© ESO, 2017

1. Introduction

The redistribution function in radiative transfer expresses the probability of appearance of an emitted or scattered photon of frequency ν and direction Ω in the presence of an incoming photon of frequency ν₁ and direction Ω₁. The incoming and the outgoing photons can be polarized. In this work, we follow the convention of describing polarized light through the Stokes vector (S₀,S₁,S₂,S₃) ≡ (I,Q,U,V), where I is the radiation intensity, Q and U are two linearly independent states of linear polarization, and V represents circular polarization.

The first expressions of the redistribution function in different cases were heuristically derived by Hummer (1962), without taking into account polarization. He distinguished frequency coherent scattering, denoted as case II with corresponding redistribution function R_II which includes a δ(ν−ν₁) frequency conservation function, and incoherent scattering, denoted as case III with redistribution function R_III, where the frequencies of the incoming and outgoing photons are independent. Omont et al. (1972) gave a quantum mechanical basis to this formulation, and they were able to provide the branching ratios that weight both coherent and incoherent scattering contributions (see their well-known Eq. (60) for the case of an infinitely sharp lower level). The general case where both frequency coherent and incoherent scattering contribute is known as partial frequency redistribution (PRD). When only incoherent scattering contributes, which is the case of scattering in spectral line cores, it is known as complete frequency redistribution (CRD).

The redistribution function governs the scattering term of the radiative transfer equation (see, e.g., Mihalas 1978). As a consequence, redistribution functions are widely used in the modeling of radiation from stellar atmospheres out of local thermodynamical equilibrium (NLTE). However, the mechanism modeled by the redistribution function is scattering, where a photon is absorbed and then reemitted in a spectral line, possibly coherently in the far wings. The description of the two coupled processes (absorption and emission) requires the solution of the statistical equilibrium of the atomic levels as a preliminary step. For a two-level atom, there is formally only one statistical equilibrium equation expressing the atomic density matrix of the upper level in terms of that of the lower level. Its analytic solution is readily derived (even in the presence of polarization), and the redistribution function follows as a result. In the case of a greater number of levels, the solution becomes much more involved. Attempts were done for the three-level atom. An analytic solution is still possible in this case. This is the equivalent two-level approximation (Hubený et al. 1983). For more than three levels the solution can only be achieved numerically.

The case of the two-term atom with unpolarized and infinitely sharp lower levels is a particular case of multilevel atom. Each level is defined by its quantum numbers (L,S,J) with a given orbital quantum number for each term, L_ℓ in the lower term and L_u in the upper term. S is the spin angular momentum quantum number, which for allowed transitions is the same for both terms; and J is the total angular momentum quantum number defined by J = L + S. Due to this vector addition of angular momenta, there can be multiple values of J for both the upper (${\mathit{J}}_{\mathrm{u}}\mathit{,}{\mathit{J}}_{\mathrm{u}}^{\mathrm{\prime }}\mathit{,}\mathit{...}$) and lower (${\mathit{J}}_{\mathit{\ell }}\mathit{,}{\mathit{J}}_{\mathit{\ell }}^{\mathrm{\prime }}\mathit{,}\mathit{...}$) terms. This structure is responsible for D₁-D₂ pairs of lines with L_ℓ = 0,L_u = 1 and S = 1/2, which results in one single lower level J_ℓ = 1/2 and two upper levels J_u = 1/2 and ${\mathit{J}}_{\mathrm{u}}^{\mathrm{\prime }}\mathrm{=}\mathrm{3}\mathit{/}\mathrm{2}$, as for the Na i D₁-D₂ lines. Other atoms or ions can be seen in such a structure, and their interest was recently increased by the observations by Stenflo et al. (2000a,b), who recorded some linear polarization in the so-called second solar spectrum of some D₁ lines, which are in principle unpolarizable having J_u = 1/2 < 1. The second solar spectrum is the spectrum of the linear polarization formed by scattering and observed close to the solar limb, as defined in Stenflo & Keller (1997).

The interest raised by this problem led several teams to investigate the question by modeling the linear polarization formed by scattering in such lines. However, traditional methods of radiative transfer with partial redistribution based on the works of Hummer (1962) and Omont et al. (1972, Eq. (60)) cannot be readily adapted to the treatment of polarized line formation in these complex two-term transitions, which need to take into account the presence of fine structure as well as the effect of the magnetic field. Empirical attempts at generalizing the redistribution function to the polarized case, also including the effects of fine structure and magnetic fields (Smitha et al. 2011b, 2013a; Belluzzi & Trujillo Bueno 2014) have been proposed. Hyperfine structure was also considered (Smitha et al. 2012b; Sowmya et al. 2014b). These generalizations were exploited in a series of papers for comparison with the observations (Smitha et al. 2011a, 2012a, 2013b, 2014; Belluzzi et al. 2015, 2012; Belluzzi & Trujillo Bueno 2012, 2013; Sowmya et al. 2014a).

However, when compared with the results of a formal theoretical derivation, it appears that these empirical expressions are not fully correct. The aim of the present paper is to publish the correct expressions derived from first principles. The correction concerns essentially the branching ratios that weight the frequency coherent R_II and incoherent R_III terms. In fact, we find a contribution of the magnetic field (Zeeman effect) and of the fine (or hyperfine) structure inside these branching ratios, which were missed by Smitha et al. (2013a) and partly missed by Belluzzi & Trujillo Bueno (2014).

Alsina Ballester et al. (2016) recently studied the polarization of the Mg II k line, which is a D₂-type line, including the effects of a magnetic field. Because their study was spectrally close to the line, they applied the two-level formalism of Bommier (1997b). del Pino Alemán et al. (2016) studied the polarization of the full Mg II h-k doublet, also taking into account the effects of fine structure and quantum interference in the upper term. They apply the formalism of Casini et al. (2014), who derived a theoretical redistribution function including polarization, fine and hyperfine structure. However, the final result of Casini et al. (2014) can be applied only in the collisionless regime (pure R_II). del Pino Alemán et al. (2016) made use of a generalization of the formalism of Casini et al. (2014) to include collisional effects by modifying the branching ratios in a “physically consistent” manner (following their words at the top of the left column, p. 2), but without providing explicit forms of these branching ratios. Providing these ratios is the object of the present paper. Recently, Casini et al. (2017) reinvestigated this question in greater detail. Their final result is given in a compact form by their Eq. (20), which is the same as Eq. (1) of del Pino Alemán et al. (2016). It is comprised of three different added contributions, as is our final result (see Eq. (30) and seq.), which is, however, presented here in its fully explicit algebraic form.

The object of the present paper is to derive branching ratios and associated redistribution functions from the multilevel/multiline general formalism of statistical equilibrium of the atomic density matrix and radiative transfer equations recently published by Bommier (2016a), derived itself from first principles following the method outlined in Bommier (1997a). For the derivation of the redistribution function, a necessary first step is to formally solve the statistical equilibrium problem for the atomic density matrix of the upper state. To this end, the lower state is assumed to be unpolarized and infinitely sharp (Sect. 2); polarization and magnetic fields are considered there. Section 3 provides the redistribution function in the presence of a magnetic field. Section 4 provides instead the redistribution function in the particular case of zero magnetic field. In the concluding Sect. 5, we outline the main limitation of this formalism, which is that collisional transfer between the two fine-structure upper levels cannot be taken into account. In the solar atmosphere, these collisional transfers are due to collisions with neutral hydrogen atoms, which are also responsible for level depolarization due to collisional transitions between the Zeeman sublevels inside a given level. The two rates for the depolarizing collisions in a given level and the collisional transfer between the upper term fine-structure levels are, unfortunately, of the same order of magnitude (see, e.g., Kerkeni & Bommier 2002, Eqs. (11)–(13)). The effects of level depolarizing collision rates can be taken into account in the non-magnetic case (which is why the results of Sect. 4 are not trivially an extension of the magnetic case of Sect. 3). However, collisional transfers cannot be taken into account at all because their presence prevents the derivation of a formal solution of the statistical equilibrium system of equations because several upper levels are coupled by these collisional transfers. A numerical solution of the statistical equilibrium system of equations is required for the solution of the full problem where all required rates are considered. A numerical solution of the statistical equilibrium equations coupled with the radiative transfer equation is under way (Bommier 2016b). This new numerical approach to the solution of the coupled problem does not make use of a redistribution function.

It should be noted that our present derivation is free of any flat spectrum approximation about the incident radiation field.

