text:
\section{Introduction}
The diagnosis of magnetic fields in astrophysical plasmas relies on the
measurement and interpretation of the polarization signature of the
magnetic fields in spectral lines. The physical conditions of these
plasmas vary greatly depending on the type of astronomical objects
considered. As a consequence a variety of simultaneous, and sometimes
competing, effects -- radiative, collisional, and from plasma electric
and magnetic fields -- need to be taken into consideration in the
modeling of the observed radiation.

In dense plasmas, such as those characteristic of stellar
atmospheres, strong gradients of temperature and density, as well as of
magnetic and velocity fields, often occur (e.g., due to atmosphere
stratification under the effect of gravitation and pressure gradients,
or because of the presence of convective motions and dynamo actions),
which need to be taken into account in modeling the
transport of polarized radiation through the plasma. This is certainly
the case for the strongest absorption lines of the solar spectrum,
giving rise to a great diversity of line profile shapes and degrees of
polarization, especially when observed with high temporal and spatial
resolution \cite[e.g.,][]{BL09}

Besides these non-local, radiative transfer effects, the particular
excitation conditions of a magnetized plasma away from local
thermodynamic equilibrium often represents a challenging problem for
the description of the interaction between matter and polarized
radiation. Apart from the well-studied
Zeeman effect, subtle quantum
processes associated with the polarized state of the atoms are often at
play in stellar atmospheres, such as the Hanle effect, or other
quantum effects associated with level-crossing physics. While these
processes offer an opportunity for a highly refined diagnostics of
magnetized plasmas, they also present theoretical and computational
difficulties that have commonly prevented their full exploitation
in the astrophysical context.

Modeling of partial redistribution of the photon's frequency in
the scattering of polarized radiation is perhaps one of the most
difficult problems to treat at the fundamental level, as this generally
requires to describe the interaction of radiation with matter beyond the
lowest order of approximation, which is instead adequate in the case
of absorption and emission lines that are formed under conditions of
complete frequency redistribution. For example, when isotropic
plasma collisions are effective at destroying the coherence of
multi-photon interactions, the scattering of radiation can
safely be described as the incoherent succession of single-photon
absorption and emission. It can also be shown that irradiation of the
atom by a spectrally flat radiation allows a description of the
scattering process that is phenomenologically identical to that
provided by such two-step,
single-photon processes of absorption and incoherent re-emission
(e.g., \citealt{He54}, \S20; see also \citealt{Sa67}, \S2.6, and
Section~\ref{sec:balance} of this paper).
Therefore, the regime
for which the frequency coherence effects characteristic of radiation
scattering are instead important is typically one of low plasma densities
(i.e., low collisional rates) and highly spectrally modulated radiation, such
as that emerging from stellar atmospheres in correspondence of very deep
absorption lines.

This paper introduces a general frequency redistribution function for
the polarized two-term atom, and then considers particular cases of it,
which are relevant to the investigation of strong resonance lines in
the solar spectrum, for which partial redistribution effects are deemed
to be
important. However, the results presented here are also valid in the case
of subordinate lines, when both the upper and lower terms are
radiatively broadened by spontaneous de-excitation. This study is based
on a novel formalism \cite[][work in progress]{Ca15} that, within the
framework of non-relativistic Quantum Electrodynamics, describes the
time evolution of the atomic system and the radiation field in terms of
\emph{propagators}.
This formalism relies on a diagrammatic representation of the
atom-photon interaction, which allows to correctly identify and enumerate
the types of processes that contribute to the scattering of polarized
radiation. At the lowest order of interaction, it reproduces
the theory of polarized line formation for complete redistribution of
frequency \citep{LL04}.

Over the past four decades, several authors have addressed the
problem of partial redistribution in polarized spectral lines. We will
not provide a detailed account of the progress in this area, but we
limit ourselves to simply tracing the main lines of work.
Notable are the works of \cite{Om72,Om73}, and of Heinzel and
collaborators \cite[e.g.,][]{He81,HH82,Hu82}, all of which derive
essentially from the seminal work of \cite{FV62}, who had extended
the impact theory of pressure broadening developed by \citeauthor{An49}
(\citeyear{An49}; see also \citealt{Ba58}) to second order, in order to
study the line shape of molecular Raman scattering.
\cite{FV62} assumed the diagonality of the density matrix of the
initial (lower) state, as did the subsequent works that were based on
the same formalism. This assumption corresponds to
the hypothesis of non-coherent lower term (see Sect~\ref{sec:special}).
More recently, \cite{St94} followed a heuristic approach built upon the
Kramers-Heisenberg scattering amplitude in order to derive a
semi-classical theory of partial redistribution, which has
enabled the derivation of redistribution functions similar to those
presented in this paper \cite[see, e.g.,][]{Sm13},
again for atoms with non-coherent lower term.

Other works have approached the general problem of spectral line
formation, investigating frequency redistribution effects in the
statistical equilibrium of the atom and in the transport of polarized
radiation through an absorptive and scattering medium.
\cite{LT71} followed an approach based on \cite{He54}, which allowed for
the presence of atomic coherence (i.e., non-diagonal density matrix) in
the lower term.
Remarkably their work led to results for the scattering redistribution
function that are in full agreement with ours.
\cite{Co82} investigated the redistribution problem relying on
the quantum-regression theorem \cite[e.g.,][]{Lo73}, and derived
results that agree with those of \cite{Om72}.
\cite{La97} extended the first-order theory of polarized line formation
of \cite{La83} to include partial redistribution effects in a two-term
atom. This was done by describing the atomic states in terms of energy
\emph{metalevels} \citep[or energy \emph{sub-states};][]{WS53},
and proposing a heuristic generalization of the atomic density
matrix based upon the ``metalevel'' idea. Their formalism naturally
reproduces the results of the first-order theory of polarized
line formation, in the limit of complete redistribution (e.g., for a
flat-spectrum illumination of the atom), and it was applied successfully
to the problem of the scattering polarization in the \ion{Na}{1} D-doublet
\citep{La98}.
Finally, \cite{Bo97a,Bo97b} formally extended the line formation
theory of \cite{La83} to higher orders of perturbation, at the same time
relying also on the results of \cite{Ba58} to describe relaxation
effects on the atomic system from the interaction with a thermal bath of
colliding perturbers. The redistribution function for the
two-level atom with unpolarized lower level derived from that theory was
shown to be in agreement with the results of \cite{Om72,Om73},
and of the heuristic approaches of \cite{St94} and \cite{La97} in the
absence of collisions.

In this paper, we introduce in Section~\ref{sec:theory} some fundamental
theoretical
results that lie at the basis of our treatment of partial frequency
redistribution of polarized radiation. In Section~\ref{sec:special} we
re-derive several special forms of the
redistribution function known from the past literature, and also
present a generalization of those previous results that allows for
atomic coherence in the lower term. In Section~\ref{sec:lifetimes}
we briefly discuss the definition of the radiative lifetimes of
atomic states perturbed by the presence of external fields. This is an
important clarification for the applicability of the results here
presented. (As usual, collisional widths can be added to the natural
widths of the atomic level in a phenomenological way, in the limit of
the impact approximation. See, e.g., \citealt{An49,LT71}.)
In Section~\ref{sec:application} we specialize the equation of
radiative transfer with partial redistribution to the general case of a
polarized two-term atom with hyperfine structure. The cases of simpler
atomic structures can be obtained from this more general case, and as an
illustration we re-derive the expression of the scattering emissivity for
the two-term atom with unpolarized lower levels and in the limit
of zero magnetic field, which has been derived previously through the metalevel
approach \citep{La97}. Finally, in
Section~\ref{sec:balance} we verify that the generalized
radiative transfer equation with the inclusion of coherent scattering
satisfies the fundamental condition that the energy flux of the
radiation through
a closed surface containing the scatterer must be zero, in the absence
of collisions.

\section{Theory} \label{sec:theory}
We consider a two-term atom with upper levels $\{u,u',u'',\ldots\}$
and lower levels $\{l,l',l'',\ldots\}$. These levels represent the
energy eigenstates of the atomic Hamiltonian in the presence of the
external fields. In general, we allow for these levels to be
arbitrarily polarized, and therefore we assume the existence of
atomic coherence among the levels, described by the (complex)
non-diagonal elements of the atomic density matrix, such as
$\rho_{uu'}$ and $\rho_{ll'}$. We want to study the scattering of
polarized radiation
by an ensemble of such atoms, the radiation being described by
an arbitrary incident beam of wave vector $\bm k$ and polarization
state $(\lambda,\mu)$, and an emerging beam of given wave vector
$\bm k'$ and polarization state $(\lambda',\mu')$. In particular, we
want to focus on the \emph{coherent} part of the evolution equation of
the radiation field, where the upper levels $u,u'$ enter only as
virtual states. This part only appears when we push the development of
the formal theory to include multi-photon effects \cite[e.g.,][]{Bo97a}.

We thus find that the equation for the transport of polarized
radiation \emph{for the two-term atom}, expressed in the atomic
reference frame, can be written in the form
\begin{equation} \label{eq:RT}
\frac{d}{dt}\,I_{\lambda'\mu'}(\bm{k'})
=
	-\sum_{\lambda\mu}\kappa_{\lambda'\mu'}^{\lambda\mu}(\bm{k'})\,
	I_{\lambda\mu}(\bm{k'})
	+\varepsilon\apx{1}_{\lambda'\mu'}(\bm{k'})
	+\varepsilon\apx{2}_{\lambda'\mu'}(\bm{k'})\;.
\end{equation}
The first-order terms, $\varepsilon\apx{1}_{\lambda'\mu'}(\bm{k'})$ and
$\kappa_{\lambda'\mu'}^{\lambda\mu}(\bm{k'})$, whose explicit expressions
we omit for the moment (see Sect.~\ref{sec:application}),
describe, respectively, the emission of radiation
of wave vector $\bm k'$ and polarization state $(\lambda',\mu')$ due to
the spontaneous de-excitation of the atom from all possible states
$\rho_{uu'}$, and the absorption rate of the
incident radiation propagating \emph{along the same vector} $\bm k'$
(but in all possible polarization states) when the atom is in any
of the possible states $\rho_{ll'}$. We neglect,
here and in the following, any contribution of
stimulated emission to the expression of
$\kappa_{\lambda'\mu'}^{\lambda\mu}(\bm{k'})$. We make this choice in
order to facilitate comparison of our results with those in the former
literature on the subject of partial redistribution.
The second-order term $\varepsilon\apx{2}_{\lambda'\mu'}(\bm{k'})$
describes instead the scattering of radiation by
the atom from all possible lower states $\rho_{ll'}$, via all
possible intermediate (virtual) levels $u,u'$, and final levels $l''$.

Thus, the first-order terms in Equation~(\ref{eq:RT}) account for the
transfer of radiation along the direction $\bm k'$, while the
second-order term describes the phenomenon of coherent scattering.
We note that we use here the term ``coherent'' in the broader sense of
``memory preserving'', rather than in the usual restricted sense of
``frequency preserving''. This seems to be a better interpretation in
light of the theoretical development that led to Equation~(\ref{eq:RT})
\citep[][work in progress]{Ca15}.

The coherent-scattering term has the following expression,
\begin{eqnarray} \label{eq:2emiss}
\varepsilon\apx{2}_{\lambda'\mu'}(\bm{k'})
&\equiv&
	\frac{1}{\hbar^4}\sum_{ll'}\rho_{ll'}\sum_{uu'l''}
	\sum_{\lambda\mu,\bm{k}}
	Q^\ast_{ul''}(\lambda',\bm{k'})
	Q_{u'l''}(\mu',\bm{k'})
	Q_{ul}(\lambda,\bm{k})
	Q_{u'l'}^\ast(\mu,\bm{k})\,
      I_{\lambda\mu}(\bm{k}) \nonumber \\
\noalign{\vskip -6pt}
&&\hphantom{+\frac{1}{\hbar^4}\sum_{ll'}\rho_{ll'}\sum_{uu'l''}
	\sum_{\lambda\mu,\bm{k}}} \times
	\Bigl( \Psi_{u'l',l''ul}^{-k,+k'-k} +
	\bar\Psi_{ul,l''u'l'}^{-k,+k'-k} \Bigr)\;.
\end{eqnarray}
First we note that the sum over $\bm k$ actually stands for a
continuous integration on both frequency and direction of propagation,
so we can operate the usual substitution, which holds in the atomic
frame of rest,
\begin{equation} \label{eq:cont.rad}
\sum_{\bm{k}} I_{\lambda\mu}(\bm{k}) \longrightarrow
	\frac{1}{2\pi^2}\frac{{\cal V}}{c^3}
	\int_0^\infty d\omega_k\,\omega_k^2
	\oint \frac{d\bm{\hat{k}}}{4\pi}\,
	I_{\lambda\mu}(\bm{k})\;,
\end{equation}
where $\cal V$ is the volume of the quantization box. The ``vertex''
form factor, $Q_{ab}(\lambda,\bm{k})$, in the electric-dipole
approximation, is given by \citep[e.g.,][]{LL04}
\begin{equation} \label{eq:rdipole}
Q_{ab}(\lambda,\bm{k})
	=\sqrt{\frac{2\pi e_0^2\hbar}{{\cal V}}\,
	\omega_k}\,
	\sum_q(-1)^q(r_q)_{ab}\,
	(\bm{e}_{\bm{\hat{k}}\lambda})_{-q}\;,
\end{equation}
where we adopted the spherical-tensor representation of the
electric-dipole and polarization unit vectors, $\bm{r}$ and
$\bm{e}_{\bm{\hat{k}}\alpha}$, respectively.

Finally, the complex line profiles $\Psi_{ab,cde}^{\pm h,\pm k\pm l}$
are given by
\begin{eqnarray} \label{eq:Psi}
\Psi_{ab,cde}^{\pm h,\pm k\pm l}
&\equiv&
\frac{-{\rm i}}{
(\omega_{ac}\pm\omega_h\mp\omega_l\mp\omega_k
	+{\rm i}\epsilon_a+{\rm i}\epsilon_c)
(\omega_{ad}\pm\omega_h\mp\omega_l+{\rm i}\epsilon_a+{\rm i}\epsilon_d)
(\omega_{ae}\pm\omega_h+{\rm i}\epsilon_a+{\rm i}\epsilon_e)
} \nonumber \\
&+&\frac{-{\rm i}}{
(\omega_{ac}\pm\omega_h\mp\omega_l\mp\omega_k
	+{\rm i}\epsilon_a+{\rm i}\epsilon_c)
(\omega_{bc}\mp\omega_l\mp\omega_k+{\rm i}\epsilon_b+{\rm i}\epsilon_c)
(\omega_{cd}\pm\omega_k-{\rm i}\epsilon_c+{\rm i}\epsilon_d)
} \nonumber \\
&-&\frac{-{\rm i}}{
(\omega_{ad}\pm\omega_h\mp\omega_l+{\rm i}\epsilon_a+{\rm i}\epsilon_d)
(\omega_{bd}\mp\omega_l+{\rm i}\epsilon_b+{\rm i}\epsilon_d)
(\omega_{cd}\pm\omega_k-{\rm i}\epsilon_c+{\rm i}\epsilon_d)
}\;.
\end{eqnarray}
For notational convenience, we introduced the Bohr frequency,
$\omega_{pq}=\omega_p-\omega_q$, for any two levels $p$ and $q$. The
quantity $\epsilon_p$ represents the width of the level $p$. In
Equation~(\ref{eq:2emiss}), $\bar\Psi$ indicates complex conjugation of the
profile $\Psi$.

\begin{figure}[t!]
\centering
\includegraphics[width=2.65in]{diag1.eps}\vspace{0pt}
\includegraphics[width=\hsize]{diag2.eps}
\caption{\label{fig:diagrams}
Feynman diagrams representing the first-order (i.e., single photon)
processes of absorption and emission (top row), and the second-order
(i.e., two-photon) processes describing scattering, two-photon
absorption, and two-photon emission (bottom row). The central diagram
in the bottom row represents the lowest-order radiative correction to
the atomic propagator. Its main effect is that of ``dressing'' a given
atomic state with a finite lifetime associated with spontaneous
de-excitation.}
\end{figure}

Both Equations~(\ref{eq:2emiss}) and (\ref{eq:Psi}) are derived from
first principles. The supporting theory, which is still work in
progress, follows a diagrammatic approach to the derivation of the
evolution equation for a general atomic system interacting with a
polarized radiation field.
This approach leads to the same perturbation series for the evolution
equation of the coupled system atom+radiation as that of \cite{Bo97a},
and the initial conditions for the system's density matrix are also
handled in a similar way. Somewhat surprisingly, the fundamental
results that we derive separately for the statistical equilibrium of
the atomic system and for the transport of polarized radiation do not
agree with the work of \cite{Bo97a,Bo97b}, although both approaches
lead to the same redistribution function for the two-level atom with
unpolarized lower level and in the absence of collisions.

