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# Master equation theory applied to the redistribution of polarized radiation in the weak radiation field limit: V - The two-term atom (Corrigendum)

Véronique Bommier

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Véronique Bommier. Master equation theory applied to the redistribution of polarized radiation in the weak radiation field limit: V - The two-term atom (Corrigendum). Astronomy & Astrophysics - A&A, 2018, 619, pp.C1. 10.1051/0004-6361/201630169e. hal-02341544

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Submitted on 9 Aug 2025

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# Master equation theory applied to the redistribution of polarized radiation in the weak radiation field limit

## V. The two-term atom
(Corrigendum)

Véronique Bommier

LESIA, Observatoire de Paris, Université PSL, CNRS, Sorbonne Université, Univ. Paris Diderot, Sorbonne Paris Cité 5, Place Jules Janssen, 92190 Meudon, France
e-mail: v.bommier@obspm.fr

A&A 607, A50 (2017), https://doi.org/10.1051/0004-6361/201630169

**Key words.** atomic processes – line: formation – line: profiles – magnetic fields – polarization – errata, addenda

A sign factor was lacking in the expressions of the redistribution matrix in the case of incomplete Paschen-Back effect, Eqs. (A.1) and (A.6) of Bommier (2017). This sign factor is unity in the absence of incomplete Paschen-Back effect. A (2I + 1) denominator was also missing in Eqs. (A6) and (40), and some typos occurred in Eq. (A.6). The correct formulæ are provided below. The corrected Eq. (A.1) is:

$$ \begin{aligned} \mathcal{R}_{ij}(v, \nu_1, \Omega, \Omega_1; \mathbf{B}) = & \sum_{J_u \bar{J}_u J_u' \bar{J}_u' M_u J_u' \bar{J}_u' M_u' J_\ell \bar{J}_\ell' M_\ell' \bar{J}_\ell' J_\ell'' M_\ell'' \bar{J}_\ell'' M_\ell'' \mathbf{KK}' \mathbf{Q}} \int f(v) d^3v (-1)^Q \mathcal{T}_{-\mathbf{Q}}^{K'}(j, \Omega_1) \mathcal{T}_{\mathbf{Q}}^K(i, \Omega) \\ & \times 3 \frac{2L_u + 1}{2S + 1} \sqrt{(2K+1)(2K'+1)} (-1)^{M_\ell - M_\ell'} (-1)^{J_u + \bar{J}_u + J_u' + \bar{J}_u'} (-1)^{J_\ell + \bar{J}_\ell + J_\ell' + \bar{J}_\ell'} \\ & \times \sqrt{(2J_u + 1)(2\bar{J}_u + 1)(2J_u' + 1)(2\bar{J}_u' + 1)(2J_\ell + 1)(2\bar{J}_\ell + 1)(2J_\ell' + 1)(2\bar{J}_\ell' + 1)} \\ & \times C_{J_u' M_u}^{J_u} (B) C_{J_u' M_u}^{\bar{J}_u} (B) C_{J_u'' M_u'}^{J_u'} (B) C_{J_u'' M_u'}^{\bar{J}_u''} (B) C_{J_\ell' M_\ell}^{J_\ell} (B) C_{J_\ell' M_\ell}^{\bar{J}_\ell} (B) C_{J_\ell'' M_\ell''}^{J_\ell''} (B) C_{J_\ell'' M_\ell''}^{\bar{J}_\ell''} (B) \\ & \times \begin{Bmatrix} J_u & 1 & J_\ell \\ L_\ell & S & L_u \end{Bmatrix} \begin{Bmatrix} J_u' & 1 & \bar{J}_\ell \\ L_\ell & S & L_u \end{Bmatrix} \begin{Bmatrix} \bar{J}_u & 1 & J_\ell' \\ L_\ell & S & L_u \end{Bmatrix} \begin{Bmatrix} \bar{J}_u' & 1 & \bar{J}_\ell' \\ L_\ell & S & L_u \end{Bmatrix} \\ & \times \begin{pmatrix} J_u & 1 & J_\ell \\ -M_u & p & M_\ell \end{pmatrix} \begin{pmatrix} J_u' & 1 & \bar{J}_\ell \\ -M_u' & p' & M_\ell \end{pmatrix} \begin{pmatrix} \bar{J}_u & 1 & J_\ell' \\ -M_u & p''' & M_\ell'' \end{pmatrix} \begin{pmatrix} \bar{J}_u' & 1 & \bar{J}_\ell' \\ -M_u' & p'' & M_\ell'' \end{pmatrix} \\ & \times \begin{pmatrix} 1 & 1 & K' \\ -p & p' & Q \end{pmatrix} \begin{pmatrix} 1 & 1 & K \\ -p''' & p'' & Q \end{pmatrix} \\ & \times \left\{ \frac{\Gamma_R}{\Gamma_R + \Gamma_I + \Gamma_E + \frac{i\Delta E_{M_u M_u'}}{\hbar}} \delta(\tilde{v} - \tilde{v}_1 - \nu_{M_\ell M_\ell'}) \left[ \frac{1}{2}\Phi_{ba}(\nu_{M_u M_\ell} - \tilde{v}_1) + \frac{1}{2}\Phi_{ba}^*(\nu_{M_u M_\ell} - \tilde{v}_1) \right] \right. \\ & \left. + \left[ \frac{\Gamma_R}{\Gamma_R + \Gamma_I + \frac{i\Delta E_{M_u M_u'}}{\hbar}} - \frac{\Gamma_R}{\Gamma_R + \Gamma_I + \Gamma_E + \frac{i\Delta E_{M_u M_u'}}{\hbar}} \right] \right. \\ & \left. \times \left[ \frac{1}{2}\Phi_{ba}(\nu_{M_u M_\ell} - \tilde{v}_1) + \frac{1}{2}\Phi_{ba}^*(\nu_{M_u M_\ell} - \tilde{v}_1) \right] \left[ \frac{1}{2}\Phi_{ba}(\nu_{M_u M_\ell} - \tilde{v}_1) + \frac{1}{2}\Phi_{ba}^*(\nu_{M_u M_\ell} - \tilde{v}_1) \right] \right\} \end{aligned} \quad (A.1) $$
The corrected Eq. (A.6) is:

$$
\begin{equation}
\begin{aligned}
\mathcal{R}_{ij}(v, v_1, \Omega, \Omega_1; \mathbf{B}) = & \sum_{J_u \bar{J}_u J'_u F_u \bar{F}_u F_u^* M_u J_u \bar{J}_u J_u^* F_u \bar{F}_u' F_u' M_u J_\ell J'_\ell F_\ell \bar{F}_\ell F_\ell^* M_\ell J'_\ell \bar{J}_\ell' \bar{F}_\ell' \bar{F}_\ell' F_\ell^* M_\ell' \bar{K}_\ell' \Omega} \\
& \times \int f(\mathbf{v}) d^3\mathbf{v} (-1)^Q \mathcal{T}_{-Q}^{K'}(j, \Omega_1) \mathcal{T}_Q^K(i, \Omega) \\
& \times 3 \frac{2L_u + 1}{(2I + 1)(2S + 1)} \sqrt{(2K + 1)(2K' + 1)} (-1)^{M_\ell - M_\ell'} \\
& \times (-1)^{J_u + \bar{J}_u + J_u' + \bar{J}_u} (-1)^{J_\ell + \bar{J}_\ell + J_\ell' + \bar{J}_\ell} (-1)^{F_u + \bar{F}_u + F_u' + \bar{F}_u'} (-1)^{F_\ell + \bar{F}_\ell + F_\ell' + \bar{F}_\ell'} \\
& \times \sqrt{(2J_u + 1)(2\bar{J}_u + 1)(2J_u' + 1)(2\bar{J}_u' + 1)(2J_\ell + 1)(2\bar{J}_\ell + 1)(2J_\ell' + 1)(2\bar{J}_\ell' + 1)} \\
& \times \sqrt{(2F_u + 1)(2\bar{F}_u + 1)(2F_u' + 1)(2\bar{F}_u' + 1)(2F_\ell + 1)(2\bar{F}_\ell + 1)(2F_\ell' + 1)(2\bar{F}_\ell' + 1)} \\
& \times C_{J_u \bar{J}_u M_u (F_u^* M_u)}^{J_u} (B) C_{J_u M_u (F_u^* M_u)}^{\bar{J}_u} (B) C_{J_u^* M_u' (F_u^* M_u')}^{J_u'} (B) C_{J_u M_u' (F_u^* M_u')}^{\bar{J}_u'} (B) \\
& \times C_{J_\ell M_\ell (F_\ell^* M_\ell)}^{J_\ell} (B) C_{J_\ell^* M_\ell' (F_\ell^* M_\ell')}^{J_\ell'} (B) C_{J_\ell^* M_\ell' (F_\ell^* M_\ell')}^{J_\ell'} (B) C_{J_\ell M_\ell' (F_\ell^* M_\ell')}^{\bar{J}_\ell'} (B) \\
& \times C_{F_u^* (J_u M_u) M_u}^{F_u} (B) C_{\bar{F}_u^* (J_u M_u) M_u}^{\bar{F}_u} (B) C_{F_u^{\prime\prime} (J_u' M_u') M_u'}^{F_u'} (B) C_{\bar{F}_u^{\prime\prime} (J_u' M_u') M_u'}^{\bar{F}_u'} (B) \\
& \times C_{F_\ell^* (J_\ell M_\ell) M_\ell}^{F_\ell} (B) C_{\bar{F}_\ell^* (J_\ell M_\ell) M_\ell}^{\bar{F}_\ell} (B) C_{F_\ell^{\prime\prime} (J_\ell' M_\ell') M_\ell'}^{F_\ell'} (B) C_{\bar{F}_\ell^{\prime\prime} (J_\ell' M_\ell') M_\ell'}^{\bar{F}_\ell'} (B) \\