2. Two-term statistical equilibrium analytical resolution

2.1. Expression of the statistical equilibrium equation

We write the statistical equilibrium equation for an excited-level coherence ${}^{{\mathit{\alpha }}_{\mathrm{u}}{\mathit{J}}_{\mathrm{u}}{\mathit{J}}_{\mathrm{u}}^{\mathrm{\prime }}}\mathit{\rho }_{{\mathit{M}}_{\mathrm{u}}{\mathit{M}}_{\mathrm{u}}^{\mathrm{\prime }}}$ where α_u represents the set of quantum numbers necessary to specify the atomic term configuration, including L_uS. The lower term is assumed to be infinitely sharp, i.e. its lifetime is infinite. In other words, the absorption probability B(α_ℓL_ℓ → α_uL_u)I_ν and the inelastic collision excitation probability C(α_ℓL_ℓ → α_uL_u) are very small with respect to the radiative de-excitation probability A(α_uL_u → α_ℓL_ℓ). This assumes that the radiation field in the medium is weak (see the definition in Sect. 2 of Bommier 1997a). Because of the stated condition of the infinite lifetime of the lower term, we can assume that the lower levels are completely unpolarized, since the presence of depolarizing collisions will effectively destroy any atomic polarization in the lower term. Thus, only diagonal elements ${}^{{\mathit{\alpha }}_{\mathit{\ell }}{\mathit{J}}_{\mathit{\ell }}{\mathit{J}}_{\mathit{\ell }}}\mathit{\rho }_{{\mathit{M}}_{\mathit{\ell }}{\mathit{M}}_{\mathit{\ell }}}$ of the lower-term atomic density matrix must be considered. From Eq. (9) of Bommier (2016a) one then has $\begin{array}{c}\\ \frac{\mathrm{d}}{\mathrm{d}\mathit{t}}{}^{{\mathit{\alpha }}_{\mathrm{u}}{\mathit{J}}_{\mathrm{u}}{\mathit{J}}_{\mathrm{u}}^{\mathrm{\prime }}}{\mathit{\rho }}_{{\mathit{M}}_{\mathrm{u}}{\mathit{M}}_{\mathrm{u}}^{\mathrm{\prime }}}\mathrm{(}{r}\mathit{,}{{v}}^{\mathrm{)}}\mathrm{=}\mathrm{-}\frac{\mathrm{i\Delta }{\mathit{E}}_{{\mathit{M}}_{\mathrm{u}}{\mathit{M}}_{\mathrm{u}}^{\mathrm{\prime }}}}{ħ}{}^{{\mathit{\alpha }}_{\mathrm{u}}{\mathit{J}}_{\mathrm{u}}{\mathit{J}}_{\mathrm{u}}^{\mathrm{\prime }}}{\mathit{\rho }}_{{\mathit{M}}_{\mathrm{u}}{\mathit{M}}_{\mathrm{u}}^{\mathrm{\prime }}}\mathrm{(}{r}\mathit{,}{{v}}^{\mathrm{)}}\\ \mathrm{+}\begin{array}{c}\sum \end{array}_{{\mathit{J}}_{\mathit{\ell }}{\mathit{M}}_{\mathit{\ell }}}^{}{}^{{\mathit{\alpha }}_{\mathit{\ell }}{\mathit{J}}_{\mathit{\ell }}{\mathit{J}}_{\mathit{\ell }}}{\mathit{\rho }}_{{\mathit{M}}_{\mathit{\ell }}{\mathit{M}}_{\mathit{\ell }}}\mathrm{(}{r}\mathit{,}{{v}}^{\mathrm{)}}\mathrm{3}\mathit{B}\mathrm{(}{\mathit{\alpha }}_{\mathit{\ell }}{\mathit{L}}_{\mathit{\ell }}\mathrm{\to }{\mathit{\alpha }}_{\mathrm{u}}{\mathit{L}}_{\mathrm{u}}\mathrm{)}\mathrm{(}\mathrm{2}{\mathit{L}}_{\mathit{\ell }}\mathrm{+}\mathrm{1}\mathrm{)}\mathrm{(}\mathrm{2}{\mathit{J}}_{\mathit{\ell }}\mathrm{+}\mathrm{1}\mathrm{)}\sqrt{\mathrm{(}\mathrm{2}{\mathit{J}}_{\mathrm{u}}\mathrm{+}\mathrm{1}\mathrm{)}\mathrm{(}\mathrm{2}{\mathit{J}}_{\mathrm{u}}^{\mathrm{\prime }}\mathrm{+}\mathrm{1}\mathrm{)}}\\ \mathrm{\times }\left\{\begin{array}{c}\\ {\mathit{J}}_{\mathrm{u}}\mathrm{1}{\mathit{J}}_{\mathit{\ell }}\\ {\mathit{L}}_{\mathit{\ell }}\mathit{S}{\mathit{L}}_{\mathrm{u}}\end{array}\right\}\left\{\begin{array}{c}\\ {\mathit{J}}_{\mathrm{u}}^{\mathrm{\prime }}\mathrm{1}{\mathit{J}}_{\mathit{\ell }}\\ {\mathit{L}}_{\mathit{\ell }}\mathit{S}{\mathit{L}}_{\mathrm{u}}\end{array}\right\}\left(\begin{array}{c}\\ {\mathit{J}}_{\mathrm{u}}\mathrm{1}{\mathit{J}}_{\mathit{\ell }}\\ \mathrm{-}{\mathit{M}}_{\mathrm{u}}\mathit{p}{\mathit{M}}_{\mathit{\ell }}\end{array}\right)\left(\begin{array}{c}\\ {\mathit{J}}_{\mathrm{u}}^{\mathrm{\prime }}\mathrm{1}{\mathit{J}}_{\mathit{\ell }}\\ \mathrm{-}{\mathit{M}}_{\mathrm{u}}^{\mathrm{\prime }}{\mathit{p}}^{\mathrm{\prime }}{\mathit{M}}_{\mathit{\ell }}\end{array}\right)\\ \mathrm{\times }\mathrm{(}\mathrm{-}\mathrm{1}{\mathrm{)}}^{{\mathit{J}}_{\mathrm{u}}\mathrm{-}{\mathit{J}}_{\mathrm{u}}^{\mathrm{\prime }}}\mathrm{\int }\mathrm{d}{\mathit{\nu }}_{\mathrm{1}}\mathrm{􏽉}\frac{\mathrm{d}{{\Omega }}_{\mathrm{1}}}{\mathrm{4}\mathit{\pi }}{\mathrm{ℐ}}_{\mathrm{-}\mathit{p}\mathrm{-}{\mathit{p}}^{\mathrm{\prime }}}\mathrm{(}{\mathit{\nu }}_{\mathrm{1}}\mathit{,}{{\Omega }}_{\mathrm{1}}\mathrm{)}\left[\frac{\mathrm{1}}{\mathrm{2}}{\mathrm{\Phi }}_{\mathit{ba}}\mathrm{(}{\mathit{\nu }}_{{\mathit{M}}_{\mathrm{u}}^{\mathrm{\prime }}{\mathit{M}}_{\mathit{\ell }}}\mathrm{-}{\begin{array}{c}\mathrm{˜}\\ {\mathit{\nu }}_{\mathrm{1}}\end{array}}^{\mathrm{)}}\mathrm{+}\frac{\mathrm{1}}{\mathrm{2}}{\mathrm{\Phi }}_{\mathit{ba}}^{\mathrm{\ast }}\left({\mathit{\nu }}_{{\mathit{M}}_{\mathrm{u}}{\mathit{M}}_{\mathit{\ell }}}\mathrm{-}\begin{array}{c}\mathrm{˜}\\ {\mathit{\nu }}_{\mathrm{1}}\end{array}\right)\right]\\ \mathrm{-}\frac{\mathrm{1}}{\mathrm{2}}\begin{array}{c}\sum \end{array}_{{\mathit{J}}_{\mathrm{u}}^{\mathrm{\prime \prime }}}\{\begin{array}{c}\sum \end{array}_{{\mathit{J}}_{\mathit{\ell }}{\mathit{M}}_{\mathit{\ell }}}{}^{{\mathit{\alpha }}_{\mathrm{u}}{\mathit{J}}_{\mathrm{u}}^{\mathrm{\prime \prime }}{\mathit{J}}_{\mathrm{u}}^{\mathrm{\prime }}}{\mathit{\rho }}_{{\mathit{M}}_{\mathrm{u}}{\mathit{M}}_{\mathrm{u}}^{\mathrm{\prime }}}\mathrm{(}{r}\mathit{,}{{v}}^{\mathrm{)}}\mathit{A}\mathrm{(}{\mathit{\alpha }}_{\mathrm{u}}{\mathit{L}}_{\mathrm{u}}\mathrm{\to }{\mathit{\alpha }}_{\mathit{\ell }}{\mathit{L}}_{\mathit{\ell }}\mathrm{)}\mathrm{(}\mathrm{2}{\mathit{L}}_{\mathrm{u}}\mathrm{+}\mathrm{1}\mathrm{)}\mathrm{(}\mathrm{2}{\mathit{J}}_{\mathit{\ell }}\mathrm{+}\mathrm{1}\mathrm{)}\sqrt{\mathrm{(}\mathrm{2}{\mathit{J}}_{\mathrm{u}}\mathrm{+}\mathrm{1}\mathrm{)}\mathrm{(}\mathrm{2}{\mathit{J}}_{\mathrm{u}}^{\mathrm{\prime \prime }}\mathrm{+}\mathrm{1}\mathrm{)}}\\ \mathrm{\times }\mathrm{(}\mathrm{-}\mathrm{1}{\mathrm{)}}^{{\mathit{J}}_{\mathrm{u}}^{\mathrm{\prime \prime }}\mathrm{-}{\mathit{J}}_{\mathrm{u}}}\left\{\begin{array}{c}\\ {\mathit{J}}_{\mathrm{u}}\mathrm{1}{\mathit{J}}_{\mathit{\ell }}\\ {\mathit{L}}_{\mathit{\ell }}\mathit{S}{\mathit{L}}_{\mathrm{u}}\end{array}\right\}\left\{\begin{array}{c}\\ {\mathit{J}}_{\mathrm{u}}^{\mathrm{\prime \prime }}\mathrm{1}{\mathit{J}}_{\mathit{\ell }}\\ {\mathit{L}}_{\mathit{\ell }}\mathit{S}{\mathit{L}}_{\mathrm{u}}\end{array}\right\}\left(\begin{array}{c}\\ {\mathit{J}}_{\mathit{\ell }}\mathrm{1}{\mathit{J}}_{\mathrm{u}}\\ \mathrm{-}{\mathit{M}}_{\mathit{\ell }}\mathrm{-}\mathit{p}{\mathit{M}}_{\mathrm{u}}\end{array}\right)\left(\begin{array}{c}\\ {\mathit{J}}_{\mathit{\ell }}\mathrm{1}{\mathit{J}}_{\mathrm{u}}^{\mathrm{\prime \prime }}\\ \mathrm{-}{\mathit{M}}_{\mathit{\ell }}\mathrm{-}\mathit{p}{\mathit{M}}_{\mathrm{u}}\end{array}\right)\\ \mathrm{+}\begin{array}{c}\sum \end{array}_{{\mathit{J}}_{\mathit{\ell }}{\mathit{M}}_{\mathit{\ell }}}{}^{{\mathit{\alpha }}_{\mathrm{u}}{\mathit{J}}_{\mathrm{u}}{\mathit{J}}_{\mathrm{u}}^{\mathrm{\prime \prime }}}{\mathit{\rho }}_{{\mathit{M}}_{\mathrm{u}}{\mathit{M}}_{\mathrm{u}}^{\mathrm{\prime }}}\mathrm{(}{r}\mathit{,}{{v}}^{\mathrm{)}}\mathit{A}\mathrm{(}{\mathit{\alpha }}_{\mathrm{u}}{\mathit{L}}_{\mathrm{u}}\mathrm{\to }{\mathit{\alpha }}_{\mathit{\ell }}{\mathit{L}}_{\mathit{\ell }}\mathrm{)}\mathrm{(}\mathrm{2}{\mathit{L}}_{\mathrm{u}}\mathrm{+}\mathrm{1}\mathrm{)}\mathrm{(}\mathrm{2}{\mathit{J}}_{\mathit{\ell }}\mathrm{+}\mathrm{1}\mathrm{)}\sqrt{\mathrm{(}\mathrm{2}{\mathit{J}}_{\mathrm{u}}\mathrm{+}\mathrm{1}\mathrm{)}\mathrm{(}\mathrm{2}{\mathit{J}}_{\mathrm{u}}^{\mathrm{\prime \prime }}\mathrm{+}\mathrm{1}\mathrm{)}}\\ \mathrm{\times }\mathrm{(}\mathrm{-}\mathrm{1}{\mathrm{)}}^{{\mathit{J}}_{\mathrm{u}}^{\mathrm{\prime \prime }}\mathrm{-}{\mathit{J}}_{\mathrm{u}}^{\mathrm{\prime }}}\left\{\begin{array}{c}\\ {\mathit{J}}_{\mathrm{u}}^{\mathrm{\prime }}\mathrm{1}{\mathit{J}}_{\mathit{\ell }}\\ {\mathit{L}}_{\mathit{\ell }}\mathit{S}{\mathit{L}}_{\mathrm{u}}\end{array}\right\}\left\{\begin{array}{c}\\ {\mathit{J}}_{\mathrm{u}}^{\mathrm{\prime \prime }}\mathrm{1}{\mathit{J}}_{\mathit{\ell }}\\ {\mathit{L}}_{\mathit{\ell }}\mathit{S}{\mathit{L}}_{\mathrm{u}}\end{array}\right\}\left(\begin{array}{c}\\ {\mathit{J}}_{\mathit{\ell }}\mathrm{1}{\mathit{J}}_{\mathrm{u}}^{\mathrm{\prime }}\\ \mathrm{-}{\mathit{M}}_{\mathit{\ell }}\mathrm{-}{\mathit{p}}^{\mathrm{\prime }}{\mathit{M}}_{\mathrm{u}}^{\mathrm{\prime }}\end{array}\right)\left(\begin{array}{c}\\ {\mathit{J}}_{\mathit{\ell }}\mathrm{1}{\mathit{J}}_{\mathrm{u}}^{\mathrm{\prime \prime }}\\ \mathrm{-}{\mathit{M}}_{\mathit{\ell }}\mathrm{-}{\mathit{p}}^{\mathrm{\prime }}{\mathit{M}}_{\mathrm{u}}^{\mathrm{\prime }}\end{array}\right)\}\mathrm{·}\end{array}$(1)The first line of this equation accounts for the oscillation effect due to the possible presence of a magnetic field (the Hanle effect). Lines 2–4 describe the creation of the upper level atomic coherence under the effect of radiation absorption from the lower term, and lines 5–8 describe the coherence relaxation under the effect of spontaneous emission. Collisions are neglected for the moment. The atomic density matrix element depends on the atom position r and velocity v.