It is worth spending a few words on the notation adopted for the
profiles $\Psi_{ab,cde}^{\pm h,\pm k\pm l}$. The subscripts identify
the atomic levels involved in a transition, while the superscripts
represent the frequencies of the corresponding emitted ($+$) and
absorbed ($-$) photons. These profiles originate in our theory from
the formal product
of a first-order Feynman diagram with a second-order diagram. The two
diagrams are distinguished by the comma separators in both the subscript
and superscript lists. The time ordering of these Feynman diagrams is
from right to left, in the profile's notation. Thus, in
Equation~(\ref{eq:Psi}), we read the contribution
of a first-order process involving the transition $b\to a$ with exchange
of a photon $\bm{h}$, and at the same time the contribution of a second-order
process involving the transition $e\to d\to c$ with exchange of two
photons, $\bm{l}$ and $\bm{k}$.

A pictorial representation of the
Feynman diagrams for atom-photon interactions up to second order is given
in Figure~\ref{fig:diagrams}. In all diagrams, time is flowing upward.
The first-order diagrams (top row) account for the processes of
single-photon absorption and emission, while the second-order diagrams
(bottom row) describe the scattering of photons (first two diagrams),
as well as two-photon absorption and emission (last two diagrams). The
central diagram in the bottom row represents a typical atomic self-energy
insertion, corresponding to the lowest-order radiative correction to the
energy of the atomic state. This diagram is accounted for in the theory
by ``dressing'' the atomic propagator, which results in the atomic levels
acquiring a finite lifetime because of the possibility of spontaneous
de-excitation.

The process of two-photon absorption is the simplest example of
non-linear optical effect, and can typically be neglected for highly
diluted radiation fields, such as in the case of solar applications.
Diagrams of that type give rise to radiation observables that are
related to \emph{second-order coherence} \cite[e.g.,][]{Lo83,MW95},
and which we explicitly neglect in our treatment, although the impact
of this choice for the inner consistency of the formalism is still to be
fully assessed.

We will also ignore the two-photon emission diagram of
Fig.~\ref{fig:diagrams}, as this does not contribute significantly to
the scattering of radiation in a two-term atom model. In fact, because
of energy conservation, the initial and final states in that diagram,
$p$ and $q$, must belong respectively to the upper and lower terms of
the model atom, and so the intermediate transition through the
virtual state $r$ is constrained by the set of selection rules
that apply to the $p$--$q$ transition. This typically results in a very
low rate for the two-photon emission process. An instructive example
is that of the Lyman-$\alpha$ transition of hydrogen, between the atomic
terms of principal quantum numbers $n=2$ and $n=1$. The Lyman-$\alpha$
emission is dominated by the electric-dipole transition from
$2\,P_{1/2,3/2}$ to $1\,S_{1/2}$, with a lifetime
$\approx 1.6\times 10^{-9}$\,s. The transition between $2\,S_{1/2}$ and
$1\,S_{1/2}$ is forbidden to the lowest order of approximation, but it
can occur via two-photon emission, corresponding to the last diagram
of Fig.~\ref{fig:diagrams}, through a virtual state $r$ belonging to
a $n\,P_J$ term.
Such transition has indeed been observed in the laboratory, and it is
responsible for a measurable lifting of the metastability of the
$1\,S_{1/2}$ state, with a lifetime $\approx 0.14$\,s \citep{BS57}.
However, its contribution compared to the dominant term of
Lyman-$\alpha$ is completely negligible for any practical application
to the polarized line diagnostics of astrophysical plasmas.

In the general case of a multi-term atom, there are additional terms that
must be considered, which originate from different combinations of
Feynman diagrams (involving up to third-order processes, to the degree
of approximation of the theory). \emph{For the two-term
atom, and neglecting stimulation effects, the profiles of
Equation~(\ref{eq:Psi}) are the only ones bringing a contribution to the
scattering of polarized radiation.

After a rather involved algebraic manipulation, the sum of complex
profiles in Equation~(\ref{eq:2emiss}) is shown to be proportional to the
following redistribution function,
\begin{eqnarray}	\label{eq:red.gen}
&&\kern -1cm
{\cal R}(\Omega_u,\Omega_{u'};
	\Omega_l,\Omega_{l'},\Omega_{l''};
	\omega_k,\omega_{k'})
	\equiv
	(\epsilon_{uu'}+{\rm i}\omega_{uu'})
	\left(
	\Psi_{u'l',l''ul}^{-k,+k'-k} + \bar\Psi_{ul,l''u'l'}^{-k,+k'-k}
	\right)
\nonumber \\
&=&
\frac{2\epsilon_{l''}(\epsilon_{ll'}+{\rm i}\omega_{ll'})}{%
	(\omega_k-\omega_{ul'}+{\rm i}\epsilon_{ul'})
       (\omega_k-\omega_{u'l}-{\rm i}\epsilon_{u'l})
	(\omega_{k'}-\omega_{ul''}+{\rm i}\epsilon_{ul''})
	(\omega_{k'}-\omega_{u'l''}-{\rm i}\epsilon_{u'l''})}
\nonumber \\
&+&
\frac{2\epsilon_{l''}
	(\epsilon_{uu'}+{\rm i}\omega_{uu'})}{%
	(\omega_k-\omega_{ul'}+{\rm i}\epsilon_{ul'})
	(\omega_k-\omega_{u'l}-{\rm i}\epsilon_{u'l})
	(\omega_k-\omega_{k'}+\omega_{l'l''}+{\rm i}\epsilon_{l'l''})
	(\omega_k-\omega_{k'}+\omega_{ll''}-{\rm i}\epsilon_{ll''})}
\nonumber \\
&+&
\frac{(\epsilon_{ll'}+{\rm i}\omega_{ll'})
	(\epsilon_{uu'}+{\rm i}\omega_{uu'})}{%
	(\omega_{k'}-\omega_{ul''}+{\rm i}\epsilon_{ul''})
	(\omega_{k'}-\omega_{u'l''}-{\rm i}\epsilon_{u'l''})
	(\omega_k-\omega_{k'}+\omega_{l'l''}+{\rm i}\epsilon_{l'l''})
	(\omega_k-\omega_{k'}+\omega_{ll''}-{\rm i}\epsilon_{ll''})}
\nonumber \\
&+&
\frac{2\epsilon_{l''}
	(\epsilon_{ll'}+{\rm i}\omega_{ll'})
	(\epsilon_{uu'}+{\rm i}\omega_{uu'})}{%
	(\omega_k-\omega_{ul'}+{\rm i}\epsilon_{ul'})
	(\omega_k-\omega_{u'l}-{\rm i}\epsilon_{u'l})
	(\omega_{k'}-\omega_{ul''}+{\rm i}\epsilon_{ul''})
	(\omega_{k'}-\omega_{u'l''}-{\rm i}\epsilon_{u'l''})}
\nonumber \\
&&\kern 1cm \times
\frac{2\epsilon_{l''}+\epsilon_{ll'}+\epsilon_{uu'}
      +{\rm i}(\omega_{ll'}+\omega_{uu'})}{%
	(\omega_k-\omega_{k'}+\omega_{l'l''}+{\rm i}\epsilon_{l'l''})
	(\omega_k-\omega_{k'}+\omega_{ll''}-{\rm i}\epsilon_{ll''})}\;,
\end{eqnarray}
where we introduced the sum of the level widths,
$\epsilon_{ab}=\epsilon_a+\epsilon_b$, and in the argument list of the
function ${\cal R}$ we adopted the shorthand notation
$\Omega_a=\omega_a-{\rm i}\epsilon_a$, to indicate the complex energy of
the level $a$, including its width.

Equation~(\ref{eq:red.gen}) is one of the preliminary results of our
theory of partial redistribution with polarization. Another result
suggests that, \emph{in the absence of collisions
(i.e., at zero temperature), the process that populates the upper term
by radiative absorption is inhibited in the limit of infinitely sharp
lower levels.} In this limit, the first-order contribution to the
emissivity in Equation~(\ref{eq:RT}) vanishes, and all radiation processes
contributing to the emergent radiation depend exclusively on the population
distribution and coherence within the lower term.
The self-consistency of this last result has not yet been fully
demonstrated, and for this reason we cannot provide here a rigorous proof
of the phenomenon of inhibition of one-photon absorption in the limit of
infinitely sharp lower levels. However, such picture is supported also by the following
intuitive physical argument. Since the ideal limit of sharp lower levels
is equivalent to that of infinite lifetime of the same
levels, any measurement aimed at determining the population distribution
of the atomic system would always find the system in the lower state.
In other words, any radiative transition involving the upper levels can
only be virtual, without actually populating the upper state (in the
sense specified above). Thus the corresponding contribution to the emitted
radiation must be fully accounted for by the coherent-scattering
emissivity given in Equation~(\ref{eq:2emiss}), in which only the
density matrix of the lower state explicitly appears.

Equation (\ref{eq:red.gen}) represents the most general form
of redistribution function for a two-term atom with polarized levels
that are partially degenerate or non degenerate. It includes, as
particular cases, the redistribution
functions for the two- and three-level atoms that have formerly been
described in the literature (see Section~\ref{sec:special}).
While it is possible to cast
${\cal R}(\Omega_u,\Omega_{u'}; \Omega_l,\Omega_{l'},\Omega_{l''};
	\omega_k,\omega_{k'})$ in various equivalent forms (see, e.g.,
Equation~(\ref{eq:red.gen.alt}) below) -- all of them descending directly
from Equation~(\ref{eq:Psi}) and the definition in Equation~(\ref{eq:red.gen}) --
the particular form that we adopted for Equation~(\ref{eq:red.gen}) is
best suited for re-deriving those former results in the partial
redistribution literature, as we are now going to show.

\section{Special forms of the redistribution function} \label{sec:special}
We consider in this section a few particular cases of
Equation~(\ref{eq:red.gen}), some of which correspond to well-known results
from the past literature on partial redistribution.
The simplest case is that of a transition between two atomic levels with
fully degenerate sublevels (i.e., for zero magnetic field), so that
$\Omega_u=\Omega_{u'}$ and
$\Omega_l=\Omega_{l'}=\Omega_{l''}$. From Equation~(\ref{eq:red.gen}),
we immediately find
\begin{eqnarray}	\label{eq:red.0}
{\cal R}_0(\Omega_u; \Omega_l; \omega_k,\omega_{k'})
&=&
\frac{4\epsilon_l^2}{%
	\left[(\omega_k-\omega_{ul})^2+\epsilon_{ul}^2\right]\!\!
	\left[(\omega_{k'}-\omega_{ul})^2+\epsilon_{ul}^2\right]}
\nonumber \\
&&\kern -1in
\mathop{+}
\frac{4\epsilon_l \epsilon_u}{%
	\left[(\omega_k-\omega_{ul})^2+\epsilon_{ul}^2\right]\!\!
	\left[(\omega_k-\omega_{k'})^2+4\epsilon_l^2\right]}
	+\frac{4\epsilon_l \epsilon_u}{%
	\left[(\omega_{k'}-\omega_{ul})^2+\epsilon_{ul}^2\right]\!\!
	\left[(\omega_k-\omega_{k'})^2+4\epsilon_l^2\right]}
\nonumber \\
&&\kern -1in
\mathop{+}
\frac{16\epsilon_l^2\epsilon_u(2\epsilon_l+\epsilon_u)}{%
	\left[(\omega_k-\omega_{ul})^2+\epsilon_{ul}^2\right]\!\!
	\left[(\omega_{k'}-\omega_{ul})^2+\epsilon_{ul}^2\right]\!\!
	\left[(\omega_k-\omega_{k'})^2+4\epsilon_l^2\right]}\;,
\end{eqnarray}
which is identical to the redistribution function derived by
\citet[][cf.~their Equation~(VIII.3.33); see also \citealt{Om72},
and \citealt{Mi78}]{WS53} using a metalevel model of the atomic system.

The case of a three-level ``$\Lambda$-type'' atomic system \cite[see,
e.g.,][Figure~2]{LT71}
with fully degenerate sublevels, is obtained by letting
$\Omega_u=\Omega_{u'}$ and
$\Omega_l=\Omega_{l'}\ne\Omega_{l''}$ in Equation~(\ref{eq:red.gen}),
which gives at once
\begin{eqnarray}	\label{eq:red.1}
{\cal R}_1(\Omega_u; \Omega_l,\Omega_{l''};
	\omega_k,\omega_{k'})
&=&
\frac{4\epsilon_l\epsilon_{l''}}{%
      \left[(\omega_k-\omega_{ul})^2+\epsilon_{ul}^2\right]\!\!
	\left[(\omega_{k'}-\omega_{ul''})^2+\epsilon_{ul''}^2\right]}
\nonumber \\
&&\kern -4cm
\mathop{+}
\frac{4\epsilon_{l''}\epsilon_u}{%
	\left[(\omega_k-\omega_{ul})^2+\epsilon_{ul}^2\right]\!\!
	\left[(\omega_k-\omega_{k'}+\omega_{ll''})^2+\epsilon_{ll''}^2\right]}
+\frac{4\epsilon_l\epsilon_u}{%
	\left[(\omega_{k'}-\omega_{ul''})^2+\epsilon_{ul''}^2\right]\!\!
	\left[(\omega_k-\omega_{k'}+\omega_{ll''})^2+\epsilon_{ll''}^2\right]}
\nonumber \\
&&\kern -4cm
\mathop{+}
\frac{16\epsilon_l\epsilon_{l''}\epsilon_u(\epsilon_{ll''}+\epsilon_u)}{%
	\left[(\omega_k-\omega_{ul})^2+\epsilon_{ul}^2\right]\!\!
	\left[(\omega_{k'}-\omega_{ul''})^2+\epsilon_{ul''}^2\right]\!\!
	\left[(\omega_k-\omega_{k'}+\omega_{ll''})^2+\epsilon_{ll''}^2\right]}\;.
\end{eqnarray}
This problem, which describes, for example, the Raman scattering in
subordinate lines, has also been previously considered in the
literature,
e.g., by \citet[][Equation~(3.2); see also \citealt{WS53}, and
\citealt{We33}]{Hu82}.