& \times \left\{ \begin{array}{c} \left\{ \begin{array}{cc} J_u & J_\ell \\ L_\ell & S \end{array} \right\} \left\{ \begin{array}{cc} J_u' & J_\ell' \\ L_\ell & S \end{array} \right\} \left\{ \begin{array}{cc} \bar{J}_u & \bar{J}_\ell \\ L_\ell & S \end{array} \right\} \left\{ \begin{array}{cc} \bar{J}_u & \bar{J}_\ell' \\ L_\ell & S \end{array} \right\} \left\{ \begin{array}{cc} \bar{J}_u' & \bar{J}_\ell' \\ L_\ell & S \end{array} \right\} \\ \\ \left\{ \begin{array}{cc} F_u & F_\ell \\ J_\ell & I \end{array} \right\} \left\{ \begin{array}{cc} F_u' & \bar{F}_\ell \\ \bar{J}_\ell & I \end{array} \right\} \left\{ \begin{array}{cc} \bar{F}_u & 1 \\ J_\ell & I \end{array} \right\} \left\{ \begin{array}{cc} \bar{F}_u' & 1 \\ \bar{J}_\ell & I \end{array} \right\} \left\{ \begin{array}{cc} \bar{F}_u' & 1 \\ \bar{J}_\ell & I \end{array} \right\} \\ \\ \left( \begin{array}{ccc} F_u & 1 & F_\ell \\ -M_u & p & M_\ell \end{array} \right) \left( \begin{array}{ccc} F_u' & 1 & \bar{F}_\ell \\ -M_u' & p' & M_\ell \end{array} \right) \left( \begin{array}{ccc} \bar{F}_u & 1 & F_\ell' \\ -M_u & p'' & M_\ell' \end{array} \right) \left( \begin{array}{ccc} \bar{F}_u' & 1 & \bar{F}_\ell' \\ -M_u' & p'' & M_\ell' \end{array} \right) \\ \\ \left( \begin{array}{ccc} 1 & 1 & K' \\ -p & p' & Q \end{array} \right) \left( \begin{array}{ccc} 1 & 1 & K \\ -p'' & p'' & Q \end{array} \right) \\ \\ \times \left\{ \frac{\Gamma_R}{\Gamma_R + \Gamma_I + \Gamma_E + \frac{i\Delta E_{M_u M_u'}}{\hbar}} \delta(\tilde{v} - \tilde{v}_1 - \nu_{M_\ell M_\ell'}) \left[ \frac{1}{2}\Phi_{ba}(\nu_{M_u M_\ell} - \tilde{v}_1) + \frac{1}{2}\Phi_{ba}^*(\nu_{M_u M_\ell} - \tilde{v}_1) \right] \right. \\ \\ \left. + \left[ \frac{\Gamma_R}{\Gamma_R + \Gamma_I + \frac{i\Delta E_{M_u M_u'}}{\hbar}} - \frac{\Gamma_R}{\Gamma_R + \Gamma_I + \Gamma_E + \frac{i\Delta E_{M_u M_u'}}{\hbar}} \right] \times \left[ \frac{1}{2}\Phi_{ba}(\nu_{M_u M_\ell} - \tilde{v}_1) + \frac{1}{2}\Phi_{ba}^*(\nu_{M_u M_\ell} - \tilde{v}_1) \right] \left[ \frac{1}{2}\Phi_{ba}(\nu_{M_u M_\ell} - \tilde{v}) + \frac{1}{2}\Phi_{ba}^*(\nu_{M_u M_\ell} - \tilde{v}) \right] \right\}
\end{array} \right\}
\end{aligned}
\tag{A.6}
\end{equation}
$$