We denote $\mathrm{\Delta }{\mathit{E}}_{{\mathit{M}}_{\mathrm{u}}{\mathit{M}}_{\mathrm{u}}^{\mathrm{\prime }}}$ the energy difference between the two levels α_uJ_uM_u and ${\mathit{\alpha }}_{\mathrm{u}}{\mathit{J}}_{\mathrm{u}}^{\mathrm{\prime }}{\mathit{M}}_{\mathrm{u}}^{\mathrm{\prime }}$$\mathrm{\Delta }{\mathit{E}}_{{\mathit{M}}_{\mathrm{u}}{\mathit{M}}_{\mathrm{u}}^{\mathrm{\prime }}}\mathrm{=}\mathit{E}\mathrm{\left(}{\mathit{\alpha }}_{\mathrm{u}}{\mathit{J}}_{\mathrm{u}}{\mathit{M}}_{\mathrm{u}}\mathrm{\right)}\mathrm{-}\mathit{E}\mathrm{\left(}{\mathit{\alpha }}_{\mathrm{u}}{\mathit{J}}_{\mathrm{u}}^{\mathrm{\prime }}{\mathit{M}}_{\mathrm{u}}^{\mathrm{\prime }}\mathrm{\right)}\mathit{,}$(2)where $\mathit{E}\mathrm{\left(}{\mathit{\alpha }}_{\mathrm{u}}{\mathit{J}}_{\mathrm{u}}{\mathit{M}}_{\mathrm{u}}\mathrm{\right)}\mathrm{=}\mathit{E}\mathrm{(}{\mathit{\alpha }}_{\mathrm{u}}{\mathit{J}}_{\mathrm{u}}\mathrm{;}\mathit{B}\mathrm{=}\mathrm{0}\mathrm{)}\mathrm{+}\mathit{h}{\mathit{\nu }}_{\mathrm{L}}{\mathit{g}}_{{\mathit{J}}_{\mathrm{u}}}{\mathit{M}}_{\mathrm{u}}\mathit{,}$(3)where E(α_uJ_u;B = 0) is the fine structure level energy in the absence of a magnetic field, ν_L is the Larmor frequency ${\mathit{\nu }}_{\mathrm{L}}\mathrm{=}\frac{\left|\mathit{e}\right|\mathit{B}}{\mathrm{2}\mathit{\pi }{\mathit{m}}_{\mathit{e}}}\mathit{,}$(4)where |e| is the electron charge absolute value, m_e is the electron mass, B is the magnetic field strength, and g_Ju is the Landé factor of the upper level α_uJ_u. In Eq. (3) above, the incomplete Paschen-Back effect is ignored; it is addressed below in Sect. 3.3. This effect has to be taken into account when the Zeeman splitting and the fine or hyperfine splitting are of the same order of magnitude.

Similarly, ν_MuMℓ denotes the frequency of the transition between the two levels α_uJ_uM_u and α_ℓJ_ℓM_ℓ, and ${\mathit{\nu }}_{{\mathit{M}}_{\mathrm{u}}^{\mathrm{\prime }}{\mathit{M}}_{\mathit{\ell }}}$ that of the transition between the two levels ${\mathit{\alpha }}_{\mathrm{u}}{\mathit{J}}_{\mathrm{u}}^{\mathrm{\prime }}{\mathit{M}}_{\mathrm{u}}^{\mathrm{\prime }}$ and α_ℓJ_ℓM_ℓ$\{\begin{array}{c}\\ {\mathit{\nu }}_{{\mathit{M}}_{\mathrm{u}}{\mathit{M}}_{\mathit{\ell }}}\mathrm{=}\left[\mathit{E}\mathrm{\left(}{\mathit{\alpha }}_{\mathrm{u}}{\mathit{J}}_{\mathrm{u}}{\mathit{M}}_{\mathrm{u}}\mathrm{\right)}\mathrm{-}\mathit{E}\mathrm{\left(}{\mathit{\alpha }}_{\mathit{\ell }}{\mathit{J}}_{\mathit{\ell }}{\mathit{M}}_{\mathit{\ell }}\mathrm{\right)}\right]\mathit{/}\mathit{h}\\ {\mathit{\nu }}_{{\mathit{M}}_{\mathrm{u}}^{\mathrm{\prime }}{\mathit{M}}_{\mathit{\ell }}}\mathrm{=}\left[\mathit{E}\mathrm{\left(}{\mathit{\alpha }}_{\mathrm{u}}{\mathit{J}}_{\mathrm{u}}^{\mathrm{\prime }}{\mathit{M}}_{\mathrm{u}}^{\mathrm{\prime }}\mathrm{\right)}\mathrm{-}\mathit{E}\mathrm{\left(}{\mathit{\alpha }}_{\mathit{\ell }}{\mathit{J}}_{\mathit{\ell }}{\mathit{M}}_{\mathit{\ell }}\mathrm{\right)}\right]\mathit{/}\mathit{h,}\end{array}$(5)where h is the Planck constant. The expression ${\mathit{I}}_{\mathrm{-}\mathit{p}\mathrm{-}{\mathit{p}}^{\mathrm{\prime }}}\mathrm{\left(}\begin{array}{c}{}_{\mathrm{˜}}\\ {\mathit{\nu }}_{\mathrm{1}}\end{array}\mathit{,}{{\Omega }}_{\mathrm{1}}\mathrm{\right)}$ denotes the tensor of the incident radiation of frequency ν₁ and direction Ω₁. This tensor is defined in Sect. 5.11 of the monograph by Landi Degl’Innocenti & Landolfi (2004), devoted to the spherical tensors for polarimetry. It is related to the Stokes parameters of the incident radiation by ${\mathrm{ℐ}}_{\mathit{q}{\mathit{q}}^{\mathrm{\prime }}}\mathrm{\left(}{\mathit{\nu }}_{\mathrm{1}}\mathit{,}{{\Omega }}_{\mathrm{1}}\mathrm{\right)}\mathrm{=}\begin{array}{c}\sum \end{array}_{\mathit{j}\mathrm{=}\mathrm{0}}^{\mathrm{3}}{\mathrm{𝒯}}_{\mathit{q}{\mathit{q}}^{\mathrm{\prime }}}\left(\mathit{j,}{{\Omega }}_{\mathrm{1}}\right){\mathit{S}}_{\mathit{j}}\mathrm{\left(}{\mathit{\nu }}_{\mathrm{1}}\mathit{,}{{\Omega }}_{\mathrm{1}}\mathrm{\right)}\mathit{,}$(6)where S_j (j = 0,1,2,3) is one of the Stokes parameters, and T_qq′(j,Ω₁) is the spherical tensor for polarimetry defined and tabulated by Landi Degl’Innocenti & Landolfi (2004, Table 5.3, p. 206). Given the direction Ω₁ of the incident radiation, the Doppler effect has to be accounted for in the absorption profiles by the atom with velocity v$\begin{array}{c}\mathrm{˜}\\ {\mathit{\nu }}_{\mathrm{1}}\end{array}\mathrm{=}{\mathit{\nu }}_{\mathrm{1}}\left(\mathrm{1}\mathrm{-}\frac{{{\Omega }}_{\mathrm{1}}\mathrm{·}{v}}{\mathit{c}}\right)\mathit{,}$(7)see also Sect. 3.1 of Bommier (2016a) about Doppler (or velocity) redistribution. The parameter Φ_ba denotes the Lorentz absorption profile, which has the same width 2γ_ba (full width at half maximum) for all the components of the same multiplet. The definition of the width of the profile, which includes the effects of collisions, can be found in Sect. 3.4 of Bommier (2016a).

Accounting for $\sum _{{\mathit{M}}_{\mathit{\ell }}\mathit{p}}\mathrm{(}\mathrm{2}{\mathit{J}}_{\mathrm{u}}\mathrm{+}\mathrm{1}\mathrm{)}\left(\begin{array}{c}\\ {\mathit{J}}_{\mathit{\ell }}& \mathrm{1}& {\mathit{J}}_{\mathrm{u}}\\ \mathrm{-}{\mathit{M}}_{\mathit{\ell }}& \mathrm{-}\mathit{p}& {\mathit{M}}_{\mathrm{u}}\end{array}\right)\left(\begin{array}{c}\\ {\mathit{J}}_{\mathit{\ell }}& \mathrm{1}& {\mathit{J}}_{\mathrm{u}}^{\mathrm{\prime \prime }}\\ \mathrm{-}{\mathit{M}}_{\mathit{\ell }}& \mathrm{-}\mathit{p}& {\mathit{M}}_{\mathrm{u}}\end{array}\right)\mathrm{=}{\mathit{\delta }}_{{\mathit{J}}_{\mathrm{u}}{\mathit{J}}_{\mathrm{u}}^{\mathrm{\prime \prime }}}$(8)and analogously for ${\mathit{J}}_{\mathrm{u}}^{\mathrm{\prime }}{\mathit{M}}_{\mathrm{u}}^{\mathrm{\prime }}$, and $\sum _{{\mathit{J}}_{\mathit{\ell }}}\mathrm{(}\mathrm{2}{\mathit{L}}_{\mathrm{u}}\mathrm{+}\mathrm{1}\mathrm{)}\mathrm{(}\mathrm{2}{\mathit{J}}_{\mathit{\ell }}\mathrm{+}\mathrm{1}\mathrm{)}{\left\{\begin{array}{c}\\ {\mathit{J}}_{\mathrm{u}}& \mathrm{1}& {\mathit{J}}_{\mathit{\ell }}\\ {\mathit{L}}_{\mathit{\ell }}& \mathit{S}& {\mathit{L}}_{\mathrm{u}}\end{array}\right\}}^{\mathrm{2}}\mathrm{=}\mathrm{1}\mathit{,}$(9)lines 5–8 of the above Eq. (1), which account for the coherence destruction processes, simply reduce into $\mathrm{-}\mathit{A}\mathrm{(}{\mathit{\alpha }}_{\mathrm{u}}{\mathit{L}}_{\mathrm{u}}\mathrm{\to }{\mathit{\alpha }}_{\mathit{\ell }}{\mathit{L}}_{\mathit{\ell }}\mathrm{)}{}^{{\mathit{\alpha }}_{\mathrm{u}}{\mathit{J}}_{\mathrm{u}}{\mathit{J}}_{\mathrm{u}}^{\mathrm{\prime }}}{\mathit{\rho }}_{{\mathit{M}}_{\mathrm{u}}{\mathit{M}}_{\mathrm{u}}^{\mathrm{\prime }}}\mathrm{(}{r}\mathit{,}{{v}}^{\mathrm{)}}\mathit{,}$(10)which is also $\mathrm{-}\sum _{{\mathit{J}}_{\mathit{\ell }}}\frac{\mathrm{1}}{\mathrm{2}}\left[\mathit{A}\mathrm{(}{\mathit{\alpha }}_{\mathrm{u}}{\mathit{L}}_{\mathrm{u}}{\mathit{J}}_{\mathrm{u}}\mathrm{\to }{\mathit{\alpha }}_{\mathit{\ell }}{\mathit{L}}_{\mathit{\ell }}{\mathit{J}}_{\mathit{\ell }}\mathrm{)}\mathrm{+}\mathit{A}\mathrm{(}{\mathit{\alpha }}_{\mathrm{u}}{\mathit{L}}_{\mathrm{u}}{\mathit{J}}_{\mathrm{u}}^{\mathrm{\prime }}\mathrm{\to }{\mathit{\alpha }}_{\mathit{\ell }}{\mathit{L}}_{\mathit{\ell }}{\mathit{J}}_{\mathit{\ell }}\mathrm{)}\right]{}^{{\mathit{\alpha }}_{\mathrm{u}}{\mathit{J}}_{\mathrm{u}}{\mathit{J}}_{\mathrm{u}}^{\mathrm{\prime }}}{\mathit{\rho }}_{{\mathit{M}}_{\mathrm{u}}{\mathit{M}}_{\mathrm{u}}^{\mathrm{\prime }}}\mathrm{(}{r}\mathit{,}{{v}}^{\mathrm{)}}\mathit{,}$(11)from $\mathit{A}\mathrm{(}{\mathit{\alpha }}_{\mathrm{u}}{\mathit{L}}_{\mathrm{u}}{\mathit{J}}_{\mathrm{u}}\mathrm{\to }{\mathit{\alpha }}_{\mathit{\ell }}{\mathit{L}}_{\mathit{\ell }}{\mathit{J}}_{\mathit{\ell }}\mathrm{)}\mathrm{=}\mathrm{(}\mathrm{2}{\mathit{L}}_{\mathrm{u}}\mathrm{+}\mathrm{1}\mathrm{)}\mathrm{(}\mathrm{2}{\mathit{J}}_{\mathit{\ell }}\mathrm{+}\mathrm{1}\mathrm{)}{\left\{\begin{array}{c}\\ {\mathit{J}}_{\mathrm{u}}& \mathrm{1}& {\mathit{J}}_{\mathit{\ell }}\\ {\mathit{L}}_{\mathit{\ell }}& \mathit{S}& {\mathit{L}}_{\mathrm{u}}\end{array}\right\}}^{\mathrm{2}}\mathit{A}\mathrm{(}{\mathit{\alpha }}_{\mathrm{u}}{\mathit{L}}_{\mathrm{u}}\mathrm{\to }{\mathit{\alpha }}_{\mathit{\ell }}{\mathit{L}}_{\mathit{\ell }}\mathrm{)}$(12)and similarly for ${\mathit{J}}_{\mathrm{u}}^{\mathrm{\prime }}$.

