1. Lecture 4: Coupled Channel Approximation and the R-Matrix Codes Recall: To solve the (e+ion) problem we compute ion wavefunctions first, independently using Superstructure or similar atomic structure code The coupled-channel (CC) approximation couples the free electron and ion wavefunctions R-Matrix method is the most efficient for most atomic processes in plasmas

2. “Stages” of the R-Matrix Codes Superstructure (SS)  one-electron orbitals (sspnl), optmized over target ion wavefunctions One and two-electron Slater integrals SS (sspnl)  STG1 Angular algebra: STG2 reconstructs the target ion and couplings for the (e+ion) system STG1  STG2 (non-relativistic LS coupling) STG2  RECUPD: Intermediate re-coupling LS  LSJ (Breit-Pauli approximation) RECUPD  STGH: (e+ion) Hamiltonian Diagonalization

3. Flow Chart: Sets of R-Matrix Codes Non-relativistic R-Matrix and relativistic Breit- Pauli R-Matrix (BPRM): LS and LSJ coupling R-Matrix II codes: “Complete” (e+ion) angular treatment; large number of levels Dirac R-Matrix Codes (DARC): Use GRASP for target ion wavefunctions for high-Z systems Fig. 3.9  Flow chart

4. “Asymptotic” R-Matrix Codes Following (e+ion) hamiltonian diagonalization, STGH produces an H.DAT file which is utilized by subsidiary codes to calculate:  electron-ion cross sections (STGF)  (e+ion) bound state energy levels (STGB)  bound-bound transition probabities (STGBB)  bound-free (photoionization) cross sections (STGBF)

5. Astrophysical Quantities Absorption oscillator strengths and photoionization cross sections  Opacities Line emissivities  Emission Line Diagnostics All atomic parameters  Non-LTE radiative transfer models