Of more general interest is the redistribution function for a two-term
atom with non-degenerate levels, and completely relaxed atomic
coherence in the lower term (\textit{non-coherent lower term};
n.c.l.t.). Such a model can
describe the formation of many resonance lines observed in the quiet-Sun
atmosphere, in a regime of magnetic strengths such that the Hanle effect
of the lower levels is saturated, but not so large as to induce
level-crossing interference in the lower term. For example, the
formation
of the \ion{Na}{1} D$_1$-D$_2$ doublet at $\lambda 589$\,nm falls into
this category, for $0.1\,\mathrm{G}\lesssim B\lesssim 300\,\textrm{G}$
\citep{TB02,Ca02}. In order to derive this form of the redistribution
function, we observe that the diagonality condition
$\rho_{ll'}=\delta_{ll'}\rho_{ll}$ for the case of n.c.l.t.\ implies
$\Omega_l=\Omega_{l'}$ in Equation~(\ref{eq:red.gen}). We thus obtain
\begin{eqnarray}	\label{eq:red.nclt}
&&\kern -1cm
{\cal R}(\Omega_u,\Omega_{u'};
	\Omega_l,\Omega_{l''};
	\omega_k,\omega_{k'})_{\hbox{\footnotesize n.c.l.t.}}
\nonumber \\
&=&
\frac{4\epsilon_l\epsilon_{l''}}{%
	(\omega_k-\omega_{ul}+{\rm i}\epsilon_{ul})
       (\omega_k-\omega_{u'l}-{\rm i}\epsilon_{u'l})
	(\omega_{k'}-\omega_{ul''}+{\rm i}\epsilon_{ul''})
	(\omega_{k'}-\omega_{u'l''}-{\rm i}\epsilon_{u'l''})}
\nonumber \\
&+&
\frac{2\epsilon_{l''}
	(\epsilon_{uu'}+{\rm i}\omega_{uu'})}{%
	(\omega_k-\omega_{ul}+{\rm i}\epsilon_{ul})
	(\omega_k-\omega_{u'l}-{\rm i}\epsilon_{u'l})
	\left[(\omega_k-\omega_{k'}+\omega_{ll''})^2+\epsilon_{ll''}^2\right]}
\nonumber \\
&+&
\frac{2\epsilon_l
	(\epsilon_{uu'}+{\rm i}\omega_{uu'})}{%
	(\omega_{k'}-\omega_{ul''}+{\rm i}\epsilon_{ul''})
	(\omega_{k'}-\omega_{u'l''}-{\rm i}\epsilon_{u'l''})
	\left[(\omega_k-\omega_{k'}+\omega_{ll''})^2+\epsilon_{ll''}^2\right]}
\nonumber \\
&+&
\frac{4\epsilon_l\epsilon_{l''}
	(\epsilon_{uu'}+{\rm i}\omega_{uu'})}{%
	(\omega_k-\omega_{ul}+{\rm i}\epsilon_{ul})
	(\omega_k-\omega_{u'l}-{\rm i}\epsilon_{u'l})
	(\omega_{k'}-\omega_{ul''}+{\rm i}\epsilon_{ul''})
	(\omega_{k'}-\omega_{u'l''}-{\rm i}\epsilon_{u'l''})}
\nonumber \\
&&\kern 1cm \times
\frac{2\epsilon_{l''}+2\epsilon_l+\epsilon_{uu'}
	+{\rm i}\omega_{uu'}}{%
	(\omega_k-\omega_{k'}+\omega_{ll''})^2+\epsilon_{ll''}^2}\;.
\end{eqnarray}
Often the initial and final levels of the scattering process, $l$ and
$l''$, represent metastable states, hence characterized by very small
radiative level widths. In this limit of \emph{sharp lower levels} (s.l.l.),
$\epsilon_l,\epsilon_{l''}\to 0$, the first and fourth contributions
in Equation~(\ref{eq:red.nclt}) vanish, and the redistribution function
tends to the well-known result \cite[e.g.,][]{La97}
\begin{eqnarray}	\label{eq:red.sll}
{\cal R}_2(\Omega_u,\Omega_{u'};
	\Omega_l,\Omega_{l''};
	\omega_k,\omega_{k'})
&=& \frac{2\pi\,
	(\epsilon_{uu'}+{\rm i}\omega_{uu'})\,
	\delta(\omega_k-\omega_{k'}+\omega_{ll''})}{%
	(\omega_k-\omega_{ul}+{\rm i}\epsilon_{u})
	(\omega_k-\omega_{u'l}-{\rm i}\epsilon_{u'})}\;,
\end{eqnarray}
where we also considered that
$\omega_{k'}+\omega_{l''}=\omega_k+\omega_l$
(because of the Dirac-$\delta$ function) in order to combine the two
non-vanishing contributions of Equation~(\ref{eq:red.nclt}).

A more general case is that of infinitely sharp lower levels with
the possibility of atomic coherence in the lower term.
The derivation of the corresponding redistribution function from
Equation~(\ref{eq:red.gen}) is not trivial. It is however easily
accomplished if we first recast the general redistribution function
of Equation~(\ref{eq:red.gen}) in the alternative but equivalent form
\begin{eqnarray}	\label{eq:red.gen.alt}
&&\kern -1cm
{\cal R}(\Omega_u,\Omega_{u'};
	\Omega_l,\Omega_{l'},\Omega_{l''};
	\omega_k,\omega_{k'}) \nonumber \\
&=&(\epsilon_{uu'}+{\rm i}\omega_{uu'})\biggl[
	\frac{{\rm i}}{%
	(\omega_k-\omega_{ul}+{\rm i}\epsilon_{u}-{\rm i}\epsilon_{l})
	(\omega_k-\omega_{u'l}-{\rm i}\epsilon_{u'l})
	(\omega_{k'}-\omega_k-\omega_{ll''}+{\rm i}\epsilon_{ll''})}
	\nonumber \\
&&\hphantom{(\epsilon_{uu'}+{\rm i}\omega_{uu'})\biggl[}-
	\frac{{\rm i}}{%
	(\omega_k-\omega_{ul'}+{\rm i}\epsilon_{ul'})
	(\omega_k-\omega_{u'l'}-{\rm i}\epsilon_{u'}+{\rm i}\epsilon_{l'})
	(\omega_{k'}-\omega_k-\omega_{l'l''}-{\rm i}\epsilon_{l'l''})}
	\biggr] \nonumber \\
&+&(\epsilon_{ll'}+{\rm i}\omega_{ll'})\biggl[
	\frac{{\rm i}}{%
	(\omega_k-\omega_{ul}+{\rm i}\epsilon_{u}-{\rm i}\epsilon_{l})
	(\omega_k-\omega_{ul'}+{\rm i}\epsilon_{ul'})
	(\omega_{k'}-\omega_{ul''}+{\rm i}\epsilon_{ul''})}
	\nonumber \\
&&\hphantom{(\epsilon_{ll'}+{\rm i}\omega_{ll'})\biggl[}-
	\frac{{\rm i}}{%
	(\omega_k-\omega_{u'l}-{\rm i}\epsilon_{u'l})
	(\omega_k-\omega_{u'l'}-{\rm i}\epsilon_{u'}+{\rm i}\epsilon_{l'})
	(\omega_{k'}-\omega_{u'l''}-{\rm i}\epsilon_{u'l''})}
	\biggr]\;.
\end{eqnarray}
We also define the generalized function\footnote{This definition
differs from the $\zeta(x)$ function introduced by \cite{He54} for
the presence of ${\rm i}$ at the numerator.}
\begin{equation} \label{eq:zeta}
\zeta(x)\equiv\lim_{\sigma\to 0}\,\frac{{\rm i}}{x+{\rm i}\,\sigma}
	=\pi\,\delta(x)+{\rm i}\;{\rm Pv}\,\frac{1}{x}\;.
\end{equation}
Then, in the limit of $\epsilon_l,\epsilon_{l'},\epsilon_{l''}\to 0$,
the redistribution function for sharp lower levels
follows at once from Equation~(\ref{eq:red.gen.alt}):
\begin{eqnarray} \label{eq:red.sharp}
&&\kern -1cm
{\cal R}(\Omega_u,\Omega_{u'};
	\Omega_l,\Omega_{l'},\Omega_{l''};
	\omega_k,\omega_{k'})_{\rm s.l.l.} \nonumber \\
\noalign{\vskip 3pt}
&=&(\epsilon_{uu'}+{\rm i}\omega_{uu'})
	\biggl[
	\frac{\zeta(\omega_k-\omega_{k'}+\omega_{ll''})}{%
		(\omega_k-\omega_{ul}+{\rm i}\epsilon_{u})
		(\omega_k-\omega_{u'l}-{\rm i}\epsilon_{u'})}
	+\frac{\zeta^*(\omega_k-\omega_{k'}+\omega_{l'l''})}{%
		(\omega_k-\omega_{ul'}+{\rm i}\epsilon_{u})
		(\omega_k-\omega_{u'l'}-{\rm i}\epsilon_{u'})}
	\biggr] \nonumber \\
\noalign{\vskip 6pt}
&-&\omega_{ll'} \biggl[
	\frac{1}{%
		(\omega_k-\omega_{ul}+{\rm i}\epsilon_{u})
		(\omega_k-\omega_{ul'}+{\rm i}\epsilon_{u})
		(\omega_{k'}-\omega_{ul''}+{\rm i}\epsilon_{u})}
	\nonumber \\
&&\hphantom{\omega_{ll'} \biggl[} -
	\frac{1}{%
		(\omega_k-\omega_{u'l}-{\rm i}\epsilon_{u'})
		(\omega_k-\omega_{u'l'}-{\rm i}\epsilon_{u'})
		(\omega_{k'}-\omega_{u'l''}-{\rm i}\epsilon_{u'})}
	\biggr]\;.
\end{eqnarray}
We note that Equation~(\ref{eq:red.sharp}) properly tends to the
redistribution function ${\cal R}_2$ of Equation~(\ref{eq:red.sll}) when
$\omega_{ll'}\to 0$, since there is no contribution from the second square
bracket in this case, and $\zeta(x)+\zeta^*(x)=2\pi\,\delta(x)$.

We conclude this section with the evaluation of the integral norm
of the general redistribution function, Equation~(\ref{eq:red.gen}) or
(\ref{eq:red.gen.alt}). By contour
integration,\footnote{\label{note:domainextend}%
Because $\omega_{ul}\gg\epsilon_{ul}$, the
frequency domain of integration $[0,+\infty)$ can always be extended
backward to $-\infty$ with no appreciable modification of the value of
the integral.} we find
\begin{eqnarray} \label{eq:integral2}
\int_{-\infty}^\infty d\omega_{k'}\;
	{\cal R}(\Omega_u,\Omega_{u'};
	\Omega_l,\Omega_{l'},\Omega_{l''};
	\omega_k,\omega_{k'})
&=& 2\pi\,
	\frac{\epsilon_{uu'}+\epsilon_{ll'}
	+{\rm i}(\omega_{uu'}+\omega_{ll'})}{%
	(\omega_k-\omega_{ul'}+{\rm i}\epsilon_{ul'})
      (\omega_k-\omega_{u'l}-{\rm i}\epsilon_{u'l})
	}\;, \\
\int_{-\infty}^\infty d\omega_{k}\;
	{\cal R}(\Omega_u,\Omega_{u'};
	\Omega_l,\Omega_{l'},\Omega_{l''};
	\omega_k,\omega_{k'})
&=& 2\pi\,
	\frac{\epsilon_{uu'}+2\epsilon_{l''}
	+{\rm i}\omega_{uu'}}{%
	(\omega_{k'}-\omega_{ul''}+{\rm i}\epsilon_{ul''})
      (\omega_{k'}-\omega_{u'l''}-{\rm i}\epsilon_{u'l''})
	}\;,
\end{eqnarray}
and from either of these two expressions it follows immediately that
\begin{equation} \label{eq:red.norm}
\int_{-\infty}^\infty d\omega_k
\int_{-\infty}^\infty d\omega_{k'}\,
	{\cal R}(\Omega_u,\Omega_{u'};
	\Omega_l,\Omega_{l'},\Omega_{l''};
	\omega_k,\omega_{k'})
= 4\pi^2\;.
\end{equation}

The results expressed by Equations~(\ref{eq:integral2})--(\ref{eq:red.norm})
differ from those derived in the theory of \cite{Bo97a,Bo97b}, where the
line shape function in the radiative transfer equation contributed by
higher orders is shown to vanish, when integrated over the frequency of
either the incoming or the outgoing photon. On the other hand, in Bommier's
formalism, the contribution of the first-order emissivity is always present,
even when the lifetime of the lower term is assumed to be infinite and thus
no upper-term excitation should be expected, based on the physical argument
presented at the end of Section~\ref{sec:theory}. Thus, in the work of
\cite{Bo97a}, the redistribution function always results from
the combination of both first- and second-order emissivity terms, even
in the limit of infinitely sharp lower levels.
Accordingly,
redistribution effects are accounted for through a modified emission
coefficient where the distinction between true absorption and
scattering is lost \cite[see also][]{Co82}.

Despite this difference
between our respective formalisms,
in the collisionless regime and for an unpolarized lower
term,
we verified that the redistribution function
${\cal R}(\Omega_u,\Omega_{u'};
	\Omega_l,\Omega_{l''};
	\omega_k,\omega_{k'})_{\hbox{\footnotesize n.c.l.t.}}$ of
Equation~(\ref{eq:red.nclt}) coincides with that derived by
\citeauthor{Bo97b} (\citeyear{Bo97b}, Equations~(43)--(45); for this
demonstration, all level widths in those formulas that correspond
to our $\epsilon_u$ must be replaced by $\epsilon_{ul}$).

The general redistribution function, Equation~(\ref{eq:red.gen})
or (\ref{eq:red.gen.alt}), with all the particular cases considered
in this section, is expressed in the reference frame of the atom. In
realistic applications to the plasma diagnostics of astrophysical
objects, we must generalize this function to the laboratory frame, so
to include the effects of Doppler broadening through the convolution of
the atomic velocity distribution (typical, a Maxwellian). In order not
to burden the presentation of the following discussion, we provide the
details of such derivation in the Appendix.

\section{Interlude: the problem of radiative lifetimes}
\label{sec:lifetimes}

One critical question, in the application of Equations~(\ref{eq:2emiss}) and
(\ref{eq:red.gen}) (or (\ref{eq:red.gen.alt})) to the study of
partial redistribution effects in
the presence of external fields, is the proper definition of the level
widths, $\epsilon_a$, i.e., of the lifetimes of atomic states that are
in general energetically perturbed by the presence of external fields.
If we restrict our considerations to the case of a two-term atom
with sharp lower levels, the problem of such definition concerns only the
excited states $u$ and $u'$ (cf.~Equation~(\ref{eq:red.sharp})). To
the lowest order of approximation, their widths are determined exclusively
by the external fields and by the process of spontaneous de-excitation
(in particular, we are neglecting in this treatment the effects of stimulated
emission). Within such an approximation, the complex frequencies $\Omega_u$
correspond to the eigenvalues of the non-Hermitian operator
\cite[e.g.,][]{CT92}
\begin{equation} \label{eq:non-Hermitean}
K_u=\mathscr{P}_u H_{\rm A}\mathscr{P}_u-{\rm i}\hbar\mathit{\Gamma}_u\;,
\end{equation}
where $H_{\rm A}$ is the Hamiltonian of the atomic system in the
presence of the external fields, $\mathscr{P}_u$ is the projection
operator over the subspace of the states $u$ of that Hamiltonian,
and finally
\begin{equation} \label{eq:damping}
\mathit{\Gamma}_u=\frac{2}{3}\frac{e_0^2}{\hbar c^3}\,\omega_{ul}^3
	\sum_q \mathscr{P}_u\,r_q\,\mathscr{P}_l\,r_q^\dagger\,
      \mathscr{P}_u\;,
\end{equation}
where evidently $\mathscr{P}_l$ is the projection operator over the
subspace of the lower terms, which is radiatively connected to the
subspace of the excited states $u$.