The lower level is assumed to be infinitely sharp and unpolarized. Accordingly, the Zeeman sublevels are equally populated as ${{\mathit{\alpha }}_{\mathit{\ell }}{\mathit{J}}_{\mathit{\ell }}{\mathit{J}}_{\mathit{\ell }}}^{}{\mathit{\rho }}_{{\mathit{M}}_{\mathit{\ell }}{\mathit{M}}_{\mathit{\ell }}}\mathrm{(}{r}\mathit{,}{{v}}^{\mathrm{)}}\mathrm{=}\frac{\mathrm{1}}{\mathrm{(}\mathrm{2}{\mathit{L}}_{\mathit{\ell }}\mathrm{+}\mathrm{1}\mathrm{)}\mathrm{(}\mathrm{2}\mathit{S}\mathrm{+}\mathrm{1}\mathrm{)}}\mathrm{·}$(13)This also assumes that the lower term remains much more populated than the upper term. Indeed, the ratio of the upper term population to the lower term population is at most on the order of the number of photons per mode , and in stellar atmosphere physics this number is usually such that , which is the weak radiation field condition (Bommier 1997a, Sect. 2). As a consequence, the population ratio is ≪1. By simply neglecting the upper term population with respect to the lower term one, both the above condition and the normalization condition of the atomic density matrix Trρ = 1 are satisfied. Outside the weak radiation field condition, the whole present formalism cannot be applied, as discussed in Bommier (1997a). Moreover, the effect of the approximation is even weaker when one finally considers the ratios of the Stokes parameters Q/I,U/I,V/I instead of the Stokes parameters themselves. The important point is that all the lower sublevel populations are equal, and that there are no off-diagonal elements (coherences) in the lower level density matrix. Coherences or unequal sublevel populations in the lower level would invalidate the results of the present paper because they would prevent the analytical solution of the statistical equilibrium as is done in the following section. When these elements are non-negligible with respect to the lower level total population, the statistical equilibrium equations form a system that has to be numerically solved. Apart from this case, a global scaling factor could be applied to these lower sublevel populations without changing the results of the present paper, to which the scaling factor would also have to be applied.

Another consequence of the large lower term population is that the velocity distribution in this term remains Maxwellian. Lines 2–4 of the above Eq. ( 1), which account for the coherence creation processes, can then be rewritten as $\begin{array}{c}\\ \begin{array}{c}\sum \end{array}_{{\mathit{J}}_{\mathit{\ell }}{\mathit{M}}_{\mathit{\ell }}}^{}\mathrm{3}\mathit{B}\mathrm{(}{\mathit{\alpha }}_{\mathit{\ell }}{\mathit{L}}_{\mathit{\ell }}\mathrm{\to }{\mathit{\alpha }}_{\mathrm{u}}{\mathit{L}}_{\mathrm{u}}\mathrm{)}\frac{\mathrm{2}{\mathit{J}}_{\mathit{\ell }}\mathrm{+}\mathrm{1}}{\mathrm{2}\mathit{S}\mathrm{+}\mathrm{1}}\sqrt{\mathrm{(}\mathrm{2}{\mathit{J}}_{\mathrm{u}}\mathrm{+}\mathrm{1}\mathrm{)}\mathrm{(}\mathrm{2}{\mathit{J}}_{\mathrm{u}}^{\mathrm{\prime }}\mathrm{+}\mathrm{1}\mathrm{)}}\mathrm{(}\mathrm{-}\mathrm{1}{\mathrm{)}}^{{\mathit{J}}_{\mathrm{u}}\mathrm{-}{\mathit{J}}_{\mathrm{u}}^{\mathrm{\prime }}}\\ \mathrm{\times }\left\{\begin{array}{c}\\ {\mathit{J}}_{\mathrm{u}}\mathrm{1}{\mathit{J}}_{\mathit{\ell }}\\ {\mathit{L}}_{\mathit{\ell }}\mathit{S}{\mathit{L}}_{\mathrm{u}}\end{array}\right\}\left\{\begin{array}{c}\\ {\mathit{J}}_{\mathrm{u}}^{\mathrm{\prime }}\mathrm{1}{\mathit{J}}_{\mathit{\ell }}\\ {\mathit{L}}_{\mathit{\ell }}\mathit{S}{\mathit{L}}_{\mathrm{u}}\end{array}\right\}\left(\begin{array}{c}\\ {\mathit{J}}_{\mathrm{u}}\mathrm{1}{\mathit{J}}_{\mathit{\ell }}\\ \mathrm{-}{\mathit{M}}_{\mathrm{u}}\mathit{p}{\mathit{M}}_{\mathit{\ell }}\end{array}\right)\left(\begin{array}{c}\\ {\mathit{J}}_{\mathrm{u}}^{\mathrm{\prime }}\mathrm{1}{\mathit{J}}_{\mathit{\ell }}\\ \mathrm{-}{\mathit{M}}_{\mathrm{u}}^{\mathrm{\prime }}{\mathit{p}}^{\mathrm{\prime }}{\mathit{M}}_{\mathit{\ell }}\end{array}\right)\\ \mathrm{\times }\mathrm{\int }\mathrm{d}{\mathit{\nu }}_{\mathrm{1}}\mathrm{􏽉}\frac{\mathrm{d}{{\Omega }}_{\mathrm{1}}}{\mathrm{4}\mathit{\pi }}\begin{array}{c}\sum \end{array}_{\mathit{j}\mathrm{=}\mathrm{0}}^{\mathrm{3}}{\mathrm{𝒯}}_{\mathrm{-}\mathit{p}\mathrm{-}{\mathit{p}}^{\mathrm{\prime }}}\left(\mathit{j,}{{\Omega }}_{\mathrm{1}}\right){\mathit{S}}_{\mathit{j}}\mathrm{(}{\mathit{\nu }}_{\mathrm{1}}\mathit{,}{{\Omega }}_{\mathrm{1}}\mathrm{)}\left[\frac{\mathrm{1}}{\mathrm{2}}{\mathrm{\Phi }}_{\mathit{ba}}\mathrm{(}{\mathit{\nu }}_{{\mathit{M}}_{\mathrm{u}}^{\mathrm{\prime }}{\mathit{M}}_{\mathit{\ell }}}\mathrm{-}{\begin{array}{c}\mathrm{˜}\\ {\mathit{\nu }}_{\mathrm{1}}\end{array}}^{\mathrm{)}}\mathrm{+}\frac{\mathrm{1}}{\mathrm{2}}{\mathrm{\Phi }}_{\mathit{ba}}^{\mathrm{\ast }}\left({\mathit{\nu }}_{{\mathit{M}}_{\mathrm{u}}{\mathit{M}}_{\mathit{\ell }}}\mathrm{-}\begin{array}{c}\mathrm{˜}\\ {\mathit{\nu }}_{\mathrm{1}}\end{array}\right)\right]\mathit{.}\end{array}$(14)

