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2 Theory

The theoretical scheme is described in earlier works (Hummer et al. 1993; Nahar & Pradhan 1999, 2000). We sketch the basic points below. In the coupled channel or close coupling (CC) approximation an atom (ion) is described in terms of an (e + ion) complex that comprises of a "target'' ion, with N bound electrons, and a "free'' electron that may be either bound or continuum. The total energy of the system is either negative or positive; negative eigenvalues of the (N + 1)-electron Hamiltonian correspond to bound states of the (e + ion) system. In the CC approximation the wavefunction expansion, $\Psi(E)$, for a total spin and angular symmetry $SL\pi $ or $J\pi $, of the (N+1) electron system is represented in terms of the target ion states or levels as:

\begin{displaymath}\Psi_{\rm E}({\rm e+ion}) = A \sum_{i} \chi_{i}({\rm ion})\theta_{i} + \sum_{j} c_{j}
\end{displaymath} (1)

where $\chi_{i}$ is the target ion wave function in a specific state $S_iL_i\pi_i$ or level $J_i\pi_i$, and $\theta_{i}$ is the wave function for the (N+1)th electron in a channel labeled as $S_iL_i(J_i)\pi_i \ k_{i}^{2}\ell_i(SL\pi) \ [J\pi]$; ki2 is the incident kinetic energy. In the second sum the $\Phi_j$'s are correlation wavefunctions of the (N+1) electron system that (a) compensate for the orthogonality conditions between the continuum and the bound orbitals, and (b) represent additional short-range correlation that is often of crucial importance in scattering and radiative CC calculations for each $SL\pi $.

The BPRM method yields the solutions of the relativistic CC equations using the Breit-Pauli Hamiltonian for the (N+1)-electron system to obtain the total wavefunctions $\Psi_{\rm E}({\rm e+ion})$ (Hummer et al.1993). The BP Hamiltonian is

\begin{displaymath}H_{N+1}^{\rm BP}=H_{N+1}+H_{N+1}^{\rm mass} + H_{N+1}^{\rm Dar}
+ H_{N+1}^{\rm so},
\end{displaymath} (2)

where HN+1 is the nonrelativistic Hamiltonian,

\begin{displaymath}H_{N+1} = \sum_{i=1}\sp{N+1}\left\{-\nabla_i\sp 2 - \frac{2Z}{r_i}
+ \sum_{j>i}\sp{N+1} \frac{2}{r_{ij}}\right\},
\end{displaymath} (3)

and the additional terms are the one-body terms, the mass correction, the Darwin and the spin-orbit terms respectively. The spin-orbit interaction splits the LS terms into fine-structure levels $J\pi $, where J is the total angular momentum. The positive and negative energy states (Eq. 1) define continuum or bound (e + ion) states,

\begin{displaymath}\begin{array}{l} E = k^2 > 0 \longrightarrow
{\rm continuum~(...
...{z^2}{\nu^2} < 0
\longrightarrow {\rm bound~state}, \end{array}\end{displaymath} (4)

where $\nu$ is the effective quantum number relative to the core level. Determination of the quantum defect ( $\mu(\ell))$, defined as $\nu_i = n - \mu(\ell)$ where $\nu_i$ is relative to the core level $S_iL_i\pi_i$, is helpful in establishing the $\ell$-value associated with a given channel level.

The $\Psi_{\rm E}$ represents a CI-type wavefunction over a large number of electronic configurations depending on the target levels included in the eigenfunction expansion (Eq. 1). Transition matrix elements may be calculated with these wavefunctions, and the electron dipole (E1), electric quadrupole (E2), magnetic dipole (M1) or other operators to obtain the corresponding transition probabilities. The present version of the BPRM codes implements the E1 operator to enable the calculation of dipole allowed and intercombination transition probabilities. The oscillator strength is proportional to the generalized line strength defined, in either length form or velocity form, by the equations

 \begin{displaymath}S_{\rm L}=
\left\vert\left\langle{\mit\Psi}_{\rm f}
...}^{N+1} r_j\vert
{\mit\Psi}_{\rm i}\right\rangle\right\vert^2
\end{displaymath} (5)


 \begin{displaymath}S_{\rm V}=\omega^{-2}
\left\vert\left\langle{\mit\Psi}_{\rm ...
...rtial r_j}\vert
{\mit\Psi}_{\rm i}\right\rangle\right\vert^2.
\end{displaymath} (6)

In these equations $\omega$ is the incident photon energy in Rydberg units, and $\mit\Psi_{\rm i}$ and $\mit\Psi_{\rm f}$ are the bound wave functions representing the initial and final states respectively. The line strengths are energy independent quantities.

Using the energy difference, Eji, between the initial and final states, the oscillator strength, fij, for the transition can be obtained from S as

\begin{displaymath}f_{ij} = {E_{ji}\over {3g_i}}S,
\end{displaymath} (7)

and the Einstein's A-coefficient, Aji, as

\begin{displaymath}A_{ji}({\rm a.u.}) = {1\over 2}\alpha^3{g_i\over g_j}E_{ji}^2f_{ij},
\end{displaymath} (8)

where $\alpha$ is the fine structure constant, and gi, gj are the statistical weight factors of the initial and final states, respectively. In cgs units,

\begin{displaymath}A_{ji}({\rm s}^{-1}) = {A_{ji}({\rm a.u.})\over \tau_0},
\end{displaymath} (9)

where $\tau_0 = 2.4191~10^{-17}$ s is the atomic unit of time.

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