We note how in Equation~(\ref{eq:damping}) we have assumed that the energy
separation between the levels of the upper and lower terms can be
approximated by a single value $\omega_{ul}$. This is an adequate
approximation in most cases of interest for magnetic studies
of the solar atmosphere, where the field strengths at play, and the fine
structure of the atomic terms, are such that the associated energy
separations of the levels are much smaller than $\omega_{ul}$.
Correspondingly, level mixing and
quantum interference are also triggered between relatively close
levels, such that the energy span of the atomic terms (intended here
as the sets of the interfering atomic levels) is much smaller than
their average separation $\omega_{ul}$. Under this approximation, and
in the presence of a magnetic field, it is possible to show that the
\textit{damping matrix} $\mathit{\Gamma}_u$ is diagonal on the basis of the
eigenvectors of $H_{\rm A}$ \cite[e.g.,][]{RC61}. In this case, $K_u$
is also diagonal on the same basis.
Thus, \emph{the level widths for radiative de-excitation correspond to
the diagonal elements of the damping matrix} $\mathit{\Gamma}_u$.

The use of Equation~(\ref{eq:damping}) for determining the line shape of
the scattered radiation is adequate when the frequency of the emitted
photon is near the atomic resonance. On the other hand, away from the
resonance condition (e.g., in the case of Rayleigh scattering in the
far wings of a line), it is questionable whether the lifetime of the
excited state should contain the resonance frequency $\omega_{ul}$.
The treatment of the natural line shape given by \cite{Po64} suggests
in fact that one should replace $\omega_{k'}$ for $\omega_{ul}$ in the
expression of $\mathit{\Gamma}_u$ (cf., in particular, Equations~(8.53) and (8.61)
in \citealt{Po64}). Of course, in the proximity of the atomic resonance
(i.e., on the energy shell), one retrieves the usual form of the
damping matrix as given by Equation~(\ref{eq:damping}).

The problem of the lifetime of the lower term is significantly more
involved. First of all, it is important to remark that, because of the
limitations of our current theory of radiation scattering in the
absence of collisions, no new contributions can arise that are able
to ``dress'' a metastable lower term, and determine thus its radiative
lifetime for transitions towards the upper level. Nonetheless, the
energy widths $\epsilon_l$ of the lower term
properly appear in the general form of the redistribution function,
since those widths are associated with the complex frequencies
$\Omega_l$. In the case of subordinate lines, for example, we can
expect that those widths will have an important contribution from
spontaneous de-excitation, which is again determined through
Equation~(\ref{eq:damping}).
In the case of a metastable state, instead, the radiative level widths can
only be specified \emph{ad hoc}, since they are not provided
self-consistently by the theory. In real cases, such as for a collisional
plasma at a temperature $T>0$, we can expect that the level widths of
a metastable term will predominantly be determined by collisional processes.
In the following, we will then assume that the radiative broadening of
metastable levels can be neglected, and therefore that the radiative
lifetime of such levels is practically infinite.

\section{The two-term atom with hyperfine structure}
\label{sec:application}

Standard manipulations \cite[e.g.,][]{LL04} allow us to express
Equation~(\ref{eq:RT}) by means of the irreducible spherical tensor
formalism, and the emerging polarized radiation $I_{\lambda'\mu'}(\bm{k'})$
in terms of the corresponding Stokes vector,
\begin{equation} \label{eq:RT.1}
\frac{1}{c}\,\frac{d}{dt}\,S_i(\omega_{k'},\bm{\hat{k}'}) =
	-\sum_j \kappa_{ij}(\omega_{k'},\bm{\hat{k}'})\,
		S_j(\omega_{k'},\bm{\hat{k}'})
	+\varepsilon\apx{1}_i(\omega_{k'},\bm{\hat{k}'})
	+\varepsilon\apx{2}_i(\omega_{k'},\bm{\hat{k}'})\;,\qquad (i=0,1,2,3)
\end{equation}
where
\begin{eqnarray} \label{eq:2emiss.1}
\varepsilon\apx{2}_i(\omega_{k'},\bm{\hat{k}'})
&\equiv&\frac{4}{3}\frac{e_0^4}{\hbar^2 c^4}\,{\cal N}\omega_{k'}^4
	\sum_{ll'}\rho_{ll'}\sum_{uu'l''}
	\sum_{qq'}\sum_{pp'}(-1)^{q'+p'}\,
	(r_q)_{ul}(r_{q'})^\ast_{u'l'}
	(r_p)_{u'l''}(r_{p'})^\ast_{ul''} \\
&&\times
	\sum_{KQ}\sum_{K'Q'}\sqrt{(2K+1)(2K'+1)}\,
	\thrj{1}{1}{K}{-q}{q'}{-Q}
	\thrj{1}{1}{K'}{-p}{p'}{-Q'}\,
	T^{K'}_{Q'}(i,\bm{\hat{k}'}) \nonumber \\
&&\times \int_0^\infty d\omega_k
	\left(
	\Psi_{u'l',l''ul}^{-k,+k'-k} + \bar\Psi_{ul,l''u'l'}^{-k,+k'-k}
	\right)
	J^K_Q(\omega_k)\;.\qquad (i=0,1,2,3) \nonumber
\end{eqnarray}
The geometric tensors $T^K_Q(i,\bm{\hat{k}})$ were introduced by
\cite{La84}, and are conveniently tabulated by
\citeauthor{Bo97b} (\citeyear{Bo97b}; see also \citealt{LL04}).
The radiation tensors $J^K_Q(\omega_k)$ are defined in terms of
the incident Stokes vector as follows,
\begin{equation} \label{eq:Jtens}
J^K_Q(\omega_k)=\oint \frac{d\bm{\hat{k}}}{4\pi}\,
      \sum_{j=0}^3 T^K_Q(j,\bm{\hat{k}})\,S_j(\omega_k,\bm{\hat{k}})\;.
\end{equation}

It is instructive to compare the form of Equation~(\ref{eq:RT.1}) with
the usual expression of the radiative transfer equation for unpolarized
light \cite[e.g.,][]{Mi78,Sh91} including the contribution from
non-monochromatic scattering,
\begin{eqnarray} \label{eq:scalar.RT}
\frac{1}{c}\,\frac{d}{dt}\,I(\omega_{k'},\bm{\hat{k}'}) =
	&-& \bigl[\kappa^\mathrm{abs}(\omega_{k'}) +
		 \kappa^\mathrm{sca}(\omega_{k'})\bigr]
		I(\omega_{k'},\bm{\hat{k}'}) \nonumber \\
	&+&\varepsilon(\omega_{k'},\bm{\hat{k}'})
	+\oint \frac{d\bm{\hat{k}}}{4\pi} \int_0^\infty d\omega_k\;
	 \chi(\omega_{k'},\bm{\hat{k}'};\omega_k,\bm{\hat{k}})\,
	I(\omega_k,\bm{\hat{k}})\;.
\end{eqnarray}
We see that the last term with the double integral corresponds to
$\varepsilon\apx{2}_i(\omega_{k'},\bm{\hat{k}'})$ of
Equation~(\ref{eq:2emiss.1}) for the case of unpolarized
radiation, as it becomes evident when we
specialize Equation~(\ref{eq:Jtens}) to that case (i.e.,
$S_1=S_2=S_3=0$). In particular, for monochromatic
(e.g., Rayleigh) scattering,
$\chi(\omega_{k'},\bm{\hat{k}'};\omega_k,\bm{\hat{k}})\equiv
\chi_{\omega_{k'}}(\bm{\hat{k}'},\bm{\hat{k}})\,\delta(\omega_k-\omega_{k'})$,
and Equation~(\ref{eq:scalar.RT}) becomes formally identical to Equation~(1.19) of
\cite{Sh91}.

In the particular case of sharp lower levels, and in the
absence of collisions, the 1st-order emission term in the transfer
equation vanishes. As we anticipated in the discussion following
Equation~(\ref{eq:red.gen}), this condition follows from the derivation
of the statistical equilibrium equations according to the formalism on
which the results presented in this paper are based.
Thus, in the case of sharp lower levels, and in the absence of
collisions, the only 1st-order contribution to Equation~(\ref{eq:RT.1})
is represented by the absorption term.
Because there is no excitation of the upper levels (due to the infinite
lifetime of the lower levels), in this case the absorption term does not
originate from true photon absorption, and corresponds instead to the atom's
cross-section for the coherent scattering of radiation
(see Sect.~\ref{sec:balance}). In contrast, when
the lifetime of the lower levels for transitions towards
the upper term is finite, one must also account for the emissivity
term of the first order in Equation~(\ref{eq:RT.1}), which is associated
with the spontaneous de-excitation of the atom following the
excitation of the upper term.

In this section we want to specialize Equation~(\ref{eq:RT.1}) to the case of a
two-term atom with hyperfine structure, in the presence of a magnetic
field. This is a general enough model to encompass most of the
chromospheric lines of interest for magnetic diagnostics. The formalism is
similar to that of \cite{CM05}, so we extensively use the same notation of
that paper.

If we indicate
with $\alpha_l$ and $\alpha_u$ the electronic configurations of the
lower and upper terms, respectively, and assume the direction of the
magnetic field as the quantization axis ($z$-axis), then the atomic
states involved in Equation~(\ref{eq:RT.1}) are of the form
\begin{eqnarray*}
l&\equiv&\alpha_l I\mu_l M_l\;,\quad
l'\equiv\alpha_l I\mu_l' M_l'\;,\quad
l''\equiv\alpha_l I\mu_l'' M_l''\;,\\
u&\equiv&\alpha_u I\mu_u M_u\;,\quad
u'\equiv\alpha_u I\mu_u' M_u'\;,
\end{eqnarray*}
where $M$ is the projection of the total angular momentum $\bm{F}$ on
the $z$-axis, $I$ is the quantum number of the nuclear spin, while
$\mu$ is the index of the atomic Hamiltonian eigenbasis spanning the
subspace of all the quantum numbers $J$ and $F$ that are associated with
a given value of $M$.  We then can write
\begin{displaymath}
\ket{\alpha I\mu M}=\sum_{JF} C^{JF}_\mu(\alpha I M)\,
	\ket{\alpha IJFM}\;,
\end{displaymath}
with the orthogonality conditions
\begin{equation} \label{eq:diag.prop}
\sum_{JF} C^{JF}_\mu(\alpha I M)\,C^{JF}_{\mu'}(\alpha I M)
	=\delta_{\mu\mu'}\;,\qquad
\sum_\mu C^{JF}_\mu(\alpha I M)\,C^{J'F'}_\mu(\alpha I M)
	=\delta_{JJ'}\delta_{FF'}\;.
\end{equation}

The density matrix element for the lower state, $\rho_{ll'}$, can be
written in terms of the irreducible spherical tensor components of the
statistical operator,
\begin{eqnarray}
\rho_{ll'}
&\equiv&\bra{\alpha_l I\mu_l M_l}\rho\ket{\alpha_l I \mu_l' M_l'}
	\nonumber \\
\noalign{\vspace{6pt}}
&=&\sum_{J_l F_l}\sum_{J_l' F_l'}
	C^{J_l F_l}_{\mu_l}(\alpha_l I M_l)\,
	C^{J_l' F_l'}_{\mu_l'}(\alpha_l I M_l') \nonumber \\
\noalign{\vspace{-6pt}}
&&\hphantom{\sum_{J_l F_l}\sum_{J_l' F_l'}} \times
	\sum_{K_l Q_l} (-1)^{F_l-M_l}\sqrt{2K_l+1}\,
	\thrj{F_l}{F_l'}{K_l}{M_l}{-M_l'}{-Q_l}\,
	{}^{\alpha_l I}\!\rho^{K_l}_{Q_l}(J_l F_l,J_l' F_l')\;.
\end{eqnarray}
Finally, using the Wigner-Eckart theorem and its corollaries
\citep[e.g.,][]{BS93}, we
derive the following expression of the dipole matrix element,
\begin{eqnarray}
(r_q)_{ul}
&\equiv&\bra{\alpha_u I\mu_u M_u}r_q\ket{\alpha_l I \mu_l M_l}
	\nonumber \\
\noalign{\vspace{6pt}}
&=&\sum_{J_u F_u}\sum_{J_l F_l}
	C^{J_u F_u}_{\mu_u}(\alpha_u I M_u)\,
	C^{J_l F_l}_{\mu_l}(\alpha_l I M_l)\,
	\sqrt{(2J_u+1)(2F_u+1)(2F_l+1)}\;
	\redmat{\alpha_u J_u}{\bm{r}}{\alpha_l J_l} \nonumber \\
\noalign{\vspace{-6pt}}
&&\hphantom{\sum_{J_u F_u}\sum_{J_l F_l}} \times
	(-1)^{I+J_u-M_u}
	\thrj{F_u}{F_l}{1}{-M_u}{M_l}{q}
	\sixj{F_u}{F_l}{1}{J_l}{J_u}{I}\;.
\end{eqnarray}

To further proceed, we will assume that the two-term atom is
adequately described within the $LS$-coupling scheme, so that
$\alpha\equiv\beta LS$, where $\beta$ identifies a particular $LS$
term of the atom. We then can write additionally
\begin{eqnarray}
\redmat{\alpha_u J_u}{\bm{r}}{\alpha_l J_l}
&\equiv&
	\redmat{\beta_u L_u S J_u}{\bm{r}}{\beta_l L_l S J_l}\;.
	\nonumber \\
&=&(-1)^{1+L_u+S+J_l} \sqrt{(2L_u+1)(2J_l+1)}\;
	\sixj{J_u}{J_l}{1}{L_l}{L_u}{S}\,
	\redmat{\beta_u L_u}{\bm{r}}{\beta_l L_l}\;.
\end{eqnarray}
The reduced matrix element in the last line can be expressed
in terms of the Einstein $B_{lu}$ coefficient for absorption from the
lower to the upper term of the atom,
\begin{equation}
B_{lu}=\frac{16\pi^3}{3}\frac{e_0^2}{\hbar^2 c}\,
	\frac{2L_u+1}{2L_l+1}\,
	|\redmat{\beta_u L_u}{\bm{r}}{\beta_l L_l}|^2\;,
\end{equation}
or alternatively through the Einstein coefficient for spontaneous emission
(cf.~Equation~(\ref{eq:damping})),
\begin{equation} \label{eq:Einstein A}
A_{ul}
=\frac{4}{3}\frac{e_0^2}{\hbar c^3}\,
	\omega_{ul}^3\,
	|\redmat{\beta_u L_u}{\bm{r}}{\beta_l L_l}|^2
\equiv\frac{4}{3}\frac{e_0^2}{\hbar c^3}\,
	\omega_{ul}^3
	\sum_{ql}|(r_q)_{ul}|^2
\;.
\end{equation}

It is relevant at this point to follow up on the discussion
presented at the end of Section~\ref{sec:lifetimes}, with regard to the
proper expression that must be adopted for the natural width of the
excited state $u$, under excitation conditions that are away from the
atomic resonance.
Because of the conclusions of that discussion, we note that we cannot
make the usual identification $A_{ul}=2\epsilon_u$, which is adequate
only near the atomic resonance, and we must adopt instead for
$\epsilon_u$ the more general expression
\begin{equation} \label{eq:gen_width}
\epsilon_u\equiv\epsilon_u(\omega_{k'})=\frac{1}{2}\,A_{ul}\,
	\frac{\omega_{k'}^3}{\omega_{ul}^3}\;.
\end{equation}
This result will be used in Section~\ref{sec:balance}. It is worth
noting that \cite{He54} proposed that one should rather use
\begin{displaymath}
\epsilon_u\equiv\epsilon_u(\omega_{k'})=\frac{1}{2}\,A_{ul}\,
	\frac{\omega_{k'}}{\omega_{ul}}\;.
\end{displaymath}
However, \cite{Po64} pointed out that such alternative expression of the
line width is a consequence of adopting the $\bm{p\cdot A}$ form for
the atom-photon interaction Hamiltonian (\emph{velocity gauge}), rather
than the dipolar form $\bm{d\cdot E}$ (\emph{length gauge}; see footnote
at p.~116 of \citealt{Po64}).
Since in this paper we consistently use the electric-dipole interaction
Hamiltonian in the length gauge, Equation~(\ref{eq:gen_width}) represents
the appropriate form of the line width that must be adopted in our
treatment of radiation scattering.