2.2. Solution of the statistical equilibrium equation

2.2.1. In the presence of a magnetic field

Introducing the spherical tensors for polarimetry (Landi Degl’Innocenti & Landolfi 2004, see Eq. (5.156)) ${\mathrm{𝒯}}_{\mathrm{-}\mathit{p}\mathrm{-}{\mathit{p}}^{\mathrm{\prime }}}\left(\mathit{j,}{{\Omega }}_{\mathrm{1}}\right)\mathrm{=}\sum _{\mathit{KQ}}\mathrm{(}\mathrm{-}\mathrm{1}{\mathrm{)}}^{\mathrm{1}\mathrm{-}\mathit{p}}\frac{\sqrt{\mathrm{2}\mathit{K}\mathrm{+}\mathrm{1}}}{\sqrt{\mathrm{3}}}\left(\begin{array}{c}\\ \mathrm{1}& \mathrm{1}& \mathit{K}\\ \mathrm{-}\mathit{p}& {\mathit{p}}^{\mathrm{\prime }}& \mathit{Q}\end{array}\right){\mathrm{𝒯}}_{\mathrm{-}\mathit{Q}}^{\mathit{K}}\left(\mathit{j,}{{\Omega }}_{\mathrm{1}}\right)\mathit{,}$(15)the statistical equilibrium Eq. (1) can be resolved into $\begin{array}{c}\\ {}^{{\mathit{\alpha }}_{\mathrm{u}}{\mathit{J}}_{\mathrm{u}}{\mathit{J}}_{\mathrm{u}}^{\mathrm{\prime }}}{\mathit{\rho }}_{{\mathit{M}}_{\mathrm{u}}{\mathit{M}}_{\mathrm{u}}^{\mathrm{\prime }}}\mathrm{(}{r}\mathit{,}{{v}}^{\mathrm{)}}\mathrm{=} \frac{\mathrm{1}}{\mathit{A}\mathrm{(}{\mathit{\alpha }}_{\mathrm{u}}{\mathit{L}}_{\mathrm{u}}\mathrm{\to }{\mathit{\alpha }}_{\mathit{\ell }}{\mathit{L}}_{\mathit{\ell }}\mathrm{)}\mathrm{+}\frac{\mathrm{i\Delta }{\mathit{E}}_{{\mathit{M}}_{\mathrm{u}}{\mathit{M}}_{\mathrm{u}}^{\mathrm{\prime }}}}{ħ}}\\ \mathrm{\times }\begin{array}{c}\sum \end{array}_{{\mathit{J}}_{\mathit{\ell }}{\mathit{M}}_{\mathit{\ell }}}^{}\begin{array}{c}\sum \end{array}_{{\mathit{K}}^{\mathrm{\prime }}\mathit{Q}}\mathit{B}\mathrm{(}{\mathit{\alpha }}_{\mathit{\ell }}{\mathit{L}}_{\mathit{\ell }}\mathrm{\to }{\mathit{\alpha }}_{\mathrm{u}}{\mathit{L}}_{\mathrm{u}}\mathrm{)}\frac{\mathrm{2}{\mathit{J}}_{\mathit{\ell }}\mathrm{+}\mathrm{1}}{\mathrm{2}\mathit{S}\mathrm{+}\mathrm{1}}\sqrt{\mathrm{(}\mathrm{2}{\mathit{J}}_{\mathrm{u}}\mathrm{+}\mathrm{1}\mathrm{)}\mathrm{(}\mathrm{2}{\mathit{J}}_{\mathrm{u}}^{\mathrm{\prime }}\mathrm{+}\mathrm{1}\mathrm{)}}\sqrt{\mathrm{3}\mathrm{(}\mathrm{2}{\mathit{K}}^{\mathrm{\prime }}\mathrm{+}\mathrm{1}\mathrm{)}}\mathrm{(}\mathrm{-}\mathrm{1}{\mathrm{)}}^{{\mathit{J}}_{\mathrm{u}}\mathrm{-}{\mathit{J}}_{\mathrm{u}}^{\mathrm{\prime }}\mathrm{+}\mathrm{1}\mathrm{-}\mathit{p}}\\ \mathrm{\times }\left\{\begin{array}{c}\\ {\mathit{J}}_{\mathrm{u}}\mathrm{1}{\mathit{J}}_{\mathit{\ell }}\\ {\mathit{L}}_{\mathit{\ell }}\mathit{S}{\mathit{L}}_{\mathrm{u}}\end{array}\right\}\left\{\begin{array}{c}\\ {\mathit{J}}_{\mathrm{u}}^{\mathrm{\prime }}\mathrm{1}{\mathit{J}}_{\mathit{\ell }}\\ {\mathit{L}}_{\mathit{\ell }}\mathit{S}{\mathit{L}}_{\mathrm{u}}\end{array}\right\}\left(\begin{array}{c}\\ {\mathit{J}}_{\mathrm{u}}\mathrm{1}{\mathit{J}}_{\mathit{\ell }}\\ \mathrm{-}{\mathit{M}}_{\mathrm{u}}\mathit{p}{\mathit{M}}_{\mathit{\ell }}\end{array}\right)\left(\begin{array}{c}\\ {\mathit{J}}_{\mathrm{u}}^{\mathrm{\prime }}\mathrm{1}{\mathit{J}}_{\mathit{\ell }}\\ \mathrm{-}{\mathit{M}}_{\mathrm{u}}^{\mathrm{\prime }}{\mathit{p}}^{\mathrm{\prime }}{\mathit{M}}_{\mathit{\ell }}\end{array}\right)\left(\begin{array}{c}\\ \mathrm{1}\mathrm{1}{\mathit{K}}^{\mathrm{\prime }}\\ \mathrm{-}\mathit{p}{\mathit{p}}^{\mathrm{\prime }}\mathit{Q}\end{array}\right)\\ \mathrm{\times }\mathrm{\int }\mathrm{d}{\mathit{\nu }}_{\mathrm{1}}\mathrm{􏽉}\frac{\mathrm{d}{{\Omega }}_{\mathrm{1}}}{\mathrm{4}\mathit{\pi }}\begin{array}{c}\sum \end{array}_{\mathit{j}\mathrm{=}\mathrm{0}}^{\mathrm{3}}{\mathrm{𝒯}}_{\mathrm{-}\mathit{Q}}^{{\mathit{K}}^{\mathrm{\prime }}}\mathrm{\left(}\mathit{j,}{{\Omega }}_{\mathrm{1}}\mathrm{\right)}{\mathit{S}}_{\mathit{j}}\mathrm{(}{\mathit{\nu }}_{\mathrm{1}}\mathit{,}{{\Omega }}_{\mathrm{1}}\mathrm{)}\left[\frac{\mathrm{1}}{\mathrm{2}}{\mathrm{\Phi }}_{\mathit{ba}}\mathrm{(}{\mathit{\nu }}_{{\mathit{M}}_{\mathrm{u}}^{\mathrm{\prime }}{\mathit{M}}_{\mathit{\ell }}}\mathrm{-}{\begin{array}{c}\mathrm{˜}\\ {\mathit{\nu }}_{\mathrm{1}}\end{array}}^{\mathrm{)}}\mathrm{+}\frac{\mathrm{1}}{\mathrm{2}}{\mathrm{\Phi }}_{\mathit{ba}}^{\mathrm{\ast }}\left({\mathit{\nu }}_{{\mathit{M}}_{\mathrm{u}}{\mathit{M}}_{\mathit{\ell }}}\mathrm{-}\begin{array}{c}\mathrm{˜}\\ {\mathit{\nu }}_{\mathrm{1}}\end{array}\right)\right]\mathit{.}\end{array}$(16)Introducing the spherical components of the density matrix ${}^{{\mathit{\alpha }}_{\mathrm{u}}{\mathit{J}}_{\mathrm{u}}{\mathit{J}}_{\mathrm{u}}^{\mathrm{\prime }}}\mathit{\rho }_{\mathit{Q}}^{\mathit{K}}{}^{\mathrm{(}}{r}\mathit{,}{{v}}^{\mathrm{)}}$ with $\mathit{K}\mathrm{=}\mathrm{0}\mathit{,}\mathrm{1}\mathit{,}\mathrm{2}\mathit{,}\mathit{...}\mathit{,}{\mathit{J}}_{\mathrm{u}}\mathrm{+}{\mathit{J}}_{\mathrm{u}}^{\mathrm{\prime }}$ and Q = −K,−K + 1,..., K−1,K as defined by (Landi Degl’Innocenti & Landolfi 2004, see Eq. (3.97)) ${\mathit{\alpha J}{\mathit{J}}^{\mathrm{\prime }}}^{}{\mathit{\rho }}_{\mathit{Q}}^{\mathit{K}}\mathrm{=}\sum _{\mathit{M}{\mathit{M}}^{\mathrm{\prime }}}\mathrm{(}\mathrm{-}\mathrm{1}{\mathrm{)}}^{\mathit{J}\mathrm{-}\mathit{M}}\sqrt{\mathrm{2}\mathit{K}\mathrm{+}\mathrm{1}}\left(\begin{array}{c}\\ \mathit{J}& {\mathit{J}}^{\mathrm{\prime }}& \mathit{K}\\ \mathit{M}& \mathrm{-}{\mathit{M}}^{\mathrm{\prime }}& \mathrm{-}\mathit{Q}\end{array}\right){}^{\mathit{\alpha J}{\mathit{J}}^{\mathrm{\prime }}}{\mathit{\rho }}_{\mathit{M}{\mathit{M}}^{\mathrm{\prime }}}\mathit{,}$(17)the spherical component can be written as $\begin{array}{c}\\ {}^{{\mathit{\alpha }}_{\mathrm{u}}{\mathit{J}}_{\mathrm{u}}{\mathit{J}}_{\mathrm{u}}^{\mathrm{\prime }}}{\mathit{\rho }}_{\mathit{Q}}^{\mathit{K}}\mathrm{(}{r}\mathit{,}{{v}}^{\mathrm{)}}\mathrm{=}\begin{array}{c}\sum \end{array}_{{\mathit{M}}_{\mathrm{u}}{\mathit{M}}_{\mathrm{u}}^{\mathrm{\prime }}}\frac{\mathrm{1}}{\mathit{A}\mathrm{(}{\mathit{\alpha }}_{\mathrm{u}}{\mathit{L}}_{\mathrm{u}}\mathrm{\to }{\mathit{\alpha }}_{\mathit{\ell }}{\mathit{L}}_{\mathit{\ell }}\mathrm{)}\mathrm{+}\frac{\mathrm{i\Delta }{\mathit{E}}_{{\mathit{M}}_{\mathrm{u}}{\mathit{M}}_{\mathrm{u}}^{\mathrm{\prime }}}}{ħ}}\\ \mathrm{\times }\begin{array}{c}\sum \end{array}_{{\mathit{J}}_{\mathit{\ell }}{\mathit{M}}_{\mathit{\ell }}}^{}\begin{array}{c}\sum \end{array}_{{\mathit{K}}^{\mathrm{\prime }}}\mathit{B}\mathrm{(}{\mathit{\alpha }}_{\mathit{\ell }}{\mathit{L}}_{\mathit{\ell }}\mathrm{\to }{\mathit{\alpha }}_{\mathrm{u}}{\mathit{L}}_{\mathrm{u}}\mathrm{)}\frac{\mathrm{2}{\mathit{J}}_{\mathit{\ell }}\mathrm{+}\mathrm{1}}{\mathrm{2}\mathit{S}\mathrm{+}\mathrm{1}}\sqrt{\mathrm{(}\mathrm{2}{\mathit{J}}_{\mathrm{u}}\mathrm{+}\mathrm{1}\mathrm{)}\mathrm{(}\mathrm{2}{\mathit{J}}_{\mathrm{u}}^{\mathrm{\prime }}\mathrm{+}\mathrm{1}\mathrm{)}}\sqrt{\mathrm{3}\mathrm{(}\mathrm{2}\mathit{K}\mathrm{+}\mathrm{1}\mathrm{)}\mathrm{(}\mathrm{2}{\mathit{K}}^{\mathrm{\prime }}\mathrm{+}\mathrm{1}\mathrm{)}}\mathrm{(}\mathrm{-}\mathrm{1}{\mathrm{)}}^{{\mathit{J}}_{\mathrm{u}}^{\mathrm{\prime }}\mathrm{-}{\mathit{M}}_{\mathrm{u}}^{\mathrm{\prime }}\mathrm{+}\mathrm{1}\mathrm{-}{\mathit{p}}^{\mathrm{\prime }}}\\ \mathrm{\times }\left\{\begin{array}{c}\\ {\mathit{J}}_{\mathrm{u}}\mathrm{1}{\mathit{J}}_{\mathit{\ell }}\\ {\mathit{L}}_{\mathit{\ell }}\mathit{S}{\mathit{L}}_{\mathrm{u}}\end{array}\right\}\left\{\begin{array}{c}\\ {\mathit{J}}_{\mathrm{u}}^{\mathrm{\prime }}\mathrm{1}{\mathit{J}}_{\mathit{\ell }}\\ {\mathit{L}}_{\mathit{\ell }}\mathit{S}{\mathit{L}}_{\mathrm{u}}\end{array}\right\}\\ \mathrm{\times }\left(\begin{array}{c}\\ {\mathit{J}}_{\mathrm{u}}\mathrm{1}{\mathit{J}}_{\mathit{\ell }}\\ \mathrm{-}{\mathit{M}}_{\mathrm{u}}\mathit{p}{\mathit{M}}_{\mathit{\ell }}\end{array}\right)\left(\begin{array}{c}\\ {\mathit{J}}_{\mathrm{u}}^{\mathrm{\prime }}\mathrm{1}{\mathit{J}}_{\mathit{\ell }}\\ \mathrm{-}{\mathit{M}}_{\mathrm{u}}^{\mathrm{\prime }}{\mathit{p}}^{\mathrm{\prime }}{\mathit{M}}_{\mathit{\ell }}\end{array}\right)\left(\begin{array}{c}\\ \mathrm{1}\mathrm{1}{\mathit{K}}^{\mathrm{\prime }}\\ \mathrm{-}\mathit{p}{\mathit{p}}^{\mathrm{\prime }}\mathit{Q}\end{array}\right)\left(\begin{array}{c}\\ {\mathit{J}}_{\mathrm{u}}\mathit{K}{\mathit{J}}_{\mathrm{u}}^{\mathrm{\prime }}\\ \mathrm{-}{\mathit{M}}_{\mathrm{u}}\mathit{Q}{\mathit{M}}_{\mathrm{u}}^{\mathrm{\prime }}\end{array}\right)\\ \mathrm{\times }\mathrm{\int }\mathrm{d}{\mathit{\nu }}_{\mathrm{1}}\mathrm{􏽉}\frac{\mathrm{d}{{\Omega }}_{\mathrm{1}}}{\mathrm{4}\mathit{\pi }}\begin{array}{c}\sum \end{array}_{\mathit{j}\mathrm{=}\mathrm{0}}^{\mathrm{3}}{\mathrm{𝒯}}_{\mathrm{-}\mathit{Q}}^{{\mathit{K}}^{\mathrm{\prime }}}\mathrm{\left(}\mathit{j,}{{\Omega }}_{\mathrm{1}}\mathrm{\right)}{\mathit{S}}_{\mathit{j}}\mathrm{(}{\mathit{\nu }}_{\mathrm{1}}\mathit{,}{{\Omega }}_{\mathrm{1}}\mathrm{)}\left[\frac{\mathrm{1}}{\mathrm{2}}{\mathrm{\Phi }}_{\mathit{ba}}\mathrm{(}{\mathit{\nu }}_{{\mathit{M}}_{\mathrm{u}}^{\mathrm{\prime }}{\mathit{M}}_{\mathit{\ell }}}\mathrm{-}{\begin{array}{c}\mathrm{˜}\\ {\mathit{\nu }}_{\mathrm{1}}\end{array}}^{\mathrm{)}}\mathrm{+}\frac{\mathrm{1}}{\mathrm{2}}{\mathrm{\Phi }}_{\mathit{ba}}^{\mathrm{\ast }}\left({\mathit{\nu }}_{{\mathit{M}}_{\mathrm{u}}{\mathit{M}}_{\mathit{\ell }}}\mathrm{-}\begin{array}{c}\mathrm{˜}\\ {\mathit{\nu }}_{\mathrm{1}}\end{array}\right)\right]\mathit{.}\end{array}$(18)In the case of the two-level atom, i.e., when there is one single J_u and one single J_ℓ, this equation can be reduced in terms of generalized profiles as defined in Landi Degl’Innocenti et al. (1991). Equation (37) of this paper is thus obtained, which is also Eq. (7) of Bommier (1997b). This is made possible owing to the following relation, which is valid only if ${\mathit{\alpha }}_{\mathrm{u}}{\mathit{J}}_{\mathrm{u}}\mathrm{=}{\mathit{\alpha }}_{\mathrm{u}}{\mathit{J}}_{\mathrm{u}}^{\mathrm{\prime }}$$\mathrm{\Delta }{\mathit{E}}_{{\mathit{M}}_{\mathrm{u}}{\mathit{M}}_{\mathrm{u}}^{\mathrm{\prime }}}\mathrm{=}\mathit{h}{\mathit{\nu }}_{\mathrm{L}}{\mathit{g}}_{{\mathit{J}}_{\mathrm{u}}}\left({\mathit{M}}_{\mathrm{u}}\mathrm{-}{\mathit{M}}_{\mathrm{u}}^{\mathrm{\prime }}\right)\mathrm{=}\mathit{h}{\mathit{\nu }}_{\mathrm{L}}{\mathit{g}}_{{\mathit{J}}_{\mathrm{u}}}\mathit{Q}\mathit{.}$(19)The factorization in Q of $\mathrm{\Delta }{\mathit{E}}_{{\mathit{M}}_{\mathrm{u}}{\mathit{M}}_{\mathrm{u}}^{\mathrm{\prime }}}$ permits restricting the summation over M_u and ${\mathit{M}}_{\mathrm{u}}^{\mathrm{\prime }}$ to just the second addendum of Eq. (18), leading to the generalized profile and also making possible the introduction of the depolarizing collision factors D^(K), as obtained in Bommier (1997b). In the general case of fine structure, when there are several α_uJ_u levels, it is not possible to factorize $\mathrm{\Delta }{\mathit{E}}_{{\mathit{M}}_{\mathrm{u}}{\mathit{M}}_{\mathrm{u}}^{\mathrm{\prime }}}$ in terms of Q when ${\mathit{\alpha }}_{\mathrm{u}}{\mathit{J}}_{\mathrm{u}}\ne {\mathit{\alpha }}_{\mathrm{u}}{\mathit{J}}_{\mathrm{u}}^{\mathrm{\prime }}$ because the form of $\mathrm{\Delta }{\mathit{E}}_{{\mathit{M}}_{\mathrm{u}}{\mathit{M}}_{\mathrm{u}}^{\mathrm{\prime }}}$ is more involved. However, when the magnetic field is zero, $\mathrm{\Delta }{\mathit{E}}_{{\mathit{M}}_{\mathrm{u}}{\mathit{M}}_{\mathrm{u}}^{\mathrm{\prime }}}$ does not depend on the magnetic quantum numbers M_u and it simply coincides with $\mathrm{\Delta }{\mathit{E}}_{{\mathit{J}}_{\mathrm{u}}{\mathit{J}}_{\mathrm{u}}^{\mathrm{\prime }}}$, which again makes it possible to introduce the depolarizing collision factors D^(K), as is done in the following section. However, the depolarizing effect of collisions is thus only partly taken into account (see Sect. 5), and this can be done only in the absence of a magnetic field. The outcome is the numerical solution of the statistical equilibrium equations.

