ are the dimensionless quantities

$$\begin{eqnarray} E_{\rm r}& = & 1-\frac{\ln(c)}{\ln(\frac{E}{\Delta E}+c)}\\ \Om... ...E_{\rm r}) & = &\frac{\Omega(E)}{\ln(\frac{E}{\Delta E}+{\rm e})}\end{eqnarray}$$ (6) (7)

with $\Delta E$ being the transition energy, E the electron energy with respect to the reaction threshold and c is an adjustable scaling parameter. A key aspect of the approach by Burgess & Tully lies in the fact that the low as well as the high energy limits $\Omega_{\rm r}(0)$ and $\Omega_{\rm r}(1)$ are both finite and can be computed. For an electric dipole transition these limits are

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

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

$$\Upsilon(T)=\int_0\sp\infty\Omega(E){\rm\ e}\sp{-E/(\kappa T)} {\rm d}(E/\kappa T)$$ (10)

through the analogous relations

$$\begin{eqnarray} T_{\rm r}&=&1-\frac{\ln(c)}{\ln(\frac{\kappa T}{\Delta E}+c)}\\... ...&=&\frac{\Upsilon(T)} {\ln(\frac{\kappa T}{\Delta E}+{\rm e})}\,,\end{eqnarray}$$ (11) (12)

where T is the electron temperature and $\kappa$ the Boltzmann constant; the limits are

$$\begin{eqnarray} \Upsilon_{\rm r}(0)&=&\Omega(0) \\ \Upsilon_{\rm r}(1)&=& \Omega_{\rm r}(1)\ .\end{eqnarray}$$ (13) (14)

For a forbidden transition similar scaling relations are introduced by

$$\begin{eqnarray} E_{\rm r}&=&\frac{\frac{E}{\Delta E}}{\frac{E}{\Delta E}+c}\\ \... ...kappa T}{\Delta E}+c}\\ \Upsilon_{\rm r}(T_{\rm r}) &=&\Upsilon(E)\end{eqnarray}$$ (15) (16) (17) (18)

with the following limits

$$\begin{eqnarray} \Omega_{\rm r}(0)&=&\Omega(0) \\ \Omega_{\rm r}(1)&=&\Omega_{\r... ...on_{\rm r}(0)&=&\Omega(0) \\ \Upsilon_{\rm r}(1)&=&\Omega_{\rm CB}\end{eqnarray}$$ (19) (20) (21) (22)

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