Using Equations~(21)--(25), Equation~(\ref{eq:2emiss.1}) becomes
\begin{eqnarray} \label{eq:RT.JF}
\varepsilon\apx{2}_i(\omega_{k'},\bm{\hat{k}'})
&=& \frac{3}{16\pi^3}\,
	{\cal N}\hbar\,\frac{\omega_{k'}^4}{\omega_{ul}^3}\,
	\Pi_{L_u L_l}^2 A_{ul} B_{lu} \\
&&\kern -1.5cm\times
	\sum_{J_u J_u' J_u'' J_u'''}
	\sum_{J_l J_l' J_l'' J_l'''}
	\sum_{F_u F_u' F_u'' F_u'''}
	\sum_{F_l F_l' F_l'' F_l'''}
	(-1)^{J_u+J_u'+J_u''+J_u'''}
	(-1)^{J_l+J_l'+J_l''+J_l'''}
	\nonumber \\
&&\mathop{\times}
	\Pi_{J_u J_u' J_u'' J_u'''}\,
	\Pi_{J_l J_l' J_l'' J_l'''}\,
	\sixj{J_u}{J_l}{1}{L_l}{L_u}{S}
	\sixj{J_u'}{J_l'}{1}{L_l}{L_u}{S}
	\sixj{J_u''}{J_l''}{1}{L_l}{L_u}{S}
	\sixj{J_u'''}{J_l'''}{1}{L_l}{L_u}{S} \nonumber \\
&&\mathop{\times}
	\Pi_{F_u F_u' F_u'' F_u'''}\,
	\Pi_{F_l F_l' F_l'' F_l'''}\,
	\sixj{F_u}{F_l}{1}{J_l}{J_u'}{I}
	\sixj{F_u'}{F_l'}{1}{J_l'}{J_u}{I}
	\sixj{F_u''}{F_l''}{1}{J_l''}{J_u''}{I}
	\sixj{F_u'''}{F_l'''}{1}{J_l'''}{J_u'''}{I} \nonumber \\
&&\kern -1.5cm \times
	\sum_{\mu_u M_u}\sum_{\mu_u' M_u'}\sum_{\mu_l'' M_l''}
	C^{J_u F_u}_{\mu_u}(M_u)\,
	C^{J_u'' F_u''}_{\mu_u}(M_u)\,
	C^{J_u' F_u'}_{\mu_u'}(M_u')\,
	C^{J_u''' F_u'''}_{\mu_u'}(M_u')\,
	C^{J_l'' F_l''}_{\mu_l''}(M_l'')\,
	C^{J_l''' F_l'''}_{\mu_l''}(M_l'') \nonumber \\
&&\kern -1.5cm \times
	\sum_{\bar J_l \bar J_l'}
	\sum_{\bar F_l \bar F_l'}
	\sum_{\mu_l M_l}\sum_{\mu_l' M_l'}
	C^{J_l F_l}_{\mu_l}(M_l)\,
	C^{\bar J_l \bar F_l}_{\mu_l}(M_l)\,
	C^{J_l' F_l'}_{\mu_l'}(M_l')\,
	C^{\bar J_l' \bar F_l'}_{\mu_l'}(M_l') \nonumber \\
&&\kern -1.5cm \times
	\sum_{KQ}\sum_{K'Q'}\sum_{K_l Q_l}
	\sum_{qq'} \sum_{pp'}
	(-1)^{\bar F_l-M_l+q'+p'}
	\thrj{1}{1}{K}{-q}{q'}{-Q}
	\thrj{1}{1}{K'}{-p}{p'}{-Q'}
	\thrj{\bar F_l}{\bar F_l'}{K_l}{M_l}{-M_l'}{-Q_l}
	\nonumber \\
&&\mathop{\times}
	\thrj{F_u}{F_l}{1}{-M_u}{M_l}{q}
	\thrj{F_u'}{F_l'}{1}{-M_u'}{M_l'}{q'}
	\thrj{F_u'''}{F_l'''}{1}{-M_u'}{M_l''}{p}
	\thrj{F_u''}{F_l''}{1}{-M_u}{M_l''}{p'}
	\nonumber \\
&&\mathop{\times}
	\Pi_{KK'K_l}\,
	T^{K'}_{Q'}(i,\bm{\hat{k}'})\,
	\rho^{K_l}_{Q_l}(\bar J_l \bar F_l,\bar J_l' \bar F_l')
	\nonumber \\
&&\kern -1.5cm\times
      \sum_{j=0}^3 \oint \frac{d\bm{\hat{k}}}{4\pi}\,
	T^K_Q(j,\bm{\hat{k}})
	\int_0^\infty d\omega_k
	\left(
	\Psi_{u'l',l''ul}^{-k,+k'-k} + \bar\Psi_{ul,l''u'l'}^{-k,+k'-k}
	\right)
	S_j(\omega_k,\bm{\hat{k}})\;,\qquad (i=0,1,2,3)
	\nonumber
\end{eqnarray}
where for simplicity of notation we removed the information of the
term from the density matrix element and from the argument of the
projection coefficients $C^{JF}_\mu$ of the atomic eigenstates, as
well as defining
\begin{equation}
\Pi_{ab\ldots}\equiv\sqrt{(2a+1)(2b+1)\cdots}\;.
\end{equation}

The quantities of the first order appearing in Equation~(\ref{eq:RT.1}),
which represent single-photon absorption and spontaneous emission, for
the case of the multi-term atom with hyperfine structure, have been given
by \citeauthor{CM05}
(\citeyear{CM05}; cf.~their Equations~(32a,b) and (33a,b)). For
convenience, we reproduce those expressions using the same notation adopted
in this paper:
\begin{eqnarray} \label{eq:emiss.JF}
\varepsilon\apx{1}_i(\omega_{k'},\bm{\hat{k}'})
&=&\frac{\sqrt3}{8\pi^2}\,
	{\cal N}\hbar\,\frac{\omega_{k'}^4}{\omega_{ul}^3}\,
	\Pi_{L_u}^2 A_{ul}
	\sum_{J_u J_u'}
	\sum_{J_l J_l'}
	\sum_{F_u F_u'}
	\sum_{F_l F_l'}
	(-1)^{J_u'-J_u+J_l'-J_l} \\
&&\mathop{\times}
	\Pi_{J_u J_u'} \Pi_{J_l J_l'}
	\Pi_{F_u F_u'} \Pi_{F_l F_l'}\,
	\sixj{J_u}{J_l}{1}{L_l}{L_u}{S}
	\sixj{J_u'}{J_l'}{1}{L_l}{L_u}{S}
	\sixj{F_u}{F_l}{1}{J_l}{J_u}{I}
	\sixj{F_u'}{F_l'}{1}{J_l'}{J_u'}{I}
	\nonumber \\
&&\kern -1.5cm\times
	\sum_{\bar J_u \bar J_u'}
	\sum_{\bar F_u \bar F_u'}
	\sum_{\mu_u M_u}\sum_{\mu_u' M_u'}\sum_{\mu_l M_l}
	C^{J_u F_u}_{\mu_u}(M_u)\,
	C^{\bar J_u \bar F_u}_{\mu_u}(M_u)\,
	C^{J_u' F_u'}_{\mu_u'}(M_u')\,
	C^{\bar J_u' \bar F_u'}_{\mu_u'}(M_u')\,
	C^{J_l F_l}_{\mu_l}(M_l)\,
	C^{J_l' F_l'}_{\mu_l}(M_l) \nonumber \\
\noalign{\eject}
&&\kern -1.5cm\times
	\sum_{KQ}\sum_{K_u Q_u}
	\sum_{qq'}
	(-1)^{\bar F_u-M_u'+q'+1}
	\thrj{1}{1}{K}{-q}{q'}{-Q}\,
	\thrj{\bar F_u}{\bar F_u'}{K_u}{M_u}{-M_u'}{-Q_u}
	\thrj{F_u'}{F_l'}{1}{-M_u'}{M_l}{q}
	\thrj{F_u}{F_l}{1}{-M_u}{M_l}{q'}
	\nonumber \\
&&\mathop{\times}
	\Pi_{K K_u}\,
	T^K_Q(i,\bm{\hat{k}'})\,
	\rho^{K_u}_{Q_u}(\bar J_u \bar F_u,\bar J_u' \bar F_u')
	\left(
	\Phi_{lu'}^{+k'} + \bar\Phi_{lu}^{+k'}
	\right)\;,\qquad (i=0,1,2,3) \nonumber
\end{eqnarray}
\begin{eqnarray} \label{eq:absorb.JF}
\eta_i(\omega_{k'},\bm{\hat{k}'})
&=&\frac{\sqrt3}{8\pi^2}\,
	{\cal N}\hbar\omega_{k'}\,
	\Pi_{L_l}^2 B_{lu}
	\sum_{J_u J_u'}
	\sum_{J_l J_l'}
	\sum_{F_u F_u'}
	\sum_{F_l F_l'}
	(-1)^{J_u'-J_u+J_l'-J_l} \\
&&\mathop{\times}
	\Pi_{J_u J_u'} \Pi_{J_l J_l'}
	\Pi_{F_u F_u'} \Pi_{F_l F_l'}\,
	\sixj{J_u}{J_l}{1}{L_l}{L_u}{S}
	\sixj{J_u'}{J_l'}{1}{L_l}{L_u}{S}
	\sixj{F_u}{F_l}{1}{J_l}{J_u}{I}
	\sixj{F_u'}{F_l'}{1}{J_l'}{J_u'}{I}
	\nonumber \\
&&\kern -1.5cm\times
	\sum_{\bar J_l \bar J_l'}
	\sum_{\bar F_l \bar F_l'}
	\sum_{\mu_l M_l}\sum_{\mu_l' M_l'}\sum_{\mu_u M_u}
	C^{J_l F_l}_{\mu_l}(M_l)\,
	C^{\bar J_l \bar F_l}_{\mu_l}(M_l)\,
	C^{J_l' F_l'}_{\mu_l'}(M_l')\,
	C^{\bar J_l' \bar F_l'}_{\mu_l'}(M_l')\,
	C^{J_u F_u}_{\mu_u}(M_u)\,
	C^{J_u' F_u'}_{\mu_u}(M_u) \nonumber \\
&&\kern -1.5cm\times
	\sum_{KQ}\sum_{K_l Q_l}
	\sum_{qq'}
	(-1)^{\bar F_l-M_l+q'+1}
	\thrj{1}{1}{K}{-q}{q'}{-Q}\,
	\thrj{\bar F_l}{\bar F_l'}{K_l}{M_l}{-M_l'}{-Q_l}
	\thrj{F_u}{F_l}{1}{-M_u}{M_l}{q}
	\thrj{F_u'}{F_l'}{1}{-M_u}{M_l'}{q'}
	\nonumber \\
&&\mathop{\times}
	\Pi_{K K_l}\,
	T^K_Q(i,\bm{\hat{k}'})\,
	\rho^{K_l}_{Q_l}(\bar J_l \bar F_l,\bar J_l' \bar F_l')
	\left(
	\Phi_{ul'}^{-k'} + \bar\Phi_{ul}^{-k'}
	\right)\;,\qquad (i=0,1,2,3) \nonumber
\end{eqnarray}
having also defined
\begin{equation} \label{eq:Phi}
\Phi_{ab}^{\pm k}
\equiv \frac{\rm i}{\omega_{ba}\mp\omega_k+{\rm i}\epsilon_{ba}}\;,
\end{equation}
where $\ket{a}$ and $\ket{b}$ are in general eigenstates of
the atomic Hamiltonian including the contributions of the external
fields. Again, we used the notation $\bar{\Phi}$ to indicate the
complex conjugate of the profile $\Phi$.

The magneto-optical coefficients $\rho_i(\omega_{k'},\bm{\hat{k}'})$,
for $i=1,2,3$ \cite[see Equation~(32a) of][]{CM05}, are obtained
from the expression of $\eta_i(\omega_{k'},\bm{\hat{k}'})$,
Equation~(\ref{eq:absorb.JF}), through the following substitution,
\begin{displaymath}
	\left(
	\Phi_{ul'}^{-k'} + \bar\Phi_{ul}^{-k'}
	\right) \longrightarrow
	{\rm i} \left(
	\Phi_{ul'}^{-k'} - \bar\Phi_{ul}^{-k'}
	\right)\;.
\end{displaymath}

The elements of the absorption matrix
$\kappa_{ij}(\omega_{k'},\bm{\hat{k}'})$, which appear in
Equation~(\ref{eq:RT.1}), are related to the coefficients
$\eta_i(\omega_{k'},\bm{\hat{k}'})$ and
$\rho_i(\omega_{k'},\bm{\hat{k}'})$ via the following relations
\begin{gather}
\label{eq:dichroic}
\kappa_{ii}=\eta_0\;,\qquad
\kappa_{0i}=\kappa_{i0}=\eta_i\;,\qquad(i=0,1,2,3) \\
\label{eq:magnetooptical}
\kappa_{ij}=\epsilon_{ijk}\,\rho_k\;.\qquad(i,j,k=1,2,3)
\end{gather}
where $\epsilon_{ijk}$ is the fully antisymmetric Levi-Civita tensor.