2.2.2. In zero magnetic field

In the absence of a magnetic field, the M indexes in ΔE and ν can be replaced by the J indexes and the sums over the M indexes can be performed. Then, Eq. (16) reduces to $\begin{array}{c}\\ {}^{{\mathit{\alpha }}_{\mathrm{u}}{\mathit{J}}_{\mathrm{u}}{\mathit{J}}_{\mathrm{u}}^{\mathrm{\prime }}}{\mathit{\rho }}_{{\mathit{M}}_{\mathrm{u}}{\mathit{M}}_{\mathrm{u}}^{\mathrm{\prime }}}\mathrm{(}{r}\mathit{,}{{v}}^{\mathrm{)}}\mathrm{=}\frac{\mathrm{1}}{\mathit{A}\mathrm{(}{\mathit{\alpha }}_{\mathrm{u}}{\mathit{L}}_{\mathrm{u}}\mathrm{\to }{\mathit{\alpha }}_{\mathit{\ell }}{\mathit{L}}_{\mathit{\ell }}\mathrm{)}\mathrm{+}\frac{\mathrm{i\Delta }{\mathit{E}}_{{\mathit{J}}_{\mathrm{u}}{\mathit{J}}_{\mathrm{u}}^{\mathrm{\prime }}}}{ħ}}\\ \mathrm{\times }\begin{array}{c}\sum \end{array}_{{\mathit{J}}_{\mathit{\ell }}}^{}\begin{array}{c}\sum \end{array}_{\mathit{KQ}}\mathit{B}\mathrm{(}{\mathit{\alpha }}_{\mathit{\ell }}{\mathit{L}}_{\mathit{\ell }}\mathrm{\to }{\mathit{\alpha }}_{\mathrm{u}}{\mathit{L}}_{\mathrm{u}}\mathrm{)}\frac{\mathrm{2}{\mathit{J}}_{\mathit{\ell }}\mathrm{+}\mathrm{1}}{\mathrm{2}\mathit{S}\mathrm{+}\mathrm{1}}\sqrt{\mathrm{(}\mathrm{2}{\mathit{J}}_{\mathrm{u}}\mathrm{+}\mathrm{1}\mathrm{)}\mathrm{(}\mathrm{2}{\mathit{J}}_{\mathrm{u}}^{\mathrm{\prime }}\mathrm{+}\mathrm{1}\mathrm{)}}\sqrt{\mathrm{3}\mathrm{(}\mathrm{2}{\mathit{K}}^{\mathrm{\prime }}\mathrm{+}\mathrm{1}\mathrm{)}}\mathrm{(}\mathrm{-}\mathrm{1}{\mathrm{)}}^{\mathrm{1}\mathrm{+}{\mathit{J}}_{\mathit{\ell }}\mathrm{+}\mathrm{2}{\mathit{J}}_{\mathrm{u}}\mathrm{-}{\mathit{M}}_{\mathrm{u}}^{\mathrm{\prime }}}\\ \mathrm{\times }\left\{\begin{array}{c}\\ {\mathit{J}}_{\mathrm{u}}\mathrm{1}{\mathit{J}}_{\mathit{\ell }}\\ {\mathit{L}}_{\mathit{\ell }}\mathit{S}{\mathit{L}}_{\mathrm{u}}\end{array}\right\}\left\{\begin{array}{c}\\ {\mathit{J}}_{\mathrm{u}}^{\mathrm{\prime }}\mathrm{1}{\mathit{J}}_{\mathit{\ell }}\\ {\mathit{L}}_{\mathit{\ell }}\mathit{S}{\mathit{L}}_{\mathrm{u}}\end{array}\right\}\left\{\begin{array}{c}\\ {\mathit{K}}^{\mathrm{\prime }}{\mathit{J}}_{\mathrm{u}}{\mathit{J}}_{\mathrm{u}}^{\mathrm{\prime }}\\ {\mathit{J}}_{\mathit{\ell }}\mathrm{1}\mathrm{1}\end{array}\right\}\left(\begin{array}{c}\\ {\mathit{J}}_{\mathrm{u}}{\mathit{K}}^{\mathrm{\prime }}{\mathit{J}}_{\mathrm{u}}^{\mathrm{\prime }}\\ \mathrm{-}{\mathit{M}}_{\mathrm{u}}\mathit{Q}{\mathit{M}}_{\mathrm{u}}^{\mathrm{\prime }}\end{array}\right)\\ \mathrm{\times }\mathrm{\int }\mathrm{d}{\mathit{\nu }}_{\mathrm{1}}\mathrm{􏽉}\frac{\mathrm{d}{{\Omega }}_{\mathrm{1}}}{\mathrm{4}\mathit{\pi }}\begin{array}{c}\sum \end{array}_{\mathit{j}\mathrm{=}\mathrm{0}}^{\mathrm{3}}{\mathrm{𝒯}}_{\mathrm{-}\mathit{Q}}^{{\mathit{K}}^{\mathrm{\prime }}}\mathrm{\left(}\mathit{j,}{{\Omega }}_{\mathrm{1}}\mathrm{\right)}{\mathit{S}}_{\mathit{j}}\mathrm{(}{\mathit{\nu }}_{\mathrm{1}}\mathit{,}{{\Omega }}_{\mathrm{1}}\mathrm{)}\left[\frac{\mathrm{1}}{\mathrm{2}}{\mathrm{\Phi }}_{\mathit{ba}}\mathrm{(}{\mathit{\nu }}_{{\mathit{J}}_{\mathrm{u}}^{\mathrm{\prime }}{\mathit{J}}_{\mathit{\ell }}}\mathrm{-}{\begin{array}{c}\mathrm{˜}\\ {\mathit{\nu }}_{\mathrm{1}}\end{array}}^{\mathrm{)}}\mathrm{+}\frac{\mathrm{1}}{\mathrm{2}}{\mathrm{\Phi }}_{\mathit{ba}}^{\mathrm{\ast }}\left({\mathit{\nu }}_{{\mathit{J}}_{\mathrm{u}}{\mathit{J}}_{\mathit{\ell }}}\mathrm{-}\begin{array}{c}\mathrm{˜}\\ {\mathit{\nu }}_{\mathrm{1}}\end{array}\right)\right]\mathit{,}\end{array}$(20)where only the sum over M_ℓ has been performed, whereas Eq. (18) becomes $\begin{array}{c}\\ {}^{{\mathit{\alpha }}_{\mathrm{u}}{\mathit{J}}_{\mathrm{u}}{\mathit{J}}_{\mathrm{u}}^{\mathrm{\prime }}}{\mathit{\rho }}_{\mathit{Q}}^{\mathit{K}}\mathrm{(}{r}\mathit{,}{{v}}^{\mathrm{)}}\mathrm{=}\frac{\mathrm{1}}{\mathit{A}\mathrm{(}{\mathit{\alpha }}_{\mathrm{u}}{\mathit{L}}_{\mathrm{u}}\mathrm{\to }{\mathit{\alpha }}_{\mathit{\ell }}{\mathit{L}}_{\mathit{\ell }}\mathrm{)}\mathrm{+}\frac{\mathrm{1}}{\mathrm{2}}\left[{\mathit{D}}^{\mathrm{\left(}\mathit{K}\mathrm{\right)}}\mathrm{\left(}{\mathit{\alpha }}_{\mathrm{u}}{\mathit{J}}_{\mathrm{u}}\mathrm{\right)}\mathrm{+}{\mathit{D}}^{\mathrm{\left(}\mathit{K}\mathrm{\right)}}\mathrm{\left(}{\mathit{\alpha }}_{\mathrm{u}}{\mathit{J}}_{\mathrm{u}}^{\mathrm{\prime }}\mathrm{\right)}\right]\mathrm{+}\frac{\mathrm{i\Delta }{\mathit{E}}_{{\mathit{J}}_{\mathrm{u}}{\mathit{J}}_{\mathrm{u}}^{\mathrm{\prime }}}}{ħ}}\\ \mathrm{\times }\mathit{B}\mathrm{(}{\mathit{\alpha }}_{\mathit{\ell }}{\mathit{L}}_{\mathit{\ell }}\mathrm{\to }{\mathit{\alpha }}_{\mathrm{u}}{\mathit{L}}_{\mathrm{u}}\mathrm{)}\frac{\mathrm{2}{\mathit{J}}_{\mathit{\ell }}\mathrm{+}\mathrm{1}}{\mathrm{2}\mathit{S}\mathrm{+}\mathrm{1}}\sqrt{\mathrm{3}\mathrm{(}\mathrm{2}{\mathit{J}}_{\mathrm{u}}\mathrm{+}\mathrm{1}\mathrm{)}\mathrm{(}\mathrm{2}{\mathit{J}}_{\mathrm{u}}^{\mathrm{\prime }}\mathrm{+}\mathrm{1}\mathrm{)}}\mathrm{(}\mathrm{-}\mathrm{1}{\mathrm{)}}^{\mathrm{1}\mathrm{+}{\mathit{J}}_{\mathit{\ell }}\mathrm{+}{\mathit{J}}_{\mathrm{u}}\mathrm{+}\mathit{Q}}\\ \mathrm{\times }\left\{\begin{array}{c}\\ {\mathit{J}}_{\mathrm{u}}\mathrm{1}{\mathit{J}}_{\mathit{\ell }}\\ {\mathit{L}}_{\mathit{\ell }}\mathit{S}{\mathit{L}}_{\mathrm{u}}\end{array}\right\}\left\{\begin{array}{c}\\ {\mathit{J}}_{\mathrm{u}}^{\mathrm{\prime }}\mathrm{1}{\mathit{J}}_{\mathit{\ell }}\\ {\mathit{L}}_{\mathit{\ell }}\mathit{S}{\mathit{L}}_{\mathrm{u}}\end{array}\right\}\left\{\begin{array}{c}\\ \mathit{K}{\mathit{J}}_{\mathrm{u}}{\mathit{J}}_{\mathrm{u}}^{\mathrm{\prime }}\\ {\mathit{J}}_{\mathit{\ell }}\mathrm{1}\mathrm{1}\end{array}\right\}\\ \mathrm{\times }\mathrm{\int }\mathrm{d}{\mathit{\nu }}_{\mathrm{1}}\mathrm{􏽉}\frac{\mathrm{d}{{\Omega }}_{\mathrm{1}}}{\mathrm{4}\mathit{\pi }}\begin{array}{c}\sum \end{array}_{\mathit{j}\mathrm{=}\mathrm{0}}^{\mathrm{3}}{\mathrm{𝒯}}_{\mathrm{-}\mathit{Q}}^{\mathit{K}}\mathrm{\left(}\mathit{j,}{{\Omega }}_{\mathrm{1}}\mathrm{\right)}{\mathit{S}}_{\mathit{j}}\mathrm{(}{\mathit{\nu }}_{\mathrm{1}}\mathit{,}{{\Omega }}_{\mathrm{1}}\mathrm{)}\left[\frac{\mathrm{1}}{\mathrm{2}}{\mathrm{\Phi }}_{\mathit{ba}}\mathrm{(}{\mathit{\nu }}_{{\mathit{J}}_{\mathrm{u}}^{\mathrm{\prime }}{\mathit{J}}_{\mathit{\ell }}}\mathrm{-}{\begin{array}{c}\mathrm{˜}\\ {\mathit{\nu }}_{\mathrm{1}}\end{array}}^{\mathrm{)}}\mathrm{+}\frac{\mathrm{1}}{\mathrm{2}}{\mathrm{\Phi }}_{\mathit{ba}}^{\mathrm{\ast }}\left({\mathit{\nu }}_{{\mathit{J}}_{\mathrm{u}}{\mathit{J}}_{\mathit{\ell }}}\mathrm{-}\begin{array}{c}\mathrm{˜}\\ {\mathit{\nu }}_{\mathrm{1}}\end{array}\right)\right]\mathit{,}\end{array}$(21)where the sums over ${\mathit{M}}_{\mathit{\ell }}\mathit{,}{\mathit{M}}_{\mathrm{u}}\mathit{,}{\mathit{M}}_{\mathrm{u}}^{\mathrm{\prime }}$ have been performed, and where one has applied $\begin{array}{c}\sum \end{array}_{\mathit{M}{\mathit{M}}^{\mathrm{\prime }}}\left(\begin{array}{c}\\ \mathit{J}& {\mathit{J}}^{\mathrm{\prime }}& \mathit{K}\\ \mathit{M}& \mathrm{-}{\mathit{M}}^{\mathrm{\prime }}& \mathrm{-}\mathit{Q}\end{array}\right)\left(\begin{array}{c}\\ \mathit{J}& {\mathit{J}}^{\mathrm{\prime }}& {\mathit{K}}^{\mathrm{\prime }}\\ \mathit{M}& \mathrm{-}{\mathit{M}}^{\mathrm{\prime }}& \mathrm{-}\mathit{Q}\end{array}\right)\mathrm{=}\frac{\mathrm{1}}{\mathrm{2}\mathit{K}\mathrm{+}\mathrm{1}}{\mathit{\delta }}_{\mathit{K}{\mathit{K}}^{\mathrm{\prime }}}\mathit{.}$(22)The depolarizing collisions have been taken into account here, by adding their contribution in the K-order coherence inverse lifetime contribution and similarly to Eq. (11). This is possible only for the tensorial component ${}^{{\mathit{\alpha }}_{\mathrm{u}}{\mathit{J}}_{\mathrm{u}}{\mathit{J}}_{\mathrm{u}}^{\mathrm{\prime }}}\mathit{\rho }_{\mathit{Q}}^{\mathit{K}}$ but not for the dyadic component ${}^{{\mathit{\alpha }}_{\mathrm{u}}{\mathit{J}}_{\mathrm{u}}{\mathit{J}}_{\mathrm{u}}^{\mathrm{\prime }}}\mathit{\rho }_{{\mathit{M}}_{\mathrm{u}}{\mathit{M}}_{\mathrm{u}}^{\mathrm{\prime }}}$, as explained below. In addition, this does not account for all the elastic or quasi-elastic collision effects. In the solar atmosphere, these effects are due to collisions with neutral hydrogen atoms. These collisions induce both transitions between Zeeman sublevels and between fine or hyperfine structure levels. Here we only account for the transitions between the Zeeman sublevels and we ignore the other transitions. However, as detailed in Sect. 5, both contributions are of the same order of magnitude. Thus, by so doing we include only a part of the effect of the elastic or quasi-elastic collisions.

