3 Generating ${\sl \Omega}(E,Z)$ or ${\sl \Upsilon}(T,Z)$ from the S_mn matrices

Since we have type 2 transitions (instead of type 1) and include relativistic corrections, Eqs. (62) to (65) of B&T become

$${\sl \Upsilon} = (Z+1)^{-2} \, (1 + \epsilon E_{ij} + \mu T (2 + \epsilon E_{ij} + 2 \mu T ) ) \; {\rm spline}$$

$(P_1,P_2,P_3,P_4,P_5,T_{\rm r}) \, ,$(62^*)

where

$$P_n = {\rm spline}(S_{1n},S_{2n},S_{3n},S_{4n},S_{5n},Z_{\rm r}) \; \; (1 \leq n \leq 5) \, , \eqno(\rm 63^*)$$

$$T_{\rm r} = {{kT/E_{ij} } \over {kT/E_{ij} + C_T}} \, , \eqno(\rm 64^*)$$

$$Z_{\rm r} = {Z \over {Z + C_Z}} \, . \eqno(\rm 65^*)$$

Correspondingly, for ${\sl \Omega}$, we have

$${\sl \Omega} = (Z+1)^{-2} \, ( 1 + \epsilon \, E_i)( 1 + \epsilon \, E_j) \; {\rm spline}$$

$(P_1,P_2,P_3,P_4,P_5,E_{\rm r}) \, ,$(62^*')

where

$$E_{\rm r} = {{E_j/E_{ij} } \over {E_j/E_{ij} + C_E}}. \eqno(\rm 64^{*'})$$