As a special case, we consider the limit of zero magnetic field for
Equations~(\ref{eq:RT.JF}), (\ref{eq:emiss.JF}), and (\ref{eq:absorb.JF}).
In that case, there is no dependence of the line profiles
on the $\mu$-indices, and so
we can use the orthogonality properties (\ref{eq:diag.prop}) to
perform the trivial summations over those indices. In addition, it is
possible to sum over all magnetic quantum numbers, as well as over the
indices $(q,q')$ and $(p,p')$. After some tedious Racah-algebra
manipulations \cite[e.g.,][]{Va88}, we find
\begin{eqnarray}	\label{eq:RT.JF.noB}
\varepsilon\apx{2}_i(\omega_{k'},\bm{\hat{k}'})\biggr|_{B=0}
&=& \frac{3}{16\pi^3}\,
	{\cal N}\hbar\,\frac{\omega_{k'}^4}{\omega_{ul}^3}\,
	\Pi_{L_u L_l}^2 A_{ul} B_{lu}
	\sum_{J_l J_l'J_l''}
	\sum_{J_u J_u'}
	\sum_{F_l F_l'F_l''}
	\sum_{F_u F_u'}
	(-1)^{F_u'+F_l''+1}\,
	\\
&&\mathop{\times}
	\Pi_{J_l J_l'}
	\Pi_{J_l''J_u J_u'}^2
	\sixj{J_u}{J_l}{1}{L_l}{L_u}{S}
	\sixj{J_u'}{J_l'}{1}{L_l}{L_u}{S}
	\sixj{J_u}{J_l''}{1}{L_l}{L_u}{S}
	\sixj{J_u'}{J_l''}{1}{L_l}{L_u}{S}
	\nonumber \\
&&\mathop{\times}
	\Pi_{F_l F_l'}
	\Pi_{F_l''F_u F_u'}^2
	\sixj{F_u}{F_l}{1}{J_l}{J_u}{I}
	\sixj{F_u'}{F_l'}{1}{J_l'}{J_u'}{I}
	\sixj{F_u}{F_l''}{1}{J_l''}{J_u}{I}
	\sixj{F_u'}{F_l''}{1}{J_l''}{J_u'}{I}
	\nonumber \\
&&\kern -2.5cm\times
	\sum_{KQ}\sum_{K'Q'}\sum_{K_l Q_l}
	(-1)^{K_l-Q_l}\,\Pi_{KK'K_l}
	\thrj{K}{K'}{K_l}{Q}{Q'}{-Q_l}
	\sixj{K'}{F_u}{F_u'}{F_l''}{1}{1}
	\ninj{K}{K'}{K_l}{1}{F_u'}{F_l'}{1}{F_u}{F_l}
	\nonumber \\
\noalign{\vspace{-3pt}}
&&\mathop{\times}
	T^{K'}_{Q'}(i,\bm{\hat{k}'})\,
	\rho^{K_l}_{Q_l}(J_l F_l,J_l'F_l')
	\nonumber \\
&&\kern -2.5cm\times
      \sum_{j=0}^3 \oint \frac{d\bm{\hat{k}}}{4\pi}\,
	T^K_Q(j,\bm{\hat{k}})
	\int_0^\infty d\omega_k
	\left(
	\Psi_{u'l',l''ul}^{-k,+k'-k} + \bar\Psi_{ul,l''u'l'}^{-k,+k'-k}
	\right)
	S_j(\omega_k,\bm{\hat{k}})\;,\qquad (i=0,1,2,3)
	\nonumber
\end{eqnarray}
whereas the contributions of the first-order terms for this
special case of vanishing magnetic fields are given by\footnote{%
We take here the opportunity to correct a statement given by \cite{CM05}.
Equations~(34) in
that paper are only valid when both the fine and hyperfine structures
of the levels -- and not just the Zeeman splitting, as originally
claimed in the paper -- can also be neglected with respect to the width
of the line profile. In particular, those equations hold for the
frequency-integrated polarization. The case of zero magnetic field is
instead properly represented by Equations~(\ref{eq:emiss1}) and
(\ref{eq:absorb1}) given here.}
\begin{eqnarray} \label{eq:emiss1}
\varepsilon\apx{1}_i(\omega_{k'},\bm{\hat{k}'})\biggr|_{B=0}
&=&\frac{\sqrt3}{8\pi^2}\,
	{\cal N}\hbar\,\frac{\omega_{k'}^4}{\omega_{ul}^3}\,
	\Pi_{L_u}^2 A_{ul}
	\sum_{J_u J_u'}
	\sum_{F_u F_u'}
	\sum_{J_l F_l}
	(-1)^{J_u-J_u'+F_u+F_l+1}\,
	\Pi_{J_u J_u'F_u F_u'}\Pi_{J_l F_l}^2 \\
&&\mathop{\times}
	\sixj{J_u}{J_l}{1}{L_l}{L_u}{S}
	\sixj{J_u'}{J_l}{1}{L_l}{L_u}{S}
	\sixj{F_u}{F_l}{1}{J_l}{J_u}{I}
	\sixj{F_u'}{F_l}{1}{J_l}{J_u'}{I}
	\nonumber \\
&&\kern -1cm\times
	\sum_{KQ}\,
	\sixj{K}{F_u}{F_u'}{F_l}{1}{1}\;
	T^K_Q(i,\bm{\hat{k}'})\,
	\rho^K_Q(J_u F_u,J_u'F_u')
	\left(\Phi^{+k'}_{lu'}+\bar\Phi^{+k'}_{lu}\right)\;,
	\qquad (i=0,1,2,3) \nonumber
\end{eqnarray}
\begin{eqnarray} \label{eq:absorb1}
\eta_i(\omega_{k'},\bm{\hat{k}'})\biggr|_{B=0}
&=&\frac{\sqrt3}{8\pi^2}\,
	{\cal N}\hbar\omega_{k'}\,
	\Pi_{L_l}^2 B_{lu}
	\sum_{J_l J_l'}
	\sum_{F_l F_l'}
	\sum_{J_u F_u}
	(-1)^{J_l'-J_l+F_u+F_l+1}\,
	\Pi_{J_l J_l'F_l F_l'}\Pi_{J_u F_u}^2 \\
&&\mathop{\times}
	\sixj{J_u}{J_l}{1}{L_l}{L_u}{S}
	\sixj{J_u}{J_l'}{1}{L_l}{L_u}{S}
	\sixj{F_u}{F_l}{1}{J_l}{J_u}{I}
	\sixj{F_u}{F_l'}{1}{J_l'}{J_u}{I}
	\nonumber \\
&&\kern -1cm\times
	\sum_{KQ}(-1)^K\,
	\sixj{K}{F_l}{F_l'}{F_u}{1}{1}\;
	T^K_Q(i,\bm{\hat{k}'})\,
	\rho^K_Q(J_l F_l,J_l'F_l')
	\left(\Phi^{-k'}_{ul'}+\bar\Phi^{-k'}_{ul}\right)\;.
	\qquad (i=0,1,2,3) \nonumber
\end{eqnarray}

In the absence of hyperfine structure ($I=0$), Equation~(\ref{eq:RT.JF.noB})
reduces to
\begin{eqnarray}
\varepsilon\apx{2}_i(\omega_{k'},\bm{\hat{k}'})\biggr|_{B=0}
&=&\frac{3}{16\pi^3}\,
	{\cal N}\hbar\,\frac{\omega_{k'}^4}{\omega_{ul}^3}\,
	\Pi_{L_u L_l}^2 A_{ul} B_{lu} \\
&&\kern -1.8cm\times
	\sum_{J_l J_l' J_l''}
	\sum_{J_u J_u'}
	(-1)^{J_u'+J_l''+1}\,
	\Pi_{J_l J_l'}
	\Pi_{J_l''J_u J_u'}^2
	\sixj{J_u}{J_l}{1}{L_l}{L_u}{S}
	\sixj{J_u'}{J_l'}{1}{L_l}{L_u}{S}
	\sixj{J_u}{J_l''}{1}{L_l}{L_u}{S}
	\sixj{J_u'}{J_l''}{1}{L_l}{L_u}{S}
	\nonumber \\
&&\kern -1.8cm\times
	\sum_{KQ}\sum_{K'Q'}\sum_{K_l Q_l}
	(-1)^{K_l-Q_l}\,\Pi_{KK'K_l}
	\thrj{K}{K'}{K_l}{Q}{Q'}{-Q_l}
	\sixj{K'}{J_u}{J_u'}{J_l''}{1}{1}
	\ninj{K}{K'}{K_l}{1}{J_u'}{J_l'}{1}{J_u}{J_l}
	\nonumber \\
&&\mathop{\times}
	T^{K'}_{Q'}(i,\bm{\hat{k}'})\,
	\rho^{K_l}_{Q_l}(J_l,J_l') \nonumber \\
&&\kern -1.8cm\times
      \sum_{j=0}^3 \oint \frac{d\bm{\hat{k}}}{4\pi}\,
	T^K_Q(j,\bm{\hat{k}})
	\int_0^\infty d\omega_k
	\left(
	\Psi_{u'l',l''ul}^{-k,+k'-k} + \bar\Psi_{ul,l''u'l'}^{-k,+k'-k}
	\right)
	S_j(\omega_k,\bm{\hat{k}})\;,\qquad (i=0,1,2,3)
	\nonumber
\end{eqnarray}
and finally, in the additional case of unpolarized lower term
(u.l.s.: $K_l=0$, $J_l'=J_l$),
\begin{eqnarray} \label{eq:RT.J.metalevel}
\varepsilon\apx{2}_i(\omega_{k'},\bm{\hat{k}'})\biggr|_{B=0}^\textrm{u.l.s.}
&=&\frac{3}{16\pi^3}\,
	{\cal N}\hbar\,\frac{\omega_{k'}^4}{\omega_{ul}^3}\,
	\Pi_{L_u L_l}^2 A_{ul} B_{lu} \\
&&\kern -2.1cm\times
	\sum_{J_l J_l'}
	\sum_{J_u J_u'}
	(-1)^{J_l-J_l'}\,
	\frac{%
	\Pi_{J_l J_l'J_u J_u'}^2
	}{%
	\epsilon_{J_u J_u'}+{\rm i}\omega_{J_u J_u'}
	}\,
	\sixj{J_u}{J_l}{1}{L_l}{L_u}{S}
	\sixj{J_u'}{J_l'}{1}{L_l}{L_u}{S}
	\sixj{J_u'}{J_l}{1}{L_l}{L_u}{S}
	\sixj{J_u}{J_l'}{1}{L_l}{L_u}{S}
	\nonumber \\
&&\kern -2.1cm\times
	\sum_{KQ} (-1)^Q\,
	\sixj{K}{J_u}{J_u'}{J_l}{1}{1}
	\sixj{K}{J_u}{J_u'}{J_l'}{1}{1}\,
	\frac{1}{\sqrt{2J_l+1}}\,
	T^K_{-Q}(i,\bm{\hat{k}'})\,
	\rho^{0}_{0}(J_l,J_l) \nonumber \\
&&\kern -2.1cm\times
      \sum_{j=0}^3 \oint \frac{d\bm{\hat{k}}}{4\pi}\,
	T^K_Q(j,\bm{\hat{k}})
	\int_0^\infty d\omega_k\,
	{\cal R}(\Omega_u,\Omega_{u'};
	\Omega_l,\Omega_{l'};
	\omega_k,\omega_{k'})_\textrm{n.c.l.t.}\,
	S_j(\omega_k,\bm{\hat{k}})\;,\qquad (i=0,1,2,3)
	\nonumber
\end{eqnarray}
where we recalled Equation~(\ref{eq:red.nclt}), and renamed $l''$ into $l'$.

In the limit of sharp lower levels, this last expression can be compared
with the one for the redistribution matrix derived for the same model
atom by \cite{La97}, using the metalevel approach to the description
of the atomic density matrix. We note that the redistribution function
${\cal R}(\Omega_u,\Omega_{u'};\Omega_l,\Omega_{l'};
	\omega_k,\omega_{k'})_\textrm{n.c.l.t.}$ in the integral
must be replaced by the function ${\cal R}_2$ of Equation~(\ref{eq:red.sll})
for this comparison. Once this substitution is made, and taking into
account the vanishing of the 1st-order emissivity in the limit of sharp
lower levels,
we can demonstrate the formal identity of Equation~(\ref{eq:RT.J.metalevel})
with the result of that former derivation.

Similarly, imposing $S=I=0$ in Equation~(\ref{eq:RT.JF}), we can derive
the coherent scattering emissivity for the two-level atom in the
presence of a magnetic field. This also turns out to be identical with the
analogous result derived by \cite{La97}, in the limit of sharp and
unpolarized lower level considered by those authors.

\section{Verification of the energy-balance condition for coherent scattering}
\label{sec:balance}

A fundamental property of the transport of radiation in a
collisionless gas of atoms at the statistical equilibrium (i.e.,
under stationary conditions,
such that the internal energy of the atomic ensemble does not change
with time) is that the radiation's energy flux through a closed surface
containing the gas must be zero. It is thus important to
verify that
Equation~(\ref{eq:RT.1}) --  which generalizes the usual radiative transfer
equation for polarized radiation \cite[e.g.,][]{LL04} via the inclusion
of coherent scattering -- satisfies such fundamental energy-balance
condition.

Because our present theory cannot account for the broadening of the
lower levels due to photon absorption
(see discussion at the end of Section~\ref{sec:lifetimes}),
in order to guarantee the self-consistency of this verification, we
only consider the case of infinitely sharp lower levels.
It must be noted that this choice is consistent with the approximation
of neglecting stimulated emission. In fact, if the incident radiation
field is strong enough to induce any appreciable broadening of the lower
levels, then one should also expect that stimulation effects will be
comparatively important, and therefore should be included in the picture.
This is certainly the case when the objective is to verify the energy balance
among the various radiation processes.

As a consequence of the assumption of infinite lifetime of the lower
level, we then must drop the contribution of the 1st-order
emissivity to the radiative energy balance. Nonetheless, we will retain a
finite width of the lower levels in the expressions of the line profiles,
for the sake of this demonstration.
This is because, in real cases, these levels may still be broadened by
collisional processes, or by the possibility of spontaneous de-excitation
towards other lower terms, in the case of subordinate lines in a
multi-term atom.