An attempt to account for the depolarizing collisions was heuristically proposed by Smitha et al. (2013a), but introducing the same D^(K) rate for both α_uJ_u and ${\mathit{\alpha }}_{\mathrm{u}}{\mathit{J}}_{\mathrm{u}}^{\mathrm{\prime }}$ states (with D⁽⁰⁾ = 0). The above expression is the result of a formal derivation and accounts for the fine structure as far as possible, i.e., in the absence of a magnetic field and in tensorial components. This is the only case where the effect of the transitions between the Zeeman sublevels can be simply accounted for through the D^(K)(α_uJ_u) rates. In the general case, this is not possible due to the presence of $\mathrm{\Delta }{\mathit{E}}_{{\mathit{M}}_{\mathrm{u}}{\mathit{M}}_{\mathrm{u}}^{\mathrm{\prime }}}$ because the explicit dependence on M_u and ${\mathit{M}}_{\mathrm{u}}^{\mathrm{\prime }}$ excessively complicates the formalism. In addition, the transitions between the fine structure levels cannot be accounted for in any case. This was pointed out by Belluzzi & Trujillo Bueno (2014) in their Sect. 3.1, and was the reason why they discarded the whole depolarizing effect of collisions from their formalism. As pointed out by these authors, taking into account all the effects of the collisions with neutral hydrogen requires the numerical solution of the statistical equilibrium.

In the general case of fine structure, as stated in the above, it is not possible to factorize $\mathrm{\Delta }{\mathit{E}}_{{\mathit{M}}_{\mathrm{u}}{\mathit{M}}_{\mathrm{u}}^{\mathrm{\prime }}}$ in terms of Q when ${\mathit{\alpha }}_{\mathrm{u}}{\mathit{J}}_{\mathrm{u}}\ne {\mathit{\alpha }}_{\mathrm{u}}{\mathit{J}}_{\mathrm{u}}^{\mathrm{\prime }}$ and in non-zero magnetic fields. The statistical equilibrium equations are then simple only in the dyadic basis. In this basis, the introduction of the depolarizing collisions effect, even limited to those responsible for the D^(K) coefficients, leads to the coupling of all the upper level Zeeman sublevel coherences, as visible in the unnumbered equation before Eq. (7.99) of Landi Degl’Innocenti & Landolfi (2004). This coupling prevents an analytical solution of the statistical equilibrium in the dyadic basis from being attained. Transforming into the irreducible tensors basis leads to other couplings, also preventing an analytical solution, due to the non-factorization of $\mathrm{\Delta }{\mathit{E}}_{{\mathit{M}}_{\mathrm{u}}{\mathit{M}}_{\mathrm{u}}^{\mathrm{\prime }}}$ in terms of Q in the presence of a fine structure. This is why it is possible to simply introduce D^(K) coefficients only in the zero magnetic field case, when there is a fine structure. The outcome of this problem is the numerical solution of the statistical equilibrium equations.

3. Redistribution function in the presence of a magnetic field

The expression of the redistribution function follows by porting the solution of the statistical equilibrium in the emissivity, which is given in Eqs. (15), (16) of Bommier (2016a). The emissivity represents the emitted light quantity per unit volume and is given by ${\mathit{\epsilon }}_{\mathit{i}}\mathrm{(}\mathit{\nu ,}{{\Omega }}^{\mathrm{)}}\mathrm{=}{\mathit{k}}_{\mathit{L}}\mathrm{\int }\mathrm{d}{\mathit{\nu }}_{\mathrm{1}}\mathrm{􏽉}\frac{\mathrm{d}{{\Omega }}_{\mathrm{1}}}{\mathrm{4}\mathit{\pi }}\begin{array}{c}\sum \end{array}_{\mathit{j}\mathrm{=}\mathrm{0}}^{\mathrm{3}}{\mathrm{ℛ}}_{\mathit{ij}}\mathrm{(}\mathit{\nu ,}{\mathit{\nu }}_{\mathrm{1}}\mathit{,}{\Omega }\mathit{,}{{\Omega }}_{\mathrm{1}}\mathrm{;}{{B}}^{\mathrm{)}}{\mathit{S}}_{\mathit{j}}\mathrm{\left(}{\mathit{\nu }}_{\mathrm{1}}\mathit{,}{{\Omega }}_{\mathrm{1}}\mathrm{\right)}\mathit{,}$(23)where the redistribution function R_ij⁽ν,ν₁,Ω,Ω₁;B⁾ is the joint probability of observing a photon of frequency ν and polarization i emitted in the Ω direction, given a photon of frequency ν₁ and polarization j incident along the Ω₁ direction, in the presence of a magnetic field B. The parameter k_L is the line absorption coefficient ${\mathit{k}}_{\mathit{L}}\mathrm{=}\frac{\mathit{h\nu }}{\mathrm{4}\mathit{\pi }}\mathrm{𝒩}\mathit{B}\mathrm{(}{\mathit{\alpha }}_{\mathit{\ell }}{\mathit{L}}_{\mathit{\ell }}\mathrm{\to }{\mathit{\alpha }}_{\mathrm{u}}{\mathit{L}}_{\mathrm{u}}\mathrm{)}\mathit{,}$(24)where is the emitter atom or ion density and B(α_ℓL_ℓ → α_uL_u) is the Einstein coefficient for absorption in the line. We recall that the atom lower term is here assumed to be overpopulated, i.e., the entire atomic population is assumed to be in the ground state as defined in Eq. (13); is then also the lower term atom or ion density.

The emission can possibly end in level $\mathrm{\left(}{\mathit{\alpha }}_{\mathit{\ell }}{\mathit{L}}_{\mathit{\ell }}\mathit{S}{\mathit{J}}_{\mathit{\ell }}^{\mathrm{\prime }}{\mathit{M}}_{\mathit{\ell }}^{\mathrm{\prime }}\mathrm{\right)}$ of the lower term, which may be different from the initial (α_ℓL_ℓSJ_ℓM_ℓ) level.

The contribution of inelastic collisions has to be considered. For de-excitation, this adds a collisional contribution C(α_uL_u → α_ℓL_ℓ) to the radiative de-excitation probability A(α_uL_u → α_ℓL_ℓ). For excitation, it adds a contribution term besides the radiative excitation term in Eq. (1). This collisional excitation term is at the origin of the well-known Planck source term of the radiative transfer equation, which we will not rewrite here (see for instance Eq. (25) of Belluzzi & Trujillo Bueno 2014).