This equation is in excellent agreement as for the Racah algebra with Eq. (30) of Casini et al. (2014). The product of two coefficients $C_{J' M_J (F^* M)}^J$ (B) $C_{F^{\prime\prime} (J_M J_M) M}^F$ (B) is equal to the coefficient $C_\mu^{JF} (M)$ of Casini et al. (2014), because these coefficients all result from matrix diagonalization, performed in one step (FS + HFS) in Casini et al. (2014) and in two steps (FS and HFS) in our case. A similar coefficient is visible in Eq. (3.58) of Landi Degl'Innocenti & Landolfi (2004).
The following equation replaces Eq. (40) of Bommier (2017), by introducing the (2I + 1) denominator

$$
\begin{equation}
\begin{aligned}
\mathcal{R}_{ij}(v, v_1, \Omega, \Omega_1; \mathbf{B} = \mathbf{0}) = & \sum_{J_u F_u J'_u F'_u J_\ell F_\ell J'_\ell F'_\ell K Q} \int f(v) d^3 v (-1)^2 \mathcal{T}_{-Q}^K(j, \Omega_1) \mathcal{T}_Q^K(i, \Omega) \\
& \times 3 \frac{2L_u + 1}{(2I + 1)(2S + 1)} (2J_u + 1)(2J'_u + 1)(2J_\ell + 1)(2J'_\ell + 1)(2F_u + 1)(2F'_u + 1)(2F_\ell + 1)(2F'_\ell + 1) (-1)^{F_\ell - F'_\ell} \\
& \times \begin{Bmatrix} J_u & 1 & J_\ell \\ L_\ell & S & L_u \end{Bmatrix} \begin{Bmatrix} J'_u & 1 & J_\ell \\ L_\ell & S & L_u \end{Bmatrix} \begin{Bmatrix} J_u & 1 & J'_\ell \\ L_\ell & S & L_u \end{Bmatrix} \begin{Bmatrix} J'_u & 1 & J'_\ell \\ L_\ell & S & L_u \end{Bmatrix} \\
& \times \begin{Bmatrix} F_u & 1 & F_\ell \\ J_\ell & I & J_u \end{Bmatrix} \begin{Bmatrix} F'_u & 1 & F_\ell \\ J_\ell & I & J'_u \end{Bmatrix} \begin{Bmatrix} F_u & 1 & F'_\ell \\ J_\ell & I & J_u \end{Bmatrix} \begin{Bmatrix} F'_u & 1 & F'_\ell \\ J_\ell & I & J'_u \end{Bmatrix} \\
& \times \begin{Bmatrix} K & F_u & F'_u \\ F_\ell & 1 & 1 \end{Bmatrix} \begin{Bmatrix} K & F_u & F'_u \\ F'_\ell & 1 & 1 \end{Bmatrix} \\
& \times \left\{ \frac{\Gamma_R}{\Gamma_R + \Gamma_I + \Gamma_E + \frac{i\Delta E_{F_u F'_u}}{\hbar}} \delta(\tilde{v} - \tilde{v}_1 - v_{F_\ell F'_\ell}) \left[ \frac{1}{2} \Phi_{ba} (v_{F'_u F_\ell} - \tilde{v}_1) + \frac{1}{2} \Phi_{ba}^* (v_{F_u F_\ell} - \tilde{v}_1) \right] \right. \\
& \left. + \left| \frac{\Gamma_R}{\Gamma_R + \Gamma_I + \frac{1}{2}[D^{(K)}(\alpha_u F_u) + D^{(K)}(\alpha_u F'_u)] + \frac{i\Delta E_{F_u F'_u}}{\hbar}} - \frac{\Gamma_R}{\Gamma_R + \Gamma_I + \Gamma_E + \frac{i\Delta E_{F_u F'_u}}{\hbar}} \right| \right. \\
& \left. \times \left[ \frac{1}{2} \Phi_{ba} (v_{F'_u F_\ell} - \tilde{v}_1) + \frac{1}{2} \Phi_{ba}^* (v_{F_u F_\ell} - \tilde{v}_1) \right] \left[ \frac{1}{2} \Phi_{ba} (v_{F'_u F_\ell} - \tilde{v}_1) + \frac{1}{2} \Phi_{ba}^* (v_{F_u F_\ell} - \tilde{v}_1) \right] \right\}
\end{aligned}
\tag{40}
\end{equation}
$$

*Acknowledgements.* The author is very grateful to Ernest Alsina Ballester for having pointed out the errors. Ernest Alsina Ballester redid the calculations in the metalevels formalism (Landi Degl'Innocenti et al. 1997).

References

Bommier, V. 2017, A&A, 607, A50

Casini, R., Landi Degl'Innocenti, M., Manso Sainz, R., Landi Degl'Innocenti, E. & Landolfi, M. 2014, ApJ, 791, 94

Landi Degl'Innocenti, E., Landi Degl'Innocenti, M., & Landolfi, M. 1997, Proc. Forum THÉMIS, Science with THÉMIS, eds. N. Mein & S. Sahal-Bréchot (Paris: Obs. Paris-Meudon), 59

Landi Degl'Innocenti, E., & Landolfi, M. 2004, Astrophys. Space Sci. Lib., 307