The verification of the radiative balance condition in the collisionless
regime can then be set up in general form starting from
Equation~(\ref{eq:RT.1}). For simplicity, we assume that the incident
radiation is unpolarized,
\begin{displaymath}
S_i(\omega_k,\bm{\hat{k}})
	=\delta_{i0}\,S_0(\omega_k,\bm{\hat{k}})\;.
\end{displaymath}
We then can ignore the contribution from the absorption terms associated
with the magneto-optical effects (see
Equation~(\ref{eq:magnetooptical})), and only need to consider
the expression of the absorption coefficient adopting the same
notations used in writing Equation~(\ref{eq:2emiss.1}),
\begin{eqnarray} \label{eq:absorb.gen}
\eta_i(\omega_{k'},\bm{\hat{k}'})
&=& \frac{2\pi}{\sqrt3}\,\frac{e_0^2}{\hbar c}\,
	{\cal N}\omega_{k'}
	\sum_{ll'}\rho_{ll'}\sum_u\sum_{qq'}(-1)^{q'+1}\,
	(r_q)_{ul}(r_{q'})_{ul'}^* \\
&&{}\times
	\sum_{KQ}\sqrt{2K+1}\,\thrj{1}{1}{K}{-q}{q'}{-Q}\,
	T^K_Q(i,\bm{\hat{k}'})
	\left(
	\Phi_{ul'}^{-k'} + \bar\Phi_{ul}^{-k'}
	\right)\;.\qquad (i=0,1,2,3) \nonumber
\end{eqnarray}

Using the tabulated expressions for
$T^K_Q(i,\bm{\hat{k}})$ \citep{Bo97b,LL04}, we can show that
\begin{equation}
\label{eq:avg1}
\oint\frac{d\bm{\hat{k}}}{4\pi}\,
	T^K_Q(i,\bm{\hat{k}})
=\delta_{Q0} \left(\delta_{i0}\,\delta_{K0}
	-\frac{1}{\sqrt{2}}\,
	\delta_{i1}\,\delta_{K2}\right)\;
\end{equation}
With the above assumptions, and for $i=0$, from
Equations~(\ref{eq:RT.1}), (\ref{eq:2emiss.1}), (\ref{eq:dichroic}),
and (\ref{eq:absorb.gen}), after some straightforward algebra, we find
\begin{eqnarray} \label{eq:balance}
&&\oint\frac{d\bm{\hat{k}}}{4\pi}
\int_0^\infty d\omega_{k}\,
\frac{1}{c}\,\frac{d}{dt}\,S_0(\omega_{k},\bm{\hat{k}}) \\
&=&{}-\frac{2\pi}{\sqrt3}\,\frac{e_0^2}{\hbar c}\,
	{\cal N}
	\sum_{ll'}\rho_{ll'}
	\sum_{KQ}\sum_{qq'}(-1)^{q'+1}\,
	\Pi_K\,\thrj{1}{1}{K}{-q}{q'}{-Q}
	\oint\frac{d\bm{\hat{k}}}{4\pi}\,T^K_Q(0,\bm{\hat{k}})
	\int_0^\infty d\omega_{k}\,S_0(\omega_k,\bm{\hat{k}}) \nonumber \\
&&{}\times\Biggl\{
	\sum_u (r_q)_{ul}(r_{q'})_{ul'}^*\,
	\omega_k \left(
	\Phi_{ul'}^{-k} + \bar\Phi_{ul}^{-k}
	\right) \nonumber \\
&&\hphantom{{}\times\Biggl\{} -
	\frac{2}{3}\frac{e_0^2}{\hbar c^3}
	\sum_{uu'} (r_q)_{ul}(r_{q'})_{u'l'}^*
	\sum_{pl''} (r_p)_{u'l''}(r_p)_{ul''}^*\,
	\frac{1}{\pi}
	\int_0^\infty d\omega_{k'}\,\omega_{k'}^4
	\left(
	\Psi_{u'l',l''ul}^{-k,+k'-k} + \bar\Psi_{ul,l''u'l'}^{-k,+k'-k}
	\right) \Biggr\} \nonumber \\
&\equiv&{}-\frac{2\pi}{\sqrt3}\,\frac{e_0^2}{\hbar c}\,
	{\cal N}
	\sum_{ll'}\rho_{ll'}
	\sum_{KQ}\sum_{qq'}(-1)^{q'+1}\,
	\Pi_K\,\thrj{1}{1}{K}{-q}{q'}{-Q}
	\oint\frac{d\bm{\hat{k}}}{4\pi}\,T^K_Q(0,\bm{\hat{k}})
	\int_0^\infty d\omega_{k}\,S_0(\omega_k,\bm{\hat{k}}) \nonumber \\
&&{}\times\Biggl\{
	\sum_u (r_q)_{ul}(r_{q'})_{ul'}^*\,
	\omega_k \left(
	\Phi_{ul'}^{-k} + \bar\Phi_{ul}^{-k}
	\right) \nonumber \\
&&\hphantom{{}\times\Biggl\{} -
	\sum_{uu'} (r_q)_{ul}(r_{q'})_{u'l'}^*
	\frac{ \sum_{pl''} (r_p)_{u'l''}(r_p)_{ul''}^* }
	     {\sum_{p\bar{l}} |(r_p)_{u\bar{l}}|^2 }\,
	\frac{1}{\pi}
	\int_0^\infty d\omega_{k'}\;\epsilon_u(\omega_{k'})\,\omega_{k'}
	\left(
	\Psi_{u'l',l''ul}^{-k,+k'-k} + \bar\Psi_{ul,l''u'l'}^{-k,+k'-k}
	\right) \Biggr\}\;, \nonumber
\end{eqnarray}
where in the last equivalence we used the definition of
$\epsilon_u(\omega_{k'})$, Equations~(\ref{eq:gen_width}) and
(\ref{eq:Einstein A}).
In the following, we neglect the (weak) dependence of
$\epsilon_u(\omega_{k'})$ on the frequency of the scattered radiation,
so that it can be taken outside the integral in the last line of
Equation~(\ref{eq:balance}).
This is justified by the fact that the redistribution function appearing
in that integral effectively limits the frequency range around the
resonance frequency $\omega_{ul}$ that contributes to the finite value
of the integral.

From Equation~(\ref{eq:red.gen}), and using the form of the redistribution
function given in Equation~(\ref{eq:red.gen.alt}), we see that the
integration over
$\omega_{k'}$ in Equation~(\ref{eq:balance}) implies the evaluation of
four divergent integrals. In order to accomplish this task, we apply a
consistent cut-off procedure, which is derived from the usual Feynman's
regularization techniques of quantum field theory \cite[e.g.,][]{BS59}.
We thus find, once again extending the integration domain to $-\infty$
(see Note~\ref{note:domainextend})
\begin{eqnarray} \label{eq:integral}
&&\kern -1cm
\frac{1}{\pi}
\int_{-\infty}^\infty d\omega_{k'}\,\omega_{k'}
\left(
\Psi_{u'l',l''ul}^{-k,+k'-k} + \bar\Psi_{ul,l''u'l'}^{-k,+k'-k}
\right) \nonumber \\
&=&
	\frac{\omega_k+\omega_{ll''}-{\rm i}\epsilon_{ll''}}{%
	(\omega_k-\omega_{ul}+{\rm i}\epsilon_{u}-{\rm i}\epsilon_{l})
	(\omega_k-\omega_{u'l}-{\rm i}\epsilon_{u'l}) } +
	\frac{\omega_k+\omega_{l'l''}+{\rm i}\epsilon_{l'l''}}{%
	(\omega_k-\omega_{ul'}+{\rm i}\epsilon_{ul'})
	(\omega_k-\omega_{u'l'}-{\rm i}\epsilon_{u'}+{\rm i}\epsilon_{l'}) }
	\nonumber \\
&+&\frac{\epsilon_{ll'}+{\rm i}\,\omega_{ll'}}
	  {\epsilon_{uu'}+{\rm i}\,\omega_{uu'}} \biggl[
	\frac{\omega_{ul''}-{\rm i}\epsilon_{ul''}}{%
	(\omega_k-\omega_{ul}+{\rm i}\epsilon_{u}-{\rm i}\epsilon_{l})
	(\omega_k-\omega_{ul'}+{\rm i}\epsilon_{ul'}) } \nonumber \\
&&\hphantom{{}+\frac{\epsilon_{ll'}+{\rm i}\,\omega_{ll'}}
	  {\epsilon_{uu'}+{\rm i}\,\omega_{uu'}} \biggl[ }+
	\frac{\omega_{u'l''}+{\rm i}\epsilon_{u'l''}}{%
	(\omega_k-\omega_{u'l}-{\rm i}\epsilon_{u'l})
	(\omega_k-\omega_{u'l'}-{\rm i}\epsilon_{u'}+{\rm i}\epsilon_{l'}) }
	\biggr]\;.
\end{eqnarray}
We note that the dependence of this integral on $\omega_{l''}$ is
confined to the numerator. We then introduce an approximation,
which is to replace $\omega_{l''}$ in Equation~(\ref{eq:integral})
by some appropriate average $\bar{\omega}_l$.
If we further assume that the conditions for the diagonality of the
damping matrix $\mathit{\Gamma}_u$ apply (see Section~\ref{sec:lifetimes}),
we are then able to formally perform the summation over $p$ and $l''$ in
Equation~(\ref{eq:balance}),
\begin{equation} \label{eq:last.result1}
\sum_{pl''} (r_p)_{u'l''}(r_p)_{ul''}^*
	=\delta_{uu'} \sum_{pl''} |(r_p)_{ul''}|^2\;.
\end{equation}
Under these assumptions, after some more tedious algebra, we obtain
\begin{eqnarray} \label{eq:last.result2}
\frac{\epsilon_u}{\pi}
\int_{-\infty}^\infty d\omega_{k'}\,\omega_{k'}
\left(
\Psi_{ul',l''ul}^{-k,+k'-k} + \bar\Psi_{ul,l''ul'}^{-k,+k'-k}
\right)
&=& \frac{\epsilon_u+{\rm i}\,(\omega_u-\bar{\omega}_l)}{%
	\omega_k-\omega_{ul'}+{\rm i}\epsilon_{ul'}} +
  \frac{\epsilon_u-{\rm i}\,(\omega_u-\bar{\omega}_l)}{%
	\omega_k-\omega_{ul}-{\rm i}\epsilon_{ul}} \nonumber \\
&\approx&
	\omega_k \left(
	\Phi_{ul'}^{-k} + \bar\Phi_{ul}^{-k}
	\right)\;,
\end{eqnarray}
where, for the last approximation, we took into account the hypothesis
of sharp lower levels, and recalled the definition of the profiles
$\Phi^{\pm k}_{ab}$, Equation~(\ref{eq:Phi}).
In particular, \emph{this approximation becomes exact in the absence of
lower-level polarization, or if the magnetic field is zero.}
Using then the results of Equations~(\ref{eq:last.result1}) and
(\ref{eq:last.result2}), the quantity within curly braces in
Equation~(\ref{eq:balance}) vanishes, thus verifying the expected
condition of radiative balance in a collisionless gas of atoms.

It is important to note that the approximations involved in this
verification are dominated by the assumption that the energy separation
among the lower levels $\{l,l',l'',\ldots\}$ is much smaller than the
frequency of the incident radiation, $\omega_k$, so that the energy values
of those levels can all be approximated by the average $\bar{\omega}_l$.
In the presence of a magnetic field, the energy separation of the atomic
levels is proportional to the Larmor frequency for the applied field,
$\omega_B$.
For typical observing conditions in the solar atmosphere, the ratio
$\omega_B/\omega_k$ is of the order of $10^{-4}$ or less
(e.g., for $\lambda\sim 1\,\rm\mu m$, and $B\sim 10^3\,\rm G$,
$\omega_B/\omega_k\approx 5\times 10^{-6}$), so the above approximation is
well justified.
For transitions in complex atoms, the energy span of the various
$\omega_{l''}$ may be dominated by the fine-structure (FS) separation of
the lower term, in which case the above approximation is justified if
$\omega_{\rm FS}/\omega_k\ll 1$. For example, in the case of the
H$\alpha$ line, $\omega_{\rm FS}/\omega_k\approx 2.4\times 10^{-5}$.

To conclude this section, we formally demonstrate that the illumination
of the atom by a spectrally flat radiation $S_j(\omega_0,\bm{\hat{k}})$
produces a line profile that corresponds to the one obtained in the
limit of complete redistribution (see, e.g., \citealt{He54}, \S20,
or \citealt{Sa67}, \S2.6).
In order to see this, we consider again the emissivity term associated
with the coherent scattering of radiation (cf.\
Equations~(\ref{eq:2emiss.1}) and (\ref{eq:Jtens})), along with the
integral norm of
Equation~(\ref{eq:integral2}),
\begin{eqnarray} \label{eq:coherent.flat}
\varepsilon\apx{2}_i(\omega_{k'},\bm{\hat{k}'})
&\equiv&
	\frac{4}{3}\frac{e_0^4}{\hbar^2 c^4}\,{\cal N}\omega_{k'}^4
	\sum_{ll'}\rho_{ll'}\sum_{uu'l''}
	\sum_{qq'}\sum_{pp'}(-1)^{q'+p'}\,
	(r_q)_{ul}(r_{q'})^\ast_{u'l'}
	(r_p)_{u'l''}(r_{p'})^\ast_{ul''} \nonumber \\
&&{}\times
	\sum_{KQ}\sum_{K'Q'}\Pi_{KK'}\,
	\thrj{1}{1}{K}{-q}{q'}{-Q}
	\thrj{1}{1}{K'}{-p}{p'}{-Q'}\,
	T^{K'}_{Q'}(i,\bm{\hat{k}'}) \nonumber \\
&&{}\times
	\frac{2\pi}{\epsilon_{uu'}+{\rm i}\omega_{uu'}}
	\left(
	\Phi_{l''u'}^{+k'} + \bar\Phi_{l''u}^{+k'}
	\right)
	\oint \frac{d\bm{\hat{k}}}{4\pi}\,
      \sum_{j=0}^3 T^K_Q(j,\bm{\hat{k}})\,S_j(\omega_0,\bm{\hat{k}})
\nonumber \\
\noalign{\eject}
&=&
	\frac{1}{2\pi^2\sqrt3}\,\frac{e_0^2}{c^3}\,
	{\cal N}\omega_{k'}^4
	\sum_{uu'} \Biggl\{
	\frac{16\pi^3}{\sqrt3}\frac{e_0^2}{\hbar^2 c}\,
	\frac{1}{\epsilon_{uu'}+{\rm i}\omega_{uu'}}
	\sum_{ll'}\rho_{ll'}
	\sum_{qq'}(-1)^{q'+1}\,
	(r_q)_{ul}(r_{q'})^\ast_{u'l'} \nonumber \\
&&\kern 3.5cm\times
	\sum_{KQ}\Pi_{K}
	\thrj{1}{1}{K}{-q}{q'}{-Q}
	J^K_Q(\omega_0)
	\Biggr\}
	\sum_{l''}\sum_{pp'}(-1)^{p'+1}\,
	(r_p)_{u'l''}(r_{p'})^\ast_{ul''} \nonumber \\
&&{}\times
	\sum_{K'Q'}\Pi_{K'}
	\thrj{1}{1}{K'}{-p}{p'}{-Q'}\,
	T^{K'}_{Q'}(i,\bm{\hat{k}'})
	\left(
	\Phi_{l''u'}^{+k'} + \bar\Phi_{l''u}^{+k'}
	\right)\;.\qquad (i=0,1,2,3)
\end{eqnarray}
On the other hand, using the same notation, the emissivity of the first
order is given by
\begin{eqnarray} \label{eq:incoherent.flat}
\varepsilon\apx{1}_i(\omega_{k'},\bm{\hat{k}'})
&=& \frac{1}{2\pi^2\sqrt3}\,\frac{e_0^2}{c^3}\,
	{\cal N}\omega_{k'}^4
	\sum_{uu'}\rho_{uu'}\sum_{l''}\sum_{pp'}(-1)^{p'+1}\,
	(r_p)_{u'l''}(r_{p'})_{ul''}^* \\
&&{}\times
	\sum_{K'Q'}\Pi_{K'}\,\thrj{1}{1}{K'}{-p}{p'}{-Q'}\,
	T^{K'}_{Q'}(i,\bm{\hat{k}'})
	\left(
	\Phi_{l''u'}^{+k'} + \bar\Phi_{l''u}^{+k'}
	\right)\;.\qquad (i=0,1,2,3) \nonumber
\end{eqnarray}
We then see that Equations~(\ref{eq:coherent.flat}) and
(\ref{eq:incoherent.flat}) coincide \emph{in the case
of spectrally flat illumination}, once we identify the quantity within
curly brackets in Equation~(\ref{eq:coherent.flat})
with the upper-term density matrix $\rho_{uu'}$ that appears in
the expression of $\varepsilon\apx{1}_i(\omega_{k'},\bm{\hat{k}'})$.
In fact, this identification derives directly from the solution of the
statistical equilibrium problem for the two-term atom illuminated by
spectrally flat radiation (and neglecting stimulated emission), which
is obtained within the lowest-order approximation of the theory of
polarized line formation \citep{LL04}.
In turn, \emph{this proves the validity of that former theory
for the description of the scattering of polarized radiation in the limit
of complete redistribution.}

It is important to remark that the above result does \emph{not} imply that
the emitted radiation gets counted twice when we include the
coherent-scattering term in the radiative transfer equation, as this would
evidently violate energy conservation. We have previously noted that
the excitation of $\rho_{uu'}$ is inhibited in the limit of infinitely
sharp lower levels, and therefore all the emitted radiation must
come from the coherent-scattering term in that case.
Then, if the illumination of the atom is spectrally flat,
the above demonstration shows that the scattered radiation has exactly
the same spectral structure \emph{as if} the atom had absorbed and
re-emitted incoherently the incident radiation \cite[cf.][Sect.~20]{He54}.