To simplify notations, we introduce the usual Γ probabilities in the excited term $\{\begin{array}{c}\\ {\mathrm{\Gamma }}_{\mathrm{R}}\mathrm{=}\mathit{A}\mathrm{(}{\mathit{\alpha }}_{\mathrm{u}}{\mathit{L}}_{\mathrm{u}}\mathrm{\to }{\mathit{\alpha }}_{\mathit{\ell }}{\mathit{L}}_{\mathit{\ell }}\mathrm{)}\\ {\mathrm{\Gamma }}_{\mathrm{I}}\mathrm{=}\mathit{C}\mathrm{(}{\mathit{\alpha }}_{\mathrm{u}}{\mathit{L}}_{\mathrm{u}}\mathrm{\to }{\mathit{\alpha }}_{\mathit{\ell }}{\mathit{L}}_{\mathit{\ell }}\mathrm{)}\\ {\mathrm{\Gamma }}_{\mathrm{E}}\mathrm{=}\mathrm{2}{\mathit{\gamma }}_{\mathit{ba}}^{\mathrm{\left(}\mathit{c}\mathrm{\right)}}\mathit{,}\end{array}$(25)where Γ_R is the radiative deexcitation coefficient, Γ_I is the inelastic collisional deexcitation coefficient, and Γ_E is the elastic collision contribution to line broadening as described in Sahal-Bréchot & Bommier (2014, 2017). The expression of ${\mathit{\gamma }}_{\mathit{ba}}^{\mathrm{\left(}\mathit{c}\mathrm{\right)}}$ in terms of the collisional S and T = 1−S matrices is also given in Eqs. (28), (29) of Bommier (2016a).

The redistribution function as defined in Eq. (23) is normalized to $\mathrm{\int }\mathrm{d}\mathit{\nu }\mathrm{\int }\mathrm{d}{\mathit{\nu }}_{\mathrm{1}}\mathrm{􏽉}\frac{\mathrm{d}{\Omega }}{\mathrm{4}\mathit{\pi }}\mathrm{􏽉}\frac{\mathrm{d}{{\Omega }}_{\mathrm{1}}}{\mathrm{4}\mathit{\pi }}\mathrm{\int }\frac{\mathrm{d}\mathit{\gamma }}{\mathrm{2}\mathit{\pi }}\mathrm{\int }\frac{\mathrm{d}{\mathit{\gamma }}_{\mathrm{1}}}{\mathrm{2}\mathit{\pi }}{\mathrm{ℛ}}_{\mathit{ij}}\mathrm{(}\mathit{\nu ,}{\mathit{\nu }}_{\mathrm{1}}\mathit{,}{\Omega }\mathit{,}{{\Omega }}_{\mathrm{1}}\mathrm{;}{{B}}^{\mathrm{)}}\mathrm{=}{\mathit{\delta }}_{\mathit{i,}\mathrm{0}}{\mathit{\delta }}_{\mathit{j,}\mathrm{0}}\frac{{\mathrm{\Gamma }}_{\mathrm{R}}}{{\mathrm{\Gamma }}_{\mathrm{R}}\mathrm{+}{\mathrm{\Gamma }}_{\mathrm{I}}}\mathit{,}$(26)and not to unity. This is the same for the two-level atom redistribution function defined in Bommier (1997b). This is also the same in Omont et al. (1972); the normalization condition is given in their Eq. (64), which is the same normalization condition as above, and γ is the angle with the reference axis for the linear polarization definition (the direction of positive polarization) in the plane perpendicular to the line of sight (see Fig. 2 in Landi Degl’Innocenti 1983). The redistribution function is not normalized to unity because here it accounts for the total emitted radiation. However, the photon emission generally results from two contributions, scattering of the incident photon on the one hand, and collisional excitation followed by radiative de-excitation on the other. This explains why the result of the scattering contribution is not normalized to unity. However, other authors prefer to normalize their redistribution function to unity (see, e.g., Mihalas 1978, Eqs. (2)–(7)). In this case, the effect of the competing collisional excitation is accounted for by the (1−ϵ) coefficient, which factorizes the scattering contribution in the transfer equation (see Eqs. (2)–(36), (2)–(41) and (13)–(92)). Here, $\mathit{ϵ}\mathrm{=}{\mathrm{\Gamma }}_{\mathrm{I}}\mathit{/}{}^{\mathrm{(}}\mathrm{\Gamma }_{\mathrm{R}}\mathrm{+}{\mathrm{\Gamma }}_{\mathrm{I}}{}^{\mathrm{)}}$. In order to apply the formulae we present in the following, it is convenient to multiply our redistribution functions by ${}^{\mathrm{(}}\mathrm{\Gamma }_{\mathrm{R}}\mathrm{+}{\mathrm{\Gamma }}_{\mathrm{I}}{}^{\mathrm{)}}\mathit{/}{\mathrm{\Gamma }}_{\mathrm{R}}$. Belluzzi & Trujillo Bueno (2014) apply the same normalization as ours to their redistribution functions (see their Eqs. (1), (7)–(12) and following unnumbered equations). As the results of Casini et al. (2014) can only be applied in the collisionless regime, the two possible normalizations described above are identical and are both equal to unity in this regime.

The redistribution function normalization condition is based on the following normalization property of the irreducible spherical tensors for polarimetry (Landi Degl’Innocenti 1984) $\mathrm{􏽉}\frac{\mathrm{d}{\Omega }}{\mathrm{4}\mathit{\pi }}\mathrm{\int }\frac{\mathrm{d}\mathit{\gamma }}{\mathrm{2}\mathit{\pi }}{\mathrm{𝒯}}_{\mathit{Q}}^{\mathit{K}}\mathrm{\left(}\mathit{i,}{\Omega }\mathrm{\right)}\mathrm{=}{\mathit{\delta }}_{\mathit{i,}\mathrm{0}}{\mathit{\delta }}_{\mathit{K,}\mathrm{0}}{\mathit{\delta }}_{\mathit{Q,}\mathrm{0}}\mathit{,}$(27)as can be derived for instance from Table 1 in Bommier (1997b).

Considering both the second-order $\mathit{\epsilon }\begin{array}{c}\left(\mathrm{2}\right)\\ \mathit{i}\end{array}$ and fourth-order $\mathit{\epsilon }\begin{array}{c}\left(\mathrm{4}\right)\\ \mathit{i}\end{array}$ contributions to the emissivity as defined in Eqs. (15), (16) of Bommier (2016a), the following expressions can be derived, which contain both frequency coherent R_II and frequency incoherent R_III and their branching ratios.

3.1. Considering only fine structure

Let us first provide a few formulae useful for the calculation. In the fourth-order emissivity, there is a profile ${\mathrm{\Phi }}_{\mathit{ca}}{}^{\mathrm{(}}\begin{array}{c}{}_{\mathrm{˜}}\\ \mathit{\nu }\end{array}\mathrm{-}\begin{array}{c}{}_{\mathrm{˜}}\\ {\mathit{\nu }}_{\mathrm{1}}\end{array}\mathrm{-}{\mathit{\nu }}_{{\mathit{\alpha }}_{\mathit{\ell }}{\mathit{J}}_{\mathit{\ell }}{\mathit{M}}_{\mathit{\ell }}\mathit{,}{\mathit{\alpha }}_{\mathit{\ell }}{\mathit{J}}_{\mathit{\ell }}^{\mathrm{\prime }}{\mathit{M}}_{\mathit{\ell }}^{\mathrm{\prime }}}{}^{\mathrm{)}}$. In the case of the two-term atom, c = a and the profile width is the lower term width γ_a, which tends towards zero because we have assumed that the lower level is infinitely sharp γ_a− → 0. Therefore, $\frac{\mathrm{1}}{\mathrm{2}}{\mathrm{\Phi }}_{\mathit{a}}\mathrm{(}\begin{array}{c}\mathrm{˜}\\ \mathit{\nu }\end{array}\mathrm{-}\begin{array}{c}\mathrm{˜}\\ {\mathit{\nu }}_{\mathrm{1}}\end{array}\mathrm{-}{\mathit{\nu }}_{{\mathit{M}}_{\mathit{\ell }}{\mathit{M}}_{\mathit{\ell }}^{\mathrm{\prime }}}\mathrm{)}\mathrm{=}\frac{\mathrm{1}}{{\mathit{\gamma }}_{\mathit{a}}\mathrm{-}\mathrm{2}\mathrm{i}\mathit{\pi }\mathrm{(}\begin{array}{c}\mathrm{˜}\\ \mathit{\nu }\end{array}\mathrm{-}\begin{array}{c}\mathrm{˜}\\ {\mathit{\nu }}_{\mathrm{1}}\end{array}\mathrm{-}{\mathit{\nu }}_{{\mathit{M}}_{\mathit{\ell }}{\mathit{M}}_{\mathit{\ell }}^{\mathrm{\prime }}}\mathrm{)}}\mathrm{-}\mathrm{\to }\frac{\mathrm{1}}{\mathrm{2}}\mathit{\delta }\mathrm{(}\begin{array}{c}\mathrm{˜}\\ \mathit{\nu }\end{array}\mathrm{-}\begin{array}{c}\mathrm{˜}\\ {\mathit{\nu }}_{\mathrm{1}}\end{array}\mathrm{-}{\mathit{\nu }}_{{\mathit{M}}_{\mathit{\ell }}{\mathit{M}}_{\mathit{\ell }}^{\mathrm{\prime }}}\mathrm{)}\mathrm{+}\frac{\mathrm{i}}{\mathrm{2}\mathit{\pi }}\mathrm{𝒫}\frac{\mathrm{1}}{\begin{array}{c}\mathrm{˜}\\ \mathit{\nu }\end{array}\mathrm{-}\begin{array}{c}\mathrm{˜}\\ {\mathit{\nu }}_{\mathrm{1}}\end{array}\mathrm{-}{\mathit{\nu }}_{{\mathit{M}}_{\mathit{\ell }}{\mathit{M}}_{\mathit{\ell }}^{\mathrm{\prime }}}}\mathit{,}$(28)where stands for the Cauchy principal value.

Equation (3) in Bommier (1997b), which transforms a profile product into a profile sum, is also useful for the calculation. In the present notation, this is $\frac{\mathrm{1}}{\mathrm{2}}{\mathrm{\Phi }}_{\mathit{ba}}\mathrm{(}{\mathit{\nu }}_{{\mathit{M}}_{\mathrm{u}}^{\mathrm{\prime }}{\mathit{M}}_{\mathit{\ell }}}\mathrm{-}{\begin{array}{c}\mathrm{˜}\\ {\mathit{\nu }}_{\mathrm{1}}\end{array}}^{\mathrm{)}}\frac{\mathrm{1}}{\mathrm{2}}{\mathrm{\Phi }}_{\mathit{ba}}^{\mathrm{\ast }}\left({\mathit{\nu }}_{{\mathit{M}}_{\mathrm{u}}{\mathit{M}}_{\mathit{\ell }}}\mathrm{-}\begin{array}{c}\mathrm{˜}\\ {\mathit{\nu }}_{\mathrm{1}}\end{array}\right)\mathrm{=}\frac{\mathrm{1}}{\mathrm{2}{\mathit{\gamma }}_{\mathit{ba}}\mathrm{+}\frac{\mathrm{i\Delta }{\mathit{E}}_{{\mathit{M}}_{\mathrm{u}}{\mathit{M}}_{\mathrm{u}}^{\mathrm{\prime }}}}{ħ}}\left[\frac{\mathrm{1}}{\mathrm{2}}{\mathrm{\Phi }}_{\mathit{ba}}\mathrm{(}{\mathit{\nu }}_{{\mathit{M}}_{\mathrm{u}}^{\mathrm{\prime }}{\mathit{M}}_{\mathit{\ell }}}\mathrm{-}{\begin{array}{c}\mathrm{˜}\\ {\mathit{\nu }}_{\mathrm{1}}\end{array}}^{\mathrm{)}}\mathrm{+}\frac{\mathrm{1}}{\mathrm{2}}{\mathrm{\Phi }}_{\mathit{ba}}^{\mathrm{\ast }}\left({\mathit{\nu }}_{{\mathit{M}}_{\mathrm{u}}{\mathit{M}}_{\mathit{\ell }}}\mathrm{-}\begin{array}{c}\mathrm{˜}\\ {\mathit{\nu }}_{\mathrm{1}}\end{array}\right)\right]\mathit{,}$(29)where 2γ_ba is the full line width at half maximum (see Sect. 3.4 of Bommier 2016a), and 2γ_ba = Γ_R + Γ_I + Γ_E.