\section{Conclusion}
\label{sec:conclusion}

We presented a general expression of the redistribution function
for polarized radiation in the case of a transition between two atomic
terms $l$ and $u$, which can both be partially degenerate and polarized,
and subject to a magnetic field of arbitrary strength. In particular,
this function applies to the description of resonance or subordinate
lines of atoms with both fine and hyperfine structures. This function is
derived from first principles, and it represents a preliminary result
of a diagrammatic theory of scattering polarization that we have been
developing, on and off, over the past decade. The full theory is
still incomplete, lacking a self-consistent set of accompanying
equations for the statistical equilibrium of the atomic
system. Part of the problem is caused by a mild violation of the
positivity condition for the atomic density matrix in the presence of
magnetic fields.
Following a revision of
our diagrammatic approach, inspired by certain literature dealing with
the problem of the treatment of the initial conditions in perturbative
master equations \cite[see, e.g.,][and references within; a very clear
discussion of this problem can also be found in a recent paper by
\citealt{Ko10}]{Wh08}, we have identified new terms, previously
unaccounted for, in the perturbative expansion of the
evolution equation
for the atomic density matrix. These terms are in the process of being
evaluated, with the perspective that they should restore the
self-consistency of the statistical-equilibrium problem.

Because of the approximations of the theory, we cannot rigorously model
the radiative broadening of the atomic levels due to photon
absorption. Therefore, in the absence of collisions, the lower term
of an atomic transition remains infinitely sharp, and the scattered
radiation only comes from the coherent contribution to the emissivity,
Equation~(\ref{eq:2emiss}).
However, we presented the general form of the redistribution
function, Equation~(\ref{eq:red.gen}), where the widths of the lower
levels appear explicitly. This function can then be applied to treat
phenomenologically the frequency redistribution of radiation in the
presence of collisional broadening.

Despite these formal difficulties, the results we presented are
relevant enough for the spectro-polarimetric diagnostics of the
upper layers of the Sun's atmosphere, that we decided to propose them,
already at this early stage, to the astrophysical community.

In this paper, we limited ourselves to provide the algebraic formulas
for the scattered radiation in a complex two-term atom with hyperfine
structure, and in the presence of a magnetic field (relevant, for
instance, to the case of the resonant \ion{Na}{1} D-doublet, or of
complex subordinate lines such as H$\alpha$). Numerical applications
of this formalism will be considered in future papers.

Finally, it is important to note that the general redistribution
function presented in this paper, Equation~(\ref{eq:red.gen}) or
(\ref{eq:red.gen.alt}), coincides with a rarely quoted result by
\cite{LT71}, which was derived through a completely different approach
to the problem of resonance scattering (based on \citealt{He54}).
More precisely, these authors provided the expression
of a ``resonance matrix'' for scattering (see their Equation~(7.7)),
which turns out to be equivalent to
\begin{displaymath}
[(\epsilon_{uu'}+{\rm i}\omega_{uu'})
  (\epsilon_{ll'}+{\rm i}\omega_{ll'})]^{-1}\,
 {\cal R}(\Omega_u,\Omega_{u'};\Omega_l,\Omega_{l'},\Omega_{l''};
	\omega_k,\omega_{k'})\;,
\end{displaymath}
in our notation.
This correspondence allows us to derive yet another form for the general
redistribution function presented in this paper, which is fully
equivalent to Equations~(\ref{eq:red.gen}) and (\ref{eq:red.gen.alt}),
\begin{eqnarray} \label{eq:r_l&th}
{\cal R}(\Omega_u,\Omega_{u'};
	\Omega_l,\Omega_{l'},\Omega_{l''};
	\omega_k,\omega_{k'}) \\
&&\kern -1.75in
{}=
\Phi^{+k}_{lu'}\,\bar\Phi^{+k'-k}_{l''l} +
\bar\Phi^{+k'-k}_{l''l}\,\bar\Phi^{+k'}_{l''u} +
\bar\Phi^{+k}_{l'u}\,\bar\Phi^{+k'}_{l''u} +
\Phi^{+k'-k}_{l''l'}\,\bar\Phi^{+k}_{l'u} +
\Phi^{+k}_{lu'}\,\Phi^{+k'}_{l''u'} +
\Phi^{+k'-k}_{l''l'}\,\Phi^{+k'}_{l''u'} \nonumber \\
&&\kern -1.75in
{}=
\Phi^{+k}_{lu'}\,\Phi^{+k-k'}_{ll''} +
\Phi^{+k-k'}_{ll''}\,\Phi^{-k'}_{ul''} +
\Phi^{-k}_{ul'}\,\Phi^{-k'}_{ul''} +
\Phi^{+k'-k}_{l''l'}\,\Phi^{-k}_{ul'} +
\Phi^{+k}_{lu'}\,\Phi^{+k'}_{l''u'} +
\Phi^{+k'-k}_{l''l'}\,\Phi^{+k'}_{l''u'}\;, \nonumber
\end{eqnarray}
where in the last line we used the conjugation property of the
profiles $\Phi^{\pm k}_{ab}$ (cf.~Equation~(\ref{eq:Phi})),
\begin{equation}
\bar\Phi^{\pm k}_{ab}=\Phi^{\mp k}_{ba}\;.
\end{equation}

The equivalence between the general redistribution function presented
in this paper and the form (\ref{eq:r_l&th}) derived by \cite{LT71}
gives us additional confidence of the solidity of our results and of
the formalism on which they are based. It is regrettable that such
fundamental result of \cite{LT71} has essentially gone unnoticed
by the literature published over the past
forty years on the subject of partial redistribution for polarized
radiation. We hope that
our paper will at least succeed in bringing once again to the
attention of the solar polarimetry community the beauty and
relevance, still to this day, of that seminal work.

redpajama_set_name: RedPajamaArXiv


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Pascal Hulin Productions will be presenting live at the Mediterranean Conference Centre in Valletta on 3, 4 and 5 June what has been described as a "new tap dance sensation" and also as being the "best thing that has ever happened to the world of tap": Tap Reloaded 2.
Just when you thought that a dance show does not get better than Pascal's previous world wide success Tap Reloaded, Pascal has managed to come up with something even bigger and better, Tap Reloaded 2, with even more incredible acts from around the world as well as a Live Band from the USA whose musicians tap while they play!
Following the huge success of Tap Reloaded last year selling over 300,000 tickets in theatres all over Europe, Asia and the USA, Pascal Hulin (pictured) has managed to come up with yet another block buster dance show which is currently touring all over Europe this spring.
With world famous soloists like Chloe Arnold, Freddy Corado and the 2010 World Champion Tap Dance Company, The Nancy Chippendales from the USA as well as a live band, Pascal has managed to make his best dance show ever, even better!
Tap Reloaded 2, together with its cast of internationally-acclaimed tap dancers and live band from the USA, are taking the world of dance entertainment by storm filling up some of the largest and prestigious of theatres and arenas world wide and attracting millions from all corners of the globe.
Tap Reloaded 2 has a cast of 39 artists from the USA, Central America, Germany, France, Spain, Russia and Ukraine plus a live band all the way from the USA.
Tap Reloaded is an explosive, humorous family variety show which combines international world class tap dance acts, acrobatics, comedy and entertainment at the highest level. A show for all the family!
There is no comparable show to date! Tickets can be purchased from The Mediterranean Conference Centre or The Manoel Theatre, Valletta. Box office opening hours for both The MCC and The Manoel Theatre are from 10am to 1pm and from 5 to 7pm Monday to Fridays, and from 10am to 1pm on Saturdays
By MCC Content Writer|2020-11-04T02:42:10+01:0029th April 2011|Comments Off on Tap Reloaded 2

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Duke's Den A Farewell
Well, this is awkward, a supposed wordsmith struggling to find adequate words. I hadn't realized how difficult it would be to say goodbye to the countless readers who have enjoyed my work on the pages of Motorcycle.com for nearly 11 years.
I'm willing to bet that just about everyone reading this will wonder how anyone could possibly leave what they see as the best job in the world. Believe me, the internal struggles I've gone through when making this decision were nearly debilitating. After all, being a motojournalist is literally my dream job, and rising through the ranks to become chief editor at perhaps the biggest motorcycle website on the planet was even beyond my initial aspirations.
It's almost always fun riding a motorcycle, even if it rains!
When I was pondering the possibility of leaving my job, I was often reminded of the pithy statement of a former motojourno colleague, Aaron Frank, who once said about being a moto-J: "It's the best job in the world… three days a month."
Duke's Den – A Decade At The Helm
Using the power of deduction informs that some of the other days of a month weren't quite as fabulous. After all, traveling to a new and/or exotic area to test a motorcycle before anyone else in the world is an exclusive treat, and anything after that would have to feel dull in comparison. Deadlines always loom, especially when we can publish 24 hours a day, and the internet is never (ever!) full, resulting in having to flit from one project to the next with little opportunity for the gratification and relief felt from a job well done.
Duke's Den – You Got A Problem With Free (online content)?
The State Of Moto Publishing, According To Bonnier Motorcycle Group
I'm happy to have checked off Phillip Island from my racetrack bucket list last year.
Duke's Den – Racetrack Bucket Lists
Duke's Den – My Tour Of Racetracks Around The World: Part 4
I've been writing about motorcycles in a full-time capacity for more than two decades, and I literally couldn't have even dreamed of all the fantastic opportunities on and around motorcycles that I've enjoyed. I've been able to travel across the globe to ride and test the latest motorbikes, produced and starred in videos that have been viewed millions of times, looked behind the closed doors of several motorcycle factories, taken part in the world's biggest motorcycle shows, and talked to many of the biggest movers and shakers in the business. Frankly, I've pretty much done it all.
Duke's Den: Inside EICMA, The World's Biggest Motorcycle Show
Inside Yamaha's MotoGP Race Shop
If there's a Moby Dick missing from my motojourno career, it's not ever having the chance to ride a legit GP/MotoGP bike. Perhaps in my next life…
So, rather than risk stagnating on that perch, I've decided, with much consternation, to hang up my motojournalist hat. I'll soon be moving into a gig in which I'll be producing videos to support the marketing efforts of different companies. Some of them are powersports enterprises, so I'll still be hanging around behind the scenes.
I've been fortunate to have worked with some exceptional people at MO over the years, and I will miss my brothers enormously as I move on to new challenges. I leave behind Evans Brasfield, surely among the nicest fellas in the biz and a sharp shooter behind the lens, whose broad shoulders will be assuming extra weight in addition to his managing editor duties. It's been a privilege to be part of a writing team that includes the estimable John Burns, one of the cleverest motojournos that has ever put pen to paper or screen – always worth the time to read his musings.
Duke's Den – Courtship Of Editors
From sportbikes to cruisers (and dirtbikes when they don't try to break me!), I love riding on two wheels.
Brent Jaswinski, our newest hire, has already exceeded expectations for his dedication to improving his craft and providing a new and desirable voice to these pages. A regret I'll carry with me is not having the opportunity to mentor Ryan Adams for a little longer, as he's the epitome of what an enthusiastic 20-something can offer to the world of motojournalism.
A special call-out goes to Dennis Chung. Dennis has been by my virtual side for almost the entirety of my tenure at MO, even though he works 2,500 miles away from where I do, and he's helped me and the site in countless thousands of ways for more than a decade. He's our unsung hero and deserves way more praise than he receives. This entire crew will remain my friends as the wheels continue to turn.
My EiC chair was enough to convince a few TV networks to put me on the airways over the years.
And to you, my cherished readers, the relationships I've built with you via this outlet have been more rewarding than I could've imagined. I'm truly humbled by the faith you've invested in me and my words over the years, and I'm always flattered when I'm recognized out in public.
I'll treasure being trusted to provide impressions of motorcycles I've ridden enough to actually purchase motorcycles I've written about. I'll hold dear the opportunities I've had to share my stories with you, knowing that the insights and experiences I've gained on motorbikes have been passed along to millions of you over the years and into the future. Your readership has kept me inspired through the years to maintain the quest for excellence at MO.
I hope you've relished the rides I've taken you on. It's been my privilege to serve you and your appreciation of the world on two wheels. I'll still be working within it, and I hope to see you out there somewhere around motorcycles.
Until we meet again, peace out.
Some Related Reading Highlights
Duke's Den – My Favorite Wheelie Photos
Duke's Den – Most Memorable Motorcycles
Kawasaki H2R 200 mph Review
Duke's Den – Ride More!
Duke's Den – Risk/Reward
Duke's Den – Kids On Bikes
Duke's Den – Father's Day
Duke's Den – The Joy Of Riding Slow-ish Bikes Fast
Duke's Den: The WFO Nut
Duke's Den – You Can't Help Getting Older
Duke's Den – Is Electricity The Savior Of Dirtbikes?
Duke's Den – Motorcycle Riding vs. Car Driving
Exploring Lightweight Materials On Motorcycles
Serendipity Finding A Vintage Motorcycle: Starr-Crossed Lovers
Duke's Den – A Farewell appeared first on Motorcycle.com.
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credit: Timothy Jagielo
Tunde Olaniran - Namesake
The Afro-futurist mix of Hip-Hop/R&B/Punk that is Tunde Olaniran returns with his first full-length album on Quite Scientific entitled Transgressor. The record showcases Olaniran's masterful approach to mixing intelligent and conscious commentary with a slew of personal revelations and heartfelt declarations. Propulsive entries like "Namesake" and "KYBM" instantly signify the hi-energy dance-pop feel that has brought attention to Olaniran in the past, while tracks like "Let Me Go", "Up & Down", and "Everyone's Missing" make clear his abilities to be at once gut-wrenchingly emotional, candidly playful, or deadly-serious.
Currently calling the United States his home after various stints in Nigeria, Germany, and England, Olaniran has in a very short time become one of the most beloved fixtures of an ever evolving Detroit music scene, as well as a driving force within the growing artistic community in Flint, MI (where he currently resides). His previous EPs have led to plenty of praise from outlets such as The New York Times, Pitchfork, NPR, Stereogum, Paste, BlackBook, Paste, EW and countless others while his penchant for channeling his energies into additional projects as an author, choreographer, and community organizer has made Olaniran's voice one that can't help but be heard.
Genre: R & B
Subgenre: Afro Pop
From: Flint, MI
Events featuring Tunde Olaniran
Tunde Olaniran at The Sidewinder Outside